Graph theory and lie algebra

1969 ◽  
pp. 149-153 ◽  
Author(s):  
Ronald C. Hamelink
Keyword(s):  
Graph Theory ◽  
Lie Algebra

scholarly journals Controllability and Observability of a Large Scale Thermodynamical System via Connectability Approach

2010 ◽  
Author(s):  
Virdiansyah Permana ◽  
Rahmat Shoureshi
Keyword(s):  
Graph Theory ◽  
Lie Algebra ◽  
Large Scale ◽  
Point Of View ◽  
Rank Condition ◽  
Computational Point ◽  
New Approach ◽  
Class System ◽  
Necessary And Sufficient ◽  
Controllability And Observability

This study presents a new approach to determine the controllability and observability of a large scale nonlinear dynamic thermal system using graph-theory. The novelty of this method is in adapting graph theory for nonlinear class and establishing a graphic condition that describes the necessary and sufficient terms for a nonlinear class system to be controllable and observable, which equivalents to the analytical method of Lie algebra rank condition. The directed graph (digraph) is utilized to model the system, and the rule of its adaptation in nonlinear class is defined. Subsequently, necessary and sufficient terms to achieve controllability and observability condition are investigated through the structural property of a digraph called connectability. It will be shown that the connectability condition between input and states, as well as output and states of a nonlinear system are equivalent to Lie-algebra rank condition (LARC). This approach has been proven to be easier from a computational point of view and is thus found to be useful when dealing with a large system.

scholarly journals Classification of Filiform Lie Algebras up to dimension 7 Over Finite Fields

2016 ◽  
Vol 24 (2) ◽  
pp. 185-204
Author(s):  
Óscar J. Falcón ◽  
Raúl M. Falcón ◽  
Juan Núñez ◽  
Ana M. Pacheco ◽  
M. Trinidad Villar
Keyword(s):  
Graph Theory ◽  
Lie Algebra ◽  
Lie Algebras ◽  
Finite Fields ◽  
Discrete Mathematics ◽  
Main Interest ◽  
Filiform Lie Algebras ◽  
Suitable Technique ◽  
New Strategies

Abstract This paper tries to develop a recent research which consists in using Discrete Mathematics as a tool in the study of the problem of the classification of Lie algebras in general, dealing in this case with filiform Lie algebras up to dimension 7 over finite fields. The idea lies in the representation of each Lie algebra by a certain type of graphs. Then, some properties on Graph Theory make easier to classify the algebras. As main results, we find out that there exist, up to isomorphism, six, five and five 6-dimensional filiform Lie algebras and fifteen, eleven and fifteen 7-dimensional ones, respectively, over ℤ/pℤ, for p = 2, 3, 5. In any case, the main interest of the paper is not the computations itself but both to provide new strategies to find out properties of Lie algebras and to exemplify a suitable technique to be used in classifications for larger dimensions.

Connectability-Based Controllability of Large Scale Nonlinear Dynamic Thermal Systems

2009 ◽  
Author(s):  
Rahmat Shoureshi ◽  
Virdi Permana
Keyword(s):  
Graph Theory ◽  
Nonlinear Systems ◽  
Lie Algebra ◽  
Large Scale ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Rank Condition ◽  
Thermal Systems ◽  
Necessary And Sufficient ◽  
Controllability And Observability

A new approach using graph-theory to determine the controllability and observability of large scale nonlinear dynamic thermal systems is presented. The novelty of this method is in adapting graph theory for a nonlinear class and establishing graphic conditions that describe the necessary and sufficient conditions for a class of nonlinear systems to be controllable and observable which is equivalent to the analytical method of Lie algebra rank condition. Graph theory of directed graph (digraph) is utilized to model the system and its adaptation to nonlinear problems is defined. The necessary and sufficient conditions for controllability are investigated through the structural property of a digraph called connectability. In comparison to the Lie Algebra, this approach has proven to be easier, from a computational point of view, thus it is found to be useful when dealing with large scale systems. This paper presents the problem statement, properties of structured system, and analytical method of Lie algebra rank condition for controllability and observability of bilinear systems. The main results of graphical approach which describe the necessary and sufficient conditions for controllability of nonlinear systems are presented and applied to the problem of a coupled two heat exchangers, connected in an arbitrary fashion.

Pattern Recognition in Histo-Pathology: Basic Considerations

1982 ◽  
Vol 21 (01) ◽  
pp. 15-22 ◽  
Author(s):  
W. Schlegel ◽  
K. Kayser
Keyword(s):  
Pattern Recognition ◽  
Graph Theory ◽  
Normal Tissue ◽  
Diagnostic Procedures ◽  
Minimal Distance ◽  
Malignant Growth ◽  
Colon Tissue ◽  
First Results ◽  
Tissue Structures ◽  
The Given

A basic concept for the automatic diagnosis of histo-pathological specimen is presented. The algorithm is based on tissue structures of the original organ. Low power magnification was used to inspect the specimens. The form of the given tissue structures, e. g. diameter, distance, shape factor and number of neighbours, is measured. Graph theory is applied by using the center of structures as vertices and the shortest connection of neighbours as edges. The algorithm leads to two independent sets of parameters which can be used for diagnostic procedures. First results with colon tissue show significant differences between normal tissue, benign and malignant growth. Polyps form glands that are twice as wide as normal and carcinomatous tissue. Carcinomas can be separated by the minimal distance of the glands formed. First results of pattern recognition using graph theory are discussed.

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