We formulate and prove the archimedean period relations for Rankin–Selberg convolutions for ${\mathrm {GL}}(n)\times {\mathrm {GL}}(n-1)$. As a consequence, we prove the period relations for critical values of the Rankin–Selberg L-functions for ${\mathrm {GL}}(n)\times {\mathrm {GL}}(n-1)$ over arbitrary number fields.

The Lusternik–Schnirelmann category of the quaternionic Grassmannian

is known to be k(n − k). In this paper we show that this result can be deduced from Morse theory.

In the manufacturing industry, many product characteristics are of one-sided tolerances. The process capability indices Cpu (u, v) and Cpl (u, v) can be used to measure process performance. Most research work related to capability indices assumes no gauge measurement errors. This assumption insufficiently reflects real situations even when advanced measuring instruments are used. In this paper we show that using a critical value without taking into account these errors, severely underestimates the α-risk which causes a less accurate testing capacity. In order to improve the results we suggest the use of an adjusted critical value, and we give a Maple program to get it. An example in a polymer granulates manufactory is presented to illustrate this approach.

Most research works related to process capability indices assume no gauge measurement errors. However, such an assumption inadequately reflects real situations even when advanced measuring instruments are employed. If we do not take into account these errors, conclusions drawn from process capability are therefore unreliable. In this paper we study the sampling distribution of capability indices Cp''(u,v) in the presence of measurements errors, and when small subsamples data are collected from past “in-control”. We show that using a critical value without taking into account these errors, severely underestimates the α-risk which causes a less accurate testing capacity. To improve the results we suggest the use of an adjusted critical value, and we give a Maple program to get it. An example in a nougat manufactory is presented to illustrate this approach.

A continuum percolation model on is considered. Using a renormalization technique developed by Grimmett and Marstrand, we show a continuum analogue of their results. We prove the critical value of the percolation equals the limit of the critical value of a slice as the thickness of the slice tends to infinity. We also prove that the effective conductivity in the model is bounded from below by a positive constant in the supercritical case.

Instead of the basic contact process on with infection rate λ we consider for m ≧ 0 the Markov process starting with ξ0(k) = 1 for k ≧ 0 and ξ0(k)= 0 for k < 0 and with changing only those k which are at most m places to the right of the left-most infected cell. For m = 0, 1,· ··, 14 direct computations give critical values which are lower bounds for the critical value of the original basic contact process.