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Physical Sciences and Mathematics Commons™
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- Keyword
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- (Abstract Harmonic Analysis) Explicit machine computation and programs (not the theory of computation or programming) (1)
- 20C30 (1)
- 34B16 Singular nonlinear boundary value problems (1)
- 43-04 (1)
- 43A30 (1)
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- 49 Calculus of variations and optimal control; optimization (1)
- 53 Differential geometry (1)
- 58J05 Elliptic equations on manifolds general theory (1)
- 58J32 Boundary value problems on manifolds (1)
- 68 Computer science (1)
- Algebraic reconstruction technique (1)
- Block Kaczmarz (1)
- Fourier and Fourier-Stieltjes transforms on nonabelian groups and on semigroups (1)
- Kaczmarz (1)
- Representations of finite symmetric groups (1)
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Articles 1 - 6 of 6
Full-Text Articles in Physical Sciences and Mathematics
Fast Algorithms For Analyzing Partially Ranked Data, Matthew Mcdermott
Fast Algorithms For Analyzing Partially Ranked Data, Matthew Mcdermott
HMC Senior Theses
Imagine your local creamery administers a survey asking their patrons to choose their five favorite ice cream flavors. Any data collected by this survey would be an example of partially ranked data, as the set of all possible flavors is only ranked into subsets of the chosen flavors and the non-chosen flavors. If the creamery asks you to help analyze this data, what approaches could you take? One approach is to use the natural symmetries of the underlying data space to decompose any data set into smaller parts that can be more easily understood. In this work, I describe …
Infinitely Many Rotationally Symmetric Solutions To A Class Of Semilinear Laplace-Beltrami Equations On The Unit Sphere, Emily M. Fischer
Infinitely Many Rotationally Symmetric Solutions To A Class Of Semilinear Laplace-Beltrami Equations On The Unit Sphere, Emily M. Fischer
HMC Senior Theses
I show that a class of semilinear Laplace-Beltrami equations has infinitely many solutions on the unit sphere which are symmetric with respect to rotations around some axis. This equation corresponds to a singular ordinary differential equation, which we solve using energy analysis. We obtain a Pohozaev-type identity to prove that the energy is continuously increasing with the initial condition and then use phase plane analysis to prove the existence of infinitely many solutions.
A Mathematical Framework For Unmanned Aerial Vehicle Obstacle Avoidance, Sorathan Chaturapruek
A Mathematical Framework For Unmanned Aerial Vehicle Obstacle Avoidance, Sorathan Chaturapruek
HMC Senior Theses
The obstacle avoidance navigation problem for Unmanned Aerial Vehicles (UAVs) is a very challenging problem. It lies at the intersection of many fields such as probability, differential geometry, optimal control, and robotics. We build a mathematical framework to solve this problem for quadrotors using both a theoretical approach through a Hamiltonian system and a machine learning approach that learns from human sub-experts' multiple demonstrations in obstacle avoidance. Prior research on the machine learning approach uses an algorithm that does not incorporate geometry. We have developed tools to solve and test the obstacle avoidance problem through mathematics.
Experiments On Surfactants And Thin Fluid Films, Peter Megson
Experiments On Surfactants And Thin Fluid Films, Peter Megson
HMC Senior Theses
We investigate the spatiotemporal dynamics of a surfactant monolayer on a thin fluid film spreading inward into a region devoid of surfactant, a system motivated by the alveolus of the human lung. We perform experiments that simultaneously measure the fluid height profile and the fluorescence intensity due to our fluorescent surfactant, NBD-PC. We perform experiments on both a Newtonian layer of glycerol and a shear-thinning fluid layer consisting of xanthan gum mixed with glycerol. We can very successfully extract height profiles on the xanthan gum fluid, although the simultaneous measurement of fluorescent intensity profiles proved problematic, as the laser tended …
Energy-Driven Pattern Formation In Planar Dipole-Dipole Systems, Jaron P. Kent-Dobias
Energy-Driven Pattern Formation In Planar Dipole-Dipole Systems, Jaron P. Kent-Dobias
HMC Senior Theses
A variety of two-dimensional fluid systems, known as dipole-mediated systems, exhibit a dipole-dipole interaction between their fluid constituents. The com- petition of this repulsive dipolar force with the cohesive fluid forces cause these systems to form intricate and patterned structures in their boundaries. In this thesis, we show that the microscopic details of any such system are irrelevant in the macroscopic limit and contribute only to a constant offset in the system’s energy. A numeric model is developed, and some important stable domain morphologies are characterized. Previously unresolved bifurcating branches are explored. Finally, by applying a random energy background to …
Block Kaczmarz Method With Inequalities, Jonathan Briskman
Block Kaczmarz Method With Inequalities, Jonathan Briskman
CMC Senior Theses
The Kaczmarz method is an iterative algorithm that solves overdetermined systems of linear equalities. This paper studies a system of linear equalities and inequalities. We use the block version of the Kaczmarz method applied towards the equalities with the simple randomized Kaczmarz scheme for the inequalities. This primarily involves combining Needell and Tropp's work on the block Kaczmarz method with the application of a randomized Kaczmarz approach towards a system of equalities and inequalities performed by Leventhal and Lewis. We give an expected linear rate of convergence for this kind of system and find that using the block Kaczmarz scheme …