on the stability of the functional equation f x y xy f x f y xf y yf x abstract in this paper we study the hyers ulam stability and the superstability of the functional equation f x y xy f x f y xf y yf x

character expansion method for supergroups and extended superversions of the leutwyler smilga and berezin karpelevich integrals we introduce an extension of the character expansion method to the case of supergroups this method allows us to calculate a superversion of the leutwyler smilga integral which to the best of our knowledge has not been calculated before we also use the method to generalize a previously calculated superversion of the berezin karpelevich integral our character expansion method should also allow for the calculation of other supergroup integrals

on some integrals over the u n unitary group and their large n limit the integral over the u n unitary group i du exp tr au bu is re examined various approaches and extensions are first reviewed the second half of the paper deals with more recent developments relation with integrable toda lattice hierarchy diagrammatic expansion and combinatorics and what they teach us on the large n limit of log i

angular gelfand tzetlin coordinates for the supergroup uosp k 1 2k 2 we construct gelfand tzetlin coordinates for the unitary orthosymplectic supergroup uosp k1 2k2 this extends a previous construction for the unitary supergroup u k1 k2 we focus on the angular gelfand tzetlin coordinates i e our coordinates stay in the space of the supergroup we also present a generalized gelfand pattern for the supergroup uosp k1 2k2 and discuss various implications for representation theory

classical and quantum superintegrability with applications a superintegrable system is roughly speaking a system that allows more integrals of motion than degrees of freedom this review is devoted to finite dimensional classical and quantum superintegrable systems with scalar potentials and integrals of motion that are polynomials in the momenta we present a classification of second order superintegrable systems in two dimensional riemannian and pseudo riemannian spaces it is based on the study of the quadratic algebras of the integrals of motion and on the equivalence of different systems under coupling constant metamorphosis the determining equations for the existence of integrals of motion of arbitrary order in real euclidean space e2 are presented and partially solved for the case of third order integrals a systematic exposition is given of systems in two and higher dimensional space that allow integrals of arbitrary order the algebras of integrals of motions are not necessarily quadratic but close polynomially or rationally the relation between superintegrability and the classification of orthogonal polynomials is analyzed

on the superdeterminant function for supermatrices a rigorous proof is given of the multiplicative property of the superdeterminant berezinian of a supermatrix the proof obviates the need to consider the domain of existence of the logarithm function for supermatrices and devolves on the identity det 1 pq det 1 qp 1 where p and q are compatible rectangular matrices over the odd part of a grassmann algebra

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

gelfand tzetlin coordinates for the unitary supergroup the gelfand tzetlin method provides explicit coordinates on the parameter space of the unitary groupu k which make direct evaluations of group integrals possible it is closely related to the gelfand construction of finite dimensional irreducible representations we generalize the gelfand tzetlin method to the unitary supergroupu k1 k2 the coordinates on the parameter space for supergroup integrals and the invariant haar measure are evaluated as an example the supersymmetric harish chandra itzykson zuber integral is calculated a generalized gelfand pattern containing anticommuting variables is introduced which determines the representation

stability of a functional equation deriving from cubic and quadratic functions abstract in this paper we establish the general solution of the functional equation 6 f x y 6 f x y 4 f 3 y 3 f x 2 y 3 f x 2 y 9 f 2 y and investigate the hyers ulam rassias stability of this equation

recursive construction for a class of radial functions ii superspace we extend the recursion formula for matrix bessel functions which we obtained previously to superspace it is sufficient to do this for the unitary orthosymplectic supergroup by direct computations we show that fairly explicit results can be obtained at least up to dimension 8 8 for the supermatrices since we introduce a new technique we discuss various of its aspects in some detail

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

orthogonality relations and supercharacter formulas of u m n representations in this paper we obtain the orthogonality relations for the supergroup u m n which are remarkably different from the ones for the u n case we extend our results for ordinary representations obtained some time ago to the case of complex conjugated and mixed representations our results are expressed in terms of the young tableaux notation for irreducible representations we use the supersymmetric harish chandra itzykson zuber integral and the character expansion technique as mathematical tools for deriving these relations as a byproduct we also obtain closed expressions for the supercharacters and dimensions of some particular irreducible u m n representations a new way of labeling the u m n irreducible representations in terms of m n numbers is proposed finally as a corollary of our results new identities among the dimensions of the irreducible representations of the unitary group u n are presented

on the hyers ulam stability of the functional equations that have the quadratic property abstract the hyers ulam stability of the quadratic functional equation 1 on a restricted domain shall be investigated and the result shall be applied to the study of an asymptotic behavior of that equation furthermore the hyers ulam stability problems of another quadratic equation 4 on a restricted domain shall also be treated under the approximately even or odd condition and some asymptotic behaviors of the quadratic mappings and the additive mappings shall be investigated

arbitrary rotation invariant random matrix ensembles and supersymmetry orthogonal and unitary symplectic case recently the supersymmetry method was extended from gaussian ensembles to arbitrary unitarily invariant matrix ensembles by generalizing the hubbard stratonovich transformation here we complete this extension by including arbitrary orthogonally and unitary symplectically invariant matrix ensembles the results are equivalent to but the approach is different from the superbosonization formula we express our results in a unifying way we also give explicit expressions for all one point functions and discuss features of the higher order correlations

supersymmetry for systems with unitary disorder circular ensembles a generalized hubbard stratonovitch transformation relating an integral over random unitary matrices to an integral over efetov s unitary model manifold is introduced this transformation adapts the supersymmetry method to disordered and chaotic systems that are modelled not by a hamiltonian but by their scattering matrix or time evolution operator in contrast to the standard method no saddle point approximation is made and no massive modes have to be eliminated this first paper on the subject applies the generalized hubbard stratonovitch transformation to dyson s circular unitary ensemble it is shown how a supersymmetric variant of the harish chandra itzykson zuber formula can be used to compute in the large n limit the n level correlation function for any n non trivial applications to random network models quantum chaotic maps and lattice gauge theory are expected

derivation of the supersymmetric harish chandra integral for uosp k 1 2k 2 the previous supersymmetric generalization of the unitary harish chandra integral prompted the conjecture that the harish chandra formula should have an extension to superspaces we prove this conjecture for the unitary orthosymplectic supermanifold uosp k1 2k2 to this end we construct and solve an eigenvalue equation

new correlation functions for random matrices and integrals over supergroups the averages of ratios of characteristic polynomials det x of n n random matrices x are investigated in the large n limit for the gue goe and gse ensembles the density of states and the two point correlation function are derived from these ratios the method relies on an extension of the harish chandra itzykson zuber integrals to the goe ensemble and to supergroups which are explicitly computed through heat kernel differential equations an external matrix source linearly coupled to the random matrices may also be added to the gaussian distribution and allows for a discussion of the universality of the goe results in the large n limit

on the stability of functional equations in banach spaces the paper is devoted to some results on the problem of s m ulam for the stability of functional equations in banach spaces the problem was posed by ulam 60 years ago

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

the dilute potts model on random surfaces we present a new solution of the asymmetric two matrix model in the large n limit which only involves a saddle point analysis the model can be interpreted as ising in the presence of a magnetic field on random dynamical lattices with the topology of the sphere resp the disk for closed resp open surfaces we elaborate on the resulting phase diagram the method can be equally well applied to a more general q 1 matrix model which represents the dilute potts model on random dynamical lattices we discuss in particular duality of boundary conditions for open random surfaces

integration of grassmann variables over invariant functions on flat superspaces we study integration over functions on superspaces these functions are invariant under a transformation which maps the whole superspace onto the part of the superspace which only comprises purely commuting variables we get a compact expression for the differential operator with respect to the commuting variables which results from berezin integration over all grassmann variables also we derive cauchy like integral theorems for invariant functions on supervectors and symmetric supermatrices this extends theorems partly derived by other authors as a physical application we calculate the generating function of the one point correlation function in random matrix theory furthermore we give another derivation of supermatrix bessel functions for u k1 k2

estimates of optimal storage conditions in neural network memories based on random matrix theory the authors formulate a method for estimating the critical conditions for storage of sets of data in neural network memory this variational method is based on random matrix theory and calculating the average spectrum of a matrix whose elements are given by overlaps of the stored patterns several generic cases of random overlap matrices are considered they investigate the cases of simply uncorrelated random patterns and spatially and semantically correlated ones they obtain bounds of the critical curve in the control parameters space which determine the stability of the stored data sets

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

general random matrix approach to account for the effect of static disorder on the spectral properties of light harvesting systems we develop a random matrix model approach to study static disorder in pigment protein complexes in photosynthetic organisms as a case study we examine the ring of b850 bacteriochlorophylls in the peripheral light harvesting complex of rhodospirillum molischianum formulated in terms of an effective hamiltonian describing the collective electronic excitations of the system we numerically examine and compare various models of disorder and observe that both the density of states and the absorption spectrum of the model show remarkable spectral universality for the case of unitary disorder we develop a method to analytically evaluate the density of states of the ensemble using the supersymmetric formulation of random matrix theory succinct formulas that can be readily applied in future studies are provided in an appendix

asymptotics of skew orthogonal polynomials exact integral expressions of the skew orthogonal polynomials involved in orthogonal 1 and symplectic 4 random matrix ensembles are obtained the even rank skew orthogonal polynomials are average characteristic polynomials of random matrices from there asymptotics of the skew orthogonal polynomials are derived

duality biorthogonal polynomials and multi matrix models the statistical distribution of eigenvalues of pairs of coupled random matrices can be expressed in terms of integral kernels having a generalized christoffel darboux form constructed from sequences of biorthogonal polynomials for measures involving exponentials of a pair of polynomials v1 v2 in two different variables these kernels may be expressed in terms of finite dimensional windows spanned by finite subsequences having length equal to the degree of one or the other of the polynomials v1 v2 the vectors formed by such subsequences satisfy dual pairs of first order systems of linear differential equations with polynomial coefficients having rank equal to one of the degrees of v1 or v2 and degree equal to the other they also satisfy recursion relations connecting the consecutive windows and deformation equations determining how they change under variations in the coefficients of the polynomials v1 and v2 viewed as overdetermined systems of linear difference differential deformation equations these are shown to be compatible and hence to admit simultaneous fundamental systems of solutions the main result is the demonstration of a spectral duality property namely that the spectral curves defined by the characteristic equations of the pair of matrices defining the dual differential systems are equal upon interchange of eigenvalue and polynomial parameters

a generalization of the hyers ulam rassias stability of approximately additive mappings abstract in this paper we prove a generalization of the stability of approximately additive mappings in the spirit of hyers ulam and rassias

asymptotic coincidence of the statistics for degenerate and non degenerate correlated real wishart ensembles the correlated wishart model provides the standard benchmark when analyzing time series of any kind unfortunately the real case which is the most relevant one in applications poses serious challenges for analytical calculations often these challenges are due to square root singularities which cannot be handled using common random matrix techniques we present a new way to tackle this issue using supersymmetry we carry out an anlaytical study which we support by numerical simulations for large but finite matrix dimensions we show that statistical properties of the fully correlated real wishart model generically approach those of a correlated real wishart model with doubled matrix dimensions and doubly degenerate empirical eigenvalues this holds for the local and global spectral statistics with monte carlo simulations we show that this is even approximately true for small matrix dimensions we explicitly investigate the k point correlation function as well as the distribution of the largest eigenvalue for which we find a surprisingly compact formula in the doubly degenerate case moreover we show that on the local scale the k point correlation function exhibits the sine and the airy kernel in the bulk and at the soft edges respectively we also address the positions and the fluctuations of the possible outliers in the data

characteristic functions and invariants of supermatrices the characteristic functions and the invariants of supermatrices are studied it is shown that the euclidean algorithm is useful in obtaining a system of invariants

on the stability of the orthogonal pexiderized cauchy equation abstract we investigate the stability of pexiderized mappings in banach modules over a unital banach algebra as a consequence we establish the hyers ulam stability of the orthogonal cauchy functional equation of pexider type f 1 x y f 2 x f 3 y x y in which is the orthogonality in the sense of ratz

a riemann hilbert problem for biorthogonal polynomials we characterize the biorthogonal polynomials that appear in the theory of coupled random matrices via a riemann hilbert problem our riemann hilbert problem is different from the ones that were proposed recently by ercolani and mclaughlin kapaev and bertola et al we believe that our formulation may be tractable to asymptotic analysis

large deviations asymptotics for spherical integrals consider the spherical integral i n dn en exp n tr udnu en dm n u where m n denote the haar measure on the orthogonal group on when 1 and on the unitary group un when 2 and dn en are diagonal real matrices whose spectral measures converge to d e in this paper we prove the existence and represent as solution to a variational problem the limit i d e lim n 2 log i n dn en this limit appears in so called matrix models but also in the evaluation of large deviations of the spectral measure of generalized wishart matrices our technique is based on stochastic calculus large deviations and elements from free probability

the integral theorem for supersymmetric invariants a supersymmetric integral theorem that extends results of parisi sourlas efetov wegner and others is rigorously proved in particular arbitrary generators are allowed in the integrand instead of canonical ones and the invariance condition is very much relaxed the connection with cauchy s integral formula is made transparent in passing the unitary lie supergroup is studied by using elementary methods applications in the theory of disordered systems are discussed

character expansions for the orthogonal and symplectic groups formulas for the expansion of arbitrary invariant group functions in terms of the characters for the sp 2n so 2n 1 and so 2n groups are derived using a combinatorial method the method is similar to one used by balantekin to expand group functions over the characters of the u n group all three expansions have been checked for all n by using them to calculate the known expansions of the generating function of the homogeneous symmetric functions an expansion of the exponential of the traces of group elements appearing in the finite volume gauge field partition functions is worked out for the orthogonal and symplectic groups

replica method for wide correlators in gaussian orthogonal unitary and symplectic random matrix ensembles we calculate connected correlators in gaussian orthogonal unitary and symplectic random matrix ensembles by the replica method in the 1 n expansion we obtain averaged one point green s functions up to the next to leading order wide two level correlators up to the first non trivial order and wide three level correlators up to the first non trivial order by carefully treating fluctuations in saddle point evaluation

arbitrary unitarily invariant random matrix ensembles and supersymmetry we generalize the supersymmetry method in random matrix theory to ensembles which are unitarily invariant but otherwise arbitrary our exact approach extends a previous contribution in which we constructed a supersymmetric representation for the class of norm dependent random matrix ensembles here we derive a supersymmetric formulation under very general circumstances a reduced probability density and a projector are identified that map the probability density from ordinary to superspace furthermore it is demonstrated that setting up the theory in fourier superspace has considerable advantages general and exact expressions for the correlation functions are given we also show how the use of hyperbolic symmetry can be circumvented in the present context in which the nonlinear model is not used we construct exact supersymmetric integral representations of the correlation functions for arbitrary positions of the imaginary increments in the green s functions

a supersymmetry approach to billiards with randomly distributed scatterers ii correlations in a previous contribution stockmann h j 2002 j phys a math gen 35 5165 the density of states was calculated for a billiard with randomly distributed delta like scatterers doubly averaged over the positions of the impurities and the billiard shape this result is now extended to the k point correlation function using supersymmetric methods we show that the correlations in the bulk are always identical to those of the gaussian unitary ensemble gue of random matrices in passing from the band centre to the tail states the density of states is depleted considerably and the two point correlation function shows a gradual change from the gue behaviour to that found for completely uncorrelated eigenvalues this can be viewed as similar to a mobility edge

the itzykson zuber integral for u m n the itzykson zuber iz integral is computed for the superunitary group u m n as a consequence the nonzero correlations of superunitary matrices can be found

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

dyson s correlation functions and graded symmetry a new derivation of dyson s k level correlation functions of the gaussian unitary ensemble gue is given the method uses matrices with graded symmetry the number of integrations needed for the ensemble average becomes independent of the level number n for arbitrary level number n the k level correlation function is expressed as an integral involving the eigenvalues of a 2k 2k graded matrix the limit of infinitely many levels n is calculated by a simple saddle point approximation of this integral avoiding the introduction of hermite polynomials and oscillator wave functions

on the graded group u 1 1 a noncanonical parametrization for the graded group u 1 1 is introduced similar to the euler angles for the ordinary group su 2 two differential representations for the underlying algebra of u 1 1 are constructed on the full and the restricted i e coset parameter space respectively a space of functions living on the latter is found exhibiting close formal similarities to a hilbert space remarkably the indices of those functions and thus the orthogonality and completeness relations involve anticommuting variables using this hilbert space a representation of u 1 1 is evaluated which shows analogies to the wigner functions for su 2

the k point random matrix kernels obtained from one point supermatrix models the k point correlation functions of the gaussian random matrix ensembles are certain determinants of functions which depend on only two arguments they are referred to as kernels since they are the building blocks of all correlations we show that the kernels are obtained for arbitrary level number directly from supermatrix models for one point functions more precisely the generating functions of the one point functions are equivalent to the kernels this is surprising because it implies that already the one point generating function holds essential information about the k point correlations this also establishes a link to the averaged ratios of spectral determinants i e of characteristic polynomials

stability for the functional equation of cubic type abstract in this article we establish the stability of the orthogonally cubic type functional equation 1 2 for all x 1 x 2 x 3 with x i x j i j 1 2 3 where is the orthogonality in the sense of ratz and investigate the stability of the n dimensional cubic type functional equation 1 3 where n 3 is an integer

norm dependent random matrix ensembles in external field and supersymmetry the class of norm dependent random matrix ensembles is studied in the presence of an external field the probability density of the ensemble depends on the trace of the squared random matrices but is otherwise arbitrary an exact mapping to superspace is performed a transformation formula is derived which gives the probability density in superspace as a single integral over the probability density in ordinary space this is done for orthogonal unitary and symplectic symmetries in the case of unitary symmetry some explicit results for the correlation functions are derived

the calogero moser equation system and the ensemble average in the gaussian ensembles from random matrix theory it is known that for special values of the coupling constant the calogero moser cm equation system is nothing but the radial part of a generalized harmonic oscillator schrodinger equation this allows an immediate construction of the solutions by means of a rodriguez relation the results are easily generalized to arbitrary values of the coupling constant by this the cm equations become nearly trivial as an application an expansion for eitr xy in terms of eigenfunctions of the cm equation system is obtained where x and y are matrices taken from one of the gaussian ensembles and the brackets denote an average over the angular variables

the stability of a cubic type functional equation with the fixed point alternative abstract in this note we investigate the generalized hyers ulam rassias stability for the new cubic type functional equation f x y 2 z f x y 2 z f 2 x f 2 y 2 f x y 2 f x z 2 f x z 2 f y z 2 f y z by using the fixed point alternative the first systematic study of fixed point theorems in nonlinear analysis is due to g isac and th m rassias internat j math math sci 19 1996 219 228

wigner dyson statistics from the replica method we compute the correlation functions of the eigenvalues in the gaussian unitary ensemble using the fermionic replica method we show that non trivial saddle points which break replica symmetry must be included in the calculation in order to correctly reproduce the exact asymptotic behaviour of the correlation functions at a large distance

a riemann hilbert problem for skew orthogonal polynomials we find a local d 1 x d 1 riemann hilbert problem characterizing the skew orthogonal polynomials associated to the partition function of orthogonal ensembles of random matrices with a potential function of degree d our riemann hilbert problem is similar to a local dxd riemann hilbert problem found by kuijlaars and mclaughlin characterizing the bi orthogonal polynomials this gives more motivation for finding methods to compute asymptotics of high order riemann hilbert problems and brings us closer to finding full asymptotic expansions of the skew orthogonal polynomials

the generalized hyers ulam rassias stability of a cubic functional equation abstract in this paper we obtain the general solution and the generalized hyers ulam stability for a cubic functional equation f 2x y f 2x y 2f x y 2f x y 12f x

a method of integration over matrix variables ii the integral over twon n hermitan matricesz g c dadbexp tr a2 b2 2cab g n a4 b4 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

on the stability of an n dimensional cubic functional equation abstract let n 2 be an integer number in this paper we investigate the generalized hyers ulam rassias stability in banach spaces and also banach modules over a banach algebra and a c algebra and the stability using the alternative fixed point of an n dimensional cubic functional equation in banach spaces f j 1 n 1 x j 2 x n f j 1 n 1 x j 2 x n j 1 n 1 f 2 x j 2 f j 1 n 1 x j 4 j 1 n 1 f x j x n f x j x n

complex curve of the two matrix model and its tau function we study the hermitian and normal two matrix models in planar approximation for an arbitrary number of eigenvalue supports its planar graph interpretation is given the study reveals a general structure of the underlying analytic complex curve different from the hyperelliptic curve of the one matrix model the matrix model quantities are expressed through the periods of meromorphic generating differential on this curve and the partition function of the multiple support solution as a function of filling numbers and coefficients of the matrix potential is shown to be a quasiclassical tau function the relation to 1 supersymmetric yang mills theories is discussed a general class of solvable multi matrix models with tree like interactions is considered

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