On Sets of Planes in Projective Spaces Intersecting Mutually in One Point

Let be a projective space. In this paper we consider sets of planes of such that any two planes of intersect in exactly one point. Our investigation will lead to a classification of these sets in most cases. There are the following two main results:- If is a set of planes of a projective space intersecting mutually in one point, then the set of intersection points spans a subspace of dimension 6. There are up to isomorphism only three sets where this dimension is 6. These sets are related to the Fano plane.- If is a set of planes of PG(d,q) intersecting mutually in one point, and if q3, 3(q²+q+1), then is either contained in a Klein quadric in PG(5,q), or is a dual partial spread in PG(4,q), or all elements of pass through a common point.     

A study on Markovian maximality,change of probability and regularity

LetE be a rigid separable Banach space andm a bounded Borel measure onE. Let Ext denote the family of all gradient type Dirichlet forms onL ²(E, m) such that the domain of their extended generators (cf. Definition 1.1) contain the smooth functions. We prove three results. First, we prove the existence of the maximum element in Ext whenever Ext is not empty. Secondly, let be the maximum element in Ext (when Ext Ø) and let be a positive function in D(). We define a new measure =²·m and we consider the family Ext associated with the measure . We prove that if is associated with a diffusion process, Ext is not empty and its maximum element is also associated with a diffusion process. Finally, whenm is a centered Gaussian measure onE, we can prove that Ext contains exactly one element.     

Unisecant curves on a general ruled surface

Let X be an irreducible algebraic curve of genus g smooth and proper over an algebraically closed field k, a locally free sheaf of rank 2 over X, F=P() the projective bundle associated to and :FX the canonical projection. Aunisecant curve on F is a curve (effective divisor) C on F such that the intersection number (C,–¹(x))=1, x X. Notice that a section of F over X or alternatively a sub-line bundle of means simply an irreducible unisecant curve. We give here some results on unisecant curves on F. In particular we are able to prove C.Segre's result regarding his general surfaces [8]. A more ample account including all the details will appear later.     

Monads on projective spaces

Let be a vector bundle on P ⁿ . There is a strong relationship between and its intermediate cohomology modules. In the case where has low rank, we exploit this relationship to provide various splitting criteria for . In particular, we give a splitting criterion for in terms of the vanishing of certain intermediate cohomology modules. We also show that the Horrocks-Mumford bundle is the only non-split rank two bundle on P ⁴ with a Buchsbaum second cohomology module.Partially supported by NSF Grants.     

Probability and expectation inequalities

Summary This paper introduces a mathematical framework within which a wide variety of known and new inequalities can be viewed from a common perspective. Probability and expectation inequalities of the following types are considered: (a)P(ZA) P(ZA) for some class of setsA, (b)k(Z)k(Z) for some class of functionsk, and (c)l(Z)k(Z) for some class of pairs of functionsl andk. It is shown, sometimes using explicit constructions ofZ andZ, that, in several cases, (a) (b) (c); included here are cases of normal and elliptically contoured distributions. A case where (a) (b) (c) is studied and is expressed in terms ofn monotone functions for (some of) which integral representations are obtained. Also, necessary and sufficient conditions for (c) are given.Research supported by the Air Force Office of Scientific Research under Grants AFOSR-75-2796 and AFOSR-80-0080Research supported by the National Science Foundation under Grants MCS78-01240 and MCS81-00748     

Piecewise polynomial approximation

For best piecewise polynomial approximation _n=_n (f; [0, 1]) of a functionf, which is continuous on the interval [0, 1] and admits a bounded analytic continuation onto the disk K=z:¦z–1¦<, the relation _n=o[ _f (e ^–n )] is valid.Translated from Matematicheskie Zametki, Vol. 11, No. 2, pp. 129–134, February, 1972.     

Closability and resolvent of Dirichlet forms perturbed by jumps

We give a necessary and sufficient condition for the closability of a form _J =+J, where is a Dirichlet form on someL ² (E, m) andJ is a symmetric measure onE×E. We then exhibit the resolvent family of such a form.     

Bemerkungen zum Vergleich linearer normaler Experimente

The information comparison _A > _B between two linear normal experiments _A , _B is studied in terms of the Loewner orderingAB between their defining design matricesA, B respectively. The result due toKiefer [3],Takeuchi [5]Hansen, Torgersen [1] and others is related to the comparison of certain translation invariant experiments via a reduction principle. The analysis given here completes the discussion in § 25 of [2].     

Continuity of martingales in the Brownian excursion filtration

Summary Given a Brownian motionB, we consider the filtration ( _{x xR} ), where _x is defined as the -field generated by the excursions ofB belowx. In this paper we prove a conjecture of Walsh which says that all -martingales are continuous.     

Poisson kernel is the unique approximate identity,asymptotically multiplicative on H∞

Let for anyf H(R), where (x): = ^–1(x^–1). Then (x) P (x + h) for some h R and > 0; P denotes the Poisson kernel.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova Akademii Nauk SSSR, Vol. 170, pp. 82–89, 1989.     

How to build a barricade

Given a packing of congruent copies of a strictly convex planar setC in a parallel stripS of widthw and a straight linel, let |l| denote the sum of the lengths of the segmentslC _i over alliI It is proved that Where (C) is the density of the densest packing of congruent copies ofC in the plane. Some generalizations are also considered.     

Quasi-coherence of Hardy spaces in several complex variables

In this paper, we prove that the Hardy spaceH ^p (), 1p<, over a strictly pseudoconvex domain in ⁿ with smooth boundary is quasi-coherent. More precisely, we show that Toeplitz tuplesT with suitable symbols onH ^p () have property (). This proof is based on a well known exactness result for the tangential Cauchy-Riemann complex.     

Representations of tensor product l.m.c.*-algebras

If E is a l.m.c.*-algebra with a b.a.i., (E), (E) denote the enveloping algebra and the space of representations of E respectively, while (E) stands for the non-zero extreme points of the continuous positive linear forms on E. Thus, for suitable l.m.c.*-algebras E, F and an admissible topology on E F, (E F) is given by the completed-tensor product of (E), (F) (where is the projective tensorial l.m.c.C*-topology), while (E F) by the cartesian product of (E), (F). An analogous decomposition of (E F) is not valid in general.This paper is partly based on the author's Ph.D. Thesis (Univ. of Athens)     

Fortsetzungssätze und Moduln über Semifields

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Herrn Professor Dr. Frank Terpe zum 60. Geburtstag gewidmet     

On a Property of Subexponential Distributions

Let F be a distribution function (d.f.) on [0, ) with finite first moment m >0. We define the integrated tail distribution function F ₁ of F by F ₁(t)=m^-1 ₀ ^t (1- F(u))du, t0. In this paper, we obtain sufficient conditions under which implications FSF ₁S and F ₁S FS hold, where S is the class of subexponential distributions.     

Local completeness I

An infinite algebra is locally complete if its local closure is the set of all finitary operations. We give a local completeness criterion in terms of a system R of finitary relations on A such that the polymorph Pol of each R is locally incomplete and for every locally incomplete algebra A; F we have F Pol for some R. This system consists of (i) certain natural relations whose polymorphs are best possible in the sense that they are co-atoms in the lattice of locally closed incomplete algebras, (ii) five types of binary relations, (iii) one type of ternary relations and (iv) at least ternary totally reflexive and symmetric relations that are not locally central.Presented by K. A. Baker.     

A decision theoretical characterization of weak ergodicity

Summary The relation between the ergodic coefficient and deficiency relative to the least informative experiment is investigated. The result is applied to nonhomogeneous Markov chains (NMC's). Our main result can be described as follows: Given an NMC, define the experiments _n ^(j) for n1 consisting in observing the (n+j)-th state of the chain, the j-th state being the unknown parameter. Then the chain is weakly ergodic if and only if for any j, _n ^(j) converges as n (with respect to deficiencies) to the least informative experiment. It is finally shown that in the homogeneous case, the rate of convergence is always exponential.     

Asymptotic behaviour of Gaussian random fields

Summary Let X={X(t), t ^N} be a centred Gaussian random field with covariance X(t)X(s)=r(t–s) continuous on ^N×^N and r(0)=1. Let (t,s)=((X(t)–X(s)) ²)^1/2; (t,s) is a pseudometric on ^N. Assume X is -separable. Let D ₁ be the unit cube in ^N and for 0<k, D _k= {x^N: k ^–1 xD₁}, Z(k)=sup{X(t),tD _k}. If X is sample continuous and ¦r(t)¦ =o(1/log¦t¦) as ¦t¦8 then Z(k)-(2Nlogk) ^1/20 as k a.s.     

On convergence of types and processes in Euclidean space

Let be an Euclidean space; Y _n , Z, U random vectors in ; h _n , g _n affine transformations and let þ be a subgroup of the group G of all the in vertible affine transformations, closed relative to G. Suppose that g_n and where Z is nonsingular. The behaviour of _n = h _n g _n ^–1 as n is discussed first. The results are used then to prove that if for all t(0, ), where h _n þ and Z ₁ is nonsingular and nonsymmetric with respect to þ then H, for all t(0,) and is a continuous homomorphism of the multiplicative group of (0, ) into þ. The explicit forms of the possible are shown.