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.

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).

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.

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.

MATRIX STUDY OF THE EQUATION OF SOLID RIGID MOTIONS

2021 ◽  
Vol 10 (9) ◽  
pp. 3195-3207
Author(s):  
K. Atchonouglo ◽  
K. Nwuitcha
Keyword(s):  
Symmetric Matrix ◽  
Equations Of Motion ◽  
Center Of Mass ◽  
Positive Definite ◽  
Matrix Formulation ◽  
Total Body Mass ◽  
Positive Definite Symmetric Matrix ◽  
Rigid Motions ◽  
Definite Symmetric Matrix ◽  
Total Body

In this article, we described the equations of motion of a rigid solid by a matrix formulation. The matrices contained in our movement description are homogeneous to the same unit. Inertial characteristics are met in a 4x4 positive definite symmetric matrix called "tensor generalized Poinsot." This matrix consists of 3x3 positive definite symmetric matrix called "inertia tensor Poinsot", the coordinates of the center of mass multiplied by the total body mass and the total mass of the rigid body. The equations of motion are formulated as a gender skew 4x4 matrices. They summarize the "principle of fundamental dynamics". The Poinsot generalized tensor appears linearly in this equality as required by the linear dependence of the equations of motion with the ten characteristics inertia of the rigid solid.

A Dual Ellipse Is a Cylindroid

1998 ◽  
Author(s):  
Ashitava Ghosal ◽  
Bahram Ravani
Keyword(s):  
Rigid Body ◽  
Velocity Vector ◽  
Symmetric Matrix ◽  
Positive Definite ◽  
Inner Product ◽  
Dual Numbers ◽  
The Real ◽  
Unit Speed ◽  
Definite Symmetric Matrix ◽  
Two Degree Of Freedom

Abstract In this paper, we take a relook at two-degree-of-freedom instantaneous rigid body kinematics in terms of dual numbers and vectors, and show that a dual ellipse is a cylindroid. The instantaneous angular and linear velocities of a rigid body is expressed as a dual velocity vector, and the inner product of two dual vectors, as a dual number, is used. We show that the tip of a dual velocity vector lies on a dual ellipse, and the maximum and minimum magnitude of the dual velocity vector, for a unit speed motion, can be obtained as eigenvalues of a positive definite, symmetric matrix whose elements are the dual numbers from the inner products. From the real and dual parts of the equation of the dual ellipse, we derive the equation of a cylindroid (Ball,1900).

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