scholarly journals Matrices whose inverses are tridiagonal, band or block-tridiagonal and their relationship with the covariance matrices of a random Markov process

Filomat ◽  
2019 ◽  
Vol 33 (5) ◽  
pp. 1335-1352
Author(s):  
Ulan Brimkulov
Keyword(s):  
Markov Random Fields ◽  
Covariance Matrices ◽  
Tridiagonal Matrix ◽  
Matrix Elements ◽  
Banded Matrix ◽  
Markov Random ◽  
First Case ◽  
The Matrix ◽  
Full Dimension ◽  
Product Of Matrices

The article discusses the matrices of the form A1n, Amn, AmN, whose inverses are: tridiagonal matrix A-1n (n - dimension of the A-mn matrix), banded matrix A-mn (m is the half-width band of the matrix) or block-tridiagonal matrix A-m N (N = n x m - full dimension of the block matrix; m - the dimension of the blocks) and their relationships with the covariance matrices of measurements with ordinary (simple) Markov Random Processes (MRP), multiconnected MRP and vector MRP, respectively. Such covariance matrices frequently occur in the problems of optimal filtering, extrapolation and interpolation of MRP and Markov Random Fields (MRF). It is shown, that the structures of the matrices A1n, Amn, AmN have the same form, but the matrix elements in the first case are scalar quantities; in the second case matrix elements represent a product of vectors of dimension m; and in the third case, the off-diagonal elements are the product of matrices and vectors of dimension m. The properties of such matrices were investigated and a simple formulas of their inverses were found. Also computational efficiency in the storage and the inverse of such matrices have been considered. To illustrate the acquired results, an example on the covariance matrix inversions of two-dimensional MRP is given.

Sampling from Gaussian Markov random fields conditioned on linear constraints

2008 ◽  
Vol 48 ◽  
pp. 1041 ◽  
Author(s):  
Daniel Peter Simpson ◽  
Ian W. Turner ◽  
A. N. Pettitt
Keyword(s):  
Random Fields ◽  
Markov Random Fields ◽  
Linear Constraints ◽  
Gaussian Markov Random Fields ◽  
Markov Random

scholarly journals A Methodology for Redesigning Networks by Using Markov Random Fields

Mathematics ◽  
2021 ◽  
Vol 9 (12) ◽  
pp. 1389
Author(s):  
Julia García Cabello ◽  
Pedro A. Castillo ◽  
Maria-del-Carmen Aguilar-Luzon ◽  
Francisco Chiclana ◽  
Enrique Herrera-Viedma
Keyword(s):  
Decision Making ◽  
Random Fields ◽  
Markov Random Fields ◽  
Application Performance ◽  
Cross Border ◽  
Markov Random ◽  
User Groups ◽  
Universal Definition ◽  
Definition Of ◽  
Decision Making Model

Standard methodologies for redesigning physical networks rely on Geographic Information Systems (GIS), which strongly depend on local demographic specifications. The absence of a universal definition of demography makes its use for cross-border purposes much more difficult. This paper presents a Decision Making Model (DMM) for redesigning networks that works without geographical constraints. There are multiple advantages of this approach: on one hand, it can be used in any country of the world; on the other hand, the absence of geographical constraints widens the application scope of our approach, meaning that it can be successfully implemented either in physical (ATM networks) or non-physical networks such as in group decision making, social networks, e-commerce, e-governance and all fields in which user groups make decisions collectively. Case studies involving both types of situations are conducted in order to illustrate the methodology. The model has been designed under a data reduction strategy in order to improve application performance.

scholarly journals On the Modular Operator of Mutli-component Regions in Chiral CFT

2021 ◽  
Author(s):  
Stefan Hollands
Keyword(s):  
Field Theory ◽  
Integral Equation ◽  
Hilbert Problem ◽  
Matrix Elements ◽  
New Approach ◽  
Conformal Field ◽  
The Matrix ◽  
Kms Condition ◽  
Modular Operator ◽  
Riemann Hilbert Problem

AbstractWe introduce a new approach to find the Tomita–Takesaki modular flow for multi-component regions in general chiral conformal field theory. Our method is based on locality and analyticity of primary fields as well as the so-called Kubo–Martin–Schwinger (KMS) condition. These features can be used to transform the problem to a Riemann–Hilbert problem on a covering of the complex plane cut along the regions, which is equivalent to an integral equation for the matrix elements of the modular Hamiltonian. Examples are considered.

scholarly journals Analytical angular solutions for the atom–diatom interaction potential in a basis set of products of two spherical harmonics: two approaches

2021 ◽  
Author(s):  
Mariusz Pawlak ◽  
Marcin Stachowiak
Keyword(s):  
Spherical Harmonics ◽  
Interaction Potential ◽  
Chem Phys ◽  
Basis Set ◽  
Matrix Elements ◽  
Trigonometric Functions ◽  
Variational Theory ◽  
Molecular Collision ◽  
The Matrix ◽  
Analytical Expressions

AbstractWe present general analytical expressions for the matrix elements of the atom–diatom interaction potential, expanded in terms of Legendre polynomials, in a basis set of products of two spherical harmonics, especially significant to the recently developed adiabatic variational theory for cold molecular collision experiments [J. Chem. Phys. 143, 074114 (2015); J. Phys. Chem. A 121, 2194 (2017)]. We used two approaches in our studies. The first involves the evaluation of the integral containing trigonometric functions with arbitrary powers. The second approach is based on the theorem of addition of spherical harmonics.

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