Analytical Solution of the Three-Dimensional Laplace Equation in Terms of Linear Combinations of Hypergeometric Functions

We present some solutions of the three-dimensional Laplace equation in terms of linear combinations of generalized hyperogeometric functions in prolate elliptic geometry, which simulates the current tokamak shapes. Such solutions are valid for particular parameter values. The derived solutions are compared with the solutions obtained in the standard toroidal geometry.

KONFLIK KOGNITIF MAHASISWA DALAM MEMAHAMI KONSEP GEOMETRI HIPERBOLIK DAN ELLIPTIK

Using schemes of Euclid's geometrical concepts in long-term memory to understand hyperbolic geometry and elliptic geometry concepts with assimilation and accommodation allows for cognitive conflict. This study aims to reduce the occurrence of cognitive conflict by understanding the mathematical content of the three of Euclidean geometries, hyperbolic and elliptic. The research was conduct used descriptive exploratory. The results indicate that Euclid's geometry representation is still used in representing hyperbolic and elliptic geometry so that cognitive conflict occurs. Cognitive conflicts that occur are related to the position of two lines, parallels, two triangles with the same that correspond angles, intersects one of two parallel lines, the number of angles in a triangle, and Sacherri's valid hypothesis. The efforts that can be made to reduce the occurrence of cognitive conflict are to change existing schemes or create new schemes so that the information obtained can be combined into existing schemes in a deductive axiomatic approach to material content through the accommodation process

On Insolvability of the 4-th Hilbert Problem for Hyperbolic Geometries

The article proves the insolvability of the 4-th Hilbert Problem for hyperbolic geometries. It has been hypothesized that this fundamental mathematical result (the insolvability of the 4-th Hilbert Problem) holds for other types of non-Euclidean geometry (geometry of Riemann (elliptic geometry), non-Archimedean geometry, and Minkowski geometry). The ancient Golden Section, described in Euclid’s Elements (Proposition II.11) and the following from it Mathematics of Harmony, as a new direction in geometry, are the main mathematical apparatus for this fundamental result. By the way, this solution is reminiscent of the insolvability of the 10-th Hilbert Problem for Diophantine equations in integers. This outstanding mathematical result was obtained by the talented Russian mathematician Yuri Matiyasevich in 1970, by using Fibonacci numbers, introduced in 1202 by the famous Italian mathematician Leonardo from Pisa (by the nickname Fibonacci), and the new theorems in Fibonacci numbers theory, proved by the outstanding Russian mathematician Nikolay Vorobyev and described by him in the third edition of his book “Fibonacci numbers”.

Symmetry in Regular Polyhedra Seen as 2D Möbius Transformations: Geodesic and Panel Domes Arising from 2D Diagrams

This paper shows a methodology for reducing the complex design process of space structures to an adequate selection of points lying on a plane. This procedure can be directly implemented in a bi-dimensional plane when we substitute (i) Euclidean geometry by bi-dimensional projection of the elliptic geometry and (ii) rotations/symmetries on the sphere by Möbius transformations on the plane. These graphs can be obtained by sites, specific points obtained by homological transformations in the inversive plane, following the analogous procedure defined previously in the three-dimensional space. From the sites, it is possible to obtain different partitions of the plane, namely, power diagrams, Voronoi diagrams, or Delaunay triangulations. The first would generate geo-tangent structures on the sphere; the second, panel structures; and the third, lattice structures.

3D Effects in the Wake Vortex Formation of an Elliptic Flat Plate in Hover

A Numerical Analysis is conducted to investigate the Leading Edge Vortex (LEV) dynamics of an elliptic flat plate undergoing 2 dimensional symmetric flapping motion in hover. The plate is modeled with an aspect ratio of 3 and a flapping trajectory resulting in Reynolds number 225 is studied. The leading edge vortex stability is analyzed as a function of the non dimensional formation number and a vorticity transport analysis is carried to understand the flux budgets present. The LEV formation number is found to be 2.6. The results of vorticity analysis show the highly three dimensional nature of the LEV growth for an elliptic geometry.

Determination of effect of elliptic notches and grooves on stress concentration factors on notched bar in tension and grooved shaft under torsion

Stress concentration of structural members can be reduced considerably by the judicious choice of elliptic shaped stress raisers like notch and groove. Substantial effort has been given by numerous researchers to accurately measure the effect of such stress raisers, particularly of semicircular shaped notch and groove. An exhaustive bibliographical study proved that there is scope to investigate further and establish an alternative design criteria; concerning the elliptic geometry. Computational method, primarily the finite element method has been used to analyze the models under loading. This paper suggests the use of a modified elliptic shape which gives less stress concentration when compared to semicircular notch and groove. The ratio of minor and major half axes of the ellipse should be between 0.3 and 0.4. The introduction of shoulder with elliptic notch and groove even reduces the stress concentration. The results obtained from FEM analysis propose optimal values of geometrical design parameters. The study represents not only a precise view of stress distribution, but also to develop charts that can be used by designer for practical purposes.DOI: http://dx.doi.org/10.3329/jname.v10i1.13675