on leja sequences some results and applications in this paper we collect and review some results concerning leja sequences on one and two dimensional domains then we present two applications and discuss some open problems on the asymptotic distribution of leja points on two dimensional domains

approximation of quantum graph vertex couplings by scaled schrodinger operators on thin branched manifolds we discuss approximations of vertex couplings of quantum graphs using families of thin branched manifolds we show that if a neumann type laplacian on such manifolds is amended by suitable potentials the resulting schrodinger operators can approximate non trivial vertex couplings the latter include not only the couplings but also those with wavefunctions discontinuous at the vertex we work out the example of the symmetric couplings and make a conjecture that the same method can be applied to all couplings invariant with respect to the time reversal we conclude with a result that certain vertex couplings cannot be approximated by a pure laplacian

determinant bundles manifolds with boundary and surgery we define determinant bundles associated to the following data i a family of generalized dirac operators on even dimensional manifolds with boundary ii the choice of a spectral section for the family of dirac operators induced on the boundary under the assumption that the operators of the boundary family have null spaces of constant dimension we define through the notion ofb zeta function a quillen metric we also introduce the analogue of the bismut freed connection we prove that the curvature of a natural perturbation of the bismut freed connection equals the 2 form piece in the right hand side of the family index formula thus extending to manifolds with boundary results of quillen bismut and freed given a closed fibration we investigate the behaviour of the quillen metric and of the bismut freed connection under the operation of surgery along a fibering hypersurface we prove in particular additivity formulae for the curvature and for the logarithm of the holonomy

sparse second moment analysis for elliptic problems in stochastic domains we consider the numerical solution of elliptic boundary value problems in domains with random boundary perturbations assuming normal perturbations with small amplitude and known mean field and two point correlation function we derive using a second order shape calculus deterministic equations for the mean field and the two point correlation function of the random solution for a model dirichlet problem which are 3rd order accurate in the boundary perturbation size using a variational boundary integral equation formulation on the unperturbed nominal boundary and a wavelet discretization we present and analyze an algorithm to approximate the random solution s mean and its two point correlation function at essentially optimal order in essentially mathcal o n work and memory where n denotes the number of unknowns required for consistent discretization of the boundary of the nominal domain

zeta determinants on manifolds with boundary abstract through a general theory for relative spectral invariants we study the determinant of global boundary problems of aps type in particular we compute the determinant ratio for dirac laplacian boundary problems in terms of a scattering fredholm determinant over the boundary

eta invariant and holonomy the even dimensional case abstract in previous work we introduced eta invariants for even dimensional manifolds it plays the same role as the eta invariant of atiyah patodi singer which is for odd dimensional manifolds it is associated to k 1 representatives on even dimensional manifolds and is closely related to the so called wzw theory in physics in fact it is an intrinsic interpretation of the wess zumino term without passing to the bounding 3 manifold spectrally the eta invariant is defined on a finite cylinder rather than on the manifold itself thus it is an interesting question to find an intrinsic spectral interpretation of this new invariant we address this issue here using adiabatic limit technique the general formulation relates the mod z reduction of eta invariant for even dimensional manifolds with the holonomy of the determinant line bundle of a natural family of dirac type operators in this sense our result might be thought of as an even dimensional analogue of witten s holonomy theorem proved by bismut freed and cheeger independently

spectra of graph neighborhoods and scattering let g e e 0 be a family of e thin riemannian manifolds modeled on a finite metric graph g for example the e neighborhood of an embedding of g in some euclidean space with straight edges we study the asymptotic behavior of the spectrum of the laplace beltrami operator on g e as e 0 for various boundary conditions we obtain complete asymptotic expansions for the kth eigenvalue and the eigenfunctions uniformly for k ce 1 in terms of scattering data on a non compact limit space we then use this to determine the quantum graph which is to be regarded as the limit object in a spectral sense of the family g e our method is a direct construction of approximate eigenfunctions from the scattering and graph data and the use of a priori estimates to show that all eigenfunctions are obtained in this way

eta invariants and regularized determinants for odd dimensional hyperbolic manifolds with cusps we study eta invariants of dirac operators and regularized determinants of dirac lapla cians over hyperbolic manifolds with cusps and their relations with selberg zeta functions using the selberg trace formula and a detailed analysis of the unipotent orbital integral we show that the eta and zeta functions defined by the relative traces are regular at the origin so that we can define the eta invariant and the regularized determinant we also show that the selberg zeta function of odd type has a meromorphic extension over c prove a relation of the eta invariant and a certain value of the selberg zeta function of odd type and derive a corresponding functional equation these results generalize the earlier work of john millson to hyperbolic manifolds with cusps we also prove that the selberg zeta function of even type has a meromorphic extension over c relate it to the regularized determinant and obtain a corresponding functional equation

fundamental solution of the heat and schrodinger equations with point interaction abstract we find explicit formulae for the fundamental solution for the heat and time dependent schrodinger equations with point interaction in dimension n 3 we discuss in detail the small time asymptotics of these fundamental solutions the case n 2 has the peculiarity of involving logarithmic terms in the expansion

coupling in the singular limit of thin quantum waveguides we analyze the problem of approximating a smooth quantum waveguide with a quantum graph we consider a planar curve with compactly supported curvature and a strip of constant width around the curve we rescale the curvature and the width in such a way that the strip can be approximated by a singular limit curve consisting of one vertex and two infinite straight edges i e a broken line we discuss the convergence of the laplacian with dirichlet boundary conditions on the strip in a suitable sense and we obtain two possible limits the laplacian on the line with dirichlet boundary conditions in the origin and a nontrivial family of point perturbations of the laplacian on the line the first case generically occurs and corresponds to the decoupling of the two components of the limit curve while in the second case a coupling takes place we present also two families of curves which give rise to coupling

quantitative tomography for continuous variable quantum systems we present a continuous variable tomography scheme that reconstructs the husimi q function wigner function by lagrange interpolation using measurements of the q function wigner function at the padua points conjectured to be optimal sampling points for two dimensional reconstruction our approach drastically reduces the number of measurements required compared to using equidistant points on a regular grid although reanalysis of such experiments is possible the reconstruction algorithm produces a reconstructed function with exponentially decreasing error and quasilinear runtime in the number of padua points moreover using the interpolating polynomial of the q function we present a technique to directly estimate the density matrix elements of the continuous variable state with only a linear propagation of input measurement error furthermore we derive a state independent analytical bound on this error such that our estimate of the density matrix is accompanied by a measure of its uncertainty

optimal systems of nodes for lagrange interpolation on bounded intervals a survey in this brief survey special attention is paid to some recent procedures for constructing optimal interpolation processes i e with lebesgue constant having logarithmic behaviour a new result on lagrange interpolation based on the zeros of the associated jacobi polynomials and on suitable additional nodes is given

hyperinterpolation in the cube we construct an hyperinterpolation formula of degree n in the three dimensional cube by using the numerical cubature formula for the product chebyshev measure given by the product of a near minimal formula in the square with gauss chebyshev lobatto quadrature the underlying function is sampled at n n 3 2 points whereas the hyperinterpolation polynomial is determined by its n 1 n 2 n 3 6 n 3 6 coefficients in the trivariate chebyshev orthogonal basis the effectiveness of the method is shown by a numerical study of the lebesgue constant which turns out to increase like log 3 n and by the application to several test functions

quantum ergodicity of boundary values of eigenfunctions suppose that n is a bounded piecewise smooth domain we prove that the boundary values cauchy data of eigenfunctions of the laplacian on with various boundary conditions are quantum ergodic if the classical billiard map on the ball bundle b is ergodic our proof is based on the classical observation that the boundary values of an interior eigenfunction 2 is an eigenfunction of an operator fh on the boundary of with h 1 in the case of the neumann boundary condition fh is the boundary integral operator induced by the double layer potential we show that fh is a semiclassical fourier integral operator quantizing the billiard map plus a small remainder the quantum dynamics defined by fh can be exploited on the boundary much as the quantum dynamics generated by the wave group were exploited in the interior of domains with corners and ergodic billiards in the work of zelditch zworski 1996 novelties include the facts that fh is not unitary and consequently the boundary values are equidistributed by measures which are not invariant under and which depend on the boundary conditions ergodicity of boundary values of eigenfunctions on domains with ergodic billiards was conjectured by s ozawa 1993 and was almost simultaneously proved by gerard leichtnam 1993 in the case of convex c1 1 domains with continuous tangent planes and with dirichlet boundary conditions our methods seem to be quite different motivation to study piecewise smooth domains comes from the fact that almost all known ergodic domains are of this form

quantum graphs ii some spectral properties of quantum and combinatorial graphs the paper deals with some spectral properties of mostly infinite quantum and combinatorial graphs quantum graphs have been intensively studied lately due to their numerous applications to mesoscopic physics nanotechnology optics and other areas a schnol type theorem is proven that allows one to detect that a point belongs to the spectrum when a generalized eigenfunction with an sub exponential growth integral estimate is available a theorem on spectral gap opening for decorated quantum graphs is established its analogue is known for the combinatorial case it is also shown that if a periodic combinatorial or quantum graph has a point spectrum it is generated by compactly supported eigenfunctions scars

a general solution procedure for the forced vibrations of a continuous system with cubic nonlinearities primary resonance case abstract nonlinear vibrations of a general model of continuous system is considered the model consists of arbitrary linear and cubic operators the equation of motion is solved by the method of multiple scales a perturbation method the primary resonances of external excitation is analysed the amplitude and phase modulation equations are presented approximate analytical solution is derived steady state solutions and their stability are discussed finally the solution algorithm is applied to two different engineering problems one of the application is the transverse vibration of an axially moving euler bernoulli beam and the other is a viscoelastic beam

branched quantum wave guides with dirichlet boundary conditions the decoupling case we consider a family of open sets m which shrinks with respect to an appropriate parameter to a graph under the additional assumption that the vertex neighbourhoods are small we show that the appropriately shifted dirichlet spectrum of m converges to the spectrum of the differential laplacian on the graph with dirichlet boundary conditions at the vertices i e a graph operator without coupling between different edges the smallness is expressed by a lower bound on the first eigenvalue of a mixed eigenvalue problem on the vertex neighbourhood the lower bound is given by the first transversal mode of the edge neighbourhood we also allow curved edges and show that all bounded eigenvalues converge to the spectrum of a laplacian acting on the edge with an additional potential coming from the curvature

the ζ determinant and the additivity of the η invariant on the smooth self adjoint grassmannian in this paper we discuss the existence of the determinant of a dirac operator on a compact manifold with boundary we show that the determinant is well defined in the case of the operator with a domain determined by a boundary condition from the smooth self adjoint grassmannian discussed in the papers 5 13 29 we prove a generalization of a pasting formula for the invariant see 34 the results of the paper are used in the recent proof of the projective equality of the determinant and quillen determinant on see 30 31

convergence of spectra of uniformly fattened open book structures abstract we consider a stratified compact surface m in r 3 which we call an open book structure whose singular points form simple closed smooth curves and its neighborhood m which we call a fattened open book structure assuming a transversality condition at the singularities and absence of zero dimensional strata i e corners we show that the discrete spectrum of the neumann laplacian in m converges when 0 to the spectrum of a differential operator on m similar results have been obtained before for the case of fattened graphs i e m being one dimensional in the case of a smooth surfaces m the problem has been studied well however having singularities along strata of lower dimensions significantly complicates considerations as in the quantum graph case such considerations are triggered by various applications

a remark on krein s resolvent formula and boundary conditions we prove an analogue of krein s resolvent formula expressing the resolvents of self adjoint extensions in terms of boundary conditions applications to quantum graphs and systems with point interactions are discussed

an algorithm for computing fekete points in the triangle on the line and its tensor products fekete points are known to be the gauss lobatto quadrature points but unlike high order quadrature fekete points generalize to non tensor product domains such as the triangle thus fekete points might serve as an alternative to the gauss lobatto points for certain applications in this work we present a new algorithm to compute fekete points and give results up to degree 19 for the triangle for degree d 10 these points have the smallest lebesgue constant currently known the computations validate a conjecture of bos j approx theory 64 1991 pp 271 280 that fekete points along the boundary of the triangle are the one dimensional gauss lobatto points

computing multivariate fekete and leja points by numerical linear algebra we discuss and compare two greedy algorithms that compute discrete versions of fekete like points for multivariate compact sets by basic tools of numerical linear algebra the first gives the so called approximate fekete points by qr factorization with column pivoting of vandermonde like matrices the second computes discrete leja points by lu factorization with row pivoting moreover we study the asymptotic distribution of such points when they are extracted from weakly admissible meshes

the burghelea friedlander kappeler gluing formula for zeta determinants on a warped product manifold and a product manifold the burghelea friedlander kappeler bfk gluing formula for the regularized zeta determinants of laplacians contains a constant which is expressed by the constant term in the asymptotic expansion of the regularized zeta determinants of a one parameter family of the dirichlet to neumann operators when the dimension of a cutting hypersurface is odd or the metric is a product one near a cutting hypersurface this constant is well known in this paper we discuss this constant in two cases one is when a warped product metric is given near a cutting hypersurface and the other is when a manifold is a product manifold especially in the first case we use the result of fucci and kirsten commun math phys 317 635 665 2013 in which the regularized zeta determinant of the laplacian defined on a warped product manifold is computed

relative index pairing and odd index theorem for even dimensional manifolds abstract we prove an analogue for even dimensional manifolds of the atiyah patodi singer twisted index theorem for trivialized flat bundles we show that the eta invariant appearing in this result coincides with the eta invariant by dai and zhang up to an integer we also obtain the odd dimensional counterpart for manifolds with boundary of the relative index pairing by lesch moscovici and pflaum

spectral boundary conditions for generalizations of laplace and dirac operators spectral boundary conditions for laplace type operators on a compact manifold x with boundary are partly dirichlet partly oblique neumann conditions where the partitioning is provided by a pseudodifferential projection they have an interest in string and brane theory relying on pseudodifferential methods we give sufficient conditions for the existence of the associated resolvent and heat operator and show asymptotic expansions of their traces in powers and power log terms allowing a smearing function the leading log coefficient is identified as a non commutative residue which vanishes when 1 the study has new consequences for well posed spectral boundary problems for first order dirac like elliptic operators generalizing the atiyah patodi singer problem it is found e g that the zeta function is always regular at zero in the selfadjoint case there is a stability of the zeta function value and the eta function regularity at zero under perturbations of the boundary projection of order dim x

uniform weighted approximation on the square by polynomial interpolation at chebyshev nodes abstract the paper deals with de la vallee poussin type interpolation on the square at tensor product chebyshev zeros of the first kind the approximation is studied in the space of locally continuous functions with possible algebraic singularities on the boundary equipped with weighted uniform norms in particular simple necessary and sufficient conditions are proved for the uniform boundedness of the related lebesgue constants error estimates in some sobolev type spaces are also given pros and cons of such a kind of filtered interpolation are analyzed in comparison with the lagrange polynomials interpolating at the same chebyshev grid or at the equal number of padua nodes the advantages in reducing the gibbs phenomenon are shown by means of some numerical experiments

quantum graphs as holonomic constraints we consider the dynamics on a quantum graph as the limit of the dynamics generated by a one particle hamiltonian in r2 with a potential having a deep strict minimum on the graph when the width of the well shrinks to zero for a generic graph we prove convergence outside the vertices to the free dynamics on the edges for a simple model of a graph with two edges and one vertex we prove convergence of the dynamics to the one generated by the laplacian with dirichlet boundary conditions in the vertex

computing fekete and lebesgue points simplex square disk we have computed point sets with maximal absolute value of the vandermonde determinant fekete points or minimal lebesgue constant lebesgue points on three basic bidimensional compact sets the simplex the square and the disk using routines of the matlab optimization toolbox we have obtained some of the best bivariate interpolation sets known so far

corners in m theory m theory can be defined on closed manifolds as well as on manifolds with boundary as an extension we show that manifolds with corners appear naturally in m theory we illustrate this with four situations the lift to bounding 12 dimensions of m theory on anti de sitter spaces ten dimensional heterotic string theory in relation to 12 dimensions and the two m branes within m theory in the presence of a boundary the m2 brane is taken with or as a boundary and the worldvolume of the m5 brane is viewed as a tubular neighborhood we then concentrate on the variant of the heterotic theory as a corner and explore analytical and geometric consequences in particular we formulate and study the phase of the partition function in this setting and identify the corrections due to the corner s the analysis involves considering m theory on disconnected manifolds and makes use of the extension of the atiyah patodi singer index theorem to manifolds with corners and the b calculus of melrose

hyperinterpolation on the square we show that hyperinterpolation at near minimal cubature points for the product chebyshev measure along with xu compact formula for the corresponding reproducing kernel provide a simple and powerful polynomial approximation formula in the uniform norm on the square the lebesgue constant of the hyperinterpolation operator grows like log 2 of the degree as that of quasi optimal interpolation sets recently proposed in the literature moreover we give an accurate implementation of the hyperinterpolation formula with linear cost in the number of cubature points and we compare it with interpolation formulas at the same set of points

spectral geometry of singular riemannian spaces on considere la trace du noyau de la chaleur et l application de la methode de l equation de la chaleur au theoreme d indice pour la caracteristique d euler et des complexes signatures

spectral determinants on mandelstam diagrams we study the regularized determinant of the laplacian as a functional on the space of mandelstam diagrams noncompact translation surfaces glued from finite and semi infinite cylinders a mandelstam diagram can be considered as a compact riemann surface equipped with a conformal flat singular metric omega 2 where omega is a meromorphic one form with simple poles such that all its periods are pure imaginary and all its residues are real the main result is an explicit formula for the determinant of the laplacian in terms of the basic objects on the underlying riemann surface the prime form theta functions the canonical meromorphic bidifferential and the divisor of the meromorphic form omega as an important intermediate result we prove a decomposition formula of the type of burghelea friedlander kappeler for the determinant of the laplacian for flat surfaces with cylindrical ends and conical singularities

eigenvalue inequalities and absence of threshold resonances for waveguide junctions abstract let r d be a domain consisting of several cylinders attached to a bounded center one says that admits a threshold resonance if there exists a non trivial bounded function u solving u u in and vanishing at the boundary where is the bottom of the essential spectrum of the dirichlet laplacian in we give a sufficient condition for the absence of threshold resonances in terms of the laplacian eigenvalues on the center the proof is elementary and is based on the min max principle some two and three dimensional examples and applications to the study of laplacians on thin networks are discussed

aps boundary conditions eta invariants and adiabatic limits we prove an adiabatic limit formula for the eta invariant of a manifold with boundary the eta invariant is defined using the atiyah patodisinger boundary condition and the underlying manifold is fibered over a manifold with boundary our result extends the work of bismut cheeger to manifolds with boundary

bivariate lagrange interpolation at the node points of lissajous curves the degenerate case in this article we study bivariate polynomial interpolation on the node points of degenerate lissajous figures these node points form chebyshev lattices of rank 1 and are generalizations of the well known padua points we show that these node points allow unique interpolation in appropriately defined spaces of polynomials and give explicit formulas for the lagrange basis polynomials further we prove mean and uniform convergence of the interpolating schemes for the uniform convergence the growth of the lebesgue constant has to be taken into consideration it turns out that this growth is of logarithmic nature

three to one internal resonances in a general cubic non linear continuous system a general continuous system with an arbitrary cubic non linearity is considered the non linearity is expressed in terms of an arbitrary cubic operator three to one internal resonance case is considered a general approximate solution is presented for the system amplitude and phase modulation equations are derived steady state solutions and their stability are discussed in the general sense the sufficiency condition for such resonances to occur is derived finally the algorithm is applied to a beam resting on a non linear elastic foundation

bivariate polynomial interpolation on the square at new nodal sets abstract as known the problem of choosing good nodes is a central one in polynomial interpolation while the problem is essentially solved in one dimension all good nodal sequences are asymptotically equidistributed with respect to the arc cosine metric in several variables it still represents a substantially open question in this work we consider new nodal sets for bivariate polynomial interpolation on the square first we consider fast leja points for tensor product interpolation on the other hand for interpolation in p n 2 on the square we experiment four families of points which are asymptotically equidistributed with respect to the dubiner metric which extends to higher dimension the arc cosine metric one of them nicknamed padua points gives numerically a lebesgue constant growing like log square of the degree

a general solution procedure for vibrations of systems with cubic nonlinearities and nonlinear time dependent internal boundary conditions abstract the subject of this paper is the development of a general solution procedure for the vibrations primary resonance and nonlinear natural frequency of systems with cubic nonlinearities subjected to nonlinear and time dependent internal boundary conditions this is a commonly occurring situation in the vibration analysis of continuous systems with intermediate elements the equations of motion form a set of nonlinear partial differential equations with nonlinear time dependent and coupled internal boundary conditions the method of multiple timescales an approximate analytical method is applied directly to each partial differential equation of motion as well as coupled boundary conditions i e on each sub domain and the corresponding internal boundary conditions for a continuous system with intermediate elements which ultimately leads to approximate analytical expressions for the frequency response relation and nonlinear natural frequencies of the system these closed form solutions provide direct insight into the relationship between the system parameters and vibration characteristics of the system moreover the suggested solution procedure is applied to a sample problem which is discussed in detail

computing approximate fekete points by qr factorizations of vandermonde matrices we propose a numerical method implemented in matlab for computing approximate fekete points on compact multivariate domains it relies on the search of maximum volume submatrices of vandermonde matrices computed on suitable discretization meshes and uses a simple greedy algorithm based on qr factorization with column pivoting the method gives also automatically an algebraic cubature formula provided that the moments of the underlying polynomial basis are known numerical tests are presented for the interval and the square which show that approximate fekete points are well suited for polynomial interpolation and cubature

discrete weighted least squares method for the poisson and biharmonic problems on domains with smooth boundary in this article a discrete weighted least squares method for the numerical solution of elliptic partial differential equations exhibiting smooth solution is presented it is shown how to create well conditioned matrices of the resulting system of linear equations using algebraic polynomials carefully selected matching points and weight factors two simple algorithms generating suitable matching points the chebyshev matching points for standard two dimensional domains and the approximate fekete points of sommariva and vianello for general domains are described the efficiency of the presented method is demonstrated by solving the poisson and biharmonic problems with the homogeneous dirichlet boundary conditions defined on circular and annular domains using basis functions in the form satisfying and in the form not satisfying the prescribed boundary conditions

hyper2d a numerical code for hyperinterpolation on rectangles hyperinterpolation at morrow patterson xu cubature points for the product chebyshev measure provides a simple and powerful polynomial approximation method on rectangles here we present an accurate and efficient matlab octave implementation of the hyperinterpolation formula accompanied by several numerical tests

geometric weakly admissible meshes discrete least squares approximations and approximate fekete points using the concept of geometric weakly admissible meshes see 2 below together with an algorithm based on the classical qr factorization of matrices we compute efficient points for discrete multivariate least squares approximation and lagrange interpolation

a general solution procedure for the forced vibrations of a system with cubic nonlinearities three to one internal resonances with external excitation a general vibrational model of a continuous system with arbitrary linear and cubic operators is considered approximate analytical solutions are found using the method of multiple scales the primary resonances of the external excitation and three to one internal resonances between two arbitrary natural frequencies are treated the amplitude and phase modulation equations are derived the steady state solutions and their stability are discussed the solution algorithm is applied to two specific problems 1 axially moving euler bernoulli beam and 2 axially moving viscoelastic beam

the index formula and the spectral shift function for relatively trace class perturbations we compute the fredholm index index da of the operator da d dt a on l2 r h associated with the operator path a t t where af t a t f t for a e t r and appropriate f l2 r h via the spectral shift function a a associated with the pair a a of asymptotic operators a a on the separable complex hilbert space h in the case when a t is generally an unbounded relatively trace class perturbation of the unbounded self adjoint operator a we derive a formula an extension of a formula due to pushnitski relating the spectral shift function a a for the pair a a and the corresponding spectral shift function h2 h1 for the pair of operators h2 h1 dada da da in this relative trace class context h2 h1 1 1 2 1 2 a a d 2 1 2for a e 0 this formula is then used to identify the fredholm index of da with 0 a a in addition we prove that index da coincides with the spectral flow spflow a t t of the family a t t r and also relate it to the fredholm perturbation determinant for the pair a a index da spflow a t t 0 a a 1lime 0im ln deth a iei a iei 1 0 h2 h1 with the choice of the branch of ln deth on c such that limim z ln deth a zi a zi 1 0 we also provide some applications in the context of supersymmetric quantum mechanics to zeta function and heat kernel regularized spectral asymmetries and the eta invariant

an index theorem for toeplitz operators on odd dimensional manifolds with boundary we establish an index theorem for toeplitz operators on odd dimensional spin manifolds with boundary it may be thought of as an odd dimensional analogue of the atiyah patodi singer index theorem for dirac operators on manifolds with boundary in particular there occurs naturally an invariant of type associated to k 1 representatives on even dimensional manifolds which should be of independent interests for example it gives an intrinsic interpretation of the so called wess zumino term in the wzw theory in physics

bivariate lagrange interpolation at the padua points the ideal theory approach the padua points are a family of points on the square 1 1 2 given by explicit formulas that admits unique lagrange interpolation by bivariate polynomials interpolation polynomials and cubature formulas based on the padua points are studied from an ideal theoretic point of view which leads to the discovery of a compact formula for the interpolation polynomials the lp convergence of the interpolation polynomials is also studied

principal parametric resonances of a general continuous system with cubic nonlinearities abstract a generalized vibrational model of cubic nonlinear continuous system with arbitrary parametric excitation is considered the method of multiple scales a perturbation method is used to find an approximate analytical solution the primary parametric resonance of the parametric excitation is considered the amplitude and phase modulation equations are derived steady state solutions and their stability are discussed the solution algorithm is applied to the problem of nonlinear vibrations of viscoelastic pipes conveying fluids natural frequencies of viscoelastic pipes are found frequency response curves and bifurcation points are drawn stable and unstable regions of trivial and nontrivial solutions are obtained

kirchhoff s rule for quantum wires we formulate and discuss one particle quantum scattering theory on an arbitrary finite graph with n open ends and where we define the hamiltonian to be minus the laplace operator with general boundary conditions at the vertices this results in a scattering theory with n channels the corresponding on shell s matrix formed by the reflection and transmission amplitudes for incoming plane waves of energy e 0 is given explicitly in terms of the boundary conditions and the lengths of the internal lines it is shown to be unitary which may be viewed as the quantum version of kirchhoff s law we exhibit covariance and symmetry properties it is symmetric if the boundary conditions are real also there is a duality transformation on the set of boundary conditions and the lengths of the internal lines such that the low energy behaviour of one theory gives the high energy behaviour of the transformed theory finally we provide a composition rule by which the on shell s matrix of a graph is factorizable in terms of the s matrices of its subgraphs all proofs use only known facts from the theory of self adjoint extensions standard linear algebra complex function theory and elementary arguments from the theory of hermitian symplectic forms

fast calculation of scattering by 3 d inhomogeneities in uniaxial anisotropic multilayers a fast algorithm to calculate the response of uniaxial layered media on a rectilinear mesh is proposed in order to handle scattering phenomena due to inhomogeneities within them the uniaxial layered media are with their optical axes parallel to the interfaces of the layered structure the generalized poisson summation formula is used to efficiently calculate the discrete fourier transform of spatial responses of the layeredmedia from its continuous fourier spectrum using the windowing technique the integral involved in constructing the discrete fourier spectrum can be performed in a small domain without compromising the accuracy of the result furthermore a fast numerical integration method based on the recently proposed padua points is used which together with the windowing technique yields a fast solution to such scattering problems numerical simulations illustrate the accuracy and efficiency of the proposed algorithm

bivariate lagrange interpolation at the padua points the so called padua points give a simple geometric and explicit construction of bivariate polynomial interpolation in the square moreover the associated lebesgue constant has minimal order o

bivariate lagrange interpolation at the padua points computational aspects the so called padua points give a simple geometric and explicit construction of bivariate polynomial interpolation in the square moreover the associated lebesgue constant has minimal order of growth o log 2 n here we show four families of padua points for interpolation at any even or odd degree n and we present a stable and efficient implementation of the corresponding lagrange interpolation formula based on the representation in a suitable orthogonal basis we also discuss extension of non polynomial padua like interpolation to other domains such as triangles and ellipses we give complexity and error estimates and several numerical tests

scattering solutions in networks of thin fibers small diameter asymptotics small diameter asymptotics is obtained for scattering solutions in a network of thin fibers the asymptotics is expressed in terms of solutions of related problems on the limiting quantum graph we calculate the lagrangian gluing conditions at vertices v in gamma for the problems on the limiting graph if the frequency of the incident wave is above the bottom of the absolutely continuous spectrum the gluing conditions are formulated in terms of the scattering data for each individual junction of the network

convergence of spectra of mesoscopic systems collapsing onto a graph abstract let m be a finite graph in the plane and let me be a domain that looks like the e fattened graph m exact conditions on the domain are given it is shown that the spectrum of the neumann laplacian on me converges when e 0 to the spectrum of an ode problem on m the presence of an electromagnetic field is also allowed considerations of this kind arise naturally in mesoscopic physics and other areas of physics and chemistry the results of the paper extend the ones previously obtained by j rubinstein and m schatzman

relative zeta functions relative determinants and scattering theory we use the method of zeta function regularization to regularize the ratio det a det a0 of the determinants of two elliptic self adjoint operators a a0 satisfying certain natural assumptions this is of interest especially if the regularized determinants of the individual operators don t exist as for example in the case of elliptic operators on a noncompact manifold

bivariate lagrange interpolation at the node points of non degenerate lissajous curves motivated by an application in magnetic particle imaging we study bivariate lagrange interpolation at the node points of lissajous curves the resulting theory is a generalization of the polynomial interpolation theory developed for a node set known as padua points with appropriately defined polynomial spaces we will show that the node points of non degenerate lissajous curves allow unique interpolation and can be used for quadrature rules in the bivariate setting an explicit formula for the lagrange polynomials allows to compute the interpolating polynomial with a simple algorithmic scheme compared to the already established schemes of the padua and xu points the numerical results for the proposed scheme show similar approximation errors and a similar growth of the lebesgue constant

gaussian cubature and bivariate polynomial interpolation gaussian cubature is used to study bivariate polynomial interpolation based on the common zeros of quasi orthogonal polynomials

minimal numerical differentiation formulas we investigate numerical differentiation formulas on irregular centers in two or more variables that are exact for polynomials of a given order and minimize an absolute seminorm of the weight vector error bounds are given in terms of a growth function that carries the information about the geometry of the centers specific forms of weighted ell 1 and weighted least squares minimization are proposed that produce numerical differentiation formulas with particularly good performance in numerical experiments the results are of interest in particular for meshless generalized finite difference methods as they provide a consistency error analysis for such methods

vibrations of continuous systems with a general operator notation suitable for perturbative calculations the operator notation previously developed to analyze vibrations of continuous systems has been further generalized to model a system with an arbitrary number of coupled differential equations linear parts of the equations are expressed with an arbitrary linear differential and or integral operators and non linear parts are expressed with arbitrary quadratic and cubic operators equations of motion are solved in their general form using the method of multiple scales a perturbation technique the case of primary resonances of the external excitation and one to one internal resonances between the natural frequencies of the equations is considered the algorithm developed is applied to a non linear cable vibration problem having small sag to span ratios

tensor product gauss lobatto points are fekete points for the cube tensor products of gauss lobatto quadrature points are frequently used as collocation points in spectral element methods unfortunately it is not known if gauss lobatto points exist in non tensor product domains like the simplex in this work we show that the n dimensional tensor product of gauss lobatto quadrature points are also fekete points this suggests a way to generalize spectral methods based on gauss lobatto points to non tensor product domains since fekete points are known to exist and have been computed in the triangle and tetrahedron in one dimension this result was proved by fejer in 1932 but the extension to higher dimensions in non trivial

least squares polynomial approximation on weakly admissible meshes disk and triangle we construct symmetric polar wams weakly admissible meshes with low cardinality for least squares polynomial approximation on the disk these are then mapped to an arbitrary triangle numerical tests show that the growth of the least squares projection uniform norm is much slower than the theoretical bound and even slower than that of the lebesgue constant of the best known interpolation points for the triangle as opposed to good interpolation points such meshes are straightforward to compute for any degree the construction can be extended to polygons by triangulation

bound states in curved quantum waveguides a free quantum particle living on a curved planar strip of a fixed width d with dirichlet boundary conditions is studied it can serve as a model for electrons in thin films on a cylinder type substrate or in a curved quantum wire assuming that the boundary of is infinitely smooth and its curvature decays fast enough at infinity it is proved that a bound state with energy below the first transversal mode exists for all sufficiently small d a lower bound on the critical width is obtained using the birman schwinger technique

a comparison of different versions of the method of multiple scales for partial differential equations abstract applications of the methods of multiple scales a perturbation method to partial differential systems arising in non linear vibrations of continuous systems are considered two different versions of the method of multiple scales are applied to two general non linear models in one of the models the small parameter multiplies an arbitrary non linear cubic operator whereas in the other model arbitrary quadratic and cubic non linearities exist the linear parts of both models are represented by arbitrary operators general solutions are found by applying different versions of the method of multiple scales results of the first version reconstitution method and the second version proposed by rahman and burton 8 are compared for both models from the comparisons of both methods it is found that the second version yields better results applications of the general models to specific problems are also presented a final recommendation is to use the second version of the method of multiple scales combined with the direct perturbation method in finding steady state solutions of partial differential equations

variational problems on multiply connected thin strips i basic estimates and convergence of the laplacian spectrum let m be a planar embedded graph whose arcs meet transversally at the vertices let be a strip shaped domain around m of width except in a neighborhood of the singular points assume that the boundary of is smooth we define comparison operators between functions on and on m and we derive energy estimates for the compared functions we define a laplace operator on m which is in a certain sense the limit of the laplace operator on with neumann boundary conditions in particular we show that the p th eigenvalue of the laplacian on converges to the p th eigenvalue of the laplacian on m as tends to 0 a similar result holds for the magnetic schrodinger operator