Mathematics Commons^™

Cheeger Constants Of Two Related Hyperbolic Riemann Surfaces, Ronald E. Hoagland

Masters Theses

This thesis concerns the study of the Cheeger constant of two related hyperbolic Riemann surfaces. The first surface R is formed by taking the quotient U²/Γ(4), where U² is the upper half-plane model of the hyperbolic plane and Γ(4) is a congruence subgroup of PSL2(Z), an isometry group of U² . This quotient is shown to form a Riemann surface which is constructed by gluing sides of a fundamental domain for Γ(4) together according to certain specified side pairings. To form the related Riemann surface R' , we follow a similar procedure, this time taking the …

Using A Discursive Framework To Analyze Geometric Learning And Instruction, Peter Wiles, Rick Anderson

Faculty Research and Creative Activity

In this study we applied a discursive perspective of learning (Sfard, 2008) to a sequence of 21 geometry mini-lessons taught in a fourth grade classroom. From this perspective, learning is defined as changes in mathematical discourse. We first characterize and then compare discourse from the beginning and the end of the mini-lesson sequence. We identify shifts in the discourse that occurred during the sequence. We then discuss how the characteristics of the st geometric discourse informed task design and instruction. This perspective provided a useful means for linking instruction to student learning in an operationalized manner.

Hidden Symmetries In Classical Mechanics And Related Number Theory Dynamical System, Mohsin Md Abdul Karim

Masters Theses

Classical Mechanics consists of three parts: Newtonian, Lagrangian and Hamiltonian Mechanics, where each part is a special extension of the previous part. Each part has explicit symmetries (the explicit Laws of Motion), which, in turn, generate implicit or hidden symmetries (like the Law of Conservation of Energy, etc). In this Master's Thesis, different types of hidden symmetries are considered; they are reflected in the Noether Theorem and the Poincare Recurrence Theorem applied to Lagrangian and Hamiltonian Systems respectively.

The Poincare Recurrence Theorem is also applicable to some number theory problems, which can be considered as dynamical systems. In …

An Exposition Of The Eisenstein Integers, Sarada Bandara

Masters Theses

In this thesis, we will give a brief introduction to number theory and prime numbers. We also provide the necessary background to understand how the imaginary ring of quadratic integers behaves.

An example of said ring are complex numbers of the form ℤ[ω] = {a+bω ∣ a, b ∈ ℤ} where ω² + ω + 1 = 0. These are known as the Eisenstein integers, which form a triangular lattice in the complex plane, in contrast with the Gaussian integers, ℤ[i] = {a + bi ∣ a, b ∈ …

Hyperbolic Geometry With And Without Models, Chad Kelterborn

Masters Theses

We explore the development of hyperbolic geometry in the 18th and early 19th following the works of Legendre, Lambert, Saccheri, Bolyai, Lobachevsky, and Gauss. In their attempts to prove Euclid's parallel postulate, they developed hyperbolic geometry without a model. It was not until later in the 19th century, when Felix Klein provided a method (which was influenced by projective geometry) for viewing the hyperbolic plane as a disk in the Euclidean plane, appropriately named the "Klein disk model". Later other models for viewing the hyperbolic plane as a subset of the Euclidean plane were created, namely the Poincaré disk model, …

A Glance At Tropical Operations And Tropical Linear Algebra, Semere Tsehaye Tesfay

Masters Theses

The tropical semiring is ℝ ∪ {∞} with the operations x ⊕ y = min{x, y}, x ⊕ ∞ = ∞ ⊕ x = x, x ⊙ y = x + y, x ⊙ ∞ = ∞ ⊙ y = ∞. This paper explores how ideas from classical algebra and linear algebra over the real numbers such as polynomials, roots of polynomials, lines, matrices and matrix operations, determinants, eigen values and eigen vectors would appear in tropical mathematics. It uses numerous computed examples to illustrate these concepts and explores the relationship between certain tropical matrices and graph …

A Gentle Introduction To Pythontex, Andrew Mertz, William A. Slough

William A. Slough

No abstract provided.

Symmetric Random Walks On Homeo+(R), B. Deroin, V. Klepstyn, A. Navas, Kamlesh Parwani

Kamlesh Parwani

We study symmetric random walks on finitely generated groups of orientation-preserving homeomorphisms of the real line. We establish an oscillation property for the induced Markov chain on the line that implies a weak form of recurrence. Except for a few special cases, which can be treated separately, we prove a property of "global stability at a finite distance": roughly speaking, there exists a compact interval such that any two trajectories get closer and closer whenever one of them returns to the compact interval. The probabilistic techniques employed here lead to interesting results for the study of group actions on the …

Symmetric Random Walks On Homeo+(R), B. Deroin, V. Klepstyn, A. Navas, Kamlesh Parwani

Faculty Research and Creative Activity

We study symmetric random walks on finitely generated groups of orientation-preserving homeomorphisms of the real line. We establish an oscillation property for the induced Markov chain on the line that implies a weak form of recurrence. Except for a few special cases, which can be treated separately, we prove a property of "global stability at a finite distance": roughly speaking, there exists a compact interval such that any two trajectories get closer and closer whenever one of them returns to the compact interval. The probabilistic techniques employed here lead to interesting results for the study of group actions on the …

A Gentle Introduction To Pythontex, Andrew Mertz, William Slough

Faculty Research and Creative Activity

No abstract provided.

Symmetric Random Walks On Homeo+(R), B. Deroin, V. Klepstyn, A. Navas, Kamlesh Parwani

Faculty Research and Creative Activity

We study symmetric random walks on finitely generated groups of orientation-preserving homeomorphisms of the real line. We establish an oscillation property for the induced Markov chain on the line that implies a weak form of recurrence. Except for a few special cases, which can be treated separately, we prove a property of "global stability at a finite distance": roughly speaking, there exists a compact interval such that any two trajectories get closer and closer whenever one of them returns to the compact interval. The probabilistic techniques employed here lead to interesting results for the study of group actions on the …

Near Minimum Energy Distributions On The Sphere Using Voronoi Cells, Benedictus Sitou Mensah

Masters Theses

No abstract provided.

Of Music, Mathematics, And Magic: Why Math Is All Made Up And Why It Works So Well, Gregory A. Leach

Masters Theses

No abstract provided.

Harmonic Functions On R-Covered Foliations And Group Actions On The Circle, Sergio Fenley, Renato Feres, Kamlesh Parwani

Kamlesh Parwani

Let (M,F) be a compact codimension-one foliated manifold whose leaves are equipped with Riemannian metrics, and consider continuous functions on M that are harmonic along the leaves of F. If every such function is constant on leaves we say that (M,F) has the Liouville property. Our main result is that codimension-one foliated bundles over compact negatively curved manifolds satisfy the Liouville property. Related results for R-covered foliations, as well as for discrete group actions and discrete harmonic functions, are also established.

Harmonic Functions On R-Covered Foliations And Group Actions On The Circle, Sergio Fenley, Renato Feres, Kamlesh Parwani

Faculty Research and Creative Activity

Let (M,F) be a compact codimension-one foliated manifold whose leaves are equipped with Riemannian metrics, and consider continuous functions on M that are harmonic along the leaves of F. If every such function is constant on leaves we say that (M,F) has the Liouville property. Our main result is that codimension-one foliated bundles over compact negatively curved manifolds satisfy the Liouville property. Related results for R-covered foliations, as well as for discrete group actions and discrete harmonic functions, are also established.

Harmonic Functions On R-Covered Foliations And Group Actions On The Circle, Sergio Fenley, Renato Feres, Kamlesh Parwani

Faculty Research and Creative Activity

Let (M,F) be a compact codimension-one foliated manifold whose leaves are equipped with Riemannian metrics, and consider continuous functions on M that are harmonic along the leaves of F. If every such function is constant on leaves we say that (M,F) has the Liouville property. Our main result is that codimension-one foliated bundles over compact negatively curved manifolds satisfy the Liouville property. Related results for R-covered foliations, as well as for discrete group actions and discrete harmonic functions, are also established.

A Tikz Tutorial: Generating Graphics In The Spirit Of Tex, Andrew Mertz, William Slough

Andrew Mertz

TikZ is a system which can be used to specify graphics of very high quality. For example, accurate place- ment of picture elements, use of TEX fonts, ability to incorporate mathematical typesetting, and the possi- bility of introducing macros can be viewed as positive factors of this system. The syntax uses an amal- gamation of ideas from METAFONT, METAPOST, PSTricks, and SVG, allowing its users to \program" their desired graphics. The latest revision to TikZ introduces many new features to an already feature- packed system, as evidenced by its 560-page user manual. Here, we present a tutorial overview of this …

The Two Covering Radius Of The Two Error Correcting Bch Code, Andrew Klapper, Andrew Mertz

Andrew Mertz

The m-covering radii of codes are natural generalizations of the covering radii of codes. In this paper we analyze the 2-covering radii of double error correcting BCH code.

A Tikz Tutorial: Generating Graphics In The Spirit Of Tex, Andrew Mertz, William Slough

Faculty Research and Creative Activity

TikZ is a system which can be used to specify graphics of very high quality. For example, accurate place- ment of picture elements, use of TEX fonts, ability to incorporate mathematical typesetting, and the possi- bility of introducing macros can be viewed as positive factors of this system. The syntax uses an amal- gamation of ideas from METAFONT, METAPOST, PSTricks, and SVG, allowing its users to \program" their desired graphics. The latest revision to TikZ introduces many new features to an already feature- packed system, as evidenced by its 560-page user manual. Here, we present a tutorial overview of this …

C^1 Actions Of The Mapping Class Group On The Circle, Kamlesh Parwani

Kamlesh Parwani

Let S be a connected orientable surface with finitely many punctures, finitely many boundary components, and genus at least 6. Then any C^1 action of the mapping class group of S on the circle is trivial. The techniques used in the proof of this result permit us to show that products of Kazhdan groups and certain lattices cannot have C^1 faithful actions on the circle. We also prove that for n > 5, any C^1 action of Aut(F_n) or Out(F_n) on the circle factors through an action of Z/2Z.

The Two Variable Substitution Problem For Free Products Of Groups, Leo P. Comerford, Charles C. Edmunds

Leo Comerford

We consider equations of the form W(x,y) = U with U an element of a free product G of groups. We show that with suitable algorithmic conditions on the free factors of G, one can effectively determine whether or not the equations have solutions in G. We also show that under certain hypotheses on the free factors of G and the equation itself, the equation W(x,y) = U has only finitely many solutions, up to the action of the stabilizer of W(x,y) in Aut().

The Two Variable Substitution Problem For Free Products Of Groups, Leo P. Comerford, Charles C. Edmunds

Faculty Research and Creative Activity

We consider equations of the form W(x,y) = U with U an element of a free product G of groups. We show that with suitable algorithmic conditions on the free factors of G, one can effectively determine whether or not the equations have solutions in G. We also show that under certain hypotheses on the free factors of G and the equation itself, the equation W(x,y) = U has only finitely many solutions, up to the action of the stabilizer of W(x,y) in Aut().

C^1 Actions Of The Mapping Class Group On The Circle, Kamlesh Parwani

Faculty Research and Creative Activity

Let S be a connected orientable surface with finitely many punctures, finitely many boundary components, and genus at least 6. Then any C^1 action of the mapping class group of S on the circle is trivial. The techniques used in the proof of this result permit us to show that products of Kazhdan groups and certain lattices cannot have C^1 faithful actions on the circle. We also prove that for n > 5, any C^1 action of Aut(F_n) or Out(F_n) on the circle factors through an action of Z/2Z.

C^1 Actions Of The Mapping Class Group On The Circle, Kamlesh Parwani

Faculty Research and Creative Activity

Let S be a connected orientable surface with finitely many punctures, finitely many boundary components, and genus at least 6. Then any C^1 action of the mapping class group of S on the circle is trivial. The techniques used in the proof of this result permit us to show that products of Kazhdan groups and certain lattices cannot have C^1 faithful actions on the circle. We also prove that for n > 5, any C^1 action of Aut(F_n) or Out(F_n) on the circle factors through an action of Z/2Z.

The Two Variable Substitution Problem For Free Products Of Groups, Leo Comerford, Charles Edmunds

Faculty Research and Creative Activity

We consider equations of the form W(x,y) = U with U an element of a free product G of groups. We show that with suitable algorithmic conditions on the free factors of G, one can effectively determine whether or not the equations have solutions in G. We also show that under certain hypotheses on the free factors of G and the equation itself, the equation W(x,y) = U has only finitely many solutions, up to the action of the stabilizer of W(x,y) in Aut().

Casimir Effect In Quantum Physics, Matthew James Urfer

Masters Theses

No abstract provided.

Numerical Approximations Of Differential Equations And Applications In Maple, Joyce Zimmerman

Masters Theses

No abstract provided.

Fixed Points Of Abelian Actions On S2, John Franks, Michael Handel, Kamlesh Parwani

Kamlesh Parwani

We prove that if $F$ is a finitely generated abelian group of orientation preserving $C^1$ diffeomorphisms of $R^2$ which leaves invariant a compact set then there is a common fixed point for all elements of $F.$ We also show that if $F$ is any abelian subgroup of orientation preserving $C^1$ diffeomorphisms of $S^2$ then there is a common fixed point for all elements of a subgroup of $F$ with index at most two.

Fixed Points Of Abelian Actions On S2, John Franks, Michael Handel, Kamlesh Parwani

Faculty Research and Creative Activity

We prove that if $F$ is a finitely generated abelian group of orientation preserving $C^1$ diffeomorphisms of $R^2$ which leaves invariant a compact set then there is a common fixed point for all elements of $F.$ We also show that if $F$ is any abelian subgroup of orientation preserving $C^1$ diffeomorphisms of $S^2$ then there is a common fixed point for all elements of a subgroup of $F$ with index at most two.

Fixed Points Of Abelian Actions On S2, John Franks, Michael Handel, Kamlesh Parwani

Faculty Research and Creative Activity

We prove that if $F$ is a finitely generated abelian group of orientation preserving $C^1$ diffeomorphisms of $R^2$ which leaves invariant a compact set then there is a common fixed point for all elements of $F.$ We also show that if $F$ is any abelian subgroup of orientation preserving $C^1$ diffeomorphisms of $S^2$ then there is a common fixed point for all elements of a subgroup of $F$ with index at most two.