A Khovanov homology-style construction extended to biquandle brackets, and associated Mathematica packages for computations

In their paper entitled "Quantum Enhancements and Biquandle Brackets," Nelson, Orrison, and Rivera introduced biquandle brackets, which are customized skein invariants for biquandle-colored links. These invariants generalize the Jones polynomial, which is categorified by Khovanov homology. At the end of their paper, Nelson, Orrison, and Rivera asked if the methods of Khovanov homology could be extended to obtain a categorification of biquandle brackets. We outline herein a Khovanov homology-style construction that is an attempt to obtain such a categorification of biquandle brackets. The resulting knot invariant does generalize Khovanov homology, but the biquandle bracket is not always recoverable, meaning the construction is not a true categorification of biquandle brackets. However, the construction does lead to a definition that gives a "canonical" biquandle 2-cocycle associated to a biquandle bracket, which, to the authors' knowledge, was not previously known. Additionally, the authors have created multiple Mathematica packages that can be used for experimental computations with biquandles, biquandle brackets, biquandle 2-cocycles, and the newly-discovered canonical biquandle 2-cocycle associated to a biquandle bracket. We provide herein an explanation of these Mathematica packages, including example computations and an appendix containing the full source code. The packages may also be downloaded from vilas.us/biquandles.No embargoAcademic Major: Computer and Information ScienceAcademic Major: Mathematic

Human gravity-gradient noise in interferometric gravitational-wave detectors

Among all forms of routine human activity, the one which produces the strongest gravity-gradient noise in interferometric gravitational-wave detectors (e.g. LIGO) is the beginning and end of weight transfer from one foot to the other during walking. The beginning and end of weight transfer entail sharp changes (time scale τ∼20 msec) in the horizontal jerk (first time derivative of acceleration) of a person’s center of mass. These jerk pairs, occurring about twice per second, will produce gravity-gradient noise in LIGO in the frequency band 2.5 Hz≲f≲1/(2τ)≃25 Hz with the form sqrt[Sh(f)]∼0.6×10-23 Hz-1/2(f/10 Hz)-6[∑i(ri/10 m)-6]1/2. Here the sum is over all the walking people, ri is the distance of the i’th person from the nearest interferometer test mass, and we estimate this formula to be accurate to within a factor 3. To ensure that this noise is negligible in advanced LIGO interferometers, people should be prevented from coming nearer to the test masses than r≃10 m. A r≃10 m exclusion zone will also reduce to an acceptable level gravity gradient noise from the slamming of a door and the striking of a fist against a wall. The dominant gravity-gradient noise from automobiles and other vehicles is probably that from decelerating to rest. To keep this below the sensitivity of advanced LIGO interferometers will require keeping vehicles at least 30 m from all test masses

Engineering a campus-wide accessible music library

Thesis (M. Eng.)--Massachusetts Institute of Technology, Dept. of Electrical Engineering and Computer Science, 2005.Includes bibliographical references.The Library Access to Music Project has created a new kind of music library at the Massachusetts Institute of Technology. The library is always open and available in dormitory rooms and classrooms, because it transmits music on demand over the Institute's cable television system. By using the analog cable television system, LAMP differs from existing commercial offerings in that essentially any musical recording may be added to the collection - not just recordings where "digital rights" have been obtained. Additionally, LAMP is orders of magnitude less expensive than existing commercial offerings, and it is compatible with a much wider range of receiving apparatuses. With these advantages come unfortunate limitations that spring from LAMP's technical architecture and posture under copyright law. Nonetheless, LAMP has been a moderate success since its opening in October 2004, playing an average of 580 songs per day.by Keith J. Winstein.M.Eng

Transport Architectures for an Evolving Internet

In the Internet architecture, transport protocols are the glue between an application’s needs and the network’s abilities. But as the Internet has evolved over the last 30 years, the implicit assumptions of these protocols have held less and less well. This can cause poor performance on newer networks—cellular networks, datacenters—and makes it challenging to roll out networking technologies that break markedly with the past. Working with collaborators at MIT, I have built two systems that explore an objective-driven, computer-generated approach to protocol design. My thesis is that making protocols a function of stated assumptions and objectives can improve application performance and free network technologies to evolve. Sprout, a transport protocol designed for videoconferencing over cellular networks, uses probabilistic inference to forecast network congestion in advance. On commercial cellular networks, Sprout gives 2-to-4 times the throughput and 7-to-9 times less delay than Skype, Apple Facetime, and Google Hangouts. This work led to Remy, a tool that programmatically generates protocols for an uncertain multi-agent network. Remy’s computer-generated algorithms can achieve higher performance and greater fairness than some sophisticated human-designed schemes, including ones that put intelligence inside the network. The Remy tool can then be used to probe the difficulty of the congestion control problem itself—how easy is it to “learn” a network protocol to achieve desired goals, given a necessarily imperfect model of the networks where it ultimately will be deployed? We found weak evidence of a tradeoff between the breadth of the operating range of a computer-generated protocol and its performance, but also that a single computer-generated protocol was able to outperform existing schemes over a thousand-fold range of link rates

Transport Architectures for an Evolving Internet

In the Internet architecture, transport protocols are the glue between an application’s needs and the network’s abilities. But as the Internet has evolved over the last 30 years, the implicit assumptions of these protocols have held less and less well. This can cause poor performance on newer networks—cellular networks, datacenters—and makes it challenging to roll out networking technologies that break markedly with the past. Working with collaborators at MIT, I have built two systems that explore an objective-driven, computer-generated approach to protocol design. My thesis is that making protocols a function of stated assumptions and objectives can improve application performance and free network technologies to evolve. Sprout, a transport protocol designed for videoconferencing over cellular networks, uses probabilistic inference to forecast network congestion in advance. On commercial cellular networks, Sprout gives 2-to-4 times the throughput and 7-to-9 times less delay than Skype, Apple Facetime, and Google Hangouts. This work led to Remy, a tool that programmatically generates protocols for an uncertain multi-agent network. Remy’s computer-generated algorithms can achieve higher performance and greater fairness than some sophisticated human-designed schemes, including ones that put intelligence inside the network. The Remy tool can then be used to probe the difficulty of the congestion control problem itself—how easy is it to “learn” a network protocol to achieve desired goals, given a necessarily imperfect model of the networks where it ultimately will be deployed? We found weak evidence of a tradeoff between the breadth of the operating range of a computer-generated protocol and its performance, but also that a single computer-generated protocol was able to outperform existing schemes over a thousand-fold range of link rates

Additions of carbenium ions to nonconjugated dienes. The retarding (−I)-effect of the second double bond.

Kinetics for the addition of the p-methoxybenzhydryl cation (AnPhCH+, 10) towards nonconjugated dienes 11 [H2C=C(CH3)-(CH2)n-C(CH3)=CH2] have been determined in CH2Cl2 at −30° to −70°C. Reactivity increases with increasing number of methylene groups separating the two double bonds. For N = 4, reactivity reaches the value for saturated alkyl substituents, and nucleophilic assistance of the second double bond is never observed

Categorifying Biquandle Brackets

In their paper entitled "Quantum Enhancements and Biquandle Brackets," Nelson, Orrison, and Rivera introduced biquandle brackets, which are customized skein invariants for biquandle-colored links. These invariants generalize the Jones polynomial, which is categorified by Khovanov homology. At the end of their paper, Nelson, Orrison, and Rivera asked if the methods of Khovanov homology could be extended to obtain a categorification of biquandle brackets. We outline herein a Khovanov homology-style construction that is an attempt to obtain such a categorification of biquandle brackets. The resulting knot invariant generalizes Khovanov homology, but the biquandle bracket is not always recoverable, meaning the construction is not a true categorification of biquandle brackets. However, the construction does lead to a definition that gives a "canonical" biquandle 2-cocycle associated to a biquandle bracket, which, to the authors' knowledge, was not previously known.Comment: We also provide Mathematica packages that can be used to do computations with biquandles, biquandle brackets, biquandle 2-cocycles, and, in particular, the canonical biquandle 2-cocycle associated to a biquandle bracket. These Mathematica packages can be found at http://www.vilas.us/biquandles