A note on the Radon-Nikodym theorem

In his book Transversal Theory (4) Mirsky introduces the problem of finding a necessary and sufficient condition in order that a finite family of sets have a unique independent transversal. The purpose of this note is to give one necessary and sufficient condition. More generally, we give a necessary and sufficient condition in order that a common basis of two rank-finite matroids be unique. We then specialize this result to give an answer to the above question.

In this article we give necessary and sufficient conditions in order a closed cone of a Banach space E to be isomorphic to the positive cone l 1 +(Γ) of an l 1(Γ) space. This problem has applications in the theory of Banach spaces as in the characterization of reflexive spaces and also on the problem of the embeddability of 1in Banach spaces.

In this paper we provide sufficient conditions in order that Uryson's equation has a nontrivial solution. Using a modified version of. Krasnoselskii's fixed point theorem on operators expanding a cone in a Banach space we get two new existence results for superlinear equations. The special structure of cones used provides bounds of the solution.

Let E be a real Banach space, C a closed, convex subset of E and f : [ 0 , 1 ] × E × E → E f:[0,1] \times E \times E \to E be continuous. Let u 0 , u 1 ∈ C {u_0},{u_1} \in C and consider the boundary value problem ( ∗ ) u = f ( t , u , u ′ ) , u ( 0 ) = u 0 , u ( 1 ) = u 1 . \begin{equation}\tag {$\ast $} u = f(t,u,u’),\quad u(0) = {u_0},\quad u(1) = {u_1}.\end{equation} We establish sufficient conditions in order that ( ∗ ) (\ast ) have a solution u : [ 0 , 1 ] → C u:[0,1] \to C .

the domain of T being the set of the functions u(x) EFx such that the first integral exists and is finite for almost all y, and the domain of T* being analogously defined. We are interested in the study of necessary and sufficient conditions in order that T restricted to LP(X) be a bounded transformation of LP(X) into Lq(Y) with bound ?co. Obviously such a necessary and sufficient condition is given by:

A map of sufficient conditions for the symmetric nonnegative inverse eigenvalue problem

Associated with a given suitable function, or a measure, on R \mathbb {R} , we introduce a correlation function so that the Wronskian of the Fourier transform of the function is the Fourier transform of the corresponding correlation function, and the same holds for the Laplace transform. We obtain two types of results. First, we show that Wronskians of the Fourier transform of a nonnegative function on R \mathbb {R} are positive definite functions and that the Wronskians of the Laplace transform of a nonnegative function on R + \mathbb {R}_+ are completely monotone functions. Then we establish necessary and sufficient conditions in order that a real entire function, defined as a Fourier transform of a positive kernel K K , belongs to the Laguerre–Pólya class, which answers an old question of Pólya. The characterization is given in terms of a density property of the correlation kernel related to K K , via classical results of Laguerre and Jensen and employing Wiener’s L 1 L^1 Tauberian theorem. As a consequence, we provide a necessary and sufficient condition for the Riemann hypothesis in terms of a density of the translations of the correlation function related to the Riemann ξ \xi -function.

We investigate the Pompeiu property for subsets of the real line, under no assumption of connectedness. In particular we focus our study on finite unions of bounded (disjoint) intervals, and we emphasize the different results corresponding to the cases where the function in question is supposed to have constant integral on all isometric images, or just on all the translation-images of the domain. While no set of the previous kind enjoys the Pompeiu property in the latter sense, we provide a necessary and sufficient condition in order a union of two intervals to have the Pompeiu property in the former sense, and we produce some examples to give an insight of the complexity of the problem for three-interval sets.

Growth of vertically aligned arrays of carbon nanotubes for high field emission

In the present paper, using the result of Bennett [1] on characterization of factorable matrices, we give necessary and sufficient conditions in order that Σλ_{n}x_{n} is summable |R,p_{n}|_{s} whenever Σμ_{n}x_{n} is summable |C,0|_{k}, and Σλ_{n}x_{n} is summable |C,0|_{s} whenever Σμ_{n}x_{n} is summable |R,p_{n}|_{r},. where 1

L. Leindler obtained a necessary and sufficient condition in order to a function

Let X be a Hausdorff topological space. We consider various locally convex spaces of continuous real valued functions on X and give necessary and sufficient conditions in order that (i) they contain an absolutely convex weakly compact total subset and (ii) they contain an absolutely convex total subset which is an Eberlein compact, when given the weak topology.

Let (P,V) be an irreducible imprimitivity system for a group H based on a dual group  of an Abelian group A and acting on a Hilbert space ℋ. Given ψ∈H, we find necessary and sufficient conditions in order that the set of vectors {∫Â〈â,a〉¯Vh−1 dP(â)ψ : a∈A, h∈H} be a frame in ℋ. Moreover, we apply these results to some examples that are considered in the literature in the context of square-integrability modulo a coset space.

We study the metric geometry of homogeneous reductive spaces of the unitary group of a finite von Neumann algebra with a non complete Riemannian metric. The main result gives an abstract sufficient condition in order that the geodesics of the Levi-Civita connection are locally minimal. Then, we show how this result applies to several examples.

The criterion of equal area [4, 5] is known as a necessary and sufficient condition in order that a unidimensional superconductor of infinite length, submitted to fixed current density and magnetic field, could not evolve towards a resistive state after an accidental local overheating; this criterion relates only to the normalized source term of heating (competition between the heat power generated by joule effect and the one which is absorbed by the cryogenic bath). We give here an optimal criterion, valid for any length (case of an uncooled region situated between two well cooled ones); it allows the engineers to play upon a non dimensional grouping of parameters (characterizing the intensity of the source term), related to the conductor, in order to extend the field of its superconducting properties. We give also some intermediate complementary criteria which are only sufficient but much more easy to check