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[3] viXra:2502.0130 [pdf] submitted on 2025-02-17 23:13:29
Authors: James A. Smith
Comments: 8 Pages.
Using GA's capacities for formulating reflections, we solve an interesting multiple-tangency problem. The solution is obtained in two ways, the easiest of which transforms the relevant reflections into a single rotation. The solution is validated via a GeoGebra worksheet.
Category: Geometry
[2] viXra:2502.0071 [pdf] submitted on 2025-02-11 16:23:13
Authors: Pier Paolo Conti
Comments: 37 Pages.
Throughout history, humanity has sought to limit error in order to gain more precise insights and optimize its tools, be they physical or conceptual. This work explores the intricate relationship between geometry, mathematics, and the propagation of uncertainty, with particular attention to methods that either mitigate or avoid the amplification of uncertainty in mathematical, geometric, and applied contexts. The study begins by examining the foundational concepts underlying uncertainty in mathematical models, exploring how various geometric and topological structures can be leveraged to better understand and control the flow of uncertainty across different domains. A new number convention, the so-called empirical number, is introduced, enabling a more accessible assessment of uncertainty propagation. Particular focus is placed on those constructions that actively work to reduce uncertainty, offering insights into techniques that prevent the cascading effect of errors, a challenge often encountered in both theoretical and applied mathematics.Through the use of geometric principles, this work provides novel approaches to managing the inherent uncertainties in complex systems, ranging from simple algebraic problems to intricate applications. It highlights methods such as error propagation reduction, geometrically optimized models, and innovative adaptations to traditional methods that reduce computational or conceptual uncertainty. These techniques are of significant theoretical importance and are also crucial in practical applications, where precision and reliability are paramount, particularly in applied mathematics.By addressing both the philosophical and practical dimensions of uncertainty, this work paves the way for a refined understanding of the interaction between mathematical theory and real-world applications, offering tools to navigate complex, uncertain environments with greater confidence and precision. The exploration of these ideas provides new insights into the role of uncertainty in mathematical systems, particularly in those constructions that prioritize stability and error mitigation over mere approximation.The exposition begins with the definition of the empirical number, followed by the wide application of the Monte Carlo method to assess the functionality presented. It also recalls the theory of error propagation, addresses basic segment operations, and reexamines the Pythagorean theorem, which plays a crucial role in limiting error propagation. Further, the work discusses the applications of these ideas in calculus, specifically in the propagation of uncertainty through differentiation and integration.
Category: Geometry
[1] viXra:2502.0011 [pdf] submitted on 2025-02-01 21:00:12
Authors: Urs Frauenfelder, Joa Weber
Comments: 19 Pages.
In this article we introduce the notion of Floer function which has the property that the Hessianis a Fredholm operator of index zero in a scale of Hilbert spaces. Since the Hessian has a complicated transformation under chart transition, in general this is not an intrinsic condition. Therefore we introduce the concept of Floerfolds for which we show that the notion of Floer function is intrinsic.
Category: Geometry
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