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A spherical conical metric g g on a surface Σ \Sigma is a metric of constant curvature 1 1 with finitely many isolated conical singularities. The uniformization problem for such metrics remains largely open when at least one of the cone angles exceeds 2 π 2\pi . The eigenfunctions of the Friedrichs Laplacian Δ g \Delta _g with eigenvalue λ = 2 \lambda =2 play a special role in this problem, as they represent local obstructions to deformations of the metric g g in the class of spherical conical metrics. In the present paper we apply the theory of multivalued harmonic maps to spheres to the question of existence of such eigenfunctions. In the first part we establish a new criterion for the existence of 2 2 -eigenfunctions, given in terms of a certain meromorphic data on Σ \Sigma . As an application we give a description of all 2 2 -eigenfunctions for metrics on the sphere with at most three conical singularities. The second part is an algebraic construction of metrics with large number of 2 2 -eigenfunctions via the deformation of multivalued harmonic maps. We provide new explicit examples of metrics with many 2 2 -eigenfunctions via both approaches, and describe the general algorithm to find metrics with arbitrarily large number of 2 2 -eigenfunctions. 

Abstract We continue our study, initiated in [34], of Riemann surfaces with constant curvature and isolated conic singularities. Using the machinery developed in that earlier paper of extended configuration families of simple divisors, we study the existence and deformation theory for spherical conic metrics with some or all of the cone angles greater than $2\pi $. Deformations are obstructed precisely when the number $2$ lies in the spectrum of the Friedrichs extension of the Laplacian. Our main result is that, in this case, it is possible to find a smooth local moduli space of solutions by allowing the cone points to split. This analytic fact reflects geometric constructions in [37, 38]. 

We study the statistical properties of tidal weather (variability period <30 days) of DW1 amplitude using the extended Canadian Middle Atmospheric Model (eCMAM) and Sounding of the Atmosphere using Broadband Emission Radiometry (SABER). A hierarchy of statistical models, for example, the autoregressive (AR), vector AR, and parsimonious vector AR models, are built to predict tidal weather. The quasi 23‐day oscillation found in the tidal weather is a key parameter in the statistical models. Comparing to the more complex vector AR and parsimonious vector AR models, which consider the spatial correlations of tidal weather, the simplest AR model can predict one‐day tidal weather with an accuracy of 89% (R²: correlation coefficient squared). In the AR model, 23 coefficients at each latitude and height are obtained from seven years of eCMAM data. Tidal weather is predicted via a linear combination of 23 days of tidal weather data prior to the prediction day. Different sensitivity tests are performed to prove the robustness of these coefficients. These coefficients obtained from eCMAM are in very good agreement with those from SABER. SABER tidal weather is predicted with an accuracy of 86% and 87% at one day by the AR models with coefficients from eCMAM and SABER, respectively. The five‐day forecast accuracy is between 60 and 65%.