The Third Quadratic Transformation on Elliptic Functions

1970 ◽  
Vol 2 (3) ◽  
pp. 287-289 ◽  
Author(s):  
Patrick du Val
Keyword(s):  
Elliptic Functions ◽  
Quadratic Transformation ◽  
The Third

scholarly journals A dual quadratic transformation associated with the Hessian conics of a pencil

1940 ◽  
Vol 32 ◽  
pp. xiv-xvi
Author(s):  
T. Scott
Keyword(s):  
Quadratic Transformation ◽  
The Third

1. The invariants and covariants of a system of two conics have been much studied2 but little has been said about those of three conies. Three conics have a symmetrical invariant Ω123, or in symbolical notation (a b c)2. According to Ciamberlini3 the vanishing of this invariant signifies that the Φ conic of any two of f1, f2, f3 is inpolar with respect to the third; and in a previous paper4 I have derived by symbolical methods a more symmetrical result, viz., if Ω123 vanishes, then u being any line in the plane, u1, u2, u3 are concurrent, where ui is the polar with respect to fi of the pole of u with respect to Φjk.

scholarly journals IX. The third elliptic integral and the ellipsotomic problem

1904 ◽  
Vol 203 (359-371) ◽  
pp. 217-304
Keyword(s):  
Elliptic Integral ◽  
Great Influence ◽  
Elliptic Functions ◽  
Mechanical Theory ◽  
The Third ◽  
History Of ◽  
Modern Analysis

The Abel Centennial Ceremony, held in Christiania, September, 1902, has directed the attention of mathematicians to the great influence of Abel on modern analysis, and. to the history of elliptic functions, and of the foundation by Crelle of the “ Journal für die reine und angewandte Mathematik.” Abel’s article in the first volume of ‘ Crelle’s Journal,' 1826, " Ueber die Integration der Differential-Formel ρdx ⁄ √R (A), wenn R und ρ gauze Functionen sind,” is of great importance as indicating the existence of what is now called the pseudo-elliptic integral; the present memoir is intended to show the utility of this integral in its application to mechanical theory.

scholarly journals XIII. On a new method of approximation applicable to elliptic and ultra-elliptic functions

1860 ◽  
Vol 150 ◽  
pp. 223-227
Keyword(s):  
Geometric Mean ◽  
Elliptic Functions ◽  
New Method ◽  
Geometric Means ◽  
The Third ◽  
Harmonic Means ◽  
General Method

The difficulty of finding approximate values of elliptic functions of the third kind has led me to consider a general method of approximation, which I believe to be new, at least in its application to the evaluation of integrals of irrational functions. It depends on the known principle that the geometric mean between two quantities is also a geometric mean between their arithmetic and harmonic means. If we take any two positive quantities, we may approximate to their geometric means as follows:— Take the arithmetic and harmonic means of the two quantities, then again take the arithmetic and harmonic means of those means, and so on: the successive means will approximate with great rapidity to the geometric mean.

Reflection and Transmission of Circularly Polarized Elastic Waves of Finite Amplitude

1979 ◽  
Vol 46 (4) ◽  
pp. 867-872 ◽  
Author(s):  
M. M. Carroll
Keyword(s):  
Elastic Waves ◽  
Finite Elasticity ◽  
Elliptic Functions ◽  
Finite Amplitude ◽  
Plane Boundary ◽  
Elastic Solids ◽  
Jacobian Elliptic Functions ◽  
Reflection And Transmission ◽  
The Third ◽  
Fixed Boundaries

Finite amplitude standing wave solutions, obtained previously, are specialized to the case of incompressible isotropic elastic solids with cubic or quintic shear response. This allows closed-form expressions for the motion and stress field, in terms of Jacobian elliptic functions and elliptic integrals and furnishes solutions for approximate finite elasticity theories in which terms up to sixth degree in the stress and strains are retained. The solutions for reflection from free or fixed boundaries, for resonant standing waves in a plate, and for reflection and transmission at a plane boundary are examined in the context of the third and fourth-order approximations.

XVIII. Researches on the geometrical properties of elliptic integrals

1852 ◽  
Vol 142 ◽  
pp. 311-416 ◽  
Keyword(s):  
Elliptic Integral ◽  
Plane Curve ◽  
Elliptic Functions ◽  
Elliptic Integrals ◽  
Mathematical Science ◽  
Third Order ◽  
Geometrical Properties ◽  
First Order ◽  
The Third ◽  
The Right

I. In placing before the Royal Society the following researches on the geometrical types of elliptic integrals, which nearly complete my investigations on this interesting subject, I may be permitted briefly to advert to what bad already been effected in this department of geometrical research. Legendre, to whom this important branch of mathematical science owes so much, devised a plane curve, whose rectification might be effected by an elliptic integral of the first order. Since that time many other geometers have followed his example, in contriving similar curves, to represent, either by their quadrature or rectification, elliptic functions. Of those who have been most successful in devising curves which should possess the required properties, may be mentioned M. Gudermann, M. Verhulst of Brussels, and M. Serret of Paris. These geometers however have succeeded in deriving from those curves scarcely any of the properties of elliptic integrals, even the most elementary. This barrenness in results was doubtless owing to the very artificial character of the genesis of those curves, devised, as they were, solely to satisfy one condition only of the general pro­blem. In 1841 a step was taken in the right direction. MM. Catalan and Gudermann, in the journals of Liouville and Crelle, showed how the arcs of spherical conic sec­tions might be represented by elliptic integrals of the third order and circular form. They did not, however, extend their investigations to the case of elliptic integrals of the third order and logarithmic form; nor even to that of the first order. These cases still remained, without any analogous geometrical representative, a blemish to the theory.

A quadratic mapping with invariant cubic curve

1981 ◽  
Vol 89 (1) ◽  
pp. 89-105 ◽  
Author(s):  
Roger Penrose ◽  
Cedric A. B. Smith
Keyword(s):  
Projective Plane ◽  
Linear Transformation ◽  
Elliptic Functions ◽  
Quadratic Mapping ◽  
Cremona Transformation ◽  
Quadratic Transformation ◽  
Cubic Curve

AbstractIn the projective plane, if H is a harmonic homology (linear transformation with H2 = I), and G a general inversion (quadratic transformation projectively equivalent to an inversion), then under a certain condition there is a pencil of cubics each of which is invariant under G, H separately. These are related to transformations discovered by Mandel, Todd and Lyness. As a near converse, we find that, given a Pascal configuration, there is a quadratic Cremona transformation under which each cubic passing through the vertices of the configuration is invariant. As a by-product, parametric expressions are found for elliptic functions of a fifth of a period.

Interaction Behaviours Between Soliton and Cnoidal Periodic Waves for the Cubic Generalised Kadomtsev–Petviashvili Equation

2015 ◽  
Vol 70 (7) ◽  
pp. 539-544 ◽  
Author(s):  
Bo Ren ◽  
Ji Lin
Keyword(s):  
Elliptic Integral ◽  
Elliptic Functions ◽  
Jacobi Elliptic Functions ◽  
Periodic Waves ◽  
The Third ◽  
Interaction Solutions ◽  
Petviashvili Equation ◽  
Kink Soliton

AbstractThe consistent tanh expansion (CTE) method is applied to the cubic generalised Kadomtsev–Petviashvili (CGKP) equation. The interaction solutions between one kink soliton and the cnoidal periodic waves are explicitly given. Some special concrete interaction solutions in terms of the Jacobi elliptic functions and the third type of incomplete elliptic integral are discussed both in analytical and graphical ways.

Residual Symmetry and Explicit Soliton–Cnoidal Wave Interaction Solutions of the (2+1)-Dimensional KdV–mKdV Equation

2016 ◽  
Vol 71 (4) ◽  
pp. 351-356 ◽  
Author(s):  
Wenguang Cheng ◽  
Biao Li
Keyword(s):  
Elliptic Integral ◽  
Wave Interaction ◽  
Elliptic Functions ◽  
Mkdv Equation ◽  
Jacobi Elliptic Functions ◽  
Cnoidal Wave ◽  
Residual Symmetry ◽  
The Third ◽  
Interaction Solutions ◽  
Consistent Riccati Expansion

AbstractThe truncated Painlevé method is developed to obtain the nonlocal residual symmetry and the Bäcklund transformation for the (2+1)-dimensional KdV–mKdV equation. The residual symmetry is localised after embedding the (2+1)-dimensional KdV–mKdV equation to an enlarged one. The symmetry group transformation of the enlarged system is computed. Furthermore, the (2+1)-dimensional KdV–mKdV equation is proved to be consistent Riccati expansion (CRE) solvable. The soliton–cnoidal wave interaction solution in terms of the Jacobi elliptic functions and the third type of incomplete elliptic integral is obtained by using the consistent tanh expansion (CTE) method, which is a special form of CRE.

Progress report on 21-cm observations at low latitude between longitudes (l 11) 0° and 66°

1967 ◽  
Vol 31 ◽  
pp. 177-179
Author(s):  
W. W. Shane
Keyword(s):  
Preliminary Report ◽  
Progress Report ◽  
Galactic Centre ◽  
Low Latitude ◽  
The Third ◽  
Introductory Lecture ◽  
Centre Region

In the course of several 21-cm observing programmes being carried out by the Leiden Observatory with the 25-meter telescope at Dwingeloo, a fairly complete, though inhomogeneous, survey of the regionl11= 0° to 66° at low galactic latitudes is becoming available. The essential data on this survey are presented in Table 1. Oort (1967) has given a preliminary report on the first and third investigations. The third is discussed briefly by Kerr in his introductory lecture on the galactic centre region (Paper 42). Burton (1966) has published provisional results of the fifth investigation, and I have discussed the sixth in Paper 19. All of the observations listed in the table have been completed, but we plan to extend investigation 3 to a much finer grid of positions.

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