Mathematics Commons^™

Characterizing Forced Communication In Networks, Samuel C. Gutekunst

HMC Senior Theses

This thesis studies a problem that has been proposed as a novel way to disrupt communication networks: the load maximization problem. The load on a member of a network represents the amount of communication that the member is forced to be involved in. By maximizing the load on an important member of the network, we hope to increase that member's visibility and susceptibility to capture. In this thesis we characterize load as a combinatorial property of graphs and expose possible connections between load and spectral graph theory. We specifically describe the load and how it changes in several canonical classes …

Symmetries Of Embedded Complete Bipartite Graphs, Erica Flapan, Nicole Lehle '09, Blake Mellor, Matt Pittluck, Xan Vongsathorn '09

Pomona Faculty Publications and Research

We characterize which automorphisms of an arbitrary complete bipartite graph K_n,m can be induced by a homeomorphism of some embedding of the graph in S³.

Knotted And Linked Products Of Recombination On T(2,N)#T(2,M) Substrates, Erica Flapan, Jeremy Grevet, Qi Li, Chen Daisy Sun, Helen Wong

Pomona Faculty Publications and Research

We develop a topological model of site-specific recombination that applies to substrates which are the connected sum of two torus links of the form T(2,n)#T(2,m). Then we use our model to prove that all knots and links that can be produced by site-specific recombination on such substrates are contained in one of two families, which we illustrate.

An Extremal Problem For Characteristic Functions, Stephan Ramon Garcia, Isabelle Chalendar, Williams T. Ross, Dan Timotin

Pomona Faculty Publications and Research

Suppose E is a subset of the unit circle T and H^infinity C L^infinity is the Hardy subalgebra. We examine the problem of finding the distance from the characteristic function of E to zⁿH^infinity. This admits an alternate description as a dual extremal problem. Precise solutions are given in several important cases. The techniques used involve the theory of Toeplitz and Hankel operators as well as the construction of certain conformal mappings.

Mathematical And Physical Aspects Of Complex Symmetric Operators, Stephan Ramon Garcia, Emil Prodan, Mihai Putinar

Pomona Faculty Publications and Research

Recent advances in the theory of complex symmetric operators are presented and related to current studies in non-hermitian quantum mechanics. The main themes of the survey are: the structure of complex symmetric operators, C-selfadjoint extensions of C-symmetric unbounded operators, resolvent estimates, reality of spectrum, bases of C-orthonormal vectors, and conjugate-linear symmetric operators. The main results are complemented by a variety of natural examples arising in field theory, quantum physics, and complex variables.

On Approximation Schemes And Compactness, Asuman Güven Aksoy, Jose M. Almira

CMC Faculty Publications and Research

We present an overview of some results about characterization of compactness in which the concept of approximation scheme has had a role. In particular, we present several results that were proved by the second author, jointly with Luther, a decade ago, when these authors were working on a very general theory of approximation spaces. We then introduce and show the basic properties of a new concept of compactness, which was studied by the first author in the eighties, by using a generalized concept of approximation scheme and its associated Kolmogorov numbers, which generalizes the classical concept of compactness.

The Apple Doesn’T Fall Far From The (Metric) Tree: Equivalence Of Definitions, Asuman Güven Aksoy, Sixian Jin

CMC Faculty Publications and Research

In this paper we prove the equivalence of definitions for metric trees and for δ-Hperbolic spaces. We point out how these equivalences can be used to understand the geometric and metric properties of δ-Hperbolic spaces and its relation to CAT(κ) spaces.

Existence Of Positive Solutions For A Superlinear Elliptic System With Neumann Boundary Condition, Alfonso Castro, Juan C. Cardeño

All HMC Faculty Publications and Research

We prove the existence of a positive solution for a class of nonlin- ear elliptic systems with Neumann boundary conditions. The proof combines extensive use of a priori estimates for elliptic problems with Neumann boundary condition and Krasnoselskii's compression-expansion theorem

Infinitely Many Rotationally Symmetric Solutions To A Class Of Semilinear Laplace-Beltrami Equations On The Unit Sphere, Emily M. Fischer

HMC Senior Theses

I show that a class of semilinear Laplace-Beltrami equations has infinitely many solutions on the unit sphere which are symmetric with respect to rotations around some axis. This equation corresponds to a singular ordinary differential equation, which we solve using energy analysis. We obtain a Pohozaev-type identity to prove that the energy is continuously increasing with the initial condition and then use phase plane analysis to prove the existence of infinitely many solutions.

The Archbishop's Odyssey, Leonard Sprague

Undergraduate Review

For centuries, scholars have analyzed a collection of problems that, nowadays, has been defined as NP-complete. Currently, NP-complete problems have no known efficient solutions. The Clay Mathematics Institute has offered a reward of one million dollars for a solution. The problem of finding Hamilton paths and cycles has been shown to be in this category. Knight’s tours, where the knight must visit every square of a chessboard exactly once, are examples of Hamilton paths and cycles.

This research revolves around the creation of a new branch of the tour problems, through a new piece: the Archbishop. Chess Grandmaster Jose Capablanca …

A Mathematical Analysis Of A Game Of Craps, Yaqin Sun

Undergraduate Review

The game of craps is an extremely popular game offered by casino operators. There are some 40 different types of bets that one can place each time the game is played. One of the best bets from a player’s point of view is the Pass Line bet. The probability of winning a Pass Line bet is almost the same as the probability of losing (²⁴⁴⁄₄₉₅ versus ²⁵¹⁄₄₉₅) as we will derive rigorously in this article. Since the “house” has such a small advantage over the players, many players possess the illusion that they have …

Composition Of Integers With Bounded Parts, Darren B. Glass

Math Faculty Publications

In this note, we consider ordered partitions of integers such that each entry is no more than a fixed portion of the sum. We give a method for constructing all such compositions as well as both an explicit formula and a generating function describing the number of k-tuples whose entries are bounded in this way and sum to a fixed value g.

A Survey Of Distance Magic Graphs, Rachel Rupnow

Dissertations, Master's Theses and Master's Reports - Open

In this report, we survey results on distance magic graphs and some closely related graphs. A distance magic labeling of a graph G with magic constant k is a bijection l from the vertex set to {1, 2, . . . , n}, such that for every vertex x

Σ ^{l(y) = k,}
_y∈NG(x)

where N_G(x) is the set of vertices of G adjacent to x. If the graph G has a distance magic labeling we say that G is a distance magic graph.

In Chapter 1, we explore the background of …

Topological Symmetry Groups Of Small Complete Graphs, Erica Flapan, Dwayne Chambers

Pomona Faculty Publications and Research

Topological symmetry groups were originally introduced to study the symmetries of non-rigid molecules, but have since been used to study the symmetries of any graph embedded in R³. In this paper, we determine for each complete graph K_n with n ≤ 6, what groups can occur as topological symmetry groups or orientation preserving topological symmetry groups of some embedding of the graph in R³.

Tiling Properties Of Spectra Of Measures, John Haussermann

Electronic Theses and Dissertations

We investigate tiling properties of spectra of measures, i.e., sets Λ in R such that {e 2πiλx : λ ∈ Λ} forms an orthogonal basis in L 2 (µ), where µ is some finite Borel measure on R. Such measures include Lebesgue measure on bounded Borel subsets, finite atomic measures and some fractal Hausdorff measures. We show that various classes of such spectra of measures have translational tiling properties. This lead to some surprizing tiling properties for spectra of fractal measures, the existence of complementing sets and spectra for finite sets with the Coven-Meyerowitz property, the existence of complementing Hadamard …

Well-Rounded Zeta-Function Of Planar Arithmetic Lattices, Lenny Fukshansky

CMC Faculty Publications and Research

We investigate the properties of the zeta-function of well-rounded sublattices of a fixed arithmetic lattice in the plane. In particular, we show that this function has abscissa of convergence at s=1 with a real pole of order 2, improving upon a result of Stefan Kühnlein. We use this result to show that the number of well-rounded sublattices of a planar arithmetic lattice of index less than or equal to N is O(N log N) as N → ∞. To obtain these results, we produce a description of integral well-rounded sublattices of a fixed planar integral well-rounded lattice and investigate convergence …

On The Geometry Of Cyclic Lattices, Lenny Fukshansky, Xun Sun

CMC Faculty Publications and Research

Cyclic lattices are sublattices of ZN that are preserved under the rotational shift operator. Cyclic lattices were introduced by D.~Micciancio and their properties were studied in the recent years by several authors due to their importance in cryptography. In particular, Peikert and Rosen showed that on cyclic lattices in prime dimensions, the shortest independent vectors problem SIVP reduces to the shortest vector problem SVP with a particularly small loss in approximation factor, as compared to general lattices. In this paper, we further investigate geometric properties of cyclic lattices. Our main result is a counting estimate for the number of well-rounded …

Small Zeros Of Quadratic Forms Outside A Union Of Varieties, Wai Kiu Chan, Lenny Fukshansky, Glenn R. Henshaw

CMC Faculty Publications and Research

Let be a quadratic form in variables defined on a vector space over a global field , and be a finite union of varieties defined by families of homogeneous polynomials over . We show that if contains a nontrivial zero of , then there exists a linearly independent collection of small-height zeros of in , where the height bound does not depend on the height of , only on the degrees of its defining polynomials. As a corollary of this result, we show that there exists a small-height maximal totally isotropic subspace of the quadratic space such that is not …

Arithmetical Graphs, Riemann-Roch Structure For Lattices, And The Frobenius Number Problem, Jeremy Usatine

HMC Senior Theses

If R is a list of positive integers with greatest common denominator equal to 1, calculating the Frobenius number of R is in general NP-hard. Dino Lorenzini defines the arithmetical graph, which naturally arises in arithmetic geometry, and a notion of genus, the g-number, that in specific cases coincides with the Frobenius number of R. A result of Dino Lorenzini's gives a method for quickly calculating upper bounds for the g-number of arithmetical graphs. We discuss the arithmetic geometry related to arithmetical graphs and present an example of an arithmetical graph that arises in this context. We also discuss the …

Reed's Conjecture And Cycle-Power Graphs, Alexa Serrato

HMC Senior Theses

Reed's conjecture is a proposed upper bound for the chromatic number of a graph. Reed's conjecture has already been proven for several families of graphs. In this paper, I show how one of those families of graphs can be extended to include additional graphs and also show that Reed's conjecture holds for a family of graphs known as cycle-power graphs, and also for their complements.