double scaling limit for matrix models with nonanalytic potentials we study the double scaling limit for unitary invariant ensembles of random matrices with nonanalytic potentials and find the asymptotic expansion for the entries of the corresponding jacobi matrix our approach is based on the perturbation expansion for the string equations the first order perturbation terms of the jacobi matrix coefficients are expressed through the hastings mcleod solution of the painleve ii equation the limiting reproducing kernel is expressed in terms of solutions of the dirac system of differential equations with a potential defined by the first order terms of the expansion

uniform asymptotics for polynomials orthogonal with respect to varying exponential weights and applications to universality questions in random matrix theory we consider asymptotics for orthogonal polynomials with respect to varying exponential weights wn x dx e nv x dx on the line as n the potentials v are assumed to be real analytic with sufficient growth at infinity the principle results concern plancherel rotach type asymptotics for the orthogonal polynomials down to the axis using these asymptotics we then prove universality for a variety of statistical quantities arising in the theory of random matrix models some of which have been considered recently in 31 and also in 4 an additional application concerns the asymptotics of the recurrence coefficients and leading coefficients for the orthonormal polynomials see also 4 the orthogonal polynomial problem is formulated as a riemann hilbert problem following 19 20 the riemann hilbert problem is analyzed in turn using the steepest descent method introduced in 12 and further developed in 11 13 a critical role in our method is played by the equilibrium measure d v for v as analyzed in 8 1999 john wiley sons inc

universality at the edge of the spectrum for unitary orthogonal and symplectic ensembles of random matrices we prove universality at the edge of the spectrum for unitary 2 orthogonal 1 and symplectic 4 ensembles of random matrices in the scaling limit for a class of weights w x e v x where v is a polynomial v x 2mx2m 2m 0 the precise statement of our results is given in theorem 1 1 and corollaries 1 2 and 1 4 below for the same class of weights a proof of universality in the bulk of the spectrum is given in 12 for the unitary ensembles and in 9 for the orthogonal and symplectic ensembles our starting point in the unitary case is 12 and for the orthogonal and symplectic cases we rely on our recent work 9 which in turn depends on the earlier work of widom 46 and tracy and widom 42 as in 9 the uniform plancherel rotach type asymptotics for the orthogonal polynomials found in 12 plays a central role the formulae in 46 express the correlation kernels for 1 4 as a sum of a christoffel darboux cd term as in the case 2 together with a correction term in the bulk scaling limit 9 the correction term is of lower order and does not contribute to the limiting form of the correlation kernel by contrast in the edge scaling limit considered here the cd term and the correction term contribute to the same order this leads to additional technical difficulties over and above 49 2006 wiley periodicals inc

correlation functions cluster functions and spacing distributions for random matrices the usual formulas for the correlation functions in orthogonal and symplectic matrix models express them as quaternion determinants from this representation one can deduce formulas for spacing probabilities in terms of fredholm determinants of matrix valued kernels the derivations of the various formulas are somewhat involved in this article we present a direct approach which leads immediately to scalar kernels for the unitary ensembles and matrix kernels for the orthogonal and symplectic ensembles and the representations of the correlation functions cluster functions and spacing distributions in terms of them

learning closed horn expressions the paper studies the learnability of horn expressions within the framework of learning from entailment where the goal is to exactly identify some pre fixed and unknown expression by making queries to membership and equivalence oracles it is shown that a class that includes both range restricted horn expressions where terms in the conclusion also appear in the condition of a horn clause and constrained horn expressions where terms in the condition also appear in the conclusion of a horn clause is learnable this extends previous results by showing that a larger class is learnable with better complexity bounds a further improvement in the number of queries is obtained when considering the class of horn expressions with inequalities on all syntactically distinct terms

universality in several matrix models via approximate transport maps we construct approximate transport maps for perturbative several matrix models as a consequence we deduce that local statistics have the same asymptotic as in the case of independent gue or goe matrices i e they are given by the sine kernel in the bulk and the tracy widom distribution at the edge and we show averaged energy universality i e universality for averages of m points correlation functions around some energy level e in the bulk as a corollary these results yield universality for self adjoint polynomials in several independent gue or goe matrices which are close to the identity

strong asymptotics of orthogonal polynomials with respect to exponential weights we consider asymptotics of orthogonal polynomials with respect to weights w x dx e q x dx on the real line where q x 2m k 0 qkx k q2m 0 denotes a polynomial of even order with positive leading coefficient the orthogonal polynomial problem is formulated as a riemann hilbert problem following 22 23 we employ the steepest descent type method introduced in 18 and further developed in 17 19 in order to obtain uniform plancherel rotach type asymptotics in the entire complex plane as well as asymptotic formulae for the zeros the leading coefficients and the recurrence coefficients of the orthogonal polynomials c 1999 john wiley sons inc

on random matrix averages involving half integer powers of goe characteristic polynomials correlation functions involving products and ratios of half integer powers of characteristic polynomials of random matrices from the gaussian orthogonal ensemble goe frequently arise in applications of random matrix theory rmt to physics of quantum chaotic systems and beyond we provide an explicit evaluation of the large n limits of a few non trivial objects of that sort within a variant of the supersymmetry formalism and via a related but different method as one of the applications we derive the distribution of an off diagonal entry k ab of the resolvent or wigner k matrix of goe matrices which among other things is of relevance for experiments on chaotic wave scattering in electromagnetic resonators

transport maps for beta matrix models and universality we construct approximate transport maps for non critical beta matrix models that is maps so that the push forward of a non critical beta matrix model with a given potential is a non critical beta matrix model with another potential up to a small error in the total variation distance one of the main features of our construction is that these maps enjoy regularity estimates that are uniform in the dimension in addition we find a very useful asymptotic expansion for such maps which allows us to deduce that local statistics have the same asymptotic behavior for both models

bosonic fock representations of the affine virasoro algebra bosonic fock representations of the virasoro algebra and the affine lie algebras attached to classical lie algebras with non zero centres are explicitly presented based on these fock representations of the affine virasoro algebra are constructed an automorphism of the affine virasoro algebra is also given

rank 1 real wishart spiked model in this paper we consider n dimensional real wishart matrices y in the class input amssym w bbb r sigma m in which all but one eigenvalue of is 1 let the nontrivial eigenvalue of be 1 then as n m with m n 2 finite and nonzero the eigenvalue distribution of y will converge into the marchenko pastur distribution inside a bulk region when increases from 0 one starts to see a stray eigenvalue of y outside of the support of the marchenko pastur density as this stray eigenvalue leaves the bulk region a phase transition will occur in the largest eigenvalue distribution of the wishart matrix in this paper we will compute the asymptotics of the largest eigenvalue distribution when the phase transition occurs we will first establish the results that are valid for all n and m and will use them to carry out the asymptotic analysis in particular we have derived a contour integral formula for the harish chandra itzykson zuber integral when x and y are real symmetric and y is a rank 1 matrix this allows us to write down a fredholm determinant formula for the largest eigenvalue distribution and analyze it using orthogonal polynomial techniques as a result we obtain an integral formula for the largest eigenvalue distribution in the large n limit characterized by painleve transcendents the approach used in this paper is very different from a recent paper by bloemenal and virag in which the largest eigenvalue distribution was obtained using a stochastic operator method in particular the painleve formula for the largest eigenvalue distribution obtained in this paper is new 2012 wiley periodicals inc

reasoning with models we develop a model based approach to reasoning in which the knowledge base is represented as a set of models satisfying assignments rather than a logical formula and the set of queries is restricted we show that for every propositional knowledge base kb there exists a set of characteristic models with the property that a query is true in kb if and only if it is satisfied by the models in this set we characterize a set of functions for which the model based representation is compact and provides efficient reasoning these include cases where the formula based representation does not support efficient reasoning in addition we consider the model based approach to abductive reasoning and show that for any propositional kb reasoning with its model based representation yields an abductive explanation in time that is polynomial in its size some of our technical results make use of the monotone theory a new characterization of boolean functions recently introduced the notion of restricted queries is inherent in our approach this is a wide class of queries for which reasoning is efficient and exact even when the model based representation kb provides only an approximate representation of the domain in question moreover we show that the theory developed here generalizes the model based approach to reasoning with horn expressions and captures even the notion of reasoning with horn approximations

universality of the double scaling limit in random matrix models we study unitary random matrix ensembles in the critical case where the limiting mean eigenvalue density vanishes quadratically at an interior point of the support we establish universality of the limits of the eigenvalue correlation kernel at such a critical point in a double scaling limit the limiting kernels are constructed out of functions associated with the second painleve equation this extends a result of bleher and its for the special case of a critical quartic potential the two main tools we use are equilibrium measures and riemann hilbert problems in our treatment of equilibrium measures we allow a negative density near the critical point which enables us to treat all cases simultaneously the asymptotic analysis of the riemann hilbert problem is done with the deift zhou steepest descent analysis for the construction of a local parametrix at the critical point we introduce a modification of the approach of baik deift and johansson so that we are able to satisfy the required jump properties exactly c 2005 wiley periodicals inc

learnability and the vapnik chervonenkis dimension valiant s learnability model is extended to learning classes of concepts defined by regions in euclidean space e n the methods in this paper lead to a unified treatment of some of valiant s results along with previous results on distribution free convergence of certain pattern recognition algorithms it is shown that the essential condition for distribution free learnability is finiteness of the vapnik chervonenkis dimension a simple combinatorial parameter of the class of concepts to be learned using this parameter the complexity and closure properties of learnable classes are analyzed and the necessary and sufficient conditions are provided for feasible learnability

on the relation between orthogonal symplectic and unitary matrix ensembles for the unitary ensembles of n n hermitian matrices associated with a weight function w there is a kernel expressible in terms of the polynomials orthogonal with respect to the weight function which plays an important role for the orthogonal and symplectic ensembles of hermitian matrices there are 2 2 matrix kernels usually constructed using skew orthogonal polynomials which play an analogous role these matrix kernels are determined by their upper left hand entries we derive formulas expressing these entries in terms of the scalar kernel for the corresponding unitary ensembles we also show that whenever w w is a rational function the entries are equal to the scalar kernel plus some extra terms whose number equals the order of w w general formulas are obtained for these extra terms we do not use skew orthogonal polynomials in the derivations

orthogonal and symplectic matrix models universality and other properties we study orthogonal and symplectic matrix models with polynomial potentials and multi interval supports of the equilibrium measures for these models we find the bounds similar to those for the hermitian matrix models for the rate of convergence of linear eigenvalue statistics and for the variance of linear eigenvalue statistics and find the logarithms of partition functions up to the order o 1 we prove also the universality of local eigenvalue statistics in the bulk

first order jk clausal theories are pac learnable abstract we present positive pac learning results for the nonmonotonic inductive logic programming setting in particular we show that first order range restricted clausal theories that consist of clauses with up to k literals of size at most j each are polynomial sample polynomial time pac learnable with one sided error from positive examples only in our framework concepts are clausal theories and examples are finite interpretations we discuss the problems encountered when learning theories which only have infinite nontrivial models and propose a way to avoid these problems using a representation change called flattening finally we compare our results to pac learnability results for the normal inductive logic programming setting

on self similar lie algebras and virtual endomorphisms we introduce the notion of virtual endomorphisms of lie algebras and use it as an approach for constructing self similarity of lie algebras this is done in particular for a class of metabelian lie algebras having homological type fp n which are lie algebra analogues of lamplighter groups we establish several criteria when the existence of virtual endomorphism implies a self similar lie structure furthermore we prove that the classical lie algebra sl n k where char k does not divide n affords non trivial faithful self similarity

conjunctions of unate dnf formulas learning and structure abstract a central topic in query learning is to determine which classes of boolean formulas are efficiently learnable with membership and equivalence queries we consider the class r kconsisting of conjunctions ofkunate dnf formulas this class generalizes the class ofk clause cnf formulas and the class of unate dnf formulas both of which are known to be learnable in polynomial time with membership and equivalence queries we prove that r 2can be properly learned with a polynomial number of polynomial size membership and equivalence queries but can be properly learned in polynomial time with such queries if and only if p np thus the barrier to properly learning r 2with membership and equivalence queries is computational rather than informational few results of this type are known in our proofs we use recent results of hellersteinet al 1997 j assoc comput mach 43 5 840 862 characterizing the classes that are polynomial query learnable together with work of bshouty on the monotone dimension of boolean functions we extend some of our results to r kand pose open questions on learning dnf formulas of small monotone dimension we also prove structural results for r k we construct for any fixedk 2 a class of functionsfthat cannot be represented by any formula in r k but which cannot be easily shown to have this property more precisely for any functionfonnvariables in the class the value offon any polynomial size set of points in its domain is not a witness thatfcannot be represented by a formula in r k our construction is based on bch codes

on the 1 n expansion for some unitary invariant ensembles of random matrices we present a version of the 1 n expansion for random matrix ensembles known as matrix models the case where the support of the density of states of an ensemble consists of one interval and the case where the density of states is even and its support consists of two symmetric intervals is treated in these cases we construct the expansion scheme for the jacobi matrix determining a large class of expectations of symmetric functions of eigenvalues of random matrices prove the asymptotic character of the scheme and give an explicit form of the first two terms this allows us in particular to clarify certain theoretical physics results on the variance of the normalized traces of the resolvent of random matrices we also find the asymptotic form of several related objects such as smoothed squares of certain orthogonal polynomials the normalized trace and the matrix elements of the resolvent of the jacobi matrices etc

edge universality for orthogonal ensembles of random matrices we prove edge universality of local eigenvalue statistics for orthogonal invariant matrix models with real analytic potentials and one interval limiting spectrum our starting point is the result of shcherbina commun math phys 285 957 974 2009 on the representation of the reproducing matrix kernels of orthogonal ensembles in terms of scalar reproducing kernel of corresponding unitary ensemble

polynomial certificates for propositional classes this paper studies the complexity of learning classes of expressions in propositional logic from equivalence queries and membership queries in particular we focus on bounding the number of queries that are required to learn the class ignoring computational complexity this quantity is known to be captured by a combinatorial measure of concept classes known as the certificate complexity the paper gives new constructions of polynomial size certificates for monotone expressions in conjunctive normal form cnf for unate cnf functions where each variable affects the function either positively or negatively but not both ways and for horn cnf functions lower bounds on certificate size for these classes are derived showing that for some parameter settings the new certificate constructions are optimal finally the paper gives an exponential lower bound on the certificate size for a natural generalization of these classes known as renamable horn cnf functions thus implying that the class is not learnable from a polynomial number of queries

on the statistical mechanics approach in the random matrix theory integrated density of states we consider the ensemble of random symmetricn n matrices specified by an orthogonal invariant probability distribution we treat this distribution as a gibbs measure of a mean field type model this allows us to show that the normalized eigenvalue counting function of this ensemble converges in probability to a nonrandom limit asn and that this limiting distribution is the solution of a certain self consistent equation

on universality for orthogonal ensembles of random matrices we prove universality of local eigenvalue statistics in the bulk of the spectrum for orthogonal invariant matrix models with real analytic potentials with one interval limiting spectrum our starting point is the tracy widom formula for the matrix reproducing kernel the key idea of the proof is to represent the differentiation operator matrix written in the basis of orthogonal polynomials as a product of a positive toeplitz matrix and a two diagonal skew symmetric toeplitz matrix

universality for orthogonal and symplectic laguerre type ensembles we give a proof of the universality conjecture for orthogonal 1 and symplectic 4 random matrix ensembles of laguerre type in the bulk of the spectrum as well as at the hard and soft spectral edges our results are stated precisely in the introduction theorems 1 1 1 4 1 6 and corollaries 1 2 1 5 1 7 they concern the appropriately rescaled kernels k n correlation and cluster functions gap probabilities and the distributions of the largest and smallest eigenvalues corresponding results for unitary 2 laguerre type ensembles have been proved by the fourth author in ref 23 the varying weight case at the hard spectral edge was analyzed in ref 13 for 2 in this paper we do not consider varying weights our proof follows closely the work of the first two authors who showed in refs 7 8 analogous results for hermite type ensembles as in refs 7 8 we use the version of the orthogonal polynomial method presented in refs 22 25 to analyze the local eigenvalue statistics the necessary asymptotic information on the laguerre type orthogonal polynomials is taken from ref 23

universality of the local eigenvalue statistics for a class of unitary invariant random matrix ensembles this paper is devoted to the rigorous proof of the universality conjecture of random matrix theory according to which the limiting eigenvalue statistics ofn n random matrices within spectral intervals ofo n 1 is determined by the type of matrix real symmetric hermitian or quaternion real and by the density of states we prove this conjecture for a certain class of the hermitian matrix ensembles that arise in the quantum field theory and have the unitary invariant distribution defined by a certain function the potential in the quantum field theory satisfying some regularity conditions

how many queries are needed to learn we investigate the query complexity of exact learning in the membership and proper equivalence query model we give a complete characterization of concept classes that are learnable with a polynomial number of polynomial sized queries in this model we give applications of this characterization including results on learning a natural subclass of dnf formulas and on learning with membership queries alone query complexity has previously been used to prove lower bounds on the time complexity of exact learning we show a new relationship between query complexity and time complexity in exact learning if any honest class is exactly and properly learnable with polynomial query complexity but not learnable in polynomial time then p np in particular we show that an honest class is exactly polynomial query learnable if and only if it is learnable using an oracle for p 4

logical settings for concept learning abstract three different formalizations of concept learning in logic as well as some variants are analyzed and related it is shown that learning from interpretations reduces to learning from entailment which in turn reduces to learning from satisfiability the implications of this result for inductive logic programming and computational learning theory are then discussed and guidelines for choosing a problem setting are formulated

bulk universality and related properties of hermitian matrix models we give a new proof of universality properties in the bulk of spectrum of the hermitian matrix models assuming that the potential that determines the model is globally c 2 and locally c 3 function see theorem 3 1 the proof as our previous proof in pastur and shcherbina in j stat phys 86 109 147 1997 is based on the orthogonal polynomial techniques but does not use asymptotics of orthogonal polynomials rather we obtain the sin kernel as a unique solution of a certain non linear integro differential equation that follows from the determinant formulas for the correlation functions of the model we also give a simplified and strengthened version of paper boutet de monvel et al in j stat phys 79 585 611 1995 on the existence and properties of the limiting normalized counting measure of eigenvalues we use these results in the proof of universality and we believe that they are of independent interest