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

Unique-Neighbor-Like Expansion and Group-Independent Cosystolic Expansion

In recent years, high dimensional expanders have been found to have a variety of applications in theoretical computer science, such as efficient CSPs approximations, improved sampling and list-decoding algorithms, and more. Within that, an important high dimensional expansion notion is cosystolic expansion, which has found applications in the construction of efficiently decodable quantum codes and in proving lower bounds for CSPs. Cosystolic expansion is considered with systems of equations over a group where the variables and equations correspond to faces of the complex. Previous works that studied cosystolic expansion were tailored to the specific group ??. In particular, Kaufman, Kazhdan and Lubotzky (FOCS 2014), and Evra and Kaufman (STOC 2016) in their breakthrough works, who solved a famous open question of Gromov, have studied a notion which we term "parity" expansion for small sets. They showed that small sets of k-faces have proportionally many (k+1)-faces that contain an odd number of k-faces from the set. Parity expansion for small sets could then be used to imply cosystolic expansion only over ??. In this work we introduce a stronger unique-neighbor-like expansion for small sets. We show that small sets of k-faces have proportionally many (k+1)-faces that contain exactly one k-face from the set. This notion is fundamentally stronger than parity expansion and cannot be implied by previous works. We then show, utilizing the new unique-neighbor-like expansion notion introduced in this work, that cosystolic expansion can be made group-independent, i.e., unique-neighbor-like expansion for small sets implies cosystolic expansion over any group

Double Balanced Sets in High Dimensional Expanders

Recent works have shown that expansion of pseudorandom sets is of great importance. However, all current works on pseudorandom sets are limited only to product (or approximate product) spaces, where Fourier Analysis methods could be applied. In this work we ask the natural question whether pseudorandom sets are relevant in domains where Fourier Analysis methods cannot be applied, e.g., one-sided local spectral expanders. We take the first step in the path of answering this question. We put forward a new definition for pseudorandom sets, which we call "double balanced sets". We demonstrate the strength of our new definition by showing that small double balanced sets in one-sided local spectral expanders have very strong expansion properties, such as unique-neighbor-like expansion. We further show that cohomologies in cosystolic expanders are double balanced, and use the newly derived strong expansion properties of double balanced sets in order to obtain an exponential improvement over the current state of the art lower bound on their minimal distance

Absorption of Nickel, Chromium, and Iron by the Root Surface of Primary Molars Covered with Stainless Steel Crowns

Objective. The purpose of this study was to analyze the absorption of metal ions released from stainless steel crowns by root surface of primary molars. Study Design. Laboratory research: The study included 34 primary molars, exfoliated or extracted during routine dental treatment. 17 molars were covered with stainless-steel crowns for more than two years and compared to 17 intact primary molars. Chemical content of the mesial or distal root surface, 1 mm apically to the crown or the cemento-enamel junction (CEJ), was analyzed. An energy dispersive X-ray spectrometer (EDS) was used for chemical analysis. Results. Higher amounts of nickel, chromium, and iron (5-6 times) were found in the cementum of molars covered with stainless-steel crowns compared to intact molars. The differences between groups were highly significant (P < .001). Significance. Stainless-steel crowns release nickel, chromium, and iron in oral environment, and the ions are absorbed by the primary molars roots. The additional burden of allergenic metals should be reduced if possible

Effects of texturization due to chemical etching and laser on the optical properties of multicrystalline silicon for applications in solar cells

In this work we carried out the texturization of surfaces of multicrystalline silicon type-p in order to decrease the reflection of light on the surface, using the chemical etching method and then a treatment with laser. In the first method, it was immersed in solutions of HF:HNO3:H2O, HF:HNO3:CH3COOH, HF:HNO3:H3PO4, in the proportion 14:01:05, during 30 seconds, 1, 2 and 3 minutes. Subsequently with a laser (ND:YAG) grids were generated beginning with parallel lines separated 50μm. The samples were analyzed by means of diffuse spectroscopy (UV-VIS) and scanning electron micrograph (SEM) before and after the laser treatment. The lowest result of reflectance obtained by HF:HNO3:H2O during 30 seconds, was of 15.5%. However, after applying the treatment with laser the reflectance increased to 17.27%. On the other hand, the samples treated (30 seconds) with acetic acid and phosphoric acid as diluents gives as a result a decrease in the reflectance values after applying the laser treatment from 21.97% to 17.79% and from 27.73% to 20.03% respectively. The above indicates that in some cases it is possible to decrease the reflectance using jointly the method of chemical etching and then a laser treatment

The Repeatability of System Level ESD Test and Relevant ESD Generator Parameters

Some system level ESD tests do not repeat well if different ESD generators are used. For improving the test repeatability, ESD generator specifications were considered to be changed and a world wide Round Robin test were performed in 2006 to compare the modified and unmodified ESD generators. The test results show the failure level variations up to 1:3 for an EUT among eight different ESD generators. Multiple ESD parameters including discharge currents and transient fields have been measured. This paper tries to find which parameters would predict the failure level the best in general. The transient fields show large variations among different ESD generators. The voltage induced in a semi-circular loop and the ringing after first discharge current peak show the best correlation to failure levels. The regulation on the transient field is expected to improve the test repeatability

Policies for promoting agricultural development. Report of a Conference on Productivity and Innovation in Agriculture in the Underdeveloped Countries / David Hapgood, Editor ; Max F. Millikan, Conference Chairman

Conference held at the Massachusetts Institute of Technology, Endicott House, Dedham, Massachusetts, June 29 through August 7, 1964Conference financed by A.E.D"1514"--handwritten on cove