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

reciprocity theorems for two way and one way wave vectors a comparison for acoustic applications in which there is a preferred direction of propagation the axial direction it is useful to arrange the two way and one way wave equations into the same matrix vector formalism in this formalism axial variations of the wave vector are expressed in terms of lateral variations of the same wave vector the two way wave vector contains the field quantities pressure and velocity axial component only whereas the one way wave vector contains waves propagating in the positive and negative axial direction by exploiting the equivalent form of the two way and one way matrix vector equations it appears to be possible to derive two way and one way reciprocity theorems that have an equivalent form but a different interpretation the main differences appear in the boundary integrals for unbounded media in the contrast terms and for the correlation type theorems in the handling of evanescent waves

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

a recursive formula for characters of simple lie algebras abstract in 1964 antoine and speiser published succinct and elegant formulae for the characters of the irreducible highest weight modules for the lie algebras a2 and b2 the result for a2 may have been known as early as 1957 these can be derived from a recursive formula for characters valid for all simple complex kac moody lie algebras for which the weyl kac character formula holds

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

linear and nonlinear inverse scattering in this paper we discuss one dimensional scattering and inverse scattering for the helmholtz equation on the half line from the point of view of the layer stripping by full or nonlinear scattering we mean the transformation between the index of refraction actually half of its logarithmic derivative and the reflection coefficient we refer to this mapping as nonlinear scattering because the mapping itself is nonlinear another appropriate name is multiple scattering as this model includes the effects of multiple reflections by linear scattering we mean the born or single scattering approximation this is the frechet derivative of the full scattering transform at the constant index of refraction which can be calculated to be exactly the fourier transform in j sylvester d p winebrenner and f gylys colwell siam j appl math 56 1996 pp 736 754 we introduced a variant of layer stripping based on causality and the riesz transform rather than on trace formulas see a brickstein and

the electromagnetic scattering problem in the time domain for a dissipative slab and a point source using invariant imbedding a vertical magnetic dipole source above an inhomogeneous slab is considered where the permittivity e e z and conductivity z vary with the depth z using the axial symmetry of the problem a hankel transform is introduced besides physical scattering kernels nonphysical scattering kernels are introduced and the direct and inverse transformed problem is solved by means of wave splitting in an invariant imbedding approach the imbedding equations for the nonphysical scattering kernels are simpler than the corresponding equations for the physical kernels and they are solved using existing techniques for the imbedding equations for normally incident plane waves the solution for the physical kernels is obtained by establishing relations between the two sets of kernels these relations are given by volterra equations of the second kind and thus are very easy to solve numerically some applications and extensions of the results derived in the paper are discussed

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

differential and integral methods for multidimensional inverse scattering problems a layer stripping procedure for solving three dimensional schrodinger equation inverse scattering problems is developed this procedure operates by recursively reconstructing the radon transform of the potential from the jump in the radon transform of the scattered field at the wave front this reconstructed potential is then used to propagate the wave front and scattered field differentially further into the support of the potential the connections between this differential procedure and integral equation procedures are then illustrated by the derivations of two well known exact integral equation procedures using the radon transform and a generalized radon transform these procedures as well as the layer stripping procedure are then reduced to the familiar born approximation result for this problem by neglecting multiple scattering events this illustrates the central role of the radon transform in both exact and approximate inversion procedures

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

signal restoration after transmission through an advective and diffusive medium this paper considers an inverse problem associated with mass transport in a pipe it illustrates how wave splitting techniques can be used for an inverse problem associated with one dimensional mass transport processes this is done by using a generalization of fick s law which introduces a relaxation parameter into the problem so converting the parabolic partial differential equation by a singular perturbation into a hyperbolic one this generalized law by ensuring finite mass flux propagation speeds enables a stable equation to be used to reconstruct the interior boundary condition so providing a regularized solution to the inverse problem theoretical results for the solution of the inverse problem are also developed

factorization of a dissipative wave equation and the green functions technique for axially symmetric fields in a stratified slab the time domain scattering problem for axially symmetric fields and a dissipative stratified slab is considered explicit representations are derived for the wave splitting operators that factorize the dissipative hankel transformed wave equation the green functions approach associated with this splitting is developed partial differential equations pdes for the green functions are derived and used to derive the imbedding equations for the reflection and transmission kernels numerical results for the internal fields are given in the direct problem and the phase velocity is reconstructed in the inverse problem

abelian ideals of borel subalgebras and affine weyl groups abstract this paper is devoted to a detailed study of certain remarkable posets which form a natural partition of all abelian ideals of a borel subalgebra our main result is a nice uniform formula for the dimension of maximal ideals in these posets we also obtain results on ad nilpotent ideals which complete the analysis started in j algebra 225 2000 130 258 2002 112

spherical orbits and abelian ideals let g be a connected reductive complex algebraic group with lie algebra lie g g let b be a borel subgroup of g with unipotent radical bu we denote the lie algebra of b and bu by b and bu respectively the group b acts on any ideal of b by means of the adjoint representation after a preliminary section we study a relationship between spherical nilpotent orbits and abelian ideals a of b using the structure theory for these orbits from 10 the principal result of this section is that for an abelian ideal a of b any nilpotent orbit meeting a is a spherical g variety see theorem 2 3 as a consequence of this we obtain a short conceptual proof of a finiteness theorem from 14 namely for a parabolic subgroup p of g and an abelian ideal a of p lie p in the nilpotent radical pu lie pu the group p operates on a with finitely many orbits the proof of this fact in 14 involved long and tedious case by case considerations we also prove a partial converse to the result just mentioned following 4 we say that an ideal of b is ad nilpotent whenever it consists of nilpotent elements in case g is simply laced we show that an ad nilpotent ideal c of b is abelian provided any nilpotent orbit meeting c is spherical see proposition 2 7 in section 3 we consider some properties of ad nilpotent ideals of b in theorem 3 2 we give a description of the normaliser of such ideals this applies in particular to abelian ideals of b a remarkable theorem of doi 10 1006 aima 2000 1959 available online at http www idealibrary com on

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

a monte carlo evaluation of the power of some tests for heteroscedasticity abstract szroeter s asymptotically normal test outperforms the goldfeld qu2ndt test the breusch pagan lagrange multiplier test and bamset when it is possible to order the observations according to increasing variance with no prior information on variance ordering bamset is best some observations concerning degree of heteroscedasticity and model specification are made

the riemann green function and the invariant imbedding equations for hyperbolic systems of first order abstract in the present paper we deal with a boundary value problem for a hyperbolic system of first order in one space variable an explicit integral representation for the riemann green function is first obtained from which a set of discontinuities at both the initial time and along certain characteristics are calculated next the reflection and transmission functions are then defined by simply evaluating this function at certain boundary points and finally a set of integro partial differential equations satisfied by the reflection and transmission functions is derived there are the time dependent invariant imbedding equations for the systems we consider here and these extend the results obtained by corones and krueger for second order hyperbolic equations

a further class of tests for heteroscedasticity abstract this paper considers the problem of testing for heteroscedasticity in the linear regression model when one is willing to postulate only the ranking of the disturbance variances under the alternative hypothesis a new class of tests is proposed and methods of finding the tests critical values are discussed an empirical power comparison with other tests that have found favor in the literature demonstrates the potential power advantage of these new tests and results in particular versions being recommended for practical use

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

a banach space ode framework for inverse scattering via invariant imbedding abstract using invariant imbedding corones davison and krueger j math anal appl 2 1983 393 415 j acoust soc amer 74 1983 1535 1541 inverse optics proc spie 413 a j devaney ed pp 102 106 spie bellingham wash 1983 have approached the inverse scattering problem for an inhomogeneous medium by deriving an integro differential equation whose initial data is the reflection kernel for the medium and whose solution yields the material properties of the medium in this paper we pose this integro differential equation as an ode in the banach space c 0 1 to obtain local existence uniqueness continuous dependence and differentiability of the solution with respect to the initial data in addition we show that the frechet derivative of the mapping taking initial data to the material properties has the form 4 i k where i is the identity and k is a compact operator on the banach space

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

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

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

determination of the size of an object and its location in a rectangular cavity by eigenfrequency shifts first order approximation abstract the acoustic properties of a cavity are modified when objects are placed in the cavity such objects modify the wave equation or boundary conditions and give rise to volumetric and scattering effects these effects cause changes in the eigenfrequencies of the cavity this paper describes a method of using eigenfrequency shifts for determination of the object size and location in the cavity a theoretical and experimental determination of the object size and position in a 2 d rectangular cavity has been carried out it is shown that the object location and size can be calculated from the measured eigenfrequencies and their mode orders under four boundary conditions when the object is small the object function can be expressed as a 2 d spatial fourier series the spatial frequencies of which are related to the eigenfrequency shifts caused by the object the theory can also be applied to determine the position of multiple objects in the cavity experimental results show that the accuracy of the calculated object function is dependent on the number of eigenfrequency shifts used in the calculation the difficulties involved in the mode order identification are discussed

infinite dimensional reductive monoids associated to highest weight representations of kac moody groups abstract starting with a highest weight representation of a kac moody group over the complex numbers we construct a monoid whose unit group is the image of the kac moody group under the representation multiplied by the nonzero complex numbers we show that this monoid has similar properties to those of a j irreducible reductive linear algebraic monoid in particular the monoid is unit regular and has a bruhat decomposition and the idempotent lattice of the generalized renner monoid of the bruhat decomposition is isomorphic to the face lattice of the convex hull of the weyl group orbit of the highest weight

a compact green function approach to the time domain direct and inverse problems for a stratified dissipative slab a new type of green function approach is introduced for time domain scattering for a stratified dissipative slab the new green functions are defined as mappings from the transmitted field to the split fields i e the right moving and left moving fields at any point within the stratified slab unlike the usual green functions the present green functions have compact support in the time variable the new technique is illustrated for the case of normally incident electromagnetic waves on a stratified slab where the permittivity permeability and conductivity vary with the depth the linear and homogeneous partial differential equations pdes for these compact green functions are derived the pdes together with the initial and boundary conditions are well suited for a numerical treatment numerical results for the compact green functions are presented in the direct problem while in the inverse problem the permittivity and conductivity or the permeability and conductivity are reconstructed simult

root polytopes triangulations and the subdivision algebra i the type a n root polytope p a n is the convex hull in r n 1 of the origin and the points e i e j for 1 i j n 1 given a tree t on the vertex set n 1 the associated root polytope p t is the intersection of p a n with the cone generated by the vectors e i e j where i j e t i j the reduced forms of a certain monomial m t in commuting variables x ij under the reduction x ij x jk x ik x ij x jk x ik x ik can be interpreted as triangulations of p t using these triangulations the volume and ehrhart polynomial of p t are obtained if we allow variables x ij and x kl to commute only when i j k l are distinct then the reduced form of m t is unique and yields a canonical triangulation of p t in which each simplex corresponds to a noncrossing alternating forest most generally in the noncommutative case which was introduced in the form of a noncommutative quadratic algebra by kirillov the reduced forms of all monomials are unique

acoustic impulse responses for nonuniform media abstract the acoustic impulse responses of a family of nonuniform lossless one dimensional media are determined the nonuniform portions of the media are confined to layers of either finite or infinite thicknesses the acoustic impedances of the nonuniform layers as functions of travel time are assumed known the impulse responses are determined by evaluating the fourier transforms of their reflection coefficients using infinite series expansions and contour integrations

robustness of size of tests of autocorrelation and heteroscedasticity to nonnormality abstract a comprehensive empirical examination is made of the sensitivity of tests of disturbance covariance in the linear regression model to nonnormal disturbance behaviour tests of autocorrelation appear to be quite robust except for extreme nonnormality but tests for heteroscedasticity are highly susceptible to kurtosis

inverse scattering for discrete transmission line models this paper presents several methods for the identification of the impedance and reflection coefficient profile of a nonuniform discrete transmission line from its response to a given forcing function this problem is the prototype for a wealth of one dimensional inverse scattering problems arising in various fields a unified straightforward approach is presented providing all known inversion procedures and some novel ones the derivations readily follow from causality of signal propagation on the transmission line it is shown that a direct exploitation of the signal propagation model leads to recursive computationally efficient and reliable inversion algorithms in particular the relationships between layer peeling schur type difference equation and layer adjoining levinson type integral equation algorithms are clearly displayed

the hodge theory of soergel bimodules we prove soergel s conjecture on the characters of indecomposable soergel bimodules we deduce that kazhdan lusztig polynomials have positive coecients for arbitrary coxeter systems using results of soergel one may deduce an algebraic proof of the kazhdan lusztig conjecture

faces and maximizer subsets of highest weight modules abstract in this paper we study general highest weight modules v over a complex finite dimensional semisimple lie algebra g we present three formulas for the set of weights of a large family of modules v which include but are not restricted to all simple modules and all parabolic verma modules these formulas are direct and do not involve cancellations and were not previously known in the literature our results extend the notion of the weyl polytope to general highest weight g modules v we also show that for all simple modules the convex hull of the weights is a w j invariant polyhedron for some parabolic subgroup w j we compute its vertices faces and symmetries more generally we also do this for all parabolic verma modules and for all modules v with highest weight not on a simple root hyperplane to show our results we extend the notion of convexity to arbitrary additive subgroups a r of coefficients our techniques enable us to completely classify weak a faces of the support sets wt v in the process extending classical results of satake borel tits vinberg and casselman as well as modern variants by chari dolbin ridenour and cellini marietti to general highest weight modules

lie algebra modules with finite dimensional weight spaces i let g denote a reductive lie algebra over an algebraically closed field of characteristic zero and let 4 denote a cartan subalgebra of g in this paper we study finitely generated g modules that decompose into direct sums of finite dimensional 4 weight spaces we show that the classification of irreducible modules in this category can be reduced to the classification of a certain class of irreducible modules those we call torsion free modules we also show that if g is a simple lie algebra that admits a torsion free module then g is of type a or c

layer stripping solutions of multidimensional inverse scattering problems a layer stripping procedure for solving three dimensional schrodinger equation inverse scattering problems is developed this method operates by recursively reconstructing the potential from the jump in the scattered field at the wave front and then using the reconstructed potential to propagate the wave front and the scattered field further into the inhomogeneous region it is thus a generalization of algorithms that have been developed for one dimensional inverse scattering problems although the procedure has not yet been numerically tested the corresponding one dimensional algorithms have performed well on synthetic data the procedure is applied to a two dimensional inverse seismic problem connections between simplifications of this method and born approximation inverse scattering methods are also noted

generalization of the bremmer coupling series an operator formalism is developed to expand the acoustic wave field in a multi dimensionally smoothly varying medium generated by a source localized in space and time into a sum of constituents each of which can be interpreted as a wave that has traveled up and down with respect to a direction of preference a definite number of times this expansion is a generalization of the bremmer coupling series the condition of smoothness of the medium relates to the width of the signature of the source in the configuration both the existence and the convergence in the weak sense of the expansion are discussed the operator calculus involved leads to a natural generalization of the concept of slowness surface to multi dimensionally smoothly varying media the operator associated with the corresponding generalized vertical slowness induces the full one way wave operator in the type of media under consideration in addition a wavefield decomposition operator as well as an interaction operator that couples the decomposed constituents are derived

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

on the variation in the cohomology of the symplectic form of the reduced phase space is called the momentum mapping of the hamiltonian t action given 1 1 the condition 1 2 just means that t acts along the fibers of j for the basic definitions and properties of non commutative hamiltonian group actions see am the results of this paper can easily be extended to hamiltonian actions of arbitrary compact connected lie groups by applying our results to the action of its maximal toms and using the equivariance of the momentum mapping for some more details see the remarks at the end of sect 2 we will assume throughout this paper that the momentum map is proper that is j i u is compact for each compact subset u of t now let r be a regular value of j that is t j tmm t is surjective for all meye j 1 4 then ye is a smooth submanifold of m compact because

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

inverse problem in nonstationary multidimensional medium the problem of a scalar wave propagation from the point impulsive source in the layer of a nonstationary multidimensional medium is considered the boundary problem for the wave equation is reformulated in the problem with the initial condition using the invariant imbedding method the integral differential inverse procedures of the various orders were obtained from the imbedding equations using the singularities method the order of inverse procedure is defined by the degree of a polynomial in the analytical representation of the medium characteristic near the layer boundary it was shown that the coefficients of the polynomial are calculated with the help of the differential characteristics of the point impulsive source in the inhomogeneous medium the cause and character of the multidimensional inverse problem overdefiniteness are considered the application of the proposed procedure for a statistical problem is discussed

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

a family of koszul algebras arising from finite dimensional representations of simple lie algebras abstract let g be a finite dimensional simple lie algebra and let s g be the locally finite part of the algebra of invariants end c v s g g where v is the direct sum of all simple finite dimensional modules for g and s g is the symmetric algebra of g given an integral weight let be the subset of roots which have maximal scalar product with given a dominant integral weight and such that is a subset of the positive roots we construct a finite dimensional subalgebra s g of s g and prove that the algebra is koszul of global dimension at most the cardinality of using this we construct naturally an infinite dimensional non commutative koszul algebra of global dimension equal to the cardinality of the results and the methods are motivated by the study of the category of finite dimensional representations of the affine and quantum affine algebras

factorization of the wave equation in a nonplanar stratified medium the wave equation is considered for a stratified medium where the stratifications are in the form of a family of nested c2 surfaces along which the velocity c is constant c varying only in a direction normal to the surfaces on each surface c is constant the solution u of the wave equation is decomposed into an outgoing wave component u and an incoming wave component u the associated outgoing and incoming wave conditions are expressed in terms of integral operators kernels being time dependent single and double layer potential type terms operating on u and the normal derivative u n on each surface using the decomposition the scalar wave equation is split into a vector system involving the components u and u the vector system decoupling in a region where c is constant such a splitting is useful for the inverse problem where a reflection operator relating the outgoing wave to an incoming wave can be defined and this in turn can be used to determine c

faces of highest weight modules and the universal weyl polyhedron abstract let v be a highest weight module over a kac moody algebra g and let conv v denote the convex hull of its weights we determine the combinatorial isomorphism type of conv v i e we completely classify the faces and their inclusions in the special case where g is semisimple this brings closure to a question studied by cellini and marietti 2015 7 for the adjoint representation and by khare 2016 2017 17 18 for most modules the determination of faces of finite dimensional modules up to the weyl group action and some of their inclusions also appears in previous works of satake 1960 25 borel and tits 1965 3 vinberg 1990 26 and casselman 1997 6 for any subset of the simple roots we introduce a remarkable convex cone which we call the universal weyl polyhedron which controls the convex hulls of all modules parabolically induced from the corresponding levi factor namely the combinatorial isomorphism type of the cone stores the classification of faces for all such highest weight modules as well as how faces degenerate as the highest weight gets increasingly singular to our knowledge this cone is new in finite and infinite type we further answer a question of michel brion by showing that the localization of conv v along a face is always the convex hull of the weights of a parabolically induced module finally as we determine the inclusion relations between faces representation theoretically from the set of weights without recourse to convexity we answer a similar question for highest weight modules over symmetrizable quantum groups

a study of several new and existing tests for heteroscedasticity in the general linear model abstract several optimum non parametric tests for heteroscedasticity are proposed and studied along with the tests introduced in the literature in terms of power and robustness properties it is found that all tests are reasonably robust to the ordinary least squares ols residual estimates number and character of the regressors only a few are robust to both the distributional and independence assumptions about the errors the power of tests can be improved with the ols residual estimates the increased sample size and the variability of the regressors it can be substantially reduced if the observations are not normally distributed and may increase or decrease if the errors are dependent each test is optimum to detect a specific form of heteroscedasticity and a serious power loss may occur if the underlying heteroscedasticity assumption in the data generation deviates from it

the accurate dynamic modelling of contaminant dispersion in channels throw some dye into a flowing river how is the dye going to disperse along the river the dynamics may be modelled by taylor s one dimensional model of shear dispersion we discuss the creation and utility of models of the dispersion of the dye using invariant manifold theory the models are effective exponentially quickly we describe a one mode centre manifold and a two mode and a three mode invariant manifold model from the analysis we decide under what conditions these three models are valid using the geometric picture provided by the theory of invariant manifolds we find a complete set of initial conditions for the models so that the evolution of the full and approximate systems are identical after a short time throughout a comparison of the three models is made to identify the practical situations when each model will be preferred

a point optimal test for heteroscedastic disturbances abstract a new test for additive heteroscedasticity in the disturbances of the linear regression model is proposed power functions of various forms of the new test are compared empirically with those of currently favoured tests for a range of heteroscedastic models the results highlight the power advantage of a test which is mp1 at a central point in the alternative hypothesis parameter space the main conclusion is that the recommended version of the new test is generally more powerful than existing tests against medium and severe heteroscedasticity whereas the king and szroeter tests perform better against weak heteroscedasticity

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

a vector partition function for the multiplicities of slkc abstract we use gelfand tsetlin diagrams to write down the weight multiplicity function for the lie algebra sl k c type ak 1 as a single partition function this allows us to apply known results about partition functions to derive interesting properties of the weight diagrams we relate this description to that of the duistermaat heckman measure from symplectic geometry which gives a large scale limit way to look at multiplicity diagrams we also provide an explanation for why the weight polynomials in the boundary regions of the weight diagrams exhibit a number of linear factors using symplectic geometry we prove that the partition of the permutahedron into domains of polynomiality of the duistermaat heckman function is the same as that for the weight multiplicity function and give an elementary proof of this for sl 4 c a3

abelian ideals in a borel subalgebra of a complex simple lie algebra let mathfrak g be a complex simple lie algebra and mathfrak b a fixed borel subalgebra of mathfrak g we describe the abelian ideals in mathfrak b in a uniform way that is independent of the classification of complex simple lie algebras as an application we derive a formula for the maximal dimension of a commutative lie subalgebra of mathfrak g

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

reconstruction of the velocity and density in a stratified acoustic half space using a short pulse point source the reconstruction of the velocity and density in a stratified half space using the reflection data generated by a short pulse point source is considered transverse spatial hankel transform is applied to the transient reflected and incident pressures on the surface the reflection kernel in the hankel transformed space is obtained through a deconvolution the time domain green function approach based on the wave splitting is then used to reconstruct the parameters this time domain approach to the inverse problem is exact and noniterative numerical results for some synthetical examples are presented to illustrate the hankel transform deconvolution and the reconstruction the finite difference time domain method is used to generate the transient reflection data in the physical space in a numerical simulation the reconstruction using obliquely incident plane waves is also considered in an appendix

locally best invariant tests of the error covariance matrix of the linear regression model summary this paper considers a class of hypothesis testing problems concerning the covariance matrix of the disturbances in the classical linear regression model a test that is locally best invariant against one sided alternative hypotheses is constructed and shown to be identical to a one sided version of the lagrange multiplier test

modelling variance heterogeneity residual maximum likelihood and diagnostics the assumption of equal variance in the normal regression model is not always appropriate to attempt to eliminate unequal variance a transformation is often used but if the transformation is not successful or the variances are of intrinsic interest it may be necessary to model the variances in some way we consider the normal regression model when log linear dependence of the variances on explanatory variables is suspected detection of the dependence estimation and tests of homogeneity based on full and residual maximum likelihood are discussed as are regression diagnostic methods based on case deletion and log likelihood displacement