On a strong limit-point condition and an integral inequality associated with a symmetric matrix differential expression

1977 ◽  
Vol 76 (2) ◽  
pp. 155-159 ◽  
Author(s):  
D. A. R. Rigler
Keyword(s):  
Differential Expression ◽  
Vector Function ◽  
Real Line ◽  
Integral Inequality ◽  
Limit Point ◽  
Symmetric Matrix ◽  
Strong Limit ◽  
The Matrix ◽  
Point Condition ◽  
Matrix Differential

SynopsisThis paper is concerned with some properties of an ordinary symmetric matrix differential expression M, denned on a certain class of vector-functions, each of which is defined on the real line. For such a vector-function F we have M[F] = −F“ + QF on R, where Q is an n × n matrix whose elements are reasonably behaved on R. M is classified in an equivalent of the limit-point condition at the singular points ± ∞, and conditions on the matrix coefficient Q are given which place M, when n> 1, in the equivalent of the strong limit-point for the case n = 1. It is also shown that the same condition on Q establishes the integral inequality for a certain class of vector-functions F.

A numerical method for the determination of the Titchmarsh-Weyl m -coefficient

1991 ◽  
Vol 435 (1895) ◽  
pp. 535-549 ◽  
Keyword(s):  
Numerical Method ◽  
Error Estimates ◽  
Complex Plane ◽  
Real Line ◽  
Integral Inequality ◽  
Limit Point ◽  
Computational Techniques ◽  
Limit Point Case ◽  
Null Set

A numerical method for determining the Titchmarsh-Weyl m ( λ ) function for the singular eigenvalue equation – ( py' )' + qy = λwy on [ a ,∞), where a is finite, is presented. The algorithm, based on Weyl’s theory, utilizes a result first used by Atkinson to map a point on the real line onto the Weyl circle in the complex plane. In the limit-point case these circles ‘nest’ and tend to the limit-point m ( λ ). Using Weyl’s result for the diameter of the circles, error estimates for m ( λ ) are obtained. In 1971, W. N. Everitt obtained an extension of an integral inequality of Hardy-Littlewood, namely the help inequality. He showed that the existence of that inequality is determined by the properties of the null set of Im[ λ 2 m ( λ )]. In view of the major difficulties in analysing m ( λ ) even in the rare cases when it is given explicitly, very few examples of the help inequality are known. The computational techniques discussed in this paper have been applied to the problem of finding best constants in these inequalities.

Oscillation Criteria for Matrix Differential Equations

1967 ◽  
Vol 19 ◽  
pp. 184-199 ◽  
Author(s):  
H. C. Howard
Keyword(s):  
Existence Theorems ◽  
Symmetric Matrix ◽  
Order System ◽  
Matrix Differential Equation ◽  
First Order ◽  
The Matrix ◽  
Matrix Differential Equations ◽  
Matrix Differential ◽  
Image Position

We shall be concerned at first with some properties of the solutions of the matrix differential equation1.1whereis an n × n symmetric matrix whose elements are continuous real-valued functions for 0 < x < ∞, and Y(x) = (yij(x)), Y″(x) = (y″ ij(x)) are n × n matrices. It is clear such equations possess solutions for 0 < x < ∞, since one can reduce them to a first-order system and then apply known existence theorems (6, Chapter 1).

Pell Numbers, Pell–Lucas Numbers and Modular Group

2007 ◽  
Vol 14 (01) ◽  
pp. 97-102 ◽  
Author(s):  
Q. Mushtaq ◽  
U. Hayat
Keyword(s):  
Modular Group ◽  
Symmetric Matrix ◽  
Lucas Numbers ◽  
Lucas Number ◽  
Pell Numbers ◽  
The Matrix

We show that the matrix A(g), representing the element g = ((xy)2(xy2)2)m (m ≥ 1) of the modular group PSL(2,Z) = 〈x,y : x2 = y3 = 1〉, where [Formula: see text] and [Formula: see text], is a 2 × 2 symmetric matrix whose entries are Pell numbers and whose trace is a Pell–Lucas number. If g fixes elements of [Formula: see text], where d is a square-free positive number, on the circuit of the coset diagram, then d = 2 and there are only four pairs of ambiguous numbers on the circuit.

scholarly journals An induced real quaternion spherical ensemble of random matrices

2017 ◽  
Vol 06 (01) ◽  
pp. 1750001
Author(s):  
Anthony Mays ◽  
Anita Ponsaing
Keyword(s):  
Real Line ◽  
Relative Size ◽  
Functional Differentiation ◽  
The Real ◽  
Real Quaternion ◽  
Large Matrix ◽  
Wall Boundary Conditions ◽  
The Matrix ◽  
Soft Wall ◽  
Point Correlation

We study the induced spherical ensemble of non-Hermitian matrices with real quaternion entries (considering each quaternion as a [Formula: see text] complex matrix). We define the ensemble by the matrix probability distribution function that is proportional to [Formula: see text] These matrices can also be constructed via a procedure called ‘inducing’, using a product of a Wishart matrix (with parameters [Formula: see text]) and a rectangular Ginibre matrix of size [Formula: see text]. The inducing procedure imposes a repulsion of eigenvalues from [Formula: see text] and [Formula: see text] in the complex plane with the effect that in the limit of large matrix dimension, they lie in an annulus whose inner and outer radii depend on the relative size of [Formula: see text], [Formula: see text] and [Formula: see text]. By using functional differentiation of a generalized partition function, we make use of skew-orthogonal polynomials to find expressions for the eigenvalue [Formula: see text]-point correlation functions, and in particular the eigenvalue density (given by [Formula: see text]). We find the scaled limits of the density in the bulk (away from the real line) as well as near the inner and outer annular radii, in the four regimes corresponding to large or small values of [Formula: see text] and [Formula: see text]. After a stereographic projection, the density is uniform on a spherical annulus, except for a depletion of eigenvalues on a great circle corresponding to the real axis (as expected for a real quaternion ensemble). We also form a conjecture for the behavior of the density near the real line based on analogous results in the [Formula: see text] and [Formula: see text] ensembles; we support our conjecture with data from Monte Carlo simulations of a large number of matrices drawn from the [Formula: see text] induced spherical ensemble. This ensemble is a quaternionic analog of a model of a one-component charged plasma on a sphere, with soft wall boundary conditions.

17.—The Limit-point and Limit-circle Cases for Polynomials in a Differential Operator

1975 ◽  
Vol 72 (3) ◽  
pp. 219-224 ◽  
Author(s):  
Anton Zettl
Keyword(s):  
Differential Operator ◽  
Differential Expression ◽  
Limit Point ◽  
Half Line ◽  
Limit Circle

SynopsisThis paper is concerned with the L2 classification of ordinary symmetrical differential expressions defined on a half-line [0, ∞) and obtained from taking formal polynomials of symmetric differential expression. The work generalises results in this area previously obtained by Chaudhuri, Everitt, Giertz and the author.

Limit-point and limit-circle criteria for Sturm-Liouville equations with intermittently negative principal coefficients

1986 ◽  
Vol 103 (3-4) ◽  
pp. 215-228 ◽  
Author(s):  
W. N. Everitt ◽  
I. W. Knowles ◽  
T. T. Read
Keyword(s):  
Differential Expression ◽  
Limit Point ◽  
Limit Circle ◽  
Almost Everywhere ◽  
Sturm Liouville ◽  
Image Position

SynopsisLimit-point and limit-circle criteria are given for the generalised Sturm-Liouville differential expressionwhere(i) p, q, and w are real-valued on [a, b),(ii) p−1, q, w are locally Lebesgue integrable on [a, b),(iii) w > 0 almost everywhere on [a, b) and the principal coefficient p is allowed toassume both positive and negative values.

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