Connectedness Of Sets Research Articles

Solarity and connectedness of sets in the space and in finite-dimensional polyhedral spaces

Generalized -piecewise functions constructed from given monotone path-connected boundedly compact subsets of the space are studied. They are shown to be monotone path-connected suns. In finite-dimensional polyhedral spaces, luminosity points of sets admitting a lower semicontinuous selection of the metric projection operator are investigated. An example of a non--connected sun in a four-dimensional polyhedral normed space is constructed. Bibliography: 14 titles.

Solarity and connectedness of sets in the space $C[a,b]$ and in finite-dimensional polyhedral spaces

Изучаются свойства обобщенных $n$-ломаных относительно монотонно линейных ограниченно компактных множеств в пространстве $C[a,b]$. Доказывается, что такие множества монотонно линейно связны и являются солнцами. Изучаются точки светимости множеств в полиэдральных пространствах, допускающих полунепрерывную снизу выборку из метрической проекции. Строится пример четырехмерного полиэдрального пространства и не $B$-связного солнца в нем. Библиография: 14 названий.

Quantitative expressions for the connectedness of sets in ℝ n

We prove that, for two arbitrary points a and b of a connected set E ⊂ Rn (n ≥ 2) and for any e > 0, there exist points x0 = a, x2,..., xp = b in E such that $$\parallel {x_1} - {x_0}{\parallel ^n} + \cdots + \parallel {x_p} - {x_{p - 1}}{\parallel ^n} < \varepsilon$$ . We prove that the exponent n in this assertion is sharp. The nonexistence of a chain of points in E with $$\parallel {x_1} - {x_0}{\parallel ^\alpha } + \cdots + \parallel {x_p} - {x_{p - 1}}{\parallel ^\alpha } < \varepsilon$$ for some α ∈ (1, n) proves to be equivalent to the existence of a nonconstant function f: E → R in the class Lipα(E). For each such α, we construct a curve E(α) of Hausdorff dimension α in Rn and a nonconstant function f: E(α) → R such that f ∈ Lipα(E(α)).

A parametric type of KKM theorem in FC-spaces with applications

The paper first proves a characteristic property in FC-spaces. By the use of the connectedness of sets, a parametric type of KKM theorem is then established in noncompact FC-spaces by introducing a linear ordered space. As a consequence, some recent results, such as noncompact minimax inequalities, saddle point theorem, and section theorem, are improved. The results generalize the corresponding results in the literatures.

On connectedness of sets in the real spectra of polynomial rings

Let R be a real closed field. The Pierce–Birkhoff conjecture says that any piecewise polynomial function f on R n can be obtained from the polynomial ring R[x 1,..., x n ] by iterating the operations of maximum and minimum. The purpose of this paper is threefold. First, we state a new conjecture, called the Connectedness conjecture, which asserts, for every pair of points $${{\alpha,\beta\in\,{\rm {Sper}}\ R[x_1,\ldots,x_n]}}$$ , the existence of connected sets in the real spectrum of R[x 1,..., x n ], satisfying certain conditions. We prove that the Connectedness conjecture implies the Pierce–Birkhoff conjecture. Secondly, we construct a class of connected sets in the real spectrum which, though not in itself enough for the proof of the Pierce–Birkhoff conjecture, is the first and simplest example of the sort of connected sets we really need, and which constitutes the first step in our program for a proof of the Pierce–Birkhoff conjecture in dimension greater than 2. Thirdly, we apply these ideas to give two proofs that the Connectedness conjecture (and hence also the Pierce–Birkhoff conjecture in the abstract formulation) holds locally at any pair of points $${{\alpha,\beta\in\,{\rm {Sper}}\ R[x_1,\ldots,x_n]}}$$ , one of which is monomial. One of the proofs is elementary while the other consists in deducing this result as an immediate corollary of the main connectedness theorem of this paper.

Vlasov-Wulbert connectedness of sets in Hilbert space

is the sphere of the same radius also centered at x. A closed ball can also be denoted by a single symbol. For a ball denoted by B, the notation B means that this is an open ball of the same radius centered at the same point. Suppose that k ∈ N ∪ {0}, L is an affine subspace of X, codimL = k, and x ∈ L. The set {y ∈ L : ‖x− y‖ < r} ⊂ L is called a disk of codimension k and is denoted by D(x, r) = DL(x, r). The closure D(x, r) = clD(x, r) is called a closed disk of codimension k, and

Arcwise Connectedness of Sets of Solutions to Differential Inclusions

Arcwise Connectedness of Sets of Solutions to Differential Inclusions

On the connectedness of the set of efficient points n convex optimization problems with multiple or random Objectives

In this paper sufficient conditions for the connectedness of certain efficient point sets of convex optimization problems with multiple or random objectives are provided. Known concepts of efficiency like functional efficiency and proper efficiency (with respect to vector optimization problems) as well as efficiency with probability 1 (with respect to stochastic optimization problems with fixed feasible domain) are treated. By this the number of objectives that have to be minimized on the feasible domain (a subset of a Banach-space) can be infinite also.