A Phase-Field Perspective on Mereotopology

Mereotopology is a concept rooted in analytical philosophy. The phase-field concept is based on mathematical physics and finds applications in materials engineering. The two concepts seem to be disjoint at a first glance. While mereotopology qualitatively describes static relations between things like x isConnected y (topology) or x isPartOf y (mereology) by first order logic and Boolean algebra, the phase-field concept describes the geometric shape of things and its dynamic evolution by drawing on a scalar field. The geometric shape of any thing is defined by its boundaries to one or more neighboring things. The notion and description of boundaries thus provides a bridge between mereotopology and the phase-field concept. The present article aims to relate phase-field expressions describing boundaries and especially triple junctions to their Boolean counterparts in mereotopology and contact algebra. An introductory overview on mereotopology is followed by an introduction to the phase-field concept already indicating first relations to mereo- topology. Mereotopological axioms and definitions are then discussed in detail from a phase-field perspective. A dedicated section introduces and discusses further notions of the isConnected relation emerging from the phase-field perspective like isSpatiallyConnected, isTemporallyConnected, isPhysicallyConnected, isPathConnected and wasConnected. Such relations introduce dynamics and thus physics into mereotopology as transitions from isDisconnected to isPartOf can be described.

A study on Contact Algebra by Browerian Logic

There is a close relation between Boolean logic or two-valued logic and an electric di-contact algebra. Two-valued logic is concerned with propositions which are either true or false and which can be combined in various ways. Similarly, the switches of circuits are activated by contacts which, open or closed, can be combined in analogous ways. But there are positions which are not two-valued - a generalisation of truth values of a proposition leads to an n-valued logic. It is then natural to raise the query whether it is possible to generalise the notion of switching contact analogous to the generalisation of truth value of a proposition. If it is so, does there exist an isomorphism between propositional algebra in n-valued logic and a structure in switching circuits based on contact values? The solution of the problems leads to a new algebra. Here we have reviewed this contact algebra by Browerian logic.

A Note on the Rank of a Restricted Lie Algebra

In this short note, we study the rank of a restricted Lie algebra (𝔤, [p]), and give some applications which concern the dimensions of non-trivial irreducible modules. We also compute the rank of the restricted contact algebra.

On dimension andweight of a local contact algebra

As proved in [16], there exists a duality ?t between the category HLC of locally compact Hausdorff spaces and continuous maps, and the category DHLC of complete local contact algebras and appropriate morphisms between them. In this paper, we introduce the notions of weight wa and of dimension dima of a local contact algebra, and we prove that if X is a locally compact Hausdorff space then w(X) = wa(?t(X)), and if, in addition, X is normal, then dim(X) = dima(?t(X)).

Relative Darboux Theorem for Singular Manifolds and Local Contact Algebra

In 1999 V. Arnol’d introduced the local contact algebra: studying the problem of classification of singular curves in a contact space, he showed the existence of the ghost of the contact structure (invariants which are not related to the induced structure on the curve). Our main result implies that the only reason for existence of the local contact algebra and the ghost is the difference between the geometric and (defined in this paper) algebraic restriction of a 1-form to a singular submanifold. We prove that a germ of any subset N of a contact manifold is well defined, up to contactomorphisms, by the algebraic restriction to N of the contact structure. This is a generalization of the Darboux-Givental’ theoremfor smooth submanifolds of a contactmanifold. Studying the difference between the geometric and the algebraic restrictions gives a powerful tool for classification of stratified submanifolds of a contact manifold. This is illustrated by complete solution of three classification problems, including a simple explanation of V. Arnold's results and further classification results for singular curves in a contact space. We also prove several results on the external geometry of a singular submanifold N in terms of the algebraic restriction of the contact structure to N. In particular, the algebraic restriction is zero if and only if N is contained in a smooth Legendrian submanifold of M.

First Steps of Local Contact Algebra

We consider germs of mappings of a line to contact space and classify the first simple singularities up to the action of contactomorphisms in the target space and diffeomorphisms of the line. Even in these first cases there arises a new interesting interaction of local commutative algebra with contact structure.

THE ROLE OF THE CONTACT ALGEBRA IN MULTIMATRIX MODELS

In this paper we investigate the implications of the topological nature of the (1, q) matrix models. We show that the algebraic structure is enough to derive the correlation functions of pure gravity. A straightforward generalization for several primaries is presented. We further find that the recursion relations in the multimatrix models may be derived from the contact algebra alone. We show that the contact algebra, using the ghost number assignment derived from the KdV hierarchy, is consistent only for contacts with the puncture operator and its descendents. We discuss the role of the contact algebra and its consistency in general.