Strongly Peaking Representations and Compressions of Operator Systems

We use Arveson’s notion of strongly peaking representation to generalize uniqueness theorems for free spectrahedra and matrix convex sets that admit minimal presentations. A fully compressed separable operator system necessarily generates the $C^*$-envelope and is such that the identity is the direct sum of strongly peaking representations. In particular, a fully compressed presentation of a separable operator system is unique up to unitary equivalence. Under various additional assumptions, minimality conditions are sufficient to determine a separable operator system uniquely.

Presentations of a numerical semigroup

In this paper, we mainly study the minimal presentations of numerical semigroups. Moreover, we examine the concept of gluing, complete intersection, catenary degree, elasticity of some numerical semigroups.

Parabolic double cosets in Coxeter groups

International audience Parabolic subgroups WI of Coxeter systems (W,S) and their ordinary and double cosets W/WI and WI\W/WJ appear in many contexts in combinatorics and Lie theory, including the geometry and topology of generalized flag varieties and the symmetry groups of regular polytopes. The set of ordinary cosets wWI , for I ⊆ S, forms the Coxeter complex of W , and is well-studied. In this extended abstract, we look at a less studied object: the set of all double cosets WIwWJ for I,J ⊆ S. Each double coset can be presented by many different triples (I, w, J). We describe what we call the lex-minimal presentation and prove that there exists a unique such choice for each double coset. Lex-minimal presentations can be enumerated via a finite automaton depending on the Coxeter graph for (W, S). In particular, we present a formula for the number of parabolic double cosets with a fixed minimal element when W is the symmetric group Sn. In that case, parabolic subgroups are also known as Young subgroups. Our formula is almost always linear time computable in n, and the formula can be generalized to any Coxeter group.

Isolated factorizations and their applications in simplicial affine semigroups

We introduce the concept of isolated factorizations of an element of a commutative monoid and study its properties. We give several bounds for the number of isolated factorizations of simplicial affine semigroups and numerical semigroups. We also generalize [Formula: see text]-rectangular numerical semigroups to the context of simplicial affine semigroups and study their isolated factorizations. As a consequence of our results, we characterize those complete intersection simplicial affine semigroups with only one Betti minimal element in several ways. Moreover, we define Betti sorted and Betti divisible simplicial affine semigroups and characterize them in terms of gluings and their minimal presentations. Finally, we determine all the Betti divisible numerical semigroups, which turn out to be those numerical semigroups that are free for any arrangement of their minimal generators.

Bearing Witness, Moral Responsibility and Distant Suffering

Media analyses of the aesthetic presentation of news frequently and persuasively demonstrate the skewed ethics of news production, where those in need who are distant and dissimilar are often presented in ways that do not fully humanize their condition. However, it is argued here, they also locate too fully the locus of moral responsibility in the production and presentation of the media text. This article explores the moral demands inherent in two kinds of media presentation: the explicit focus on shock and awe over intimacy; and minimal presentations that substitute bare information (such as statistics or maps) for affective stories. It is argued that in both cases a moral encounter with the other can be motivated despite the news presentation and it is concluded that an excessive emphasis on the role of the news in generating moral responsibility justifies a spectatorish inertia that should be resisted.

Minimal presentations of shifted numerical monoids

A numerical monoid is an additive submonoid of the non-negative integers. Given a numerical monoid [Formula: see text], consider the family of “shifted” monoids [Formula: see text] obtained by adding [Formula: see text] to each generator of [Formula: see text]. In this paper, we examine minimal relations among the generators of [Formula: see text] when [Formula: see text] is sufficiently large, culminating in a description that is periodic in the shift parameter [Formula: see text]. We explore several applications to computation and factorization theory, and improve a recent result of Thanh Vu from combinatorial commutative algebra.