scholarly journals Linear differential equations for the resolvents of the classical matrix ensembles

2020 ◽  
pp. 2250003
Author(s):  
Anas A. Rahman ◽  
Peter J. Forrester
Keyword(s):  
Differential Equations ◽  
Spectral Density ◽  
Spectral Moments ◽  
Absolute Value ◽  
First Order ◽  
Jacobi Weights ◽  
Matrix Formula ◽  
Linear Differential ◽  
Gaussian Case ◽  
Systematic Derivation

The spectral density for random matrix [Formula: see text] ensembles can be written in terms of the average of the absolute value of the characteristic polynomial raised to the power of [Formula: see text], which for even [Formula: see text] is a polynomial of degree [Formula: see text]. In the cases of the classical Gaussian, Laguerre, and Jacobi weights, we show that this polynomial, and moreover, the spectral density itself, can be characterized as the solution of a linear differential equation of degree [Formula: see text]. This equation, and its companion for the resolvent, are given explicitly for [Formula: see text] and [Formula: see text] for all three classical cases, and also for [Formula: see text] in the Gaussian case. Known dualities for the spectral moments relating [Formula: see text] to [Formula: see text] then imply corresponding differential equations in the case [Formula: see text], and for the Gaussian ensemble, the case [Formula: see text]. We apply the differential equations to give a systematic derivation of recurrences satisfied by the spectral moments and by the coefficients of their [Formula: see text] expansions, along with first-order differential equations for the coefficients of the [Formula: see text] expansions of the corresponding resolvents. We also present the form of the differential equations when scaled at the hard or soft edges.

The Wave Theory of the Electron

1928 ◽  
Vol 24 (4) ◽  
pp. 501-505 ◽  
Author(s):  
J. M. Whittaker
Keyword(s):  
Wave Function ◽  
Differential Equations ◽  
Fourth Order ◽  
Commutative Algebra ◽  
Wave Theory ◽  
Wave Functions ◽  
Linear Differential Equations ◽  
Wave Mechanics ◽  
First Order ◽  
Linear Differential

In two recent papers Dirac has shown how the “duplexity” phenomena of the atom can be accounted for without recourse to the hypothesis of the spinning electron. The investigation is carried out by the methods of non-commutative algebra, the wave function ψ being a matrix of the fourth order. An alternative presentation of the theory, using the methods of wave mechanics, has been given by Darwin. The four-rowed matrix ψ is replaced by four wave functions ψ1, ψ2, ψ3, ψ4 satisfying four linear differential equations of the first order. These functions are related to one particular direction, and the work can only be given invariance of form at the expense of much additional complication, the four wave functions being replaced by sixteen.

scholarly journals Hyers-Ulam Stability of the First-Order Matrix Differential Equations

2015 ◽  
Vol 2015 ◽  
pp. 1-7
Author(s):  
Soon-Mo Jung
Keyword(s):  
Differential Equations ◽  
Variable Coefficients ◽  
Linear Differential Equations ◽  
Ulam Stability ◽  
Order Linear ◽  
First Order ◽  
Matrix Differential Equations ◽  
Homogeneous Matrix ◽  
Linear Differential ◽  
Matrix Differential

We prove the generalized Hyers-Ulam stability of the first-order linear homogeneous matrix differential equationsy→'(t)=A(t)y→(t). Moreover, we apply this result to prove the generalized Hyers-Ulam stability of thenth order linear differential equations with variable coefficients.

On reducible non-linear differential equations occurring in mechanics

1953 ◽  
Vol 217 (1130) ◽  
pp. 327-343 ◽  
Keyword(s):  
Differential Equations ◽  
Basic Problem ◽  
Order Differential Equation ◽  
Linear Differential Equations ◽  
Qualitative Information ◽  
First Order ◽  
Internal Ballistics ◽  
Integral Curves ◽  
Non Linear ◽  
Linear Differential

The purpose of this paper is to present a new method of approach to certain problems in mechanics which give rise to ordinary non-linear differential equations of the second order. The method, which is based on the topology of the integral curves of a first-order differential equation, aims at providing qualitative information which can be used, if necessary, in guiding numerical calculations of the solutions. Among the equations discussed are those of Emden and Blasius, which occur in astrophysics and in boundary-layer theory respectively; these, together with the equation of a basic problem of internal ballistics, are shown to be reducible to different forms of the same first-order equation, which is itself of a type studied originally by Poincaré in another connexion.

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