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[5] viXra:2410.0109 [pdf] submitted on 2024-10-19 13:50:43
Authors: Thomas Günther
Comments: 14 Pages. Published in International Journal of Engineering and Applied Physics - Vol. 4, no. 3, pp. 1041—1054, 2024.
The present article provides a detailed mathematical treatise on the geometry of skateboard transition ramps. These usually form a circular segment shape and are used in skateboarding for transitioning from a horizontal plane to another angle of incline. Transitions play a crucial role in the flow of skateparks and are required, for example, for quarter pipes, mini ramps, vert ramps, or jump ramps. Skateboarding has been an Olympic sport since 2023, yet many publicly funded skateparks still do not meet the demands of the sport. On one hand, there is often a lack of willingness from authorities to engage with the athletes beforehand; on the other hand, the people involved in the planning process often lack the necessary experience and mathematical expertise to perfectly fulfill the users' needs. Thoughtful planning is particularly crucial for ramps with curved surfaces to ensure the flow of the skatepark. In addition to the mathematical analysis, source codes for programs are provided to make the calculations as convenient as possible for all users.
Category: Geometry
[4] viXra:2410.0075 [pdf] submitted on 2024-10-12 14:51:23
Authors: Andrey V. Voron
Comments: 10 Pages.
The classification of Pythagorean triples is based on differences in the tangents of their triangles in a certain group, the value of which tends to one divided by a prime number and has the form of a fraction. The corresponding tables containing primitive and non-primitive Pythagorean triples are constructed in accordance with the proposed visual classification. Based on the presented classification, the concept of the "parent" Pythagorean triple is introduced — a primitive triple underlying a series (table) of Pythagorean triples derived from it. The first "parent" Pythagorean triple among the set of primitive tangent values of their triangles is defined by us as "119, 120, 169". It is shown that the first "parent" Pythagorean triple on the basis of the increase of the smaller legs of their triangles — 3, 4, 5 — can be considered as the initial structural unit of a two—dimensional planar construction of a figure — a right triangle, and the product of the numbers 3, 4, 5 — equal to 60 - it is advisable to consider, in turn, as a structural unit of a three-dimensional A three—dimensional figure is a parallelogram.
Category: Geometry
[3] viXra:2410.0073 [pdf] submitted on 2024-10-13 02:19:57
Authors: Vladislav Koshchakov
Comments: 9 Pages. In Russian (An abstract is required in the article)
A geometric approach to the formation of Pythagorean triples is proposed, which allows not only to limit oneself to the second degree of the equation, but also to expand it to any value.
Category: Geometry
[2] viXra:2410.0062 [pdf] submitted on 2024-10-11 14:22:38
Authors: Berndt Gensel, Theophilus Agama
Comments: 16 Pages.
In this paper we continue the development of the circles of partition by introducing a certain geometry of the axes of complex circles of partition. We use this geometry to verify the condition in the squeeze principle in special cases with regards to the orientation of the axes of complex circles of partition.
Category: Geometry
[1] viXra:2410.0059 [pdf] submitted on 2024-10-12 03:36:46
Authors: Parker Emmerson
Comments: 12 Pages.
In this paper, we explore the properties and optimization techniques related to polyhedral cones and energy numbers with a focus on the cone of positive nxn semidefinite matrices and efficient computation strategies for kernels. In Part (a), we examine the polyhedral nature of the cone of positive semidefinite matrices, S^n , establishing that it does not form a polyhedral cone for due to its infinite dimensional characteristics. In Part (b), we present an algorithm for efficiently computing the kernel function K(x, x prime) = (latex formatting in wysiwig)on-the-fly, leveraging a polyhedral description of the convex hull generated by the feature mappings phi and phi prime. By restructuring the problem and using gradient-based optimization techniques, our approach minimizes memory usage and computational overhead, thus enabling scalable computation. Through examples and visualizations, we demonstrate the practical applications and efficiency of the proposed algorithm in optimizing these kernel computations.
Category: Geometry
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