on a misconception involving point collocation and the rayleigh hypothesis it is shown that the rayleigh hypothesis does not govern convergence of the simple point collocation approach to the numerical solution of scattering by a sinusoidal grating a previously developed numerical technique interval arithmetic is employed to perform some decisive numerical experiments which not only support but guarantee the validity of this conclusion

accuracy analysis and optimization of the method of auxiliary sources mas for scattering by a circular cylinder this paper presents a rigorous accuracy analysis of the method of auxiliary sources mas when applied to scattering problems a benchmark canonical geometry consisting of a perfectly conducting infinite circular cylinder is chosen for clarity and simplicity for this particular structure it is shown that the mas square matrix can be inverted analytically yielding exact mathematical expressions for the discretization error and the condition number of the pertinent linear system it is also demonstrated that the error increases very abruptly for source locations associated with the characteristic eigenvalues of the scattering geometry precisely as predicted in theory various plots depict comparisons between analytical and computational data for the boundary condition error and all occurring discrepancies are fully explained among several important results of the analysis the fundamental mas question concerning the optimal location of the auxiliary sources is thoroughly investigated and resolved on the grounds of error minimization

expected behavior of quantum thermodynamic machines with prior information we estimate the expected behavior of the quantum model of a heat engine when we have incomplete information about external macroscopic parameters such as the magnetic field controlling the intrinsic energy scales of the working medium we explicitly derive the prior probability distribution for these unknown parameters a i phantom rule 0 28em 0ex phantom rule 4pt 0ex i 1 2 based on a few simple assumptions the prior probability distribution is found to be of the form ensuremath pi a i ensuremath propto 1 a i by calculating the expected values of various physical quantities related to this engine we find that the expected behavior of the quantum model exhibits thermodynamiclike features this leads us to a surprising proposal that incomplete information quantified as an appropriate prior distribution can lead us to expect classical thermodynamic behavior in quantum models

a novel scheme for the solution of the time domain integral equations of electromagnetics a new method to numerically solve time domain integral equations pertinent to electromagnetic surface scattering phenomena is presented the method uses approximate prolate spheroidal wave functions and standard rao wilton glisson basis functions to effect the temporal and spatial discretization of the integral equations respectively because the temporal basis functions are noncausal an extrapolation scheme is used to construct a system of equations that can be solved by marching on in time numerical results show that the proposed method is stable and that its solutions converge exponentially fast with the time bandwidth product of the approximate prolate spheroidal wave functions to results from a frequency domain method of moments solver that uses spatial basis functions and integration rules identical to those in the time domain solver

on the behaviour of time discretisations of the electric field integral equation we derive a separation of variables solution for time domain electromagnetic scattering from a perfectly conducting infinite flat plate the time dependent part of the equations are then used as a model problem in order to study the effects of various time discretisations on the full scattering problem we examine and explain how exponential and polynomial instabilities arise in the approximation schemes and show that the time averaging which is often used in an attempt to stabilise solutions of the full problem acts to destabilise some of the schemes our results show that two of the time discretisations can produce good results when coupled with a space exact approximation and indicate that they will be useful when coupled with an accurate enough spatial approximation

inversion of block circulant matrices and circular array approach closed form expressions of the inverse of a block circulant matrix bcm are well known such expressions are derived here following two alternative procedures which are based on the discrete fourier transform dft and appear to be simple and brief when compared to previously published procedures in the case of circulant matrices of scalars the formulas of inversion are derived using the theory of circular arrays and this approach suggested an extension of the dft to the bcm case

equivariant deformations of homogeneous spaces abstract c algebraic deformations of homogeneous spaces g are constructed by completing dense subspaces of c 0 g in a different multiplication and c norm these deformations are equivariant in the sense that they still carry a natural action of g by left translation the motivating examples are the noncommutative heisenberg manifolds of rieffel but the construction given here is based on the authors twisted dual group algebras which are equivariant deformations of c 0 g the procedure is general enough to give other recently studied families of noncommutative spaces and some interesting new examples of simple c algebras

green function discretization for numerical solution of the helmholtz equation abstract a discretization method based on free space green function interpolation with a minimuml2norm property has been developed for numerical solution of the helmholtz equation it has been shown to produce accurate solutions with less than three points per wave which implies a manyfold improvement in solution efficiency over previous methods the accuracy of the method does not break down formally until the grid resolution approaches two points per wave

stability and convergence of the method of fundamental solutions for helmholtz problems on analytic domains the method of fundamental solutions mfs is a popular tool to solve laplace and helmholtz boundary value problems its main drawback is that it often leads to ill conditioned systems of equations in this paper we investigate for the interior helmholtz problem on analytic domains how the singularities charge points of the mfs basis functions have to be chosen such that approximate solutions can be represented by the mfs basis in a numerically stable way for helmholtz problems on the unit disc we give a full analysis which includes the high frequency short wavelength limit for more difficult and nonconvex domains such as crescents we demonstrate how the right choice of charge points is connected to how far into the complex plane the solution of the boundary value problem can be analytically continued which in turn depends on both domain shape and boundary data using this we develop a recipe for locating charge points which allows us to reach error norms of typically 10 1 1 on a wide variety of analytic domains at high frequencies of order only 3 points per wavelength are needed which compares very favorably to boundary integral methods

a mesh adapted closed form regular kernel for 3d singular integral equations the green s functions employed in the method of moments mom diverge when observation and source points coincide this is at the origin of the difficulties in computing the mom matrix entries and in handling the near field interactions in fast fourier transform fft based fast methods and other sampling based methods in this paper we show that this singularity can be avoided and a modified regular green s function can be used instead this latter is obtained from the spectral representation of the usual green s function via windowing of its spectrum the width of the spectral window depends on the size of the mesh employed for discretizing the problem so that the proposed regular green s function is a mesh adapted regular kernel we address a general 3d problem we relate the mom reaction integrals to the 2d fourier spectrum of the green s function that allows to discuss the necessary spectral bandwidth for the windowed green s function we employ a tapered window and present a closed form expression for the spatial green s function numerical results are presented for 3d antenna and scattering problems discretized with rao wilton glisson rwg functions and for uniform and nonuniform meshing they show that the proposed method yields accurate solutions also for the antenna input impedance the meaning of the regularized green s function is also discussed and put in perspective

detecting non uniqueness of solutions to biharmonic integral equations through svd we consider the singular values of an integral operator and of a corresponding square matrix derived from the integral operator by means of a quadrature formula and a collocation the integral operator and also the matrix depend on a real parameter which may also enter the singular values of the operator and the matrix when a singular value drops to zero for a certain critical value of the parameter the corresponding homogeneous integral equation or matrix equation has a nontrivial solution based on several examples with biharmonic integral operators we conjecture that the order of approximation of the critical value for the matrix is at least equal to the order of the quadrature formula used it is therefore possible with a reasonable accuracy to detect such critical values for the integral operator simply through a singular value decomposition of the matrix derived by a quadrature and collocation

on convergence and inherent oscillations within computational methods employing fictitious sources recent studies of the method of auxiliary sources mas and the method of fundamental solutions mfs have shown that it is possible to have convergence of the field which is the finally desired quantity together with divergence of the fictitious sources which are intermediate quantities the aforementioned studies concern a simple scattering problem where the scattered field satisfies the helmholtz equation and the fictitious sources are placed symmetrically in the present paper we extend the previous results in two directions one having to do with the geometry and the other with frequency more precisely we first show that the above convergence divergence phenomena can also occur when the fictitious sources are placed asymmetrically and then study the nature of this divergence in detail second we show that all findings carry over to the case of laplace s equation we additionally point out certain differences of the discussed phenomena with the well known ones of internal resonances

symmetries of the car algebra a automorphism of period two of the uhf c algebra of type 2 is constructed for which the fixed point algebra is not approximately finite dimensional the construction also yields a diagonal maximal commutative c subalgebra whose spectrum is not totally disconnected variations and consequences of the construction are discussed

an improved accuracy version of the method of auxiliary sources for computational electromagnetics the method of auxiliary sources mas is normally applicable to electromagnetic problems involving structures of significant thickness so that an adequately large distance between source and collocation points is guaranteed for thin or open geometries the accuracy of the method is depleted due to numerical instabilities caused by the highly singular terms of the dyadic green s function dgf in this paper a modified mas mmas is developed to circumvent this particular difficulty higher order terms of the dgf are numerically calculated by introducing a canonical grid where derivatives can be accurately computed via a discrete scheme unlike standard mas where the dgf analytical problematic expression is invoked instead this procedure is equivalent to the involvement of auxiliary currents and charges in the solution instead of the elementary source fields used in standard mas

transient scattering from two dimensional dielectric cylinders of arbitrary shape the problem of transient scattering by arbitrarily shaped two dimensional dielectric cylinders is solved using the marching on in time technique the dielectric problem is approached via the surface equivalence principle two coupled integral equations are derived by enforcing the continuity of the electric and magnetic fields which are solved by using the method of moments numerical results are presented for two cross sections viz a circle and a square and compared with inverse discrete fourier transform idft techniques in each case good agreement is obtained with the idft solution

on the oscillations appearing in numerical solutions of solvable and nonsolvable integral equations for thin wire antennas differences between certain solvable and nonsolvable ill posed integral equations with the same nonsingular kernel are discussed the main results come from constructing a solvable equation in the context of straight thin wire antennas the kernel of this equation is the usual approximate also called reduced kernel while its exact solution is the familiar sinusoidal current numerical solutions to this solvable equation are compared to corresponding numerical solutions of the usual halle n and pocklington equations with the approximate kernel it is known from previous publications that these last two equations are nonsolvable and that their numerical solutions present severe oscillations when the number of basis functions is sufficiently large it is found that the difficulties encountered in the former solvable equation are much less important compared to those of the nonsolvable ones the same conclusion is brought out from other integral equations arising in different contexts thin wire circular loop antenna method of auxiliary sources and straight wire antenna of infinite length we discuss the consistency of our results with picard s theorem the results in this paper supplement previous publications regarding the difficulties of numerically solving thin wire integral equations with the approximate kernel

systemes dynamiques non commutatifs et moyennabilite on introduit la notion de fonction de type positif relativement a un systeme dynamique c dont les proprietes sont analogues a celles des fonctions de type positif classiques on montre que dans le cas d un groupe discret g agissant sur une algebre de von neumann m la moyennabilite de l action est equivalente a l existence d une suite bornee de fonctions de type positif sur g a support fini a valeurs dans le centre z m de m qui converge vers 1 pour la topologie de la convergence simple ultrafaible

on the choice of expansion and weighting functions in the numerical solution of operator equations one of the objectives of this paper is to discuss the mathematical requirements that the expansion functions must satisfy in the method of moments mm solution of an operator equation a simple differential equation is solved to demonstrate these requirements the second objective is to study the numerical stability of point matching method galerkin s method and the method of least squares pocklington s integral equation is considered and numerical results are presented to illustrate the effect of various choices of weighting functions on the rate of convergence finally it is shown that certain choices of expansion and weighting functions yield numerically acceptable results even though they are not admissible from a strictly mathematical point of view the reason for this paradox is outlined

on the k theory of twisted higher rank graph c algebras abstract we investigate the k theory of twisted higher rank graph algebras by adapting parts of elliott s computation of the k theory of the rotation algebras we show that each 2 cocycle on a higher rank graph taking values in an abelian group determines a continuous bundle of twisted higher rank graph algebras over the dual group we use this to show that for a circle valued 2 cocycle on a higher rank graph obtained by exponentiating a real valued cocycle the k theory of the twisted higher rank graph algebra coincides with that of the untwisted one

finitely generated nilpotent group c algebras have finite nuclear dimension we show that group c algebras of finitely generated nilpotent groups have finite nuclear dimension it then follows from a string of deep results that the c algebra a generated by an irreducible representation of such a group has decomposition rank at most 3 if in addition a satisfies the universal coefficient theorem another string of deep results shows it is classifiable by its ordered ktheory and is approximately subhomogeneous we observe that all c algebras generated by faithful irreducible representations of finitely generated torsion free nilpotent groups satisfy the universal coefficient theorem

generalizing effective spatial bandwidth for pole location in mas the elliptic cylinder the location of poles in the method of auxiliary sources for 2 d scatterers can be assisted using the effective spatial bandwidth ebw of the incident and scattered fields previously only circular boundaries were considered here ebw is extended to noncircular boundaries calculations presented for a cylinder with elliptical cross section verify ebw as an additional tool for pole placement for more general geometries

on the use of spatio temporal multiresolution analysis in method of moments solutions of transient electromagnetic scattering a novel method of moments approach to the solution of time domain integral equation formulation of electromagnetic scattering problems is presented the method is based on a spatio temporal multiresolution analysis this analysis facilitates a basis from which a small number of expansion functions is selected via an iterative procedure and utilized to model the unknown current distribution in contrast to marching on in time sequential procedures the proposed method models the unknown current simultaneously at all the time steps within the time frame of interest this new method is applied to a one dimensional 1 d problem of electromagnetic plane wave interaction with a dielectric slab a comparison of the computed results with results based on the analytic solution demonstrates that the method is capable of attaining accurate results while achieving substantial reduction in computation time and resources

on the application of numerical methods to hallen s equation the so called hallen integral equation for the current on a finite linear antenna center driven by a delta function generator takes two forms depending on the choice of kernel the two kernels are usually referred to as the exact and the approximate or reduced kernel with the approximate kernel the integral equation has no solution nevertheless the same numerical method is often applied to both forms of the integral equation in this paper the behavior of the numerical solutions thus obtained is investigated and the similarities and differences between the two numerical solutions are discussed the numerical method is galerkin s method with pulse functions we first apply this method to the two corresponding forms of the integral equation for the current on a linear antenna of infinite length in this case the method yields an infinite toeplitz system of algebraic equations in which the width of the pulse basis functions enters as a parameter the infinite system is solved exactly for nonzero pulse width the exact solution is then developed asymptotically for the case where the pulse width is small when the asymptotic expressions for the case of the infinite antenna are used as a guide for the behavior of the solutions of the finite antenna the latter problem is greatly facilitated for the approximate kernel the main results of this paper carry over to a certain numerical method applied to the corresponding equation of the pocklington type

metric aspects of noncommutative homogeneous spaces for a closed cocompact subgroup of a locally compact group g given a compact abelian subgroup k of g and a homomorphism k g satisfying certain conditions landstad and raeburn constructed equivariant noncommutative deformations c g of the homogeneous space g generalizing rieffel s construction of quantum heisenberg manifolds we show that when g is a lie group and g is connected given any norm on the lie algebra of g the seminorm on c g induced by the derivation map of the canonical g action defines a compact quantum metric furthermore it is shown that this compact quantum metric space depends on continuously with respect to quantum gromov hausdorff distances

continuous fields of c algebras coming from group cocycles and actions recently i have been attempting to formulate a suitable c algebraic framework for the subject of deformation quantization 3 19 continuous fields of c algebras provide one of the key elements for this framework the main examples of deformation quantizations which i have constructed up to now in this c algebra framework come from letting either cocycles on groups or actions of groups vary it has thus become necessary to show that one obtains in this way fields of c aigebras that are indeed continuous since this material is of a general nature and can be useful in other situations 4 6 11 13 it has seemed appropriate to give a separate exposition of it in the present article section 1 of this article contains a review of the published results on continuous fields which we will need as well as a discussion of the fact that the approach which we will take involves treating upper and lower semi continuity separately in sect 2 we discuss the continuity of fields of c algebras which arise from varying cocycles on groups while in sect 3 we do the same for actions which vary

surface wave and superdirectivity aspects of effective current for linear antennas previous papers have defined the concept of effective current for linear antennas as a method of postprocessing the unphysical solutions obtained by applying discretization galerkin methods to the usual integral equations that employ the approximate or reduced kernel such an unphysical solution can be thought of as an auxiliary rapidly oscillating current located on the axis of the linear antenna while the effective current is the magnetic field due to the oscillating current at a distance equal to the antenna radius multiplied by a factor equal to the antenna circumference in the present paper we focus on the rapid decrease of the aforementioned magnetic field and point out similarities to surface waves to the well known phenomenon of superdirectivity and to the method of auxiliary sources mas our study sheds light on the unusual nature of electromagnetic waves generated by currents with very rapid spatial variations subject to the condition that the discretization is fine we treat th

the use of the frill generator in thin wire integral equations for a frill generator feed certain fundamental mathematical properties of hallen s and pocklington s equations with the approximate kernel are stated and derived for a particular moment method procedure galerkin s method with pulse functions applied to hallen s equation the consequences to the numerical solutions are carefully examined generalizations to other numerical methods with subsectional basis functions are discussed many of our results come from studying the simpler problem of the infinite dipole analytically and applying the understanding thus obtained to the case of the finite dipole

universal efficiency at optimal work with bayesian statistics if the work per cycle of a quantum heat engine is averaged over an appropriate prior distribution for an external parameter a the work becomes optimal at curzon ahlborn ca efficiency more general priors of the form a 1 a yield optimal work at an efficiency which stays close to ca value in particular near equilibrium the efficiency scales as one half of the carnot value this feature is analogous to the one recently observed in literature for certain models of finite time thermodynamics further the use of bayes theorem implies that the work estimated with posterior probabilities also bears close analogy with the classical formula these findings suggest that the notion of prior information can be used to reveal thermodynamic features in quantum systems thus pointing to a connection between thermodynamic behavior and the concept of information

a numerical solution for the general radiation problem based on the combined methods of superposition and singular value decomposition in the method of wave superposition the acoustic field due to a complex radiator is expressed in terms of a fredholm integral equation of the first kind called the superposition integral equation in general fredholm integral equations of the first kind are ill posed and therefore difficult to solve numerically in this paper it will be shown that a simple collocation procedure when combined with the singular value decomposition can yield accurate results for the numerical solution of the superposition integral as an example of the application of the method the acoustic radiation from a circular cylinder will be analyzed using this numerical procedure and compared to the exact solution it is shown that for this problem the accuracy of the numerical solution can be judged by evaluating how well the superposition solution approximates the specified boundary condition on the surface of the radiator an example is also given of a problem which has no exact solution in this situation it is suggested without proof that the accuracy of the numerical solution can be judged in a similar manner by evaluating the error in the superposition solution s satisfaction of the boundary condition

transient scattering by conducting cylinders te case transient scattering transverse electric case by two dimensional cylinders both open and closed is investigated using the marching on in time technique a simple averaging scheme is also presented to control the late time oscillations numerical results are presented for certain representative geometries and compared with inverse fourier transform techniques good agreement is noted in each case

protecting efie based scattering computations from effects of interior resonances a numerical method based on the standard matrix formulation of the electric field integral equation efie for calculating the scattering from conducting bodies near resonant frequencies is given for both stabilizing the numerical calculations and finding the form of the resonant fields when this method is used along with a direct solution to the original matrix equation it requires insignificant additional computation an additional advantage of this approach is that it can easily be implemented in existing computer codes by using a single standard but general plug in module

basis function considerations for the method of moments using the fictitious current model in the method of moments mom using the fictitious current model fcm an impulsive basis function is principally employed to expand the fictitious currents for various electromagnetic scattering problems in this paper we introduce the subdomain basis functions and consider the dependence of the solution accuracy and computation time of the mom fcm on the basis functions by means of a canonical two dimensional 2 d scattering problem

analysis of an infinite current source above a semi infinite lossy ground using fictitious current auxiliary sources in conjunction with complex image theory techniques the canonical problem of an infinitely long electric current line radiating above a lossy infinite half space is examined the solution is based on the method of auxiliary sources mas in this method one can in general apply a numerical solution by introducing sets of fictitious current sources whose fields are elementary analytical solutions to the boundary value problem in order to approximately describe the actual electromagnetic em fields in each domain in general the convergence rate and the accuracy of the mas solution depend on the spatial distributions of the fictitious current sources sets and their locations in regard to the singularities of the actual em field simulated by each set here both the accuracy and the convergence rate of the method are examined investigating complex image approximations in order to optimally choose the auxiliary sources placements it is proved that the convergence rate and the accuracy of the method are significantly improved by utilizing the complex images as locations of the auxiliary sources the main contribution of the paper consists in the application of mas to an open structure which involves lossy dielectrics excited by a nonuniform em field as well as in the optimal choice of the locations of the fictitious current sources according to complex image techniques

morita equivalence of smooth noncommutative tori we show that in the generic case the smooth noncommutative tori associated with two n n real skew symmetric matrices are morita equivalent if and only if the matrices are in the same orbit of the natural so n n z action

non commutative spheres iii irrational rotations leta be the irrational rotation algebra i e thec algebra generated by two unitariesu v satisfyingvu e 2 i uv with irrational and consider the fixed point subalgebrab under the flip automorphismu u 1 v v 1 we prove thatb is an af algebra

symmetries on the discrete heisenberg group c algebra we consider symmetries on the c algebra of the discrete heisenberg group and their generalization to the generalized discrete heisenberg groups and study the structure of the crossed products by the symmetries and their fixed point algebras and also compute the stable rank of the crossed products and their fixed point algebras

on the nature of oscillations in discretizations of the extended integral equation using analytical methods previous studies have shown that it is possible for oscillations to occur in the auxiliary surface current determined by applying the method of auxiliary sources mas to problems of scattering by perfect conductors of a very simple shape such oscillations are inherent to mas and would occur even in a hypothetical computer with ideal hardware and software because the integral equation relevant to mas very much resembles the extended integral equation in which the unknown is the actual surface current on the conductor one might surmise that similar oscillations also occur in discretizations of the latter equation in this communication we use analytical means to show that this is not the case therefore any oscillations that do occur in discretizations of the extended integral equation are at least for sufficiently simple problems likely to be due to matrix ill conditioning which magnifies errors that would otherwise be unimportant

on the use of spatio temporal wavelet expansions for transient analysis of wire antennas and scatterers to analyze a wire antenna excited by a time varying voltage source or a wire scatterer excitated by transient electromagnetic incident wave the problem is formulated in terms of a time domain integral equation for the induced current to solve the integral equation we reduce it to matrix equation via the method of moments using the known to be stable implicit scheme however rather than directly constructing and solving the relatively large matrix equation we propose an iterative procedure which allows us to gradually obtain a solution of refined accuracy both everywhere and simultaneously at any time instance to render this procedure rapidly converging we use a basis of spatio temporal wavelet functions this basis facilitates a good approximation of the induced current using far less basis functions than would be needed if other expansions such as standard pulse or fourier basis functions were chosen the use of this basis further enables the iterative procedure to increase the temporal and spatial resolutions where required without unnecessarily affecting their levels elsewhere

acoustic radiation impedances and impulse responses for elliptical cylinders using internal source density and singular value decomposition methods abstract a new internal source distribution method is combined with the singular value decomposition method to determine the acoustic impedance and associated impulse response for an infinite cylinder with a plane of symmetry and a normal velocity with a specified circumferential variation an internal linear distribution of unknown monopole and or dipole line sources along the plane of symmetry is determined to match the specified harmonic normal velocity on the surface in a least mean square error sense after the discretization of the source distributions the subsequent solution of a set or sets of linear algebraic equations for the internal source distributions is obtained using the singular value decomposition method line integrals for the associated surface pressure the acoustic impedance and the corresponding impulse response are then simply evaluated by quadrature methods numerical results for the acoustic impedances and associated impulse responses are presented for circular cylinders vibrating with various natural in vacuo modal shapes numerical results for acoustic impedances and associated impulse responses for elliptical cylinders vibrating with various natural in vacuo modal shapes are then presented and their characteristics are discussed and compared to the corresponding results for circular cylinders

asymptotic stability of the fundamental solution method abstract in a previous paper 1988 we studied the numerical stability of the fundamental solution method applied to the dirichlet problem of laplace s equation and presented a practical scheme for examining the stability of the method using the singular value decomposition in the present paper we give an asymptotic estimate of the stability with respect to the number of collocation points

a stable solution of time domain electric field integral equation for thin wire antennas using the laguerre polynomials in this paper a numerical method to obtain an unconditionally stable solution of the time domain electric field integral equation for arbitrary conducting thin wires is presented the time domain electric field integral equation td efie technique has been employed to analyze electromagnetic scattering and radiation problems from thin wire structures however the most popular method to solve the td efie is typically the marching on in time mot method which sometimes may suffer from its late time instability instead we solve the time domain integral equation by expressing the transient behaviors in terms of weighted laguerre polynomials by using these orthonormal basis functions for the temporal variation the time derivatives can be handled analytically and stable results can be obtained even for late time furthermore the excitation source in most scattering and radiation analysis of electromagnetic systems is typically done using a gaussian shaped pulse in this paper both a gaussian pulse and other waveshapes like a rectangular pulse or a ramp like function have been used as excitations for the scattering and radiation of thin wire antennas with and without junctions the time domain results are compared with the inverse discrete fourier transform idft of a frequency domain analysis

dimension and stable rank in the k theory of c algebras in topological k theory which can be viewed as the algebraic side of the theory of vector bundles some of the interesting properties which one investigates are for example the conditions under which bundles must possess trivial direct summands or the extent to which the cancellation property for direct whitney sums of bundles holds such properties turn out to be controlled in part by the dimension of the base space and results describing the nature of this control are among what are frequently called stability results 17 during the past few years there has been a sudden flowering of a theory for c algebras see surveys 35 36 37 since c algebras are profitably thought of as non commutative locally compact spaces with the finitely generated projective modules being the appropriate generalization of vector bundles according to swan s theorem 40 it would be natural to look for stability results for c algebras but until now the theory has been focussed on the c groups of c algebras and there has been little discussion of stability properties presumably in part for lack of an appropriate concept of dimension for c algebras one of the main objectives of this paper is to introduce a concept of dimension for c algebras which directly generalizes the classical concept of dimension for compact spaces 16 20 22 and to develop some techniques for calculating it notably for c algebras which are obtained as crossed product algebras for an action of the group of integers in algebraic k theory which is the generalization of topological c theory to rings a substantial number of stability properties have been discovered 2 39 42 48 within that theory the concept which has played a role most analogous to that of dimension is the bass stable rank but despite its many successes the bass stable rank has often been difficult to calculate even for relatively uncomplicated rings another main objective of the present paper is to make accessible for application to c algebras some of the stability results from algebraic c theory this comes about from the fact that the concept of dimension which we introduce dominates the bass stable rank i was led to seek stability results for c algebras by my desire to determine whether the cancellation property for projective modules holds for the irrational rotation c algebras discussed in 30 which as elliott 11 has suggested can appropriately be called 2 dimensional non commutative tori in more concrete terms the question is whether two projections in such an algebra which have the same trace must be unitarily equivalent the present paper provides the general theory which is needed for the affirmative answer which i obtained in a subsequent paper i will derive the

a space time discretization criterion for a stable time marching solution of the electric field integral equation numerical techniques based on a time domain recursive solution of the electric field integral equation efie may exhibit instability phenomena induced by the joint space time discretization the above problem is addressed with specific reference to the evaluation of electromagnetic scattering from perfectly conducting bodies of arbitrary shape we analyze a particular formulation of the method of moments which relies on a triangular patch geometrical model of the exterior surface of the scattering body and operates according to a marching on in time scheme whereby the surface current distribution at a given time step is recursively evaluated as a function of the current distribution at previous steps a heuristic stability condition is devised which allows us to define a proper time step as well as a geometrical discretization criterion ensuring convergence of the numerical procedure and therefore eliminating insurgence of late time oscillations the stability condition is discussed and validated by means of a few working examples

structure of group c algebras of the generalized disconnected mautner groups abstract we construct finite composition series of group c algebras of the generalized disconnected mautner groups such that their subquotients are tensor products of commutative c algebras and extensions by ah algebras as applications we consider the real cases of the generalized disconnected mautner groups and estimate the stable rank and connected stable rank of these group c algebras

work extremum principle structure and function of quantum heat engines we consider a class of quantum heat engines consisting of two subsystems interacting with a work source and coupled to two separate baths at different temperatures t h g t c the purpose of the engine is to extract work due to the temperature difference its dynamics is not restricted to the near equilibrium regime the engine structure is determined by maximizing the extracted work under various constraints when this maximization is carried out at finite power the engine dynamics is described by well defined temperatures and satisfies the local version of the second law in addition its efficiency is bounded from below by the curzon ahlborn value 1 ensuremath sqrt t c t h and from above by the carnot value 1 ensuremath t c t h the latter is reached at finite power for a macroscopic engine while the former is achieved in the equilibrium limit t h ensuremath rightarrow t c the efficiency that maximizes the power is strictly larger than the curzon ahloborn value when the work is maximized at a zero power even a small few level engine extracts work right at the carnot efficiency

wave field singularity aspects in large size scatterers and inverse problems it is known that any scattered wave field carrying energy into infinity must have source singularity centers within a bounded space otherwise the scattered field should be identically equal to zero everywhere kupradze 1935 in this paper attention is paid to localization of these singularities under the assumption that every scattered wave is determined uniquely by its own singularities investigations have shown that these singularities are distributed as bright centers and the distance between them depends on frequency to determine the position localization of the scattered wave field singularities the functions describing converging and diverging waves are used based on these concepts and the method of auxiliary sources an efficient numerical method to reconstruct a field up to its singularities is suggested the localization of singularities is used for partial representation of the scattered fields which reduces significantly the number of unknowns in describing the scattering process and leading into optimized inverse scattering problem solutions

stability and convergence of collocation schemes for retarded potential integral equations time domain boundary integral formulations of transient scattering problems involve retarded potential integral equations solving such equations numerically is both complicated and computationally intensive and numerical methods often prove to be unstable collocation schemes are easier to implement than full finite element formulations but little appears to be known about their stability and convergence here we derive and analyze some new stable collocation schemes for the single layer equation for transient acoustic scattering and use spatial fourier and temporal laplace transform techniques to demonstrate that such stable schemes are second order convergent

on the equivalence of the trefftz method and method of fundamental solutions for laplace and biharmonic equations in this paper it is proved that the two approaches known in the literature as the method of fundamental solutions mfs and the trefftz method are mathematically equivalent in spite of their essentially minor and apparent differences in formulation in deriving the equivalence of the trefftz method and the mfs for the laplace and biharmonic problems it is interesting to find that the complete set in the trefftz method for the laplace and biharmonic problems are embedded in the degenerate kernels of the mfs the degenerate scale appears using the mfs when the geometrical matrix is singular the occurring mechanism of the degenerate scale in the mfs is also studied by using circulant the comparison of accuracy and efficiency of the two methods was addressed

deformation quantization and the baum connes conjecture alternative titles of this paper would have been index theory without index or the baum connes conjecture without baum

eliminating unphysical oscillations arising in galerkin solutions to classical integral equations of antenna theory an asymptotic study previous works have discussed in detail the difficulties occurring when one applies numerical methods to hallen s and pocklington s integral equations for the current distribution along a linear antenna when the so called approximate kernel is used the main difficulty is the appearance of unphysical oscillations near the driving point and or near the ends of the antenna another work has proposed an easy to apply possible remedy to overcome these unnatural oscillations the basic idea is to define a new current from the near magnetic field produced by the original oscillating current in the present paper for the case of an antenna center driven by a delta function generator we place this remedy on a much firmer basis by means of an asymptotic study for the case of the linear antenna of infinite length we provide specific connections of our study to actual finite length antennas

extension and optimization of the axisymmetric 2 5 d eigensolver toward far field calculations in stratified backgrounds we extend and optimize the axisymmetric 2 5 d eigensolver to calculate the far field results of axisymmetric structures in stratified backgrounds sbs which is useful and practical for optical design and engineering two kinds of near field to far field transformations nfffts are proposed one is called closed surface transformation which is based on the dipole radiations in sbs the other is called open surface transformation which is based on the analytic continuation this transformation can provide the far field results of any axisymmetric structure including backgrounds the far field results achieved by these two transformations are in good agreement with each other and with analytical ones moreover in order to obtain accurate near field results we optimize the perfectly matched layer pml setting of the 2 5 d eigensolver corresponding optimization rules are proposed and discussed

application of a new theoretical treatment to an old problem sinusoidal pressure release boundary scattering the scattering of sound from a sinusoidal pressure release boundary has long been the subject of study among researchers recently r l holford j acoust soc am 70 1116 1128 1981 outlined an exact solution to a variant of the helmholtz integral equation that permits calculation of the unknown velocity potential on the surface by the solution to a matrix equation application of this solution is made to the sinusoid to generate plane wave reflection coefficients the reflection coefficients are then compared with two sets of model tank data as well as with the approximate solutions of kirchhoff eckart rayleigh and brekhovskikh the exact solution compares most favorably with the data particularly on the rougher surfaces the principal errors produced by the approximate solutions are a shift in the location of the reflection nulls an inability to correctly predict unity when all side orders are evanescent and the absence of rayleigh reflection anomalies these errors appear regardless of the

estimating performance of feynman s ratchet with limited information we estimate the performance of feynman s ratchet at given values of the ratio of cold to hot reservoir temperatures and the figure of merit efficiency in the case of engine and coefficienct of performance in the case of refrigerator the latter implies that only the ratio of two intrinsic energy scales is known to the observer but their exact values are completely uncertain the prior probability distribution for the uncertain energy parameters is argued to be jeffreys prior we define an average measure for performance of the model by averaging over the prior distribution the power output heat engine or the criterion refrigerator which is the product of rate of heat absorbed from the cold reservoir and the coefficient of performance cop we observe that the figure of merit at optimal performance close to equilibrium is reproduced by the prior averaging procedure further we obtain the well known expressions of finite time thermodynamics for the efficiency at optimal power and the cop at optimal criterion given by and respectively this analogy is explored further and we point out that the expected heat flow from and to the reservoirs behaves as an effective newtonian flow we also show in a class of quasi static models of quantum heat engines how curzon ahlborn efficiency emerges in asymptotic limit with the use of jeffreys prior

geometric invariants of the quantum hall effect we study the two dimensional hall effect with a random potential the hall conductivity is identified as a geometric invariant associated with an algebra of observables using the pairing betweenk theory and cyclic cohomology theory we identify this geometric invariant with a topological index thereby giving the hall conductivity a new interpretation

numerical stability and convergence of approximations of retarded potential integral equations stability and convergence results for approximations of retarded potential integral equations are given such integral approximations arise in time marching schemes for direct scattering problems and are both extremely complicated and computationally intensive they are often observed to have exponentially increasing unstable solutions a stability analysis is performed which is shown to provide a way of predicting whether a given scheme is stable or not the proposed test is supported by numerical experiments and requires much less computation than the scheme itself it is also shown that if an approximation for this type of integral equation is both stable and consistent then it is convergent

on the use of nonsingular kernels in certain integral equations for thin wire circular loop antennas fundamental properties of certain popular integral equations for thin wire circular loop antennas with nonsingular kernels are studied the cornerstone of our study is a large asymptotic formula for the n th fourier coefficient of the nonsingular kernel four different methods of driving the circular loop are considered namely the delta function generator the finite gap generator a certain type of frill generator and the case of plane wave incidence the excitation model is crucial to our discussions since it determines the behavior of the solution convergence also discussed are associated difficulties that may arise when moment methods are applied to the aforementioned equations as well as a simple improvement to the fourier series method for determining the current in the case of the frill generator the main results herein closely parallel recent results for the case of the straight antenna and similarities and differences between the straight and circular cases are discussed

towards a stable two dimensional time domain source model solution by use of a combined source formulation the application of the source model technique to the solution of two dimensional 2 d transient scattering problems directly in the time domain is described special attention is given to the accuracy and the stability of the numerical solution scheme a detailed comparison is made between the more often used single type source solution scheme and a newly proposed combined source solution scheme it is found that the combined source scheme can offer a significant advantage in terms of stability without any appreciable loss in accuracy over the single type source scheme

coefficient of performance at maximum figure of merit and its bounds for low dissipation carnot like refrigerators the figure of merit for refrigerators performing finite time carnot like cycles between two reservoirs at temperature t h and t c t h is optimized it is found that the coefficient of performance at maximum figure of merit is bounded between 0 and sqrt 9 8e c 3 2 for the low dissipation refrigerators where e c t c t h t c is the carnot coefficient of performance for reversible refrigerators these bounds can be reached for extremely asymmetric low dissipation cases when the ratio between the dissipation constants of the processes in contact with the cold and hot reservoirs approaches to zero or infinity respectively the observed coefficients of performance for real refrigerators are located in the region between the lower and upper bounds which is in good agreement with our theoretical estimation

on two types of convergence in the method of auxiliary sources the method of auxiliary sources mas is often applied to problems involving an externally illuminated smooth perfect electric conductor pec one seeks to approximately satisfy the boundary condition on the pec surface using auxiliary sources located inside the pec surface usually the underlying auxiliary surface is also smooth and closed the currents on the auxiliary sources mas currents are the initial unknowns once they have been found one can easily determine the field due to them mas field at all points external to the auxiliary surface and in particular at all points external to the pec scatterer we show that in the limit n it is possible to have a convergent mas field together with divergent mas currents and that this phenomenon is accompanied by an abrupt behavior of the limiting value of the mas field we show this possibility through an analytical study of a two dimensional scattering problem involving a circular cylinder in which mas fields and currents can be determined explicitly the analytical study proceeds from first principles it involves fundamental electromagnetics and relatively simple mathematical manipulations numerical results supplement the analytical study and demonstrate the nature of the aforementioned divergence our study sheds light on other aspects of mas in particular it establishes interesting similarities and differences between mas and its continuous version and reveals many similarities between mas currents and numerical solutions of hallen s integral equation with the approximate kernel

on the phenomenon of oscillations in the method of auxiliary sources when one applies the method of auxiliary sources to scattering problems involving perfect conductors one first seeks fictitious auxiliary currents located inside the conductor and then determines the field from these currents for a simple two dimensional problem involving an infinite circular cylinder illuminated by an electric current filament it has recently been shown analytically that it is possible to have divergent auxiliary currents to make this statement precise one must properly normalize the currents together with a convergent field it was also shown through numerical investigations that the aforementioned divergence appears as abnormal rapid oscillations in the present paper we investigate such phenomena in more detail with particular emphasis on oscillations for a perfectly conducting ground plane illuminated by an electric current filament we once again demonstrate the possibility of having divergent oscillating currents producing a convergent field we develop an asymptotic formula for the oscillating current values which sheds light on the nature of the oscillations we revisit the circular cylinder problem to develop a similar asymptotic formula we also discuss roundoff errors and possible generalizations to scatterers of other shapes the present study is to a great extent analytical with the analytical predictions confirmed and supplemented by numerical results

on the structure of twisted group c algebras we first give general structural results for the twisted group algebras c g of a locally compact group g with large abelian subgroups in particular we use a theorem of williams to realise c g as the sections of a c bundle whose fibres are twisted group algebras of smaller groups and then give criteria for the simplicity of these algebras next we use a device of rosenberg to show that when is a discrete subgroup of a solvable lie group g the k groups k c are isomorphic to certain twisted k groups k g of the homogeneous space g and we discuss how the twisting class h 3 g z depends on the cocycle

prior information and inference of optimality in thermodynamic processes we propose a bayesian inference rule to derive the prior distribution function for a constrained thermodynamic process with incomplete information based on this prior we develop procedures to estimate the work extracted from a heat engine operating between two finite reservoirs in particular we find that the optimal work extractable can be inferred with very good agreement which extends to the far from equilibrium regime the estimate for efficiency is shown to follow a universal behavior beyond the linear response term c 2 c 2 8 where c is the carnot bound estimation of this feature can be ascribed to a symmetry with respect to different allowed inferences with each assigned an equal weight in contrast to finite time irreversible models considered in the literature this universality holds for a reversible model of a heat engine but with incomplete information

on acoustic and electric faraday cages two dimensional problems involving many identical small circles are considered the circles are the cross sections of parallel wires modelling a cage or a grating both electrostatic and acoustic fields are considered the main emphasis is on periodic configurations of n circles distributed evenly around a large circle a ring foldy s theory is used for acoustic problems and then adapted for electrostatic problems in both cases circulant matrices are encountered the fields can be calculated explicitly then the limit n is studied a connection between the n circle problem and the limiting problem fields exterior to the ring is established using known results on the convergence of a defective form of the trapezoidal rule defective in that endpoint contributions are ignored because the integrand has logarithmic singularities at those points this shows that the solution of the limiting problem is approached very slowly as n 1 log n as n

on the use of svd improved point matching in the current model method em scattering an approach which uses the singular value decomposition svd to improve the accuracy of the numerical solution obtained with fictitious current models is introduced in this approach the svd is essentially facilitating a systematic way to optimally reduce the generalized inverse matrix used in the solution to a submatrix of smaller rank this reduction strikes a balance between the fulfillment of the boundary conditions at the matching points and that between them clearly the boundary conditions errors at the matching points are no longer strictly zero however the previously discernible errors between the matching points are markedly suppressed the approach is efficacious not only when the impedance matrix is inherently singular or highly ill conditioned but also when this matrix is entirely well conditioned it can be generalized and implemented in any method of moments code which uses point matching for testing the approach has been incorporated into an existing solution based on the current model method for the problem of scattering from periodic sinusoidal surfaces and is shown to render the solution more accurate

efficient electromagnetic scattering computation using the random auxiliary sources method for multiple composite 3 d arbitrary objects the electromagnetic em scattering problem from three dimensional 3 d arbitrary composite objects is proposed using the random auxiliary sources ras method based on direct application of the boundary conditions with the uniqueness theorem and the use of random equivalent problems concept more degrees of freedom to the sources positions are added resulting in significantly efficient solutions with lower memory requirements the technique does not require any singularity treatment due to placing the equivalent sources away from the boundaries while boundary conditions are not enforced exactly an iterative framework is introduced that can achieve an acceptable level of error in their satisfaction for an arbitrary randomly generated set of equivalent sources the presented technique promises a significant reduction in the execution time and memory requirements compared to the surface equivalent based method of moments mom the solution stability repeatability and numerical noise susceptibility are investigated thoroughly through this work also a novel edge correction scheme has been implemented to extend the capabilities of this procedure to structures with sharp edges the results of the presented technique are compared to series solutions for conducting spheres and a commercially available mom code for arbitrarily shaped objects and combinations of different materials

analytic continuation considerations when using generalized formulations for scattering problems a generalized operator equation has been developed for a simple scattering problem for which a closed form solution exists the operator equation has been solved numerically via the method of moments using spatially impulsive fictitious sources as expansion functions together with a simple point matching testing procedure the study focused on the convergence and accuracy of the solution and examined how they are dependent on the location and distance of the fictitious sources relative to the area containing the singularities of the actual field simulated by these sources as expected it was found that when the actual field simulated by the fictitious impulsive sources has singularities lying between the physical boundary and the closed surface over which the sources are placed the impulsive expansion does not yield a uniformly convergent solution in this case instabilities are encountered as the number of sources increases in the sense that a small improvement of the boundary error requires a considerable change in the currents the moment matrix is then difficult to invert and easily susceptible to large round off errors conversely if the actual field has no singularities lying between the physical boundary and the closed surface over which the sources are placed the impulsive expansion does yield a uniformly convergent solution to any degree of precision

complex multipole beam approach to electromagnetic scattering problems a novel approach to reducing the matrix size associated with the method of moments mom solution of the problem of electromagnetic scattering from arbitrary shaped closed bodies is presented the key step in this approach is to represent the scattered field in terms of a series of beams produced by multipole sources located in a complex space on the scatterer boundary the fields generated by these multipole sources resemble the gabor basis functions by utilizing the properties of the gabor series guidelines for selecting the orders as well as locations of the multipole sources are developed it is shown that the present approach not only reduces the number of unknowns but also generates a generalized impedance matrix with a banded structure and a low condition number the accuracy of the proposed method is verified by comparing the numerical results with those derived by using the method of moments

on inductive limits of matrix algebras over the two torus it will be shown in this paper that certain real rank zero c algebras which are inductive limits of c algebras of the form i m k i c inline graphic xmlns xlink http www w3 org 1999 xlink xlink href 01i can be expressed as inductive limits of c algebras of the form i m k i c s 1 in particular if both a and b are of real rank zero and are inductive limits of c algebras of the form i m k i c s 1 then also a b is an inductive limit of c algebras of the form i m k i c s 1 hence a b can be classified by its k theory this is a key step in the general classification theory of inductive limit c algebras

treating the instabilities in marching on in time method from a different perspective electromagnetic scattering a general method is presented to treat the instabilities which are frequently observed in the electromagnetic transient solutions using the marching on in time method the basic idea is to apply an finite impulse response fir filter with a constant group delay during the course of marching in time an electric field integral equation efie formulation for perfectly conducting bodies is used as a vessel to demonstrate the method sample numerical results are presented and discussed the computed results while showing good agreement with the data obtained from other methods present great stability improvement

highly directive current distributions general theory a theoretical scheme for studying the properties of localized monochromatic and highly directive classical current distributions in two and three dimensions is formulated and analyzed for continuous current distributions it is shown that maximizing the directivity d in the far field while constraining c5n t where n is the integral of the square of the magnitude of the current density and t is proportional to the total radiated power leads to a fredholm integral equation of the second kind for the optimum current this equation is a useful analytical tool for studying currents that produce optimum directivities above the directivity of a uniform distribution various consequences of the present formulation are examined analytically for essentially arbitrary geometries of the current carrying region in particular certain properties of the optimum directivity are derived and differences between the continuous and discrete cases are pointed out when c the directivity tends to infinity monotonically in accord with oseen s einstein needle radiation s1063 651x 98 02108 4

fmm accelerated source model technique for many scatterer problems motivated by the lack of scalability of the source model technique smt to many scatterer problems a new smt based method for 3 d many scatterer problems is developed the large time complexity of the smt is reduced by deriving an implicit compact representation of the scattering problem via a fast multipole formulation which divides the interactions into far and near ones while grouping and approximating the far interactions this results in significant acceleration of classical smt which is demonstrated in a variety of 3 d many body scattering problems the improvement is notable already at the minimal number of unknowns necessary and grows with the size of the problem

on the stable rank of algebras of operator fields over metric spaces let be a finitely generated torsion free two step nilpotent group let c denote the universal c algebra of we show that sr c sr c 1 where for a unital c algebra a sr a is the stable rank of a and where 1 is the space of one dimensional representations of in process we give a stable rank estimate for maximal full algebras of operator fields over metric spaces