A Remark on a Theorem of Lyapunov

1970 ◽  
Vol 13 (1) ◽  
pp. 141-143 ◽  
Author(s):  
James S. W. Wong
Keyword(s):  
Symmetric Matrix ◽  
Stability Result ◽  
Positive Definite ◽  
Linear Ordinary Differential Equation ◽  
Dimensional Euclidean Space ◽  
Constant Matrix ◽  
Exponentially Stable ◽  
Definite Symmetric Matrix ◽  
The Stability ◽  
Image Position

Consider the linear ordinary differential equation1where x ∊ En, the n-dimensional Euclidean space and A is an n × n constant matrix. Using a matrix result of Sylvester and a stability result of Perron, Lyapunov [4] established the following theorem which is basic in the stability theory of ordinary differential equations:Theorem (Lyapunov). The following three statements are equivalent:(I) The spectrum σ(A) of A lies in the negative half plane.(II) Equation (1) is exponentially stable, i.e. there exist μ, K>0 such that every solution x(t) of (1) satisfies2where ∥ ∥ denotes the Euclidean norm.(III) There exists a positive definite symmetric matrix Q, i.e. Q=Q* and there exist q1,q2>0 such that3satisfying4where I is the identity matrix.

A Two-Point Boundary Problem for Ordinary Self-Adjoint Differential Equations of Fourth Order

1954 ◽  
Vol 6 ◽  
pp. 416-419 ◽  
Author(s):  
H. M. Sternberg ◽  
R. L. Sternberg
Keyword(s):  
Boundary Problem ◽  
Fourth Order ◽  
Symmetric Matrix ◽  
Positive Definite ◽  
Matrix Functions ◽  
Point Boundary ◽  
Vector Identity ◽  
Class C ◽  
Image Position ◽  
Homogeneous Vector

The purpose of this note is to establish Theorem A below for the two-point homogeneous vector boundary problemwhere the Pi(x) are given real m × m symmetric matrix functions of x with P0(x) positive definite and Pi(x) of class C2−i on an infinite interval [a, ∞), and where by a solution of (1.1) — (1.2) for a ≤ x1 < x2 < ∞ we understand a real m-dimensional column vector u = u(x) of class C2 on [a, ∞) which is such that Pi(x)u(2−i) is of class C2−i on [a, ∞) and which satisfies (1.1) — (1.2) with the former a vector identity on [a, ∞).

scholarly journals G2-metrics arising from non-integrable special Lagrangian fibrations

2019 ◽  
Vol 6 (1) ◽  
pp. 348-365 ◽  
Author(s):  
Ryohei Chihara
Keyword(s):  
Dynamical Systems ◽  
Symmetric Matrix ◽  
Positive Definite ◽  
Special Lagrangian ◽  
3 Dimensional ◽  
Dimensional Group ◽  
Constrained Dynamical Systems ◽  
Definite Symmetric Matrix ◽  
Type 3 ◽  
Torsion Free

AbstractWe study special Lagrangian fibrations of SU(3)-manifolds, not necessarily torsion-free. In the case where the fiber is a unimodular Lie group G, we decompose such SU(3)-structures into triples of solder 1-forms, connection 1-forms and equivariant 3 × 3 positive-definite symmetric matrix-valued functions on principal G-bundles over 3-manifolds. As applications, we describe regular parts of G2-manifolds that admit Lagrangian-type 3-dimensional group actions by constrained dynamical systems on the spaces of the triples in the cases of G = T3 and SO(3).

Decomposition of the Multivariate Beta Distribution with Applications

1972 ◽  
Vol 15 (2) ◽  
pp. 225-228 ◽  
Author(s):  
D. G. Kabe ◽  
R. P. Gupta
Keyword(s):  
Beta Distribution ◽  
Symmetric Matrix ◽  
Positive Definite ◽  
Arbitrary Rank ◽  
Positive Definite Symmetric Matrix ◽  
Multivariate Beta ◽  
Explicit Procedure ◽  
Definite Symmetric Matrix ◽  
Multivariate Beta Distribution

SummaryLet L be a positive definite symmetric matrix having a noncentral multivariate beta density of an arbitrary rank, see, e.g. Hayakawa ([2, p. 12, Equation 38]). Then an explicit procedure is given for decomposing the density of L in terms of densities of independent beta variates.

STABILITY ANALYSIS OF THE BANDLIMITED AND BANDPASS LINEAR PREDICTION MODELS

2005 ◽  
Vol 14 (05) ◽  
pp. 1007-1014
Author(s):  
YAN WU
Keyword(s):  
Linear Prediction ◽  
Prediction Models ◽  
Symmetric Matrix ◽  
Positive Definite ◽  
Constructive Method ◽  
Companion Matrix ◽  
Minimum Phase ◽  
Phase Property ◽  
The Matrix ◽  
The Stability

A constructive method of proof is presented for the minimum-phase property of two important linear prediction models, i.e., the bandlimited and bandpass linear prediction models. A generic functional symmetric matrix is first constructed. It is shown that the matrix is positive definite under a mild condition. The result is used, along with a special eigen-structure of the associated companion matrix, to establish the stability of the two linear prediction models.

A theorem in thermoelasticity and its application to linear stability

1989 ◽  
Vol 424 (1866) ◽  
pp. 143-153 ◽  
Keyword(s):  
Small Amplitude ◽  
Wave Speed ◽  
Final Section ◽  
Stability Result ◽  
Positive Definite ◽  
Fourier Synthesis ◽  
Acoustic Tensor ◽  
Thermoelastic Body ◽  
The Stability ◽  
Constant Deformation

Equations are written down governing the propagation of plane sinusoidal waves of small amplitude through a homogeneously prestrained equilibrium state of a materially homogeneous thermoelastic body of arbitrary elastic and thermal symmetry. The symmetric isothermal and isentropic acoustic tensors are defined in the usual way and it is assumed that the former is positive definite, so that it has three real and positive eigenvalues. It is shown, under the usual assumption that the specific heat at constant deformation is positive, that the three real and positive eigenvalues of the isentropic acoustic tensor are interlaced with those of the isothermal acoustic tensor, the smallest eigenvalue belonging to the isothermal and the largest to the isentropic acoustic tensor. Under the additional assumption that the symmetrized thermal conductivity tensor is positive definite, it is further shown that this result on the interlacing of the eigenvalues is sufficient to guarantee, for all positive values of the frequency of the sinusoidal waves, that the material is linearly stable in the sense that sinusoidal waves may not increase without bound in the direction of propagation. In the final section, the wide diversity in behaviour of the complex squared wave speed as a function of frequency is illustrated graphically. The stability result is extended to negative frequencies as these would be required in any Fourier synthesis of the sinusoidal wave solutions. A connection with Whitham’s wave hierarchy approach is mentioned.

scholarly journals A new type of CGO solutions and its applications in corner scattering

2022 ◽  
Author(s):  
Jingni Xiao
Keyword(s):  
Compact Support ◽  
Symmetric Matrix ◽  
Small Neighborhood ◽  
Scalar Function ◽  
Positive Definite ◽  
Incident Field ◽  
Identity Matrix ◽  
Geometric Optics ◽  
New Type ◽  
Definite Symmetric Matrix

Abstract We consider corner scattering for the operator ∇ · γ(x)∇ + k2ρ(x) in R2, with γ a positive definite symmetric matrix and ρ a positive scalar function. A corner is referred to one that is on the boundary of the (compact) support of γ(x) − I or ρ(x) − 1, where I stands for the identity matrix. We assume that γ is a scalar function in a small neighborhood of the corner. We show that any admissible incident field will be scattered by such corners, which are allowed to be concave. Moreover, we provide a brief discussion on the existence of non-scattering waves when γ − I has a jump across the corner. In order to prove the results, we construct a new type of complex geometric optics (CGO) solutions.

Strong Oscillation of Elliptic Equations in General Domains

1973 ◽  
Vol 16 (1) ◽  
pp. 105-110 ◽  
Author(s):  
C. A. Swanson
Keyword(s):  
Differential Equation ◽  
Elliptic Equations ◽  
General Type ◽  
Unbounded Domains ◽  
Elliptic Partial Differential Equation ◽  
Continuous Functions ◽  
Positive Definite ◽  
Dimensional Euclidean Space ◽  
The Matrix ◽  
Image Position

Strong oscillation criteria will be obtained for the linear elliptic partial differential equation(1)in unbounded domains R of general type in n-dimensional Euclidean space En. It will be assumed throughout that B and each Aij are real-valued continuous functions in R, and that the matrix (Aij(x)) is symmetric and positive definite in R.

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