Lecture Hall Theorems, q-series and Truncated Objects

We show here that the refined theorems for both lecture hall partitions and anti-lecture hall compositions can be obtained as straightforward consequences of two q-Chu Vandermonde identities, once an appropriate recurrence is derived. We use this approach to get new lecture hall-type theorems for truncated objects. We compute their generating function and give two different multivariate refinements of these new results : the q-calculus approach gives (u,v,q)-refinements, while a completely different approach gives odd/even (x,y)-refinements. From this, we are able to give a combinatorial characterization of truncated lecture hall partitions and new finitizations of refinements of Euler's theorem

Extraplanar Dust in the Edge-On Spiral NGC 891

We present high-resolution (<0.65") optical broad-band images of the edge-on Sb galaxy NGC 891 obtained with the WIYN 3.5-m telescope. These BVR images reveal a complex network of hundreds of dust absorbing structures far from the mid-plane of the galaxy. The dust structures have a wide range of morphologies and are clearly visible to |z|<1.5 kpc from the mid-plane. In this paper we discuss the general characteristics of the population of absorbing structures, as well as physical properties of 12 individual features. These 12 structures are characterised by N_H >10^21 cm^-2, with masses estimated to be more than 2x10^5 - 5x10^6 solar masses, assuming Galactic gas-to-dust relationships. The gravitational potential energies of the individual dust structures, given their observed heights and derived masses, lie in the range of 20-200x10^51 ergs. Rough number counts of extraplanar dust features suggest the mass of high-z gas associated with extraplanar dust in NGC 891 likely exceeds 2x10^8 solar masses, or ~2% of the total neutral ISM mass of the galaxy. We discuss several mechanisms which may produce high-z dust structures such as those seen in the images presented here. It is not yet known which of these mechanisms are primarily responsible for the extensive extraplanar dust structures seen in our images. The data presented are part of a larger program to search for and characterize off-plane dust structures in edge-on systems. (Abstract Abridged)Comment: To appear in the Astronomical Journal: 37 pages, Latex; 9 separate figures; the paper and high-resolution figures are also available at http://www.astro.wisc.edu/~howk/Papers/papers.htm

Pattern avoidance in compositions and multiset permutations

We study pattern avoidance by combinatorial objects other than permutations, namely by ordered partitions of an integer and by permutations of a multiset. In the former case we determine the generating function explicitly, for integer compositions of n that avoid a given pattern of length 3 and we show that the answer is the same for all such patterns. We also show that the number of multiset permutations that avoid a given three-letter pattern is the same for all such patterns, thereby extending and refining earlier results of Albert, Aldred et al., and by Atkinson, Walker and Linton. Further, the number of permutations of a multiset S, with a_i copies of i for i = 1, ..., k, that avoid a given permutation pattern in S_3 is a symmetric function of the a_i's, and we will give here a bijective proof of this fact first for the pattern (123), and then for all patterns in S_3 by using a recently discovered bijection of Amy N. Myers.Comment: 8 pages, no figur

Transmission overhaul estimates for partial and full replacement at repair

Timely transmission overhauls increase in-flight service reliability greater than the calculated design reliabilities of the individual aircraft transmission components. Although necessary for aircraft safety, transmission overhauls contribute significantly to aircraft expense. Predictions of a transmission's maintenance needs at the design stage should enable the development of more cost effective and reliable transmissions in the future. The frequency is estimated of overhaul along with the number of transmissions or components needed to support the overhaul schedule. Two methods based on the two parameter Weibull statistical distribution for component life are used to estimate the time between transmission overhauls. These methods predict transmission lives for maintenance schedules which repair the transmission with a complete system replacement or repair only failed components of the transmission. An example illustrates the methods

Transmission overhaul and replacement predictions using Weibull and renewel theory

A method to estimate the frequency of transmission overhauls is presented. This method is based on the two-parameter Weibull statistical distribution for component life. A second method is presented to estimate the number of replacement components needed to support the transmission overhaul pattern. The second method is based on renewal theory. Confidence statistics are applied with both methods to improve the statistical estimate of sample behavior. A transmission example is also presented to illustrate the use of the methods. Transmission overhaul frequency and component replacement calculations are included in the example