How proofs are prepared at Camelot

We study a design framework for robust, independently verifiable, and workload-balanced distributed algorithms working on a common input. An algorithm based on the framework is essentially a distributed encoding procedure for a Reed--Solomon code, which enables (a) robustness against byzantine failures with intrinsic error-correction and identification of failed nodes, and (b) independent randomized verification to check the entire computation for correctness, which takes essentially no more resources than each node individually contributes to the computation. The framework builds on recent Merlin--Arthur proofs of batch evaluation of Williams~[{\em Electron.\ Colloq.\ Comput.\ Complexity}, Report TR16-002, January 2016] with the observation that {\em Merlin's magic is not needed} for batch evaluation---mere Knights can prepare the proof, in parallel, and with intrinsic error-correction. The contribution of this paper is to show that in many cases the verifiable batch evaluation framework admits algorithms that match in total resource consumption the best known sequential algorithm for solving the problem. As our main result, we show that the $k$-cliques in an $n$-vertex graph can be counted {\em and} verified in per-node $O(n^{(\omega+\epsilon)k/6})$ time and space on $O(n^{(\omega+\epsilon)k/6})$ compute nodes, for any constant $\epsilon>0$ and positive integer $k$ divisible by $6$, where $2\leq\omega<2.3728639$ is the exponent of matrix multiplication. This matches in total running time the best known sequential algorithm, due to Ne{\v{s}}et{\v{r}}il and Poljak [{\em Comment.~Math.~Univ.~Carolin.}~26 (1985) 415--419], and considerably improves its space usage and parallelizability. Further results include novel algorithms for counting triangles in sparse graphs, computing the chromatic polynomial of a graph, and computing the Tutte polynomial of a graph.Comment: 42 p

On sets of irreducible polynomials closed by composition

Let $\mathcal S$ be a set of monic degree $2$ polynomials over a finite field and let $C$ be the compositional semigroup generated by $\mathcal S$. In this paper we establish a necessary and sufficient condition for $C$ to be consisting entirely of irreducible polynomials. The condition we deduce depends on the finite data encoded in a certain graph uniquely determined by the generating set $\mathcal S$. Using this machinery we are able both to show examples of semigroups of irreducible polynomials generated by two degree $2$ polynomials and to give some non-existence results for some of these sets in infinitely many prime fields satisfying certain arithmetic conditions

Benchmark of FEM, Waveguide and FDTD Algorithms for Rigorous Mask Simulation

An extremely fast time-harmonic finite element solver developed for the transmission analysis of photonic crystals was applied to mask simulation problems. The applicability was proven by examining a set of typical problems and by a benchmarking against two established methods (FDTD and a differential method) and an analytical example. The new finite element approach was up to 100 times faster than the competing approaches for moderate target accuracies, and it was the only method which allowed to reach high target accuracies.Comment: 12 pages, 8 figures (see original publication for images with a better resolution

Molecular gas in extreme star-forming environments: the starbursts Arp220 and NGC6240 as case studies

We report single-dish multi-transition measurements of the 12^CO, HCN, and HCO^+ molecular line emission as well as HNC J=1-0 and HNCO in the two ultraluminous infra-red galaxies Arp220 and NGC6240. Using this new molecular line inventory, in conjunction with existing data in the literature, we compiled the most extensive molecular line data sets to date for such galaxies. The many rotational transitions, with their different excitation requirements, allow the study of the molecular gas over a wide range of different densities and temperatures with significant redundancy, and thus allow good constraints on the properties of the dense gas in these two systems. The mass (~(1-2) x 10^10 Msun) of dense gas (>10^5-6 cm^-3) found accounts for the bulk of their molecular gas mass, and is consistent with most of their IR luminosities powered by intense star bursts while self-regulated by O,B star cluster radiative pressure onto the star-forming dense molecular gas. The highly excited HCN transitions trace a gas phase ~(10-100)x denser than that of the sub-thermally excited HCO^+ lines (for both galaxies). These two phases are consistent with an underlying density-size power law found for Galactic GMCs (but with a steeper exponent), with HCN lines tracing denser and more compact regions than HCO^+. Whether this is true in IR-luminous, star forming galaxies in general remains to be seen, and underlines the need for observations of molecular transitions with high critical densities for a sample of bright (U)LIRGs in the local Universe -- a task for which the HI-FI instrument on board Herschel is ideally suited to do.Comment: 38 pages (preprint ApJ style), 3 figures, accepted for Ap

Close Pairs as Proxies for Galaxy Cluster Mergers

Galaxy cluster merger statistics are an important component in understanding the formation of large-scale structure. Unfortunately, it is difficult to study merger properties and evolution directly because the identification of cluster mergers in observations is problematic. We use large N-body simulations to study the statistical properties of massive halo mergers, specifically investigating the utility of close halo pairs as proxies for mergers. We examine the relationship between pairs and mergers for a wide range of merger timescales, halo masses, and redshifts (0<z<1). We also quantify the utility of pairs in measuring merger bias. While pairs at very small separations will reliably merge, these constitute a small fraction of the total merger population. Thus, pairs do not provide a reliable direct proxy to the total merger population. We do find an intriguing universality in the relation between close pairs and mergers, which in principle could allow for an estimate of the statistical merger rate from the pair fraction within a scaled separation, but including the effects of redshift space distortions strongly degrades this relation. We find similar behavior for galaxy-mass halos, making our results applicable to field galaxy mergers at high redshift. We investigate how the halo merger rate can be statistically described by the halo mass function via the merger kernel (coagulation), finding an interesting environmental dependence of merging: halos within the mass resolution of our simulations merge less efficiently in overdense environments. Specifically, halo pairs with separations less than a few Mpc/h are more likely to merge in underdense environments; at larger separations, pairs are more likely to merge in overdense environments.Comment: 12 pages, 9 figures; Accepted for publication in ApJ. Significant additions to text and two figures changed. Added new findings on the universality of pair mergers and added analysis of the effect of FoF linking length on halo merger

On the Triality Theory for a Quartic Polynomial Optimization Problem

This paper presents a detailed proof of the triality theorem for a class of fourth-order polynomial optimization problems. The method is based on linear algebra but it solves an open problem on the double-min duality left in 2003. Results show that the triality theory holds strongly in a tri-duality form if the primal problem and its canonical dual have the same dimension; otherwise, both the canonical min-max duality and the double-max duality still hold strongly, but the double-min duality holds weakly in a symmetrical form. Four numerical examples are presented to illustrate that this theory can be used to identify not only the global minimum, but also the largest local minimum and local maximum.Comment: 16 pages, 1 figure; J. Industrial and Management Optimization, 2011. arXiv admin note: substantial text overlap with arXiv:1104.297

Covariant entropy conjecture and concordance cosmological models

Recently a covariant entropy conjecture has been proposed for dynamical horizons. We apply this conjecture to concordance cosmological models, namely, those cosmological models filled with perfect fluids, in the presence of a positive cosmological constant. As a result, we find this conjecture has a severe constraint power. Not only does this conjecture rule out those cosmological models disfavored by the anthropic principle, but also it imposes an upper bound $10^{-60}$ on the cosmological constant for our own universe, which thus provides an alternative macroscopic perspective for understanding the long-standing cosmological constant problem.Comment: 10 pages, 1 figure, JHEP style, references added, published versio

Water-soluble SOA from Alkene ozonolysis: composition and droplet activation kinetics inferences from analysis of CCN activity

Cloud formation characteristics of the water-soluble organic fraction (WSOC) of secondary organic aerosol (SOA) formed from the ozonolysis of alkene hydrocarbons (terpinolene, 1-methlycycloheptene and cycloheptene) are studied. Based on size-resolved measurements of CCN activity (of the pure and salted WSOC samples) we estimate the average molar volume and surface tension depression associated with the WSOC using Köhler Theory Analysis (KTA). Consistent with known speciation, the results suggest that the WSOC are composed of low molecular weight species, with an effective molar mass below 200 g mol^(−1). The water-soluble carbon is also surface-active, depressing surface tension 10–15% from that of pure water (at CCN-relevant concentrations). The inherent hygroscopicity parameter, κ, of the WSOC ranges between 0.17 and 0.25; if surface tension depression and molar volume effects are considered in κ, a remarkably constant "apparent" hygroscopicity ~0.3 emerges for all samples considered. This implies that the volume fraction of soluble material in the parent aerosol is the key composition parameter required for prediction of the SOA hygroscopicity, as shifts in molar volume across samples are compensated by changes in surface tension. Finally, using "threshold droplet growth analysis", the water-soluble organics in all samples considered do not affect CCN activation kinetics