Using TPA to count linear extensions

A linear extension of a poset $P$ is a permutation of the elements of the set that respects the partial order. Let $L(P)$ denote the number of linear extensions. It is a #P complete problem to determine $L(P)$ exactly for an arbitrary poset, and so randomized approximation algorithms that draw randomly from the set of linear extensions are used. In this work, the set of linear extensions is embedded in a larger state space with a continuous parameter ?. The introduction of a continuous parameter allows for the use of a more efficient method for approximating $L(P)$ called TPA. Our primary result is that it is possible to sample from this continuous embedding in time that as fast or faster than the best known methods for sampling uniformly from linear extensions. For a poset containing $n$ elements, this means we can approximate $L(P)$ to within a factor of $1 + \epsilon$ with probability at least $1 - \delta$ using an expected number of random bits and comparisons in the poset which is at most $O(n^3(ln n)(ln L(P))\epsilon^{-2}\ln \delta^{-1}).$Comment: 12 pages, 4 algorithm

Vampire Statistics and Other Mathematical Oddities

The world tends to trust mathematicians and their numbers. By extension, the numbers generated by polls and surveys command much respect, sometimes beyond their deserved due. Thus, when an especially juicy statistic enters the public consciousness, it can take on a life of its own, long after new data superseded the old survey and should have driven a stake through its heart

Mapping out the time-evolution of exoplanet processes

There are many competing theories and models describing the formation, migration and evolution of exoplanet systems. As both the precision with which we can characterize exoplanets and their host stars, and the number of systems for which we can make such a characterization increase, we begin to see pathways forward for validating these theories. In this white paper we identify predicted, observable correlations that are accessible in the near future, particularly trends in exoplanet populations, radii, orbits and atmospheres with host star age. By compiling a statistically significant sample of well-characterized exoplanets with precisely measured ages, we should be able to begin identifying the dominant processes governing the time-evolution of exoplanet systems.Comment: Astro2020 white pape

Optimal Token Allocations in Solitaire Knock \u27M Down

In the game Knock ’m Down, tokens are placed in N bins. At each step of the game, a bin is chosen at random according to a fixed probability distribution. If a token remains in that bin, it is removed. When all the tokens have been removed, the player is done. In the solitaire version of this game, the goal is to minimize the expected number of moves needed to remove all the tokens. Here we present necessary conditions on the number of tokens needed for each bin in an optimal solution, leading to an asymptotic solution. MR Subject Classifications: primary: 91A6

Optimal Token Allocations in Solitaire Knock\u27m Down

In the game Knock \u27m Down, tokens are placed in N bins. At each step of the game, a bin is chosen at random according to a fixed probability distribution. If a token remains in that bin, it is removed. When all the tokens have been removed, the player is done. In the solitaire version of this game, the goal is to minimize the expected number of moves needed to remove all the tokens. Here we present necessary conditions on the number of tokens needed for each bin in an optimal solution, leading to an asymptotic solution

The effect of radial edge lift variation on the speed of RGP lens adaptation

This project was designed to determine if the speed of adaptation to rigid gas permeable (RGP) lenses could be increased by initially fitting low edge lift lenses to reduce lid sensation, and subsequently switching the subject to the higher edge lift lens for long term wear. Thirty-two subjects were dispensed lenses and twenty-nine successfully wore the lenses for the entire eight week period. Half of the subjects wore a low edge design for four weeks, followed by a high edge design for the final four weeks. The remaining subjects wore identical pairs of high edge lift designs for both four week periods to serve as the control group. There were no significant differences in the speed of adaptation between the groups as measured by responses to a questionnaire completed by the subjects at each visit; however, large variations in staining and fitting performance for individual patients demonstrated the importance of customizing the peripheral curve system and the edge lift for each patient

Welcome to the Journal of Humanistic Mathematics

Welcome to the first issue of the Journal of Humanistic Mathematics