Let $G$ be a $k$-edge-connected graph and let $L$ denote the subset of all vertices having odd degree in $G$. For every subset $K=\{u_1,u_2,\dots ,u_k\}$ of $L$ with $\left|K\right|\le \frac{\left|L\right|}{2}$, and for every function $h$ defined on $K$ having the property that $h(u_i)\in \left\{\lceil \frac{d_G(u_i)}{2}\rceil ,\lfloor \frac{d_G(u_i)}{2}\rfloor \right\}$ for all $u_i\in K$, there exists an orientation $D$ of $G$ such that $d_D^{+}(x)=h(x)$ when $x\in K$ and $\lfloor \frac{d_G(x)}{2}\rfloor \le d_D^{+}(x)\le \lceil \frac{d_G(x)}{2}\rceil$ when $x\in V\left(G\right)-K.$

Some new bounds for the Čebyšev functional of a pair of vectors in inner product spaces are pointed out. Reverses for the celebrated Jensen’s inequality for convex functions defined on inner product spaces are given as well.

In this note, we improve upon results of Steiner [9] and Mimuro [8] concerning the structure of non-trivial cycles in Collatz’s problem.

To investigate conditions for strongly continuous semigroups to be chaotic or hypercyclic, we consider a strongly continuous semigroup $\left\{T_t\right\}$ on a function space $C_0([0,\infty ),ℂ)$ or $L^p([0,\infty ),ℂ)$ expressed by $T_{t}f(x)=g(x,t)f(x+t)$. We also consider a strongly continuous semigroup $\left\{T_t\right\}$ on a function space $C_0([0,1],ℂ)$ or $L^p([0,1],ℂ)$ expressed by $T_{t}f(x)=q(x,t)f(e^{\gamma t}x)$ with , which have the relation to the solution semigroups to an initial value problem.

A basis of the cycle space, $\mathcal{C}(G)$, of a graph $G$ is called a $d$-fold if each edge of $G$ occurs in at most $d$ cycles of the basis. The basis number, $b\left(G\right)$, of a graph $G$ is defined to be the least integer $d$ such that $G$ has a $d$-fold basis for its cycle space. MacLane proved that a graph $G$ is planar if and only if $b\left(G\right)\le 2$. Schmeichel showed that for $n\ge 5$, $b(K_n•P_2)\le 1+b(K_n)$ Ali proved that for $n,m\ge 5$, $b(K_n•K_m)\le 3+b(K_m)+b(K_n)$. Jaradat proved that for any two bipartite graphs $G$ and $H$, $b(G\text{Λ}H)\le 5+b(G)+b(H)$. In this paper we give an upper bound of the basis number of the semi-strong product of bipartite graphs. Also, we give an example where the bound is achieved.

Certain basic inequalities, involving the squared mean curvature and one of the scalar curvature, the sectional curvature and the Ricci curvature for a submanifold of any Riemannian manifold, are obtained. Applying these results we obtain the corresponding inequalities for different kinds of submanifolds of a locally conformal Kaehler space form. Equality cases are also discussed. Finally, we also find a sufficient condition for a Lagrangian submanifold of a locally conformal Kaehler space form to be minimal.