We propose a general technique related to the polytopal Sperner lemma for proving old and new multilabeled versions of Sperner’s lemma. A notable application of this technique yields a cake-cutting theorem where the number of players and the number of pieces can be independently chosen. We also prove multilabeled versions of Fan’s lemma, a combinatorial analogue of the Borsuk-Ulam theorem, and ...

Thi spaperdi scussesstochasti cextensi onsofasi mpl eprocessal gebrai nacausal i ty-based setti ng. Atomi c acti ons are supposed to happen after a del ay that i s deter-mi ned bya stochasti c vari abl e wi th acertai n di stri buti on. Asi mpl e stochasti c type of event structures i s di scussed, restri cti ng the di stri buti on functi ons to be expo-nenti al. Acorrespondi ngoperati onalsema...

Let {Xi},i = l, • • • , be a sequence of independent and identically distributed random variables with density function f(x). We shall assume throughout that EXi=p>0. Let {Sn}, n = l, • • • , be the sequence of cumulative sums Sn= 23"-1 -^«'Let H(x) = 23»= i P(Sn ^x) and h(x)=H'(x). Since H(x) is nondecreasing, h(x) is defined for almost all x. Let A be any Borel set of positive real numbers an...