Browsing Mathematics, Department of by Title
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Fast equilibrium selection by rational players living in a changing world
(The Econometric Society, 200101)We study a coordination game with randomly changing payoffs and small frictions in changing actions. Using only backwards induction, we find that players must coordinate on the riskdominant equilibrium. More precisely, a ... 
Fiber Brownian motion and the "hot spots" problem
(Duke University Press, 200010)We show that in some planar domains both extrema of the second Neumann eigenfunction lie strictly inside the domain. The main technical innovation is the use of "fiber Brownian motion," a process which switches between ... 
Fibrations and Sheaves
(20121213)The purpose of this book is to give a systematic treatment of fibration theory and sheaf theory, the emphasis being on the foundational essentials. 
A FlemingViat particle representation of Dirichlet Laplacian
(SpringerVerlag GmbH, 200011)We consider a model with a large number N of particles which move according to independent Brownian motions. A particle which leaves a domain D is killed; at the same time, a different particle splits into two particles. ... 
A Gaussian oscillator
(Institute of Mathematical Statistics, 20041006)We present a stochastic process with sawtooth paths whose distribution is given by a simple rule and whose stationary distribution is Gaussian. The process arose in a natural way in research on interaction of an inert ... 
Geometric Properties of 2dimensional Brownian Paths
(SpringerVerlag GmbH, 1989)Let A be the set of all points of the plane C, visited by twodimensional Brownian motion before time 1. With probability 1, all points of A are "twist points" except a set of harmonic measure zero. "Twist points" may be ... 
Graphs of polyhedra; polyhedra as graphs
(2005)Relations between graph theory and polyhedra are presented in two contexts. In the first, the symbiotic dependence between 3connected planar graphs and convex polyhedra is described in detail. In the second, a theory of ... 
The Heat Equation and Reflected Brownian Motion in TimeDependent Domains
(Institute of Mathematical Statistics, 200401)The paper is concerned with reflecting Brownian motion (RBM) in domains with deterministic moving boundaries, also known as "noncylindrical domains," and its connections with partial differential equations. Construction ... 
The heat equation in time dependent domains with insulated boundaries
(Academic Press (Elsevier), 200410)The paper studies, among other things, two types of possible singularities of the solution to the heat equation at the boundary of a moving domain. Several explicit results on "heat atoms" and "heat singularities" are given. 
Hitting a boundary point with reflected Brownian motion
(SpringerVerlag, 1992)An explicit integral test involving the reflection angle is given for the reflected Brownian motion in a halfplane to hit a fixed boundary point. 
Homotopical Topos Theory
(201205)The purpose of this book is twofold: (1) To give a systematic introduction to topos theory from a purely categorical point of view, thus ignoring all logical and algebraic issues. (2) To give an account of the homotopy ... 
The "hot spots" problem in planar domains with one hole.
(Duke University Press, 2005)There exists a planar domain with piecewise smooth boundary and one hole such that the second eigenfunction for the Laplacian with Neumann boundary conditions attains its maximum and minimum inside the domain. 
Hölder domains and the boundary Harnack principle
(Duke University Press, 199110)A version of the boundary Harnack principle is proven. 
Intersection local time for points of infinite multiplicity
(Institute of Mathematical Statistics, 199404)For each a [is an element of the set] (0, 1/2), there exists a random measure [beta] [subscript] a which is supported on the set of points where twodimensional Brownian motion spends a units of local time. The measure ... 
Isogonal prismatoids
(Springer New York, 1997)A prismatoid is a polyhedron with all vertices in two parallel planes. A polyhedron P is isogonal if all its vertices form one transitivity class under isometric symmetries of P. Although these restrictions appear very ... 
Iterated law of iterated logarithm
(Institute of Mathematical Statistics, 199510)Suppose [epsilon] [is a member of the set] [0, 1) and let theta [subscipt epsilon] (t) = (1 − [epsilon]) [square root of] (2tln [subscript] 2 t). Let L [to the power of epsilon] [subscript] t denote the amount of local ... 
Ito formula for an asymptotically 4stable process
(Institute of Mathematical Statistics, 199602)An Itotype formula is given for an asymptotically 4stable process. 
Labyrinth dimension of Brownian trace
(Institute of Mathematics, 1995)Suppose that X is a twodimensional Brownian motion. The trace X[0, 1] contains a selfavoiding continuous path whose Hausdorff dimension is equal to 2. 
Lagrangian Mechanics
(20090112)My original set of lectures on Mechanics was divided into three parts: Lagrangian Mechanics, Hamiltonian Mechanics, Equivariant Mechanics. The present text is an order of magnitude expansion of the first part and is ...