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.

Coefficients of Symmetric Functions of Bounded Boundary Rotation

1974 ◽  
Vol 26 (6) ◽  
pp. 1351-1355 ◽  
Author(s):  
Ronald J. Leach
Keyword(s):  
Unit Disc ◽  
Symmetric Functions ◽  
The Family ◽  
Bounded Boundary Rotation ◽  
Boundary Rotation ◽  
Image Position

Let denote the family of all functions of the formthat are analytic in the unit disc U, f′(z) ≠ 0 in U and f maps U onto a domain of boundary rotation at most . Recently Brannan, Clunie and Kirwan [2] and Aharonov and Friedland [1] have solved the problem of estimating |amp+1| for all , provided m = 1.

Inequalities for Generalized Symmetric Functions

1969 ◽  
Vol 12 (5) ◽  
pp. 615-623 ◽  
Author(s):  
K.V. Menon
Keyword(s):  
Symmetric Function ◽  
Symmetric Functions ◽  
Elementary Symmetric Function ◽  
Generating Series ◽  
Image Position ◽  
Complete Symmetric Function

The generating series for the elementary symmetric function Er, the complete symmetric function Hr, are defined byrespectively.

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 POSITIVE SOLUTIONS FOR A CLASS OF p(x)-LAPLACIAN PROBLEMS

2009 ◽  
Vol 51 (3) ◽  
pp. 571-578 ◽  
Author(s):  
G. A. AFROUZI ◽  
H. GHORBANI
Keyword(s):  
Positive Solution ◽  
Positive Solutions ◽  
Symmetric Functions ◽  
Symmetric Domain ◽  
Sign Conditions ◽  
Image Position

AbstractWe consider the system where p(x), q(x) ∈ C1(RN) are radial symmetric functions such that sup|∇ p(x)| < ∞, sup|∇ q(x)| < ∞ and 1 < inf p(x) ≤ sup p(x) < ∞, 1 < inf q(x) ≤ sup q(x) < ∞, where −Δp(x)u = −div(|∇u|p(x)−2∇u), −Δq(x)v = −div(|∇v|q(x)−2∇v), respectively are called p(x)-Laplacian and q(x)-Laplacian, λ1, λ2, μ1 and μ2 are positive parameters and Ω = B(0, R) ⊂ RN is a bounded radial symmetric domain, where R is sufficiently large. We prove the existence of a positive solution when for every M > 0, $\lim_{u \rightarrow +\infty} \frac{h(u)}{u^{p^--1}} = 0$ and $\lim_{u \rightarrow +\infty} \frac{\gamma(u)}{u^{q^--1}} = 0$. In particular, we do not assume any sign conditions on f(0), g(0), h(0) or γ(0).

A note on lemmas of Green and Kondo

1967 ◽  
Vol 63 (1) ◽  
pp. 83-85 ◽  
Author(s):  
A. O. Morris
Keyword(s):  
Symmetric Functions ◽  
Rational Numbers ◽  
Infinite Sets ◽  
Image Position ◽  
Countably Infinite

Let R be the field of rational numbers, {x} = {x1, z2, …}, {y} = {y1, y2, …} be two countably infinite sets of variables and t an indeterminate. Let (λ) = (λ1, λ2, …, λm) be a partition of n. Then Littlewood (5) has shown thatcan be expressed in the formwhere Qλ(x, t) and Qλ(y, t) denote certain symmetric functions on the sets {x} and {y} respectively. In additionwhere is the partition of n conjugate to (λ). In fact, Littlewood (5) showed thatwhere the summation is over all terms obtained by permutations of the variables xi (i = 1, 2, …) and.

Two inequalities for the complete symmetric functions

1978 ◽  
Vol 84 (1) ◽  
pp. 1-3 ◽  
Author(s):  
V. J. Baston
Keyword(s):  
Symmetric Function ◽  
Symmetric Functions ◽  
Positive Definite ◽  
Even Order ◽  
Image Position ◽  
Complete Symmetric Function ◽  
Special Case

In (l) Hunter proved that the complete symmetric functions of even order are positive definite by obtaining the inequalitywhere ht denotes the complete symmetric function of order t. In this note we show that the inequality can be strengthened, which, in turn, enables theorem 2 of (l) to be sharpened. We also obtain a special case of an inequality conjectured by McLeod(2).

III.—Double Binary Forms

1924 ◽  
Vol 43 ◽  
pp. 43-50 ◽  
Author(s):  
H. W. Turnbull
Keyword(s):  
Integral Function ◽  
Quarterly Journal ◽  
Plane Curves ◽  
Binary Forms ◽  
Cartesian Coordinates ◽  
German School ◽  
Symbolic Methods ◽  
Image Position ◽  
Rational Integral

In the Quarterly Journal, No. 162, 1910, Professor A. R. Forsyth considered some of the problems arising from a homographic transformation of plane curves whose equations could be written in the formwhere F is a rational integral function of z and z′, and where z = x + iy, z′ = x − iy determine the rectangular Cartesian coordinates of the plane. It was suggested that the theory could be developed algebraically by using the symbolic methods of the German school which proved so powerful in furthering the theory of binary forms.

Equally Inclined Spheres

1962 ◽  
Vol 58 (2) ◽  
pp. 420-421 ◽  
Author(s):  
J. G. Mauldon
Keyword(s):  
Symmetric Functions ◽  
Linear Equations ◽  
Quadratic Equation ◽  
Cartesian Coordinates ◽  
Elementary Symmetric Functions ◽  
Equation Solving ◽  
Image Position

If five spheres in 3-space are such that each pair is inclined at the same non-zero angle θ, then where b1, …, b5 (the ‘bends’ (1) of the spheres) are the reciprocals of their radii. To prove this result, establish a system of rectangular cartesian coordinates (x, y, z) and let the spheres have centres (xi, yi, zi) and radii , where i = 1,…, 5. Then for x5, y5, z5, r5 we have the equations which, on subtraction, yield three linear equations and one quadratic equation. Solving the three linear equations for x5, y5, z5 and substituting, we see that the required relation is algebraic (indeed quadratic) in r5 and hence in b5. Since it is also symmetric in b1,…, b5, it follows that it can be expressed as a polynomial relation in the elementary symmetric functions p1, …, p5 in b5, …, b5.

Linear Transformations on Algebras of Matrices: The Invariance of the Elementary Symmetric Functions

1959 ◽  
Vol 11 ◽  
pp. 383-396 ◽  
Author(s):  
Marvin Marcus ◽  
Roger Purves
Keyword(s):  
Symmetric Function ◽  
Symmetric Functions ◽  
Similarity Transformation ◽  
Elementary Symmetric Function ◽  
Linear Transformations ◽  
Elementary Symmetric Functions ◽  
Image Position

In this paper we examine the structure of certain linear transformations T on the algebra of w-square matrices Mn into itself. In particular if A ∈ Mn let Er(A) be the rth elementary symmetric function of the eigenvalues of A. Our main result states that if 4 ≤ r ≤ n — 1 and Er(T(A)) = Er(A) for A ∈ Mn then T is essentially (modulo taking the transpose and multiplying by a constant) a similarity transformation:No such result as this is true for r = 1,2 and we shall exhibit certain classes of counterexamples. These counterexamples fail to work for r = 3 and the structure of those T such that E3(T(A)) = E3(A) for all ∈ Mn is unknown to us.

An Inequality for Elementary Symmetric Functions

1972 ◽  
Vol 15 (1) ◽  
pp. 133-135 ◽  
Author(s):  
K. V. Menon
Keyword(s):  
Symmetric Function ◽  
Symmetric Functions ◽  
Geometric Mean ◽  
Binomial Coefficient ◽  
Elementary Symmetric Function ◽  
Elementary Symmetric Functions ◽  
Image Position

Let Er denote the rth elementary symmetric function on α1 α2,…,αm which is defined by1E0 = 1 and Er=0(r>m).We define the rth symmetric mean by2where denote the binomial coefficient. If α1 α2,…,αm are positive reals thenwe have two well-known inequalities3and4In this paper we consider a generalization of these inequalities. The inequality (4) is known as Newton's inequality which contains the arithmetic and geometric mean inequality.

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