existence of weak solutions for a hyperbolic model of chemosensitive movement a hyperbolic model for chemotaxis and chemosensitive movement in one space dimension is considered in contrast to parabolic models for chemotaxis the hyperbolic model allows us to take the dependence of the particle speed on external stimuli explicitly into account this qualitatively covers recent experiments on chemotaxis in which it has been shown that particles adapt their speed to the surrounding environment the model presented here consists of two hyperbolic differential equations of first order coupled with an elliptic equation we assume that the speed depends on the external stimulus only and not on its gradients in that case solutions with steep gradients are expected which have the interpretation of moving swarms a notion of weak solutions for this hyperbolic chemotaxis model is presented and the global existence of weak solutions is shown the proof relies on the vanishing viscosity method i e we obtain the weak solution as the limit of classical solutions of an associated parabolically regularized problem for vanishing viscosity parameter numerical simulations demonstrate phenomena like swarming behaviour and formation of steep gradients

patterns in eigenvalues the 70th josiah willard gibbs lecture typical large unitary matrices show remarkable patterns in their eigenvalue distribution these same patterns appear in telephone encryption the zeros of riemann s zeta function a variety of physics problems and in the study of toeplitz operators this paper surveys these applications and what is currently known about the patterns

subdeterminant approach for pseudo orbit expansions of spectral determinants in quantum maps and quantum graphs we study the implications of unitarity for pseudo orbit expansions of the spectral determinants of quantum maps and quantum graphs in particular we advocate to group pseudo orbits into subdeterminants we show explicitly that the cancellation of long orbits is elegantly described on this level and that unitarity can be built in using a simple subdeterminant identity which has a nontrivial interpretation in terms of pseudo orbits this identity yields much more detailed relations between pseudo orbits of different lengths than was known previously we reformulate newton identities and the spectral density in terms of subdeterminant expansions and point out the implications of the subdeterminant identity for these expressions we analyze furthermore the effect of the identity on spectral correlation functions such as the autocorrelation and parametric cross correlation functions of the spectral determinant and the spectral form factor

characterization of quantum chaos by the autocorrelation function of spectral determinants the autocorrelation function of spectral determinants is proposed as a convenient tool for the characterization of spectral statistics in general and for the study of the intimate link between quantum chaos and the random matrix theory in particular for this purpose the correlation functions of spectral determinants are evaluated for various random matrix ensembles and are compared with the corresponding semiclassical expressions the method is demonstrated by applying it to the spectra of the quantized sinai billiards in two and three dimensions

an accurate and efficient algorithm for the computation of the characteristic polynomial of a general square matrix an algorithm is presented for the efficient and accurate computation of the coefficients of the characteristic polynomial of a general square matrix the algorithm is especially suited for the evaluation of canonical traces in determinant quantum monte carlo methods

uniform stability of switched linear systems extensions of lasalle s invariance principle this paper addresses the uniform stability of switched linear systems where uniformity refers to the convergence rate of the multiple solutions that one obtains as the switching signal ranges over a given set we provide a collection of results that can be viewed as extensions of lasalle s invariance principle to certain classes of switched linear systems using these results one can deduce asymptotic stability using multiple lyapunov functions whose lie derivatives are only negative semidefinite depending on the regularity assumptions placed on the switching signals one may be able to conclude just asymptotic stability or uniform exponential stability we show by counter example that the results obtained are tight

universal fluctuations of zeros of chaotic wavefunctions wavefunctions of one and two dimensional quantum systems can be parametrized by a finite number of zeros lying in phase space we study correlations of these zeros for fully chaotic systems in terms of a statistical model based on random polynomials excellent agreement is found for the two point correlation function and nearest neighbour spacing distribution of this model and the results obtained for wavefunctions of dynamical systems we conjecture that these correlation functions are valid for any chaotic system after rescaling the phase space distances unfolding some consequences for the distribution of zeros due to time reversal symmetry are also discussed

correlations between zeros of a random polynomial we obtain exact analytical expressions for correlations between real zeros of the kac random polynomial we show that the zeros in the interval 1 1 are asymptotically independent of the zeros outside of this interval and that the straightened zeros have the same limit translation invariant correlations then we calculate the correlations between the straightened zeros of theo 1 random polynomial

optimal reconstruction of images from localized phase the importance of localized phase in signal representation is investigated the convergence rate of the pocs algorithm projection onto convex sets used for image reconstruction from spectral phase is defined and analyzed and the characteristics of images optimally reconstructed from phase only information are presented it is concluded that images of geometric form are most efficiently reconstructed from their spectral phase whereas images of symmetric form have the poorest convergence characteristics the transition between the two extremes is shown to be continuous the results provide a new approach and analysis of the previously reported advantages of the localized phase representation over the global approach and suggest possible compression schemes

an iterative method for conformal mapping abstract in this paper we discuss an iterative method for the calculation of the boundary values of the conformal mapping of a simply connected region g onto a region h with smooth boundary the method is based on a certain riemann hilbert problem it turns out that this problem is the linearized version of a singular integral equation of the second kind hence the method is a newton method whenever the boundaries of g and h are sufficiently smooth its convergence is locally quadratic if g is the unit circle the solution of the linearized problem can be represented explicitly in terms of integral transforms from this one derives a quadratically convergent newton like numerical method that avoids the numerical solution of systems of linear equations and therefore in comparison with other methods based on integral equations is quite economical in terms of computer time and storage requirements

a fredholm integral equation of the second kind for conformal mapping abstract we present a fredholm integral equation of the second kind for the derivative of the boundary correspondence function of the conformal mapping of a jordan region with piecewise twice differentiable boundary onto the unit disk one of our derivations of the equation leads to a more general regularisation procedure for some singular integral equations of the first kind the integral equation can be solved numerically by applying the fourier method the convergence to in l 2 of the solution of the discrete equation so obtained follows by simple theorems examples show that the equation often yields better numerical results than both a related method proposed by henrici and warschawski s equation

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

a nonoverlapping domain decomposition method for symm s equation for conformal mapping symm s equation is a first kind integral equation for computing conformal maps of simply connected regions the package confpack solves symm s equation by an indirect boundary element method using an accurate corner representation this solution technique is extended here to include nonoverlapping domain decomposition degrees of freedom are introduced on one or more interfaces and different unknowns are used leading to a system of second kind equations the corresponding linear system can be expressed in schur complement form the accurate treatment of corners is preserved in the new formulation the results of serial and parallel matlab implementations of the new algorithm show significant speedups as the number of unknowns grows

the distributions of the determinant of fixed trace ensembles of real symmetric and of hermitian random matrices probability densities of the determinant are obtained for fixed trace ensembles of n x n hermitian and real symmetric matrices from the exact determinant densities of the gaussian orthogonal ensemble goe and of the gaussian unitary ensemble gue respectively

fluctuations and ergodicity of the form factor of quantum propagators and random unitary matrices we consider the spectral form factor of random unitary matrices as well as of floquet matrices of kicked tops as given by the squared moduli of the traces tn tr fn with the integer time n 0 1 2 for a typical matrix f the time dependence of the form factor tn 2 looks erratic only after a local time average over a suitably large time window n does a systematic time dependence become manifest for matrices drawn from the circular unitary ensemble we prove ergodicity in the limits of large matrix dimension and time window n the local time average has vanishingly small ensemble fluctuations and may be identified with the ensemble average by numerically diagonalizing floquet matrices of kicked tops with a globally chaotic classical limit we find the same ergodicity as a byproduct we find that the traces tn of random matrices from the circular ensembles behave very much like independent gaussian random numbers again floquet matrices of chaotic tops share that universal behaviour it becomes clear that the form factor of chaotic dynamical systems can be fully faithful to random matrix theory not only in its locally time averaged systematic time dependence but also in its fluctuations

impact of localization on dyson s circular ensemble a wide variety of complex physical systems described by unitary matrices have been shown numerically to satisfy level statistics predicted by dyson s circular ensemble we argue that the impact of localization in such systems is to impose certain restrictions on the eigenvalues we consider a solvable model which takes into account such restrictions qualitatively and find that within the model a gap is created in the spectrum and there is a transition from the universal wigner distribution towards a poisson distribution with increasing localization

density of complex critical points of a real random so m 1 polynomial we study the density of complex critical points of a real random so m 1 polynomial in m variables in a previous paper macdonald in j stat phys 136 5 807 2009 the author used the poincare lelong formula to show that the density of complex zeros of a system of these real random polynomials rapidly approaches the density of complex zeros of a system of the corresponding complex random polynomials the su m 1 polynomials in this paper we use the kac rice formula to prove an analogous result the density of complex critical points of one of these real random polynomials rapidly approaches the density of complex critical points of the corresponding complex random polynomial in one variable we give an exact formula and a scaling limit formula for the density of critical points of the real random so 2 polynomial as well as for the density of critical points of the corresponding complex random su 2 polynomial

the fixed trace β hermite ensemble of random matrices and the low temperature distribution of the determinant of an n n β hermite matrix the hermite ensemble he of tridiagonal n n random matrices of dumitriu and edelman 2002 j math phys 43 5830 is a continuum of ensembles in which the reciprocal of the temperature in the 2d electrostatic interpretation of the eigenvalue characteristics can take any value the eigenvalue distributions coincide with those of the classical gaussian ensembles goe gue gse for 1 2 4 a fixed trace hermite ensemble fthe is defined from the he and is used to extend the spherical ensembles of classical symmetries to spherical ensembles at low temperature when for a fixed value of n the asymptotic distributions of reduced determinants dn of random n n h and fth matrices are shown to be standard gaussians accordingly the fluctuations of the potential at the origin ln dn have a generalized gumbel distribution at low temperature for large n and large a ln n variance results from the strongly correlated fluctuations of eigenvalues around their equilibrium positions

a charge simulation method for the numerical conformal mapping of interior exterior and doubly connected domains abstract a simple numerical method is described for computing the following three conformal maps a from a domain interior to a closed jordan curve onto the interior of the unit disk b from a domain exterior to a closed jordan curve onto the exterior of the unit disk and c from a doubly connected domain bounded by two closed jordan curves onto a circular annulus the numerical method is based on the charge simulation method or the method of fundamental solutions applied to the dirichlet problem of laplace s equation in which a pair of conjugate harmonic functions are approximated by a linear combination of complex logarithmic potentials without integration typical examples show that numerical results of high accuracy can be obtained by the simple method if the domains have no concave corners

the zeros of random polynomials further results and applications we consider the density of the zeros of random polynomials with nonzero mean correlated gaussian coefficients we show that for most communication problems the original result by hammersley 1956 can be expressed in a simpler form in order to illustrate the usefulness of the presented theory two applications are considered first the zeros of frequency selective ricean fading channels are investigated second the zeros of the transfer function corresponding to channel impulse response estimates obtained by a least squares approach are calculated for both applications implications for equalizer design are discussed

secular determinants of random unitary matrices we consider the characteristic polynomials of random unitary matrices u drawn from various circular ensembles in particular the statistics of the coefficients of these polynomials are studied the variances of these secular coefficients are given explicitly for arbitrary dimension and continued analytically to arbitrary values of the level repulsion exponent the latter secular coefficients are related to the traces of powers of u by newton s well known formulae while the traces tend to have gaussian distributions and to be statistically independent among one another in the limit as the matrix dimension grows large the secular coefficients exhibit strong mutual correlations due to newton s mixing of traces to coefficients these results might become relevant for current efforts at combining semiclassics and random matrix theory in quantum treatments of classically chaotic dynamics

random analytic chaotic eigenstates the statistical properties of random analytic functions z are investigated as a phase space model for eigenfunctions of fully chaotic systems we generalize to the plane and to the hyperbolic plane a theorem concerning the equidistribution of the zeros of z previously demonstrated for a spherical phase space su 2 polynomials for systems with time reversal symmetry the number of real roots is computed for the three geometries in the semiclassical regime the local correlation functions are shown to be universal independent of the system considered or the geometry of phase space in particular the autocorrelation function of is given by a gaussian function the connections between this model and the gaussian random function hypothesis as well as the random matrix theory are discussed

distribution of roots of random polynomials we consider polynomials of high degree with random coefficients which appear in the context of quantum chaotic dynamics and investigate various conditions under which their roots tend to concentrate near the unit circle in the complex plane correlation functions of roots are computed analytically we also investigate a certain class of random polynomials whose roots cover in a uniform way the riemann sphere special emphasis is devoted to the influence of symmmetries

eigenvalue correlations in the circular ensembles dyson 1972 introduced two types of brownian motion ensembles of random matrices for studying approximate symmetries in complex quantum systems the magnitude of symmetry breaking plays the role of a fictitious time t or 0 the authors study the eigenvalue correlations in the circular type ensembles which serve as models for the evolution operators of quantum maps with chaotic classical limits in two cases involving time reversal symmetry breaking they evaluate explicitly the eigenangle density correlation functions of all orders for all t and for all values of the matrix dimensionality n the general case is described by a hierarchic set of relations among the correlation functions as a function of t the transition in the correlations is found to be rapid for large n discontinuous for n to infinity as a function of a local parameter lambda which measures the mean square symmetry admixing matrix element in units of the local average spacing the transition is found to be smooth the same lambda dependent results were found earlier for the gaussian type ensembles which serve as models for the hamiltonian operators of autonomous chaotic systems

chaotic analytic zero points exact statistics for those of a random spin state a natural statistical ensemble of 2j points on the unit sphere can be associated via the majorana representation with a random quantum state of spin j and an exact expression is obtained here for the general k point correlation function in this ensemble the pair correlation in the large j limit takes the relatively simple form where and is the angular separation of the pair of points on the sphere it appears from the numerical work of others that in this limit these statistics are typical of the zero points of analytic functions associated with chaotic quantum dynamical systems

mesenchymal motion models in one dimension mesenchymal motion denotes a form of cell movement through tissue which can be observed for certain cancer metastases in t hillen j math biol 53 2006 pp 585 616 a mathematical model for this form of movement was introduced in the current paper we present a comprehensive analysis of the one dimensional mesenchymal motion model we establish the global existence of classical solutions and rigorously carry out the parabolic limit of the model we discuss the stationary solutions prove the existence of traveling wave solutions and use numerical simulations to illustrate the results finally we discuss the biological implications of the results

parametric statistics of zeros of husimi representations of quantum chaotic eigenstates and random polynomials local parametric statistics of zeros of husimi representations of quantum eigenstates are introduced it is conjectured that for a classically fully chaotic system one should use the model of parametric statistics of complex roots of gaussian random polynomials which is exactly solvable as demonstrated below for example the velocities derivatives of zeros of husimi function with respect to an external parameter are predicted to obey a universal non maxwellian distribution where is the mean square velocity the conjecture is demonstrated numerically in a generic chaotic system with two degrees of freedom dynamical formulation of the zero flow in terms of an integrable many body dynamical system is given as well

lyndon word decompositions and pseudo orbits on q nary graphs abstract a foundational result in the theory of lyndon words words that are strictly earlier in lexicographic order than their cyclic permutations is the chen fox lyndon theorem which states that every word has a unique non increasing decomposition into lyndon words this article extends this factorization theorem obtaining the proportion of these decompositions that are strictly decreasing this result is then used to count primitive pseudo orbits sets of primitive periodic orbits on q nary graphs as an application we obtain a diagonal approximation to the variance of the characteristic polynomial coefficients of q nary quantum graphs

shape from moments an estimation theory perspective this paper discusses the problem of recovering a planar polygon from its measured complex moments these moments correspond to an indicator function defined over the polygon s support previous work on this problem gave necessary and sufficient conditions for such successful recovery process and focused mainly on the case of exact measurements being given in this paper we extend these results and treat the same problem in the case where a longer than necessary series of noise corrupted moments is given similar to methods found in array processing system identification and signal processing we discuss a set of possible estimation procedures that are based on the prony and the pencil methods relate them one to the other and compare them through simulations we then present an improvement over these methods based on the direct use of the maximum likelihood estimator exploiting the above methods as initialization finally we show how regularization and thus maximum a posteriori probability estimator could be applied to reflect prior knowledge about the recovered polygon

joint densities of secular coefficients for unitary matrices we determine the joint density of secular coefficients and of the traces of powers for random unitary matrices from dyson s circular ensembles a byproduct are workable unitarity conditions necessary and sufficient for all roots of a polynomial to be unimodular and different these findings will be useful for combinations of random matrix theory with semiclassical approximations in calculating spectral properties of quantum systems whose classical limit is globally chaotic in a first such application we show that reasonable quasi energy spectra result by throwing dice a la random matrix theory to complement restricted knowledge rigorous or from short periodic orbits of the secular polynomial of a unitary floquet matrix

exact statistics of complex zeros for gaussian random polynomials with real coefficients k point correlations of complex zeros for gaussian ensembles of random polynomials of order n with real coefficients grprc are calculated exactly following an approach of hannay for the case of gaussian random polynomials with complex coefficients grpcc it is shown that in the thermodynamic limit of gaussian random holomorphic functions all the statistics converge to their grpcc counterparts as one moves off the real axis while close to the real axis the two cases are essentially different special emphasis is given to one and two point correlation functions in various regimes

interpolating ensembles of random unitary matrices we investigate transitions between ensembles of unitary random matrices modeling changes of statistical properties of quantum chaotic systems which are periodically time dependent the transitions simulating change of integrability are modeled with the help of numerically generated one parameter families of ensembles interpolating between an ensemble of diagonal unitary matrices and a circular unitary or circular orthogonal ensemble in an analogous manner we described transitions between circular orthogonal and unitary ensembles corresponding to the time reversal symmetry breaking perturbation of a chaotic system in all cases we present results concerning statistics of the quasi energy levels and eigenvectors textcopyright 1996 the american physical society

numerical conformal mapping using cross ratios and delaunay triangulation we propose a new algorithm for computing the riemann mapping of the unit disk to a polygon also known as the schwarz christoffel transformation the new algorithm crdt for cross ratios of the delaunay triangulation is based on cross ratios of the prevertices and also on cross ratios of quadrilaterals in a delaunay triangulation of the polygon the crdt algorithm produces an accurate representation of the riemann mapping even in the presence of arbitrary long thin regions in the polygon unlike any previous conformal mapping algorithm we believe that crdt solves all difficulties with crowding and global convergence although these facts depend on conjectures that we have so far not been able to prove we demonstrate convergence with computational experiments the riemann mapping has applications in two dimensional potential theory and mesh generation we demonstrate crdt on problems in long thin regions in which no other known algorithm can perform comparably

distribution of roots of random real generalized polynomials the average density of zeros for monic generalized polynomials p n z phi z sum nolimits k 1 n c k f k z with real holomorphic f k and real gaussian coefficients is expressed in terms of correlation functions of the values of the polynomial and its derivative we obtain compact expressions for both the regular component generated by the complex roots and the singular one real roots of the average density of roots the density of the regular component goes to zero in the vicinity of the real axis like lmz we present the low and high disorder asymptotic behaviors then we particularize to the large n limit of the average density of complex roots of monic algebraic polynomials of the form p n z z n sum nolimits k 1 n c k z n k with real independent identically distributed gaussian coefficients having zero mean and dispersion delta 1 sqrt n lambda the average density tends to a simple universal function of 2nlog z and in the domain coth 2 n sin arg z where nearly all the roots are located for largen

the treatment of corner and pole type singularities in numerical conformal mapping techniques abstract this paper is a report of recent developments concerning the nature and the treatment of singularities that affect certain numerical conformal mapping techniques the paper also includes some new results on the nature of singularities that the mapping function may have in the complement of the closure of the domain under consideration

quantum chaos and random matrix theory some new results abstract new insight into the correspondence between quantum chaos and random matrix theory is gained by developing a semiclassical theory for the autocorrelation function of spectral determinants we study in particular the unitary operators which are the quantum versions of area preserving maps the relevant random matrix ensembles are the circular ensembles the resulting semiclassical expressions depend on the symmetry of the system with respect to time reversal and on a classical parameter tru 1 where u is the classical one step evolution operator for system without time reversal symmetry we are able to reproduce the exact random matrix predictions in the limit 0 for systems with time reversal symmetry we can reproduce only some of the features of random matrix theory for both classes we obtain the leading corrections in the semiclassical theory for integrable systems is also developed resulting in expressions which reproduce the theory for the poissonian ensemble to leading order in the semiclassical limit

on dilute unitary random matrices we study random dilution of random matrices where are uniformly distributed over the group of unitary matrices and are non random hermitian matrices we show that the eigenvalue distribution function of dilute random matrices converges to the semicircle wigner distribution in the limit where p is the dilution parameter this convergence can be explained by the observation that the dilution eliminates statistical dependence between the entries of the same statement is valid for the entries of our results support the conjecture that the wigner law is valid for wide classes of dilute hermitian random matrices

images of eigenvalue distributions under power maps in 9 it was shown that if u is a random n n unitary matrix then for any p n the eigenvalues of u p are i i d uniform similar results were also shown for general compact lie groups we study what happens when p n instead for the classical groups we find that we can describe the eigenvalue distribution of u p in terms of the eigenvalue distributions of smaller classical groups the earlier result is then a special case the proofs rely on the fact that a certain subgroup of the weyl group is itself a weyl group we generalize this fact and use it to study the power map problem for general compact lie groups

signed zeros of gaussian vector fields density correlation functions and curvature we calculate correlation functions of the signed density of zeros of gaussian distributed vector fields we are able to express correlation functions of arbitrary order through the curvature tensor of a certain abstract riemann cartan or riemannian manifold as an application we discuss one and two point functions the zeros of a two dimensional gaussian vector field model the distribution of topological defects in the high temperature phase of two dimensional systems with orientational degrees of freedom such as superfluid films thin superconductors and liquid crystals

numerical conformal mapping methods based on function conjugation abstract a unifying treatment of methods for computing conformal maps from the unit disk onto a jordan region is presented integral and integro differential equations involving the conjugation operator for the boundary correspondence function are first derived using an arbitrary auxiliary function having certain properties then various iteration methods for solving these equations are described in this generality so that the basic ideas become manifest specific methods are then treated as examples of the general theory among them are in particular the successive conjugation methods of theodorsen melentiev and kulisch timman and friberg the projection method of bergstrom and the newton methods of vertgeim wegmann and hubner which make use of the easy construction of the solutions of riemann hilbert problems many of these methods are treated in greater generality than in the literature the connections with the methods of fornberg menikoff zemach chakravarthy anderson and challis burley are also outlined

invariance principles for hyperbolic random walk systems abstract reaction random walk systems are hyperbolic models for the description of spatial motion in one dimension and reaction of particles in contrast to reaction diffusion equations particles have finite propagation speed for parabolic systems invariance results and maximum principles are well known a convex set is positively invariant if at each boundary point an outer normal is a left eigenvector of the diffusion matrix and if the vector field defined by the pure reaction equation points inward at the boundary here we show a corresponding result for random walk systems the model parameters are the particle speeds the rates of change in direction and the reaction vector field a convex domain is invariant if at each boundary point an outer normal is a left eigenvector of the speed matrix and if a vector field given by the reaction equation combined with the turning rates points inward finally a positivity result is shown

real roots of random polynomials universality close to accumulation points we identify the scaling region of a width o n 1 in the vicinity of the accumulation points t 1 of the real roots of a random kac like polynomial of large degree n we argue that the density of the real roots in this region tends to a universal form shared by all polynomials with independent identically distributed coefficients ci as long as the second moment e c2i is finite in particular we reveal a gradual in contrast to the previously reported abrupt and quite nontrivial suppression of the number of real roots for coefficients with a nonzero mean value n e ci scaled as n n 1 2

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

the chaotic analytic function in contrast to a stationary gaussian random function of a real variable which is free to have any correlation function the closest analogous analytic random function in the complex plane has no true freedom it is statistically unique since it has arisen only recently as an apparently universal feature in the physical context of quantum chaos i refer to it here as the chaotic analytic function i note that it is implied by the assumption that a quantum chaotic wavefunction has gaussian randomness and has a constant value for the average of its wigner function in phase space interpreted literally this shows that the chaotic analytic function is the bargmann function of a pure white noise wavefunction more physically if constant is replaced by smooth on the scale of a planck area these assumptions are the semiclassical ones made by berry for chaotic eigenstates the analysis shows that the chaotic analytic function is still obtained semiclassically

universal hypotrochoidic law for random matrices with cyclic correlations the celebrated elliptic law describes the distribution of eigenvalues of random matrices with correlations between off diagonal pairs of elements having applications to a wide range of physical and biological systems here we investigate the generalization of this law to random matrices exhibiting higher order cyclic correlations between k tuples of matrix entries we show that the eigenvalue spectrum in this ensemble is bounded by a hypotrochoid curve with k fold rotational symmetry this hypotrochoid law applies to full matrices as well as sparse ones and thereby holds with remarkable universality we further extend our analysis to matrices and graphs with competing cycle motifs which are described more generally by polytrochoid spectral boundaries

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

rate of convergence of linear functions on the unitary group we study the rate of convergence to a normal random variable of the real and imaginary parts of where u is an n n random unitary matrix and an is a deterministic complex matrix we show that the rate of convergence is o n 2 b with 0 b 1 depending only on the asymptotic behaviour of the singular values of an for example if the singular values are non degenerate different from zero and o 1 as n then b 0 the proof uses a berry esseen inequality for linear combinations of eigenvalues of random unitary matrices and so appropriate for strongly dependent random variables

the autocorrelation function for spectral determinants of quantum graphs this paper considers the spectral determinant of quantum graph families with chaotic classical limit the secular coefficients of the spectral determinant are found to follow distributions with zero mean and variance approaching a constant in the limit of large network size for graphs without symmetries this constant is in general different from the random matrix result and depends on the classical limit a closed expression for this system dependent constant is given here explicitly in terms of the spectrum of an underlying markov process related results for graphs with time reversal symmetry are given

continuum percolation for gaussian zeroes and ginibre eigenvalues we study continuum percolation on certain negatively dependent point processes on r 2 specifically we study the ginibre ensemble and the planar gaussian zero process which are the two main natural models of translation invariant point processes on the plane exhibiting local repulsion for the ginibre ensemble we establish the uniqueness of infinite cluster in the supercritical phase for the gaussian zero process we establish that a non trivial critical radius exists and we prove the uniqueness of infinite cluster in the supercritical regime

numerical conformal mapping via the szego kernel abstract a new method to compute the riemann mapping function is numerically implemented and tested on examples the method expresses the szego kernel as the solution of a second kind integral equation the equation has its origin in an earlier non numerical work of stein and one of the authors 8 the experimental results show the algorithm to be effective and stable

a domain decomposition method for approximating the conformal modules of long quadrilaterals this paper is concerned with the study of a domain decomposition method for approximating the conformal modules of long quadrilaterals the method has been studied already by us and also by d gaier and w k hayman but only in connection with a special class of quadrilaterals viz quadrilaterals where a the defining domain is bounded by two parallel straight lines and two jordan arcs and b the four specified boundary points are the four corners where the arcs meet the straight lines our main purpose here is to explain how the method may be extended to a wider class of quadrilaterals than that indicated above

on the distribution of zeros of chaotic wavefunctions wavefunctions in a phase space representation can be expressed as entire functions of their zeros if the phase space is compact these zeros seem to hide a lot of relevant and explicit information about the underlying classical dynamics besides an understanding of their statistical properties may prove to be useful in the analytical calculations of the wavefunctions in quantum chaotic systems this motivates us to pursue the present study which by a numerical statistical analysis shows that both long range correlations as well as short range correlations exist between zeros while the latter turn out to be universal and parametric independent the former seem to be system dependent and are significantly affected by various parameters i e symmetry localization etc furthermore for the delocalized quantum dynamics the distribution of these zeros seem to mimic that of the zeros of the random functions as well as random polynomials

random unitary matrices methods of constructing random matrices typical of circular unitary and circular orthogonal ensembles are presented we generate numerically random unitary matrices and show that the statistical properties of their spectra level spacing distribution number variance and eigenvectors entropy participation ratio eigenvector statistics confer to the predictions of the random matrix theory for both cue and coe

periodic orbit quantization of baker s map we consider the resummation of the semiclassical selberg zeta function for quantized maps on compact phase space specifically the quantized baker s map in particular we demonstrate that the semiclassical periodic orbit expansion leads to an effectively finite polynomial for the spectrum of the quantum map however the coefficients are not self inverse as required by unitarity an extension of the semiclassical approximation by including corrections due to the discontinuities of the classical map improves the situation the improved polynomial is closer to being self inverse but the eigenphases remain complex slightly better results are obtained if the functional equation is imposed on the semiclassical expansion

how many zeros of a random polynomial are real we provide an elementary geometric derivation of the kac integral formula for the expected number of real zeros of a random polynomial with independent standard normally distributed coefficients we show that the expected number of real zeros is simply the length of the moment curve 1 t tn projected onto the surface of the unit sphere divided by the probability density of the real zeros is proportional to how fast this curve is traced out we then relax kac s assumptions by considering a variety of random sums series and distributions and we also illustrate such ideas as integral geometry and the fubini study metric

zeros of the i i d gaussian power series a conformally invariant determinantal process consider the zero set of the random power series f z p anz n with i i d complex gaussian coefficientsan we show that these zeros form a determinantal process more precisely their joint intensity can be written as a minor of the bergman kernel we show that the number of zeros of f in a disk of radius r about the origin has the same distribution as the sum of independent 0 1 valued random variables xk where p xk 1 r 2k moreover the set of absolute values of the zeros of f has the

density of complex zeros of a system of real random polynomials we study the density of complex zeros of a system of real random so m 1 polynomials in m variables we show that the density of complex zeros of this random polynomial system with real coefficients rapidly approaches the density of complex zeros in the complex coefficients case we also show that the behavior the scaled density of complex zeros near m of the system of real random polynomials is different in the m 2 case than in the m 1 case the density approaches infinity instead of tending linearly to zero

a fast method for solving certain integral equations of the first kind with application to conforma mapping abstract we describe a fast iterative method for solving fredholm integral equations of the first kind whose kernels have a logarithmic principal part the method is a fourier galerkin method and due to the singularity of the kernel the linear system of simultaneous equations is block diagonally dominant and can be solved rapidly by an iterative method we present examples of the computation of conformal mappings of simply and multiply connected domains

bifurcations of periodic orbits and uniform approximations we derive uniform approximations for contributions to gutzwiller s periodic orbit sum for the spectral density which are valid close to bifurcations of periodic orbits in systems with mixed phase space there orbits lie close together and give collective contributions while the individual contributions of gutzwiller s type would diverge at the bifurcation new results for the tangent the period doubling and the period tripling bifurcation are given they are obtained by going beyond the local approximation and including higher order terms in the normal form of the action the uniform approximations obtained are tested on the kicked top and are found to be in excellent agreement with exact quantum results

an integral equation method for the numerical conformal mapping of interior exterior and doubly connected domains a numerical method based on the integral equation formulation of symm is described for computing approximations to the mapping functions which accomplish the following conformal maps a the mapping of a domain interior to a closed jordan curve onto the interior of the unit disc b the mapping of a domain exterior to a closed jordan curve onto the exterior of the unit disc c the mapping of a doubly connected domain bounded by two closed jordan curves onto a circular annulus the numerical method is based on approximating the unknown source density by cubic splines and singular functions and is particularly suited for the mapping of difficult domains having sharp corners

the monge metric on the sphere and geometry of quantum states topological and geometrical properties of the set of mixed quantum states in the n dimensional hilbert space are analysed assuming that the corresponding classical dynamics takes place on the sphere we use the vector su 2 coherent states and the generalized husimi distributions to define the monge distance between two arbitrary density matrices the monge metric has a simple semiclassical interpretation and induces a non trivial geometry among all pure states the distance from the maximally mixed state proportional to the identity matrix admits the largest value for the coherent states while the delocalized chaotic states are close to this contrasts the geometry induced by the standard trace hilbert schmidt or bures metrics for which the distance from is the same for all pure states we discuss possible physical consequences including unitary time evolution and the process of decoherence we introduce also a simplified monge metric defined in the space of pure quantum states and more suitable for numerical computation

correlation functions for random complex zeroes strong clustering and local universality we prove strong clustering of k point correlation functions of zeroes of gaussian entire functions in the course of the proof we also obtain universal local bounds for k point functions of zeroes of arbitrary nondegenerate gaussian analytic functions in the second part of the paper we show that strong clustering yields the asymptotic normality of fluctuations of some linear statistics of zeroes of gaussian entire functions in particular of the number of zeroes in measurable domains of large area this complements our recent results from the paper fluctuations in random complex zeroes

exact statistical properties of the zeros of complex random polynomials the zeros of complex gaussian random polynomials with coefficients such that the density in the underlying complex space is uniform are known to have the same statistical properties as the zeros of the coherent state representation of one dimensional chaotic quantum systems we extend the interpretation of these polynomials by showing that they also arise as the wavefunction for a quantum particle in a magnetic field constructed from a random superposition of states in the lowest landau level a study of the statistical properties of the zeros is undertaken using exact formulae for the one and two point distribution functions attention is focused on the moments of the two point correlation in the bulk the variance of a linear statistic and the asymptotic form of the two point correlation at the boundary a comparison is made with the same quantities for the eigenvalues of complex gaussian random matrices

intrinsic ordering of quasienergy states for mixed regular chaotic quantum systems zeros of the husimi distribution a method is proposed which allows an intrinsic ordering of the quasienergy states for a time periodic quantum system according to the number of essential zeros of the husimi distribution in phase space the corresponding classical system shows a typical mixed dynamics with coexisting regular and chaotic regions in phase space the ordering of states allows a straightforward comparison with semiclassical quantities moreover a direct connection of the organization of zeros with the underlying classical phase space dynamics is demonstrated

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