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Physical Sciences and Mathematics Commons™
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- 00A65 Mathematics and Music (1)
- 05A19 Combinatorial identities bijective combinatorics (1)
- 05C10 Planar graphs; geometric and topological aspects of graph theory (1)
- 05C15 Coloring of graphs and hypergraphs (1)
- 05C35 Extremal problems (1)
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- 11A55 Continued fractions (1)
- 11B39 Fibonacci and Lucas numbers and polynomials and generalizations (1)
- 11B65 Binomial coefficients; factorials; q-identities (1)
- 11J70 Continued fractions and generalizations (1)
- 13P10 Commutative Rings and Algebras/Computational Aspects/Grobner Bases (1)
- 14C20 Divisors linear systems invertible sheaves (1)
- 14L30 Group actions on varieties or schemes (quotients) (1)
- 14T05 Tropical geometry (1)
- 34C15 Nonlinear oscillations coupled oscillators (1)
- 37N25 Dynamical systems in biology (1)
- 50C21 Flows in graphs (1)
- 60J20 Probability theory and stochastic processes/Markov processes/Applications of Markov chains and discrete-time Markov processes on general state spaces (1)
- 62M09 Non-Markovian processes: estimation (1)
- 90B10 Network models deterministic (1)
- 90B20 Traffic problems (1)
- 92B20 Neural networks artificial life and related topics (1)
- 92B25 Biological rhythms and synchronization (1)
Articles 1 - 8 of 8
Full-Text Articles in Physical Sciences and Mathematics
Finding The Beat In Music: Using Adaptive Oscillators, Kate M. Burgers
Finding The Beat In Music: Using Adaptive Oscillators, Kate M. Burgers
HMC Senior Theses
The task of finding the beat in music is simple for most people, but surprisingly difficult to replicate in a robot. Progress in this problem has been made using various preprocessing techniques (Hitz 2008; Tomic and Janata 2008). However, a real-time method is not yet available. Methods using a class of oscillators called relay relaxation oscillators are promising. In particular, systems of forced Hopf oscillators (Large 2000; Righetti et al. 2006) have been used with relative success. This work describes current methods of beat tracking and develops a new method that incorporates the best ideas from each existing method and …
Markov Bases For Noncommutative Harmonic Analysis Of Partially Ranked Data, Ann Johnston
Markov Bases For Noncommutative Harmonic Analysis Of Partially Ranked Data, Ann Johnston
HMC Senior Theses
Given the result $v_0$ of a survey and a nested collection of summary statistics that could be used to describe that result, it is natural to ask which of these summary statistics best describe $v_0$. In 1998 Diaconis and Sturmfels presented an approach for determining the conditional significance of a higher order statistic, after sampling a space conditioned on the value of a lower order statistic. Their approach involves the computation of a Markov basis, followed by the use of a Markov process with stationary hypergeometric distribution to generate a sample.This technique for data analysis has become an accepted tool …
Extending List Colorings Of Planar Graphs, Sarah Loeb
Extending List Colorings Of Planar Graphs, Sarah Loeb
HMC Senior Theses
In the study of list colorings of graphs, we assume each vertex of a graph has a specified list of colors from which it may be colored. For planar graphs, it is known that there is a coloring for any list assignment where each list contains five colors. If we have some vertices that are precolored, can we extend this to a coloring of the entire graph? We explore distance constraints when we allow the lists to contain an extra color. For lists of length five, we fix $W$ as a subset of $V(G)$ such that all vertices in $W$ …
Group Actions And Divisors On Tropical Curves, Max B. Kutler
Group Actions And Divisors On Tropical Curves, Max B. Kutler
HMC Senior Theses
Tropical geometry is algebraic geometry over the tropical semiring, or min-plus algebra. In this thesis, I discuss the basic geometry of plane tropical curves. By introducing the notion of abstract tropical curves, I am able to pass to a more abstract metric-topological setting. In this setting, I discuss divisors on tropical curves. I begin a study of $G$-invariant divisors and divisor classes.
Verification Of Solutions To The Sensor Location Problem, Chandler May
Verification Of Solutions To The Sensor Location Problem, Chandler May
HMC Senior Theses
Traffic congestion is a serious problem with large economic and environmental impacts. To reduce congestion (as a city planner) or simply to avoid congested channels (as a road user), one might like to accurately know the flow on roads in the traffic network. This information can be obtained from traffic sensors, devices that can be installed on roads or intersections to measure traffic flow. The sensor location problem is the problem of efficiently locating traffic sensors on intersections such that the flow on the entire network can be extrapolated from the readings of those sensors. I build on current research …
Combinatorial Interpretations Of Fibonomial Identities, Elizabeth Reiland
Combinatorial Interpretations Of Fibonomial Identities, Elizabeth Reiland
HMC Senior Theses
The Fibonomial numbers are defined by \[ \begin{bmatrix}n \\ k \end{bmatrix} = \frac{\prod_{i=n-k+1} ^{n} F_i}{\prod_{j=1}^{k} F_j} \] where $F_i$ is the $i$th Fibonacci number, defined by the recurrence $F_n=F_{n-1}+F_{n-2}$ with initial conditions $F_0=0,F_1=1$. In the past year, Sagan and Savage have derived a combinatorial interpretation for these Fibonomial numbers, an interpretation that relies upon tilings of a partition and its complement in a given grid.In this thesis, I investigate previously proven theorems for the Fibonomial numbers and attempt to reinterpret and reprove them in light of this new combinatorial description. I also present combinatorial proofs for some identities I did …
Continued Fractions: A New Form, Donald Lee Wiyninger Iii
Continued Fractions: A New Form, Donald Lee Wiyninger Iii
HMC Senior Theses
While the traditional form of continued fractions is well-documented, a new form, designed to approximate real numbers between 1 and 2, is less well-studied. This report first describes prior research into the new form, describing the form and giving an algorithm for generating approximations for a given real number. It then describes a rational function giving the rational number represented by the continued fraction made from a given tuple of integers and shows that no real number has a unique continued fraction. Next, it describes the set of real numbers that are hardest to approximate; that is, given a positive …
Noise, Delays, And Resonance In A Neural Network, Austin Quan
Noise, Delays, And Resonance In A Neural Network, Austin Quan
HMC Senior Theses
A stochastic-delay differential equation (SDDE) model of a small neural network with recurrent inhibition is presented and analyzed. The model exhibits unexpected transient behavior: oscillations that occur at the boundary of the basins of attraction when the system is bistable. These are known as delay-induced transitory oscillations (DITOs). This behavior is analyzed in the context of stochastic resonance, an unintuitive, though widely researched phenomenon in physical bistable systems where noise can play in constructive role in strengthening an input signal. A method for modeling the dynamics using a probabilistic three-state model is proposed, and supported with numerical evidence. The potential …