an all frequency weakly singular surface integral equation for electromagnetism in dielectric media reformulation and well posedness analysis abstract we develop and analyze a surface integral equation sie whose solution pertains to numerical simulations of propagating time harmonic electromagnetic waves in three dimensional dielectric media the formulae to evaluate the far field pattern and propagation of the electric and magnetic fields in the interior and exterior of a dielectric body through surface integrals require the solution of a 2 2 system of weakly singular sies for the two unknown electric and magnetic fields at the interface surface of the dielectric body the sie is governed by an operator that is of the classical identity plus compact form the tangential surface currents and normal surface charges of the dielectric model can be easily computed from the surface electric and magnetic fields a novel feature of that second kind sie is that when augmented with two stabilization scalar equations the corresponding reformulation does not suffer from spurious resonances or low frequency breakdown we analyze this novel feature of the sie in both the sobolev and continuous function space settings without any tangential constraints by proving the uniqueness of the stabilized solution of the sie for all positive incident wave frequencies and physically appropriate real complex dielectric media constants using numerical examples we also demonstrate that without our stabilization constraints the 2 2 second kind sie suffers from some spurious resonances and incorporating the two additional scalar equations removes these spurious resonances at the zero frequency limit the two stabilization scalar equations are automatically satisfied and the 2 2 formulation nicely decouples to a well posed system at the zero frequency limit our analysis to establish the equivalence of the classical dielectric model governed by the maxwell equations and the all frequency sie based representation provides a natural decomposition of the surface unknowns into their tangential currents and normal charges the salient features of the representation and well posedness analysis provide a tool to develop a new class of efficient and high order algorithms with numerical analysis for simulation of the three dimensional dielectric media model from low to high frequencies

on single integral equations for the transmission problem of acoustics the transmission problem namely scattering of time harmonic waves in a compressible fluid by a fluid inclusion with different material properties is usually formulated as a pair of coupled boundary integral equations over the interface s between the inclusion and the exterior fluid in this paper however we consider methods for solving the transmission problem using a single integral equation over s for a single unknown function in fact we derive four different integral equations using a hybrid of the direct green s theorem and indirect layer ansatz methods and give conditions for the unique solvability of each and for the subsequent construction of the solution to the transmission problem some of our single integral equations are fredholm integral equations of the second kind with weakly singular kernels thus these equations all of which appear to be new are attractive computationally

the determination of the surface conductivity of a partially coated dielectric a variational method is given for determining the essential supremum of the surface conductivity of a partially coated anisotropic dielectric medium from a knowledge of the far field pattern of the time harmonic electric field at fixed frequency corresponding to an incident plane wave it is assumed that the shape of the scatterer has been determined e g by solving the far field equation and using the linear sampling method numerical examples are given for the scalar case with constant surface conductivity

the zeta function of the laplacian on certain fractals we prove that the zeta function of the laplacian a on self similar fractals with spectral decimation admits a meromorphic continuation to the whole complex plane we characterise the poles compute their residues and give expressions for some special values of the zeta function furthermore we discuss the presence of oscillations in the eigenvalue counting function thereby answering a question posed by j kigami and m lapidus for this class of fractals

strongly elliptic boundary integral equations for electromagnetic transmission problems we study a boundary integral equation method for transmission problems for strongly elliptic differential operators which yields a strongly elliptic system of pseudodifferential operators and which therefore can be used for numerical computations with galerkin s procedure the method is shown to work for the vector helmholtz equation in 3 with electromagnetic transmission conditions we propose a slightly modified system of boundary values in order for the corresponding bilinear form to be coercive over h 1 we analyse the boundary integral equations using the calculus of pseudodifferential operators here the concept of the principal symbol is used to derive existence and regularity results for the solution

nearly cloaking the electromagnetic fields the approximate cloaking is investigated for time harmonic maxwell s equations via the approach of transformation optics the problem is reduced to certain boundary effect estimates due to an inhomogeneous electromagnetic inclusion with an asymptotically small support but an arbitrary content enclosed by a thin high conducting layer sharp estimates are established in terms of the asymptotic parameter which are independent of the material tensors of the small electromagnetic inclusion the result implies that the blow up a small region construction via the transformation optics approach yields a near cloak for the electromagnetic waves a novelty lies in the fact that the geometry of the cloaking construction of this work can be very general moreover by incorporating the conducting layer developed in the present paper right between the cloaked region and the cloaking region arbitrary electromagnetic contents can be nearly cloaked our mathematical technique extends the general one developed in h y

the spectrum of infinite regular line graphs let g be an infinite d regular graph and l g its line graph we consider discrete laplacians on g and l g and show the exact relation between the spectrum of ag and that of al g our method is also applicable to di d2 semiregular graphs subdivision graphs and para line graphs

expansion in generalized eigenfunctions for laplacians on graphs and metric measure spaces we consider an arbitrary selfadjoint operator on a separable hilbert space to this operator we construct an expansion in generalized eigenfunctions in which the original hilbert space is decomposed as a direct integral of hilbert spaces consisting of general eigenfunctions this automatically gives a plancherel type formula for suitable operators on metric measure spaces we discuss some growth restrictions on the generalized eigenfunctions for laplacians on locally finite graphs the generalized eigenfunctions are exactly the solutions of the cor responding difference equation

brownian motion on the sierpinski gasket we construct a brownian motion taking values in the sierpinski gasket a fractal subset of 2 and study its properties this is a diffusion process characterized by local isotropy and homogeneity properties we show for example that the process has a continuous symmetric transition density p t x y with respect to an appropriate hausdorff measure and obtain estimates on p t x y

integrals and potentials of differential 1 forms on the sierpinski gasket abstract we provide a definition of the integral along paths in the sierpinski gasket k for differential smooth 1 forms associated to the standard dirichlet form e on k we show how this tool can be used to study the potential theory on k in particular we prove i a de rham reconstruction of a 1 form from its periods around lacunas in k ii a hodge decomposition of 1 forms with respect to the hilbertian energy norm iii the existence of potentials of smooth 1 forms on a suitable covering space of k we finally show that this framework provides versions of the de rham duality theorem for the fractal k

weak uncertainty principle for fractals graphs and metric measure spaces we develop a new approach to formulate and prove the weak uncertainty inequality which was recently introduced by okoudjou and strichartz we assume either an appropriate measure growth condition with respect to the effective resistance metric or in the absence of such a metric we assume the poincare inequality and reverse volume doubling property we also consider the weak uncertainty inequality in the context of nash type inequalities our results can be applied to a wide variety of metric measure spaces including graphs fractals and manifolds

harmonic analysis for resistance forms abstract in this paper we define the green functions for a resistance form by using effective resistance and harmonic functions then the green functions and harmonic functions are shown to be uniformly lipschitz continuous with respect to the resistance metric making use of this fact we construct the green operator and the measure valued laplacian the domain of the laplacian is shown to be a subset of uniformly lipschitz continuous functions while the domain of the resistance form in general consists of uniformly 1 2 holder continuous functions

theory of oscillations in average crisis induced transient lifetimes analytical and numerical study of the roughly periodic oscillations emerging on the background of the well known power law governing the scaling of the average lifetimes of crisis induced chaotic transients is presented the explicit formula giving the amplitude of normal oscillations in terms of the eigenvalues of unstable orbits involved in the crisis is obtained using a simple geometrical model we also discuss the commonly encountered situation when normal oscillations appear together with anomalous ones caused by the fractal structure of basins of attraction

analysis on products of fractals for a class of post critically finite p c f fractals which includes the sierpinski gasket sg there is a satisfactory theory of analysis due to kigami including energy harmonic functions and laplacians in particular the laplacian coincides with the generator of a stochastic process constructed independently by probabilistic methods the probabilistic method is also available for non p c f fractals such as the sierpinski carpet in this paper we show how to extend kigami s construction to products of p c f fractals since the products are not themselves p c f this gives the first glimpse of what the analytic theory could accomplish in the non p c f setting there are some important differences that arise in this setting it is no longer true that points have positive capacity so functions of finite energy are not necessarily continuous also the boundary of the fractal is no longer finite so boundary conditions need to be dealt with in a more involved manner all in all the theory resembles pde theory while in the p c f case it is much closer to ode theory

boundary integral equations for magnetic screens in ℝ 3 on developpe une methode d integrale limite pour la diffusion d ondes electromagnetiques sur des objets minces le probleme aux limites exterieur pour l equation de helmholtz vectorielle avec donnees de neumann sur un morceau de surface ouvert est converti en un systeme d equations integrales pour les sauts de la composante tangentielle du champ et sa divergence a travers l ecran une legere modification des donnees de cauchy donne un systeme fortement elliptique d equations pseudodifferentielles sur s

on the kleinman martin integral equation method for electromagnetic scattering by a dielectric body the interface problem describing the scattering of time harmonic electromagnetic waves by a dielectric body is often formulated as a pair of coupled boundary integral equations for the electric and magnetic current densities on the interface gamma in this paper following an idea developed by kleinman and martin siam j appl math 48 1988 pp 307 325 for acoustic scattering problems we consider methods for solving the dielectric scattering problem using a single integral equation over gamma for a single unknown density one knows that such boundary integral formulations of the maxwell equations are not uniquely solvable when the exterior wave number is an eigenvalue of an associated interior maxwell boundary value problem we obtain four different families of integral equations for which we can show that by choosing some parameters in an appropriate way they become uniquely solvable for all real frequencies we analyze the well posedness of the integral equations in the space of finite ener

crisis and stochastic resonance in shinriki s circuit abstract in nonlinear electronic oscillators of the chua type the merging of two precritical attractors leads to crisis induced chaos chaos intermittency this behavior can be induced either by addition of noise to the precritical system or by variation of a control parameter close to such a crisis small periodic signals can be enhanced by the addition of noise via the mechanism of stochastic resonance we present an experimental observation of this effect as well as a thorough description of the critical behavior of the system

spectral dimension and bohr s formula for schrodinger operators on unbounded fractal spaces we establish an asymptotic formula for the eigenvalue counting function of the schrodinger operator for some unbounded potentials v on several types of unbounded fractal spaces we give sufficient conditions for bohr s formula to hold on metric measure spaces which admit a cellular decomposition and then verify these conditions for fractafolds and fractal fields based on nested fractals in particular we partially answer a question of fan khandker and strichartz regarding the spectral asymptotics of the harmonic oscillator potential on the infinite blow up of a sierpinski gasket

electromagnetic waves in an inhomogeneous medium abstract in this paper we consider the electromagnetic wave problem in an inhomogeneous medium we first prove uniqueness of the solution using rellich and cauchy kowalewska theorems then we explicitly compute the dirichlet neumann operator on the sphere we reduce the equations to a problem on a truncated domain and we give a variational formulation this formulation reads as a compact perturbation of a coercive operator which leads to the existence of the solution according to fredholm s alternative

boundary integral equations for the scattering of electromagnetic waves by a homogeneous dielectric obstacle time harmonic electromagnetic waves are scattered by a homogeneous dielectric obstacle the corresponding electromagnetic transmission problem is reduced to a single integral equation over s for a single unknown tangential vector field where s is the interface between the obstacle and the surrounding medium in fact several different integral equations are derived and analysed including two previously known equations due to e marx and j r mautz and two new singular integral equations mautz s equation is shown to be uniquely solvable at all frequencies a new uniquely solvable singular integral equation is also found the paper also includes a review of methods using pairs of coupled integral equations over s it is these methods that are usually used in practice although single integral equations seem to offer some computational advantages

on the structures of generating iterated function systems of cantor sets abstract a generating ifs of a cantor set f is an ifs whose attractor is f for a given cantor set such as the middle 3rd cantor set we consider the set of its generating ifss we examine the existence of a minimal generating ifs i e every other generating ifs of f is an iterating of that ifs we also study the structures of the semi group of homogeneous generating ifss of a cantor set f in r under the open set condition osc if dim h f 1 we prove that all generating ifss of the set must have logarithmically commensurable contraction factors from this logarithmic commensurability theorem we derive a structure theorem for the semi group of generating ifss of f under the osc we also examine the impact of geometry on the structures of the semi groups several examples will be given to illustrate the difficulty of the problem we study

spectral analysis on infinite sierpinski gaskets we study the spectral properties of the laplacian on infinite sierpin ski gaskets we prove that the laplacian with the neumann boundary condition has pure point spectrum moreover the set of eigenfunctions with compact support is complete the same is true if the infinite sierpin ski gasket has no boundary but is false for the laplacian with the dirichlet boundary condition in all these cases we describe the spectrum of the laplacian and all the eigenfunctions with compact support to obtain these results first we prove certain new formulas for eigenprojectors of the laplacian on finite sierpin ski pre gaskets then we prove that the spectrum of the discrete laplacian on a sierpin ski lattice is pure point and the eigenfunctions are localized 1998 academic press

calculus on the sierpinski gasket i polynomials exponentials and power series abstract we study the analog of power series expansions on the sierpinski gasket for analysis based on the kigami laplacian the analog of polynomials are multiharmonic functions which have previously been studied in connection with taylor approximations and splines here the main technical result is an estimate of the size of the monomials analogous to x n n we propose a definition of entire analytic functions as functions represented by power series whose coefficients satisfy exponential growth conditions that are stronger than what is required to guarantee uniform convergence we present a characterization of these functions in terms of exponential growth conditions on powers of the laplacian of the function these entire analytic functions enjoy properties such as rearrangement and unique determination by infinite jets that one would expect however not all exponential functions eigenfunctions of the laplacian are entire analytic and also many other natural candidates such as the heat kernel do not belong to this class nevertheless we are able to use spectral decimation to study exponentials and in particular to create exponentially decaying functions for negative eigenvalues

type i intermittency in nonstationary systems models and human heart rate variability abstract type i intermittency occurs whenever a discrete dynamical system in a chaotic state is about to undergo a tangential bifurcation then long stretches of almost periodic behavior laminar phases occur these laminar phases are interspersed with chaotic behavior in type i intermittency simple models show that a characteristic u shaped probability distribution is obtained for the laminar phase length we have shown elsewhere that in some cases of pathology the laminar phase length distribution characteristic for type i intermittency may be obtained in human heart rate variability data the heart and its regulatory systems are usually presumed to be noisy although the effect of additive noise on the laminar phase distribution in type i intermittency is well known the effect of multiplicative noise has not been studied in this presentation we discuss the properties of classes of models of type i intermittency in which the control parameter is changed randomly within the specified parameter range we show that the properties of these models are importantly different from those obtained for type i intermittency in the presence of additive noise we also show how the two models help explain some of the features seen in the intermittency in human heart rate variability some technical problems with measuring laminar phase lengths in time series will also be discussed

calculus on the sierpinski gasket ii point singularities eigenfunctions and normal derivatives of the heat kernel this paper continues the study of fundamental properties of elementary functions on the sierpinski gasket sg related to the laplacian defined by kigami harmonic functions multiharmonic functions and eigenfunctions of the laplacian we describe the possible point singularities of such functions and we use the description at certain periodic points to motivate the definition of local derivatives at these points we study the global behavior of eigenfunctions on all generic infinite blow ups of sg and construct eigenfunctions that decay at infinity in certain directions we study the asymptotic behavior of normal derivatives of dirichlet eigenfunctions at boundary points and give experimental evidence for the behavior of the normal derivatives of the heat kernel at boundary points

boundary integral method for multi layered electromagnetic scattering problems abstract in this paper we consider the scattering of time harmonic electromagnetic plane waves by an impenetrable obstacle which is embedded in a piecewise homogeneous medium applying potential theory the problem can be reformulated as a boundary integral system we obtain the existence and uniqueness of the solution to the system by using the generalized garding inequality and the fredholm theory

the determination of the surface impedance of a partially coated obstacle from far field data a variational method is given for determining the essential supremum of the surface impedance of a partiallycoated perfect conductor from a knowledge of the far field pattern of the time harmonic electric field at fixed frequency it is assumed that the shape of the scatterer has been determined e g bysolving the far field equation and using the linear sampling method numerical examples are given for the scalar case with constant surface impedance

derivations and dirichlet forms on fractals abstract we study derivations and fredholm modules on metric spaces with a local regular conservative dirichlet form in particular on finitely ramified fractals we show that there is a non trivial fredholm module if and only if the fractal is not a tree i e not simply connected this result relates fredholm modules and topology refines and improves known results on p c f fractals we also discuss weakly summable fredholm modules and the dixmier trace in the cases of some finitely and infinitely ramified fractals including non self similar fractals if the so called spectral dimension is less than 2 in the finitely ramified self similar case we relate the p summability question with estimates of the lyapunov exponents for harmonic functions and the behavior of the pressure function

fractafolds based on the sierpinski gasket and their spectra we introduce the notion of fractafold which is to a fractal what a manifold is to a euclidean half space we specialize to the case when the fractal is the sierpinski gasket sg we show that each such compact fractafold can be given by a cellular construction based on a finite cell graph g which is 3 regular in the case that the fractafold has no boundary we show explicitly how to obtain the spectrum of the fractafold from the spectrum of the graph using the spectral decimation method of fukushima and shima this enables us to obtain isospectral pairs of nonhomeomorphic fractafolds we also show that although sg is topologically rigid there are fractafolds based on sg that are not topologically rigid