Tombs and cairns present a dating challenge when the human remains are unstratified, incomplete and dispersed. By considering the distribution of time intervals between deaths as a possible a priori condition of multiple burials of select groups, a Bayesian model is suggested that may constrain the uncertainty date range of the group. The method may also address the wide uncertainties seen in radiocarbon calibration on a calibration curve plateau. The mathematical justification for the choice of Log Normal intervals, between death events, is first presented, followed by worked examples that compare the treatment of groups of 22 dates using Phase then Sequence with interval gaps. Finally, scenarios of potential Select Groups are examined, to demonstrate the efficacy of this alternative heuristic model to current model treatments.

Mapping every simplex in the Delaunay mosaic of a discrete point set to the radius of the smallest empty circumsphere gives a generalized discrete Morse function. Choosing the points from a Poisson point process in ℝn, we study the expected number of simplices in the Delaunay mosaic as well as the expected number of critical simplices and nonsingular intervals in the corresponding generalized discrete gradient. Observing connections with other probabilistic models, we obtain precise expressions for the expected numbers in low dimensions. In particular, we obtain the expected numbers of simplices in the Poisson–Delaunay mosaic in dimensions n ≤ 4.

We estimate double sums

$$\begin{eqnarray}S_{{\it\chi}}(a,{\mathcal{I}},{\mathcal{G}})=\mathop{\sum }\limits_{x\in {\mathcal{I}}}\mathop{\sum }\limits_{{\it\lambda}\in {\mathcal{G}}}{\it\chi}(x+a{\it\lambda}),\quad 1\leq a<p-1,\end{eqnarray}$$

${\it\chi}$

$p$

${\mathcal{I}}=\{1,\dots ,H\}$

${\mathcal{G}}$

$T$

$p$

$S_{{\it\chi}}(a,{\mathcal{I}},{\mathcal{G}})$

$H\geq p^{1/4+{\it\varepsilon}}$

$T\geq p^{1/2+{\it\varepsilon}}$

${\it\varepsilon}>0$

$H$

$T$

$$\begin{eqnarray}T_{{\it\chi}}(a,{\mathcal{G}})=\mathop{\sum }\limits_{{\it\lambda},{\it\mu}\in {\mathcal{G}}}{\it\chi}(a+{\it\lambda}+{\it\mu}),\quad 1\leq a<p-1,\end{eqnarray}$$

$p$

This paper extends previously developed generalized set propagation operations to work over relationships among an arbitrary number of variables, thereby expanding the domain of engineering design problems the theory can address. It then narrows its scope to a class of functions and sets useful to designers solving engineering problems: monotonic algebraic functions and closed intervals of real numbers, proving formulas for computing the operations under these conditions. The work is aimed at the automated optimal design of electro-mechanical systems from catalogs of parts; an electronic example illustrates.

The document is published in collaboration with the following organisations: the European Society of Emergency Medicine (Leuven); the European Resuscitation Council (Antwerpen); the Air Medical Physician Association (Salt Lake City, US); the German Interdisciplinary Association of Critical Care Medicine; and the German Society of Anaesthesiology and Intensive Care. The document is to be published jointly in the following journals: 1) Trauma Care (ITACCS); 2) Resuscitation; 3) Prehospital and Disaster Medicine; 4) European Journal of Emergency Medicine; 5) Trauma and Emergency Medicine Journal (SA); 6) Emergency Medicine(Norway); 7) JEUR; and 8) Notfall und Rettungsmedizin (Germany).