A bilinear version of Orlicz-Pettis theorem

Given three Banach spaces X, Y and Z and a bounded bilinear map , a sequence x=n(xn)⊆X is called B-absolutely summable if is finite for any y∈Y. Connections of this space with are presented. A sequence x=n(xn)⊆X is called B-unconditionally summable if is finite for any y∈Y and z^∗∈Z^∗ and for any M⊆N there exists xM∈X for which _∑n∈M〈B(xn,y),z^∗〉=〈B(xM,y),z^∗〉 for all y∈Y and z^∗∈Z^∗. A bilinear version of Orlicz-Pettis theorem is given in this setting and some applications are presented.     

Indecomposability graph and critical vertices of an indecomposable graph

Given a directed graph G=(V,A), the induced subgraph of G by a subset X of V is denoted by G[X]. A subset X of V is an interval of G provided that for a,b∈X and x∈V?X, (a,x)∈A if and only if (b,x)∈A, and similarly for (x,a) and (x,b). For instance, 0?, V and {x}, x∈V, are intervals of G, called trivial intervals. A directed graph is indecomposable if all its intervals are trivial, otherwise it is decomposable. Given an indecomposable directed graph G=(V,A), a vertex x of G is critical if G[V?{x}] is decomposable. An indecomposable directed graph is critical when all its vertices are critical. With each indecomposable directed graph G=(V,A) is associated its indecomposability directed graph defined on V by: given x≠y∈V, (x,y) is an arc of if G[V?{x,y}] is indecomposable. All the results follow from the study of the connected components of the indecomposability directed graph. First, we prove: if G is an indecomposable directed graph, which admits at least two non critical vertices, then there is x∈V such that G[V?{x}] is indecomposable and non critical. Second, we characterize the indecomposable directed graphs G which have a unique non critical vertex x and such that G[V?{x}] is critical. Third, we propose a new approach to characterize the critical directed graphs.     

Parametric bing and Krasinkiewicz maps

It is shown that if is a perfect map between metrizable spaces and Y is a C-space, then the function space C(X,I) with the source limitation topology contains a dense Gδ-subset of maps g such that every restriction map gy=g|f⁻¹(y), y∈Y, satisfies the following condition: all fibers of gy are hereditarily indecomposable and any continuum in f⁻¹(y) either contains a component of a fiber of gy or is contained in a fiber of gy.     

Locally compact abelian groups admitting non-trivial quasi-convex null sequences

In this paper, we show that, for every locally compact abelian group G, the following statements are equivalent:

(i)

G contains no sequence such that {0}∪{±x_n∣n∈N} is infinite and quasi-convex in G, and x_n?0;

(ii)

one of the subgroups {g∈G∣2g=0} or {g∈G∣3g=0} is open in G;

(iii)

G contains an open compact subgroup of the form or for some cardinal κ.

     

Strongly smooth paths of idempotents

It is shown that a curve q(t), t∈I (0∈I) of idempotent operators on a Banach space X, which verifies that for each ξ∈X, the map t?q(t)ξ∈X is continuously differentiable, can be lifted by means of a regular curve Gt, of invertible operators in X:

     

Modular metric spaces, I: Basic concepts

The notion of a modular is introduced as follows. A (metric) modular on a set X is a function w:(0,∞)×X×X→[0,∞] satisfying, for all x,y,z∈X, the following three properties: x=y if and only if w(λ,x,y)=0 for all λ>0; w(λ,x,y)=w(λ,y,x) for all λ>0; w(λ+μ,x,y)≤w(λ,x,z)+w(μ,y,z) for all λ,μ>0. We show that, given x₀∈X, the set X_w={x∈X:lim_λ→∞w(λ,x,x₀)=0} is a metric space with metric , called a modular space. The modular w is said to be convex if (λ,x,y)?λw(λ,x,y) is also a modular on X. In this case X_w coincides with the set of all x∈X such that w(λ,x,x₀)<∞ for some λ=λ(x)>0 and is metrizable by . Moreover, if or , then ; otherwise, the reverse inequalities hold. We develop the theory of metric spaces, generated by modulars, and extend the results by H. Nakano, J. Musielak, W. Orlicz, Ph. Turpin and others for modulars on linear spaces.     

Addition theorems and representations of topological semigroups

Let a representation T of semigroup G on linear space X be given. We call x∈Xa finite vector if its orbit T(G) is contained in a finite-dimensional subspace. In this paper some statements on finite vectors will be proved and applied to functional equations

     

Soluble groups with their centralizer factor groups of bounded rank

For a group class X, a group G is said to be a CX-group if the factor group G/C_G(g^G)∈X for all g∈G, where C_G(g^G) is the centralizer in G of the normal closure of g in G. For the class F_f of groups of finite order less than or equal to f, a classical result of B.H. Neumann [Groups with finite classes of conjugate elements, Proc. London Math. Soc. 1 (1951) 178-187] states that if G∈CF_f, the commutator group G^′ belongs to F_f^′ for some f^′ depending only on f. We prove that a similar result holds for the class , the class of soluble groups of derived length at most d which have Prüfer rank at most r. Namely, if , then for some r^′ depending only on r. Moreover, if , then for some r^′ and f^′ depending only on r,d and f.     

The Mahler measure of parametrizable polynomials

Our aim is to explain instances in which the value of the logarithmic Mahler measure m(P) of a polynomial P∈Z[x,y] can be written in an unexpectedly neat manner. To this end we examine polynomials defining rational curves, which allows their zero-locus to be parametrized via x=f(t), y=g(t) for f,g∈C(t). As an illustration of this phenomenon, we prove the equality

     

Factors and vertex-deleted subgraphs

A relationship is considered between an f-factor of a graph and that of its vertex-deleted subgraphs. Katerinis [Some results on the existence of 2n-factors in terms of vertex-deleted subgraphs, Ars Combin. 16 (1983) 271-277] proved that for even integer k, if G-x has a k-factor for each x∈V(G), then G has a k-factor. Enomoto and Tokuda [Complete-factors and f-factors, Discrete Math. 220 (2000) 239-242] generalized Katerinis’ result to f-factors, and proved that if G-x has an f-factor for each x∈V(G), then G has an f-factor for an integer-valued function f defined on V(G) with even. In this paper, we consider a similar problem to that of Enomoto and Tokuda, where for several vertices x we do not have to know whether G-x has an f-factor. Let G be a graph, X be a set of vertices, and let f be an integer-valued function defined on V(G) with even, |V(G)-X|?2. We prove that if and if G-x has an f-factor for each x∈V(G)-X, then G has an f-factor. Moreover, if G excludes an isolated vertex, then we can replace the condition with . Furthermore the condition will be when |X|=1.     

Sum of graphs of continuous functions and boundedness of additive operators

Assume that and are uniformly continuous functions, where D₁,D₂⊂X are nonempty open and arc-connected subsets of a real normed space X. We prove that then either f and g are affine functions, that is f(x)=x^∗(x)+a and g(x)=x^∗(x)+b with some x^∗∈X^∗ and a,b∈R or the algebraic sum of graphs of functions f and g has a nonempty interior in a product space X×R treated as a normed space with a norm .     

On some geometric parameters in Banach spaces

Let X be a normed linear space and be the unit sphere of X. Let , , and J(X)=sup{‖x+y‖∧‖x−y‖}, x and y∈S(X) be the modulus of convexity, the modulus of smoothness, and the modulus of squareness of X, respectively. Let . In this paper we proved some sufficient conditions on δ(?), ρX(?), J(X), E(X), and , where the supremum is taken over all the weakly null sequence xn in X and all the elements x of X for the uniform normal structure.     

A note on the class of paracompact spaces whose product with every paracompact space is paracompact

We prove that if X is a paracompact space which has a neighborhood assignment x→Hx such that for each y∈X the closure of the set is compact then the products T×X, for every paracompact space T, and Xω are paracompact. The first result answers a problem of H. Junnila.     

On the reconstruction problem for factorizable homeomorphism groups and foliated manifolds

For a group G of homeomorphisms of a regular topological space X and a subset U⊆X, set . We say that G is a factorizable group of homeomorphisms, if for every open cover U of X, _?U∈UG generates G.

Theorem I. Let G, H be factorizable groups of homeomorphisms of X and Y respectively, and suppose that G, H do not have fixed points. Let φ be an isomorphism between G and H. Then there is a homeomorphism τ between X and Y such thatφ(g)=τ○g○τ−1for everyg∈G.     

Solutions of polynomial Pell's equation

Let D=F²+2G be a monic quartic polynomial in Z[x], where . Then for F/G∈Q[x], a necessary and sufficient condition for the solution of the polynomial Pell's equation X²−DY²=1 in Z[x] has been shown. Also, the polynomial Pell's equation X²−DY²=1 has nontrivial solutions X,Y∈Q[x] if and only if the values of period of the continued fraction of are 2, 4, 6, 8, 10, 14, 18, and 22 has been shown. In this paper, for the period of the continued fraction of is 4, we show that the polynomial Pell's equation has no nontrivial solutions X,Y∈Z[x].     

On solvability of two-dimensional Lp-Minkowski problem

Given p≠0 and a positive continuous function g, with g(x+T)=g(x), for some 0<T<1 and all real x, it is shown that for suitable choice of a constant C>0 the functional has a minimizer in the class of positive functions u∈C¹(R) for which u(x+T)=u(x) for all x∈R. This minimizer is used to prove the existence of a positive periodic solution y∈C²(R) of two-dimensional L_p-Minkowski problem y^1−p(x)(y″(x)+y(x))=g(x), where p∉{0,2}.