Staticity and regularity for zero rest-mass fields near spatial infinity on flat spacetime

Linear zero-rest-mass fields generically develop logarithmic singularities at the critical sets where spatial infinity meets null infinity. Friedrich's representation of spatial infinity is ideally suited to study this phenomenon. These logarithmic singularities are an obstruction to the smoothness of the zero-rest-mass field at null infinity and, in particular, to peeling. In the case of the spin-2 field it has been shown that these logarithmic singularities can be precluded if the initial data for the field satisfies a certain regularity condition involving the vanishing, at spatial infinity, of a certain spinor (the linearised Cotton spinor) and its totally symmetrised derivatives. In this article we investigate the relation between this regularity condition and the staticity of the spin-2 field. It is shown that while any static spin-2 field satisfies the regularity condition, not every solution satisfying the regularity condition is static. This result is in contrast with what happens in the case of General Relativity where staticity in a neighbourhood of spatial infinity and the smoothness of the field at future and past null infinities are much more closely related.

On Capacity-Filling and Substitutable Choice Rules

Each capacity-filling and substitutable choice rule is known to have a maximizer-collecting representation: There exists a list of priority orderings such that from each choice set that includes more alternatives than the capacity, the choice is the union of the priority orderings’ maximizers. We introduce the notion of a critical set and constructively prove that the number of critical sets for a choice rule determines its smallest-size maximizer-collecting representation. We show that responsive choice rules require the maximal number of priority orderings in their smallest-size maximizer-collecting representations among all capacity-filling and substitutable choice rules. We also analyze maximizer-collecting choice rules in which the number of priority orderings equals the capacity. We show that if the capacity is greater than three and the number of alternatives exceeds the capacity by at least two, then no capacity-filling and substitutable choice rule has a maximizer-collecting representation of the size equal to the capacity.

A Discussion of a Cryptographical Scheme Based in F-Critical Sets of a Latin Square

This communication provides a discussion of a scheme originally proposed by Falcón in a paper entitled “Latin squares associated to principal autotopisms of long cycles. Applications in cryptography”. Falcón outlines the protocol for a cryptographical scheme that uses the F-critical sets associated with a particular Latin square to generate access levels for participants of the scheme. Accompanying the scheme is an example, which applies the protocol to a particular Latin square of order six. Exploration of the example itself, revealed some interesting observations about both the structure of the Latin square itself and the autotopisms associated with the Latin square. These observations give rise to necessary conditions for the generation of the F-critical sets associated with certain autotopisms of the given Latin square. The communication culminates with a table which outlines the various access levels for the given Latin square in accordance with the scheme detailed by Falcón.

The size of some vanishing and critical sets

We prove that the vanishing sets of all top forms on a non-orientable manifold are at least 1-dimensional in the general case and at most $1$-codimen\-sional in the compact case. We apply these facts to show that the critical sets of some differentiable maps are at least 1-dimensional in the general case and at most 1-codimensional when the source manifold is compact.

On Hasse diagrams connected with the poset (1, 2, 7)

Representations of posets introduced in 1972 by L. O. Nazarova and A. V. Roiter, arise when solving many problems in various fields of mathematics. One of the most important problem in the theory of representations of any objects is a description of the cases of representation finite type and representation tame type. The first of these problems for posets was solved by M. M. Kleiner, and the second L. O, Nazarova. M. M. Kleiner proved that a poset has finite type if and only if it does not contain subsets of the form (1, 1, 1, 1), (2, 2, 2), (1, 3, 3), (1, 2, 5) and (И, 4), which are called the critical sets. A generalization of this criterion to the tame case was obtained by L. O. Nazarova. The corresponding sets are called supercritical and they consist of the posets (1, 1, 1, 1, 1), (1, 1, 1, 2), (2, 2, 3), (1, 3, 4), (1, 2, 6) and (И, 5). V. M. Bondarenko proposed a generalization of the critical and supercritical posets, calling them 1-oversupercritical. This paper studies the combinatorial properties of one of such sets.