On the degree of points and lines in a restricted linear space

In this paper we show that, with the exception of a few easily characterized linear spaces, all restricted linear spaces of square order n have the maximal degree of the lines equal to n + 1, the degree of every point at least n + 1, and further we show that p ? n² + n + 1 ? q, where p is the number of points and q the number of lines.     

Weakly K-analytic spaces and the three-space property for analyticity

Let (E,E^′) be a dual pair of vector spaces. The paper studies general conditions which allow to lift analyticity (or K-analyticity) from the weak topology σ(E,E^′) to stronger ones in the frame of (E,E^′). First we show that the Mackey dual of a space Cp(X) is analytic iff the space X is countable. This yields that for uncountable analytic spaces X the Mackey dual of Cp(X) is weakly analytic but not analytic. We show that the Mackey dual E of an (LF)-space of a sequence of reflexive separable Fréchet spaces with the Heinrich density condition is analytic, i.e. E is a continuous image of the Polish space NN. This extends a result of Valdivia. We show also that weakly quasi-Suslin locally convex Baire spaces are metrizable and complete (this extends a result of De Wilde and Sunyach). We provide however a large class of weakly analytic but not analytic metrizable separable Baire topological vector spaces (not locally convex!). This will be used to prove that analyticity is not a three-space property but we show that a metrizable topological vector space E is analytic if E contains a complete locally convex analytic subspace F such that the quotient E/F is analytic. Several questions, remarks and examples are included.     

Planar and affine spaces

In this note, we characterize finite three-dimensional affine spaces as the only linear spaces endowed with set Ω of proper subspaces having the properties (1) every line contains a constant number of points, say n, with n>2; (2) every triple of noncollinear points is contained in a unique member of Ω; (3) disjoint or coincide is an equivalence relation in Ω with the additional property that every equivalence class covers all points. We also take a look at the case n=2 (in which case we have a complete graph endowed with a set Ω of proper complete subgraphs) and classify these objects: besides the affine 3-space of order 2, two small additional examples turn up. Furthermore, we generalize our result in the case of dimension greater than three to obtain a characterization of all finite affine spaces of dimension at least 3 with lines of size at least 3.     

A Characterization of Certain Families of Strong Parapolar Spaces

A broad class of strong parapolar spaces is characterized by the properties of having all symplecta of rank at least 3, having finite singular rank, and possessing a geometric property (WH) concerning 6-circuits which are embedded in so as to contain a distance-3 pair of antipodal points. The geometries in question belong to a class F of strong parapolar spaces which entered the scene as a class for which the existence of Veldkamp lines could be proved. The paper attempts to transcend the characterization theorem, by giving a natural context in which one might expect to encounter the basic hypothesis of the Cohen–Cooperstein characterization theorem – namely a gap in the spectrum of projective ranks of subspaces x S, where x is a point not in the symplecton S. The theorem of the previous paragraph is then seen simply as an application of the principles advanced in this one.     

On the reconstruction of the degree sequence

Harary's edge reconstruction conjecture states that a graph G=(V,E) with at least four edges is uniquely determined by the multiset of its edge-deleted subgraphs, i.e. the graphs of the form G−e for e∈E. It is well-known that this multiset uniquely determines the degree sequence of a graph with at least four edges. In this note we generalize this result by showing that the degree sequence of a graph with at least four edges is uniquely determined by the set of the degree sequences of its edge-deleted subgraphs with one well-described class of exceptions. Moreover, the multiset of the degree sequences of the edge-deleted subgraphs always allows one to reconstruct the degree sequence of the graph.     

A de Bruijn-Erdős theorem for 1–2 metric spaces

A special case of a combinatorial theorem of De Bruijn and Erd?s asserts that every noncollinear set of n points in the plane determines at least n distinct lines. Chen and Chvátal suggested a possible generalization of this assertion in metric spaces with appropriately defined lines. We prove this generalization in all metric spaces where each nonzero distance equals 1 or 2.     

Isometries between spaces of homogeneous polynomials

We derive Banach-Stone theorems for spaces of homogeneous polynomials. We show that every isometric isomorphism between the spaces of homogeneous approximable polynomials on real Banach spaces E and F is induced by an isometric isomorphism of E^′ onto F^′. With an additional geometric condition we obtain the analogous result in the complex case. Isometries between spaces of homogeneous integral polynomials and between the spaces of all n-homogeneous polynomials are also investigated.     

Entire Functions on Banach Spaces with a Separable Dual

Let E and F be complex Banach spaces. We show that if E has a separable dual, then every holomorphic function from E into F which is bounded on weakly compact sets is bounded on bounded sets.     

Extending bounded type holomorphic mappings on a Banach space

We show that a Riemann domain Ω over a symmetrically regular Banach space E admits holomorphic extension to pseudo-convex domains over E″ with respect to two natural spaces of holomorphic functions of bounded type on Ω.     

Embedding a Latin square with transversal into a projective space

A Latin square of side n defines in a natural way a finite geometry on 3n points, with three lines of size n and n² lines of size 3. A Latin square of side n with a transversal similarly defines a finite geometry on 3n+1 points, with three lines of size n, n²−n lines of size 3, and n concurrent lines of size 4. A collection of k mutually orthogonal Latin squares defines a geometry on kn points, with k lines of size n and n² lines of size k. Extending the work of Bruen and Colbourn [A.A. Bruen, C.J. Colbourn, Transversal designs in classical planes and spaces, J. Combin. Theory Ser. A 92 (2000) 88-94], we characterise embeddings of these finite geometries into projective spaces over skew fields.     

Separable anisotropic differential operators in weighted abstract spaces and applications

In this paper operator-valued multiplier theorems in Banach-valued weighted Lp spaces are studied. Also weighted Sobolev-Lions type spaces are discussed when E₀, E are two Banach spaces and E₀ is continuously and densely embedded on E. Several conditions are found that ensure the continuity of the embedding operators that are optimally regular in these spaces in terms of interpolations of E₀. These results permit us to show the separability of the anisotropic differential operators in an E-valued weighted Lp space. By using these results the maximal regularity of infinite systems of quasi elliptic partial differential equations are established.     

Enlargements of positive sets

In this paper we introduce the notion of enlargement of a positive set in SSD spaces. To a maximally positive set A we associate a family of enlargements E(A) and characterize the smallest and biggest element in this family with respect to the inclusion relation. We also emphasize the existence of a bijection between the subfamily of closed enlargements of E(A) and the family of so-called representative functions of A. We show that the extremal elements of the latter family are two functions recently introduced and studied by Stephen Simons. In this way we extend to SSD spaces some former results given for monotone and maximally monotone sets in Banach spaces.     

A one-dimensional embedding complex

We give the first explicit computations of rational homotopy groups of spaces of “long knots” in Euclidean spaces. We define a spectral sequence which converges to these rational homotopy groups whose E¹ term is defined in terms of familiar Lie algebras. For odd k we establish a vanishing line for this spectral sequence, show the Euler characteristic of the rows of this E¹ term is zero, and make calculations of E² in a finite range.     

Minimal linear spaces

A decent linear space DLS(k) is a linear space with minimal line size at least three and with maximal line size exactly k. Denote by v_k (resp. b_k) the minimum number of points (resp. lines) in a DLS(k). We determine the numbers v_k and b_k for all k and prove that each DLS(k) with b_k lines has v_k points. Thus the DLS(k)'s with b_k lines are the minimal linear spaces.     

Polarized non-abelian representations of slim near-polar spaces

In (Bull Belg Math Soc Simon Stevin 4:299–316, 1997), Shult introduced a class of parapolar spaces, the so-called near-polar spaces. We introduce here the notion of a polarized non-abelian representation of a slim near-polar space, that is, a near-polar space in which every line is incident with precisely three points. For such a polarized non-abelian representation, we study the structure of the corresponding representation group, enabling us to generalize several of the results obtained in Sahoo and Sastry (J Algebraic Comb 29:195–213, 2009) for non-abelian representations of slim dense near hexagons. We show that with every polarized non-abelian representation of a slim near-polar space, there is an associated polarized projective embedding.     

On the weak topology of Banach spaces over non-archimedean fields

It is known that within metric spaces analyticity and K-analyticity are equivalent concepts. It is known also that non-separable weakly compactly generated (shortly WCG) Banach spaces over R or C provide concrete examples of weakly K-analytic spaces which are not weakly analytic. We study the case which totally differs from the above one. A general theorem is provided which shows that a Banach space E over a locally compact non-archimedean non-trivially valued field is weakly Lindelöf iff E is separable iff E is WCG iff E is weakly web-compact (in the sense of Orihuela). This provides a non-archimedean version of a remarkable Amir-Lindenstrauss theorem.     

Orlicz-Sobolev extensions and measure density condition

We study the extension properties of Orlicz-Sobolev functions both in Euclidean spaces and in metric measure spaces equipped with a doubling measure. We show that a set E⊂R satisfying a measure density condition admits a bounded linear extension operator from the trace space W1,Ψ(Rn)E_| to W1,Ψ(Rn). Then we show that a domain, in which the Sobolev embedding theorem or a Poincaré-type inequality holds, satisfies the measure density condition. It follows that the existence of a bounded, possibly non-linear extension operator or even the surjectivity of the trace operator implies the measure density condition and hence the existence of a bounded linear extension operator.     

Strictly singular and strictly co-singular inclusions between symmetric sequence spaces

Strict singularity and strict co-singularity of inclusions between symmetric sequence spaces are studied. Suitable conditions are provided involving the associated fundamental functions. The special case of Lorentz and Marcinkiewicz spaces is characterized. It is also proved that if E?F are symmetric sequence spaces with E≠?₁ and F≠c₀ and ?_∞ then there exist a intermediate symmetric sequence space G such that E?G?F and both inclusions are not strictly singular. As a consequence new characterizations of the spaces c₀ and ?₁ inside the class of all symmetric sequence spaces are given.     

On the set of orthogonally additive functions with orthogonally additive second iterate

Let E be a real inner product space of dimension at least 2. We show that both the set of all orthogonally additive functions mapping E into E having orthogonally additive second iterate and its complement are dense in the space of all orthogonally additive functions from E into E with the Tychonoff topology.