相关论文: The Modular Tree of Pythagorus
We prove that every 2-sphere graph different from a prism can be vertex 4-colored in such a way that all Kempe chains are forests. This implies the following three tree theorem: the arboricity of a discrete 2-sphere is 3. Moreover, the…
The twined almost commutative structure of the standard spectral triple on the noncommutative torus with rational parameter is exhibited, by showing isomorphisms with a spectral triple on the algebra of sections of certain bundle of…
We propose a novel approach for studying rooted trees by using functions that we will call descent functions. We provide a construction method for rooted trees that allows to study their properties through the use of descent functions.…
Phylogenetic networks generalise phylogenetic trees and allow for the accurate representation of the evolutionary history of a set of present-day species whose past includes reticulate events such as hybridisation and lateral gene transfer.…
In this paper, we regard the smooth quadric threefold $Q_{3}$ as Lagrangian Grassmannian and search for fixed rational curves of low degree in $Q_{3}$ with respect to a torus action, which is the maximal subgroup of the special linear group…
The structure of topological spaces is analysed here through the lenses of fibrous preorders. Each topological space has an associated fibrous preorder and those fibrous preorders which return a topological space are called spacial. A…
Motivated by geometric origami and the straight skeleton construction, we outline a map between spaces of phylogenetic trees and spaces of planar polygons. The limitations of this map is studied through explicit examples, culminating in…
We study spherical tetrahedra with rational dihedral angles and rational volumes. Such tetrahedra occur in the Rational Simplex Conjecture by Cheeger and Simons, and we supply vast families, discovered by computational efforts, of positive…
We discuss certain ternary algebraic structures appearing more or less naturally in various domains of theoretical and mathematical physics. Far from being exhaustive, this article is intended above all to draw attention to these algebras,…
The Tulczyjew triple on a principal bundle with connection is constructed in a convenient trivialisation. A reduction by the structure group is performed leading to the triple on the trivialised Atiyah algebroid and a presentation of this…
Generalised Pythagorean triples are integer tuples $(x,y,z)$ satisfying the equation $E_{a,b,c}: ax^2+by^2+cz^2=0$. A significant amount of research has been devoted towards understanding generalised Pythagorean triples and, in particular,…
A rational triangle is a triangle with rational side lengths. We consider three different families of rational triangles having a fixed side and whose vertices are rational points in the plane. We display a one-to-one correspondence between…
We consider the triples of integer numbers that are solutions of the equation $x^2+qy^2=z^2$, where $q$ is a fixed, square-free arbitrary positive integer. The set of equivalence classes of these triples forms an abelian group under the…
We study the homogeneous coordinate rings of real multiplication noncommutative tori as defined by A. Polishchuk. Our aim is to understand how these rings give rise to an arithmetic structure on the noncommutative torus. We start by giving…
We introduce a linear algebraic object called a bidiagonal triad. A bidiagonal triad is a modification of the previously studied and similarly defined concept of bidiagonal triple. A bidiagonal triad and a bidiagonal triple both consist of…
A semiregular tree is a tree where all non-pendant vertices have the same degree. Belardo et al. (MATCH Commun. Math. Chem. 61(2), pp. 503-515, 2009) have shown that among all semiregular trees with a fixed order and degree, a graph with…
We investigate Cayley graphs of finite semigroups and monoids. First, we look at semigroup digraphs, i.e., directed Cayley graphs of semigroups, and give a Sabidussi-type characterization in the case of monoids. We then correct a proof of…
Phylogenetic networks are a generalisation of phylogenetic trees that allow for more complex evolutionary histories that include hybridisation-like processes. It is of considerable interest whether a network can be considered `tree-like' or…
We construct the quaternion algebra [10] "geometrically" by a three dimensional analogue of the classic two dimensional geometric description of the complex field. The algebraic description of the multiplication operation in three…
Casual structure can take the form of cone bundles on a manifold, more general local preorders on a topological space, or simplicial orientations implicit in a simplicial set. This note takes a triangulation of a conal manifold M to mean an…