In this paper we introduce the interpolationdegeneration strategy to study KhlerEinstein metrics on a smooth Fano manifold with cone singularities along a smooth divisor that is proportional to the anti-canonical divisor. By interpolation we show the angles in (0, 2] that admit a conical KhlerEinstein metric form a connected interval, and by degeneration we determine the boundary of the interval in some important cases. As a first application, we show that there exists a KhlerEinstein metric on P 2 with cone singularity along a smooth conic (degree 2) curve if and only if the angle is in (/2, 2]. When the angle is 2/3 this proves the existence of a SasakiEinstein metric on the link of a three dimensional <i>A</i> ₂ singularity, and thus answers a question posed by GauntlettMartelliSparksYau. As a second application we prove a version of Donaldsons conjecture about conical Khler

We establish an identity for closed hyperbolic surfaces whose terms depend on the dilogarithms of the lengths of simple closed geodesics in all 3-holed spheres and 1-holed tori in the surface.

This paper is devoted to developing and analysing a highly accurate conservative method for solving the quantum Zakharov system. The scheme is based on a linearly-implicit compact finite difference discretization and conserve the mass as well as energy in discrete level. Detailed numerical analysis is presented which shows the method is fourth-order accurate in space and second-order accurate in time. Several numerical examples are reported to confirm the conservation properties and high accuracy of the proposed scheme. Finally the compact scheme is applied to study the convergence rate of the quantum Zakharov system to its limiting model in the semi-classical limit.

The aim of this paper is to develop a mathematical framework for opto-elastography. In opto-elastography, a mechanical perturbation of the medium produces a decorrelation of optical speckle patterns due to the displacements of optical scatterers. To model this, we consider two optically random media, with the second medium obtained by shifting the first medium in some local region. We derive the radiative transfer equation for the cross-correlation of the wave fields in the media. Then we derive its diffusion approximation. In both the radiative transfer and the diffusion regimes, we relate the correlation of speckle patterns to the solutions of the radiative transfer and the diffusion equations. We present numerical simulations based on our model which are in agreement with recent experimental measurements.

Let π be a cuspidal automorphic representation of PGL_2(AQ) of arithmetic conductor C and archimedean parameter T, and let ϕ be an L2-normalized automorphic form in the space of π. The sup-norm problem asks for bounds on ∥ϕ∥_∞ in terms of C and T. The quantum unique ergodicity (QUE) problem concerns the limiting behavior of the L2-mass |ϕ|^2(g)dg of ϕ. Previous work on these problems in the conductor-aspect has focused on the case that ϕ is a newform. In this work, we study these problems for a class of automorphic forms that are not newforms. Precisely, we assume that for each prime divisor p of C, the local component π_p is supercuspidal (and satisfies some additional technical hypotheses), and consider automorphic forms ϕ for which the local components ϕ_p∈π_p are "minimal" vectors. Such vectors may be understood as non-archimedean analogues of lowest weight vectors in holomorphic discrete series representations of PGL_2(R). For automorphic forms as above, we prove a sup-norm bound that is sharper than what is known in the newform case. In particular, if π_∞ is a holomorphic discrete series of lowest weight k, we obtain the optimal bound C^{1/8−ϵ} k^{1/4−ϵ} ≪_ϵ |ϕ|_∞ ≪_ϵ C^{1/8+ϵ} k^{1/4+ϵ}. We prove also that these forms give analytic test vectors for the QUE period, thereby demonstrating the equivalence between the strong QUE and the subconvexity problems for this class of vectors. This finding contrasts the known failure of this equivalence [31] for newforms of powerful level.