Generating the Matrix Coefficients

2010 ◽  
pp. 289-352
Author(s):  
Stefan A. Sauter ◽  
Christoph Schwab
Keyword(s):  
Matrix Coefficients ◽  
The Matrix

Orthogonal basis for the Shapovalov form on $U_q(\mathfrak{sl}(n + 1))$

2015 ◽  
Vol 27 (02) ◽  
pp. 1550004 ◽  
Author(s):  
Andrey Mudrov
Keyword(s):  
Lie Algebra ◽  
Cartan Subalgebra ◽  
Verma Module ◽  
Orthogonal Basis ◽  
Enveloping Algebra ◽  
Pbw Basis ◽  
Inverse Form ◽  
Matrix Coefficients ◽  
The Matrix ◽  
Quantized Universal Enveloping Algebra

Let U be either the classical or quantized universal enveloping algebra of the Lie algebra [Formula: see text] extended over the field of fractions of the Cartan subalgebra. We suggest a PBW basis in U over the extended Cartan subalgebra diagonalizing the contravariant Shapovalov form on generic Verma module. The matrix coefficients of the form are calculated and the inverse form is explicitly constructed.

A method for determination of the matrix coefficients of the A-parameters of a multipole that simulates the impact of the adjacent track circuit

2017 ◽  
Vol 88 (3) ◽  
pp. 99-104 ◽  
Author(s):  
E. M. Tarasov ◽  
D. V. Zheleznov ◽  
A. G. Isaicheva ◽  
S. V. Kopeikin
Keyword(s):  
Matrix Coefficients ◽  
The Matrix ◽  
Track Circuit ◽  
The Impact

Darboux operators for linear first-order multi-component equations in arbitrary dimensions

Open Physics ◽  
2013 ◽  
Vol 11 (4) ◽  
Author(s):  
Axel Schulze-Halberg
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Partial Differential ◽  
First Order ◽  
Matrix Coefficients ◽  
The Matrix ◽  
Arbitrary Dimensions ◽  
Weyl Equation

AbstractWe construct Darboux operators for linear, multi-component partial differential equations of first order. The number of variables and the dimension of the matrix coefficients in our equations are arbitrary. The Darboux operator and the transformed equation are worked out explicitly. We present an application of our formalism to the (1+2)-dimensional Weyl equation.

scholarly journals Matrix coefficients of the large discrete series representations of Sp(2; R) as hypergeometric series of two variables

2012 ◽  
Vol 208 ◽  
pp. 201-263
Author(s):  
Takayuki Oda
Keyword(s):  
Differential Equations ◽  
Hypergeometric Series ◽  
Discrete Series ◽  
Higher Order ◽  
Holonomic System ◽  
Radial Part ◽  
Matrix Coefficients ◽  
Series Representations ◽  
The Matrix ◽  
Discrete Series Representations

AbstractWe investigate the radial part of the matrix coefficients with minimal K-types of the large discrete series representations of Sp(2; R). They satisfy certain difference-differential equations derived from Schmid operators. This system is reduced to a holonomic system of rank 4, which is finally found to be equivalent to higher-order hypergeometric series in the sense of Appell and Kampé de Fériet.

scholarly journals A Numerical Method for Solving 3D Elasticity Equations with Sharp-Edged Interfaces

2013 ◽  
Vol 2013 ◽  
pp. 1-13 ◽  
Author(s):  
Liqun Wang ◽  
Songming Hou ◽  
Liwei Shi
Keyword(s):  
Finite Element ◽  
Piecewise Linear ◽  
Main Idea ◽  
Three Dimensions ◽  
Test Function ◽  
Linear System Of Equations ◽  
Elasticity Equations ◽  
Matrix Coefficients ◽  
The Matrix ◽  
3D Elasticity

Interface problems occur frequently when two or more materials meet. Solving elasticity equations with sharp-edged interfaces in three dimensions is a very complicated and challenging problem for most existing methods. There are several difficulties: the coupled elliptic system, the matrix coefficients, the sharp-edged interface, and three dimensions. An accurate and efficient method is desired. In this paper, an efficient nontraditional finite element method with nonbody-fitting grids is proposed to solve elasticity equations with sharp-edged interfaces in three dimensions. The main idea is to choose the test function basis to be the standard finite element basis independent of the interface and to choose the solution basis to be piecewise linear satisfying the jump conditions across the interface. The resulting linear system of equations is shown to be positive definite under certain assumptions. Numerical experiments show that this method is second order accurate in the L∞ norm for piecewise smooth solutions. More than 1.5th order accuracy is observed for solution with singularity (second derivative blows up).

scholarly journals DESIGNING SHIP COURSE MOVEMENT USING PSEUDOINVERSE MATRICES FOR MEASURING STATE VECTOR OF CONTROLLED OBJECT

2021 ◽  
Vol 2021 (1) ◽  
pp. 61-69
Author(s):  
Alexander Aleksandrovich Dyda ◽  
Kseniya Chumakova ◽  
Van Thanh Nguyen
Keyword(s):  
Control Systems ◽  
Matrix Model ◽  
State Vector ◽  
Matrix Coefficients ◽  
Course Control ◽  
The Matrix ◽  
Nomoto Model ◽  
Pseudo Inverse ◽  
Higher Order Models ◽  
Controlled Object

To configure the control systems of the ship movement along the trajectory, it is necessary to be concerned with the parameters of its controllability. There has been proposed building a matrix model based on measurements of the state vector of a controlled object. The construction of the model is considered on the example of the problem of ship course control. An algorithm for determining the matrix coefficients of the selected model is proposed. The operation of the considered algorithm has been checked for square matrices by finding their inverse matrices, as well as for rectangular matrices for which the pseudo inverse matrix was found. The illustration of the proposed approach is carried out using the example of a simple 1-order linear Nomoto model. The considered approach is quite universal and can be applied to higher order models, including nonlinear ones.

MODELING BY FITTING A UNION OF POLYNOMIAL FUNCTIONS TO DATA IN AN ERRORS-IN-VARIABLES CONTEXT

2013 ◽  
Vol 27 (02) ◽  
pp. 1350004 ◽  
Author(s):  
LEVENTE HUNYADI ◽  
ISTVÁN VAJK
Keyword(s):  
Covariance Matrices ◽  
Polynomial Functions ◽  
Errors In Variables ◽  
Polynomial Eigenvalue Problem ◽  
Asymmetric Distance ◽  
Matrix Coefficients ◽  
Cut Method ◽  
The Matrix ◽  
Noise Covariance ◽  
Local Fitting

We present a model construction method based on a local fitting of polynomial functions to noisy data and building the entire model as a union of regions explained by such polynomial functions. Local fitting is shown to reduce to solving a polynomial eigenvalue problem where the matrix coefficients are data covariance and approximated noise covariance matrices that capture distortion effects by noise. By defining the asymmetric distance between two points as the projection of one onto the function fitted to the neighborhood of the other, we use a best weighted cut method to find a proper partitioning of the entire set of data into feasible regions. Finally, the partitions are refined using a modified version of a k-planes algorithm.

scholarly journals Programming with parametric elements of the matrix coefficients

1977 ◽  
Vol 11 (2) ◽  
pp. 233-238
Author(s):  
Ramon M. Arana
Keyword(s):  
Matrix Coefficients ◽  
The Matrix

scholarly journals On some aspects of the matrix data perturbation in linear program

2003 ◽  
Vol 13 (2) ◽  
pp. 153-164 ◽  
Author(s):  
Margita Kon-Popovska
Keyword(s):  
Linear Program ◽  
Linear Functions ◽  
Basic Solution ◽  
System Matrix ◽  
Data Perturbation ◽  
Matrix Coefficients ◽  
The Matrix ◽  
Base Matrix ◽  
Technology Resources ◽  
Optimal Basic Solution

Linear program under changes in the system matrix coefficients has proved to be more complex than changes of the coefficients in objective functions and right hand sides. The most of the previous studies deals with problems where only one coefficient, a row (column), or few rows (columns) are linear functions of a parameter. This work considers a more general case, where all the coefficients are polynomial (in the particular case linear) functions of the parameter tT??R. For such problems, assuming that some non-singularity conditions hold and an optimal base matrix is known for some particular value t of the parameter, corresponding explicit optimal basic solution in the neighborhood of t is determined by solving an augmented LP problem with real system matrix coefficients. Parametric LP can be utilized for example to model the production problem where, technology, resources, costs and similar categories vary with time. .

scholarly journals REPRESENTATION OF SOLUTIONS OF SYSTEMS OF LINEAR DIFFERENTIAL EQUATIONS WITH MULTIPLE DELAYS AND NONPERMUTABLE VARIABLE COEFFICIENTS

2020 ◽  
Vol 25 (2) ◽  
pp. 303-322
Author(s):  
Michal Pospíšil
Keyword(s):  
Differential Equations ◽  
Variable Coefficients ◽  
Linear Differential Equations ◽  
Multiple Delays ◽  
Matrix Coefficients ◽  
Representation Of Solutions ◽  
The Matrix ◽  
Constant Coefficients ◽  
Linear Differential ◽  
Delay Differential

Solutions of nonhomogeneous systems of linear differential equations with multiple constant delays are explicitly stated without a commutativity assumption on the matrix coefficients. In comparison to recent results, the new formulas are not inductively built, but depend on a sum of noncommutative products in the case of constant coefficients, or on a sum of iterated integrals in the case of time-dependent coefficients. This approach shall be more suitable for applications.Representation of a solution of a Cauchy problem for a system of higher order delay differential equations is also given.

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