In this article, we formalize the definition of divisible Z-module and its properties in the Mizar system [3]. We formally prove that any non-trivial divisible Z-modules are not finitely-generated. We introduce a divisible Z-module, equivalent to a vector space of a torsion-free Z-module with a coefficient ring Q. Z-modules are important for lattice problems, LLL (Lenstra, Lenstra and Lovász) b...

Relativizing an idea from multiplicity theory, we say that an element x of a von Neumann algebra M is n-divisible if W (x) ∩ M unitally contains a factor of type In. We decide the density of the n-divisible operators, for various n, M, and operator topologies. The most sensitive case is σ-strong density in II1 factors, which is closely related to the McDuff property. We make use of Voiculescu’s...

Existence of a monetary steady state is established in a random matching model with divisible goods, divisible money, an arbitrary bound on individual money holdings, and take-it-or-leave-it offers by consumers. The monetary steady state shown to exist has nice properties: the value function, defined on money holdings, is strictly increasing and strictly concave, and the distribution over money...

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.