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.

Numerical schemes provably preserving the positivity of density and pressure are highly desirable for ideal magnetohydrodynamics (MHD), but the rigorous positivity-preserving (PP) analysis remains challenging. The difficulties mainly arise from the intrinsic complexity of the MHD equations as well as the indeterminate relation between the PP property and the divergence-free condition on the magnetic field. This paper presents the first rigorous PP analysis of conservative schemes with the Lax--Friedrichs (LF) flux for 1D and multidimensional ideal MHD. The significant innovation is the discovery of the theoretical connection between the PP property and a discrete divergence-free (DDF) condition. This connection is established through the generalized LF splitting properties, which are alternatives to the usually expected LF splitting property that does not hold for ideal MHD. The generalized LF splitting properties involve a number of admissible states strongly coupled by the DDF condition, making their derivation very difficult. We derive these properties via a novel equivalent form of the admissible state set and an important inequality, which is skillfully constructed by technical estimates. Rigorous PP analysis is then presented for finite volume and discontinuous Galerkin schemes with the LF flux on uniform Cartesian meshes. In the 1D case, the PP property is proved for the first-order scheme with proper numerical viscosity, and also for arbitrarily high-order schemes under conditions accessible by a PP limiter. In the 2D case, we show that the DDF condition is necessary and crucial for achieving the PP property. It is observed that even slightly violating the proposed DDF condition may cause failure to preserve the positivity of pressure. We prove that the 2D LF type scheme with proper numerical viscosity preserves both the positivity and the DDF condition. Sufficient conditions are derived for 2D PP high-order schemes, and extension to 3D is discussed. Numerical examples provided in the supplementary material further confirm the theoretical findings.

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We introduce and study mutation of torsion pairs, as a generalization of mutation of cluster tilting objects, rigid objects and maximal rigid objects. It is proved that any mutation of a torsion pair is again a torsion pair. A geometric realization of mutation of torsion pairs in the cluster category of type A_n or A_\infty is given via rotation of Ptolemy diagrams.

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