Nabil Obeid - Academia.edu (original) (raw)

Papers by Nabil Obeid

Tensors are mathematical objects that generalize vectors and matrices. They describe geometrical ... more Tensors are mathematical objects that generalize vectors and matrices. They describe geometrical quantities and they are used in various applied settings including mathematical physics. The indicial notation of tensors permits us to write an expression in a compact manner and to use simplifying mathematical operations. In a large number of problems in differential geometry and general relativity, the time consuming and straightforward algebraic manipulation is obviously very important. Thus, tensor computation came into existence and became necessary and desirable at the same time. Over the past 25 years, few algorithms have appeared for simplifying tensor expressions. Among the most important tensor computation systems, we can mention SHEEP, Macsyma !Tensor Package, MathTensor and GRTensorII. Meanwhile, graph theory, which had been lying almost dormant for hundreds of years since the time of Euler, started to explode by the turn of the 20th century. It has now grown into a major discipline in mathematics, which has branched off today in various directions such as coloring problems, Ramsey theory, factorization theory and optimization, with problems permeating into many scientific areas such as physics, chemistry, engineering, psychology, and of course computer science. Investigating some of the tensor computation packages will show that they have some deficiencies. Thus, rather than building a new system and adding more features to it, it was an objective in this thesis to express an efficient algorithms by removing most, if not all, restrictions compared to other packages, using graph theory. A summary of the implementation and the advantages of this system is also included. iii I am thankful to my supervisor, Professor Stephen Watt, who taught me everything I needed to know about tensor expressions, for accepting to supervise me, for his kindness and generous contributions of time and for his careful commentary of my thesis. Without him, this work would never been completed. Also, I would like to thank every person in the SCL lab for their helpful advice and useful comments on this thesis. Furthermore, I am sincerely grateful to my parents who kept supporting me regardless of the consequences and to my family, especially my wife, for their endless support and love. I shall never forget that.

Absolutely continuous invariant measures for the family of maps x → rxe-bxwith application to the Belousov-Zhabotinski reaction

Dynamics and Stability of Systems, 1990

For the family of transformations τ(r,b,x)=rxe-bx which models the Poincare sections of the Belou... more For the family of transformations τ(r,b,x)=rxe-bx which models the Poincare sections of the Belousov-Zhabotinski reaction we prove that there is an uncountable set A such that for each τξA,τ(r,b,˙) has an absolutely continuous invariant measure. We also approximate the density of this measure and thereby the Lyapunov exponent of τ(r,b,˙) .This is accomplished by proving that τ(r,b,˙) is conjugate, via an absolutely continuous homeomorphism, to a transormation T, where Tx is piecewise expanding, for some integers

Tensors are mathematical objects that generalize vectors and matrices. They describe geometrical ... more Tensors are mathematical objects that generalize vectors and matrices. They describe geometrical quantities and they are used in various applied settings including mathematical physics. The indicial notation of tensors permits us to write an expression in a compact manner and to use simplifying mathematical operations. In a large number of problems in differential geometry and general relativity, the time consuming and straightforward algebraic manipulation is obviously very important. Thus, tensor computation came into existence and became necessary and desirable at the same time. Over the past 25 years, few algorithms have appeared for simplifying tensor expressions. Among the most important tensor computation systems, we can mention SHEEP, Macsyma !Tensor Package, MathTensor and GRTensorII. Meanwhile, graph theory, which had been lying almost dormant for hundreds of years since the time of Euler, started to explode by the turn of the 20th century. It has now grown into a major di...

Absolutely continuous invariant measures for a class of meromorphic functions

We are going to consider a meromorphic function g: RetoRe,\Re \to \Re,RetoRe, which has a constant sign in th... more We are going to consider a meromorphic function g: RetoRe,\Re \to \Re,RetoRe, which has a constant sign in the upper half plan. We will show that it has a special form$${\rm g(z) = A} + \varepsilon\left\lbrack Bz- \sum\sb{s} p\sb{s} ({1\over{z-c\sb{s}}}{+}{1\over {c\sb{s}}})\right\rbrack$$where the poles are real and simples. Subsequently, we will demonstrate that it has an absolutely continuous invariant measure. Finally, we will present an example to emphasise the use of this transformation.

Tensors are mathematical objects that generalize vectors and matrices. They describe geometrical ... more Tensors are mathematical objects that generalize vectors and matrices. They describe geometrical quantities and they are used in various applied settings including mathematical physics. The indicial notation of tensors permits us to write an expression in a compact manner and to use simplifying mathematical operations. In a large number of problems in differential geometry and general relativity, the time consuming and straightforward algebraic manipulation is obviously very important. Thus, tensor computation came into existence and became necessary and desirable at the same time. Over the past 25 years, few algorithms have appeared for simplifying tensor expressions. Among the most important tensor computation systems, we can mention SHEEP, Macsyma !Tensor Package, MathTensor and GRTensorII. Meanwhile, graph theory, which had been lying almost dormant for hundreds of years since the time of Euler, started to explode by the turn of the 20th century. It has now grown into a major discipline in mathematics, which has branched off today in various directions such as coloring problems, Ramsey theory, factorization theory and optimization, with problems permeating into many scientific areas such as physics, chemistry, engineering, psychology, and of course computer science. Investigating some of the tensor computation packages will show that they have some deficiencies. Thus, rather than building a new system and adding more features to it, it was an objective in this thesis to express an efficient algorithms by removing most, if not all, restrictions compared to other packages, using graph theory. A summary of the implementation and the advantages of this system is also included. iii I am thankful to my supervisor, Professor Stephen Watt, who taught me everything I needed to know about tensor expressions, for accepting to supervise me, for his kindness and generous contributions of time and for his careful commentary of my thesis. Without him, this work would never been completed. Also, I would like to thank every person in the SCL lab for their helpful advice and useful comments on this thesis. Furthermore, I am sincerely grateful to my parents who kept supporting me regardless of the consequences and to my family, especially my wife, for their endless support and love. I shall never forget that.

Absolutely continuous invariant measures for the family of maps x → rxe-bxwith application to the Belousov-Zhabotinski reaction

Dynamics and Stability of Systems, 1990

For the family of transformations τ(r,b,x)=rxe-bx which models the Poincare sections of the Belou... more For the family of transformations τ(r,b,x)=rxe-bx which models the Poincare sections of the Belousov-Zhabotinski reaction we prove that there is an uncountable set A such that for each τξA,τ(r,b,˙) has an absolutely continuous invariant measure. We also approximate the density of this measure and thereby the Lyapunov exponent of τ(r,b,˙) .This is accomplished by proving that τ(r,b,˙) is conjugate, via an absolutely continuous homeomorphism, to a transormation T, where Tx is piecewise expanding, for some integers

Tensors are mathematical objects that generalize vectors and matrices. They describe geometrical ... more Tensors are mathematical objects that generalize vectors and matrices. They describe geometrical quantities and they are used in various applied settings including mathematical physics. The indicial notation of tensors permits us to write an expression in a compact manner and to use simplifying mathematical operations. In a large number of problems in differential geometry and general relativity, the time consuming and straightforward algebraic manipulation is obviously very important. Thus, tensor computation came into existence and became necessary and desirable at the same time. Over the past 25 years, few algorithms have appeared for simplifying tensor expressions. Among the most important tensor computation systems, we can mention SHEEP, Macsyma !Tensor Package, MathTensor and GRTensorII. Meanwhile, graph theory, which had been lying almost dormant for hundreds of years since the time of Euler, started to explode by the turn of the 20th century. It has now grown into a major di...

Absolutely continuous invariant measures for a class of meromorphic functions

We are going to consider a meromorphic function g: RetoRe,\Re \to \Re,RetoRe, which has a constant sign in th... more We are going to consider a meromorphic function g: RetoRe,\Re \to \Re,RetoRe, which has a constant sign in the upper half plan. We will show that it has a special form$${\rm g(z) = A} + \varepsilon\left\lbrack Bz- \sum\sb{s} p\sb{s} ({1\over{z-c\sb{s}}}{+}{1\over {c\sb{s}}})\right\rbrack$$where the poles are real and simples. Subsequently, we will demonstrate that it has an absolutely continuous invariant measure. Finally, we will present an example to emphasise the use of this transformation.