In this paper, we generalize properties for single-valued functions in homotopy theory to multi-valued functions. We present new definitions such as locally m-pathwise connected, m-simply m-covering space, and regular space. apply some topological from algebraic topology the

We begin with formulating the stationary equations of the Kuramoto model as a system of polynomial equations in a novel way. Then, based on an algebraic geometric root count, we give a prescription of computing an upper bound on the number of equilibria of the Kuramoto model for the most general case, i.e., defined on an arbitrary graph and having generic values of natural frequencies and inhom...

This note establishes that the familiar internal characterizations of the Tychonoff spaces whose rings of continuous real-valued functions are complete, or σ-complete, as lattice ordered rings already hold in the larger setting of pointfree topology. In addition, we prove the corresponding results for rings of integer-valued functions.

In this paper we consider a general class of systems determined by operator valued measures which are assumed to be countably additive in the strong operator topology. This replaces our previous assumption of countable additivity in the uniform operator topology by the weaker assumption. Under the relaxed assumption plus an additional assumption requiring the existence of a dominating measure, ...

Transfer entropy (TE) is a model-free approach based on information theory to capture causality between variables, which has been used for the modeling and monitoring of, and fault diagnosis in, complex industrial processes. It is able to detect the causality between variables without assuming any underlying model, but it is computationally burdensome. To overcome this limitation, a hybrid meth...

1. We begin with some abstract definitions. Suppose X is a topological space and F is a collection of real valued (or complex valued) functions on X . We say that F defines an Fstructure (X , F) on X if the topology on X is the weakest topology so that each element in F is continuous, i.e., the collection {f(U)| U open in R, f ∈ F} forms a subbasis for the topology on X . For instance, take a s...

In this paper we present a topology on the space of real-valued functions defined on a functionally Hausdorff space X that is finer than the topology of pointwise convergence and for which (1) the closure of the set of continuous functions C(X) is the set of upper semicontinuous functions on X , and (2) the pointwise convergence of a net in C(X) to an upper semicontinuous limit automatically en...

We provide a natural topology for a cone metric space and a noncompactness measure is defined for this space, which enables us to extend existing results for mappings and set-valued mappings defined on classical metric spaces. Moreover it is proved that the topology of any uniform topological space is generated by a cone metric. c ©2016 All rights reserved.