The Multiplication of an Alternant by a Symmetric Function of the Variables

1899 ◽  
Vol 22 ◽  
pp. 539-542
Author(s):  
Thomas Muir
Keyword(s):  
Symmetric Function ◽  
Non Linear ◽  
Two Indices ◽  
Image Position

(1) As is well known, the simplest form of alternant isand the problem of multiplying it by any symmetric function of a, b, c, d, … has been in a manner fully solved.(2) When the symmetric function is linear in each of the variables—that is to say, when it takes any of the forms Σa, Σab, Σabc, ….—the result is an alternant got from the multiplicand by increasing the last index, the last two indices, the last three indices, …. respectively by 1. Thus, writing for shortness' sake five variables only, we haveThis was first established in 1825 by Schweins in his Theorie der Differenzen und Differentiale, p. 378; but it is also barely possible that it was known to Prony in 1795 (see Journ. de l'Ec. Polyt., i. pp. 264, 265), and Cauchy in 1812 (see Journ. de l'Ec. Polyt., x. pp. 49, 50).(3) When the symmetric function is non-linear, the result takes the form not of one alternant, but of an aggregate of alternants. These cannot be so readily specified, but the mode of obtaining them can be made clear without any difficulty. Let us take the case of the function Σa3b the multiplicand being ∣a0b1c2d3∣.

The Static Aeroelasticity of Slender Configurations

1963 ◽  
Vol 14 (1) ◽  
pp. 75-104 ◽  
Author(s):  
G. J. Hancock
Keyword(s):  
Static Stability ◽  
Aerodynamic Forces ◽  
Flexible Aircraft ◽  
The Past ◽  
Plate Models ◽  
Non Linear ◽  
The Future ◽  
Static Aeroelasticity ◽  
Definition Of ◽  
Image Position

SummaryThe validity and applicability of the static margin (stick fixed) Kn,where as defined by Gates and Lyon is shown to be restricted to the conventional flexible aircraft. Alternative suggestions for the definition of static margin are put forward which can be equally applied to the conventional flexible aircraft of the past and the integrated flexible aircraft of the future. Calculations have been carried out on simple slender plate models with both linear and non-linear aerodynamic forces to assess their static stability characteristics.

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.

scholarly journals Non-linear integral equations for Heun functions

1969 ◽  
Vol 16 (4) ◽  
pp. 281-289 ◽  
Author(s):  
B. D. Sleeman
Keyword(s):  
Integral Equations ◽  
Heun Equation ◽  
Mathieu Functions ◽  
Special Cases ◽  
Mathieu’S Equation ◽  
Heun Functions ◽  
Non Linear ◽  
Linear Integral ◽  
Image Position ◽  
Linear Integral Equations

Some years ago Lambe and Ward (1) and Erdélyi (2) obtained integral equations for Heun polynomials and Heun functions. The integral equations discussed by these authors were of the formFurther, as is well known, the Heun equation includes, among its special cases, Lamé's equation and Mathieu's equation and so (1.1) may be considered a generalisation of the integral equations satisfied by Lamé polynomials and Mathieu functions. However, integral equations of the type (1.1) are not the only ones satisfied by Lamé polynomials; Arscott (3) discussed a class of non- linear integral equations associated with these functions. This paper then is concerned with discussing the existence of non-linear integral equations satisfied by solutions of Heun's equation.

scholarly journals Simple eigenvalues and bifurcation for a multiparameter problem

1988 ◽  
Vol 31 (1) ◽  
pp. 77-88 ◽  
Author(s):  
D. F. McGhee ◽  
M. H. Sallam
Keyword(s):  
Banach Spaces ◽  
Simple Eigenvalue ◽  
Linear Mapping ◽  
Spectral Parameters ◽  
Non Linear ◽  
Multiparameter Problem ◽  
Image Position ◽  
Bifurcation Of Solutions

We are concerned with the problem of bifurcation of solutions of a non-linear multiparameter problem at a simple eigenvalue of the linearised problem.Let X and Y be real Banach spaces, and let A, Bi, i = 1, …, n∈B(X, Y). Let : Rn × X → Y be a non-linear mapping. We consider the equationwhereand λ=(λ1, λ2,…,λn) ∈ Rn is an n-tuple of spectral parameters.

A Note on the Bending of Beams Made of Non-Linear Material

1963 ◽  
Vol 67 (626) ◽  
pp. 124-125
Author(s):  
C. R. Calladine
Keyword(s):  
Creep Behaviour ◽  
General Behaviour ◽  
Channel Section ◽  
Linear Elastic ◽  
Simple Tension ◽  
Non Linear ◽  
Walled Channel ◽  
Linear Material ◽  
Image Position ◽  
Bending Of Beams

Patel And Venkatraman have studied the general behaviour in bending of a thin-walled channel section (Fig. 1) made of non-linear elastic material which deforms in simple tension according to the law where ε represents strain, σ direct stress and λ, n the relevant material properties. (It is understood that a negative sign is inserted in where this is necessary to make σ and ε have the same sign.) As Patel and Venkatraman explain, the results are, by a trivial substitution, applicable to steady creep behaviour of the channel section.

Iterative bounds for the stable solutions of convex non-linear boundary value problems

1977 ◽  
Vol 76 (2) ◽  
pp. 81-94 ◽  
Author(s):  
J. W. Mooney ◽  
G. F. Roach
Keyword(s):  
Boundary Value Problems ◽  
Differential Expression ◽  
Boundary Value ◽  
Linear Boundary ◽  
Non Linear ◽  
Unstable Solutions ◽  
Neumann Type ◽  
Thermal Combustion ◽  
Stable Solutions ◽  
Image Position

SynopsisWe consider a class of convex non-linear boundary value problems of the formwhere L is a linear, uniformly elliptic, self-adjoint differential expression, f is a given non-linear function, B is a boundary differential expression of either Dirichlet or Neumann type and D is a bounded open domain with boundary ∂D. Particular problems of this class arise in the process of thermal combustion [8].In this paper we show that stable solutions of this class can be bounded from below (above) by a monotonically increasing (decreasing) sequence of Newton (Picard) iterates. The possibility of using these schemes to construct unstable solutions is also considered.

scholarly journals Properties of Alfvén Solitons in a Finite-Beta Plasma J. Plasma Phys. vol. 27, 1982, p. 193

1984 ◽  
Vol 32 (2) ◽  
pp. 347-347 ◽  
Author(s):  
Steven R. Spangler ◽  
James P. Sheerin
Keyword(s):  
Soliton Solutions ◽  
Alfven Waves ◽  
Plasma Phys ◽  
Alfvén Waves ◽  
Envelope Soliton ◽  
Translational Invariance ◽  
Non Linear ◽  
Image Position

In the aforementioned paper we obtained an equation for non-linear Alfvén waves in a finite-β plasma, and investigated envelope soliton solutions thereof. The purpose of this note is to point out an error in the derivation of the soliton envelopes, and present corrected expressions for these solitons.The error arises from our assumption of translational invariance of both the envelope and phase of an envelope soliton expressed in equations (16) and (17). Rather, the phase is related to the amplitude by where y ≡ x — VEt is a comoving co-ordinate, and all other quantities are defined in the above paper.

Finite bang-bang controllability for certain non-linear systems

1977 ◽  
Vol 77 (1-2) ◽  
pp. 137-144 ◽  
Author(s):  
Gunnar Aronsson
Keyword(s):  
Control System ◽  
Sufficient Conditions ◽  
Linear Control ◽  
Usual Sense ◽  
Unperturbed System ◽  
Control Functions ◽  
Bounded Controls ◽  
Non Linear ◽  
Non Linear Systems ◽  
Image Position

SynopsisThis paper gives sufficient conditions ensuring that a non-linear control system of the formis controllable by means of control functions u(t), such that each ui(t) only takes two values, with a finite number of switches. It is assumed that the ‘unperturbed’ system ẋ = A(t)x + B(t)u is controllable in the usual sense, i.e. by measurable and bounded controls.

Solutions of a non-linear differential equation. I

1967 ◽  
Vol 63 (3) ◽  
pp. 743-754 ◽  
Author(s):  
C. E. Billigheimer
Keyword(s):  
Differential Equation ◽  
Linear Differential Equation ◽  
Non Linear ◽  
Linear Differential ◽  
Image Position

We consider in this paper solutions of the equationwhere the primes indicate differentiation with respect to s, and a, b, c are constants.

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).

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