This paper establishes a real integral representation of the reciprocal $\Gamma$ function in terms of a regularized hypersingular integral. The equivalence with the usual complex representation is demonstrated. A regularized complex representation along the usual Hankel path is derived.

In the paper, the authors survey integral representations (including the Lévy--Khintchine representations) and applications of some bivariate means (including the logarithmic mean, the identric mean, Stolarsky's mean, the harmonic mean, the (weighted) geometric means and their reciprocals, and the Toader--Qi mean) and the multivariate (weighted) geometric means and their reciprocals, derive integral representations of bivariate complex geometric mean and its reciprocal, and apply these newly-derived integral representations to establish integral representations of Heronian mean of power 2 and its reciprocal.

Industrial manipulators and parallel robots are often used for tasks like drilling or milling, that require three translational, but only two rotational degrees of freedom (“3T2R”). While kinematic models for specific mechanisms for these tasks exist, a general kinematic model for parallel robots is still missing. This paper presents the definition of the rotational component of kinematic constraints equations for parallel robots based on two reciprocal sets of Euler angles for the end-effector orientation and the orientation residual. The method allows to completely remove the redundant coordinate in 3T2R tasks and to solve the inverse kinematics for general serial and parallel robots with the gradient-descent algorithm. The functional redundancy of robots with full mobility is exploited using nullspace projection.

Reciprocal Personal/Public Protection (RPPP) featured with source control is introduced, Mask Discriminating Mouth and Nose (MDMN) is employed to serve the purpose, which includes polymer based mouth cover with optional nose cover. The new knowledge that mouth is a primary, active and dominant source of the virus has been well established, which is the base of MDMN. Source classification and related source control tools are discussed, mouth cover is recommended as the tool prioritized to use. Liquid droplets is identified as a hard issue related to mask, liquid droplets, mask fitting, comfort and facial recognition constitute real challenges of mask in addition to efficiency, All of these have been addressed with MDMN. Comparisons between MDMN and masks/face covering are taken on four aspects: efficiency and efficacy, tolerance and comfort, cost and waste, and civil rights and public interest. Mouth cover is recommended to replace the face covering and act as both a personal tool and a public utensil, mouth cover with nose cover can provide better protection than N95 etc. RPPP with MDMN, could be an alternative for lockdown, a parallel strategy to vaccine, and a collectively living way during the pandemic era. MDMN, featured with high efficiency protection, high degree comfort, easy wearing, tight-fitting, easy facial recognition & communication, reusability, cost-effective, environment friendly and scale manufacturing more readily and widely etc., is a simple and sustainable solution, which is essential for ordinary people to keep wearing it properly for protection.