On two problems in graph Ramsey theory

We study two classical problems in graph Ramsey theory, that of determining the Ramsey number of bounded-degree graphs and that of estimating the induced Ramsey number for a graph with a given number of vertices. The Ramsey number r(H) of a graph H is the least positive integer N such that every two-coloring of the edges of the complete graph $K_N$ contains a monochromatic copy of H. A famous result of Chv\'atal, R\"{o}dl, Szemer\'edi and Trotter states that there exists a constant c(\Delta) such that r(H) \leq c(\Delta) n for every graph H with n vertices and maximum degree \Delta. The important open question is to determine the constant c(\Delta). The best results, both due to Graham, R\"{o}dl and Ruci\'nski, state that there are constants c and c' such that 2^{c' \Delta} \leq c(\Delta) \leq 2^{c \Delta \log^2 \Delta}. We improve this upper bound, showing that there is a constant c for which c(\Delta) \leq 2^{c \Delta \log \Delta}. The induced Ramsey number r_{ind}(H) of a graph H is the least positive integer N for which there exists a graph G on N vertices such that every two-coloring of the edges of G contains an induced monochromatic copy of H. Erd\H{o}s conjectured the existence of a constant c such that, for any graph H on n vertices, r_{ind}(H) \leq 2^{c n}. We move a step closer to proving this conjecture, showing that r_{ind} (H) \leq 2^{c n \log n}. This improves upon an earlier result of Kohayakawa, Pr\"{o}mel and R\"{o}dl by a factor of \log n in the exponent.Comment: 18 page

Maximum union-free subfamilies

An old problem of Moser asks: how large of a union-free subfamily does every family of m sets have? A family of sets is called union-free if there are no three distinct sets in the family such that the union of two of the sets is equal to the third set. We show that every family of m sets contains a union-free subfamily of size at least \lfloor \sqrt{4m+1}\rfloor - 1 and that this bound is tight. This solves Moser's problem and proves a conjecture of Erd\H{o}s and Shelah from 1972. More generally, a family of sets is a-union-free if there are no a+1 distinct sets in the family such that one of them is equal to the union of a others. We determine up to an absolute multiplicative constant factor the size of the largest guaranteed a-union-free subfamily of a family of m sets. Our result verifies in a strong form a conjecture of Barat, F\"{u}redi, Kantor, Kim and Patkos.Comment: 10 page

Preliminary assessment, restoration and aquaculture support for a small wetland

In line with the strategy of regional wetland datasets integration to a common national digital platform, map of small wetlands less than 2.2 ha in Kochi Taluk was prepared. A representative small wetland at Edakochi village of Kerala was selected through maps and field visits for preliminary assessment and restoration. Shuttle Radar Topography Mission’s Digital Elevation Model (DEM) was used to assess the general elevation, slope and flow accumulation pattern of the selected wetland along with assessment of the catchment area and drainage pattern. Restoration works of the selected wetland was carried out vis-a-vis side bund strengthening and sluice gate fortification. The comparative analysis of water quality assessment of wetland before and after restoration revealed improvement in water quality parameters as well as increase in water level. The Dissolved Oxygen level of the aquatic system was found to have increased substantially along with other several favourable changes in water parameters due to the restoration activities. The restored wetland at Edakochi was further utilised for multispecies farming of prawns, Pearl spot, Milk fish and Grey mullet and the harvest indicated sustainable yield. Aquaculture practice in wetlands with real time scientific advisories could ensure continuous data generation and village level climate resilience

Magnetic Impurity in a Metal with Correlated Conduction Electrons: An Infinite Dimensions Approach

We consider the Hubbard model with a magnetic Anderson impurity coupled to a lattice site. In the case of infinite dimensions, one-particle correlations of the impurity electron are described by the effective Hamiltonian of the two-impurity system. One of the impurities interacts with a bath of free electrons and represents the Hubbard lattice, and the other is coupled to the first impurity by the bare hybridization interaction. A study of the effective two-impurity Hamiltonian in the frame of the 1/N expansion and for the case of a weak conduction-electron interaction (small U) reveals an enhancement of the usual exponential Kondo scale. However, an intermediate interaction (U/D = 1 - 3), treated by the variational principle, leads to the loss of the exponential scale. The Kondo temperature T_K of the effective two-impurity system is calculated as a function of the hybridization parameter and it is shown that T_K decreases with an increase of U. The non-Fermi-liquid character of the Kondo effect in the intermediate regime at the half filling is discussed.Comment: 12 pages with 8 PS figures, RevTe

A quantitative analysis of measures of quality in science

Condensing the work of any academic scientist into a one-dimensional measure of scientific quality is a difficult problem. Here, we employ Bayesian statistics to analyze several different measures of quality. Specifically, we determine each measure's ability to discriminate between scientific authors. Using scaling arguments, we demonstrate that the best of these measures require approximately 50 papers to draw conclusions regarding long term scientific performance with usefully small statistical uncertainties. Further, the approach described here permits the value-free (i.e., statistical) comparison of scientists working in distinct areas of science.Comment: 11 pages, 8 figures, 4 table

Dynamics, correlations and phases of the micromaser

The micromaser possesses a variety of dynamical phase transitions parametrized by the flux of atoms and the time-of-flight of the atom within the cavity. We discuss how these phases may be revealed to an observer outside the cavity using the long-time correlation length in the atomic beam. Some of the phase transitions are not reflected in the average excitation level of the outgoing atom, which is the commonly used observable. The correlation length is directly related to the leading eigenvalue of the time evolution operator, which we study in order to elucidate the phase structure. We find that as a function of the time-of-flight the transition from the thermal to the maser phase is characterized by a sharp peak in the correlation length. For longer times-of-flight there is a transition to a phase where the correlation length grows exponentially with the flux. We present a detailed numerical and analytical treatment of the different phases and discuss the physics behind them.Comment: 60 pages, 18 figure files, Latex + \special{} for the figures, (some redundant figures are eliminated and others are changed

Consequences of marine barriers for genetic diversity of the coral-specialist yellowbar angelfish from the Northwestern Indian Ocean

© 2019 The Authors. Ecology and Evolution published by John Wiley & Sons Ltd. Ocean circulation, geological history, geographic distance, and seascape heterogeneity play an important role in phylogeography of coral-dependent fishes. Here, we investigate potential genetic population structure within the yellowbar angelfish (Pomacanthus maculosus) across the Northwestern Indian Ocean (NIO). We then discuss our results with respect to the above abiotic features in order to understand the contemporary distribution of genetic diversity of the species. To do so, restriction site-associated DNA sequencing (RAD-seq) was utilized to carry out population genetic analyses on P. maculosus sampled throughout the species’ distributional range. First, genetic data were correlated to geographic and environmental distances, and tested for isolation-by-distance and isolation-by-environment, respectively, by applying the Mantel test. Secondly, we used distance-based and model-based methods for clustering genetic data. Our results suggest the presence of two putative barriers to dispersal; one off the southern coast of the Arabian Peninsula and the other off northern Somalia, which together create three genetic subdivisions of P. maculosus within the NIO. Around the Arabian Peninsula, one genetic cluster was associated with the Red Sea and the adjacent Gulf of Aden in the west, and another cluster was associated with the Arabian Gulf and the Sea of Oman in the east. Individuals sampled in Kenya represented a third genetic cluster. The geographic locations of genetic discontinuities observed between genetic subdivisions coincide with the presence of substantial upwelling systems, as well as habitat discontinuity. Our findings shed light on the origin and maintenance of genetic patterns in a common coral reef fish inhabiting the NIO, and reinforce the hypothesis that the evolution of marine fish species in this region has likely been shaped by multiple vicariance events