A star with sufficiently large mass will collapse in its senectitude because of gravitation. Many researchers have tried to describe this collapse process. Investigations have been conducted in the case that the star has spherical symmetry and the initial density is uniform. In this article, the case that the initial density is not uniform will be considered. When the density function is high in the center and decreasing with the radius, the collapse process will be described, and in this case, the singularity will only come out in the center because of collision. If the density is not monotonic and there is a crust with high density around the star, it is proved that the non-central collision singularity may come out either in the Schwarzschild sphere or outside of it, i.e., the naked singularize may come out.

The Einstein gravitational equations in the spherically symmetric case and for the dust model (i.e., p = 0) have been studied by several authors. However, the solutions obtained by them are not completed yet, and the corresponding metric is written in implicit forms which is inconvenient for physical problems. In the present paper we make the following improvements: (1) We obtain all spherical solutions for the dust model with explicit expressions which consist of three classes and an exceptional case; (2) All these solutions contain singularities which are analyzed together with their physical properties.

In this paper, a survey on the index iteration theory for symplectic paths is given. Three applications of this theory are presented including closed characteristics on convex hypersurfaces and brake orbits on bounded domains.

The asymptotic expressions of the covariance matrices for both the least square estimates _Lα_T and Markov (best linear) estimates $\bar \alpha _T $$ are obtained, based on a sample in a finite interval (0, T) of the regression co-efficients α = (α₁, …, α_{m 0})′ of a parameter-continuous process with a stationary residual. We assume that the regression variables φ_ν(t), t ⩾ 0, ν = 1, …, m₀, are continuous in t, and satisfy conditions (3.1)–(3.3). For the residual, we assume that it is a stationary process that possesses a bounded continuous spectral density f(λ). Under these assumptions, it is proven that $\mathop {\lim }\limits_{T \to \infty } D_T E_0 \left( {L\alpha _T L\alpha _T^* } \right)D_T = 2\pi \left[ {B\left( 0 \right)} \right]^{ - 1} \int_{ - \infty }^\infty {f( - \lambda )d\alpha (\lambda )\left[ {B\left( 0 \right)} \right]^{ - 1} } ,$ where the matrices D_T, B(0), α(λ) are defined in Section 3.

Under the assumptions mentioned above, if, furthermore, there exist some positive integer m and a constant C such that g(λ)(1 + λ²)^m ⩾ C > 0, where g(λ) is the spectral density of the residual, and for every N > 0, $\frac{{\lim _{T \to \infty } \smallint _0^T \overline {\varphi _\mu ^{(k)} (t + h)} \varphi _\nu ^{(j)} (t + l)dt}}{{\sqrt {\Phi _\mu (T)\Phi _\nu (T)} }},0 \leqslant k,j \leqslant m$ converge uniformly in h, l ∈ (−N, N), then the following formula holds. $\mathop {\lim }\limits_{T \to \infty } D_T E_0 \hat \alpha _T \hat \alpha _T^ * D_T = 2\pi \left[ {\int_{ - \infty }^\infty {\frac{1}{{g( - \lambda )}}d\alpha (\lambda )} } \right]^{ - 1} .$

The asymptotic equivalence of the least square estimates and the Markov estimates is also discussed.

A modified version of the natural power method (NP) for fast estimation and tracking of the principal eigenvectors of a vector sequence is Presented. It is an extension of the natural power method because it is a solution to obtain the principal eigenvectors and not only for tracking of the principal subspace. As compared with some power-based methods such as Oja method, the projection approximation subspace tracking (PAST) method, and the novel information criterion (NIC) method, the modified natural power method (MNP) has the fastest convergence rate and can be easily implemented with only O(np) flops of computation at each iteration, where n is the dimension of the vector sequence and p is the dimension of the principal subspace or the number of the principal eigenvectors. Furthermore, it is guaranteed to be globally and exponentially convergent in contrast with some non-power-based methods such as MALASE and OPERA.

Denote by B_2σ,p (1 < p < ∞) the bandlimited class p-integrable functions whose Fourier transform is supported in the interval [−σ, σ]. It is shown that a function in B_2σ,p can be reconstructed in L_p(ℝ) by its sampling sequences {f (κπ / σ)}_κ∈ℤ and {f’ (κπ / σ)}_κ∈ℤ using the Hermite cardinal interpolation. Moreover, it will be shown that if f belongs to L_p ^r (ℝ), 1 < p < ∞, then the exact order of its aliasing error can be determined.

The purpose of this paper is to discuss the pure scalar characterization of the automorphism group Aut (L₅(2)) and the linear group L₆(2). It is proved that Aut(L₅(2)) and L₆(2) can be characterized quantitatively by the set of element orders. The main results are obtained by using William’s work on prime graph components of finite groups and Brauer characters in trivializing the possible 2-subgroups.

The main purpose of this paper is to discuss the stationarity of two classes of nonstationary processes after wavelet transformations. Owing to the fact that the wavelet transformation possesses localization and implicit differencing property, the authors show that after wavelet transformation, the fractionally differenced process and the harmonizable periodically correlated process may be changed into stationary processes.

This paper is concerned with the composition operators between the area-type Nevanlinna classes. Some sufficient and necessary conditions are given in terms of the concept of Carleson measure and the standard techniques of Montel Theorem for the composition operator C_φ: N_a ^p → N_a ^q to be bounded or compact, where 1 < p ⩽ q. Moreover, the inducing maps which induce invertible or Fredholm composition operators on N_a ^p are characterized.

In this paper, we consider the empirical Bayes (EB) test problem for the scale parameters in the scale exponential family with a weighted linear loss function. The EB test rules are constructed by the kernel estimation method. The asymptotical optimality and convergence rates of the EB test rules are obtained. The main results are illustrated by applying the proposed test to type II censored data from the exponential distribution and to the test problem for the dispersion parameter in the linear regression model.

This paper investigates the relationship between the normality and the shared values for a meromorphic function on the unit disc Δ. Based on Marty’s normality criterion and through a detailed analysis of the meromorphic functions, it is shown that if for every f ∈ $\mathcal{F}$$, f and f^(k) share a and b on Δ and the zeros of f(z) − a are of multiplicity k ⩾ 3, then $\mathcal{F}$$ is normal on Δ, where $\mathcal{F}$$ is a family of meromorphic functions on the unit disc Δ, and a and b are distinct values.