General Topology-I

In recent times, composite indicators have gained astounding popularity in a wide variety of research areas. Their adoption by global institutions has further captured the attention of the media and policymakers around the globe, and their number of applications has surged ever since. This increase in their popularity has solicited a plethora of methodological contributions in response to the substantial criticism surrounding their underlying framework. In this paper, we put composite indicators under the spotlight, examining the wide variety of methodological approaches in existence. In this way, we offer a more recent outlook on the advances made in this field over the past years. Despite the large sequence of steps required in the construction of composite indicators, we focus particularly on two of them, namely weighting and aggregation. We find that these are where the paramount criticism appears and where a promising future lies. Finally, we review the last step of the robustness analysis that follows their construction, to which less attention has been paid despite its importance. Overall, this study aims to provide both academics and practitioners in the field of composite indices with a synopsis of the choices available alongside their recent advances.

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On the Methodological Framework of Composite Indices: A Review of the Issues of Weighting, Aggregation, and Robustness

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The Foundations Of Differential Geometry

Dynamical systems: stability, symbolic dynamics and chaos

Harmonic analysis on semi‐simple lie groups—i

/pdf/mathematics-without-numbers-an-introduction-to-the-study-of-1sqsjd3myn.pdf

Mathematics without numbers: an introduction to the study of logic

Let \({\mathcal{S}}\) be a semigroup acting on a topological space M. We define attractors for the action of \({\mathcal{S}}\) on M. This concept depends on a family \({\mathcal{F}}\) of subsets of \({\mathcal{S}}\) . For certain semigroups and families it recovers the concept of attractors for flows or semiflows. We define and study the complementary repeller of an attractor. We also characterize the set of chain recurrent points in terms of attractors.

Attractors and Chain Recurrence for Semigroup Actions

Let \({\mathcal{S}}\) be a semigroup acting on a topological space M. We study finest Morse decompositions for the action of \({\mathcal{S}}\) on M. This concept depends on a family of subsets of \({\mathcal{S}}\) . For certain semigroups and families it recovers the concept of Morse decomposition for flows and semiflows. This paper also studies the behaviour of Morse decompositions for semigroup actions on principal bundles and their associated bundles. The emphasis is put on the study of those decompositions considering their projections onto the base space and their intersections with the fibers.

Finest Morse Decompositions for Semigroup Actions on Fiber Bundles

This paper is produced in response to the questioning of Morse decomposition for semigroup actions on noncompact spaces. We show how the limit behavior can be studied in arbitrary topological spaces by using powerful tools such as the Stone-Čech compactification and shadowing semigroups. We extend Conley’s characterization of chain recurrence in terms of attractors from the setting of flows on compact metric spaces to the setting of semigroup actions on any topological space.

/pdf/on-limit-behavior-of-semigroup-actions-on-noncompact-spaces-2qoh80oj8m.pdf

Josiney A. Souza

Papers

Mathematics without numbers: an introduction to the study of logic

Attractors and Chain Recurrence for Semigroup Actions

Finest Morse Decompositions for Semigroup Actions on Fiber Bundles

On limit behavior of semigroup actions on noncompact spaces

Prolongational limit sets of control systems