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In this paper, we focus on the stochastic inverse eigenvalue problem of reconstructing a stochastic matrix from the prescribed spectrum. We directly reformulate the stochastic inverse eigenvalue problem as a constrained optimization problem over several matrix manifolds to minimize the distance between isospectral matrices and stochastic matrices. Then we propose a geometric Polak–Ribi`ere–Polyak-based nonlinear conjugate gradient method for solving the constrained optimization problem. The global convergence of the proposed method is established. Our method can also be extended to the stochastic inverse eigenvalue problem with prescribed entries. An extra advantage is that our models yield new isospectral flow methods. Finally, we report some numerical tests to illustrate the efficiency of the proposed method for solving the stochastic inverse eigenvalue problem and the case of prescribed entries.

Zika virus (ZIKV) is a mosquito-borne flavivirus. It was first isolated from Uganda in 1947 and has become an emergent event since 2007. However, because of the inconsistency of alignment methods, the evolution of ZIKV remains poorly understood. In this study, we first use the complete protein and an alignment-free method to build a phylogenetic tree of 87 Zika strains in which Asian, East African, and West African lineages are characterized. We also use the NS5 protein to construct the genetic relationship among 44 Zika strains. For the first time, these strains are divided into two clades: African 1 and African 2. This result suggests that ZIKV originates from Africa, then spread to Asia, Pacific islands, and throughout the Americas. We also perform the phylogeny analysis for 53 viruses in genus Flavivirus to which ZIKV belongs using complete proteins. Our conclusion is consistent with the classification by the hosts and transmission vectors.

We give a proof of the parabolic/singular Koszul duality for the category O of affine Kac–Moodyalgebras. The main new tool is a relation between moment graphs and finite codimensional affine Schubert varieties. We apply this duality to q-Schur algebras and to cyclotomic rational double affine Heckealgebras.This yields a proof of a conjecture of Chuang–Miyachi relating the level-rank duality with the Ringel–Koszul duality of cyclotomic rational double affine Hecke algebras.