Summability results for operator matrices on topological vector spaces

Basic summability results are established for matrices of linear and some nonlinear mappings between topological vector spaces.     

Some fixed-points theorems for multi-valued mappings in topological vector spaces

Sunto Si danno dei teoremi di punto fisso der trasformazioni multivoche non compatte, definite in un sottoinsieme convesso e chiuso di uno spazio lineare topologico, localmente convesso e di Hausdorff. Tali teoremi estendono alcuni noti risultati dovuti a Darbo, Sadovskij, Daneš ed altri.

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Work performed under the auspices of the National Research Council (C.N.R.).

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Entrata in Redazione il 18 maggio 1971.     

Some results on Hu's conjecture concerning binary trees

Given n weights, w₁, w₂,…, w_n, such that 0?w₁?w₂???w₁, we examine a property of permutation π¹, where π¹=(w₁, w_n, w₂, w_n?1,…), concerning alphabetical binary trees.For each permutation π of these n weights, there is an optimal alphabetical binary tree corresponding to π, we denote it's cost by V(π). There is also an optimal almost uniform alphabetical binary tree, corresponding to π, we denote it's cost by V_u(π).This paper asserts that V_u(π¹)?V_u(π)?V(π) for all π. This is a preliminary result concerning the conjecture of T.C. Hu. Hu's conjecture is V(π¹)?V(π) for all π.     

Markowitz’s model with Euclidean vector spaces

In this paper a new approach of the Markowitz’s model is presented. Indeed, using an inner product, a quantitative and explicit solution for optimal portfolio selection is given. To do this, a scalar product is defined in Rⁿ${\mathfrak{R}}^n$ which allows us to calculate the composition of the optimal portfolio and the variance for a given expected return by means of the distance between the subspace of feasible solutions and the origin of the affine space.     

Efficacy of cones on topological vector spaces and application to common fixed points of multifunctions

Let (E, τ) be a topological vector space and P a cone in E. We shall define a topology τ _P on E so that (E, τ _P ) is a normable topological vector space and P is a normal cone with normal constant M = 1. Then by using the norm, we shall give some results about common fixed points of two multifunctions on cone metric spaces.     

Rigorous integration of smooth vector fields around spiral saddles with an application to the cubic Chua's attractor

In this paper, we present a general mathematical framework for integrating smooth vector fields in the vicinity of a fixed point with a spiral saddle. We restrict our study to the three-dimensional setting, where the stable manifold is of spiral type (and thus two-dimensional), and the unstable manifold is one-dimensional. The aim is to produce a general purpose set of bounds that can be applied to any system of this type. The existence (and explicit computation) of such bounds is important when integrating along the flow near the spiral saddle fixed point. As an application, we apply our work to a concrete situation: the cubic Chua's equations. Here, we present a computer assisted proof of the existence of a trapping region for the flow.     

Isoperimetric inequality for radial probability measures on Euclidean spaces

We generalize the Poincaré limit which asserts that the n-dimensional Gaussian measure is approximated by the projections of the uniform probability measure on the Euclidean sphere of appropriate radius to the first n-coordinates as the dimension diverges to infinity. The generalization is done by replacing the projections with certain maps. Using this generalization, we derive a Gaussian isoperimetric inequality for an absolutely continuous probability measure on Euclidean spaces with respect to the Lebesgue measure, whose density is a radial function.     

Some calculations on the conditional densities of well-admissible measures on linear spaces

When a probability measurem on a topological vector spaceE is well-admissible in a directionk E, the conditional law in the directionk given the other directions is absolutely continuous with respect to the Lebesgue measure. We shall prove that its density function is differentiable (in the sense precised below) and we shall calculate their derivatives. We given then two applications of such calculations.     

Radon-Nikodým derivatives for vector measures belonging to Köthe function spaces

Let m, n be a couple of vector measures with values on a Banach space. We develop a separation argument which provides a characterization of when the Radon-Nikodým derivative of n with respect to m—in the sense of the Bartle-Dunford-Schwartz integral—exists and belongs to a particular sublattice Z(μ) of the space of integrable functions L¹(m). We show that this theorem is in fact a particular feature of our separation argument, which can be applied to prove other results in both the vector measure and the function space settings.     

An alternative approach to multiply self-decomposable probability measures on Banach spaces

Summary For every >0 we define -times self-decomposable probability measures on Banach spaces by random power series. We prove the Urbanik and integral representation for such measures and discuss some related limit problems.     

Operator-semistable, operator semi-selfdecomposable probability measures and related nested classes on p-adic vector spaces

Let V be a finite dimensional p-adic vector space and let τ be an operator in GL(V). A probability measure μ on V is called τ-decomposable or m ? [(L)\tilde]₀(t)\mu\in {\tilde L}_0(\tau) if μ = τ(μ)* ρ for some probability measure ρ on V. Moreover, when τ is contracting, if ρ is infinitely divisible, so is μ, and if ρ is embeddable, so is μ. These two subclasses of [(L)\tilde]₀(t){\tilde L}_0(\tau) are denoted by L ₀(τ) and L ₀ ^#(τ) respectively. When μ is infinitely divisible τ-decomposable for a contracting τ and has no idempotent factors, then it is τ-semi-selfdecomposable or operator semi-selfdecomposable. In this paper, sequences of decreasing subclasses of the above mentioned three classes, [(L)\tilde]_m(t) é L_m(t) é L^#_m(t), 1 ￡ m ￡ ￥{\tilde L}_m(\tau)\supset L_m(\tau) \supset L^\#_m(\tau), 1\le m\le \infty , are introduced and several properties and characterizations are studied. The results obtained here are p-adic vector space versions of those given for probability measures on Euclidean spaces.     

An application of rosenthal's moment inequality to the strong law of large numbers

We apply a moment inequality of H.P. Rosenthal to get various generalizations of a theorem of Brunk-Chung-Prohoroff. In particular, we show that the strong law hols provided $\text{∑}\text{n=1}\text{∞}\text{n}{}^{\mathrm{?2p}}\text{E(X}{}_{\mathrm{n}}\text{}{}^{\mathrm{2p}}\text{) < ∞}$ and $\text{∑}\text{n=1}\text{∞}\text{n}{}^{\mathrm{?2?p}}\text{(E(}\text{X}\text{n}\text{2}\text{))}{}^{\mathrm{p}}\text{< ∞}$ for one p ? 1.     

Some results on Gaussian Besov–Lipschitz spaces and Gaussian Triebel–Lizorkin spaces

In this paper we define Besov–Lipschitz and Triebel–Lizorkin spaces in the context of Gaussian harmonic analysis, the harmonic analysis of Hermite polynomial expansions. We study inclusion relations among them, some interpolation results and continuity results of some important operators (the Ornstein–Uhlenbeck and the Poisson–Hermite semigroups and the Bessel potentials) on them. We also prove that the Gaussian Sobolev spaces are contained in them. The proofs are general enough to allow extensions of these results to the case of Laguerre or Jacobi expansions and even further in the general framework of diffusion semigroups.     

Some basic observations on Kelly's conjecture for graphs

P.J. Kelly first mentioned the possibility of determining a graph from subgraphs obtained by deleting several points. While such problems have received a great deal of attention in the case of deletions of single points, the problem for several points is virtually untouched. This paper contains some basic results on that problem, including the negative observation that for every k, there exist two non-isomorphic graphs with the same collection of k-point subgraphs.