In 1918 RamanujanJ made the following conjecture: If ç = 5, 7, or 11, and if 24^ — 1 is divisible by q, then the number p(n) of unrestricted partitions of n is divisible by q. Ramanujan himself proved this conjecture to be true in casej q = S, 7, 5, and 7, and also§ for q=ll and l l 2 . It has since been proved|| for q = 5. Some modification of the conjecture is necessary, however, since, as Ch...

An element d of a semigroup S is called divisible if it has roots of arbitrary order; that is, for every n ∈ N there is an element dn in S such that dn = d . If the elements dn can be taken in a prescribed subset D of S then d is said to be divisible in D . In the algebraic as well as in the topological theory of groups and semigroups divisibility is the major basic concept which allows the int...

The procurement decision problem of divisible goods is investigated, and an optimization decision method for procuring a kind of divisible goods (such as coal, oil, electricity, gas, etc.) is presented based on multi-attribute auction. Firstly, the procurement decision problem of divisible goods is converted into a multi-attribute auction of divisible goods. Correspondingly, several important a...

We prove that a stationary max–infinitely divisible process is mixing (ergodic) iff its dependence function converges to 0 (is Cesaro summable to 0). These criteria are applied to some classes of max–infinitely divisible processes.

Clinical interviews administered to third- to sixth-graders explored children's conceptualizations of rational number and of certain extensive physical quantities. We found within child consistency in reasoning about diverse aspects of rational number. Children's spontaneous acknowledgement of the existence of numbers between 0 and 1 was strongly related to their induction that numbers are infi...

We study the freely infinitely divisible distributions that appear as the laws of free subordinators. This is the free analog of classically infinitely divisible distributions supported on [0,∞), called the free regular measures. We prove that the class of free regular measures is closed under the free multiplicative convolution, t boolean power for 0 ≤ t ≤ 1, t free multiplicative power for t ...

A divisible load (job, task) can be arbitrarily divided into independent small load fractions which are assigned to processors in a parallel or distributed computing system for simultaneous processing. Given an arbitrarily divisible load without precedence constraint and a parallel/distributed computing system with communication delays, the problem of divisible load distribution is to find the ...