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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.

## 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.

## 2-Complexes with Large 2-Girth

### Discrete & Computational Geometry (2017-09-05): 1-30 , September 05, 2017

The 2-girth of a 2-dimensional simplicial complex *X* is the minimum size of a non-zero 2-cycle in
$$H_2(X, {\mathbb {Z}}/2)$$
. We consider the maximum possible girth of a complex with *n* vertices and *m* 2-faces. If
$$m = n^{2 + \alpha }$$
for
$$\alpha < 1/2$$
, then we show that the 2-girth is at most
$$4 n^{2 - 2 \alpha }$$
and we prove the existence of complexes with 2-girth at least
$$c_{\alpha , \epsilon } n^{2 - 2 \alpha - \epsilon }$$
. On the other hand, if
$$\alpha > 1/2$$
, the 2-girth is at most
$$C_{\alpha }$$
. So there is a phase transition as
$$\alpha $$
passes 1 / 2. Our results depend on a new upper bound for the number of combinatorial types of triangulated surfaces with *v* vertices and *f* faces.

## 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.

## Staircase transportation problems with superadditive rewards and cumulative capacities

### Mathematical Programming (1993-02-01) 62: 199-213 , February 01, 1993

A cumulative-capacitated transportation problem is studied. The supply nodes and demand nodes are each chains. Shipments from a supply node to a demand node are possible only if the pair lies in a sublattice, or equivalently, in a staircase disjoint union of rectangles, of the product of the two chains. There are (lattice) superadditive upper bounds on the cumulative flows in all leading subrectangles of each rectangle. It is shown that there is a greatest cumulative flow formed by the natural generalization of the South-West Corner Rule that respects cumulative-flow capacities; it has maximum reward when the rewards are (lattice) superadditive; it is integer if the supplies, demands and capacities are integer; and it can be calculated myopically in linear time. The result is specialized to earlier work of Hoeffding (1940), Fréchet (1951), Lorentz (1953), Hoffman (1963) and Barnes and Hoffman (1985). Applications are given to extreme constrained bivariate distributions, optimal distribution with limited one-way product substitution and, generalizing results of Derman and Klein (1958), optimal sales with age-dependent rewards and capacities.

## An algorithm for a piecewise linear model of trade and production with negative prices and bankruptcy

### Mathematical Programming (1979-12-01) 16: 190-209 , December 01, 1979

The general equilibrium model is approximated as a piecewise linear convex model and solved from the point of view of welfare economics using linear programming and fixed point methods.