Numerical Algorithms for Dual Bases of Positive-Dimensional Ideals

An ideal of a local polynomial ring can be described by calculating a standard basis with respect to a local monomial ordering. However standard basis algorithms are not numerically stable. Instead we can describe the ideal numerically by finding the space of dual functionals that annihilate it, reducing the problem to one of linear algebra. There are several known algorithms for finding the truncated dual up to any specified degree, which is useful for describing zero-dimensional ideals. We present a stopping criterion for positive-dimensional cases based on homogenization that guarantees all generators of the initial monomial ideal are found. This has applications for calculating Hilbert functions.Comment: 19 pages, 4 figure

Small-world MCMC and convergence to multi-modal distributions: From slow mixing to fast mixing

We compare convergence rates of Metropolis--Hastings chains to multi-modal target distributions when the proposal distributions can be of ``local'' and ``small world'' type. In particular, we show that by adding occasional long-range jumps to a given local proposal distribution, one can turn a chain that is ``slowly mixing'' (in the complexity of the problem) into a chain that is ``rapidly mixing.'' To do this, we obtain spectral gap estimates via a new state decomposition theorem and apply an isoperimetric inequality for log-concave probability measures. We discuss potential applicability of our result to Metropolis-coupled Markov chain Monte Carlo schemes.Comment: Published at http://dx.doi.org/10.1214/105051606000000772 in the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org

UPMASK: unsupervised photometric membership assignment in stellar clusters

We develop a method for membership assignment in stellar clusters using only photometry and positions. The method, UPMASK, is aimed to be unsupervised, data driven, model free, and to rely on as few assumptions as possible. It is based on an iterative process, principal component analysis, clustering algorithm, and kernel density estimations. Moreover, it is able to take into account arbitrary error models. An implementation in R was tested on simulated clusters that covered a broad range of ages, masses, distances, reddenings, and also on real data of cluster fields. Running UPMASK on simulations showed that it effectively separates cluster and field populations. The overall spatial structure and distribution of cluster member stars in the colour-magnitude diagram were recovered under a broad variety of conditions. For a set of 360 simulations, the resulting true positive rates (a measurement of purity) and member recovery rates (a measurement of completeness) at the 90% membership probability level reached high values for a range of open cluster ages ($10^{7.1}-10^{9.5}$ yr), initial masses ($0.5-10\times10^3$M_{\sun}) and heliocentric distances ($0.5-4.0$ kpc). UPMASK was also tested on real data from the fields of the open cluster Haffner~16 and of the closely projected clusters Haffner~10 and Czernik~29. These tests showed that even for moderate variable extinction and cluster superposition, the method yielded useful cluster membership probabilities and provided some insight into their stellar contents. The UPMASK implementation will be available at the CRAN archive.Comment: 12 pages, 13 figures, accepted for publication in Astronomy and Astrophysic

Opinion strength influences the spatial dynamics of opinion formation

Opinions are rarely binary; they can be held with different degrees of conviction, and this expanded attitude spectrum can affect the influence one opinion has on others. Our goal is to understand how different aspects of influence lead to recognizable spatio-temporal patterns of opinions and their strengths. To do this, we introduce a stochastic spatial agent-based model of opinion dynamics that includes a spectrum of opinion strengths and various possible rules for how the opinion strength of one individual affects the influence that this individual has on others. Through simulations, we find that even a small amount of amplification of opinion strength through interaction with like-minded neighbors can tip the scales in favor of polarization and deadlock

Noetherianity for infinite-dimensional toric varieties

We consider a large class of monomial maps respecting an action of the infinite symmetric group, and prove that the toric ideals arising as their kernels are finitely generated up to symmetry. Our class includes many important examples where Noetherianity was recently proved or conjectured. In particular, our results imply Hillar-Sullivant's Independent Set Theorem and settle several finiteness conjectures due to Aschenbrenner, Martin del Campo, Hillar, and Sullivant. We introduce a matching monoid and show that its monoid ring is Noetherian up to symmetry. Our approach is then to factorize a more general equivariant monomial map into two parts going through this monoid. The kernels of both parts are finitely generated up to symmetry: recent work by Yamaguchi-Ogawa-Takemura on the (generalized) Birkhoff model provides an explicit degree bound for the kernel of the first part, while for the second part the finiteness follows from the Noetherianity of the matching monoid ring.Comment: 20 page