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For each natural odd number n ≥ 3, we exhibit a maximal family of n-dimensional Calabi–Yau manifolds whose Yukawa coupling length is 1. As a consequence, Shafarevich’s conjecture holds true for these families. Moreover, it follows from Deligne and Mostow (Publ. Math. IHÉS, 63:5–89, 1986) and Mostow (Publ. Math. IHÉS, 63:91–106, 1986; J. Am. Math. Soc., 1(3):555–586, 1988) that, for n = 3, it can be partially compactified to a Shimura family of ball type, and for n = 5,9, there is a sub Q-PVHS of the family uniformizing a Zariski open subset of an arithmetic ball quotient

We propose a semiparametric latent Gaussian copula model for modelling mixed multivariate data, which contain a combination of both continuous and binary variables. The model assumes that the observed binary variables are obtained by dichotomizing latent variables that satisfy the Gaussian copula distribution. The goal is to infer the conditional independence relationship between the latent random variables, based on the observed mixed data. Our work has two main contributions: we propose a unified rankbased approach to estimate the correlation matrix of latent variables; we establish the concentration inequality of the proposed rankbased estimator. Consequently, our methods achieve the same rates of convergence for precision matrix estimation and graph recovery, as if the latent variables were observed. The methods proposed are numerically assessed through extensive simulation studies, and real

Consider a system of periodic pendulum lattice with analytic weak coupling: i x ̈i +sinxi =−ε 􏰀 ∂xiβα(xj,xj+1,xj+2), xi =xi+N, i∈Z, j =i−2 where N 􏰃 3 is an integer, ε > 0 is a small parameter and the function βα is an analytic function of a certain form. It is shown in this paper that for small enough ε, the system admits motions such that the energy transfers between the pendulums in any predetermined order.