moment matrices and multi component kp with applications to random matrix theory questions on random matrices and non intersecting brownian motions have led to the study of moment matrices with regard to several weights the main result of this paper is to show that the determinants of such moment matrices satisfy upon adding one set of time deformations for each weight the multi component kp hierarchy these determinants are thus tau functions for these integrable hierarchies the tau functions so obtained with appropriate shifts of the time parameters forward and backwards will be expressed in terms of multiple orthogonal polynomials for these weights and their cauchy transforms the main result is a vast generalization of a known fact about infinitesimal deformations of orthogonal polynomials it concerns an identity between the orthogonality of polynomials on the real line the bilinear identity in kp theory and a generating functional for the full kp theory an additional fact not discussed in this paper is that these functions satisfy virasoro constraints with respect to these time parameters

differential equations for multiple orthogonal polynomials with respect to classical weights raising and lowering operators we obtain a lowering operator for multiple orthogonal polynomials having orthogonality conditions with respect to r n classical weights these multiple orthogonal polynomials are generalizations of the classical orthogonal polynomials combining the lowering operator with the raising operators which have been obtained earlier in the literature we then also obtain a linear differential equation of order r 1

ladder operators and differential equations for multiple orthogonal polynomials in this paper we obtain the ladder operators and associated compatibility conditions for types i and ii multiple orthogonal polynomials these ladder equations extend known results for orthogonal polynomials and can be used to derive the differential equations satisfied by multiple orthogonal polynomials our approach is based on riemann hilbert problems and the christoffel darboux formula for multiple orthogonal polynomials and the nearest neighbor recurrence relations as an illustration we give several explicit examples involving multiple hermite and laguerre polynomials and multiple orthogonal polynomials with exponential weights and cubic potentials

block orthogonal polynomials i definitions and properties constrained orthogonal polynomials have been recently introduced in the study of the hohenberg kohn functional to provide basis functions satisfying particle number conservation for an expansion of the particle density more generally we define block orthogonal bo polynomials which are orthogonal with respect to a first euclidean inner product to a given dimensional subspace e i of polynomials associated with the constraints in addition they are mutually orthogonal with respect to a second euclidean inner product we recast the determination of these polynomials into a general problem of finding particular orthogonal bases in an euclidean vector space endowed with distinct inner products an explicit two step gram schmidt orthogonalization g so process to determine these bases is given by definition the standard block orthogonal sbo polynomials are associated with a choice of e i equal to the subspace of polynomials of degree less than i we investigate their properties emphasizing similarities to and differences from the standard orthogonal polynomials applications to classical orthogonal polynomials will be given in forthcoming papers

a general and exact method for determining limit cycles in decentralized relay systems an exact method for evaluating the periods and stability of limit cycles in decentralized relay systems is presented the method can cope with relays having dead zones and hystereses the underlying idea is to convert the continuous decentralized relay system under a limit cycle to an equivalent fictitious sampled data system with synchronous samplers consequently regular sampled data tools can be applied in a straightforward manner to derive closed form necessary conditions as well as stability conditions and other properties of such systems this paper extends and generalises recent results obtained by the authors

a remark concerning percolation thresholds in polydisperse systems of finite diameter rods a lattice based analysis of the percolation threshold for randomly distributed cylindrical particles is generalized to consider arbitrary joint distributions over the radii and lengths of the rods effects due to the finite hard core diameter of the particles are accounted for an analogy to site percolation on a modified bethe lattice is exploited to yield a result for the percolation threshold that is equivalent to one that has been obtained from integral equation methods in the limit of large aspect ratios for the rods

multiple correction and continued fraction approximation abstract the main aim of this paper is to further develop a multiple correction method formulated in a previous work 6 as applications we find a kind of hybrid type finite continued fraction approximations in two cases of landau constants and lebesgue constants in addition we refine the previous results of lu 29 and xu and you 48 for the euler mascheroni constant

zeta function regularization the multiplicative anomaly and the wodzicki residue the multiplicative anomaly associated with the zeta function regularized determinant is computed for the laplace type operators v1 v1 and v2 v2 with v1 v2 constant in a d dimensional compact smooth manifold md making use of several results due to wodzicki and by direct calculations in some explicit examples it is found that the multiplicative anomaly is vanishing for d odd and for d 2 an application to the one loop effective potential of the o 2 self interacting scalar model is outlined

generalizations of orthogonal polynomials we give a survey of recent generalizations of orthogonal polynomials that includes multidimensional matrix and vector orthogonal polynomials and multivariate versions multipole orthogonal rational functions variants and extensions of the orthogonality conditions multiple orthogonality most of these generalizations are inspired by the applications in which they are applied we also give a glimpse of these applications which are usually generalizations of applications where classical orthogonal polynomials also play a fundamental role moment problems numerical quadrature rational approximation linear algebra recurrence relations and random matrices

the operator moment problem vector continued fractions and an explicit form of the favard theorem for vector orthogonal polynomials abstract we study the operator moment problem for a p 2 diagonal nonsymmetric matrix operator and give an explicit form of the favard theorem for vector orthogonal polynomials as a main tool we use the development of a system of resolvent functions of the operator in a vector continued fraction by a modified jacobi perron algorithm

percolation thresholds and excluded area for penetrable rectangles in two dimensions the percolation threshold for penetrable rectangles in two dimensions is examined by way of an analogy to a lattice model that employs the averaged pairwise excluded area between particles as a key ingredient percolation thresholds are calculated as functions of i the aspect ratio of the rectangles for the case of a random isotropic orientational distribution and ii the orientational order parameter for situations in which the particles exhibit varying degrees of alignment the threshold volume fraction is found to rise with increases in the degree of particle alignment and our results for isotropically distributed systems are in close agreement with findings from recent monte carlo mc simulations

series gevrey de type arithmetique i theoremes de purete et de dualite gevrey series are ubiquitous in analysis any series satisfying some possibly non linear analytic differential equation is gevrey of some rational order the present work stems from two observations 1 the classical gevrey series e g generalized hypergeometric series with rational parameters enjoy arithmetic counterparts of the archimedean gevrey condition 2 the differential operators which occur in classical treatises on special functions have a rather simple structure they are either fuchsian or have only two singularities 0 and infinity one of them regular the other irregular with a single slope the main idea of the paper is that the arithmetic property 1 accounts for the global analytic property 2 the existence of an injective arithmetic gevrey solution at one point determines to a large extent the global behaviour of a differential operator with polynomial coefficients proofs use both p adic and complex analysis and a detailed arithmetic study of the laplace transform

on askey scheme and d orthogonality i a characterization theorem this is the first in a series of papers dealing with generalized hypergeometric d orthogonal polynomials extending the polynomial families in the askey scheme in this paper we give a characterization theorem to introduce new examples of generalized hypergeometricd orthogonal polynomials to be studied in the forthcoming works for d 1 we obtain an unification of some known characterization theorems in the orthogonal polynomials theory

g functions and multisum versus holonomic sequences abstract the purpose of the paper is three fold a we prove that every sequence which is a multidimensional sum of a balanced hypergeometric term has an asymptotic expansion of gevrey type 1 with rational exponents b we construct a class of g functions that come from enumerative combinatorics and c we give a counterexample to a question of zeilberger that asks whether holonomic sequences can be written as multisums of balanced hypergeometric terms the proofs utilize the notion of a g function introduced by siegel and its analytic arithmetic properties shown recently by andre

gaussian quadrature for multiple orthogonal polynomials we study multiple orthogonal polynomials of type i and type ii which have orthogonality conditions with respect to r measures these polynomials are connected by their recurrence relation of order r 1 first we show a relation with the eigenvalue problem of a banded lower hessenberg matrix l n containing the recurrence coefficients as a consequence we easily find that the multiple orthogonal polynomials of type i and type ii satisfy a generalized christoffel darboux identity furthermore we explain the notion of multiple gaussian quadrature for proper multi indices which is an extension of the theory of gaussian quadrature for orthogonal polynomials and was introduced by borges in particular we show that the quadrature points and quadrature weights can be expressed in terms of the eigenvalue problem of l n

limit cycles in decentralized relay systems abstract the problem of limit cycles in decentralized relay systems is revisited an exact method for evaluating the periods of limit cycles and determining their stability in such systems is presented though the necessary conditions derived in this paper are equivalent to the existing and well known infinite series counterpart conditions the new conditions are constructive and easier to apply since they are given in closed form the new method is applicable to any system and uses the z transform familiar to most control engineers and for which extensive tables can be found in most textbooks on discrete time control examples demonstrate the usefulness of the new conditions the new method also reveals an interesting property of decentralized relay systems

euler s constant euler s work and modern developments this paper has two parts the first part surveys euler s work on the constant 0 57721 bearing his name together with some of his related work on the gamma function values of the zeta function and divergent series the second part describes various mathematical developments involving euler s constant as well as another constant the euler gompertz constant these developments include connections with arithmetic functions and the riemann hypothesis and with sieve methods random permutations and random matrix products it also includes recent results on diophantine approximation and transcendence related to euler s constant

d orthogonality of little q laguerre type polynomials abstract in this paper we solve a characterization problem in the context of the d orthogonality that allows us on one hand to provide a q analog for the d orthogonal polynomials of laguerre type introduced by the first author and douak and on the other hand to derive new l q classical d orthogonal polynomials generalizing the little q laguerre polynomials various properties of the resulting basic hypergeometric polynomials are singled out for d 1 we obtain a characterization theorem involving as far as we know new l q classical orthogonal polynomials for which we give the recurrence relation and the difference equation

global stability of relay feedback systems for a large class of relay feedback systems rfs there will be limit cycle oscillations conditions to check existence and local stability of limit cycles for these systems are well known global stability conditions however are practically nonexistent the paper presents conditions in the form of linear matrix inequalities lmis that when satisfied guarantee global asymptotic stability of limit cycles induced by relays with hysteresis in feedback with linear time invariant lti stable systems the analysis consists in finding quadratic surface lyapunov functions for poincare maps associated with rfs these results are based on the discovery that a typical poincare map induced by an lti flow between two hyperplanes can be represented as a linear transformation analytically parametrized by a scalar function of the state moreover level sets of this function are convex subsets of linear manifolds the search for quadratic lyapunov functions on switching surfaces is done by solving a set of lmis although this analysis methodology yields only a sufficient criterion of stability it has proved very successful in globally analyzing a large number of examples with a unique locally stable symmetric unimodal limit cycle in fact it is still an open problem whether there exists an example with a globally stable symmetric unimodal limit cycle that could not be successfully analyzed with this new methodology examples analyzed include minimum phase systems systems of relative degree larger than one and of high dimension such results lead us to believe that globally stable limit cycles of rfs frequently have quadratic surface lyapunov functions

microsolutions of differential operators and values of arithmetic gevrey series we continue our investigation of e operators in particular their connection with g operators these differential operators are fundamental in understanding the dio phantine properties of siegel s e and g functions we study in detail microsolutions in kashiwara s sense of fuchsian differential operators and apply this to the construction of basis of solutions at 0 and of any e operator from microsolutions of a g operator this provides a constructive proof of a theorem of andre we also focus on the arithmetic nature of connection constants and stokes constants between different bases of solutions of e operators for this we introduce and study in details an arithmetic inverse laplace transform that enables one to get rid of transcendental numbers inherent to andre s original approach as an application we define a set of special values of arithmetic gevrey series and discuss its conjectural relation with the ring of exponential periods of kontsevich zagier

multiple correction and continued fraction approximation ii the main aim of this paper is to further develop the multiple correction method formulated in our previous works cao et al 2015 cao 2015 as applications we establish a kind of hybrid type finite continued fraction approximations related to bbp type series of the constant and other classical constants such as catalan constant or 2

precise percolation thresholds of two dimensional random systems comprising overlapping ellipses abstract this work explores the percolation thresholds of continuum systems consisting of randomly oriented overlapping ellipses high precision percolation thresholds for various homogeneous ellipse systems with different aspect ratios are obtained from extensive monte carlo simulations based on the incorporation of vieillard baron s contact function of two identical ellipses with our efficient algorithm for continuum percolation in addition we generalize vieillard baron s contact function from identical ellipses to unequal ellipses and extend the monte carlo algorithm to heterogeneous ellipse systems where the ellipses have different dimensions and or aspect ratios based on the concept of modified excluded area a general law is verified for precise prediction of percolation threshold for many heterogeneous ellipse systems in particular the study of heterogeneous ellipse systems gains insight into the apparent percolation threshold symmetry observed earlier in systems comprising unequal circles consiglio et al 2004

new results on the stieltjes constants asymptotic and exact evaluation abstract the stieltjes constants k a are the expansion coefficients in the laurent series for the hurwitz zeta function about s 1 we present new asymptotic summatory and other exact expressions for these and related constants

rational approximations for values of derivatives of the gamma function the arithmetic nature of euler s constant is still unknown and even getting good rational approximations to it is difficult recently aptekarev managed to find a third order linear recurrence with polynomial coefficients which admits two rational solutions a n and b n such that a n b n converges sub exponentially to viewed as 1 where r is the usual gamma function although this is not yet enough to prove that it is a major step in this direction in this paper we present a different but related approach based on simultaneous pade approximants to euler s functions from which we construct and study a new third order recurrence that produces a sequence in z whose height grows like the factorial and that converges sub exponentially to log z for any complex number z 0 where log is defined by its principal branch we also show how our approach yields in theory rational approximations of numbers related to s 1 for any integer s 1 in particular we construct a sixth order recurrence which provides simultaneous rational approximations of factorial height converging sub exponentially to the numbers and 1 2 1 2 2 2

stieltjes continued fraction and qd algorithm scalar vector and matrix cases the definition in previous studies of vector stieltjes continued fractions in connection with spectral properties of band operators with intermediate zero diagonals left unsolved the question of a direct definition of their coefficients in terms of the original data a vector of stieltjes series the subject was more undefined in the matrix case a new version of the qd algorithm for matrix problem allows to extend to the vector and matrix cases the result of stieltjes expansion of a scalar function in terms of a stieltjes continued fraction beside this connection it solves the inverse miura transform and gives interesting identities between general band matrix and sparse band matrix finally as a consequence we extend to some dynamical systems a method known for toda lattices

exponential sums and lattice points iii the gauss circle problem and the dirichlet divisor problem are special cases of the problem of counting the points of the integer lattice in a planar domain bounded by a piecewise smooth curve in the bombieri iwaniec mozzochi exponential sums method we must count the number of pairs of arcs of the boundary curve which can be brought into coincidence by an automorphism of the integer lattice these coincidences are parametrised by integer points close to certain plane curves the resonance curves this paper sets up an iteration step from a strong hypothesis about integer points close to curves to a bound for the discrepancy the number of integer points minus the area as in the latest work on single exponential sums the bombieri iwaniec mozzochi method itself gives bounds for the number of integer points close to a curve in part of the required range and it can in principle be used iteratively we use a bound obtained by swinnerton dyer s approximation determinant method in the discrepancy estimate o r k log r lambda in terms of the maximum radius of curvature r we reduce k from 2 3 classical and 46 73 paper ii in this series to 131 208 the corresponding exponent in the dirichlet divisor problem becomes k 2 131 416

multiple orthogonal polynomials for classical weights a new set of special functions which has a wide range of applications from number theory to integrability of nonlinear dynamical systems is described we study multiple orthogonal polynomials with respect to p 1 weights satisfying pearson s equation in particular we give a classification of multiple orthogonal polynomials with respect to classical weights which is based on properties of the corresponding rodrigues operators we show that the multiple orthogonal polynomials in our classification satisfy a linear differential equation of order p 1 we also obtain explicit formulas and recurrence relations for these polynomials

globally nilpotent differential operators and the square ising model we recall various multiple integrals with one parameter related to the isotropic square ising model and corresponding respectively to the n particle contributions of the magnetic susceptibility to the lattice form factors to the two point correlation functions and to their extensions the univariate analytic functions defined by these integrals are holonomic and even g functions they satisfy fuchsian linear differential equations with polynomial coefficients and have some arithmetic properties we recall the explicit forms found in previous work of these fuchsian equations as well as their russian doll and direct sum structures these differential operators are selected fuchsian linear differential operators and their remarkable properties have a deep geometrical origin they are all globally nilpotent or sometimes even have zero p curvature we also display miscellaneous examples of globally nilpotent operators emerging from enumerative combinatorics problems for which no integral representation is yet known focusing on the factorized parts of all these operators we find out that the global nilpotence of the factors resp p curvature nullity corresponds to a set of selected structures of algebraic geometry elliptic curves modular curves curves of genus five six and even a remarkable weight 1 modular form emerging in the three particle contribution 3 of the magnetic susceptibility of the square ising model noticeably this associated weight 1 modular form is also seen in the factors of the differential operator for another n fold integral of the ising class 3 h for the staircase polygons counting and in ap ry s study of 3 g functions naturally occur as solutions of globally nilpotent operators in the case where we do not have g functions but hamburger functions one irregular singularity at 0 or that correspond to the confluence of singularities in the scaling limit the p curvature is also found to verify new structures associated with simple deformations of the nilpotent property

the generalized euler constant function γ z and a generalization of somos s quadratic recurrence constant abstract we define the generalized euler constant function z n 1 z n 1 1 n log n 1 n when z 1 its values include both euler s constant 1 and the alternating euler constant log 4 1 we extend euler s two zeta function series for to polylogarithm series for z integrals for z provide its analytic continuation to c 1 we prove several other formulas for z including two functional equations one is an inversion relation between z and 1 z we generalize somos s quadratic recurrence constant and sequence to cubic and other degrees give asymptotic estimates and show relations to z and to an infinite nested radical due to ramanujan we calculate z and z at roots of unity in particular 1 involves the glaisher kinkelin constant a several related series infinite products and double integrals are evaluated the methods used involve the kinkelin bendersky hyperfactorial k function the weierstrass products for the gamma and barnes g functions and jonquiere s relation for the polylogarithm

some discrete multiple orthogonal polynomials in this paper we extend the theory of discrete orthogonal polynomials on a linear lattice to polynomials satisfying orthogonality conditions with respect to r positive discrete measures first we recall the known results of the classical orthogonal polynomials of charlier meixner kravchuk and hahn t s chihara an introduction to orthogonal polynomials gordon and breach new york 1978 r koekoek and r f swarttouw reports of the faculty of technical mathematics and informatics no 98 17 delft 1998 a f nikiforov et al classical orthogonal polynomials of a discrete variable springer berlin 1991 these polynomials have a lowering and raising operator which give rise to a rodrigues formula a second order difference equation and an explicit expression from which the coefficients of the three term recurrence relation can be obtained then we consider r positive discrete measures and define two types of multiple orthogonal polynomials the continuous case jacobi laguerre hermite etc was studied by van assche and coussement j comput appl math 127 2001 317 347 and aptekarev et al multiple orthogonal polynomials for classical weights manuscript the families of multiple orthogonal polynomials of type ii that we will study have a raising operator and hence a rodrigues formula this will give us an explicit formula for the polynomials finally there also exists a recurrence relation of order r 1 for these multiple orthogonal polynomials of type ii we compute the coefficients of the recurrence relation explicitly when r 2

multiple orthogonal polynomials results on multiple orthogonal polynomials will be surveyed multiple orthogonal polynomials are intimately related to hermite pade approximants and often they are also called hermite pade polynomials special attention will be paid to an application of multiple orthogonal polynomials and to analytic theory of two model families of general multiple orthogonal polynomials referred to as angelesco and nikishin systems among the applications the number theory special functions and spectral analysis of nonsymmetric band operators will be highlighted in the analytic theory results and methods for the study of multiple orthogonal polynomials asymptotics will be reviewed new results on strong asymptotics of multiple orthogonal polynomials for nikishin system will be presented

a riemann hilbert problem for biorthogonal polynomials we characterize the biorthogonal polynomials that appear in the theory of coupled random matrices via a riemann hilbert problem our riemann hilbert problem is different from the ones that were proposed recently by ercolani and mclaughlin kapaev and bertola et al we believe that our formulation may be tractable to asymptotic analysis

local stability of limit cycles for mimo relay feedback systems abstract this paper concerns with the local stability of limit cycles for decentralized relay feedback systems it presents a sufficient condition for the local stability based on the well known poincare map method the effectiveness of the presented result is illustrated by a numerical example

percolation thresholds of two dimensional continuum systems of rectangles the present paper introduces an efficient monte carlo algorithm for continuum percolation composed of randomly oriented rectangles by conducting extensive simulations we report high precision percolation thresholds for a variety of homogeneous systems with different rectangle aspect ratios this paper verifies and extends the excluded area theory it is confirmed that percolation thresholds are dominated by the average excluded areas for both homogeneous and heterogeneous rectangle systems except for some special heterogeneous systems where the rectangle lengths differ too much from one another in terms of the excluded areas generalized formulas are proposed to effectively predict precise percolation thresholds for all these rectangle systems this paper is therefore helpful for both practical applications and theoretical studies concerning relevant systems

legendre modified moments for euler s constant polynomial moments are often used for the computation of gauss quadrature to stabilize the numerical calculation of the orthogonal polynomials see w gautschi computational aspects of orthogonal polynomials in p nevai ed orthogonal polynomials theory and practice nato asi series series c mathematical and physical sciences vol 294 kluwer dordrecht 1990 pp 181 216 6 w gautschi on the sensitivity of orthogonal polynomials to perturbations in the moments numer math 48 4 1986 369 382 5 w gautschi on generating orthogonal polynomials siam j sci statist comput 3 3 1982 289 317 4 or numerical resolution of linear systems c brezinski pade type approximation and general orthogonal polynomials isnm vol 50 basel boston stuttgart birkhauser 1980 3 these modified moments can also be used to accelerate the convergence of sequences to a real or complex numbers if the error satisfies some properties as done in c brezinski acceleration de la convergence en analyse numerique lecture notes in mathematics vol 584 springer berlin new york 1977 m prevost pade type approximants with orthogonal generating polynomials j comput appl math 9 4 1983 333 346 in this paper we use legendre modified moments to accelerate the convergence of the sequence h n log n 1 to the euler s constant c a formula for the error is given it is proved that it is a totally monotonic sequence at last we give applications to the arithmetic property of c

finite size scaling in stick percolation this work presents the generalization of the concept of universal finite size scaling functions to continuum percolation a high efficiency algorithm for monte carlo simulations is developed to inv

two differential difference equations arising in number theory we survey many old and new results on solutions of the following pair of adjoint differential difference equations 1 up u ap u bp u 1 2 uq u aq u bq u 1 we bring together scattered results usually proved only for specific a b pairs while emphasizing the connections between the two equations we also point out some of the ways these two equations are used in number theory we give several new integral relationships between 1 and 2 and use them to prove a new application of 2 in number theory namely e log p n u tf u x log x k x oo u i a er i n x p2 n p n i u where p1 n and p2 n are the first and second largest prime divisors of n and f u satisfies 2 with a b 1 c 1

further results on generalized multiple fractional part integrals for complex values in this paper the following multiple fractional part integrals i n 1 2 n 0 1 n j 1 n x j j s n 1 d x 1 d x n and j n 0 1 n s n s n 1 d x 1 d x n are studied for positive integer n and complex values of j j 1 2 n where u denotes the fractional part of u r s denotes the real part of s and s n x 1 x 2 x n it is proved that i 1 can be represented as a linear combination of the riemann zeta function the beta function and euler s constant as r 1 moreover i n 1 2 n can be expressed by i n 1 1 2 n 1 the beta function and the incomplete beta function for n 2 3 in addition the recurrence formula of j n n 2 3 is established and j n can be expressed by i 1 logarithmic function and some binomial coefficients

d orthogonality of humbert and jacobi type polynomials in this paper we treat three questions related to the d orthogonality of the humbert polynomials the first one consists to determinate the explicit expression of the d dimensional functional vector for which the d orthogonality holds the second one is the investigation of the components of humbert polynomial sequence that allows us to introduce as far as we know new d orthogonal polynomials generalizing the classical jacobi ones the third one consists to solve a characterization problem related to a generalized hypergeometric representation of the humbert polynomials

some generalized hypergeometric d orthogonal polynomial sets in this paper we characterize the d orthogonal polynomial sets given by their explicit expressions in a specific basis as application we consider the generalized hypergeometric case to characterize d orthogonal polynomial sets of laguerre type meixner type meixner pollaczek type krawtchouk type continuous dual hahn type and dual hahn type for d 1 we obtain a unification of some characterization theorems in the orthogonal polynomials theory

the stieltjes function definition and properties definition et proprietes d une fonction liee au developpement en serie de laurent de la fonction zeta de riemann au voisinage du point singulier s 1 fonction de stieltjes

arithmetical properties of some series with logarithmic coefficients we prove approximation formulas for the logarithms of some infinite products in particular for euler s constant log frac 4 pi and log where is somos s quadratic recurrence constant in terms of classical legendre polynomials and partial sums of their series expansions we also give conditional irrationality and linear independence criteria for these numbers the main tools are euler type integrals hypergeometric series and laplace method

a differential delay equation arising from the sieve of eratosthenes consideration is given to the differential delay equation introduced by buchstab 1937 in connection with an asymptotic formula for the uncanceled terms in the sieve of eratosthenes maier 1985 used this result to show there is unexpected irreqularity in the distribution of primes in short intervals the function omega u is studied in this paper using numerical and analytical techniques the results are applied to give some numerical constants in maier s theorem

pdes for the gaussian ensemble with external source and the pearcey distribution the present paper studies a gaussian hermitian random matrix ensemble with external source given by a fixed diagonal matrix with two eigenvalues a as a first result the probability that the eigenvalues of the ensemble belong to an interval e satisfies a fourth order pde with quartic nonlinearity the variables are the eigenvalue a and the boundary of e this equation enables one to find a pde for the pearcey distribution the latter describes the statistics of the eigenvalues near the closure of a gap i e when the support of the equilibrium measure for large size random matrices has a gap that can be made to close the gaussian hermitian random matrix ensemble with external source described above has this feature the pearcey distribution is shown to satisfy a fourth order pde with cubic nonlinearity this also gives the pde for the transition probability of the pearcey process a limiting process associated with nonintersecting brownian motions on r c 2006 wiley periodicals inc

exponential sums and the riemann zeta function v a van der corput exponential sum is s sigma exp 2 pi i f m where m has size m the function f x has size t and alpha log m log t 1 there are different bounds for s in different ranges for alpha in the middle range where alpha is near 1 over 2 s o sqrt m t theta epsilon this theta bounds the exponent of growth of the riemann zeta function on its critical line rm re s 1 over 2 van der corput used an iteration which changed alpha at each step the bombieri iwaniec method whilst still based on mean squares introduces number theoretic ideas and problems the second spacing problem is to count the number of resonances between short intervals of the sum when two arcs of the graph of y f x coincide approximately after an automorphism of the integer lattice in the previous paper in this series proc london math soc 3 66 1993 1 40 and the monograph area lattice points and exponential sums we saw that coincidence implies that there is an integer point close to some resonance curve one of a family of curves in some dual space now calculated accurately in the paper resonance curves in the bombieri iwaniec method which is to appear in funct approx comment math we turn the whole bombieri iwaniec method into an axiomatised step an upper bound for the number of integer points close to a plane curve gives a bound in the second spacing problem and a small improvement in the bound for s ends and cusps of resonance curves are treated separately bounds for sums of type s lead to bounds for integer points close to curves and another branching iteration luckily swinnerton dyer s method is stronger we improve theta from 0 156140 in the previous paper and monograph to 0 156098 in fact 32 205 epsilon 269 410 epsilon is an exponent pair for every epsilon 0

multiple orthogonal polynomials on the semicircle and applications in this paper two types of multiple orthogonal polynomials on the semicircle with respect to a set of r different weight functions are defined such polynomials are generalizations of polynomials orthogonal on the semicircle with respect to a complex valued inner product f g email protected 0 pfe i qge i qwe i email protected the existence and uniqueness of introduced multiple orthogonal polynomials for certain classes of weight functions are proved some properties of multiple orthogonal polynomials on the semicircle including certain recurrence relations of order r 1 are presented finally an application in numerical integration is given

large n limit of gaussian random matrices with external source part i we continue the study of the hermitian random matrix ensemble with external source open image in new window where a has two distinct eigenvalues a of equal multiplicity this model exhibits a phase transition for the value a 1 since the eigenvalues of m accumulate on two intervals for a 1 and on one interval for 0 1 was treated in part i where it was proved that local eigenvalue correlations have the universal limiting behavior which is known for unitarily invariant random matrices that is limiting eigenvalue correlations are expressed in terms of the sine kernel in the bulk of the spectrum and in terms of the airy kernel at the edge in this paper we establish the same results for the case 0 a 1 as in part i we apply the deift zhou steepest descent analysis to a 3 3 matrix riemann hilbert problem due to the different structure of an underlying riemann surface the analysis includes an additional step involving a global opening of lenses which is a new phenomenon in the steepest descent analysis of riemann hilbert problems

renormalization of multiple zeta values abstract multiple zeta values mzvs in the usual sense are the special values of multiple variable zeta functions at positive integers their extensive studies are important in both mathematics and physics with broad connections and applications in contrast very little is known about the special values of multiple zeta functions at non positive integers since the values are usually undefined we define and study multiple zeta functions at integer values by adapting methods of renormalization from quantum field theory and following the hopf algebra approach of connes and kreimer this definition of renormalized mzvs agrees with the convergent mzvs and extends the work of ihara kaneko zagier on renormalization of mzvs with positive arguments we further show that the important quasi shuffle stuffle relation for usual mzvs remains true for the renormalized mzvs

an effective asymptotic formula for the stieltjes constants the stieltjes constants k appear in the coefficients in the regular part of the laurent expansion of the riemann zeta function s about its only pole at s 1 we present an asymptotic expression for k for k 1 this form encapsulates both the leading rate of growth and the oscillations with k furthermore our result is effective for computation consistently in close agreement for both magnitude and sign for even moderate values of k comparison to some earlier work is made

random graphs with arbitrary degree distributions and their applications recent work on the structure of social networks and the internet has focused attention on graphs with distributions of vertex degree that are significantly different from the poisson degree distributions that have been widely studied in the past in this paper we develop in detail the theory of random graphs with arbitrary degree distributions in addition to simple undirected unipartite graphs we examine the properties of directed and bipartite graphs among other results we derive exact expressions for the position of the phase transition at which a giant component first forms the mean component size the size of the giant component if there is one the mean number of vertices a certain distance away from a randomly chosen vertex and the average vertex vertex distance within a graph we apply our theory to some real world graphs including the worldwide web and collaboration graphs of scientists and fortune 1000 company directors we demonstrate that in some cases random graphs with appropriate distributions of vertex degree predict with surprising accuracy the behavior of the real world while in others there is a measurable discrepancy between theory and reality perhaps indicating the presence of additional social structure in the network that is not captured by the random graph

bifurcation of periodic orbits in a class of planar filippov systems abstract in this paper we discuss the perturbations of a general planar filippov system with exactly one switching line when the system has a limit cycle we give a condition for its persistence when the system has an annulus of periodic orbits we give a condition under which limit cycles are bifurcated from the annulus we also further discuss the stability and bifurcations of a nonhyperbolic limit cycle when the system has an annulus of periodic orbits we show via an example how the number of limit cycles bifurcated from the annulus is affected by the switching

generalized euler constants for arithmetical progressions the work of lehmer and briggs on euler constants in arithmetical progressions is extended to the generalized euler constants that arise in the laurent expansion of g s about s 1 the results are applied to the summation of several classes of slowly converging series a table of the constants is provided

four parametric linear euler sums abstract in this paper firstly we aim to investigate analytic continuations of altogether four types of parametric linear euler sums by using the euler maclaurin summation formula and the euler boole summation formula secondly we show how nicely three shuffle relations among the four parametric linear euler sums can be derived by using an l summing formula due to hassani and rahimpour thirdly using one of the shuffle relations we present certain series representations of product of riemann zeta functions some special cases and relevant connections of the results presented here with those involving known identities are also demonstrated and indicated

a pde for non intersecting brownian motions and applications abstract consider n n 1 n 2 n p non intersecting brownian motions on the real line starting from the origin at t 0 with n i particles forced to reach p distinct target points i at time t 1 with 1 2 p this can be viewed as a diffusion process in a sector of r n this work shows that the transition probability that is the probability for the particles to pass through windows e k at times t k satisfies in a new set of variables a non linear pde which can be expressed as a near wronskian that is a determinant of a matrix of size p 1 with each row being a derivative of the previous except for the last column it is an interesting open question to understand those equations from a more probabilistic point of view as an application of these equations let the number of particles forced to the extreme points 1 and p tend to infinity keep the number of particles forced to intermediate points fixed inliers but let the target points themselves go to infinity according to a proper scale a new critical process appears at the point of bifurcation where the bulk of the particles forced to n depart from those going to n these statistical fluctuations near that point of bifurcation are specified by a kernel which is a rational perturbation of the pearcey kernel this work also shows that such equations are an essential tool in obtaining certain asymptotic results finally the paper contains a conjecture

the approximate invariance of the average number of connections for the continuum percolation of squares at criticality we perform monte carlo simulations to determine the average excluded area aex of randomly oriented squares randomly oriented widthless sticks and aligned squares in two dimensions we find significant differences between our results for randomly oriented squares and previous analytical results for the same the sources of these differences are explained using our results for aex and monte carlo simulation results for the percolation threshold we estimate the mean number of connections per object bc at the percolation threshold for squares in 2 d we study systems of squares that are allowed random orientations within a specified angular interval our simulations show that the variation in bc is within 1 6 when the angular interval is varied from 0 to 2

maximal lyapunov exponents for random matrix products in this article we study the lyapunov exponent for random matrix products of positive matrices and express them in terms of associated complex functions this leads to new explicit formulae for the lyapunov exponents and to an efficient method for their computation

d orthogonality via generating functions in this paper starting from a suitable generating function of a polynomial set we show how to decide whether the considered polynomial set is d orthogonal and if it is so how to determine the corresponding d dimensional functional vector then we apply the obtained results to some known and new d orthogonal polynomial sets for the known ones we give new proofs for some already obtained results

some discrete d orthogonal polynomial sets in this paper we characterize all polynomial sets which are at the same time d orthogonal and appell the resulting polynomials reduce to charlier polynomials for d 1 1 various properties of the obtained polynomials are singled out generating function recurrence relation of order d 1 and a difference equation of order d 1 we also explicitly express the d dimensional functional for which the d orthogonality holds