spectral asymptotics for the third order operator with periodic coefficients abstract we consider the self adjoint third order operator with 1 periodic coefficients on the real line the spectrum of the operator is absolutely continuous and covers the real line we determine the high energy asymptotics of the periodic antiperiodic eigenvalues and of the branch points of the lyapunov function furthermore in the case of small coefficients we show that either whole spectrum has multiplicity one or the spectrum has multiplicity one except for a small spectral nonempty interval with multiplicity three in the last case the asymptotics of this small interval is determined

geometric quantization of the brst charge in the first half of this paper sects 1 4 we generalise the standard geometric quantization procedure to symplectic supermanifolds in the second half sects 5 6 we apply this to two examples that exhibit classical brst symmetry i e we quantize the brst charge and the ghost number more precisely in the first example we consider the reduced symplectic manifold obtained by symplectic reduction from a free group action with ad equivariant moment map in the second example we consider a foliated configuration space whose cotangent bundle admits the construction of a brst charge associated to this foliation we show that the classical brst symmetry can be described in terms of a hamiltonian supergroup action on the extended phase space and that geometric quantization gives us a super unitary representation of this supergroup finally we point out how these results are related to reduction at the quantum level as compared with the reduction at the classical level

susy vertex algebras and supercurves given a strongly conformal susy vertex algebra v and a supercurve x we construct a vector bundle fancyscript v r x on x the fiber of which is isomorphic to v moreover the state field correspondence of v canonically gives rise to local sections of these vector bundles we also define chiral algebras on any supercurve x and show that the vector bundle fancyscript v r x corresponding to a susy vertex algebra carries the structure of a chiral algebra

trace formulae for the matrix schrodinger equation with energy dependent potential abstract in this paper we consider the eigenvalue problems for the matrix schrodinger equation with energy dependent potential and with separated boundary conditions on the finite interval and find new trace formulae for the matrix schrodinger operator

riemannian symmetric superspaces and their origin in random matrix theory gaussian random matrix ensembles defined over the tangent spaces of the large families of cartan s symmetric spaces are considered such ensembles play a central role in mesoscopic physics as they describe the universal ergodic limit of disordered and chaotic single particle systems the generating function for the spectral correlations of each ensemble is reduced to an integral over a riemannian symmetric superspace in the limit of large matrix dimension such a space is defined as a pair g h m r where g h is a complex analytic graded manifold homogeneous with respect to the action of a complex lie supergroup g and m r is a maximal riemannian submanifold of the support of g h

on discontinuous galerkin methods discontinuous galerkin methods have received considerable attention in recent years for problems in which advection and diffusion terms are present several alternatives for treating the diffusion and advective fluxes have been introduced this report summarizes some of the methods that have been proposed several numerical examples are included in the paper these present discontinuous galerkin solutions of one dimensional problems with a scalar variable results are presented for diffusion reaction problems and advection diffusion problems we discuss the performance of various formulations with respect to accuracy as well as stability of the method copyright 2003 john wiley sons ltd

toward a theory of the integer quantum hall transition continuum limit of the chalker coddington model an n channel generalization of the network model of chalker and coddington is considered the model for n 1 is known to describe the critical behavior at the plateau transition in systems exhibiting the integer quantum hall effect using a recently discovered equality of integrals the network model is transformed into a lattice field theory defined over efetov s model space with unitary symmetry the transformation is exact for all n no saddle point approximation is made and no massive modes have to be eliminated the naive continuum limit of the lattice theory is shown to be a supersymmetric version of pruisken s nonlinear model with couplings xx n 4 and xy n 2 at the symmetric point it follows that the model for n 2 which describes a spin degenerate landau level and the random flux problem is noncritical on the basis of symmetry considerations and inspection of the hamiltonian limit a modified network model is formulated which still lies in the quantum hall universality class the prospec

fourier analysis on a hyperbolic supermanifold with constant curvature the fourier inversion theorem is proved for a rank one noncompact homogeneous space the hyperbolic superplane the proof makes use of some novel features of perfectly graded superspaces which are not encountered in classical geometric analysis an application to quasi one dimensional disordered one electron systems is given

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

eigenvalue and eigenfunction asymptotics for regular sturm liouville problems abstract in this paper we obtain asymptotic formulas for eigenvalues eigenfunctions and the reciprocals of the eigenfunction norms for eigenvalue problems associated with the general sturm liouville equation pu r q u 0 having regular endpoints the method is based on a conversion to liouville normal form and an iterative procedure of solving the associated volterra integral equation producing an asymptotic expansion of the solution in higher powers of 1 1 2 as

borg type theorems for matrix valued schrodinger operators abstract a borg type uniqueness theorem for matrix valued schrodinger operators is proved more precisely assuming a reflectionless potential matrix and its spectrum a half line 0 we derive the triviality of the potential matrix our approach is based on trace formulas and matrix valued herglotz representation theorems as a by product of our techniques we obtain an extension of borg s classical result from the class of periodic scalar potentials to the class of reflectionless matrix valued potentials

on the determinant of elliptic differential and finite difference operators in vector bundles over s 1 for an elliptic differential operatora overs 1 a sum limits k 0 n a k x d k witha k x in end r and as a principal angle the regularized determinant det a is computed in terms of the monodromy mapp a associated toa and some invariant expressed in terms ofa n anda n 1 a similar formula holds for finite difference operators a number of applications and implications are given in particular we present a formula for the signature ofa whena is self adjoint and show that the determinant ofa is the limit of a sequence of computable expressions involving determinants of difference approximation ofa

a triangulated vortex method for the axisymmetric euler equations a new method is presented for the prediction of unsteady axisymmetric inviscid flows by combining a triangulated vortex approach with a novel evaluation technique for the biot savart integrals a lagrangian vortex method is developed which eliminates the singularities usually present in axisymmetric methods without recourse to normalizations or other approximations furthermore the computational effort scales as the number of control points n and in the large n limit depends only on the order of quadrature employed the accuracy and computational effort are assessed by comparison with the velocity field of a gaussian core vortex ring and the use of the technique is illustrated by computation of the motion of norbury rings and of vortex ring pairing

a stochastic particle mesh scheme for uncertainty propagation in vortical flows abstract a new mesh particle scheme is constructed for uncertainty propagation in vortical flow the scheme is based on the incorporation of polynomial chaos pc expansions into a lagrangian particle approximation of the navier stokes equations the main idea of the method is to use a unique set of particles to transport the stochastic modes of the solution the particles are transported by the mean velocity field while their stochastic strengths are updated to account for diffusive and convective effects induced by the coupling between stochastic modes an integral treatment is used for the evaluation of the coupled stochastic terms following the framework of the particle strength exchange pse methods which yields a conservative algorithm it is also shown that it is possible to apply solution algorithms used in deterministic setting including particle mesh techniques and particle remeshing thus the method combines the advantages of particles discretizations with the efficiency of pc representations validation of the method on uncertain diffusion and convection problems is first performed an example is then presented of natural convection of a hot patch of fluid in infinite domain and the computations are used to illustrate the effectiveness of the approach for both large number of particles and high order pc expansions

line bundles on super riemann surfaces we give the elements of a theory of line bundles their classification and their connections on super riemann surfaces there are several salient departures from the classical case for example the dimension of the picard group is not constant and there is no natural hermitian form on pic furthermore the bundles with vanishing chern number aren t necessarily flat nor can every such bundle be represented by an antiholomorphic connection on the trivial bundle nevertheless the latter representation is still useful in investigating questions of holomorphic factorization we also define a subclass of all connections those which are compatible with the superconformal structure the compatibility conditions turn out to be constraints on the curvature 2 form

an inverse problem for the matrix schrodinger equation abstract by modifying and generalizing some old techniques of n levinson a uniqueness theorem is established for an inverse problem related to periodic and sturm liouville boundary value problems for the matrix schrodinger equation

the xi function despite the fact that spectral and inverse spectral properties of one dimensional schrsdinger operators h d2 dx2 have been extensively studied for seventy five years there remain large areas where our knowledge is limited for example while the inverse theory for operators on l2 co co is well understood in case v is periodic 12 24 25 35 39 42 49 it is not understood in case limjx o v x co and h has discrete spectrum our goal here is to introduce a special function x a on r associated to h which we believe will be a valuable tool in the spectral and inverse spectral theory in a sense we will make precise it complements the weyl m functions m x a a main application of which we will make here concerns a generalization of the trace formula for schrsdinger operators to general v s recall the well known trace formula for periodic potentials let v x v x t 1 then by floquet theory see e g 10 37 44

consistent superspace integration the standard berezin method for integration over odd variables is combined in a new way with de witt s contour method for integration over even grassmann variables to give a new method of superspace integration it is shown that this integral unlike the standard superspace integral is invariant under coordinate transformations in superspace the relation between the new method and the standard method is discussed

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

a note on the global structure of supermoduli spaces we recall some deformation theory of susy curves and study obstructions to projectedness of supermoduli spaces both from a general standpoint and by means of the local coordinate charts most commonly used in the physical literature we prove that these give rise to a projected atlas for genusg 2 only

holomorphic supercurves and supersymmetric sigma models we introduce a natural generalisation of holomorphic curves to morphisms of supermanifolds referred to as holomorphic supercurves more precisely supercurves are morphisms from a riemann surface endowed with the structure of a supermanifold which is induced by a holomorphic line bundle to an ordinary almost complex manifold they are called holomorphic if a generalised cauchy riemann condition is satisfied we show by means of an action identity that holomorphic supercurves are special extrema of a supersymmetric action functional

characters of topological n 2 vertex algebras are jacobi forms on the moduli space of elliptic supercurves abstract we show that trace functions on modules of topological n 2 super vertex algebras give rise to conformal blocks on elliptic supercurves we show that they satisfy a system of linear partial differential equations with respect to the modular parameters of the supercurves under some finiteness condition on the vertex algebra these differential equations can be interpreted as a connection on the vector bundle of conformal blocks we show that this connection is equivariant with respect to a natural action of the jacobi modular group on the modular parameters and the trace functions in the appendix we prove the convergence of the trace functions

the geometry of super riemann surfaces we define super riemann surfaces as smooth 2 2 dimensional supermanifolds equipped with a reduction of their structure group to the group of invertible upper triangular 2 2 complex matrices the integrability conditions for such a reduction turn out to be most of the torsion constraints of 2d supergravity we show that they are both necessary and sufficient for a frame to admit local superconformal coordinates the other torsion constraints are merely conditions to fix some of the gauge freedom in this description or to specify a particular connection on such a manifold analogous to the levi civita connection in riemannian geometry unlike ordinary riemann surfaces a super riemann surface cannot be regarded as having only one complex dimension nevertheless in certain important aspects super riemann surfaces behave as nicely as if they had only one dimension in particular they posses an analog widehat partial of the cauchy riemann operator on ordinary riemann surfaces a differential operator taking values in the bundle of half volume forms this operator furnishes a short resolution of the structure sheaf making possible a quillen theory of determinant line bundles finally we show that the moduli space of super riemann surfaces is embedded in the larger space of complex curves of dimension 1 1

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

numerical simulation of axisymmetric viscous flows by means of a particle method a vortex particle method for the simulation of axisymmetric viscous flow is presented the flow is assumed to be laminar and incompressible the navier stokes equations are expressed in an integral velocity vorticity formulation the inviscid scheme is based on nitsche s method for axisymmetric vortex sheets meanwhile two techniques are proposed for dealing with the viscous term the first uses an integral green s function method while the second is based on a diffusion velocity approach both are obtained by extension of existing methods for 2d flows the problem of satisfying boundary conditions along the axis of symmetry is specifically addressed the problem is solved by using cut off functions that are derived from the green s function of the axisymmetric diffusion equation the scheme is applied to simulate the evolution of vortex rings at intermediate reynolds number the processes of entrainment and wake formation are evident in the calculations as well as the extension of the support of vorticity due to viscous diffusion

a trace formula for multidimensional schrodinger operators we prove multidimensional analogs of the trace formula obtained previously for one dimensional schrodinger operators for example letvbe a continuous function on 0 1 r fora 1 let abe the laplace operator on 0 1 with mixed dirichlet neumann boundary conditions formula let a number of points ina then we ll prove that formula with v the average ofvat the 2 corners of 0 1

the periodic euler bernoulli equation we continue the study of the floquet spectral theory of the beam equation namely the fourth order eigenvalue problem a x u x x u x x where the functions a and p are periodic and strictly positive this equation models the transverse vibrations of a thin straight periodic beam whose physical characteristics are described by a and p here we develop a theory analogous to the theory of the hill operator d dx 2 q x we first review some facts and notions from our previous works including the concept of the pseudospectrum or spectrum our new analysis begins with a detailed study of the zeros of the function f a k for any given quasimomentum k c where f k 0 is the floquet bloch variety of the beam equation the hill quantity corresponding to f k is 2 cos kb where a a is the discriminant and b the period of q we show that the multiplicity m of any zero of f a k can be one or two and m 2 for some k if and only if is also a zero of another entire function d a independent of k furthermore we show that d a has exactly one zero in each gap of the spectrum and two zeros counting multiplicities in each gap if is a double zero of f a k it may happen that there is only one floquet solution with quasimomentum k thus there are exceptional cases where the algebraic and geometric multiplicities do not agree next we show that if is an open gap of the pseudospectrum i e then the floquet matrix t a has a specific jordan anomaly at a a and a we then introduce a multipoint dirichlet type eigenvalue problem which is the analogue of the dirichlet problem for the hill equation we denote by n n z the eigenvalues of this multipoint problem and show that n n z is also characterized as the set of values of a for which there is a proper floquet solution f x a such that f 0 a 0 we also show theorem 7 that each gap of the l 2 r spectrum contains exactly one n and each gap of the pseudospectrum contains exactly two n s counting multiplicities here when we say gap or gap we also include the endpoints so that when two consecutive bands or bands touch the in between collapsed gap or gap is a point we believe that n n z can be used to formulate the associated inverse spectral problem as an application of theorem 7 we show that if is a collapsed closed gap then the floquet matrix t is diagonalizable some of the above results were conjectured in our previous works however our conjecture that if all the gaps are closed then the beam operator is the square of a second order hill type operator is still open

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 lyapunov function for schrodinger operators with a periodic 2 2 matrix potential abstract we consider the schrodinger operator on the real line with a 2 2 matrix valued 1 periodic potential the spectrum of this operator is absolutely continuous and consists of intervals separated by gaps we define a lyapunov function which is analytic on a two sheeted riemann surface on each sheet the lyapunov function has the same properties as in the scalar case but it has branch points which we call resonances we prove the existence of real as well as non real resonances for specific potentials we determine the asymptotics of the periodic and the anti periodic spectrum and of the resonances at high energy we show that there exist two type of gaps 1 stable gaps where the endpoints are the periodic and the anti periodic eigenvalues 2 unstable resonance gaps where the endpoints are resonances i e real branch points of the lyapunov function we also show that periodic and anti periodic spectrum together determine the spectrum of the matrix hill operator

modules and morita theorem for operads associative rings a b are called morita equivalent when the categories of left modules over them are equivalent we call two classical linear operads p q morita equivalent if the categories of algebras over them are equivalent we transport a part of morita theory to the operadic context by studying modules over operads as an application of this philosophy we consider an operadic version of the sheaf of linear differential operators on a super manifold m and give a comparison theorem between algebras over this sheaf on m and m red

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

discontinuous galerkin and mimetic finite difference methods for coupled stokes darcy flows on polygonal and polyhedral grids we study locally mass conservative approximations of coupled darcy and stokes flows on polygonal and polyhedral meshes the discontinuous galerkin dg finite element method is used in the stokes region and the mimetic finite difference method is used in the darcy region dg finite element spaces are defined on polygonal and polyhedral grids by introducing lifting operators mapping mimetic degrees of freedom to functional spaces optimal convergence estimates for the numerical scheme are derived results from computational experiments supporting the theory are presented

arbitrary high order discontinuous galerkin schemes based on the grp method for compressible euler equations combined with the discontinuous galerkin dg framework the generalized riemann problem grp method is applied to design a grp dg scheme with high order accuracy for compressible euler equations since numerical fluxes with second order accuracy in time are derived by the grp method the reconstruction steps for physical variables in the new scheme are halved compared with the traditional runge kutta discontinuous galerkin rk dg scheme the numerical results are also improved due to more introduced physical information several numerical examples verify the validity of the proposed schemes

a reduction method for atypical representations of classical lie superalgebras abstract we introduce a reduction method for studying representations of classical lie superalgebras with atypical central character we show that the atypical quotient of universal enveloping algebra has a non trivial jacobson radical the factor by this radical has a new center which is calculated for sl 1 n and psl 2 2 using this center we obtain new character formulae generalization of borel weil bott and beilinson bernstein localization theorems

trace formulae for the schrodinger equation with energy dependent potential we consider the eigenvalue problems for the schrodinger equation with energy dependent potential subject to the eigenparameter boundary conditions and quasiperiodic boundary conditions respectively two new formulae of the regularized trace for this class of schrodinger equation with boundary conditions are obtained in an explicit form

eigenvalue estimates and trace formulas for the matrix hill s equation abstract estimates for floquet multipliers and periodic eigenvalues are developed for the matrix hill s equation basic estimates and a description of the periodic eigenvalues as roots of an entire function are adequate for establishing trace formulas with a residue computation refined estimates and power sum formulas are available for restricted pairs of hill s operators

conformal spectral theory for the monodromy matrix for any n x n monodromy matrix we define the lyapunov function which is analytic on an associated n sheeted riemann surface on each sheet the lyapunov function has the standard properties of the lyapunov function for the hill operator the lyapunov function has real or complex branch points which we call resonances we determine the asymptotics of the periodic anti periodic spectrum and of the resonances at high energy we show that the endpoints of each gap are periodic anti periodic eigenvalues or resonances real branch points moreover the following results are obtained 1 we define the quasimomentum as an analytic function on the riemann surface of the lyapunov function various properties and estimates of the quasimomentum are obtained 2 we construct the conformal mapping with imaginary part given by the lyapunov exponent and we obtain various properties of this conformal mapping which are similar to the case of the hill operator 3 we determine various new trace formulae for potentials and the lyapunov exponent 4 we obtain a priori estimates of gap lengths in terms of the dirichlet integral we apply these results to the schrodinger operators and to first order periodic systems on the real line with a matrix valued complex self adjoint periodic potential