Incubators vs Zombies: Fault-Tolerant, Short, Thin and Lanky Spanners for Doubling Metrics

Recently Elkin and Solomon gave a construction of spanners for doubling metrics that has constant maximum degree, hop-diameter O(log n) and lightness O(log n) (i.e., weight O(log n)w(MST). This resolves a long standing conjecture proposed by Arya et al. in a seminal STOC 1995 paper. However, Elkin and Solomon's spanner construction is extremely complicated; we offer a simple alternative construction that is very intuitive and is based on the standard technique of net tree with cross edges. Indeed, our approach can be readily applied to our previous construction of k-fault tolerant spanners (ICALP 2012) to achieve k-fault tolerance, maximum degree O(k^2), hop-diameter O(log n) and lightness O(k^3 log n)

A note on Elkin's improvement of Behrend's construction

We provide a short proof of a recent result of Elkin in which large subsets of the integers 1 up to N free of 3-term progressions are constructed.Comment: 4 pages, submitted to volume in honour of Mel Nathanso

Improved Distributed Algorithms for Exact Shortest Paths

Computing shortest paths is one of the central problems in the theory of distributed computing. For the last few years, substantial progress has been made on the approximate single source shortest paths problem, culminating in an algorithm of Becker et al. [DISC'17] which deterministically computes $(1+o(1))$-approximate shortest paths in $\tilde O(D+\sqrt n)$ time, where $D$ is the hop-diameter of the graph. Up to logarithmic factors, this time complexity is optimal, matching the lower bound of Elkin [STOC'04]. The question of exact shortest paths however saw no algorithmic progress for decades, until the recent breakthrough of Elkin [STOC'17], which established a sublinear-time algorithm for exact single source shortest paths on undirected graphs. Shortly after, Huang et al. [FOCS'17] provided improved algorithms for exact all pairs shortest paths problem on directed graphs. In this paper, we present a new single-source shortest path algorithm with complexity $\tilde O(n^{3/4}D^{1/4})$. For polylogarithmic $D$, this improves on Elkin's $\tilde{O}(n^{5/6})$ bound and gets closer to the $\tilde{\Omega}(n^{1/2})$ lower bound of Elkin [STOC'04]. For larger values of $D$, we present an improved variant of our algorithm which achieves complexity $\tilde{O}\left( n^{3/4+o(1)}+ \min\{ n^{3/4}D^{1/6},n^{6/7}\}+D\right)$, and thus compares favorably with Elkin's bound of $\tilde{O}(n^{5/6} + n^{2/3}D^{1/3} + D )$ in essentially the entire range of parameters. This algorithm provides also a qualitative improvement, because it works for the more challenging case of directed graphs (i.e., graphs where the two directions of an edge can have different weights), constituting the first sublinear-time algorithm for directed graphs. Our algorithm also extends to the case of exact $\kappa$-source shortest paths...Comment: 26 page

Anchorage in Aboriginal affairs: A. P. Elkin on religious continuity and civic obligation

In Australian Aboriginal affairs, the acculturative strand of assimilation developed in large part from Elkin’s religious and Idealist commitment, for which in the years 1928 to 1933 he won social-scientific authority. In competition with both an eliminationist politics of race and a segregationist politics of territory, Elkin drew upon religious experience, apologetics, sociology, and networks to establish a ‘positive policy’ as an enduring ideal in Aboriginal affairs. His leadership of the 1930s reform movement began within the Anglican Church, became national through civic-religious organs of publicity, and gained scientific authority as Elkin made religious themes a central concern in Australian anthropology. But from the 1960s until recently, most scholars have lost sight of the centrality of Idealism and religion in our protagonist’s seminal project of acculturative assimilation. This thesis aims to show how Elkin dealt with problems fundamental to twentieth century Aboriginal affairs and indeed to Australian modernity more generally – problems of faith and science, morality and expediency – in developing his positive policy towards Aborigines

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Hyperelliptic jacobians with real multiplication

Let $K$ be a field of characteristic $p \neq 2$, and let $f(x)$ be a sextic polynomial irreducible over $K$ with no repeated roots, whose Galois group is isomorphic to \A_5. If the jacobian $J(C)$ of the hyperelliptic curve $C:y^2=f(x)$ admits real multiplication over the ground field from an order of a real quadratic field $D$, then either its endomorphism algebra is isomorphic to $D$, or $p > 0$ and $J(C)$ is a supersingular abelian variety. The supersingular outcome cannot occur when $p$ splits in $D$.Comment: Corrected typos; clarified proofs; added more examples in positive characteristi

Anchorage in Aboriginal affairs: A. P. Elkin on religious continuity and civic obligation

In Australian Aboriginal affairs, the acculturative strand of assimilation developed in large part from Elkin’s religious and Idealist commitment, for which in the years 1928 to 1933 he won social-scientific authority. In competition with both an eliminationist politics of race and a segregationist politics of territory, Elkin drew upon religious experience, apologetics, sociology, and networks to establish a ‘positive policy’ as an enduring ideal in Aboriginal affairs. His leadership of the 1930s reform movement began within the Anglican Church, became national through civic-religious organs of publicity, and gained scientific authority as Elkin made religious themes a central concern in Australian anthropology. But from the 1960s until recently, most scholars have lost sight of the centrality of Idealism and religion in our protagonist’s seminal project of acculturative assimilation. This thesis aims to show how Elkin dealt with problems fundamental to twentieth century Aboriginal affairs and indeed to Australian modernity more generally – problems of faith and science, morality and expediency – in developing his positive policy towards Aborigines

Time resolved spectroscopy of the cool Ap star HD 213637

We present an analysis of high time resolution spectra of the chemically peculiar Ap star HD 213637. The star shows rapid radial velocity variations with a period close to the photometric pulsation period. Radial velocity pulsation amplitudes vary significantly for different rare earth elements. The highest pulsation amplitudes belong to lines of Tb III (∼360 m s−1), Pr II (∼250 m s−1) and Pr III (∼230 m s−1).We did not detect any pulsations from spectral lines of Eu II and in Hα, in contrast to many other roAp stars. We also did not find radial velocity pulsations using spectral lines of other chemical elements, including Mg, Si, Ca, Sc, Cr, Fe, Ni, Y and Ba. There are phase shifts between the maxima of pulsation amplitudes of different rare earth elements and ions, which is evidence of an outwardly running magneto-acoustic wave propagating through the upper stellar atmosphere

Magnetic stars from a FEROS cool Ap star survey

New magnetic Ap stars with split Zeeman components are presented. These stars were discovered from observations with the Fibre-fed Extended Range Optical Spectrograph (FEROS) spectrograph at the European Southern Observatory (ESO) 2.2-m telescope. 15 new magnetic stars are analysed here. Several stars with very strong magnetic fields were found, including HD 70702 with a 15-kG magnetic field strength, and HD 168767 with a 16.5-kG magnetic field strength measured using split Zeeman components of spectral lines and by comparison with synthetic calculations. The physical parameters of the stars were estimated from photometric and spectroscopic data. Together with previously published results for stars with strong magnetic fields, the relationship between magnetic field strength and rotation period is discussed