Nonlinear desingularization in certain free-boundary problems

We consider a nonlinear, elliptic, free-boundary problem involving an initially unknown setA that represents, for example, the cross-section of a steady vortex ring or of a confined plasma in equilibrium. The solutions are characterized by a variational principle which allows us to describe their behaviour under a limiting process such that the diameter ofA tends to zero, while the solutions degenerate to the solution of a related linear problem. This limiting solution is the sum of the Green function of the linear operator and of a smooth function satisfying the boundary conditions. Mathematically speaking, this limiting process, that we call nonlinear desingularization, is a novel kind of bifurcation phenomenon since the nonlinear effect here involves smoothing the singularity of the associated linear problem.Research partially supported by A FOSR and NSF grants     

The vile,dopey, evil and odious game players

Many of my friends celebrate anniversaries these days, be it 50th birthday, or 60th, 70th, 80th, even 90th. It is to be deplored that they all accumulate together, rather than distribute themselves uniformly over my lifetime! It is now Gert Sabidussi’s turn, who just joined the octogenarians club, and it is a pleasure to dedicate a paper to him, since he did, and continues to do, excellent algebraic graph theory, including insights into the automorphism group of graphs, studies of stable graphs, Sabidussi representation theorems for symmetric graphs, Sabidussi’s compatibility conjecture, Sabidussi graphs, etc. Many leading mathematicians throughout the world are working on problems and insights initiated by Gert. He does all this in a relaxed playful way, as I witnessed when I acquired my own Sabidussi number 1. Hence it is only natural to relate here the fun that Gert and his friend Wil had while playing games in Dubrovnik, where a grand conference took place in 2009 honoring Gert. Unfortunately, I had to skip that conference, but, unknown to them, I planted a listening device.     

Transfer reactions induced by 17, 18O on 12, 13C at c.m. energies between 12.6 TO 14.0 MeV

Angular distributions for one-neutron and many-nucleon transfer reactions were measured in the systems ^{17, 18}O and ^{12, 13}C at c.m. energies between 12.6 to 14.0 MeV. All the cross sections were analyzed in terms of the full-recoil finite-range DWBA model. For multi-nucleon transfer processes an inert cluster transfer was assumed. The sensitivity of DWBA calculations to various parameter sets is discussed, and the effect of the no-recoil approximation on the phase and magnitude of the transfer amplitude is studied. For one-neutron transfer reactions the forward part of the angular distributions was reproduced reasonably well by the DWBA model, yielding satisfactory spectroscopic information. The rise of the cross section at the backward angles was not reproduced by the DWBA model. Possible contributions of cluster-exchange and compound-nucleus reactions are discussed. For multi-nucleon transfer reactions poorer fits were obtained, particularly in the ¹⁸O+¹²C system.     

NMR studies of structure and reorganization dynamics of an [6Li]-allylic lithium compound complexed to [14N,15N]-N,N,N',N'-tetramethylethylenediamine: inversion and ligand lithium exchange

Proton, 13C, 6Li, and 15N NMR line-shape studies of exo,exo-1-trimethylsilyl-3-(dimethylethylsilyl)allyllithium-6Li complexed to [14N,15N]-N,N,N',N'-tetramethylethylenediamine (TMEDA) 2 as a function of temperature and of added diamine reveal the dynamics of three fast equilibrium reorganization processes. These are (with DeltaH values in kilocalories per mole and DeltaS values in entropic units): mutual exchange of lithium between two 2 molecules (6.3, -21), exchange of TMEDA between its free and complexed states (5.0 and -22), and first-order transfer of complexed ligand between the allyl faces (7.0 and -20). Intermediates that are dimeric in TMEDA are proposed for the first two of these reorganization processes.     

Universal pion freeze-out in heavy-ion collisions

Based on an evaluation of data on pion interferometry and on particle yields at midrapidity, we propose a universal condition for thermal freeze-out of pions in heavy-ion collisions. We show that freeze-out occurs when the mean free path of pions lambda(f) reaches a value of about 1 fm, which is much smaller than the spatial extent of the system at freeze-out. This critical mean free path is independent of the centrality of the collision and beam energy from the Alternating Gradient Synchrotron to the Relativistic Heavy Ion Collider.     

Measurement of the energy penetration depth into solid targets irradiated by ultrashort laser pulses

The energy penetration depth of a short (100 fs) Ti-sapphire laser pulse (0.8 &mgr;m) of intensity 3x10(16) W/cm(2), in solid density materials has been measured. High-Z (BaF2) and low-Z (MgF2) solid layers targets were used. The penetration depth was determined from the measurement of the x-ray emission spectra, as a function of the target thickness. The investigation of these spectra showed that in the low-Z case, solid density material to a depth of 50 nm was heated to a peak electron temperature of approximately 150 eV. For the high-Z material, the penetration depth corresponding to this temperature exceeded 100 nm. This is evidence of a larger heat penetration depth in a high-Z material in comparison to a low-Z material. A model based on electron heat conduction is used to estimate the energy penetration depth. It is suggested that the larger heat penetration in high-Z material is due to heating of the material, caused by the radiation flux, generated by the electron heat conduction.     

A non-analytic proof of the newman—znám result for disjoint covering systems

A direct combinatorial proof is given to a generalization of the fact that the largest modulusN of a disjoint covering system appears at leastp times in the system, wherep is the smallest prime dividingN. The method is based on geometric properties of lattice parallelotopes. This research was supported by grant 85-00368 from the United States-Is rael Binational Science Foundation, Jerusalem, Israel.