Narrow Results

We consider a minimal, free action, φ, of the group ℤ^d on the Cantor set X, for d ≥ 1. We introduce the notion of small positive cocycles for such an action. We show that the existence of such cocycles allows the construction of finite Kakutani–Rohlin approximations to the action. In the case, d = 1, small positive cocycles always exist and the approximations provide the basis for the Bratteli–Vershik model for a minimal homeomorphism of X. Finally, we consider two classes of examples when d = 2 and show that such cocycles exist in both.

In this paper, we study the interesting open problem of classifying the minimal Lagrangian submanifolds of dimension $n$ in complex space forms with semi-parallel second fundamental form. First, we completely solve the problem in cases $n=2,3,4$. Second, supposing further that the scalar curvature is constant for $n\ge 5$, we also give an answer to the problem by applying the classification theorem of [F. Dillen, H. Li, L. Vrancken and X. Wang, Lagrangian submanifolds in complex projective space with parallel second fundamental form, Pacific J. Math. 255 (2012) 79–115]. Finally, for such Lagrangian submanifolds in the above problem with $n\ge 3$, we establish an inequality in terms of the traceless Ricci tensor, the squared norm of the second fundamental form and the scalar curvature. Moreover, this inequality is optimal in the sense that all the submanifolds attaining the equality are completely determined.

There are quasi-ideals of an ordered semigroup which cannot be expressed as an intersection of a left ideal and a right ideal. A quasi-ideal with this intersection property is called strong quasi-ideal. We show that strong quasi-simplicity is equivalent to t-simplicity of an ordered semigroup; and hence it turns out to be the case that the ordered semigroups which are complete semilattices (chains) of t-simple subsemigroups can be characterized by their strong quasi-ideals. An ordered semigroup S is complete semilattice (chain) of t-simple subsemigroups if and only if every strong quasi-ideal of S is a completely semiprime (prime) ideal. Also we introduce and characterize the minimal strong quasi-ideals.

In a graph $G=(V,E)$, a module is a vertex subset $M$ of $V$ such that every vertex outside $M$ is adjacent to all or none of $M$. For example, $\varnothing$, $\{x\}$$(x\in V)$ and $V$ are modules of $G$, called trivial modules. A graph, all the modules of which are trivial, is prime; otherwise, it is decomposable. A vertex $x$ of a prime graph $G$ is critical if $G-x$ is decomposable. Moreover, a prime graph with $k$ noncritical vertices is called $(-k)$-critical graph. A prime graph $G$ is $k$-minimal if there is some $k$-vertex set $X$ of vertices such that there is no proper induced subgraph of $G$ containing $X$ is prime. From this perspective, Boudabbous proposes to find the $(-k)$-critical graphs and $k$-minimal graphs for some integer $k$ even in a particular case of graphs. This research paper attempts to answer Boudabbous’s question. First, we describe the $(-k)$-critical tree. As a corollary, we determine the number of nonisomorphic $(-k)$-critical tree with $n$ vertices where $k\in \{1,2,\lfloor \frac{n}{2}\rfloor \}$. Second, we provide a complete characterization of the $k$-minimal tree. As a corollary, we determine the number of nonisomorphic $k$-minimal tree with $n$ vertices where $k\le 3$.