dsc time domain solution of maxwell s equations a new computational algorithm the discrete singular convolution dsc is introduced for solving scattering and guided wave problems described by time domain maxwell s equations the dsc algorithm is utilized for the spatial discretization and the fourth order runge kutta scheme is used for the time advancing staggered meshes are used for electromagnetic fields four standard test problems a hollow air filled waveguide a dielectric slab loaded rectangular waveguide a shield microstrip line and a dielectric square are employed to illustrate the usefulness to test the accuracy and to explore the limitation of the dsc algorithm results are compared with those of finite difference scaling function multi resolution time domain and finite element based high frequency structure simulator numerical experiments indicate that the present algorithm is a promising approach for achieving high accuracy in electromagnetic wave computations

multimodal admittance method in waveguides and singularity behavior at high frequencies this work presents a multimodal method for the propagation in a waveguide with varying height and its relation to trapped modes or quasi trapped modes the coupled mode equations are obtained by projecting the helmholtz equation on the local transverse modes to solve this problem we integrate the riccati equation governing the admittance matrix dirichlet to neumann operator for many propagating modes i e at high frequencies the numerical integration of the riccati equation shows that the rule is that this matrix has quasi singularities associated to quasi trapped modes

an invariant imbedding analysis of general wave scattering problems the invariant imbedding technique via the solution of a riccati type equation is modified to calculate the wave fields inside and scattered from a strongly laterally and vertically heterogenous anisotropic inclusion which may be large but remains compact the factorization underlying this approach is carried out with respect to direction of average power flow rather than the more conventional factorization with respect to local direction of propagation the solution of the operator riccati equation is related to the dirichlet to neumann map the formulation is robust in the sense that it can handle a rather extreme range of modal wave speeds and allows continuous as well as discontinuous medium variations on different wave length scales it also inherently takes care of critical angle phenomena the algorithm based on the invariant imbedding approach yields the internal fields for a full survey of sources and receivers simultaneously the wave field solution in the inclusion is coupled to the

exact and operator rational approximate solutions of the helmholtz weyl composition equation in underwater acoustics the quadratic profile the application of phase space and path integral methods to both mathematical and computational direct and inverse wave propagation modeling at the level of the scalar one way helmholtz equation is briefly reviewed the role of operator symbols is stressed and their properties briefly discussed the construction of the operator symbol requires the exact or approximate solution of the helmholtz weyl composition equation in the weyl pseudodifferential operator calculus the exact symbols for several quadratic profiles are presented and briefly analyzed the symbols exactly corresponding to a family of operator rational approximations are also presented for the quadratic case these results are used to illustrate several points pertinent to wide angle propagation modeling and the refractive index profile reconstruction problem in underwater acoustics

a formal proof of the deadline driven scheduler in pptl axiomatic system this paper presents an approach for verifying the correctness of the feasibility theorem on the deadline driven scheduler dds with the axiom system of propositional projection temporal logic pptl to do so the deadline driven scheduling algorithm is modeled by an msvl modeling simulation and verification language program and the feasibility theorem is formulated by pptl formulas with two parts a necessary part and a sufficient part then several lemmas are abstracted and proved by means of the axiom system of pptl with the help of the lemmas two parts of the theorem are deduced respectively this case study convinces us that some real time properties of systems can be formally verified by theorem proving using the axiom system of pptl

numerical solution of the coupled mode equations in duct acoustics we present a fast solver for the coupled mode equations in duct acoustics it is based on a partitioning of the resulting system of ordinary differential equations into separate subsystems the number of which increases by the separability of the problem this structure is obtained by transformations of the dependent variable such that weakly interacting modes are separated the decoupling process requires the numerical solution of algebraic riccati equations however these computations can be done on a spatial scale much larger than the characteristic wavelength of the problem 23 refs 7 figs 1 tab

computing the topological pressure for intermittent maps the topological pressure is obtained as the leading zero of a dynamical zeta function we consider the problem of computing this zero when it is close to a singularity in particular we study a family of intermittent maps which we argue exhibit a branch point singularity in its zeta functions the convergence of the cycle expansion close to this point is extremely slow to deal with this problem we consider a resummation of the cycle expansion the idea is quite similar to that of pade approximants but the ansatz is a generalized series expansion around the branch point rather than a rational function the improvement on convergence of the leading zero is considerable we also briefly discuss the relation between correlation decay and the nature of the branch point

an explicit fourth order orthogonal curvilinear staggered grid fdtd method for maxwell s equations the explicit fourth order staggered finite difference time domain scheme previously proposed for a cartesian grid is extended to maxwell s equations in an orthogonal curvilinear coordinate system and applied to electromagnetic wave problems a simple technique is also presented for generating orthogonal curvilinear grids that conform to the material boundaries and interfaces of the problem numerical experiments are presented to illustrate the efficiency and accuracy of the method

an operator marching method for inverse problems in range dependent waveguides abstract for large scale inverse problems in acoustics and electromagnetics numerical schemes based on direct methods e g fem and meshless methods often result in huge linear systems and thus are not feasible in terms of computing speed and memory storage this work proposes the inverse fundamental operator marching method based on the dirichlet to neumann map for solving large scale inverse boundary value problems in range dependent waveguides truncated singular value decomposition is employed to solve ill conditioned linear systems arising in marching process and the number of propagating modes in the waveguide assumes the role of a regularization parameter numerical results show that the method is computationally efficient highly accurate stable with respect to data noise for retrieving the propagating part of the starting field it particularly suits to long range wave propagation in slowly varying stratified waveguide

factorization of second order elliptic boundary value problems by dynamic programming we present a method to factorize a second order boundary value problem in a cylindrical domain in a system of uncoupled first order initial value problems together with a nonlinear riccati type equation for functional operators this uncoupling is obtained by a space invariant embedding technique along the axis of the cylinder this method can be viewed as an infinite dimensional generalization of the block gauss lu factorization

trace formulas for stochastic evolution operators weak noise perturbation theory periodic orbit theory is all effective tool for the analysis of classical and quantum chaotic systems in this paper we extend this approach to stochastic systems in particular to mappings with additive noise the theory is cast in the standard field theoretic formalism and weak noise perturbation theory written in terms of feynman diagrams the result is a stochastic analog of the next to leading h corrections to the gutzwiller trace formula with long time averages calculated from periodic orbits of the deterministic system the perturbative corrections are computed analytically and tested numerically on a simple 1 dimensional system

exact constructions of square root helmholtz operator symbols the focusing quadratic profile operator symbols play a pivotal role in both the exact well posed one way reformulation of solving the elliptic helmholtz equation and the construction of the generalized bremmer coupling series the inverse square root and square root helmholtz operator symbols are the initial quantities of interest in both formulations in addition to providing the theoretical framework for the development and implementation of the parabolic equation pe method in wave propagation modeling exact standard left and weyl symbol constructions are presented for both the inverse square root and square root helmholtz operators in the case of the focusing quadratic profile in one transverse spatial dimension extending and ultimately unifying the previously published corresponding results for the defocusing quadratic case j math phys 33 1887 1914 1992 both i spectral modal summation representations and ii contour integral representations exploiting the underlying periodicity of the associated quantum mechanical harmonic oscillator problem are derived and ultimately related through the propagating and nonpropagating contributions to the operator symbol high and low frequency asymptotic operator symbol expansions are given along with the exact symbol representations for the corresponding operator rational approximations which provide the basis for the practical computational realization of the pe method moreover while the focusing quadratic profile is in some respects nonphysical the corresponding helmholtz operator symbols nevertheless establish canonical symbol features for more general profiles containing locally quadratic wells

periodic orbit quantization of bound chaotic systems the authors present periodic orbit quantizations of the hyperbola billiard and the x2y2 potential these two systems may be considered as belonging to the one parameter family of potentials x2y2 1a the quantum states are determined by means of the zeros of an expanded and truncated selberg zeta function the symmetries of the problem are considered and the selberg zeta function is factorized into the irreducible representations of the symmetry group the thus calculated eigenenergies are in good agreement with quantum mechanical calculations and converge when the number of terms in the expansion is increased the results strongly indicate that the trace formula provides individual quantum eigenstates for chaotic systems

helmholtz and parabolic equation solutions to a benchmark problem in ocean acoustics the helmholtz equation he describes wave propagation in applications such as acoustics and electromagnetics for realistic problems solving the he is often too expensive instead approximations like the parabolic wave equation pe are used for low frequency shallow water environments one persistent problem is to assess the accuracy of the pe model in this work a recently developed he solver that can handle a smoothly varying bathymetry variable material properties and layered materials is used for an investigation of the errors in pe solutions in the he solver a preconditioned krylov subspace method is applied to the discretized equations the preconditioner combines domain decomposition and fast transform techniques a benchmark problem with upslope downslope propagation over a penetrable lossy seamount is solved the numerical experiments show that for the same bathymetry a soft and slow bottom gives very similar he and pe solutions whereas the pe model is far from accurate for a hard and fast bottom a first attempt to estimate the error is made by computing the relative deviation from the energy balance for the pe solution this measure gives an indication of the magnitude of the error but cannot be used as a strict error bound

boundary integral equation method for source localization with a continuous wave sonar in this paper matched field processing is combined with the boundary integral equation method biem of scattering theory to study a sound source localization problem in a perturbed shallow ocean it is assumed that there is a known inclusion embedded in a shallow water waveguide continuous waves cw produced by a sound source are scattered by the inclusion and then received by a hydrophone array because the symmetry of the waveguide has been destroyed by the existence of the inclusion a proper procedure is required to avoid the mismatching a numerical scheme is presented that makes use of the separation of the source and the detection array and greatly reduces the computation a numerical simulation using this method is presented

applications and time domain solution of higher order parabolic equations in underwater acoustics a higher order parabolic equation and the corresponding higher order time domain parabolic equation are derived from a pade series and solved numerically these models provide accurate solutions for problems involving very wide angle propagation e g propagation in the nearfield or over a hard ocean bottom propagation in domains in which sound speed variations are large e g propagation in deep water deep within the ocean bottom in high speed ocean bottoms and possibly of different wave types and propagation out to very long ranges the possibility of modeling elastic wave propagation with a similar approach is considered

the riccati method for the helmholtz equation the operator riccati equation for the dirichlet to neumann map is derived from the exact operator factorization of the two dimensional variable coefficient helmholtz equation numerical schemes are developed for the operator riccati equation and its variant using a local eigenfunction expansion this leads to a practical computational method for acoustic wave propagation over large range distances since the boundary value problem of the helmholtz equation is reduced to initial value problems that are solved by marching in the range the efficiency and accuracy of the method is demonstrated by numerical experiments including the plane parallel range dependent waveguide benchmark problem proposed by the acoustical society of america

factorization and path integration of the helmholtz equation numerical algorithms the propagator for the reduced scalar helmholtz equation plays a significant role in both analytical and computational studies of acoustic direct wave propagation path functional integrals are taken to provide the principal representation of the propagator and are computed directly the path integral is the primary tool in extending the classical fourier methods so appropriate for wave propagation in homogeneous media to inhomogeneous media for transversely inhomogeneous environments the n dimensional helmholtz equation can be exactly factored into separate forward and backward one way wave equations a parabolic based one way phase space path integral construction provides the generalization of the tappert hardin split step fft algorithm to the full one way factored helmholtz wave equation these extended marching algorithms can readily accommodate density profiles and range updating and further in conjunction with imbedding methods provide the basis for incorporating backscatter effects in a complementary manner for general range dependent environments elliptic based two way path integral constructions lead to an approximate representation of the propagator feynman garrod and a natural statistical monte carlo means of evaluation taken together the path integrals provide the basis for a global analysis in addition to providing a unifying framework for dynamical approximations resolution of the square root operator and the concept of an underlying stochastic process the one way marching algorithms are applied to ocean acoustic environments seismological environments and extreme model environments designed to establish their range of validity and manner of breakdown

index set expressions can represent temporal logic formulas abstract in temporal logic tl a well formed formula is generally formed by applying rules of its syntax finitely many times however under some circumstances although formulas such as ones expressed by index set expressions are constructed via applying rules of the syntax infinitely many times they are possibly still well formed since their equivalent concise syntax formulas can be found with this motivation this paper investigates the relationship between formulas specified by index set expressions and concise syntax by means of fixed point approach firstly we present two kinds of formulas namely i n 0 i q and i n 0 q i and prove they are indeed well formed by proving they are equivalent to formulas q and q respectively further we generalize i n 0 i q to i n 0 p i i q and explore the least and greatest fixed points of an abstract equation x q p x based on these some well formed special instances of i n 0 p i i q are obtained besides with the index set expression technique we equivalently represent u strong until and w weak until constructs of propositional linear temporal logic ltl within propositional projection temporal logic pptl

one way large range step methods for helmholtz waveguides a useful approach for long range computation of the helmholtz equation in a waveguide is to re formulate it as the operator differential riccati equation for the dirichlet to neumann map for waveguides with slow range dependence the piecewise range independent approximation is used to derive a second order range stepping method for this one way re formulation the range marching formulas are exact for each range independent piece and a large range step is possible if the range dependence is gradual based on a fourth order conservative exponential method for linear evolution equations a fourth order method that admits even larger range steps is developed for the one way re formulation numerical examples are used to demonstrate the improved accuracy of the fourth order method

inversion versus migration a new perspective to an old discussion seismic imaging techniques can be subdivided into inversion and migration the object functions for inversion and migration are respectively the medium contrast parameters and reflectivity in this paper the relationship between inversion and migration is approached by analyzing the underlying representations the forward models it appears that the two way representation which underlies inversion as well as the one way representation which underlies migration can both be expressed in terms of a volume integral over the appropriate object function in their linearized form these representations account for primaries only in this case the one way representation in terms of reflectivity is the most accurate of the two which implies that proper migration is more accurate than linearized inversion internal multiples can be taken into account by the nonlinear representations as an alternative however the generalized primary representation is introduced in its explicit form this one way

on the effect of pruning on the singularity structure of zeta functions we investigate the topological zeta function for unimodal maps in general and dynamical zeta functions for the tent map in particular for the generic situation when the kneading sequence is aperiodic it is shown that the zeta functions have a natural boundary along its radius of convergence beyond which the function lacks analytic continuation we make a detailed study of the function n 0 1 z2n associated with sequences of period doublings it is demonstrated that this function has a dense set of singular points and zeros on the unit circle exhibiting a rich number theoretical structure

a fast algorithm for the electromagnetic scattering from a large cavity a fast algorithm is presented for solving electromagnetic scattering from a rectangular open cavity embedded in an infinite ground plane the medium inside the cavity is assumed to be vertically layered by introducing a transparent artificial boundary condition the problem in the open cavity is reduced to a bounded domain problem a simple finite difference method is then applied to solve the model helmholtz equation the fast algorithm is designed for solving the resulting discrete system in terms of the discrete fourier transform in the horizontal direction a gaussian elimination in the vertical direction and a preconditioning conjugate gradient method with a complex diagonal preconditioner for the indefinite interface system the existence and uniqueness of the finite difference solution are established for arbitrary wave numbers our numerical experiments for large numbers of mesh points up to 16 million unknowns and for large wave numbers e g between 100 and 200 wavelengths show that the algorithm is extremely efficient the cost for calculating the radar cross section which is of significant interest in practice is o m2 for an m times m mesh the proposed algorithm may be extended easily to solve discrete systems from other discretization methods of the model problem

a domain decomposition method for the helmholtz equation in a multilayer domain the two dimensional helmholtz equation for problems where the physical domain consists of layers with different material properties is studied an efficient preconditioner for iterative solution of the problem is constructed the problem is discretized with fourth order accurate finite difference operators for the construction of the radiation boundary conditions a fourth order finite element method also is used the large sparse complex indefinite and ill conditioned system of equations that arises is solved with preconditioned restarted gmres a domain decomposition method is used in which the preconditioning is based on the schur complement algorithm with fast poisson type preconditioners for the subdomains the memory requirements for the preconditioner are nearly linear in the number of unknowns the arithmetic complexity for each iteration is low whereas the construction of the preconditioner is a bit more expensive electromagnetic wave propagation in a three layered waveguide is used as a model problem numerical experiments show that convergence is achieved in a few iterations compared with banded gaussian elimination which is a standard solution method the iterative method shows significant gain in both memory requirements and arithmetic complexity furthermore the relative gain grows when the problem size increases

a complete proof system for propositional projection temporal logic the paper presents a proof system for propositional projection temporal logic pptl with projection plus the syntax semantics and logical laws of pptl are introduced together with an axiom system consisting of axioms and inference rules to facilitate proofs some of the frequently used theorems are proved a normal form of pptl formulas is presented and the soundness and completeness of the proof system are demonstrated to show how the axiom system works a full omega regular property for the mutual exclusion problem is specified by a pptl formula and then a deductive proof of the property is performed

separation of variables in deformed cylinders 327 abstract we study the laplace operator subject to dirichlet boundary conditions in a two dimensional domain that is one to one mapped onto a cylinder rectangle or infinite strip as a result of this transformation the original eigenvalue problem is reduced to an equivalent problem for an operator with variable coefficients taking advantage of the simple geometry we separate variables by means of the fourier decomposition method the ode system obtained in this way is then solved numerically yielding the eigenvalues of the operator the same approach allows us to find complex resonances arising in some noncompact domains we discuss numerical examples related to quantum waveguide problems the aim of these experiments is to compare the method based on the separation of variables with the standard finite volume procedure for the most computationally difficult examples related to domains with narrow throats one can clearly see the advantages of the proposed method

derivation and application of extended parabolic wave theories ii path integral representations the n dimensional reduced scalar helmholtz equation for a transversely inhomogeneous medium is naturally related to parabolic propagation models through 1 the n dimensional extended parabolic weyl pseudodifferential equation and 2 an imbedding in an n 1 dimensional parabolic schrodinger equation the first relationship provides the basis for the parabolic based hamiltonian phase space path integral representation of the half space propagator the second relationship provides the basis for the elliptic based path integral representations associated with feynman and fradkin feynman and garrod and feynman and dewitt morette exact and approximate path integral constructions are derived for the homogeneous and transversely inhomogeneous cases corresponding to both narrow and wide angle extended parabolic wave theories the path integrals allow for a global perspective of the transition from elliptic to parabolic wave theory in addition to providing a unifying framework for dynamical approximations resolution of the square root operator and the concept of an underlying stochastic process

periodic orbit sum rules for billiards accelerating cycle expansions we show that the periodic orbit sums for two dimensional billiards satisfy an infinity of exact sum rules we demonstrate their utility by using the flow conservation sum rule to accelerate the convergence of cycle expansions for the overlapping three disc billiard the effectiveness of the approach is studied by applying the method on averages known explicitly by other sum rules the method is then applied to the lyapunov exponent

a modified schwarz christoffel mapping for regions with piecewise smooth boundaries a method where polygon corners in schwarz christoffel mappings are rounded is used to construct mappings from the upper half plane to regions bounded by arbitrary piecewise smooth curves from a given curve a polygon is constructed by taking tangents to the curve in a number of carefully chosen so called tangent points the schwarz christoffel mapping for that polygon is then constructed and modified to round the corners since such a modification causes effects on the polygon outside the rounded corners the parameters in the mapping have to be redetermined this is done by comparing side lengths in tangent polygons to the given curve and the curve produced by the modified schwarz christoffel mapping the set of equations that this comparison gives can normally be solved using a quasi newton method the resulting function maps the upper half plane on a region bounded by a curve that apart from possible vertices is smooth i e one time continuously differentiable that passes through the tangent points on the given curve has the same direction as the given curve in these points and changes direction monotonically between them furthermore where the original curve has a vertex the constructed curve has a vertex with the same inner angle the method is especially useful for unbounded regions with smooth boundary curves that pass infinity as straight lines such as channels with parallel walls at the ends these properties are kept in the region produced by the constructed mapping

propagation modelling based on wave field factorization and invariant imbedding the proposed propagation models address the propagation of cw acoustic fields through a transition region separating two different range independent oceans wave field factorization refers to the conceptual decoupling of the radiation fields in the two range independent oceans into positive and negative propagating components invariant imbedding refers to the reformulation of the propagation problem from a two point boundary value problem to an initial value problem in the reformulation the extent of the transition region is an independent variable the formulation is expressed in terms of reflection and transmission operators which apply to the transition region the proposed models are generalizations of existent models in reflection seismology these latter models allow for no cross range variability of wave speed in either the two now homogeneous oceans nor in the transition region the seismology models do accommodate p s coupling which is neglected in the acoustic application the incorporat

efficient numerical simulation for long range wave propagation we develop an efficient algorithm for simulating wave propagation over long distances with both weak and strong scatterers in domains with weak heterogeneities the wave field is decomposed into forward propagating and back scattered modes using two coupled parabolic equations in the region near strong scatterers the helmholtz equation is used to capture the strong scattering events the key idea in our method is to combine these two regimes using a combined domain decomposition and wave decomposition method a transparent interface condition is derived to couple these two regions together numerical examples show that the simulated field is close to the field obtained using the full helmholtz equation in the whole domain

acoustic transmission in curved ducts with varying cross sections methods for predicting the acoustic propagation properties of ducts are developed and analysed for ducts that are asymptotically straight with constant crosssection with application to splitter si

consistent coupled mode theory of sound propagation for a class of nonseparable problems this article examines the effects of boundary condition approximations that arise whenever the coupled mode theories of a d pierce j acoust soc am 37 19 27 1965 and d m milder j acoust soc am 46 1259 1263 1969 are applied to propagation problems involving range dependent boundaries this boundary condition approximation requires that the depth derivative rather than the normal derivative of the field be continuous across a sloping interface the approximation is necessary in order to carry out the partial separation of depth and range variables effected in the mathematical formulation of the theory this article will show that a consequence of this approximation is that conventional coupled mode theory applied to dissipationless media with nonhorizontal boundaries does not conserve energy it is shown that a correction to coupled mode theory can be derived such that the proper boundary conditions are satisfied and energy is conserved to first order in the local bottom slope moreover the corrections are not prohibitive in terms of added computational complexity numerical examples are presented which illustrate the nonconservation of energy effect and the corrections to the theory

marching schemes for inverse scattering problems in waveguides with curved boundaries abstract a marching scheme is developed for inverse scattering problems of the helmholtz equation in waveguides with curved boundaries we implement a local orthogonal transform to transform the irregular waveguide in physical plane into a regular rectangle in computing plane then the modified helmholtz system in computational domain is piecewise solved through a second order numerical marching scheme and we propose a spectral projector based on the truncated local propagating eigenfunction expansion to regularize the marching scheme in the end the marching scheme is verified by extensive numerical experiments and it is shown that the scheme is efficient stable and accurate in rapidly varying waveguides with curved boundaries even when the number of propagating modes in the main propagation direction is variable

a split step pade solution for the parabolic equation method a split step pade solution is derived for the parabolic equation pe method higher order pade approximations are used to reduce both numerical errors and asymptotic errors e g phase errors due to wide angle propagation this approach is approximately two orders of magnitude faster than solutions based on pade approximations that account for asymptotic errors but not numerical errors in contrast to the split step fourier solution which achieves similar efficiency for some problems the split step pade solution is valid for problems involving very wide propagation angles large depth variations in the properties of the waveguide and elastic ocean bottoms the split step pade solution is practical for global scale problems

higher order paraxial wave equation approximations in heterogenous media a new family of paraxial wave equation approximations is derived these approximations are of higher order accuracy than the parabolic approximation and they can be applied to the same computationa

a fourth order magnus scheme for helmholtz equation for wave propagation in a slowly varying waveguide it is necessary to solve the helmholtz equation in a domain that is much larger than the typical wavelength standard finite difference and finite element methods must resolve the small oscillatory behavior of the wave field and are prohibitively expensive for practical applications a popular method is to approximate the waveguide by segments that are uniform in the propagation direction and use separation of variables in each segment for a slowly varying waveguide it is possible that the length of such a segment is much larger than the typical wavelength to reduce memory requirements it is advantageous to reformulate the boundary value problem of the helmholtz equation as an initial value problem using a pair of operators such an operator marching scheme can also be solved with the piecewise uniform approximation of the waveguide this is related to the second order midpoint exponential method for a system of linear odes in this paper we develop a fourth order operator marching scheme for the helmholtz equation using a fourth order magnus method

entire fredholm determinants for evaluation of semiclassical and thermodynamical spectra proofs that fredholm determinants of transfer operators for hyperbolic flows are entire can be extended to a large new class of multiplicative evolution operators we construct such operators both for the gutzwiller semiclassical quantum mechanics and for classical thermodynamic formalism and introduce a new functional determinant which is expected to be entire for axiom a flows and whose zeros coincide with the zeros of the gutzwiller voros zeta function

a generalized impedance method for application of the parabolic approximation to underwater acoustics the parabolic equation pe has become an important computational tool for approximating underwater acoustic path losses in inhomogeneous environments the numerical solution of this equation is performed on a grid over the propagation medium which includes the water column and has also included a homogeneous false bottom below the actual ocean bottom this situation is viewed as one in which the propagation medium can be divided into a horizontally inhomogeneous upper layer and a horizontally homogeneous lower layer the pressure field in the latter layer is found using standard techniques valid for homogeneous media and in the former layer is found using an implicit finite difference scheme these fields and their normal derivatives are matched along the interface between the layers the result is referred to as a generalized impedance boundary condition gibc which serves to eliminate the need for a false bottom predictions obtained using the standard parabolic equation and a completely homoge

stability ordering of cycle expansions may be made to only a short time of order l 21 1 in the future where l1 is the largest lyapunov exponent despite this unpredictability much can be said about the average behavior of the system in many cases using only a small number of unstable periodic orbits or cycles this can be achieved by using cycle expansions 1 2 of ruelle s dynamical zeta function 3 cycle expansions have proved very useful in both classical and quantum chaos giving accurate estimates of the escape rate of open billiard systems 4 and the energy levels of helium 5 using a surprisingly small number of classical cycles the main idea of this approach is that a long generic trajectory may be approximated by various periodic orbits at different times and that longer periodic orbits may often be shadowed by shorter fundamental cycles closely following the shorter cycles along different sections of its length thus averages are calculated using fundamental cycles with small corrections due to longer cycles periodic orbits which are exact repetitions of smaller cycles are explicitly summed so that all expressions are written in terms of the remaining prime cycles the expansions work best when the symbolic dynamics is well understood and long periodic orbits are well shadowed by shorter ones in this paper we investigate a system in which neither of these conditions holds the strong field lorentz gas in spite of these difficulties reasonable results may be obtained by ordering the expansion in terms of stability rather than the length of periodic orbits this approach should be valid wherever cycle expansions can be applied including flows for which a natural topological length is difficult to define the origin of cycle expansions is a well developed theory of trace formulas and dynamical zeta functions 3 6 here we will need only the expression for the classical time average of some quantity a in a closed system 2

derivation and application of extended parabolic wave theories i the factorized helmholtz equation the reduced scalar helmholtz equation for a transversely inhomogeneous half space supplemented with an outgoing radiation condition and an appropriate boundary condition on the initial value plane defines a direct acoustic propagation model this elliptic formulation admits a factorization and is subsequently equivalent to a first order weyl pseudodifferential equation which is recognized as an extended parabolic propagation model perturbation treatments of the appropriate weyl composition equation result in a systematic development of approximate wave theories which extend the narrow angle weak inhomogeneity and weak gradient ordinary parabolic schrodinger approximation the analysis further provides for the formulation and exact solution of a multidimensional nonlinear inverse problem appropriate for ocean acoustic and seismic studies the wave theories foreshadow computational algorithms the inclusion of range dependent effects and the extension to 1 the vector formulation appropriate for elas