It is shown that certain weak-base structures on a topological space give a D-space. This solves the question by A.V. Arhangel’skii of when quotient images of metric spaces are D-spaces. A related result about symmetrizable spaces also answers a question of Arhangel’skii. Theorem. Any symmetrizable space X is a D-space (hereditarily). Hence, quotient mappings, with compact fibers, from metric s...

Cluster algebras are a class of commutative rings introduced by Fomin and Zelevinsky. It is well-known that these algebras are closely related with different areas of mathematics. A particular analogy exists between combinatorial aspects of cluster algebras and Kac-Moody algebras: roughly speaking, cluster algebras are associated with skew-symmetrizable matrices while Kac-Moody algebras corresp...

Some skew-symmetrizable integer exchange matrices are associated to ideal (tagged) triangulations of marked bordered surfaces. These exchange matrices admit unfoldings to skew-symmetric matrices. We develop a combinatorial algorithm that determines if a given skew-symmetrizable matrix is of such type. This algorithm generalizes the one in [1]. As a corollary, we use this algorithm to determine ...

The necessary and sufficient condition for stability of K-symmetrizable interval matrix is given. An algorithm for checking stability of K-symmetrizable interval matrix is proposed.

The paper is motivated by an analogy between cluster algebras and Kac-Moody algebras: both theories share the same classification of finite type objects by familiar Cartan-Killing types. However the underlying combinatorics beyond the two classifications is different: roughly speaking, Kac-Moody algebras are associated with (symmetrizable) Cartan matrices, while cluster algebras correspond to s...

The paper is motivated by an analogy between cluster algebras and Kac-Moody algebras: both theories share the same classification of finite type objects by familiar Cartan-Killing types. However the underlying combinatorics beyond the two classifications is different: roughly speaking, Kac-Moody algebras are associated with (symmetrizable) Cartan matrices, while cluster algebras correspond to s...

The symmetrizable and converged Laplace–Beltrami operator ( M) is an indispensable tool for spectral geometrical analysis of point clouds. The M, introduced by Liu et al. [LPG12] is guaranteed to be symmetrizable, but its convergence degrades when it is applied to models with sharp features. In this paper, we propose a novel M, which is not only symmetrizable but also can handle the point-sampl...