The authors give a stochastic maximum principle for square-integrable optimal control of linear stochastic systems. The control domain is not necessarily convex and the cost functional can have a quadratic growth. In particular, they give a stochastic maximum principle for the linear quadratic optimal control problem.

Considering the bundle of 2-jets as a realization of the holomorphic manifold over 3-dimensional nilpotent algebra, the authors introduce a new class of lifts of connections in the bundle of 2-jets which is a generalization of the complete lifts.

This paper investigates the stabilization of a Bresse system with internal damping and logarithmic source. The authors use the potential well theory. For initial data in the stability set created by the Nehari surface, the existence of a global solution is proved by using Faedo-Galerkin’s approximation. The Nakao theorem gives the exponential decay. A numerical approach is presented to illustrate the results obtained.

By constructing counterexamples, the authors show that the fixed subgroups are not compressed in direct products of free and surface groups, and hence negate a conjecture in [Zhang, Q., Ventura, E. and J. Wu, Fixed subgroups are compressed in surface groups, Internat. J. Algebra Comput., 25, 2015, 865–887].

A linear forest is a graph consisting of paths. In this paper, the authors determine the maximum number of edges in an (m, n)-bipartite graph which does not contain a linear forest consisting of paths on at least four vertices for n ≥ m when m is sufficiently large.

Edge-to-edge tilings of the sphere by congruent a²bc-quadrilaterals are classified as 3 classes: (1) A 1-parameter family of quadrilateral subdivisions of the octahedron with 24 tiles, and a flip modification for one special parameter; (2) a 2-parameter family of 2-layer earth map tilings with 2n tiles for each n ≥ 3; (3) a 3-layer earth map tiling with 8n tiles for each n ≥ 2, and two flip modifications for each odd n. The authors also describe the moduli of parameterized tilings and provide the full geometric data for all tilings.

Let ℤ/mℤ be the ring of residual classes modulo m, and let A and B be nonempty subsets of ℤ/mℤ. In this paper, the authors give the structure of A and B for which ∣A + B∣ = ∣A∣ + ∣B∣ − 1 = m − 2.

This paper aims to study the Berger type deformed Sasaki metric g_BS on the second order tangent bundle T²M over a bi-Kählerian manifold M. The authors firstly find the Levi-Civita connection of the Berger type deformed Sasaki metric g_BS and calculate all forms of Riemannian curvature tensors of this metric. Also, they study geodesics on the second order tangent bundle T²M and bi-unit second order tangent bundle $T_{1,1}^{2}M$, and characterize a geodesic of the bi-unit second order tangent bundle in terms of geodesic curvatures of its projection to the base. Finally, they present some conditions for a section σ: M → T²M to be harmonic and study the harmonicity of the different canonical projections and inclusions of (T²M, g_BS). Moreover, they search the harmonicity of the Berger type deformed Sasaki metric g_BS and the Sasaki metric g_S with respect to each other.

Consider a branching random walk with a random environment in time in the d-dimensional integer lattice. The branching mechanism is governed by a supercritical branching process, and the particles perform a lazy random walk with an independent, non-identical increment distribution. For A ⊂ ℤ^d, let ℤ_n(A) be the number of offsprings of generation n located in A. The exact convergence rate of the local limit theorem for the counting measure Z_n(·) is obtained. This partially extends the previous results for a simple branching random walk derived by Gao (2017, Stoch. Process Appl.).