optimal estimates and asymptotics for the stress concentration between closely located stiff inclusions if stiff inclusions are closely located then the stress which is the gradient of the solution may become arbitrarily large as the distance between two inclusions tends to zero in this paper we investigate the asymptotic behavior of the stress concentration factor which is the normalized magnitude of the stress concentration as the distance between two inclusions tends to zero for that purpose we show that the gradient of the solution to the case when two inclusions are touching decays exponentially fast near the touching point we also prove a similar result when two inclusions are closely located and there is no potential difference on boundaries of two inclusions we then use these facts to show that the stress concentration factor converges to a certain integral of the solution to the touching case as the distance between two inclusions tends to zero we then present an efficient way to compute this integral

hyperdeterminantal computation for the laughlin wavefunction the decomposition of the laughlin wavefunction in the slater orthogonal basis appears in the discussion on the second quantized form of the laughlin states and is straightforwardly equivalent to the decomposition of the even powers of the vandermonde determinants in the schur basis such a computation is notoriously difficult and the coefficients of the expansion have not yet been interpreted in our paper we give an expression of these coefficients in terms of hyperdeterminants of sparse tensors we use this result to construct an algorithm allowing us to compute one coefficient of the development without computing the others thanks to a program in c we performed the calculation for the square of the vandermonde up to an alphabet of 11 letters

configurational invariants of hamiltonian systems in this paper we explore the general conditions in order that a two dimensional natural hamiltonian system possess a second invariant which is a polynomial in the momenta and is therefore liouville integrable we examine the possibility that the invariant is preserved by the hamiltonian flow on a given energy hypersurface only weak integrability and derive the additional requirement necessary to have conservation at arbitrary energy strong integrability using null complex coordinates we show that the leading order coefficient of the polynomial is an arbitrary holomorphic function in the case of weak integrability and a polynomial in the coordinates in the strongly integrable one we review the results obtained so far with strong invariants up to degree four and provide some new examples of weakly integrable systems with linear and quadratic invariants

gradient estimates for solutions to the conductivity problem in this paper we derive very precise gradient estimates for solutions to the conductivity problem in the case where two circular conductivity inclusions are very close but not touching the novelty of these estimates is that they give very specific information about the blow up of the gradient as the conductivities of the inclusions degenerate

distribution of bipartite entanglement for random pure states we calculate analytic expressions for the distribution of bipartite entanglement for pure random quantum states all moments of the purity distribution are derived and an asymptotic expansion for the distribution itself is deduced an approximate expression for moments and distribution of meyer wallach entanglement for random pure states is then obtained

a theory of exact and approximate configurational invariants abstract a theory of integrable hamiltonian systems in two dimensions is formulated and applied the four dimensional phase space problem that is the vanishing of the poisson bracket between another invariant and the hamiltonian is here reduced to the solution of a series of two dimensional configuration space equations the theory is also applicable to more than two dimensions the constraints are found which admit to integrability of the orbits for magnetic or coriolis forces as well as for forces derivable from a potential when a system admits a given invariant the invariant is found for example by quadrature a number of examples including known and apparently previously unknown invariants are given the theory of exact integrals of the motion also can be extended to the derivation of approximate invariants the orbital structure of integrable or approximately integrable systems correlates with the degree maximum power of the velocity of a standardized invariant the theory admits a variational principle among other approximation techniques for the computation of a best approximate invariant the problem of the general cubic potential with one symmetric coordinate v 1 2 ax 2 1 2 by 2 cx 2 y 1 3 dy 3 of which the well studied henon heiles potential is the special case for a b and c d is examined in detail

higher dimensional gcd matrices abstract we evaluate the higher dimensional determinants of the greatest common divisor matrix defined on a gcd closed set and on an arbitrary set of distinct positive integers

an optimal estimate for electric fields on the shortest line segment between two spherical insulators in three dimensions abstract we consider a gradient estimate for a conductivity problem whose inclusions are two neighboring insulators in three dimensions when inclusions with an extreme conductivity insulators or perfect conductors are closely located the gradient can be concentrated in between inclusions and then becomes arbitrarily large as the distance between inclusions approaches zero the gradient estimate in between insulators in three dimensions has been regarded as a challenging problem while the optimal blow up rates in terms of the distance were successfully obtained for the other extreme conductivity problems in two and three dimensions and are attained on the shortest line segment between inclusions in this paper we establish upper and lower bounds of gradients on the shortest line segment between two insulating unit spheres in three dimensions these bounds present the optimal blow up rate of gradient on the line segment which is substantially different from the rates in the other problems

from random matrices to random analytic functions we consider two families of random matrix valued analytic functions 1 g 1 zg 2 and 2 g 0 zg 1 z 2 g 2 where g i are n x n random matrices with independent standard complex gaussian entries the random set of z where these matrix analytic functions become singular is shown to be determinantal point processes in the sphere and the hyperbolic plane respectively the kernels of these determinantal processes are reproducing kernels of certain hilbert spaces bargmann fock spaces of holomorphic functions on the corresponding surfaces along with the new results this also gives a unified framework in which to view a theorem of peres and virag n 1 in the second setting above and a well known result of ginibre on gaussian random matrices which may be viewed as an analogue of our results in the whole plane

a few remarks on colour flavour transformations truncations of random unitary matrices berezin reproducing kernels and selberg type integrals we investigate diverse relations of the colour flavour transformations cft introduced by zirnbauer in supersymmetry for systems with unitary disorder circular ensembles 1996 j phys a math gen 29 7113 36 to various topics in random matrix theory and multivariate analysis such as measures on truncations of unitary random matrices jacobi ensembles of random matrices berezin reproducing kernels and a generalization of the selberg integral due to kaneko kadell and yan involving the schur functions apart from suggesting explicit formulae for bosonic cft for the unitary group in the range of parameters beyond that in zirnbauer s paper we also suggest an alternative variant of the transformation with integration going over an unbounded domain of a pair of hermitian matrices the latter makes possible the evaluation of certain averages in random matrix theory

atlas of two dimensional irreversible conservative lagrangian mechanical systems with a second quadratic integral this paper aims at the most comprehensive and systematic construction and tabulation of mechanical systems that admit a second invariant quadratic in velocities other than the hamiltonian the configuration space is in general a two dimensional riemannian or pseudo riemannian manifold and the determination of its geometry is a part of the process of solution forces acting on the system include a part derived from a scalar potential and a part derived from a vector potential associated with terms linear in velocities in the lagrangian function of the system the last caused time irreversibility of the system we constructed 41 multiparameter integrable systems of the type described in the title mostly on riemannian manifolds they are mostly new and cover all previously known systems as special cases corresponding to special values of the parameters those include all known cases of motion of a particle in the plane and all known cases in the dynamics of rigid body in the last field we introduce a n

integrable hamiltonian systems with vector potentials we investigate integrable two dimensional hamiltonian systems with scalar and vector potentials admitting second invariants which are linear or quadratic in the momenta in the case of a linear second invariant we provide some examples of weakly integrable systems in the case of a quadratic second invariant we recover the classical strongly integrable systems in cartesian and polar coordinates and provide some new examples of integrable systems in parabolic and elliptical coordinates

quantitative estimates for enhancement of the field excited by an emitter due to presence of two closely located spherical inclusions abstract a field in a homogeneous medium can be amplified or enhanced by inserting closely located perfectly conducting inclusions into the medium in this paper precise quantitative estimates for such enhancement are derived when the given field is the one excited by an emitter of a dipole type and inclusions are spheres of the same radii in three dimensions derived estimates reveal the difference as well as the similarity between enhancement of the field excited by the emitter and that of the smooth back ground field in particular an estimate shows that when the enhancement occurs the factor of enhancement is log 1 which is different from that for the smooth background field which is known to be log 1 is the distance between two inclusions

a new class of integrable systems we present a family of dynamical systems associated with the motion of a particle in two space dimensions these systems possess a second integral of motion quadratic in velocities apart from the hamiltonian and are thus completely integrable they were found through the derivation and subsequent resolution of the integrability condition in the form of a partial differential equation pde for the potential a most important point is that the same pde was derived through considerations on the analytic structure of the singularities of the solutions weak painleve property

marginal distribution of the s matrix elements for dyson s measure and some applications in recent attempts to construct a statistical theory of nuclear reactions by doing statistics directly on the s matrix elements dyson s measure which remains invariant under an automorphism that maps the space of unitary and symmetric matrices into itself is of fundamental importance the authors study some of the marginal distributions of the individual s matrix elements or of groups of them that arise from dyson s measure to understand the problem better a similar discussion is first carried out for haar s measure of unitary matrices which do not have the restriction of symmetry and some of the effects of this restriction are thus exhibited some applications of these mathematical results to reaction theory are discussed

on asymptotic behavior of multilinear eigenvalue statistics of random matrices we prove the law of large numbers and the central limit theorem for analogs of u and v von mises statistics of eigenvalues of random matrices as their size tends to infinity we show first that for a certain class of test functions kernels determining the statistics the validity of these limiting laws reduces to the validity of analogous facts for certain linear eigenvalue statistics we then check the conditions of the reduction statements for several most known ensembles of random matrices the reduction phenomenon is well known in statistics dealing with i i d random variables it is of interest that an analogous phenomenon is also the case for random matrices whose eigenvalues are strongly dependent even if the entries of matrices are independent

asymptotic mean density of sub unitary ensembles the asymptotic behaviour of the mean spectral density for the product of n n random unitary matrices distributed according to the haar measure and a fixed diagonal matrix is investigated the large n limit of the exact expression derived by wei and fyodorov 2008 j phys a math theor 41 502001 is calculated by a modification of the saddle point method it is shown that in the bulk the result coincides with the one obtained within the free probability theory by haagerup and larsen 2000 j funct anal 176 331 close to the edge points of the bulk asymptotics the mean density for large n is described by a universal function depending on a certain scaling variable the large deviation formulae valid outside the bulk are also derived obtained formulae agree well with the results of direct numerical calculations

product of random projections jacobi ensembles and universality problems arising from free probability we consider the product of two independent randomly rotated projectors the square of its radial part turns out to be distributed as a jacobi ensemble we study its global and local properties in the large dimension scaling relevant to free probability theory we establish asymptotics for one point and two point correlation functions as well as properties of largest and smallest eigenvalues

truncations of random unitary matrices we analyse properties of non hermitian matrices of size m constructed as square submatrices of unitary orthogonal random matrices of size n m distributed according to the haar measure in this way we define ensembles of random matrices and study the statistical properties of the spectrum located inside the unit circle in the limit of large matrices this ensemble is characterized by the ratio m n for the truncated cue we analytically derive the joint density of eigenvalues and all correlation functions in the strongly non unitary case universal ginibre behaviour is found for n m fixed and n to the universal resonance width distribution with n m open channels is recovered

hankel hyperdeterminants rectangular jack polynomials and even powers of the vandermonde abstract we investigate the link between rectangular jack polynomials and hankel hyperdeterminants as an application we give an expression of the even power of the vandermonde in terms of jack polynomials

on the spectrum of the poincare variational problem for two close to touching inclusions in 2d we study the spectrum of the poincare variational problem for two close to touching inclusions in r2 we derive the asymptotics of its eigenvalues as the distance between the inclusions tends to zero

a unified treatment of quartic invariants at fixed and arbitrary energy two dimensional hamiltonian systems admitting second invariants which are quartic in the momenta are investigated using the jacobi geometrization of the dynamics this approach allows for a unified treatment of invariants at both arbitrary and fixed energy in the differential geometric picture the quartic invariant corresponds to the existence of a fourth rank killing tensor expressing the jacobi metric in terms of a kahler potential the integrability condition for the existence of the killing tensor at fixed energy is a nonlinear equation involving the kahler potential at arbitrary energy further conditions must be imposed which lead to an overdetermined system with isolated solutions we obtain several new integrable and superintegrable systems in addition to all previously known examples

canonical transformations of the extended phase space toda lattices and the stackel family of integrable systems we consider some examples of canonical transformations of the extended phase space which map a completely integrable system into another completely integrable system the proposed transformations are closed to the known maupertuis jacobi mappings and to the kepler change of the time using the kepler transformation we construct new integrable systems related to the toda lattices and the stackel family of integrable systems

averages on the unitary group and applications to the problem of disordered conductors in a recent approach to the study of disordered conductors a maximum entropy criterion is used to perform statistical calculations directly on the transfer matrix for the full physical system quantities of great importance in that approach are averages of products of the u and u matrix elements u being a unitary matrix evaluated with the invariant measure of the unitary group the authors evaluate those averages that contain up to four u and four u matrix elements since they are the ones needed in the calculation of the average and covariance of transmission and reflection coefficients the evaluation involves no direct integration at all but makes use solely of the invariance of the measure the fact that the matrix elements ua alpha are commuting c numbers and the unitarity of u the physical consequences for the average and covariance of transmission and reflection coefficients are briefly discussed

s matrix poles for chaotic quantum systems as eigenvalues of complex symmetric random matrices from isolated to overlapping resonances we study complex eigenvalues of large symmetric random matrices of the form where both and are real symmetric is a random gaussian and is such that when when the model can be used to describe the universal statistics of s matrix poles resonances in the complex energy plane we derive the ensuing distribution of the resonance widths which generalizes the well known distribution to the case of overlapping resonances we also consider a different class of almost real matrices when is random and uncorrelated with

randomizing quantum states constructions and applications the construction of a perfectly secure private quantum channel in dimension d is known to require 2 log d shared random key bits between the sender and receiver we show that if only near perfect security is required the size of the key can be reduced by a factor of two more specifically we show that there exists a set of roughly d log d unitary operators whose average effect on every input pure state is almost perfectly randomizing as compared to the d2 operators required to randomize perfectly aside from the private quantum channel variations of this construction can be applied to many other tasks in quantum information processing we show for instance that it can be used to construct locc data hiding schemes for bits and qubits that are much more efficient than any others known allowing roughly log d qubits to be hidden in 2 log d qubits the method can also be used to exhibit the existence of quantum states with locked classical correlations an arbitrarily large amplification of the correlation being accomplished by sending a negligibly small classical key our construction also provides the basic building block for a method of remotely preparing arbitrary d dimensional pure quantum states using approximately log d bits of communication and log d ebits of entanglement

full counting statistics of chaotic cavities from classical action correlations we present a trajectory based semiclassical calculation of the full counting statistics of quantum transport through chaotic cavities in the regime of many open channels our method to obtain the mth moment of the density of transmission eigenvalues requires two correlated sets of m classical trajectories therefore generalizing previous works on conductance and shot noise the semiclassical results agree for all values of m with the corresponding predictions from random matrix theory

hankel hyperdeterminants and selberg integrals we investigate the simplest class of hyperdeterminants defined by cayley in the case of hankel hypermatrices tensors of the form ai1i2 ik f i1 i2 ik it is found that many classical properties of hankel determinants can be generalized and a connection with selberg type integrals is established in particular selberg s original formula amounts to the evaluation of all hankel hyperdeterminants built from the moments of the jacobi polynomials many higher dimensional analogues of classical hankel determinants are evaluated in closed form the toeplitz case is also briefly discussed in physical terms both cases are related to the partition functions of one dimensional coulomb systems with logarithmic potential

fluctuation formula for complex random matrices a gaussian fluctuation formula is proved for linear statistics of complex random matrices in the case where the statistic is rotationally invariant for a general linear statistic without this symmetry coulomb gas theory is used to predict that the distribution will again be a gaussian with a specific mean and variance the variance splits naturally into a bulk and surface contibution the latter resulting from the long range correlations at the boundary of the support of the eigenvalue density

invariants at fixed and arbitrary energy a unified geometric approach invariants at arbitrary and fixed energy strongly and weakly conserved quantities for two dimensional hamiltonian systems are treated in a unified way this is achieved by utilizing the jacobi metric geometrization of the dynamics using killing tensors we obtain an integrability condition for quadratic invariants which involves an arbitrary analytic function s z for invariants at arbitrary energy the function s z is a second degree polynomial with real second derivative the integrability condition then reduces to darboux s condition for quadratic invariants at arbitrary energy the four types of classical quadratic invariants for positive definite two dimensional hamiltonians are shown to correspond to certain conformal transformations we derive the explicit relation between invariants in the physical and jacobi time gauges in this way knowledge about the invariant in the physical time gauge enables one to write down directly the components of the corresponding killing tensor for the jacobi metric we also discuss the possibility of searching for linear and quadratic invariants at fixed energy and its connection with the problem of the third integral in galactic dynamics in our approach linear and quadratic invariants at fixed energy can be found by solving a linear ordinary differential equation of the first or second degree respectively

gradient estimates for the perfect conductivity problem we establish both upper and lower bounds of the gradient estimates for the perfect conductivity problem in the case where perfect stiff conductors are closely spaced inside an open bounded domain and away from the boundary these results give the optimal blow up rates of the stress for conductors with arbitrary shape and in all dimensions

a matrix model for the β jacobi ensemble this note presents a random matrix model for general 0 jacobi ensembles this generalizes the well known manova models for 1 2 4 and eliminates the quantization of and other parameters present in the previously known models this model is a partial answer to an open problem presented by dumitriu and edelman where they also presented models for the laguerre and hermite ensembles

powers of the vandermonde determinant and the quantum hall effect the expansion of the laughlin ansatz for describing the ground state wavefunction for the fractional quantum hall effect as a linear combination of slater determinantal wavefunctions for n particles is discussed in terms of the corresponding expansion of even powers of the vandermonde alternant into schur functions two new algorithms for computing the coefficients of the complete expansion are given they appear to be substantially more efficient than other methods and avoid any use of symmetric group characters a number of examples are given and the results obtained for n 7 8 and 9 reviewed the separate calculation of individual coefficients is also discussed

hierarchical dobinski type relations via substitution and the moment problem we consider the transformation properties of integer sequences arising from the normal ordering of exponentiated boson a a 1 monomials of the form exp a ra r 1 2 under the composition of their exponential generating functions they turn out to be of sheffer type we demonstrate that two key properties of these sequences remain preserved under substitutional composition a the property of being the solution of the stieltjes moment problem and b the representation of these sequences through infinite series dobinski type relations we present a number of examples of such composition satisfying properties a and b we obtain new dobinski type formulae and solve the associated moment problem for several hierarchically defined combinatorial families of sequences

optimal estimates of the field enhancement in presence of a bow tie structure of perfectly conducting inclusions in two dimensions abstract this paper deals with the field enhancement due to insertion of a bow tie structure of perfectly conducting inclusions into the two dimensional space with a given field the field enhancement is represented by the gradient blow up of a solution to the conductivity problem the bow tie structure consists of two disjoint bounded domains which have corners with possibly different aperture angles the domains are parts of cones near the vertices which are nearly touching to each other we construct functions explicitly which characterize the field enhancement as consequences we derive optimal estimates of the gradient in terms of the distance between two inclusions and aperture angles of the corners the estimates show in quantitatively precise way that the field is enhanced beyond the corner singularities due to the interaction between two inclusions and the blow up rate is much higher than the one for the case of inclusions with smooth boundaries

the calogero sutherland model and generalized classical polynomials multivariable generalizations of the classical hermite laguerre and jacobi polynomials occur as the polynomial part of the eigenfunctions of certain schrodinger operators for calogero sutherland type quantum systems for the generalized hermite and laguerre polynomials the multidimensional analogues of many classical results regarding generating functions differentiation and integration formulas recurrence relations and summation theorems are obtained we use this and related theory to evaluate the global limit of the ground state density obtaining in the hermite case the wigner semi circle law and to give an explicit solution for an initial value problem in the hermite and laguerre case

transmission eigenvalue densities and moments in chaotic cavities from random matrix theory we point out that the transmission eigenvalue density and higher order correlation functions in chaotic cavities for an arbitrary number of incoming and outgoing leads n1 n 2 are analytically known from the jacobi ensemble of random matrix theory using this result and a simple linear statistic we give an exact and non perturbative expression for moments of the form m for m n1 n2 1 and 2 thus improving the existing results in the literature secondly we offer an independent derivation of the average density and higher order correlation function for 2 4 which does not make use of the orthogonal polynomials technique this result may be relevant for an efficient numerical implementation avoiding determinants pacs numbers 73 23 b 73 50 td 05 45 mt 73 63 kv

on the mean density of complex eigenvalues for an ensemble of random matrices with prescribed singular values given any fixed n n positive semi definite diagonal matrix g 0 we derive the explicit formula for the density of complex eigenvalues for random matrices a of the form where the random unitary matrices u are distributed on the group u n according to the haar measure

quantum chaotic dynamics and random polynomials we investigate the distribution of roots of polynomials of high degree with random coefficients which among others appear naturally in the context of quantum chaotic dynamics it is shown that under quite general conditions their roots tend to concentrate near the unit circle in the complex plane in order to further increase this tendency we study in detail the particular case of self inversive random polynomials and show that for them a finite portion of all roots lies exactly on the unit circle correlation functions of these roots are also computed analytically and compared to the correlations of eigenvalues of random matrices the problem of ergodicity of chaotic wavefunctions is also considered for that purpose we introduce a family of random polynomials whose roots spread uniformly over phase space while these results are consistent with random matrix theory predictions they provide a new and different insight into the problem of quantum ergodicity special attention is devoted to the role of symmetries in the distribution of roots of random polynomials

spectral analysis of the neumann poincare operator and characterization of the stress concentration in anti plane elasticity when holes or hard elastic inclusions are closely located stress which is the gradient of the solution to the anti plane elasticity equation can be arbitrarily large as the distance between two inclusions tends to zero it is important to precisely characterize the blow up of the gradient of such an equation in this paper we show that the blow up of the gradient can be characterized by a singular function defined by the single layer potential of an eigenfunction corresponding to the eigenvalue 1 2 of a neumann poincare type operator defined on the boundaries of the inclusions by comparing the singular function with the one corresponding to two disks osculating to the inclusions we quantitatively characterize the blow up of the gradient in terms of explicit functions in electrostatics our results apply to the electric field which is the gradient of the solution to the conductivity equation in the case where perfectly conducting or insulating inclusions are closely located

hyperdeterminantal expressions for jack functions of rectangular shapes abstract we derive a jacobi trudi type formula for jack functions of rectangular shapes in this formula we make use of a hyperdeterminant which is cayley s simple generalization of the determinant in addition after developing the general theory of hyperdeterminants we give summation formulas for schur functions involving hyperdeterminants and evaluate toeplitz type hyperdeterminants by using jack function theory

integral formulae for the eigenvalue density of complex random matrices we show that the density of complex eigenvalues for unitary invariant ensembles of complex matrices a can be written as an integral over the eigenvalues gj of aa for the standard random matrix ensembles with matrix density of the form jw gj this integral can be further reduced to a twofold integral involving the christoffel darboux kernel for the orthogonal polynomials associated with weight w

composed ensembles of random unitary matrices composed ensembles of random unitary matrices are defined via products of matrices each pertaining to a given canonical circular ensemble of dyson we investigate statistical properties of spectra of some composed ensembles and demonstrate their physical relevance we also discuss the methods of generating random matrices distributed according to invariant haar measure on the orthogonal and unitary group

new algorithms for linear k matroid intersection and matroid k parity problems we present algorithms for thek matroid intersection problem and for the matroidk parity problem when the matroids are represented over the field of rational numbers andk 2 the computational complexity of the algorithms is linear in the cardinality and singly exponential in the rank of the matroids as an application we describe new polynomially solvable cases of thek dimensional assignment problem and of thek dimensional matching problem the algorithms use some new identities in multilinear algebra including the generalized binet cauchy formula and its analogue for the pfaffian these techniques extend known methods developed earlier fork 2

positive semi definiteness and sum of squares property of fourth order four dimensional hankel tensors a symmetric positive semi definite psd tensor which is not sum of squares sos is called a psd non sos pns tensor is there a fourth order four dimensional pns hankel tensor the answer for this question has both theoretical and practical significance under the assumptions that the generating vector v of a hankel tensor a is symmetric and the fifth element v 4 of v is fixed at 1 we show that there are two surfaces m 0 and n 0 with the elements v 2 v 6 v 1 v 3 v 5 of v as variables such that m 0 n 0 a is sos if and only if v 0 m 0 and a is psd if and only if v 0 n 0 where v 0 is the first element of v if m 0 n 0 for a point p v 2 v 6 v 1 v 3 v 5 there are no fourth order four dimensional pns hankel tensors with symmetric generating vectors for such v 2 v 6 v 1 v 3 v 5 then we call such p a pns free point we prove that a 45 degree planar closed convex cone a segment a ray and an additional point are pns free numerical tests check various grid points and report that they are all pns free

1 1 dimensional separation of variables in this paper we explore general conditions which guarantee that the geodesic flow on a two dimensional manifold with indefinite signature is locally separable this is equivalent to showing that a two dimensional natural hamiltonian system on the hyperbolic plane possesses a second integral of motion which is a quadratic polynomial in the momenta associated with a secind rank killing tensor we examine the possibility that the integral is preserved by the hamiltonian flow on a given energy hypersurface only weak integrability and derive the additional requirement necessary to have conservation at arbitrary values of the hamiltonian strong integrability using null coordinates we show that the leading order coefficients of the invariant are arbitrary functions of one variable in the case of weak integrability these functions are quadratic polynomials in the coordinates in the case of strong integrability we show that for 1 1 dimensional systems there are three possible types of conformal killing

spectral radii of truncated circular unitary matrices abstract consider a truncated circular unitary matrix which is a p n by p n submatrix of an n by n circular unitary matrix by deleting the last n p n columns and rows jiang and qi 11 proved that the maximum absolute value of the eigenvalues known as spectral radius of the truncated matrix after properly normalized converges in distribution to the gumbel distribution if p n n is bounded away from 0 and 1 in this paper we investigate the limiting distribution of the spectral radius under one of the following four conditions 1 p n and p n n 0 as n 2 n p n n 0 and n p n log n 3 as n 3 n p n and n p n log n 0 as n and 4 n p n k 1 is a fixed integer we prove that the spectral radius converges in distribution to the gumbel distribution under the first three conditions and to a reversed weibull distribution under the fourth condition

a complete set of covariants of the four qubit system we obtain a complete and minimal set of 170 generators for the algebra of sl 2 4 covariants of a binary quadrilinear form interpreted in terms of a four qubit system this describes in particular the algebraic varieties formed by the orbits of local filtering operations in its projective hilbert space also this sheds some light on the local unitary invariants and provides all the possible building blocks for the construction of entanglement measures for such a system

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

the square of the vandermonde determinant and its q generalization the vandermonde determinant plays a crucial role in the quantum hall effect via laughlin s wavefunction ansatz herein the properties of the square of the vandermonde determinant as a symmetric function are explored in detail important properties satisfied by the coefficients arising in the expansion of the square of the vandermonde determinant in terms of schur functions are developed and generalized to q dependent coefficients via the q discriminant algorithms for the efficient calculation of the q dependent coefficients as finite polynomials in q are developed the properties such as the factorization of the q dependent coefficients are exposed further light is shed upon the vanishing of certain expansion coefficients at q 1 the q generalization of the sum rule for the squares of the coefficients is derived a number of compelling conjectures are stated

commutative law for products of infinitely large isotropic random matrices ensembles of isotropic random matrices are defined by the invariance of the probability measure under the left and right multiplication by an arbitrary unitary matrix we show that the multiplication of large isotropic random matrices is spectrally commutative and self averaging in the limit of infinite matrix size n the notion of spectral commutativity means that the eigenvalue density of a product abc of such matrices is independent of the order of matrix multiplication for example the matrix abcd has the same eigenvalue density as adcb in turn the notion of self averaging means that the product of n independent but identically distributed random matrices which we symbolically denote by aaa has the same eigenvalue density as the corresponding power a n of a single matrix drawn from the underlying matrix ensemble for example the eigenvalue density of abccabc is the same as that of a 2 b 2 c 3 we also discuss the singular behavior of the eigenvalue and singular value densities of isotropic matrices and their products for small eigenvalues 0 we show that the singularities at the origin of the eigenvalue density and of the singular value density are in one to one correspondence in the limit n the eigenvalue density of an isotropic random matrix has a power law singularity at the origin s with a power se 0 2 when and only when the density of its singular values has a power law singularity with a power s 4 s these results are obtained analytically in the limit n we supplement these results with numerical simulations for large but finite n and discuss finite size effects for the most common ensembles of isotropic random matrices

nonlinear random matrix statistics symmetric functions and hyperdeterminants nonlinear statistics i e statistics of permanents on the eigenvalues of invariant random matrix models are considered for the three dyson s symmetry classes 1 2 4 general formulas in terms of hyperdeterminants are found for 2 for specific cases and all s more computationally efficient results are obtained based on symmetric functions expansions as an application we consider the case of quantum transport in chaotic cavities extending results from savin et al 2008 phys rev b 77 125332

orthogonal polynomials through the invariant theory of binary forms abstract we present an algebraic theory of orthogonal polynomials in several variables that includes classical orthogonal polynomials as a special case our bottom line is a straightforward connection between apolarity of binary forms and the inner product provided by a linear functional defined on a polynomial ring explicit determinantal formulae and multivariable extension of the heine integral formula are stated moreover a general family of covariants that includes transvectants is introduced such covariants turn out to be the average value of classical basis of symmetric polynomials over a set of roots of suitable orthogonal polynomials

precise estimates of the field excited by an emitter in presence of closely located inclusions of a bow tie shape abstract this paper studies in a quantitatively precise manner the field enhancement due to presence of an emitter of the dipole type near the bow tie structure of perfectly conducting inclusions in the two dimensional space we put special emphasis on field enhancement near vertices of the bow tie structure and derive upper and lower bounds of the gradient blow up there all three different kinds of symmetries are considered by varying locations and directions of the emitter and a different estimate is derived for each case

resurgence in quasi classical scattering in quasi classical studies of closed systems e g a billiard resurgence means that the contribution of long periodic orbits to the spectral determinant can be expressed in terms of composites of short orbits and the resulting expression for the determinant is manifestly real the question has thus long been posed whether something like resurgence applies to a scattering system with its resonances we find here a resurgent expression for wigner s r matrix which gives the s matrix by a cayley transform in which long scattering pseudo orbits are expressed in terms of composites of short pseudo orbits both scattering and periodic and the result is manifestly hermitian giving a unitary expression for the s matrix this is particularly useful in the case that the resonance width is comparable with the resonance spacing the pseudo orbits are defined in terms of a fictitious and to some extent arbitrary closed reference system we relate the results to other formulations we give a simple but non trivial approximation for a particular example which illustrates the phenomenon of resonance trapping

approximation of haar distributed matrices and limiting distributions of eigenvalues of jacobi ensembles we develop a tool to approximate the entries of a large dimensional complex jacobi ensemble with independent complex gaussian random variables based on this and the author s earlier work in this direction we obtain the tracy widom law of the largest singular values of the jacobi emsemble moreover the circular law the marchenko pastur law the central limit theorem and the laws of large numbers for the spectral norms are also obtained

lecture hall tableaux abstract we introduce lecture hall tableaux which are fillings of a skew young diagram satisfying certain conditions lecture hall tableaux generalize both lecture hall partitions and anti lecture hall compositions and also contain reverse semistandard young tableaux as a limit case we show that the coefficients in the schur expansion of multivariate little q jacobi polynomials are generating functions for lecture hall tableaux using a selberg type integral we show that moments of multivariate little q jacobi polynomials which are equal to generating functions for lecture hall tableaux of a young diagram have a product formula we also explore various combinatorial properties of lecture hall tableaux

probability density of the determinant of a random hermitian matrix the probability density function for the determinant of a nn random hermitian matrix taken from the gaussian unitary ensemble is calculated it is found to be a meijer g function or a linear combination of two meijer g functions depending on the parity of n the integer moments of this probability density are also given

the importance of the selberg integral it has been remarked that a fair measure of the impact of atle selberg s work is the number of mathematical terms that bear his name one of these is the selberg integral an n dimensional generalization of the euler beta integral we trace its sudden rise to prominence initiated by a question to selberg from enrico bombieri more than thirty years after its initial publication in quick succession the selberg integral was used to prove an outstanding conjecture in random matrix theory and cases of the macdonald conjectures it further initiated the study of q analogues which in turn enriched the macdonald conjectures we review these developments and proceed to exhibit the sustained prominence of the selberg integral as evidenced by its central role in random matrix theory calogero sutherland quantum many body systems knizhnik zamolodchikov equations and multivariable orthogonal polynomial theory

an efficient tree decomposition method for permanents and mixed discriminants abstract we present an efficient algorithm to compute permanents mixed discriminants and hyperdeterminants of structured matrices and multidimensional arrays tensors we describe the sparsity structure of an array in terms of a graph and we assume that its treewidth denoted as is small our algorithm requires o n 2 arithmetic operations to compute permanents and o n 2 n 3 for mixed discriminants and hyperdeterminants we finally show that mixed volume computation continues to be hard under bounded treewidth assumptions

advanced determinant calculus a complement abstract this is a complement to my previous article advanced determinant calculus c krattenthaler advanced determinant calculus seminaire lotharingien combin 42 1999 the andrews festschrift article b42q 67 pp in the present article i share with the reader my experience of applying the methods described in the previous article in order to solve a particular problem from number theory g almkvist c krattenthaler j petersson some new formulas for experiment math 12 2003 441 456 moreover i add a list of determinant evaluations which i consider as interesting which have been found since the appearance of the previous article or which i failed to mention there including several conjectures and open problems

statistics of real eigenvalues in ginibre s ensemble of random real matrices the integrable structure of ginibre s orthogonal ensemble of random matrices is looked at through the prism of the probability p n k to find exactly k real eigenvalues in the spectrum of an n ifmmode times else texttimes fi n real asymmetric gaussian random matrix the exact solution for the probability function p n k is presented and its remarkable connection to the theory of symmetric functions is revealed an extension of the dyson integration theorem is a key ingredient of the theory presented

algebraic invariants of five qubits we consider the action of the group on fifth rank tensors on a two dimensional space that is the hilbert space of a five qubit system the hilbert series of the algebra of polynomial invariants of five qubits pure states is obtained and the simplest invariants are computed

probability density of determinants of random matrices in this brief paper the probability density of a random real complex and quaternion determinant is rederived using the singular values the behaviour of suitably rescaled random determinants is studied in the limit of infinite order of the matrices

the selberg jack symmetric functions abstract k aomoto has recently given a simple proof of an extension of a selberg s integral we prove the following generalization of aomoto s theorem for eachk 0 there exists a family sk t of homogeneous symmetric functions such that if the integrand of selberg s integral is multiplied bysk t1 tn then the integral has a certain closed form sincesk t is a renormalized jack symmetric function with 1 k we call sk t the selberg jack symmetric functions r p stanley and i g macdonald have extended many of the basic properties of the i j schur functions to the jack symmetric functions stanley s extensions of the pieri formula in a form obtained by macdonald s extension of the duality property and the combinatorial representation arise naturally in our proof we give proofs of these and other results of stanley and macdonald we give two constant term identities which are equivalent to the integration formula these results extend w g morris theorem foranand the macdonald morris constant term identity forbcn by exploring selberg s proof of his integral we give a recurrence relation outline a method for calculating the selberg jack symmetric functions and show that a constant term orthogonality relation recently proved by macdonald arises naturally we give an expansion which allows us to evaluate the normalization of this orthogonality thus giving an alternative proof of the integration formula

universality of eigenvector statistics of kicked tops of different symmetries the authors show that the eigenvectors of the floquet operators of periodically kicked tops with orthogonal unitary and symplectic canonical transformations conform to the predictions of the respective circular ensembles of random matrices

variable separation theory for the null hamilton jacobi equation the theory of the separation of variables for the null hamilton jacobi equation h 0 is systematically revisited and based on levi civita separability conditions with lagrangian multipliers the separation of the null equation is shown to be equivalent to the ordinary separation of the image of the original hamiltonian under a generalized jacobi maupertuis transformation the general results are applied to the special but fundamental case of the orthogonal separation of a natural hamiltonian with a fixed value of the energy the separation is then related to conditions which extend those of stackel and kalnins and miller for the null geodesic case and it is characterized by the existence of conformal killing two tensors of special kind

quantum conductance problems and the jacobi ensemble in one dimensional transport problems the scattering matrix s is decomposed into a block structure corresponding to reflection and transmission matrices at the two ends for s a random unitary matrix the singular value probability distribution function of these blocks is calculated the same is done when s is constrained to be symmetric or to be self dual quaternion real as is relevant in the presence of particular time reversal symmetries the latter results are shown to be very similar to those obtained under the unphysical assumption that s has real elements or has real quaternion elements respectively three methods are used metric forms a variant of the ingham seigel matrix integral and a theorem specifying the jacobi random matrix ensemble in terms of wishart distributed matrices

truncations of random orthogonal matrices statistical properties of nonsymmetric real random matrices of size m obtained as truncations of random orthogonal n n matrices are investigated we derive an exact formula for the density of eigenvalues which consists of two components finite fraction of eigenvalues are real while the remaining part of the spectrum is located inside the unit disk symmetrically with respect to the real axis in the case of strong nonorthogonality m n const the behavior typical to real ginibre ensemble is found in the case m n l with fixed l a universal distribution of resonance widths is recovered

asymptotics of selberg like integrals the unitary case and newton s interpolation formula we investigate the asymptotic behavior of the selberg like integral 1n 0 1 nx1p i j xi xj 2 ixia 1 1 xi b 1dxi as n for different scalings of the parameters a and b with n integrals of this type arise in the random matrix theory of electronic scattering in chaotic cavities supporting n channels in the two attached leads making use of newton s interpolation formula we show that an asymptotic limit exists and we compute it explicitly

random matrix theory and ζ 1 2 it we study the characteristic polynomials z u of matrices u in the circular unitary ensemble cue of random matrix theory exact expressions for any matrix size n are derived for the moments of z and z z and from these we obtain the asymptotics of the value distributions and cumulants of the real and imaginary parts of log z as n in the limit we show that these two distributions are independent and gaussian costin and lebowitz 15 previously found the gaussian limit distribution for im log z using a different approach and our result for the cumulants proves a conjecture made by them in this case we also calculate the leading order n asymptotics of the moments of z and z z these cue results are then compared with what is known about the riemann zeta function s on its critical line re s 1 2 assuming the riemann hypothesis equating the mean density of the non trivial zeros of the zeta function at a height t up the critical line with the mean density of the matrix eigenvalues gives a connection between n and t invoking this connection our cue results coincide with a theorem of selberg for the value distribution of log 1 2 it in the limit t they are also in close agreement with numerical data computed by odlyzko 29 for large but finite t this leads us to a conjecture for the moments of 1 2 it finally we generalize our random matrix results to the circular orthogonal coe and circular symplectic cse ensembles

eigenvector statistics of the product of ginibre matrices we develop a method to calculate left right eigenvector correlations of the product of m independent n n complex ginibre matrices for illustration we present explicit analytical results for the vector overlap for a couple of examples for small m and n we conjecture that the integrated overlap between left and right eigenvectors is given by the formula o 1 m 2 n 1 and support this conjecture by analytical and numerical calculations we derive an analytical expression for the limiting correlation density as n for the product of ginibre matrices as well as for the product of elliptic matrices in the latter case we find that the correlation function is independent of the eccentricities of the elliptic laws

the single ring theorem we study the empirical measure lan of the eigenvalues of non normal square matrices of the form an untnvn with un vn independent haar distributed on the unitary group and tn real diagonal we show that when the empirical measure of the eigenvalues of tn converges and tn satisfies some technical conditions lan converges towards a rotationally invariant measure on the complex plane whose support is a single ring in particular we provide a complete proof of feinberg zee single ring theorem 6 we also consider the case where un vn are independent haar distributed on the orthogonal group

time reversal symmetry and random polynomials we analyse the density of roots of random polynomials where each complex coefficient is constructed of a random modulus and a fixed deterministic phase the density of roots is shown to possess a singular component only in the case for which the phases increase linearly with the index of coefficients this means that contrary to earlier belief eigenvectors of a typical quantum chaotic system with some antiunitary symmetry will not display a clustering curve in the stellar representation moreover a class of time reverse invariant quantum systems is shown for which spectra display fluctuations characteristic of orthogonal ensemble while eigenvectors confer to predictions of unitary ensemble