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Physical Sciences and Mathematics Commons™
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- Combinatorics (5)
- Mathematics (2)
- Optimization and Control (2)
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- Algebraic geometry (1)
- Approximate Dynamic Programming (1)
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Articles 1 - 15 of 15
Full-Text Articles in Physical Sciences and Mathematics
Dynamic Server Allocation At Parallel Queues, Susan E. Martonosi
Dynamic Server Allocation At Parallel Queues, Susan E. Martonosi
All HMC Faculty Publications and Research
We explore whether dynamically reassigning servers to parallel queues in response to queue imbalances can reduce average waiting time in those queues. We use approximate dynamic programming methods to determine when servers should be switched, and we compare the performance of such dynamic allocations to that of a pre-scheduled deterministic allocation. Testing our method on both synthetic data and data from airport security checkpoints at Boston Logan International Airport, we find that in situations where the uncertainty in customer arrival rates is significant, dynamically reallocating servers can substantially reduce waiting time. Moreover, we find that intuitive switching strategies that are …
Combinatorics Of Two-Toned Tilings, Arthur T. Benjamin, Phyllis Chinn, Jacob N. Scott '11, Greg Simay
Combinatorics Of Two-Toned Tilings, Arthur T. Benjamin, Phyllis Chinn, Jacob N. Scott '11, Greg Simay
All HMC Faculty Publications and Research
We introduce the function a(r, n) which counts tilings of length n + r that utilize white tiles (whose lengths can vary between 1 and n) and r identical red squares. These tilings are called two-toned tilings. We provide combinatorial proofs of several identities satisfied by a(r, n) and its generalizations, including one that produces kth order Fibonacci numbers. Applications to integer partitions are also provided.
Characteristics Of Optimal Solutions To The Sensor Location Problem, David R. Morrison '08, Susan E. Martonosi
Characteristics Of Optimal Solutions To The Sensor Location Problem, David R. Morrison '08, Susan E. Martonosi
All HMC Faculty Publications and Research
In Bianco et al. (2001), the authors present the Sensor Location Problem: that of locating the minimum number of traffic sensors at intersections of a road network such that the traffic ow on the entire network can be determined. They offer a necessary and sufficient condition on the set of monitored nodes in order for the ow everywhere to be determined. In this paper, we present a counterexample that demonstrates that the condition is not actually sufficient (though it is still necessary). We present a stronger necessary condition for ow calculability, and show that it is a sufficient condition in …
Squaring, Cubing, And Cube Rooting, Arthur T. Benjamin
Squaring, Cubing, And Cube Rooting, Arthur T. Benjamin
All HMC Faculty Publications and Research
We present mentally efficient algorithms for mentally squaring and cubing 2-digit and 3-digit numbers and for finding cube roots of numbers with 2-digit or 3-digit answers.
Toric Symmetry Of Cp3, Dagan Karp, Dhruv Ranganathan, Paul Riggins '12, Ursula Whitcher
Toric Symmetry Of Cp3, Dagan Karp, Dhruv Ranganathan, Paul Riggins '12, Ursula Whitcher
All HMC Faculty Publications and Research
We exhaustively analyze the toric symmetries of CP^3 and its toric blowups. Our motivation is to study toric symmetry as a computational technique in Gromov-Witten theory and Donaldson-Thomas theory. We identify all nontrivial toric symmetries. The induced nontrivial isomorphisms lift and provide new symmetries at the level of Gromov-Witten Theory and Donaldson-Thomas Theory. The polytopes of the toric varieties in question include the permutohedron, the cyclohedron, the associahedron, and in fact all graph associahedra, among others.
A New Framework For Network Disruption, Susan E. Martonosi, Doug Altner, Michael Ernst, Elizabeth Ferme, Kira Langsjoen, Danika Lindsay, Sean S. Plott '08, Andrew S. Ronan
A New Framework For Network Disruption, Susan E. Martonosi, Doug Altner, Michael Ernst, Elizabeth Ferme, Kira Langsjoen, Danika Lindsay, Sean S. Plott '08, Andrew S. Ronan
All HMC Faculty Publications and Research
Traditional network disruption approaches focus on disconnecting or lengthening paths in the network. We present a new framework for network disruption that attempts to reroute flow through critical vertices via vertex deletion, under the assumption that this will render those vertices vulnerable to future attacks. We define the load on a critical vertex to be the number of paths in the network that must flow through the vertex. We present graph-theoretic and computational techniques to maximize this load, firstly by removing either a single vertex from the network, secondly by removing a subset of vertices.
Existence Of Solutions For A Wave Equation With Non-Monotone Nonlinearity And A Small Parameter, Jose F. Caicedo, Alfonso Castro, Rodrigo Duque
Existence Of Solutions For A Wave Equation With Non-Monotone Nonlinearity And A Small Parameter, Jose F. Caicedo, Alfonso Castro, Rodrigo Duque
All HMC Faculty Publications and Research
We provide sufficient conditions for the existence of solutions to a semilinear wave equation with non-monotone nonlinearity involving a small parameter. Our results are based on the analysis of a an operator that characterizes the projection onto the kernel of the wave operator subject to periodic-Dirichlet boundary conditions. Such a kernel is infinite dimensional which makes standard compactness arguments inapplicable.
The Combinatorialization Of Linear Recurrences, Arthur T. Benjamin, Halcyon Derks, Jennifer J. Quinn
The Combinatorialization Of Linear Recurrences, Arthur T. Benjamin, Halcyon Derks, Jennifer J. Quinn
All HMC Faculty Publications and Research
We provide two combinatorial proofs that linear recurrences with constant coefficients have a closed form based on the roots of its characteristic equation. The proofs employ sign-reversing involutions on weighted tilings.
Third And Fourth Binomial Coefficients, Arthur T. Benjamin, Jacob N. Scott '11
Third And Fourth Binomial Coefficients, Arthur T. Benjamin, Jacob N. Scott '11
All HMC Faculty Publications and Research
While formulas for the sums of kth binomial coefficients can be shown inductively or algebraically, these proofs give little insight into the combinatorics involved. We prove formulas for the sums of 3rd and 4th binomial coefficients via purely combinatorial arguments.
A Census Of Vertices By Generations In Regular Tessellations Of The Plane, Alice Paul '12, Nicholas Pippenger
A Census Of Vertices By Generations In Regular Tessellations Of The Plane, Alice Paul '12, Nicholas Pippenger
All HMC Faculty Publications and Research
We consider regular tessellations of the plane as infinite graphs in which q edges and q faces meet at each vertex, and in which p edges and p vertices surround each face. For 1/p + 1/q = 1/2, these are tilings of the Euclidean plane; for 1/p + 1/q < 1/2, they are tilings of the hyperbolic plane. We choose a vertex as the origin, and classify vertices into generations according to their distance (as measured by the number of edges in a shortest path) from the origin. For all p ≥ 3 and q ≥ 3 with 1/p + 1/q ≤ 1/2, we give simple combinatorial derivations of the rational generating functions for the number of vertices in each generation.
A Space-Filling, Nonregular Tetrahedron, Margaret Cagle, Joyce Frost, Christine Latulippe, Darryl H. Yong
A Space-Filling, Nonregular Tetrahedron, Margaret Cagle, Joyce Frost, Christine Latulippe, Darryl H. Yong
All HMC Faculty Publications and Research
This activity is an investigation of a special nonregular tetrahedron that can be arranged to fill space without leaving any internal gaps in the same way that certain planar figures tessellate the plane. These tetrahedra can be connected together with hinges to make fun and interesting puzzles. More background information can be found in the paper "An Amazing, Space-Filling, Non-Regular Tetrahedron" by Joyce Frost and Peg Cagle, published by the IAS/Park City Mathematics Institute (available at mathforum.org/pcmi/hstp/resources/dodeca/).
Applications Of Convex And Algebraic Geometry To Graphs And Polytopes, Mohamed Omar
Applications Of Convex And Algebraic Geometry To Graphs And Polytopes, Mohamed Omar
All HMC Faculty Publications and Research
No abstract provided.
A Primer Of Swarm Equilibria, Andrew J. Bernoff, Chad M. Topaz
A Primer Of Swarm Equilibria, Andrew J. Bernoff, Chad M. Topaz
All HMC Faculty Publications and Research
We study equilibrium configurations of swarming biological organisms subject to exogenous and pairwise endogenous forces. Beginning with a discrete dynamical model, we derive a variational description of the corresponding continuum population density. Equilibrium solutions are extrema of an energy functional and satisfy a Fredholm integral equation. We find conditions for the extrema to be local minimizers, global minimizers, and minimizers with respect to infinitesimal Lagrangian displacements of mass. In one spatial dimension, for a variety of exogenous forces, endogenous forces, and domain configurations, we find exact analytical expressions for the equilibria. These agree closely with numerical simulations of the underlying …
Evidence Of The Harmonic Faraday Instability In Ultrasonic Atomization Experiments With A Deep, Inviscid Fluid, Andrew P. Higginbotham '09, Aaron Guillen '11, Nathan C. Jones '10, Thomas D. Donnelly, Andrew J. Bernoff
Evidence Of The Harmonic Faraday Instability In Ultrasonic Atomization Experiments With A Deep, Inviscid Fluid, Andrew P. Higginbotham '09, Aaron Guillen '11, Nathan C. Jones '10, Thomas D. Donnelly, Andrew J. Bernoff
All HMC Faculty Publications and Research
A popular method for generating micron-sized aerosols is to submerge ultrasonic (ω~MHz) piezoelectric oscillators in a water bath. The submerged oscillator atomizes the fluid, creating droplets with radii proportional to the wavelength of the standing wave at the fluid surface. Classical theory for the Faraday instability predicts a parametric instability driving a capillary wave at the subharmonic (ω/2) frequency. For many applications it is desirable to reduce the size of the droplets; however, using higher frequency oscillators becomes impractical beyond a few MHz. Observations are presented that demonstrate that smaller droplets may also be created by …
Strong Nonnegativity And Sums Of Squares On Real Varieties, Mohamed Omar, Brian Osserman
Strong Nonnegativity And Sums Of Squares On Real Varieties, Mohamed Omar, Brian Osserman
All HMC Faculty Publications and Research
Motivated by scheme theory, we introduce strong nonnegativity on real varieties, which has the property that a sum of squares is strongly nonnegative. We show that this algebraic property is equivalent to nonnegativity for nonsingular real varieties. Moreover, for singular varieties, we reprove and generalize obstructions of Gouveia and Netzer to the convergence of the theta body hierarchy of convex bodies approximating the convex hull of a real variety.