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Poncelet Theorem
, M.Sc. Thesis Sharif University of Technology ; Jafari, Amir (Supervisor)
Abstract
In this thesis the theorems of Poncelet and Cayley will be discussed. We will use algebraic geometry and in particular the theory of elliptic curves to present a proof following the works of Griffiths and Harris (refrence 1).
The P2-version of poncelet’s theorem that we prove in our thesis says:let C , D be smooth conic in general position. Suppose there is an n-sided olygon inscribed in C and circumscribed about D.Then for any point of C, there exists an n-sided polygen, also inscribed in C and circumscribe about D, which has this point for one of its vertics.the Cayley’s theorem provides a criterion on the existence of such an n-gon given the conics C and D as above
The P2-version of poncelet’s theorem that we prove in our thesis says:let C , D be smooth conic in general position. Suppose there is an n-sided olygon inscribed in C and circumscribed about D.Then for any point of C, there exists an n-sided polygen, also inscribed in C and circumscribe about D, which has this point for one of its vertics.the Cayley’s theorem provides a criterion on the existence of such an n-gon given the conics C and D as above
On Curves and Surfaces Defined Over Number Fields
, Ph.D. Dissertation Sharif University of Technology ; Fanai, Hamid Reza (Supervisor) ; Shahshahani, Mehrdad (Co-Advisor)
Abstract
The main purpose of this thesis is to study computational questions related to compact Riemann surfaces (algebraic curves) with main emphasis on the theory of Grothendieck’s dessins d’enfants; where recovering the explicit formula of a Belyi map from the corresponding dessin on a topological surface is important. Some applications such as investigating the modular j-function and also certain classes of modular curves are included in this thesis along with a summary of attempts toward generalizing the Belyi theorem to complex dimension two.First two chapters contain necessary prerequisites on compact Riemann surfaces and a short introduction to the theory of dessins d’enfants and Belyi...
A control method for a class of minimum-phase fractional-delay systems based on shaping the sensitivity function
, Article 2009 6th International Conference on Electrical Engineering/Electronics, Computer, Telecommunications and Information Technology, ECTI-CON 2009, Chonburi, 6 May 2009 through 9 May 2009 ; Volume 1 , 2009 , Pages 372-375 ; 9781424433889 (ISBN) ; Karimi Ghartemani, M ; Sharif University of Technology
2009
Abstract
This paper deals with the boundary control problem for a certain class of linear infinite-dimensional systems commonly known as fractional-delay systems. It is assumed that the systems under consideration are, in general, described by multi-valued transfer functions. In this paper, we restrict our studies to a class of multi-valued transfer functions which are defined on a Riemann surface with limited number of Riemann sheets where the origin is a branch point. The proposed controller design algorithm is based on shaping the sensitivity function. Contrary to many other methods which approximate the underlying infinite-dimensional system by a finite-dimensional one and then apply a classical...
An efficient numerical algorithm for stability testing of fractional-delay systems
, Article ISA Transactions ; Volume 48, Issue 1 , 2009 , Pages 32-37 ; 00190578 (ISSN) ; Karimi Ghartemani, M ; Sharif University of Technology
2009
Abstract
This paper presents a numerical algorithm for BIBO stability testing of a certain class of the so-called fractional-delay systems. The characteristic function of the systems under consideration is a multi-valued function of the Laplace variable s which is defined on a Riemann surface with finite number of Riemann sheets where the origin is a branch point. The stability analysis of such systems is not straightforward because there is no universally applicable analytical method to find the roots of the characteristic equation on the right half-plane of the first Riemann sheet. The proposed method is based on the Rouche's theorem which provides the number of the zeros of a given function in a...