Pseudorandom Generators for Width-3 Branching Programs

We construct pseudorandom generators of seed length $\tilde{O}(\log(n)\cdot \log(1/\epsilon))$ that $\epsilon$-fool ordered read-once branching programs (ROBPs) of width $3$ and length $n$. For unordered ROBPs, we construct pseudorandom generators with seed length $\tilde{O}(\log(n) \cdot \mathrm{poly}(1/\epsilon))$. This is the first improvement for pseudorandom generators fooling width $3$ ROBPs since the work of Nisan [Combinatorica, 1992]. Our constructions are based on the `iterated milder restrictions' approach of Gopalan et al. [FOCS, 2012] (which further extends the Ajtai-Wigderson framework [FOCS, 1985]), combined with the INW-generator [STOC, 1994] at the last step (as analyzed by Braverman et al. [SICOMP, 2014]). For the unordered case, we combine iterated milder restrictions with the generator of Chattopadhyay et al. [CCC, 2018]. Two conceptual ideas that play an important role in our analysis are: (1) A relabeling technique allowing us to analyze a relabeled version of the given branching program, which turns out to be much easier. (2) Treating the number of colliding layers in a branching program as a progress measure and showing that it reduces significantly under pseudorandom restrictions. In addition, we achieve nearly optimal seed-length $\tilde{O}(\log(n/\epsilon))$ for the classes of: (1) read-once polynomials on $n$ variables, (2) locally-monotone ROBPs of length $n$ and width $3$ (generalizing read-once CNFs and DNFs), and (3) constant-width ROBPs of length $n$ having a layer of width $2$ in every consecutive $\mathrm{poly}\log(n)$ layers.Comment: 51 page

On rr-Simple kk-Path

An $r$-simple $k$-path is a {path} in the graph of length $k$ that passes through each vertex at most $r$ times. The $r$-SIMPLE $k$-PATH problem, given a graph $G$ as input, asks whether there exists an $r$-simple $k$-path in $G$. We first show that this problem is NP-Complete. We then show that there is a graph $G$ that contains an $r$-simple $k$-path and no simple path of length greater than $4\log k/\log r$. So this, in a sense, motivates this problem especially when one's goal is to find a short path that visits many vertices in the graph while bounding the number of visits at each vertex. We then give a randomized algorithm that runs in time $$\mathrm{poly}(n)\cdot 2^{O( k\cdot \log r/r)}$$ that solves the $r$-SIMPLE $k$-PATH on a graph with $n$ vertices with one-sided error. We also show that a randomized algorithm with running time $\mathrm{poly}(n)\cdot 2^{(c/2)k/ r}$ with $c<1$ gives a randomized algorithm with running time \poly(n)\cdot 2^{cn} for the Hamiltonian path problem in a directed graph - an outstanding open problem. So in a sense our algorithm is optimal up to an $O(\log r)$ factor

High Adenylyl Cyclase Activity and \u3cem\u3eIn Vivo\u3c/em\u3e cAMP Fluctuations in Corals Suggest Central Physiological Role

Corals are an ecologically and evolutionarily significant group, providing the framework for coral reef biodiversity while representing one of the most basal of metazoan phyla. However, little is known about fundamental signaling pathways in corals. Here we investigate the dynamics of cAMP, a conserved signaling molecule that can regulate virtually every physiological process. Bioinformatics revealed corals have both transmembrane and soluble adenylyl cyclases (AC). Endogenous cAMP levels in live corals followed a potential diel cycle, as they were higher during the day compared to the middle of the night. Coral homogenates exhibited some of the highest cAMP production rates ever to be recorded in any organism; this activity was inhibited by calcium ions and stimulated by bicarbonate. In contrast, zooxanthellae or mucus had \u3e1000-fold lower AC activity. These results suggest that cAMP is an important regulator of coral physiology, especially in response to light, acid/base disturbances and inorganic carbon levels

Absolutely Sound Testing of Lifted Codes

In this work we present a strong analysis of the testability of a broad, and to date the most interesting known, class of “affine-invariant ” codes. Affine-invariant codes are codes whose coordinates are associated with a vector space and are invariant under affine transformations of the coordinate space. Affine-invariant linear codes form a natural abstraction of algebraic properties such as linearity and low-degree, which have been of significant interest in theoretical computer science in the past. The study of affine-invariance is motivated in part by its relationship to property testing: Affine-invariant linear codes tend to be locally testable under fairly minimal and almost necessary conditions. Recent works by Ben-Sasson et al. (CCC 2011) and Guo et al. (ITCS 2013) have introduced a new class of affine-invariant linear codes based on an operation called “lifting”. Given a base code over a t-dimensional space, its m-dimensional lift consists of all words whose restriction to every t-dimensional affine subspace is a codeword of the base code. Lifting not only captures the most familiar codes, which can be expressed as lifts of low-degree polynomials, it also yields new codes when lifting “medium-degree ” polynomials whose rate is better than that of corresponding polynomial codes, and all other combinatorial qualities are no worse. In this work we show that codes derived from lifting are also testable in an “absolutely sound” way. Specifically, we consider the natural test: Pick a random affine subspace of base dimension and verify that a given word is a codeword of the base code when restricted to the chosen subspace. We show that this test accepts codewords with probability one, while rejecting words at constant distance from the code with constant probability (depending only on the alphabet size). This work thus extends the results of Bhattacharyya et al. (FOCS 2010) and Haramaty et al. (FOCS 2011), while giving concrete new codes of higher rate that have absolutely sound testers.

Aragonite Precipitation by “<em>Proto-Polyps</em>” in Coral Cell Cultures

<div><p>The mechanisms of coral calcification at the molecular, cellular and tissue levels are poorly understood. In this study, we examine calcium carbonate precipitation using novel coral tissue cultures that aggregate to form “<em>proto-polyps</em>”. Our goal is to establish an experimental system in which calcification is facilitated at the cellular level, while simultaneously allowing <em>in vitro</em> manipulations of the calcifying fluid. This novel coral culturing technique enables us to study the mechanisms of biomineralization and their implications for geochemical proxies. Viable cell cultures of the hermatypic, zooxanthellate coral, <em>Stylophora pistillata</em>, have been maintained for 6 to 8 weeks. Using an enriched seawater medium with aragonite saturation state similar to open ocean surface waters (Ω_arag∼4), the primary cell cultures assemble into “<em>proto-polyps</em>” which form an extracellular organic matrix (ECM) and precipitate aragonite crystals. These extracellular aragonite crystals, about 10 µm in length, are formed on the external face of the <em>proto-polyps</em> and are identified by their distinctive elongated crystallography and X-ray diffraction pattern. The precipitation of aragonite is independent of photosynthesis by the zooxanthellae, and does not occur in control experiments lacking coral cells or when the coral cells are poisoned with sodium azide. Our results demonstrate that <em>proto-polyps</em>, aggregated from primary coral tissue culture, function (from a biomineralization perspective) similarly to whole corals. This approach provides a novel tool for investigating the biophysical mechanism of calcification in these organisms.</p> </div

Relative amino acid composition of skeletal and cell culture ECM as determined by HPLC fluorometric analysis.

<p>Cell culture ECM is derived from the NaOH-treated pellet.</p

Relief contrast images of coral cell cultures.

<p>(A) Relief contrast of individual cells in <i>Stylophora pistillata</i> primary cell culture at T_0. Assembly of <i>proto-polyps</i> after (B) 40 h (C) 50 h and (D) 73 h in culture (magnification 20×).</p

Aragonite mineralogy.

<p>X-ray powder diffraction pattern of (A) tissue culture sample and (B) and of a powdered mother colony skeleton showing the characteristic diffraction peaks of aragonite. Composite minerals were determined by peak matching of XRD data in Jade software (MDI Products, Inc.). Peaks at 26.3 and 45.9 degree are characteristic aragonite peaks. Peaks at 21.6, 30.4, 37.4, 43.5, 49, and 54 degrees are due to LaB₆, added to ensure correct peak identification of very small samples (C).</p

SEM images and elemental composition.

<p>SEM images of typical columnar aggregates of aragonite crystals, elongated along the c axis, formed in the dorsal surface of the cell culture (A–B) and in the <i>S. pistillata</i> skeleton (C). In both aggregates one can see the orthorhombic structure, characteristic of aragonite crystals. (D) ECM of a <i>proto-polyp</i> after cells removed by 1 M NaOH. Red circle on the SEM image indicates the sample point of the EDS. Energy-dispersive X-ray spectrum showing calcium carbonate composition of the aragonite crystal in (a) <i>proto-polyp</i>, in (c) coral skeleton and (d) in the ECM scaffold while (b) the composition in the cells is C, O, N and P. The Au and Pd peaks are from the gold coating.</p