Coupling with the stationary distribution and improved sampling for colorings and independent sets

We present an improved coupling technique for analyzing the mixing time of Markov chains. Using our technique, we simplify and extend previous results for sampling colorings and independent sets. Our approach uses properties of the stationary distribution to avoid worst-case configurations which arise in the traditional approach. As an application, we show that for $k/\Delta >1.764$, the Glauber dynamics on $k$-colorings of a graph on $n$ vertices with maximum degree $\Delta$ converges in $O(n\log n)$ steps, assuming $\Delta =\Omega(\log n)$ and that the graph is triangle-free. Previously, girth $\ge 5$ was needed. As a second application, we give a polynomial-time algorithm for sampling weighted independent sets from the Gibbs distribution of the hard-core lattice gas model at fugacity $\lambda <(1-\epsilon)e/\Delta$, on a regular graph $G$ on $n$ vertices of degree $\Delta =\Omega(\log n)$ and girth $\ge 6$. The best known algorithm for general graphs currently assumes $\lambda <2/(\Delta -2)$.Comment: Published at http://dx.doi.org/10.1214/105051606000000330 in the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org

Sampling Random Colorings of Sparse Random Graphs

We study the mixing properties of the single-site Markov chain known as the Glauber dynamics for sampling $k$-colorings of a sparse random graph $G(n,d/n)$ for constant $d$. The best known rapid mixing results for general graphs are in terms of the maximum degree $\Delta$ of the input graph $G$ and hold when $k>11\Delta/6$ for all $G$. Improved results hold when $k>\alpha\Delta$ for graphs with girth $\geq 5$ and $\Delta$ sufficiently large where $\alpha\approx 1.7632\ldots$ is the root of $\alpha=\exp(1/\alpha)$; further improvements on the constant $\alpha$ hold with stronger girth and maximum degree assumptions. For sparse random graphs the maximum degree is a function of $n$ and the goal is to obtain results in terms of the expected degree $d$. The following rapid mixing results for $G(n,d/n)$ hold with high probability over the choice of the random graph for sufficiently large constant~$d$. Mossel and Sly (2009) proved rapid mixing for constant $k$, and Efthymiou (2014) improved this to $k$ linear in~$d$. The condition was improved to $k>3d$ by Yin and Zhang (2016) using non-MCMC methods. Here we prove rapid mixing when $k>\alpha d$ where $\alpha\approx 1.7632\ldots$ is the same constant as above. Moreover we obtain $O(n^{3})$ mixing time of the Glauber dynamics, while in previous rapid mixing results the exponent was an increasing function in $d$. As in previous results for random graphs our proof analyzes an appropriately defined block dynamics to "hide" high-degree vertices. One new aspect in our improved approach is utilizing so-called local uniformity properties for the analysis of block dynamics. To analyze the "burn-in" phase we prove a concentration inequality for the number of disagreements propagating in large blocks

Biology of primate relaxin: A paracrine signal in early pregnancy?

Relaxin is a peptide hormone that exerts numerous effects in a variety of tissues across a broad range of species. Although first identified more than 75 years ago interest in relaxin biology has waxed and waned over the years consistent with peaks and troughs of new experimental data on its wide-ranging biological effects and advances in relaxin enabling technologies. Recent insights into species-dependent differences in relaxin biology during pregnancy have once again stimulated a relative surge of interest in the study of relaxin's reproductive biology. Identification and pharmacological characterization of orphaned relaxin receptors and exploration of its paracrine effects on pregnancy using genomic and proteomic technologies have succeeded in fueling current interest in relaxin research. Primates and non-primate vertebrates exhibit very disparate profiles of relaxin genomics, proteomics and functional biology. Non-human primates appear to exhibit a very close similarity to humans with respect to relaxin reproductive biology but the similarities and subtle differences are only just beginning to be understood. We, and others, have shown that relaxin produces significant changes to the non-human primate endometrium during the peri-implantation period that are consistent with relaxin's long perceived role as a paracrine modulator of pregnancy. The purpose of this review is to summarize the reproductive biology of relaxin in non-human primates with a specific emphasis on the paracrine role of ovarian and endometrial relaxin during embryo implantation and early pregnancy

Ketone Hydrogenation with Iridium Complexes with “non N–H” Ligands: The Key Role of the Strong Base

Ferrocenyl phosphine thioether ligands (PS), not containing deprotonatable functions, efficiently support the iridium catalyzed ketone hydrogenation in combination with a strong base co-catalyst. Use of an internal base ([Ir(OMe)(COD)]2 in place of [IrCl(COD)]2) is not sufficient to insure activity and a strong base is still necessary, suggesting that the active catalyst is an anionic hydride complex. Computational investigations that include solvent effects demonstrate the thermodynamically accessible generation of the tetrahydrido complex [IrH4(PS)]-and suggest an operating cycle via a [Na+(MeOH)3∙∙∙Ir-H4(PS)] contact ion pair with an energy span of 18.2 kcal/mol. The cycle involves an outer sphere stepwise H-/H+ transfer, the proton originating from H2 after coordination and heterolytic activation. The base plays the dual role of generating the anionic complex and providing the Lewis acid co-catalyst for ketone activation. The best cycle for the neutral system, on the other hand, requires an energy span of 26.3 kcal/mol. This work highlights, for the first time, the possibility of outer sphere hydrogenation in the presence of non deprotonatable ligands and the role of the strong base in the activation of catalytic systems with such type of ligands

First Report of a Population of Western Blacknose Dace (\u3ci\u3eRhinichthys obtusus\u3c/i\u3e) in the Brushy Creek System of the Black Warrior River Drainage, Alabama

Alabama is home to the southernmost populations of Rhinichthys obtusus, the Western Blacknose Dace. Within Alabama, R. obtusus is found in the Tennessee, Coosa, and Black Warrior River basins, but its presence in the Black Warrior River drainage has been limited. Until now, R. obtusus in the Black Warrior drainage has only been reported as collections of 1 to 4 specimens at a time in the Sipsey Fork drainage. Herein, we report two novel occurrences of R. obtusus in the headwaters of the Brushy Creek system in the Black Warrior River drainage including a singleton and a large population. The population observed in Collier Creek is located over 21 km from the Tennessee River divide, on the divide between the Brushy Creek system and the Sipsey Fork drainage, and is in close proximity to an artificial lake, any of which may be the origin of this population. The presence of a population of R. obtusus in the system necessitates further sampling of the headwaters of the Black Warrior River drainage to understand its natural prevalence in the region

Predicting Structural and Optical Properties of Hollow-Core Photonic Bandgap Fibers from Second Stage Preforms

We propose a simple theory based on mass conservation that allows accurate prediction of guidance properties in hollow-core photonic bandgap fibers (HC-PBGF) from knowledge of the second stage preforms from which the fibers are drawn

Magellan/MMIRS near-infrared multi-object spectroscopy of nebular emission from star forming galaxies at 2<z<3

To investigate the ingredients, which allow star-forming galaxies to present Lyalpha line in emission, we studied the kinematics and gas phase metallicity (Z) of the interstellar medium. We used multi-object NIR spectroscopy with Magellan/MMIRS to study nebular emission from z=2-3 star-forming galaxies discovered in 3 MUSYC fields. We detected emission lines from four active galactic nuclei and 13 high-z star-forming galaxies, including Halpha lines down to a flux of 4.E-17 erg/sec/cm^2. This yielded 7 new redshifts. The most common emission line detected is [OIII]5007, which is sensitive to Z. We were able to measure Z for 2 galaxies and to set upper(lower) limits for another 2(2). The Z values are consistent with 0.3<Z/Zsun<1.2. Comparing the Lyalpha central wavelength with the systemic redshift, we find Delta_v(Lyalpha-[OIII])=70-270 km/sec. High-redshift star-forming galaxies, Lyalpha emitting (LAE) galaxies, and Halpha emitters appear to be located in the low mass, high star-formation rate (SFR) region of the SFR versus stellar mass diagram, confirming that they are experiencing burst episodes of star formation, which are building up their stellar mass. Their Zs are consistent with the relation found for z<2.2 galaxies in the Z versus stellar mass plane. The measured Delta_v(Lyalpha-[OIII]) values imply that outflows of material, driven by star formation, could be present in the z=2-3 LAEs of our sample. Comparing with the literature, we note that galaxies with lower Z than ours are also characterized by similar Delta_v(Lyalpha-[OIII]) velocity offsets. Strong [OIII] is detected in many Lyalpha emitters. Therefore, we propose the Lyalpha/[OIII] flux ratio as a tool for the study of high-z galaxies; while influenced by Z, ionization, and Lyalpha radiative transfer in the ISM, it may be possible to calibrate this ratio to primarily trace one of these effects.Comment: 22 pages, 13 figures, 6 table

Optimal Mixing via Tensorization for Random Independent Sets on Arbitrary Trees

We study the mixing time of the single-site update Markov chain, known as the Glauber dynamics, for generating a random independent set of a tree. Our focus is obtaining optimal convergence results for arbitrary trees. We consider the more general problem of sampling from the Gibbs distribution in the hard-core model where independent sets are weighted by a parameter ? > 0; the special case ? = 1 corresponds to the uniform distribution over all independent sets. Previous work of Martinelli, Sinclair and Weitz (2004) obtained optimal mixing time bounds for the complete ?-regular tree for all ?. However, Restrepo et al. (2014) showed that for sufficiently large ? there are bounded-degree trees where optimal mixing does not hold. Recent work of Eppstein and Frishberg (2022) proved a polynomial mixing time bound for the Glauber dynamics for arbitrary trees, and more generally for graphs of bounded tree-width. We establish an optimal bound on the relaxation time (i.e., inverse spectral gap) of O(n) for the Glauber dynamics for unweighted independent sets on arbitrary trees. Moreover, for ? ? .44 we prove an optimal mixing time bound of O(n log n). We stress that our results hold for arbitrary trees and there is no dependence on the maximum degree ?. Interestingly, our results extend (far) beyond the uniqueness threshold which is on the order ? = O(1/?). Our proof approach is inspired by recent work on spectral independence. In fact, we prove that spectral independence holds with a constant independent of the maximum degree for any tree, but this does not imply mixing for general trees as the optimal mixing results of Chen, Liu, and Vigoda (2021) only apply for bounded degree graphs. We instead utilize the combinatorial nature of independent sets to directly prove approximate tensorization of variance/entropy via a non-trivial inductive proof