Conjugacy in Garside Groups III: Periodic braids

An element in Artin's braid group B_n is said to be periodic if some power of it lies in the center of B_n. In this paper we prove that all previously known algorithms for solving the conjugacy search problem in B_n are exponential in the braid index n for the special case of periodic braids. We overcome this difficulty by putting to work several known isomorphisms between Garside structures in the braid group B_n and other Garside groups. This allows us to obtain a polynomial solution to the original problem in the spirit of the previously known algorithms. This paper is the third in a series of papers by the same authors about the conjugacy problem in Garside groups. They have a unified goal: the development of a polynomial algorithm for the conjugacy decision and search problems in B_n, which generalizes to other Garside groups whenever possible. It is our hope that the methods introduced here will allow the generalization of the results in this paper to all Artin-Tits groups of spherical type.Comment: 33 pages, 13 figures. Classical references implying Corollaries 12 and 15 have been added. To appear in Journal of Algebr

Conjugacy in Garside groups I: Cyclings, powers, and rigidity

In this paper a relation between iterated cyclings and iterated powers of elements in a Garside group is shown. This yields a characterization of elements in a Garside group having a rigid power, where 'rigid' means that the left normal form changes only in the obvious way under cycling and decycling. It is also shown that, given X in a Garside group, if some power X^m is conjugate to a rigid element, then m can be bounded above by ||\Delta||^3. In the particular case of braid groups, this implies that a pseudo-Anosov braid has a small power whose ultra summit set consists of rigid elements. This solves one of the problems in the way of a polynomial solution to the conjugacy decision problem (CDP) and the conjugacy search problem (CSP) in braid groups. In addition to proving the rigidity theorem, it will be shown how this paper fits into the authors' program for finding a polynomial algorithm to the CDP/CSP, and what remains to be done.Comment: 41 page

Unbiased Cosmological Parameter Estimation from Emission Line Surveys with Interlopers

The galaxy catalogs generated from low-resolution emission line surveys often contain both foreground and background interlopers due to line misidentification, which can bias the cosmological parameter estimation. In this paper, we present a method for correcting the interloper bias by using the joint-analysis of auto- and cross-power spectra of the main and the interloper samples. In particular, we can measure the interloper fractions from the cross-correlation between the interlopers and survey galaxies, because the true cross-correlation must be negligibly small. The estimated interloper fractions, in turn, remove the interloper bias in the cosmological parameter estimation. For example, in the Hobby-Eberly Telescope Dark Energy Experiment (HETDEX) low-redshift ($z<0.5$) [O II] $\lambda3727${\AA} emitters contaminate high-redshift ($1.9<z<3.5$) Lyman-$\alpha$ line emitters. We demonstrate that the joint-analysis method yields a high signal-to-noise ratio measurement of the interloper fractions while only marginally increasing the uncertainties in the cosmological parameters relative to the case without interlopers. We also show the same is true for the high-latitude spectroscopic survey of Wide-Field Infrared Survey Telescope (WFIRST) mission where contamination occurs between the Balmer-$\alpha$ line emitters at lower redshifts ($1.1<z<1.9$) and Oxygen ([O III] $\lambda5007${\AA}) line emitters at higher redshifts ($1.7<z<2.8$).Comment: 36 pages, 26 figure

Phenotypic plasticity for life-history traits in Drosophila melanogaster. III. Effect of the environment on genetic parameters

We estimated genetic and environmental variance components for developmental time and dry weight at eclosion in Drosophila melanogaster raised in ten different environments (all combinations of 22, 25 and 28°C and 0·5, 1 and 4% yeast concentration, and 0·25% yeast at 25°C). We used six homozygous lines derived from a natural population for complete diallel crosses in each environment. Additive genetic variances were consistently low for both traits (h2 around 10%). The additive genetic variance of developmental time was larger at lower yeast concentrations, but the heritability did not increase because other components were also larger. The additive genetic effects of the six parental lines changed ranks across environments, suggesting a mechanism for the maintenance of genetic variation in heterogenous environments. The variance due to non-directional dominance was small in most environments. However, there was directional dominance in the form of inbreeding depression for both traits. It was pronounced at high yeast levels and temperatures but disappeared when yeast or temperature were decreased. This meant that the heterozygous flies were more sensitive to environmental differences than homozygous flies. Because dominance effects are not heritable, this suggests that the evolution of plasticity can be constrained when dominance effects are important as a mechanism for plasticit