Bounds for the Roots of a Trinomial Equation

1940 ◽  
Vol 47 (7) ◽  
pp. 468 ◽  
Author(s):  
E. C. Kennedy
Keyword(s):  
Trinomial Equation

scholarly journals X. On Hamilton's numbers

1887 ◽  
Vol 178 ◽  
pp. 285-312
Keyword(s):  
General Equation ◽  
Minimum Degree ◽  
A Priori ◽  
Senior Author ◽  
System Of Equations ◽  
Absolute Minimum ◽  
Doctor Of Philosophy ◽  
The University ◽  
The Given ◽  
Trinomial Equation

In the year 1786 Erland Samuel Bring, Professor at the University of Lund in Sweden, showed how by an extension of the method of Tschirnhausen it was possible to deprive the general algebraical equation of the 5th degree of three of its terms without solving an equation higher than the 3rd degree. By a well-understood, however singular, academical fiction, this discovery was ascribed by him to one of his own pupils, a certain Sven Gustaf Sommelius, and embodied in a thesis humbly submitted to himself for approval by that pupil, as a preliminary to his obtaining his degree of Doctor of Philosophy in the University. The process for effecting this reduction seems to have been overlooked or forgotten, and was subsequently re-discovered many years later by Mr. Jerrard. In a report contained in the ‘Proceedings of the British Association’ for 1836, Sir William Hamilton showed that Mr. Jerrard was mistaken in supposing that the method was adequate to taking away more than three terms of the equation of the 5th degree, but supplemented this somewhat unnecessary refutation of a result, known à priori to be impossible, by an extremely valuable discussion of a question raised by Mr. Jerrard as to the number of variables required in order that any system of equations of given degrees in those variables shall admit of being satisfied without solving any equation of a degree higher than the highest of the given degrees. In the year 1886 the senior author of this memoir showed in a paper in Kronecker'e (better known as Crelle’s ) ‘Journal that the trinomial equation of the 5th degree, upon which by Bring’s method the general equation of that degree can be made to depend, has necessarily imagmaiy coefficients except in the case where four of the roots of the original equation are imaginary, and also pointed out method of obtaining the absolute minimum degree M of an equation from which an given number of specified terms can be taken away subject to the condition of no having to solve any equation of a degree higher than M. The numbers furnished be Hamilton’s method, it is to be observed, are not minima unless a more stringer condition than this is substituted, viz., that the system of equations which have to be resolved in order to take away the proposed terms shall be the simplest possible i. e ., of the lowest possible weight and not merely of the lowest order; in the memo: in ‘Crelle,’ above referred to, he has explained in what sense the words weight an order are here employed. He has given the name of Hamilton’s Numbers to these relative minima (minima, i. e ., in regard to weight) for the case where the terms to be taken away from the equation occupy consecutive places in it, beginning with the second.

Series for all the Roots of a Trinomial Equation

1939 ◽  
Vol 46 (7) ◽  
pp. 422 ◽  
Author(s):  
Albert Eagle
Keyword(s):  
Trinomial Equation

scholarly journals Hausa vowels and dipthongs

1985 ◽  
Vol 16 (2) ◽  
pp. 161-182
Author(s):  
Mona Lindau-Webb
Keyword(s):  
Speech Production ◽  
Acoustic Properties ◽  
Formant Frequencies ◽  
Short Vowels ◽  
Trinomial Equation

Acoustic properties of Hausa vowels and diphthongs from several speakers are investigated. The results show that Hausa is best described as having a five vowel system, where the five basic vowels have the qualities of the long vowels. Long vowels are derived as double basic vowels. The qualities of the short vowels are significantly different from those of the long vowels, but these quality differences can be accounted for by an undershoot mechanism in the speech production. The diphthong /au/ is modeled, using the formant frequencies of /a/ and /u/ with an interpolation in accordance with a trinomial equation. The diphthong /ai/ is mostly realized as a long [e:] phonetically.

scholarly journals Behavior of the Trinomial ArcsB(n,k,r)when0<α<1

2007 ◽  
Vol 2007 ◽  
pp. 1-8
Author(s):  
Kaoutar Lamrini Uahabi ◽  
Mohammed Zaoui
Keyword(s):  
Real Number ◽  
Unit Disk ◽  
Complex Number ◽  
Polar Coordinates ◽  
The Family ◽  
Trinomial Equation

We deal with the familyB(n,k,r)of trinomial arcs defined as the set of roots of the trinomial equationzn=αzk+(1−α), wherez=ρeiθis a complex number,nandkare two integers such that0<k<n, andαis a real number between0and1. These arcsB(n,k,r)are continuous arcs inside the unit disk, expressed in polar coordinates(ρ,θ). The question is to prove thatρ(θ)is a decreasing function, for each trinomial arcB(n,k,r).

scholarly journals Trinomial equation: the Hypergeometric way

2021 ◽  
Vol 5 (1) ◽  
pp. 236-247
Author(s):  
Daniele Ritelli ◽  
◽  
Giulia Spaletta ◽  
Keyword(s):  
Nineteenth Century ◽  
Power Series ◽  
Computer Algebra ◽  
Free Parameter ◽  
Elliptic Functions ◽  
Late Nineteenth Century ◽  
Algebraic Equations ◽  
Analytical Treatment ◽  
Late Nineteenth ◽  
Trinomial Equation

This paper is devoted to the analytical treatment of trinomial equations of the form \(y^n+y=x,\) where \(y\) is the unknown and \(x\in\mathbb{C}\) is a free parameter. It is well-known that, for degree \(n\geq 5,\) algebraic equations cannot be solved by radicals; nevertheless, roots are described in terms of univariate hypergeometric or elliptic functions. This classical piece of research was founded by Hermite, Kronecker, Birkeland, Mellin and Brioschi, and continued by many other Authors. The approach mostly adopted in recent and less recent papers on this subject (see [<a href="#1">1</a>,<a href="#2">2</a>] for example) requires the use of power series, following the seminal work of Lagrange [<a href="#3">3</a>]. Our intent is to revisit the trinomial equation solvers proposed by the Italian mathematician Davide Besso in the late nineteenth century, in consideration of the fact that, by exploiting computer algebra, these methods take on an applicative and not purely theoretical relevance.

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