On relations betweenR a Rb transformation and canonical diagonal form of λ-matrix

1997 ◽  
Vol 40 (3) ◽  
pp. 258-268 ◽  
Author(s):  
Renji Tao ◽  
Peirong Feng
Keyword(s):  
Diagonal Form ◽  
Canonical Diagonal Form

On common monic divisors having a given canonical diagonal form for matrix polynomials

1996 ◽  
Vol 79 (6) ◽  
pp. 1402-1405
Author(s):  
V. M. Petrichkovich ◽  
V. M. Prokip ◽  
F. A. Prukhnits'kii
Keyword(s):  
Diagonal Form ◽  
Matrix Polynomials ◽  
Canonical Diagonal Form

scholarly journals Petrographic structures: Khibiny ijolites and urtites

Vestnik MGTU ◽  
2021 ◽  
Vol 24 (2) ◽  
pp. 160-167
Author(s):  
Yuri Leonidovich Voytekhovsky ◽  
Alena Alexandrovna Zakharova
Keyword(s):  
Crystalline Rocks ◽  
Diagonal Form ◽  
Standard Description ◽  
Khibiny Massif ◽  
Constitutional Principle ◽  
Hardy Weinberg Equilibrium ◽  
Definition Of ◽  
Canonical Diagonal Form ◽  
Strict Definition

In addition to the standard description of the structures and textures of crystalline rocks the mathematical approaches have been proposed based on a rigorous determination of the petrographic structure through the probabilities of binary intergrain contacts. In general, the petrographic structure is defined as an invariant aspect of rock organization, algebraically expressed by the canonical diagonal form of the symmetric Pij matrix and geometrically visualized by structural indicatrices - surfaces of the 2nd order. The agreed nomenclature of possible petrographic structures for an n-mineral rock is simple: the symbol Snm means that there are exactly m positive numbers in the canonical diagonal form of the Pij matrix. New types of barycentric diagrams have been proposed. To describe the massive texture, the concept of Hardy - Weinberg equilibrium has been proposed. This boundary classifies barycentric diagrams into areas within which canonical types of Рij matrices and topological types of structural indicatrices are preserved. The change in the organization of the rock within a type is quantitative, the transition from one type to another means structural restructuring. The methods are used to describe ijolites and urtites of the Khibiny massif, the Kola Peninsula. In the modern taxonomy of rocks, the boundaries between them are mostly conditional and are drawn according to the contents of rock-forming minerals, for example, between ijolites and urtites - according to the contents of nepheline and pyroxene. The strict definition of the petrographic structure proposed by the authors makes it possible to introduce into petrography the constitutional principle (structure + composition), which is successfully acting in mineralogy.

scholarly journals Petrographic structures and Hardy – Weinberg equilibrium

2020 ◽  
Vol 242 ◽  
pp. 133
Author(s):  
Yury VOYTEKHOVSKY ◽  
Alena ZAKHAROVA
Keyword(s):  
Quadratic Forms ◽  
Dimensional Space ◽  
Complete Classification ◽  
Algebraic Expression ◽  
Diagonal Form ◽  
Hardy Weinberg Equilibrium ◽  
Algebraic Forms ◽  
Canonical Diagonal Form

The article is devoted to the most narrative side of modern petrography – the definition, classification and nomenclature of petrographic structures. We suggest a mathematical formalism using the theory of quadratic forms (with a promising extension to algebraic forms of the third and fourth orders) and statistics of binary (ternary and quaternary, respectively) intergranular contacts in a polymineralic rock. It allows constructing a complete classification of petrographic structures with boundaries corresponding to Hardy – Weinberg equilibria. The algebraic expression of the petrographic structure is the canonical diagonal form of the symmetric probability matrix of binary intergranular contacts in the rock. Each petrographic structure is uniquely associated with a structural indicatrix – the central quadratic surface in n-dimensional space, where n is the number of minerals composing the rock. Structural indicatrix is an analogue of the conoscopic figure used for optical recognition of minerals. We show that the continuity of changes in the organization of rocks (i.e., the probabilities of various intergranular contacts) does not contradict a dramatic change in the structure of the rocks, neighboring within the classification. This solved the problem, which seemed insoluble to A.Harker and E.S.Fedorov. The technique was used to describe the granite structures of the Salminsky pluton (Karelia) and the Akzhailau massif (Kazakhstan) and is potentially applicable for the monotonous strata differentiation, section correlation, or wherever an unambiguous, reproducible determination of petrographic structures is needed. An important promising task of the method is to extract rocks' genetic information from the obtained data.

A new version of the Fast Multipole Method for the Laplace equation in three dimensions

Acta Numerica ◽  
1997 ◽  
Vol 6 ◽  
pp. 229-269 ◽  
Author(s):  
Leslie Greengard ◽  
Vladimir Rokhlin
Keyword(s):  
Laplace Equation ◽  
Fast Multipole Method ◽  
High Accuracy ◽  
Three Dimensions ◽  
Potential Fields ◽  
Diagonal Form ◽  
Fast Multipole ◽  
Reasonable Cost ◽  
Multipole Method ◽  
Translation Operators

We introduce a new version of the Fast Multipole Method for the evaluation of potential fields in three dimensions. It is based on a new diagonal form for translation operators and yields high accuracy at a reasonable cost.

Convergence to diagonal form of block Jacobi-type methods

2014 ◽  
Vol 129 (3) ◽  
pp. 449-481 ◽  
Author(s):  
Vjeran Hari
Keyword(s):  
Diagonal Form

scholarly journals Wave Propagation Analysis in Composite Laminates Containing a Delamination Using a Three-Dimensional Spectral Element Method

2012 ◽  
Vol 2012 ◽  
pp. 1-19 ◽  
Author(s):  
Fucai Li ◽  
Haikuo Peng ◽  
Xuewei Sun ◽  
Jinfu Wang ◽  
Guang Meng
Keyword(s):  
Wave Propagation ◽  
Lamb Wave ◽  
Composite Laminates ◽  
Composite Structures ◽  
Spectral Element Method ◽  
Three Dimensional ◽  
Spectral Element ◽  
Wave Mode ◽  
Diagonal Form ◽  
Element Method

A three-dimensional spectral element method (SEM) was developed for analysis of Lamb wave propagation in composite laminates containing a delamination. SEM is more efficient in simulating wave propagation in structures than conventional finite element method (FEM) because of its unique diagonal form of the mass matrix. Three types of composite laminates, namely, unidirectional-ply laminates, cross-ply laminates, and angle-ply laminates are modeled using three-dimensional spectral finite elements. Wave propagation characteristics in intact composite laminates are investigated, and the effectiveness of the method is validated by comparison of the simulation results with analytical solutions based on transfer matrix method. Different Lamb wave mode interactions with delamination are evaluated, and it is demonstrated that symmetric Lamb wave mode may be insensitive to delamination at certain interfaces of laminates while the antisymmetric mode is more suited for identification of delamination in composite structures.

Solution of two-qubit Rabi model with extended generalized rotating-wave approximation

2021 ◽  
pp. 2150213
Author(s):  
Zhanyuan Yan ◽  
Peihua Qu ◽  
BingBing Xu ◽  
Shihui Zhang ◽  
Jinying Ma
Keyword(s):  
Order Approximation ◽  
Photon Number ◽  
Energy Spectra ◽  
Hamiltonian Matrix ◽  
Diagonal Form ◽  
Coupling Region ◽  
Rotating Wave Approximation ◽  
Wave Approximation ◽  
Rabi Model ◽  
Rotating Wave

The generalized rotating-wave approximation (GRWA) method is extended to the two-qubit quantum Rabi model. In the first-order approximation (one photon exchange), the Hamiltonian matrix in photon number space is simplified by introducing two variational parameters. However, the Hamiltonian matrix is not a diagonalizable matrix yet. Furthermore, by presenting a constraint condition on coupling strength and atomic transition frequency, the Hamiltonian matrix is simplified and an effective solvable Hamiltonian with block diagonal form is obtained. In the even and odd parity space, the energy spectra and eigenstates of the two-qubit quantum Rabi model are achieved analytically. Most of the energy spectra, especially the lower energy levels, agree well with the numerical exact results in ultra-strong coupling region, and the ground state wave function can gives a fairly accurate result of mean photon number.

scholarly journals On the reduction of pairs of hermitian or symmetric matrices to diagonal form by congruence

1986 ◽  
Vol 73 ◽  
pp. 213-226 ◽  
Author(s):  
Yoo Pyo Hong ◽  
Roger A. Horn ◽  
Charles R. Johnson
Keyword(s):  
Diagonal Form ◽  
Symmetric Matrices
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