Jack superpolynomials with negative fractional parameter: clustering properties and super-Virasoro ideals

The Jack polynomials P_\lambda^{(\alpha)} at \alpha=-(k+1)/(r-1) indexed by certain (k,r,N)-admissible partitions are known to span an ideal I^{(k,r)}_N of the space of symmetric functions in N variables. The ideal I^{(k,r)}_N is invariant under the action of certain differential operators which include half the Virasoro algebra. Moreover, the Jack polynomials in I^{(k,r)}_N admit clusters of size at most k: they vanish when k+1 of their variables are identified, and they do not vanish when only k of them are identified. We generalize most of these properties to superspace using orthogonal eigenfunctions of the supersymmetric extension of the trigonometric Calogero-Moser-Sutherland model known as Jack superpolynomials. In particular, we show that the Jack superpolynomials P_{\Lambda}^{(\alpha)} at \alpha=-(k+1)/(r-1) indexed by certain (k,r,N)-admissible superpartitions span an ideal {\mathcal I}^{(k,r)}_N of the space of symmetric polynomials in N commuting variables and N anticommuting variables. We prove that the ideal {\mathcal I}^{(k,r)}_N is stable with respect to the action of the negative-half of the super-Virasoro algebra. In addition, we show that the Jack superpolynomials in {\mathcal I}^{(k,r)}_N vanish when k+1 of their commuting variables are equal, and conjecture that they do not vanish when only k of them are identified. This allows us to conclude that the standard Jack polynomials with prescribed symmetry should satisfy similar clustering properties. Finally, we conjecture that the elements of {\mathcal I}^{(k,2)}_N provide a basis for the subspace of symmetric superpolynomials in N variables that vanish when k+1 commuting variables are set equal to each other.Comment: 36 pages; the main changes in v2 are : 1) in the introduction, we present exceptions to an often made statement concerning the clustering property of the ordinary Jack polynomials for (k,r,N)-admissible partitions (see Footnote 2); 2) Conjecture 14 is substantiated with the extensive computational evidence presented in the new appendix C; 3) the various tests supporting Conjecture 16 are reporte

IMECE2005-81300 EVALUATION OF HEAVY TRUCK ROLLOVER CRASHWORTHINESS

ABSTRACT Heavy trucks (those having a gross vehicle weight rating of greater than 10,000 pounds) are an essential part of the United States economy and account for 4% of all registered vehicles. The large size and weight of these vehicles can pose a serious safety threat to the vehicle's occupants in the event of a rollover collision. The rollover crashworthiness of heavy trucks, in particular the structural integrity of the cab, is analyzed in this paper. An actual rollover accident was analyzed and the cab design of an exemplar vehicle was evaluated. Modifications were made to the exemplar and an inverted drop test onto the roof of the cab was conducted. Recommendations for improving the rollover crashworthiness of heavy trucks are provided. An analysis of heavy truck rollover accidents was also conducted for data available from 1994-2002 by submitting queries to the Fatality Analysis Reporting System (FARS), which is administered by the National Highway Safety Administration (NHTSA), in order to determine the number of incapacitating and fatal injuries that occurred when the occupants were contained in the cab during a rollover accident. The percentage of incapacitating and fatal injuries for restrained occupants was determined by analyzing the rollover data obtained from the FARS rollover query that was used and was found to be 35%. Therefore, restrained occupants in heavy trucks can sustain significant injuries during rollover accidents, in part, due to insufficient rollover crashworthiness

On the Mixing of Diffusing Particles

We study how the order of N independent random walks in one dimension evolves with time. Our focus is statistical properties of the inversion number m, defined as the number of pairs that are out of sort with respect to the initial configuration. In the steady-state, the distribution of the inversion number is Gaussian with the average ~N^2/4 and the standard deviation sigma N^{3/2}/6. The survival probability, S_m(t), which measures the likelihood that the inversion number remains below m until time t, decays algebraically in the long-time limit, S_m t^{-beta_m}. Interestingly, there is a spectrum of N(N-1)/2 distinct exponents beta_m(N). We also find that the kinetics of first-passage in a circular cone provides a good approximation for these exponents. When N is large, the first-passage exponents are a universal function of a single scaling variable, beta_m(N)--> beta(z) with z=(m-)/sigma. In the cone approximation, the scaling function is a root of a transcendental equation involving the parabolic cylinder equation, D_{2 beta}(-z)=0, and surprisingly, numerical simulations show this prediction to be exact.Comment: 9 pages, 6 figures, 2 table

Random walk generated by random permutations of {1,2,3, ..., n+1}

We study properties of a non-Markovian random walk $X^{(n)}_l$, $l =0,1,2, >...,n$, evolving in discrete time $l$ on a one-dimensional lattice of integers, whose moves to the right or to the left are prescribed by the \text{rise-and-descent} sequences characterizing random permutations $\pi$ of $[n+1] = \{1,2,3, ...,n+1\}$. We determine exactly the probability of finding the end-point $X_n = X^{(n)}_n$ of the trajectory of such a permutation-generated random walk (PGRW) at site $X$, and show that in the limit $n \to \infty$ it converges to a normal distribution with a smaller, compared to the conventional P\'olya random walk, diffusion coefficient. We formulate, as well, an auxiliary stochastic process whose distribution is identic to the distribution of the intermediate points $X^{(n)}_l$, $l < n$, which enables us to obtain the probability measure of different excursions and to define the asymptotic distribution of the number of "turns" of the PGRW trajectories.Comment: text shortened, new results added, appearing in J. Phys.

Numerical Estimation of the Asymptotic Behaviour of Solid Partitions of an Integer

The number of solid partitions of a positive integer is an unsolved problem in combinatorial number theory. In this paper, solid partitions are studied numerically by the method of exact enumeration for integers up to 50 and by Monte Carlo simulations using Wang-Landau sampling method for integers up to 8000. It is shown that, for large n, ln[p(n)]/n^(3/4) = 1.79 \pm 0.01, where p(n) is the number of solid partitions of the integer n. This result strongly suggests that the MacMahon conjecture for solid partitions, though not exact, could still give the correct leading asymptotic behaviour.Comment: 6 pages, 4 figures, revtex

A novel protamine variant reversal of heparin anticoagulation in human blood in vitro

AbstractPurpose: Protamine reversal of heparin anticoagulation during cardiovascular surgery may cause severe hypotension and pulmonary hypertension. A novel protamine variant, [+18RGD], has been developed that effectively reverses heparin anticoagulation without toxicity in canine experiments. Heretofore, human studies have not been undertaken. This investigation hypothesized that [+18RGD] would effectively reverse heparin anticoagulation of human blood in vitro. Methods: Fifty patients who underwent anticoagulation therapy during vascular surgery had blood sampled at baseline and 30 minutes after receiving heparin (150 IU/kg). Activated clotting times were used to define specific quantities of [+18RGD] or protamine necessary to completely reverse heparin anticoagulation in the blood sample of each patient. These defined amounts of [+18RGD] or protamine were then administered to the heparinized blood samples, and percent reversals of activated partial thromboplastin time, thrombin clotting time, and antifactor Xa/IIa levels were determined. In addition, platelet aggregation assays, as well as platelet and white blood cell counts were performed. Results: [+18RGD] and protamine were equivalent in reversing heparin as assessed by thrombin clotting time, antifactor Xa, antifactor IIa levels, and white blood cell changes. [+18RGD], when compared with protamine, was superior in this regard, as assessed by activated partial thromboplastin time (94.5 ± 1.0 vs 86.5 ± 1.3%δ, respectively; p < 0.001) and platelet declines (–3.9 ± 2.9 vs –12.8 ± 3.4 per mm3, respectively; p = 0.048). Platelet aggregation was also decreased for [+18RGD] compared with protamine (23.6 ± 1.5 vs 28.5 ± 1.9%, respectively; p = 0.048). Conclusions: [+18RGD] was as effective as protamine for in vitro reversal of heparin anticoagulation by most coagulation assays, was statistically more effective at reversal than protamine by aPTT assay, and was associated with lesser platelet reductions than protamine. [+18RGD], if less toxic than protamine in human beings, would allow for effective clinical reversal of heparin anticoagulation. (J Vasc Surg 1997;26:1043-8.

Equilibria of `Discrete' Integrable Systems and Deformations of Classical Orthogonal Polynomials

The Ruijsenaars-Schneider systems are `discrete' version of the Calogero-Moser (C-M) systems in the sense that the momentum operator p appears in the Hamiltonians as a polynomial in e^{\pm\beta' p} (\beta' is a deformation parameter) instead of an ordinary polynomial in p in the hierarchies of C-M systems. We determine the polynomials describing the equilibrium positions of the rational and trigonometric Ruijsenaars-Schneider systems based on classical root systems. These are deformation of the classical orthogonal polynomials, the Hermite, Laguerre and Jacobi polynomials which describe the equilibrium positions of the corresponding Calogero and Sutherland systems. The orthogonality of the original polynomials is inherited by the deformed ones which satisfy three-term recurrence and certain functional equations. The latter reduce to the celebrated second order differential equations satisfied by the classical orthogonal polynomials.Comment: 45 pages. A few typos in section 6 are correcte

On the asymptotics of higher-dimensional partitions

We conjecture that the asymptotic behavior of the numbers of solid (three-dimensional) partitions is identical to the asymptotics of the three-dimensional MacMahon numbers. Evidence is provided by an exact enumeration of solid partitions of all integers <=68 whose numbers are reproduced with surprising accuracy using the asymptotic formula (with one free parameter) and better accuracy on increasing the number of free parameters. We also conjecture that similar behavior holds for higher-dimensional partitions and provide some preliminary evidence for four and five-dimensional partitions.Comment: 30 pages, 8 tables, 4 figures (v2) New data (63-68) for solid partitions added; (v3) published version, new subsection providing an unbiased estimate of the leading for the leading coefficient added, some tables delete

Fluctuations of the one-dimensional asymmetric exclusion process using random matrix techniques

The studies of fluctuations of the one-dimensional Kardar-Parisi-Zhang universality class using the techniques from random matrix theory are reviewed from the point of view of the asymmetric simple exclusion process. We explain the basics of random matrix techniques, the connections to the polynuclear growth models and a method using the Green's function.Comment: 41 pages, 10 figures, minor corrections, references adde

Nonequilibrium Steady States of Matrix Product Form: A Solver's Guide

We consider the general problem of determining the steady state of stochastic nonequilibrium systems such as those that have been used to model (among other things) biological transport and traffic flow. We begin with a broad overview of this class of driven diffusive systems - which includes exclusion processes - focusing on interesting physical properties, such as shocks and phase transitions. We then turn our attention specifically to those models for which the exact distribution of microstates in the steady state can be expressed in a matrix product form. In addition to a gentle introduction to this matrix product approach, how it works and how it relates to similar constructions that arise in other physical contexts, we present a unified, pedagogical account of the various means by which the statistical mechanical calculations of macroscopic physical quantities are actually performed. We also review a number of more advanced topics, including nonequilibrium free energy functionals, the classification of exclusion processes involving multiple particle species, existence proofs of a matrix product state for a given model and more complicated variants of the matrix product state that allow various types of parallel dynamics to be handled. We conclude with a brief discussion of open problems for future research.Comment: 127 pages, 31 figures, invited topical review for J. Phys. A (uses IOP class file