For natural numbers $r$ and $n\ge 2$ a complete classification of natural affinors on the natural bundle ${\left(J^r{T}^{*}\right)}^{*}$ dual to $r$-jet prolongation $J^r{T}^{*}$ of the cotangent bundle over $n$-manifolds is given.

Let ${\lambda }_1\left(Q\right)$ be the first eigenvalue of the Sturm-Liouville problem $$y^{\text{'}\text{'}}-Q\left(x\right)y+\lambda y=0,y\left(0\right)=y\left(1\right)=0,0<x<1.$$ We give some estimates for $m_{\alpha ,\beta ,\gamma }={inf}_{Q\in T_{\alpha ,\beta ,\gamma }}{\lambda }_1\left(Q\right)$ and $M_{\alpha ,\beta ,\gamma }={sup}_{Q\in T_{\alpha ,\beta ,\gamma }}{\lambda }_1\left(Q\right)$, where $T_{\alpha ,\beta ,\gamma }$ is the set of real-valued measurable on $\left[0,1\right]$ $x^{\alpha }{(1-x)}^{\beta }$-weighted $L_{\gamma }$-functions $Q$ with non-negative values such that ${\int }_0^1{x}^{\alpha }{(1-x)}^{\beta }{Q}^{\gamma }\left(x\right)\mathrm{d}x=1$ $(\alpha ,\beta ,\gamma \in ℝ,\gamma \ne 0)$.