On the matrix formulation of the theory of partial polarization in terms of observables

1960 ◽  
Vol 15 (3) ◽  
pp. 370-388 ◽  
Author(s):  
G. B. Parrent ◽  
P. Roman
Keyword(s):  
Matrix Formulation ◽  
The Matrix ◽  
Partial Polarization

The Convex Polytopes and Homogeneous Coordinate Rings of Bivariate Polynomials

2019 ◽  
Vol 16 (2) ◽  
pp. 1
Author(s):  
Shamsatun Nahar Ahmad ◽  
Nor’Aini Aris ◽  
Azlina Jumadi
Keyword(s):  
Convex Polytopes ◽  
Polynomial Systems ◽  
Matrix Formulation ◽  
Bivariate Polynomials ◽  
Homogeneous Coordinates ◽  
The Matrix ◽  
Projective Vector ◽  
Coordinate Rings ◽  
Resultant Matrix ◽  
The Given

Concepts from algebraic geometry such as cones and fans are related to toric varieties and can be applied to determine the convex polytopes and homogeneous coordinate rings of multivariate polynomial systems. The homogeneous coordinates of a system in its projective vector space can be associated with the entries of the resultant matrix of the system under consideration. This paper presents some conditions for the homogeneous coordinates of a certain system of bivariate polynomials through the construction and implementation of the Sylvester-Bèzout hybrid resultant matrix formulation. This basis of the implementation of the Bèzout block applies a combinatorial approach on a set of linear inequalities, named 5-rule. The inequalities involved the set of exponent vectors of the monomials of the system and the entries of the matrix are determined from the coefficients of facets variable known as brackets. The approach can determine the homogeneous coordinates of the given system and the entries of the Bèzout block. Conditions for determining the homogeneous coordinates are also given and proven.

scholarly journals Homogenization of very rough three-dimensional interfaces for the poroelasticity theory with Biot's model

2019 ◽  
Author(s):  
Nguyen Thi Kieu ◽  
Pham Chi Vinh ◽  
Do Xuan Tung
Keyword(s):  
Incident Angle ◽  
Three Dimensional ◽  
Homogenization Method ◽  
Reflection And Transmission Coefficients ◽  
Biot Theory ◽  
Rough Interface ◽  
Matrix Formulation ◽  
Reflection And Transmission ◽  
Transmission Coefficients ◽  
The Matrix

In this paper, we carry out the homogenization of a very rough three-dimensional interface separating  two dissimilar generally anisotropic poroelastic solids modeled by the Biot theory. The very rough interface is assumed to be a cylindrical surface that rapidly oscillates between two parallel planes, and the motion is time-harmonic. Using the homogenization method with the matrix formulation of the poroelasicity theory, the explicit  homogenized equations have been derived. Since the obtained  homogenized equations are totally explicit, they are very convenient for solving various practical problems. As an example proving this, the reflection and transmission of SH waves at a very rough interface of tooth-comb type is considered. The closed-form analytical expressions of the reflection and transmission coefficients have been  derived. Based on them, the effect of the incident angle and some material parameters  on the reflection and transmission coefficients are examined numerically.

scholarly journals The matrix formulation of type IIB five-branes

1998 ◽  
Vol 523 (1-2) ◽  
pp. 158-170 ◽  
Author(s):  
Savdeep Sethi
Keyword(s):  
Matrix Formulation ◽  
The Matrix

Closed Shell Hartree Theory with Orthonormal Orbitals

1980 ◽  
Vol 35 (4) ◽  
pp. 408-411 ◽  
Author(s):  
Yasuyuki Ishikawa
Keyword(s):  
Closed Shell ◽  
Natural Orbital ◽  
Matrix Formulation ◽  
The Matrix ◽  
Polyatomic Systems

Abstract A natural orbital multiconfigurational SCF formalism has been applied to the closed shell Hartree theory with orthonormal orbitals. A prescription is given for constructing a single, one-electron Hamiltonian with which one can determine all the occupied orbitals. The formalism is suited to the matrix formulation of the orthogonalized Hartree theory for polyatomic systems.

scholarly journals The Convex Polytopes and Homogeneous Coordinate Rings of Bivariate Polynomials

2019 ◽  
Vol 16 (2) ◽  
pp. 1
Author(s):  
Shamsatun Nahar Ahmad ◽  
Nor’ Aini Aris ◽  
Azlina Jumadi
Keyword(s):  
Convex Polytopes ◽  
Polynomial Systems ◽  
Matrix Formulation ◽  
Bivariate Polynomials ◽  
Homogeneous Coordinates ◽  
The Matrix ◽  
Projective Vector ◽  
Coordinate Rings ◽  
Resultant Matrix ◽  
The Given

Concepts from algebraic geometry such as cones and fans are related to toric varieties and can be applied to determine the convex polytopes and homogeneous coordinate rings of multivariate polynomial systems. The homogeneous coordinates of a system in its projective vector space can be associated with the entries of the resultant matrix of the system under consideration. This paper presents some conditions for the homogeneous coordinates of certain system of bivariate polynomials through the construction and implementation of the Sylvester-Bèzout hybrid resultant matrix formulation. This basis of the implementation of the Bèzout block applies a combinatorial approach on a set of linear inequalities, named 5-rule. The inequalities involved the set of exponent vectors of the monomials of the system and the entries of the matrix are determined from the coefficients of facets variable known as brackets. The approach can determine the homogeneous coordinates of the given system and the entries of the Bèzout block. Conditions for determining the homogeneous coordinates are also given and proven.   

scholarly journals SUPERSYMMETRIC RELATIVISTIC QUANTUM MECHANICS

2006 ◽  
Vol 21 (06) ◽  
pp. 1333-1340 ◽  
Author(s):  
YOSHINOBU HABARA ◽  
HOLGER B. NIELSEN ◽  
MASAO NINOMIYA
Keyword(s):  
Quantum Mechanics ◽  
Relativistic Quantum ◽  
Equations Of Motion ◽  
Relativistic Quantum Mechanics ◽  
Chiral Multiplet ◽  
Second Quantization ◽  
Matrix Formulation ◽  
The Matrix ◽  
Particle Level ◽  
Dirac Sea

We present an attempt to formulate the supersymmetric and relativistic quantum mechanics in the sense of realizing supersymmetry on the single particle level, by utilizing the equations of motion which is equivalent to the ordinary second quantization of the chiral multiplet. The matrix formulation is used to express the operators such as supersymmetry generators and fields of the chiral multiplets. We realize supersymmetry prior to filling the Dirac sea.

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