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Articles 1 - 9 of 9
Full-Text Articles in Mathematics
Mixing Measures For Trees Of Fixed Diameter, Ari Holcombe Pomerance
Mixing Measures For Trees Of Fixed Diameter, Ari Holcombe Pomerance
Mathematics, Statistics, and Computer Science Honors Projects
A mixing measure is the expected length of a random walk in a graph given a set of starting and stopping conditions. We determine the tree structures of order n with diameter d that minimize and maximize for a few mixing measures. We show that the maximizing tree is usually a broom graph or a double broom graph and that the minimizing tree is usually a seesaw graph or a double seesaw graph.
Generalizations Of The Arcsine Distribution, Rebecca Rasnick
Generalizations Of The Arcsine Distribution, Rebecca Rasnick
Electronic Theses and Dissertations
The arcsine distribution looks at the fraction of time one player is winning in a fair coin toss game and has been studied for over a hundred years. There has been little further work on how the distribution changes when the coin tosses are not fair or when a player has already won the initial coin tosses or, equivalently, starts with a lead. This thesis will first cover a proof of the arcsine distribution. Then, we explore how the distribution changes when the coin the is unfair. Finally, we will explore the distribution when one person has won the first …
On Passing The Buck, Adam J. Hammett, Anna Joy Yang
On Passing The Buck, Adam J. Hammett, Anna Joy Yang
The Research and Scholarship Symposium (2013-2019)
Imagine there are n>1 people seated around a table, and person S starts with a fair coin they will flip to decide whom to hand the coin next -- if "heads" they pass right, and if "tails" they pass left. This process continues until all people at the table have "touched" the coin. Curiously, it turns out that all people seated at the table other than S have the same probability 1/(n-1) of being last to touch the coin. In fact, Lovasz and Winkler ("A note on the last new vertex visited by a random walk," J. Graph Theory, …
Unequal Edge Inclusion Probabilities In Link-Tracing Network Sampling With Implications For Respondent-Driven Sampling, Miles Q. Ott, Krista J. Gile
Unequal Edge Inclusion Probabilities In Link-Tracing Network Sampling With Implications For Respondent-Driven Sampling, Miles Q. Ott, Krista J. Gile
Statistical and Data Sciences: Faculty Publications
Respondent-Driven Sampling (RDS) is a widely adopted linktracing sampling design used to draw valid statistical inference from samples of populations for which there is no available sampling frame. RDS estimators rely upon the assumption that each edge (representing a relationship between two individuals) in the underlying network has an equal probability of being sampled. We show that this assumption is violated in even the simplest cases, and that RDS estimators are sensitive to the violation of this assumption.
Random Search Models Of Foraging Behavior: Theory, Simulation, And Observation., Ben C. Nolting
Random Search Models Of Foraging Behavior: Theory, Simulation, And Observation., Ben C. Nolting
Department of Mathematics: Dissertations, Theses, and Student Research
Many organisms, from bacteria to primates, use stochastic movement patterns to find food. These movement patterns, known as search strategies, have recently be- come a focus of ecologists interested in identifying universal properties of optimal foraging behavior. In this dissertation, I describe three contributions to this field. First, I propose a way to extend Charnov's Marginal Value Theorem to the spatially explicit framework of stochastic search strategies. Next, I describe simulations that compare the efficiencies of sensory and memory-based composite search strategies, which involve switching between different behavioral modes. Finally, I explain a new behavioral analysis protocol for identifying the …
Random Walks On The Torus With Several Generators, Timothy Prescott '02, Francis E. Su
Random Walks On The Torus With Several Generators, Timothy Prescott '02, Francis E. Su
All HMC Faculty Publications and Research
Given n vectors {i} ∈ [0, 1)d, consider a random walk on the d-dimensional torus d = ℝd/ℤd generated by these vectors by successive addition and subtraction. For certain sets of vectors, this walk converges to Haar (uniform) measure on the torus. We show that the discrepancy distance D(Q*k) between the kth step distribution of the walk and Haar measure is bounded below by D(Q*k) ≥ C1k−n/2, where C1 = C(n, d) is …
Discrepancy Convergence For The Drunkard's Walk On The Sphere, Francis E. Su
Discrepancy Convergence For The Drunkard's Walk On The Sphere, Francis E. Su
All HMC Faculty Publications and Research
We analyze the drunkard's walk on the unit sphere with step size θ and show that the walk converges in order C/sin2(θ) steps in the discrepancy metric (C a constant). This is an application of techniques we develop for bounding the discrepancy of random walks on Gelfand pairs generated by bi-invariant measures. In such cases, Fourier analysis on the acting group admits tractable computations involving spherical functions. We advocate the use of discrepancy as a metric on probabilities for state spaces with isometric group actions.
A Leveque-Type Lower Bound For Discrepancy, Francis E. Su
A Leveque-Type Lower Bound For Discrepancy, Francis E. Su
All HMC Faculty Publications and Research
A sharp lower bound for discrepancy on R / Z is derived that resembles the upper bound due to LeVeque. An analogous bound is proved for discrepancy on Rk / Zk. These are discussed in the more general context of the discrepancy of probablity measures. As applications, the bounds are applied to Kronecker sequences and to a random walk on the torus.
Convergence Of Random Walks On The Circle Generated By An Irrational Rotation, Francis E. Su
Convergence Of Random Walks On The Circle Generated By An Irrational Rotation, Francis E. Su
All HMC Faculty Publications and Research
Fix . Consider the random walk on the circle which proceeds by repeatedly rotating points forward or backward, with probability , by an angle . This paper analyzes the rate of convergence of this walk to the uniform distribution under ``discrepancy'' distance. The rate depends on the continued fraction properties of the number . We obtain bounds for rates when is any irrational, and a sharp rate when is a quadratic irrational. In that case the discrepancy falls as (up to constant factors), where is the number of steps in the walk. This is the first example of a sharp …