Local-To-Global Agreement Expansion via the Variance Method

Agreement expansion is concerned with set systems for which local assignments to the sets with almost perfect pairwise consistency (i.e., most overlapping pairs of sets agree on their intersections) implies the existence of a global assignment to the ground set (from which the sets are defined) that agrees with most of the local assignments. It is currently known that if a set system forms a two-sided or a partite high dimensional expander then agreement expansion is implied. However, it was not known whether agreement expansion can be implied for one-sided high dimensional expanders. In this work we show that agreement expansion can be deduced for one-sided high dimensional expanders assuming that all the vertices\u27 links (i.e., the neighborhoods of the vertices) are agreement expanders. Thus, for one-sided high dimensional expander, an agreement expansion of the large complicated complex can be deduced from agreement expansion of its small simple links. Using our result, we settle the open question whether the well studied Ramanujan complexes are agreement expanders. These complexes are neither partite nor two-sided high dimensional expanders. However, they are one-sided high dimensional expanders for which their links are partite and hence are agreement expanders. Thus, our result implies that Ramanujan complexes are agreement expanders, answering affirmatively the aforementioned open question. The local-to-global agreement expansion that we prove is based on the variance method that we develop. We show that for a high dimensional expander, if we define a function on its top faces and consider its local averages over the links then the variance of these local averages is much smaller than the global variance of the original function. This decreasing in the variance enables us to construct one global agreement function that ties together all local agreement functions

High Dimensional Random Walks and Colorful Expansion

Random walks on bounded degree expander graphs have numerous applications, both in theoretical and practical computational problems. A key property of these walks is that they converge rapidly to their stationary distribution. In this work we {\em define high order random walks}: These are generalizations of random walks on graphs to high dimensional simplicial complexes, which are the high dimensional analogues of graphs. A simplicial complex of dimension $d$ has vertices, edges, triangles, pyramids, up to $d$-dimensional cells. For any $0 \leq i < d$, a high order random walk on dimension $i$ moves between neighboring $i$-faces (e.g., edges) of the complex, where two $i$-faces are considered neighbors if they share a common $(i+1)$-face (e.g., a triangle). The case of $i=0$ recovers the well studied random walk on graphs. We provide a {\em local-to-global criterion} on a complex which implies {\em rapid convergence of all high order random walks} on it. Specifically, we prove that if the $1$-dimensional skeletons of all the links of a complex are spectral expanders, then for {\em all} $0 \le i < d$ the high order random walk on dimension $i$ converges rapidly to its stationary distribution. We derive our result through a new notion of high dimensional combinatorial expansion of complexes which we term {\em colorful expansion}. This notion is a natural generalization of combinatorial expansion of graphs and is strongly related to the convergence rate of the high order random walks. We further show an explicit family of {\em bounded degree} complexes which satisfy this criterion. Specifically, we show that Ramanujan complexes meet this criterion, and thus form an explicit family of bounded degree high dimensional simplicial complexes in which all of the high order random walks converge rapidly to their stationary distribution.Comment: 27 page

A Report on the X-ray Properties of the tau Sco Like Stars

An increasing number of OB stars have been shown to possess magnetic fields. Although the sample remains small, it is surprising that the magnetic and X-ray properties of these stars appear to be far less correlated than expected. This contradicts model predictions, which generally indicate that the X-rays from magnetic stars to be harder and more luminous than their non-magnetic counterparts. Instead, the X-ray properties of magnetic OB stars are quite diverse. $\tau$ Sco is one example where the expectations are better met. This bright main sequence, early B star has been studied extensively in a variety of wavebands. It has a surface magnetic field of around 500 G, and Zeeman Doppler tomography has revealed an unusual field configuration. Furthermore, tau Sco displays an unusually hard X-ray spectrum, much harder than similar, non-magnetic OB stars. In addition, the profiles of its UV P Cygni wind lines have long been known to possess a peculiar morphology. Recently, two stars, HD 66665 and HD 63425, whose spectral types and UV wind line profiles are similar to those of $\tau$ Sco, have also been determined to be magnetic. In the hope of establishing a magnetic field - X-ray connection for at least a sub-set of the magnetic stars, we obtained XMM-Newton EPIC spectra of these two objects. Our results for HD 66665 are somewhat inconclusive. No especially strong hard component is detected; however, the number of source counts is insufficient to rule out hard emission. longer exposure is needed to assess the nature of the X-rays from this star. On the other hand, we do find that HD 63425 has a substantial hard X-ray component, thereby bolstering its close similarity to tau Sco.Comment: MNRAS, accepte

A study of mother daughter relationships, Family Service Society of Quincy, Massachusetts

Thesis (M.S.)--Boston Universit

A study of the school adjustments of fourteen girls paroled to the community from the Industrial School for Girls at Lancaster, Massachusetts

Thesis (M.S.)--Boston Universit

Spartan Daily, April 25, 2017

Volume 148, Issue 35https://scholarworks.sjsu.edu/spartan_daily_2017/1033/thumbnail.jp

Spartan Daily, March 7, 2018

Volume 150, Issue 18https://scholarworks.sjsu.edu/spartan_daily_2018/1017/thumbnail.jp

Spartan Daily, March 27, 2019

Volume 152, Issue 27https://scholarworks.sjsu.edu/spartan_daily_2019/1026/thumbnail.jp

Spartan Daily, November 29, 2016

Volume 147, Issue 36https://scholarworks.sjsu.edu/spartan_daily_2016/1076/thumbnail.jp