Volume 7 (2023)

We construct the concrete categories \(\mathbf{I\text{-}Loc}\) and \(\mathbf{\mathfrak h\text{-}Loc}\) over the category \(\mathbf{Loc}\) of locales and we deduce that they are topological categories, where \(\mathbf I\) and \(\mathfrak h\) denote respectively the classes of interior and \(h\) operators of the category \(\mathbf{Loc}\) of locales.

We consider Gibbs’ definition of chemical equilibrium and connect it with dynamic equilibrium, in terms of no substance formed. We determine the activity coefficient as a function of temperature and pressure, in reactions with or without interaction of a solvent, incorporating the error terms from Raoult’s Law and Henry’s Law, if necessary. We compute the maximal reaction paths and apply the results to electrochemistry, using the Nernst equation.

In this paper, we interested in Wilker inequalities. We provide finer bounds than known previous. Moreover, bounds are obtained for the following trigonometric function
\[g_n(x) = \left(\frac{\sin(x)}{x}\right)^2 \left( 1 – \frac{2\left(\frac{2 x}{\pi}\right)^{2n+2}}{1-(\frac{2x}{\pi})^2}\right) +\frac{\tan(x)}{x}, \ n\geq 0.\]

In this work, generalized Euler’s \(\Phi_w\)-function of edge weighted graphs is defined which consists of the sum of the Euler’s \(\varphi\)-function of the weight of edges of a graph and we denote it by \(\Phi_w(G)\) and the general form of Euler’s \(\Phi_w\)-function of some standard edge weighted graphs is determined. Also, we define the divisor sum \(T_{k_w}\)-function \(T_{k_w}(G)\) of the graph \(G\), which is counting the sum of the sum of the positive divisor \(\sigma_k\)-function for the weighted of edges of a graph \(G\). It is determined a relation between generalized Euler’s \(\Phi_w\)-function and generalized divisor sum \(T_{k_w}\)-function of edge weighted graphs.