19 research outputs found
Pseudorandom Generators for Width-3 Branching Programs
We construct pseudorandom generators of seed length that -fool ordered read-once branching programs
(ROBPs) of width and length . For unordered ROBPs, we construct
pseudorandom generators with seed length . This is the first improvement for pseudorandom
generators fooling width 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
for the classes of: (1) read-once polynomials on
variables, (2) locally-monotone ROBPs of length and width
(generalizing read-once CNFs and DNFs), and (3) constant-width ROBPs of length
having a layer of width in every consecutive
layers.Comment: 51 page
On -Simple -Path
An -simple -path is a {path} in the graph of length that passes
through each vertex at most times. The -SIMPLE -PATH problem, given a
graph as input, asks whether there exists an -simple -path in . We
first show that this problem is NP-Complete. We then show that there is a graph
that contains an -simple -path and no simple path of length greater
than . 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 that solves the -SIMPLE -PATH on a graph with
vertices with one-sided error. We also show that a randomized algorithm
with running time with 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 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.
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Viral Glycosphingolipids Induce Lytic Infection and Cell Death in Marine Phytoplankton
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<sub>arag</sub>∼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
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 LaB<sub>6</sub>, added to ensure correct peak identification of very small samples (C).</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<sub>0.</sub> Assembly of <i>proto-polyps</i> after (B) 40 h (C) 50 h and (D) 73 h in culture (magnification 20×).</p