Related papers: Cluster algebras and continued fractions
We review some important results by Gross, Hacking, Keel, and Kontsevich on cluster algebra theory, namely, the column sign-coherence of $C$-matrices and the Laurent positivity, both of which were conjectured by Fomin and Zelevinsky. We…
The cluster category is a triangulated category introduced for its combinatorial similarities with cluster algebras. We prove that a cluster algebra A of finite type can be realized as a Hall algebra, called the exceptional Hall algebra, of…
In the combinatorial theory of continued fractions, the Foata--Zeilberger bijection and its variants have been extensively used to derive various continued fractions enumerating several (sometimes infinitely many) simultaneous statistics on…
For an unpunctured marked surface $\Sigma$, we consider a skein algebra $\mathscr{S}_{\mathfrak{sl}_{3},\Sigma}^{q}$ consisting of $\mathfrak{sl}_3$-webs on $\Sigma$ with the boundary skein relations at marked points. We construct a quantum…
The aim of the article is to understand the combinatorics of snake graphs by means of linear algebra. In particular, we apply Kasteleyn's and Temperley--Fisher's ideas about spectral properties of weighted adjacency matrices of planar…
In 2011 Musiker, Schiffler and Williams obtained expansion formulae for cluster algebras from orientable surfaces. For singly and doubly notched arcs these formulae required the notion of $\gamma$-symmetric perfect matchings and…
A well known method for convergence acceleration of continued fraction $\K(a_n/b_n)$ is to use the modified approximants $S_n(\omega_n)$ in place of the classical approximants $S_n(0)$, where $\omega_n$ are close to tails $f^{(n)}$ of…
The continued fraction mapping maps a number in the interval $[0,1)$ to the sequence of its partial quotients. When restricted to the set of irrationals, which is a subspace of the Euclidean space $\mathbb{R}$, the continued fraction…
One of the remarkable properties of cluster algebras is that any cluster, obtained from a sequence of mutations from an initial cluster, can be written as a Laurent polynomial in the initial cluster (known as the "Laurent phenomenon").…
A basic result in the elementary theory of continued fractions says that two real numbers share the same tail in their continued fraction expansions iff they belong to the same orbit under the projective action of PGL(2,Z). This result was…
In a cluster algebra, a subset of initial cluster variables can be specialised in such a way that all elements of the resulting algebra become polynomial in the remaining variables.
We introduce the notion of a lower bound cluster algebra generated by projective cluster variables as a polynomial ring over the initial cluster variables and the so-called projective cluster variables. We show that under an acyclicity…
Ongoing efforts that span over decades show a rise of AI methods for accelerating scientific discovery, yet accelerating discovery in mathematics remains a persistent challenge for AI. Specifically, AI methods were not effective in creation…
Our main result is that any real cubic algebraic number has a continued fraction expansion with polynomial coefficients. Some generalizations are mentioned.
Holm and Jorgensen have shown the existence of a cluster structure on a certain category $D$ that shares many properties with finite type $A$ cluster categories and that can be fruitfully considered as an infinite analogue of these. In this…
We mainly introduce an abstract pattern to study cluster algebras. Cluster algebras, generalized cluster algebras and Laurent phenomenon algebras are unified in the language of generalized Laurent phenomenon algebras (briefly, GLP algebras)…
Consistency is a key property of all statistical procedures analyzing randomly sampled data. Surprisingly, despite decades of work, little is known about consistency of most clustering algorithms. In this paper we investigate consistency of…
Graph clustering is a fundamental technique in data analysis with applications in many different fields. While there is a large body of work on clustering undirected graphs, the problem of clustering directed graphs is much less understood.…
Adler, Keane, and Smorodinsky showed that if one concatenates the finite continued fraction expansions of the sequence of rationals \[ \frac{1}{2}, \frac{1}{3}, \frac{2}{3}, \frac{1}{4}, \frac{2}{4}, \frac{3}{4}, \frac{1}{5}, \cdots \] into…
We study explicit continued fraction expansions for certain series. Some of these expansions have symmetry that generalizes some remarkable examples discovered independently by Kmosek and Shallit. Furthermore, we prove the following…