The moving average (MA) chart, the exponentially weighted moving average (EWMA) chart and the cumulative sum (CUSUM) chart are the most popular schemes for detecting shifts in a relevant process parameter. Any control chart system of span $k$ is specified by a partition of the space ${\mathbb{R}}^{k}$ into three disjoint parts. We call this partition as the control chart frame of span $k.$ A shift in the process parameter is signalled at time $t$ by having the vector of the last $k$ sample characteristics fall out of the...

This paper deals with conditions of compatibility of a system of copulas and with bounds of general Fréchet classes. Algebraic search for the bounds is interpreted as a solution to a linear system of Diophantine equations. Classical analytical specification of the bounds is described.

Algebraic bounds of Fréchet classes of copulas can be derived from the fundamental attributes of the associated copulas. A minimal system of algebraic bounds and related basic bounds can be defined using properties of pointed convex polyhedral cones and their relationship with non-negative solutions of systems of linear homogeneous Diophantine equations, largely studied in Combinatorics. The basic bounds are an algebraic improving of the Fréchet-Hoeffding bounds. We provide conditions of compatibility...

In this note, we show that the model obtained by finite support iteration of a sequence of generic extensions of models of ZFC of length $\omega $ is sometimes the smallest common extension of this sequence and very often it is not.

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