M\"obius Polynomials

We introduce the M\"obius polynomial $M_n(x) = \sum_{d|n} \mu\left( \frac nd \right) x^d$, which gives the number of aperiodic bracelets of length $n$ with $x$ possible types of gems, and therefore satisfies $M_n(x) \equiv 0$ (mod $n$) for all $x \in \mathbb Z$. We derive some key properties, analyze graphs in the complex plane, and then apply M\"obius polynomials combinatorially to juggling patterns, irreducible polynomials over finite fields, and Euler's totient theorem.Comment: 10 pages, 2 figure

Bilinear Forms on Frobenius Algebras

We analyze the homothety types of associative bilinear forms that can occur on a Hopf algebra or on a local Frobenius $k$-algebra $R$ with residue field $k$. If $R$ is symmetric, then there exists a unique form on $R$ up to homothety iff $R$ is commutative. If $R$ is Frobenius, then we introduce a norm based on the Nakayama automorphism of $R$. We show that if two forms on $R$ are homothetic, then the norm of the unit separating them is central, and we conjecture the converse. We show that if the dimension of $R$ is even, then the determinant of a form on $R$, taken in $\dot k/\dot k^2$, is an invariant for $R$. \textit{Key words}: bilinear form, Frobenius algebra, homothety, Hopf algebra, isometry, local algebra, Nakayama automorphism, Ore extension, symmetric algebr

Markov Chains for Collaboration

Consider a system of $n$ players in which each initially starts on a different team. At each time step, we select an individual winner and an individual loser randomly and the loser joins the winner's team. The resulting Markov chain and stochastic matrix clearly have one absorbing state, in which all players are on the same team, but the combinatorics along the way are surprisingly elegant. The expected number of time steps until each team is eliminated is a ratio of binomial coefficients. When a team is eliminated, the probabilities that the players are configured in various partitions of $n$ into $t$ teams are given by multinomial coefficients. The expected value of the time to absorbtion is $(n-1)^2$ steps. The results depend on elementary combinatorics, linear algebra, and the theory of Markov chains

Nakayama automorphisms of Frobenius algebras

AbstractWe show that the Nakayama automorphism of a Frobenius algebra R over a field k is independent of the field (Theorem 4). Consequently, the k-dual functor on left R-modules and the bimodule isomorphism type of the k-dual of R, and hence the question of whether R is a symmetric k-algebra, are independent of k. We give a purely ring-theoretic condition that is necessary and sufficient for a finite-dimensional algebra over an infinite field to be a symmetric algebra (Theorem 7)

Biologically Inspired Feedback Design for Drosophila Flight

We use a biologically motivated model of the Drosophila's flight mechanics and sensor processing to design a feedback control scheme to regulate forward flight. The model used for insect flight is the grand unified fly (GUF) [3] simulation consisting of rigid body kinematics, aerodynamic forces and moments, sensory systems, and a 3D environment model. We seek to design a control algorithm that will convert the sensory signals into proper wing beat commands to regulate forward flight. Modulating the wing beat frequency and mean stroke angle produces changes in the flight envelope. The sensory signals consist of estimates of rotational velocity from the haltere organs and translational velocity estimates from visual elementary motion detectors (EMD's) and matched retinal velocity filters. The controller is designed based on a longitudinal model of the flight dynamics. Feedforward commands are generated based on a desired forward velocity. The dynamics are linearized around this operating point and a feedback controller designed to correct deviations from the operating point. The control algorithm is implemented in the GUF simulator and achieves the desired tracking of the forward reference velocities and exhibits biologically realistic responses

Symptomology Associated with in Utero Exposures to Polysubstance in an Appalachian Population.

Neonatal abstinence syndrome (NAS) is seen as a very high rate at our institution in Huntington, West Virginia, and the majority of exposures are polysubstance in nature. Polysubstance can have different meaning for each region. At our institution, polysubstance is any combination of opioids, gabapentin, methamphetamine, cocaine, marijuana, benzodiazepines, nicotine or other neuroactive substances with 3-4 substances being the norm. Rapidly changing combinations of drug use and the lack of literature create a difficult situation for clinicians who are often reliant on treatment recommendations that lack references or conclusive data supporting the clinical approaches. Elucidating withdrawal symptoms consistent with in utero exposures to particular drug combinations is difficult. Many substances induce similar withdrawal symptoms in neonates and the vast majority of cases present as polysubstance exposure. Standard methodology often leads to a research approach which isolates populations and substance of exposure to determine the individual effects on the neonate. In some drug combinations, like opioid and gabapentin exposure, the substances in concert create symptoms and complications that are not observed with either drug alone. The history of responses to substance use epidemics has been to handle each drug as a separate disease process, this is no longer a viable option. The following is a review of the literature available discussing individual substance withdrawal characteristics in neonates combined with the clinical insight gained at our hospital from treating such high rates of complex polysubstance exposure

Dynamical orbital effects of General Relativity on the satellite-to-satellite range and range-rate in the GRACE mission: a sensitivity analysis

We numerically investigate the impact of GTR on the orbital part of the satellite-to-satellite range \rho and range-rate \dot\rho of the twin GRACE A/B spacecrafts through their dynamical equations of motion integrated in an Earth-centered frame over a time span \Delta t=1 d. Instead, the GTR effects connected with the propagation of the electromagnetic waves linking the spacecrafts are neglected. The present-day accuracies in measuring the GRACE biased range and range-rate are \sigma_\rho\sim 1-10 \mum, \sigma_\dot\rho\sim 0.1-1 \mum s^-1; studies for a follow-on of such a mission points toward a range-rate accuracy of the order of \sigma_\dot\rho\sim 1 nm s^-1 or better. The GTR range and range-rate effects turn out to be \Delta\rho=80 \mum and \Delta\dot\rho=0.012 \mum s^-1 (Lense-Thirring), and \Delta\rho=6000 \mum and \Delta\dot\rho=10 \mum s^-1 (Schwarzschild). We also compute the dynamical range and range-rate perturbations caused by the first six zonal harmonic coefficients J_L, L=2,3,4,5,6,7 of the classical multipolar expansion of the terrestrial gravitational potential in order to evaluate their aliasing impact on the relativistic effects. Conversely, we also quantitatively, and preliminarily, assess the possible a-priori \virg{imprinting} of GTR itself, not solved-for in all the GRACE-based Earth's gravity models produced so far, on the estimated values of the low degree zonals of the geopotential. The present sensitivity analysis can also be extended, in principle, to different orbital configurations in order to design a suitable dedicated mission able to accurately measure the relativistic effects considered.Comment: LaTex, 24 pages, 5 figures, 9 tables. Accepted for publication in Advances in Space Research (ASR

A shorter working week: A radical and pragmatic proposal

Transition to a shorter working week