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We develop a rigidity theory for frameworks in $\mathbb{R}^3$ which have two coincident points but are otherwise generic and only infinitesimal motions which are tangential to a family of cylinders induced by the realisation are considered.…
Given a residually connected incidence geometry $\Gamma$ that satisfies two conditions, denoted $(B_1)$ and $(B_2)$, we construct a new geometry $H(\Gamma)$ with properties similar to those of $\Gamma$. This new geometry $H(\Gamma)$ is…
Fix a finite group $G$. We study the computational complexity of counting problems of the following flavor: given a group $\Gamma$, count the number of homomorphisms $\Gamma \to G$. Our first result establishes that this problem is…
An N -tiling of triangle ABC by triangle T is a way of writing ABC as a union of N triangles congruent to T, overlapping only at their boundaries. The triangle T is the "tile". The tile may or may not be similar to ABC . We wish to…
Consider a fixed connected, finite graph $\Gamma$ and equip its vertices with weights $p_i$ which are non-negative integers. We show that there is a finite number of possibilities for the coefficients of the canonical cycle of a numerically…
This paper develops the algebraic foundation required to build a Zariski-type geometry for \emph{commutative ternary $\Gamma$-semirings}, where multiplication is an inherently triadic, multi-parametric interaction…
In this article, we look into the tree gonality of genus $3$ metric graphs $\Gamma$ which is defined as the minimum of degrees of all tropical morphisms from any tropical modification of $\Gamma$ to any metric tree. It is denoted by…
There are four non-isomorphic configurations of triples that can form a triangle in a $3$-uniform hypergraph. Forbidding different combinations of these four configurations, fifteen extremal problems can be defined, several of which already…
The trinomial transform of a sequence is a generalization of the well-known binomial transform, replacing binomial coefficients with trinomial coefficients. We examine Pascal-like triangles under trinomial transform, focusing on the ternary…
Given a permutation group $G$, the derangement graph $\Gamma_G$ of $G$ is the Cayley graph with connection set the set of all derangements of $G$. We prove that, when $G$ is transitive of degree at least $3$, $\Gamma_G$ contains a triangle.…
A graph $\Gamma$ is $G$-symmetric if it admits $G$ as a group of automorphisms acting transitively on the set of arcs of $\Gamma$, where an arc is an ordered pair of adjacent vertices. Let $\Gamma$ be a $G$-symmetric graph such that its…
Congruences, or $2$-parameter families of lines in $3$-space are of interest in many situations, in particular in geometric optics. In this paper we consider elements of their geometry which are invariant under affine changes of…
A method to define the complex structure and separate the conformal mode is proposed for a surface constructed by two-dimensional dynamical triangulation. Applications are made for surfaces coupled to matter fields such as $n$ scalar fields…
A system of three particles undergoing inelastic collisions in arbitrary spatial dimensions is studied with the aim of establishing the domain of ``inelastic collapse''---an infinite number of collisions which take place in a finite time.…
Trilinear mappings appear naturally when performing spatial isogeometric discretizations of degree $p = 1$. Among them, birational maps are characterized by the property that both the mapping and the associated inverse map are rational and…
We use group theory to construct infinite families of maps on surfaces which are invariant under Wilson's map operations of order 3 but not under the operations of order 2, such as duality and Petrie duality.
A dual approach to defining the triangle sequence (a type of multidimensional continued fraction algorithm, initially developed in NT/9906016) for a pair of real numbers is presented, providing a new, clean geometric interpretation of the…
A method of ``blocking'' triangulations that rests on the self-similarity feature of dynamically triangulated random manifolds is proposed. The method is used to define the renormalization group for random geometries. As an illustration,…
We describe complex twistor spaces over inner 3-symmetric spaces $G/H$, such that $H$ acts transitively on the fibre. Like in the symmetric case, these are flag manifolds $G/K$ where $K$ is the centralizer of a torus in $G$. Moreover, they…
There are various results connecting ranks of incidence matrices of graphs and hypergraphs with their combinatorial structure. Here, we consider the generalized incidence matrix $N_2$ (defined by inclusion of pairs in edges) for one natural…