Tomographic resolution of ray and finite-frequency methods: A membrane-wave investigation

The purpose of this study is to evaluate the resolution potential of current finite-frequency approaches to tomography, and to do that in a framework similar to that of global scale seismology. According to our current knowledge and understanding, the only way to do this is by constructing a large set of ‘ground-truth' synthetic data computed numerically (spectral elements, finite differences, etc.), and then to invert them using the various available linearized techniques. Specifically, we address the problem of using surface wave data to map phase-velocity distributions. Our investigation is strictly valid for the propagation of elastic waves on a spherical, heterogeneous membrane, and a good analogue for the propagation of surface waves within the outermost layers of the Earth. This amounts to drastically reducing the computational expense, with a certain loss of accuracy if very short-wavelength features of a strongly heterogeneous Earth are to be modelled. Our analysis suggests that a single-scattering finite-frequency approach to tomography, with sensitivity kernels computed via the adjoint method, is significantly more powerful than ray-theoretical methods, as a tool to image the fine structure of the Eart

Effects of off great-circle propagation on the phase of long-period surface waves

Surface wave phase corrections for departures from great-circle propagation are computed using two-point ray-tracing through the aspherical earth model M84C of Woodhouse & Dziewonski (1984). For Rayleigh and Love waves with periods in the range 100–250 s, we determine whether these corrections provide significant variance reductions in source determinations compared with corrections calculated assuming great-circle propagation through the heterogeneous structure. For most source-receiver geometries, the off great-circle travel-time effects are small (< 10 s) for second and third orbits (e.g. R2 and R3), and their application in source determinations does not significantly reduce the data variance. This suggests that for the loworder heterogeneous models currently available the geometrical optics approximation is valid for long-period low orbit surface waves. Off great-circle phase anomalies increase quasi-linearly with increasing orbit number, indicating that the geometrical optics approximation degrades for higher orbits, which emphasizes the importance of developing higher order approximations for free-oscillation studies.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/73147/1/j.1365-246X.1987.tb05217.x.pd

Surface wave tomography: global membrane waves and adjoint methods

We implement the wave equation on a spherical membrane, with a finite-difference algorithm that accounts for finite-frequency effects in the smooth-Earth approximation, and use the resulting ‘membrane waves' as an analogue for surface wave propagation in the Earth. In this formulation, we derive fully numerical 2-D sensitivity kernels for phase anomaly measurements, and employ them in a preliminary tomographic application. To speed up the computation of kernels, so that it is practical to formulate the inverse problem also with respect to a laterally heterogeneous starting model, we calculate them via the adjoint method, based on backpropagation, and parallelize our software on a Linux cluster. Our method is a step forward from ray theory, as it surpasses the inherent infinite-frequency approximation. It differs from analytical Born theory in that it does not involve a far-field approximation, and accounts, in principle, for non-linear effects like multiple scattering and wave front healing. It is much cheaper than the more accurate, fully 3-D numerical solution of the Earth's equations of motion, which has not yet been applied to large-scale tomography. Our tomographic results and trade-off analysis are compatible with those found in the ray- and analytical-Born-theory approache

Abelian BF theory and Turaev-Viro invariant

The U(1) BF Quantum Field Theory is revisited in the light of Deligne-Beilinson Cohomology. We show how the U(1) Chern-Simons partition function is related to the BF one and how the latter on its turn coincides with an abelian Turaev-Viro invariant. Significant differences compared to the non-abelian case are highlighted.Comment: 47 pages and 6 figure

Surface wave tomography: global membrane waves and adjoint methods

We implement the wave equation on a spherical membrane, with a finite-difference algorithm that accounts for finite-frequency effects in the smooth-Earth approximation, and use the resulting 'membrane waves' as an analogue for surface wave propagation in the Earth. In this formulation, we derive fully numerical 2-D sensitivity kernels for phase anomaly measurements, and employ them in a preliminary tomographic application. To speed up the computation of kernels, so that it is practical to formulate the inverse problem also with respect to a laterally heterogeneous starting model, we calculate them via the adjoint method, based on backpropagation, and parallelize our software on a Linux cluster. Our method is a step forward from ray theory, as it surpasses the inherent infinite-frequency approximation. It differs from analytical Born theory in that it does not involve a far-field approximation, and accounts, in principle, for non-linear effects like multiple scattering and wave front healing. It is much cheaper than the more accurate, fully 3-D numerical solution of the Earth's equations of motion, which has not yet been applied to large-scale tomography. Our tomographic results and trade-off analysis are compatible with those found in the ray- and analytical-Born-theory approaches

The existence of time

Of those gauge theories of gravity known to be equivalent to general relativity, only the biconformal gauging introduces new structures - the quotient of the conformal group of any pseudo-Euclidean space by its Weyl subgroup always has natural symplectic and metric structures. Using this metric and symplectic form, we show that there exist canonically conjugate, orthogonal, metric submanifolds if and only if the original gauged space is Euclidean or signature 0. In the Euclidean cases, the resultant configuration space must be Lorentzian. Therefore, in this context, time may be viewed as a derived property of general relativity.Comment: 21 pages (Reduced to clarify and focus on central argument; some calculations condensed; typos corrected

Conformal Spinning Quantum Particles in Complex Minkowski Space as Constrained Nonlinear Sigma Models in U(2,2) and Born's Reciprocity

We revise the use of 8-dimensional conformal, complex (Cartan) domains as a base for the construction of conformally invariant quantum (field) theory, either as phase or configuration spaces. We follow a gauge-invariant Lagrangian approach (of nonlinear sigma-model type) and use a generalized Dirac method for the quantization of constrained systems, which resembles in some aspects the standard approach to quantizing coadjoint orbits of a group G. Physical wave functions, Haar measures, orthonormal basis and reproducing (Bergman) kernels are explicitly calculated in and holomorphic picture in these Cartan domains for both scalar and spinning quantum particles. Similarities and differences with other results in the literature are also discussed and an extension of Schwinger's Master Theorem is commented in connection with closure relations. An adaptation of the Born's Reciprocity Principle (BRP) to the conformal relativity, the replacement of space-time by the 8-dimensional conformal domain at short distances and the existence of a maximal acceleration are also put forward.Comment: 33 pages, no figures, LaTe

Resonance modes in a 1D medium with two purely resistive boundaries: calculation methods, orthogonality and completeness

Studying the problem of wave propagation in media with resistive boundaries can be made by searching for "resonance modes" or free oscillations regimes. In the present article, a simple case is investigated, which allows one to enlighten the respective interest of different, classical methods, some of them being rather delicate. This case is the 1D propagation in a homogeneous medium having two purely resistive terminations, the calculation of the Green function being done without any approximation using three methods. The first one is the straightforward use of the closed-form solution in the frequency domain and the residue calculus. Then the method of separation of variables (space and time) leads to a solution depending on the initial conditions. The question of the orthogonality and completeness of the complex-valued resonance modes is investigated, leading to the expression of a particular scalar product. The last method is the expansion in biorthogonal modes in the frequency domain, the modes having eigenfrequencies depending on the frequency. Results of the three methods generalize or/and correct some results already existing in the literature, and exhibit the particular difficulty of the treatment of the constant mode

The Euler-Lagrange Cohomology and General Volume-Preserving Systems

We briefly introduce the conception on Euler-Lagrange cohomology groups on a symplectic manifold $(\mathcal{M}^{2n}, \omega)$ and systematically present the general form of volume-preserving equations on the manifold from the cohomological point of view. It is shown that for every volume-preserving flow generated by these equations there is an important 2-form that plays the analog role with the Hamiltonian in the Hamilton mechanics. In addition, the ordinary canonical equations with Hamiltonian $H$ are included as a special case with the 2-form $\frac{1}{n-1} H \omega$. It is studied the other volume preserving systems on $({\cal M}^{2n}, \omega)$. It is also explored the relations between our approach and Feng-Shang's volume-preserving systems as well as the Nambu mechanics.Comment: Plain LaTeX, use packages amssymb and amscd, 15 pages, no figure

Learning dynamical information from static protein and sequencing data

Many complex processes, from protein folding to neuronal network dynamics, can be described as stochastic exploration of a high-dimensional energy landscape. While efficient algorithms for cluster detection in high-dimensional spaces have been developed over the last two decades, considerably less is known about the reliable inference of state transition dynamics in such settings. Here, we introduce a flexible and robust numerical framework to infer Markovian transition networks directly from time-independent data sampled from stationary equilibrium distributions. We demonstrate the practical potential of the inference scheme by reconstructing the network dynamics for several protein folding transitions, gene-regulatory network motifs and HIV evolution pathways. The predicted network topologies and relative transition time scales agree well with direct estimates from time-dependent molecular dynamics data, stochastic simulations and phylogenetic trees, respectively. Owing to its generic structure, the framework introduced here will be applicable to high-throughput RNA and protein sequencing datasets and future cryo-electronmicroscopy data