1. D. A. Darling, On a class of problems related to the random division of an interval, Ann. Math. Statist. 24 (1953) 239–253. 2. B. V. Gnedenko, The Theory of Probability, 4th ed., Chelsea, New York, 1968. 3. R. Honsberger, Ingenuity in Mathematics, Mathematical Association of America, Washington, D.C., 1970. 4. M. G. Kendall and P. A. P. Moran, Geometrical Probability, Griffin, London, 1963. ...

We study a geometrically constrained coalescence model derived from spin systems. Given two probability distributions PR and PB on the positive reals with finite means, colour the real line alternately with red and blue intervals so that the lengths of the red intervals have distribution PR, the lengths of the blue intervals have distribution PB , and distinct intervals have independent lengths...

We study the problem of designing prediction markets for random variables with continuous or countably infinite outcomes on the real line. Our interval betting languages allow traders to bet on any interval of their choice. Both the call market mechanism and two automated market maker mechanisms, logarithmic market scoring rule (LMSR) and dynamic parimutuel markets (DPM), are generalized to han...

In this paper, we define and study the notion of the real-valued functions on a frame $L$. We show that $F(L) $, consisting of all frame homomorphisms from the power set of $mathbb{R}$ to a frame $ L$, is an $f$-ring, as a generalization of all functions from a set $X$ into $mathbb R$. Also, we show that $F(L) $ is isomorphic to a sub-$f$-ring of $mathcal{R}(L)$, the ring of real-valued continu...

Given a positive real Hermitian holomorphic line bundle L over a smooth real projective manifold X , the space of real holomorphic sections of the bundle L inherits for every d ∈ N∗ a L scalar product which induces a Gaussian measure. When X is a curve or a surface, we estimate the volume of the cone of real sections whose vanishing locus contains many real components. In particular, the volume...