In this paper, according to research results such as the asymptotic sums of a special class of sine and cosine series and the monotonicity of trigonometric series, we obtain some conclusions by establishing corresponding concepts compared with the discrete case and studying a class of asymptotic relations for special sine and cosine integrals. In addition, for the last theorem in the paper, we show that under the conditions {\mathrm{l}\mathrm{i}\mathrm{m}}_{t\to \mathrm{\infty }}\frac{f(t)}{\omega \left(t^{-1}\right)}=A and some assumptions on the modulus of continuity, these generalized monotonicity concepts are equivalent except for the second class of supremum bounded variation functions.

In this paper, we discuss submersions of general submanifolds in nearly Kähler manifolds and prove that if \pi :M\to B is a submersion from a submanifold M of a nearly Kähler manifold \overline{M} to an almost Hermitian manifold B, then Bis a nearly Kähler manifold. Furthermore, this paper presents some properties concerning the decomposition theory for such submersions and studies the relationship between the holomorphic sectional curvatures of \overline{M} and B.

In this paper, the empirical Bayes (EB) test problem for the shape parameter of the Lomax distribution family is investigated. By using the recursive kernel estimation of the density function and monotonicity, the empirical Bayes test functions are constructed. Under suitable conditions, the convergence rates of the EB tests are obtained. The convergence rates can be arbitrarily close to O(n^{-1}). Finally, an example is given to verify that prior distributions satisfying the conditions of the theorem exist.

The index theory of combinatorial matrix is the central research of combinatorial matrix theory. Combinatorial matrix index theory not only has important theoretical value, but also has a wide range of practical applications in many aspects, e.g., computer science, ergodic theory, communications theory, sociology and economics. In this paper, we summarize a number of classic indices of combinatorial matrix, and adopt a unified view to sum them up as structural indices of combinatorial matrix. This theoretical system opens up a new field for the research of index theory.