Zur Theorie der Maxima und Minima einer Function vonn Veränderlichen

Ohne Zusammenfassung     

Lines of Minima and Teichmüller Geodesics

For two measured laminations ν⁺ and ν⁻ that fill up a hyperbolizable surface S and for , let be the unique hyperbolic surface that minimizes the length function e ^t l(ν⁺) + e ^-t l(ν⁻) on Teichmüller space. We characterize the curves that are short in and estimate their lengths. We find that the short curves coincide with the curves that are short in the surface on the Teichmüller geodesic whose horizontal and vertical foliations are respectively, e ^t ν⁺ and e ^t ν⁻. By deriving additional information about the twists of ν⁺ and ν⁻ around the short curves, we estimate the Teichmüller distance between and . We deduce that this distance can be arbitrarily large, but that if S is a once-punctured torus or four-times-punctured sphere, the distance is bounded independently of t. Received: May 2006, Revision: November 2006, Accepted: February 2007     

Über einige Fragen betreffend die Theorie der Maxima und Minima mehrfacher Integrale

Ohne Zusammenfassung     

Zur Theorie der Maxima und Minima einer Function von zwei Veränderlichen

Ohne Zusammenfassung     

Local Geometry of Levi-Forms Associated with the Existence of Complex Submanifolds and the Minimality of Generic CR Manifolds

Let M be a generic CR manifold in \BbbC^m+d\Bbb{C}^{m+d} of codimension d, locally given as the common zero set of real-valued functions r ¹,…,r ^d . Given an integer δ=1,…,d, we find a necessary and sufficient condition for M to contain a real submanifold of codimension δ with the same CR structure. We also find a necessary and sufficient condition and several sufficient conditions for M to admit a complex submanifold of complex dimension n, for any n=1,…,m. We use the method of prolongation of an exterior differential system. The conditions are systems of partial differential equations on r ¹,…,r ^d of third order.     

Density of the polynomials in Hardy and Bergman spaces of slit domains

It is shown that for any t, 0<t<∞, there is a Jordan arc Γ with endpoints 0 and 1 such that G\{1} í \mathbbD:={z:|z| < 1}\Gamma\setminus\{1\}\subseteq\mathbb{D}:=\{z:|z|<1\} and with the property that the analytic polynomials are dense in the Bergman space \mathbbA^t(\mathbbD\G)\mathbb{A}^{t}(\mathbb{D}\setminus\Gamma) . It is also shown that one can go further in the Hardy space setting and find such a Γ that is in fact the graph of a continuous real-valued function on [0,1], where the polynomials are dense in H^t(\mathbbD\G)H^{t}(\mathbb{D}\setminus\Gamma) ; improving upon a result in an earlier paper.     

Discreteness of the Spectrum of the Laplacian and Stochastic Incompleteness

We propose geometric conditions equivalent to the discreteness of the spectrum of the Laplacian on a class of Riemannian manifolds with ends close to warped products. For this class of manifolds we establish a relationship between discreteness of the spectrum and stochastic incompleteness.      

On the Existence of Optimal Unions of Subspaces for Data Modeling and Clustering

Given a set of vectors F={f ₁,…,f _m } in a Hilbert space H\mathcal {H}, and given a family C\mathcal {C} of closed subspaces of H\mathcal {H}, the subspace clustering problem consists in finding a union of subspaces in C\mathcal {C} that best approximates (is nearest to) the data F. This problem has applications to and connections with many areas of mathematics, computer science and engineering, such as Generalized Principal Component Analysis (GPCA), learning theory, compressed sensing, and sampling with finite rate of innovation. In this paper, we characterize families of subspaces C\mathcal {C} for which such a best approximation exists. In finite dimensions the characterization is in terms of the convex hull of an augmented set C⁺\mathcal {C}^{+}. In infinite dimensions, however, the characterization is in terms of a new but related notion; that of contact half-spaces. As an application, the existence of best approximations from π(G)-invariant families C\mathcal {C} of unitary representations of Abelian groups is derived.     

Regularity of the Laws of Shot Noise Series and of Related Processes

We investigate the regularity of shot noise series and of Poisson integrals. We give conditions for the absolute continuity of their law with respect to the Lebesgue measure and for their continuity in total-variation norm. In particular, the case of truncated series in addressed. Our method relies on a disintegration of the probability space based on a mere conditioning by the first jumps of the underlying Poisson process.     

Deformation of canonical morphisms and the moduli of surfaces of general type

In this article we study the deformation of finite maps and show how to use this deformation theory to construct varieties with given invariants in a projective space. Among other things, we prove a criterion that determines when a finite map can be deformed to a one-to-one map. We use this criterion to construct new surfaces of general type with birational canonical map, for different c₁²c_{1}^{2} and χ (the canonical map of the surfaces we construct is in fact a finite, birational morphism). Our general results enable us to describe some new components of the moduli of surfaces of general type. We also find infinitely many moduli spaces having one component whose general point corresponds to a canonically embedded surface and another component whose general point corresponds to a surface whose canonical map is a degree 2 morphism.     

Symmetric functions,codes of partitions and the KP hierarchy

We consider an operator of Bernstein for symmetric functions and give an explicit formula for its action on an arbitrary Schur function. This formula is given in a remarkably simple form when written in terms of some notation based on the code of a partition. As an application, we give a new and very simple proof of a classical result for the KP hierarchy, which involves the Plücker relations for Schur function coefficients in a τ-function for the hierarchy. This proof is especially compact because we are able to restate the Plücker relations in a form that is symmetrical in terms of partition code notation.     

A Measure of Non-convexity in the Plane and the Minkowski Sum

In this paper a measure of non-convexity for a simple polygonal region in the plane is introduced. It is proved that for “not far from convex” regions this measure does not decrease under the Minkowski sum operation, and guarantees that the Minkowski sum has no “holes”.     

Unmixed bipartite graphs and sublattices of the Boolean lattices

The correspondence between unmixed bipartite graphs and sublattices of the Boolean lattice is discussed. By using this correspondence, we show existence of squarefree quadratic initial ideals of toric ideals arising from minimal vertex covers of unmixed bipartite graphs.     

Markov Processes on the Adeles and Representations of Euler Products

In this article we construct Markov processes on the ring of adeles. Their transition probabilities are given as solutions of Kolmogorov’s differential equations, and the resulting processes have independent p-components which are p-adic-valued additive processes investigated in the last two decades.     

Energy-Preserving Integrators and the Structure of B-series

B-series are a powerful tool in the analysis of Runge–Kutta numerical integrators and some of their generalizations (“B-series methods”). A general goal is to understand what structure-preservation can be achieved with B-series and to design practical numerical methods that preserve such structures. B-series of Hamiltonian vector fields have a rich algebraic structure that arises naturally in the study of symplectic or energy-preserving B-series methods and is developed in detail here. We study the linear subspaces of energy-preserving and Hamiltonian modified vector fields which admit a B-series, their finite-dimensional truncations, and their annihilators. We characterize the manifolds of B-series that are conjugate to Hamiltonian and conjugate to energy-preserving and describe the relationships of all these spaces.     

The Relation of the Morse Index of Closed Geodesics with the Maslov-type Index of Symplectic Paths

In this paper, we consider the relation of the Morse index of a closed geodesic with the Maslov–type index of a path in a symplectic group. More precisely, for a closed geodesic c on a Riemannian manifold M with its linear Poincaré map P (a symplectic matrix), we construct a symplectic path γ(t) starting from identity I and ending at P, such that the Morse index of the closed geodesic c equals the Maslov–type index of γ. As an application of this result, we study the parity of the Morse index of any closed geodesic. Project 10071040 supported by NNSF, 200014 supported by Excellent. Ph.D. Funds of ME of China, and PMC Key Lab. of ME of China     

On Tightness and Weak Convergence in the Approximation of the Occupation Measure of Fractional Brownian Motion

In this paper we consider approximations of the occupation measure of the Fractional Brownian motion by means of some functionals defined on regularizations of the paths. In a previous article Berzin and León proved a cylindrical convergence to a Wiener process of conveniently rescaled functionals. Here we show the tightness of the approximation in the space of continuous functions endowed with the topology of uniform convergence on compact sets. This allows us to simplify the identification of the limit.