The Theory of Matrices. By F R. Gantmacher. Two volumes, pp. 374 and 276. 1959. (Translated from the Russian by K. A. Hirsch; Chelsea Publishing Company, New York)

I describe the properties of a fourth-order accurate space, second-order accurate time two-dimensional P-Sk’ finite-difference scheme based on the MadariagaVirieux staggered-grid formulation. The numerical scheme is developed from the first-order system of hyperbolic elastic equations of motion and constitutive laws expressed in particle velocities and stresses. The Madariaga-Virieux staggered-grid scheme has the desirable quality that it can correctly model any variation in material properties, including both large and small Poisson’s ratio materials, with minimal numerical dispersion and numerical anisotropy. Dispersion analysis indicates that the shortest wavelengths in the model need to be sampled at 5 gridpoints/wavelength. The scheme can be used to accurately simulate wave propagation in mixed acoustic-elastic media, making it ideal for modeling marine problems. Explicitly calculating both velocities and stresses makes it relatively simple to initiate a source at the free-surface or within a layer and to satisfy free-surface boundary conditions. Benchmark comparisons of finite-difference and analytical solutions to Lamb’s problem are almost identical, as are comparisons of finite-difference and reflectivity solutions for elastic-elastic and acoustic-elastic layered models.

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Fourth-order finite-difference P-SV seismograms

This article provides an overview of the application of the staggered-grid finite-difference technique to model wave propagation problems in 3D elastic media. In addition to presenting generalized, discrete representations of the differential equations of motion using the staggered-grid approach, we also provide detailed formulations that describe the incorporation of moment-tensor sources, the implementation of a stable and accurate representation of a planar free-surface boundary for 3D models, and the derivation and implementation of an approximate technique to model spatially variable anelastic attenuation within time-domain finite-difference computations. The comparison of results obtained using the staggered-grid technique with those obtained using a frequency-wavenumber algorithm shows excellent agreement between the two methods for a variety of models. In addition, this article also introduces a memory optimization procedure that allows large-scale 3D finite-difference problems to be computed on a conventional, single-processor desktop workstation. With this technique, model storage is accommodated using both external (hard-disk) and internal (core) memory. To reduce system overhead, a cascaded time update procedure is utilized to maximize the number of computations performed between I/O operations. This formulation greatly expands the applicability of the 3D finite-difference technique by providing an efficient and practical algorithm for implementation on commonly available workstation platforms.

Simulating seismic wave propagation in 3D elastic media using staggered-grid finite differences

We review the quantum fidelity approach to quantum phase transitions in a pedagogical manner. We try to relate all established but scattered results on the leading term of the fidelity into a systematic theoretical framework, which might provide an alternative paradigm for understanding quantum critical phenomena. The definition of the fidelity and the scaling behavior of its leading term, as well as their explicit applications to the one-dimensional transverse-field Ising model and the Lipkin–Meshkov–Glick model, are introduced at the graduate-student level. Besides, we survey also other types of fidelity approach, such as the fidelity per site, reduced fidelity, thermal-state fidelity, operator fidelity, etc; as well as relevant works on the fidelity approach to quantum phase transitions occurring in various many-body systems.

Fidelity approach to quantum phase transitions

We analyze the problem of a heterogeneous formulation of the equation of motion and propose a new 3D fourth-order staggered-grid finite-difference (FD) scheme for modeling seismic motion and seismic-wave propagation.

We first consider a 1D problem for a welded planar interface of two half-spaces. A simple physical model of the contact of two media and mathematical considerations are shown to give an averaged medium representing the contact of two media. An exact heterogeneous formulation of the equation of motion is a basis for constructing the corresponding heterogeneous FD scheme.

In a much more complicated 3D problem we analyze a planar-interface contact of two isotropic media (both with interface parallel to a coordinate plane and interface in general position in the Cartesian coordinate system) and a nonplanar-interface contact of two isotropic media. Because in the latter case 21 elastic coefficients at each point are necessary to describe the averaged medium, we consider simplified boundary conditions for which the averaged medium can be described by only two elastic coefficients.

Based on the simplified approach we construct the explicit heterogeneous 3D fourth-order displacement-stress FD scheme on a staggered grid with the volume harmonic averaging of the shear modulus in grid positions of the stress-tensor components, volume harmonic averaging of the bulk modulus in grid positions of the normal stress-tensor components, and volume arithmetic averaging of density in grid positions of the displacement components.

Our displacement-stress FD scheme can be easily modified into the velocity-stress or displacement-velocity-stress FD schemes.

The scheme allows for an arbitrary position of the material discontinuity in the spatial grid. Numerical tests for 12 configurations in four types of models show that our scheme is more accurate than the staggered-grid schemes used so far.

Numerical examples also show that differences in thickness of a soft surface or interior layer smaller than one grid spacing can cause considerable changes in seismic motion. The results thus underline the importance of having a FD scheme with sufficient sensitivity to heterogeneity of the medium.

Manuscript received 21 May 2001.

3D Heterogeneous Staggered-grid Finite-difference Modeling of Seismic Motion with Volume Harmonic and Arithmetic Averaging of Elastic Moduli and Densities

Gyrogroups Gyrocommutative Gyrogroups Gyrogroup Extension Gyrovectors and Cogyrovectors Gyrovector Spaces Rudiments of Differential Geometry Gyrotrigonometry Bloch Gyrovector of Quantum Information and Computation Special Theory of Relativity: The Analytic Hyperbolic Geometric Viewpoint Relativistic Gyrotrigonometry Stellar and Particle Aberration.

Analytic Hyperbolic Geometry and Albert Einstein's Special Theory of Relativity

Two successive pure Lorentz transformations are equivalent to a pure Lorentz transformation preceded by a 3×3 space rotation, called a Thomas rotation. When applied to the gyration of the rotation axis of a spinning mass, Thomas rotation gives rise to the well-knownThomas precession. A 3×3 parametric, unimodular, orthogonal matrix that represents the Thomas rotation is presented and studied. This matrix representation enables the Lorentz transformation group to be parametrized by two physical observables: the (3-dimensional) relative velocity and orientation between inertial frames. The resulting parametrization of the Lorentz group, in turn, enables the composition of successive Lorentz transformations to be given by parameter composition. This composition is continuously deformed into a corresponding, well-known Galilean transformation composition by letting the speed of light approach infinity. Finally, as an application the Lorentz transformation with given orientation parameter is uniquely expressed in terms of an initial and a final time-like 4-vector.

Thomas rotation and the parametrization of the Lorentz transformation group

Beyond the Einstein Addition Law and its Gyroscopic Thomas Precession

This is the first book on analytic hyperbolic geometry, fully analogous to analytic Euclidean geometry. Analytic hyperbolic geometry regulates relativistic mechanics just as analytic Euclidean geometry regulates classical mechanics. The book presents a novel gyrovector space approach to analytic hyperbolic geometry, fully analogous to the well-known vector space approach to Euclidean geometry. A gyrovector is a hyperbolic vector. Gyrovectors are equivalence classes of directed gyrosegments that add according to the gyroparallelogram law just as vectors are equivalence classes of directed segments that add according to the parallelogram law. In the resulting "gyrolanguage" of the book one attaches the prefix "gyro" to a classical term to mean the analogous term in hyperbolic geometry. The prefix stems from Thomas gyration, which is the mathematical abstraction of the relativistic effect known as Thomas precession. Gyrolanguage turns out to be the language one needs to articulate novel analogies that the classical and the modern in this book share.The scope of analytic hyperbolic geometry that the book presents is cross-disciplinary, involving nonassociative algebra, geometry and physics. As such, it is naturally compatible with the special theory of relativity and, particularly, with the nonassociativity of Einstein velocity addition law. Along with analogies with classical results that the book emphasizes, there are remarkable disanalogies as well. Thus, for instance, unlike Euclidean triangles, the sides of a hyperbolic triangle are uniquely determined by its hyperbolic angles. Elegant formulas for calculating the hyperbolic side-lengths of a hyperbolic triangle in terms of its hyperbolic angles are presented in the book.The book begins with the definition of gyrogroups, which is fully analogous to the definition of groups. Gyrogroups, both gyrocommutative and non-gyrocommutative, abound in group theory. Surprisingly, the seemingly structureless Einstein velocity addition of special relativity turns out to be a gyrocommutative gyrogroup operation. Introducing scalar multiplication, some gyrocommutative gyrogroups of gyrovectors become gyrovector spaces. The latter, in turn, form the setting for analytic hyperbolic geometry just as vector spaces form the setting for analytic Euclidean geometry. By hybrid techniques of differential geometry and gyrovector spaces, it is shown that Einstein (Moebius) gyrovector spaces form the setting for Beltrami-Klein (Poincare) ball models of hyperbolic geometry. Finally, novel applications of Moebius gyrovector spaces in quantum computation, and of Einstein gyrovector spaces in special relativity, are presented.

Analytic Hyperbolic Geometry: Mathematical Foundations and Applications

The mere mention of hyperbolic geometry is enough to strike fear in the heart of the undergraduate mathematics and physics student. Some regard themselves as excluded from the profound insights of hyperbolic geometry so that this enormous portion of human achievement is a closed door to them. The mission of this book is to open that door by making the hyperbolic geometry of Bolyai and Lobachevsky, as well as the special relativity theory of Einstein that it regulates, accessible to a wider audience in terms of novel analogies that the modern and unknown share with the classical and familiar. These novel analogies that this book captures stem from Thomas gyration, which is the mathematical abstraction of the relativistic effect known as Thomas precession. Remarkably, the mere introduction of Thomas gyration turns Euclidean geometry into hyperbolic geometry, and reveals mystique analogies that the two geometries share. Accordingly, Thomas gyration gives rise to the prefix "gyro" that is extensively used in the gyrolanguage of this book, giving rise to terms like gyrocommutative and gyroassociative binary operations in gyrogroups, and gyrovectors in gyrovector spaces. Of particular importance is the introduction of gyrovectors into hyperbolic geometry, where they are equivalence classes that add according to the gyroparallelogram law in full analogy with vectors, which are equivalence classes that add according to the parallelogram law. A gyroparallelogram, in turn, is a gyroquadrilateral the two gyrodiagonals of which intersect at their gyromidpoints in full analogy with a parallelogram, which is a quadrilateral the two diagonals of which intersect at their midpoints. Table of Contents: Gyrogroups / Gyrocommutative Gyrogroups / Gyrovector Spaces / Gyrotrigonometry

Abraham A. Ungar

Papers

Analytic Hyperbolic Geometry and Albert Einstein's Special Theory of Relativity

Thomas rotation and the parametrization of the Lorentz transformation group

Beyond the Einstein Addition Law and its Gyroscopic Thomas Precession

Analytic Hyperbolic Geometry: Mathematical Foundations and Applications

A Gyrovector Space Approach to Hyperbolic Geometry