scholarly journals Isomorphic Operators and Functional Equations for the Skew-Circulant Algebra

2014 ◽  
Vol 2014 ◽  
pp. 1-8 ◽  
Author(s):  
Zhaolin Jiang ◽  
Tingting Xu ◽  
Fuliang Lu
Keyword(s):  
Differential Equations ◽  
Vector Space ◽  
Ordinary Differential Equations ◽  
Cyclic Group ◽  
Functional Equations ◽  
Circulant Matrix ◽  
Circulant Matrices ◽  
Linear Vector ◽  
Skew Circulant ◽  
Linear Vector Space

The skew-circulant matrix has been used in solving ordinary differential equations. We prove that the set of skew-circulants with complex entries has an idempotent basis. On that basis, a skew-cyclic group of automorphisms and functional equations on the skew-circulant algebra is introduced. And different operators on linear vector space that are isomorphic to the algebra ofn×ncomplex skew-circulant matrices are displayed in this paper.

Multibody Dynamics on Differentiable Manifolds

2021 ◽  
Vol 16 (4) ◽  
Author(s):  
Edward J. Haug
Keyword(s):  
Euclidean Space ◽  
Differential Equations ◽  
Vector Space ◽  
Ordinary Differential Equations ◽  
Multibody Dynamics ◽  
Equations Of Motion ◽  
Differentiable Manifold ◽  
Kinematics And Dynamics ◽  
Variational Equations ◽  
Differentiable Manifolds

Abstract Topological and vector space attributes of Euclidean space are consolidated from the mathematical literature and employed to create a differentiable manifold structure for holonomic multibody kinematics and dynamics. Using vector space properties of Euclidean space and multivariable calculus, a local kinematic parameterization is presented that establishes the regular configuration space of a multibody system as a differentiable manifold. Topological properties of Euclidean space show that this manifold is naturally partitioned into disjoint, maximal, path connected, singularity free domains of kinematic and dynamic functionality. Using the manifold parameterization, the d'Alembert variational equations of multibody dynamics yield well-posed ordinary differential equations of motion on these domains, without introducing Lagrange multipliers. Solutions of the differential equations satisfy configuration, velocity, and acceleration constraint equations and the variational equations of dynamics, i.e., multibody kinematics and dynamics are embedded in these ordinary differential equations. Two examples, one planar and one spatial, are treated using the formulation presented. Solutions obtained are shown to satisfy all three forms of kinematic constraint to within specified error tolerances, using fourth-order Runge–Kutta numerical integration methods.

Grouping Miller–Nicely by linear vector space rotations

2004 ◽  
Vol 115 (5) ◽  
pp. 2427-2427
Author(s):  
Suvrat Budhlakoti ◽  
Jont B. Allen ◽  
Erik Larsen
Keyword(s):  
Vector Space ◽  
Linear Vector ◽  
Linear Vector Space

scholarly journals On Rotor calculus, I

1966 ◽  
Vol 6 (4) ◽  
pp. 402-423 ◽  
Author(s):  
H. A. Buchdahl
Keyword(s):  
Vector Space ◽  
Lorentz Transformation ◽  
Real World ◽  
Three Dimensional ◽  
Linear Vector ◽  
Complex Rotation ◽  
Starting Point ◽  
Complex Linear ◽  
Definition Of ◽  
Linear Vector Space

SummaryIt is known that to every proper homogeneous Lorentz transformation there corresponds a unique proper complex rotation in a three-dimensional complex linear vector space, the elements of which are here called “rotors”. Equivalently one has a one-one correspondence between rotors and self- dual bi-vectors in space-time (w-space). Rotor calculus fully exploits this correspondence, just as spinor calculus exploits the correspondence between real world vectors and hermitian spinors; and its formal starting point is the definition of certain covariant connecting quantities τAkl which transform as vectors under transformations in rotor space (r-space) and as tensors of valence 2 under transformations in w-space.

scholarly journals The Determinants, Inverses, Norm, and Spread of Skew Circulant Type Matrices Involving Any Continuous Lucas Numbers

2014 ◽  
Vol 2014 ◽  
pp. 1-10 ◽  
Author(s):  
Jin-jiang Yao ◽  
Zhao-lin Jiang
Keyword(s):  
Circulant Matrix ◽  
Circulant Matrices ◽  
Lucas Numbers ◽  
Transformation Matrices ◽  
Skew Circulant ◽  
The Relationship

We consider the skew circulant and skew left circulant matrices with any continuous Lucas numbers. Firstly, we discuss the invertibility of the skew circulant matrices and present the determinant and the inverse matrices by constructing the transformation matrices. Furthermore, the invertibility of the skew left circulant matrices is also discussed. We obtain the determinants and the inverse matrices of the skew left circulant matrices by utilizing the relationship between skew left circulant matrices and skew circulant matrix, respectively. Finally, the four kinds of norms and bounds for the spread of these matrices are given, respectively.

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