cyclic homology comodules and mixed complexes relating cyclic and hochschild homology one can see that any formula for hc a a will not only involve hc a and hc a but also the periodicity operator s and the hochschild groups fortunately if the map s brings in some complications it also endows hc a with a comodule structure over the cyclic homology of the ground ring k actually the comodule structure exists already on the complex level a convenient complex on which s has the form of a canonical surjection is connes s double complex with differentials b and b now the idea is to view the cyclic homology of an algebra as a composite functor

a numerical study on the solution of the cauchy problem in elasticity this work deals with the cauchy problem in two dimensional linear elasticity the equations of the problem are discretized through a standard fem approach and the resulting ill conditioned discrete problem is solved within the frame of the tikhonov approach the choice of the required regularization parameter is accomplished through the generalized cross validation criterion on this basis a numerical experimentation has been performed and the calculated solutions have been used to highlight the sensitivity to the amount of known data the noise always present in the data the regularity of boundary conditions and the choice of the regularization parameter the aim of the numerical study is to implicitly device some guidelines to be used in the solution of this kind of problems

an operator splitting radial basis function method for the solution of transient nonlinear poisson problems abstract this paper presents an operator splitting radial basis function os rbf method as a generic solution procedure for transient nonlinear poisson problems by combining the concepts of operator splitting radial basis function interpolation particular solutions and the method of fundamental solutions the application of the operator splitting permits the isolation of the nonlinear part of the equation that is solved by explicit adams bashforth time marching for half the time step this leaves a nonhomogeneous modified helmholtz type of differential equation for the elliptic part of the operator to be solved at each time step the resulting equation is solved by an approximate particular solution and by using the method of fundamental solution for the fitting of the boundary conditions radial basis functions are used to construct approximate particular solutions and a grid free dimension independent method with high computational efficiency is obtained this method is demonstrated for some prototypical nonlinear poisson problems in heat and mass transfer and for a problem of transient convection with diffusion the results obtained by the os rbf method compare very well with those obtained by other traditional techniques that are computationally more expensive the new os rbf method is useful for both general irregular two and three dimensional geometry and provides a mesh free technique with many mathematical flexibilities and can be used in a variety of engineering applications

fourier integral operators fractal sets and the regular value theorem we prove that if e r2d for d 2 is an ahlfors david regular product set of sufficiently large hausdorff dimension denoted by dimh e and is a sufficiently regular function then the upper minkowski dimension of the set w e l w tl 1 l m does not exceed dimh e m in line with the regular value theorem from the elementary differential geometry our arguments are based on the mapping properties of the underlying fourier integral operators and are intimately connected with the falconer distance conjecture in geometric measure theory we shall see that our results are in general sharp in the sense that if the hausdorff dimension is smaller than a certain threshold then the dimensional inequality fails in a quantifiable way the constructions used to demonstrate this are based on the distribution of lattice points on convex surfaces and have connections with combinatorial geometry

the collocation points of the fundamental solution method for the potential problem we propose an algorithm for computing harmonic functions in a two dimensional domain with prescribed dirichlet data our algorithm is a variant of what is called a fundamental solution method 1 3 this method requires us to select 2n points in the two dimensional plane n of which are called collocation points and the remaining n are called charge points representing the position of the singularities of the fundamental solution it is known that there exists a set of 2n points by which the error is exponentially small 1 3 6 however these papers are concerned mainly with existence and as far as the authors know few fast and reliable algorithms are known for good position of the points in this paper we propose a new rule for the position of the points and examine its efficiency by numerical experiments the new rule uses fft effectively

finite element solution of two dimensional inverse elastic problems using spatial smoothing a finite element method fem is presented for calculating the surface tractions on a body from internal measurements of displacement at discrete sensor locations the solution algorithm employs a sensitivity analysis which minimizes the difference between the calculated and measured displacements at each sensor location spatial regularization is one technique employed to stabilize the minimization process by imposing various degrees of smoothness on the solution it also allows the problem to be solved with fewer sensors than traction boundary nodes as an alternative to spatial regularization a method based on keynodes is introduced which assumes that each component of the boundary traction distribution can be described by a polynomial of specified order the methods are applied to several two dimensional examples including a rolling contact problem the effects of parameters such as the number of sensors the location of the sensors and the error in the sensor displacements are discussed

the method of fundamental solutions for the numerical solution of the biharmonic equation abstract the method of fundamental solutions mfs is a relatively new technique for the numerical solution of certain elliptic boundary value problems it falls in the class of methods generally called boundary methods and like the well known boundary integral equation method is applicable when a fundamental solution of the differential equation is known in the mfs the approximate solution is a linear combination of fundamental solutions with singularities placed outside the domain of the problem the locations of the singularities are either preassigned or determined along with the coefficients of the fundamental solutions so that the approximate solution satisfies the boundary conditions as well as possible in many applications these quantities are determined by a least squares fit of the boundary conditions a nonlinear problem which is solved using standard software in this paper the mfs is formulated for biharmonic problems and is applied to a variety of standard test problems as well as to problems arising in elasticity and fluid flow

an iterative method for the cauchy problem in linear elasticity with fading regularization effect in this paper an iterative method for solving the cauchy problem in linear elasticity is introduced this problem consists in recovering missing data displacements and forces on some parts of a domain boundary from the knowledge of overspecified data displacements and forces on the remaining parts the algorithm reads as a least square fitting of the given data with a regularization term whose effect fades as the iterations go on so the algorithm converges to the solution of the cauchy problem numerical simulations using the finite element method highlight the algorithm s efficiency accuracy robustness to noisy data as well as its ability to deblur noisy data

an implicit function theorem for banach spaces and some applications we prove a generalized implicit function theorem for banach spaces without the usual assumption that the subspaces involved being complemented then we apply it to the problem of parametrization of fibers of differentiable maps the lie subgroup problem for banach lie groups as well as weil s local rigidity for homomorphisms from finitely generated groups to banach lie groups

superpotential algebras and manifolds we study a special class of calabi yau algebras in the sense of ginzburg those arising as the fundamental group algebras of acyclic manifolds motivated partly by the usefulness of superpotential descriptions in motivic donaldson thomas theory we investigate the question of whether these algebras admit superpotential presentations we establish that the fundamental group algebras of a wide class of acyclic manifolds including all hyperbolic manifolds do not admit such descriptions disproving a conjecture of ginzburg regarding them we also describe a class of manifolds that do admit such descriptions and discuss a little their motivic donaldson thomas theory finally some links with topological field theory are described

the minimal error method for the cauchy problem in linear elasticity numerical implementation for two dimensional homogeneous isotropic linear elasticity abstract in this paper yet another iterative procedure namely the minimal error method mem for solving stably the cauchy problem in linear elasticity is introduced and investigated furthermore this method is compared with another two iterative algorithms i e the conjugate gradient cgm and landweber fridman methods lfm previously proposed by marin et al marin l hao d n lesnic d 2002b conjugate gradient boundary element method for the cauchy problem in elasticity quarterly journal of mechanics and applied mathematics 55 227 247 and marin and lesnic marin l lesnic d 2005 boundary element landweber method for the cauchy problem in linear elasticity ima journal of applied mathematics 18 817 825 respectively in the case of two dimensional homogeneous isotropic linear elasticity the inverse problem analysed in this paper is regularized by providing an efficient stopping criterion that ceases the iterative process in order to retrieve stable numerical solutions the numerical implementation of the aforementioned iterative algorithms is realized by employing the boundary element method bem for two dimensional homogeneous isotropic linear elastic materials

a meshless method for solving the cauchy problem in three dimensional elastostatics the application of the method of fundamental solutions to the cauchy problem in three dimensional isotropic linear elasticity is investigated the resulting system of linear algebraic equations is ill conditioned and therefore its solution is regularized by employing the first order tikhonov functional while the choice of the regularization parameter is based on the l curve method numerical results are presented for both under and equally determined cauchy problems in a piece wise smooth geometry the convergence accuracy and stability of the method with respect to increasing the number of source points and the distance between the source points and the boundary of the solution domain and decreasing the amount of noise added into the input data respectively are analysed

the method of fundamental solutions and condition number analysis for inverse problems of laplace equation this paper investigates the applications of the method of fundamental solutions together with the conditional number analysis to solve various inverse 2d laplace problems involving under specified and or over specified boundary conditions through the method of fundamental solutions and the condition number analysis it is numerically found that solutions of inverse laplace problems can be obtained without iteration or regularization for small noise levels since the method of fundamental solutions is a boundary type meshless numerical method that can automatically satisfy the governing equation however for larger values of noise levels regularization is still necessary to obtain promising result the present paper mainly focuses on the two types of numerical predictions of inverse 2d laplace problems 1 cauchy problem and 2 shape identification problem good quantitative agreement with the analytical solutions and other numerical methods for small perturbed boundary data is observed by using present meshless numerical scheme

non iterative regularized mfs solution of inverse boundary value problems in linear elasticity numerical reconstruction of the missing boundary data in 2d and 3d linear elasticity noisy over prescribed data are available on the remaining accessible boundary the inverse problem is solved by a meshless method i e the method of fundamental solutions mfs stabilization is achieved via several svd based regularization methods the regularization parameter is selected via the criteria used for inverse problems the numerical reconstruction of the missing dirichlet and neumann data on an inaccessible part of the boundary in the case of two and three dimensional linear isotropic elastic materials from the knowledge of over prescribed noisy measurements taken on the remaining accessible boundary part is investigated this inverse problem is solved using the method of fundamental solutions mfs whilst its stabilization is achieved through several singular value decomposition svd based regularization methods such as the tikhonov regularization method 48 the damped svd and the truncated svd 18 the regularization parameter is selected according to the discrepancy principle 40 generalized cross validation criterion 14 and hansen s l curve method 20

quantum harmonic analysis and geometric invariants we develop two topics in parallel and show their inter relation the first centers on the notion of a fractional differentiable structure on a commutative or a non commutative space we call this studyquantum harmonic analysis the second concerns homotopy invariants for these spaces and is an aspect of non commutative geometry we study an algebra a which will be a banach algebra with unit represented as an algebra of operators on a hilbert space h in order to obtain a geometric interpretation of a we define a derivative on elements of a we do this in a hilbert space context takingdaas a commutatorda q a hereqis a basic self adjoint operator with discrete spectrum increasing sufficiently rapidly that exp q 2has a trace whenever 0 we can define fractional differentiability of order with 0 0 we concentrate on subalgebras a of j which have certain additional covariance properties under a group z2 g acting on h by a unitary representation u g in addition the derivativeqis assumed to be g invariant the geometric interpretation flows from the assumption that elements of a possess an arbitrarily small fractional derivative we study homotopy invariants of a in terms of equivariant entire cyclic cohomology in fact the existence of a fractional derivative on a allows the construction of the cochain jlo which plays the role of the integral of differential forms we give a simple expression for a homotopy invariant zq a g determined by pairing jlo with a g invariant elementa a such thatais a square root of the identity this invariant is zq a g 1 e t2tr u g ae q2 it da dt this representation of the pairing is reminiscent of the heat kernel representation for an index in fact this quantity is an invariant in the following sense we isolate a simple condition on a familyq of differentiations that yields a continuously differentiable family jlo of cochains since zq a g need not be an integer continuity of jlo in is insufficient to prove the constancy of the pairing however the existence of the derivative leads to the existence of the homotopy as a vanishes for a coboundary and asd jlo d is a coboundary our condition onq ensures that zq a g is independent of hence it is a homotopy invariant the theory of zq a g reduces to the study of the radon transform of sequences of certain functions the fractional differentiability properties of elements of a translate into properties of the asymptotics of the sequences of radon transforms the condition that jlofit into the framework of entire cyclic cohomology translates to the existence of some fractional derivative for functions in the algebra under study and in particular the assumption 1 thus the study of fractionally differentiable structures dove tails naturally with the theory of homotopy invariants in our study of quantum harmonic analysis we introduce spaces t of operator valued distributions these spaces are bounded linear operators between sobolev spaces the elements of the interpolation spaces the banach algebras j have derivativesdawhich belong to the spaces t for a certain range of and we extend the theory of the radon transform from products of regularized bounded operators to products of regularized operator valued distributions we sometimes wish to evaluate such an invariant at the endpoint of an interval such as 0 1 where zq a g becomes singular as 0 we discuss in brief a procedure to regularize the endpoint and a method to recover zq a g fully from certain partial information at the endpoint finally we generalize this approach to cover the case whenqcan be split into the sum of independent partsq1 q2 such that q1 q2 2 q1 2 q2 2 here we assume thatq1and q2 2are g invariant but not necessarilyq2 with further assumptions ona the most important being that q1 2 q2 2commutes witha we obtain a modified formula for an invariant namely z qj a g 1 e t2tr u g ae q21 q22 2 itd1a dt

numerical solution of cauchy problems in linear elasticity in axisymmetric situations abstract an iterative method for solving axisymmetric cauchy problems in linear elasticity is presented this kind of problem consists in recovering missing displacements and forces data on one part of a domain boundary from the knowledge of overspecified displacements and forces data on another part of this boundary numerical simulations using the finite element method highlight the algorithm s efficiency accuracy and robustness to noisy data as well as its ability to deblur noisy data an application of the inverse technique to the identification of a friction coefficient is also presented

the mfs for the cauchy problem in two dimensional steady state linear thermoelasticity we study the reconstruction of the missing thermal and mechanical fields on an inaccessible part of the boundary for two dimensional linear isotropic thermoelastic materials from over prescribed noisy cauchy data on the remaining accessible boundary this problem is solved with the method of fundamental solutions mfs together with the method of particular solutions mps via the mfs based particular solution for two dimensional problems in uncoupled thermoelasticity developed in marin and karageorghis 2012a 2013 the stabilisation regularization of this inverse problem is achieved by using the tikhonov regularization method tikhonov and arsenin 1986 whilst the optimal value of the regularization parameter is selected by employing hansen s l curve method hansen 1998

analysis of lavrentiev finite element methods for data completion problems the variational finite element solution of cauchy s problem expressed in the steklov poincare framework and regularized by the lavrentiev method has been introduced and computationally assessed in azaiez et al inverse probl sci eng 18 1063 1086 2011 the present work concentrates on the numerical analysis of the semi discrete problem we perform the mathematical study of the error to rigorously establish the convergence of the global bias variance error

operations on cyclic homology the x complex and a conjecture of deligne we prove that there is a product on the hochschild and cyclic chain complex of a homotopy gerstenhaber algebra by restricting to the special case of the algebra of hochschild cochains the so called deformation complex we obtain operations on cyclic homology of associative algebras

local structure of schelter procesi smooth orders in this paper we give the etale local classification of schelterprocesi smooth orders in central simple algebras in particular we prove that if is a central simple k algebra of dimension n2 where k is a field of trancendence degree d then there are only finitely many etale local classes of smooth orders in this result is a non commutative generalization of the fact that a smooth variety is analytically a manifold and so has only one type of etale local behaviour

a particular solution trefftz method for non linear poisson problems in heat and mass transfer this paper proposes a computational procedure based on the trefftz method for the solution of non linear poisson problems the problem is solved by finding an approximate particular solution to the poisson equation and using boundary collocation to solve the resulting laplace equation this method results in a global collocation and hence eliminates the need for discretization of the domain in both two and three dimensions the solution procedure proposed earlier in the literature based upon the above method has been reformulated for increased computational efficiency a quasi newton iteration method along with a new heuristic for source point location is used for efficient convergence of the numerical scheme the efficacy of the new formulation has been demonstrated for two classes of problems viz the thermal explosion problem and the diffusion reaction problem in a partially wetted catalyst pellet a brief error analysis of the method as well as future research directions is presented

regularized method of fundamental solutions for boundary identification in two dimensional isotropic linear elasticity we investigate the stable numerical reconstruction of an unknown portion of the boundary of a two dimensional domain occupied by an isotropic linear elastic material from a prescribed boundary condition on this part of the boundary and additional displacement and traction measurements i e cauchy data on the remaining known portion of the boundary this inverse geometric problem is approached by combining the method of fundamental solutions mfs and the tikhonov regularization method whilst the optimal value of the regularization parameter is chosen according to the discrepancy principle various geometries are considered i e convex and non convex domains with a smooth or piecewise smooth boundary in order to show the numerical stability convergence consistency and computational efficiency of the proposed method

the fundamental solution of mindlin plates resting on an elastic foundation in the laplace domain and its applications abstract the aim of this study is to investigate the method of fundamental solution mfs applied to a shear deformable plate reissner mindlin s theories resting on the elastic foundation under either a static or a dynamic load the complete expressions for internal point kernels i e fundamental solutions by the boundary element method for the mindlin plate theory are derived in the laplace transform domain for the first time on employing the mfs the boundary conditions are satisfied at collocation points by applying point forces at source points outside the domain all variables in the time domain can be obtained by durbin s laplace transform inversion method numerical examples are presented to demonstrate the accuracy of the mfs and comparisons are made with other numerical solutions in addition the sensitivity and convergence of the method are discussed for a static problem the proposed mfs is shown to be simple to implement and gives satisfactory results for shear deformable plates under static and dynamic loads

the method of fundamental solutions for inverse boundary value problems associated with the steady state heat conduction in anisotropic media in this paper the method of fundamental solutions is applied to solve some inverse boundary value problems associated with the steady state heat conduction in an anisotropic medium since the resulting matrix equation is severely ill conditioned a regularized solution is obtained by employing the truncated singular value decomposition while the optimal regularization parameter is chosen according to the l curve criterion numerical results are presented for both two and three dimensional problems as well as exact and noisy data the convergence and stability of the proposed numerical scheme with respect to increasing the number of source points and the distance between the fictitious and physical boundaries and decreasing the amount of noise added into the input data respectively are analysed a sensitivity analysis with respect to the measure of the accessible part of the boundary and the distance between the internal measurement points and the boundary is also performed the numerical results obtained show that the proposed numerical method is accurate convergent stable and computationally efficient and hence it could be considered as a competitive alternative to existing methods for solving inverse problems in anisotropic steady state heat conduction copyright 2005 john wiley sons ltd

the periodic cyclic homology of crossed products of finite type algebras we study the periodic cyclic homology groups of the cross product of a finite type algebra a by a discrete group in case a is commutative and the group is finite our results are complete and given in terms of the singular cohomology of the sets of fixed points these groups identify our cyclic homology groups with the orbifold cohomology of the underlying algebraic orbifold the proof is based on a careful study of localization at fixed points and of the resulting koszul complexes this is achieved by extending to our class of noncommutative algebras the concept of an infinitesimal neighborhood that plays such an important role in commutative algebra we provide examples of azumaya algebras for which this identification is however no longer valid as an example we discuss some affine weyl groups

error estimate procedure in the finite element method and applications statically and kinematically admissible fields are explicitly derived from the finite element solution of the primal form of linear models the error on constitutive law for these fields yields an expression of the finite element error moreover the contribution of each element to this error allows to implement an automatic mesh refinement procedure leading to a uniform distribution of a given accuracy

an invariant method of fundamental solutions for two dimensional isotropic linear elasticity abstract in this paper we propose an invariant method of fundamental solutions for solving boundary value problems in two dimensional isotropic linear elasticity the invariant method of fundamental solutions keeps a very basic natural property the so called invariance property under trivial coordinate changes in the problem description e g dilations and or contractions the problems are solved by the tikhonov regularization method in conjunction with morozov s discrepancy principle finally five examples of boundary value problems are investigated to show the efficiency of this method the numerical convergence accuracy and stability of the proposed method with respect to the number of source points the distance between the sources and the boundary and the amount of noise added into the input data respectively are also analyzed

bivariant chern character and longitudinal index in this paper we consider a family of dirac type operators on fibration p b equivariant with respect to an action of an etale groupoid such a family defines an element in the bivariant k theory we compute the action of the bivariant chern character of this element on the image of connes map in the cyclic cohomology a particular case of this result is connes index theorem for etale groupoids a connes noncommutative geometry academic press 1994 in the case of fibrations

exponential basis functions in solution of static and time harmonic elastic problems in a meshless style in this paper exponential basis functions ebfs are used in a boundary collocation style to solve engineering problems whose governing partial differential equations pdes are of constant coefficient type complex valued exponents are considered for the ebfs two dimensional elasto static and time harmonic elasto dynamic problems are chosen in this paper the solution procedure begins with first finding a set of appropriate ebfs and then considering the solution as a summation of such ebfs with unknown coefficients the unknown coefficients are determined by the satisfaction of the boundary conditions through a collocation method with the aid of a consistent and complex discrete transformation technique the basis and various forms of the transformation have been addressed and discussed we shall propose several strategies for selection of ebfs with the aid of the basis explained for the transformation while using the transformation the number of ebfs should not necessarily be equal to or less than the number of boundary information data a library of ebfs has also been presented for further use the effect of body forces is included in the solution via construction of particular solution by the use of the discrete transformation and another series of ebfs a number of sample problems are solved to demonstrate the capabilities of the method it has been shown that the time harmonic problems with high wave number can be solved without much effort the method categorized in meshless methods can be applied to many other problems in engineering mechanics and general physics since ebfs can easily be found for almost all problems with constant coefficient pdes

quasi homomorphismes cohomologie cyclique et positivite we show that cyclic cohomology of an algebraa is obtained from traces with suitable domains on the algebraqa of the second author whena is ac algebra so isqa and the notion of positive trace makes sense we hence get a notion of positivity for cyclic cocycles we prove that a positive trace onqa defines a type i or ii fredholm module ona

an integral equations method combined minimum norm solution for 3d elastostatics cauchy problem abstract in this paper we establish new density results for the equilibrium equations based on the denseness result of the elastic potential functions the cauchy problem for the equilibrium equations is investigated for this ill posed problem we construct a regularizing solution using the single layer potential function the well posedness of the regularizing solution as well as the convergence property is rigorously analyzed the advantage of the proposed scheme is that the regularizing solution is of the explicit analytic solution and therefore is easy to be implemented the method combines minimum norm solution with morozov discrepancy principle to solve an inverse problem convergence and stability estimates are then given with some examples for numerical verification on the efficiency of the proposed method the numerical convergence accuracy and stability of the method with respect to the discretisation about the integral equations on pseudo boundary and the distance between the pseudo boundary and the real boundary of the solution domain and decreasing the amount of noise added into the input data respectively are also analysed with some examples

extensions and renormalized traces it has been shown by nistor doc math j dmv 2 263 295 1997 that given any extension of associative algebras over mathbb c the connecting morphism in periodic cyclic homology is compatible under the chern connes character with the index morphism in lower algebraic k theory the proof relies on the abstract properties of cyclic theory essentially excision which does not provide explicit formulas a priori avoiding the use of excision we explain in this article how to get explicit formulas in a wide range of situations the method is connected to the renormalization procedure introduced in our previous work on the bivariant chern character for quasihomomorphisms perrot j geom phys 60 1441 1473 2010 leading to local index formulas in the sense of non commutative geometry we illustrate these principles with the example of the classical family index theorem we find that the characteristic numbers of the index bundle associated to a family of elliptic pseudodifferential operators are expressed in terms of the fiberwise wodzicki residue

matrix decomposition mfs algorithms for elasticity and thermo elasticity problems in axisymmetric domains in this work we propose an efficient matrix decomposition algorithm for the method of fundamental solutions when applied to three dimensional boundary value problems governed by elliptic systems of partial differential equations in particular we consider problems arising in linear elasticity in axisymmetric domains the proposed algorithm exploits the block circulant structure of the coefficient matrices and makes use of fast fourier transforms the algorithm is also applied to problems in thermo elasticity several numerical experiments are carried out

algebra extensions and nonsingularity this paper is concerned with a notion of nonsingularity for noncommutative algebras which arises naturally in connection with cyclic homology let us consider associative unital algebras over the complex numbers we call an algebra a quasi free when it behaves like a free algebra with respect to nilpotent extensions in the sense that any homomorphism a r i where i is a nilpotent ideal in r can be lifted to a homomorphism a r if we restrict to the category of finitely generated commutative algebras then this lifting property characterizes smooth algebras the ones corresponding to nonsingular affine varieties in this way quasi free algebras appear as noncommutative analogues of smooth algebras stretching the analogy we can even regard quasi free algebras as analogues of manifolds one of the aims of this paper is to develop the analogy further by showing that quasi free algebras provide a natural setting for noncommutative versions of certain aspects of manifolds to give an example let us consider the analogue of an embedding an extension a r i where a and r are quasi free algebras playing the role of the submanifold and ambient manifold respectively in the manifold situation i i2 is the module of linear functions on the nor2 mal bundle and the symmetric algebra sa iii is the algebra of polynomial functions now in passing from commutative to noncommutative algebras the symmetric algebra of a module is replaced by the tensor algebra of a bimodule

a bivariant chern character for families of spectral triples in this paper we construct a bivariant chern character defined on families of spectral triples such families should be viewed as a version of unbounded kasparov bimodules adapted to the category of bornological algebras the chern character then takes its values in the bivariant entire cyclic cohomology of meyer the basic idea is to work within quillen s algebra cochains formalism and construct the chern character from the exponential of the curvature of a superconnection leading to a heat kernel regularization of traces the obtained formula is a bivariant generalization of the jlo cocycle

the method of fundamental solutions for the cauchy problem in two dimensional linear elasticity in this paper the application of the method of fundamental solutions to the cauchy problem in two dimensional isotropic linear elasticity is investigated the resulting system of linear algebraic equations is ill conditioned and therefore its solution is regularised by employing the first order tikhonov functional while the choice of the regularisation parameter is based on the l curve method numerical results are presented for both smooth and piecewise smooth geometries as well as for constant and linear stress states the convergence and the stability of the method with respect to increasing the number of source points and the distance between the source points and the boundary of the solution domain and decreasing the amount of noise added into the input data respectively are analysed

fundamental solutions method for elliptic boundary value problems we consider a procedure for solving boundary value problems for elliptic homogeneous equations known as the fundamental solutions method we prove its applicability for some second order operators as well as for fourth order ones the boundary conditions of an elliptic problem are approximated by using fundamental solutions of the corresponding operator with singularities located outside the domain of interest in a specific case of the laplacian an estimate shows that the method discussed possesses convergence properties as good as those of any method using harmonic polynomials as trial functions numerical examples are given among which are simple signorini s problems for harmonic and lame operators

elastic scatterer reconstruction via the adjoint sampling method an inverse problem dealing with the reconstruction of voids in a uniform semi infinite solid from near field elastodynamic waveforms is investigated via the linear sampling method to cater to active imaging applications that are characterized by a limited density of illuminating sources existing formulation of the linear sampling method is advanced in terms of its adjoint statement that features integration over the receiver surface rather than its source counterpart to deal with an ill posedness of the integral equation that is used to reconstruct the obstacle the problem is solved by alternative means of tikhonov regularization and a preconditioned conjugate gradient method through a set of numerical examples it is shown i that the adjoint statement elevates the performance of the linear sampling method when dealing with scarce illuminating sources and ii that a combined use of the existing formulation together with its adjoint counterpart represents an effective tool for exposing an undersam

an energy error based method for the resolution of the cauchy problem in 3d linear elasticity a new method is described for the problem of expanding known displacement fields at the boundary of a solid together with the surface tractions on it towards the solid interior up to the inaccessible part of the boundary the solid is supposedly linearly elastic with known elastic moduli but not necessarily homogeneous nor isotropic the problem is the cauchy problem for the lame operator a new form of this cauchy problem suited for applications associated with surface tangential fields measurements is also stated and studied the method is based on the splitting of the elastic fields into two separate solutions of wellposed problems the gap between these fields is subsequently minimised with respect to the unknown boundary data in order to produce the desired expanded elastic fields the gap used here is an energy error associated with the elastic energy of the system various 3d applications are given including non linear boundary conditions on the unreachable boundary

excision in cyclic homology theories in this paper we establish excision for z 2z graded cyclic homology theories including periodic entire asymptotic and local bivariant cyclic cohomology our work is a modification of the approach of cuntz quillen cq to excision in bivariant periodic cyclic homology and makes also use of the work of wodzicki wo we study the dimension shift of periodic cyclic cohomology under the boundary map associated to an extension of algebras and obtain the possible values of this dimension shift excision in entire cyclic cohomology is used to calculate these groups in a number of hitherto unknown cases finally a bivariant and multiplicative chern connes character on kasparov s bivariant k theory ka with values in bivariant local cyclic cohomology is constructed to put our approach into perspective we recall the previous work on excision in cyclic homology ever since the invention of cyclic homology by connes and independently by tsygan the problem of excision played a central role in the theory it is concerned with the question of to what extent an extension

inverse cauchy method with conformal mapping application to latent flatness defect detection during rolling process abstract in this paper a semi analytical inverse cauchy problem is presented for a finite cylinder with a cylindrical hole under the surface called the detecting roll applications to latent flatness defect detection during rolling process are considered a steel strip wraps at a known angle around the detecting roll the presented inverse method aims at evaluating the residual stress profile of the strip the inputs are the displacements and free surface traction at the hole surface and the outputs are the resultant forces per unit length in the contact between the strip and the roll thus by equilibrium of the strip the residual stress profile is inferred this paper is based on plane complex elasticity with conformal mapping techniques several 2d computations are performed at several axial positions this work is theoretical the measurement device is not available and technical issues are not broached some numerical examples with synthetic data are provided in order to evaluate accuracy and noise sensitivity results encourage further investigations and technical developments

fading regularization mfs algorithm for inverse boundary value problems in two dimensional linear elasticity abstract we investigate the numerical reconstruction of the missing displacements dirichlet data and tractions neumann data on an inaccessible part of the boundary in the case of two dimensional linear isotropic elastic materials from the knowledge of over prescribed noisy measurements taken on the remaining accessible boundary part this inverse problem is solved using the fading regularization method originally proposed by cimetiere et al 2000 2001 for the laplace equation in conjunction with a meshless method namely the method of fundamental solutions mfs the stabilisation of the numerical method proposed herein is achieved by stopping the iterative procedure according to morozov s discrepancy principle morozov 1966

investigation of regularization parameters and error estimating in inverse elasticity problems the method of tarantola1 based on bayesian statistical theory for solving general inverse problems is applied to inverse elasticity problems and is compared to the spatial regularization technique presented in schnur and zabaras 2 it is shown that when normal gaussian distributions are assumed and the error in the data is uncorrelated the bayesian statistical theory takes a form similar to the deterministic regularization method presented earlier in schnur and zabaras 2 as such the statistical theory can be used to provide a statistical interpretation of regularization and to estimate error in the solution of the inverse problem examples are presented to demonstrate the effect of the regularization parameters and the error in the initial data on the solution

a short survey of noncommutative geometry we give a survey of selected topics in noncommutative geometry with some emphasis on those directly related to physics including our recent work with dirk kreimer on renormalization and the riemann hilbert problem we discuss at length two issues the first is the relevance of the paradigm of geometric space based on spectral considerations which is central in the theory as a simple illustration of the spectral formulation of geometry in the ordinary commutative case we give a polynomial equation for geometries on the four sphere with fixed volume the equation involves an idempotent e playing the role of the instanton and the dirac operator d it is of the form e 12 d e 4 5 and determines both the sphere and all its metrics with fixed volume form the expectation x is the projection on the commutant of the algebra of 4 by 4 matrices we also show using the noncommutative analog of the polyakov action how to obtain the noncommutative metric in spectral form on the noncommutative tori fr

a relaxation method of an alternating iterative algorithm for the cauchy problem in linear isotropic elasticity we propose two algorithms involving the relaxation of either the given dirichlet data boundary displacements or the prescribed neumann data boundary tractions on the over specified boundary in the case of the alternating iterative algorithm of kozlov et al 16 applied to cauchy problems in linear elasticity a convergence proof of these relaxation methods is given along with a stopping criterion the numerical results obtained using these procedures in conjunction with the boundary element method bem show the numerical stability convergence consistency and computational efficiency of the proposed method

the method of fundamental solutions for inverse source problems associated with the steady state heat conduction this paper presents the use of the method of fundamental solutions mfs for recovering the heat source in steady state heat conduction problems from boundary temperature and heat flux measurements it is well known that boundary data alone do not determine uniquely a general heat source and hence some a priori knowledge is assumed in order to guarantee the uniqueness of the solution in the present study the heat source is assumed to satisfy a second order partial differential equation on a physical basis thereby transforming the problem into a fourth order partial differential equation which can be conveniently solved using the mfs since the matrix arising from the mfs discretization is severely ill conditioned a regularized solution is obtained by employing the truncated singular value decomposition whilst the optimal regularization parameter is determined by the l curve criterion numerical results are presented for several two dimensional problems with both exact and noisy data the sensitivity analysis with respect to two solution parameters i e the number of source points and the distance between the fictitious and physical boundaries and one problem parameter i e the measure of the accessible part of the boundary is also performed the stability of the scheme with respect to the amount of noise added into the data is analysed the numerical results obtained show that the proposed numerical algorithm is accurate convergent stable and computationally efficient for solving inverse source problems in steady state heat conduction copyright 2006 john wiley sons ltd

relaxation procedures for an iterative mfs algorithm for the stable reconstruction of elastic fields from cauchy data in two dimensional isotropic linear elasticity we investigate two numerical procedures for the cauchy problem in linear elasticity involving the relaxation of either the given boundary displacements dirichlet data or the prescribed boundary tractions neumann data on the over specified boundary in the alternating iterative algorithm of kozlov et al 1991 the two mixed direct well posed problems associated with each iteration are solved using the method of fundamental solutions mfs in conjunction with the tikhonov regularization method while the optimal value of the regularization parameter is chosen via the generalized cross validation gcv criterion an efficient regularizing stopping criterion which ceases the iterative procedure at the point where the accumulation of noise becomes dominant and the errors in predicting the exact solutions increase is also presented the mfs based iterative algorithms with relaxation are tested for cauchy problems for isotropic linear elastic materials in various geometries to confirm the numerical convergence stability accuracy and computational efficiency of the proposed method

identification of boundary conditions by solving cauchy problem in linear elasticity with material uncertainties this work is a contribution to non destructive testing in the context of uncertainties it consists in identifying boundary conditions on an inaccessible part of a solid body boundary from the knowledge of over specified data given on an accessible part of this boundary this problem is well known as cauchy problem the material properties are considered uncertain polynomial chaos expansion is used in order to solve this cauchy problem in the random context a stochastic constrained optimization problem is formulated and solved numerical experiments are presented

nonabelian cohomology with coefficients in lie groups in this paper we prove some properties of the nonabelian cohomology h 1 a g of a group a with coefficients in a connected lie group g when a is finite we show that for every a submodule k of g which is a maximal compact subgroup of g the canonical map h1 a k h 1 a g is bijective in this case we also show that h 1 a g is always finite when a z and g is compact we show that for every maximal torus t of the identity component g z 0 of the group of invariants g z h 1 z t h 1 z g is surjective if and only if the z action on g is 1 semisimple which is also equivalent to the fact that all fibers of h 1 z t h 1 z g are finite when a z nz we show that h 1 z nz t h 1 z nz g is always surjective where t is a maximal compact torus of the identity component g z nz of g z nz when a is cyclic we also interpret some properties of h 1 a g in terms of twisted conjugate actions of g

singular boundary method for solving plane strain elastostatic problems this study documents the first attempt to apply the singular boundary method sbm a novel boundary only collocation method to two dimensional 2d elasticity problems unlike the method of fundamental solutions mfs the source points coincide with the collocation points on the physical boundary by using an inverse interpolation technique to regularize the singularity of the fundamental solution of the equation governing the problems of interest three benchmark elasticity problems are tested to demonstrate the feasibility and accuracy of the proposed method through detailed comparisons with the mfs boundary element method bem and finite element method fem

d branes rr fields and duality on noncommutative manifolds we develop some of the ingredients needed for string theory on noncommutative spacetimes proposing an axiomatic formulation of t duality as well as establishing a very general formula for d brane charges this formula is closely related to a noncommutative grothendieck riemann roch theorem that is proved here our approach relies on a very general form of poincare duality which is studied here in detail among the technical tools employed are calculations with iterated products in bivariant mathsf k theory and cyclic theory which are simplified using a novel diagram calculus reminiscent of feynman diagrams

boundary element solution for the cauchy problem in linear elasticity using singular value decomposition abstract in this paper the singular value decomposition svd truncated at an optimal number is analysed for obtaining approximate solutions to ill conditioned linear algebraic systems of equations which arise from the boundary element method bem discretisation of an ill posed boundary value problem in linear elasticity the regularisation parameter namely the optimal truncation number is chosen according to the discrepancy principle the numerical results obtained confirm that the svd bem produces a convergent and stable numerical solution with respect to decreasing the mesh size discretisation and the amount of noise added into the input data

the generalized finite difference method for the inverse cauchy problem in two dimensional isotropic linear elasticity abstract in this study the generalized finite difference method gfdm was used to stably and accurately solve two dimensional 2d inverse cauchy problems in linear elasticity by using the navier equations in cauchy problems overdetermined boundary conditions are imposed on parts of the boundary whereas there are missing boundary conditions on some parts of the boundary in cauchy problems conventional numerical methods generally generate highly ill conditioned matrices and thus provide unstable numerical solutions moreover even if a slight noise is added in the boundary conditions numerical errors are evidently magnified the gfdm one of the most promising meshless methods and an extension of the classical finite difference method can avoid time consuming tasks of mesh generation and numerical quadrature the gfdm was applied in this study to stably solve the 2d cauchy problems in linear elasticity and four numerical examples are provided to illustrate the consistency and accuracy of the presented meshless numerical scheme moreover the stability of the presented scheme for inverse cauchy problems was proved by adding noise into the boundary conditions

mathematical foundation of the mfs for certain elliptic systems in linear elasticity the method of fundamental solutions mfs is a trefftz type technique in which the solution of an elliptic boundary value problem is approximated by a linear combination of translates of fundamental solutions with singularities placed on a pseudo boundary i e a surface embracing the domain of the problem under consideration in this work we develop a mathematical framework for the numerical implementation of the mfs in elliptic systems we obtain density results with respect to the c l norms which establish the applicability of the method in certain systems arising from the theory of elastostatics and thermo elastostatics the domains in our density results may possess holes and they satisfy the segment condition

regularized mfs solution of inverse boundary value problems in three dimensional steady state linear thermoelasticity abstract we investigate the numerical reconstruction of the missing thermal and mechanical boundary conditions on an inaccessible part of the boundary in the case of three dimensional linear isotropic thermoelastic materials from the knowledge of over prescribed noisy data on the remaining accessible boundary we employ the method of fundamental solutions mfs and several singular value decomposition svd based regularization methods e g the tikhonov regularization method tikhonov and arsenin 1986 the damped svd and the truncated svd hansen 1998 whilst the regularization parameter is selected according to the discrepancy principle morozov 1966 generalized cross validation criterion golub et al 1979 and hansen s l curve method hansen and o leary 1993

an effective algorithm for reconstructing boundary conditions in elastic solids this paper discusses the reconstruction of boundary conditions for linear elastic solids when there are not enough known boundary conditions but the stress tensor is known in a sufficient number of internal points starting with a refined boundary elements discretization of the elastic solid an iterative strategy based on the conjugate gradient method is presented this iterative strategy uses a monte carlo simulation of the generalized cross validation criterion to select the optimal solution the proposed algorithm possesses some advantages with respect to direct approaches indeed it furnishes almost as accurate results but at a low computational cost some numerical results allow an analysis of the algorithm s performance

numerical solution of the cauchy problem for steady state heat transfer in two dimensional functionally graded materials the application of the method of fundamental solutions to the cauchy problem for steady state heat conduction in two dimensional functionally graded materials fgms is investigated the resulting system of linear algebraic equations is ill conditioned and therefore regularization is required in order to solve this system of equations in a stable manner this is achieved by employing the zeroth order tikhonov functional while the choice of the regularization parameter is based on the l curve method numerical results are presented for both smooth and piecewise smooth geometries the convergence and the stability of the method with respect to increasing the number of source points and the distance between the source points and the boundary of the solution domain and decreasing the amount of noise added into the input data respectively are analysed