A generalization of determinant formulas for the solutions of Painlev\'e II and XXXIV equations

A generalization of determinant formulas for the classical solutions of Painlev\'e XXXIV and Painlev\'e II equations are constructed using the technique of Darboux transformation and Hirota's bilinear formalism. It is shown that the solutions admit determinant formulas even for the transcendental case.Comment: 20 pages, LaTeX 2.09(IOP style), submitted to J. Phys.

Rational Solutions of the Painleve' VI Equation

In this paper, we classify all values of the parameters $\alpha$, $\beta$, $\gamma$ and $\delta$ of the Painlev\'e VI equation such that there are rational solutions. We give a formula for them up to the birational canonical transformations and the symmetries of the Painlev\'e VI equation.Comment: 13 pages, 1 Postscript figure Typos fixe

Imprint of Gravitational Lensing by Population III Stars in Gamma Ray Burst Light Curves

We propose a novel method to extract the imprint of gravitational lensing by Pop III stars in the light curves of Gamma Ray Bursts (GRBs). Significant portions of GRBs can originate in hypernovae of Pop III stars and be gravitationally lensed by foreground Pop III stars or their remnants. If the lens mass is on the order of $10^2-10^3M_\odot$ and the lens redshift is greater than 10, the time delay between two lensed images of a GRB is $\approx 1$s and the image separation is $\approx 10 \mu$as. Although it is difficult to resolve the two lensed images spatially with current facilities, the light curves of two images are superimposed with a delay of $\approx 1$ s. GRB light curves usually exhibit noticeable variability, where each spike is less than 1s. If a GRB is lensed, all spikes are superimposed with the same time delay. Hence, if the autocorrelation of light curve with changing time interval is calculated, it should show the resonance at the time delay of lensed images. Applying this autocorrelation method to GRB light curves which are archived as the {\it BATSE} catalogue, we demonstrate that more than half light curves can show the recognizable resonance, if they are lensed. Furthermore, in 1821 GRBs we actually find one candidate of GRB lensed by a Pop III star, which may be located at redshift 20-200. The present method is quite straightforward and therefore provides an effective tool to search for Pop III stars at redshift greater than 10. Using this method, we may find more candidates of GRBs lensed by Pop III stars in the data by the {\it Swift} satellite.Comment: 13 pages, 13 figures, accepted for publication in Ap

A Criterion for Photoionization of Pregalactic Clouds Exposed to Diffuse Ultraviolet Background Radiation

To elucidate the permeation of cosmic ultraviolet (UV) background radiation into a pregalactic cloud and subsequent ionization, the frequency-dependent radiative transfer equation is solved, coupled with the ionization process. A spherical top-hat cloud which is composed of pure hydrogen is assumed. As a result, it is shown that the self-shielding, although it is often disregarded in cosmological hydrodynamic simulations, could start to emerge shortly after the maximum expansion stages of density fluctuations. The present analysis reveals that the Stro"mgren approximation leads to overestimation of the photoionization effects. Also, the self-shielded neutral core is no longer sharply separated from surrounding ionized regions; a low but noticeable degree of ionization is caused by high energy photons even in the self-shielded core. The present results may be substantial on considering the biasing by photoionization against low-mass galaxy formation.Comment: 12 pages, Latex, 3 figures, To appear in ApJ, July 20, 1998 issue, email: [email protected]

Existence and Uniqueness of Tri-tronqu\'ee Solutions of the second Painlev\'e hierarchy

The first five classical Painlev\'e equations are known to have solutions described by divergent asymptotic power series near infinity. Here we prove that such solutions also exist for the infinite hierarchy of equations associated with the second Painlev\'e equation. Moreover we prove that these are unique in certain sectors near infinity.Comment: 13 pages, Late

Formation of Large-Scale Obscuring Wall and AGN Evolution Regulated by Circumnuclear Starbursts

By considering the radiative force by a circumnuclear starburst as well as an AGN, we analyze the equilibrium configuration and the stability of dusty gas in the circumnuclear regions. It is found that the radiative force by an intensive starburst can support a stable gaseous wall with a scale-height of several hundred parsecs. Moreover, by taking the simple stellar evolution in the starburst into account, we find that the covering factor of the wall decreases on a time-scale of several $10^7$ yr. The large-scale wall, if formed, works to obscure the nucleus due to the dust opacity. Hence, it is anticipated that the index of AGN type tends to shift from higher to lower in several $10^7$ yr according as the circumnuclear starburst becomes dimmer. On the other hand, if the AGN itself is brighter than the circumnuclear starburst (e.g. quasar case), no stable large-scale wall forms. In that case, the AGN is highly probably identified as type 1. The present mechanism may provide a physical explanation for the putative correlation between AGN type and host properties that Sy2's are more frequently associated with circumnuclear starbursts than Sy1's, whereas quasars are mostly observed as type 1 regardless of star-forming activity in the host galaxies.Comment: 10 pages, 3 figures, ApJ Letters in pres

Puffing-enhanced fuel/air mixing of an evaporating n-decane/ethanol emulsion droplet and a droplet group under convective heating

Pu ng of a decane/ethanol emulsion droplet and a droplet group under convective heating and its e ects on fuel/air mixing are investigated by direct numerical simulation (DNS) that resolves all the liquid/gas and liquid/liquid interfaces. With distinct di erences in the boiling point between decane and ethanol, the embedded ethanol subdroplets can be superheated and boil explosively. Pu ng, i.e. ejection of ethanol vapour, occurs from inside the parent decane droplet, causing secondary breakup of the droplet. The ejected ethanol vapour mixes with the outer gas mixture composed of air and vapour of the primary fuel decane, and its e ects on fuel/air mixing can be characterised by the scalar dissipation rates (SDRs). For the primary fuel SDR, the cross-scalar di usion due to ethanol vapour pu ng plays a dominant role in enhancing the micromixing. When the vapour ejection direction is inclined toward the wake direction, the wake is elongated, but the shape of the stoichiometric mixture fraction iso-surface is not changed much, indicating a limited e ect on droplet grouping in a spray. On the other hand, when the ejection direction is inclined toward the transverse direction, the stoichiometric surface is pushed further away in the transverse direction and its topology is changed by the pu ng. The trajectories of ejected ethanol vapour pockets can be predicted by the correlation obtained for a jet in cross ow, and the vapour pockets may reach a few diameters away from the droplet. Therefore, in a multiple-droplet con guration, the transverse ethanol vapour ejection due to pu ng may transiently change the droplet grouping characteristics. In simulation cases with multiple droplets, the interaction changing the droplet grouping due to pu ng has been con rmed, especially for droplets in the mostupstream position in a spray. This implies that pu ng should be accurately included in the mixing and combustion modelling of such a biofuel-blended diesel spray process.Financial support from the Engineering and Physical Sciences Research Council (EPSRC), grant No. EP/J018023/