Abstract For any Liouville number $\alpha $ , all of the following are transcendental numbers: ${e}^\alpha $\log _{e}\alpha $\sin \alpha $\cos $\tan $\sinh $\cosh $\tanh $\arcsin and inverse functions evaluated at listed trigonometric hyperbolic functions, noting that wherever multiple values involved, every such value is transcendental. This remains true if ‘Liouville number’ replaced by ‘ U -...

<p style='text-indent:20px;'>In this paper we show the existence of a nonnegative solution for singular problem with logarithmic and exponential nonlinearity, namely <inline-formula><tex-math id="M1">\begin{document}$ -\Delta u = \log(u)\chi_{\{u>0\}} + \lambda f(u) $\end{document}</tex-math></inline-formula> in id="M2">\begin{document}$ \Omega id="M3"&gt...

Pseudomonas syringae pv. tomato DC3000 produces seven to eight major extracellular proteins (EXPs) in a minimal medium inducing hrp genes. Using a polyclonal antibody against DC3000 EXPs, we have determined that the production and secretion of five EXPs (EXP-60, EXP-45, EXP-43, EXP-22, and EXP-10) are under the control of the Hrp regulation and secretion system.