Distinct Volume Subsets

Suppose that a and d are positive integers with a ≥ 2. Let h[subscript a,d](n) be the largest integer t such that any set of n points in R[superscript d] contains a subset of t points for which all the nonzero volumes of the ([t over a]) subsets of order a are distinct. Beginning with Erdos in 1957, the function h[subscript 2,d](n) has been closely studied and is known to be at least a power of n. We improve the best known bound for h[subscript 2,d](n) and show that h[subscript a,d](n) is at least a power of n for all a and d.David & Lucile Packard Foundation (Fellowship)Simons Foundation (Fellowship)National Science Foundation (U.S.) (Grant DMS-1069197)Alfred P. Sloan Foundation (Fellowship)NEC Corporation (MIT Award

Plasticization and antiplasticization of polymer melts diluted by low molar mass species

An analysis of glass formation for polymer melts that are diluted by structured molecular additives is derived by using the generalized entropy theory, which involves a combination of the Adam-Gibbs model and the direct computation of the configurational entropy based on a lattice model of polymer melts that includes monomer structural effects. Antiplasticization is accompanied by a "toughening" of the glass mixture relative to the pure polymer, and this effect is found to occur when the diluents are small species with strongly attractive interactions with the polymer matrix. Plasticization leads to a decreased glass transition temperature T_g and a "softening" of the fragile host polymer in the glass state. Plasticization is prompted by small additives with weakly attractive interactions with the polymer matrix. The shifts in T_g of polystyrene diluted by fully flexible short oligomers are evaluated from the computations, along with the relative changes in the isothermal compressibility at T_g to characterize the extent to which the additives act as antiplasticizers or plasticizers. The theory predicts that a decreased fragility can accompany both antiplasticization and plasticization of the glass by molecular additives. The general reduction in the T_g and fragility of polymers by these molecular additives is rationalized by analyzing the influence of the diluent's properties (cohesive energy, chain length, and stiffness) on glass formation in diluted polymer melts. The description of glass formation at fixed temperature that is induced upon change the fluid composition directly implies the Angell equation for the structural relaxation time as function of the polymer concentration, and the computed "zero mobility concentration" scales linearly with the inverse polymerization index N.Comment: 12 pages, 15 figure

Self-consistent multi-mode lasing theory for complex or random lasing media

A semiclassical theory of single and multi-mode lasing is derived for open complex or random media using a self-consistent linear response formulation. Unlike standard approaches which use closed cavity solutions to describe the lasing modes, we introduce an appropriate discrete basis of functions which describe also the intensity and angular emission pattern outside the cavity. This constant flux (CF) basis is dictated by the Green function which arises when formulating the steady state Maxwell-Bloch equations as a self-consistent linear response problem. This basis is similar to the quasi-bound state basis which is familiar in resonator theory and it obeys biorthogonality relations with a set of dual functions. Within a single-pole approximation for the Green function the lasing modes are proportional to these CF states and their intensities and lasing frequencies are determined by a set of non-linear equations. When a near threshold approximation is made to these equations a generalized version of the Haken-Sauermann equations for multi-mode lasing is obtained, appropriate for open cavities. Illustrative results from these equations are given for single and few mode lasing states, for the case of dielectric cavity lasers. The standard near threshold approximation is found to be unreliable. Applications to wave-chaotic cavities and random lasers are discussed.Comment: 18 pages, 9 figure

Survey of micrometeorological parameters within a forest canopy at Fort Polk, Louisiana, A

CER80-81WZS-FWL-WEM44.Includes bibliographical references (page 44).February 1982.A field investigation of micrometeorological parameters inside and above a forest canopy at Fort Polk, Louisiana, was conducted in conjunction with the Atmospheric Sciences Laboratory Dusty Infrared Test IIIA. The three orthogonal components of the wind, ory- and wet-bulb temperatures and total solar radiation were measured inside this forest canopy by means of an instrumented meteorological tower. In addition, turbulence inside the forest canopy was monitored by means of hot-wire anemometers. Tethersonde balloon sounding above the forest canopy was further performed. The meteorological data was reduced by means of three different statistical methods. Single sample period values, one-minute sample averages and sequential sample values were computed. The latter two methods led to the construction of time series which can readily be used to perform advanced statistical analyses. Totals of 27 h 29 min of meteorological tower data and 2 h 50 min of balloon data were reduced. The results are presented in tabular form in 1422 tables and partially displayed in 1795 figures under separate cover in view of their large volume. Selected samples of the results are, however, presented herein. The results supply a data base for analyses of airflow in a forest canopy. Suggestions for future work of significance for mission-oriented cases and for modeling of airflow in a forest canopy are outlined.Contract DAAG29-76-D-0100 conducted for the U.S. Army Atmospheric Sciences Laboratory, White Sands Missile Range

An Upper Mass Limit on a Red Supergiant Progenitor for the Type II-Plateau Supernova SN 2006my

We analyze two pre-supernova (SN) and three post-SN high-resolution images of the site of the Type II-Plateau supernova SN 2006my in an effort to either detect the progenitor star or to constrain its properties. Following image registration, we find that an isolated stellar object is not detected at the location of SN 2006my in either of the two pre-SN images. In the first, an I-band image obtained with the Wide-Field and Planetary Camera 2 on board the Hubble Space Telescope, the offset between the SN 2006my location and a detected source ("Source 1") is too large: > 0.08", which corresponds to a confidence level of non-association of 96% from our most liberal estimates of the transformation and measurement uncertainties. In the second, a similarly obtained V-band image, a source is detected ("Source 2") that has overlap with the SN 2006my location but is definitively an extended object. Through artificial star tests carried out on the precise location of SN 2006my in the images, we derive a 3-sigma upper bound on the luminosity of a red supergiant that could have remained undetected in our pre-SN images of log L/L_Sun = 5.10, which translates to an upper bound on such a star's initial mass of 15 M_Sun from the STARS stellar evolutionary models. Although considered unlikely, we can not rule out the possibility that part of the light comprising Source 1, which exhibits a slight extension relative to other point sources in the image, or part of the light contributing to the extended Source 2, may be due to the progenitor of SN 2006my. Only additional, high-resolution observations of the site taken after SN 2006my has faded beyond detection can confirm or reject these possibilities.Comment: Minor text changes from Version 1. Appendix added detailing the determination of confidence level of non-association of point sources in two registered astronomical image

Distinct Volume Subsets

Suppose that a and d are positive integers with a ≥ 2. Let h_(a,d)(n) be the largest integer t such that any set of n points in ℝ^d contains a subset of t points for which all the nonzero volumes of the [equaton; see abstract in PDF for details] subsets of order a are distinct. Beginning with Erdős in 1957, the function h_(2,d)(n) has been closely studied and is known to be at least a power of n. We improve the best known bound for h_(2,d)(n) and show that h_(a,d)(n) is at least a power of n for all a and d

A taste of the deep-sea: The roles of gustatory and tactile searching behaviour in the grenadier fish <i>Coryphaenoides armatus</i>

The deep-sea grenadier fishes (Coryphaenoides spp.) are among the dominant predators and scavengers in the ocean basins that cover much of Earth's surface. Baited camera experiments were used to study the behaviour of these fishes. Despite the apparent advantages of rapidly consuming food, grenadiers attracted to bait spend a large proportion of their time in prolonged periods of non-feeding activity. Video analysis revealed that fish often adopted a head-down swimming attitude (mean of 21.3 degrees between the fish and seafloor), with swimming velocity negatively related to attitude. The fish also swam around and along vertical and horizontal structures of the lander with their head immediately adjacent to the structure. We initially hypothesised that this behaviour was associated with the use of the short chin barbel in foraging. Barbel histology showed numerous taste buds in the skin, and a barbel nerve with about 20,000 axons in adult fish. A tracing experiment in one undamaged animal revealed the termination fields of the barbel neurons in the trigeminal and rhombencephalic regions, indicating both a mechanoreceptory and a gustatory role for the barbel. Our conclusion was that olfactory foraging becomes ineffective at close ranges and is followed by a search phase using tactile and gustatory sensing by the barbel. The development of this sensory method probably co-evolved alongside behavioural changes in swimming mechanics to allow postural stability at low swimming speeds