779 research outputs found
Higher-dimensional models of networks
Networks are often studied as graphs, where the vertices stand for entities
in the world and the edges stand for connections between them. While relatively
easy to study, graphs are often inadequate for modeling real-world situations,
especially those that include contexts of more than two entities. For these
situations, one typically uses hypergraphs or simplicial complexes.
In this paper, we provide a precise framework in which graphs, hypergraphs,
simplicial complexes, and many other categories, all of which model higher
graphs, can be studied side-by-side. We show how to transform a hypergraph into
its nearest simplicial analogue, for example. Our framework includes many new
categories as well, such as one that models broadcasting networks. We give
several examples and applications of these ideas
Introduction to Gestural Similarity in Music. An Application of Category Theory to the Orchestra
Mathematics, and more generally computational sciences, intervene in several
aspects of music. Mathematics describes the acoustics of the sounds giving
formal tools to physics, and the matter of music itself in terms of
compositional structures and strategies. Mathematics can also be applied to the
entire making of music, from the score to the performance, connecting
compositional structures to acoustical reality of sounds. Moreover, the precise
concept of gesture has a decisive role in understanding musical performance. In
this paper, we apply some concepts of category theory to compare gestures of
orchestral musicians, and to investigate the relationship between orchestra and
conductor, as well as between listeners and conductor/orchestra. To this aim,
we will introduce the concept of gestural similarity. The mathematical tools
used can be applied to gesture classification, and to interdisciplinary
comparisons between music and visual arts.Comment: The final version of this paper has been published by the Journal of
Mathematics and Musi
Hopf algebras for matroids over hyperfields
Recently, M.~Baker and N.~Bowler introduced the notion of matroids over hyperfields as a unifying theory of various generalizations of matroids. In this paper we generalize the notion of minors and direct sums from ordinary matroids to matroids over hyperfields. Using this we generalize the classical construction of matroid-minor Hopf algebras to the case of matroids over hyperfields
Interval-valued fuzzy graphs
We define the Cartesian product, composition, union and join on
interval-valued fuzzy graphs and investigate some of their properties. We also
introduce the notion of interval-valued fuzzy complete graphs and present some
properties of self complementary and self weak complementary interval-valued
fuzzy complete graphs
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