ABSTRACT: This extensive research paper provides a thorough investigation of the algebraic structure of complex products in groups, with comprehensive analysis of both commutative and non-commutative ...
In the third century BCE, Apollonius of Perga asked how many circles one could draw that would touch three given circles at exactly one point each. It would take 1,800 years to prove the answer: eight ...
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Mathematical Large Language Models (LLMs) have demonstrated strong problem-solving capabilities, but their reasoning ability is often constrained by pattern recognition rather than true conceptual ...
Abstract: Unsupervised representation learning techniques are commonly employed to analyze high-dimensional or unstructured data. In some cases, users may have prior knowledge of the topology of the ...
This Paper addresses the limitations of classical machine learning approaches primarily developed for data lying in Euclidean space. Modern machine learning increasingly encounters richly structured ...
This repo contains the working files for my personal lecture notes for Algebraic Topology 1 being taught in the Winter term of 2023/4 by Prof. Dr. Markus Hausmann at the University of Bonn. A compiled ...
In this research, we explore the properties and applications of the mapping cone and its variant, the pinched mapping cone. The mapping cone is a construction that arises naturally in algebraic ...
In the computer vision field of object detection, deep learning models are trained to identify objects of interest within an image of a scene. For example, such models can be trained to detect viruses ...
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