Open Access. Powered by Scholars. Published by Universities.®
Physical Sciences and Mathematics Commons™
Open Access. Powered by Scholars. Published by Universities.®
- Keyword
-
- 03D15 Complexity of computation (1)
- 05A19 Combinatorial identities (1)
- 05C57 Games on graphs (1)
- 05D40 Probabilistic methods in extremal combinatorics (1)
- 06F05 Ordered semigroups and monoids (1)
-
- 11B37 Recurrences (1)
- 11B39 Fibonacci and Lucas numbers and polynomials and generalizations (1)
- 13A50 Invariant Theory (1)
- 13F60 Cluster algebras (1)
- 14C17 (1)
- 14M15 Grassmannians (1)
- 14N10 (1)
- 14N10 Enumerative problems (combinatorial problems) in algebraic geometry (1)
- 14T05 (1)
- 14T20 Geometric aspects of tropical varieties (1)
- 15A69 Multilinear algebra (1)
- 20M18 Inverse semigroups (1)
- 31 Potential Theory (1)
- 34 Ordinary Differential Equations (1)
- 35 Partial Differential Equations (1)
- 37B10 Symbolic dynamics (1)
- 68Q05 Models of computation (1)
- 91A43 Games involving graphs (1)
- 91A46 Combinatorial games (1)
- Bijective combinatorics (1)
- Dynamical Systems (1)
- Enumerative problems (combinatorial problems) in algebraic geometry (1)
- Etc.) (1)
- Flag manifolds (1)
- Fractal Dimension (1)
Articles 1 - 10 of 10
Full-Text Articles in Physical Sciences and Mathematics
On Rank-Two And Affine Cluster Algebras, Feiyang Lin
On Rank-Two And Affine Cluster Algebras, Feiyang Lin
HMC Senior Theses
Motivated by existing results about the Kronecker cluster algebra, this thesis is concerned with two families of cluster algebras, which are two different ways of generalizing the Kronecker case: rank-two cluster algebras, and cluster algebras of type An,1. Regarding rank-two cluster algebras, our main result is a conjectural bijection that would prove the equivalence of two combinatorial formulas for cluster variables of rank-two skew-symmetric cluster algebras. We identify a technical result that implies the bijection and make partial progress towards its proof. We then shift gears to study certain power series which arise as limits of ratios of …
Radial Singular Solutions To Semilinear Partial Differential Equations, Marcelo A. Almora Rios
Radial Singular Solutions To Semilinear Partial Differential Equations, Marcelo A. Almora Rios
HMC Senior Theses
We show the existence of countably many non-degenerate continua of singular radial solutions to a p-subcritical, p-Laplacian Dirichlet problem on the unit ball in R^N. This result generalizes those for the 2-Laplacian to any value p and extends recent work on the p-Laplacian by considering solutions both radial and singular.
The Slice Rank Polynomial Method, Thomas C. Martinez
The Slice Rank Polynomial Method, Thomas C. Martinez
HMC Senior Theses
Suppose you wanted to bound the maximum size of a set in which every k-tuple of elements satisfied a specific condition. How would you go about this? Introduced in 2016 by Terence Tao, the slice rank polynomial method is a recently developed approach to solving problems in extremal combinatorics using linear algebraic tools. We provide the necessary background to understand this method, as well as some applications. Finally, we investigate a generalization of the slice rank, the partition rank introduced by Eric Naslund in 2020, along with various discussions on the intuition behind the slice rank polynomial method and …
The Complexity Of Symmetry, Matthew Lemay
The Complexity Of Symmetry, Matthew Lemay
HMC Senior Theses
One of the main goals of theoretical computer science is to prove limits on how efficiently certain Boolean functions can be computed. The study of the algebraic complexity of polynomials provides an indirect approach to exploring these questions, which may prove fruitful since much is known about polynomials already from the field of algebra. This paper explores current research in establishing lower bounds on invariant rings and polynomial families. It explains the construction of an invariant ring for whom a succinct encoding would imply that NP is in P/poly. It then states a theorem about the circuit complexity partial …
On The Tropicalization Of Lines Onto Tropical Quadrics, Natasha Crepeau
On The Tropicalization Of Lines Onto Tropical Quadrics, Natasha Crepeau
HMC Senior Theses
Tropical geometry uses the minimum and addition operations to consider tropical versions of the curves, surfaces, and more generally the zero set of polynomials, called varieties, that are the objects of study in classical algebraic geometry. One known result in classical geometry is that smooth quadric surfaces in three-dimensional projective space, $\mathbb{P}^3$, are doubly ruled, and those rulings form a disjoint union of conics in $\mathbb{P}^5$. We wish to see if the same result holds for smooth tropical quadrics. We use the Fundamental Theorem of Tropical Algebraic Geometry to outline an approach to studying how lines lift onto a tropical …
Tiling Representations Of Zeckendorf Decompositions, John Lentfer
Tiling Representations Of Zeckendorf Decompositions, John Lentfer
HMC Senior Theses
Zeckendorf’s theorem states that every positive integer can be decomposed uniquely into a sum of non-consecutive Fibonacci numbers (where f1 = 1 and f2 = 2). Previous work by Grabner and Tichy (1990) and Miller and Wang (2012) has found a generalization of Zeckendorf’s theorem to a larger class of recurrent sequences, called Positive Linear Recurrence Sequences (PLRS’s). We apply well-known tiling interpretations of recurrence sequences from Benjamin and Quinn (2003) to PLRS’s. We exploit that tiling interpretation to create a new tiling interpretation specific to PLRS’s that captures the behavior of the generalized Zeckendorf’s theorem.
Towards Tropical Psi Classes, Jawahar Madan
Towards Tropical Psi Classes, Jawahar Madan
HMC Senior Theses
To help the interested reader get their initial bearings, I present a survey of prerequisite topics for understanding the budding field of tropical Gromov-Witten theory. These include the language and methods of enumerative geometry, an introduction to tropical geometry and its relation to classical geometry, an exposition of toric varieties and their correspondence to polyhedral fans, an intuitive picture of bundles and Euler classes, and finally an introduction to the moduli spaces of n-pointed stable rational curves and their tropical counterparts.
Exploring Winning Strategies For The Game Of Cycles, Kailee Lin
Exploring Winning Strategies For The Game Of Cycles, Kailee Lin
HMC Senior Theses
This report details my adventures exploring the Game of Cycles in search of winning strategies. I started by studying combinatorial game theory with hopes to use the Sprague-Grundy Theorem and the structure of Nimbers to gain insight for the Game of Cycles. In the second semester, I pivoted to studying specific types of boards instead. In this thesis I show that variations of the mirror-reverse strategy developed by Alvarado et al. in the original Game of Cycles paper can be used to win on additional game boards with special structure, such as lollipops, steering wheel locks, and 3-spoke trees. Additionally …
On The Inverse Hull Of A One-Sided Shift Of Finite Type, Aria Beaupre
On The Inverse Hull Of A One-Sided Shift Of Finite Type, Aria Beaupre
HMC Senior Theses
Let S be the semigroup constructed from a one-sided shift of finite type. In this thesis, we will provide the construction of H(S), the inverse hull of S, explore the properties of H(S), and begin to characterize the structure of H(S). We will also focus on a kind of one-sided shift of finite type, Markov shifts, and prove an invariant on isomorphic inverse hulls of Markov shifts.
Fractals, Fractional Derivatives, And Newton-Like Methods, Eleanor Byrnes
Fractals, Fractional Derivatives, And Newton-Like Methods, Eleanor Byrnes
HMC Senior Theses
Inspired by the fractals generated by the discretizations of the Continuous Newton Method and the notion of a fractional derivative, we ask what it would mean if such a fractional derivative were to replace the derivatives in Newton's Method. This work, largely experimental in nature, examines these new iterative methods by generating their Julia sets, computing their fractal dimension, and in certain tractable cases examining the behaviors using tools from dynamical systems.