Here we study arithmetically Cohen-Macaulay rank $2$ vector bundles with trivial determinant on $K3$ and Enriques surfaces.

In this paper, we prove a common fixed point theorem for a family of selfmappings satisfying a general contractive condition of operator type.

In this paper, we consider the classes of analytic functions which are defined by conditions joining ideas of analytic functions with generalized bounded turning and bounded boundary rotation. Inclusion and radii results for these classes are studied.

We apply term algebraic graphs for an infinite family of graphs for which the vertex set and the neighbourhood of each vertex are quasiprojective varieties over the commutative ring $K$. For each integral domain $K$ with unity of characteristic $\ne 2$ and integral $m\ge 2$ we construct an edge transitive graph ${\mathrm{\Gamma }}_m\left(K\right)$ of girth $\ge m$ and diameter bounded by the constant independent on $K$. In particular, for each $m$ we have a family of algebraic small world graphs $\mathrm{\Gamma }(m,F_{p^s}),s=1,2,\dots$ over $F_p$, where $p$ is prime, of girth $\ge m$.

In this paper, a class of complementarity problem and three classes of variational inequalities in real Banach spaces are introduced, and the equivalence among them are established under certain conditions. Several coercivity conditions are introduced for the existence of solutions of the generalized complementarity problem. Our results can be viewed as extension and generalization of the recent paper [N. J. Huang, J. Li, D. O’Regan, Generalized f-complementarity problems in Banach spaces, vol. 142, pg. 3828-3840, 2008].

Convexity and log convexity results are established for sums involving ratios of binomial coefficients. We utilise recent results in which integral identities have been given to represent sums involving ratios of binomial coefficients.

Let ${\tau }_M$ be any preradical for $\sigma \left[M\right]$ and $N$ any module in $\sigma \left[M\right]$. A module $N$ is called ${\tau }_M$-lifting if for every submodule $K$ of $N$, there is a decomposition $K=A\oplus B$, such that $A$ is a direct summand of $N$ and $B\subseteq {\tau }_M\left(N\right)$. We call $N$ is (strongly) $FI‐{\tau }_M$-lifting if for every fully invariant submodule $K$ of $N$, there is a decomposition $K=A\oplus B$, such that $A$ is a (fully invariant) direct summand of $N$ and $B\subseteq {\tau }_M\left(N\right)$. The class of $\mathrm{FI}‐{\tau }_M$-lifting modules properly contains the class of ${\tau }_M$-lifting modules and the class of strongly $\mathrm{FI}‐{\tau }_M$-lifting modules. In this paper we investigate whether the class of (strongly) $\mathrm{FI}‐{\tau }_M$-lifting modules are closed under particular class of submodules, direct summands and direct sums.