Acta Mathematica Sinica-English Series最新文献

Regression models are often transformed into certain alternative forms in statistical inference theory. In this paper, we assume that a general linear model (GLM) is transformed into two different forms, and our aim is to study some comparison problems under the two transformed general linear models (TGLMs). We first construct a general vector composed of all unknown parameters under the two different TGLMs, derive exact expressions of best linear minimum bias predictors (BLMBPs) by solving a constrained quadratic matrix-valued function optimization problem in the Löwner partial ordering, and describe a variety of mathematical and statistical properties and performances of the BLMBPs. We then approach some algebraic characterization problems concerning relationships between the BLMBPs under two different TGLMs. As applications, two specific cases are presented to illustrate the main contributions in the study.

We consider the following fractional prescribed curvature problem

where (s in (0,, {1 over 2})) for N = 3, s ∈ (0, 1) for N ≥ 4 and (2_{s}^{*}={2N over N-2s}) is the fractional critical Sobolev exponent, K(y) has a local maximum point in r ∈ (r₀ − δ, r₀ + δ). First, for any sufficient large k, we construct a 2k bubbling solution to (0.1) of some new type, which concentrates on an upper and lower surfaces of an oblate cylinder through the Lyapunov–Schmidt reduction method. Furthermore, a non-degeneracy result of the multi-bubbling solutions is proved by use of various Pohozaev identities, which is new in the study of the fractional problems.

For a piecewise monotone function F of height 1, an open question was raised: Does F have an iterative root f of order n ≤ N(F) + 1 if the ‘characteristic endpoints condition’ is not satisfied? This question was answered partly in the case that F is strictly increasing on its characteristic interval K(F) but f is strictly decreasing on K(F). In this paper we discuss the question for F increasing on K(F) in some remaining cases, giving the necessary and sufficient conditions for the existence of continuous iterative roots f decreasing on K(F) of order n = N(F) > 2 with H(f) = n − 1.

In this paper, complete convergence and complete moment convergence for maximal weighted sums of ρ⁻-mixing random variables are investigated, and some sufficient conditions for the convergence are provided. The relationships among the weights of the partial sums, boundary function and weight function are in a sense revealed. Additionally, a Marcinkiewicz–Zygmund type strong law of large numbers for maximal weighted sums of ρ⁻-mixing random variables is established. The results obtained extend the corresponding ones for random variables with independence structure and some dependence structures. As an application, the strong consistency for the tail-value-at-risk (TVaR) estimator in the financial and actuarial fields is established.

We study Berry–Esseen bounds and Cramér-type moderate deviations of a jump-type Cox–Ingersoll–Ross (CIR) process driven by a standard Wiener process and a subordinator. In the subcritical case, we obtain the best Berry–Esseen bound of the sample mean and the MLE of the growth rate if the Lévy measure of the subordinator has finite third order moment. Under the Cramér condition, we establish the Cramér-type moderate deviations of the MLE of the growth rate. We first derive a Berry–Esseen bound, a deviation inequality and the Cramér-type moderate deviations for the sample mean of the CIR process by analyzing the asymptotic behaviors of the characteristic function and the moment generating function of the sample mean. Then we analyze a type of additive functional of the jump-type CIR process and use a transformation to study the Berry–Esseen bound and the Cramér-type moderate deviations for the MLE of the growth rate.

The main purpose of this article is using the elementary techniques and the properties of the character sums to study the computational problem of one kind products of Gauss sums, and give an interesting triplication formula for them.

In this paper, the semirings with invariant basis numbers are investigated. First, we give some properties of a semiring which has an invariant basis number, and then give some necessary and sufficient conditions that the direct sum of two semirings has an invariant basis number. As an application, we prove that division semirings, quasilocal semirings and stably finite semirings have invariant basis numbers, respectively.

In this paper, we study a class of Finsler metrics of cohomogeneity two on ℝ × ℝⁿ. They are called weakly orthogonally invariant Finsler metrics. These metrics not only contain spherically symmetric Finsler metrics and Marcal–Shen’s warped product metrics but also partly contain another “warping” introduced by Chen–Shen–Zhao. We obtain differential equations that characterize weakly orthogonally invariant Finsler metrics with vanishing Douglas curvature, and therefore we provide a unifying frame work for Douglas equations due to Liu–Mo, Mo–Solórzano–Tenenblat and Solórzano. As an application, we obtain a lot of new examples of weakly orthogonally invariant Douglas metrics.

This article deals with the sharp discrete isoperimetric inequalities in analysis and geometry for planar convex polygons. First, the analytic isoperimetric inequalities based on the Schur convex function are established. In the wake of the analytic isoperimetric inequalities, Bonnesen-style isoperimetric inequalities and inverse Bonnesen-style inequalities for the planar convex polygons are obtained.

For any real number x, [x] denotes the integer part of x. (cal{F})₁, (cal{F})₂ denote two multiplicative function classes which are small in numerical sense. In this paper, we study the summation (sumnolimits_{{nleq x}}f([x/n])) for f ∈ (cal{F})₁. As specific cases, we take d^(e)(n), β(n), a(n), μ₂(n) denoting the number of exponential divisors of n, the number of square-full divisors of n, the number of non-isomorphic Abelian groups of order n, and the characteristic function of the square-free integers, respectively. In the case of μ₂(n), we improved the result of Liu, Wu and Yang. The sums shaped like (sumnolimits_{{nleq x}}f([x/n]+f([x/n]))) for f ∈ (cal{F})₂ are also researched.