Über die Tonneliertheit von lokalkonvexen Tensorprodukten

It is known that the inductive tensor product of two barrelled spaces is barrelled and that the projective tensor product of two barrelled metrizable spaces or barrelled (DF)-spaces is barrelled. In this note it will be shown by a counterexample that the projective tensor product EF of two barrelled spaces E and F in general is not barrelled, even if E is (DF)-Montel-space and F (F)-Montel-space. Furthermore we show that the -tensor product of two (B)-spaces in general is not barrelled. It follows from the fact that an (F)-space E is nuclear if and only if the -tensor product E l ⁴ is barrelled.     

Categories equivalent to the category of rational H-spaces

The category of rationalH-spaces is shown to be equivalent to the category of commutative Hopf algebras over , the category of cocommutative Hopf algebras over , and the categoryL of graded Lie algebras over by the rational cohomology, homology, and homotopy functors, respectively. Several consequences of these equivalences are derived. It is also proved that the loop-space functor is an equivalence from the category of coformal rational spaces to . Dually, the category of rationalcoH-spaces is shown to be equivalent to the comonoid category ofL and to the category of cocommutative coalgebras over . The suspension functor is an equivalence from the category of formal, rational spaces to the category of 2-connected, rationalcoH-spaces.     

On representing sylvester-gallai designs

A misstated conjecture in [3] leads to an interesting (1, 3) representation of the 7-point projective plane inR ⁴ where points are represented by lines and planes by 3-spaces. The corrected form of the original conjecture will be negated if there is a (1, 3) representation of the 13-point projective plane inR ⁴ but that matter is not settled.     

Regular homomorphisms of generalized projective planes

A generalized projective plane is an incidence structure together with a relation distant on the set of points and also on the set of lines, such that any two distant points A,B (lines a,b) have a unique common line (A,B) (common point (a,b)) and three further axioms hold. Every commutative ring with 1 supplies a model. A homomorphism of into an incidence structure is called regular if the following condition and its dual are valid: A distant B and c IA,B implies c=(A,B). We shall prove the following two theorems. Let be a generalized projective plane satisfying a richness condition called (U). Let M I m. If and are regular homomorphisms of such that X = M X = M for each point X of the line m then A = B A = B for any two points A,B. If is a projective plane over a commutative ring such that (U) holds then the surjective regular homomorphisms of are induced by the ideals of the ring; in particular, the image of under a regular homomorphism is again a projective plane over a ring, and preserves distant.     

A generalization of Macaev's theorem to non-commutative LP-spaces

For 1P-spaces associated with a von Neumann algebra are shown to belong to the class UMD (that is, to possess the unconditionality property for martingale differences). With the aid of a recent result of the authors, which permits the classical Hilbert transform to be transferred to UMD spaces, a generalization of Macaev's theorem to non-commutative L^P-spaces is introduced. This generalization utilizes the Hilbert kernel in a central role, broadens the harmonic conjugation aspects of Macaev's theorem, and provides a universal bound depending only on p.     

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 “build-down” scheme for linear programming

We propose a build-down scheme for Karmarkar's algorithm and the simplex method for linear programming. The scheme starts with an optimal basis candidate set including all columns of the constraint matrix, then constructs a dual ellipsoid containing all optimal dual solutions. A pricing rule is developed for checking whether or not a dual hyperplane corresponding to a column intersects the containing ellipsoid. If the dual hyperplane has no intersection with the ellipsoid, its corresponding column will not appear in any of the optimal bases, and can be eliminated from. As these methods iterate, is eventually built-down to a set that contains only the optimal basic columns.     

A characterization of semimartingales on nuclear spaces

Summary This work is devoted to prove the following fact: Suppose that is a nuclear space whose dual is nuclear under the strong topology. IfX is a weakly adapted mapping with values in such that for any,X'() has a modification which is a semimartingale then there exists a unique projective system of Hubert space-valued semimartingales indexed by the Hilbert-Schmidt neighbourhood base of the dual space whose projective limit isX.In the last part we study in detail a semimartingale defined as the convolution of a distribution by a random Dirac measure whose support is determined by the trajectories of a real-valued semimartingale.     

Folgenräume als universelle Generatoren für Klassen lokalkonvexer Räume

Motivated by the known characterizations of equicontinuity in the dual of a Schwartz space, a nuclear space, or a strongly nuclear space,we introduce the concepts of a -sequence and of a ()-sequence in the dual of an arbitrary lcs [E,], and we investigate the corresponding topologies and ₍₎ on E of uniform convergence on these sequences. Here is a normal sequence space such that . Under favorable enough conditions on , including the nuclearity of its normal topology , [,] acts as a universal generator for those lcs [E,] which satisfy =. Under somewhat weaker assumptions on , [,₍₎] is a universal generator for the lcs [E,] with =₍₎. These results cover e.g. the cases of -nuclear spaces and of nuclear spaces known from the recent literature. As an application we show that every non-trivial ultrabornological lcs is representable as an inductive limit of isomorphic copies of [, ( , )], where is any nuclear power series space of infinite type with stable exponent sequence.     

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.     

Projektive G-Faserräume

Let > 0 be an integer. A projective -fibre space is formed by a covering of a projective geometry with -1 isomorphic geometries. The double elliptic space (Sphärischer Raum) is an example of a 2-fibre space. This note deals with projective -fibre spaces which are structured by multy valued orderfunctions (This notion was introduced by W. JUNKERS [2] for projective geometries) the range of which is a group G. If such an ordered -fibre space has the property ¦G¦=, it is called projective G-fibre space. It is proved that the desarguesian projective G-fibre spaces V are exactly those, which are induced by a vector space S over a field K (commutative or not) having a normal subgroup P K^*(·) with K^* P such that GK^*/P and SV^*/P. This theorem is a generalization of the well-known case P=K^*.     

On a Type of Mixed Norm Spaces

In this paper the mixed norm spaces L(B,p,q) and their duals are investigated. In the case p,q < it is proved that the dual of L(B,p,q) is L(B,p,q), where p-1 + p-1 = 1 and q-1 + q-1 = 1. For p = 2 and q = an isometric isomorphism is discussed between the mixed norm space L(B,2,) and L(B,2), the L-space of 2-valued functions. Here a measurability theorem is proved for 2-valued functions. The dual of an important subspace of L(B,2,) is characterized as a space of vector measures. Finally, as an application we show that if B is finitely generated then the dual of L(B,2,) is L(B,2,1).     

Bourbaki spaces of topological groups

A study is presented of the relationship between the topological and uniformity properties of a group G and the spaces (G), (G) of all nonempty closed subsets and closed subgroups of G. A base for the neighborhood system of a closed subset X of G is formed by the sets S(X, U)={Y Y XU, X YU}, where U ranges over all neighborhoods of the identity in G. Criteria are obtained for the space (G) and some of its subspaces to be totally bounded and locally totally bounded. Some classes of groups with compact spaces (G) are described. It is proved that the spaces (G), (G) are complete in the case of projective metrizable groups G.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 42, No. 4, pp. 542–549, April, 1990.     

The classification of compact punctally cohesive Desarguesian projective Klingenberg planes

It is shown that for a compact Desarguesian projective Klingenberg plane P with incidence structure P=#x2119;, , I and neighbour relation , where two distinct points always lie on some line, exactly one of the following holds: P is a non-discrete connected or totally disconnected ordinary projective plane with =id, P is a finite projective plane with =id, P is a finite projective Hjelmslev plane with id, or P is a non-discrete totally disconnected ordinary projective plane with id.Dedicated to H. Salzmann on his 60th birthdayThe author wishes to thank the Natural Sciences and Engineering Research Council of Canada (NSERC) for their financial assistance in the writing of this paper.     

A Second Descent Problem for Quadratic Forms

Let F be a field of characteristic different from 2. We discuss a new descent problem for quadratic forms, complementing the one studied by Kahn and Laghribi. More precisely, we conjecture that for any quadratic form q over F and any Im(W(F) W(F(q))), there exists a quadratic form W(F) such that dim 2 dim and _F(q), where F(q) is the function field of the projective quadric defined by q = 0. We prove this conjecture for dim 3 and any q, and get partial results for dim {4, 5,6}. We also give other related results.     

Projective Modules and Complete Intersections

Let A be a Noetherian ring of dimension n and P be a projective A module of rank n having trivial determinant. It is proved that if n is even and the image of a generic element g P^* is a complete intersection, then [P] = [Q A] in K₀(A) for some projective A module Q of rank n – 1. Further, it is proved that if n is odd, A is Cohen–Macaulay and [P] = [Q A] in K₀(A) for some projective A module Q of rank n – 1, then P has a unimodular element.     

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)     

Visibility and parallelotopes in projective space

It is shown that if is a finite set of points in an ordered d-dimensional projective space, d2 and if all the hyperplanes spanned by points of are considered, then every such a hyperplane is visible from at most 2^d distinct points of ; if a hyperplane bounds 2^d distinct residences, then it contains exactly d points of . These results have been proved previously for d=2 and d=3.This paper is a revised version of results presented at the Conference on Geometry held at the University of Haifa (Israel), March. 10–13 (1975), [6].     

Weakly compact operators onH ∞

We prove that a weakly compact operator fromH or any of its even duals into an arbitrary Banach space is uniformly convexifying. By using this, we establish three dicothomies: (1) every operator defined onH or any of its even duals either fixes a copy ofl or factors through a Banach space having the Banach-Saks property; (2) every quotient ofH or any of its even duals either contains a copy ofl or is super-reflexive; (3) every subspace ofL ¹/H ₀ ¹ or any of its even duals either contains a complemented copy ofl ¹ or is super-reflexive.