Mather β-Function for Ellipses and Rigidity

Bialy, Misha

doi:10.3390/e24111600

Cited by 4 publications

(2 citation statements)

References 25 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…One more important peculiarity of the approach in this paper should be mentioned. This paper does not deal with the Jacobi elliptic functions (Alquran & Jarrah, 2019;Dragović & Radnović, 2011), or the Weierstrass elliptic functions (Bialy, 2022;Reznik et al, 2021), which refer to practical notions of complex analysis, in expansion method and periodic wave solutions the of ellipse, in particular Wang & Jin, 2022). The conic functions derived in this paper are referred to as elliptic identities in trigonometry.…”

Section: Methodsmentioning

confidence: 99%

An elementary treatise on elliptic functions as trigonometry

Al-Ossmi

2023

Alifmatika J. Pendidik. dan Pembelajaran Mat.

View full text Add to dashboard Cite

This article concerns the examination of trigonometric identities from an elliptic perspective. The treatment of elliptic functions presented herein adheres to a structure analogous to the traditional exposition of trigonometric functions, with the exception that an ellipse replaces the unit circle. The degree of similarity between the elliptic functions and their trigonometric counterparts is moderated by the periodicity of the so-called El- functions. These identities not only establish the values of the functions, but also establish a correlation between their ratios and the major and minor axes of the underlying ellipse. The resemblance between the functions is somewhat modified by the periodic nature of the El-identities, whereby each ratio is associated with the major and minor axis of the ellipse. This article adopts the notation (E) to denote the El- functions and distinguish them from the opposite circular functions.

show abstract

Section: Methodsmentioning

confidence: 99%

An elementary treatise on elliptic functions as trigonometry

Al-Ossmi

2023

Alifmatika J. Pendidik. dan Pembelajaran Mat.

View full text Add to dashboard Cite

show abstract

“…While Vertex-Focus Method focuses on constructing the parabola using the vertex and the focus of the parabola. In Focus-Directrix Method [14-15], the construction process involves drawing the directrix perpendicular to the given line and finding points on the parabola equidistant from the focus and the directrix, whereas Vertex-Focus Method [16] as this method begins with marking the vertex and the focus, followed by constructing points on the parabola equidistant from the vertex and the line connecting the vertex and the midpoint between the focus and the vertex In this part of the article, a new form of two-dimensional curve is produced by shifting the focus point along the equilibrium axis to be located at the vertex position which means a Semi Parabolic Curve (SPC) whose focus is the vertex point (the SPC's head point). An SPC is the locus of a point which moves in a plane such that its distance from a fixed point, called the focus, is greater than or equal to the distance from a fixed straight line, called the directrix.…”

Section: Introductionmentioning

confidence: 99%

The Fifth Section, the Semi Parabolic Curves, when the Focus equals the Vertex

H.M. Al-ossmi

2024

Al-Kitab J. Pure Sci.

View full text Add to dashboard Cite

This article introduces a unique case study involving open curves of parabolic form situated within two-dimensional spaces. It presents a new form of a two-dimensional curve achieved by repositioning the focal point to coincide with the vertex position, resulting in what is termed a Semi-Parabolic Curve (SPC) where the focal point acts as the vertex referred to as the SPC head point. In essence, the SPC represents the path traced by a point on a plane, where its distance from a fixed point (the focus), is always greater than or equal to its distance from a fixed straight line (the directrix). Furthermore, the article provides the coordinate equations that govern the points along these curves. With the potential to pave the way for exploring additional geometric aspects relevant to this class of curves, and to enabling comparative analyses across diverse mathematical and geometric domains, particularly in three-dimensional contexts in the future.

show abstract