Maximal Spacings in Several Dimensions

Programming in several dimensions

Writing a program to solve a problem is a process that can be divided into two phases: first, we invent a mental model of the solution; secondly, we map the mental model onto a physical representation. The mental model is multidimensional and syntax-free; in today’s textual programming languages, the physical representation is singledimensional and syntax-burdened. In fact, it hasn’t changed gr...

in several dimensions

We consider solutions to the Euler equations in the whole space from a certain class, which can be characterized, in particular, by finiteness of mass, total energy and momentum. We prove that for a large class of right-hand sides, including the viscous term, such solutions, no matter how smooth initially, develop a singularity within a finite time. We find a sufficient condition for the singul...

A Log Log Law for Maximal Uniform Spacings

McGill University Let X1 , X2, • . . be a sequence of independent uniformly distributed random variables on [0, 1] and Kn be the kth largest spacing induced by X 1 , X12 . We show that P(Kn < (log n log3n log 2)/n i .o .) = 1 where log, is the j times iterated logarithm. This settles a question left open in Devroye (1981) . Thus, we have lim inf(nKn log n + log3n) _ -log 2 almost surely, and li...

Contact Processes in Several Dimensions

This paper deals primarily with the basic contact processes introduced and studied by Harris [12t4] . These are random evolutions on the state space S={a l l subsets of Z a} (Zd=the d-dimensional integer lattice) with extremely simple local dynamics. Namely, if we think of the contact process (~) as representing the spread of an infection, { t eS denoting the set of infected sites at time t, th...

Companding and random quantization in several dimensions