On formation of long-living states

The motion of a particle in the potential well is studied when the particle is attached to the infinite elastic string. This is generic with the problem of dissipative quantum mechanics investigated by Caldeira and Leggett. Besides the dissipative motion there is another scenario of interaction of the string with the particle attached. Stationary particle-string states exist with string deformations accompanying the particle. This is like polaronic states in solids. Our polaronic states in the well are non-decaying and with continuous energy spectrum. Perhaps these states have a link to quantum electrodynamics. Quantum mechanical wave function, singular on some line, is smeared out by electron "vibrations" due to the interaction with photons. In those anomalous states the smeared singularity position would be analogous to the place where the particle is attached to the string

The model of neutrino vacuum flavour oscillations and quantum mechanics

It is shown that the model of vacuum flavour oscillations is in disagreement with quantum mechanics theorems and postulates. Features of the model are analyzed. It is noted that apart from the number of mixed mass states neutrino oscillations are forbidden by Fock-Krylov theorem. A possible reason of oscillation model inadequacy is discussed.Comment: 12 page

How to infer relative fitness from a sample of genomic sequences

Mounting evidence suggests that natural populations can harbor extensive fitness diversity with numerous genomic loci under selection. It is also known that genealogical trees for populations under selection are quantifiably different from those expected under neutral evolution and described statistically by Kingman's coalescent. While differences in the statistical structure of genealogies have long been used as a test for the presence of selection, the full extent of the information that they contain has not been exploited. Here we shall demonstrate that the shape of the reconstructed genealogical tree for a moderately large number of random genomic samples taken from a fitness diverse, but otherwise unstructured asexual population can be used to predict the relative fitness of individuals within the sample. To achieve this we define a heuristic algorithm, which we test in silico using simulations of a Wright-Fisher model for a realistic range of mutation rates and selection strength. Our inferred fitness ranking is based on a linear discriminator which identifies rapidly coalescing lineages in the reconstructed tree. Inferred fitness ranking correlates strongly with actual fitness, with a genome in the top 10% ranked being in the top 20% fittest with false discovery rate of 0.1-0.3 depending on the mutation/selection parameters. The ranking also enables to predict the genotypes that future populations inherit from the present one. While the inference accuracy increases monotonically with sample size, samples of 200 nearly saturate the performance. We propose that our approach can be used for inferring relative fitness of genomes obtained in single-cell sequencing of tumors and in monitoring viral outbreaks

On the modulus of continuity for spectral measures in substitution dynamics

The paper gives first quantitative estimates on the modulus of continuity of the spectral measure for weak mixing suspension flows over substitution automorphisms, which yield information about the "fractal" structure of these measures. The main results are, first, a Hoelder estimate for the spectral measure of almost all suspension flows with a piecewise constant roof function; second, a log-Hoelder estimate for self-similar suspension flows; and, third, a Hoelder asymptotic expansion of the spectral measure at zero for such flows. Our second result implies log-Hoelder estimates for the spectral measures of translation flows along stable foliations of pseudo-Anosov automorphisms. A key technical tool in the proof of the second result is an "arithmetic-Diophantine" proposition, which has other applications. In the appendix this proposition is used to derive new decay estimates for the Fourier transforms of Bernoulli convolutions.Comment: 42 pages, accepted version; to appear in Advances in Mathematic