Importance of the heavier singlet neutrinos in leptogenesis

We argue that fast interactions of the lightest singlet neutrino N1 would project part of a preexisting lepton asymmetry L{p} onto a direction that is protected from N1 washout effects, thus preventing it from being erased. In particular, we consider an asymmetry generated in N2 decays, assuming that N1 interactions are fast enough to bring N1 into full thermal equilibrium. If N1 decays occur at T > or = 10{9} GeV, that is, before the muon Yukawa interactions enter into thermal equilibrium, then generically part of L{p} survives. In this case some of the constraints implied by the standard N1 leptogenesis scenario hold only if L{p} approximately 0. For T < or = 10{9} GeV, L{p} is generally erased, unless special alignment or orthogonality conditions in flavor space are realized.     

A BKR Operation for Events Occurring for Disjoint Reasons with High Probability

Given events A and B on a product space \(S={\prod }_{i = 1}^{n} S_{i}\), the set \(A \Box B\) consists of all vectors x = (x₁,…,x_n) ∈ S for which there exist disjoint coordinate subsets K and L of {1,…,n} such that given the coordinates x_i,i ∈ K one has that x ∈ A regardless of the values of x on the remaining coordinates, and likewise that x ∈ B given the coordinates x_j,j ∈ L. For a finite product of discrete spaces endowed with a product measure, the BKR inequality

$$ P(A \Box B) \le P(A)P(B) $$

(1)

was conjectured by van den Berg and Kesten (J Appl Probab 22:556–569, 1985) and proved by Reimer (Combin Probab Comput 9:27–32, 2000). In Goldstein and Rinott (J Theor Probab 20:275–293, 2007) inequality Eq. 1 was extended to general product probability spaces, replacing \(A \Box B\) by the set Open image in new window consisting of those outcomes x for which one can only assure with probability one that x ∈ A and x ∈ B based only on the revealed coordinates in K and L as above. A strengthening of the original BKR inequality Eq. 1 results, due to the fact that Open image in new window . In particular, it may be the case that \(A \Box B\) is empty, while Open image in new window is not. We propose the further extension Open image in new window depending on probability thresholds s and t, where Open image in new window is the special case where both s and t take the value one. The outcomes Open image in new window are those for which disjoint sets of coordinates K and L exist such that given the values of x on the revealed set of coordinates K, the probability that A occurs is at least s, and given the coordinates of x in L, the probability of B is at least t. We provide simple examples that illustrate the utility of these extensions.

     

Strategies for Evaluating Boundary Fractal Dimensions by computer aided image analysis

For many profiles the values of the boundary fractals deduced by different exploration techniques are identical but the presence of deep fissures of convoluted structure can perturb the information gathered by the exploration technique. Data for different boundaries generated by different exploration techniques are used to illustrate the physical significance of the deduced fractal dimensions as evaluated using the different techniques. Heuristic programs for teaching an automated instrument to distinguish between different regions of ruggedness around a profile and to prevent “fractal rabbits” from appearing in the data are outlined. It is shown how the logic for removing fractal rabbits from the experimental data will also generate data on the presence of sharp edges on the profile. The possible occurrence of pseudo-texture fractal dimensions from projection occlusion is discussed.     

Self-assembly of nanoparticles into rings: a lattice-gas model

A coarse-grained lattice-gas model in three dimensions is developed to study the self-assembly of nanoparticles into micrometer-sized rings from a thin liquid film containing the nanoparticles. The model describes the nanoparticles as well as the solvent on length scales that are typical of the solvent bulk correlation length. Morphologies obtained from simulations of the model resemble recent experiments and provide a microscopic picture for the formation of nanoparticle rings. The role of evaporation rate, film thickness, diffusion rate, and nanoparticle coverage is discussed and compared to other continuum theories. Predictions of novel structures resulting from low nanoparticle mobility are analyzed.     

Linear differential operators related to the Jacobian Conjecture have a closed image

The main result of this work is the following theorem: LetP,QɛC[x, y] satisfy the Jacobian identity