Notes on Chern��s Affine Bernstein Conjecture

There were two famous conjectures on complete affine maximal surfaces, one due to Calabi, the other to Chern. Both were solved with different methods about one decade ago by studying the associated Euler-Lagrange equation. Here we survey recent developments and techniques in the study of certain Monge-Ampère equations associated with Chern??s Affine Bernstein Conjecture, in particular two of its proofs in our recent monograph (Li et?al. in Affine Bernstein problems and Monge-Ampère equations, 2010). In addition, we add some details of the proofs of auxiliary material that were omitted in (Li et?al. in Affine Bernstein problems and Monge-Ampère equations, 2010). The related background information is provided in the Introduction.     

Tur��n��s extremal problem on locally compact abelian groups

Let G be a locally compact abelian group (LCA group) and ?? be an open, 0-symmetric set. Let F:= F(??) be the set of all continuous functions f: G ?? ? which are supported in ?? and are positive definite. The Turán constant of ?? is then defined as $$ \mathcal{T}(\Omega ): = \sup \left\{ {\int_\Omega {f:f \in \mathcal{F}} (\Omega ),f(0) = 1} \right\} $$ . Mihalis Kolountzakis and the author has shown that structural properties ?? like spectrality, tiling or packing with a certain set ?? ?? of subsets ?? in finite, compact or Euclidean (i.e., ? ^d ) groups and in ? ^d yield estimates of T (??). However, in these estimates some notion of the size, i.e., density of ?? played a natural role, and thus in groups where we had no grasp of the notion, we could not accomplish such estimates. In the present work a recent generalized notion of asymptotic uniform upper density is invoked, allowing a more general investigation of the Turán constant in relation to the above structural properties. Our main result extends a result of Arestov and Berdysheva, (also obtained independently and along different lines by Kolountzakis and the author), stating that convex tiles of a Euclidean space necessarily have $$ \mathcal{T}_{\mathbb{R}^d } (\Omega )\left| \Omega \right|/2^d $$ . In our extension ? ^d could be replaced by any LCA group, convexity is considerably relaxed to ?? being a difference set, and the condition of tiling is also relaxed to a certain packing type condition and positive asymptotic uniform upper density of the set ??. Also our goal is to give a more complete account of all the related developments and history, because until now an exhaustive overview of the full background of the so-called Turán problem was not delivered.     

On a question of Koll��r

We show: if a bounded domain in a Stein space covers a compact complex space, it must be smooth. This give a negative answer to a question of Kollár.     

Some Recent Developments on Hopf��s Holomorphic Form

Hopf??s theorem on surfaces in ${\mathbb{R}^3}$ with constant mean curvature (Hopf in Math Nach 4:232?C249, 1950-51) was a turning point in the study of such surfaces. In recent years, Hopf-type theorems appeared in various ambient spaces, (Abresch and Rosenberg in Acta Math 193:141?C174, 2004 and Abresch and Rosenberg in Mat Contemp Sociedade Bras Mat 28:283-298, 2005). The simplest case is the study of surfaces with parallel mean curvature vector in ${M_k^n \times \mathbb{R}, n \ge 2}$ , where ${M_k^n}$ is a complete, simply-connected Riemannian manifold with constant sectional curvature k ?? 0. The case n?=?2 was solved in Abresch and Rosenberg 2004. Here we describe some new results for arbitrary n.     

De Bruijn’s question on the zeros of Fourier transforms

Letf(z) be a real entire function of genus 1^*, δ≥0, and suppose that for each ε>0, all but a finite number of the zeros off(z) lie in the strip |Imz| ≤δ+ε. Let λ be a positive constant such that . It is shown that for each ε>0, all but a finite number of the zeros of the entire function lie in the strip and if Δ² < 2λ, then all but a finite number of the zeros of e^−λD2 f(z) are real and simple. As a consequence, de Bruijn's question whether the functions e^γ t ²,λ>0, are strong universal factors is answered affirmatively. The authors wish to acknowledge the financial support of the Korea Research Foundation made in the program year of (1998–2000).     

Higman’s question revisited

We construct a two-generated group with the co-recursively enumerable word problem that has no presentation by recursive permutations. This answers Higman’s question and exemplifies a group with the minimal possible number of generators. The previous article [1], in which that question was claimed settled, contains an incorrigible error. Dedicated to the 60th birthday of Academician Yu. L. Ershov Supported by the Alexander von Humboldt Foundation. This result was obtained during my work at the University of Heidelberg (Germany) as Alexander von Humboldt Research Fellow. The term a “II-group” was proposed by A. Nies. Translated fromAlgebra i Logika, Vol. 39, No. 2, pp. 134–144, March–April, 2000.     

A note on a modification of Gajda��s equation

In the present paper we deal with an integral generalization of a d??Alembert-type equation for unknown mappings defined on a locally compact Abelian group and taking their values in the field of complex numbers.     

La conjecture de Manin g��om��trique pour une famille de quadriques intrins��ques

We prove a version of Manin??s conjecture for a certain family of intrinsic quadrics, the base field being a global field of positive characteristic. We also explain how a very slight variation of the method we use allows to establish the conjecture for a certain generalized del Pezzo surface.     

Indestructibility of Vop��nka��s Principle

Vop??nka??s Principle is a natural large cardinal axiom that has recently found applications in category theory and algebraic topology. We show that Vop??nka??s Principle and Vop??nka cardinals are relatively consistent with a broad range of other principles known to be independent of standard (ZFC) set theory, such as the Generalised Continuum Hypothesis, and the existence of a definable well-order on the universe of all sets. We achieve this by showing that they are indestructible under a broad class of forcing constructions, specifically, reverse Easton iterations of increasingly directed closed partial orders.     

Loi de Weyl presque s?re pour un Syst��me Diff��rentiel en Dimension 1

We consider quite general differential operators on the circle with a small random lower order perturbation. We embrace two points a view, the semiclassical and the high energy limits. We show (a) in the semiclassical limit, that the eigenvalues inside a subdomain of the pseudospectrum are distributed according to a Weyl law with a probability close to 1, (b) that the large eigenvalues obey a Weyl law almost surely.     

A note on ��- and ��-modifications

The notions of ??(?? ₁,?? ₂), ??(?? ₁,?? ₂) and (??(?? ₁,?? ₂),??(?? ₁,?? ₂))-continuity were introduced in [2]. In this paper, the characterization of the continuity is investigated, and we introduce the (??(?? ₁,?? ₂),??(?? ₁,?? ₂))-continuity on generalized topological spaces. Finally, we investigate the relations between (??(?? ₁,?? ₂),??(?? ₁,?? ₂))-continuity and (??(?? ₁,?? ₂),??(?? ₁,?? ₂))-continuity on generalized topological spaces     

Khinchin��s theorem for approximation by integral points on quadratic varieties

We prove an analogue of the Khinchin??s theorem for the Diophantine approximation by integer vectors lying on a quadratic variety. The proof is based on the study of a dynamical system on a homogeneous space of the orthogonal group. We show that in this system, generic trajectories of a certain geodesic flow visit a family of shrinking subsets infinitely often.