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## A functional limit theorem for waves reflected by a random medium

### Applied Mathematics and Optimization (1994-11-01) 30: 307-334 , November 01, 1994

We introduce a class of distribution-valued stochastic processes that arise in the study of pulse reflection from random media and we analyze their asymptotic properties when they are scaled in a natural way.

## Deferred Corrections Software and Its Application to Seismic Ray Tracing

### Defect Correction Methods (1984-01-01) 5: 211-226 , January 01, 1984

We give first a historical account of the various stages of development of iterated deferred corrections software, mainly for ordinary two-point boundary value problems, but mentioning also some work on partial differential equations. Then we describe the latest code on the PASVA series (No. 4), which extends the earlier one to problems with discontinuous data and mixed systems of differential and algebraic conditions. Finally, an example of application of this code to two-point ray tracing on piece-wise continuous media is given.

## MORSE THEORY, GRAPHS, AND STRING TOPOLOGY

### Morse Theoretic Methods in Nonlinear Analysis and in Symplectic Topology (2006-01-01) 217: 149-184 , January 01, 2006

###
*Abstract*

In these lecture notes we discuss a body of work in which Morse theory is used to construct various homology and cohomology operations. In the classical setting of algebraic topology this is done by constructing a moduli space of graph flows, using homotopy theoretic methods to construct a virtual fundamental class, and evaluating cohomology classes on this fundamental class. By using similar constructions based on “fat„ or ribbon graphs, we describe how to construct string topology operations on the loop space of a manifold, using Morse theoretic techniques. Finally, we discuss how to relate these string topology operations to the counting of *J*-holomorphic curves in the cotangent bundle. We end with speculations about the relationship between the absolute and relative Gromov –Witten theory of the cotangent bundle, and the open-closed string topology of the underlying manifold.

## Finitely determined implies very weak Bernoulli

### Israel Journal of Mathematics (1974-03-01) 17: 94-104 , March 01, 1974

It is shown that if a process is finitely determined then it is very weak Bernoulli (VWB). Combined with known results this says that a process is isomorphic to a Bernoulli shift if and only if it satisfies an asymptotic independence condition, namely that of being VWB.

## Holomorphic families of injections

### Acta Mathematica (1986-12-01) 157: 259-286 , December 01, 1986

## Elliptic Partial Differential Equations of Second Order

### Elliptic Partial Differential Equations of Second Order (2001-01-01): 224 , January 01, 2001

## Extremal Fields

### Superconcentration and Related Topics (2014-01-01): 73-85 , January 01, 2014

This chapter introduces the notion of extremal Gaussian fields and proves superconcentration in extremal fields. The method is then used to prove superconcentration in certain kinds of spin glass models and in the discrete Gaussian free field.

## Simulation of Dynamic Earthquake Ruptures in Complex Geometries Using High-Order Finite Difference Methods

### Journal of Scientific Computing (2013-04-01) 55: 92-124 , April 01, 2013

We develop a stable and high-order accurate finite difference method for problems in earthquake rupture dynamics in complex geometries with multiple faults. The bulk material is an isotropic elastic solid cut by pre-existing fault interfaces that accommodate relative motion of the material on the two sides. The fields across the interfaces are related through friction laws which depend on the sliding velocity, tractions acting on the interface, and state variables which evolve according to ordinary differential equations involving local fields.

The method is based on summation-by-parts finite difference operators with irregular geometries handled through coordinate transforms and multi-block meshes. Boundary conditions as well as block interface conditions (whether frictional or otherwise) are enforced weakly through the simultaneous approximation term method, resulting in a provably stable discretization.

The theoretical accuracy and stability results are confirmed with the method of manufactured solutions. The practical benefits of the new methodology are illustrated in a simulation of a subduction zone megathrust earthquake, a challenging application problem involving complex free-surface topography, nonplanar faults, and varying material properties.

## Topological Proof of Cartan’s Theorem

### Lie Groups (2004-01-01) 225: 107-111 , January 01, 2004

We will give another proof of Cartan’s Theorem 16.5. Since this was already proved in the last chapter, *the reader can skip this chapter with no loss of continuity*. As a by-product of this second proof, we will obtain some topological insight into the “flag manifold” *G*/*T*, where *T* is a maximal torus in the compact Lie group *T*, a topic that we will take up in the final chapter.

## The Universal Enveloping Algebra

### Lie Groups (2004-01-01) 225: 54-57 , January 01, 2004

We have seen that elements of the Lie algebra of a Lie group *G* are derivations of *C*^{∞} (*G*); that is, differential operators that are left-invariant. The universal enveloping algebra is the ring of all left-invariant differential operators, including higher-order ones. There is a purely algebraic construction of this ring.