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Miniature of Computing word length in alternate presentations of thompson's group F
Computing word length in alternate presentations of thompson's group F

Date: 2009-12-01

Creator: Matthew Horak, Melanie Stein, Jennifer Taback

Access: Open access

We introduce a new method for computing the word length of an element of Thompson's group F with respect to a "consecutive" generating set of the form Xn = {x0,x1, ⋯, xn}, which is a subset of the standard infinite generating set for F. We use this method to show that (F, Xn) is not almost convex, and has pockets of increasing, though bounded, depth dependent on n. © 2009 World Scientific Publishing Company.
Miniature of Basins of Attraction and Metaoptimization for Particle Swarm Optimization Methods
Basins of Attraction and Metaoptimization for Particle Swarm Optimization Methods

Date: 2024-01-01

Creator: David Ma

Access: Open access

Particle swarm optimization (PSO) is a metaheuristic optimization method that finds near- optima by spawning particles which explore within a given search space while exploiting the best candidate solutions of the swarm. PSO algorithms emulate the behavior of, say, a flock of birds or a school of fish, and encapsulate the randomness that is present in natural processes. In this paper, we discuss different initialization schemes and meta-optimizations for PSO, its performances on various multi-minima functions, and the unique intricacies and obstacles that the method faces when attempting to produce images for basins of attraction, which are the sets of initial points that are mapped to the same minima by the method. This project compares the relative strengths and weaknesses of the Particle Swarm with other optimization methods, namely gradient-descent, in the context of basin mapping and other metrics. It was found that with proper parameterization, PSO can amply explore the search space regardless of initialization. For all functions, the swarm was capable of finding, within some tolerance, the global minimum or minima in fewer than 60 iterations by having sufficiently well chosen parameters and parameterization schemes. The shortcomings of the Particle Swarm method, however, are that its parameters often require fine-tuning for different search spaces to most efficiently optimize and that the swarm cannot produce the analytical minimum. Overall, the PSO is a highly adaptive and computationally efficient method with few initial restraints that can be readily used as the first step of any optimization task.
Miniature of Mathematical Notions of Resilience: The Effects of DisturbanceI in One-Dimensional Nonlinear Systems
Mathematical Notions of Resilience: The Effects of DisturbanceI in One-Dimensional Nonlinear Systems

Date: 2015-05-01

Creator: Stephen Ligtenberg

Access: Open access

Miniature of Exploring Random Walks on Graphs for Protein Function Prediction
Exploring Random Walks on Graphs for Protein Function Prediction

Date: 2018-05-01

Creator: Angela M Dahl

Access: Open access

Ordinary Differential Equations and Applied Dynamics Applied Mathematics Domino tableaux Dynamic Systems Thompson's group F Other Applied Mathematics Partial Differential Equations Physical Sciences and Mathematics L-functions Minimization Numerical Analysis and Computation Optimization Robinson-Schensted algorithm Unequal parameter Iwahori-Hecke algebra Analysis Automatic group Basins of Attraction Bifurcations C-graph automatic group Geometry and Topology Harmonic Analysis and Representation Hilbert modular forms Mathematical model Non-linear Dynamics Nonlinear Schrödinger equation Nonlinear lattices Number Theory Sensitivity Analysis Solitons Thompson's group automatic group breather graph automatic group (Cayley) graph automatic group 1-level density 2D Hexagonal Acoustic transistor Acoustics, Dynamics, and Controls Algebra Almost convex Almost convexity Analytic continuation Apiculture Approximation Approximation Theory Asymptotic density Asymptotic group theory Backtest Overfitting Basins of Attrac- tion Baumslag-Solitar group Bayesian statistics Bendixson-Dulac Binary tree Bioinformatics Biostatistics Breather Brownian motion COVID-19 Calcium dynamics Cantilever array Cardiac Ganglion Cardiac muscle cell Carrying simplex Cayley automatic group Cayley graph automatic group Cellular and Molecular Physiology Central Pattern Generators Central pattern generator Comparative and Evolutionary Physiology Competition Complex reflection groups Computational Neuroscience Cone type Context-free grammar Context-sensitive language Counter automata Counter language Covering Spaces Current Support Theorem Cycles D-finite generating function Data Science Diestel-Leader graphs Diestel-Leader groups Diestel–Leader group Dirichlet L-functions Dirichlet beta function Discrete Mathematics and Combinatorics Disease Modeling Dispersive shock waves Dynamical Systems Entomology Epidemiology Error estimates Euler product Extinction Extreme Value Theory FLUPSY 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abstract algebra acoustic filter acoustics algebraic number theory almost convexity analytic number theory aquaculture automorphisms breathers bumblebee ethology cayley graphs continuous-time Markov chain critical values decision support tool defect differential equation differential equations dirichlet L functions dispersive shock waves domino tableaux dynamical ecological ecology modeling explicit formula f-ray class L-functions functional equation game theory granular chains graph theory group theory hermitian hexagonal lattice hierarchical regression lamplighter group. logistic regression long-range interactions magnetic magnetic interaction magnetic lattice managment mathematics mathmatical modeling mixture model modular forms non-autonomous non-local nonlinear acoustics nonlinear dynamics nonlinear lattice nonlinear localized mode nonlinear waves orthonormal basis oyster farm oyster seed property R ∞ protein-protein interaction networks radial random walks ratios conjecture representation restricted wreath product riemann zeta function stablilty tangential traveling waves twisted conjugacy classes unitary upweller virtually infinite cyclic group