right limits and reflectionless measures for cmv matrices we study cmv matrices by focusing on their right limit sets we prove a cmv version of a recent result of remling dealing with the implications of the existence of absolutely continuous spectrum and we study some of its consequences we further demonstrate the usefulness of right limits in the study of weak asymptotic convergence of spectral measures and ratio asymptotics for orthogonal polynomials by extending and refining earlier results of khrushchev to demonstrate the analogy with the jacobi case we recover corresponding previous results of simon using the same approach

the birth of e8 out of the spinors of the icosahedron e8 is prominent in mathematics and theoretical physics and is generally viewed as an exceptional symmetry in an eight dimensional 8d space very different from the space we inhabit for instance the lie group e8 features heavily in 10d superstring theory contrary to that point of view here we show that the e8 root system can in fact be constructed from the icosahedron alone and can thus be viewed purely in terms of 3d geometry the 240 roots of e8 arise in the 8d clifford algebra of 3d space as a double cover of the 120 elements of the icosahedral group generated by the root system h3 as a by product by restricting to even products of root vectors spinors in the 4d even subalgebra of the clifford algebra one can show that each 3d root system induces a root system in 4d which turn out to also be exactly the exceptional 4d root systems the spinorial point of view explains their existence as well as their unusual automorphism groups this spinorial approach thus in fact allows one to construct all exceptional root systems within the geometry of three dimensions which opens up a novel interpretation of these phenomena in terms of spinorial geometry

coupling of finite elements and boundary elements for some nonlinear interface problems abstract this paper considers the symmetric coupling method with boundary elements and finite elements analyzed by costabel and stephan we present experimental rates of convergence for the h version where we achieve the accuracy of the galerkin approximation by refining the mesh in our computations we use as finite elements piecewise bilinear functions and as boundary elements piecewise constant functions

on isospectral sets of jacobi operators we consider the inverse spectral problem for a class of reflectionless bounded jacobi operators with empty singularly continuous spectra our spectral hypotheses admit countably many accumulation points in the set of eigenvalues as well as in the set of boundary points of intervals of absolutely continuous spectrum the corresponding isospectral set of jacobi operators is explicitly determined in terms of dirichlet type data

a fully discrete bem fem for the exterior stokes problem in the plane we reformulate the johnson nedelec approach for the exterior two dimensional stokes problem taking advantage of the parameterization of the artificial boundary the main aim of this paper is the presentation and analysis of a fully discrete numerical method for this problem this one responds to the needs of having efficient approximate quadratures for the weakly singular boundary integrals we give a complete error analysis of both the galerkin and fully discrete galerkin methods

coupling of fem and bem in shape optimization in the present paper we consider the numerical solution of shape optimization problems which arise from shape functionals of integral type over a compact region of the unknown shape especially l 2 tracking type functionals the underlying state equation is assumed to satisfy a poisson equation with dirichlet boundary conditions we proof that the shape hessian is not strictly h 1 2 coercive at the optimal domain which implies ill posedness of the optimization problem under consideration since the adjoint state depends directly on the state we propose a coupling of finite element methods fem and boundary element methods bem to realize an efficient first order shape optimization algorithm fem is applied in the compact region while the rest is treated by bem the coupling of fem and bem essentially retains all the structural and computational advantages of treating the free boundary by boundary integral equations

catalan solids derived from three dimensional root systems and quaternions catalan solids are the duals of the archimedean solids the vertices of which can be obtained from the coxeter dynkin diagrams a3 b3 and h3 whose simple roots can be represented by quaternions the respective weyl groups w a3 w b3 and w h3 acting on the highest weights generate the orbits corresponding to the solids possessing these symmetries vertices of the platonic and archimedean solids result from the orbits derived from fundamental weights the platonic solids are dual to each other however the duals of the archimedean solids are the catalan solids whose vertices can be written as the union of the orbits up to some scale factors obtained by applying the above weyl groups on the fundamental highest weights 100 010 and 011 for each diagram the faces are represented by the orbits derived from the weights 010 110 101 011 and 111 which correspond to the vertices of the archimedean solids representations of the weyl groups w a3 w b3 and w h3 by the quaternions sim

quaternionic root systems and subgroups of the aut f4 cayley dickson doubling procedure is used to construct the root systems of some celebrated lie algebras in terms of the integer elements of the division algebras of real numbers complex numbers quaternions and octonions starting with the roots and weights of su 2 expressed as the real numbers one can construct the root systems of the lie algebras of so 4 sp 2 so 5 so 8 so 9 f4 and e8 in terms of the discrete elements of the division algebras the roots themselves display the groups structures besides the octonionic roots of e8 which form a closed octonion algebra the automorphism group aut f4 of the dynkin diagram of f4 of order 2304 the largest crystallographic group in four dimensional euclidean space is realized as the direct product of two binary octahedral group of quaternions preserving the quaternionic root system of f4 the weyl groups of many lie algebras such as g2 so 7 so 8 so 9 su 3 xsu 3 and sp 3 su 2 have been constructed as the subgroups of aut f4 we have also class

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

coupling regularization and adaptive hp bem for the solution of a delamination problem in this paper we couple regularization techniques with the adaptive hp version of the boundary element method hp bem for the efficient numerical solution of linear elastic problems with nonmonotone contact boundary conditions as a model example we treat the delamination of composite structures with a contaminated interface layer this problem has a weak formulation in terms of a nonsmooth variational inequality the resulting hemivariational inequality is first regularized and then discretized by an adaptive hp bem we give conditions for the uniqueness of the solution and provide an a priori error estimate furthermore we prove the very first a posteriori error estimate for the nonsmooth variational problem utilizing a novel mixed regularized formulation thus enabling hp adaptivity various numerical experiments illustrate the behavior strengths and limitations of the proposed high order approximation scheme

fem and bem coupling for a nonlinear transmission problem with signorini contact this paper is concerned with the interface problem consisting of a nonlinear uniformly monotone partial differential equation in some bounded lipschitz domain omega in reals n n 2 or n geq 3 and the laplace equation with some radiation condition in the unbounded exterior domain omega c reals n setminus overline omega the two problems are coupled by transmission conditions and signorini contact conditions on the interface gamma partial omega the exterior part of the interface problem is rewritten in terms of boundary integral operators this leads to a variational inequality with a nonlinear monotone operator by this approach existence and uniqueness of a solution in appropriate sobolev spaces are obtained its approximation is performed by coupling the finite element method fem in omega and the boundary element method bem the latter living on the interface gamma yielding a discrete monotone variational inequality we present an abstract cea type error estimate and derive asymptotic error estimates

asymptotic expansions of the error of spline galerkin boundary element methods in this paper we analyse the existence of asymptotic expansions of the error of galerkin methods with splines of arbitrary degree for the approximate solution of integral equations with logarithmic kernels these expansions are obtained in terms of an interpolation operator and are useful for the application of richardson extrapolation and for obtaining sharper error bounds we also present and analyse a family of fully discrete spline galerkin methods for the solution of the same equations following the analysis of galerkin methods we show the existence of asymptotic expansions of the error

the transformation operator for schrodinger operators on almost periodic infinite gap backgrounds abstract we investigate the kernels of the transformation operators for one dimensional schrodinger operators with potentials which are asymptotically close to bohr almost periodic infinite gap potentials

on a fem bem formulation for an exterior quasilinear problem in the plane we use a version of the fem bem method introduced by costabel boundary elements ix vol 1 c a brebbia et al eds springer verlag 1987 and han j comput math 8 1990 pp 223 232 to discretize an exterior quasilinear problem we provide error estimates for the galerkin method and propose a fully discrete scheme based on simple quadrature formulas furthermore we show that these numerical integration schemes preserve the optimal rates of convergence finally we present results of numerical experiments involving our discretization method

biorthogonal wavelet approximation for the coupling of fem bem we apply multiscale methods to the coupling of finite and boundary element methods to solve an exterior dirichlet boundary value problem for the two dimensional poisson equation adopting biorthogonal wavelet matrix compression to the boundary terms with n degrees of freedom we show that the resulting compression strategy fits the optimal convergence rate of the coupling galerkin methods while the number of nonzero entries in the corresponding stiffness matrices is considerably smaller than n 2

weyl titchmarsh m function asymptotics local uniqueness results trace formulas and borg type theorems for dirac operators we explicitly determine the high energy asymptotics for weyl titchmarsh matrices associated with general dirac type operators on half lines and on r we also prove new local uniqueness results for dirac type operators in terms of exponentially small differences of weyl titchmarsh matrices as concrete applications of the asymptotic high energy expansion we derive a trace formula for dirac operators and use it to prove a borg type theorem

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

on the coupling of vem and bem in two and three dimensions this paper introduces and analyzes the combined use of the virtual element method vem and the boundary element method bem to numerically solve linear transmission problems in two and three dime

adiabatic invariants and asymptotic behavior of lyapunov exponents of the schrodinger equation we give an upper bound for the high energy behavior of the lyapunov exponent of the one dimensional schrodinger equation we relate this behavior to the diffrentiability properties of the potential as an application this result provides an upper bound for the asymptotic length of the gaps of the schrodinger equation

maximal subgroups of the coxeter group w h4 and quaternions abstract the largest finite subgroup of o 4 is the non crystallographic coxeter group w h 4 of order 14 400 its derived subgroup is the largest finite subgroup w h 4 z 2 of so 4 of order 7200 moreover up to conjugacy it has five non normal maximal subgroups of orders 144 two 240 400 and 576 two groups w h 2 w h 2 z 4 and w h 3 z 2 possess non crystallographic structures with orders 400 and 240 respectively the groups of orders 144 240 and 576 are the extensions of the weyl groups of the root systems of su 3 su 3 su 5 and so 8 respectively we represent the maximal subgroups of w h 4 with sets of quaternion pairs acting on the quaternionic root systems

weak solvability of interior transmission problems via mixed finite elements and dirichlet to neumann mappings we study the weak solvability of an interior linear nonlinear transmission problem arising in steady heat transfer and potential theory for the variational formulation we use a dirichlet to neumann mapping on the interface which is obtained from the application of the boundary integral method to the linear domain and we utilize a mixed finite element method in the nonlinear region existence and uniqueness of solution for the continuous formulation are provided and general approximation results for a fully discrete galerkin method are derived in particular a compatibility condition between the mesh sizes involved is deduced in order to conclude the solvability and stability of this galerkin scheme

grand antiprism and quaternions vertices of the four dimensional 4d semi regular polytope the grand antiprism and its symmetry group of order 400 are represented in terms of quaternions with unit norm it follows from the icosian representation of the e8 root system which decomposes into two copies of the root system of h4 the symmetry of the grand antiprism is a maximal subgroup of the coxeter group w h4 it is the group aut h2 h 2 which is constructed in terms of 20 quaternionic roots of the coxeter diagram h2 h 2 the root system of h4 represented by the binary icosahedral group i of order 120 constitutes the regular 4d polytope 600 cell when its 20 quaternionic vertices corresponding to the roots of the diagram h2 h 2 are removed from the vertices of the 600 cell the remaining 100 quaternions constitute the vertices of the grand antiprism we give a detailed analysis of the construction of the cells of the grand antiprism in terms of quaternions the dual polytope of the grand antiprism has also been constructed

a wavelets method for plane elasticity problem a neumann boundary value problem of plane elasticity problem in the exterior circular domain is reduced into an equivalent natural boundary integral equation and a poisson integral formula with the dtn method using the trigonometric wavelets and galerkin method we obtain a fast numerical method for the natural boundary integral equation which has an unique solution in the quotient space we decompose the stiffness matrix in our numerical method into four circulant and symmetrical or antisymmetrical submatrices and hence the solution of the associated linear algebraic system can be solved with the fast fourier transform fft and the inverse fast fourier transform ifft instead of the inverse matrix examples are given for demonstrating our method has good accuracy of our method even though the exact solution is almost singular

on the kotani last and schrodinger conjectures in the theory of ergodic one dimensional schrodinger operators ac spectrum has been traditionally expected to be very rigid two key conjectures in this direction state on one hand that ac spectrum demands almost periodicity of the potential and on the other hand that the eigenfunctions are almost surely bounded in the essential suport of the ac spectrum we show how the repeated slow deformation of periodic potentials can be used to break rigidity and disprove both conjectures

quaternionic representation of the coxeter group w h4 and the polyhedra the vertices of the four dimensional polytope 3 3 5 and its dual 5 3 3 admitting the symmetry of the non crystallographic coxeter group w h4 of order 14 400 are represented in terms of quaternions with unit norm where the polytope 3 3 5 is represented by the elements of the binaryicosahedral group of quaternions of order 120 we projected the polytopes to three dimensional euclidean space where the quaternionic vertices are the orbits of the coxeter group w h3 icosahedral group with inversion where w h3 z2 is one of the maximal subgroups of the coxeter group w h4 the orbits of the icosahedral group w h3 in the polytope 3 3 5 are the conjugacy classes of the binary icosahedral group and represent a number of icosahedrons dodecahedrons and one icosidodecahedron in three dimensions the 15 orbits of the icosahedral group w h3 in the polytope 5 3 3 represent the dodecahedrons icosidodecahedrons small rhombicosidodecahedrons and some convex solids possessing the icosahedral symmetry one of the convex solids with 60 vertices is very similar to the truncated icosahedron soccer ball but with two different edge lengths which can be taken as a realistic model of the c60 molecule at extreme temperature and pressure

symmetries of the octonionic root system of e8 octonionic root system of e8 is decomposed as the 9 sets of hurwitz integers each set satisfying the binary tetrahedral group structure and the 12 sets of quaternionic units each set obeying the dicylic group structure of order 12 this fact is used to prove that the automorphism group of the octonionic root system of e7 is the finite subgroup of g2 of order 12 096 an explicit 7 7 matrix realization of which is constructed possible use of the octonionic representation of the e6 root system as orbifolds and the relevance of the binary tetrahedral structures with the statistical mechanics models are suggested

coupling of finite and boundary element methods for an elastoplastic interface problem a class of transmission problems is considered in which a nonlinear variational problem in one domain is coupled with a linear elliptic problem in a second domain a typical example is a problem from three dimensional elasticity theory where an elastoplastic material is embedded into a linear elastic material the nonlinear problem is given in variational form with a strictly convex functional the linear elliptic problem is described by boundary integral equations on the coupling boundary the typical saddle point structure of such problems is analyzed galerkin approximations are studied which consist of a finite element approximation in the first domain coupled with a boundary element method on the coupling boundary the convergence of the galerkin approximation is based on the saddle point structure which is shown to hold for the exact as well as the discretized problems

on the coupled bem and fem for a nonlinear exterior dirichlet problem in r2 in this paper we apply the coupling of boundary integral and finite element methods to solve a nonlinear exterior dirichlet problem in the plane specifically the boundary value problem consists of a nonlinear second order elliptic equation in divergence form in a bounded inner region and the laplace equation in the corresponding unbounded exterior region in addition to appropriate boundary and transmission conditions the main feature of the coupling method utilized here consists in the reduction of the nonlinear exterior boundary value problem to an equivalent monotone operator equation we provide sufficient conditions for the coefficients of the nonlinear elliptic equation from which existence uniqueness and approximation results are established then we consider the case where the corresponding operator is strongly monotone and lipschitz continuous and derive asymptotic error estimates for a boundary finite element solution we prove the unique solvability of the discrete operator equations and based on a strang type abstract error estimate we show the strong convergence of the approximated solutions moreover under additional regularity assumptions on the solution of the continous operator equation the asymptotic rate of convergenceo h is obtained

counterexamples to the kotani last conjecture for continuum schrodinger operators via character automorphic hardy spaces abstract the kotani last conjecture states that every ergodic operator in one space dimension with non empty absolutely continuous spectrum must have almost periodic coefficients this statement makes sense in a variety of settings for example discrete schrodinger operators jacobi matrices cmv matrices and continuum schrodinger operators in the main body of this paper we show how to construct counterexamples to the kotani last conjecture for continuum schrodinger operators by adapting the approach developed by volberg and yuditskii to construct counterexamples to the kotani last conjecture for jacobi matrices this approach relates the reflectionless operators associated with the prescribed spectrum to a family of character automorphic hardy spaces and then relates the shift action on the space of operators to the resulting action on the associated characters the key to our approach is an explicit correspondence between the space of continuum reflectionless schrodinger operators associated with a given set and the space of reflectionless jacobi matrices associated with a derived set once this correspondence is established we can rely to a large extent on the previous work of volberg and yuditskii to produce the resulting action on the space of characters we analyze this action and identify situations where we can observe absolute continuity without almost periodicity in the appendix we show how to implement this strategy and obtain analogous results for extended cmv matrices

automorphism groups of pure integral octonions a matrix representation of the automorphism group of pure integral octonions constituting the root system of e7 is constructed it is shown that it is a finite subgroup of the exceptional group of g2 of order 12096 called the adjoint chevalley group g2 2 its four maximal subgroups of orders 432 192 192 and 336 preserve respectively the octonionic root systems of e6 so 12 su 2 3 so 8 and su 8 it is also shown explicitly that the full automorphism group of the pure octonions or ei i 1 7 constituting the roots of su 2 7 is a group of order 1344 possible implications in physics are discussed

quaternionic roots of so 8 so 9 f4 and the related weyl groups the root systems of so 8 so 9 and f4 are constructed by quaternions triality manifests itself as permutations of pure quaternion units e1 e2 and e3 it is shown that the automorphism groups of the associated root systems are the finite subgroups of o 4 generated by left right actions of unit quaternions on the root systems the relevant finite groups of quaternions the binary tetrahedral and binary octahedral groups play essential roles in the construction of the weyl groups and their conjugacy classes the relations between the dynkin indices standard orthogonal vector and the quaternionic weights are obtained

an optimal iterative process for the johnson nedelec method of coupling boundary and finite elements we reformulate the discretization of the johnson nedelec method 11 of coupling boundary elements and finite elements for an exterior bidimensional laplacian this new formulation leads to optimal error estimates and allows the use of simple quadrature formulas for calculation of the boundary element matrix we show that if the parameter of discretization is sufficiently small the fully discrete scheme is well posed and the error estimates remain unaltered the rest of the paper is devoted to the study of an efficient algorithm for solving the resulting discrete linear systems

the h p boundary element method for solving 2 and 3 dimensional problems abstract in this survey paper we report on recent developments of the hp version of the boundary element method we consider weakly singular and hypersingular integral equations on polygons and polyhedral surfaces we show that the galerkin solutions computed with the hp version on geometric meshes converge exponentially fast towards the exact solutions of the integral equations the implementation of the hp version of the boundary element method is discussed and adaptive algorithms are given model problems for the coupling of finite elements and boundary elements are also considered numerical results are presented which underline the theoretical results

coupling of finite element and boundary integral methods for electromagnetic scattering in a two layered medium consider a time harmonic electromagnetic plane wave incident on an inhomogeneity embedded in a two layered medium in this paper a method of coupling of finite element and boundary integral equation methods is presented for the solutions of electromagnetic scattering in both transverse electric and magnetic polarization cases the well posedness of the continuous and discrete problems as well as optimal error estimates for the coupled variational approximations are obtained numerical results are included to illustrate the accuracy with the optimal convergence property of the proposed method and to show the wave features in a two layered medium

polyhedra obtained from coxeter groups and quaternions we note that all regular and semiregular polytopes in arbitrary dimensions can be obtained from the coxeter dynkin diagrams the vertices of a regular or semiregular polytope are the weights obtained as the orbit of the coxeter weyl group acting on the highest weight representing a selected irreducible representation of the lie group this paper in particular deals with the determination of the vertices of the platonic and archimedean solids from the coxeter diagrams a3 b3 and h3 in the context of the quaternionic representations of the root systems and the coxeter weyl groups we use lie algebraic techniques in the derivation of vertices of the polyhedra and show that the polyhedra possessing the tetrahedral octahedral and icosahedral symmetries are related to the coxeter weyl groups representing the symmetries of the diagrams of a3 b3 and h3 respectively this technique leads to the determination of the vertices of all platonic and archimedean solids except two chiral polyhedra snubcuboctahedr

equality of the spectral and dynamical definitions of reflection for full line jacobi matrices schrodinger operators and cmv matrices we show that being reflectionless in the sense of the well known property of m functions is equivalent to a lack of reflection in the dynamics in the sense that any state that goes entirely to x as t goes entirely to x as t this allows us to settle a conjecture of deift and simon from 1983 regarding ergodic jacobi matrices

localization in one dimensional random media a scattering theoretic approach we use scattering theoretic methods to prove exponential localization for random displacement models in one dimension the operators we consider model both quantum and classical wave propagation our main tools are the reflection and transmission coefficients for compactly supported single site perturbations we show that randomly displaced non reflectionless single sites lead to localization

on the adaptive coupling of fem and bem in 2 d elasticity this paper concerns the combination of the finite element method fem and the boundary element method bem using the symmetric coupling as a model problem in two dimensions we consider the hencky material a certain nonlinear elastic material in a bounded domain with navier lame differential equation in the unbounded complementary domain using some boundary integral operators the problem is rewritten such that the galerkin procedure leads to a fem bem coupling and quasi optimally convergent discrete solutions beside this a priori information we derive an a posteriori error estimate which allows up to a constant factor the error control in the energy norm since information about the singularities of the solution is not available a priori in many situation and having in mind the goal of an automatic mesh refinement we state adaptive algorithms for the h version of the fem bem coupling illustrating numerical results are included

a dual dual formulation for the coupling of mixed fem and bem in hyperelasticity we combine a dual mixed finite element method with the boundary integral equation method to study the solvability and galerkin approximations of a linear nonlinear transmission problem arising in plane elastostatics our approach is based on the introduction of the strain tensor as an auxiliary unknown in the finite element domain and it leads to what we call a dual dual mixed formulation since the corresponding operator equation becomes of a twofold saddle point structure we derive existence and uniqueness of solution for the continuous and discrete formulations and provide the associated error analysis in particular we extend the finite element space for plane elasticity given by the peers element and define an explicit finite element boundary element subspace satisfying all the required compatibility conditions most of our analysis makes use of an extension of the classical babuska brezzi theory to a class of nonlinear saddle point problems

fem bem coupling methods for tokamak plasma axisymmetric free boundary equilibrium computations in unbounded domains abstract incorporating boundary conditions at infinity into simulations on bounded computational domains is a repeatedly occurring problem in scientific computing the combination of finite element methods fem and boundary element methods bem is the obvious instrument and we adapt here for the first time the two standard fem bem coupling approaches to the free boundary equilibrium problem the johnson nedelec coupling and the bielak maccamy coupling we recall also the classical approach for fusion applications dubbed according to its first appearance von hagenow lackner coupling and present the less used alternative introduced by albanese blum and de barbieri in 2 we show that the von hagenow lackner coupling suffers from undesirable non optimal convergence properties that suggest that other coupling schemes in particular johnson nedelec or albanese blum de barbieri are more appropriate for non linear equilibrium problems moreover we show that any of such coupling methods requires newton like iteration schemes for solving the corresponding non linear discrete algebraic systems

least squares methods for the coupling of fem and bem in the present paper we propose least squares formulations for the numerical solution of exterior boundary value problems the partial differential equation is a first order system in a bounded subdomain and the unbounded subdomain is treated by means of boundary integral equations the first order system is derived from a strongly elliptic second order system the analysis of the present least squares formulations is reduced to the analysis of the galerkin method for the coupling of finite element and boundary element methods fem and bem of the second order problem the least squares approach requires no stability condition however it requires the computation of negative as well as of half integer sobolev norms the arising linear systems can be preconditioned to have condition numbers sim 1 the present methods benefit strongly from the use of biorthogonal wavelets on the coupling boundary and the computation of corresponding equivalent norms in sobolev spaces in particular the application of green s formula leads to an efficient discretization of least squares formulations

a fem bem coupling method for a nonlinear transmission problem modelling coulomb friction contact a nonlinear transmission problem modelling elastoplastic interface problems with contact and coulomb friction is reduced to a boundary domain variational inequality we present a corresponding fem bem coupling procedure and derive for its galerkin solution an a priori error estimate furthermore we reformulate the problem as an equivalent saddle point problem whose discretization can be solved by the uzawa algorithm convergence of the fem bem coupling method is proved and numerical results are given

local spectral properties of reflectionless jacobi cmv and schrodinger operators we prove that jacobi cmv and schrodinger operators which are reflectionless on a homogeneous set e in the sense of carleson under the assumption of a blaschke type condition on their discrete spectra accumulating at e have purely absolutely continuous spectrum on e

a fully discrete bem fem method for an exterior elasticity system in the plane abstract we present a modified version of the usual bem fem coupling for the exterior elasticity problem in the plane cf costabel and stephan siam j numer anal 27 1990 1212 1226 this new formulation allows us to take advantage of techniques from hsiao et al computing 25 1980 557 566 to compute the boundary integral terms using simple quadrature formulas we provide error estimates for the galerkin method and prove that the corresponding fully discrete scheme preserves the optimal rates of convergence

the trace formula for schrodinger operators on the line this paper discusses certain aspects of the spectral and inverse spectral problems for the schrodinger operator l q frac d 2 dx 2 q x for q x c the space of bounded continuous functions the trace formula of the title is the relation q 0 lambda 0 sum limits j 1 infty lambda 2j lambda 2j 1 2 mu j with appropriate choices of j j 0 and j j 1 which is a familiar relation in the theory of hill s equation we characterize the set subseteq c of potentials for which this holds further extensions of the theory of hill s equation are also obtained from the spectrum l q a torust q is constructed which is in general infinite dimensjional everyq x can be mapped to a continuous path ont q described by the auxiliary spectrum j j 1 under certain geometrical conditions on l q this path is the orbit of ac 1 vector field ont q and the mapping extends to one from the hull q overline q x xi xi in mathbb r to the closure of this orbit in particular q is compact these results have applications in the theory of schrodinger operators with ergodic potentials

variational formulations of transmission problems via fem bem and dtn mappings the purpose of this work is to review some of the main results on the weak solvability and galerkin approximations of interior and exterior transmission problems arising in potential theory and elastostatics both linear and nonlinear model problems are included in our presentation we consider variational formulations based on the combination of mixed finite elements with either boundary integral equations or dirichlet to neumann dtn mappings we provide existence and uniqueness of solution for the continuous formulations and give general approximation estimates for the associated galerkin schemes some open questions are also mentioned

the coupling of boundary integral and finite element methods for the exterior steady oseen problem abstract in this article we present a new numerical method for solving the steady oseen equations in an unbounded plane domain the technique consists in coupling the boundary integral and the finite element methods an artificial smooth boundary is introduced separating an interior inhomogeneous region from an exterior homogeneous one the solution in exterior region is represented by an integral equation over the artificial boundary this integral equation is incorporated into a velocity pressure formulation for the interior region and a finite element method is used to approximate the resulting variational problem finally the optimal error estimates of the numerical solution are derived computer results will be discussed in a forthcoming paper

existence results for a nonlinear transmission problem abstract let o and i be open bounded regular subsets of r n such that the closure of i is contained in o let f o be a regular function on o and let f and g be continuous functions from i r to r by exploiting an argument based on potential theory and on the leray schauder principle we show that under suitable and completely explicit conditions on f and g there exists at least one pair of continuous functions u o u i such that u o 0 in o cl i u i 0 in i u o x f o x for all x o u o x f x u i x for all x i i u o x i u i x g x u i x for all x i where the last equality is attained in certain weak sense a simple example shows that such a pair of functions u o u i is in general neither unique nor locally unique if instead the fourth condition of the problem is obtained by a small nonlinear perturbation of a homogeneous linear condition then we prove the existence of at least one classical solution which is in addition locally unique

group theoretical analysis of 600 cell and 120 cell 4d polytopes with quaternions cell 3 3 5 and 120 cell 5 3 3 four dimensional dual polytopes relevant to quasicrystallography have been studied with the quaternionic representation of the coxeter group w h4 the maximal subgroups w su 5 z2 and w h3 z2 of w h4 play important roles in the analysis of cell structures of the dual polytopes in particular the weyl coxeter group w su 4 is used to determine the tetrahedral cells of the polytope 3 3 5 and the coxeter group w h3 is used for the dodecahedral cells of 5 3 3 using the lie algebraic techniques in terms of quaternions we explicitly construct cell structures forming the vertices of the 4d polytopes

absolute summability of the trace relation for certain schrodinger operators a recently established general trace formula for one dimensional schrodinger operators is systematically studied in the context of short range potentials potentials which approach different spatial asymptotes sufficiently fast and appropriate impurity defect interactions in one dimensional solids we prove the absolute summability of the trace formula and establish its connections with scattering quantities such as reflection coefficients in each case

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

almost periodicity of some random potentials in this paper we study some random potentials with purely absolutely continuous spectrum in one dimension there are no examples of random potentials with absolutely continuous spectrum that are not almost periodic though it is known from kotani 3 that such potentials are deterministic hence one could conjecture that any random potential with purely absolutely continuous spectrum is almost periodic we show this result for random potentials with infinitely many bands in their spectrum such that the gaps behave well in the finite bands case concini and johnson l discussed the same problem a precise definition of the behavior of the gaps could be found in condition 2 3 we make use of the results of levitan 7 kotani 3 4 and moser 9 for doing this we also discuss the behavior of the lyapunov exponent asymptotically and its relation to the smoothness of the potentials

uniqueness theorems in inverse spectral theory for one dimensional schrodinger operators new unique characterization results for the potential v x in connection with schrodinger operators on r and on the half line 0 are proven in terms of appropriate krein spectral shift functions particular results obtained include a generalization of a well known uniqueness theorem of borg and marchenko for schrodinger operators on the half line with purely discrete spectra to arbitrary spectral types and a new uniqueness result for schrodinger operators with confining potentials on the entire real line

spectral properties of a class of reflectionless schrodinger operators abstract we prove that one dimensional reflectionless schrodinger operators with spectrum a homogeneous set in the sense of carleson belonging to the class introduced by sodin and yuditskii have purely absolutely continuous spectra this class includes all earlier examples of reflectionless almost periodic schrodinger operators in addition we construct examples of reflectionless schrodinger operators with more general types of spectra given by the complement of a denjoy widom type domain in c which exhibit a singular component

wavelet least squares methods for boundary value problems this paper is concerned with least squares methods for the numerical solution of operator equations our primary focus is the discussion of the following conceptual issues the selection of appropriate least squares functionals their numerical evaluation in the special light of recent developments of wavelet methods and a natural way of preconditioning the resulting systems of linear equations we describe first a general format of variational problems that are well posed in a certain natural topology in order to illustrate the scope of these problems we identify several special cases such as second order elliptic boundary value problems their formulation as a first order system transmission problems the system of stokes equations or more general saddle point problems particular emphasis is placed on the separate treatment of essential nonhomogeneous boundary conditions we propose a unified treatment based on wavelet expansions in particular we exploit the fact that weighted sequence norms of wavelet coefficients are equivalent to relevant function norms arising in the least squares context this provides access to difficult norms efficient preconditioners and in the case of first order systems optimal l2 error estimates

symmetric coupling of boundary elements and raviart thomas type mixed finite elements in elastostatics both mixed finite element methods and boundary integral methods are important tools in computational mechanics according to a good stress approximation recently even low order mixed methods of raviart thomas type became available for problems in elasticity since either methods are robust for critical poisson ratios it appears natural to couple the two methods as proposed in this paper the symmetric coupling changes the elliptic part of the bilinear form only hence the convergence analysis of mixed finite element methods is applicable to the coupled problem as well specifically we couple boundary elements with a family of mixed elements analyzed by stenberg the locking free implementation is performed via lagrange multipliers numerical examples are included