In this paper, the authors apply $\overline \partial $ steepest descent method to study the Cauchy problem for the derivative nonlinear Schrödinger equation with finite density type initial data $\matrix{{{\rm{i}}q + {q_{xx}} + {\rm{i}}{{\left( {{{\left| q \right|}^2}q} \right)}_x} = 0,} \hfill \cr {q\left( {x,0} \right) = {q_0}\left( x \right),} \hfill \cr } $ where $\mathop {\lim }\limits_{x \to \pm \infty } {q_0}\left( x \right) = {q_ \pm }\,{\rm{and}}\,\,\left| {{q_ \pm }} \right| = 1$. Based on the spectral analysis of the Lax pair, they express the solution of the derivative Schrödinger equation in terms of solutions of a Riemann-Hilbert problem. They compute the long time asymptotic expansion of the solution q(x, t) in different space-time regions. For the region $\xi = {x \over t}$ with ∣ξ + 2∣ < 1, the long time asymptotic is given by $q\left( {x,t} \right) = T{\left( \infty \right)^{ - 2}}q_\Lambda ^r\left( {x,t} \right) + {\cal O}\left( {{t^{ - {3 \over 4}}}} \right),$ in which the leading term is N(I) solitons, the second term is a residual error from a $\overline \partial $ equation. For the region ∣ξ + 2∣ > 1, the long time asymptotic is given by $q\left( {x,t} \right) = T{\left( \infty \right)^{ - 2}}q_\Lambda ^r\left( {x,t} \right) - {t^{ - {1 \over 2}}}{\rm{i}}{f_{11}} + {\cal O}\left( {{t^{ - {3 \over 4}}}} \right),$ in which the leading term is N(I) solitons, the second ${t^{ - {1 \over 2}}}$ order term is soliton-radiation interactions and the third term is a residual error from a $\overline \partial $ equation. These results are verification of the soliton resolution conjecture for the derivative Schrödinger equation. In their case of finite density type initial data, the phase function θ(z) is more complicated that in finite mass initial data. Moreover, two triangular decompositions of the jump matrix are used to open jump lines on the whole real axis and imaginary axis, respectively.

In this paper, the authors establish a generalized maximum principle for pseudo-Hermitian manifolds. As corollaries, Omori-Yau type maximum principles for pseudo-Hermitian manifolds are deduced. Moreover, they prove that the stochastic completeness for the heat semigroup generated by the sub-Laplacian is equivalent to the validity of a weak form of the generalized maximum principles. Finally, they give some applications of these generalized maximum principles.

In this paper, the authors consider the local well-posedness for the derivative Schrödinger equation in higher dimension ${u_t} - {\rm{i}}\Delta u + {\left| u \right|^2}\left( {\overrightarrow \gamma \cdot \nabla u} \right) + {u^2}\left( {\overrightarrow \lambda \cdot \nabla \overline u } \right) = 0,\,\,\,\,\left( {x,t} \right) \in {\mathbb{R}^n} \times \mathbb{R},\,\,\overrightarrow \gamma ,\overrightarrow \lambda \in {\mathbb{R}^n};\,\,n \ge 2.$

It is shown that the Cauchy problem of the derivative Schrödinger equation in higher dimension is locally well-posed in ${H^s}\left( {{\mathbb{R}^n}} \right)\,\,\left( {s > {n \over 2}} \right)$ for any large initial data. Thus this result can compare with that in one dimension except for the endpoint space ${H^{{n \over 2}}}$.

This paper analyzes the limiting behavior of stochastic linear-quadratic optimal control problems in finite time-horizon [0, T] as T → ∞. The so-called turnpike properties are established for such problems, under stabilizability condition which is weaker than the controllability, normally imposed in the similar problem for ordinary differential systems. In dealing with the turnpike problem, a crucial issue is to determine the corresponding static optimization problem. Intuitively mimicking the deterministic situations, it seems to be natural to include both the drift and the diffusion expressions of the state equation to be zero as constraints in the static optimization problem. However, this would lead us to a wrong direction. It is found that the correct static problem should contain the diffusion as a part of the objective function, which reveals a deep feature of the stochastic turnpike problem.

In this paper, the authors introduce the index of subgeneral position for closed subschemes and obtain a second main theorems based on this notion. They also give the corresponding Schmidt’s subspace type theorem via the analogue between Nevanlinna theory and Diophantine approximation.

The conditional nonlinear optimal perturbation (CNOP for short) approach is a powerful tool for predictability and targeted observation studies in atmosphere-ocean sciences. By fully considering nonlinearity under appropriate physical constraints, the CNOP approach can reveal the optimal perturbations of initial conditions, boundary conditions, model parameters, and model tendencies that cause the largest simulation or prediction uncertainties. This paper reviews the progress of applying the CNOP approach to atmosphere-ocean sciences during the past five years. Following an introduction of the CNOP approach, the algorithm developments for solving the CNOP are discussed. Then, recent CNOP applications, including predictability studies of some high-impact ocean-atmospheric environmental events, ensemble forecast, parameter sensitivity analysis, uncertainty estimation caused by errors of model tendency or boundary condition, are reviewed. Finally, a summary and discussion on future applications and challenges of the CNOP approach are presented.

In this paper, the authors propose a novel smoothing descent type algorithm with extrapolation for solving a class of constrained nonsmooth and nonconvex problems, where the nonconvex term is possibly nonsmooth. Their algorithm adopts the proximal gradient algorithm with extrapolation and a safe-guarding policy to minimize the smoothed objective function for better practical and theoretical performance. Moreover, the algorithm uses a easily checking rule to update the smoothing parameter to ensure that any accumulation point of the generated sequence is an (affine-scaled) Clarke stationary point of the original nonsmooth and nonconvex problem. Their experimental results indicate the effectiveness of the proposed algorithm.

In this paper, the authors study the 1D steady Boltzmann flow in a channel. The walls of the channel are assumed to have vanishing velocity and given temperatures θ ₀ and θ ₁. This problem was studied by Esposito-Lebowitz-Marra (1994, 1995) where they showed that the solution tends to a local Maxwellian with parameters satisfying the compressible Navier-Stokes equation with no-slip boundary condition. However, a lot of numerical experiments reveal that the fluid layer does not entirely stick to the boundary. In the regime where the Knudsen number is reasonably small, the slip phenomenon is significant near the boundary. Thus, they revisit this problem by taking into account the slip boundary conditions. Following the lines of [Coron, F., Derivation of slip boundary conditions for the Navier-Stokes system from the Boltzmann equation, J. Stat. Phys., 54(3–4), 1989, 829–857], the authors will first give a formal asymptotic analysis to see that the flow governed by the Boltzmann equation is accurately approximated by a superposition of a steady CNS equation with a temperature jump condition and two Knudsen layers located at end points. Then they will establish a uniform L ^∞ estimate on the remainder and derive the slip boundary condition for compressible Navier-Stokes equations rigorously.

Convergence and analytic extension are of fundamental importance in the mathematical construction and study of conformal field theory. The author reviews some main convergence results, conjectures and problems in the construction and study of conformal field theories using the representation theory of vertex operator algebras. He also reviews the related analytic extension results, conjectures and problems. He discusses the convergence and analytic extensions of products of intertwining operators (chiral conformal fields) and of q-traces and pseudo-q-traces of products of intertwining operators. He also discusses the convergence results related to the sewing operation and the determinant line bundle and a higher-genus convergence result. He then explains conjectures and problems on the convergence and analytic extensions in orbifold conformal field theory and in the cohomology theory of vertex operator algebras.

Given a bounded symmetric domain Ω the author considers the geometry of its totally geodesic complex submanifolds S ⊂ Ω. In terms of the Harish-Chandra realization Ω ⋐ ℂ ⁿ and taking S to pass through the origin 0 ∈ Ω, so that S = E ⋂ Ω for some complex vector subspace of ℂ ⁿ, the author shows that the orthogonal projection ρ: Ω → E maps Ω onto S, and deduces that S ⊂ Ω is a holomorphic isometry with respect to the Carathéodory metric. His first theorem gives a new derivation of a result of Yeung’s deduced from the classification theory by Satake and Ihara in the special case of totally geodesic complex submanifolds of rank 1 and of complex dimension ≥ 2 in the Siegel upper half plane ${{\cal H}_g}$, a result which was crucial for proving the nonexistence of totally geodesic complex suborbifolds of dimension ≥ 2 on the open Torelli locus of the Siegel modular variety ${{\cal A}_g}$ by the same author. The proof relies on the characterization of totally geodesic submanifolds of Riemannian symmetric spaces in terms of Lie triple systems and a variant of the Hermann Convexity Theorem giving a new characterization of the Harish-Chandra realization in terms of bisectional curvatures.

Let W be a closed area enlargeable manifold in the sense of Gromov-Lawson and M be a noncompact spin manifold, the authors show that the connected sum M#W admits no complete metric of positive scalar curvature. When W = T ⁿ, this provides a positive answer to the generalized Geroch conjecture in the spin setting.