The concept of compactness is a necessary condition of any system that is going to call itself a finitary method of proof. However, it can also apply to predicates of sets of formulas in general and in that manner it can be used in relation to level functions, a flavor of measure functions. In what follows we will tie these concepts of measure and compactness together and expand some concepts w...

We develop a measure of compactness based on the distance between voters within the same district relative to the minimum distance achievable, which we coin the relative proximity index. Any compactness measure that satisfies three desirable properties (anonymity of voters, efficient clustering, and invariance to scale, population density, and number of districts) ranks districting plans identi...

Abstract The aim of this paper is to provide necessary and sufficient conditions on the generator a multiplication operator acting in spaces functions bounded Young Riesz variation so that it is, among other things, invertible, continuous, finite rank, compact, Fredholm or has closed range. Furthermore, we characterize various spectra such operators give some estimates their measure non-compact...

It has been argued that convexity in the Euler lattice can be interpreted in terms of consonance (Honingh and Bod 2005). In this paper, a second hypothesis is presented that states that compactness in the Euler-lattice is an indication of consonance. It will be investigated if, and to which degree convexity and compactness are in agreement with the consonance of chords according to Euler’s meas...

Our aim in this paper is to study the existence of solution sets and its topological structure for non-localfractional differential equations on half-line a Banach space using Riemann-Liouville definition. Themain result based Meir-Keeler fixed point theorem condensing operators combined with measure ofnon-compactness. An example given illustrate feasibility our main result.