We provide a unified approach for constructing Wick words in mixed q-Gaussian algebras, which are generated by sj = aj +a ∗ j , j = 1, · · · , N, where aia ∗ j −qija ∗ j ai = δij . Here we also allow equality in −1 ≤ qij = qji ≤ 1. This approach relies on Speicher’s central limit theorem and the ultraproduct of von Neumann algebras. We also use the unified argument to show that the Ornstein–Uhlenbeck semigroup is hypercontractive, the Riesz transform associated to the number operator is bounded, and the number operator satisfies the Lp Poincar´e inequalities with constants C √p. Finally we prove that the mixed q-Gaussian algebra is weakly amenable and strongly solid in the sense of Ozawa and Popa. Our approach is mainly combinatorial and probabilistic. The results in this paper can be regarded as generalizations of previous results due to Speicher, Biane, Lust-Piquard, Avsec, et al.
For the physical vacuum free boundary problem with the sound speed being C 1/2 - H¨older continuous near vacuum boundaries of the compressible Euler equations with damping, the global existence of solutions and convergence to Barenblatt self-similar solutions of the porous media equation was recently proved in  for 1-d motions by Luo and the author. This paper generalizes the results for 1-d motions to 3-d spherically symmetric motions. Compared with the 1-d theory, besides the high degeneracy of the equations near the physical vacuum boundary, the analytical difficulties lie in the complexity of equations and the coordinates singularity in the center of symmetry which is resolved by constructing suitable weights. The results obtained in this work contribute to the theory of global solutions to free boundary problems of compressible inviscid fluids in the presence of vacuum states, for which the currently available results are mainly for the local-in-time well-posedness theory, also to the theory of global smooth solutions of dissipative hyperbolic systems which fail to be strictly hyperbolic.
Benjamin AllenHarvard University, Emmanuel CollegeChristine SampleEmmanuel CollegeYulia DementievaEmmanuel CollegeRuben C. MedeirosEmmanuel CollegeChristopher PaolettiEmmanuel CollegeMartin A. NowakHarvard University
Publications of CMSA of Harvardmathscidoc:1702.38002
Over time, a population acquires neutral genetic substitutions as a consequence of random
drift. A famous result in population genetics asserts that the rate, K, at which these substitutions
accumulate in the population coincides with the mutation rate, u, at which they arise in
individuals: K = u. This identity enables genetic sequence data to be used as a “molecular
clock” to estimate the timing of evolutionary events. While the molecular clock is known to
be perturbed by selection, it is thought that K = u holds very generally for neutral evolution.
Here we show that asymmetric spatial population structure can alter the molecular clock
rate for neutral mutations, leading to either K<u or K>u. Our results apply to a general class
of haploid, asexually reproducing, spatially structured populations. Deviations from K = u
occur because mutations arise unequally at different sites and have different probabilities of
fixation depending on where they arise. If birth rates are uniform across sites, then K u. In
general, K can take any value between 0 and Nu. Our model can be applied to a variety of
population structures. In one example, we investigate the accumulation of genetic mutations
in the small intestine. In another application, we analyze over 900 Twitter networks to
study the effect of network topology on the fixation of neutral innovations in social evolution.
This paper proves the nonlinear asymptotic stability of the Lane-Emden solutions for spherically symmetric motions of viscous gaseous stars if the adiabatic constant γ lies in the stability range (4/3, 2). It is shown that for small perturbations of a LaneEmden solution with same mass, there exists a unique global (in time) strong solution to the vacuum free boundary problem of the compressible Navier-Stokes-Poisson system with spherical symmetry for viscous stars, and the solution captures the precise physical behavior that the sound speed is C 1/2 -H¨older continuous across the vacuum boundary provided that γ lies in (4/3, 2). The key is to establish the global-in-time regularity uniformly up to the vacuum boundary, which ensures the large time asymptotic uniform convergence of the evolving vacuum boundary, density and velocity to those of the Lane-Emden solution with detailed convergence rates, and detailed large time behaviors of solutions near the vacuum boundary. In particular, it is shown that every spherical surface moving with the fluid converges to the sphere enclosing the same mass inside the domain of the Lane-Emden solution with a uniform convergence rate and the large time asymptotic states for the vacuum free boundary problem (1.1.2) are determined by the initial mass distribution and the total mass. To overcome the difficulty caused by the degeneracy and singular behavior near the vacuum free boundary and coordinates singularity at the symmetry center, the main ingredients of the analysis consist of combinations of some new weighted nonlinear functionals (involving both lower-order and higher-order derivatives) and space-time weighted energy estimates. The constructions of these weighted nonlinear functionals and space-time weights depend crucially on the structures of the Lane-Emden solution, the balance of pressure and gravitation, and the dissipation. Finally, the uniform boundedness of the acceleration of the vacuum boundary is also proved.
For a self-similar measure in d-dimensional Euclidean space with overlaps but satisfies the so-called bounded measure type condition introduced by Tang and the authors, we set up a framework for deriving a closed formula for the Lq-spectrum of the measure for nonnegative q. The framework allows us to include iterated function systems that have different contraction ratios and those in higher dimension. For self-similar measures with overlaps, closed formulas for the Lq-spectrum have only been obtained earlier for measures satisfying Strichartz second-order identities. We illustrate how to use our results to prove the differentiability of the Lq-spectrum, obtain the multifractal dimension spectrum, and compute the Hausdorff dimension of the measure.