Tropical Convexity Over Max-Min Semiring, 2016 West Chester University of Pennsylvania

#### Tropical Convexity Over Max-Min Semiring, Viorel Nitica, Sergei Sergeev

*Viorel Nitica*

No abstract provided.

A Metric On Max-Min Algebra, 2016 University of Oregon

#### A Metric On Max-Min Algebra, Jonathan Eskeldson, Miriam Jaffe, Viorel Nitica

*Viorel Nitica*

No abstract provided.

On Hyperplanes And Semispaces In Max-Min Convex Geometry, 2016 West Chester University of Pennsylvania

#### On Hyperplanes And Semispaces In Max-Min Convex Geometry, Viorel Nitica, Sergeĭ Sergeev

*Viorel Nitica*

No abstract provided.

A Metric On Max-Min Algebra, 2016 University of Oregon

#### A Metric On Max-Min Algebra, Jonathan Eskeldson, Miriam Jaffe, Viorel Nitica

*Viorel Nitica*

No abstract provided.

On Hyperplanes And Semispaces In Max-Min Convex Geometry, 2016 West Chester University of Pennsylvania

#### On Hyperplanes And Semispaces In Max-Min Convex Geometry, Viorel Nitica, Sergeĭ Sergeev

*Viorel Nitica*

No abstract provided.

Topological And Hq Equivalence Of Prime Cyclic P-Gonal Actions On Riemann Surfaces, 2016 Rose-Hulman Institute of Technology

#### Topological And Hq Equivalence Of Prime Cyclic P-Gonal Actions On Riemann Surfaces, Sean A. Broughton

*Mathematical Sciences Technical Reports (MSTR)*

Two Riemann surfaces *S*_{1} and *S*_{2} with conformal *G*-actions have topologically equivalent actions if there is a homeomorphism *h :* *S _{1} -> S_{2} *which intertwines the actions. A weaker equivalence may be defined by comparing the representations of

*G*on the spaces of holomorphic

*q-*differentials

*H*and

^{q}(S_{1})*H*In this note we study the differences between topological equivalence and

^{q}(S_{2}).*H*equivalence of prime cyclic actions, where

^{q}*S*and

_{1}/G*S*have genus zero.

_{2}/GThe Implicit Function Theorem And Free Algebraic Sets, 2016 Washington University in St Louis

#### The Implicit Function Theorem And Free Algebraic Sets, Jim Agler, John E. Mccarthy

*Mathematics Faculty Publications*

We prove an implicit function theorem for non-commutative functions. We use this to show that if p ( X;Y ) is a generic non-commuting polynomial in two variables, and X is a generic matrix, then all solutions Y of p ( X;Y ) = 0 will commute with X

Klein Bottle Queries, 2016 Georgia State University

#### Klein Bottle Queries, Austin Lowe

*Georgia State Undergraduate Research Conference*

No abstract provided.

Aspects Of Non-Commutative Function Theory, 2016 Washington University in St Louis

#### Aspects Of Non-Commutative Function Theory, Jim Agler, John E. Mccarthy

*Mathematics Faculty Publications*

We discuss non commutative functions, which naturally arise when dealing with functions of more than one matrix variable.

The Log-Exponential Smoothing Technique And Nesterov’S Accelerated Gradient Method For Generalized Sylvester Problems, 2016 Thua Thien Hue College of Education

#### The Log-Exponential Smoothing Technique And Nesterov’S Accelerated Gradient Method For Generalized Sylvester Problems, N. T. An, Daniel J. Giles, Nguyen Mau Nam, R. Blake Rector

*Mathematics and Statistics Faculty Publications and Presentations*

The Sylvester or smallest enclosing circle problem involves finding the smallest circle enclosing a finite number of points in the plane. We consider generalized versions of the Sylvester problem in which the points are replaced by sets. Based on the log-exponential smoothing technique and Nesterov’s accelerated gradient method, we present an effective numerical algorithm for solving these problems.

Minimizing Differences Of Convex Functions With Applications To Facility Location And Clustering, 2016 Portland State University

#### Minimizing Differences Of Convex Functions With Applications To Facility Location And Clustering, Mau Nam Nguyen, R. Blake Rector, Daniel J. Giles

*Mathematics and Statistics Faculty Publications and Presentations*

In this paper we develop algorithms to solve generalized Fermat-Torricelli problems with both positive and negative weights and multifacility location problems involving distances generated by Minkowski gauges. We also introduce a new model of clustering based on squared distances to convex sets. Using the Nesterov smoothing technique and an algorithm for minimizing differences of convex functions called the DCA introduced by Tao and An, we develop effective algorithms for solving these problems. We demonstrate the algorithms with a variety of numerical examples.

Adinkras And Arithmetical Graphs, 2016 Harvey Mudd College

#### Adinkras And Arithmetical Graphs, Madeleine Weinstein

*HMC Senior Theses*

Adinkras and arithmetical graphs have divergent origins. In the spirit of Feynman diagrams, adinkras encode representations of supersymmetry algebras as graphs with additional structures. Arithmetical graphs, on the other hand, arise in algebraic geometry, and give an arithmetical structure to a graph. In this thesis, we will interpret adinkras as arithmetical graphs and see what can be learned.

Our work consists of three main strands. First, we investigate arithmetical structures on the underlying graph of an adinkra in the specific case where the underlying graph is a hypercube. We classify all such arithmetical structures and compute some of the corresponding ...

Convexity Of Neural Codes, 2016 Harvey Mudd College

#### Convexity Of Neural Codes, Robert Amzi Jeffs

*HMC Senior Theses*

An important task in neuroscience is stimulus reconstruction: given activity in the brain, what stimulus could have caused it? We build on previous literature which uses neural codes to approach this problem mathematically. A neural code is a collection of binary vectors that record concurrent firing of neurons in the brain. We consider neural codes arising from place cells, which are neurons that track an animal's position in space. We examine algebraic objects associated to neural codes, and completely characterize a certain class of maps between these objects. Furthermore, we show that such maps have natural geometric implications related ...

Topology Of The Affine Springer Fiber In Type A, 2016 University of Massachusetts - Amherst

#### Topology Of The Affine Springer Fiber In Type A, Tobias Wilson

*Doctoral Dissertations*

We develop algorithms for describing elements of the affine Springer fiber in type

A for certain 2 g(C[[t]]). For these , which are equivalued, integral, and regular,

it is known that the affine Springer fiber, X, has a paving by affines resulting from

the intersection of Schubert cells with X. Our description of the elements of Xallow

us to understand these affine spaces and write down explicit dimension formulae. We

also explore some closure relations between the affine spaces and begin to describe the

moment map for the both the regular and extended torus action.

Algebraicity Of Rational Hodge Isometries Of K3 Surfaces, 2016 University of Massachusetts Amherst

#### Algebraicity Of Rational Hodge Isometries Of K3 Surfaces, Nikolay Buskin

*Doctoral Dissertations*

Consider any rational Hodge isometry

$\psi:H^2(S_1,\QQ)\rightarrow H^2(S_2,\QQ)$ between any two K\"ahler $K3$

surfaces $S_1$ and $S_2$. We prove that the cohomology class of $\psi$ in $H^{2,2}(S_1\times S_2)$

is a polynomial in Chern classes of coherent analytic sheaves

over $S_1 \times S_2$. Consequently, the cohomology class of $\psi$ is algebraic

whenever $S_1$ and $S_2$ are algebraic.

Skein Theory And Algebraic Geometry For The Two-Variable Kauffman Invariant Of Links, 2016 University of Massachusetts - Amherst

#### Skein Theory And Algebraic Geometry For The Two-Variable Kauffman Invariant Of Links, Thomas Shelly

*Doctoral Dissertations*

We conjecture a relationship between the Hilbert schemes of points on a singular plane curve and the Kauffman invariant of the link associated to the singularity. Specifcally, we conjecture that the generating function of certain weighted Euler characteristics of the Hilbert schemes is given by a normalized specialization of the difference between the Kauffman and HOMFLY polynomials of the link. We prove the conjecture for torus knots. We also develop some skein theory for computing the Kauffman polynomial of links associated to singular points on plane curves.

Spherical Tropicalization, 2016 University of Massachusetts Amherst

#### Spherical Tropicalization, Anastasios Vogiannou

*Doctoral Dissertations*

In this thesis, I extend tropicalization of subvarieties of algebraic tori over a trivially valued algebraically closed field to subvarieties of spherical homogeneous spaces. I show the existence of tropical compactifications in a general setting. Given a tropical compactification of a closed subvariety of a spherical homogeneous space, I show that the support of the colored fan of the ambient spherical variety agrees with the tropicalization of the closed subvariety. I provide examples of tropicalization of subvarieties of GL(n), SL(n), and PGL(n).

Quasi-Platonic Psl2(Q)-Actions On Closed Riemann Surfaces, 2015 Rose-Hulman Institute of Technology

#### Quasi-Platonic Psl2(Q)-Actions On Closed Riemann Surfaces, Sean A. Broughton

*Mathematical Sciences Technical Reports (MSTR)*

This paper is the first of two papers whose combined goal is to explore the dessins d'enfant and symmetries of quasi-platonic actions of *PSL _{2}(q)*. A quasi-platonic action of a group

*G*on a closed Riemann

*S*surface is a conformal action for which

*S/G*is a sphere and

*S->S/G*is branched over

*{0, 1,infinity}*. The unit interval in

*S/G*may be lifted to a dessin d'enfant

*D*, an embedded bipartite graph in

*S.*The dessin forms the edges and vertices of a tiling on

*S*by dihedrally symmetric polygons, generalizing the ...

Quasi-Platonic Psl2(Q)-Actions On Closed Riemann Surfaces, 2015 Rose-Hulman Institute of Technology

#### Quasi-Platonic Psl2(Q)-Actions On Closed Riemann Surfaces, Sean A. Broughton

*S. Allen Broughton*

This paper is the first of two papers whose combined goal is to explore the dessins d'enfant and symmetries of quasi-platonic actions of *PSL _{2}(q)*. A quasi-platonic action of a group

*G*on a closed Riemann

*S*surface is a conformal action for which

*S/G*is a sphere and

*S->S/G*is branched over

*{0, 1,infinity}*. The unit interval in

*S/G*may be lifted to a dessin d'enfant

*D*, an embedded bipartite graph in

*S.*The dessin forms the edges and vertices of a tiling on

*S*by dihedrally symmetric polygons, generalizing the ...

Integrability And Regularity Of Rational Functions, 2015 Washington University in St. Louis

#### Integrability And Regularity Of Rational Functions, Greg Knese

*Mathematics Faculty Publications*

Motivated by recent work in the mathematics and engineering literature, we study integrability and non-tangential regularity on the two-torus for rational functions that are holomorphic on the bidisk. One way to study such rational functions is to fix the denominator and look at the ideal of polynomials in the numerator such that the rational function is square integrable. A concrete list of generators is given for this ideal as well as a precise count of the dimension of the subspace of numerators with a specified bound on bidegree. The dimension count is accomplished by constructing a natural pair of commuting ...