Let N ≥ 2 be an integer and let 1 ≤ i ≤ N − 1. Consider a gambler who starts with an initial fortune of $i and then on each successive gamble either wins $1 or loses $1 independent of the past with probabilities p and q = 1 − p respectively. Let Xn denote the total fortune after the nth gamble. The gambler’s objective is to reach a total fortune of $N , without first getting ruined (running out...

We shall see what Locke means by this in more detail below, but a brief note is sufficient to show that there is a prima facie conflict. Locke explains that “[w]henever the Mind refers any of its Ideas to any things extraneous to them, they are then capable to be called true or false ... the Mind in such a reference makes a tacit Supposition of their Conformity to that Thing” (2.32.2). That is, an

We discuss our recent solution to Siegel’s 1943 problem concerning the smallest co-volume lattices of hyperbolic 3-space. Over the last few decades the theory of Kleinian groups—discrete groups of isometries of hyperbolic 3-space—has flourished because of its intimate connectionswith low-dimensional topology and geometry and has been inspired by thediscoveries ofW. P. Thurston. The culminationm...

SECTION 1: INTRODUCTION. Lagrange proved that any positive integer was the sum of four or fewer numbers of the form x with x a positive integer. Waring asked if given an n ≥ 2, there is an f = f(n) such that every positive integer is the sum of f or fewer numbers of the form x with x a positive integer. Hilbert showed the answer was yes, via a very difficult and sophisticated proof. Subsequentl...

This paper deals with optimal investment and redistribution of the free reserves connected to life and pension insurance contracts in form of dividends and bonus. Formulated appropriately this problem can be viewed as a modification of Merton’s problem of optimal consumption and investment with a very particular form of consumption and utility hereof. Both are linked to a finite state Markov ch...

We deal with a recently proposed method of Chubanov [1], for solving linear homogeneous systems with positive variables. We use Nesterov's excessive gap method in the basic procedure. As a result, the iteration bound for the basic procedure is reduced by the factor $nsqrt{n}$. The price for this improvement is that the iterations are more costly, namely $O(n^2 )$ instead of $O(n)$. The overall ...