We consider the inverse eigenvalue problem of reconstructing a doubly stochastic matrix from the given spectrum data. We reformulate this inverse problem as a constrained nonlinear least squares problem over several matrix manifolds, which minimizes the distance between isospectral matrices and doubly stochastic matrices. Then a Riemannian Fletcher-Reeves conjugate gradient method is proposed for solving the constrained nonlinear least squares problem, and its global convergence is established. An extra gain is that a new Riemannian isospectral flow method is obtained. Our method is also extended to the case of prescribed entries. Finally, some numerical tests are reported to illustrate the efficiency of the proposed method.

Based on the inverse scattering method, the formulae of one higher-order pole solitons and multiple higherorder poles solitons of the nonlinear Schrödinger equation (NLS) equation are obtained. Their denominators are expressed as det(+ Ω * Ω), where Ω is a matrix frequently constructed for solving the Riemann-Hilbert problem, and the asterisk denotes complex conjugate. We take two methods for proving + Ω * Ω is invertible. The first one shows matrix Ω is equivalent to a self-adjoint Hankel matrix Δ, proving det(+ Ω * Ω) = det(+ Δ † Δ) ≥ 1. The second one considers the blockmatrix form of det(+ Ω * Ω), proving | det(+ Ω * Ω)| ≥ 1. In addition, we prove that the dimension of Ω is equivalent to the sum of the orders of pole points of the transmission coefficient and its diagonal entries compose a set of basis.

Despite the remarkable success of immunotherapy in the treatment of melanoma, resistance to these agents still affects patient prognosis and response to therapies. Beta-2-microglobulin (β2M), an important subunit of major histocompatibility complex (MHC) class I, has important biological functions and roles in tumor immunity. In recent years, increasing studies have shown that B2M gene deficiency can inhibit MHC class I antigen presentation and lead to cancer immune evasion by affecting β2M expression. Based on this, B2M gene defect and T cell-based immunotherapy can interact to affect the efficacy of melanoma treatment. Taking into account the many recent advances in B2M-related melanoma immunity, here we discuss the immune function of the B2M gene in tumors, its common genetic alteration in melanoma, and its impact on and related improvements in melanoma immunotherapy. Our comprehensive review of β2M biology and its role in tumor immunotherapy contributes to understanding the potential of B2M gene as a promising melanoma therapeutic target.

This paper is concerned with the problem of finding a zero of a tangent vector field on a Riemannian manifold. We first reformulate the problem as an equivalent Riemannian optimization problem. Then we propose a Riemannian derivative-free Polak-Ribiére-Polyak method for solving the Riemannian optimization problem, where a non-monotone line search is employed. The global convergence of the proposed method is established under some mild assumptions. To further improve the efficiency, we also provide a hybrid method, which combines the proposed geometric method with the Riemannian Newton method. Finally, some numerical experiments are reported to illustrate the efficiency of the proposed method.