In 2011, Guillera [‘A new Ramanujan-like series for $1/\pi ^2$’, Ramanujan J. 26 (2011), 369–374] introduced a remarkable rational ${}_{7}F_{6}( \frac {27}{64} )$-series for ${1}/{\pi ^2}$ using the Wilf–Zeilberger (WZ) method, and Chu and Zhang later proved this evaluation using an acceleration method based on Dougall’s ${}_{5}F_{4}$-sum. Another proof of Guillera’s ${}_{7}F_{6}( \frac {27}{64} )$-series was given by Guillera in 2018, and this subsequent proof used a recursive argument involving Dougall’s sum together with the WZ method. Subsequently, Chen and Chu introduced a q-analogue of Guillera’s ${}_{7}F_{6}( \frac {27}{64} )$-series. The many past research articles concerning Guillera’s ${}_{7}F_{6}( \frac {27}{64} )$-series for ${1}/{\pi ^2}$ naturally lead to questions about similar results for other mathematical constants. We apply a WZ-based acceleration method to prove new rational ${}_{7}F_{6}( \frac {27}{64} )$- and ${}_{6}F_{5}( \frac {27}{64} )$-series for $\sqrt {2}$.

Cossette, Marceau, and Perreault derived formulas for aggregation and capital allocation based on risks following two bivariate exponential distributions. Here, we derive formulas for aggregation and capital allocation for 18 mostly commonly known families of bivariate distributions. This collection of formulas could be a useful reference for financial risk management.

In this study, a methodology to design frame-like periodic solids for isotropic symmetry by appropriate sizing of unit-cell struts is presented. The methodology utilizes the closed-form effective elastic constants of 2D frame-like periodic solids with square symmetry and 3D frame-like periodic solids with cubic symmetry, derived using the homogenization method based on equivalent strain energy. By using the closed-form effective elastic constants, an equation to enforce isotropic symmetry can be analytically constructed. Thereafter, the equation can be used to determine relative unit-cell strut sizes that are required for isotropic symmetry. The methodology is tested with 2D and 3D frame-like periodic solids with some common unit-cell topologies. Satisfactory results are observed.

This paper presents a closed-form characterization of the allocation of resources in an overlapping generations model of two-sided, partial altruism. Three assumptions are made: (i) parents and children play Markov strategies, (ii) utility takes the CRRA form, and (iii) the income of children is stochastic but proportional to the saving of parents. In families where children are rich relative to their parents, saving rates—measured as a function of the family's total resources—are higher than when children are poor relative to their parents. Income redistribution from the old to the young, therefore, leads to an increase in aggregate saving.