scholarly journals On Appell's Function P (θ, φ)

1932 ◽  
Vol 3 (1) ◽  
pp. 53-55
Author(s):  
Pierre Humbert
Keyword(s):  
Third Order ◽  
The Third ◽  
Direct Generalization ◽  
Circular Functions ◽  
Image Position

§1. Appell's functions, P (θ, φ), Q (θ, φ) and R (θ, φ) are defined by the expansionwhere j3 = 1, affording, both for the third order and the field of two variables, a very direct generalization of the circular functions, as

On Stability of Solutions of Certain Differential Equations of the Third Order

1967 ◽  
Vol 10 (5) ◽  
pp. 681-688 ◽  
Author(s):  
B.S. Lalli
Keyword(s):  
Differential Equation ◽  
Asymptotic Stability ◽  
Lyapunov Function ◽  
Differential Equations ◽  
Global Asymptotic Stability ◽  
Sufficient Conditions ◽  
Stability Of Solutions ◽  
Third Order ◽  
The Third ◽  
Image Position

The purpose of this paper is to obtain a set of sufficient conditions for “global asymptotic stability” of the trivial solution x = 0 of the differential equation1.1using a Lyapunov function which is substantially different from similar functions used in [2], [3] and [4], for similar differential equations. The functions f1, f2 and f3 are real - valued and are smooth enough to ensure the existence of the solutions of (1.1) on [0, ∞). The dot indicates differentiation with respect to t. We are taking a and b to be some positive parameters.

scholarly journals Asymptotic formulae for linear equations

1972 ◽  
Vol 13 (2) ◽  
pp. 147-152 ◽  
Author(s):  
Don B. Hinton
Keyword(s):  
Asymptotic Behaviour ◽  
Fourth Order ◽  
Linear Equations ◽  
Order Equation ◽  
Third Order ◽  
Fourth Order Equation ◽  
The Third ◽  
Asymptotic Formulae ◽  
Image Position

Numerous formulae have been given which exhibit the asymptotic behaviour as t → ∞solutions ofwhere F(t) is essentially positive and Several of these results have been unified by a theorem of F. V. Atkinson [1]. It is the purpose of this paper to establish results, analogous to the theorem of Atkinson, for the third order equationand for the fourth order equation

A further method for the evaluation of zeros of Bessel functions and some new asymptotic expansions for zeros of functions of large order

1951 ◽  
Vol 47 (4) ◽  
pp. 699-712 ◽  
Author(s):  
F. W. J. Olver
Keyword(s):  
Differential Equation ◽  
Bessel Functions ◽  
Numerical Technique ◽  
Third Order ◽  
New Approach ◽  
The Third ◽  
Zeros Of Bessel Functions ◽  
Zeros Of Functions ◽  
Linear Differential ◽  
Image Position

In a recent paper (1) I described a method for the numerical evaluation of zeros of the Bessel functions Jn(z) and Yn(z), which was independent of computed values of these functions. The essence of the method was to regard the zeros ρ of the cylinder functionas a function of t and to solve numerically the third-order non-linear differential equation satisfied by ρ(t). It has since been successfully used to compute ten-decimal values of jn, s, yn, s, the sth positive zeros* of Jn(z), Yn(z) respectively, in the ranges n = 10 (1) 20, s = 1(1) 20. During the course of this work it was realized that the least satisfactory feature of the new method was the time taken for the evaluation of the first three or four zeros in comparison with that required for the higher zeros; the direct numerical technique for integrating the differential equation satisfied by ρ(t) becomes unwieldy for the small zeros and a different technique (described in the same paper) must be employed. It was also apparent that no mere refinement of the existing methods would remove this defect and that a new approach was required if it was to be eliminated. The outcome has been the development of the method to which the first part (§§ 2–6) of this paper is devoted.

scholarly journals 4. Note on Professor Tait's “Quaternion Path” to Determinants of the Third Order

1869 ◽  
Vol 6 ◽  
pp. 121-125
Author(s):  
Hugh Martin
Keyword(s):  
Third Order ◽  
The Third ◽  
Image Position ◽  
Special Case

I have read with much interest Professor Tait's “Note on Determinants of the Third Order” in the Proceedings of this Session (pp. 59–61), and admire the method of discovering new properties of Determinants. I am not sure, however, that the properties, when discovered, are more difficult of proof by Determinant methods, and I venture to submit the following as simple and elementary:—The first property, namely,is true under greater generality, and the Determinant proof is the same as for the special case.

scholarly journals 1. On the Extraction of the Square Root of a Matrix of the Third Order

1872 ◽  
Vol 7 ◽  
pp. 675-682 ◽  
Author(s):  
Cayley
Keyword(s):  
Second Order ◽  
Square Root ◽  
Third Order ◽  
The Third ◽  
Image Position

Professor Tait has considered the question of finding the square root of a strain, or what is the same thing, that of a matrix of the third order— A mode of doing this is indicated in my “Memoir on the Theory of Matrices” (Phil. Trans., 1858, pp. 17–37), and it is interesting to work out the solution.The notation and method will be understood from the simple case of a matrix of the second order. I write to denote the two equations, x1 = ax + by, y1 = cx + dy.

Disconjugacy Conditions for the Third Order Linear Differential Equation

1969 ◽  
Vol 12 (5) ◽  
pp. 603-613 ◽  
Author(s):  
Lynn Erbe
Keyword(s):  
Differential Equation ◽  
Linear Differential Equation ◽  
Sufficient Conditions ◽  
Order Linear Differential Equation ◽  
Third Order ◽  
Order Linear ◽  
The Third ◽  
Counting Multiplicity ◽  
Linear Differential ◽  
Image Position

An nth order homogeneous linear differential equation is said to be disconjugate on the interval I of the real line in case no non-trivial solution of the equation has more than n - 1 zeros (counting multiplicity) on I. It is the purpose of this paper to establish several necessary and sufficient conditions for disconjugacy of the third order linear differential equation(1.1)where pi(t) is continuous on the compact interval [a, b], i = 0, 1, 2.

scholarly journals On induced permutation matrices and the symmetric group

1936 ◽  
Vol 5 (1) ◽  
pp. 1-13 ◽  
Author(s):  
A. C. Aitken
Keyword(s):  
Symmetric Group ◽  
Matrix Representation ◽  
Zero Element ◽  
Natural Order ◽  
Third Order ◽  
The Third ◽  
Permutation Matrices ◽  
The Matrix ◽  
Matrix Relation ◽  
Image Position

The n! operations Ai of permutations upon n different ordered symbols correspond to n! matrices Ai of the nth order, which have in each row and in each column only one non-zero element, namely a unit. Such matrices Ai are called permutation matrices, since their effect in premultiplying an arbitrary column vector x = {x1x2….xn} is to impress the permutation Ai upon the elements xi. For example the six matrices of the third orderare permutation matrices. It is convenient to denote them bywhere the bracketed indices refer to the permutations of natural order. Clearly the relation Ai Aj = Ak entails the matrix relation AiAj = Ak; in other words, the n! matrices Ai, give a matrix representation of the symmetric group of order n!.

Morphological effects on the third-order nonlinear optical response of polydiacetylene nanofibers

2019 ◽  
Vol 9 (3) ◽  
pp. 1087-1092 ◽  
Author(s):  
Haruki Maki ◽  
Rie Chiba ◽  
Tsunenobu Onodera ◽  
Hitoshi Kasai ◽  
Rodrigo Sato ◽  
...  
Keyword(s):  
Nonlinear Optical ◽  
Optical Response ◽  
Nonlinear Optical Response ◽  
Third Order ◽  
The Third ◽  
Image Position

Abstract

scholarly journals Some Extensions of Pincherle's Polynomials

1920 ◽  
Vol 39 ◽  
pp. 21-24 ◽  
Author(s):  
Pierre Humbert
Keyword(s):  
Differential Equation ◽  
General Theory ◽  
Differential Equations ◽  
Second Order ◽  
Linear Differential Equations ◽  
Third Order ◽  
The Third ◽  
Hypergeometric Type ◽  
Linear Differential ◽  
Image Position

The polynomials which satisfy linear differential equations of the second order and of the hypergeometric type have been the object of extensive work, and very few properties of them remain now hidden; the student who seeks in that direction a subject for research is compelled to look, not after these functions themselves but after generalisations of them. Among these may be set in first place the polynomials connected with a differential equation of the third order and of the extended hypergeometric type, of which a general theory has been given by Goursat. The number of such polynomials of which properties have been studied in particular is rather small; in fact, Appell's polynomialsand Pincherle's polynomials, arising from the expansionsare, so far as I know, the only well-known ones. To show what can be done in these ways, I shall briefly give the definition and principal properties of some polynomials analogous to Pincherle's and of some allied functions.

Initial flow field over an impulsively started circular cylinder

1975 ◽  
Vol 72 (4) ◽  
pp. 625-647 ◽  
Author(s):  
M. Bar-Lev ◽  
H. T. Yang
Keyword(s):  
Circular Cylinder ◽  
Flow Field ◽  
Viscous Fluid ◽  
Stream Function ◽  
Asymptotic Expansions ◽  
Analytic Solutions ◽  
Third Order ◽  
Initial Flow ◽  
The Third ◽  
Circular Functions

The initial flow field of an incompressible, viscous fluid around a circular cylinder, set impulsively to move normal to its axis, is studied in detail. The nonlinear vorticity equation is solved by the method of matched asymptotic expansions. Analytic solutions for the stream function in terms of exponential and error functions for the inner flow field, and of circular functions for the outer, are obtained to the third order, from which a uniformly valid composite solution is found. Also presented are the vorticity, pressure, separation point and drag. These quantities agree with the numerical computations of Collins & Dennis. Extended solutions developed by Padé approximants indicate that higher than third-order approximations will yield only minor improvements.

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