Permutation patterns and statistics

Let S_n denote the symmetric group of all permutations of the set {1, 2, ...,n} and let S = \cup_{n\ge0} S_n. If Pi is a set of permutations, then we let Av_n(Pi) be the set of permutations in S_n which avoid every permutation of Pi in the sense of pattern avoidance. One of the celebrated notions in pattern theory is that of Wilf-equivalence, where Pi and Pi' are Wilf equivalent if #Av_n(Pi)=#Av_n(Pi') for all n\ge0. In a recent paper, Sagan and Savage proposed studying a q-analogue of this concept defined as follows. Suppose st:S->N is a permutation statistic where N represents the nonnegative integers. Consider the corresponding generating function, F_n^{st}(Pi;q) = sum_{sigma in Av_n(Pi)} q^{st sigma}, and call Pi,Pi' st-Wilf equivalent if F_n^{st}(Pi;q)=F_n^{st}(Pi';q) for all n\ge0. We present the first in-depth study of this concept for the inv and maj statistics. In particular, we determine all inv- and maj-Wilf equivalences for any Pi containd in S_3. This leads us to consider various q-analogues of the Catalan numbers, Fibonacci numbers, triangular numbers, and powers of two. Our proof techniques use lattice paths, integer partitions, and Foata's fundamental bijection. We also answer a question about Mahonian pairs raised in the Sagan-Savage article.Comment: 28 pages, 5 figures, tightened up the exposition, noted that some of the conjectures have been prove

A Robust Numerical Method for Integration of Point-Vortex Trajectories in Two Dimensions

The venerable 2D point-vortex model plays an important role as a simplified version of many disparate physical systems, including superfluids, Bose-Einstein condensates, certain plasma configurations, and inviscid turbulence. This system is also a veritable mathematical playground, touching upon many different disciplines from topology to dynamic systems theory. Point-vortex dynamics are described by a relatively simple system of nonlinear ODEs which can easily be integrated numerically using an appropriate adaptive time stepping method. As the separation between a pair of vortices relative to all other inter-vortex length scales decreases, however, the computational time required diverges. Accuracy is usually the most discouraging casualty when trying to account for such vortex motion, though the varying energy of this ostensibly Hamiltonian system is a potentially more serious problem. We solve these problems by a series of coordinate transformations: We first transform to action-angle coordinates, which, to lowest order, treat the close pair as a single vortex amongst all others with an internal degree of freedom. We next, and most importantly, apply Lie transform perturbation theory to remove the higher-order correction terms in succession. The overall transformation drastically increases the numerical efficiency and ensures that the total energy remains constant to high accuracy.Comment: 21 pages, 4 figure

A Group of Red, Ly-alpha Emitting, High Redshift Galaxies

We have discovered two new high redshift (z=2.38) galaxies, near the previously known z=2.38 galaxy 2139-4434 B1 (Francis et al. 1996). All three galaxies are strong Ly-alpha emitters, and have much redder continuum colors (I-K about 5) than other optically-selected high redshift galaxies. We hypothesize that these three galaxies are QSO IIs; radio-quiet counterparts of high redshift radio galaxies, containing concealed QSO nuclei. The red colors are most easily modelled by an old (> 0.5 Gyr), massive (> 10E11 solar masses) stellar population. If true, this implies that at least one galaxy cluster of mass much greater than 3E11 solar masses had collapsed before redshift five.Comment: 12 pages, 2 figures, uses aaspp4 style file. Accepted for publication in Astrophysical Journal Letter

The Design of Pumpjets for Hydrodynamic Propulsion

A procedure for use in the design of a wake adapted pumpjet mounted on the aft end of a body of revolution is presented. To this end, a pumpjet is designed for the Akron airship. The propulsor mass flow is selected to minimize kinetic energy losses through the duct and in the discharge jet. The shaft speed and disk size are selected to satisfy specified limits of cavitation performance and to provide acceptable blade loading. The streamtubes which pass through a propulsor mounted on a tapered afterbody follow essentially conical surfaces. A method is provided for defining these surfaces as a function of shroud geometry, rotor head distribution, and the energy distribution of the ingested mass flow. The three-dimensional effects to which the conical flow subjects the cylindrical blade design sections are described and a technique is presented which permits incorporation of these effects in the blade design procedure

Wind Tunnel Experiments with Flexible Plates in Transonic Flows

The evolution of adaptive shock control bump (SCB) design has seen the system flexibility increase to a point where the aerodynamic loading can affect the deformation of the plate. By studying the effects of a flexible plate subject to transonic flow the fluid structure interaction can be investigated. In this study an array of thin plates (0.4 and 0.6 mm) with different aspect ratios (1 and 1.33) are exposed to a Mach 1.4 normal shockwave. PIV is used in combination with Schlieren imaging to provide a detailed view of the flow curvature surrounding the plate as well as the global shock structure. A technique that extracts the plate deformation from the PIV images is also presented which provides fluid and structural information for each test. The relationship between plate and flow angle is discussed as well as the effect of plate stiffness and free stream influence of each plate configuration

Nesting of Red-Winged Blacbirds in Cattails and Commed Reed Grass in Mentor Harsh

Author Institution: Department of Biology, John Carrol UniversityDuring summer 1976, a study of habitat selection by the red-winged blackbird, Agelaius phoeniceus, was conducted. Mentor Marsh near Mentor, Ohio was chosen because of its unique geologic and botanical history. A swamp forest prior to 1959, the area is now a marsh composed of Typha latifolia, T. angustifolia, and Phragmites australis. Preferences of A. phoeniceus for each of these plants were investigated, and T. latifolia was significantly most chosen. Presumed advantages include rigidity, easy nest accessibility, and wide spacing to facilitate nest defense. Additional features of habitat selected such as available perches, proximity of open water, and edge effect were investigated. If rapid succession to monodominant stands of P. australis continues at the present rate, the nesting red-winged blackbird population may decrease sharply

Computerized system for translating a torch head

The system provides a constant travel speed along a contoured workpiece. It has a driven skate characterized by an elongated bed, with a pair of independently pivoted trucks connected to the bed for support. The trucks are mounted on a contoured track of arbitrary configuration in a mutually spaced relation. An axially extensible torch head manipulator arm is mounted on the bed of the carriage and projects perpendicular from the midportion. The torch head is mounted at its distal end. A real-time computerized control drive subsystem is used to advance the skate along the track of a variable rate for maintaining a constant speed for the torch head tip, and to position the torch axis relative to a preset angle to the workpiece