Related papers: Singularities and clusters
We consider nonlinear recurrences generated from the iteration of maps that arise from cluster algebras. More precisely, starting from a skew-symmetric integer matrix, or its corresponding quiver, one can define a set of mutation…
Always dealing with an arbitrary field we consider the variety $(k^{n\times n})^{p}$ under the action of $GL_{n}$ by simultaneous similarity. We define discrete and continuous invariants which completely determine the orbits. The discrete…
Correlation clustering provides a method for separating the vertices of a signed graph into the optimum number of clusters without specifying that number in advance. The main goal in this type of clustering is to minimize the number of…
Complex systems are usually represented as an intricate set of relations between their components forming a complex graph or network. The understanding of their functioning and emergent properties are strongly related to their structural…
Bidirected graphs are multigraphs where every edge has an independent direction at each end. In the paper, with an arbitrary bidirected graph we associate a non-negative integral quadratic form (called the incidence form of the graph), and…
We provide a complete classification of the singularities of cluster algebras of finite cluster type. This extends our previous work about the case of trivial coefficients. Additionally, we classify the singularities of cluster algebras for…
In this paper, we deal with plane curves with cusps. It is well known that there are various types of cusps. Among them, we investigate criteria for $(n, n+1)$ cusps with respect to several differential conditions and relations between…
The topology of symplectic 4-manifolds is related to that of singular plane curves via the concept of branched covers. Thus, various classification problems concerning symplectic 4-manifolds can be reformulated as questions about singular…
We introduce the simple notion of a "crystallographic arrangement" and prove a one-to-one correspondence between these arrangements and the connected simply connected Cartan schemes for which the real roots are a finite root system (up to…
The aim of this article is to proof a necessary and sufficient condition for the existence of a Cartan connection on a principal bundle. After collecting the essentially well known facts to fix the terminology, soldering forms and…
This note is devoted to a trick which yields almost trivial proofs that certain complexes associated to topological surfaces are connected or simply connected. Applications include new proofs that the complexes of curves, separating curves,…
We study some spectral properties of a matrix that is constructed as a combination of a Laplacian and an adjacency matrix of simple graphs. The matrix considered depends on a positive parameter, as such we consider the implications in…
The classifying spaces of cobordisms of singular maps have two fairly different constructions. We expose a homotopy theoretical connection between them. As a corollary we show that the classifying spaces in some cases have a simple product…
In this paper, we prove some fundamental theorems for holomorphic curves on angular domain intersecting a hypersurface, finite set of fixed hyperplanes in general position and finite set of fixed hypersurfaces in general position on complex…
We adapt algorithms for resolving the singularities of complex algebraic varieties to prove that the natural map of homology theories from complex bordism to the bordism theory of complex derived orbifolds splits. In equivariant stable…
Matrices are two-dimensional data structures allowing one to conceptually organize information. For example, adjacency matrices are useful to store the links of a network; correlation matrices are simple ways to arrange gene co-expression…
In this paper, the association scheme defined on the flags of a finite generalized quadrangle is considered. All possible fusions of this scheme are listed, and a full description for those of classes 2 and 3 is given. Furthermore, it is…
We enumerate the singular algebraic curves in a complete linear system on a smooth projective surface. The system must be suitably ample in a rather precise sense. The curves may have up to eight nodes, or a triple point of a given type and…
We classify compact manifolds of dimension three equipped with a path structure and a fixed contact form (which we refer to as a strict path structure) under the hypothesis that their automorphism group is non-compact. We use a Cartan…
First, we prove that the set of $n\times n$ complex matrices is the closure of a certain open subset whose elements have a very specific canonical form under congruence, which is uniquely determined up to the values of some parameters, but…