Approximate Hypergraph Coloring under Low-discrepancy and Related Promises

A hypergraph is said to be $\chi$-colorable if its vertices can be colored with $\chi$ colors so that no hyperedge is monochromatic. $2$-colorability is a fundamental property (called Property B) of hypergraphs and is extensively studied in combinatorics. Algorithmically, however, given a $2$-colorable $k$-uniform hypergraph, it is NP-hard to find a $2$-coloring miscoloring fewer than a fraction $2^{-k+1}$ of hyperedges (which is achieved by a random $2$-coloring), and the best algorithms to color the hypergraph properly require $\approx n^{1-1/k}$ colors, approaching the trivial bound of $n$ as $k$ increases. In this work, we study the complexity of approximate hypergraph coloring, for both the maximization (finding a $2$-coloring with fewest miscolored edges) and minimization (finding a proper coloring using fewest number of colors) versions, when the input hypergraph is promised to have the following stronger properties than $2$-colorability: (A) Low-discrepancy: If the hypergraph has discrepancy $\ell \ll \sqrt{k}$, we give an algorithm to color the it with $\approx n^{O(\ell^2/k)}$ colors. However, for the maximization version, we prove NP-hardness of finding a $2$-coloring miscoloring a smaller than $2^{-O(k)}$ (resp. $k^{-O(k)}$) fraction of the hyperedges when $\ell = O(\log k)$ (resp. $\ell=2$). Assuming the UGC, we improve the latter hardness factor to $2^{-O(k)}$ for almost discrepancy-$1$ hypergraphs. (B) Rainbow colorability: If the hypergraph has a $(k-\ell)$-coloring such that each hyperedge is polychromatic with all these colors, we give a $2$-coloring algorithm that miscolors at most $k^{-\Omega(k)}$ of the hyperedges when $\ell \ll \sqrt{k}$, and complement this with a matching UG hardness result showing that when $\ell =\sqrt{k}$, it is hard to even beat the $2^{-k+1}$ bound achieved by a random coloring.Comment: Approx 201

Cephalopod classification and taxonomy

Chambered nautilus, cuttlefishes, squids and octopus are the four major groups of cephalopods, which belong to the highly evolved class of phylum Mollusca. Cephalopods are the third largest molluscan class after bivalves and gastropods and consist of more than 800 species (Lindgren et al. 2004). The fossil record contains about 17,000 named species of cephalopods. Although the diversity of cephalopods is very much reduced in the modern era, cephalopods are found to occur in all the oceans of the world from the tropics to the polar seas and at all depths ranging from the surface to below 5000m. Cephalopods were dominant predators millions of years before fish appeared. The earliest cephalopods were primitive shelled nautiloids which evolved in the Late Cambrian period. The living cephalopods range in size from 25mm (Southern pygmy squid, Idiosepius notoides) to more than 12m (Colossal squid, Mesonychoteuthis hamiltoni) in length

Magnetoresistance in the superconducting state at the (111) LaAlO3_3/SrTiO3_3 interface

Condensed matter systems that simultaneously exhibit superconductivity and ferromagnetism are rare due the antagonistic relationship between conventional spin-singlet superconductivity and ferromagnetic order. In materials in which superconductivity and magnetic order is known to coexist (such as some heavy-fermion materials), the superconductivity is thought to be of an unconventional nature. Recently, the conducting gas that lives at the interface between the perovskite band insulators LaAlO$_3$ (LAO) and SrTiO$_3$ (STO) has also been shown to host both superconductivity and magnetism. Most previous research has focused on LAO/STO samples in which the interface is in the (001) crystal plane. Relatively little work has focused on the (111) crystal orientation, which has hexagonal symmetry at the interface, and has been predicted to have potentially interesting topological properties, including unconventional superconducting pairing states. Here we report measurements of the magnetoresistance of (111) LAO/STO heterostructures at temperatures at which they are also superconducting. As with the (001) structures, the magnetoresistance is hysteretic, indicating the coexistence of magnetism and superconductivity, but in addition, we find that this magnetoresistance is anisotropic. Such an anisotropic response is completely unexpected in the superconducting state, and suggests that (111) LAO/STO heterostructures may support unconventional superconductivity.Comment: 6 Pages 4 figure

Gastropod classification and taxonomy

Gastropods are a large and highly diversified class within the phylum Mollusca. Many gastropods possess a shell whereas some are without shells. The shelled gastropods are also called univalves. Some of these gastropods are terrestrial while other gastropods live in marine or freshwater habitat. There are approximately 85,000 - 100000 described species of molluscs (Strong et al., 2008) found throughout the world from the garden to deep-water hydrothermal vent colonies. Current estimates place the total number of molluscs including undescribed species as high as 240,000 species (Appeltan et al., 2011). Gastropods encompass 80 % of living molluscs species. In the conventional division of subclass, recognized species of prosobranchs (largely marine, but with terrestrial and freshwater representatives) formed 53 % followed by pulmonates (43% - terrestrial and freshwater, few marine littoral) and opisthobranchs (4% - marine) (Boss, 1971). Gastropods are considered as the oldest known fossils with their shells being evolved in rocks 540 million years ago. Many of the today’s gastropod species have continued unaltered for over 350 million years

Marine molluscan diversity in India – exploitation, conservation

The molluscs (soft bodied animals) belong to the large and diverse phylum Mollusca, which includes a variety of familiar animals well-known as decorative shells or as seafood. These range from tiny snails, clams, and abalone to larger organisms such as squid, cuttlefish and the octopus. These molluscs occupy a variety of habitats ranging from mountain forests, freshwater to more than 10 km depth in the sea. They range in size from less than 1 mm to more than 15 m (for example the giant squid) and their population density may exceed 40,000/m2 in some areas. In the tropical marine environment, molluscs occupy every trophic level, from primary producers to top carnivores. India has extensive molluscan resources along her coasts. In the numerous bays, brackish waters and estuaries and in the seas around the subcontinent; molluscs belonging to different taxonomic groups, such as, mussels, oysters, clams, pearl-oysters, window-pane oysters, ark-shells, whelks, chanks, cowries, squids and cuttlefish have been exploited since time immemorial for food, pearls and shells

भारत में समुद्रीय मृद्रुकवची की विविधता: 21 वीं शताब्दी में उपयोजन, संरक्षण एवं जलवायु परिवर्तन

कृपया पूरा लेखा पढ

Superconductivity and Frozen Electronic States at the (111) LaAlO3_3/SrTiO3_3 Interface

In spite of Anderson's theorem, disorder is known to affect superconductivity in conventional s-wave superconductors. In most superconductors, the degree of disorder is fixed during sample preparation. Here we report measurements of the superconducting properties of the two-dimensional gas that forms at the interface between LaAlO$_3$ (LAO) and SrTiO$_3$ (STO) in the (111) crystal orientation, a system that permits \emph{in situ} tuning of carrier density and disorder by means of a back gate voltage $V_g$. Like the (001) oriented LAO/STO interface, superconductivity at the (111) LAO/STO interface can be tuned by $V_g$. In contrast to the (001) interface, superconductivity in these (111) samples is anisotropic, being different along different interface crystal directions, consistent with the strong anisotropy already observed other transport properties at the (111) LAO/STO interface. In addition, we find that the (111) interface samples "remember" the backgate voltage $V_F$ at which they are cooled at temperatures near the superconducting transition temperature $T_c$, even if $V_g$ is subsequently changed at lower temperatures. The low energy scale and other characteristics of this memory effect ($<1$ K) distinguish it from charge-trapping effects previously observed in (001) interface samples.Comment: 6 pages, 5 Figure

Biochemical composition of different body parts of Gafrarium tumidum (Roding, 1798) from Mandapam, South East Coast of India

Due to its abundance in various areas along the Mandapam coast (latitude 9° 17”N; longitude 79° 11”E), Southeast coast of India, the commonly occurring intertidal bivalve Gafrarium tumidum was selected for this study to investigate its nutritive value. Tissues of different body parts such as mantle, viscera and foot were analyzed. Biochemical constituents like protein, carbohydrate and fat were estimated in different body parts and the percentage of protein was 61.74%, carbohydrate 32.64% and lipid 14.37%. The fatty acids of bivalve were analyzed by gas chromatography/mass spectrometry. In the fatty acids, polyunsaturated fatty acids (PUFA) were found to be dominant, contributing 6.99% (2.34, 2.67 and 1.98% mantle, viscera and foot, respectively), whereas saturated fatty acids (SFA) contributed 5.19% (1.70, 1.94 and 1.55%) followed by monounsaturated fatty acids (MUFA) which contributed 2.75% (0.92 0.93 0.90%). Stearic (0.81%) and palmitic acids (0.71%) were the dominant saturated fatty acid in viscera and oleic acid (0.93%) was the monounsaturated fatty acid found in the viscera. Linoleic (1.38%) and α- linolenic acids (1.07%) were the dominant polyunsaturated fatty acid in viscera. The total amino acid detected was found to be 42.97%, among them the essential amino acids (EAA) was 20.77% and the non essential amino acids (NEAA) present was found to be 22.2%. The results of this study revealed that phenylalanine (1.19%) and methionine (1.04%) in EAA and alanine (1.10%) and glutamic acid (1.10%) in NEAA were the major constituents. It could also be added that the composition of marine bivalves is a nutritional assurance to millions of malnourished people

Contribution to the knowledge of ornamental molluscs of Parangipettai, Southeast Coast of India

The ornamental shell industry in India in recent years is a multimillion dollar industry. India’s ornamental trade is currently worth US$ 278 million. The collection of molluscan shells is a popular activity in coastal areas. Ornamental molluscs are sold in the market exclusively for aquarium and ornamental purposes. Molluscan shells are used in studs like ear-rings, bangles, table lamps, spoons cups and saucers etc., and it is also used in Aquarium. A study was carried out on ornamental molluscan resources of Tamil nadu coast to identify, quantify and assess the shell resources potential for development of a small-scale shell industry and also a suitable collection techniques and methods of shell catalogue preparation during Oct 2007 to Sep 2008. Gastropods and bivalves have been collected for ornamental purpose by the fisher folks. During the survey gastropods and bivalve species belonging to 21families such as Turritellidae, Cypraeidae, Volutidae, Muricidae, Conidae, Trochidae, Olividae, Cassidae, Marginellidae, Bursidae, Strombidae, Buccinidae, Naticidae, Tonnidae, Melongenidae, Vassidae, Harpidae, Architectonidae Arcidae, Pectinidae, Veneridae, were collected. From the total molluscan diversity 85% were exclusively ornamental molluscs. The study revealed that the occurrence of gastropods and bivalves species here are most valuable shells and naturally rare and not easily available