Non-isolated Hypersurface Singularities and L\^e Cycles

In this series of lectures, I will discuss results for complex hypersurfaces with non-isolated singularities. In Lecture 1, I will review basic definitions and results on complex hypersurfaces, and then present classical material on the Milnor fiber and fibration. In Lecture 2, I will present basic results from Morse theory, and use them to prove some results about complex hypersurfaces, including a proof of L\^e's attaching result for Milnor fibers of non-isolated hypersurface singularities. This will include defining the relative polar curve. Lecture 3 will begin with a discussion of intersection cycles for proper intersections inside a complex manifold, and then move on to definitions and basic results on L\^e cycles and L\^e numbers of non-isolated hypersurface singularities. Lecture 4 will explain the topological importance of L\^e cycles and numbers, and then I will explain, informally, the relationship between the L\^e cycles and the complex of sheaves of vanishing cycles.Comment: Notes from a series of lectures from the S\~ao Carlos singularities meeting of 2014. Revision made to Exercise 3.1 (a

The Nucleon-Nucleon Potential in the 1/N_c Expansion

The nucleon-nucleon potential is analysed using the 1/N_c expansion of QCD. The NN potential is shown to have an expansion in 1/N_c^2, and the strengths of the leading order central, spin-orbit, tensor, and quadratic spin-orbit forces (including isospin dependence) are determined. Comparison with a successful phenomenological potential (Nijmegen) shows that the large-N_c analysis explains many of the qualitative features observed in the nucleon-nucleon interaction. The 1/N_c expansion implies an effective Wigner supermultiplet symmetry for light nuclei. Results for baryons containing strange quarks are presented in an appendix.Comment: 17 pages, 3 figures, TeX, macros harvmac and eps

Interfering directed paths and the sign phase transition

We revisit the question of the "sign phase transition" for interfering directed paths with real amplitudes in a random medium. The sign of the total amplitude of the paths to a given point may be viewed as an Ising order parameter, so we suggest that a coarse-grained theory for system is a dynamic Ising model coupled to a Kardar-Parisi-Zhang (KPZ) model. It appears that when the KPZ model is in its strong-coupling ("pinned") phase, the Ising model does not have a stable ferromagnetic phase, so there is no sign phase transition. We investigate this numerically for the case of {\ss}1+1 dimensions, demonstrating the instability of the Ising ordered phase there.Comment: 4 pages, 4 figure

Accelerating Universes with Scaling Dark Matter

Friedmann-Robertson-Walker universes with a presently large fraction of the energy density stored in an $X$-component with $w_X<-1/3$, are considered. We find all the critical points of the system for constant equations of state in that range. We consider further several background quantities that can distinguish the models with different $w_X$ values. Using a simple toy model with a varying equation of state, we show that even a large variation of $w_X$ at small redshifts is very difficult to observe with $d_L(z)$ measurements up to $z\sim 1$. Therefore, it will require accurate measurements in the range $1<z<2$ and independent accurate knowledge of $\Omega_{m,0}$ (and/or $\Omega_{X,0}$) in order to resolve a variable $w_X$ from a constant $w_X$.Comment: submitted to IJMPD (uses Latex, 12 pages, 6 Figures) Minor corrections, Figures 4, 6 revised. Conclusions unchange

A Comparison of Absorption and Emission Line Abundances in the Nearby Damped Lyman-alpha Galaxy SBS 1543+593

We have used the Space Telescope Imaging Spectrograph (STIS) aboard HST to measure a sulfur abundance of [S/H] = -0.41 +/-0.06 in the interstellar medium (ISM) of the nearby damped Lyman-alpha (DLA) absorbing galaxy SBS 1543+593. A direct comparison between this QSO absorption line abundance on the one hand, and abundances measured from HII region emission line diagnostics on the other, yield the same result: the abundance of sulfur in the neutral ISM is in good agreement with that of oxygen measured in an HII region 3 kpc away. Our result contrasts with those of other recent studies which have claimed order-of-magnitude differences between HI (absorption) and HII (emission) region abundances. We also derive a nickel abundance of [Ni/H] < -0.81, some three times less than that of sulfur, and suggest that the depletion is due to dust, although we cannot rule out an over-abundance of alpha-elements as the cause of the lower metallicity. It is possible that our measure of [S/H] is over-estimated if some SII arises in ionized gas; adopting a plausible star formation rate for the galaxy along the line of sight, and a measurement of the CII* 1335.7 absorption line detected from SBS 1543+593, we determine that the metallicity is unlikely to be smaller than we derive by more than 0.25 dex. We estimate that the cooling rate of the cool neutral medium is log [l_c (ergs s^{-1} H atom^{-1})] = -27.0, the same value as that seen in the high redshift DLA population.Comment: 31 pages; accepted for publication in the Ap

Observable effects caused by vacuum pair creation in the field of high-power optical lasers

We consider the possibility of an experimental proof of vacuum e+e- pair creation in the focus of two counter-propagating optical laser beams with an intensity of the order of 10^20 - 10^22 W/cm^2. Our approach is based on the collisionless kinetic equation for the distribution function of the e+e- pairs with the source term for particle production. As a possible experimental signal of vacuum pair production we consider the refraction of a high-frequency probe laser beam by the produced e+e- plasma to be observed by an interference filter. The generation of higher harmonics of the laser frequency in the self-consistent electric field is also investigated.Comment: 7 pages, 7 figures; typos corrected, Eq.(16) corrected, reference adde

Explanation for Anomalous Shock Temperatures Measured by Neutron Resonance Spectroscopy

Neutron resonance spectrometry (NRS) has been used to measure the temperature inside Mo samples during shock loading. The temperatures obtained were significantly higher than predicted assuming ideal hydrodynamic loading. The effect of plastic flow and non-ideal projectile behavior were assessed. Plastic flow was calculated self-consistently with the shock jump conditions: this is necessary for a rigorous estimate of the locus of shock states accessible. Plastic flow was estimated to contribute a temperature rise of 53K compared with hydrodynamic flow. Simulations were performed of the operation of the explosively-driven projectile system used to induce the shock in the Mo sample. The simulations predicted that the projectile was significantly curved on impact, and still accelerating. The resulting spatial variations in load, including radial components of velocity, were predicted to increase the apparent temperature that would be deduced from the width of the neutron resonance by 160K. These corrections are sufficient to reconcile the apparent temperatures deduced using NRS with the accepted properties of Mo, in particular its equation of state.Comment: near-final version, waiting for final consent from an autho