critical statistics for non hermitian matrices we introduce a generalized ensemble of non hermitian matrices interpolating between the gaussian unitary ensemble the ginibre ensemble and the poisson ensemble the joint eigenvalue distribution of this model is obtained by means of an extension of the itzykson zuber formula to general complex matrices its correlation functions are studied both in the case of weak non hermiticity and in the case of strong non hermiticity in the weak non hermiticity limit we show that the spectral correlations in the bulk of the spectrum display critical statistics the asymptotic linear behavior of the number variance is already approached for energy differences of the order of the eigenvalue spacing to lowest order its slope does not depend on the degree of non hermiticity close the edge the spectral correlations are similar to the hermitian case in the strong non hermiticity limit the crossover behavior from the ginibre ensemble to the poisson ensemble first appears close to the surface of the spectrum our model may be relevant for the description of the spectral correlations of an open disordered system close to an anderson transition

quantum graphs a simple model for chaotic scattering we connect quantum graphs with infinite leads and turn them into scattering systems we show that they display all the features which characterize quantum scattering systems with an underlying classical chaotic dynamics typical poles delay time and conductance distributions ericson fluctuations and when considered statistically the ensemble of scattering matrices reproduces quite well the predictions of the appropriately defined random matrix ensembles the underlying classical dynamics can be defined and it provides important parameters which are needed for the quantum theory in particular we derive exact expressions for the scattering matrix and an exact trace formula for the density of resonances in terms of classical orbits analogous to the semiclassical theory of chaotic scattering we use this in order to investigate the origin of the connection between random matrix theory and the underlying classical chaotic dynamics being an exact theory and due to its relative simplicity it offers new insights into this problem which is at the forefront of the research in chaotic scattering and related fields

strictly convex norms and topology we introduce a new topological property called and the corresponding class of topological spaces which includes spaces with g diagonals and gruenhage spaces using we characterize those banach spaces which admit equivalent strictly convex norms and give an internal topological characterization of those scattered compact spaces k for which the dual banach space c k admits an equivalent strictly convex dual norm we establish some relationships between and other topological concepts and the position of several well known examples in this context for instance we show that c k admits an equivalent strictly convex dual norm where k is kunen s compact s space also under additional axioms we provide examples of compact scattered non gruenhage spaces of cardinality 1 having

a posteriori error estimates for continuous discontinuous galerkin approximations of the kirchhoff love plate we present energy norm a posteriori error estimates for continuous discontinuous galerkin c dg approximations of the kirchhoff love plate problem the method is based on a continuous displacement field inserted into a symmetric discontinuous galerkin formulation of the fourth order partial differential equation governing the deflection of a thin plate we also give explicit formulas for the penalty parameter involved in the formulation

continuous functions on totally ordered spaces that are compact in their order topologies abstract the totally ordered spaces that are compact and connected in their order topologies are characterized as images of lexicographic cubes the banach space of continuous functions on the lexicographic product of compact totally ordered spaces whose spaces of continuous functions have locally uniformly convex norms has an equivalent locally uniformly convex norm if and only if the product is countable the banach space of continuous functions on a compact totally ordered space always has an equivalent kadec norm those compact totally ordered spaces for which the banach space of continuous functions has an equivalent locally uniformly convex norm are characterized in terms of the bounded decreasing interval functions that can be defined on the intervals of the totally ordered spaces some examples are discussed in detail

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

topological properties of banach spaces weakly compact sets in banach spaces are fragmentable we introduce the weaker notion of the a fragmentability of a subset of a banach space a notion that has a far wider range of applicability we prove that all weakly cech analytic subsets of a banach space are a fragmented this implies that all banach spaces having kadec norms are a fragmented however l is not fragmented

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

scattering reflection and impedance of waves in chaotic and disordered systems with absorption we review recent progress in analysing wave scattering in systems with both intrinsic chaos and or disorder and internal losses when the scattering matrix is no longer unitary by mapping the problem onto a nonlinear supersymmetric model we are able to derive closed form analytic expressions for the distribution of reflection probability in a generic disordered system one of the most important properties resulting from such an analysis is statistical independence between the phase and the modulus of the reflection amplitude in every perfectly open channel the developed theory has far reaching consequences for many quantities of interest including local green functions and time delays in particular we point out the role played by absorption as a sensitive indicator of mechanisms behind the anderson localization transition we also provide a random matrix based analysis of s matrix and impedance correlations for various symmetry classes as well as the distribution of transmitted power for systems with broken time reversal invariance completing previous works on the subject the results can be applied in particular to the experimentally accessible impedance and reflection in a microwave or an ultrasonic cavity attached to a system of antennas

new approach to the statistical properties of energy levels the joint distribution of energy eigenvalues of a hamiltonian is derived by means of the usual statistical laws of classical many body systems it makes a transition from the poisson type to the gaussian type depending on the value of a single parameter characteristic of the hamiltonian

a stability property for locally uniformly rotund renorming abstract it is proved that c k e the space of all continuous functions on a hausdorff compact space k taking values in a banach space e admits an equivalent locally uniformly rotund norm if c k and e do so moreover if the equivalent lur norms on c k and e are lower semicontinuous with respect to some weak topologies the lur norm on c k e can be chosen to be lower semicontinuous with respect to an appropriate weak topology as a consequence we prove that if x and y are two hausdorff compacta and c x c y admit equivalent pointwise lower semicontinuous lur norms then so does c x y

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

statistics of resonances and delay times in random media beyond random matrix theory we review recent developments in quantum scattering from mesoscopic systems various spatial geometries whose closed analogues show diffusive localized or critical behaviour are considered these are the features that cannot be described by the universal random matrix theory results instead one has to go beyond this approximation and incorporate them in a non perturbative way here we pay particular attention to the traces of these non universal characteristics in the distribution of the wigner delay times and resonance widths the former quantity captures time dependent aspects of quantum scattering while the latter is associated with the poles of the scattering matrix

transition between hermitian and non hermitian gaussian ensembles the transition between hermitian and non hermitian matrices of the gaussian unitary ensemble is revisited an expression for the kernel of the rescaled hermite polynomials is derived which expresses the sum in terms of the highest order polynomials from this christoffel darboux like formula some results are derived including an extension to the complex plane of the airy kernel

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

statistical properties of eigenvectors in non hermitian gaussian random matrix ensembles statistical properties of eigenvectors in non hermitian random matrix ensembles are discussed with an emphasis on correlations between left and right eigenvectors two approaches are described one is an exact calculation for ginibre s ensemble in which each matrix element is an independent identically distributed gaussian complex random variable the other is a simpler calculation using n 1 as an expansion parameter where n is the rank of the random matrix this is applied to girko s ensemble consequences of eigenvector correlations which may be of physical importance in applications are also discussed it is shown that eigenvalues are much more sensitive to perturbations than in the corresponding hermitian random matrix ensembles it is also shown that in problems with time evolution governed by a non hermitian random matrix transients are controlled by eigenvector correlations

alur dual renormings of banach spaces we give a covering type characterization for the class of dual banach spaces with an equivalent alur dual norm

borel properties of linear operators abstract given an injective bounded linear operator t x y between banach spaces we study the borel measurability of the inverse map t 1 tx x a remarkable result of saint raymond ann inst fourier grenoble 26 1976 211 256 states that if x is separable then the borel class of t 1 is if and only if x is the th iterated sequential weak closure of t y for some countable ordinal we show that saint raymond s result holds with minor changes for arbitrary banach spaces if we assume that t has certain property named co discreteness after hansell proc london math soc 28 1974 683 699 as an application we show that the borel class of the inverse of a co discrete operator t can be estimated by the image of the unit ball or the restrictions of t to separable subspaces of x our results apply naturally when x is a wcd banach space since in this case any injective bounded linear operator defined on x is automatically co discrete

density of state in a complex random matrix theory with external source the density of state for a complex random matrix coupled to an external deterministic source is considered for a finite n and a compact expression in an integral representation is obtained

eigenfunction statistics of complex systems a common mathematical formulation we derive a common mathematical formulation for the eigenfunction statistics of hermitian operators represented by a multiparametric probability density the system information in the formulation enters through two parameters only namely system size and the complexity parameter a function of all system parameters including size the behavior is contrary to the eigenvalue statistics which is sensitive to the complexity parameter only and shows a single parametric scaling the existence of a mathematical formulation of both eigenfunctions and eigenvalues common to a wide range of complex systems indicates the possibility of a similar formulation for many physical properties this also suggests the possibility to classify them in various universality classes defined by the complexity parameter

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

classical wave experiments on chaotic scattering we review recent research on the transport properties of classical waves through chaotic systems with special emphasis on microwaves and sound waves inasmuch as these experiments use antennas or transducers to couple waves into or out of the systems scattering theory has to be applied for a quantitative interpretation of the measurements most experiments concentrate on tests of predictions from random matrix theory and the random plane wave approximation in all studied examples a quantitative agreement between experiment and theory is achieved to this end it is necessary however to take absorption and imperfect coupling into account concepts that were ignored in most previous theoretical investigations classical phase space signatures of scattering are being examined in a small number of experiments

baire class 1 selectors for upper semicontinuous set valued maps let t be a metric space and x a banach space let f t x be a set valued map assuming arbitrary values and satisfying the upper semicontinuity condition t t f t c o is closed for each weakly closed set c in x then there is a sequence of norm continuous functions converging pointwise in the norm to a selection for f we prove a statement of similar precision and generality when x is a metric space

universal sum and product rules for random matrices the spectral density of random matrices is studied through a quaternionic generalization of the green s function which precisely describes the mean spectral density of a given matrix under a particular type of random perturbation exact and universal expressions are found in the high dimension limit for the quaternionic green s functions of random matrices with independent entries when summed or multiplied with deterministic matrices from these the limiting spectral density can be accurately predicted

eigenvalue spacing statistics of a four matrix model of some four by four random matrices a simple four matrix model consisting of some 4 4 real and imaginary random matrices and three parameters is introduced it is shown that the eigenvalue spacing statistics of an ensemble of such matrices can be used to describe transitions between all the wigner surmises of random matrix theory formulae for the nearest neighbour spacing distributions of the ensemble are given for various parameter values

statistical theory of multimode random lasers we present a semiclassical laser theory for multimode lasing in optical resonators with overlapping modes nonlinear saturation and mode competition are characterized in terms of left and right eigenmodes of a non hermitian operator the theory is applied to wave chaotic cavities and weakly disordered random media in the limit of sufficiently strong pumping we find that the mean number of laser peaks increases with the 1 3 power of the pump strength

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

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

non hermitian random matrix theory summation of planar diagrams the single ring theorem and the disc annulus phase transition i review aspects of work done in collaboration with a zee and r scalettar 1997 nucl phys b 504 579 1997 nucl phys b 501 643 2001 j math phys 42 5718 on complex non hermitian random matrices i open by explaining why the bag of tools used regularly in analysing hermitian random matrices cannot be applied directly to analyse non hermitian matrices and then introduce the method of hermitization which solves this problem then for rotationally invariant ensembles i derive a master equation for the average density of eigenvalues in the complex plane in the limit of infinitely large matrices this is achieved by resumming all the planar diagrams which appear in the perturbative expansion of the hermitized green function remarkably this resummation can be carried out explicitly for any rotationally invariant ensemble i prove that in the limit of infinitely large matrices the shape of the eigenvalue distribution is either a disc or an annulus this is the celebrated single ring theorem which of these shapes is realized is determined by the parameters coupling constants which determine the ensemble by varying these parameters a phase transition may occur between the two possible shapes i briefly discuss the universal features of this transition as the analysis of this problem relies heavily on summation of planar feynman diagrams i make a special effort at presenting a pedagogical exposition of the diagrammatic method which some readers may find useful

exact s matrix for n disc systems and various boundary conditions ii determination and partial classification of resonances scattering of waves and particles from two and three disc systems with discrete and symmetries is studied for various boundary conditions resonances are numerically determined and partially classified by using the irreducible representations of the symmetry groups new physical effects are expected splitting up of resonances and resonances of interaction between the scatterers some of these effects can be observed on the far field form functions

conformal deformation of equilibrium measures in normal random ensembles we investigate the eigenvalue statistics of ensembles of normal random matrices when their order n tends to infinite in the model the eigenvalues have uniform density within a region determined by a simple analytic polynomial curve we study the conformal deformations of equilibrium measures of normal random ensembles to the real line and give sufficient conditions for it to weakly converge to a wigner measure

continuous piecewise linear finite elements for the kirchhoff love plate equation a family of continuous piecewise linear finite elements for thin plate problems is presented we use standard linear interpolation of the deflection field to reconstruct a discontinuous piecewise quadratic deflection field this allows us to use discontinuous galerkin methods for the kirchhoff love plate equation three example reconstructions of quadratic functions from linear interpolation triangles are presented a reconstruction using morley basis functions a fully quadratic reconstruction and a more general least squares approach to a fully quadratic reconstruction the morley reconstruction is shown to be equivalent to the basic plate triangle bpt given a condition on the reconstruction operator a priori error estimates are proved in energy norm and l 2 norm numerical results indicate that the morley reconstruction bpt does not converge on unstructured meshes while the fully quadratic reconstruction show optimal convergence

on weakly locally uniformly rotund banach spaces abstract we show that every normed space e with a weakly locally uniformly rotund norm has an equivalent locally uniformly rotund norm after obtaining a discrete network of the unit sphere s e for the weak topology we deduce that the space e must have a countable cover by sets of small local diameter which in turn implies the renorming conclusion this solves a question posed by deville godefroy haydon and zizler for a weakly uniformly rotund norm we prove that the unit sphere is always metrizable for the weak topology despite the fact that it may not have the kadec property moreover banach spaces having a countable cover by sets of small local diameter coincide with the descriptive banach spaces studied by hansell so we present here some new characterizations of them

on determination of statistical properties of spectra from parametric level dynamics we analyse an approach aimed at determining statistical properties of spectra of a time periodic quantum chaotic system based on the parameter dynamics of their quasienergies in particular we show that application of the methods of statistical physics proposed previously in the literature taking into account appropriate integrals of motion of the parametric dynamics is fully justified even if the used integrals of motion do not determine the invariant manifold in a unique way the indetermination of the manifold is removed by applying dirac s theory of constrained hamiltonian systems and imposing appropriate primary first class constraints and a gauge transformation generated by them in the standard way the obtained results close the gap in the whole reasoning aiming at understanding statistical properties of spectra in terms of parametric dynamics

level dynamics for conservative and dissipative quantum systems we establish level dynamics for finite matrices employing a unified treatment of real symmetric complex hermitian quaternion real unitary and arbitrary complex matrices in all cases the level dynamics take the form of the classical hamiltonian flow of some fictitious many particle systems equilibrium statistical mechanics of the latter leads to the well known matrix ensembles of random matrix theory ginibre s ensemble in particular is thus associated with level dynamics of arbitrary complex matrices

microwave billiards with broken time reversal invariance we consider a microwave resonator with three single channel waveguides attached one of these serves to couple waves into and out of the resonator the remaining two are connected to form a one way handle so as to break time reversal invariance the poles of the input output scattering coefficient of such a resonator are shown to be the eigenvalues of a non hermitian effective hamiltonian heff d h i0 the anti hermitian part 0 of which has rank 1 and is responsible for the breaking of time reversal invariance all of the spectral statistics recently observed for such a microwave billiard are reproduced quantitatively by taking h and 0 as random matrices in particular the distribution of nearest neighbour spacings of the resonances is close to that of the gue when h belongs to the goe corresponding to a sinai shape of the resonator linear level repulsion results when h belongs to the poissonian ensemble as it corresponds to a rectangular resonator

spectral statistics in chaotic systems with a point interaction we consider quantum systems with a chaotic classical limit that are perturbed by a point like scatterer the spectral form factor k for these systems is evaluated semiclassically in terms of periodic and diffractive orbits it is shown for order 2 and 3 that off diagonal contributions to the form factor which involve diffractive orbits cancel exactly the diagonal contributions from diffractive orbits implying that the perturbation by the scatterer does not change the spectral statistic we further show that parametric spectral statistics for these systems are universal for small changes of the strength of the scatterer

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

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

lossy chaotic electromagnetic reverberation chambers universal statistical behavior of the vectorial field the effective hamiltonian formalism is extended to vectorial electromagnetic waves in order to describe statistical properties of the field in reverberation chambers the latter are commonly used in electromagnetic compatibility tests as a first step the distribution of wave intensities in chaotic systems with varying opening in the weak coupling limit for scalar quantum waves is derived by means of random matrix theory in this limit the only parameters are the modal overlap and the number of open channels using the extended effective hamiltonian we describe the intensity statistics of the vectorial electromagnetic eigenmodes of lossy reverberation chambers finally the typical quantity of interest in such chambers namely the distribution of the electromagnetic response is discussed by determining the distribution of the phase rigidity describing the coupling to the environment using random matrix numerical data we find good agreement between the theoretical prediction and numerical calculations of the response

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

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

on the structure of non dentable closed bounded convex sets a self contained proof is given of the following result theorem let k be a non dentable closed bounded convex nonempty subset of a banach space x so that k equals the closed convex hull of its weak to norm points of continuity there exists a nonempty subset w of k satisfying w is non dentable closed convex and the weak and norm topologies on w coincide moreover there exists a closed linear subspace y of x so that y has a finite dimensional decomposition and a nonempty convex subset w satisfying any bounded set w satisfying has no extreme points this yields the recent discovery of w schachermayer that a closed bounded convex subset of a banach space has the radon nikod m property rnp provided it has the convex point of continity property cpcp and the krein milman property kmp and the cpcp case of the earlier discovery of j bourgain that a banach space has the rnp provided every subspace with a finite dimensional decomposition has the rnp the proof is a variation and crystallization of the original arguments for these discoveries in particular the proof uses the concept of convex sets with small combinations of slices this is a refinement of the concept of strong regularity introduced by n ghoussoub g godefroy and b maurey both concepts crystallize ideas appearing in bourgain s work the proof also yields further results concerning the structure of non dentable convex sets

dentability indices with respect to measures of non compactness abstract we study the relationships between the ordinal indices of set derivations associated to several measures of non compactness we obtain applications to the szlenk index improving a result of lancien and lur renorming providing a non probabilistic proof of a result of troyanski

the joint energy distribution function for the hamiltonian for the one channel case a closed analytical expression is derived for the joint distribution function of the real and the imaginary parts of the eigenenergies of the operator for the one channel case where is taken from the poissonian or one of the gaussian ensembles with universality index and where the squared moduli of the components of w are assumed to be distributed with universality index in the strong coupling limit and for the special case the joint distribution function of the real parts of the eigenvalues of h becomes identical with the joint energy distribution function of the eigenvalues of

wave chaos in quantum systems with point interaction we study perturbations of the quantized version 0 of integrable hamiltonian systems by point interactions we relate the eigenvalues of to the zeros of a certain meromorphic function assuming the eigenvalues of 0 are poisson distributed we get detailed information on the joint distribution of the zeros of and give bounds on the probability density for the spacings of eigenvalues of our results confirm the wave chaos phenomenon as different from the quantum chaos phenomenon predicted by random matrix theory

large scale correlations in normal non hermitian matrix ensembles we compute the large scale macroscopic correlations in ensembles of normal random matrices with a general non gaussian measure and in ensembles of general non hermitian matrices with a class of non gaussian measures in both cases the eigenvalues are complex and in the large n limit they occupy a domain in the complex plane for the case when the support of eigenvalues is a connected compact domain we compute two three and four point connected correlation functions in the first non vanishing order in 1 n in a manner that the algorithm of computing higher correlations becomes clear the correlation functions are expressed through the solution of the dirichlet boundary problem in the domain complementary to the support of eigenvalues

large n eigenvalue distribution of randomly perturbed asymmetric matrices the density of complex eigenvalues of random asymmetric matrices is found in the large n limit the matrices are of the form where a is a matrix of independent identically distributed random variables with zero mean and variance the limiting density is bounded the area of the support of cannot be less than in the case of commuting with its conjugate is expressed in terms of the eigenvalue distribution of the non perturbed part

quantum limit of the laser line width in chaotic cavities and statistics of residues of scattering matrix poles he quantum limited line width of a laser cavity is enhanced above the schawlow townes value by the petermann factor k due to the non orthogonality of the cavity modes we derive the relation between the petermann factor and the residues of poles of the scattering matrix and investigate the statistical properties of the petermann factor for cavities in which the radiation is scattered chaotically for a single scattering channel we determine the complete probability distribution of k and find that the average petermann factor depends non analytically on the area of the opening and greatly exceeds the most probable value for an arbitrary number n of scattering channels we calculate as a function of the decay rate i of the lasing mode we find for n 1 that for typical values of i the average petermann factor propto sqrt n 1 is parametrically larger than unity

semiclassical theory of spectral rigidity the spectral rigidity l of a set of quantal energy levels is the mean square deviation of the spectral staircase from the straight line that best fits it over a range of l mean level spacings in the semiclassical limit 0 formulae are obtained giving l as a sum over classical periodic orbits when l l max where l max n 1 for a system of n freedoms l is shown to display the following universal behaviour as a result of properties of very long classical orbits if the system is classically integrable all periodic orbits filling tori l 1 5 l as in an uncorrelated poisson eigenvalue sequence if the system is classically chaotic all periodic orbits isolated and unstable and has no symmetry l in l 2 2 d if 1 l l max as in the gaussian unitary ensemble of random matrix theory if the system is chaotic and has time reversal symmetry l in l 2 e if 1 l l max as in the gaussian orthogonal ensemble when l l max l saturates non universally at a value determined by short classical orbits of order n 1 for integrable systems and in 1 for chaotic systems these results are obtained by using the periodic orbit expansion for the spectral density together with classical sum rules for the intensities of long orbits and a semiclassical sum rule restricting the manner in which their contributions interfere for two examples l is studied in detail the rectangular billiard integrable and the riemann zeta function assuming its zeros to be the eigenvalues of an unknown quantum system whose unknown classical limit is chaotic

rank one non hermitian perturbations of hermitian beta ensembles of random matrices we provide a tridiagonal matrix model and compute the joint eigenvalue density of a rank one non hermitian perturbation of a random matrix from the gaussian or laguerre beta ensemble

continuous discontinuous finite element approximations of fourth order elliptic problems in structural and continuum mechanics with applications to thin beams and plates and strain gradient elasticity abstract a new finite element method for fourth order elliptic partial differential equations is presented and applied to thin bending theory problems in structural mechanics and to a strain gradient theory problem the method combines concepts from the continuous galerkin cg method the discontinuous galerkin dg method and stabilization techniques a brief review of the cg method the dg method and stabilization techniques highlights the advantages and disadvantages of these methods and suggests a new approach for the solution of fourth order elliptic problems a continuous discontinuous galerkin c dg method is proposed which uses c 0 continuous interpolation functions and is formulated in the primary variable only the advantage of this formulation over a more traditional mixed approach is that the introduction of additional unknowns and related difficulties can be avoided in the context of thin bending theory the c dg method leads to a formulation where displacements are the only degrees of freedom and no rotational degrees of freedom need to be considered the main feature of the c dg method is the weak enforcement of continuity of first and higher order derivatives through stabilizing terms on interior boundaries consistency stability and convergence of the method are shown analytically numerical experiments verify the theoretical results and applications are presented for bernoulli euler beam bending poisson kirchhoff plate bending and a shear layer problem using toupin mindlin strain gradient theory

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

on banach spaces whose norm open sets are fσ sets in the weak topology we investigate non separable banach spaces whose norm open sets are countable unions of sets closed in the weak topology and a narrower class of banach spaces with a network for the norm topology which is discrete in the weak topology in particular we answer a question of arhangel skii exhibiting various examples of non separable function spaces c k with a discrete network for the pointwise topology and consistently we answer some questions of edgar and oncina concerning borel structures and kadec renormings in banach spaces

continuous functions on compact totally ordered spaces abstract the banach space c k of continuous functions on k with its pointwise topology is shown to be fragmented by pointwise closed sets when k is a compact totally ordered set when k is a dendron and when k is a compact hausdorff space with a closed crossfree subbase for its topology in some special instances c k will have an equivalent locally uniformly convex norm

average locally uniform rotundity and a class of nonlinear maps abstract we consider some topological characterizations of dual banach spaces that admit an equivalent dual average locally uniformly rotund norm and provide a criterion for such renorming which involves the class of slicely continuous maps

a generalisation of the calogero moser system abstract a generalised calogero moser many body system in which the particles possess extra internal degrees of freedom is introduced and solved by the method of olshanetsky and perelomov further the hierarchy of flows commuting with this system is obtained from a hierarchy of linear systems possessing sl n symmetry by the usual method of reduction thus they are shown to be integrable

semiclassical quantization of billiards with mixed boundary conditions the semiclassical theory for billiards with mixed boundary conditions is developed and explicit expressions for the smooth and the oscillatory parts of the spectral density are derived the parametric dependence of the spectrum on the boundary condition is shown to be a very useful diagnostic tool in the semiclassical analysis of the spectrum of billiards it is also used to check in detail some recently proposed parametric spectral statistics the methods are illustrated in the analysis of the spectrum of the sinai billiard and its parametric dependence on the boundary condition on the dispersing arc

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

interpolation between airy and poisson statistics for unitary chiral non hermitian random matrix ensembles we consider a family of chiral non hermitian gaussian random matrices in the unitarily invariant symmetry class the eigenvalue distribution in this model is expressed in terms of laguerre polynomials in the complex plane these are orthogonal with respect to a non gaussian weight including a modified bessel function of the second kind and we give an elementary proof for this in the large n limit the eigenvalue statistics at the spectral edge close to the real axis are described by the same family of kernels interpolating between airy and poisson that was recently found by one of the authors for the elliptic ginibre ensemble we conclude that this scaling limit is universal appearing for two different non hermitian random matrix ensembles with unitary symmetry as a second result we give an equivalent form for the interpolating airy kernel in terms of a single real integral similar to representations for the asymptotic kernel in the bulk and at the hard edge of the spectrum this makes its structure as a one parameter deformation of the airy kernel more transparent