The study of Lagrangian submanifolds in K¨ahler manifolds and in the nearly K¨ahler six-sphere has been a very active field over the last quarter of century. In this article we survey the main results done during that period from Riemannian geometric point of view.

Let R be a prime algebra over a commutative ring K, d anonzero derivation of R, f(x1, .., xn) a multilinear polynomial over K inn non-commuting variables, a 2 R and m 1 a fixed integer. Supposethat f(x1, .., xn) is not central on R. If a(d(f(r1, .., rn)))m = 0, for anyr1, .., rn 2 R, then a = 0.

In this note we study modules with $(W{C_{11}}^{+})$ property. We prove thatif $M$ satisfies $(W{C_{11}}^{+})$ and $M/(Soc M)$ has finite uniform dimensionthen $M=M_{1}\oplus M_{2}$ where $M_1$ is semisimple and $M_2$ with finiteuniform dimension. In particular, if $M$ satisfies $(W{C_{11}}^{+})$ andascending chain (respectively, descending chain) condition on essentialsubmodules then $M=M_{1}\oplus M_2$ for some semisimple submodule $M_1$and Noetherian (respectively, Artinian) submodule $M_2$.

For the bounded linear operators acting on a complex separable Hilbert space ${\cal H}$, we introduce a binary relation $\sim_{sas}$ called strongly approximative similarity. It lies between the similarity and the essential similarity. For a class of biquasitriangular operators and a class of quasitriangular operators, this relation is characterized respectively. As a result, the relation $\sim_{sas}$ is an equivalent relation in this two cases.

Let $M$ be a domain in the unit $n$-sphere with smooth boundary. The purpose of this paper is to describe some inequalities between Dirichlet and Neumann eigenvalues for $M$ under certain convex restrictions on the boundary. We prove that if the mean curvature of the boundary is nonpositive, then the $k$th nonzero Neumann eigenvalue is less than or equal to the $k$th Dirichlet eigenvalue for $k=1, 2,\cdots$. Furthermore, if the second fundamental form of the boundary is nonpositive, then the $(k+\left[\frac{n-1}{2}\right])$th nonzero Neumann eigenvalue is less than or equal to the $k$th Dirichlet eigenvalue for $k=1,2,\cdots$.

In this paper, the asymptotic behavior of wavelet series at a neighborhood of a point of divergence is investigated. Our results extend the works of Reyes [8, 9].

An inequality of Hadamard’s type for convex functions and convex functions on the co-ordinates defined in a rectangle from the plane and some applications are given.

In this paper, we characterize line graphs and total graphs that are hinge-free, i.e., there is no triple of vertices $x,y,z$ such that the distance between $y$ and $z$ increases after $x$ is removed. Based on our characterizations, we show that given a graph $G$ with $n$ vertices and $m$ edges, determining its line graph and total graph to be hinge-free can be solved in O($n+m$) time. Moreover, characterizations of hinge-free iterated line graphs and total graphs are also discussed.