replica treatment of non hermitian disordered hamiltonians we employ the fermionic and bosonic replicated nonlinear models to treat ginibre unitary symplectic and orthogonal ensembles of non hermitian random matrix hamiltonians using the saddle point approach combined with the borel resummation procedure we derive the exact large n results for microscopic density of states in all three ensembles we also obtain tails of the density of states as well as the two point function for the unitary ensemble

universal microscopic correlation functions for products of independent ginibre matrices we consider the product of n complex non hermitian independent random matrices each of size n n with independent identically distributed gaussian entries ginibre matrices the joint probability distribution of the complex eigenvalues of the product matrix is found to be given by a determinantal point process as in the case of a single ginibre matrix but with a more complicated weight given by a meijer g function depending on n using the method of orthogonal polynomials we compute all eigenvalue density correlation functions exactly for finite n and fixed n they are given by the determinant of the corresponding kernel which we construct explicitly in the large n limit at fixed n we first determine the microscopic correlation functions in the bulk and at the edge of the spectrum after unfolding they are identical to that of the ginibre ensemble with n 1 and thus universal in contrast the microscopic correlations we find at the origin differ for each n 1 and generalize the known bessel law in the complex plane for n 2 to a new hypergeometric kernel 0fn 1

harbourne schenck and seceleanu s conjecture abstract in 2 conjecture 5 5 2 harbourne schenck and seceleanu conjectured that for r 6 and all r 8 the artinian ideal i l 1 2 l r 1 2 k x 1 x r generated by the square of r 1 general linear forms l i fails the weak lefschetz property this paper is entirely devoted to prove this conjecture it is worthwhile to point out that half of the conjecture namely the case when the number of variables r is even was already proved in 5 theorem 6 1

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

on a class of special linear systems of mathbb p 3 in this paper we deal with linear systems of p 3 through fat points we consider the behavior of these systems under a cubic cremona transformation that allows us to produce a class of special systems which we conjecture to be the only ones

products of independent quaternion ginibre matrices and their correlation functions we discuss the product of independent induced quaternion 4 ginibre matrices and the eigenvalue correlations of this product matrix the joint probability density function for the eigenvalues of the product matrix is shown to be identical to that of a single ginibre matrix but with a more complicated weight function we find the skew orthogonal polynomials corresponding to the weight function of the product matrix and use the method of skew orthogonal polynomials to compute the eigenvalue correlation functions for product matrices of finite size the radial behavior of the density of states is studied in the limit of large matrices and the macroscopic density is discussed the microscopic limit at the origin at the edge s and in the bulk is also discussed for the radial behavior of the density of states

matroid configurations and symbolic powers of their ideals star configurations are certain unions of linear subspaces of projective space that have been studied extensively we develop a framework for studying a substantial gen eralization which we call matroid configurations whose ideals generalize stanley reisner ideals of matroids such a matroid configuration is a union of complete intersections of a fixed codimension relating these to the stanley reisner ideals of matroids and using methods of liaison theory allows us in particular to describe the hilbert function and minimal generators of the ideal of what we call a hypersurface configuration we also establish that the symbolic powers of the ideal of any matroid configuration are cohen macaulay as applications we study ideals coming from certain complete hypergraphs and ideals derived from tetrahedral curves we also consider waldschmidt constants and resurgences in particular we determine the resurgence of any star configuration and many hypersurface configurations previously the only non trivial cases for which the resurgence was known were certain monomial ideals and ideals of finite sets of points finally we point out a connection to secant varieties of varieties of reducible forms

eulerian polynomials and b splines an interrelationship between eulerian polynomials eulerian fractions and euler frobe nius polynomials euler frobenius fractions and b splines is presented the properties of eulerian polynomials and eulerian fractions and their applications in b spline interpolation and evaluation of riemann zeta function values at odd integers are given the relation between eulerian numbers and b spline values at knot points are also discussed

skew orthogonal polynomials for the real and quaternion real ginibre ensembles and generalizations there are some distinguished ensembles of non hermitian random matrices for which the joint probability density function can be written down explicitly are unchanged by rotations and furthermore which have the property that the eigenvalues form a pfaffian point process for these ensembles in which the elements of the matrices are either real or real quaternion the kernel of the pfaffian is completely determined by certain skew orthogonal polynomials which permit an expression in terms of averages over the characteristic polynomial and the characteristic polynomial multiplied by the trace we use schur polynomial theory knowledge of the value of a schur polynomial averaged against real and real quaternion gaussian matrices and the selberg integral to evaluate these averages

gap probabilities in non hermitian random matrix theory we compute the gap probability that a circle of radius r around the origin contains exactly k complex eigenvalues four different ensembles of random matrices are considered the ginibre ensembles and their chiral complex counterparts with both complex 2 or quaternion real 4 matrix elements for general non gaussian weights we give a fredholm determinant or pfaffian representation respectively depending on the non hermiticity parameter at maximal non hermiticity that is for rotationally invariant weights the product of fredholm eigenvalues for 4 follows from the 2 case by skipping every second factor in contrast to the known relation for hermitian ensembles on additionally choosing gaussian weights we give new explicit expressions for the fredholm eigenvalues in the chiral case in terms of bessel k and incomplete bessel i functions this compares with known results for the ginibre ensembles in terms of incomplete exponentials furthermore we present an asymptotic expansion of the log

eigenvalue statistics of random real matrices completing ginibre s work we determine the joint probability density of eigenvalues in a gaussian ensemble of real asymmetric matrices which is invariant under orthogonal transformations the symmetry parameter r may vary from 1 antisymmetric ensemble through 0 completely asymmetric ensemble to 1 symmetric ensemble the elliptic law for the average density of eigenvalues in the limit of large dimension is recovered matrices of the type considered appear in models for neural network dynamics and dissipative quantum dynamics

universal cubic eigenvalue repulsion for random normal matrices random matrix models consisting of normal matrices defined by the sole constraint n dag n 0 will be explored it is shown that cubic eigenvalue repulsion in the complex plane is universal with respect to the probability distribution of matrices the density of eigenvalues all correlation functions and level spacing statistics are calculated normal matrix models offer more probability distributions amenable to analytical analysis than complex matrix models where only a model wth a gaussian distribution are solvable the statistics of numerically generated eigenvalues from gaussian distributed normal matrices are compared to the analytical results obtained and agreement is seen

linear subspaces symbolic powers and nagata type conjectures abstract inspired by results of guardo van tuyl and the second author for lines in p 3 we develop asymptotic upper bounds for the least degree of a homogeneous form vanishing to order at least m on a union of disjoint r dimensional planes in p n for n 2 r 1 these considerations lead to new conjectures that suggest that the well known conjecture of nagata for points in p 2 is not an exotic statement but rather a manifestation of a much more general phenomenon which seems to have been overlooked so far

universal eigenvector correlations in quaternionic ginibre ensembles non hermitian random matrices enjoy non trivial correlations in the statistics of their eigenvectors we study the overlap among left and right eigenvectors in ginibre ensembles with quaternion val

comparison of symbolic and ordinary powers of ideals all given rings in this paper are commutative associative with identity and noetherian recently l ein r lazarsfeld and k smith els discovered a remarkable and surprising fact about the behavior of symbolic powers of ideals in affine regular rings of equal characteristic 0 if h is the largest height of an associated prime of i then i hn i n for all n 0 here if w is the complement of the union of the associated primes of i i t denotes the contraction of i t rw to r where rw is the localization of r at the multiplicative system w their proof depends on the theory of multiplier ideals including an asymptotic version and in particular requires resolution of singularities as well as vanishing theorems we want to acknowledge that without their generosity and quickness in sharing their research this manuscript would not exist our objective here is to give stronger results that can be proved by methods that are in some ways more elementary our results are valid in both equal characteristic 0 and in positive prime characteristic p but depend on reduction to characteristic p we use tight closure methods and in consequence we need neither resolution of singularities nor vanishing theorems that may fail in positive characteristic for the most basic form of the result all that we need from tight closure theory is the definition of tight closure and the fact that in a regular ring every ideal is tightly closed we note that the main argument here is closely related to a proof given in hu 5 14 16 p 45 that regular local rings in characteristic p are ufds

skew orthogonal laguerre polynomials for chiral real asymmetric random matrices we apply the method of skew orthogonal polynomials sop in the complex plane to asymmetric random matrices with real elements belonging to two different classes explicit integral representations valid for arbitrary weight functions are derived for the sop and for their cauchy transforms given as the expectation values of traces and determinants or their inverses respectively our proof uses the fact that the joint probability distribution function for all combinations of real eigenvalues and complex conjugate eigenvalue pairs can be written as a product examples for the sop are given in terms of laguerre polynomials for the chiral ensemble also called the non hermitian real wishart laguerre ensemble both without and with the insertion of characteristic polynomials such characteristic polynomials play the role of mass terms in applications to complex dirac spectra in field theory in addition for the elliptic real ginibre ensemble we recover the sop of forrester and nagao in terms of hermite polynomials

eulerian numbers a spline perspective in this paper the spline interpretations of eulerian numbers and refined eulerian numbers are presented many classical results about eulerian numbers can follow from the properties of b splines directly and some new results about the refined eulerian numbers and descent polynomials are also derived specifically the explicit and recurrence formulas for the refined eulerian numbers and descent polynomials are obtained this paper also provides a new approach to study eulerian numbers

on the weak lefschetz property for almost complete intersections generated by uniform powers of general linear forms abstract in 10 conjecture 6 6 migliore the first author and nagel conjectured that for all n 4 the artinian ideal i l 0 d l 2 n 1 d r k x 0 x 2 n generated by the d th powers of 2 n 2 general linear forms fails to have the weak lefschetz property if and only if d 1 this paper is entirely devoted to prove partially this conjecture more precisely we prove that r i fails to have the weak lefschetz property provided 4 n 8 d 4 or d 2 r 1 r 8 4 n 2 r r 2 1

limit linear series basic theory in this paper we introduce techniques for handling the degeneration of linear series on smooth curves as the curves degenerate to a certain type of reducible curves curves of compact type the technically much simpler special case of 1 dimensional series was developed by beauville 2 knudsen 21 23 harris and mumford 17 in the guise of admissible covers it has proved very useful for studying the moduli space of curves the above papers and harris 16 and the simplest sorts of weierstrass points diaz 4 with our extended tools we are able to prove for example that in this paper we present the basic theory of limit linear series necessary for proving these results the results themselves will be taken up in our forthcoming papers 8 12 simpler applications not requiring the tools developed in this paper but perhaps clarified by them have already been given in our papers 5 7

inverse system of a symbolic power i abstract the authors determine macaulay s inverse system to a symbolic power of the graded ideal of functions vanishing at a union of irreducible varieties in projective space

the solution of a chiral random matrix model with complex eigenvalues we describe in detail the solution of the extension of the chiral gaussian unitary ensemble chgue into the complex plane the correlation functions of the model are first calculated for a finite number of n complex eigenvalues where we exploit the existence of orthogonal laguerre polynomials in the complex plane when taking the large n limit we derive new correlation functions in the case of weak and strong non hermiticity thus describing the transition from the chgue to a generalized ginibre ensemble we briefly discuss applications to the dirac operator eigenvalue spectrum in quantum chromodynamics with non vanishing chemical potential this is an extended version of hep th 0204068

random matrices beyond the cartan classification it is known that hermitean random matrix ensembles can be identified with symmetric coset spaces of lie groups or else with tangent spaces of the same this results in a classification of random matrix ensembles as well as applications in practical calculations of physical observables in this paper we show that a large number of non hermitean random matrix ensembles defined by physically motivated symmetries chiral symmetry time reversal invariance space rotation invariance particle hole symmetry or different reality conditions can likewise be identified with symmetric spaces we give explicit representations of the random matrix ensembles identified with lateral algebra subspaces and of the corresponding symmetric subalgebras spanning the group of invariance among the ensembles listed we identify as special cases all the hermitean ensembles identified with the cartan classes of symmetric spaces and the three ginibre ensembles with complex eigenvalues

general eigenvalue correlations for the real ginibre ensemble we rederive in a simplified version the lehmann sommers eigenvalue distribution for the gaussian ensemble of asymmetric real matrices invariant under real orthogonal transformations as a basis for a detailed derivation of a pfaffian generating functional for n point densities this produces a simple free fermion diagram expansion for the correlations leading to quaternion determinants in each order n all will explicitly be given with the help of a very simple symplectic kernel for even dimension n the kernel is valid for both complex and real eigenvalues and describes a deep connection between both a slight modification by an artificial additional grassmannian also solves the more complicated odd n case as illustration we present some numerical results in the space of complex eigenvalue n tuples

uniform bounds and symbolic powers on smooth varieties the purpose of this note is to show how one can use multiplier ideals toestablish effective uniform bounds on the multiplicative behavior of certainfamilies of ideal sheaves on a smooth algebraic variety in particular weprove a quick but rather surprising result concerning the symbolic powersof radical ideals on such a variety let

asymptotics of skew orthogonal polynomials exact integral expressions of the skew orthogonal polynomials involved in orthogonal 1 and symplectic 4 random matrix ensembles are obtained the even rank skew orthogonal polynomials are average characteristic polynomials of random matrices from there asymptotics of the skew orthogonal polynomials are derived

monomial ideals almost complete intersections and the weak lefschetz property many algebras are expected to have the weak lefschetz property though this is often very difficult to establish we illustrate the subtlety of the problem by studying monomial and some closely related ideals our results exemplify the intriguing dependence of the property on the characteristic of the ground field and on arithmetic properties of the exponent vectors of the monomials

dimension theoretic characterization of maximal irreducible algebraic systems of plane nodal curves of a given order n and with a given number d of nodes introduction let k be an algebraically closed ground field of characteristic zero the following result is well known see severi 1 anhang f p 317 if en d is a complete i e maximal irreducible algebraic system defined over k of plane algebraic not necessarily irreducible curves of a given order n such that the general curve c of en d k has d nodes and no other singularities then the dimension of en d is equal to 3n p 1 where p n 1 n 2 2 d is the effective genus of c we note that 3n p 1 is equal to n n 3 2 d here n n n 3 2 is the dimension of the linear system of all plane curves of order n this is in agreement with the intuitive expectation that the requirement that a curve of order n possess d nodes in non assigned position imposes d independent algebraic non linear conditions on the curve the above result is easily proved and will be included by us in a more general theorem the proof of which is the main object of this paper theorem 2 section 3 and which deals with irreducible algebraic systems e of curves of order n in which the general curve c of e k has arbitrary singularities but is free from multiple components equivalently c is a reduced curve this theorem asserts that if p is the effective genus of c the definition of the effective genus of a reducible but reduced curve is given in section 3 formula 2 then dim e c 3n p 1 with equality if and only if c has only nodes as singularities it is easily seen see severi loc cit that if 0 c d n n 1 2 then there always exists curves of order n having d nodes the maximum

chudnovsky s conjecture for very general points in pkn abstract we prove a long standing conjecture of chudnovsky for very general and generic points in p k n where k is an algebraically closed field of characteristic zero and for any finite set of points lying on a quadric without any assumptions on k we also prove that for any homogeneous ideal i in the homogeneous coordinate ring r k x 0 x n chudnovsky s conjecture holds for large enough symbolic powers of i

critique of the replica trick it is shown that the replica trick fails to give the correct non perturbative result for the two point function s2 of the gaussian unitary ensemble of n n random matrices the failure arises from an incorrect description of the symmetries of the random matrix system in the limit n to infinity the correct description which involves integration over both non compact and compact degrees of freedom is obtained by using the method of superfields some implications for the localisation transition in disordered electronic systems and the theory of the quantised hall effect are suggested

pfaffian point process for the gaussian real generalised eigenvalue problem the generalised eigenvalues for a pair of n n matrices x1 x2 are defined as the solutions of the equation det x1 x2 0 or equivalently for x2 invertible as the eigenvalues of x 2 1 x 1 we consider gaussian real matrices x1 x2 for which the generalised eigenvalues have the rotational invariance of the half sphere or after a fractional linear transformation the rotational invariance of the unit disk in these latter variables we calculate the joint eigenvalue probability density function the probability pn k of finding k real eigenvalues the densities of real and complex eigenvalues the latter being related to an average over characteristic polynomials and give an explicit pfaffian formula for the higher correlation functions rho k 1 k 2 a limit theorem for pn k is proved and the scaled form of rho k 1 k 2 is shown to be identical to the analogous limit for the correlations of the eigenvalues of real gaussian matrices we show that these correlations satisfy sum rules characteristic of the underlying two component coulomb gas

random matrices close to hermitian or unitary overview of methods and results the paper discusses recent progress in understanding statistical properties of eigenvalues of weakly non hermitian and non unitary random matrices the first type of ensembles is of the form i with being a large random n n hermitian matrix with independent entries deformed by a certain anti hermitian n n matrix i satisfying in the limit of large dimension n the condition tr 2 n tr 2 here can be either a random or just a fixed given hermitian matrix ensembles of such a type with 0 emerge naturally when describing quantum scattering in systems with chaotic dynamics and serve to describe resonance statistics related models are used to mimic complex spectra of the dirac operator with chemical potential in the context of quantum chromodynamics ensembles of the second type arising naturally in scattering theory of discrete time systems are formed by n n matrices with complex entries such that i for 0 this coincides with the circular unitary ensemble and 0 i describes deviation from unitarity our result amounts to answering statistically the following old question given the singular values of a matrix describe the locus of its eigenvalues we systematically show that the obtained expressions for the correlation functions of complex eigenvalues describe a non trivial crossover from wigner dyson statistics of real unimodular eigenvalues typical of hermitian unitary matrices to ginibre statistics in the complex plane typical of ensembles with strong non hermiticity tr 2 tr 2 when n finally we discuss scarce results available on eigenvector statistics for weakly non hermitian random matrices

combinatorics and geometry of power ideals we investigate ideals in a polynomial ring which are generated by powers of linear forms such ideals are closely related to the theories of fat point ideals cox rings and box splines we pay special attention to a family of power ideals that arises naturally from a hyperplane arrangement a we prove that their hilbert series are determined by the combinatorics of a and can be computed from its tutte polynomial we also obtain formulas for the hilbert series of certain closely related fat point ideals and zonotopal cox rings our work unifies and generalizes results due to dahmen micchelli holtz ron postnikov shapiro shapiro and sturmfels xu among others it also settles a conjecture of holtz ron on the spline interpolation of functions on the lattice points of a zonotope

the weak and strong lefschetz properties for artinian k algebras abstract let a i 0ai be a standard graded artinian k algebra where chark 0 then a has the weak lefschetz property if there is an element l of degree 1 such that the multiplication l a i a i 1 has maximal rank for every i and a has the strong lefschetz property if l d a i a i d has maximal rank for every i and d the main results obtained in this paper are the following 1 every height three complete intersection has the weak lefschetz property our method surprisingly uses rank two vector bundles on p 2 and the grauert mulich theorem 2 we give a complete characterization including a concrete construction of the hilbert functions that can occur for k algebras with the weak or strong lefschetz property and the characterization is the same one 3 we give a sharp bound on the graded betti numbers achieved by our construction of artinian k algebras with the weak or strong lefschetz property and fixed hilbert function this bound is again the same for both properties some hilbert functions in fact force the algebra to have the maximal betti numbers 4 every artinian ideal in k x y possesses the strong lefschetz property this is false in higher codimension

the lefschetz properties of monomial complete intersections in positive characteristic abstract stanley proved that in characteristic zero all artinian monomial complete intersections have the strong lefschetz property we provide a positive characteristic complement to stanley s result in the case of artinian monomial complete intersections generated by monomials all of the same degree and also for arbitrary artinian monomial complete intersections in characteristic two to establish these results we first prove an a priori lower bound on the characteristics that guarantee the lefschetz properties we then use a variety of techniques to complete the classifications

almost hermitian random matrices crossover from wigner dyson to ginibre eigenvalue statistics by using the method of orthogonal polynomials we analyze the statistical properties of complex eigenvalues of random matrices describing a crossover from hermitian matrices characterized by the wigner dyson statistics of real eigenvalues to strongly non hermitian ones whose complex eigenvalues were studied by ginibre two point statistical measures as e g spectral form factor number variance and small distance behavior of the nearest neighbor distance distribution p s are studied in more detail in particular we found that the latter function may exhibit unusual behavior p s ensuremath propto s 5 2 for some parameter values

eigenvalue correlations in non hermitean symplectic random matrices a correlation function of complex eigenvalues of n n random matrices drawn from non hermitean random matrix ensembles of symplectic symmetry is given in terms of a quaternion determinant spectral properties of gaussian ensembles are studied in detail in the regimes of weak and strong non hermiticity

characteristic polynomials in real ginibre ensembles we calculate the average of two characteristic polynomials for the real ginibre ensemble of asymmetric random matrices and its chiral counterpart considered as quadratic forms they determine a skew symmetric kernel from which all complex eigenvalue correlations can be derived our results are obtained in a very simple fashion without going to an eigenvalue representation and are completely new in the chiral case they hold for gaussian ensembles which are partly symmetric with kernels given in terms of hermite and laguerre polynomials respectively depending on an asymmetry parameter this allows us to interpolate between the maximally asymmetric real ginibre and the gaussian orthogonal ensemble as well as their chiral counterparts

the geometry of the moduli space of curves of genus 23 the problem of describing the birational geometry of the moduli space mg of complex curves of genus g has a long history severi already knew in 1915 thatmg is unirational for g 10 cf sev see also ac1 for a modern proof in the same paper severi conjectured thatmg is unirational for all genera g then for a long period this problem seemed intractable mumford writes in mu p 51 whether more mg s g 11 are unirational or not is a very interesting problem but one which looks very hard too especially if g is quite large the breakthrough came in the eighties when eisenbud harris and mumford proved that mg is of general type as soon as g 24 and that the kodaira dimension of m23 is 1 see hm eh3 we note that mg is rational for g 6 see dol for problems concerning the rationality of various moduli spaces severi s proof of the unirationality of mg for small g was based on representing a general curve of genus g as a plane curve of degree d with nodes this is possible when d 2g 3 2 when the number of nodes is small i e d 1 d 2 6 the dominant map from the variety of plane curves of degree d and genus g to mg yields a rational parametrization of the moduli space the two conditions involving d and can be satisfied only when g 10 so severi s argument cannot be extended for other genera however using much more subtle ideas chang ran and sernesi proved the unirationality of mg for g 11 12 13 see cr1 se1 while for g 15 16 they proved that the kodaira dimension is see cr2 4 the remaining cases g 14 and 17 g 23 are still quite mysterious harris and morrison conjectured in hmo that mg is uniruled precisely when g 23 all these facts indicate that m23 is a very interesting transition case our main result is the following

asymptotic properties of b splines eulerian numbers and cube slicing in this paper we clarify the connections among b splines eulerian numbers and cube slicing we first show that the asymptotic formula for eulerian numbers can be considered as a special case of the asymptotic properties of b splines the volume of cube slicing can be considered as a value of box spline functions based on the connection a very simple proof for laplace and polya s formulas which were settled by probabilistic methods is given by spline theory

a real quaternion spherical ensemble of random matrices one can identify a tripartite classification of random matrix ensembles into geometrical universality classes corresponding to the plane the sphere and the anti sphere the plane is identified with ginibre type iid matrices and the anti sphere with truncations of unitary matrices this paper focusses on an ensemble corresponding to the sphere matrices of the form y a 1 b where a and b are independent n n matrices with iid standard gaussian real quaternion entries by applying techniques similar to those used for the analogous complex and real spherical ensembles the eigenvalue joint probability density function and correlation functions are calculated this completes the exploration of spherical matrices using the traditional dyson indices 1 2 4

integrable structure of ginibre s ensemble of real random matrices and a pfaffian integration theorem in the recent publication e kanzieper and g akemann in phys rev lett 95 230201 2005 an exact solution was reported for the probability pn k to find exactly k real eigenvalues in the spectrum of an n n real asymmetric matrix drawn at random from ginibre s orthogonal ensemble ginoe in the present paper we offer a detailed derivation of the above result by concentrating on the proof of the pfaffian integration theorem the key ingredient of our analysis of the statistics of real eigenvalues in the ginoe we also initiate a study of the correlations of complex eigenvalues and derive a formula for the joint probability density function of all complex eigenvalues of a ginoe matrix restricted to have exactly k real eigenvalues in the particular case of k 0 all correlation functions of complex eigenvalues are determined

matrix averages relating to ginibre ensembles the theory of zonal polynomials is used to compute the average of a schur polynomial of argument ax where a is a fixed matrix and x is from the real ginibre ensemble this generalizes a recent result of sommers and khoruzhenko 2009 j phys a math theor 42 222002 and furthermore allows analogous results to be obtained for the complex and real quaternion ginibre ensembles as applications the positive integer moments of the general variance ginibre ensembles are computed in terms of generalized hypergeometric functions these are written in terms of averages over matrices of the same size as the moment to give duality formulas and the averages of the power sums of the eigenvalues are expressed as finite sums of zonal polynomials

asymptotic resurgences for ideals of positive dimensional subschemes of projective space abstract recent work of ein lazarsfeld smith and hochster huneke raised the containment problem of determining which symbolic powers of an ideal are contained in a given ordinary power of the ideal bocci harbourne defined a quantity called the resurgence to address this problem for homogeneous ideals in polynomial rings with a focus on zero dimensional subschemes of projective space here we take the first steps toward extending this work to higher dimensional subschemes we introduce new asymptotic versions of the resurgence and obtain upper and lower bounds on them for ideals i of smooth subschemes generalizing what is done in bocci and harbourne 2010 5 we apply these bounds to ideals of unions of general lines in p n we also pose a nagata type conjecture for symbolic powers of ideals of lines in p 3

a method to calculate correlation functions for β 1 random matrices of odd size the calculation of correlation functions for 1 random matrix ensembles which can be carried out using pfaffians has the peculiar feature of requiring a separate calculation depending on the parity of the matrix size n this same complication is present in the calculation of the correlations for the ginibre orthogonal ensemble of real gaussian matrices in fact the methods used to compute the 1 n odd correlations break down in the case of n odd real ginibre matrices necessitating a new approach to both problems the new approach taken in this work is to deduce the 1 n odd correlations as limiting cases of their n even counterparts when one of the particles is removed towards infinity this method is shown to yield the correlations for n odd real gaussian matrices

a new approach to derive pfaffian structures for random matrix ensembles correlation functions for matrix ensembles with orthogonal and unitary symplectic rotation symmetry are more complicated to calculate than in the unitary case the supersymmetry method and the orthogonal polynomials are two techniques to tackle this task recently we presented a new method to average ratios of characteristic polynomials over matrix ensembles invariant under the unitary group here we extend this approach to ensembles with orthogonal and unitary symplectic rotation symmetry we show that pfaffian structures can be derived for a wide class of orthogonal and unitary symplectic rotation invariant ensembles in a unifying way this also includes those for which this structure was not known previously as the real ginibre ensemble and the gaussian real chiral ensemble with two independent matrices as well

skew orthogonal polynomials and the partly symmetric real ginibre ensemble the partly symmetric real ginibre ensemble consists of matrices formed as linear combinations of real symmetric and real anti symmetric gaussian random matrices such matrices typically have both real and complex eigenvalues for a fixed number of real eigenvalues an earlier work has given the explicit form of the joint eigenvalue probability density function we use this to derive a pfaffian formula for the corresponding summed up generalized partition function this pfaffian formula allows the probability that there are exactly k eigenvalues to be written as a determinant with explicit entries it can be used too to give the explicit form of the correlation functions provided certain skew orthogonal polynomials are computed this task is accomplished in terms of hermite polynomials and allows us to proceed to analyze various scaling limits of the correlations including that in which the matrices are only weakly non symmetric