direct sums of tensor products and non commutative independence abstract we point out the connection between the three known fundamental notions of non commutative independence in quantum probability and the three basic choices for forming an associative direct sum of alternating tensor products of two vector spaces

an optimal order collocation method for first kind boundary integral equations on polygons this paper discusses the convergence of the collocation method using splines of any order k for first kind integral equations with logarithmic kernels on closed polygonal boundaries in bbb r 2 before discretization the equation is transformed to an equivalent equation over pi pi using a nonlinear parametrization of the polygon which varies more slowly than arc length near each corner this has the effect of producing a transformed equation with a solution which is smooth on pi pi this latter integral equation is shown to be well posed in appropriate sobolev spaces the structure of the integral operator is described in detail and can be written in terms of certain non standard mellin convolution operators using this information we are able to show that the collocation method using splines of order k degree k 1 converges with optimal order o h k the collocation points are the mid points of subintervals when k is odd and the break points when k is even and stability is shown under the assumption that the method may be modified slightly using the numerical solutions to the transformed equation we construct numerical solutions of the original equation which converge optimally in a certain weighted norm finally the method is shown to produce superconvergent approximations to interior potentials such as those used to solve harmonic boundary value problems by the boundary integral method the convergence results are illustrated with some numerical examples

the discrete galerkin method for integral equations a general theory is given for discretized versions of the galerkin method for solving fredholm integral equations of the second kind the discretized galerkin method is obtained from using numerical integration to evaluate the integrals occurring in the galerkin method the theoretical framework that is given parallels that of the regular galerkin method including the error analysis of the superconvergence of the iterated galerkin and discrete galerkin solutions in some cases the iterated discrete galerkin solution is shown to coincide with the nystrom solution with the same numerical integration method the paper concludes with applications to finite element galerkin methods

high order convergence for collocation of second kind boundary integral equations on polygons we propose collocation methods with smoothest splines to solve the integral equation of the second kind on a plane polygon they are based on the bijectivity of the double layer potential between spaces of sobolev type with arbitrary high regularity and involving the singular functions generated by the corners if splines of order 2m 1 are used we get quasi optimal estimates in h m norm and optimal order convergence for the h k norm if 0 le k le m numerical experiments are presented

unification of boolean monotone anti monotone and tensor independence and levy processes the unification of the boolean and the tensor product of quantum probability spaces given by lenczewski len98 is modified to include the monotone product and its mirror image the anti monotone product and used to reduce the theories of boolean monotone and anti monotone levy processes on dual semi groups to the theory of levy processes on involutive bialgebras this leads to a representation theorem for these processes further applications are a proof of the schoenberg correspondence for boolean monotone and anti monotone convolution semi groups a realization of boolean monotone and anti monotone creation annihilation and conservation operators on boson fock spaces and a discussion of the markov structure of these levy processes

quantum levy processes on dual groups we present a theory of quantum non commutative levy processes on dual groups which generalizes the theory of levy processes on bialgebras it follows from a result of n muraki that there exist exactly 5 notions of non commutative positive stochastic independence we show that one can associate a commutative bialgebra with each pair consisting of a dual group and one of the 5 notions of independence this construction is related to a construction of u franz our construction has the advantage that the important case of free independence is included we show that levy processes are given by their generators which are precisely the conditonally positive linear functionals on the dual group

trigonometric collocation methods with product integration for boundary integral equations on closed curves considering boundary integral equations on smooth closed curves the product integration methods have turned out to be very efficient to define fully discrete schemes with a high convergence rate here we give a general approach to product integration methods in connection with the trigonometric collocation the discretization scheme has arbitrarily high convergence rate even exponential if the solution and the curve are analytic applications and numerical experiments confirming our theoretical results are discussed

stochastic integration on the cuntz algebra o abstract we develop a theory of non commutative stochastic integration with respect to the creation and annihilation process on the full fock space over l2 r our theory largely parallels the theories of non commutative stochastic ito integration on boson and fermion fock space as developed by r hudson and k r parthasarathy it provides the first example of a non commutative stochastic calculus which does not depend on the quantum mechanical commutation or anticommutation relations but it is based on the theory of reduced free products of c algebras by d voiculescu this theory shows that the creation and annihilation processes on the full fock space over l2 r which generate the cuntz algebra o can be interpreted as a generalized brownian motion we should stress the fact that in contrast to the other theories of stochastic integration our integrals converge in the c norm on o i e uniformly rather than in some state dependent strong operator topology or non commutative l2 norm

numerical methods for integral equations of mellin type we present a survey of numerical methods based on piecewise polynomial approximation for integral equations of mellin type including examples arising in boundary integral methods for partial differential equations on polygonal domains

the collocation method for mixed boundary value problems on domains with curved polygonal boundaries we consider an indirect boundary integral equation formulation for the mixed dirichlet neumann boundary value problem for the laplace equation on a plane domain with a polygonal boundary the resulting system of integral equations is solved by a collocation method which uses a mesh grading transformation and a cosine approximating space the mesh grading transformation method yields fast convergence of the collocation solution by smoothing the singularities of the exact solution a complete stability and solvability analysis of the transformed integral equations is given by use of a mellin transform technique in a setting in which each arc of the polygon has associated with it a periodic sobolev space

on the numerical solution of a logarithmic integral equation of the first kind for the helmholtz equation we describe a quadrature method for the numerical solution of the logarithmic integral equation of the first kind arising from the single layer approach to the dirichlet problem for the two dimensional helmholtz equation in smooth domains we develop an error analysis in a sobolev space setting and prove fast convergence rates for smooth boundary data

a fast fourier collocation method for second boundary integral equations in this paper we develop a fast collocation method for second boundary integral equations by the trigonometric polynomials we propose a convenient way to compress the dense matrix representation of a compact integral operator with a smooth kernel under the fourier basis and the corresponding collocation functionals the compression leads to a sparse matrix with only o nlog 2n number of nonzero entries where 2n 1 denotes the order of the matrix thus we develop a fast fourier collocation method we prove that the fast fourier collocation method gives the optimal convergence order up to a logarithmic factor moreover we design a fast scheme for solving the corresponding truncated linear system we establish that this algorithm preserves the quasi optimal convergence of the approximate solution with requiring a number of o nlog 3n multiplications

gauss jacobi quadratures for weakly strongly hyper and nearly singular integrals in boundary integral equation methods for domains with sharp edges and corners abstract we present gauss jacobi quadrature rules in terms of hypergeometric functions for the discretization of weakly singular strongly singular hypersingular and nearly singular integrals that arise in integral equation formulations of potential problems for domains with sharp edges and corners the rules are tailored to weight functions with algebraic endpoint singularities of a fairly general form thus allowing one to easily incorporate a wide class of domains into the analysis numerical examples illustrate the accuracy and stability of the proposed algorithms it is shown that the same level of high accuracy can be achieved for any choice of the external variable the usefulness of the method is exemplified by application to the solution of a singular integral equation that arises in time harmonic electromagnetic scattering by either closed or open perfectly conducting cylindrical objects with edges and corners such as polygon cylinders and bent strips some practical aspects concerning the role of nearby singularities in achieving a highly accurate solution of singular integral equations are also discussed

on type ii0 e0 semigroups induced by boundary weight doubles powers has shown that each spatial e0 semigroup can be obtained from the boundary weight map of a cp flow acting on b k l2 0 for some separable hilbert space k in this paper we define boundary weight maps through boundary weight doubles where mn c mn c is a q positive map and is a boundary weight over l2 0 these doubles induce cp flows over k for 1 1 n n

a nystrom flavored calderon calculus of order three for two dimensional waves time harmonic and transient in this paper we present and test a full discretization of all elements of the calderon calculus layer potentials and integral operators for the helmholtz equation in smooth closed curves in the plane the resulting integral equations provide approximations of order three for all variables involved tests are shown for a wide array of direct indirect and combined field integral equation at fixed frequency and for a convolution quadrature based approximation in the time domain

a numerical method and efficient preconditioner for generalized airfoil equations in this paper the generalized airfoil equations with cauchy algebraic singularity are researched by introducing the collocation method to conquer the singularity of the integral equation theoretical analysis and examples showed this method has accuracy o h m 1 with m the order of the interpolate polynomial in addition by separating the singular part into two so that they can offset against each other in the collocation points a new equivalent integrate function is also constructed moreover to accelerate the convergence of solving the obtained linear system a tridiagonal preconditioner is also presented for the singular integral operator by the operator splitting idea compared to the traditional tridiagonal preconditioner our method is more easily computed by an explicit inverse finally some numerical examples are presented to illustrate the behavior of the collocation method and this preconditioner

global polynomial approximation for symm s equation on polygons summary to solve 1d linear integral equations on bounded intervals with nonsmooth input functions and solutions we have recently proposed a quite general procedure that is essentially based on the introduction of a nonlinear smoothing change of variable into the integral equation and on the approximation of the transformed solution by global algebraic polynomials in particular the new procedure has been applied to weakly singular equations of the second kind and to solve the generalized air foil equation for an airfoil with a flap in these cases we have obtained arbitrarily high orders of convergence through the solution of very well conditioned linear systems in this paper to enlarge the domain of applicability of our technique we show how the above procedure can be successfully used also to solve the classical symm s equation on a piecewise smooth curve the collocation method we propose applied to the transformed equation and based on chebyshev polynomials of the first kind has shown to be stable and convergent a comparison with some recent numerical methods using splines or trigonometric polynomials shows that our method is highly competitive

gibbs states of the hopfield model in the regime of perfect memory we study the thermodynamic properties of the hopfield model of an autoassociative memory ifn denotes the number of neurons andm n the number of stored patterns we prove the following results ifm n 0 asn then there exists an infinite number of infinite volume gibbs measures for all temperaturest 1 concentrated on spin configurations that have overlap with exactly one specific pattern moreover the measures induced on the overlap parameters are dirac measures concentrated on a single point and the gibbs measures on spin configurations are products of bernoulli measures ifm n asn for small enough we show that for temperaturest smaller than somet 1 the induced measures can have support only on a disjoint union of balls around the previous points but we cannot construct the infinite volume measures through convergent sequences of measures

on the convergence of collocation methods for symm s integral equation on open curves on etablit la convergence des methodes de collocation pour l equation integrale de symm sur des courbes ouvertes on confirme experimentalement les resultats theoriques sur les vitesses de convergence asymptotiques

continuous spatial semigroups of endomorphisms of ℌ to each continuous semigroup of endomorphisms a of z s with an intertwining semigroup of isometries there is associated a representation 7t of the domain 2d 6 of the generator of ac it is shown that the arveson index d a is the number of times the representation 7t contains the identity representation of 2 6 this result is obtained from an analysis of the relation between two semigroups of isometries u and s satisfying the condition s t u t e ti for t 0 and 0

the convergence of even degree spline collocation solution for potential problems in smooth domains of the plane in this article we derive new error estimates for collocation solution of potential type problems by using even degree smooth splines as trial functions it turns out that for smooth potentials the assured convergence is of the same order as by using splines of the odd degreed 1 some numerical examples which conform the theoretical results are presented

high accuracy quadrature methods for solving boundary integral equations of axisymmetric elasticity problems we investigate the mechanical quadrature methods mqms for solving boundary integral equations bies of the three dimensional axisymmetric elasticity equations with dirichlet s problems the convergence of the mqms is proved based on anselone s collective compact theory moreover an asymptotic error expansion of the mqms shows that the approximation order is of o h 3 numerical examples are tested and the results verify the theoretical analysis

index theory and second quantization of boundary value problems the second quantization functor associates to each skew symmetric operator in one particle space a derivation 6 of the algebra which is based on the given commutation relations in this paper we characterize the spatial theory of 6 in the fock representation by an index which generalizes the one studied earlier by powers and arveson in connection with the spatial cohomological obstruction for semigroups of endomorphisms of b x it is well known that such semigroups corresponding to one sided boundary conditions are generated by derivations but derivations associated to iwo sided boundary conditions do not generate semigroups we show that the known index theory for semigroups generalizes to the quantization of arbitrary boundary conditions in one particle space our quantized twosided abstract boundary conditions dictate representations in a certain indetinite inner product space a krein space and our index is an isomorphism invariant for representation theory in krein spaces the representations are not unitarizable i e are not equivalent to hermitian representations in hilbert space

fast solvers of integral and pseudodifferential equations on closed curves on the basis of a fully discrete trigonometric galerkin method and two grid iterations we propose solvers for integral and pseudodifferential equations on closed curves which solve the problem with an optimal convergence order u n u c n u sobolev norms of periodic functions in o n log n arithmetical operations

self regularization by projection for noisy pseudodifferential equations of negative order it is well known that pseudodifferential equations of negative order considered in sobolev spaces with small smoothness indices are ill posed on the other hand it is known that efficient discretization schemes with properly chosen discretization parameters allow to obtain a regularization effect for such equations the main accomplishment of the present paper is the principle for the adaptive choice of the discretization parameters directly from noisy discrete data we argue that the combination of this principle with wavelet based matrix compression techniques leads to algorithms which are order optimal in the sense of complexity

invariants for e 0 semigroups on ii 1 factors we introduce four new cocycle conjugacy invariants for e0 semigroups on ii1 factors a coupling index a dimension for the gauge group a super product system and a c semiflow using noncommutative ito integrals we show that the dimension of the gauge group can be computed from the structure of the additive cocycles we do this for the clifford flows and even clifford flows on the hyperfinite ii1 factor and for the free flows on the free group factor l f infty in all cases the index is 0 which implies they have trivial gauge groups we compute the super product systems for these families and using this we show they have trivial coupling index finally using the c semiflow and the boundary representation of powers and alevras we show that the families of clifford flows and even clifford flows contain infinitely many mutually non cocycle conjugate e0 semigroups

high order methods for weakly singular integral equations with nonsmooth input functions to solve one dimensional linear weakly singular integral equations on bounded intervals with input functions which may be smooth or not we propose to introduce first a simple smoothing change of variable and then to apply classical numerical methods such as product integration and collocation based on global polynomial approximations the advantage of this approach is that the order of the methods can be arbitrarily high and that the associated linear systems one has to solve are very well conditioned

double layer potential scheme for dirichlet problems on smooth open arcs abstract in this paper we present a new numerical scheme for the dirichlet problem on the smooth open arcs in the plane by using the double layer potential to construct the solution of the problem we induce a fredholm integral equation of the second kind existence and uniqueness of the solution of this integral equation are proved in the case when the boundary arc is convex a numerical example is given to show the validity of the present scheme

an index theory for quantum dynamical semigroups w arveson showed a way of associating continuous tensor product systems of hilbert spaces with endomorphism semigroups of type i factors we do the same for general quantum dynamical semigroups through a dilation procedure the product system so obtained is the index and its dimension is a numerical invariant for the original semigroup

a nystrom method for boundary integral equations in domains with corners we give a convergence and error analysis for a nystrom method on a graded mesh for the numerical solution of boundary integral equations for the harmonic dirichlet problem in plane domains with corners

on the dirichlet problem in elasticity for a domain exterior to an arc abstract we consider here a dirichlet problem for the two dimensional linear elasticity equations in the domain exterior to an open arc in the plane it is shown that the problem can be reduced to a system of boundary integral equations with the unknown density function being the jump of stresses across the arc existence uniqueness as well as regularity results for the solution to the boundary integral equations are established in appropriate sobolev spaces in particular asymptotic expansions concerning the singular behavior for the solution near the tips of the arc are obtained by adding special singular elements to the regular splines as test and trial functions an augmented galerkin procedure is used for the corresponding boundary integral equations to obtain a quasi optimal rate of convergence for the approximate solutions

on the asymptotic convergence of collocation methods we prove quasioptimal and optimal order estimates in various sobolev norms for the approximation of linear strongly elliptic pseudodifferential equations in one independent variable by the method of nodal collocation by odd degree polynomial splines the analysis pertains in particular to many of the boundary element methods used for numerical computation in engineering applications equations to which the analysis is applied include fredholm integral equations of the second kind certain first kind fredholm equations singular integral equations involving cauchy kernels a variety of integro differential equations and two point boundary value problems for ordinary differential equations the error analysis is based on an equivalence which we establish between the collocation methods and certain nonstandard galerkin methods we compare the collocation method with a standard galerkin method using splines of the same degree showing that the galerkin method is quasioptimal in a sobolev space of lower index and furnishes optimal order approximation for a range of sobolev indices containing and extending below that for the collocation method and so the standard galerkin method achieves higher rates of convergence

the standard form of an e0 semigroup abstract we show that the conjugacy classes of continuous semigroups of endomorphisms of b h that possess a pure normal absorbing state i e standard e 0 semigroups are indexed by the orbits of the natural action of the group of local cocycles on the set of intertwining semigoups of isometries we use this to show that in passing from an arbitrary e 0 semigroup to a cocycle perturbation in standard form the boundary representation is compressed to a hereditary subalgebra of the domain d of

a polynomial collocation method for the numerical solution of weakly singular and singular integral equations on non smooth boundaries the aim of this paper is to show the efficiency of the use of smoothing changes of variable in the numerical treatment of 1d and 2d weakly singular and singular integral equations the introduction of a smoothing transformation besides smoothing the solution allows also the use of a very simple and efficient collocation method based on chebyshev polynomials of the first kind and their zeros further we propose proper smoothing changes of variable also for the numerical approximation of those collocation matrix elements which are given by weakly singular singular or nearly singular integrals several numerical tests are given to point out the efficiency of the numerical approach we propose copyright 2003 john wiley sons ltd

a fully discrete calderon calculus for the two dimensional elastic wave equation in this paper we present a full discretization of the layer potentials and boundary integral operators for the elastic wave equation on a parametrizable smooth closed curve in the plane the method can be understood as a non conforming petrov galerkin discretization with a very precise choice of testing functions by symmetrically combining elements on two staggered grids and using a look around quadrature formula unlike in the acoustic counterpart of this work the kernel of the elastic double layer operator includes a periodic hilbert transform that requires a particular choice of the mixing parameters we give mathematical justification of this fact finally we test the method on some frequency domain and time domain problems and demonstrate its applicability on smooth open arcs

spline qualocation methods for boundary integral equations this paper further develops the qualocation method for the solution of integral equations on smooth closed curves qualocation is a petrov galerkin method in which the outer integrals are performed numerically by special quadrature rules here we allow the use of splines as trial and test functions and prove stability and convergence results for certain qualocation rules when the underlying petrov galerkin method is stable in the common case of a first kind equation with the logarithmic kernel a typical qualocation method employs piecewise linear splines and performs the outer integrals using just two points per interval this method with appropriate choice of the quadrature points and weights has ordero h 5 convergence in a suitable negative norm numerical experiments suggest that these rates of convergence are maintained for integral equations on open intervals when graded meshes are used to cope with unbounded solutions

a collocation method for the generalized airfoil equation for an airfoil with a flap we are concerned with the numerical solution of the generalized airfoil equation for an airfoil with a flap by means of global algebraic polynomial approximants this problem has been considered in a recent paper by monegato and sloan siam j numer anal 34 1997 pp 2288 2305 where the authors proved the stability and convergence of a galerkin method based on high order polynomials they also presented very promising numerical results for the corresponding collocation method but left the theoretical results of stability and convergence as an open problem we have succeeded in proving the stability and convergence properties of this collocation method by slightly modifying it in the neighborhood of the flap the rate of convergence we have derived for our collocation method is very similar to that of the above galerkin method

collocation with trigonometric polynomials for integral equations to the mixed boundary value problem we consider the direct boundary integral equation formulation for the mixed dirichlet neumann boundary value problem for the laplace equation on a plane domain with a polygonal boundary the resulting system of integral equations is solved by a collocation method which uses a mesh grading transformation and trigonometric polynomials the mesh grading transformation method yields fast convergence of the collocation solution by smoothing the singularities of the exact solution special care is taken for handling the hypersingular operator as in hartmann and stephan in dick kuo wozniakowski eds festschrift for the 80th birthday of ian sloan springer berlin 2018 with the indirect method used in elschner et al numer math 76 3 355 381 1997 this was avoided using mellin transformation techniques a stability and solvability analysis of the transformed integral equations can be performed in a setting in which each arc of the polygon has associated with it a periodic sobolev space

schoenberg correspondence on dual groups as in the classical case of levy processes on a group levy processes on a voiculescu dual group are constructed from conditionally positive functionals it is essential for this construction that schoenberg correspondence holds for dual groups the exponential of a conditionally positive functional is a convolution semigroup of states

on the asymptotic convergence of collocation methods with spline functions of even degree we investigate the collocation of linear one dimensional strongly elliptic integro dif ferential or more generally pseudo differential equations on closed curves by even degree polynomial splines the equations are collocated at the respective midpoints subject to uniform nodal grids of the even degree s splines we prove quasioptimal and optimal order asymptotic error estimates in a scale of sobolev spaces the results apply in particular to boundary element methods used for numerical computations in engineering applications the equations considered include fredholm integral equations of the second and the first kind singular integral equations involving cauchy kernels and integro differential equations having convolutional or constant coefficient principal parts respectively the error analysis is based on an equivalence between the collocation and certain varia tional methods with different degree splines as trial and as test functions we further need to restrict our operators essentially to pseudo differential operators having convolutional prin cipal part this allows an explicit fourier analysis of our operators as well as of the spline spaces in terms of trigonometric polynomials providing babuska s stability condition based on strong ellipticity our asymptotic error estimates extend partly those obtained by d n arnold and w l wendland from the case of odd degree splines to the case of even degree splines 1 introduction in this paper we investigate the asymptotic convergence of the collocation method using even degree polynomial splines applied to strongly elliptic systems of pseudo differential equations on closed curves with convolutional prin cipal part the collocation here employs the gauss points of one point integration as collocation points which corresponds to the usual boundary element collocation in applications 5 12 17 19 26 31 34 this is in contrast to the method investigated by g schmidt in 29 where one collocates at the break points of even order splines the asymptotic convergence properties for the standard galerkin method with splines of arbitrary orders are well known 22 for the collocation method however asymptotic convergence up to now has been shown for strongly elliptic systems only in the case of odd order splines by d n arnold and w l wendland 7

quadrature methods for 2d and 3d problems in this paper we give an overview on well known stability and convergence results for simple quadrature methods based on low order composite quadrature rules and applied to the numerical solution of integral equations over smooth manifolds first we explain the methods for the case of second kind equations then we discuss what is known for the analysis of pseudodifferential equations we explain why these simple methods are not recommended for integral equations over domains with dimension higher than one finally for the solution of a two dimensional singular integral equation we prove a new result on a quadrature method based on product rules

a modified boundary integral method on open arcs in the plane abstract for the dirichlet problem on open acrs in an infinite plane a modified numerical scheme using the indirect boundary integral method is presented we suggest a single layer potential added by a constant as a solution of the problem then the solution for the density function of the induced boundary integral equation gives the potential which satisfies all the conditions of the original problem approximation to the unknown density function is fulfilled by using the collocation method with the chebyshev polynomials of the first kind the procedure given in this work reduces the number of steps for computation of the method introduced by atkinson and sloan 1 experimental results of some examples are given to show the convergence of our method which is the same as that of the literature 1

a discrete collocation method for symm s integral equation on curves with corners this paper is devoted to the approximate solution of the classical first kind boundary integral equation with logarithmic kernel symm s equation on a closed polygonal boundary in 2 we propose a fully discrete method with a trial space of trigonometric polynomials combined with a trapezoidal rule approximation of the integrals before discretization the equation is transformed using a nonlinear mesh grading parametrization of the boundary curve which has the effect of smoothing out the singularities at the corners and yields fast convergence of the approximate solutions the convergence results are illustrated with some numerical examples

high order collocation and quadrature methods for some logarithmic kernel integral equations on open arcs this paper is devoted to the solution of the dirichlet problem for the laplace and helmholtz equation in the complement of a smooth open curve in the plane the solution is looked for as a single layer potential the corresponding density being therefore the solution of an integral equation on the open arc this equation is transformed into an equivalent 1 periodic integral equation having existence and uniqueness of solution for any periodic data here we study the use of collocation and quadrature methods for solving this equation we show the convergence of both methods and prove the existence of an asymptotic expansion of the error in powers of the discretization parameter as a consequence we show that for some of them a superconvergence phenomenon occurs when computing the solution of the differential problem two numerical experiments are shown in order to illustrate the theoretical results introduced in this work

haar states and levy processes on the unitary dual group abstract we study states on the universal noncommutative algebra generated by the coefficients of a unitary matrix or equivalently states on the unitary dual group its structure of dual group in the sense of voiculescu allows to define five natural convolutions we prove that there exists no haar state for those convolutions however we prove that there exists a weaker form of absorbing state that we call haar trace for the free and the tensor convolutions we show that the free haar trace is the limit in distribution of the blocks of a haar unitary matrix when the dimension tends to infinity finally we study a particular class of free levy processes on the unitary dual group which are also the limit of the blocks of random matrices on the classical unitary group when the dimension tends to infinity

numerical solution of the generalized airfoil equation for an airfoil with a flap in this paper we are concerned with the generalized airfoil equation which governs the pressure across an airfoil oscillating in a wind tunnel in particular we examine the case of an airfoil which contains a flap so that the input function has a simple jump at a point x0 1 x0 1 this generates a logarithmic singularity in the solution u at the same point x0 and considerably complicates the problem of the numerical approximation of u by introducing a nonlinear transformation of the independent variable in the integral equation we improve the behavior of the unknown function u through incorporation of the jacobian of the transformation the price is that the kernel of the new integral equation becomes more complicated nevertheless we are able to show that a galerkin method based on high order polynomials is both stable and convergent we also present some promising numerical results for the corresponding collocation method for which we have not succeeded in proving stability and convergence although those results seem to suggest that it is indeed stable and has a rate of convergence similar to that we have derived for the galerkin method

an index for continuous semigroups of endomorphisms of b b continuous semigroups of endomorphism of b b are studied an index for such semigroups is defined it is shown that the set of indices corresponds to a set which properly contains the non negative integers plus a point at infinity

quadrature methods for strongly elliptic equations of negative order on smooth closed curves in this article the authors introduce and analyze a family of simple quadrature methods for strongly elliptic equations of negative order the results cover all the classical pseudodifferential equations of this kind and hence the boundary integral equations appearing in applications the quadrature methods are based on the trapezoidal approximation of the operator with respect to a uniform mesh and on varepsilon collocation they are proved to have the stability and the convergence with the maximal rate o h beta of convergence in general and o h beta 1 for some special values of varepsilon here beta is the order of the operator the method is applicable to systems of boundary integral equations as well in this connection the authors have conducted some numerical experiments which confirm their theoretical results

a spline trigonometric galerkin method and an exponentially convergent boundary integral method we consider a galerkin method for functional equations in one space variable which uses periodic cardinal splines as trial functions and trigonometric polynomials as test functions we analyze the method applied to the integral equation of the first kind arising from a single layer potential formulation of the dirichlet problem in the interior or exterior of an analytic plane curve in constrast to ordinary spline galerkin methods we show that the method is stable and so provides quasioptimal approximation in a large family of hilbert spaces including all the sobolev spaces of negative order as a consequence we prove that the approximate solution to the dirichlet problem and all its derivatives converge pointwise with exponential rate

the numerical solution of first kind logarithmic kernel integral equations on smooth open arcs consider solving the dirichlet problem au p o pem2 s u p h p pes sup u p co pes2 with s a smooth open curve in the plane we use single layer potentials to construct a solution u p this leads to the solution of equations of the form j g q og p q ds q h p pes this equation is reformulated using a special change of variable leading to a new first kind equation with a smooth solution function this new equation is split into a principal part which is explicitly invertible and a compact perturbation then a discrete galerkin method that takes special advantage of the splitting of the integral equation is used to solve the equation numerically a complete convergence analysis is given numerical examples conclude the paper with s a smooth open contour in the plane this equation arises in a variety of contexts one being the study of elasticity crack problems in the plane we propose a new numerical method for solving 1 1 and then use it to solve some potential theory problems in the plane the method takes account of the expected singularities in g at the ends of the contour in an entirely natural way and the method is shown to converge rapidly when the curve 5 and the data are sufficiently smooth we limit the functions g and h to be real although the following development extends easily to the complex case let s have a parametrization

the spline collocation method for parabolic boundary integral equations on smooth curves we consider the spline collocation method for a class of parabolic pseudodifferential operators we show optimal order convergence results in a large scale of anisotropic sobolev spaces the results cover the classical boundary integral equations for the heat equation in the general case where the spatial domain has a smooth boundary in the plane our proof is based on a localization technique for which we use our recent results proved for parabolic pseudodifferential operators for the localization we need also some special spline approximation results in anisotropic sobolev spaces

inversion formulas for the discretized hilbert transform on the unit circle a discrete version of the hilbert transform on the unit circle is considered its moore penrose inverse with respect to suitable scalar products is derived for different side conditions furthermore stability of the pseudo inverse is studied these results allow the efficient computation of approximate solutions of singular integral equations with hilbert kernel furthermore the stability analysis of such methods becomes much easier even for graded meshes which are useful for weakly singular solutions

some boundary element methods using dirac s distributions as trial functions in this article a boundary element method using dirac s distributions as trial functions is proposed as test functions we use smoother splines the computational complexity of this method is comparable with the collocation method and it converges with less regularity requirements on the boundary data asymptotic error analysis is derived an example considering solution of the potential problem is discussed in some detail finally a numerical example demonstrates the applicability of the method in practical calculations

quantum ito s formula and stochastic evolutions using only the boson canonical commutation relations and the riemann lebesgue integral we construct a simple theory of stochastic integrals and differentials with respect to the basic field operator processes this leads to a noncommutative ito product formula a realisation of the classical poisson process in fock space which gives a noncommutative central limit theorem the construction of solutions of certain noncommutative stochastic differential equations and finally to the integration of certain irreversible equations of motion governed by semigroups of completely positive maps the classical ito product formula for stochastic differentials with respect to brownian motion and the poisson process is a special case

quantum independent increment processes on superalgebras on introduit la notion de processus quantique independant a increment stationnaire sur des superalgebres et on demontre un theoreme de reconstruction qui etablit une correspondance un a un entre ces processus et leurs generateurs infinitesimaux

noncommutative brownian motion in monotone fock space an example of noncommutative brownian motion is constructed on the monotone fock space which is a kind of fock space generated by all the decreasing finite sequences of positive real numbers the probability distribution at time associated to this brownian motion is shown to be the arcsine law normalized to mean 0 and variance t

spline collocation for a boundary integral equation on polygons with cuts in this paper we construct collocation methods for a modified integral equation of the second kind on the boundary of a polygon with a finite number of cuts cuts are often met in practice and are not treated by the classical methods the double layer potential which degenerates in this situation is modified by the addition of two kinds of terms on each cut they take into account the highly singular functions generated by the cut and lead to a bijective boundary operator this operator remains bijective between spaces of sobolev types with high orders of regularity this allows the construction of collocation methods with quasi optimal estimates and high order of convergence with respect to strong norms they use smoothest splines of odd degree numerical examples of the methods are presented

local unitary cocycles of e0 semigroups abstract this paper concerns the structure of the group of local unitary cocycles also called the gauge group of an e 0 semigroup the gauge group of a spatial e 0 semigroup has a natural action on the set of units by operator multiplication arveson has characterized completely the gauge group of e 0 semigroups of type i and as a consequence it is known that in this case the gauge group action is transitive in fact if the semigroup has index k then the gauge group action is transitive on the set of k 1 tuples of appropriately normalized independent units an action of the gauge group having this property is called k 1 fold transitive we construct examples of e 0 semigroups of type ii and index 1 which are not 2 fold transitive these new examples also illustrate that an e 0 semigroup of type ii k need not be a tensor product of an e 0 semigroup of type ii 0 and another of type i k

the modified quadrature method for classical pseudodifferential equations of negative order on smooth closed curves in this article we present and analyze the modified quadrature method for strongly elliptic boundary integral equations of negative order on smooth closed curves the method employs a modification that extracts the singularity appearing in the kernel or in some of its derivatives moreover the composite trapezoidal rule is used for approximation of the integral the modified quadrature method is proved to have the maximal rate o h 1 of convergence for operators of order 0 in general and o h 2 for a large class of operators appearing in applications some numerical experiments confirming our theoretical results are also presented

a fast trigonometric collocation method for some elliptic pseudodifferential equations abstract in this paper we propose and analyze a fast trigonometric collocation method for a class of periodic elliptic pseudodifferential equations whose pseudodifferential operators can always be represented as the sum of a principal part and a smoothing operator we show that the whole matrix representation for the principal part in our discrete linear system can be generated by only computing o n entries rather than computing all entries of the matrix where 2 n or 2 n 1 is the size of the matrix the dense matrix for the smoothing operator can be compressed into a sparse matrix with only o n log n nonzero entries we also prove that our proposed method preserves the optimal convergent order the same as without compression some numerical experiments for solving three cases of boundary integral equations are presented to demonstrate its approximate accuracy and computational efficiency verifying the theoretical estimates

optimal order collocation for the mixed boundary value problem on polygons in usual boundary elements methods the mixed dirichlet neumann problem in a plane polygonal domain leads to difficulties because of the transition of spaces in which the problem is well posed we build collocation methods based on a mixed single and double layer potential this indirect method is constructed in such a way that strong ellipticity is obtained in high order spaces of sobolev type the boundary values of this potential define a bijective boundary operator if a modified capacity adapted to the problem is not 1 this condition is analogous to the one met in the use of the single layer potential and is not a problem in practical computations the collocation methods use smoothest splines and known singular functions generated by the corners if splines of order 2m 1 are used we get quasi optimal estimates in h m norm the order of convergence is optimal in the sense that it is fixed by the approximation properties of the first missed singular function

a quadrature based approach to improving the collocation method the collocation method is a popular method for the approximate solution of boundary integral equations but typically does not achieve the high order of convergence reached by the galerkin method in appropriate negative norms in this paper a quadrature based method for improving upon the collocation method is proposed and developed in detail for a particular case that case involves operators with even symbol such as the logarithmic potential operating on 1 periodic functions a smooth spline trial space of odd degree with constant mesh spacingh 1 n and a quadrature rule with 2n points where ann point quadrature rule would be equivalent to the collocation method in this setting it is shown that a special quadrature rule which depends on the degree of the splines and the order of the operator can yield a maximum order of convergence two powers ofh higher than the collocation method