For a fractal Schrodinger operator with a continuous potential and a coupling parameter, we obtain an analog of a semi-classical asymptotic formula for the number of bound states as the parameter tends to infinity. We also study Bohr's formula for Schrodinger operators on blowups of self-similar sets. For a Schrodinger operator defined by a fractal measure and a locally bounded potential that tends to infinity, we derive an analog of Bohr's formula under various assumptions. We demonstrate how these results can be applied to self-similar measures with overlaps, including the infinite Bernoulli convolution associated with the golden ratio, a family of convolutions of Cantor-type measures, and a family of measures that we call essentially of finite type.

The statistical structure on a manifold M is predicated upon a special kind of coupling between the Riemannian metric g and a torsion-free affine connection ∇ on the TM, such that ∇g is totally symmetric, forming, by definition, a “Codazzi pair” t∇, gu. In this paper, we first investigate various transformations of affine connections, including additive translation (by an arbitrary (1,2)-tensor K), multiplicative perturbation (through an arbitrary invertible operator L on TM), and conjugation (through a non-degenerate two-form h). We then study the Codazzi coupling of ∇ with h and its coupling with L, and the link between these two couplings. We introduce, as special cases of K-translations, various transformations that generalize traditional projective and dual-projective transformations, and study their commutativity with L-perturbation and h-conjugation transformations. Our derivations allow affine connections to carry torsion, and we investigate conditions under which torsions are preserved by the various transformations mentioned above. While reproducing some known results regarding Codazzi transform, conformal-projective transformation, etc., we extend much of these geometric relations, and hence obtain new geometric insights, for the general case of a non-degenerate two-form h (instead of the symmetric g) and an affine connection with possibly non-vanishing torsion. Our systematic approach establishes a general setting for the study of Information Geometry based on transformations and coupling relations of affine connections.

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.

We consider the mixed q-Gaussian algebras introduced by Speicher which are generated by the variables Xi = li + l ∗ i , i = 1, . . . , N, where l ∗ i lj − qij lj l ∗ i = δi,j and −1 < qij = qji < 1. Using the free monotone transport theorem of Guionnet and Shlyakhtenko, we show that the mixed q-Gaussian von Neumann algebras are isomorphic to the free group von Neumann algebra L(FN ), provided that maxi,j |qij | is small enough. Similar results hold in the reduced C ∗ -algebra setting. The proof relies on some estimates which are generalizations of Dabrowski’s results for the special case qij ≡ q.

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.