Hala Alaqad | Uaeu - Academia.edu (original) (raw)

Papers by Hala Alaqad

Fractal and Fractional

Many researchers have defined the q-analogous of differential and integral operators for analytic... more Many researchers have defined the q-analogous of differential and integral operators for analytic functions using the concept of quantum calculus in the geometric function theory. In this study, we conduct a comprehensive investigation to identify the uses of the Sălăgean q-differential operator for meromorphic multivalent functions. Many features of functions that belong to geometrically defined classes have been extensively studied using differential operators based on q-calculus operator theory. In this research, we extended the idea of the q-analogous of the Sălăgean differential operator for meromorphic multivalent functions using the fundamental ideas of q-calculus. With the help of this operator, we extend the family of Janowski functions by adding two new subclasses of meromorphic q-starlike and meromorphic multivalent q-starlike functions. We discover significant findings for these new classes, including the radius of starlikeness, partial sums, distortion theorems, and coe...

There are many criterion to generalize the concept of numerical radius; one of the most recent in... more There are many criterion to generalize the concept of numerical radius; one of the most recent interesting generalization is what so-called the generalized Euclidean operator radius. Simply, it is the numerical radius of multivariable operators. In this work, several new inequalities, refinements, and generalizations are established for this kind of numerical radius.

Symmetry

This paper proves several new inequalities for the Euclidean operator radius, which refine some r... more This paper proves several new inequalities for the Euclidean operator radius, which refine some recent results. It is shown that the new results are much more accurate than the related, recently published results. Moreover, inequalities for both symmetric and non-symmetric Hilbert space operators are studied.

In this Master thesis we consider the discrete groups with emphasis on the geometry of discrete g... more In this Master thesis we consider the discrete groups with emphasis on the geometry of discrete groups, which lie at the intersection between Hyperbolic Geometry, Topology, Abstract Algebra, and Complex Analysis. Fuchsian groups are discrete subgroups of the group PSL(2,R) of linear fractional transformations of one complex variable, which is isomorphic to a quotient topological group: PSL(2,R)≅SL(2,R)/{±I}. Here SL (2,R) is special linear group and I is the identity. We study discrete groups, in particular, Fuchsian groups. We present the geometric properties of Fuchsian groups such as fundamental domains, compactness, and Dilichlet tessellations. In addition, we also present some algebraic properties of Fuchsian groups.

Abstract and Applied Analysis

A two-generator Kleinian group f , g can be naturally associated with a discrete group f , ϕ with... more A two-generator Kleinian group f , g can be naturally associated with a discrete group f , ϕ with the generator ϕ of order two and where f , ϕ f ϕ − 1 = f , g f g − 1 ⊂ f , g , f , ϕ : f , g f g − 1 = 2 . This is useful in studying the geometry of the Kleinian groups since f , g will be discrete only if f , ϕ is, and the moduli space of groups f , ϕ is one complex dimension less. This gives a necessary condition in a simpler space to determine the discreteness of f , g . The dimension reduction here is realised by a projection of principal characters of the two-generator Kleinian groups. In applications, it is important to know that the image of the moduli space of Kleinian groups under this projection is closed and, among other results, we show how this follows from Jørgensen’s results on algebraic convergence.

A two-generator Kleinian group 〈f, g〉 can be naturally associated with a discrete group 〈f, φ〉 wi... more A two-generator Kleinian group 〈f, g〉 can be naturally associated with a discrete group 〈f, φ〉 with the generator φ of order 2 and where 〈f, φfφ〉 = 〈f, gfg〉 ⊂ 〈f, g〉, [〈f, gfg〉 : 〈f, φ〉] = 2 This is useful in studying the geometry of Kleinian groups since 〈f, g〉 will be discrete only if 〈f, φ〉 is, and the moduli space of groups 〈f, φ〉 is one complex dimension less. This gives a necessary condition in a simpler space to determine the discreteness of 〈f, g〉. The dimension reduction here is realised by a projection of principal characters of two-generator Kleinian groups. In applications it is important to know that the image of the moduli space of Kleinian groups under this projection is closed and, among other results, we show how this follows from Jørgensen’s results on algebraic convergence.

The principal character of a representation of the free group of rank two into PSL(2,C) is a trip... more The principal character of a representation of the free group of rank two into PSL(2,C) is a triple of complex numbers that determines an irreducible representation uniquely up to conjugacy. It is a central problem in the geometry of discrete groups and low dimensional topology to determine when such a triple represents a discrete group which is not virtually abelian, that is a Kleinian group. A classical necessary condition is Jørgensens inequality. Here we use certainly shifted Chebyshev polynomials and trace identities to determine new families of such inequalities, some of which are best possible. The use of these polynomials also shows how we can identify the principal character of some important subgroups from that of the group itself.

Prof. Frederick Gehring (famous American mathematician) and Prof. Martin developed important tech... more Prof. Frederick Gehring (famous American mathematician) and Prof. Martin developed important techniques now used in low dimensional topology and geometry. Their innovation was to investigate the structure of the moduli space of Kleinian groups using recently discovered families of polynomial trace identities giving new inequalities such as the well known Jorgensen inequality. The new techniques and ideas in the area were applied to obtain quantitative descriptions of spaces of two generator Kleinian groups. The purposes of this project is to expose a new method for finding these inequalities in a more general setting and to identify various sharp inequalities building on earlier work of Gehring et al.

Fractal and Fractional

Many researchers have defined the q-analogous of differential and integral operators for analytic... more Many researchers have defined the q-analogous of differential and integral operators for analytic functions using the concept of quantum calculus in the geometric function theory. In this study, we conduct a comprehensive investigation to identify the uses of the Sălăgean q-differential operator for meromorphic multivalent functions. Many features of functions that belong to geometrically defined classes have been extensively studied using differential operators based on q-calculus operator theory. In this research, we extended the idea of the q-analogous of the Sălăgean differential operator for meromorphic multivalent functions using the fundamental ideas of q-calculus. With the help of this operator, we extend the family of Janowski functions by adding two new subclasses of meromorphic q-starlike and meromorphic multivalent q-starlike functions. We discover significant findings for these new classes, including the radius of starlikeness, partial sums, distortion theorems, and coe...

There are many criterion to generalize the concept of numerical radius; one of the most recent in... more There are many criterion to generalize the concept of numerical radius; one of the most recent interesting generalization is what so-called the generalized Euclidean operator radius. Simply, it is the numerical radius of multivariable operators. In this work, several new inequalities, refinements, and generalizations are established for this kind of numerical radius.

Symmetry

This paper proves several new inequalities for the Euclidean operator radius, which refine some r... more This paper proves several new inequalities for the Euclidean operator radius, which refine some recent results. It is shown that the new results are much more accurate than the related, recently published results. Moreover, inequalities for both symmetric and non-symmetric Hilbert space operators are studied.

In this Master thesis we consider the discrete groups with emphasis on the geometry of discrete g... more In this Master thesis we consider the discrete groups with emphasis on the geometry of discrete groups, which lie at the intersection between Hyperbolic Geometry, Topology, Abstract Algebra, and Complex Analysis. Fuchsian groups are discrete subgroups of the group PSL(2,R) of linear fractional transformations of one complex variable, which is isomorphic to a quotient topological group: PSL(2,R)≅SL(2,R)/{±I}. Here SL (2,R) is special linear group and I is the identity. We study discrete groups, in particular, Fuchsian groups. We present the geometric properties of Fuchsian groups such as fundamental domains, compactness, and Dilichlet tessellations. In addition, we also present some algebraic properties of Fuchsian groups.

Abstract and Applied Analysis

A two-generator Kleinian group f , g can be naturally associated with a discrete group f , ϕ with... more A two-generator Kleinian group f , g can be naturally associated with a discrete group f , ϕ with the generator ϕ of order two and where f , ϕ f ϕ − 1 = f , g f g − 1 ⊂ f , g , f , ϕ : f , g f g − 1 = 2 . This is useful in studying the geometry of the Kleinian groups since f , g will be discrete only if f , ϕ is, and the moduli space of groups f , ϕ is one complex dimension less. This gives a necessary condition in a simpler space to determine the discreteness of f , g . The dimension reduction here is realised by a projection of principal characters of the two-generator Kleinian groups. In applications, it is important to know that the image of the moduli space of Kleinian groups under this projection is closed and, among other results, we show how this follows from Jørgensen’s results on algebraic convergence.

A two-generator Kleinian group 〈f, g〉 can be naturally associated with a discrete group 〈f, φ〉 wi... more A two-generator Kleinian group 〈f, g〉 can be naturally associated with a discrete group 〈f, φ〉 with the generator φ of order 2 and where 〈f, φfφ〉 = 〈f, gfg〉 ⊂ 〈f, g〉, [〈f, gfg〉 : 〈f, φ〉] = 2 This is useful in studying the geometry of Kleinian groups since 〈f, g〉 will be discrete only if 〈f, φ〉 is, and the moduli space of groups 〈f, φ〉 is one complex dimension less. This gives a necessary condition in a simpler space to determine the discreteness of 〈f, g〉. The dimension reduction here is realised by a projection of principal characters of two-generator Kleinian groups. In applications it is important to know that the image of the moduli space of Kleinian groups under this projection is closed and, among other results, we show how this follows from Jørgensen’s results on algebraic convergence.

The principal character of a representation of the free group of rank two into PSL(2,C) is a trip... more The principal character of a representation of the free group of rank two into PSL(2,C) is a triple of complex numbers that determines an irreducible representation uniquely up to conjugacy. It is a central problem in the geometry of discrete groups and low dimensional topology to determine when such a triple represents a discrete group which is not virtually abelian, that is a Kleinian group. A classical necessary condition is Jørgensens inequality. Here we use certainly shifted Chebyshev polynomials and trace identities to determine new families of such inequalities, some of which are best possible. The use of these polynomials also shows how we can identify the principal character of some important subgroups from that of the group itself.

Prof. Frederick Gehring (famous American mathematician) and Prof. Martin developed important tech... more Prof. Frederick Gehring (famous American mathematician) and Prof. Martin developed important techniques now used in low dimensional topology and geometry. Their innovation was to investigate the structure of the moduli space of Kleinian groups using recently discovered families of polynomial trace identities giving new inequalities such as the well known Jorgensen inequality. The new techniques and ideas in the area were applied to obtain quantitative descriptions of spaces of two generator Kleinian groups. The purposes of this project is to expose a new method for finding these inequalities in a more general setting and to identify various sharp inequalities building on earlier work of Gehring et al.