raney distributions and random matrix theory recent works have shown that the family of probability distributions with moments given by the fuss catalan numbers permit a simple parameterized form for their density we extend this result to the raney distribution which by definition has its moments given by a generalization of the fuss catalan numbers such computations begin with an algebraic equation satisfied by the stieltjes transform which we show can be derived from the linear differential equation satisfied by the characteristic polynomial of random matrix realizations of the raney distribution for the fuss catalan distribution an equilibrium problem characterizing the density is identified the stieltjes transform for the limiting spectral density of the singular values squared of the matrix product formed from q inverse standard gaussian matrices and s standard gaussian matrices is shown to satisfy a variant of the algebraic equation relating to the raney distribution supported on 0 infty we show that it too permits a simple functional form upon the introduction of an appropriate choice of parameterization as an application the leading asymptotic form of the density as the endpoints of the support are approached is computed and is shown to have some universal features

level spacing distributions and the bessel kernel scaling models of randomn n hermitian matrices and passing to the limitn leads to integral operators whose fredholm determinants describe the statistics of the spacing of the eigenvalues of hermitian matrices of large order for the gaussian unitary ensemble and for many others as well the kernel one obtains by scaling in the bulk of the spectrum is the sine kernel frac sin pi x y pi x y rescaling the gue at the edge of the spectrum leads to the kernel frac ai x ai y ai x ai y x y where ai is the airy function in previous work we found several analogies between properties of this airy kernel and known properties of the sine kernel a system of partial differential equations associated with the logarithmic differential of the fredholm determinant when the underlying domain is a union of intervals a representation of the fredholm determinant in terms of a painleve transcendent in the case of a single interval and also in this case asymptotic expansions for these determinants and related quantities achieved with the help of a differential operator which commutes with the integral operator in this paper we show that there are completely analogous properties for a class of kernels which arise when one rescales the laguerre or jacobi ensembles at the edge of the spectrum namely frac j alpha sqrt x sqrt y j alpha sqrt y sqrt x j alpha sqrt x j alpha sqrt y 2 x y wherej z is the bessel function of order in the cases 1 2 these become after a variable change the kernels which arise when taking scaling limits in the bulk of the spectrum for the gaussian orthogonal and symplectic ensembles in particular an asymptotic expansion we derive will generalize ones found by dyson for the fredholm determinants of these kernels

z4 kerdock codes orthogonal spreads and extremal euclidean line sets when m is odd spreads in an orthogonal vector space of type omega 2m 2 2 are related to binary kerdock codes and extremal line sets in rr 2 m 1 with prescribed angles spreads in a 2m dimensional binary symplectic vector space are related to kerdock codes over zz 4 and extremal line sets in cc 2 m with prescribed angles these connections involve binary real and complex geometry associated with extraspecial 2 groups a geometric map from symplectic to orthogonal spreads is shown to induce the gray map from a corresponding zz 4 kerdock code to its binary image these geometric considerations lead to the construction for any odd composite m of large numbers of zz 4 kerdock codes they also produce new zz 4 linear kerdock and preparata codes 1991 mathematics subject classification primary 94b60 secondary 51m15 20c99

weighted complex projective 2 designs from bases optimal state determination by orthogonal measurements we introduce the problem of constructing weighted complex projective 2 designs from the union of a family of orthonormal bases if the weight remains constant across elements of the same basis then such designs can be interpreted as generalizations of complete sets of mutually unbiased bases being equivalent whenever the design is composed of d 1 bases in dimension d we show that for the purpose of quantum state determination these designs specify an optimal collection of orthogonal measurements using highly nonlinear functions on abelian groups we construct explicit examples from d 2 orthonormal bases whenever d 1 is a prime power covering dimensions d 6 10 and 12 for example where no complete sets of mutually unbiased bases have thus far been found

formal state determination following the general results of park and band it is shown that any state of a quantal ensemble allowed by the hilbert space formulation of nonrelativistic quantum mechanics may be determined from the results of measurements under the assumption that all measurements those of energy position and spin projection are in accordance with the projection postulate

singular value correlation functions for products of wishart random matrices we consider the product of m quadratic random matrices with complex elements and no further symmetry where all matrix elements of each factor have a gaussian distribution this generalizes the classical wishart laguerre gaussian unitary ensemble with m 1 in this paper we first compute the joint probability distribution for the singular values of the product matrix when the matrix size n and the number m are fixed but arbitrary this leads to a determinantal point process which can be realized in two different ways first it can be written as a one matrix singular value model with a nonstandard jacobian or second for m 2 as a two matrix singular value model with a set of auxiliary singular values and a weight proportional to the meijer g function for both formulations we determine all singular value correlation functions in terms of the kernels of biorthogonal polynomials which we explicitly construct they are given in terms of the hypergeometric and meijer g functions generalizing the laguerre polynomials for m 1 our investigation was motivated from applications in telecommunication of multi layered scattering multiple input and multiple output channels we present the ergodic mutual information for finite n for such a channel model with m 1 layers of scatterers as an example

brief equivalence of hybrid dynamical models this paper establishes equivalences among five classes of hybrid systems mixed logical dynamical mld systems linear complementarity lc systems extended linear complementarity elc systems piecewise affine pwa systems and max min plus scaling mmps systems some of the equivalences are established under rather mild additional assumptions these results are of paramount importance for transferring theoretical properties and tools from one class to another with the consequence that for the study of a particular hybrid system that belongs to any of these classes one can choose the most convenient hybrid modeling framework

products of rectangular random matrices singular values and progressive scattering we discuss the product of m rectangular random matrices with independent gaussian entries which have several applications including wireless telecommunication and econophysics for complex matrices an explicit expression for the joint probability density function is obtained using the harish chandra itzykson zuber integration formula explicit expressions for all correlation functions and moments for finite matrix sizes are obtained using a two matrix model and the method of biorthogonal polynomials this generalizes the classical result for the so called wishart laguerre gaussian unitary ensemble or chiral unitary ensemble at m 1 and previous results for the product of square matrices the correlation functions are given by a determinantal point process where the kernel can be expressed in terms of meijer g functions we compare the results with numerical simulations and known results for the macroscopic level density in the limit of large matrices the location of the end points of support for the latter are analyzed in detail for general m finally we consider the so called ergodic mutual information which gives an upper bound for the spectral efficiency of a mimo communication channel with multifold scattering

symmetric informationally complete positive operator valued measures a new computer study we report on a new computer study of the existence of d2 equiangular lines in d complex dimensions such maximal complex projective codes are conjectured to exist in all finite dimensions and are the underlying mathematical objects defining symmetric informationally complete measurements in quantum theory we provide numerical solutions in all dimensions d 67 and moreover a putatively complete list of weyl heisenberg covariant solutions for d 50 a symmetry analysis of this list leads to new algebraic solutions in dimensions d 24 35 and 48 which are given together with algebraic solutions for d 4 15 and 19

on the τ function of the painleve equations abstract concerning the hamiltonian structure associated with painleve equations we define the function related to a hamiltonian and show that the correlation function given in the study of the two dimensional ising model is a function for the third painleve equation the list of the hamiltonians which are polynomials in the two canonical variables is given and we prove that the function related to each of them is holomorphic

evenly distributed unitaries on the structure of unitary designs we clarify the mathematical structure underlying unitary t designs these are sets of unitary matrices evenly distributed in the sense that the average of any tth order polynomial over the design equals the average over the entire unitary group we present a simple necessary and sufficient criterion for deciding if a set of matrices constitutes a design lower bounds for the number of elements of 2 designs are derived we show how to turn mutually unbiased bases into approximate 2 designs whose cardinality is optimal in leading order designs of higher order are discussed and an example of a unitary 5 design is presented we comment on the relation between unitary and spherical designs and outline methods for finding designs numerically or by searching character tables of finite groups further we sketch connections to problems in linear optics and questions regarding typical entanglement

the curious nonexistence of gaussian 2 designs ensembles of pure quantum states whose 2nd moments equal those of the unitarily uniform haar ensemble 2 designs are optimal solutions for several tasks in quantum information science especially state and process tomography we show that gaussian states cannot form a 2 design for the continuous variable quantum optical hilbert space l 2 mathbb r this is surprising because the affine symplectic group hwsp the natural symmetry group of gaussian states is irreducible on the symmetric subspace of two copies in finite dimensional hilbert spaces irreducibility guarantees that hwsp covariant ensembles such as mutually unbiased bases in prime dimensions are always 2 designs this property is violated by continuous variables for a subtle reason the well defined hwsp invariant ensemble of gaussian states does not have a density matrix because its defining integral does not converge in fact no gaussian ensemble is even close in a precise sense to being a 2 design this surprising difference between discrete and continuous quantum mechanics has important implications for optical state and process tomography

on the solvability of the quantum rabi model and its 2 photon and two mode generalizations we study the solvability of the time independent matrix schrodinger differential equations of the quantum rabi model and its 2 photon and two mode generalizations in bargmann hilbert spaces of entire functions we show that the rabi model and its 2 photon and two mode analogs are quasi exactly solvable we derive the exact closed form expressions for the energies and the allowed model parameters for all the three cases in the solvable subspaces up to a normalization factor the eigenfunctions for these models are given by polynomials whose roots are determined by systems of algebraic equations

affine constellations without mutually unbiased counterparts it has been conjectured that a complete set of mutually unbiased bases in a space of dimension d exists if and only if there is an affine plane of order d we introduce affine constellations and compare their existence properties with those of mutually unbiased constellations the observed discrepancies make a deeper relation between the two existence problems unlikely

geometrical description of quantal state determination under the assumption that every quantal measurement may give data about the post measurement state of the inspected ensemble the problem of the state determination is reconsidered it is shown that orthogonal decomposition of the set of complex n n hermitian matrices into the commutative subsets allows operators to be found such that post measurement information on these observables allows a partial in some cases total determination of the pre measurement state to be effected

mubs inequivalence and affine planes there are fairly large families of unitarily inequivalent complete sets of n 1 mutually unbiased bases mubs known in cn for various prime powers n the number of such sets is not bounded above by any polynomial as a function of n while it is standard that there is a superficial similarity between complete sets of mubs and finite affine planes there is an intimate relationship between these large families and affine planes this note briefly summarizes old results that do not appear to be well known concerning known families of complete sets of mubs and their associated planes

the weight distribution of a class of linear codes from perfect nonlinear functions in this correspondence the weight distribution of a class of linear codes based on perfect nonlinear functions also called planar functions is determined the class of linear codes under study are either optimal or among the best codes known and have nice applications in cryptography

averaging sets a generalization of mean values and spherical designs 0 1 r one cannot of course expect there to be a single point x at which every one assumes its mean value but there is an easy generalization in terms of weighted averages there exist at most m points xi and positive weights oi summing to 1 so that aifj xi i fi dt for j 1 2 m

the lie algebraic significance of symmetric informationally complete measurements examples of symmetric informationally complete positive operator valued measures sic povms have been constructed in every dimension 67 however it remains an open question whether they exist in all finite dimensions a sic povm is usually thought of as a highly symmetric structure in quantum state space however its elements can equally well be regarded as a basis for the lie algebra gl d c in this paper we examine the resulting structure constants which are calculated from the traces of the triple products of the sic povm elements and which it turns out characterize the sic povm up to unitary equivalence we show that the structure constants have numerous remarkable properties in particular we show that the existence of a sic povm in dimension d is equivalent to the existence of a certain structure in the adjoint representation of gl d c we hope that transforming the problem in this way from a question about quantum state space to a question about lie algebras may help to make the existence problem tractable

on a new non isospectral variant of the boussinesq hierarchy we give a new non isospectral extension to 2 1 dimensions of the boussinesq hierarchy such a non isospectral extension of the third order scattering problem xxx u x v 0 has not been considered previously this extends our previous results on one component hierarchies in 2 1 dimensions associated to third order non isospectral scattering problems we characterize our entire 2 1 dimensional hierarchy and its linear problem using a single partial differential equation and its corresponding non isospectral scattering problem this then allows an alternative approach to the construction of linear problems for the entire 2 1 dimensional hierarchy reductions of this hierarchy yield new integrable hierarchies of systems of ordinary differential equations together with their underlying linear problems in particular we obtain a fourth painleve hierarchy i e a hierarchy of ordinary differential equations having the fourth painleveequation as its first member we also obtain a hierarchy having as its first member a generalization of an equation defining a new transcendent due to cosgrove

universal algorithm for optimal estimation of quantum states from finite ensembles via realizable generalized measurement we present a universal algorithm for the optimal quantum state estimation of an arbitrary finite dimensional system the algorithm specifies a physically realizable i e finite positive operator valued measurement on a finite number of identically prepared systems we illustrate the general formalism by applying it to different scenarios of the state estimation of n independent and identically prepared two level systems qubits

nice error frames canonical abstract error groups and the construction of sics abstract nice error bases are generalisations of the pauli matrices which have applications in quantum information theory these orthonormal bases for the d d matrices m d c also generalise the projective action of the heisenberg group on c d here we extend nice error bases to nice error frames these are equal norm tight frames for m d c consisting of d d unitary matrices with a group indexing structure we show that each nice error frame irreducible faithful projective representation is associated with a canonical abstract error group this is calculated in number of examples e g for all nice error bases for d 14 which then allows us to investigate which nice error bases might give rise to sics symmetric informationally complete positive operator valued measures these results show that the current catalogue of nice error bases over counts in particular we give an explicit example of a sic for d 6 with a nonabelian index group and show that the hoggar lines appear for various nice error bases some of which are subgroups of the clifford group thus all known sics appear as orbits of subgroups of the clifford group

random matrices virasoro algebras and noncommutative kp department of mathematics brandeis university waltham mass 02254 usa email adler math brandeis edu the support of a national science foundation grant dms 9503246 is gratefully acknowledged department of mathematics kyoto university kyoto 606 japan the hospitality of the university of louvain and brandeis university is gratefully acknowledged department of mathematics universite de louvain 1348 louvain la neuve belgium and brandeis university waltham mass 02254 usa e mail vanmoerbeke geom ucl ac be and math brandeis edu the support of a national science foundation grant dms 9503246 a nato a fnrs and a francqui foundation grant is gratefully acknowledged a corrected version has appeared in the duke mathematical journal 94 379 431 1998

on sic povms in prime dimensions the generalized pauli group and its normalizer the clifford group have a rich mathematical structure which is relevant to the problem of constructing symmetric informationally complete povms sic povms to date almost every known sic povm fiducial vector is an eigenstate of a canonical unitary in the clifford group i show that every canonical unitary in prime dimensions p 3 lies in the same conjugacy class of the clifford group and give a class representative for all such dimensions it follows that if even one such sic povm fiducial vector is an eigenvector of such a unitary then all of them are for a given such dimension i also conjecture that in all dimensions d the number of conjugacy classes is bounded above by 3 and depends only on dmod9 and i support this claim with computer computations in all dimensions 48

mutually orthogonal latin squares from the inner products of vectors in mutually unbiased bases mutually unbiased bases mubs are important in quantum information theory while constructions of complete sets of d 1 mubs in c d are known when d is a prime power it is unknown if such complete sets exist in non prime power dimensions it has been conjectured that complete sets of mubs only exist in if a maximal set of mutually orthogonal latin squares mols of side length d also exists there are several constructions roy and scott 2007 j math phys 48 072110 paterek daki and brukner 2009 phys rev a 79 012109 of complete sets of mubs from specific types of mols which use galois fields to construct the vectors of the mubs in this paper two known constructions of mubs alltop 1980 ieee trans inf theory 26 350 354 wootters and fields 1989 ann phys 191 363 381 both of which use polynomials over a galois field are used to construct complete sets of mols in the odd prime case the mols come from the inner products of pairs of vectors in the mubs

deformation analysis of matrix models abstract in this paper we show that the non linear equations that play a role in tracy and widom s analysis of scaling limits for hermitian matrix models are special cases of the monodromy preserving deformation equations first discussed by sato miwa and jimbo in the early 1980 s

antiquantization isomonodromy and integrability an extended analysis of links between linear differential equations and the nonlinear painleve equation pv i is given for linear equations second order equations in different forms as well as various first order systems are chosen the role of an accessory parameter is explained the relationship to the schlesinger system is made clear an extended analysis of links between linear differential equations and the nonlinear painleve equation pv i is given for linear equations second order equations in different forms as well as various first order systems are chosen the role of an accessory parameter is explained the relationship to the schlesinger system is made clear

a family of alltop functions that are ea inequivalent to the cubic function sequences with optimal correlation properties are much sought after for applications in communication systems in 1980 alltop ieee trans inf theory 26 3 350 354 1980 described a set of sequences based on a cubic function and showed that these sequences were optimal with respect to the known bounds on auto and crosscorrelation subsequently these sequences were used to construct mutually unbiased bases mubs a structure of importance in quantum information theory the key feature of this cubic function is that its difference function is a planar function functions with planar difference functions have been called alltop functions this paper provides a new family of alltop functions and establishes the use of alltop functions for construction of sequence sets and mubs

on approximately symmetric informationally complete positive operator valued measures and related systems of quantum states we address the problem of constructing positive operator valued measures povms in finite dimension n consisting of n2 operators of rank one which have an inner product close to uniform this is motivated by the related question of constructing symmetric informationally complete povms sic povms for which the inner products are perfectly uniform however sic povms are notoriously hard to construct and despite some success of constructing them numerically there is no analytic construction known we present two constructions of approximate versions of sic povms where a small deviation from uniformity of the inner products is allowed the first construction is based on selecting vectors from a maximal collection of mutually unbiased bases and works whenever the dimension of the system is a prime power the second construction is based on perturbing the matrix elements of a subset of mutually unbiased bases moreover we construct vector systems in cn which are almost orthogonal and which might turn out

mutually unbiased bases and hadamard matrices of order six we report on a search for mutually unbiased bases mubs in six dimensions we find only triplets of mubs and thus do not come close to the theoretical upper bound 7 however we point out that the natural habitat for sets of mubs is the set of all complex hadamard matrices of the given order and we introduce a natural notion of distance between bases in hilbert space this allows us to draw a detailed map of where in the landscape the mub triplets are situated we use available tools such as the theory of the discrete fourier transform to organize our results finally we present some evidence for the conjecture that there exists a four dimensional family of complex hadamard matrices of order 6 if this conjecture is true the landscape in which one may search for mubs is much larger than previously thought

extreme superposition rogue waves of infinite order and the painleve iii hierarchy we study the fundamental rogue wave solutions of the focusing nonlinear schrodinger equation in the limit of large order using a recently proposed riemann hilbert representation of the rogue wave solution of arbitrary order k we establish the existence of a limiting profile of the rogue wave in the large k limit when the solution is viewed in appropriate rescaled variables capturing the near field region where the solution has the largest amplitude the limiting profile is a new particular solution of the focusing nonlinear schrodinger equation in the rescaled variables the rogue wave of infinite order which also satisfies ordinary differential equations with respect to space and time the spatial differential equations are identified with certain members of the painleve iii hierarchy we compute the far field asymptotic behavior of the near field limit solution and compare the asymptotic formulas with the exact solution using numerical methods for solving riemann hilbert problems in a certain transitional region for the asymptotics the near field limit function is described by a specific globally defined tritronquee solution of the painleve ii equation these properties lead us to regard the rogue wave of infinite order as a new special function

on the solvability of the discrete second painleve equation the inverse monodromy method for studying the riemann hilbert problem associated with classical painlev e equations is applied to the discrete second painlev e equation

integrable fredholm operators and dual isomonodromic deformations the fredholm determinants of a special class of integrable integral operators k supported on the union of m curve segments in the complex plane are shown to be the functions of an isomonodromic family of meromorphic covariant derivative operators having regular singular points at the 2m endpoints of the curve segments and a singular point of poincare index 1 at infinity the rank r of the corresponding vector bundle over the riemann sphere equals the number of distinct terms in the exponential sum defining the numerator of the integral kernel the matrix riemann hilbert problem method is used to deduce an identification of the fredholm determinant as a function in the sense of segal wilson and sato i e in terms of abelian group actions on the determinant line bundle over a loop space grassmannian an associated dual isomonodromic family of covariant derivative operators having rank n 2m and r finite regular singular points located at the values of the exponents defining the kernel of k is derived the deformation equations for this family are shown to follow from an associated dual set of riemann hilbert data in which the roles of the r exponential factors in the kernel and the 2m endpoints of its support are interchanged the operators are analogously associated to an integral operator whose fredholm determinant is equal to that of k

technical communique an efficient algorithm for optimal control of pwa systems with polyhedral performance indices we present an algorithm for the computation of explicit optimal control laws for piecewise affine pwa systems with polyhedral performance indices the algorithm is based on dynamic programming dp and represents an extension of ideas initially proposed in kerrigan and mayne 2003 optimal control of constrained piecewise affine systems with bounded disturbances in proceedings of the 41st ieee conference on decision and control las vegas nevada usa december and baotic et al 2003 a new algorithm for constrained finite time optimal control of hybrid systems with a linear performance index in proceedings of european control conference cambridge uk september specifically we show how to exploit the underlying geometric structure of the optimization problem in order to significantly improve the efficiency of the off line computations an extensive case study is provided which clearly indicates that the algorithm proposed in this paper may be preferable to other schemes published in the literature

symmetric informationally complete quantum measurements we consider the existence in arbitrary finite dimensions d of a positive operator valued measure povm comprised of d2 rank one operators all of whose operator inner products are equal such a set is called a symmetric informationally complete povm sic povm and is equivalent to a set of d2 equiangular lines in cd sic povms are relevant for quantum state tomography quantum cryptography and foundational issues in quantum mechanics we construct sic povms in dimensions two three and four we further conjecture that a particular kind of group covariant sic povm exists in arbitrary dimensions providing numerical results up to dimension 45 to bolster this claim

there is no generalization of known formulas for mutually unbiased bases in a quantum system having a finite number n of orthogonal states two orthonormal bases ai and bj are called mutually unbiased if all inner products ai bj have the same modulus 1 n this concept appears in several quantum information problems the number of pairwise mutually unbiased bases is at most n 1 and various constructions of such n 1 bases have been found when n is a power of a prime number we study families of formulas that generalize these constructions to arbitrary dimensions using finite rings we then prove that there exists a set of n 1 mutually unbiased bases described by such formulas if and only if n is a power of a prime number

the z sub 4 linearity of kerdock preparata goethals and related codes certain notorious nonlinear binary codes contain more codewords than any known linear code these include the codes constructed by nordstrom robinson 1967 kerdock 1972 preparata 1968 goethals 1974 and delsarte goethals 1975 it is shown here that all these codes can be very simply constructed as binary images under the gray map of linear codes over z sub 4 the integers mod 4 although this requires a slight modification of the preparata and goethals codes the construction implies that all these binary codes are distance invariant duality in the z sub 4 domain implies that the binary images have dual weight distributions the kerdock and preparata codes are duals over z sub 4 and the nordstrom robinson code is self dual which explains why their weight distributions are dual to each other the kerdock and preparata codes are z sub 4 analogues of first order reed muller and extended hamming codes respectively all these codes are extended cyclic codes over z sub 4 which greatly simplifies encoding and decoding an algebraic hard decision decoding algorithm is given for the preparata code and a hadamard transform soft decision decoding algorithm for the i kerdock code binary first and second order reed muller codes are also linear over z sub 4 but extended hamming codes of length n spl ges 32 and the golay code are not using z sub 4 linearity a new family of distance regular graphs are constructed on the cosets of the preparata code

tomographically complete sets of orthonormal bases in finite systems quantum systems where the position and momentum are in the ring d is an odd integer are considered a tomographically complete set of bases i e a set of bases such that probabilities from tomography experiments corresponding to these bases can be used for the calculation of an arbitrary density matrix is considered such a set of bases is constructed by considering a factorization of a system with dimension d p1p2 where p1 p2 are prime numbers in terms of two subsystems with dimensions p1 and p2 appropriate combination of the mutually unbiased bases in the subsystems leads to a tomographically complete set of bases in the full system this set leads to probabilities along all the lines through the origin 0 0 in the phase space with exactly d points it is shown that the number of these lines is d the dedekind function the general theory is exemplified with examples

brief paper on a reachability problem for affine hypersurface systems on polytopes this paper studies the problem for an affine hypersurface system with n 1 inputs to reach a polytopic target set starting from inside a polytope in the state space we present a solution which begins with a characterization of solvability by open loop control and concludes with a procedure to synthesize a feedback control our emphasis is on methods of subdivision triangulation and covers which explicitly account for the capabilities of the control system in contrast with the previous literature the partition methods are guaranteed to yield a correct feedback synthesis assuming the problem is solvable by open loop control

control of systems integrating logic dynamics and constraints this paper proposes a framework for modeling and controlling systems described by interdependent physical laws logic rules and operating constraints denoted as mixed logical dynamical mld systems these are described by linear dynamic equations subject to linear inequalities involving real and integer variables mld systems include linear hybrid systems finite state machines some classes of discrete event systems constrained linear systems and nonlinear systems which can be approximated by piecewise linear functions a predictive control scheme is proposed which is able to stabilize mld systems on desired reference trajectories while fulfilling operating constraints and possibly take into account previous qualitative knowledge in the form of heuristic rules due to the presence of integer variables the resulting on line optimization procedures are solved through mixed integer quadratic programming miqp for which efficient solvers have been recently developed some examples and a simulation case study on a complex gas supply system are reported

achieving the orthoplex bound and constructing weighted complex projective 2 designs with singer sets abstract equiangular tight frames are examples of grassmannian line packings for a hilbert space more specifically according to a bound by welch they are minimizers for the maximal magnitude occurring among the inner products of all pairs of vectors in a unit norm frame this paper is dedicated to packings in the regime in which the number of frame vectors precludes the existence of equiangular frames the orthoplex bound then serves as an alternative to infer a geometric structure of optimal designs we construct frames of unit norm vectors in k dimensional complex hilbert spaces that achieve the orthoplex bound when k 1 is a prime power we obtain a tight frame with k 2 1 vectors and when k is a prime power with k 2 k 1 vectors in addition we show that these frames form weighted complex projective 2 designs that are useful additions to maximal equiangular tight frames and maximal sets of mutually unbiased bases in quantum state tomography our construction is based on singer s family of difference sets and the related concept of relative difference sets

painleve transcendent evaluations of finite system density matrices for 1d impenetrable bosons the recent experimental realisation of a one dimensional bose gas of ultra cold alkali atoms has renewed attention on the theoretical properties of the impenetrable bose gas of primary concern is the ground state occupation of effective single particle states in the finite system and thus the tendency for bose einstein condensation this requires the computation of the density matrix for the impenetrable bose gas on a circle we evaluate the density matrix in terms of a particular painleve vi transcendent in form and furthermore show that the density matrix satisfies a recurrence relation in the number of particles for the impenetrable bose gas in a harmonic trap and with dirichlet or neumann boundary conditions we give a determinant form for the density matrix a form as an average over the eigenvalues of an ensemble of random matrices and in special cases an evaluation in terms of a transcendent related to painleve v and vi we discuss how our results can be used to compute the ground state occupations

differential equations for singular values of products of ginibre random matrices it was proved by akemann et al 2013 phys rev e 88 052118 that squared singular values of products of m complex ginibre random matrices form a determinantal point process whose correlation kernel is expressible in terms of meijer s g functions kuijlaars and zhang arxiv 1308 1003 recently showed that at the edge of the spectrum this correlation kernel has a remarkable scaling limit which can be understood as a generalization of the classical bessel kernel of random matrix theory in this paper we investigate the fredholm determinant of the operator with the kernel where j is a disjoint union of intervals and is the characteristic function of the set j this fredholm determinant is equal to the probability that j contains no particles of the limiting determinantal point process defined by the gap probability we derive the hamiltonian differential associated with the corresponding fredholm determinant and relate them with the monodromy preserving deformation equations of the jimbo miwa mori ueno and sato theory in the special case we give a formula for the gap probability in terms of a solution of a system of nonlinear ordinary differential equations

tight informationally complete quantum measurements we introduce a class of informationally complete positive operator valued measures which are in analogy with a tight frame as close as possible to orthonormal bases for the space of quantum states these measures are distinguished by an exceptionally simple state reconstruction formula which allows painless quantum state tomography complete sets of mutually unbiased bases and symmetric informationally complete positive operator valued measures are both members of this class the latter being the unique minimal rank one members recast as ensembles of pure quantum states the rank one members are in fact equivalent to weighted 2 designs in complex projective space these measures are shown to be optimal for quantum cloning and linear quantum state tomography

minimal optimal generalized quantum measurements optimal and finite positive operator valued measurements on a finite number n of identically prepared systems have been presented recently with physical realization in mind we propose here optimal and minimal generalized quantum measurements for two level systems we explicitly construct them up to n 7 and verify that they are minimal up to n 5 we finally propose an expression which gives the size of the minimal optimal measurements for arbitrary n

on linear systems and τ functions associated with lame s equation and painleve s equation vi abstract painleve s transcendental differential equation p vi may be expressed as the consistency condition for a pair of linear differential equations with 2 2 matrix coefficients with rational entries by a construction due to tracy and widom this linear system is associated with certain kernels which give trace class operators on hilbert space this paper expresses such operators in terms of hankel operators of linear systems which are realised in terms of the laurent coefficients of the solutions of the differential equations let p t l 2 0 l 2 t be the orthogonal projection then the fredholm determinant t det i p t defines the function which is here expressed in terms of the solution of a matrix gelfand levitan equation for suitable values of the parameters solutions of the hypergeometric equation give a linear system with similar properties for meromorphic transfer functions that have poles on an arithmetic progression the corresponding hankel operator has a simple form with respect to an exponential basis in l 2 0 so det i p t can be expressed as a series of finite determinants this applies to elliptic functions of the second kind such as satisfy lame s equation with l 1

integrable structure of products of finite complex ginibre random matrices abstract we consider the squared singular values of the product of m standard complex gaussian matrices since the squared singular values form a determinantal point process with a particular meijer g function kernel the gap probabilities are given by a fredholm determinant based on this kernel it was shown by strahov 2014 that a hard edge scaling limit of the gap probabilities is described by hamiltonian differential equations which can be formulated as an isomonodromic deformation system similar to the theory of the kyoto school we generalize this result to the case of finite matrices by first finding a representation of the finite kernel in integrable form as a result we obtain the hamiltonian structure for finite size matrices and formulate it in terms of a m 1 m 1 matrix schlesinger system the case m 1 reproduces the tracy and widom theory which results in the painleve v equation for the 0 s gap probability some integrals of motion for m 2 are identified and a coupled system of differential equations in two unknowns is presented which uniquely determines the corresponding 0 s gap probability

optimizing quantum process tomography with unitary 2 designs we show that weighted unitary 2 designs define optimal measurements on the system ancilla output state for ancilla assisted process tomography of unital quantum channels examples include complete sets of mutually unbiased unitary operator bases each of these specifies a minimal series of optimal orthogonal measurements general quantum channels are also considered

discrete phase space approach to mutually orthogonal latin squares we show there is a natural connection between latin squares and commutative sets of monomials defining geometric structures in finite phase space of prime power dimensions a complete set of such monomials defines a mutually unbiased basis mub and may be associated with a complete set of mutually orthogonal latin squares mols we translate some possible operations on the monomial sets into isomorphisms of latin squares and find a general form of permutations that map between latin squares corresponding to unitarily equivalent mutually unbiased sets

weak mutually unbiased bases quantum systems with variables in are considered the properties of lines in the phase space of these systems are studied weak mutually unbiased bases in these systems are defined as bases for which the overlap of any two vectors in two different bases is equal to d 1 2 or alternatively to one of the d 1 2i 0 where di is a divisor of d apart from d 1 they are designed for the geometry of the phase space in the sense that there is a duality between the weak mutually unbiased bases and the maximal lines through the origin in the special case of prime d there are no divisors of d apart from 1 d and the weak mutually unbiased bases are mutually unbiased bases

random words toeplitz determinants and integrable systems ii this paper connects the analysis of the length of the longest weakly increasing subsequence of inhomogeneous random words to a riemnn hilbert problem and an associated system of integrable partial differential equations in particular we show that the poissonization of the distribution function of this length can be identified as the jimbo miwa ueno tau function

nonlinear regulation the piecewise linear approach this paper approaches nonlinear control problems through the use of discrete time piecewise linear systems these are systems whose next state and output maps are both described by pl maps i e by maps which are affine on each of the components of a finite polyhedral partition various results on state and output feedback observers and inverses standard for linear systems are proved for pl systems many of these results are then used in the study of more general both discrete and continuous time systems using suitable approximations

an area preserving action of the modular group on cubic surfaces and the painleve vi equation we construct an area preserving action of the modular group on a general 4 parameter family of affine cubic surfaces we present a geometrical background behind this construction that is a natural symplectic structure on a moduli space of rank two linear monodromy representations over the 2 dimensional sphere with four punctures and a natural symplectic action upon it of the braid group on three strings studying this action as a discrete dynamical system will be important in discussing the monodromy of the painleve vi equation

weak commutation relations and eigenvalue statistics for products of rectangular random matrices we study the joint probability density of the eigenvalues of a product of rectangular real complex or quaternion random matrices in a unified way the random matrices are distributed according to arbitrary probability densities whose only restriction is the invariance under left and right multiplication by orthogonal unitary or unitary symplectic matrices respectively we show that a product of rectangular matrices is statistically equivalent to a product of square matrices hereby we prove a weak commutation relation of the random matrices at finite matrix sizes which previously has been discussed for infinite matrix size moreover we derive the joint probability densities of the eigenvalues to illustrate our results we apply them to a product of random matrices drawn from ginibre ensembles and jacobi ensembles as well as a mixed version thereof for these weights we show that the product of complex random matrices yields a determinantal point process while the real and quaternion matrix ensembles correspond to pfaffian point processes our results are visualized by numerical simulations furthermore we present an application to a transport on a closed disordered chain coupled to a particle bath

the partition function of the two matrix model as an isomonodromic τ function we consider the itzykson zuber eynard mehta two matrix model and prove that the partition function is an isomonodromic function in a sense that generalizes that of jimbo et al monodromy preserving deformation of linear ordinary differential equations with rational coefficients physica d 2 306 1981 in order to achieve the generalization we need to define a notion of function for isomonodromic systems where the adregularity of the leading coefficient is not a necessary requirement

geometry of the welch bounds abstract a geometric perspective involving grammian and frame operators is used to derive the entire family of welch bounds this perspective unifies a number of observations that have been made regarding tightness of the bounds and their connections to symmetric k tensors tight frames homogeneous polynomials and t designs in particular a connection has been drawn between sampling of homogeneous polynomials and frames of symmetric k tensors it is also shown that tightness of the bounds requires tight frames the lack of tight frames of symmetric k tensors in many cases however leads to consideration of sets that come as close as possible to attaining the bounds the geometric derivation is then extended in the setting of generalized or continuous frames the welch bounds for finite sets and countably infinite sets become special cases of this general setting

dropping the independence singular values for products of two coupled random matrices we study the singular values of the product of two coupled rectangular random matrices as a determinantal point process each of the two factors is given by a parameter dependent linear combination of two independent complex gaussian random matrices which is equivalent to a coupling of the two factors via an itzykson zuber term we prove that the squared singular values of such a product form a biorthogonal ensemble and establish its exact solvability the parameter dependence allows us to interpolate between the singular value statistics of the laguerre ensemble and that of the product of two independent complex ginibre ensembles which are both known we give exact formulae for the correlation kernel in terms of a complex double contour integral suitable for the subsequent asymptotic analysis in particular we derive a christoffel darboux type formula for the correlation kernel based on a five term recurrence relation for our biorthogonal functions it enables us to find its scaling limit at the origin representing a hard edge the resulting limiting kernel coincides with the universal meijer g kernel found by several authors in different ensembles we show that the central limit theorem holds for the linear statistics of the singular values and give the limiting variance explicitly

the cauchy problem for the second member of a rm p_ iv hierarchy a rigorous method the inverse monodromy transform for studying the riemann hilbert rh problem associated with the classical painleve equations is applied to the second member of a fourth painleve hierarchy we show that the cauchy problem for the second member of this hierarchy admits in general a global meromorphic solution in x moreover for a particular choice of the monodromy data the associated rh problem can be reduced to a set of scalar rh problems and a special solution which can be written in terms of the airy function is obtained

on wright s generalized bessel kernel abstract in this paper we consider the wright s generalized bessel kernel k x y defined by x 0 1 j 1 1 u x j 1 u y u d u 1 0 where j a b x j 0 x j j a b j a c b 1 is wright s generalization of the bessel function this non symmetric kernel which generalizes the classical bessel kernel corresponding to 1 in random matrix theory is the hard edge scaling limit of the correlation kernel for certain muttalib borodin ensembles we show that if is rational i e m n with m n n g c d m n 1 and m 1 m n the wright s generalized bessel kernel is integrable in the sense of its izergin korepin slavnov we then come to the fredholm determinant of this kernel over the union of several scaled intervals which can also be interpreted as the gap probability the probability of finding no particles on these intervals the integrable structure allows us to obtain a system of coupled partial differential equations associated with the corresponding fredholm determinant as well as a hamiltonian interpretation as a consequence we are able to represent the gap probability over a single interval 0 s in terms of a solution of a system of nonlinear ordinary differential equations

a sharpening of the welch bounds and the existence of real and complex spherical t designs the welch bounds for a finite set of unit vectors are a family of inequalities indexed by t 1 2 ldots which describe how evenly spread the vectors are they have important applications in signal analysis where sequences giving equality in the first welch bound are known as welch bound equality sequences or as unit norm tight frames here we consider sequences of vectors giving equality in the higher order welch bounds these are seen to correspond to tight frames for the complex symmetric t tensors which we prove always exist we show that for t 1 the welch bounds can be sharpened for real vectors and again vectors giving equality always exist we give a unified treatment of various conditions for equality in both the real and complex cases in particular we give an explicit description of the corresponding cubature rules t designs our results set up a framework for the construction and classification several configurations of vectors of recent interest these include mutually unbiased bases complex equiangular lines spherical half designs projective t designs and minimisers of the higher order frame potential one interesting consequence is a construction of sets of complex equiangular lines which were previously unknown

transformations rs42 3 of the ranks 4 and algebraic solutions of the sixth painleve equation compositions of rational transformations of independent variables of linear matrix ordinary differential equations odes with the schlesinger transformations rs transformations are used to construct algebraic solutions of the sixth painleve equation rs transformations of the ranks 3 and 4 of 2 2 matrix fuchsian odes with 3 singular points into analogous ode with 4 singular points are classified

hybrid model reference adaptive control of piecewise affine systems this paper is concerned with the derivation of a model reference adaptive control mrac scheme for multimodal piecewise affine pwa and piecewise linear systems the control allows the plant to track asymptotically the states of a multimodal piecewise affine or smooth reference model the reference model can be characterized by a number and geometry of phase space regions that can be entirely different from those of the plant numerical simulations on a set of representative examples confirm the theoretical derivation and proof of stability

monodromy preserving deformation of linear ordinary differential equations with rational coefficients iii abstract a unified treatment of monodromy and spectrum preserving deformations is presented in particular a general procedure is described to reduce the latter into the former consistently the concept of the function previously introduced for the former 2 is extended to the isospectral context it is shown that the general monodromy and spectrum preserving deformation equations can be written as hirota s bilinear differential equations by using the functions as dependent variables

the indefinite anti self dual metrics and the painleve equations we classify the su 2 invariant anti self dual metrics with a signature the metrics are specified by a solution of painleve vi v iii or ii moreover we show the geometric meaning of the metrics specified by each type of painleve function

cauchy laguerre two matrix model and the meijer g random point field we apply the general theory of cauchy biorthogonal polynomials developed in bertola et al commun math phys 287 3 983 1014 2009 and bertola et al j approx th 162 4 832 867 2010 to the case associated with laguerre measures in particular we obtain explicit formulae in terms of meijer g functions for all key objects relevant to the study of the corresponding biorthogonal polynomials and the cauchy two matrix model associated with them the central theorem we prove is that a scaling limit of the correlation functions for eigenvalues near the origin exists and is given by a new determinantal two level random point field the meijer g random field we conjecture that this random point field leads to a novel universality class of random fields parametrized by exponents of laguerre weights we express the joint distributions of the smallest eigenvalues in terms of suitable fredholm determinants and evaluate them numerically we also show that in a suitable limit the meijer g random field converges to the bessel random field and hence the behavior of the eigenvalues of one of the two matrices converges to the one of the laguerre ensemble

parameterized picard vessiot extensions and atiyah extensions abstract generalizing atiyah extensions we introduce and study differential abelian tensor categories over differential rings by a differential ring we mean a commutative ring with an action of a lie ring by derivations in particular these derivations act on a differential category a differential tannakian theory is developed the main application is to the galois theory of linear differential equations with parameters namely we show the existence of a parameterized picard vessiot extension and therefore the galois correspondence for many differential fields with possibly non differentially closed fields of constants that is fields of functions of parameters other applications include a substantially simplified test for a system of linear differential equations with parameters to be isomonodromic which will appear in a separate paper this application is based on differential categories developed in the present paper and not just differential algebraic groups and their representations

symmetric informationally complete positive operator valued measures and the extended clifford group we describe the structure of the extended clifford group defined to be the group consisting of all operators unitary and antiunitary which normalize the generalized pauli group or weyl heisenberg group as it is often called we also obtain a number of results concerning the structure of the clifford group proper i e the group consisting just of the unitary operators which normalize the generalized pauli group we then investigate the action of the extended clifford group operators on symmetric informationally complete positive operator valued measures or sic povms covariant relative to the action of the generalized pauli group we show that each of the fiducial vectors which has been constructed so far including all the vectors constructed numerically by renes et al is an eigenvector of one of a special class of order 3 clifford unitaries this suggests a strengthening of a conjecture of zauner s we give a complete characterization of the orbits and stability groups in dimensions 2 7 fina

painless nonorthogonal expansions in a hilbert space h discrete families of vectors hj with the property that f j hj f hj for every f in h are considered this expansion formula is obviously true if the family is an orthonormal basis of h but also can hold in situations where the hj are not mutually orthogonal and are overcomplete the two classes of examples studied here are i appropriate sets of weyl heisenberg coherent states based on certain non gaussian fiducial vectors and ii analogous families of affine coherent states it is believed that such quasiorthogonal expansions will be a useful tool in many areas of theoretical physics and applied mathematics

the k tacnode process the tacnode process is a universal behavior arising in nonintersecting particle systems and tiling problems for dyson brownian bridges the tacnode process describes the grazing collision of two packets of walkers we consider such a dyson sea on the unit circle with drift for any k in mathbb z we show that an appropriate double scaling of the drift and return time leads to a generalization of the tacnode process in which k particles are expected to wrap around the circle we derive winding number probabilities and an expression for the correlation kernel in terms of functions related to the generalized hastings mcleod solutions to the inhomogeneous painleve ii equation the method of proof is asymptotic analysis of discrete orthogonal polynomials with a complex weight