We introduce a family of rings of symmetric functions depending on an infinite sequence of parameters. A distinguished basis of such a ring is comprised by analogues of the Schur functions. The corresponding structure coefficients are polynomials in
the parameters which we call the Littlewood-Richardson polynomials. We give a combinatorial rule for their calculation by modifying an earlier result of B. Sagan and the author. The new rule provides a formula for these polynomials which is manifestly positive in the sense of W. Graham. We apply this formula for the calculation of the product of equivariant Schubert classes on Grassmannians which implies a stability property of the structure coefficients. The first manifestly positive formula for such an expansion was given by A. Knutson and T. Tao by using combinatorics of puzzles while the stability property was not apparent from that formula. We also use the Littlewood-Richardson polynomials to describe the multiplication rule in the algebra of the Casimir elements for the general linear Lie algebra in the basis of the quantum immanants constructed by A. Okounkov and G. Olshanski.

We show that the globular cluster mass function (GCMF) in the Milky Way depends on cluster half-mass density (rho_h) in the sense that the turnover mass M_TO increases with rho_h while the width of the GCMF decreases. We argue that this is the expect
ed signature of the slow erosion of a mass function that initially rose towards low masses, predominantly through cluster evaporation driven by internal two-body relaxation. We find excellent agreement between the observed GCMF -- including its dependence on internal density rho_h, central concentration c, and Galactocentric distance r_gc -- and a simple model in which the relaxation-driven mass-loss rates of clusters are approximated by -dM/dt = mu_ev ~ rho_h^{1/2}. In particular, we recover the well-known insensitivity of M_TO to r_gc. This feature does not derive from a literal ``universality of the GCMF turnover mass, but rather from a significant variation of M_TO with rho_h -- the expected outcome of relaxation-driven cluster disruption -- plus significant scatter in rho_h as a function of r_gc. Our conclusions are the same if the evaporation rates are assumed to depend instead on the mean volume or surface densities of clusters inside their tidal radii, as mu_ev ~ rho_t^{1/2} or mu_ev ~ Sigma_t^{3/4} -- alternative prescriptions that are physically motivated but involve cluster properties (rho_t and Sigma_t) that are not as well defined or as readily observable as rho_h. In all cases, the normalization of mu_ev required to fit the GCMF implies cluster lifetimes that are within the range of standard values (although falling towards the low end of this range). Our analysis does not depend on any assumptions or information about velocity anisotropy in the globular cluster system.

In this work, we evaluate the lifetimes of the doubly charmed baryons $Xi_{cc}^{+}$, $Xi_{cc}^{++}$ and $Omega_{cc}^{+}$. We carefully calculate the non-spectator contributions at the quark level where the Cabibbo-suppressed diagrams are also include
d. The hadronic matrix elements are evaluated in the simple non-relativistic harmonic oscillator model. Our numerical results are generally consistent with that obtained by other authors who used the diquark model. However, all the theoretical predictions on the lifetimes are one order larger than the upper limit set by the recent SELEX measurement. This discrepancy would be clarified by the future experiment, if more accurate experiment still confirms the value of the SELEX collaboration, there must be some unknown mechanism to be explored.

We discuss a universality property of any covariant field theory in space-time expanded around pp-wave backgrounds. According to this property the space-time lagrangian density evaluated on a restricted set of field configurations, called universal s
ector, turns out to be same around all the pp-waves, even off-shell, with same transverse space and same profiles for the background scalars. In this paper we restrict our discussion to tensorial fields only. In the context of bosonic string theory we consider on-shell pp-waves and argue that universality requires the existence of a universal sector of world-sheet operators whose correlation functions are insensitive to the pp-wave nature of the metric and the background gauge flux. Such results can also be reproduced using the world-sheet conformal field theory. We also study such pp-waves in non-polynomial closed string field theory (CSFT). In particular, we argue that for an off-shell pp-wave ansatz with flat transverse space and dilaton independent of transverse coordinates the field redefinition relating the low energy effective field theory and CSFT with all the massive modes integrated out is at most quadratic in fields. Because of this simplification it is expected that the off-shell pp-waves can be identified on the two sides. Furthermore, given the massless pp-wave field configurations, an iterative method for computing the higher massive modes using the CSFT equations of motion has been discussed. All our bosonic string theory analyses can be generalised to the common Neveu-Schwarz sector of superstrings.

In present paper we propose seemingly new method for finding solutions of some types of nonlinear PDEs in closed form. The method is based on decomposition of nonlinear operators on sequence of operators of lower orders. It is shown that decompositio
n process can be done by iterative procedure(s), each step of which is reduced to solution of some auxiliary PDEs system(s) for one dependent variable. Moreover, we find on this way the explicit expression of the first-order PDE(s) for first integral of decomposable initial PDE. Remarkably that this first-order PDE is linear if initial PDE is linear in its highest derivatives. The developed method is implemented in Maple procedure, which can really solve many of different order PDEs with different number of independent variables. Examples of PDEs with calculated their general solutions demonstrate a potential of the method for automatic solving of nonlinear PDEs.

We give a prescription for how to compute the Callias index, using as regulator an exponential function. We find agreement with old results in all odd dimensions. We show that the problem of computing the dimension of the moduli space of self-dual st
rings can be formulated as an index problem in even-dimensional (loop-)space. We think that the regulator used in this Letter can be applied to this index problem.

In a quantum mechanical model, Diosi, Feldmann and Kosloff arrived at a conjecture stating that the limit of the entropy of certain mixtures is the relative entropy as system size goes to infinity. The conjecture is proven in this paper for density m
atrices. The first proof is analytic and uses the quantum law of large numbers. The second one clarifies the relation to channel capacity per unit cost for classical-quantum channels. Both proofs lead to generalization of the conjecture.

The shape of the hadronic form factor f+(q2) in the decay D0 --> K- e+ nue has been measured in a model independent analysis and compared with theoretical calculations. We use 75 fb(-1) of data recorded by the BABAR detector at the PEPII electron-pos
itron collider. The corresponding decay branching fraction, relative to the decay D0 --> K- pi+, has also been measured to be RD = BR(D0 --> K- e+ nue)/BR(D0 --> K- pi+) = 0.927 +/- 0.007 +/- 0.012. From these results, and using the present world average value for BR(D0 --> K- pi+), the normalization of the form factor at q2=0 is determined to be f+(0)=0.727 +/- 0.007 +/- 0.005 +/- 0.007 where the uncertainties are statistical, systematic, and from external inputs, respectively.

I shall present three arguments for the proposition that intelligent life is very rare in the universe. First, I shall summarize the consensus opinion of the founders of the Modern Synthesis (Simpson, Dobzhanski, and Mayr) that the evolution of intel
ligent life is exceedingly improbable. Second, I shall develop the Fermi Paradox: if they existed theyd be here. Third, I shall show that if intelligent life were too common, it would use up all available resources and die out. But I shall show that the quantum mechanical principle of unitarity (actually a form of teleology!) requires intelligent life to survive to the end of time. Finally, I shall argue that, if the universe is indeed accelerating, then survival to the end of time requires that intelligent life, though rare, to have evolved several times in the visible universe. I shall argue that the acceleration is a consequence of the excess of matter over antimatter in the universe. I shall suggest experiments to test these claims.

We present semi-analytical constraint on the amount of dark matter in the merging bullet galaxy cluster using the classical Local Group timing arguments. We consider particle orbits in potential models which fit the lensing data. {it Marginally consi
stent} CDM models in Newtonian gravity are found with a total mass M_{CDM} = 1 x 10^{15}Msun of Cold DM: the bullet subhalo can move with V_{DM}=3000km/s, and the bullet X-ray gas can move with V_{gas}=4200km/s. These are nearly the {it maximum speeds} that are accelerable by the gravity of two truncated CDM halos in a Hubble time even without the ram pressure. Consistency breaks down if one adopts higher end of the error bars for the bullet gas speed (5000-5400km/s), and the bullet gas would not be bound by the sub-cluster halo for the Hubble time. Models with V_{DM}~ 4500km/s ~ V_{gas} would invoke unrealistic large amount M_{CDM}=7x 10^{15}Msun of CDM for a cluster containing only ~ 10^{14}Msun of gas. Our results are generalisable beyond General Relativity, e.g., a speed of $4500kms$ is easily obtained in the relativistic MONDian lensing model of Angus et al. (2007). However, MONDian model with little hot dark matter $M_{HDM} le 0.6times 10^{15}msun$ and CDM model with a small halo mass $le 1times 10^{15}msun$ are barely consistent with lensing and velocity data.