geometric implementation of hypercubic lattices with noninteger dimensionality by use of low lacunarity fractal lattices it is claimed that the abstract analytic continuation of hypercubic lattices to noninteger dimensionalities can be implemented explicitly by certain fractal lattices of low lacunarity these lattices are special examples of sierpinski carpets their being of low lacunarity means that they are arbitrarily close to being translationally invariant the claim is substantiated for the ising model in d 1 ensuremath epsilon dimensions and for resistor network models with 1ldl2

time dependent inverse scattering from gradient type interfaces using an exact solution in the context of one dimensional linear wave propagation a gradient type interface is a point at which the velocity profile of a scattering medium suffers a jump in first derivative in this paper a time domain approach to scattering from such an interface leads to an integrodifferential equation involving the kernel of the reflection operator impulse response which can be solved exactly if the velocity profile is piecewise parabolic this special case provides the basis for an approximate numerical scheme to solve the inverse scattering problem recover the velocity profile from reflection data for a general velocity profile the algorithm is fast and affords good results if the velocity profile is nearly parabolic or the interest is in short time only a detailed error estimate for the approximation is provided

factorization of the dissipative wave equation and inverse scattering a decomposition of the solution of the dissipative wave equation into incoming and outgoing components across a smooth surface in a homogeneous region is presented the proof of the decomposition is given only for the plane surface this is then applied to the factorization of the dissipative wave equation into incoming and outgoing components in a planar stratified medium the ricatti integral differential equation for the reflection operator that relates the two components is obtained it is shown how the zeroth and second moments with respect to the transverse variable in a planar stratified medium of the reflection kernel can be used in the inverse problem to recover the velocity and dissipation coefficient from knowledge of the scattered field

upper semicontinuity of the attractor for a singularly perturbed hyperbolic equation abstract for a smooth bounded domain r n n 3 and e a real parameter consider the hyperbolic equation eu tt u t u u g in with dirichlet boundary conditions under certain conditions on this equation has a compact attractor a e in h 0 1 l 2 for e 0 the parabolic equation also has a compact attractor which can be naturally embedded into a compact set a 0 in h 0 1 l 2 it is shown that for any neighborhood u of a 0 the set a e u for e small

the singular limit dynamics of semilinear damped wave equations on considere la dynamique qualitative d une equation d onde amortie semi lineaire a une dimension dans la limite singuliere correspondant a l annulation du coefficient de la derivee d ordre 2 en temps

wave splitting and the reflection operator for the wave equation in r3 the problem of wave splitting in a nonhomogeneous medium in r3 is considered previous results for wave splitting in a planar stratified medium can be generalized to a general nonhomogeneous medium with sufficiently smooth velocity the wave equation is factorized into an up and down going wave system using certain integral and integral differential operators the equation for the reflection operator which relates the up going wave to a down going wave is then obtained and certain properties of the reflection operator are deduced

simultaneous reconstruction of material and transient source parameters using the invariant imbedding method this paper extends the time domain wave splitting and invariant imbedding method to an inhomogeneous wave equation with a source term uxx utt a x ux 2d x i t the direct scattering and inverse source problems of this equation are studied operators j that map the source function into the scattered waves at the edges of the slab are defined a system of coupled nonlinear integrodifferential equations for these scattering operator kernels is obtained the direct scattering problem is to obtain the scattering operator kernels j and r when parameters a and d are given the inverse problem is to simultaneously reconstruct a x and d x from the scattering operator kernels r 0 t 0 t 2 and j 0 t 0 t 1 both numerical inversion algorithms and the small time approximate reconstruction method are presented a green s function technique is used to derive green s operator kernel equations for the calculation of the internal field it provides an alternative effective and fast way to compute the scatterin

solution of underdetermined electromagnetic and seismic problems by the maximum entropy method many inversion problems require solution of a fredholm integral equation of the form t bar r int d bar r bar r sigma bar r dv where t is the observable d is an operator and sigma is the unknown parameter distribution examples occur in the areas of radiation and scattering tomography and geotomography we reduce the equation to matrix form and apply a maximum entropy technique based on the first principle of data reduction to obtain a most probable sigma distribution we illustrate the technique by synthetic data examples of geotomography assuming straight rays with and without noise the examples show how sharp anomalies may be identified in grossly underdetermined situations we outline the algorithm used and describe some computational properties our method suggests a way of overcoming the ill conditioned nature of fredholm integral equation inversion

inverse scattering method for one dimensional inhomogeneous layered media an inverse scattering method to reconstruct simultaneously the permittivity profile and the conductivity profile of one dimensional inhomogeneons medium which makes use of the transverse electric te wave and or transverse magnetic tm wave is proposed the medium is illuminated by the te and or tm plane wave at oblique incidence and the data are taken as the reflection coefficients for a set of discrete frequencies and or a finite number of incident angles furthermore the reflection coefficient data contain the gaussian noise the nonlinear integral equation relating the unknown constitutive parameter of the medium to the reflection coefficient for te wave and or tm wave is solved by the newton iteration method the inverse operator in the newton method is determined by the regularization method it is demonstrated in terms of the numerical examples that this method utilizing both polarizations and the incident angle of the incident plane wave is very effective even if the reflection coefficient contains the practical measurement error or the phase of the reflection coefficient is unknown moreover the relationships between the errors of reconstructed profile and the measured reflection coefficient are also discussed

phase transitions on fractals iii infinitely ramified lattices for pt ii see ibid vol 17 p 435 1984 in the first two papers of this series the authors considered self similar fractal lattices with a finite order of ramification r in the present paper they study physical models defined on a family of fractals with r infinity in order to characterise the geometry of these systems they need the connectivity q and the lacunarity l in addition to the fractal dimensionality d it is found that discrete symmetry spin models on these lattices undergo a phase transition at tc 0 an approximate renormalisation group scheme is constructed and used to find the dependence of tc and the critical exponents on the geometrical factors they also solve the problem of resistor networks on these fractals and discuss its consequences concerning spin models with continuous symmetry

ising model on a self avoiding walk fluctuation arguments combined with basic features of the geometry of self avoiding walks saws are shown to lead to zero transition temperature tc for the ising model on a saw the ising model on a non random fractal representing the essential geometrical features of saws is treated exactly by a decimation method it also leads to tc 0 and to ising saw critical behavior different from that on an ising chain

robust exponential attractors for singularly perturbed hodgkin huxley equations abstract here we consider a singular perturbation of the hodgkin huxley system which is derived from the lieberstein s model we study the associated dynamical system on a suitable bounded phase space when the perturbation parameter e i e the axon specific inductance is sufficiently small we prove the existence of bounded absorbing sets as well as of smooth attracting sets we deduce the existence of a smooth global attractor a e finally we prove the main result that is the existence of a family of exponential attractors e e which is holder continuous with respect to e

factorization of the wave equation in higher dimensions the factorization of the wave equation into a coupled system involving up and down going wave components is obtained for the case where the field quantities are multivariate functions of spatial variables but the velocity c is a function of the z variable only the form of the reflection operator is derived and the quadratic differential integral equation satisfied by its kernel is obtained

existence uniqueness and causality theorems for wave propagation in stratified temporally dispersive complex media a mixed initial boundary value problem for a nonlocal hyperbolic equation is analyzed with respect to unique solubility and causality the regularity of the step response and impulse response the green functions is investigated and a wave front theorem is proved the problem arises e g at time varying electromagnetic plane wave excitation of stratified temporally dispersive bi isotropic or anisotropic slabs concluding the problem is uniquely solvable strict causality holds and a well defined wave front speed exists this speed is independent of dispersion and excitation and depends on the nondispersive properties of the medium only

upper semicontinuity of the global attractor for the gierer meinhardt model we are concerned with the following gierer meinhardt model on a bounded domain r n n 1 2 3 with smooth boundary which is a biological pattern formation model proposed by a gierer and h meinhardt

an electromagnetic inverse problem for dispersive media the dispersion of transient electromagnetic waves in a homogeneous medium can be characterized by expressing either the complex permittivity as a function of frequency or the susceptibility kernel as a function of time in this paper a time domain technique is used to derive a nonlinear integrodifferential equation which relates the susceptibility kernel for a one dimensional homogeneous slab to the reflection operator for the medium thus the susceptibility kernel which is a function of time can be determined from reflection data a numerical implementation of this technique is shown the more general case of a medium consisting of a stack of homogeneous dispersive layers is also addressed

one dimensional profile inversion of a half space over a two part impedance ground a method which permits one to reveal the one dimensional 1 d electromagnetic profile of a half space bounded by a two part impedance ground is established the method reduces the problem to the solution of two functional equations one of these equations is solved exactly by reducing it to a riemann hilbert problem while the other is reduced under the born approximation to a fredholm equation of the first kind whose kernel involves the solution to the first equation since this latter constitutes an ill posed problem its regularized solution in the sense of tikhonov is given an illustrative application shows the applicability and the accuracy of the theory the functional equations are valid also when the impedance of the ground varies in an arbitrary manner the theory can be applied to a quick determination of the constitutive electromagnetic parameters of the atmosphere over a nonhomogeneous impedance ground

operator theoretic and computational approaches to ill posed problems with applications to antenna theory a general framework for regularization and approximation methods for ill posed problems is developed three levels in the resolution processes are distinguished and emphasized philosophy of resolution regularization approximation schema and regularization algorithms dilemmas and methodologies of resolution of ill posed problems and their numerical implementations are examined in this framework with particular reference to the problem of finding numerically minimum weighted norm least squares solutions of first kind integral equations and more generally of linear operator equations with nonclosed range a common problem in all these methods is delineated each method reduces the problem of resolution to a nonstandard minimization problem involving an unknown critical parameter whose optimal value is crucial to the numerical realization and amenability of the method the nonstandardness results from the fact that one does not have explicitly or a priori the function to be minimized it has to built up using additional information convergence rate estimates and robustness conditions etc several results are developed that complement recent advances in numerical analysis and regularization of inverse and ill posed identification and pattern synthesis problems an emphasis is placed on the role of constraints function space methods the role of generalized inverses and reproducing kernels in the regularization and stable computational resolution of these problems the results will be applied specifically to problems of antenna synthesis and identification however the thrust of the paper is devoted to the interdisciplinary character of operator theoretic and numerical methods for ill posed problems

dilute ising model on fractal lattices the authors study dilute ising models on a family of infinitely ramified exact fractals whose fractal dimension varies from 1 to 2 they calculate the phase diagram and critical exponents as a function of the fractal dimensionality within migdal s approximation 1975

a numerical solution to the zakharov shabat inverse scattering problem a novel numerical method to solve the inverse scattering problem associated with the zakharov shabat coupled mode equations is developed in this problem the coupling coefficient for two propagating modes is determined from reflection data both codirectional and contradirectional coupling can be accommodated as well as the presence of bound states analytical results based on an independent analytical method are derived in order to demonstrate the accuracy of the proposed method which is applicable to a broad class of reflection data

transient wave propagation in reciprocal bi isotropic media at oblique incidence in this paper a new wave splitting is suggested that simplifies the analysis of wave propagation in reciprocal bi isotropic media two different methods to solve the scattering problem are analyzed the invariant imbedding and the green function approach the medium is modeled by constitutive relations in the time domain time convolutions and the slab is assumed to be inhomogeneous with respect to the depth it is shown that the cross polarized contribution to the reflected field at normal incidence is zero for the general reciprocal inhomogeneous slab moreover the rotation and the attenuation of the wave front are calculated explicitly in the general inhomogeneous slab case special attention is paid to normal incidence and to the homogeneous semi infinite medium

time singular limit of semilinear wave equations with damping the hyperbolic semilinear initial value problem utt aut bu f u 0 u 0 u0 ut 0 u1 with commuting positive selfadjoint operators a and b in a hilbert space x is considered the term aut is a damping term it is shown that the solutions converge uniformly in time in an appropriate hilbert space z to the solution of the parabolic type initial value problem ut a 1bu a 1f u 0 u 0 u00 provided u0 converge to u00 and u1 f a and b satisfy certain conditions the applications include different initial boundary value problems from continuum mechanics

technique for evaluation of profiles of a composite chiral slab through inversion and pseudospectral approximation in this paper a technique is developed to simultaneously evaluate the permeability permittivity and chirality parameter of a composite chiral slab where they are functions of depth of the slab additionally the chirality parameter is a function of frequency the developed technique is based on the use of the riccati equation for the reflection coefficient associated with electromagnetic propagation through the slab and the green functions the technique also makes use of nth degree interpolation polynomials to approximate the unknowns using legendre gauss lobatto collocation nodes a numerical example is given through which the application of the technique is utilized

convergence rates for newton s method at singular points if newton s method is employed to find a root of a map from a banach space into itself and the derivative is singular at that root the convergence of the newton iterates to the root is linear rather than quadratic in this paper we give a detailed analysis of the linear convergence rates for several types of singular problems for some of these problems we describe modifications of newton s method which will restore quadratic convergence

application of the multilevel single linkage method to one dimensional electromagnetic inverse scattering problem an inverse scattering method for the reconstruction of the permittivity and conductivity profiles of a multilayered medium and for that of the impedance profile of a nonuniform transmission line is proposed the inversion is based on the global minimization of an objective function by the multilevel single linkage method the objective function is defined as the mean square error between the measured data and the data obtained from the solution of the forward problem an exact formulation for the gradient of the objective function in closed form is derived the necessary condition for the unique solution of the inverse problem of a nonuniform transmission line is discussed reconstruction examples using both experimental and noisy synthetic data are presented

two step inverse scattering method for one dimensional permittivity profiles a numerical method to invert the dielectric permittivity profile from the riccati equation using the newton kantorovich iterative scheme is described instead of handling the equations in terms of usual geometrical depth we determine the profile as a function of the electromagnetic path length since the convergence and the stability of the solution are found to be significantly better in this case the initial profile used as a starting point for the inversion is obtained by another method employing successive reconstruction of dielectric interfaces and homogeneous layers in a step like form this method though not always accurate is fast and well suited for the approximate reconstruction of the profile thus creating ideal starting conditions for the previous approach as a result the computation time is considerably reduced without using any a priori information the approach is applicable to both continuous and discontinuous profiles of high contrast and exhibits a good stability of the solution with respect to noisy input data a lossy medium profile can also be inverted provided the overall thickness of the inhomogeneous slab and the background permittivity are known

existence and continuity of uniform exponential attractors of the singularity perturbed hodgkin huxley system abstract in order to show that the standard partially dissipative hodgkin huxley system approximates the hyperbolic hodgkin huxley system for large time we are interested in establishing the existence of exponential attractors for the hyperbolic system with estimates of the fractal dimension and the rate of attraction of trajectories which are uniform with respect to the perturbation furthermore we obtain the continuity of exponential attractors

discrete inverse method for viscoelastic medium with complete data the discrete inverse scattering problem for viscoelastic medium is studied in this paper it is assumed that the relaxation modulus varies only with time t the object of this paper is to develop a method to reconstruct the relaxation modulus with less measurement data than before the propagation operators of the viscoelastic medium are defined first and the imbedding equations governing the behavior of the propagation operators are derived with the invariant imbedding techniques using the finite difference method these equations can be discretized to obtain a system of linear algebraic equations about the propagation operators and the material modulus for the inverse scattering problem it is assumed that the reflection data obtained from the scattering experiments are only available on one side of the medium and for one round trip through the viscoelastic slab to reconstruct the unknown relaxation modulus an inversion procedure is developed using this set of data that are complete in the sense that they can be extended to arbitrary time t and the other scattering and propagation operators can also be determined by the inversion procedure described in this paper the inversion algorithm is implemented numerically on several examples at the end of the paper it can be seen that the obtained curves of the material modulus coincide with the original relaxation modulus very well

inverse scattering for inhomogeneous viscoelastic media in this paper the inverse scattering problems for the full inhomogeneous viscoelastic medium are studied via the invariant imbedding technique special attention is paid to the propagation operators of the viscoelastic medium and the imbedding equations for these operators are derived for the inverse scattering problems it is shown that the reflection data can be extended from one round trip through the iscoelastic slab to arbitrary time with the help of the propagation operators hence the reconstruction of the relaxation modulus is sufficient to be considered only in one round trip it is also shown that only one side measurement reflection data are not sufficient to reconstruct the relaxation modulus and the density of the medium simultaneously the corresponding numerical examples are presented for the case that the relaxation modulus of the medium is modeled by two independent functions an iterative inversion procedure is proposed to recover the relaxation modulus and the density simultaneously with the input two side normally reflection data

multifrequency inverse problem for the reduced wave equation with sparse data the inverse problem for the reduced wave equation u k2n2 x u 0 is considered where the quantity to be determined is the value of n x in a compact domain d in r3 the data consists of a finite set of measurements of the scattered field produced by different incident fields the measurements are made at various points exterior to d and at possibly different frequencies the mathematical problem involves solving a system of nonlinear functional equations conditions are developed which indicate when the measured body is close to a particular comparison body a linearized perturbation is valid a new higher order nonlinear iterative procedure is developed for the full nonlinear problem which reduces to the usual solution in the linearized region the method is illustrated by computational results for the one dimensional case

transient waves in nonstationary media this paper treats propagation of transient waves in nonstationary media which has many applications in for example electromagnetics and acoustics the underlying hyperbolic equation is a general homogeneous linear first order 2 2 system of equations the coefficients in this system depend on one spatial coordinate and time furthermore memory effects are modeled by integral kernels which in addition to the spatial dependence are functions of two different time coordinates these integrals generalize the convolution integrals frequently used as a model for memory effects in the medium specifically the scattering problem for this system of equations is addressed this problem is solved by a generalization of the wave splitting concept originally developed for wave propagation in media which are invariant under time translations and by an imbedding or a green s functions technique more explicitly the imbedding equation for the reflection kernel and the green s functions propagator kernels equations are derived special attention is paid to the problem of nonstationary characteristics a few numerical examples illustrate this problem

finite dimensionality and upper semicontinuity of the global attractor of singularly perturbed hodgkin huxley systems abstract in an effort to show that the standard hodgkin huxley system approximates the hyperbolic hodgkin huxley system we view the hyperbolic system as a singular perturbation of the standard system we establish the existence of global attractors for the singularly perturbed hyperbolic system and show that they have finite fractal and hausdorff dimensions we then show that the global attractors of the singularly perturbed system converge to the global attractor for the standard system as the small perturbation parameter decreases to zero

multifrequency inverse problem for the reduced wave equation resolution cell and stability the multifrequency inverse problem associated with the reduced wave equation u k2n2 x u 0 x r3 is examined for the case where the data set is sparse the resolution cell or solution set is examined in detail and is shown to be an infinite dimensional manifold the concept of stability is introduced it is shown that the intrinsic condition of structural stability to the inverse process selects out a preferred set of solutions from the solution set the structural stability of various iterative schemes used in the inverse process are examined

an initial boundary value problem of physiological significance for equations of nerve conduction a model of nerve conduction that has gained wide acceptance in the biophysical community is that for the giant axon of the squid loligo of hodgkin and huxley 5 this nonlinear parabolic partial differential equation pde when considered as part of a particular initial boundary value problem ibvp models the electrical activity of an axon under conditions found both in nature and in the laboratory this ibvp for the hodgkin huxley equations is proved to be well posed in the sense of hadamard 3 and a priori bounds on the solution are derived

a real space renormalisation group study of the ising model on self avoiding walk chains here the authors apply a small cell real space renormalisation group technique to study the critical behaviour of the nearest neighbour interacting ising model on self avoiding walk saw chains in the limit of vanishing interactions between nearest neighbours which are not connected by the walk itself the treatment reproduces exactly the critical behaviour of the linear chain ising model when all the nearest neighbour interactions are allowed the treatment gives a vanishing tc with the small cells chosen for the ising model on saws on the square and triangular lattices the study of the critical behaviour of such models indicates some interesting departures from that of the ising model on linear chains

reconstruction of source and medium parameters via wave splitting and green function equations this paper applies the time domain wave splitting and the green function concepts to study two inverse problems involving an internal transient source embedded in a nondispersive inhomogeneous medium these two inverse problems are 1 given the physical one way travel time green operator kernel bar g 0 t and the source space distribution function reconstruct the wave speed profile c z of the medium 2 given the same kernel bar g 0 t and the medium wave speed function recover the source space distribution profile p z both numerical inversion algorithms and approximate reconstruction formulas are presented several computational examples are given

eden trees on the sierpinski gasket the authors have studied eden trees on the sierpinski gasket and obtained its fractal and spectral dimensions to be ln 7 square root 5 2 ln 2 and 1 respectively

a hyperbolic system from a neural transmission model we provide a global existence result for a hyperbolic system which arises in the theory of neural conduction

direct and inverse scattering in the time domain via invariant imbedding equations this paper examines the implementation of a new approach to direct and inverse scattering problems in the time domain which is applied here to the case of a one dimensional lossless medium the method is based on an integro differential equation which is satisfied by the reflection kernel and has been derived elsewhere using invariant imbedding techniques an example of an exact solution to this equation is given numerical schemes for solving the direct and inverse problems are derived by discretizing this equation the behavior of the resulting algorithms is then tested on several examples the results suggest that the inversion method is stable in the presence of additive noise