Showing papers in "Duke Mathematical Journal in 1993"

Density of integer points on affine homogeneous varieties

Abstract: (1.2) N(T, V)= {m V(Z): Ilmll T} where we denote by V(A), for any ring A, the set of A-points of V. Hence I1\" is some Euclidean norm on R\". The only general method available for such problems is the Hardy-Littlewood circle method, which however has certain limitations, requiring roughly that the codimension of V in the ambient space A\", as well as the degree of the equations (1.1), be small relative to n. Furthermore, there are restrictions on the size of the singular sets of the related varieties:

Some geometric aspects of graphs and their eigenfunctions

Abstract: We study three mathematical notions, that of nodal regions for eigenfunctions of the Laplacian, that of covering theory, and that of fiber products, in the context of graph theory and spectral theory for graphs. We formulate analogous notions and theorems for graphs and their eigenpairs. These techniques suggest new ways of studying problems related to spectral theory of graphs. We also perform some numerical experiments suggesting that the fiber product can yield graphs with small second eigenvalue.

Evolving plane curves by curvature in relative geometries

Abstract: In (0.1) X :S × [0, ω) → IR is the position vector of a family of closed convex plane curves, kN is the curvature vector, with k being the curvature and N the inward pointing normal given by N = −(cos θ, sin θ). The weight function γ(θ) = γ(N) is a function of the normal vector to the curve at each point but does not depend on position in the plane. Equation (0.1) has two significant interpretations. It can be seen as the generalization of the “curve shortening” problem ([Ga8]) to Minkowski geometry or as a simplified model of the motion of the interface of a metal crystal as it melts ([AnGu],[Ta1] and [Ga8]). The proof illustrates most of the techniques that have been used recently in understanding geometric evolution equations as described in [Ha3]. It is not hard to show that the self-similar solutions correspond to positive, 2π periodic solutions of the equation

Qualitative properties of solutions to some nonlinear elliptic equations in r 2

Abstract: O. Introduction. elliptic equations In this paper, we investigate properties of the solutions to the -Au R(x)e u(x) x R 2 (,) for functions R(x) which are positive near infinity. Equations of this kind arise from a variety of situations, such as from prescribing Gaussian curvature in geometry I-2] and from combustion theory in physics [3]. Recently, a series ofworks have been sought to understand the existence and the qualitative properties of the solutions of (,). Ni I-5] and Ni& Cheng [4-1 considered the case where R(x)is nonpositive; McOwen [6-1 and Aviles [7] investigated the situation where R(x) --, 0 in some order as Ixl --*. In our previous paper [1], we consider a special case where R is a constant. We proved that the solutions are radially symmetric and, hence, classified all the solutions. In this paper, we consider more general functions R(x). First, we obtain the asymptotic behavior of the solution near infinity. Consequently, we prove that all the solutions satisfy an identity, which is somewhat of a generalization of the well-known Kazdan-Warner condition. Finally, using the asymptotic behavior together with the further development of the method employed in our previous paper I-1], we show that all the solutions are radially symmetric provided R is radially symmetric and nonincreasing. This part can be viewed as the completion of[1]. Throughout this paper,we assume that the function R(x) is positive near infinity. In 1, we study the asymptotic behavior of the solution u(x) of (,). Let fl 1/2n R2 R(x)e () dx. Under some appropriate conditions, we show that the solutions approach

On the Selberg class of Dirichlet series: small degrees

Abstract: In the study of Dirichlet series with arithmetic signi cance there has appeared (through the study of known examples) certain expectations, namely (i) if a functional equation and Euler product exists, then it is likely that a type of Riemann hypothesis will hold, (ii) that if in addition the function has a simple pole at the point s = 1, then it must be a product of the Riemann zeta-function and another Dirichlet series with similar properties, and (iii) that a type of converse theorem holds, namely that all such Dirichlet series can be obtained by considering Mellin transforms of automorphic forms associated with arithmetic groups. Guided by these ideas, consider the class S of Dirichlet series (introduced by Selberg [7]) : a Dirichlet series

Arrangements of hyperplanes and vector bundles on Pn

Abstract: Any arrangement of hyperplanes in general position in $P^n$ can be regarded as a divisor with normal crossing. We study the bundles of logarithmic 1-forms corresponding to such divisors` from the point of view of classification of vector bundles on $P^n$. It turns out that all such bundles are stable. The study of jumping lines of these bundles gives a unified treatment of several classical constructions associating a curve to a collection of points in $P^n$. The main result of the paper is \"Torelli theorem\" which says that the collection of hyperplanes can be recovered from the isomorphism class of the corresponding logarithmic bundle unless the hyperplanes ocsulate a rational normal curve. In this latter case our construction reduces to that of secant bundles of Schwarzenberger.

Bounded linear operators between C∗-algebras

Abstract: Let $u:A\to B$ be a bounded linear operator between two $C^*$-algebras $A,B$. The following result was proved by the second author. Theorem 0.1. There is a numerical constant $K_1$ such that for all finite sequences $x_1,\ldots, x_n$ in $A$ we have $$\leqalignno{&\max\left\{\left\|\left(\sum u(x_i)^* u(x_i)\right)^{1/2}\right\|_B, \left\|\left(\sum u(x_i) u(x_i)^*\right)^{1/2}\right\|_B\right\}&(0.1)_1\cr \le &K_1\|u\| \max\left\{\left\|\left(\sum x^*_ix_i\right)^{1/2}\right\|_A, \left\|\left(\sum x_ix^*_i\right)^{1/2}\right\|_A\right\}.}$$ A simpler proof was given in [H1]. More recently an other alternate proof appeared in [LPP]. In this paper we give a sequence of generalizations of this inequality.

Free products of hyperfinite von Neumann algebras and free dimension

Abstract: The free product of an arbitrary pair of finite hyperfinite von Neumann algebras is examined, and the result is determined to be the direct sum of a finite dimensional algebra and an interpolated free group factor $L(\freeF_r)$. The finite dimensional part depends on the minimal projections of the original algebras and the "dimension", r, of the free group factor part is found using the notion of free dimension. For discrete amenable groups $G$ and $H$ this implies that the group von Neumann algebra $L(G*H)$ is an interpolated free group factor and depends only on the orders of $G$ and $H$.

Fluxes, Laplacians, and Kasteleyn’s theorem

Abstract: The genesis of this paper was an attempt to understand a problem in condensed matter physics related to questions about electron correlations, superconductivity, and electron-magnetic field interactions. The basic idea, which was proposed a few years ago, is that a magnetic field can lower the energy of electrons when the electron density is not small. Certain very specific and very interesting mathematical conjectures about eigenvalues of the Laplacian were made, and the present paper contains a proof of some of them. Furthermore, those conjectures lead to additional natural conjectures about determinants of Laplacians which we both present and prove here. It is not clear whether these determinantal theorems have physical applications but they might, conceivably in the context of quantum field theory. Some, but not all, of the results given here were announced earlier in [LE].