scholarly journals A NOTE ON THE GAUSS DECOMPOSITION OF THE ELLIPTIC CAUCHY MATRIX

2011 ◽  
Vol 18 (2) ◽  
pp. 179-182 ◽  
Author(s):  
L. FEHÉR ◽  
C. KLIMČÍK ◽  
S. RUIJSENAARS
Keyword(s):  
Cauchy Matrix ◽  
Gauss Decomposition

scholarly journals Cauchy matrix and Liouville formula of quaternion impulsive dynamic equations on time scales

2020 ◽  
Vol 18 (1) ◽  
pp. 353-377 ◽  
Author(s):  
Zhien Li ◽  
Chao Wang
Keyword(s):  
Time Scales ◽  
Quaternion Algebra ◽  
Matrix Exponential ◽  
Dynamic Equations ◽  
Necessary Condition ◽  
Quaternion Matrix ◽  
Exponential Functions ◽  
Cauchy Matrix ◽  
Adjoint Matrix ◽  
Sufficient And Necessary Condition

Abstract In this study, we obtain the scalar and matrix exponential functions through a series of quaternion-valued functions on time scales. A sufficient and necessary condition is established to guarantee that the induced matrix is real-valued for the complex adjoint matrix of a quaternion matrix. Moreover, the Cauchy matrices and Liouville formulas for the quaternion homogeneous and nonhomogeneous impulsive dynamic equations are given and proved. Based on it, the existence, uniqueness, and expressions of their solutions are also obtained, including their scalar and matrix forms. Since the quaternion algebra is noncommutative, many concepts and properties of the non-quaternion impulsive dynamic equations are ineffective, we provide several examples and counterexamples on various time scales to illustrate the effectiveness of our results.

scholarly journals GAUSS DECOMPOSITION, WAKIMOTO REALIZATION AND GAUGED WZNW MODELS

1994 ◽  
Vol 09 (11) ◽  
pp. 1009-1023
Author(s):  
H. ARFAEI ◽  
N. MOHAMMEDI
Keyword(s):  
Field Strength ◽  
Sigma Model ◽  
Target Space ◽  
Gauge Fields ◽  
Nonlinear Sigma Model ◽  
Model Interpretation ◽  
Gauss Decomposition ◽  
Wznw Model ◽  
Wakimoto Realization

The implications of gauging the Wess-Zumino-Novikov-Witten (WZNW) model using the Gauss decomposition of the group elements are explored. We show that, contrary to the standard gauging of WZNW models, this gauging is carried out by minimally coupling the gauge fields. We find that this gauging, in the case of gauging and Abelian vector subgroup, differs from the standard one by terms proportional to the field strength of the gauge fields. We prove that gauging an Abelian vector subgroup does not have a nonlinear sigma model interpretation. This is because the target-space metric resulting from the integration over the gauge fields is degenerate. We demonstrate, however, that this kind of gauging has a natural interpretation in terms of Wakimoto variables.

Adaptive Hermite–Gauss decomposition method to analyze optical dielectric waveguides

2003 ◽  
Vol 20 (3) ◽  
pp. 557 ◽  
Author(s):  
Alejandro Ortega-Moñux ◽  
J. Gonzalo Wangüemert-Pérez ◽  
Iñigo Molina-Fernández
Keyword(s):  
Decomposition Method ◽  
Dielectric Waveguides ◽  
Gauss Decomposition ◽  
Optical Dielectric

To estimating linear functionals values over solutions of systems with aftereffect

2020 ◽  
pp. 274-283
Author(s):  
Vladimir P. Maksimov
Keyword(s):  
Boundary Conditions ◽  
Boundary Value Problems ◽  
Boundary Value ◽  
Linear Functionals ◽  
Cauchy Matrix ◽  
Time Segment ◽  
External Perturbations ◽  
Functional Differential Systems ◽  
Representation Of Solutions ◽  
Functional Differential

For a wide class of linear functional differential systems with Volterra operators, a constructive technique is proposed to obtain estimates of linear functionals values over solutions in conditions of uncertainty of external perturbations. It can be applied to solutions of boundary value problems with arbitrary number of boundary conditions as well as to description of attainability sets in control problems with respect to given on-target functionals. External perturbations are constrained by a given linear inequalities system on the main time segment. The technique is based on the results of general theory of functional differential equations about the solvability of boundary value problems with general linear boundary conditions and the representation of solutions. The problem under consideration is reduced to the generalized moment problem. Therewith the results on the properties of the Cauchy matrix to systems with aftereffect are of essential importance. The general form of functionals allows one to cover many cases being topical in applications such as multipoint, integral ones, as well as hybrids of those.

scholarly journals ALMOST PERIODIC SOLUTIONS OF IMPULSIVE INTEGRO‐DIFFERENTIAL NEURAL NETWORKS

2010 ◽  
Vol 15 (4) ◽  
pp. 505-516 ◽  
Author(s):  
Gani Tr. Stamov ◽  
Jehad O. Alzabut
Keyword(s):  
Neural Networks ◽  
Periodic Solutions ◽  
Contraction Mapping ◽  
Sufficient Conditions ◽  
Impulsive Differential Equations ◽  
Contraction Mapping Principle ◽  
Almost Periodic Solutions ◽  
Almost Periodic ◽  
Cauchy Matrix ◽  
Bellman’S Inequality

In this paper, sufficient conditions are established for the existence of almost periodic solutions for system of impulsive integro‐differential neural networks. Our approach is based on the estimation of the Cauchy matrix of linear impulsive differential equations. We shall employ the contraction mapping principle as well as Gronwall‐Bellman's inequality to prove our main result. The research of Gani Tr. Stamov is partially supported by the Grand 100ni087–16 from Technical University–Sofia

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