The authors consider a stochastic heat equation in dimension d = 1 driven by an additive space time white noise and having a mild nonlinearity. It is proved that the functional law of its solution is absolutely continuous and possesses a smooth density with respect to the functional law of the corresponding linear SPDE.

In this paper, the mean curvature flow of complete submanifolds in Euclidean space with convex Gauss image and bounded curvature is studied. The confinable property of the Gauss image under the mean curvature flow is proved, which in turn helps one to obtain the curvature estimates. Then the author proves a long time existence result. The asymptotic behavior of these solutions when t → ∞ is also studied.

The author, motivated by his results on Hermitian metric rigidity, conjectured in [4] that a proper holomorphic mapping ƒ: Ω → Ω′ from an irreducible bounded symmetric domain Ω of rank ≥ 2 into a bounded symmetric domain Ω′ is necessarily totally geodesic provided that r′:= rank(Ω′) ≤ rank(Ω):= r. The Conjecture was resolved in the affirmative by I.-H. Tsai [8]. When the hypothesis r′ ≤ r is removed, the structure of proper holomorphic maps ƒ: Ω → Ω′ is far from being understood, and the complexity in studying such maps depends very much on the difference r′ − r, which is called the rank defect. The only known nontrivial non-equidimensional structure theorems on proper holomorphic maps are due to Z.-H. Tu [10], in which a rigidity theorem was proven for certain pairs of classical domains of type I, which implies nonexistence theorems for other pairs of such domains. For both results the rank defect is equal to 1, and a generalization of the rigidity result to cases of higher rank defects along the line of arguments of [10] has so far been inaccessible. In this article, the author produces nonexistence results for infinite series of pairs of (Ω, Ω′) of irreducible bounded symmetric domains of type I in which the rank defect is an arbitrarily prescribed positive integer. Such nonexistence results are obtained by exploiting the geometry of characteristic symmetric subspaces as introduced by N. Mok and I.-H Tsai [6] and more generally invariantly geodesic subspaces as formalized in [8]. Our nonexistence results motivate the formulation of questions on proper holomorphic maps in the non-equirank case.

A new approach to construct a new 4 × 4 matrix spectral problem from a normal 2 sx 2 matrix spectral problem is presented. AKNS spectral problem is discussed as an example. The isospectral evolution equation of the new 4 × 4 matrix spectral problem is nothing but the famous AKNS equation hierarchy. With the aid of the binary nonlinearization method, the authors get new integrable decompositions of the AKNS equation. In this process, the r-matrix is used to get the result.

The purpose of this paper is to study the mapping properties of the singular Radon transforms with rough kernels. Such singular integral operators are proved to be bounded on Lebesgue spaces.

For a compact complex spin manifold M with a holomorphic isometric embedding into the complex projective space, the authors obtain the extrinsic estimates from above and below for eigenvalues of the Dirac operator, which depend on the data of an isometric embedding of M. Further, from the inequalities of eigenvalues, the gaps of the eigenvalues and the ratio of the eigenvalues are obtained.

Let u = u(x, t, u ₀) represent the global solution of the initial value problem for the one-dimensional fluid dynamics equation $u_t - \varepsilon u_{xxt} + \delta u_x + \gamma Hu_{xx} + \beta u_{xxx} + f(u)_x = \alpha u_{xx} ,u(x,0) = u_0 (x),$ where α > 0, β ≥ 0, γ ≥ 0, δ ≥ 0 and ε ≥ 0 are constants. This equation may be viewed as a one-dimensional reduction of n-dimensional incompressible Navier-Stokes equations. The nonlinear function satisfies the conditions f(0) = 0, |f(u)| → ∞ as |u| → ∞, and f ∈ C ¹ (ℝ), and there exist the following limits $L_0 = \mathop {lim sup}\limits_{u \to 0} \frac{{f(u)}}{{u^3 }}andL_\infty = \mathop {lim sup}\limits_{u \to \infty } \frac{{f(u)}}{{u^5 }}.$. Suppose that the initial function u ₀ ∈ L ¹(ℝ) ∩ H ²(ℝ). By using energy estimates, Fourier transform, Plancherel’s identity, upper limit estimate, lower limit estimate and the results of the linear problem $v_t - \varepsilon v_{xxt} + \delta v_x + \gamma Hv_{xx} + \beta v_{xxx} = \alpha v_{xx} ,v(x,0) = v_0 (x),$ the author justifies the following limits (with sharp rates of decay) $\mathop {\lim }\limits_{t \to \infty } \left[ {(1 + t)^{m + {1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2}} \smallint _\mathbb{R} |u_{x^m } (x,t)|^2 dx} \right] = \frac{1}{{2\pi }}\left( {\frac{\pi }{{2\alpha }}} \right)^{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2}} \frac{{m!!}}{{(4\alpha )^m }}\left[ {\smallint _\mathbb{R} u_0 (x)dx} \right]^2 ,$ if $\smallint _\mathbb{R} u_0 (x)dx \ne 0,$ where 0!! = 1, 1!! = 1 and m!! = 1 · 3 ⋯ · (2m–3) · (2m − 1). Moreover $ \mathop {\lim }\limits_{t \to \infty } \left[ {(1 + t)^{m + {3 \mathord{\left/ {\vphantom {3 2}} \right. \kern-\nulldelimiterspace} 2}} \smallint _\mathbb{R} |u_{x^m } (x,t)|^2 dx} \right] = \frac{1} {{2\pi }}\left( {\frac{\pi } {{2\alpha }}} \right)^{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2}} \frac{{(m + 1)!!}} {{(4\alpha )^{m + 1} }}\left[ {\smallint _\mathbb{R} \rho _0 (x)dx} \right]^2 , $ if the initial function u ₀(x) = ρ ₀′ (x), for some function ρ ₀ ∈ C ¹(ℝ) ∩ L ¹(ℝ) and $ \smallint _\mathbb{R} \rho _0 (x)dx \ne 0. $.

For an ergodic continuous-time Markov process with a particular state in its space, the authors provide the necessary and sufficient conditions for exponential and strong ergodicity in terms of the moments of the first hitting time on the state. An application to the queue length process of M/G/1 queue with multiple vacations is given.

In this paper, the authors investigate three aspects of statistical inference for the partially linear regression models where some covariates are measured with errors. Firstly, a bandwidth selection procedure is proposed, which is a combination of the difference-based technique and GCV method. Secondly, a goodness-of-fit test procedure is proposed, which is an extension of the generalized likelihood technique. Thirdly, a variable selection procedure for the parametric part is provided based on the nonconcave penalization and corrected profile least squares. Same as “Variable selection via nonconcave penalized likelihood and its oracle properties” (J. Amer. Statist. Assoc., 96, 2001, 1348–1360), it is shown that the resulting estimator has an oracle property with a proper choice of regularization parameters and penalty function. Simulation studies are conducted to illustrate the finite sample performances of the proposed procedures.