The authors obtain various versions of the Omori-Yau’s maximum principle on complete properly immersed submanifolds with controlled mean curvature in certain product manifolds, in complete Riemannian manifolds whose k-Ricci curvature has strong quadratic decay, and also obtain a maximum principle for mean curvature flow of complete manifolds with bounded mean curvature. Using the generalized maximum principle, an estimate on the mean curvature of properly immersed submanifolds with bounded projection in N ₁ in the product manifold N ₁ × N ₂ is given. Other applications of the generalized maximum principle are also given.

The author introduces the w-function defined on the considered spacelike graph M. Under the growth conditions w = o(log z) and w = o(r), two Bernstein type theorems for M in ℝ _m ^n+m are got, where z and r are the pseudo-Euclidean distance and the distance function on M to some fixed point respectively. As the ambient space is a curved pseudo-Riemannian product of two Riemannian manifolds (Σ₁, g ₁) and (Σ₂, g ₂) of dimensions n and m, a Bernstein type result for n = 2 under some curvature conditions on Σ₁ and Σ₂ and the growth condition w = o(r) is also got. As more general cases, under some curvature conditions on the ambient space and the growth condition w = o(r) or $w = 0\left( {\sqrt r } \right)$, the author concludes that if M has parallel mean curvature, then M is maximal.

This paper concerns a system of nonlinear wave equations describing the vibrations of a 3-dimensional network of elastic strings. The authors derive the equations and appropriate nodal conditions, determine equilibrium solutions, and, by using the methods of quasilinear hyperbolic systems, prove that for tree networks the natural initial, boundary value problem has classical solutions existing in neighborhoods of the “stretched” equilibrium solutions. Then the local controllability of such networks near such equilibrium configurations in a certain specified time interval is proved. Finally, it is proved that, given two different equilibrium states satisfying certain conditions, it is possible to control the network from states in a small enough neighborhood of one equilibrium to any state in a suitable neighborhood of the second equilibrium over a sufficiently large time interval.

The authors consider a family of smooth immersions F (·, t): M ⁿ → ℝ ⁿ⁺¹ of closed hypersurfaces in ℝ ⁿ⁺¹ moving by the mean curvature flow $\frac{{\partial F(p,t)}}{{\partial t}} = - H(p,t) \cdot \nu (p,t)$ for t ∈ [0, T). They show that if the norm of the second fundamental form is bounded above by some power of mean curvature and the certain subcritical quantities concerning the mean curvature integral are bounded, then the flow can extend past time T. The result is similar to that in [6–9].

The authors prove an almost sure central limit theorem for partial sums based on an irreducible and positive recurrent Markov chain using logarithmic means, which realizes the extension of the almost sure central limit theorem for partial sums from an i.i.d. sequence of random variables to a Markov chain.

The Wielandt subgroup of a group G, denoted by w(G), is the intersection of the normalizers of all subnormal subgroups of G. In this paper, the authors show that for a p-group of maximal class G, either w _i(G) = ζ _i(G) for all integer i or w _i(G) = ζ _i+1(G) for every integer i, and w(G/K) = ζ(G/K) for every normal subgroup K in G with K ≠ 1. Meanwhile, a necessary and sufficient condition for a regular p-group of maximal class satisfying w(G) = ζ ₂(G) is given. Finally, the authors prove that the power automorphism group PAut(G) is an elementary abelian p-group if G is a non-abelian p-group with elementary $\zeta (G) \cap \mho _1 (G)$.

This paper deals with the problem of sharp observability inequality for the 1-D plate equation w _tt + w _xxxx + q(t, x)w = 0 with two types of boundary conditions w = w _xx = 0 or w = w _x = 0, and q(t, x) being a suitable potential. The author shows that the sharp observability constant is of order $\exp \left( {C\left\| q \right\|_\infty ^{\tfrac{2}{7}} } \right)$ for ‖q‖_∞ ≥ 1. The main tools to derive the desired observability inequalities are the global Carleman inequalities, based on a new point wise inequality for the fourth order plate operator.

This paper deals with a coupled system of fourth-order parabolic inequalities |u| _t ≥ −Δ² u + |v| ^q, |v| _t ≥ −Δ² v + |u| ^p in $\mathbb{S} = \mathbb{R}^n \times \mathbb{R}^ +$ with p, q > 1, n ≥ 1. A Fujita-Liouville type theorem is established that the inequality system does not admit nontrivial nonnegative global solutions on $\mathbb{S}$ whenever $\tfrac{n}{4} \leqslant \max \left( {\tfrac{{p + 1}}{{pq - 1}} - \tfrac{{q + 1}}{{pq - 1}}} \right)$. Since the general maximum-comparison principle does not hold for the fourth-order problem, the authors use the test function method to get the global non-existence of nontrivial solutions.

A vertex of a graph is said to dominate itself and all of its neighbors. A double dominating set of a graph G is a set D of vertices of G, such that every vertex of G is dominated by at least two vertices of D. The double domination number of a graph G is the minimum cardinality of a double dominating set of G. For a graph G = (V,E), a subset D ⊆ V (G) is a 2-dominating set if every vertex of V (G) \ D has at least two neighbors in D, while it is a 2-outer-independent dominating set of G if additionally the set V (G)\D is independent. The 2-outer-independent domination number of G is the minimum cardinality of a 2-outer-independent dominating set of G. This paper characterizes all trees with the double domination number equal to the 2-outer-independent domination number plus one.

Backward doubly stochastic differential equations driven by Brownian motions and Poisson process (BDSDEP) with non-Lipschitz coefficients on random time interval are studied. The probabilistic interpretation for the solutions to a class of quasilinear stochastic partial differential-integral equations (SPDIEs) is treated with BDSDEP. Under non-Lipschitz conditions, the existence and uniqueness results for measurable solutions to BDSDEP are established via the smoothing technique. Then, the continuous dependence for solutions to BDSDEP is derived. Finally, the probabilistic interpretation for the solutions to a class of quasilinear SPDIEs is given.

Let G be a permutation group on a set Ω with no fixed points in Ω, and m be a positive integer. Then the movement of G is defined as $move(G): = \mathop {\sup }\limits_\Gamma \left\{ {\left| {\Gamma ^g \backslash \Gamma } \right|\left| {g \in G} \right.} \right\}$. It was shown by Praeger that if move(G) = m, then |Ω| ≤ 3m+ t−1, where t is the number of G-orbits on Ω. In this paper, all intransitive permutation groups with degree 3m+t−1 which have maximum bound are classified. Indeed, a positive answer to her question that whether the upper bound |Ω| = 3m + t − 1 for |Ω| is sharp for every t > 1 is given.

The paper deals with the controllability of a heat equation. It is well-known that the heat equation y _t − Δ _y = uχ _E in (0, T)×Ω with homogeneous Dirichlet boundary conditions is null controllable for any T > 0 and any open nonempty subset E of Ω. In this note, the author studies the case that E is an arbitrary measurable set with positive measure.