2 geometries and the hamilton jacobi equation by using two different procedures we show that on the space of solutions of a certain class of second order ordinary differential equations u s u u a two dimensional definite or indefinite metric gab can be constructed such that the two dimensional hamilton jacobi equation gabu au b 1 holds furthermore we show that this structure is invariant under a certain subset of contact transformations canonical transformations two examples are given

einstein metrics on s 2 bundles new einstein metrics are constructed on the associated bbb r bbb p 2 s 2 and bbb r 2 bundles of principal circle bundles with base a product of kahler einstein manifolds with positive first chern class and with euler class a rational linear combination of the first chern classes these einstein metrics represent different generalizations of the well known einstein metrics found by berard bergery d page c pope n koiso and y sakane corresponding new einstein weyl structures are also constructed

a matched peak inversion approach for ocean acoustic travel time tomography a new approach for the inversion of travel time data is proposed based on the matching between model arrivals and observed peaks using the linearized model relations between sound speed and arrival time perturbations about a set of background states arrival times and associated errors are calculated on a fine grid of model states discretizing the sound speed parameter space each model state can explain identify a number of observed peaks in a particular reception lying within the uncertainty intervals of the corresponding predicted arrival times the model states that explain the maximum number of observed peaks are considered as the more likely parametric descriptions of the reception these model states can be described in terms of mean values and variances providing a statistical answer matched peak solution to the inversion problem a basic feature of the matched peak inversion approach is that each reception can be treated independently i e no constraints are posed from previous reception identification or inversion results accordingly there is no need for initialization of the inversion procedure and furthermore discontinuous travel time data can be treated the matched peak inversion method is demonstrated by application to 9 month long travel time data from the thetis 2 tomography experiment in the western mediterranean sea

differential invariants and curved bernstein gelfand gelfand sequences we give a simple construction of the bernstein gelfand gelfand sequences of natural differential operators on a manifold equipped with a parabolic geometry this method permits us to define the additional structure of a bilinear differential cup product on this sequence satisfying a leibniz rule up to curvature terms it is not associative but is part of an a algebra of multilinear differential operators which we also obtain explicitly we illustrate the construction in the case of conformal differential geometry where the cup product provides a wide reaching generalization of helicity raising and lowering for conformally invariant field equations

a hamiltonian method for finding broadband modal eigenvalues for shallow water waveguides over a layered elastic bottom modal eigenvalues can be determined by searching the locations in the complex plane of the horizontal wave number at which the complex phase function is a multiple of c t tindle and n r chapman j acoust soc am 96 1777 1782 1994 in this paper a hamiltonian method is introduced for tracing the path in the complex plane along which the phase function keeps real the hamiltonian method can also be extended to compute the broadband modal eigenvalues or the modal dispersion curves in the pekeris waveguide with fluid elastic bottoms for each proper or leaky normal mode a different hamiltonian is constructed in the complex plane and used to trace automatically the complex dispersion curve with the eigenvalue in a reference frequency as the initial value in contrast to the usual methods the dispersion curve for each mode is determined individually the hamiltonian method shows good performance by comparing with kraken

einstein weyl spaces and third order differential equations the three dimensional null surface formalism of tanimoto m tanimoto on the null surface formalism report no gr qc 9703003 1997 and forni et al forni et al null surfaces formation in 3d j math phys submitted are extended to describe einstein weyl spaces following cartan e cartan les espaces generalisees et l integration de certaines classes d equations differentielles c r acad sci 206 1425 1429 1938 la geometria de las ecuaciones diferenciales de tercer order rev mat hispano am 4 1 31 1941 in the resulting formalism einstein weyl spaces are obtained from a particular class of third order differential equations some examples of the construction which include some new einstein weyl spaces are given

3 geometries and the hamilton jacobi equation in the first part of this work we show that on the space of solutions of a certain class of systems of three second order pde s u u u u u u u u and u u u u a three dimensional definite or indefinite metric gab can be constructed such that the three dimensional hamilton jacobi equation gabu au b 1 holds furthermore we remark that this structure is invariant under a subset of contact transformations in the second part we obtain analogous results for a certain class of third order ordinary differential equation ode s u s u u u in both cases we apply our general results to the cental force problem

lorentzian metrics from characteristic surfaces the following issue is raised and discussed when do families of foliations by hypersurfaces on a given four dimensional manifold without further structure become the null surfaces of some unknown but to be determined metric gab x explicit conditions for these surfaces are found so that they do define a unique conformal metric with the surfaces themselves being characteristics of that metric by giving an additional function to be the conformal factor full knowledge of the metric is determined it is clear from these results that one can use these surfaces and the conformal factor as fundamental variables for describing any lorentzian geometry and in particular for its use in general relativity

global and singular solutions to the generalized proudman johnson equation abstract we show that there is a class of solutions to the generalized proudman johnson equation which exist globally for all parameters a having the form n 3 n 1 for n n thereby extending a result of bressan and constantin 2005 2 furthermore we present new proofs of existence of solutions developing spontaneous singularities and compute the corresponding blow up rates

linearized einstein theory via null surfaces recently there has been developed a reformulation of general relativity gr referred to as the null surface version of gr where instead of the metric field as the basic variable of the theory families of three surfaces in a four manifold become basic from these surfaces themselves a conformal metric conformal to an einstein metric can be constructed a choice of conformal factor turns it into an einstein metric the surfaces are then automatically characteristic surfaces of this metric in the present paper we explore the linearization of this null surface theory and compare it with the standard linear gr this allows a better understanding of many of the subtle mathematical issues and sheds light on some of the obscure points of the null surface theory it furthermore permits a very simple solution generating scheme for the linear theory and the beginning of a perturbation scheme for the full theory

ocean acoustic tomography based on peak arrivals the recently introduced notion of peak arrivals athanassoulis and skarsoulis j acoust soc am 97 3575 3588 1995 defined as the significant local maxima of the arrival pattern is studied here as a modeling basis for performing ocean tomography peak arrivals constitute direct theoretical counterparts of experimentally observed peaks and offer a complete modeling of experimental observables even in cases where ray or modal arrivals cannot be resolved the coefficients of the resulting peak inversion system relating travel time with sound speed perturbations are explicitly calculated in the case of range independent environments using normal mode theory to apply the peak inversion scheme to tomography the peak identification and tracking problem is examined from a statistical viewpoint maximum likelihood and least square solutions are derived and discussed the particular approach adopted treats the identification and tracking problem in close relation to the inversion procedure all possibi

second order perturbations of peak arrival times due to sound speed variations the concept of peak arrivals is used to study the second order arrival time behavior of broadband tomographic signals due to sound speed variations peak arrivals defined as the local maxima of the pressure amplitude at the receiver in the time domain are the exact theoretical counterparts of the experimental observables in ocean acoustic tomography and they offer a uniform description of all the peaks in the arrival pattern independent of whether they can be resolved as ray modal arrivals or not closed form expressions for the first and second sound speed derivatives of peak arrival times are derived in terms of sound speed derivatives of the green s function using normal mode theory the sound speed derivatives of the green s function are analytically expressed for the case of a range independent medium in terms of eigenvalues and eigenfunctions at the reference state numerical examples are given for range independent environments motivated from the thetis 2 experiment showing different kinds of

perturbations in travel time and ray geometry due to mesoscale disturbances a comparison of exact and approximate calculations exact calculations of the changes in eigenray travel time and geometry due to range dependent mesoscale disturbances are compared to three approximations in the first approximation the sound speed perturbation is integrated along the unperturbed path in the second approximation the range average of the sound speed perturbation is integrated along the unperturbed path the third approximation uses the range average raytrace to estimate changes of travel time and geometry calculations made both for a measured section through a mesoscale eddy and for a simplified model of an eddy with variable strength position and boundary width suggest that i while none of the approximations tested is uniformly the best the first approximation is to be preferred ii the eddy position relative to the ray path is quite significant and iii the exact travel time perturbations are not sensitive to whether the eddy boundaries are assumed sharp or gradual for an eddy perturbation of 9 m s 1 9 c or rossby wave displacement of about 200 m at 700 m depth the travel time approximations are typically in error by 10 20 the associated change in turning depth from the unperturbed state to either the perturbed or the range average state is typically 100 200 m these errors are significant but not large and suggest that it will be beneficial to iterate linear inverse schemes used to estimate sound speed perturbations from travel time perturbations

a numerical method for ocean acoustic normal modes the method of normal modes is frequently used to solve acoustic propagation problems in stratified oceans the propagation numbers for the modes are the eigenvalues of the boundary value problem to determine the depth dependent normal modes errors in the numerical determination of these eigenvalues appear as phase shifts in the range dependence of the acoustic field such errors can severely degrade the accuracy of the normal mode representation particularly at long ranges in this paper we present a fast finite difference method to accurately determine these propagation numbers and the corresponding normal modes it consists of a combination of well known numerical procedures such as sturm sequences the bisection method newton s and brent s methods richardson extrapolation and inverse iteration we also introduce a modified richardson extrapolation procedure that substantially increases the speed and accuracy of the computation

rational operators for filtering the parabolic equation pe method has been applied to approximate the wave equation construct radiation conditions solve scattering problems construct initial conditions and derive energy flux conditions a new application of the pe method filters based on rational approximations of depth operators is described and tested the rational approximation is designed to act as the identity operator on the desired part of the spectrum of the depth separated wave equation and to annihilate other parts of the spectrum this approach does not require explicit knowledge of the spectrum the applications of the filter include directly solving eigenvalue problems annihilating components of the wave number spectrum generating initial conditions at the source range and eliminating gibbs oscillations that arise in energy conserving pe solutions

a note on the relationship between finite difference and shooting methods for ode eigenvalue problems finite difference methods and shooting methods are two standard classes of techniques for solving ordinary differential equation eigenvalue problems we first show that a sturm sequence method often used for solving the finite difference eigenvalue problem may be interpreted in the shooting framework then we find that the sturm sequence procedure is easily extended to more complicated problems this enables the shooting method to guarantee convergence to specified subsets of the modes of the problem without the usual requirement of an initial guess for the eigenvalues

three dimensional einstein weyl geometry abstract this paper is an investigation of various differential geometric properties of three dimensional einstein weyl spaces we show that in a suitable gauge and coordinate system these spaces are analytic we find classes of explicit examples and we consider the behaviour of godesics in general and in some of these examples although most of the material is orginal our intention is to be expository and to compare and contrast the theory of einstein weyl spaces with the theory of einstein spaces

minimal lagrangian connections on compact surfaces abstract we introduce the notion of a minimal lagrangian connection on the tangent bundle of a manifold and classify all such connections in the case where the manifold is a compact oriented surface of non vanishing euler characteristic combining our classification with results of labourie and loftin we conclude that every properly convex projective surface arises from a unique minimal lagrangian connection

higher symmetries of the laplacian we identify the symmetry algebra of the laplacian on euclidean space as an explicit quotient of the universal enveloping algebra of the lie algebra of conformal motions we construct analogues of these symmetries on a general conformal manifold

source localization in a waveguide using polynomial rooting source localization in acoustic waveguides involves a multidimensional search procedure we propose a new algorithm in which the search in the depth direction is replaced by polynomial rooting using the proposed algorithm range and depth estimation by a vertical array requires a 1 d search procedure for a 3 d localization problem i e range depth and direction of arrival doa estimation the algorithm involves a 2 d search procedure consequently the proposed algorithm requires significantly less computation than other methods that are based on a brute force search procedure over the source location parameters in order to evaluate the performance of the algorithm an error analysis is carried out and monte carlo simulations are performed the results are compared with the cramer rao bound crb and to the maximum likelihood ml simulation performance the algorithm is shown to be efficient while being computationally simpler than the ml or the bartlett processors the disadvantage of the algorithm is that its snr threshold occurs in lower snr than in the ml algorithm

three dimensional manifolds kleinian groups and hyperbolic geometry 1 a conjectural picture of 3 manifolds a major thrust of mathematics in the late 19th century in which poincare had a large role was the uniformization theory for riemann surfaces that every conformai structure on a closed oriented surface is represented by a riemannian metric of constant curvature for the typical case of negative euler characteristic genus greater than 1 such a metric gives a hyperbolic structure any small neighborhood in the surface is isometric to a neighborhood in the hyperbolic plane and the surface itself is the quotient of the hyperbolic plane by a discrete group of motions the exceptional cases the sphere and the torus have spherical and euclidean structures three manifolds are greatly more complicated than surfaces and i think it is fair to say that until recently there was little reason to expect any analogous theory for manifolds of dimension 3 or more except perhaps for the fact that so many 3 manifolds are beautiful the situation has changed so that i feel fairly confident in proposing the

an immersion theorem for vaisman manifolds a locally conformally kahler lck manifold is a complex manifold admitting a kahler covering open image in new window with monodromy acting on open image in new window by kahler homotheties a compact lck manifold is vaisman if it admits a holomorphic flow acting by non trivial homotheties on open image in new window we prove that any compact vaisman manifold admits a natural holomorphic immersion to a hopf manifold n 0 as an application we obtain that any sasakian manifold has a contact immersion to an odd dimensional sphere