vicious walks with a wall noncolliding meanders and chiral and bogoliubov de gennes random matrices spatially and temporally inhomogeneous evolution of one dimensional vicious walkers with wall restriction is studied we show that its continuum version is equivalent with a noncolliding system of stochastic processes called brownian meanders here the brownian meander is a temporally inhomogeneous process introduced by yor as a transform of the bessel process that is the motion of radial coordinate of the three dimensional brownian motion represented in spherical coordinates it is proved that the spatial distribution of vicious walkers with a wall at the origin can be described by the eigenvalue statistics of gaussian ensembles of bogoliubov de gennes hamiltonians of the mean field theory of superconductivity which have a particle hole symmetry we report that a time evolution of the present stochastic process is fully characterized by the change of symmetry classes from type c to type ci in the nonstandard classes of random matrix theory of altland and zirnbauer the relation between the noncolliding systems of the generalized meanders of yor which are associated with the even dimensional bessel processes and the chiral random matrix theory is also clarified

1 2 laplacian problems with exponential nonlinearity abstract by exploiting a suitable trudinger moser inequality for fractional sobolev spaces we obtain existence and multiplicity of solutions for a class of one dimensional nonlocal equations with fractional diffusion and nonlinearity at exponential growth

universal parametric correlations of eigenvalues of random matrix ensembles eigenvalue correlations of random matrix ensembles as a function of an external parametric perturbation are investigated via the dyson brownian motion model in the situation where the level density has a hard edge singularity by solving a linearized hydrodynamical equation a universal dependence of the density density correlator on the external field is found as an application we obtain a formula for the variance of linear statistics with the parametric dependence exhibited as a laplace transform

contour integral method for transitions to the circular unitary ensemble the representation of correlation functions as a contour integral has been useful in the study of transitions to the gaussian unitary ensemble gue we develop the formalism for transitions to the circular unitary ensemble cue and consider the general lcue to cue transition where lcue denotes a superposition of l independent cue spectra in an arbitrary ratio for large matrices we derive the two level correlation function for all l including l the poisson case the results are useful in the study of weakly broken partitioning symmetries and weakly coupled mesoscopic cavities

extremal non compactness of composition operators with linear fractional symbol abstract we realize the norms of certain composition operators c with linear fractional symbol acting on the hardy space in terms of the roots of associated hypergeometric functions this realization leads to simple necessary and sufficient conditions on for c to exhibit extremal non compactness establishes equivalence of cohyponormality and cosubnormality of composition operators with linear fractional symbol and yields a complete classification of those linear fractional that induce composition operators whose norms are determined by the action of the adjoint c on the normalized reproducing kernels in h 2

the ginibre evolution in the large n limit we analyse statistics of the real eigenvalues of gl n r valued brownian motion the ginibre evolution in the limit of large n in particular we calculate the limiting two time correlation function of spin variables associated with real eigenvalues of the ginibre evolution we also show how the formalism of spin variables can be used to compute the fixed time correlation functions of real eigenvalues discovered originally by forrester and nagao eigenvalue statistics of the real ginibre ensemble phys rev lett 99 5 050603 2007 and borodin and sinclair the ginibre ensemble of real random matrices and its scaling limits commun math phys 291 1 177 224 2009

2 2 random matrix ensembles with reduced symmetry from hermitian to pt symmetric matrices a possibly fruitful extension of conventional random matrix ensembles is proposed by imposing symmetry constraints on conventional hermitian matrices or parity time symmetric matrices to illustrate the main idea we first study 2 2 complex hermitian matrix ensembles with o 2 invariant constraints yielding novel level spacing statistics such as singular distributions the half gaussian distribution distributions interpolating between the goe gaussian orthogonal ensemble distribution and half gaussian distributions as well as the gapped goe distribution such a symmetry reduction strategy is then used to explore 2 2 symmetric matrix ensembles with real eigenvalues in particular symmetric random matrix ensembles with u 2 invariance can be constructed with the conventional complex hermitian random matrix ensemble being a special case in two examples of symmetric random matrix ensembles the level spacing distributions are found to be the standard gue gaussian unitary ensemble statistics or the truncated gue statistics this article is part of a special issue of journal of physics a mathematical and theoretical devoted to quantum physics with non hermitian operators

on the efetov wegner terms by diagonalizing a hermitian supermatrix the diagonalization of hermitian supermatrices is studied such a change of coordinates is inevitable if one wishes to find certain structures in random matrix theory however it still poses serious problems since until now the calculation of all rothstein contributions known as efetov wegner terms in physics was quite cumbersome we derive the supermatrix bessel function with all efetov wegner terms for an arbitrary rotation invariant probability density function as applications we consider representations of generating functions for hermitian random matrices with and without an external field as integrals over eigenvalues of hermitian supermatrices all results are obtained with all efetov wegner terms which were previously unknown in such an explicit and compact representation

corrigendum to on norms of composition operators acting on bergman spaces j math anal appl 291 2004 189 202 abstract a theorem and a corollary in the paper cited in the title were stated incorrectly as was pointed out by christopher hammond we now state correctly and prove both of them these results still generalize and explain the geometric meaning of the cowen hurst norm formula we also include additional references and provide an example relevant for further study

correlation kernels for discrete symplectic and orthogonal ensembles in 49 h widom derived formulae expressing correlation functions of orthogonal and symplectic ensembles of random matrices in terms of orthogonal polynomials we obtain similar results for discrete ensembles with rational discrete logarithmic derivative and compute explicitly correlation kernels associated to the classical meixner and charlier orthogonal polynomials

semiclassical theory for parametric correlation of energy levels parametric energy level correlation describes the response of the energy level statistics to an external parameter such as the magnetic field using semiclassical periodic orbit theory for a chaotic system we evaluate the parametric energy level correlation depending on the magnetic field difference the small time expansion of the spectral form factor k is shown to be in agreement with the prediction of parameter dependent random matrix theory to all orders in

derivation of determinantal structures for random matrix ensembles in a new way there are several methods to treat ensembles of random matrices in symmetric spaces circular matrices chiral matrices and others orthogonal polynomials and the supersymmetry method are particular powerful techniques here we present a new approach to calculate averages over ratios of characteristic polynomials at first sight paradoxically one can call our approach supersymmetry without supersymmetry because we use structures from supersymmetry without actually mapping onto superspaces we address two kinds of integrals which cover a wide range of applications for random matrix ensembles for probability densities factorizing in the eigenvalues we find determinantal structures in a unifying way as a new application we derive an expression for the k point correlation function of an arbitrary rotation invariant probability density over the hermitian matrices in the presence of an external field

the nehari manifold for elliptic equation involving the square root of the laplacian abstract in this paper we introduce the nehari manifold for elliptic problems involving the square root of the laplacian we use it to establish the existence of solutions and multiple solutions for some nonlinear elliptic problem with sign changing weight we also establish some bifurcation results and non existence results

a semilinear elliptic system involving nonlinear boundary condition and sign changing weight function abstract in this paper we study the combined effect of concave and convex nonlinearities on the number of solutions for a semilinear elliptic system e involving nonlinear boundary condition and sign changing weight function with the help of the nehari manifold we prove that the system has at least two nontrivial nonnegative solutions when the pair of the parameters belongs to a certain subset of r 2

a brezis nirenberg splitting approach for nonlocal fractional equations in this paper we consider problems modeled by the following nonlocal fractional equation su a x u f u in u 0in rn where s 0 1 is fixed is an open bounded subset of rn n 2s with lipschitz boundary s is the fractional laplace operator and is a real parameter under two different types of conditions on the functions a and f by using a famous critical point theorem in the presence of splitting established by brezis and nirenberg we obtain the existence of at least two nontrivial weak solutions for our problem

the norm of a composition operator with linear symbol acting on the dirichlet space abstract we obtain a representation for the norm of a composition operator on the dirichlet space induced by a map of the form z a z b we compare this result to an upper bound for c that is valid whenever is univalent our work relies heavily on an adjoint formula recently discovered by gallardo gutierrez and montes rodriguez

critical correlations in a z invariant inhomogeneous ising model abstract we study the critical spin spin correlation function in the z invariant inhomogeneous ising model which includes the rectangular triangular honeycomb and checkerboard lattices as special cases using quadratic difference equations we derive various explicit representations for the correlation function confirming and simplifying also a miraculous result of baxter

the nehari manifold for systems of nonlinear elliptic equations abstract this paper deals with the existence and multiplicity results of nonlocal positive solutions to the following system p u u p 1 2 u 1 u u 1 v 1 q v v q 2 v 1 u 1 v 1 v together with dirichlet or mixed boundary conditions under some hypotheses on the parameters p p 1 and q more precisely the system considered corresponds to a perturbed eigenvalue equation combined with a second equation having concave and convex nonlinearities the study is based on the extraction of palais smale sequences in the nehari manifold the behaviour of the energy corresponding to these positive solutions with respect to the real parameters and is established

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

random matrix theory of parametric correlations in the spectra of disordered metals and chaotic billiards abstract a random matrix theory is developed for the adiabatic response to an external perturbation of the energy spectrum of a mesoscopic system the basic assumption is that spectral correlations are governed by level repulsion following dyson the dependence of the energy levels on the perturbation parameter is modeled by a brownian motion process in a fictitious viscous fluid a fokker planck equation for the evolution of the distribution function is solved to yield the correlation of level densities at different energies and different parameter values an approximate solution is obtained by asymptotic expansion and an exact solution by mapping onto a free fermion model a generalization to multiple parameters is also considered corresponding to brownian motion in a fictitious world with multiple temporal dimensions complete agreement is obtained with microscopic theory

about the spacing functions of the three matrix ensembles three ensembles of random matrices have been extensively studied orthogonal unitary and symplectic characterized by a parameter taking values 1 2 and 4 the probability that a randomly chosen interval of length 2t contains exactly r eigenvalues of such a matrix can be expressed in terms of the rth partial derivative of a fredholm determinant using this fact we give a new proof of some known relations between and as well as a relation between odd and even spheroidal functions

doubly nonlocal system with hardy littlewood sobolev critical nonlinearity abstract this article concerns about the existence and multiplicity of weak solutions for the following nonlinear doubly nonlocal problem with critical nonlinearity in the sense of hardy littlewood sobolev inequality s u u q 2 u v y 2 x y d y u 2 2 u in s v v q 2 v u y 2 x y d y v 2 2 v in u v 0 in r n where is a smooth bounded domain in r n n 2 s s 0 1 s is the well known fractional laplacian 0 n 2 2 n n 2 s is the upper critical exponent in the hardy littlewood sobolev inequality 1 q 2 and 0 are real parameters we study the fibering maps corresponding to the functional associated with p and show that minimization over suitable subsets of nehari manifold renders the existence of at least two non trivial solutions of p for suitable range of and

singularities in the spectra of random matrices we consider singularities of the set of energy levels en x of a quantum hamiltonian obtained by varying a set of d parameters x x1 xd singularities such as minima degeneracies branch points and avoided crossings can play an important role in physical applications we discuss a general method for counting these singularities and apply it to a random matrix model for the parameter dependence of energy levels we also show how the density of avoided crossing singularities is related to a non analyticity of a correlation function describing the energy levels

spectral theory for algebraic combinations of toeplitz and composition operators abstract we determine the essential spectra of algebraic combinations of toeplitz operators with continuous symbol and composition operators induced by a class of linear fractional non automorphisms of the unit disk the operators in question act on the hardy space h 2 on the unit disk our method is to realize the c algebra that they generate as an extension of the compact operators by a concrete c algebra whose invertible elements are easily characterized

the brezis nirenberg result for the fractional laplacian the aim of this paper is to deal with the non local fractional counterpart of the laplace equation involving critical non linearities studied in the famous paper of brezis and nirenberg 1983 namely our model is the equation su u u 2 2u in u 0 in rn where s is the fractional laplace operator s 0 1 is an open bounded set of rn n 2s with lipschitz boundary 0 is a real parameter and 2 2n n 2s is a fractional critical sobolev exponent in this paper we first study the problem in a general framework indeed we consider the equation lku u u 2 2u f x u 0 in u 0 in rn where lk is a general non local integrodifferential operator of order s and f is a lower order perturbation of the critical power u 2 2u in this setting we prove an existence result through variational techniques then as a concrete example we derive a brezis nirenberg type result for our model equation that is we show that if 1 s is the first eigenvalue of the non local operator s with homogeneous dirichlet boundary datum then for any 0 1 s there exists a non trivial solution of the above model equation provided n 4s in this sense the present work may be seen as the extension of the classical brezis nirenberg result to the case of non local fractional operators

subnormality and composition operators on h2 if is a non constant analytic function defined on the unit disk d such that d d the composition operator c is the operator on the hardy space h2 of the unit disk defined by c f f in this paper we investigate the relationship between properties of the symbol and the subnormality of the operators c and c the main theorem shows under a regularity hypothesis that c is subnormal if and only if is in a restricted class of linear fractional transformations the two previously known special cases the affine transformations associated with the cesaro operator and the automorphisms are included as the extreme examples the proof of the main theorem consists of verifying a moment condition and yields in addition the construction of a measure that exhibits cgj as multiplication by an analytic function on p2 and the identification of the minimal normal extension as a weighted sum of shifts

the nehari manifold for a semilinear elliptic system involving sign changing weight functions abstract in this paper we study the combined effect of concave and convex nonlinearities on the number of solutions for a semilinear elliptic system e with sign changing weight function with the help of the nehari manifold we prove that the system has at least two nontrivial nonnegative solutions when the pair of the parameters belongs to a certain subset of r 2

density of states for gaussian unitary ensemble gaussian orthogonal ensemble and interpolating ensembles through supersymmetric approach we use the supersymmetric formalism to derive an integral formula for the density of states of the gaussian orthogonal ensemble and then apply saddle point analysis to give a new derivation of the 1 n correction to wigner s law this extends the work of disertori on the gaussian unitary ensemble we also apply our method to the interpolating ensembles of mehta pandey

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

surprising pfaffian factorizations in random matrix theory with dyson index β 2 in recent decades determinants and pfaffians were found for eigenvalue correlations of various random matrix ensembles these structures simplify the average over a large number of ratios of characteristic polynomials to integrations over one and two characteristic polynomials only up to now it was thought that determinants occur for ensembles with dyson index 2 whereas pfaffians only for ensembles with 1 4 we derive a non trivial pfaffian determinant for 2 random matrix ensembles which is similar to the one for 1 4 thus it unveils a hidden universality of this structure we also give a general relation between the orthogonal polynomials related to the determinantal structure and the skew orthogonal polynomials corresponding to the pfaffian as a particular example we consider the chiral unitary ensembles in great detail

analytic continuation of the single eigenvalue probability density function an exact expression for the single eigenvalue probability density function for any dimension of a gaussian orthogonal ensemble is found by analytic continuation it is shown that the form can be used to derive a number of exact results in a simple way in odd dimensions it is shown that the single eigenvalue probability density function can be written as a sum of two densities

diffusions on symmetric spaces of type a iii and random matrix theories for rectangular matrices isotropic diffusion processes on cosets and and their zero curvature limit are studied from a unified viewpoint as indicated in our previous works the projection of the fokker planck equation onto the maximal commutative subgroup of these cosets can be described by using the radial part of the laplace beltrami operator by taking the zero curvature limit an integral which is a natural extension of the itzykson zuber integral to the rectangular matrices is explicitly evaluated the probability density function obtained from the diffusion on is studied in detail which can be applied to the quantum transport problem the explicit expressions for the probability density function in the metallic and insulating regimes are obtained for the metallic regime the integral representation for the hypergeometric function is used and the results are exact furthermore by using the orthogonal polynomial method n point correlation functions are obtained exactly for arbitrary n

spacing distributions for some gaussian ensembles of hermitian matrices the probability e n s that an interval of length s contains exactly n eigenvalues of a random matrix is expressed in terms of their correlation functions for the gaussian ensemble of hermitian matrices with an arbitrary ratio of their symmetric and antisymmetric parts the authors can thus write e n s as a convergent infinite product multiplied by an infinite sum

correlation functions of eigenvalues of multi matrix models and the limit of a time dependent matrix the universality of correlation functions of eigenvalues of large random matrices has been observed in various physical systems and proved in some particular cases as the hermitian one matrix model with polynomial potential here we consider the more difficult case of a unidimensional chain of hermitian matrices with first neighbour couplings and polynomial potentials an asymptotic expression of the orthogonal polynomials and a generalization of the darboux christoffel theorem allow us to find new results for the correlations of eigenvalues of different matrices of the chain eventually we consider the limit of the infinite chain of matrices which can be interpreted as a time dependent hermitian one matrix model and give the correlation functions of eigenvalues at different times

existence of multiple solutions for quasilinear systems via fibering method existence and multiplicity results for a general class of radial quasilinear equations are proved using the fibering method introduced and developed by s i pohozaev

statistics of energy levels without time reversal symmetry aharonov bohm chaotic billiards a planar domain d contains a single line of magnetic flux phi switching on phi breaks time reversal symmetry t for quantal particles with charge q moving in d whilst preserving the geometry of classical billiard trajectories bouncing off the boundary delta d if delta d is such that these classical trajectories are chaotic the authors predict that t breaking will cause the local statistics of quantal energy levels to change their universality class from that of the gaussian orthogonal ensemble goe of random matrix theory to that of the gaussian unitary ensemble gue in the semiclassical limit this transition is abrupt for statistics involving the first n levels gue behaviour requires that the quantum flux alpha identical to q phi h 0 13n 14 the special flux alpha 1 2 corresponds to false t breaking and for this case goe statistics are predicted these predictions are confirmed by numerical computation of spectral statistics for a classically chaotic billiard without symmetry for which delta d is a cubic conformal image of the unit disc

moments of vicious walkers and mobius graph expansions a system of brownian motions in one dimension all started from the origin and conditioned never to collide with each other in a given finite time interval 0 t is studied the spatial distribution of such vicious walkers can be described by using the repulsive eigenvalue statistics of random hermitian matrices and it was shown that the present vicious walker model exhibits a transition from the gaussian unitary ensemble gue statistics to the gaussian orthogonal ensemble goe statistics as the time t goes on from 0 to t in the present paper we characterize this gue to goe transition by presenting the graphical expansion formula for the moments of positions of vicious walkers in the gue limit stackrel ensuremath rightarrow t 0 only the ribbon graphs contribute and the problem is reduced to the classification of orientable surfaces by genus following the time evolution of the vicious walkers however the graphs with twisted ribbons called m obius graphs increase their contribution to our expansion formula and we have to deal with the topology of nonorientable surfaces application of the recent exact result of dynamical correlation functions yields closed expressions for the coefficients in the m obius expansion using the stirling numbers of the first kind

on superquadratic elliptic systems in this article we study the existence of solutions for the elliptic system updelta u frac partial h partial v left u v x right text in upomega updelta v frac partial h partial v left u v x right text in upomega u 0 v 0 text on partial upomega where upomega is a bounded open subset of mathbb r n with smooth boundary partial upomega and the function h mathbb r 2 times overline upomega to mathbb r is of class c 1

exact and asymptotic formulas for overdamped brownian dynamics abstract exact and asymptotic formulas relating to dynamical correlations for over damped brownian motion are obtained these formulas include a generalization of the f sum rule from the theory of quantum fluids a formula relating the static current current correlation to the static density density correlation and an asymptotic formula for the small k behaviour of the dynamical structure factor known exact evaluations of the dynamical density density correlation in some special models are used to illustrate and test the formulas

symmetry of matrix valued stochastic processes and noncolliding diffusion particle systems as an extension of the theory of dyson s brownian motion models for the standard gaussian random matrix ensembles we report a systematic study of hermitian matrix valued processes and their eigenvalue processes associated with the chiral and nonstandard random matrix ensembles in addition to the noncolliding brownian motions we introduce a one parameter family of temporally homogeneous noncolliding systems of the bessel processes and a two parameter family of temporally inhomogeneous noncolliding systems of yor s generalized meanders and show that all of the ten classes of eigenvalue statistics in the altland zirnbauer classification are realized as particle distributions in the special cases of these diffusion particle systems as a corollary of each equivalence in distribution of a temporally inhomogeneous eigenvalue process and a noncolliding diffusion process a stochastic calculus proof of a version of the harish chandra itzykson zuber formula of integral over unitary group is established

random matrix theory in semiclassical quantum mechanics of chaotic systems the statistical properties of the spectrum of systems which have a chaotic classical limit have been found to be similar to those of random matrix ensembles the author explains this correspondence including the fact that long ranged spectral statistics show deviations from the results of random matrix theory the method depends on the ambiguity of quantisation of a given classical system although the energy levels depend on the particular quantisation used the spectral statistics are assumed to depend only on the classical motion the ambiguity of quantisation can be represented by a small perturbation acting on the hamiltonian this perturbation executes a random walk which causes its matrix elements to undergo a random walk if the matrix elements are completely uncorrelated the spectral statistics are those of random matrix theory in the case of classically chaotic systems the matrix elements are constrained by sum rules related to classical periodic orbits this leads to deviations of the long ranged spectral statistics from the predictions of random matrix theory

on critical cases of sobolev s inequalities abstract we present a new form of the trudinger type inequality which shows an explicit dependence moreover we give an alternative proof of the brezis gallouet wainger inequality

existence results for some nonlinear elliptic system with lack of compactness abstract we are interested in existence of solutions for some typical elliptic systems governed by two pseudo laplacian operators with critical sobolev exponents we show this result by a local minimization for an adapted variational problem a concentration compactness principle allows us to obtain some convergence results

the nehari manifold for indefinite semilinear elliptic systems involving critical exponent abstract in this paper we study the combined effect of concave and convex nonlinearities on the number of solutions for an indefinite semilinear elliptic system e involving critical exponents and sign changing weight functions using nehari manifold the system is proved to have at least two nontrivial nonnegative solutions when the pair of the parameters belongs to a certain subset of r 2

norms of linear fractional composition operators we obtain a representation for the norm of the composition operator c on the hardy space h 2 whenever is a linear fractional mapping of the form z b cz d the representation shows that for such mappings the norm of c always exceeds the essential norm of c moreover it shows that a formula obtained by cowen for the norms of composition operators induced by mappings of the form z sz t has no natural generalization that would yield the norms of all linear fractional composition operators for rational numbers s and t cowen s formula yields an algebraic number as the norm we show e g that the norm of c 1 2 z is a transcendental number our principal results are based on a process that allows us to associate with each non compact linear fractional composition operator c for which c c e an equation whose maximum real solution is c 2 our work answers a number of questions in the literature for example we settle an issue raised by cowen and maccluer concerning co hyponormality of a certain family of composition operators

exact solution of the lock step model of vicious walkers the lock step model of vicious walkers on a one dimensional lattice allows each walker at the tick of a clock to move either one lattice site to the left or one lattice site to the right the only restriction is that no two walkers may arrive at the same lattice site or pass one another in periodic boundary conditions the partition function and correlation function for this model are calculated exactly taking the continuum limit gives an exactly solvable model of vicious walkers undergoing brownian motion

noncolliding brownian motion and determinantal processes a system of one dimensional brownian motions bms conditioned never to collide with each other is realized as i dyson s bm model which is a process of eigenvalues of hermitian matrix valued diffusion process in the gaussian unitary ensemble gue and as ii the h transform of absorbing bm in a weyl chamber where the harmonic function h is the product of differences of variables the vandermonde determinant the karlin mcgregor formula gives determinantal expression to the transition probability density of absorbing bm we show from the karlin mcgregor formula if the initial state is in the eigenvalue distribution of gue the noncolliding bm is a determinantal process in the sense that any multitime correlation function is given by a determinant specified by a matrix kernel by taking appropriate scaling limits spatially homogeneous and inhomogeneous infinite determinantal processes are derived we note that the determinantal processes related with noncolliding particle systems have a feature in common such that the matrix kernels are expressed using spectral projections of appropriate effective hamiltonians on the common structure of matrix kernels continuity of processes in time is proved and general property of the determinantal processes is discussed

existence results to elliptic systems with nonstandard growth conditions abstract nonstandard growth conditions in partial differential equations have been the subject of recent developments in elastic mechanics and electrorheological fluid dynamics lecture notes in mathematics vol 1748 2000 c r acad sci paris ser i math 329 1999 393 398 math ussr izv 29 1987 33 66 in this work elliptic systems with nonstandard growth conditions are studied existence and multiplicity results under growth conditions on the reaction terms are established

on some gaussian ensembles of hermitian matrices a gaussian ensemble of hermitian matrices with an arbitrary ratio of their symmetric and anti symmetric parts is studied similarly a gaussian ensemble of hermitian quaternion matrices with an arbitrary ratio of their self dual and anti self dual parts is studied analytical expressions for the correlation and cluster functions as well as their limits when the order of the matrices is large are derived for both ensembles

statistics of wave functions in mesoscopic systems we review the results of a recent study of fluctuations of wave functions in confined chaotic systems the fluctuations can be due to a random potential or be a consequence of a chaotic scattering by the walls the entire distribution function of the local amplitudes of the wave functions f 1 and the joint two point distribution are calculated in various situations the computation is performed using the supersymmetry technique and employs the studies of a reduced version of the non linear supersymmetric model developed especially for investigating the properties of a single eigenstate in a discrete spectrum of a chaotic quantum system for not very large amplitudes the complete description can be achieved using the zero dimensional approximation of the model the distribution function calculated in the limit of various symmetry classes shows the universal behavior known as the porter thomas statistics and fluctuations at distant points do not correlate in the crossover regime between the ensembles the distribution of local amplitudes shows a somewhat more sophisticated behavior the fluctuations in this case are correlated over distances exceeding the mean free path for large amplitudes generated by the states the most affected by the localization we call them prelocalized the zero dimensional approximation is no longer valid instead the statistics of their wave functions is determined by nontrivial vacua of the reduced model which is quite similar to the liouville model known in conformal field theory in particular the vacuum state of the reduced model obeys the liouville equation which indicates that in two dimensions the prelocalized states have nearly critical properties we prove their multifractality and power law statistically averaged envelope r 2 r 2 at the intermediate range of distances below the localization length with a spectrum of exponents 1 as well as obtain a logarithmically normal tail of the distribution functionf 1 we also find an evidence of prelocalized states in quasi one dimensional wires with the length shorter than the localization length their statistically averaged envelope has power law asymptotics x 2 x 2 and the tail of the distribution function is similar to that describing localized states in the infinite wires

pfaffian expressions for random matrix correlation functions it is well known that pfaffian formulas for eigenvalue correlations are useful in the analysis of real and quaternion random matrices moreover the parametric correlations in the crossover to complex random matrices are evaluated in the forms of pfaffians in this article we review the formulations and applications of pfaffian formulas for that purpose we first present the general pfaffian expressions in terms of the corresponding skew orthogonal polynomials then we clarify the relation to eynard and mehta s determinant formula for hermitian matrix models and explain how the evaluation is simplified in the cases related to the classical orthogonal polynomials applications of pfaffian formulas to random matrix theory and other fields are also mentioned

which linear fractional composition operators are essentially normal abstract we characterize the essentially normal composition operators induced on the hardy space h2 by linear fractional maps they are either compact normal or the nontrivial case induced by parabolic nonautomorphisms these parabolic maps induce the first known examples of nontrivially essentially normal composition operators in addition we characterize those linear fractionally induced composition operators on h2 that are essentially self adjoint and present a number of results for composition operators induced by maps that are not linear fractional

level repulsion in constrained gaussian random matrix ensembles introducing sets of constraints we define new classes of random matrix ensembles the constrained gaussian unitary cgue and the deformed gaussian unitary dgue ensembles the latter interpolate between the gue and the cgue we derive a sufficient condition for gue type level repulsion to persist in the presence of constraints for special classes of constraints we extend this approach to the orthogonal and to the symplectic ensembles a generalized fourier theorem relates the spectral properties of the constraining ensembles with those of the constrained ones we find that in the dgues level repulsion always prevails at a sufficiently short distance and may be lifted only in the limit of strictly enforced constraints

distribution of the ratio of two consecutive level spacings in orthogonal to unitary crossover ensembles the ratio of two consecutive level spacings has emerged as a very useful metric in investigating universal features exhibited by complex spectra it does not require the knowledge of density of states and is therefore quite convenient to compute in analyzing the spectrum of a general system the wigner surmise like results for the ratio distribution are known for the invariant classes of gaussian random matrices however for the crossover ensembles which are useful in modeling systems with partially broken symmetries corresponding results have remained unavailable so far in this work we derive exact results for the distribution and average of the ratio of two consecutive level spacings in the gaussian orthogonal to unitary crossover ensemble using a 3 3 random matrix model this crossover is useful in modeling time reversal symmetry breaking in quantum chaotic systems although based on a 3 3 matrix model our results can also be applied in the study of large spectra provided the symmetry breaking parameter facilitating the crossover is suitably scaled we substantiate this claim by considering gaussian and laguerre crossover ensembles comprising large matrices moreover we apply our result to investigate the violation of time reversal invariance in the quantum kicked rotor system

correlations for parameter dependent random matrices correlations for parameter dependent gaussian random matrices intermediate between symmetric and hermitian and antisymmetric hermitian and hermitian are calculated the dynamical density density correlation between eigenvalues at different values of the parameter is calculated for the symmetric to hermitian transition and the scaledn limit is computed for the antisymmetric hermitian to hermitian transition the equal parametern point distribution function is calculated and the scaled limit computed a circular version of the antisymmetric hermitian to hermitian transition is formulated in the thermodynamic limit the equal parameter distribution function is shown to coincide with the scaled limit expression of this distribution for the gaussian antisymmetric hermitian to hermitian transition furthermore the thermodynamic limit of the corresponding density density correlation is computed the results for the correlations are illustrated by comparison with empirical correlations calculated from numerical data obtained from computer generated gaussian random matrices

critical growth fractional elliptic systems with exponential nonlinearity abstract we study the existence of positive solutions for the system of fractional elliptic equations of the type 1 2 u p p q f x u p 2 u v q h 1 u v e u 2 v 2 in 1 1 1 2 v q p q f x u p v q 2 v h 2 u v e u 2 v 2 in 1 1 u v 0 in 1 1 u v 0 in r 1 1 where 1 p q 2 h 1 u v 2 u 2 u 2 u v h 2 u v 2 v 2 u v 2 v and 2 here 1 2 is the fractional laplacian operator we show the existence of multiple solutions for suitable range of by analyzing the fibering maps and the corresponding nehari manifold we also study the existence of positive solutions for a superlinear system with critical growth exponential nonlinearity

time reversal symmetry breaking and the statistical properties of quantum systems the breaking of the generalized time reversal symmetry of a quantum chaotic system corresponds to a transition from orthogonal to unitary ensembles of random matrices investigating this transition for the circular ensembles applicable for time dependent periodic systems the authors demonstrate and explain that there exists a relevant difference of the transition rate in comparison with the gaussian ensembles appropriate for quantum conservative systems the above supposition is supported by a numerical study of the eigenvalues and eigenvectors of the periodically kicked top

a brownian motion model for the parameter dependence of matrix elements we introduce a brownian motion model for the parametric evolution of eigenstates of a complex quantum system modelled by a random matrix the model is analogous to dyson s model for the evolution of the eigenvalues we use this approach to analyse correlation functions describing the parameter dependence of diagonal and off diagonal matrix elements of a generic operator in the case of diagonal matrix elements we compare our results with a semiclassical approach which relates sums of matrix elements to periodic classical orbits for systems with a chaotic classical limit the semiclassical correlation function agrees exactly with the random matrix theory

positive solutions of nonlinear problems involving the square root of the laplacian abstract we consider nonlinear elliptic problems involving a nonlocal operator the square root of the laplacian in a bounded domain with zero dirichlet boundary conditions for positive solutions to problems with power nonlinearities we establish existence and regularity results as well as a priori estimates of gidas spruck type in addition among other results we prove a symmetry theorem of gidas ni nirenberg type

some exact correlations in the dyson brownian motion model for transitions to the cue abstract the dyson brownian motion model for transitions to the cue is considered for initial eigenvalue probability density functions corresponding to the coe and cse the density density correlation function between an eigenvalue at position initially 0 and an eigenvalue at position for general is calculated theoretical predictions for the asymptotic behaviour are verified and the distribution for the initial rate of change of the eigenvalues is calculated also for initial conditions corresponding to equispaced eigenvalues the n point equal parameter distribution is computed the extension of the calculation of these quantities to a multiparameter version of the theory is given

a method of integration over matrix variables the integral over twon n hermitan matricesz g c dadbexp tr a 2 b 2 2cab g n a 4 b 4 is evaluated in the limit of largen for this purpose use is made of the theory of diffusion equation and that of orthogonal polynomials with a non local weight the above integral arises in the study of the planar approximation to quantum field theory

determinantal process starting from an orthogonal symmetry is a pfaffian process when the number of particles n is finite the noncolliding brownian motion bm and the noncolliding squared bessel process with index 1 besq are determinantal processes for arbitrary fixed initial configurations in the present paper we prove that if initial configurations are distributed with orthogonal symmetry they are pfaffian processes in the sense that any multitime correlation functions are expressed by pfaffians the 2 2 skew symmetric matrix valued correlation kernels of the pfaffians processes are explicitly obtained by the equivalence between the noncolliding bm and an appropriate dilatation of a time reversal of the temporally inhomogeneous version of noncolliding bm with finite duration in which all particles start from the origin n 0 and by the equivalence between the noncolliding besq and that of the noncolliding squared generalized meander starting from n 0

adjoints of rationally induced composition operators abstract we give an elementary proof of a formula recently obtained by hammond moorhouse and robbins for the adjoint of a rationally induced composition operator on the hardy space h 2 christopher hammond jennifer moorhouse marian e robbins adjoints of composition operators with rational symbol j math anal appl 341 2008 626 639 we discuss some variants and implications of this formula and use it to provide a sufficient condition for a rationally induced composition operator adjoint to be a compact perturbation of a weighted composition operator

the crossover between orthogonal and unitary symmetry in small disordered systems a supersymmetry approach the authors consider the spectral statistics of independent electrons moving at zero temperature in a weakly disordered metallic ring threaded by a magnetic flux the analysis is based on the supersymmetry method involving both commuting and anticommuting variables besides they consider an ensemble of gaussian distributed symmetric random matrices gaussian orthogonal ensemble which are perturbed by a small time reversal symmetry breaking contribution for energies smaller than the inverse diffusion time around the ring ec the spectral correlation functions of both models can be represented in terms of supermatrix integrals of identical structure in conformity with recent numerical results this implies that the spectral properties of the two models coincide these matrix integrals are to a large extent universal i e they depend only on two physical parameters the mean level spacing and a symmetry breaking parameter which is identified as the typical sensitivity of levels to the time reversal symmetry breaking perturbation the authors parametrize the relevant matrix coset space of the nonlinear sigma model in a novel way which is particularly convenient for treating models in the crossover between the two symmetry classes as an example they present a detailed calculation of the level level correlation function the basic formalism however applies quite generally and can be used for the investigation of different types of correlation functions and system geometries as well

norms of composition operators with rational symbol we compute the norms of composition operators with rational symbols that satisfy certain properties extending christopher hammond s methods on operators with linear fractional symbols this leads to a host of new examples of composition operators whose norms are calculable

gaussian point processes and two by two random matrix theory the statistics of the multidimensional gaussian point process are discussed in connection with the spacing statistics of eigenvalues of 2x2 random matrices we consider the three dimensional gaussian point process when two of the coordinates of a point are randomly chosen from a gaussian distribution having a mean of zero and a variance of sigma 2 1 but the third coordinate is chosen from a gaussian distribution having a variance in the range of 0 or sigma 3 2 or 1 the probability density function associated with a random point being at a distance r from the origin is shown to be closely related to the nearest neighbor spacing distribution of eigenvalues coming from an ensemble of 2x2 matrices defined by the french kota pandey mehta two matrix model of random matrix theory an elementary explanation of this result is given

gaussian ensembles of random hermitian matrices intermediate between orthogonal and unitary ones a gaussian ensemble of hermitian matrices depending on a parameter is considered when 0 the ensemble is gaussian orthogonal and when 1 it is gaussian unitary an analytic expression for then level correlation and cluster functions is given for anyn and 0 1 this ensemble is of relevance in the study of time reversal symmetry breaking of nuclear interactions

norms and spectral radii of composition operators acting on the dirichlet space we obtain new upper bounds on the norms of univalently induced composition operators acting on the dirichlet space and compute explicitly the norms for univalent symbols whose range is the disk minus a set of measure zero as an application we show that the spectral radius of every univalently induced composition operator on the dirichlet space is equal to one

brownian motion ensembles correlation functions of determinantal processes we calculate the n point correlation function for a large class of brownian motion ensembles of random matrix theory the corresponding fokker planck equation describes a determinantal process in the theory of matrix valued stochastic differential equations and can be mapped onto a schodinger equation of non interacting electrons in one dimension the correlation functions are obtained explicitly via a suitable generalization of the method of biorthogonal functions

vicious walkers friendly walkers and young tableaux iii between two walls we derive exact and asymptotic results for the number of star and watermelon configurations of vicious walkers confined to lie between two impenetrable walls as well as corresponding results for the analogous problem of friendly walkers our proofs make use of results from symmetric function theory and the theory of basic hypergeometric series

parametric spectral statistics in unitary random matrix ensembles from distribution functions to intra level correlations we establish a general framework to explore parametric statistics of individual energy levels in unitary random matrix ensembles for a generic confinement potential tr w h we i find the joint distribution functions of the eigenvalues of h and h h v for an arbitrary fixed v both for finite matrix size n and in the thermodynamic n limit ii derive many point parametric correlation functions of the two sets of eigenvalues and show that they are naturally parametrized by the eigenvalues of the reactance matrix for scattering off the potential v iii prove the universality of the correlation functions in unitary ensembles with non gaussian non invariant confinement potential tr w h v iv establish a general scheme for exact calculation of level number dependent parametric correlation functions and apply the scheme to the calculation of intra level velocity autocorrelation function and the distribution of parametric level shifts

fourier bessel analysis for ordinary and graded 2 2 hermitian matrices chainlike integrals in matrix spaces play an important part in high energy and solid state physics and in general random matrix theory in the special case of ordinary and graded 2 2 hermitian matrices a method is proposed to integrate out all angular variables the essence of this method is a fourier bessel analysis in these matrix spaces which is formulated in this paper close formal similarities are found between the ordinary and the graded case the main differences arise from the fact that the ordinary case can be reduced to the study of a vector space whereas no analogous feature is present in the graded case

spectral correlations in the crossover between gue and poisson regularity on the identification of scales motivated by questions of present interest in nuclear and condensed matter physics we consider the superposition of a diagonal matrix with independent random entries and a gue the relative strength of the two contributions is determined by a parameter suitably defined on the unfolded scale using results for the spectral two point correlator of this model obtained in the framework of the supersymmetry method we focus attention on two different regimes for 1 the correlations are given by dawson s integral while for 1 we derive a novel analytical formula for the two point function in both cases the energy scales in units of the mean level spacing at which deviations from pure gue behavior become noticeable can be identified we also derive an exact expansion of the local level density for a finite level number