Browsing Mathematics, Department of by Title
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Neumann eigenfunctions and Brownian couplings
(Mathematical Society of Japan, 2004)This is a review of research on geometric properties of Neumann eigenfunctions related to the "hot spots" conjecture of Jeff Rauch. The paper also presents, in an informal way, some probabilistic techniques used in the proofs. 
No triple point of planar Brownian motion is accessible
(Institute of Mathematical Statistics, 199601)We show that the boundary of a connected component of the complement of a planar Brownian path on a fixed timeinterval contains almost surely no triple point of this Brownian path. 
Nonintersection exponents for Brownian paths. Part I: Existence and an invariance principle
(SpringerVerlag GmbH, 1990)Let X and Y be independent threedimensional Brownian motions, X(0) = (0; 0; 0), Y (0) = (1; 0; 0) and let p [subscript]r = P(X[0; r] [intersected with] Y [0; r] = [empty set]. Then the "non intersection exponent" [from] ... 
Nonintersection exponents for Brownian paths. Part II: Estimations and applications to a random fractal.
(Institute of Mathematical Statistics, 199007)Let X and Y be independent twodimensional Brownian motions, X(0) = (0; 0); Y(0) = ([epsilon]; 0), and let p([epsilon]) = P(X[0; 1] [intersected with] Y [0; 1] = [empty set], q([epsilon]) = {Y [0; 1] does not contain a ... 
Nonpolar points for reflected Brownian motion
(Elsevier, 1993)Our main results are (i) a new construction of reflected Brownian motion X in a halfplane with nonsmooth angle of oblique reflection and (ii) a theorem on existence of some "exceptional" points on the paths of the ... 
Omittable lines
(2005)Every finite family of (straight) lines in the projective plane, not forming a pencil, is well know to have at least one "ordinary point" –– that is, a point common to precisely two of the lines. A line of a family is ... 
On Brownian Excursions in Lipschitz Domains. Part II: Local Asymptotic Distributions
(Birkhäuser Boston, Inc., 1989)In this paper, we continue the study initiated in Burdzy and Williams (1986) of the local properties of Brownian excursions in Lipschitz domains. The focus in part I was on local path properties of such excursions. In ... 
On domain monotonicity of the Neumann heat kernel
(Academic Press (Elsevier), 19930815)Some examples are given of convex domains for which domain monotonicity of the Neumann heat kernel does not hold. 
On minimal parabolic functions and timehomogenous parabolic htransforms
(American Mathematical Society, 19990329)Does a minimal harmonic function h remain minimal when it is viewed as a parabolic function? The question is answered for a class of long thin semiinfinite tubes D [is an element of the subset of real numbers to the power ... 
On Neumann eigenfunctions in lip domains
(American Mathematical Society, 2004)A "lip domain" is a planar set lying between graphs of two Lipschitz functions with constant 1. We show that the second Neumann eigenvalue is simple in every lip domain except the square. The corresponding eigenfunction ... 
On nodal lines of Neumann eigenfunctions
(Institute of Mathematical Statistics, 20020603)We present a new method for locating the nodal line of the second eigenfunction for the Neumann problem in a planar domain. 
On nonincrease of Brownian motion
(Institute of Mathematical Statistics, 199007)A new proof of the nonincrease of Brownian paths is given. 
On the "hot spots" conjecture of J. Rauch
(Academic Press (Elsevier), 19990510)We will state several rigorous versions of J. Rauch's "hot spots" conjecture, review some known results, and prove the conjecture under some additional assumptions. Let us, however, first observe that the conclusion ... 
On the Robin problem in fractal domains
(2005)We study the solution to the Robin boundary problem for the Laplacian in a Euclidean domain. We present some families of fractal domains where the infimum is greater than 0, and some other families of domains where it is ... 
On the time and direction of stochastic bifurcation
(Elsevier, 1998)This paper is a mathematical companion to an article introducing a new economics model, by Burdzy, Frankel and Pauzner (1997). The motivation of this paper is applied, but the results may have some mathematical interest ... 
Percolation dimension of fractals
(Academic Press (Elsevier), 199001)"Percolation dimension" is introduced in this note. It characterizes certain fractals and its definition is based on the Hausdorff dimension. It is shown that percolation dimension and "boundary dimension" are in a sense ... 
Positivity
(20091223)These notes provide a systematic account of certain aspects of the statistical structure of quantum theory. Here the all prevailing notion is that of a completely positive map and Stinespring's famous characterization ... 
Positivity of Brownian transition densities
(Electronic Journal of Probability, 19970924)Let B be a Borel subset of R [to the power of] d and let p(t, x, y) be the transition densities of Brownian motion killed on leaving B. Fix x and y in B. If p(t, x, y) is positive for one t, it is positive for every value ... 
A probabilistic proof of the boundary Harnack principle
(Birkhauser Boston, Inc., 1990)The main purpose of this paper is to give a probabilistic proof of Theorem 1.1, one using elementary properties of Brownian motion. We also obtain the fact that the Martin boundary equals the Euclidean boundary as an easy ... 
Reconstruction Theory
(201101)Suppose that G is a compact group. Denote by \underline{Rep} G the category whose objects are the continuous finite dimensional unitary representations of G and whose morphisms are the intertwining operatorsthen ...