Height Uniformity for Algebraic Points on Curves

We recall the main result of L. Caporaso, J. Harris, and B. Mazur's 1997 paper of Uniformity of rational points. It says that the Lang conjecture on the distribution of rational points on varieties of general type implies the uniformity for the numbers of rational points on curves of genus at least 2. In this paper we will investigate its analogue for their heights under the assumption of the Vojta conjecture. Basically, we will show that the Vojta conjecture gives a naturally expected simple uniformity for their heights.     

Elliptic curves retaining their rank in finite extensions and Hilbert's Tenth Problem for rings of algebraic numbers

Using Poonen's version of the ``weak vertical method' we produce new examples of ``large' and ``small' rings of algebraic numbers (including rings of integers) where and/or the ring of integers of a subfield are existentially definable and/or where the ring version of Mazur's conjecture on the topology of rational points does not hold.

     

The Mazur product map on Hardy-type sequence spaces

We study certain Hardy-type sequence spaces Hp and , 1?p?∞, which are analogues of ?^∞ and c₀, respectively. We show that the Mazur product is not onto for every p∈(1,∞) with q=p⁻¹(p−1). We present corollaries for spaces defined via weighted ?p seminorms and for c₀. The latter corollary provides a new solution of Mazur's Problem 8 in the Scottish Book.     

The Parabolic Harnack Inequality for the Time Dependent Ginzburg–Landau Type SPDE and its Application

The main purpose of this paper is to establish the parabolic Harnack inequality for the transition semigroup associated with the time dependent Ginzburg–Landau type stochastic partial differential equation (=SPDE, in abbreviation). In view of quantum field theory, this dynamics is called a P()₁-time evolution. We prove the main result by adopting a stochastic approach which is different from Bakry–Emerys ₂-method. As an application of our result, we study some estimates on the transition probability for our dynamics. We also discuss the Varadhan type asymptotics.     

Multiplicity for strongly indefinite semilinear elliptic system

In this paper we study a multiplicity result for a strongly indefinite semilinear elliptic system

     

A positivity property of the Satake isomorphism

Let G be an unramified reductive group over a local field. We consider the matrix describing the Satake isomorphism in terms of the natural bases of the source and the target. We prove that all coefficients of this matrix which are not obviously zero are in fact positive numbers. The result is then applied to an existence problem of F-crystals which is a partial converse to Mazur's theorem relating the Hodge polygon and the Newton polygon. Received: 29 June 1999 / Revised version: 7 September 1999     

On the existence of chaos in a class of two-degree-of-freedom,damped, strongly parametrically forced mechanical systems with brokenO(2) symmetry

In this paper we study some aspects of the global dynamics associated with a normal form that arises in the study of a class of two-degree-of-freedom, damped, parametrically forced mechanical systems. In our analysis the amplitude of the forcing is an (1) quantity, hence of the same order as the nonlinearity. The normal form is relevant to the study of modal interactions in parametrically excited surface waves in nearly square tanks, parametrically excited, nearly square plates, and parametrically excited beams with nearly square cross sections. These geometrical constraints result in a normal form with brokenO(2) symmetry and the two interacting modes have nearly equal frequencies. Our main result is a method for determining the parameter values for which a Silnikov type homoclinic orbit exists. Such a homoclinic orbit gives rise to a well-described type of chaos. In this problem chaos arises as a result of a balance between symmetry breaking and dissipative terms in the normal form. We use a new global perturbation technique developed by Kovai and Wiggins that is a combination of higher dimensional Melnikov methods and geometrical singular perturbation methods.Dedicated to Klaus Kirchgässner on the occasion of the 60th birthdayThis research was partially supported by an NSF Presidential Young Investigator Award and an ONR Young Investigator Award.     

Unified Approach to Robust Performance of a Class of Transfer Functions with Multilinearly Correlated Perturbations

In this paper, we study the envelope of the Nyquist plots generated by a family of stable transfer functions with multilinearly correlated perturbations and show that the outer Nyquist envelope is generated by the Nyquist plots of the vertices of this family. We then apply this result to calculating the maximal H -norm and verifying the strict positive-realness condition for uncertain transfer function families. Vertex results for robust performance analysis are established. We also study the collection of Popov plots of this transfer function family and show that a large portion of its outer boundary comes from the vertices of this family. This result is then applied to the interval transfer function family to obtain a strong Kharitonov-like theorem.     

Rigidity of proper holomorphic mappings between equidimensional bounded symmetric domains

We prove that any proper holomorphic mapping between two equidimensional irreducible bounded symmetric domains with rank is a biholomorphism. The proof of the main result in this paper will be achieved by a differential-geometric study of a special class of complex geodesic curves on the bounded symmetric domains with respect to their Bergman metrics.

     

Local properties of factored Fourier series

By using a convex sequence Bor [H. Bor, A study on local properties of factored Fourier series, Nonlinear Anal. 57 (2004) 191-197] has obtained a result dealing with local properties of factored Fourier series for summability. In this paper, we have generalized that result for summability. Some new results have also been obtained.     

Asymptotic Behavior of the Density in a Parabolic SPDE

Consider the density of the solution X(t, x) of a stochastic heat equation with small noise at a fixed t[0, T], x[0, 1]. In this paper we study the asymptotics of this density as the noise vanishes. A kind of Taylor expansion in powers of the noise parameter is obtained. The coefficients and the residue of the expansion are explicitly calculated. In order to obtain this result some type of exponential estimates of tail probabilities of the difference between the approximating process and the limit one is proved. Also a suitable iterative local integration by parts formula is developed.     

Mazur's Incidence Structure for Projective Varieties (I)

Let X be an m dimensional smooth projective variety with a Kähler metric. We construct a metrized line bundle with a rational section s over the product of Chow varieties such that

Topologien auf Schnittmoduln kohärenter komplex-und reell-analytischer Garben

In this paper we consider coherent complex-analytic sheaves F on a complex-analytic space X, and study two canonical topologies, inductive resp. projective locally convex, on F(A,F) for subsets ax. We are interested in conditions on A for which these topologies coincide, and get as a main result that this is the case for real analytic spaces which can be imbedded in some ^l and have the original X as a complexification. By complexification we apply our results to coherent real-analytic sheaves.     

Random walk loop soup

The Brownian loop soup introduced by Lawler and Werner (2004) is a Poissonian realization from a -finite measure on unrooted loops. This measure satisfies both conformal invariance and a restriction property. In this paper, we define a random walk loop soup and show that it converges to the Brownian loop soup. In fact, we give a strong approximation result making use of the strong approximation result of Komlós, Major, and Tusnády. To make the paper self-contained, we include a proof of the approximation result that we need.

     

Higher rank case of Dwork's conjecture

In this paper, we study the higher rank case of Dwork's conjecture on the -adic meromorphic continuation of the pure slope L-functions arising from the slope decomposition of an overconvergent F-crystal. Our main result is to reduce the general case of the conjecture to the special case when the pure slope part has rank one and when the base space is the simplest affine -space.

     

Zariski-Offenheit von Eigentlichen,Flachen holomorphen Abbildungen

The main result of this paper is the theorem: Proper flat morphisms of complex spaces are Zariski-open. In general a flat epimorphism of complex spaces f:X S need not be open relativ to the Zariski-topology on X and S (example 5.3). But it is shown in this paper that for flat morphisms f:X S to every x_OX there is an open neighbourhood U of x_o in X such that for any Zariski-open subset Z of X the set f (ZU) is Zariski-open in f (U). An important tool for the proof of this proposition is the notion of the dévissage relatif introduced by M. RAYNAUD and L. GRUSON in [11]. In case that f is also proper this local result about flat morphisms yields the global one stated above.     

Heegner zeros of theta functions

Heegner divisors play an important role in number theory. However, little is known on whether a modular form has Heegner zeros. In this paper, we start to study this question for a family of classical theta functions, and prove a quantitative result, which roughly says that many of these theta functions have a Heegner zero of discriminant . This leads to some interesting questions on the arithmetic of certain elliptic curves, which we also address here.

     

Rectifiability of Measures with Locally Uniform Cube Density

The conjecture that Radon measures in Euclidean space with positivefinite density are rectifiable was a central problem in GeometricMeasure Theory for fifty years. This conjecture was positivelyresolved by Preiss in 1986, using methods entirely dependenton the symmetry of the Euclidean unit ball. Since then, dueto reasons of isometric immersion of metric spaces into l andthe uncommon nature of the sup norm even in finite dimensions,a popular model problem for generalising this result to non-Euclideanspaces has been the study of 2-uniform measures in . The rectifiability or otherwise of these measureshas been a well-known question. In this paper the stronger result that locally 2-uniform measuresin are rectifiable is proved. This is the first result that proves rectifiability, from aninitial condition about densities, for general Radon measuresof dimension greater than 1 outside Euclidean space. 2000 MathematicalSubject Classification: 28A75.     

Invariant complementation and projectivity in the Fourier algebra

In this paper, we study the ideals in the Fourier algebra of a locally compact group which are complemented by an invariant projection. In particular we show that when is discrete, every ideal which is complemented by a completely bounded projection must be invariantly complemented. Perhaps surprisingly, this result does not depend of the amenability of the group or the algebra, but instead relies on the operator biprojectivity of the Fourier algebra for a discrete group.