An Arithmetic Reduction of Finite Rank 3 Geometries with Linear Spaces as Plane Residues and with Dual Linear Spaces as Point Residues

Let be a rank three incidence geometry of points, lines and planes whose planes are linear spaces and whose point residues are dual linear spaces (notice that we do not require anything on the line residues). We assume that the residual linear spaces of belong to a natural class of finite linear spaces, namely those linear spaces whose full automorphism group acts flag-transitively and whose orders are polynomial functions of some prime number. This class consists of six families of linear spaces. In the amalgamation of two such linear spaces imposes an equality on their orders leading, in particular, to a series of diophantine equations, the solutions of which provide a reduction theorem on the possible amalgams of linear spaces that can occur in .We prove that one of the following holds (up to a permutation of the words point and plane):A) the planes of and the dual of the point residues belong to the same family and have the same orders,B) the diagram of is in one of six families,C) the diagram of belongs to a list of seven sporadic cases.Finally, we consider the particular case where the line residues of are generalized digons.     

Finite semi-affine planes and generalized projective spaces

We classify the finite linear spaces for which there is an integer d₁ such that for any ordered pair (L,L) of disjoint lines and for any point x outside L and L, there are exactly d₁ lines through x intersecting L but not L. Unless all lines have size 2, such a space is necessarily a semi-affine plane or a generalized projective 3-space.     

A short proof of Totten's classification of restricted linear spaces

A restricted linear space is one which satisfies (b–v)²v, where b is the number of lines and v is the number of points. In 1976, Totten classified all restricted linear spaces as being of essentially three types. In this paper we give a short, self-contained proof of this result. Our approach is greatly simplified by the use of techniques from linear algebra.This work was supported in part by National Science Foundation Grant MCS-8217596.     

On the connectedness of the scheme of multisecants to a projective curve

Here we study the connectedness of the scheme of multisecant linear spaces to a curveC P ⁿ . In particular we prove the connectedness of the scheme of trisecant lines of a smoothC P ³ with non-special hyperplane section.     

Linear homeomorphisms of spaces of continuous functions on long Sorgenfrey lines

We carry out a linear homeomorphic classification of the spaces of continuous functions on the long Sorgenfrey lines S_α, where a is an arbitrary ordinal. The spaces of continuous functions are endowed with the topology of pointwise convergence and denoted by C_p(S_α).     

Sequential convergence in topological vector spaces

For a given linear topology , on a vector spaceE, the finest linear topology having the same convergent sequences, and the finest linear topology onE having the same precompact sets, are investigated. Also, the sequentially bornological spaces and the sequentially barreled spaces are introduced and some of their properties are studied.     

Best coapproximation in normed linear spaces

We consider, in normed linear spaces, a kind of approximation by elements of linear subspaces, introduced byC. Franchetti andM. Furi [5], which we call best coapproximation. We obtain some results on characterization and existence of elements of best coapproximation in arbitrary normed linear spaces and in spaces of continuous functions. We give some characterizations of strict convexity in terms of best coapproximation and we study some properties of the setvalued operators of best coapproximation.Work performed partially under the auspices of the GNAFA (National Group for Functional Analysis and its Applications) of the CNR (National Research Council of Italy)     

The CMP and Banach Spaces with unique metric lines

This paper is concerned with characterizing the class of Banach Spaces with unique metric lines in the class of complete metric spaces with unique lines.The concept utilized to prove the major theorem is theConsistent Midpoint Property (CMP). We define a binary operation (+) in metric spaces with unique lines and show that, under suitable assumptions, the space is a Banach Space with + as the vector addition and a=d(a, ) for some fixed .     

Banach ideals of p-compact operators

The main theme of this paper is a study in some detail of Banach ideals of continuous linear operators between Banach spaces factoring compactly through l^p (1p<) or c_o, called p-compact and -compact operators respectively. Recently operators of these types have been studied in [4] within the framework of locally convex spaces which are dense subspaces of p-compact projective limits of Banach spaces. These ideals show close resemblance to the ideals of p-nuclear operators-for the case p= they coincide. Analogously to results of Grothendieck concerning continuous linear operators, we consider vector sequence spaces isometric isomorphic to certain spaces of compact linear operators. A representation theorem for p-compact operators is deduced and isometric properties of the ideal norm are treated. The paper also includes some remarks on unconditional convergence and related operator ideals and a representation for the complete -tensor product (1p<) is given.     

A Hille-Yosida theorem for Bi-continuous semigroups

In order to treat one-parameter semigroups of linear operators on Banach spaces which are not strongly continuous, we introduce the concept of bi-continuous semigroups defined on Banach spaces with an additional locally convex topology . On such spaces we define bi-continuous semigroups as semigroups consisting of bounded linear operators which are locally bi-equicontinuous for and such that the orbit maps are -continuous. We then apply the result to semigroups induced by flows on a metric space as studied by J. R. Dorroh and J. W. Neuberger.     

Adjungierte Funktoren in der Kategorie der p-Bornologischen Räume

In the category p b of p-convex vector spaces and linear maps preserving bounded sets a p-bornological topology will be introduced on the tensor product of two spaces, likewise on the spaces of morphisms Hom(E,F). Thus one gets a pair of adjoint functors from p to p , p being the category of p-bornological spaces and continuous linear maps, and the topologies being introduced will be characterized by extreme properties with respect to the adjoint transformations.

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Dieser Arbeit liegt ein Teil der Dissertation des Autors, Kiel 1967, zugrunde.     

The unirationality of quartics over nonclosed fields revisited

Examples of smooth fourth-degree hypersurfaces which are unirational over an algebraically nonclosed field $ \Bbbk $ and contain no straight lines defined over $ \Bbbk $ are given. A method for proving the unirationality of quartics is suggested, which, unlike other methods, does not use linear spaces contained in the quartics.     

Duality Construction of Moduli Spaces

The duality of moduli spaces of coherent sheaves on curves and surfaces gives a GIT free construction of these spaces. It is a suitable generalization for topics as: strange duality, jumping lines of vector bundles on projective space, and Faltings' construction of the moduli space of semistable vector bundles on a complex curve given in his article Stable G-bundles and projective connections. Using this duality we propose a construction for the moduli space of semistable sheaves on a surface.     

Projections in minimax algebra

An axiomatic theory of linear operators can be constructed for abstract spaces defined over (R, , ), that is over the (extended) real numbersR with the binary operationsx y = max (x,y) andx y = x + y. Many of the features of conventional linear operator theory can be reproduced in this theory, although the proof techniques are quite different. Specialisation of the theory to spaces ofn-tuples provides techniques for analysing a number of well-known operational research problems, whilst specialisation to function spaces provides a natural formal framework for certain familiar problems of approximation, optimisation and duality.     

Higher derivatives of mappings of locally convex spaces

We establish sufficient conditions for n-fold bounded differentiability (b-differentiability) of mappings of locally convex spaces and sufficient conditions for n-fold Hyers-Lang differentiability (HL-differentiability) of mappings of pseudotopological linear spaces. We describe a class of locally convex spaces on which there exist everywhere infinitely b-differentiable real functions which are not everywhere continuous (and so are not everywhere HL-differentiable). Our results show, in particular, that for a wide class of locally convex spaces a significant number of the known definitions of C-mappings fall into two classes of equivalent definitions.Translated from Matematicheskie Zametki, Vol. 22, No. 5, pp. 729–744 November, 1977.     

Decomposable symmetric mappings between infinite-dimensional spaces

Decomposable mappings from the space of symmetric k-fold tensors over E, , to the space of k-fold tensors over F, , are those linear operators which map nonzero decomposable elements to nonzero decomposable elements. We prove that any decomposable mapping is induced by an injective linear operator between the spaces on which the tensors are defined. Moreover, if the decomposable mapping belongs to a given operator ideal, then so does its inducing operator. This result allows us to classify injective linear operators between spaces of homogeneous approximable polynomials and between spaces of nuclear polynomials which map rank-1 polynomials to rank-1 polynomials.     

A note on compact operators on matrix domains

We establish some identities or estimates for the operator norms and Hausdorff measures of noncompactness of linear operators given by infinite matrices that map the matrix domains of triangles in arbitrary BK spaces with AK, or in the spaces of all convergent or bounded sequences, into the spaces of all null, convergent or bounded sequences, or of all absolutely convergent series. Furthermore, we apply these results to the characterizations of compact operators on the matrix domains of triangles in the classical sequence spaces, and on the sequence spaces studied in [I. Djolovi?, Compact operators on the spaces and , J. Math. Anal. Appl. 318 (2) (2006) 658-666; I. Djolovi?, On the space of bounded Euler difference sequences and some classes of compact operators, Appl. Math. Comput. 182 (2) (2006) 1803-1811].     

\epsilon -Henig proper efficiency of set-valued optimization problems in real ordered linear spaces

The aim of this paper is to investigate \(\epsilon \) -Henig proper efficiency of set-valued optimization problems in linear spaces. Firstly, a new notion of \(\epsilon \) -Henig properly efficient point is introduced in linear spaces. Secondly, scalarization theorems of set-valued optimization problems are established in the sense of \(\epsilon \) -Henig proper efficiency. Finally, under the assumption of generalized cone subconvexlikeness, Lagrange multiplier theorems are obtained. Our results generalize some known results in the literature from topological spaces to linear spaces.     

A remark on partial linear spaces of girth 5 with an application to strongly regular graphs

We derive a lower bound on the number of points of a partial linear space of girth 5. As an application, certain strongly regular graphs with=2 are ruled out by observing that the first subconstituents are partial linear spaces.