This paper investigates the closability of closed compact linear operators on Hilbert spaces. It establishes that under certain conditions, every closed compact linear operator is closable. Additionally, it investigates the properties of closable operators and highlights their stability under limits and algebraic operations. We establish a non-closability criterion based on sequences in the domain and demonstrate that the limit of bounded compact operators need not be closable. Moreover, we exam...

Algebra 

The feasibility of physical-distancing interventions during the COVID-19 pandemic implicitly relied on structural properties of physical space. We show that minimum-distance policies (usually formulated as “individuals must remain at least δ metres apart”) require not only the Hausdorff property but a compatible metric structure that supports uniform separation with a positive lower bound. Using results from metric topology, we prove that the existence of disjoint open neighbourhoods is necessar...

Topology  Mathematics  Algebra 

This paper presents a novel framework for optimizing Carbon Release (CR) through an AI-driven approach to Fossil Fuel Intake (FFI) management. We propose a new training methodology for AI models to enhance their effectiveness in reducing FFI by learning from historical energy consumption trends and emissions data. By attributing specific quantities to key optimization components such as combustion efficiency, renewable integration, and AI impact, we derive a detailed matrix that quantifies their...

Machine Learning  Information Management  Artificial Intelligence  Grid Computing  Algebra 

$Let$G  be a compact connected Lie group with Lie algebra g. We prove that the de Rham cohomology H dR • ( G ) is isomorphic to the Lie algebra cohomology H • ( g,ℝ...

Algebra 

This article develops a framework for extending spectral theory to Frölicher licher algebras, which define smoothness via curves and functionals rather than topological or bornological structures. Motivated by classical spectral theory in locally convex algebras and its smooth extension in convenient algebras, we compare and connect these three categories through their notions of smoothness and character spectra. Beginning with the Gelfand theory for locally convex algebras, we generalize s...

Algebraic Geometry  Special Theory of Relativity  Algebra  Functional Analysis 

In this paper, we explore the inverse spectral problem for a kind of pseudo-symmetric matrix A(ε,c,s) that corresponds to a graph of a banana tree when all ε=1. The problem of reconstructing pseudo-symmetric matrices from their spectra was considered. The sufficient and necessary conditions for the existence of these matrices are provided, and an algorithm and numerical example were provided to illustrate their characteristics. In addition, the effectiveness of the solution was v...

Applied Statistical Mathematics  Algebra 

We can say for a ring R weak JN-clean ring if all elements a in R it can be written as a sum or difference of nilpotent and idempotent. Further the nilpotent element belongs to the Jacobson radical of R. The purpose of this paper is to give some characterization and basic properties of this ring. Also we will studied the relationship between weak JN-clean rings and J-reduced ring, Boolean ring, local ring and clean ring: from the main result: 1) The ring R is w...

Algebra 

If a ring R is called weak nil clean if every element in R can be expressed as the sum or difference of nilpotent element and idempotent, if further the idempotent element and nilpotent element commute the ring is called weak* nil clean. The purpose of this paper is to give some characterization and basic properties of weak nil clean rings. The main results of this work are: 1) Let R be a ring, then R is weak nil clean if and only if R/P(R) is wea...

Algebra 

Algebra 

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Algebra