A Finite Element Method for the Fractional Sturm-Liouville Problem

In this work, we propose an efficient finite element method for solving fractional Sturm-Liouville problems involving either the Caputo or Riemann-Liouville derivative of order $\alpha\in(1,2)$ on the unit interval $(0,1)$. It is based on novel variational formulations of the eigenvalue problem. Error estimates are provided for the finite element approximations of the eigenvalues. Numerical results are presented to illustrate the efficiency and accuracy of the method. The results indicate that the method can achieve a second-order convergence for both fractional derivatives, and can provide accurate approximations to multiple eigenvalues simultaneously.Comment: 30 pages, 7 figure

Orexin axon density and bouton quantification in the aging rhesus macaque thalamus: a novel methodology

Thesis (M.A.)--Boston UniversitySleep fragmentation and disturbances of normal circadian rhythms are inherent in the aging human population. The rhesus macaque (Mucaca mulatta) exhibits similar disruption in sleep patterns and is a useful model to study these disturbances. Orexin is an excitatory neuropeptide that has been implicated in the regulation of wakefulness and alertness. Specifically, it has projections from its neuronal bodies in the lateral and perifornical hypothalamus to various forebrain and brainstem regions that control arousal state. One of the regions of high innervation by these orexigenic neurons is the thalamus. This study used a semi-quantitative means to determine if there are age dependent changes in Orexin innervation in the thalamus. There was a general trend of decreasing axon density when analyzed with a novel semi-quantitative method (r = -0.515, p = 0.024, df = 17) and approached significance when assessed with a previously published method (r = -0.454, p = 0.0509, df = 17), indicating a decrease with age. Additionally, there was no significant decrease in unilateral bouton counts iii when examined with age using either method. Although the findings in this study point to an age-related decrease in axon terminals, further research should examine the total orexigenic positive content in the thalamus to explain why bouton counts do not seem to change

The Abel-Zeilberger Algorithm

We use both Abel's lemma on summation by parts and Zeilberger's algorithm to find recurrence relations for definite summations. The role of Abel's lemma can be extended to the case of linear difference operators with polynomial coefficients. This approach can be used to verify and discover identities involving harmonic numbers and derangement numbers. As examples, we use the Abel-Zeilberger algorithm to prove the Paule-Schneider identities, the Apery-Schmidt-Strehl identity, Calkin's identity and some identities involving Fibonacci numbers.Comment: 18 page

The Importance of Category Labels in Grammar Induction with Child-directed Utterances

Recent progress in grammar induction has shown that grammar induction is possible without explicit assumptions of language-specific knowledge. However, evaluation of induced grammars usually has ignored phrasal labels, an essential part of a grammar. Experiments in this work using a labeled evaluation metric, RH, show that linguistically motivated predictions about grammar sparsity and use of categories can only be revealed through labeled evaluation. Furthermore, depth-bounding as an implementation of human memory constraints in grammar inducers is still effective with labeled evaluation on multilingual transcribed child-directed utterances.Comment: The 16th International Conference on Parsing Technologies (IWPT 2020

Finding the optimal design of a passive microfluidic mixer.

The ability to thoroughly mix two fluids is a fundamental need in microfluidics. While a variety of different microfluidic mixers have been designed by researchers, it remains unknown which (if any) of these mixers are optimal (that is, which designs provide the most thorough mixing with the smallest possible fluidic resistance across the mixer). In this work, we automatically designed and rationally optimized a microfluidic mixer. We accomplished this by first generating a library of thousands of different randomly designed mixers, then using the non-dominated sorting genetic algorithm II (NSGA-II) to optimize the random chips in order to achieve Pareto efficiency. Pareto efficiency is a state of allocation of resources (e.g. driving force) from which it is impossible to reallocate so as to make any one individual criterion better off (e.g. pressure drop) without making at least one individual criterion (e.g. mixing performance) worse off. After 200 generations of evolution, Pareto efficiency was achieved and the Pareto-optimal front was found. We examined designs at the Pareto-optimal front and found several design criteria that enhance the mixing performance of a mixer while minimizing its fluidic resistance; these observations provide new criteria on how to design optimal microfluidic mixers. Additionally, we compared the designs from NSGA-II with some popular microfluidic mixer designs from the literature and found that designs from NSGA-II have lower fluidic resistance with similar mixing performance. As a proof of concept, we fabricated three mixer designs from 200 generations of evolution and one conventional popular mixer design and tested the performance of these four mixers. Using this approach, an optimal design of a passive microfluidic mixer is found and the criteria of designing a passive microfluidic mixer are established