The authors prove a sufficient stochastic maximum principle for the optimal control of a forward-backward Markov regime switching jump diffusion system and show its connection to dynamic programming principle. The result is applied to a cash flow valuation problem with terminal wealth constraint in a financial market. An explicit optimal strategy is obtained in this example.

This paper deals with strong laws of large numbers for sublinear expectation under controlled 1st moment condition. For a sequence of independent random variables, the author obtains a strong law of large numbers under conditions that there is a control random variable whose 1st moment for sublinear expectation is finite. By discussing the relation between sublinear expectation and Choquet expectation, for a sequence of i.i.d random variables, the author illustrates that only the finiteness of uniform 1st moment for sublinear expectation cannot ensure the validity of the strong law of large numbers which in turn reveals that our result does make sense.

The Boltzmann equation with external potential force exists a unique equilibrium—local Maxwellian. The author constructs the nonlinear stability of the equilibrium when the initial datum is a small perturbation of the local Maxwellian in the whole space ℝ³. Compared with the previous result [Ukai, S., Yang, T. and Zhao, H.-J., Global solutions to the Boltzmann equation with external forces, Anal. Appl. (Singap.), 3, 2005, 157–193], no smallness condition on the Sobolev norm H ¹ of the potential is needed in our arguments. The proof is based on the entropy-energy inequality and the L ² − L ^∞ estimates.

Let $\mathcal{A}$ be a von Neumann algebra with no central abelian projections. It is proved that if an additive map δ : $\mathcal{A}$ → $\mathcal{A}$ satisfies δ([[a, b], c]) = [[δ(a), b], c]+[[a, δ(b)], c]+ [[a, b], δ(c)] for any a, b, c ∈ $\mathcal{A}$ with ab = 0 (resp. ab = P, where P is a fixed nontrivial projection in $\mathcal{A}$), then there exist an additive derivation d from $\mathcal{A}$ into itself and an additive map f : $\mathcal{A}$ → $\mathcal{Z}_\mathcal{A}$ vanishing at every second commutator [[a, b], c] with ab = 0 (resp. ab = P) such that δ(a) = d(a) + f(a) for any a ∈ $\mathcal{A}$.

This paper deals with a constrained stochastic linear-quadratic (LQ for short) optimal control problem where the control is constrained in a closed cone. The state process is governed by a controlled SDE with random coefficients. Moreover, there is a random jump of the state process. In mathematical finance, the random jump often represents the default of a counter party. Thanks to the Itô-Tanaka formula, optimal control and optimal value can be obtained by solutions of a system of backward stochastic differential equations (BSDEs for short). The solvability of the BSDEs is obtained by solving a recursive system of BSDEs driven by the Brownian motions. The author also applies the result to the mean variance portfolio selection problem in which the stock price can be affected by the default of a counterparty.

To determine the stable homotopy groups of spheres π _*(S) is one of the central problems in homotopy theory. Let p be a prime greater than 5. The authors make use of the May spectral sequence and the Adams spectral sequence to prove the existence of a ϖ _n-related family of homotopy elements, β ₁ ω _n γ _s, in the stable homotopy groups of spheres, where n > 3, 3 ≤ s < p − 2 and the ϖ _n-element was detected by X. Liu.

Biharmonic maps are generalizations of harmonic maps. A well-known result on harmonic maps between surfaces shows that there exists no harmonic map from a torus into a sphere (whatever the metrics chosen) in the homotopy class of maps of Brower degree ±1. It would be interesting to know if there exists any biharmonic map in that homotopy class of maps. The authors obtain some classifications on biharmonic maps from a torus into a sphere, where the torus is provided with a flat or a class of non-flat metrics whilst the sphere is provided with the standard metric. The results in this paper show that there exists no proper biharmonic maps of degree ±1 in a large family of maps from a torus into a sphere.

Let M ⁴ be a closed minimal hypersurface in $\mathbb{S}^5$ with constant nonnegative scalar curvature. Denote by f ₃ the sum of the cubes of all principal curvatures, by g the number of distinct principal curvatures. It is proved that if both f ₃ and g are constant, then M ⁴ is isoparametric. Moreover, the authors give all possible values for squared length of the second fundamental form of M ⁴. This result provides another piece of supporting evidence to the Chern conjecture.

The authors consider a model of ferromagnetic material subject to an electric current, and prove the local in time existence of very regular solutions for this model in the scale of H ^k spaces. In particular, they describe in detail the compatibility conditions at the boundary for the initial data.

This paper deals with the function u which satisfies △ ^k u = 0, where k ≥ 2 is an integer. Such a function u is called a polyharmonic function. The author gives an upper bound of the measure of the nodal set of u, and shows some growth property of u.