Inverses and n-uncial property of Jacobian elliptic functions

2021 ◽  
pp. 1-15
Author(s):  
Leonardo Solanilla ◽  
Jhonny Andrés Leal ◽  
Diego Mauricio Tique
Keyword(s):  
Elliptic Functions ◽  
Jacobi Elliptic Functions ◽  
Numerical Implementation ◽  
Conformal Maps ◽  
Jacobian Elliptic Functions ◽  
Map Projections ◽  
Almost Everywhere ◽  
Crucial Property ◽  
Definition Of ◽  
The Way

The inverses of Jacobi elliptic functions possess an apparently-non-crucial property: they provide almost-everywhere-conformal maps on a hemisphere onto a torus and so, onto a parallelogram. Thus, they produce map projections on the sphere generalizing the famous quincuncial projection of Charles S. Peirce. Besides providing a general practical definition of n-uncial map and proving that all the considered inverse elliptic functions are n-uncial, we give operative handy formulas to calculate these maps. To the best of our knowledge, these useful formulas have not been all together published before, except for Pierce projection. We look forward to their numerical implementation. By the way, we also classify the resulting map projections according the number of singularities.

scholarly journals Equidistant map projections of a triaxial ellipsoid with the use of reduced coordinates

2017 ◽  
Vol 66 (2) ◽  
pp. 271-290 ◽  
Author(s):  
Paweł Pędzich
Keyword(s):  
Elliptic Integral ◽  
Elliptic Functions ◽  
New Method ◽  
Triaxial Ellipsoid ◽  
Polar Regions ◽  
Jacobian Elliptic Functions ◽  
Map Projections ◽  
Basic Properties ◽  
Reduced Coordinates

Abstract The paper presents a new method of constructing equidistant map projections of a triaxial ellipsoid as a function of reduced coordinates. Equations for x and y coordinates are expressed with the use of the normal elliptic integral of the second kind and Jacobian elliptic functions. This solution allows to use common known and widely described in literature methods of solving such integrals and functions. The main advantage of this method is the fact that the calculations of x and y coordinates are practically based on a single algorithm that is required to solve the elliptic integral of the second kind. Equations are provided for three types of map projections: cylindrical, azimuthal and pseudocylindrical. These types of projections are often used in planetary cartography for presentation of entire and polar regions of extraterrestrial objects. The paper also contains equations for the calculation of the length of a meridian and a parallel of a triaxial ellipsoid in reduced coordinates. Moreover, graticules of three coordinates systems (planetographic, planetocentric and reduced) in developed map projections are presented. The basic properties of developed map projections are also described. The obtained map projections may be applied in planetary cartography in order to create maps of extraterrestrial objects.

Optimized global map projections for specific applications: the triptychial projection and the Spilhaus projection

2020 ◽  
Author(s):  
Björn Grieger
Keyword(s):  
World Ocean ◽  
Elliptic Functions ◽  
Complete Information ◽  
Full Detail ◽  
Map Projections ◽  
Minimal Distortion ◽  
The World ◽  
One Step ◽  
Definition Of ◽  
First Time

<p>There is no perfect global map projection. A projection may be area preserving or conformal (shape preserving on small scales) in some regions, but it will inevitably exhibit considerable distortions in others. An oblique version of a projection (where the globe is rotated before projecting) can be optimized to avoid major distortions in specific regions of interest.<br>We present two global map projections of the Earth which either display all continents (including Antarctica) or the complete world ocean with minimal distortion and without any intersection. These are the triptychial projection and the Spilhaus projection, respectively.<br>The triptychial projection is original work and has been published by Grieger (2019). While that paper comprises complete information on the definition of the projection, the details of its application need to be collected from literature referenced therein. The triptychial projection is an oblique and rearranged version of the Peirce quincuncial projection of the world (Peirce, 1879).<br>Instances of the Spilhaus projection went viral on the internet in fall 2018. The projection is mostly attributed to a publication from 1942, but in fact it seems to appear for the first time in Spilhaus (1979). The projection is shown in that paper (and in a few later ones), but no information on its definition is provided. Developers of ArcGIS did some reverse engineering and could identify the Spilhaus projection as an oblique version of the Adams projection of the world in a square II (Adams, 1929).<br>The triptychial and the Spilhaus projection both imply several steps in their application. While the two projections look very different, they have one step in common: the conformal mapping of a hemisphere onto a square, which requires tabulated Jacobi elliptic functions. We review both projections, describe them in full detail, and provide all formulas and data needed to apply them. The algorithms employed may also be interesting for planetary applications.</p>

Motion of a single charged particle in electromagnetic fields with cyclotron resonances

1990 ◽  
Vol 43 (1) ◽  
pp. 119-139 ◽  
Author(s):  
Boris Weyssow
Keyword(s):  
Charged Particle ◽  
Phase Space ◽  
Equations Of Motion ◽  
Elliptic Functions ◽  
Particle Energy ◽  
Jacobian Elliptic Functions ◽  
Resonance Frequencies ◽  
Trapping Regions ◽  
Definition Of ◽  
Electromagnetic Perturbation

The general Hamiltonian-averaging transformation developed to study the motion of a charged particle in a strong magnetic field and various electromagnetic perturbations permits a clear definition of the dynamics of the guiding centre. In the case of a high-frequency electromagnetic perturbation, the equations of evolution of the phase-space co-ordinates are sums of guiding-centre terms, resonant terms at any of the cyclotron resonance frequencies n(r‖k‖–ω)+lΩ≈0(l and n are integers) and ponderomotive terms. In this paper we consider the n = 1 resonances giving a contribution one order stronger than the ponderomotive terms to the equations of motion. The guiding-centre transformation therefore suffices to derive the leading terms of the averaged dynamics. A simple case of isolated resonance is then considered for which, depending on the value of the external parameters (initial particle energy, amplitude of the perturbation), the phase space may possess one or two trapping regions. Even in the latter situation, the particle trajectories can be described analytically by the set of 12 Jacobian elliptic functions.

Notes on the Unconditional Convergence of Multidimensional Functional Series

2000 ◽  
Vol 7 (2) ◽  
pp. 215-220
Author(s):  
G. Bareladze
Keyword(s):  
Unconditional Convergence ◽  
Partial Sums ◽  
Almost Everywhere Convergence ◽  
Functional Series ◽  
Almost Everywhere ◽  
Definition Of ◽  
The Way

Abstract The convergence of a multidimensional functional series essentially depends on the way its partial sums are formed. Different ways of definition of partial sums lead to different kinds of convergence. In the paper relations between various kinds of unconditional almost everywhere convergence of multidimensional functional series are studied.

scholarly journals What is a firm? A reply to Jean-Philippe Robé

2021 ◽  
pp. 1-11
Author(s):  
Simon Deakin ◽  
David Gindis ◽  
Geoffrey M. Hodgson
Keyword(s):  
Theory Of The Firm ◽  
Recent Book ◽  
Everyday Language ◽  
Modern Capitalism ◽  
Legal Foundations ◽  
Common Parlance ◽  
The Law ◽  
Business Entities ◽  
Definition Of ◽  
The Way

Abstract In his recent book on Property, Power and Politics, Jean-Philippe Robé makes a strong case for the need to understand the legal foundations of modern capitalism. He also insists that it is important to distinguish between firms and corporations. We agree. But Robé criticizes our definition of firms in terms of legally recognized capacities on the grounds that it does not take the distinction seriously enough. He argues that firms are not legally recognized as such, as the law only knows corporations. This argument, which is capable of different interpretations, leads to the bizarre result that corporations are not firms. Using etymological and other evidence, we show that firms are treated as legally constituted business entities in both common parlance and legal discourse. The way the law defines firms and corporations, while the product of a discourse which is in many ways distinct from everyday language, has such profound implications for the way firms operate in practice that no institutional theory of the firm worthy of the name can afford to ignore it.

Jacobian elliptic functions and a family of bivariate peak polynomials

2021 ◽  
Vol 97 ◽  
pp. 103371
Author(s):  
Shi-Mei Ma ◽  
Jun Ma ◽  
Yeong-Nan Yeh ◽  
Roberta R. Zhou
Keyword(s):  
Elliptic Functions ◽  
Jacobian Elliptic Functions

The Place of Interests in Hobbes’s Civil Science

2016 ◽  
Vol 29 (2) ◽  
pp. 105-128
Author(s):  
Elliott Karstadt
Keyword(s):  
The Political ◽  
Crucial Importance ◽  
Political Issue ◽  
Political Ideas ◽  
Form Of Government ◽  
Civil Science ◽  
Definition Of ◽  
The Impact ◽  
The Way

Many scholars argue that Hobbes’s political ideas do not significantly develop between The Elements of Law (1640) and Leviathan (1651). This article seeks to challenge that assumption by studying the way in which Hobbes’s deployment of the vocabulary of ‘interest’ develops over the course of the 1640s. The article begins by showing that the vocabulary is newly important in Leviathan, before attempting a ‘Hobbesian definition’ of what is meant by the term. We end by looking at the impact that the vocabulary has on two key areas of Hobbes’s philosophy: his theory of counsel and his arguments in favour of monarchy as the best form of government. In both areas, Hobbes’s conception of ‘interests’ is shown to be of crucial importance in lending a new understanding of the political issue under consideration.

ANALYTICAL STUDY ON NONLINEAR DIFFERENTIAL–DIFFERENCE EQUATIONS VIA A NEW METHOD

2010 ◽  
Vol 24 (08) ◽  
pp. 761-773
Author(s):  
HONG ZHAO
Keyword(s):  
Difference Equations ◽  
Elliptic Function ◽  
Analytical Study ◽  
Elliptic Functions ◽  
Expansion Method ◽  
Periodic Wave ◽  
Jacobian Elliptic Function ◽  
Jacobian Elliptic Functions ◽  
Periodic Wave Solutions ◽  
Trigonometric Function Solutions

Based on the computerized symbolic computation, a new rational expansion method using the Jacobian elliptic function was presented by means of a new general ansätz and the relations among the Jacobian elliptic functions. The results demonstrated an effective direction in terms of a uniformed construction of the new exact periodic solutions for nonlinear differential–difference equations, where two representative examples were chosen to illustrate the applications. Various periodic wave solutions, including Jacobian elliptic sine function, Jacobian elliptic cosine function and the third elliptic function solutions, were obtained. Furthermore, the solitonic solutions and trigonometric function solutions were also obtained within the limit conditions in this paper.

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