Computational Approaches to Consecutive Pattern Avoidance in Permutations

In recent years, there has been increasing interest in consecutive pattern avoidance in permutations. In this paper, we introduce two approaches to counting permutations that avoid a set of prescribed patterns consecutively. These algoritms have been implemented in the accompanying Maple package CAV, which can be downloaded from the author's website. As a byproduct of the first algorithm, we have a theorem giving a sufficient condition for when two pattern sets are strongly (consecutively) Wilf-Equivalent. For the implementation of the second algorithm, we define the cluster tail generating function and show that it always satisfies a certain functional equation. We also explain how the CAV package can be used to approximate asymptotic constants for single pattern avoidance.Comment: 12 page

On the Asymptotic Statistics of the Number of Occurrences of Multiple Permutation Patterns

We study statistical properties of the random variables $X_{\sigma}(\pi)$, the number of occurrences of the pattern $\sigma$ in the permutation $\pi$. We present two contrasting approaches to this problem: traditional probability theory and the ``less traditional'' computational approach. Through the perspective of the first one, we prove that for any pair of patterns $\sigma$ and $\tau$, the random variables $X_{\sigma}$ and $X_{\tau}$ are jointly asymptotically normal (when the permutation is chosen from $S_{n}$). From the other perspective, we develop algorithms that can show asymptotic normality and joint asymptotic normality (up to a point) and derive explicit formulas for quite a few moments and mixed moments empirically, yet rigorously. The computational approach can also be extended to the case where permutations are drawn from a set of pattern avoiders to produce many empirical moments and mixed moments. This data suggests that some random variables are not asymptotically normal in this setting.Comment: 18 page

Automatic Generation of Theorems and Proofs on Enumerating Consecutive-Wilf classes

This article, dedicated to Herbert Saul Wilf on the occaison of his forthcoming 80-th birthday, describes two complementary approaches to enumeration, the "positive" and the "negative", each with its advantages and disadvantages. Both approaches are amenable to automation, and when applied to the currently active subarea, initiated in 2003 by Sergi Elizalde and Marc Noy, of enumerating consecutive-Wilf classes (i.e. consecutive pattern-avoidance) in permutations, were successfully pursued by DZ's two current PhD students, Andrew Baxter and Brian Nakamura. The Maple packages SERGI and ELIZALDE, implementing the algorithms enable the computer to "do research" by deriving, "all by itself", functional equations for the generating functions that enable polynomial-time enumeration for any set of patterns. In the case of ELIZALDE (the "negative" approach), these functional equations can be sometimes (automatically!) simplified, and imply "explicit" formulas, that previously were derived by humans using ad-hoc methods. We also get lots of new "explicit" results, beyond the scope of humans, but we have to admit, that we still need humans to handle "infinite families" of patterns, but this too, no doubt, will soon be automatable, and we leave it as a challenge to the (human and/or computer) reader. The Maple packages, and lots of sample output, is available from the webpage of this article: http://www.math.rutgers.edu/~zeilberg/mamarim/mamarimhtml/auto.htmlComment: 17 page

The Biermann Battery in Cosmological MHD Simulations of Population III Star Formation

We report the results of the first self-consistent three-dimensional adaptive mesh refinement magnetohydrodynamical simulations of Population III star formation including the Biermann Battery effect. We find that the Population III stars formed including this effect are both qualitatively and quantitatively similar to those from hydrodynamics-only (non-MHD) cosmological simulations. We observe peak magnetic fields of ~10^-9 G in the center of our star-forming halo at z ~ 17.55. The magnetic fields created by the Biermann Battery effect are predominantly formed early in the evolution of the primordial halo at low density and large spatial scales, and then grow through compression and by shear flows. The fields seen in this calculation are never large enough to be dynamically important (with beta >= 10^{15} at all times), and should be considered the minimum possible fields in existence during Population III star formation, and may be seed fields for the stellar dynamo or the magnetorotational instability at higher densities and smaller spatial scales.Comment: 4 pages, 3 figures; Submitted to the Astrophysical Journal Letters. Comments welcome. Typo found (and fixed) in equation

Effects of irrigation frequency on yields of winter wheat

Deliberately underirrigating a crop may yield economic benefits. The optimal use of water by an irrigator should be achieved under a deficit irrigation regime. It is important to know how water deficits affect yields and the interaction of the deficits with the scheduling of irrigations. A field experiment was conducted during the 1981 irrigation season to investigate the effects of high and low frequency deficit irrigation on yields of winter wheat. Yield and water use data were used to construct three production functions. The relationship between the level of water use and the resulting yield were determined for three irrigation frequency regimes. Field plots under daily (high frequency), weekly (normal frequency), and stress (reduced frequency) regimes were included in the field experiment as well as two dryland production plots. The relationships derived from this project were characterized by a large degree of scatter in the results. Highly favorable weather conditions offset the effects of irrigation deficits on plant yields throughout the irrigation season. At this time, a second year of data is in the process of compilation. The results of a regression analysis showed no statistically significant difference in the water use-yield relationships of the three irrigation frequency regimes. The effect of frequency did not lead to any readily apparent differences in the three production functions. The efficiency of water use of the different irrigation frequencies increases with decreasing amount of applied water for the 1981 crop year. The most efficient treatment, i.e., least water use per unit of yield, was the pre-plant irrigated, dryfarmed plots. The density of the wheat, a measure of crop quality also increased with decreased water use in this experiment. The optimal irrigation treatment (measured by production and quality) was the two-week frequency set

Formation of the Black Hole in Nova Scorpii

Israelian et al. (1999) showed that the stellar companion of the black-hole binary Nova Sco is polluted with material ejected in the supernova that accompanied the formation of the black-hole primary. Here we systematically investigate the implications of these observations for the black-hole formation process. Using a variety of supernova models, including both standard as well as hypernova models (for different helium-star masses, explosion energies, and explosion geometries) and a simple model for the evolution of the binary and the pollution of the secondary, we show that most of the observed abundance anomalies can be explained for a large range of model parameters (apart from the abundance of Ti). The best models are obtained for He star masses of 10 to 16 Msun, where spherical hypernova models are generally favoured over standard supernova ones. Aspherical hypernova models also produce acceptable fits, provided there is extensive lateral mixing. All models require substantial fallback and that the fallback material either reached the orbit of the secondary or was mixed efficiently with material that escaped. The black hole therefore formed in a two-step process, where the initial mass of the collapsed remnant was increased substantially by matter that fell back after the initial collapse. This may help to explain the high observed space velocity of Nova Sco either because of a neutrino-induced kick (if a neutron star was formed first) or by asymmetric mass ejection in an asymmetric supernova explosion.Comment: 16 pages, 3 Figures, 4 Tables. submitted to Ap

The basic ingredients of the North Atlantic storm track. Part I: land-sea contrast and orography

Understanding and predicting changes in storm tracks over longer time scales is a challenging problem, particularly in the North Atlantic. This is due in part to the complex range of forcings (land–sea contrast, orography, sea surface temperatures, etc.) that combine to produce the structure of the storm track. The impact of land–sea contrast and midlatitude orography on the North Atlantic storm track is investigated through a hierarchy of GCM simulations using idealized and “semirealistic” boundary conditions in a high-resolution version of the Hadley Centre atmosphere model (HadAM3). This framework captures the large-scale essence of features such as the North and South American continents, Eurasia, and the Rocky Mountains, enabling the results to be applied more directly to realistic modeling situations than was possible with previous idealized studies. The physical processes by which the forcing mechanisms impact the large-scale flow and the midlatitude storm tracks are discussed. The characteristics of the North American continent are found to be very important in generating the structure of the North Atlantic storm track. In particular, the southwest–northeast tilt in the upper tropospheric jet produced by southward deflection of the westerly flow incident on the Rocky Mountains leads to enhanced storm development along an axis close to that of the continent’s eastern coastline. The approximately triangular shape of North America also enables a cold pool of air to develop in the northeast, intensifying the surface temperature contrast across the eastern coastline, consistent with further enhancements of baroclinicity and storm growth along the same axis