Path-connected Sets Research Articles

Monotone path-connected sets in geometric approximation theory and their applications

Monotone path-connected sets in geometric approximation theory and their applications

Finite-Dimensional Spaces where the Class of Chebyshev Sets Coincides with the Class of Closed and Monotone Path-Connected Sets

Finite-Dimensional Spaces where the Class of Chebyshev Sets Coincides with the Class of Closed and Monotone Path-Connected Sets

A Monoparametric Family of Piecewise Linear Systems to Generate Scroll Attractors via Path-Connected Set of Polynomials

In this work, we present a monoparametric family of piecewise linear systems to generate multiscroll attractors through a polynomial family defined by path curves that connect to the roots. The idea is to define path curves where the roots of a polynomial can take values by determining an initial and a final polynomial. As a consequence, structural stability and bifurcation of the system can be obtained. Structural stability is obtained by preserving the same stability of the initial and final polynomials. However, the system bifurcates by changing the stability of the final polynomial with respect to the initial polynomial. The aim is achieved by the design of a piecewise linear controller that is applied to affine linear systems. Our results are mathematically proved and numerical examples are also provided to illustrate the approach.

Inner Bounds for the Almost Entropic Region and Network Code Construction

In the fields of information theory and network coding, a complete characterization of the almost entropic region is a fundamental but inherently difficult open problem. This paper focuses on the characterization of this region via explicit inner bounds, optimization of functions over this region, and network code construction. One approach to study the entropic region is to put a constraint on the alphabets for involved random variables, enabling us to focus on a specific set of joint distributions. We present the notion of alphabet constrained entropic set and describe its properties, namely, its a closed and path-connected set. Motivated by the properties, a random local search algorithm is designed to find an entropic vector and associated distribution near to a given target vector and extended to optimize functions of joint entropies. We present significantly improved inner bound compared to known inner bounds for the almost entropic region involving four random variables using a refinement of our algorithm. Finally, another application of the algorithm is shown to construct network codes and index codes for a given rate vector in the capacity region. This approach to constructing network codes is novel and has an entirely different flavor than classical coding theoretic construction.

Properties of monotone path-connected sets

Изучаются монотонно линейно связные множества, а также сильно и слабо связные по Менгеру множества. Вводится новое понятие $\varepsilon$-солнечности и устанавливается его связь с понятием солнечности. Доказывается, что ограниченно компактные солнца в $C(Q)$ являются монотонно линейно связными множествами. Библиография: 14 наименований.

Properties of monotone path-connected sets

We study monotone path-connected sets and also strongly and weakly Menger-connected sets. We introduce the notion of -solarity and establish a connection with the notion of solarity. We prove that boundedly compact suns in are monotone path-connected.

Continuous ε-selection and monotone path-connected sets

The sets with continuous selection from near-best approximations and the monotone path-connected sets are studied; several examples of such sets are also considered.

A monotone path-connected set with outer radially lower continuous metric projection is a strict sun

A monotone path-connected set is known to be a sun in a finite-dimensional Banach space. We show that a B-sun (a set whose intersection with each closed ball is a sun or empty) is a sun. We prove that in this event a B-sun with ORL-continuous (outer radially lower continuous) metric projection is a strict sun. This partially converses one well-known result of Brosowski and Deutsch. We also show that a B-solar LG-set (a global minimizer) is a B-connected strict sun.

Sufficient Conditions for a Path-Connected Set of Local Solutions to an Optimal Control Problem

Consider a fixed-endpoint, fixed time optimal control problem with state and input taking values in Euclidean space. We show that if the Hamiltonian function associated with this problem satisfies a scale invariance property, then the set of all local solutions to this problem---over all possible terminal state constraints---is path-connected. We also show that this result extends to problems with additional constraints such as a bound on total cost. Finally, we use this result to show that the set of all curves that can be realized by a planar elastica is path-connected, and we describe how this result can be applied to the problem of robotic manipulation for nonrigid objects.

Infinite distributive laws versus local connectedness and compactness properties

Various local connectedness and compactness properties of topological spaces are characterized by higher degrees of distributivity for their lattices of open (or closed) sets, and conversely. For example, those topological spaces for which not only the lattice of open sets but also that of closed sets is a frame, are described by the existence of web neighborhood bases, where webs are certain specific path-connected sets. Such spaces are called web spaces. The even better linked wide web spaces are characterized by F -distributivity of their topologies, and the worldwide web spaces (or C-spaces) by complete distributivity of their topologies. Similarly, strongly locally connected spaces and locally hypercompact spaces are characterized by suitable infinite distributive laws. The web space concepts are also viewed as natural extensions of spaces that are semilattices with respect to the specialization order and have continuous (unary, binary or infinitary) semilattice operations.

The s-elementary frame wavelets are path connected

An s-elementary frame wavelet is a function ψ ∈ L 2 ( R ) \psi \in L^2(\mathbb {R}) which is a frame wavelet and is defined by a Lebesgue measurable set E ⊂ R E\subset \mathbb {R} such that ψ ^ = 1 2 π χ E \hat {\psi }= \frac {1}{\sqrt {2\pi }}\chi _E . In this paper we prove that the family of s-elementary frame wavelets is a path-connected set in the L 2 ( R ) L^2(\mathbb {R}) -norm. This result also holds for s-elementary A A -dilation frame wavelets in L 2 ( R d ) L^2(\mathbb {R}^d) in general. On the other hand, we prove that the path-connectedness of s-elementary frame wavelets cannot be strengthened to uniform path-connectedness. In fact, the sets of normalized tight frame wavelets and frame wavelets are not uniformly path-connected either.

A note on Söderlind’s conjecture

Let f be a Lipschitz operator from a path-connected set D⊆ C m into C m , with the lub-Lipschitz constant L( f) and the so-called “Gerschgorim range radius” r( f) subordinate to a given vector norm ∥·∥ of C m . In 1986 [Numer. Math. 50 (1986) 27], Söderlind’s conjectured that if r( f)< L( f), then there exists a new vector norm ∥·∥ * of C m such that the induced lub-Lipschitz constant L *( f)⩽ r( f). In this paper, we affirmatively prove Söderlind’s conjecture for several class of Lipschitz operators f, whilst we construct a counterexample to disprove Söderlind’s conjecture.