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A notion of dual curve for pseudoholomorphic curves in 4--manifolds turns out to be possible only if the notion of almost complex structure structure is slightly generalized. The resulting structure is as easy (perhaps easier) to work with,…
We consider composition and division algebras over the real numbers: We note two r\^oles for the group $G_{2}$: as automorphism group of the octonions and as the isotropy group of a generic 3-form in 7 dimensions. We show why they are…
We introduce a new notation for representing labeled regular bipartite graphs of arbitrary degree. Several enumeration problems for labeled and unlabeled regular bipartite graphs have been introduced. A general algorithm for enumerating all…
Relational structures are emerging as ubiquitous mathematical machinery in the semantics of open systems of various kinds. Cartesian bicategories are a well-known categorical algebra of relations that has proved especially useful in recent…
This dissertation builds a compositional cyber-physical systems theory to develop concrete semantics relating the above diverse views necessary for safety and security assurance. In this sense, composition can take two forms. The first is…
We introduce the notion of a symmetric group parametrized by elements of a group. We show that this group is an extension of a certain subgroup of the wreath product $G \wr S_n$ by $\mathrm{H}_2(G, \mathbb{Z})$. We also discuss the…
We analyse some aspects of the notion of algebraic exponentiation introduced by the second author [16] and satisfied by the category of groups. We show how this notion provides a new approach to the categorical-algebraic question of the…
In this paper we construct two groupoids from morphisms of groupoids, with one from a categorical viewpoint and the other from a geometric viewpoint. We show that for each pair of groupoids, the two kinds of groupoids of morphisms are…
We provide a formal introduction into the classic theorems of general topology and its axiomatic foundations in set theory. In this second part we introduce the fundamental concepts of topological spaces, convergence, and continuity, as…
Combinatorial design theory studies set systems with certain balance and symmetry properties and has applications to computer science and elsewhere. This paper presents a modular approach to formalising designs for the first time using…
We propose some new method of constructing configurations, which consists in consecutive inscribing copies of one underlying configuration. A uniform characterization of the obtained class and the one introduced in our paper untitled…
The following questions are germane to our understanding of gauge-(in)variant quantities and physical possibility: how are gauge transformations and spacetime diffeomorphisms understood as symmetries, in which ways are they similar, and in…
A piecewise constant curvature manifold is a triangulated manifold that is assigned a geometry by specifying lengths of edges and stipulating that for a chosen background geometry (Euclidean, hyperbolic, or spherical), each simplex has an…
We present some of the group theoretic properties of reversing symmetry groups, and classify their structure in simple cases that occur frequently in several well-known groups of dynamical systems.
In recent years philosophers of science have explored categorical equivalence as a promising criterion for when two (physical) theories are equivalent. On the one hand, philosophers have presented several examples of theories whose…
We initiate the study of analogues of symmetric spaces for the family of finite dihedral groups. In particular, we investigate the structure of the automorphism group, characterize the involutions of the automorphism group, and determine…
Physical theories grounded in mathematical symmetries are an essential component of our understanding of a wide range of properties of the universe. Similarly, in the domain of machine learning, an awareness of symmetries such as rotation…
The probability that the commutator of two group elements is equal to a given element has been introduced in literature few years ago. Several authors have investigated this notion with methods of the representation theory and with…
The idea of pairwise paracompactness was studied by many authors in a bitopological space. Here we study the same in the setting of more general structure of a bispace using the thoughts of the same given by Bose et al[2].
We describe those group algebras over fields of characteristic different from 2 whose units symmetric with respect to the classical involution, satisfy some group identity.