an energy conserving parabolic equation for elastic media the accuracy of the parabolic equation pe method is improved for range dependent elastic media the energy conserving elastic pe is a generalization of the energy conserving acoustic pe a range dependent waveguide is approximated by a sequence of range independent regions a linear approximation is derived for the nonlinear condition corresponding to conservation of compressional energy flux at the vertical interfaces between range independent regions the rotated elastic pe is applied to generate reference solutions for demonstrating accuracy the energy conserving elastic pe is applied to solve problems involving the propagation of an interface wave along a sloping ocean bottom and a problem involving mode cutoff and coupling into shear wave beams in the ocean bottom the latter problem is analogous to a cutoff problem that was first solved with the acoustic pe f b jensen and w a kuperman j acoust soc am 67 1564 1566 1980

duality of multiple root loci abstract the multiple root loci among univariate polynomials of degree n are indexed by partitions of n we study these loci and their conormal varieties the projectively dual varieties are joins of such loci where the partitions are hooks our emphasis lies on equations and parametrizations that are useful for euclidean distance optimization we compute the ed degrees for hooks among the dual hypersurfaces are those that demarcate the set of binary forms whose real rank equals the generic complex rank

hermitian matrices with a bounded number of eigenvalues abstract conjugation covariants of matrices are applied to study the real algebraic variety consisting of complex hermitian matrices with a bounded number of distinct eigenvalues a minimal generating system of the vanishing ideal of degenerate three by three hermitian matrices is given and the structure of the corresponding coordinate ring as a module over the special unitary group is determined the method applies also for degenerate real symmetric three by three matrices for arbitrary n partial information on the minimal degree component of the vanishing ideal of the variety of n n hermitian matrices with a bounded number of eigenvalues is obtained and some known results on sum of squares presentations of subdiscriminants of real symmetric matrices are extended to the case of complex hermitian matrices

a mapping approach for handling sloping interfaces a mapping approach for handling sloping interfaces in parabolic equation solutions is developed and tested at each range the medium is rigidly translated vertically so that a sloping interface becomes horizontal to simplify the approach the slope is assumed to be small and the extra terms that arise in the wave equation under the mapping are neglected the effects of these terms can be approximately accounted for by applying a leading order correction to the phase the mapping introduces variations in topography which are relatively easy to handle for the case of a pressure release boundary condition the accuracy of the approach is demonstrated for problems involving fluid sediments the approach should also be accurate for problems involving elastic sediments and should be useful for solving three dimensional problems involving variable topography

on equations defining coincident root loci abstract we revisit an old problem in classical invariant theory viz that of giving algebraic conditions for a binary form to have linear factors with assigned multiplicities we construct a complex of sl2 representations such that the desired algebraic conditions are expressible as a specific cohomology group of this complex

brill gordan loci transvectants and an analogue of the foulkes conjecture abstract the hypersurfaces of degree d in the projective space p n correspond to points of p n where n n d d 1 now assume d 2 e is even and let x n d p n denote the subvariety of two e fold hyperplanes we exhibit an upper bound on the castelnuovo regularity of the ideal of x n d and show that this variety is r normal for r 2 the latter result is representation theoretic and says that a certain gl n 1 equivariant morphism s r s 2 e c n 1 s 2 s r e c n 1 is surjective for r 2 a statement which is reminiscent of the foulkes howe conjecture for its proof we reduce the statement to the case n 1 and then show that certain transvectants of binary forms are nonzero the latter part uses explicit calculations with feynman diagrams and hypergeometric series for ternary quartics and binary d ics we give explicit generators for the defining ideal of x n d expressed in the language of classical invariant theory

a higher order parabolic equation for wave propagation in an ocean overlying an elastic bottom a higher order elastic parabolic equation hepe is derived for wave propagation in depth dependent and weakly range dependent fluid solid media galerkin s method is used to discretize the depth operators in the hepe within layers in which depth variations in the lame constants and density are continuous discontinuities in material properties are handled with centered differences for the interface conditions between layers the numerical solution of the hepe also involves the method of alternating directions and crank nicolson integration the hepe is applied to underwater wave propagation problems involving a water column over an elastic bottom including a weakly range dependent problem the accuracy of the hepe is demonstrated with benchmark calculations

dynamic behaviour of structures in large frequency range by continuous element methods abstract the continuous element method is presented in the context of the harmonic response of beam assemblies a general formulation is described from the displacement solution of the elementary problem a direct computation of elementary dynamic stiffness matrices is presented in the present formulation distributed loadings are taken into account in the case of more complex geometries for which many coupling phenomena occur an explicit formulation is no more conceivable in this case a numerical approach is presented this approach allows an algorithmic computation of exact dynamic stiffness matrices this method called numerical continuous element allows one to consider the coupled vibrations of curved beams and those of helical beams the validation of this numerical method is achieved by comparisons with the harmonic response of various beams obtained by a finite element approach finally a comparison between eigenfrequencies obtained experimentally and numerically for a straight beam and a helical beam has been made to evaluate the performances of the method

waring like decompositions of polynomials 1 abstract let f be a homogeneous form of degree d in n variables a waring decomposition of f is a way to express f as a sum of d th powers of linear forms in this paper we consider the decompositions of a form as a sum of expressions each of which is a fixed monomial evaluated at linear forms

on a theorem of castelnuovo and the equations defining space curves introduction 491 w notat ion and conventions 493 w a regularity theorem 494 w a rationality theorem 498 w the existence of secant lines 501 w open problems 505 references 506

higher order pade approximations for accurate and stable elastic parabolic equations with application to interface wave propagation higher order pade approximations are applied to derive accurate and stable parabolic equations for sound propagation in oceans bounded below by an elastic bottom or bounded above by ice cover accuracy is achieved by placing constraints on the derivatives of the pade approximations at the point corresponding to the reference wave number stability is achieved by requiring that the pade approximations map part of the lower left quadrant of the complex plane into the upper half of the complex plane elastic parabolic equations based on these pade series can handle problems involving compressional shear and interface waves very wide propagation angles and large depth variations and weak range variations in the seismoacoustic parameters a finite difference spectral solution is developed for generating reference solutions and starting fields the rotated elastic parabolic equation is used to investigate the accuracy of the elastic parabolic equation for range dependent problems

generalization of the rotated parabolic equation to variable slopes the rotated parabolic equation j acoust soc am 87 1035 1037 1990 is generalized to problems involving ocean sediment interfaces of variable slope the approach is based on approximating a variable slope in terms of a series of constant slope regions the original rotated parabolic equation algorithm is used to march the field through each region an interpolation extrapolation approach is used to generate a starting field at the beginning of each region beyond the one containing the source for the elastic case a series of operators is applied to rotate the dependent variable vector along with the coordinate system the variable rotated parabolic equation should provide accurate solutions to a large class of range dependent seismo acoustics problems for the fluid case the accuracy of the approach is confirmed through comparisons with reference solutions for the elastic case variable rotated parabolic equation solutions are compared with energy conserving and mapping solutions

numerical solutions of range dependent benchmark problems in ocean acoustics accurate numerical solutions are presented for benchmark problems associated with propagation in a wedge with different bottom boundary conditions as well as in a plane parallel waveguide with changing profile with range the benchmark problems are solved with three different general purpose numerical codes 1 a coupled mode code providing a full spectrum two way backscatter included solution of the wave equation 2 a finite difference pe code providing a wide angle solution of the one way parabolic approximation of the wave equation and 3 a split step pe code providing a more narrow angle solution of the one way parabolic approximation of the wave equation only the two way coupled mode results can be considered accurate solutions of the benchmark problems

generalised discriminants deformed calogero moser sutherland operators and super jack polynomials abstract it is shown that the deformed calogero moser sutherland cms operators can be described as the restrictions on certain affine subvarieties called generalised discriminants of the usual cms operators for infinite number of particles the ideals of these varieties are shown to be generated by the jack symmetric functions related to the young diagrams with special geometry a general structure of the ideals which are invariant under the action of the quantum cms integrals is discussed in this context the shifted super jack polynomials are introduced and combinatorial formulas for them and for super jack polynomials are given

syzygies of canonical curves and special linear series 0 introduction 105 1 line bundles on scrolls 109 2 scrolls and pencils 113 3 syzygies of a subvariety of a scroll 115 4 canonical curves 118 5 on the equation of rational surfaces 121 6 d gonal curves for d 5 126 7 proof of the main result 131 references 136

hyperdeterminants of polynomials abstract the hyperdeterminant of a polynomial interpreted as a symmetric tensor factors into several irreducible factors with multiplicities using geometric techniques these factors are identified along with their degrees and their multiplicities the analogous decomposition for the discriminant of polynomial is found

some remarks on equations defining coincident root loci abstract let x be the projective variety of binary forms of degree d whose linear factors are distributed according to the partition of d we determine minimal sets of local generators of x y where y is the normalization of x and we show that the local jacobian matrices of x y contain the product of the identity matrix of maximal rank with a unit we use this to fill a gap in a crucial proof in chipalkatti s on equations defining coincident root loci also we give a new description of the singular locus of x and a criterion for the smoothness of x

a higher order energy conserving parabolic equqation for range dependent ocean depth sound speed and density outgoing solutions of the wave equation including parabolic equation pe and normal mode solutions are usually formulated so that pressure is continuous with range for range dependent problems the accuracy of normal mode solutions has been improved by conserving energy rather than maintaining continuity of pressure porter et al the problem of energy conservation in one way equations j acoust soc am 89 1058 1067 1991 this approach is applied to derive a higher order energy conserving pe that provides improved accuracy for problems involving large ocean bottom slopes and large range and depth variations in sound speed and density a special numerical approach and complex pade coefficients are applied to suppress gibbs oscillations the back propagated half space field an improved pe starter is applied to handle wide propagation angles reference solutions generated with a complex ray model and with the rotated pe are used for comparison

on the algebraic boundaries among typical ranks for real binary forms abstract we describe the algebraic boundaries of the regions of real binary forms with fixed typical rank and of degree at most eight showing that they are dual varieties of suitable coincident root loci

the equations of strata for binary forms let g be a fixed integer we consider the space a of binary forms of degree g we write x spec r where r s s v v being a twodimensional vector space let x c x be the subset of binary forms having a root of multiplicity 2 p we consider the ideal j of polynomial functions vanishing on x for p 3 g 2 1 xp g is a stratum in the sense of hesselink h for p g 2 1 xp g is a null cone so j becomes the radical of the ideal generated by sl v invariants of positive degree in r we describe explicitly the generators of j for p g 2 1 the surprising result is that those generators occur in degrees d 4 we also describe explicitly the hilbert functions of rr jp and the decomposition of rg jp into representations of sl v we denote by sc v the space sa b vo a2v b let r s s v v vector space over c dim v 2 and let 01 fl be the basis of v let jp be the ideal of elements of r vanishing on x p g 2 1 by the result of hesselink h the space pp g proj r j has the following desingularization y g