On the space of all regular operators from C(K) into C(K)

It is known that L^r(E, C(K)), the space of all regular operators from E into C(K), is a Riesz space for all Riesz spaces E if and only if K is Stonian. We prove that this statement holds if E is replaced by C(K), where K is a compact space, the cardinal number of which satisfies a certain condition.     

A problem ? multi-colorings of graphs

In this paper we consider colorings of the edges of the complete graph K_m with n colors such that the edges of any color form a non-trivial complete subgraph of K_m. We allow an edge of K_m to have more than one color. Such a coloring will be called r-admissible if no cycle of length r has a different color for each edge. Let E (m, n, r) be the maximum number of incidences of colors and edges, taken over all r-admissible colorings of K_m with n colors. Then for r = 3,4, and 5 we give an upper bound for E (m, n, r); as well as a lower bound for E (m, n, r) for all r. An analogue to a problem of Zarankiewicz concerning 0, 1-matrices is mentioned.     

A common property of R(E, F) and B(ℝ n , ℝ m ) and a new method for seeking a path to connect two operators

Given two Banach spaces E, F, let B(E, F) be the set of all bounded linear operators from E into F, and R(E, F) the set of all operators in B(E, F) with finite rank. It is well-known that B(? ⁿ ) is a Banach space as well as an algebra, while B(? ⁿ , ? ^m ) for m ≠ n, is a Banach space but not an algebra; meanwhile, it is clear that R(E, F) is neither a Banach space nor an algebra. However, in this paper, it is proved that all of them have a common property in geometry and topology, i.e., they are all a union of mutual disjoint path-connected and smooth submanifolds (or hypersurfaces). Let Σ _r be the set of all operators of finite rank r in B(E, F) (or B(? ⁿ , ? ^m )). In fact, we have that 1) suppose Σ _r ∈ B(? ⁿ , ? ^m ), and then Σ _r is a smooth and path-connected submanifold of B(? ⁿ , ? ^m ) and dimΣ _r = (n + m)r ? r ², for each r ∈ [0, min{n,m}; if m ≠ n, the same conclusion for Σ _r and its dimension is valid for each r ∈ [0, min{n, m}]; 2) suppose Σ _r ∈ B(E, F), and dimF = ∞, and then Σ _r is a smooth and path-connected submanifold of B(E, F) with the tangent space T _A Σ _r = {B ∈ B(E, F): BN(A) ? R(A)} at each A ∈ Σ _r for 0 ? r ? ∞. The routine methods for seeking a path to connect two operators can hardly apply here. A new method and some fundamental theorems are introduced in this paper, which is development of elementary transformation of matrices in B(? ⁿ ), and more adapted and simple than the elementary transformation method. In addition to tensor analysis and application of Thom’s famous result for transversility, these will benefit the study of infinite geometry.     

Démonstration de la conjecture de la masse positive

On a compact Riemannian manifold (Vn,g)(n>2), when the conformal Laplacian L is invertible, we show, under necessary hypotheses, that if the Green function GL of L is of the form (here r=d(P,Q)) GL(P,Q)=1/(n−2)ωn−1rn−2+H(P,Q) with H(P,Q) bounded on V, then

     

Chain conditions and weak topologies

We study conditions on Banach spaces close to separability. We say that a topological space is pcc if every point-finite family of open subsets of the space is countable. For a Banach space E, we say that E is weakly pcc if E, equipped with the weak topology, is pcc, and we also consider a weaker property: we say that E is half-pcc if every point-finite family consisting of half-spaces of E is countable. We show that E is half-pcc if, and only if, every bounded linear map E→c₀(ω₁) has separable range. We exhibit a variety of mild conditions which imply separability of a half-pcc Banach space. For a Banach space C(K), we also consider the pcc-property of the topology of pointwise convergence, and we note that the space Cp(K) may be pcc even when C(K) fails to be weakly pcc. We note that this does not happen when K is scattered, and we provide the following example:

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There exists a non-metrizable scattered compact Hausdorff space K with C(K) weakly pcc.

     

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.     

Recent Developments In Harmonic Approximation,With Applications

A theorem of J.L. Walsh (1929) says that if E is a compact subset of Rⁿ with connected complement and if u is harmonic on a neighbourhood of E, then u can be uniformly approximated on E by functions harmonic on the whole of Rⁿ. In Part I of this article we survey some generalizations of Walsh’s theorem from the period 1980–94. In Part II we discuss applications of Walsh’s theorem and its generalizations to four diverse topics: universal harmonic functions, the Radon transform, the maximum principle, and the Dirichlet problem.     

Sufficient conditions for existence of extrema in linear programming

It is shown that, on a closed convex subset X of a real Hausdorff locally convex space E, a continuous linear functional x′ on E has an extremum at an extreme point of X, provided X contains no line and X ∩ (x′)^?1 (λ₀) is non-empty and weakly compact for some real λ₀. It is also shown that any weakly locally compact closed convex subset of E that contains no line is the sum of its asymptotic cone and the closed convex hull of its extreme points.     

Non-conflicting ordering cones and vector optimization in inductive limits

Let (E, ξ)= ind (E_n, ξ_n) be an inductive limit of a sequence (E_n, ξ_n)_{n∈ N} of locally convex spaces and let every step (E_n, ξ_n) be endowed with a partial order by a pointed convex (solid) cone S_n. In the framework of inductive limits of partially ordered locally convex spaces, the notions of lastingly efficient points, lastingly weakly efficient points and lastingly globally properly efficient points are introduced. For several ordering cones, the notion of non-conflict is introduced. Under the requirement that the sequence (S_n)_{n∈ N} of ordering cones is non-conflicting, an existence theorem on lastingly weakly efficient points is presented. From this, an existence theorem on lastingly globally properly efficient points is deduced.     

Cross-intersecting families of permutations

For positive integers r and n with r?n, let Pr,n be the family of all sets {(1,y₁),(2,y₂),…,(r,yr)} such that y₁,y₂,…,yr are distinct elements of [n]={1,2,…,n}. Pn,n describes permutations of [n]. For r<n, Pr,n describes permutations of r-element subsets of [n]. Families A₁,A₂,…,Ak of sets are said to be cross-intersecting if, for any distinct i and j in [k], any set in Ai intersects any set in Aj. For any r, n and k?2, we determine the cases in which the sum of sizes of cross-intersecting sub-families A₁,A₂,…,Ak of Pr,n is a maximum, hence solving a recent conjecture (suggested by the author).     

Large semisimple groups on 16-dimensional compact projective planes are almost simple

The paper deals with the so-called Salzmann program aiming to classify special geometries according to their automorphism groups. Here, topological connected compact projective planes are considered. If finite-dimensional, such planes are of dimension 2, 4, 8, or 16. The classical example of a 16-dimensional, compact projective plane is the projective plane over the octonions with 78-dimensional automorphism group E₆(?26). A 16-dimensional, compact projective plane P admitting an automorphism group of dimension 41 or more is classical, [18] 87.5 and 87.7. For the special case of a semisimple group Δ acting on P the same result can be obtained if dim δ ≧ 37, see [16]. Our aim is to lower this bound. We show: if Δ is semisimple and dim δ ≧ 29, then P is either classical or a Moufang-Hughes plane or Δ is isomorphic to Spin₉ (?, r), r ∈ {0, 1 }. The underlying paper contains the first part of the proof showing that Δ is in fact almost simple.     

An example of a nuclear space in infinite dimensional holomorphy

LetU be an open subset of a complex locally convex spaceE, andH(U) the space of holomorphic functions fromU toC. If the dualE′ ofE is nuclear with respect to the topology generated by the absolutely convex compact subsets ofE, then it is shown thatH(U) endowed with the compact open topology is a nuclear space. In particular, ifE is the strong dual of a Fréchet nuclear space, thenH(U) is a Fréchet nuclear space.     

Topological obstructions for vertex numbers of Minkowski sums

We show that for polytopes P₁,P₂,…,Pr⊂Rd, each having ni?d+1 vertices, the Minkowski sum P₁+P₂+?+Pr cannot achieve the maximum of i_∏ni vertices if r?d. This complements a recent result of Fukuda and Weibel (2006), who show that this is possible for up to d−1 summands. The result is obtained by combining methods from discrete geometry (Gale transforms) and topological combinatorics (van Kampen-type obstructions).     

Ideals of homogeneous polynomials and weakly compact approximation property in Banach spaces

We show that a Banach space E has the weakly compact approximation property if and only if each continuous Banach-valued polynomial on E can be uniformly approximated on compact sets by homogeneous polynomials which are members of the ideal of homogeneous polynomials generated by weakly compact linear operators. An analogous result is established also for the compact approximation property.     

On double bound graphs and forbidden subposets

For a poset P=(X,≤_P), the double bound graph (DB-graph) of P is the graph DB(P)=(X,E_DB(P)), where xy∈E_DB(P) if and only if x≠y and there exist n,m∈X such that n≤_Px,y≤_Pm. We obtain that for a subposet Q of a poset P,Q is an (n, m)-subposet of P if and only if DB(Q) is an induced subgraph DB(P). Using this result, we show some characterizations of split double bound graphs, threshold double bound graphs and difference double bound graphs in terms of (n, m)-subposets and double canonical posets.     

A class of antimagic join graphs

A labeling f of a graph G is a bijection from its edge set E(G) to the set {1, 2, . . . , |E(G)|}, which is antimagic if for any distinct vertices x and y, the sum of the labels on edges incident to x is different from the sum of the labels on edges incident to y. A graph G is antimagic if G has an f which is antimagic. Hartsfield and Ringel conjectured in 1990 that every connected graph other than K 2 is antimagic. In this paper, we show that if G 1 is an n-vertex graph with minimum degree at least r, and G 2 is an m-vertex graph with maximum degree at most 2r-1 (m ≥ n), then G1 ∨ G2 is antimagic.     

Quantum effects, sequential independence and majorization

A quantum effect is a positive Hilbert space contraction operator. If {E_i}, 1?i?n, are n quantum effects (defined on some Hilbert space H), then their sequential product is the operator . It is proved that the quantum effects {E_i}, 1?i?n, are sequentially independent if and only if for every permutation r₁r₂…r_n of the set S_n={1,2,…,n}. The sequential independence of the effects E_i, 1?i?n, implies E_noE_n-1o…oE_j+1oE_jo…oE₁=(E_noE_n-1o…E_j+1)oE_jo…oE₁ for every 1?j?n. It is proved that if there exists an effect E_j, 1?j?n, such that E_j?(E_noE_n-1o…E_j+1)oE_jo…oE₁, then the effects {E_i} are sequentially independent and satisfy .     

An Ore-type condition for arbitrarily vertex decomposable graphs

Let G be a graph of order n and r, 1≤r≤n, a fixed integer. G is said to be r-vertex decomposable if for each sequence (n₁,…,n_r) of positive integers such that n₁+?+n_r=n there exists a partition (V₁,…,V_r) of the vertex set of G such that for each i∈{1,…,r}, V_i induces a connected subgraph of G on n_i vertices. G is called arbitrarily vertex decomposable if it is r-vertex decomposable for each r∈{1,…,n}.In this paper we show that if G is a connected graph on n vertices with the independence number at most ⌈n/2⌉ and such that the degree sum of any pair of non-adjacent vertices is at least n−3, then G is arbitrarily vertex decomposable or isomorphic to one of two exceptional graphs. We also exhibit the integers r for which the graphs verifying the above degree-sum condition are not r-vertex decomposable.     

Invariant convex sets in polar representations

We study a compact invariant convex set E in a polar representation of a compact Lie group. Polar representations are given by the adjoint action of K on p, where K is a maximal compact subgroup of a real semisimple Lie group G with Lie algebra g = k ⊕ p. If a ? p is a maximal abelian subalgebra, then P = E ∩ a is a convex set in a. We prove that up to conjugacy the face structure of E is completely determined by that of P and that a face of E is exposed if and only if the corresponding face of P is exposed. We apply these results to the convex hull of the image of a restricted1 momentum map.