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[2] viXra:2412.0181 [pdf] submitted on 2024-12-29 00:46:13
Authors: Richard Murdoch Montgomery
Comments: 17 Pages.
Random matrix theory (RMT) has long served as a cornerstone of modern physics, mathematics, and complex systems analysis. More recently, topological and differential topological methods have emerged as powerful tools to characterize the global structures and stability properties of ecological networks. This article develops an integrated framework bridging RMT with topology and differential topology to study ecological systems, aiming to provide novel insights into their resilience and underlying structural features. We present the relevant mathematical foundations, illustrate computational algorithms (including multiple graphical outputs), and demonstrate these methods on both synthetic and real ecological data. Our findings highlight how topological invariants, combined with the spectral properties of large random matrices, shed light on the stability and transition behaviors of communities under perturbations. Advantages and limitations are discussed, paving the way for future research directions at the intersection of mathematics and ecology.
Category: Topology
[1] viXra:2412.0103 [pdf] submitted on 2024-12-17 22:16:18
Authors: Theophilus Agama
Comments: 24 Pages. This is a consolidated work bringing together several pieces into a unified work.
This paper introduces problem theory, a new framework for studying problem and solution spaces through the lens of point-set topology and abstract algebra. We define solution spaces as topological constructs induced by the assignment of solutions to problems and establish their fundamental properties. Key results include the identification of compactness and continuity conditions in solution spaces and their algebraic interpretations within module-theoretic settings. This theory bridges abstract algebra and topology, providing new insights into the interplay between algebraic structures and topological spaces. Potential applications and directions for future research are discussed.
Category: Topology
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