zappa szep products of semigroups and their c algebras abstract zappa szep products of semigroups provide a rich class of examples of semigroups that include the self similar group actions of nekrashevych we use li s construction of semigroup c algebras to associate a c algebra to zappa szep products and give an explicit presentation of the algebra we then define a quotient c algebra that generalises the cuntz pimsner algebras for self similar actions we indicate how known examples previously viewed as distinct classes fit into our unifying framework we specifically discuss the baumslag solitar groups the binary adding machine the semigroup n n and the a x b semigroup z z

a novel symmetrical solenoidal basis for the mom analysis of closed surfaces in low frequency or dense mesh problems the electric field integral equation efie discretization requires the extraction of the solenoidal part of the current to avoid severe bad conditioning of the method of moments mom matrix the common approach to do this is using the loop basis functions for the solenoidal part here we show that the conventional loop basis has poor performances when applied to closed bodies we identify the reason of this behavior and propose a new solenoidal basis that is free of this problem in addition the associated generation algorithm also automatically detects topological complexities like handles and produces a basis with significantly improved performances as confirmed by the reported numerical examples the basis generation complexity as well as memory requirements or sparsity for the proposed basis are o n log n

a novel multilevel matrix compression method for analysis of electromagnetic scattering from pec targets we present a new multilevel matrix compression method mlmcm and its application to the analysis of scattering problems from three dimensional 3 d arbitrary shaped conductors the compression is achieved without generating the full subblocks of the matrix by the rank based method unlike the conventional rank based method incoming compression matrix and outgoing compression matrix are defined when coupling with a cluster of its far interaction groups only a small translation matrix is redefined for every two coupling groups the merits of the proposed method are 1 it is kernel function independent and can be applied to arbitrary complex media 2 it is more efficient than conventional rank based methods this paper shows numerical results to demonstrate the validity of the proposed method

on the partial differential equations of electrostatic mems devices iii refined touchdown behavior abstract this paper is a continuation of n ghoussoub y guo on the partial differential equations of electrostatic mems devices stationary case siam j math anal 38 2007 1423 1449 and n ghoussoub y guo on the partial differential equations of electrostatic mems devices ii dynamic case nodea nonlinear differential equations appl 2008 in press where we analyzed nonlinear parabolic problem u t u f x 1 u 2 on a bounded domain of r n with dirichlet boundary conditions this equation models a simple electrostatic micro electromechanical system mems device consisting of a thin dielectric elastic membrane with boundary supported at 0 above a rigid ground plate located at 1 here u is modeled to describe dynamic deflection of the elastic membrane when a voltage represented here by is applied the membrane deflects towards the ground plate and a snap through touchdown must occur when it exceeds a certain critical value pull in voltage creating a so called pull in instability which greatly affects the design of many devices in an effort to achieve better mems design the material properties of the membrane can be technologically fabricated with a spatially varying dielectric permittivity profile f x in this work some a priori estimates of touchdown behavior are established based on which the refined touchdown profiles are obtained by adapting self similar method and center manifold analysis applying various analytical and numerical techniques some properties of touchdown set such as compactness location and shape are also discussed for different classes of varying permittivity profiles

constructing nested bases approximations from the entries of non local operators in this article a method for constructing nested bases approximations to large scale fully populated discretizations of integral operators is introduced the scheme uses only few of the matrix entries for approximating the whole matrix in this sense it is similar to the adaptive cross approximation method however its computational complexity is improved

dynamics of a free boundary problem with curvature modeling electrostatic mems the dynamics of a free boundary problem for electrostatically actuated microelectromechanical systems mems is investigated the model couples the electric potential to the deformation of the membrane the deflection of the membrane being caused by application of a voltage difference across the device more precisely the electrostatic potential is a harmonic function in the angular domain that is partly bounded by the deformable membrane the gradient trace of the electric potential on this free boundary part acts as a source term in the governing equation for the membrane deformation the main feature of the model considered herein is that unlike most previous research the small deformation assumption is dropped and curvature for the deformation of the membrane is taken into account what leads to a quasilinear parabolic equation the free boundary problem is shown to be well posed locally in time for arbitrary voltage values and globally in time for small voltages values furthermore existence of asymptotically stable steady state configurations is proved in case of small voltage values as well as non existence of steady states if the applied voltage difference is large finally convergence of solutions of the free boundary problem to the solutions of the well established small aspect ratio model is shown

an effective mom solution with nested complex source beam method for electromagnetic scattering problems a nested complex source beam ncsb method based on octree structure is proposed to accelerate the method of moments mom solution for three dimensional 3 d electromagnetic scattering analysis at the finest level the far field contribution of each basis function is expressed with that of complex source beams csbs distributed on the equivalent sphere surface for every group furthermore the far field radiated by csbs of the source group at child level can be represented by that radiated by csbs of the source group at parent level similarly the received field of csbs for each observation group at the parent level can be expressed with that of csbs belong to their child level group an equivalent relationship of csbs between every two adjacent levels is built to obtain the nested equivalent process for ncsb method and only the csbs at the finest level have the direct association with the basis functions in their group the ncsb method is shown to be o nlogn computational complexity for both matrix vector product mvp time and memory storage by numerical examples with appropriate parameters and numerical results demonstrate the better efficiency of the proposed method than the current multilevel csb method

on the critical dimension of a fourth order elliptic problem with negative exponent we study the regularity of the extremal solution of the semilinear biharmonic equation 2u u 1 u 2 on a ball b rn under navier boundary conditions u u 0 on b where 0 is a parameter while 0 0 are fixed constants it is known that there exists such that for there is no solution while for 0 and it is singular supbu 1 for n 9 0 and 0 with small our proof for the singularity of extremal solutions in dimensions n 9 is based on certain improved hardy rellich inequalities

fast iterative solution of integral equations with method of moments and matrix decomposition algorithm singular value decomposition the multilevel matrix decomposition algorithm mlmda was originally developed by michielsen and boag for 2d tmz scattering problems and later implemented in 3d by rius et al the 3d mlmda was particularly efficient and accurate for piece wise planar objects such as printed antennas however for arbitrary 3d problems it was not as efficient as the multilevel fast multipole algorithm mlfma and the matrix compression error was too large for practical applications this paper will introduce some improvements in 3d mlmda like new placement of equivalent functions and svd postcompression the first is crucial to have a matrix compression error that converges to zero as the compressed matrix size increases as a result the new mda svd algorithm is comparable with the mlfma and the adaptive cross approximation aca in terms of computation time and memory requirements remarkably in high accuracy computations the mda svd approach obtains a matrix compression error one order of magnitude smaller than aca or mlfma in less computation time like the aca the mda svd algorithm can be implemented on top of an existing mom code with most commonly used green s functions but the mda svd is much more efficient in the analysis of planar or piece wise planar objects like printed antennas

nica toeplitz algebras associated with product systems over right lcm semigroups abstract we prove uniqueness of representations of nica toeplitz algebras associated to product systems of c correspondences over right lcm semigroups by applying our previous abstract uniqueness results developed for c precategories our results provide an interpretation of conditions identified in work of fowler and fowler raeburn and apply also to their crossed product twisted by a product system in the new context of right lcm semigroups as well as to a new doplicher roberts type c algebra associated to the nica toeplitz algebra as a derived construction we develop nica toeplitz crossed products by actions with completely positive maps this provides a unified framework for nica toeplitz semigroup crossed products by endomorphisms and by transfer operators we illustrate these two classes of examples with semigroup c algebras of right and left semidirect products

discrete product systems and twisted crossed products by semigroups a product systemeover a semigrouppis a family of hilbert spaces es s p together with multiplicationses et est we vieweas a unitary valued cocycle onp and consider twisted crossed productsa e pinvolvingeand an action ofpby endomorphisms of ac algebraa whenpis quasi lattice ordered in the sense of nica we isolate a class of covariant representations ofe and consider a twisted crossed productbp e pwhich is universal for covariant representations ofewhenehas finite dimensional fibres and in general is slightly larger in particular whenp nand dim e1 our algebrabn e n is a new infinite analogue of the toeplitz cuntz algebras ton our main theorem is a characterisation of the faithful representations ofbp e p

product system models for twisted c algebras of topological higher rank graphs abstract we use product systems of c correspondences to introduce twisted c algebras of topological higher rank graphs we define the notion of a continuous t valued 2 cocycle on a topological higher rank graph and present examples of such cocycles on large classes of topological higher rank graphs to every proper source free topological higher rank graph and continuous t valued 2 cocycle c on we associate a product system x of c 0 0 correspondences built from finite paths in we define the twisted cuntz krieger algebra c c to be the cuntz pimsner algebra o x and we define the twisted toeplitz algebra t c c to be the nica toeplitz algebra nt x we also associate to and c a product system y of c 0 correspondences built from infinite paths we prove that there is an embedding of t c c into nt y and an isomorphism between c c and o y

refined asymptotics for the blowup of ut δu up this work is concerned with positive blowing up solutions of the semilinear heat equation ut u up in rn our main contribution is a sort of center manifold analysis for the equation in similarity variables leading to refined asymptotics for u in a backward space time parabola near any blowup point we also explore a connection between the asymptotics of u and the local geometry of the blowup set

variational approach to mems model with fringing field abstract we study semilinear elliptic equations which are called mems model with fringing field using the variational method and dual approach we show the multiple existence of positive solutions

on the structure of positive solutions to an elliptic problem with a singular nonlinearity abstract we consider the following boundary value problem u u p u q in u 0 on where p q 0 and is a bounded smooth domain in r n n 2 it is shown that for 1 there is at least one solution u of this problem satisfying u in for any 0 the profiles of u for 0 and are analyzed for 1 it is clear that u 1 is a constant solution to the problem we also present the branch of solutions satisfying 0 u 1 in for the equation with 1 we obtain more information on the structure of positive radial solutions of the problem when is an annulus

the blow up rate of solutions of semilinear heat equations on caracterise le comportement asymptotique de la solution de l equation de la chaleur semilineaire u t u exp u dans o t u k sur o t u x 0 x k pour xe k 0 dans un voisinage d un point d explosion quand t approche du temps d explosion fini t

self similar solutions for a quenching problem with spatially dependent nonlinearity abstract this paper is devoted to the study of the self similar solutions for a semilinear parabolic equation with spatially dependent nonlinearity which arises in the model of micro electro mechanical system we first provide a result on the non existence of slow orbit for a certain range of parameters next we prove the existence of backward solutions with the desired polynomial growth condition at infinity to the associated equation by a fixed point argument then we give a detailed analysis of the behavior of global solutions at the origin finally as an application of the above results we prove a uniqueness theorem

product systems over right angled artin semigroups we build upon mac lane s definition of a tensor category to introduce the concept of a product system that takes values in a tensor groupoid g we show that the existing notions of product systems fit into our categorical framework as do the k graphs of kumjian and pask we then specialize to product systems over right angled artin semigroups these are semigroups that interpolate between free semigroups and free abelian semigroups for such a semigroup we characterize all product systems which take values in a given tensor groupoid g in particular we obtain necessary and sufficient conditions under which a collection of k 1 graphs form the coordinate graphs of a k graph

nested equivalent source approximation for the modeling of penetrable bodies a multiregion nested equivalent source approximation nesa is presented for the fast analysis of electromagnetic scattering by piecewise homogeneous media nesa was introduced for the analysis of metallic problems at low and moderate frequencies and later extended to the fully dynamic case with the introduction of a directional scheme in this communication we modify the wideband scheme for the analysis of arbitrary penetrable media the asymptotic computational costs of the algorithm are compatible with the state of the art factorizations namely linear and quasi linear o n and o n log n for moderate and high frequency problems respectively numerical examples validate the accuracy and favorable computational cost of nesa when applied to penetrable bodies

nuclearity of semigroup c algebras and the connection to amenability abstract we study c algebras associated with subsemigroups of groups for a large class of such semigroups including positive cones in quasi lattice ordered groups and left ore semigroups we describe the corresponding semigroup c algebras as c algebras of inverse semigroups groupoid c algebras and full corners in associated group crossed products these descriptions allow us to characterize nuclearity of semigroup c algebras in terms of faithfulness of left regular representations and amenability of group actions moreover we also determine when boundary quotients of semigroup c algebras are uct kirchberg algebras this leads to a unified approach to cuntz algebras and ring c algebras

a parabolic free boundary problem modeling electrostatic mems the evolution problem for a membrane based model of an electrostatically actuated microelectromechanical system is studied the model describes the dynamics of the membrane displacement and the electric potential the latter is a harmonic function in an angular domain the deformable membrane being a part of the boundary the former solves a heat equation with a right hand side that depends on the square of the trace of the gradient of the electric potential on the membrane the resulting free boundary problem is shown to be well posed locally in time furthermore solutions corresponding to small voltage values exist globally in time while global existence is shown not to hold for high voltage values it is also proven that for small voltage values there is an asymptotically stable steady state solution finally the small aspect ratio limit is rigorously justified

on the quenching of a nonlocal parabolic problem arising in electrostatic mems control abstract we consider a nonlocal parabolic model for a micro electro mechanical system specifically for a radially symmetric problem with monotonic initial data it is shown that the solution quenches so that touchdown occurs in the device in a situation where there is no steady state it is also shown that quenching occurs at a single point and a bound on the approach to touchdown is obtained numerical simulations illustrating the results are given

semigroup crossed products and the toeplitz algebras of nonabelian groups abstract we consider the quasi lattice ordered groups g p recently introduced by nica we realise their universal toeplitz algebra as a crossed product b p p by a semigroup of endomorphisms and show that the toeplitz representation is faithful precisely when an amenability condition is satisfied we then show directly that many interesting free products are amenable the final result contains theorems of cuntz and dinh on the uniqueness of toeplitz cuntz algebras as special cases

hierarchical bases for nonhierarchic 3 d triangular meshes we describe a novel basis of hierarchical multiscale functions that are linear combinations of standard rao wilton glisson rwg functions when the basis is used for discretizing the electric field integral equation efie for pec objects it gives rise to a linear system immune from low frequency breakdown and well conditioned for dense meshes the proposed scheme can be applied to any mesh with triangular facets and therefore it can be used as if it were an algebraic preconditioner the properties of the new system are confirmed by numerical results that show fast convergence rates of iterative solvers significantly better than those for the loop tree basis as a byproduct of the basis generation a generalization of the rwg functions to nonsimplex cells is introduced

on the cauchy problem for a reaction diffusion equation with a singular nonlinearity abstract we consider the following cauchy problem with a singular nonlinearity p u t u u x r n t 0 0 u t 0 c lb r n with n 3 and having a positive lower bound we find some conditions on the initial value such that the local solutions of p vanish in finite time meanwhile we obtain optimal conditions on for global existence and study the large time behavior of those global solutions in particular we prove that if 0 and n 3 x u s x 2 1 n 2 2 1 1 1 x 2 1 where u s is a singular equilibrium of p and 1 then p has a unique global classical solution u with u u s and u x t 1 1 1 1 1 1 1 t 1 1 on the other hand the structure of positive radial solutions of the steady state of p is studied and some interesting properties of the positive solutions are obtained moreover the stability and weakly asymptotic stability of the positive radial solutions of the steady state of p are also discussed

some singular equations modeling mems in the past fifteen years mathematical models for microelectromechanical systems mems have been the subject of several studies in particular due to the interesting qualitative properties they feature still most research is devoted to an illustrative but simplified model which is deduced from a more complex model when the aspect ratio of the device vanishes the so called vanishing or small aspect ratio model the analysis of the aforementioned complex model involving a moving boundary has started only recently and an outlook of the results obtained so far in this direction is provided in this survey

on a class of semilinear elliptic problems with singular nonlinearities abstract this paper is devoted to a class of semilinear elliptic problems p u f x 1 u x u 0 x with 0 r n n 1 here and are positive parameters and f x c is a nonnegative function by the sub super solutions method we show that there exists a critical parameter f such that p has at least one solution including a unique minimal solution for 0 and no solution for moreover we get the rigorous bounds on we also discuss properties of minimal solutions of p including regularity and monotonicity

the extinction versus the blow up global and non global existence of solutions of source types of degenerate parabolic equations with a singular absorption abstract we consider nonnegative solutions of degenerate parabolic equations with a singular absorption term and a source nonlinear term t u u x p 2 u x x u u 0 f u x t in i 0 t with the homogeneous zero boundary condition on i x 1 x 2 an open bounded interval in r through this paper we assume that p 2 and 0 1 to show the local existence result we prove first a sharp pointwise estimate for u x one of our main goals is to analyze conditions on which local solutions can be extended to the whole time interval t 0 the so called global solutions or by the contrary a finite time blow up 0 0 arises such that lim t 0 u t l i moreover we prove that any global solution must vanish identically after a finite time if provided that either the initial data or the source term is small enough finally we show that the condition f 0 x t 0 x t i 0 is a necessary and sufficient condition for the existence of solution of equations of this type

pure infiniteness and ideal structure of c algebras associated to fell bundles abstract we investigate structural properties of the reduced cross sectional algebra c r b of a fell bundle b over a discrete group g conditions allowing one to determine the ideal structure of c r b are studied notions of aperiodicity paradoxicality and b infiniteness for the fell bundle b are introduced and investigated by themselves and in relation to the partial dynamical system dual to b several criteria of pure infiniteness of c r b are given it is shown that they generalize and unify corresponding results obtained in the context of crossed products by the following duos laca spielberg 34 jolissaint robertson 21 sierakowski rordam 47 giordano sierakowski 18 and ortega pardo 39 for exact separable fell bundles satisfying the residual intersection property primitive ideal space of c r b is determined the results of the paper are shown to be optimal when applied to graph c algebras applications to a class of exel larsen crossed products are presented

mixed form nested approximation for wideband multiscale simulations we propose a mixed skeleton and equivalence nested approximation method to compress the impedance matrix of the method of moments for the wideband multiscale simulations we first introduce a nested skeleton approximation where the impedance matrix is expressed recursively by sampling the dominant basis functions e g skeletons with a fully algebraic implementation from the original basis functions the idea is to introduce the automatically constructed test surface around the interface between near and far field regions for each group the dominant rwgs are sampled recursively with the adaptive cross approximation to compress the matrix against the test surface second we introduce a mixed form algorithm of skeleton and equivalence nested approximation method at low levels the nested skeleton approximation is employed and it is smoothly transferred to standard wideband nested equivalence approximation wnesa at high levels an accurate number of skeletons can be always found with a predetermined threshold at a low level which will improve computation efficiency with respect to wnesa the computational complexity of the proposed algorithm is mathcal o n log n n is the number of unknowns numerical wideband multiscale simulations demonstrate the efficiency of the proposed algorithm

approximation of boundary element matrices this article considers the problem of approximating a general asymptotically smooth function in two variables typically arising in integral formulations of boundary value problems by a sum of products of two functions in one variable from these results an iterative algorithm for the low rank approximation of blocks of large unstructured matrices generated by asymptotically smooth functions is developed this algorithm uses only few entries from the original block and since it has a natural stopping criterion the approximative rank is not needed in advance

on the use of wavelet expansions in the method of moments em scattering an approach which incorporates the theory of wavelet transforms in method of moments solutions for electromagnetic wave interaction problems is presented the unknown field or response is expressed as a twofold summation of shifted and dilated forms of a properly chosen basis function which is often referred to as the mother wavelet the wavelet expansion can adaptively fit itself to the various length scales associated with the scatterer by distributing the localized functions near the discontinuities and the more spatially diffused ones over the smooth expanses of the scatterer the approach is thus best suited for the analysis of scatterers which contain a broad spectrum of length scales ranging from a subwavelength to several wavelengths using a galerkin method and subsequently applying a threshold procedure the moment method matrix is rendered sparsely populated the structure of the matrix reveals the localized scale fitting distribution long before the matrix equation is solved the performance of the proposed discretization scheme is illustrated by a numerical study of electromagnetic coupling through a double slot aperture

efie modeling of high definition multiscale structures we propose a method to significantly improve the spectral properties of the efie mom matrix for broad band analysis of structures with fine details non uniform meshes and large overall sizes a multilevel approach allows to overlay fine meshes and quasi nyquist sampled meshes the fine mesh conditioning is solved via hierarchical basis functions and the quasi nyquist sampled part is treated by an algebraic incomplete lu preconditioner numerical results show the effectiveness of the approach for several realistic structures with overall sizes of ten twenty wavelengths and levels of detail from very moderate to extending over the whole structure

wideband fast kernel independent modeling of large multiscale structures via nested equivalent source approximation we propose a wideband fast kernel independent modeling of large multiscale structures we employ a nested equivalent source approximation nesa to compress the dense system matrix the nesa was introduced by these authors for low and moderate frequency problems smaller than a few wavelengths here we introduce a high frequency nesa algorithm and propose a hybrid version with extreme wideband properties the equivalent sources of the wideband nesa wnesa are obtained by an inverse source process enforcing equivalence of radiated fields on suitably defined testing surfaces in the low frequency region the nesa is used unmodified with a complexity of mathcal o n in the high frequency region in order to obtain a fixed rank matrix compression we hierarchically divide the far coupling space into pyramids with angles related to the peer coupling group size and the nesa testing surfaces are defined as the boundaries of the pyramids this results in a directional nested low rank fixed rank approximation with mathcal o n log n computational complexity that is kernel independent overall the approach yields wideband fast solver for the modeling of large structures that inherits the efficiency and accuracy of low frequency nesa for multiscale problems numerical results and discussions demonstrate the validity of the proposed work

equivalence principle algorithm with body of revolution equivalence surface for the modeling of large multiscale structures we propose a multiscale solver of equivalence principle algorithm with a body of revolution bor equivalence surface es first the whole object is decomposed into subdomains the ess are defined as proper bors e g spheres in this work to enclose each subdomain the rao wilton glisson rwg and bor basis functions are defined on each sphere respectively second the octree is constructed in each nonzero subdomain the multilevel fast multipole algorithm mlfma is employed to solve the equivalent currents on the es individually and then the equivalent currents are projected from rwg onto the bor basis functions the couplings between neighboring subdomains are evaluated directly using mlfma and the separated subdomains are substituted by the es to es couplings and are evaluated efficiently by the bor method of moments bor mom to solve the equation system a hybrid inner and outer iterative solver is employed where the inner iteration is used to solve each local subdomains and the outer iteration is used to update the global solutions by collecting all the local solutions numerical results and discussions demonstrate the validity of the proposed work

a new model for electrostatic mems with two free boundaries abstract a moving boundary problem with two free boundaries modeling a two dimensional idealized mems device with pull in instability is discussed we use a fixed point argument to show that the model possesses stationary solutions for small source voltages we also give a rigorous evidence that solutions of the model converge towards solutions of the associated small aspect ratio equation in the vanishing aspect ratio limit

a multilevel matrix decomposition algorithm for analyzing scattering from large structures a multilevel algorithm is presented for analyzing scattering from electrically large surfaces the algorithm accelerates the iterative solution of integral equations that arise in computational electromagnetics the algorithm permits a fast matrix vector multiplication by decomposing the traditional method of moment matrix into a large number of blocks with each describing the interaction between distant scatterers the multiplication of each block by a trial solution vector is executed using a multilevel scheme that resembles a fast fourier transform fft and that only relies on well known algebraic techniques the computational complexity and the memory requirements of the proposed algorithm are o n log sup 2 n

quenching rate for a nonlocal problem arising in the micro electro mechanical system abstract in this paper we study the quenching rate of the solution for a nonlocal parabolic problem which arises in the study of the micro electro mechanical system this question is equivalent to the stabilization of the solution to the transformed problem in self similar variables first some a priori estimates are provided in order to construct a lyapunov function due to the lack of time monotonicity property we then derive some very useful and challenging estimates by a delicate analysis finally with this lyapunov function we prove that the quenching rate is self similar which is the same as the problem without the nonlocal term except the constant limit depends on the solution itself

on the quenching behavior of the solution of a semilinear parabolic equation where 0 e 0 0 6 u0 0 there exists an e b 0 such that u quenches at finite time t if e e b we say that a is a quenching point for u if there exists a sequence x t with x a and t r t such that u x t 1 as n co throughout this paper we assume that u quenches at finite time t and that u0 is smooth and satisfies

convergence asymptotic periodicity and finite point blow up in one dimensional semilinear heat equations abstract we consider the initial value problem for the semilinear heat equation u t u xx f u t 0 x l t 0 under the dirichlet the neumann or the periodic boundary conditions we show that each solution whether it exists globally for t 0 or blows up in a finite time possesses an asymptotic profile in a certain sense and tends to this profile as time increases in the special case where f u t t f u t for some t 0 among other things the above statement is interpreted as saying that any bounded global solution converges as t to a time t periodic solution having some specific spatial structure in the case where the solution blows up in a finite time say at t t 0 assuming simply that f is a smooth function satisfying some growth conditions and that the initial data is a nonconstant bounded function we prove that the blow up set is a finite set and that lim t t 0 u x t x exists with being a smooth function having at most finitely many singular points

nested equivalent source approximation for the modeling of multiscale structures we introduce a method to compress the impedance matrix of the method of moments mom for the modeling of high fidelity multiscale structures at low to moderate frequencies we start from a method recently proposed to compress static scalar problems proved to have o n computational complexity we represent far coupling between groups through equivalent source distributions on properly defined automatically generated equivalence surfaces the equivalent sources are obtained via an inverse source process that enforces equivalence of radiated fields on testing surfaces within a prescribed accuracy and are intrinsically multiscale this results in an o n complexity scaling of matrix vector products but reduces multiplicative constants i e reduces time and memory requirements for all structures it affords an improved representation of couplings in multiscale structures resulting in a fast convergence of iterative solvers our approach is green s function independent and easy to be implemented in existing mom codes the present version is based on the electric field integral equation efie numerical results prove the effectiveness of the proposed algorithm for complex multiscale structures

on a hyperbolic equation arising in electrostatic mems abstract we consider the initial boundary value problem of a damped wave equation with singular nonlinearity which describes an electrostatic micro electro mechanical system mems device the results of the pull in voltage being the critical threshold for global existence and quenching are obtained if the applied voltage then the equation admits a unique global small solution that exponentially converges to the minimal steady state while large solution may quench in finite time if then any solution quenches in finite time finally in the sense of the viscosity dominated limit the asymptotic relation of solutions between the hyperbolic equation and the parabolic one is investigated also the related error estimates in arbitrary order are derived

the zero set of a solution of a parabolic equation on etudie l ensemble nul d une solution u t x de l equation u t a x t u xx b x t u x c x t u sous des hypotheses tres generales sur les coefficients a b et c

a low frequency efie mlfma solver based on approximate diagonalization of green s function the low frequency fast multipole algorithm lffma based on the approximate diagonalization of green s function for the electric field integral equation efie is proposed for open and complex structures with high accuracy moreover the multiresolution preconditioner is employed to the mixed potential form of efie with rwg basis functions for stability herein a triangle grouping scheme is adopted for the scalar potential so that the accuracy of the lffma can be guaranteed furthermore the computational resources can be reduced largely since part of the radiation pattern of vector potential can share one of the scalar potentials by taking advantage of some mathematical deductions

a kernel independent adaptive fast multipole algorithm in two and three dimensions we present a new fast multipole method for particle simulations the main feature of our algorithm is that it does not require the implementation of multipole expansions of the underlying kernel and it is based only on kernel evaluations instead of using analytic expansions to represent the potential generated by sources inside a box of the hierarchical fmm tree we use a continuous distribution of an equivalent density on a surface enclosing the box to find this equivalent density we match its potential to the potential of the original sources at a surface in the far field by solving local dirichlet type boundary value problems the far field evaluations are sparsified with singular value decomposition in 2d or fast fourier transforms in 3d we have tested the new method on the single and double layer operators for the laplacian the modified laplacian the stokes the modified stokes the navier and the modified navier operators in two and three dimensions our numerical results indicate that our method compares very well with the best known implementations of the analytic fmm method for both the laplacian and modified laplacian kernels its advantage is the relative simplicity of the implementation and its immediate extension to more general kernels

topological aperiodicity for product systems over semigroups of ore type we prove a version of uniqueness theorem for cuntz pimsner algebras of discrete product systems over semigroups of ore type to this end we introduce doplicher roberts picture of cuntz pimsner algebras and the semigroup dual to a product system of regular c correspondences under a certain aperiodicity condition on the latter we obtain the uniqueness theorem and a simplicity criterion for the algebras in question these results generalize the corresponding ones for crossed products by discrete groups due to archbold and spielberg and for exel s crossed products due to exel and vershik they also give interesting conditions for topological higher rank graphs and p graphs and apply to the new cuntz c algebra qn arising from the ax b semigroup over n

multilevel adaptive cross approximation mlaca the multilevel adaptive cross approximation mlaca is proposed as a fast method to accelerate the matrix vector products in the iterative solution of the linear system that results from the discretization of electromagnetic integral equations ie with the method of moments mom the single level aca already described in the literature is extended with a multilevel recursive algorithm in order to improve the asymptotical complexity of both the computational cost and the memory requirements the main qualities of aca are maintained it is purely algebraic and independent of the integral equation kernel green s function as long as it produces pseudo rank deficient matrix blocks the algorithm is presented in such a way that it can be easily implemented on top of an existing mom code with most commonly used green s functions

a mixed form fast multipole algorithm the fast multipole algorithm manifests in two very different forms at low frequencies and at mid frequencies each can operate in their respective regimes but are not tenable in the other regimes the paper reports on a way to factorize the green s function for fast algorithm using a mixed form the low frequency fast multipole algorithm lf fma will be used at low frequencies or the long wavelength regime and the multilevel fast multipole algorithm mlfma will be used for the mid frequencies or the shorter wavelength regime for object modeling where both long wavelength and wave physics are important we propose a mixed form fast multipole algorithm mf fma this algorithm has no low frequency break down and it can work seamlessly from static where circuit physics is important to dynamic where wave physics is important

global solutions of singular parabolic equations arising from electrostatic mems abstract we study dynamic solutions of the singular parabolic problem p u t u x 1 u 2 x t b 0 u x 0 u 0 x 0 x b u x t 0 x b where 0 and 0 are two parameters and b is the unit ball x r n x 1 with n 2 our interest is focussed on p with 2 3 n 4 9 for which p admits a singular stationary solution in the form s x 1 x 2 3 this equation models dynamic deflection of a simple electrostatic micro electro mechanical system mems device under the assumption u 0 s x we address the existence uniqueness regularity stability or instability and asymptotic behavior of weak solutions for p given 4 6 n 3 6 n 2 4 in particular we show that if either n 8 and or 2 n 7 then the minimal compact stationary solution u of p is stable and while s x is unstable however for n 8 and 0 p has no compact minimal solution and s x is an attractor from below not from above furthermore the refined asymptotic behavior of global solutions for p is also discussed for the case where n 8 and 0 which is given by a certain matching of different asymptotic developments in the large outer region closer to the boundary and the thin inner region near the singularity

a new duality theory for compact groups the tannaka krein duality theory characterizes the category g of finite dimensional continuous unitary representations of a compact group as a subcategory of the category of hilbert spaces we prove a more powerful result characterizing g as an abstract category every strict symmetric monoidalc category with conjugates which has subobjects and direct sums and for which thec algebra of endomorphisms of the monoidal unit reduces to the complex numbers is isomorphic to a category g for a compact groupg unique up to isomorphism

regularized model of post touchdown configurations in electrostatic mems equilibrium analysis abstract in canonical models of micro electro mechanical systems mems an event called touchdown whereby the electrical components of the device come into contact is characterized by a blow up in the governing equations and a non physical divergence of the electric field in the present work we propose novel regularized governing equations whose solutions remain finite at touchdown and exhibit additional dynamics beyond this initial event before eventually relaxing to new stable equilibria we employ techniques from variational calculus dynamical systems and singular perturbation theory to obtain a detailed understanding of the properties and equilibrium solutions of the regularized family of equations

analysis of large complex structures with the synthetic functions approach an innovative procedure is presented that allows the method of moments mom analysis of large and complex antenna and scattering problems at a reduced memory and cpu cost bounded within the resources provided by a standard 32 bit personal computer the method is based on the separation of the overall geometry into smaller portions called blocks and on the degrees of freedom of the field the blocks need not be electrically unconnected on each block basis functions are generated with support on the entire block that are subsequently used as basis functions for the analysis of the complete structure only a small number of these functions is required to obtain an accurate solution therefore the overall number of unknowns is drastically reduced with consequent impact on storage and solution time these entire domain basis functions are called synthetic functions they are generated from the solution of the electromagnetic problem for the block in isolation under excitation by suitably defined sources the synthetic functions are obtained from the responses to all sources via a procedure based on the singular value decomposition because of the strong reduction of the global number of unknowns one can store the mom matrix and afford a direct solution the method is kernel free and can be implemented on existing mom codes

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

a gradient estimate to a degenerate parabolic equation with a singular absorption term the global quenching phenomena we prove global existence of nonnegative solutions to the one dimensional degenerate parabolic problems containing a singular term we also show the global quenching phenomena for l1 initial datums moreover the free boundary problem is considered in this paper

topological freeness for c correspondences abstract we study conditions that ensure uniqueness theorems of cuntz krieger type for relative cuntz pimsner algebras o j x associated to a c correspondence x over a c algebra a we give general sufficient conditions phrased in terms of a multivalued map x acting on the spectrum a of a when x j is of type i we construct a directed graph dual to x and prove a uniqueness theorem using this graph when x j is liminal we show that topological freeness of this graph is equivalent to the uniqueness property for o j x as well as to an algebraic condition which we call j acyclicity of x as an application we improve the fowler raeburn uniqueness theorem for the toeplitz algebra t x we give new simplicity criteria for o x we generalize and enhance uniqueness results for relative quiver c algebras of muhly and tomforde we also discuss applications to crossed products by endomorphisms

a hierarchical fast solver for efie mom analysis of multiscale structures at very low frequencies we present an efie fast solver that is stable down to very low frequencies for very dense meshes and multi scale problems the proposed approach uses a split potentials formulation in association to a hierarchic pre conditioner for the underlying low rank fast factorization potential splitting prevents the numerical cancellation problem that undermines the effectiveness of the low rank factorization negatively impacting on the necessary factorization tolerance the fast method presented in this work can be directly implemented into existing codes

compactness along the branch of semistable and unstable solutions for an elliptic problem with a singular nonlinearity we study the branch of semistable and unstable solutions i e those whose morse index is at most 1 of the dirichlet boundary value problem u f x 1 u 2 on a bounded domain n which models among other things a simple electrostatic microelectromechanical system mems device we extend the results of 11 relating to the minimal branch by obtaining compactness along unstable branches for 1 n 7 on any domain and for a large class of permittivity profiles f we also show the remarkable fact that powerlike profiles f x x can push back the critical dimension n 7 of this problem by establishing compactness for the semistable branch on the unit ball also for n 8 and as long as as a byproduct we are able to follow the second branch of the bifurcation diagram and prove the existence of a second solution for in a natural range in all these results the conditions on the space dimension and on the power of the profile are essentially sharp 2007 wiley periodicals inc

semigroup c algebras and amenability of semigroups we construct reduced and full semigroup c algebras for left cancellative semigroups our new construction covers particular cases already considered by a nica and also toeplitz algebras attached to rings of integers in number fields due to j cuntz moreover we show how left amenability of semigroups can be expressed in terms of these semigroup c algebras in analogy to the group case