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

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

on representations of super coalgebras the general structure of the representation theory of a z2 graded coalgebra is discussed the result contains the structure of fourier analysis on compact supergroups and quantizations thereof as a special case the general linear supergroups serve as an explicit illustration and the simplest example is carried out in detail

on representing finite lattices as intervals in subgroup lattices of finite groups let mnbe the lattice of length 2 withn 1 atoms it is an open problem to decide whether or not every such lattice or indeed whether or not every finite lattice can be represented as an interval in the subgroup lattice of some finite group we complete the work of the second author lucchini by reducing this problem to a series of questions concerning the finite non abelian simple groups

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

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

on intervals in subgroup lattices of finite groups we investigate the question of which finite lattices l are isomorphic to the lattice h g of all overgroups of a subgroup h in a finite group g we show that the structure of g is highly restricted if h g is disconnected we define the notion of a signalizer lattice in h and show for suitable disconnected lattices l if h g is minimal subject to being isomorphic to l or its dual then either g is almost simple or h admits a signalizer lattice isomorphic to l or its dual we use this theory to answer a question in functional analysis raised by watatani

overgroup lattices in finite groups of lie type containing a parabolic the main theorem is a step in a program to show there exist finite lattices that are not an interval in the lattice of subgroups of any finite group as part of the proof of the main theorem we prove a theorem on the structure of maximal parabolics in finite groups of lie type which is of independent interest

variational methods for non differentiable functionals and their applications to partial differential equations the interest in extending the theory of nonlinear partial differential equations pde to pde with discontinuous nonlinearities dnde has been increasing daily the reasons are twofold 1 many free boundary problems may be reduced to boundary value problems of dnde cf chang 16 71 chang and jiang 9 and fleishman and mahar 121 2 sometimes in dealing with a pde problem it is beneficial to put the original pde into a large category for instance dnde cf chang 7 simply speaking a dnde is as follows

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

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

three solutions for a dirichlet problem with one sided growth conditions on the nonlinearities a b s t r a c t we consider a semilinear elliptic partial differential equation depending on two positive parameters and coupled with homogeneous dirichlet boundary conditions assuming only one sided growth conditions on the nonlinearities involved we prove the existence of at least three weak solutions for and lying in convenient intervals we employ techniques of nonsmooth analysis introduced by degiovanni and zani and a theorem of ricceri for multiplicity of local minimizers

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

lattices of intermediate subfactors letnbe an irreducible subfactor of a typeii1factorm if the jones index m n is finite then the set lat n m of the intermediate subfactors for the inclusionn mforms afinitelattice the commuting and co commuting square conditions for intermediate subfactors are related to the modular identity in the lattice lat n n in particular simplicity of a finite groupgis characterized in terms of commuting square conditions of intermediate subfactors forn m n g we investigate the question of which finite lattices can be realized as intermediate subfactor lattices

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

quantum subgroups of the haagerup fusion categories we answer three related questions concerning the haagerup subfactor and its even parts the haagerup fusion categories namely we find all simple module categories over each of the haagerup fusion categories in other words we find the quantum subgroups in the sense of ocneanu we find all irreducible subfactors whose principal even part is one of the haagerup fusion categories and we compute the brauer picard groupoid of morita equivalences of the haagerup fusion categories in addition to the two even parts of the haagerup subfactor there is exactly one more fusion category which is morita equivalent to each of them this third fusion category has six simple objects and the same fusion rules as one of the even parts of the haagerup subfactor but has not previously appeared in the literature we also find the full lattice of intermediate subfactors for every irreducible subfactor whose even part is one of these three fusion categories and we discuss how our results generalize to izumi subfactors

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

subdifferential calculus and nonsmooth critical point theory a general critical point theory for continuous functions defined on metric spaces has been recently developed in this paper a new subdifferential related to that theory is introduced in particular results on the subdifferential of a sum are proved an example of application to pdes is sketched detailed applications to pdes are developed in separate papers

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

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

overgroups of primitive groups ii we continue our study of the overgroup lattices of subgroups of finite alternating and symmetric groups with applications to the question of palfy and pudlak as to whether each finite lattice is an interval in the lattice of subgroups of some finite group

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

restrictions on the structure of subgroup lattices of finite alternating and symmetric groups let g be a finite alternating or symmetric group we describe an infinite class of finite lattices none of which is isomorphic to any interval h g in the subgroup lattice of g

on a subfactor generalization of wallʼs conjecture abstract in this paper we discuss a conjecture on intermediate subfactors which is a generalization of wall s conjecture from the theory of finite groups we explore special cases of this conjecture and present supporting evidence in particular we prove special cases of this conjecture related to some finite dimensional kac algebras of izumi kosaki type which include relative version of wall s conjecture for solvable groups

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