Variations on Slavnov's scalar product

We consider the rational six-vertex model on an L-by-L lattice with domain wall boundary conditions and restrict N parallel-line rapidities, N < L/2, to satisfy length-L XXX spin-1/2 chain Bethe equations. We show that the partition function is an (L-2N)-parameter extension of Slavnov's scalar product of a Bethe eigenstate and a generic state, with N magnons each, on a length-L XXX spin-1/2 chain. Decoupling the extra parameters, we obtain a third determinant expression for the scalar product, where the first is due to Slavnov [1], and the second is due to Kostov and Matsuo [2]. We show that the new determinant is a discrete KP tau-function in the inhomogeneities, and consequently that tree-level N = 4 SYM structure constants that are known to be determinants, remain determinants at 1-loop level.Comment: 17 page

Hall-Littlewood plane partitions and KP

MacMahon's classic generating function of random plane partitions, which is related to Schur polynomials, was recently extended by Vuletic to a generating function of weighted plane partitions that is related to Hall-Littlewood polynomials, S(t), and further to one related to Macdonald polynomials, S(t,q). Using Jing's 1-parameter deformation of charged free fermions, we obtain a Fock space derivation of the Hall-Littlewood extension. Confining the plane partitions to a finite s-by-s square base, we show that the resulting generating function, S_{s-by-s}(t), is an evaluation of a tau-function of KP.Comment: 17 pages, minor changes, added a subsection and comments to clarify content, no changes made to conclusions, version to appear in IMR

On the Possibility of Quasi Small-World Nanomaterials

The possibility of materials that are governed by a fixed point related to small world networks is discussed. In particular, large-scale Monte Carlo simulations are performed on Ising ferromagnetic models on two different small-world networks generated from a one-dimensional spin chain. One has the small-world bond strengths independent of the length, and exhibits a finite-temperature phase transition. The other has small-world bonds built from atoms, and although there is no finite-temperature phase transition the system shows a slow power-law change of the effective critical temperature of a finite system as a function of the system size. An outline of a possible synthesis route for quasi small-world nanomaterials is presented.Comment: 13 pages, 9 figures, submitted to Brazilian Journal of Physics, conference proceedings for III Brazilian Meeting on Simulational Physics (2003

The shape of the urine stream — from biophysics to diagnostics

We develop a new computational model of capillary-waves in free-jet flows, and apply this to the problem of urological diagnosis in this first ever study of the biophysics behind the characteristic shape of the urine stream as it exits the urethral meatus. The computational fluid dynamics model is used to determine the shape of a liquid jet issuing from a non-axisymmetric orifice as it deforms under the action of surface tension. The computational results are verified with experimental modelling of the urine stream. We find that the shape of the stream can be used as an indicator of both the flow rate and orifice geometry. We performed volunteer trials which showed these fundamental correlations are also observed in vivo for male healthy volunteers and patients undergoing treatment for low flow rate. For healthy volunteers, self estimation of the flow shape provided an accurate estimation of peak flow rate (+-2%). However for the patients, the relationship between shape and flow rate suggested poor meatal opening during voiding. The results show that self measurement of the shape of the urine stream can be a useful diagnostic tool for medical practitioners since it provides a non-invasive method of measuring urine flow rate and urethral dilation

Refined Cauchy/Littlewood identities and six-vertex model partition functions: II. Proofs and new conjectures

We prove two identities of Hall-Littlewood polynomials, which appeared recently in a paper by two of the authors. We also conjecture, and in some cases prove, new identities which relate infinite sums of symmetric polynomials and partition functions associated with symmetry classes of alternating sign matrices. These identities generalize those already found in our earlier paper, via the introduction of additional parameters. The left hand side of each of our identities is a simple refinement of a relevant Cauchy or Littlewood identity. The right hand side of each identity is (one of the two factors present in) the partition function of the six-vertex model on a relevant domain.Comment: 34 pages, 14 figure

Aspherical Core-Collapse Supernovae in Red Supergiants Powered by Nonrelativistic Jets

We explore the observational characteristics of jet-driven supernovae by simulating bipolar-jet-driven explosions in a red supergiant progenitor. We present results of four models in which we hold the injected kinetic energy at a constant $10^{51}$ ergs across all jet models but vary the specific characteristics of the jets to explore the influence of the nature of jets on the structure of the supernova ejecta. We evolve the explosions past shock-breakout and into quasi-homologous expansion of the supernova envelope into a red supergiant wind. The oppositely-directed, nickel-rich jets give a large-scale asymmetry that may account for the non-spherical excitation and substructure of spectral lines such as H$\alpha$ and He I 10830\AA. Jets with a large fraction of kinetic to thermal energy punch through the progenitor envelope and give rise to explosions that would be observed to be asymmetric from the earliest epochs, inconsistent with spectropolarimetric measurements of Type II supernovae. Jets with higher thermal energy fractions result in explosions that are roughly spherical at large radii but are significantly elongated at smaller radii, deep inside the ejecta, in agreement with the polarimetric observations. We present shock breakout light curves that indicate that strongly aspherical shock breakouts are incompatible with recent {\it GALEX} observations of shock breakout from red supergiant stars. Comparison with observations indicates that jets must deposit their kinetic energy efficiently throughout the ejecta while in the hydrogen envelope. Thermal energy-dominated jets satisfy this criterion and yield many of the observational characteristics of Type II supernovae.Comment: 21 pages, 19 figures, submitted to ApJ on 4 Nov 200