Related papers: Characterization of complementing pairs of $({\mat…

Let $G$ be a complex, connected, reductive, algebraic group, and $\chi:\mathbb{C}^\times \to G$ be a fixed cocharacter that defines a grading on $\mathfrak{g}$, the Lie algebra of $G$. Let $G_0$ be the centralizer of…

A characteristic pair is a pair (G,C) of polynomial sets in which G is a reduced lexicographic Groebner basis, C is the minimal triangular set contained in G, and C is normal. In this paper, we show that any finite polynomial set P can be…

We show that the full group C$^*$-algebra of $PSL(n, \Z)$ is primitive when $n=2$, and not primitive when $n\geq 3$. Moreover, we show that there exists an uncountable family of pairwise inequivalent, faithful irreducible representations of…

This paper gives a structural explanation for the Z-relation by modelling pitch-class sets as complete weighted graphs and encoding their interval content in a composition of $n$ via an additivity rule. We introduce the realization number…

We introduce diagonal comparison, a regularity property of diagonal pairs where the sub-C*-algebra has totally disconnected spectrum, and establish its equivalence with the concurrence of strict comparison of the ambient C*-algebra and…

A pure pair in a graph $G$ is a pair $(Z_1,Z_2)$ of disjoint sets of vertices such that either every vertex in $Z_1$ is adjacent to every vertex in $Z_2$, or there are no edges between $Z_1$ and $Z_2$. With Maria Chudnovsky, we recently…

Let ${\cal T}=(T,w)$ be a weighted finite tree with leaves $1,..., n$.For any $I :=\{i_1,..., i_k \} \subset \{1,...,n\}$, let $D_I ({\cal T})$ be the weight of the minimal subtree of $T$ connecting $i_1,..., i_k$; the $D_{I} ({\cal T})$…

We use the group $(\Z^2,+)$ and two associated homomorphisms, $\tau_0, \tau_1$, to generate all distinct, non-zero pairs of coprime, positive integers which we describe within the context of a binary tree which we denote $T$. While this…

As a discretization of the Hodge Laplacian, the combinatorial Laplacian of simplicial complexes has garnered significant attention. In this paper, we study combinatorial Laplacians for complex pairs $(X, A)$, where $A$ is a subcomplex of a…

It is shown that the pairing of the K00 group of a C*-algebra with the densely defined traces of the algebra can be extended to a pairing with the densely defined weights. For traces the pairing can be extended to the K0 group without the…

A graph $G$ is called a pairwise compatibility graph (PCG, for short) if it admits a tuple $(T,w, d_{\min},d_{\max})$ of a tree $T$ whose leaf set is equal to the vertex set of $G$, a non-negative edge weight $w$, and two non-negative reals…

We call a pair of closed subgroups $(G_1,G_2)$ from a connected reductive algebraic group $G$ a {\it complexity $c$ pair} if the multiplication action of the pair on $G$ is of complexity $c$. The main focus of this article is on the cases…

Let $G$ be an additive abelian group and let $A,B \subseteq G$ be finite and nonempty. The pair $(A,B)$ is called critical if the sumset $A+B = {a+b \mid $a \in A$ and $b\in B$}$ satisfies $|A+B| < |A| + |B|$. Vosper proved a theorem which…

Denote the integers by Z and the positive integers by N. The groups Z^k (k a natural number) are discrete, and the classification up to isomorphism of their (topological) subgroups is trivial. But already for the countably infinite power…

We show that the full group C$^*$-algebra of the free product of two nontrivial countable amenable discrete groups, where at least one of them has more than two elements, is primitive. We also show that in many cases, this C$^*$-algebra is…

A subset $X$ of an abelian $G$ is said to be {\em complete} if every element of the subgroup generated by $X$ can be expressed as a nonempty sum of distinct elements from $X$. Let $A\subset \Z_n$ be such that all the elements of $A$ are…

The origins of the notion of matchings in groups spawn from a linear algebra problem proposed by E. K. Wakeford [24] which was tackled in 1996 [10]. In this paper, we first discuss unmatchable subsets in abelian groups. Then we formulate…

We give some properties of the subgroup $G_n(\mathbb{C})$ of the group of birational self-maps of $\mathbb{P}^n_\mathbb{C}$ generated by the standard involution and the group of automorphisms of $\mathbb{P}^n_\mathbb{C}$. We prove that…

A pure pair of size $t$ in a graph $G$ is a pair $A,B$ of disjoint sets of $t$ vertices such that $A$ is either complete or anticomplete to $B$. It is known that, for every forest $H$, every graph on $n\ge2$ vertices that does not contain…

The concept of formal duality was proposed by Cohn, Kumar and Sch\"urmann, which reflects a remarkable symmetry among energy-minimizing periodic configurations. This formal duality was later translated into a purely combinatorial property…