Cohomological characterization of pseudoconvexity in a Banach space

Let X be a Banach space with a countable unconditional basis (e.g., X=₂), X open. We show that is pseudoconvex if and only if for each affine complex line L in X the sheaf cohomology group H ¹ (,I) vanishes, where I is the ideal sheaf of all holomorphic functions on that vanish on L. We also give an example that the condition H ^q (,)=0 for all q1 unlike in finite dimensions does not imply the pseudoconvexity of . Lastly, we prove an interpolation result. Mathematics Subject Classification (2002): 32T05, 46G20.     

A sphere theorem for non-reversible Finsler metrics

For a non-reversible Finsler metric F on a compact smooth manifold M we introduce the reversibility = max {F(–X)|F(X)=1}1. We prove the following generalization of the classical sphere theorem in Riemannian geometry: A simply-connected and compact Finsler manifold of dimension n3 with reversibility and with flag curvature is homotopy equivalent to the n-sphere.Mathematics Subject Classification (2000):53C60; 53C20, 53C22     

Homogeneity in finite ordered sets

A tower in an ordered set (X, ) is defined to be a subsetS ofX which has the property that for everysS there is a maximal chainC in {xX|xs} which is wholly contained inS. An ordered set (X, ) is called tower-homogeneous if every order isomorphism between towers in (X, ) can be extended to an automorphism of (X, ). It is shown that a finite ordered set is tower-homogeneous if and only if it can be built up from singletons stepwise by constructions of three different types.     

Extremes of a Gaussian Process and the Constant H α

Consider a triangular array of standard Gaussian random variables {_n,i, i 0, n 1} such that {_n,i, i 0} is a stationary normal sequence for each n 1. Let _n,k = corr(_n,i,_n,i+k). If (1-_n,k)log n _k (0,) as n for some k, then the locations where the extreme values occur cluster and the limiting distribution of the maxima is still the Gumbel distribution as in the stationary or i.i.d. case, but shifted by a parameter measuring the clustering. Triangular arrays of Gaussian sequences are used to approximate a continuous Gaussian process X(t), t 0. The cluster behavior of the random sequence refers to the behavior of the extremes values of the continuous process. The relation is analyzed. It reveals a new definition of the constants H used for the limiting distribution of maxima of continuous Gaussian processes and provides further understanding of the limit result for these extremes.     

Convolution semigroups and central limit theorem associated with a dual convolution structure

We consider hypergroups associated with Jacobi functions ⁽⁾ (x), (–1/2). We prove the existence of a dual convolution structure on [0,+[i(]0,s ₀]{{) =++1,s ₀=min(,–+1). Next we establish a Lévy-Khintchine type formula which permits to characterize the semigroup and the infinitely divisible probabilities associated with this dual convolution, finally we prove a central limit theorem.     

Lototsky-Schnabl operators on compact convex sets and their associated limit semigroups

Given a metrizable compact convex setX of a locally convex Hausdorff space, a positive projectionT:C(X, )C(X, ) and a continuous function :X[0, 1], it is shown that under suitable assumptions there exists a positive contraction semigroup onC(X, ) that can be represented in terms of the Lototsky-Schnabl operators associated withT and . Several properties of this semigroup are investigated. In particular, its infinitesimal generator is determined in a core of its domain. WhenX ^p for somep1, then the generator is shown to be a degenerate elliptic second order differential operator.Dedicated to Professor George Maltese on the occasion of his 60^th birthday     

Regularity,partial regularity,partial information process,for a filtered statistical model

Summary We define partial regularity for a filtered statistical (semi-parametric) model indexed by ^d , as differentiability in a suitable sense of the partial likelihoods associated with a basic processX. Partial regularity turns out to be equivalent to some sort of differentiability in of the characteristics ofX. We also prove that regularity of the model implies partial regularity, and we define a partial information process, which is smaller than the complete information process. We apply these results to obtain a generalization of Cramer-Rao inequality, and to prove that partial likelihood processes are optimal among all quasi-likelihood processes which are stochastic integrals with respect to the basic processX.     

A note on characterizations of Abelian varieties by topological invariants

In this note, for any given n3 and 2mn (when m=n, we assume n divides 3 and n6), we construct examples of smooth projective varieties X of dimension n with p_g(X)=1, ₁(X)²ⁿ and the Kodaira dimension (X)=m.Mathematics Subject Classification (2000):14H45, 14H99     

Rates of convergence for the stability of large order statistics

Forr1 and eachnr, letM _nr be therth largest ofX ₁,X ₂, ...,X _n , where {X _n ,n1} is an i.i.d. sequence. Necessary and sufficient conditions are presented for the convergence of for all >0 and some –1, where {a _n } is a real sequence. Furthermore, it is shown that this series converges for all >–1, allr1 and all >0 if it converges for some >–1, somer1 and all >0.     

Convexity of Nonlinear Image of a Small Ball with Applications to Optimization

Let f: XY be a nonlinear differentiable map, X,Y are Hilbert spaces, B(a,r) is a ball in X with a center a and radius r. Suppose f (x) is Lipschitz in B(a,r) with Lipschitz constant L and f (a) is a surjection: f (a)X=Y; this implies the existence of >0 such that f (a)^* yy, yY. Then, if r,/(2L), the image F=f(B(a,)) of the ball B(a,) is convex. This result has numerous applications in optimization and control. First, duality theory holds for nonconvex mathematical programming problems with extra constraint x–a. Special effective algorithms for such optimization problems can be constructed as well. Second, the reachability set for small power control is convex. This leads to various results in optimal control.     

Darstellung einer Ordnung von Maßen durch Stoppzeiten

Summary Given a stochastic matrixP on the state spaceI an ordering for measures inI can be defined in the following way: iff(f)(f) for allf in a sufficiently rich subcone of the cone of positiveP-subharmonic functions. It is shown that, if, are probability measures with , then in theP-process (X _n)_n0 having as initial distribution there exists a stopping time such thatX is distributed according to. In addition, can be chosen in such a way, that for every positive subharmonicf with(f)< the submartingale (f(X _n))_n0 is uniformly integrable.     

An infinite dimensional analogue of the Albeverio-Röckner's theorem on Fukushima's conjecture

LetX be the solution of the SDE:dX _t = (X _t)dB _t +b(X _t)dt, with andb C _b (R) such that >0 for some constant , andB a real Brownian motion. Let be the law ofX onE=C([0, 1],R) andk E* – {0}, whereE* is the topological dual space ofE. Consider the classical form: _k (u, v)=u / kv / kd, whereu andv are smooth functions onE. We prove that, if _k is closable for anyk in a dense subset ofE* and if the smooth functions are contained in the domain of the generator of the closure of _k , must be a constant function.     

The Milnor–Totaro Theorem for Space Polygons

If denotes the curvature and the torsion of a closed, generic, and oriented polygonal space curve X in , then we show that _X (² + ²) ds = _X ds + _X | | ds > 4 if is positive. We also show that _X (² + ²) ds 2n if no four consecutive vertices lie in a plane and X has linking number n with a straight line. These extend theorems of Milnor and Totaro.     

Perversité et variation

Let S be the spectrum of a strictly henselian discrete valuation ring with residue characteristic p and =/, where is a prime number p and is an integer 1. For a scheme X of finite type over S and smooth over S along the special fiber X _s outside a closed point x, we study the vanishing cycles complex R() and the tame variation , for in the tame inertia group I _t . In particular, we show that if X is regular, flat over S of relative dimension n1, and is a topological generator of I _t , then R ^q () _x =0 for qn and is an isomorphism. Mathematics Subject Classification (2000):14F20, 14D05, 14D06     

Strong-Cut Enumerative procedure for Extreme point Mathematical Programming Problems

Summary In this paper a Strong-Cut Enumerative Procedure for solving Extreme Point Mathematical Programming Problem:MaxC X subject toAX=b and thatX is an extreme point ofDX=d, X is developed. The procedure will avoid the investigation of many of the extreme points ofDX=d,X 0 and also alternative optimas of different best extreme points ofDX=d, X 0 will not be needed. The algorithm is expected to work very efficiently.

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Zusammenfassung In dieser Arbeit Strong-Cut Enumerative Procedure for Extrem Point Mathematical Programming Problem wird ein sehr effizientes enumeratives Verfahren zur Lösung des ProblemsMax {cx}¦Ax=b;x Extremzahl vonDx=d, x 0entwickelt. Extrempunkte vonDx=d, x 0 werden in systematischer Weise abgesucht, bis Zulässigkeit inAx=b erreicht ist. Die dabei benutzten Kriterien vermeiden die Untersuchung vieler nichtoptimaler Extrempunkte und die Bestimmung alternativer Optimalpunkte vonDx=d, x 0.

     

On a Test Statistic for Linear Trend

Let {W(s)} _s 0 be a standard Wiener process. The supremum of the squared Euclidian norm Y (t)², of the R²-valued process Y(t)=(1/t W(t), {12/t ³ int_{0 t} s dW (s)– {3/t} W(t)), t [, 1], is the asymptotic, large sample distribution, of a test statistic for a change point detection problem, of appearance of linear trend. We determine the asymptotic behavior P {sup _t [, 1] Y(t)² > u as u , of this statistic, for a fixed (0,1), and for a moving = (u) 0 at a suitable rate as u . The statistical interest of our results lie in their use as approximate test levels.     

Drachennetze mit geradlinigen Hauptdiagonalen

A C^s-net of curves N (s1) [3] in a regular C^s-2-surface Eⁿ (n2) is called a C^s-kite- net [4] if N and the net N₁ of its angular bisecting curves form a pair of diagonal nets [1] in such a way that each mesh of N-curves possessing two N₁-diagonals shows, with respect to one of these (calledmain diagonal), the same symmetry of angles and lengths as a rectilinear kite in E². Referring to the fact that the main diagonals of any C^s-kite-net N (s2) are geodesics in [5], we ask in this paper for all C^s-kite-nets and, more generally, C^s-D-nets [5] (s1) withstraight main diagonals. This leads, among other results, to a characterization of the skew ruled surfaces in Eⁿ (n3) with constant parameter of distribution and the constant striction /2.

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Herrn Professor Dr. WERNER BURAU zum 70. Geburtstag gewidmet     

Tower techniques for cosimplicial resolutions

Let be a simplicial model category and J : a simplicial coaugmented functor. Given an object X, the assignment nJⁿ⁺¹X defines a cofacial resolution (an augmented cosimplicial space without its codegeneracy maps). Following Bousfield and Kan we define J^sX = tot^s([n] Jⁿ⁺¹X). An object X is called J-injective if it is a retract of JX in Ho() via the natural map. We show that certain homotopy limits of J-injective objects are J_s-injective. Our method is to use the notion of pro-weak equivalences which was first introduced in a different language and context by David Edwards and Harold Hastings. The key observation is that a cofacial resolution X (-1) X which admits a left contraction gives rise to a pro-weak equivalence of towers {X(-1)}_s0{tot^SX}_{s 0}.The first author was supported in part by National Science Foundation grant DMS-0296117     

Some Applications of Critical Point Theory of Distance Functions on Riemannian Manifolds

In this paper, we use the theory of critical points of distance functions to study the rigidity and topology of Riemannian manifolds with sectional curvature bounded below. We prove that an n-dimensional complete connected Riemannian manifold M with sectional curvature K _M 1 is isometric to an n-dimensional Euclidean unit sphere if M has conjugate radius bigger than /2 and contains a geodesic loop of length 2. We also prove that if M is an n(3)-dimensional complete connected Riemannian manifold with K _M 1 and radius bigger than /2, then any closed connected totally geodesic submanifold of dimension not less than two of M is homeomorphic to a sphere.     

Increase of stable processes

One says thatt>0 is an increase time for a real-valued path if stays above the level (t) immediately after timet, and below (t) immediately before timet. Dvoretzkyet al.,⁽¹⁰⁾ proved that Brownian motion has no increase times a.s. This result is extended here to (strictly) stable processes. Specifically, the probability that a stable processX possesses increase times is 0 if and only ifP(X ₁0)1/2.