On a Theory of the Conjugate Relations of two rational Integral Functions, comprising an application to the Theory of Sturm's Functions, and that of the greatest Algebraical Common Measure. By J. J. Sylvester, Esq., M.A., F.R.S., Barrister-at-Law.

1854 ◽  
Vol 4 (2) ◽  
pp. 142-145
Keyword(s):  
Rational Integral

The determination of units in totally real cubic fields

1960 ◽  
Vol 56 (4) ◽  
pp. 318-321 ◽  
Author(s):  
H. J. Godwin
Keyword(s):  
Trial And Error ◽  
Integral Lattice ◽  
Cubic Field ◽  
Cubic Fields ◽  
Trial And Error Method ◽  
Totally Real ◽  
Rational Integral ◽  
Fundamental Units ◽  
Error Method

The determination of a pair of fundamental units in a totally real cubic field involves two operations—finding a pair of independent units (i.e. such that neither is a power of the other) and from these a pair of fundamental units (i.e. a pair ε1; ε2 such that every unit of the field is of the form with rational integral m, n). The first operation may be accomplished by exploring regions of the integral lattice in which two conjugates are small or else by factorizing small primes and comparing different factorizations—a trial-and-error method, but often a quick one. The second operation is accomplished by obtaining inequalities which must be satisfied by a fundamental unit and its conjugates and finding whether or not a unit exists satisfying these inequalities. Recently Billevitch ((1), (2)) has given a method, of the nature of an extension of the first method mentioned above, which involves less work on the second operation but no less on the first.

Classes of Positive Definite Unimodular Circulants

1957 ◽  
Vol 9 ◽  
pp. 71-73 ◽  
Author(s):  
Morris Newman ◽  
Olga Taussky
Keyword(s):  
Positive Definite ◽  
Image Position ◽  
Rational Integral

All matrices considered here have rational integral elements. In particular some circulants of this nature are investigated. An n × n circulant is of the formThe following result concerning positive definite unimodular circulants was obtained recently (3 ; 4 ):Let C be a unimodular n × n circulant and assume that C = AA' where A is an n × n matrix and A' its transpose. Then it follows that C = C1C1', where C1 is again a circulant.

scholarly journals Note on Numerical Integration

1928 ◽  
Vol 1 (3) ◽  
pp. 139-148 ◽  
Author(s):  
C. E. Wolff
Keyword(s):  
Numerical Integration ◽  
Integral Function ◽  
Tacit Assumption ◽  
Rational Integral

All the commonly used rules for the approximate quadrature of areas, such as those of Cotes, Simpson, Tchebychef and Gauss, are based on the assumption that y can be expressed as a rational integral function of x with finite coefficients. A tacit assumption is thus made that is not infinite within the range considered, and it is therefore hardly a matter for surprise that the degree of accuracy obtainable by the use of these rules in the case of a curve which touches the end ordinates is very poor.

scholarly journals A Theorem on Rational Integral Symmetric Functions

1929 ◽  
Vol 24 ◽  
pp. i-iii
Author(s):  
John Dougall
Keyword(s):  
Symmetric Functions ◽  
Image Position ◽  
Rational Integral

An identity involving symmetric functions of n letters may in a certain class of cases be extended immediately to a greater number of letters.For example, the theoremmay be writtenand in the latter form it is true for any number of letters.

scholarly journals The unimodality of a polynomial coming from a rational integral. Back to the original proof

2014 ◽  
Vol 420 (2) ◽  
pp. 1154-1166 ◽  
Author(s):  
Tewodros Amdeberhan ◽  
Atul Dixit ◽  
Xiao Guan ◽  
Lin Jiu ◽  
Victor H. Moll
Keyword(s):  
Original Proof ◽  
Rational Integral

scholarly journals V. On the locus of singular points and lines which occur in connection the theory of the locus of ultimate intersections of a system of surfaces

1892 ◽  
Vol 183 ◽  
pp. 141-278
Keyword(s):  
Differential Equations ◽  
London Mathematical Society ◽  
Integral Function ◽  
Singular Points ◽  
Mathematical Society ◽  
First Order ◽  
Rational Integral

In a paper “On the c - and p -Discriminants of Ordinary Integrable Differential Equations of the First Order,” published in vol. 19 of the ‘Proceedings of the London Mathematical Society,' the factors which occur in the c -discriminant of an equation of the form f ( x, y, c ) = 0, where f ( x, y, c ) is a rational integral function of x, y, c , are determined analytically. It is shown that if E = 0 be the equation of the envelope locus of the curves f ( x, y, c ) = 0; if N = 0 be the equation of their node-locus; if C = 0 be the equation of their cusp-locus, then the factors of the discriminant are E, N 2 , C 3 .

scholarly journals A Theorem in Algebra

1886 ◽  
Vol 5 ◽  
pp. 59-61
Author(s):  
J. L. Mackenzie
Keyword(s):  
Integral Function ◽  
Rational Integral

If we have given two equations φ(x) = 0 and ψ(y) = 0, it is possible to express in the form of a determinant the equation whose roots are f(x, y), where f is any given rational integral function.

On a theory of the conjugate relations of two rational integral functions, comprising an application to the theory of sturm’s functions, and that of the greatest algebraical common measure

1854 ◽  
Vol 6 ◽  
pp. 324-327
Keyword(s):  
Direct Method ◽  
The Common ◽  
The Given ◽  
A Direct Method ◽  
Rational Integral ◽  
Simplified Form

The memoir consists of four sections. In the first section, the theory of the residues obtained by applying the process of the common measure to two algebraical functions is discussed. It is shown that a certain superfluous or allotrious factor enters into each, the value of which, in terms of the leading coefficients of the residues in their simplified form, is determined; and the simplified residues themselves are subsequently obtained from the given functions by a direct method.

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