Related papers: Divisibility orders in $\beta N$
We construct partitions of $\mathbb{R}^n$ into three sets $\{\mathscr{X}(1),\mathscr{X}(2),\mathscr{X}(3)\}$ that locally minimize interfacial area among compactly supported volume preserving variations and that blow down at infinity to…
Answering a question of Dobrinen and Todorcevic, we prove that below any stable ordered-union ultrafilter $\mathcal{U}$, there are exactly four nonprincipal Tukey classes: $[\mathcal{U}], [\mathcal{U}_{\operatorname{min}}],…
Consider a finite group $G$ of order $n$ with a prime divisor $p$. In this article, we establish, among other results, that if the Sylow $p$-subgroup of $G$ is neither cyclic nor generalized quaternion, then there exists a bijection $f$…
Under MA_{omega_1} every uncountable almost disjoint family is either anti-Luzin or has an uncountable Luzin subfamily. This fails under CH. Related properties are also investigated.
The IV-VI binary semiconductors and and column V elements crystalize in one of the three varieties derived from the cubic rocksalt structure. The rocksalt structure can be seen as a network of interpenetrating octahedra with an atom at the…
Working in Zermelo-Fraenkel Set Theory with Atoms over an $\omega$-categorical $\omega$-stable structure, we show how \emph{infinite} constructions over definable sets can be encoded as \emph{finite} constructions over the Stone-\v{C}ech…
In this paper I will show that it is relatively consistent with the usual axioms of mathematics (ZFC) together with a strong form of the axiom of infinity (the existence of a supercompact cardinal) that the class of uncountable linear…
We study the number of linear extensions of a partial order with a given proportion of comparable pairs of elements, and estimate the maximum and minimum possible numbers. We also consider a random interval partial order on $n$ elements,…
We say that two partial orders on $[n]$ are compatible if there exists a partial order that refines both of them. This compatibility relation induces a natural set system structure between the collection $\mathcal{F}$ of all partial orders…
It is shown why the discriminant of a maximal order within a cyclic division algebra must be minimized in order to get the densest possible matrix lattices with a prescribed nonvanishing minimum determinant. Using results from class field…
The lattice of partitions of a set and its d-divisible generalization have been much studied for their combinatorial, topological, and representation-theoretic properties. An ordered set partition is a set partition where the subsets are…
1. For many regular cardinals lambda (in particular, for all successors of singular strong limit cardinals, and for all successors of singular omega-limits), for all n in {2,3,4, ...} : There is a linear order L such that L^n has no…
The ordering temperature of a quasi-one-dimensional system, consisting of weakly interacting quantum spin-1/2 chains with antiferromagnetic spin-frustrating couplings (or zig-zag ladder) is calculated. The results show that a quantum…
For a $\beta$ ensemble on $\Sigma^{(N)}=\{(x_1,\ldots,x_N)\mathbb R^N|x_1\le\cdots\le x_N\}$ with real analytic potential and general $\beta>0$, under the assumption that its equilibrium measure is supported on $q$ intervals where $q>1$, we…
The concepts of localizable set, localization of a ring and a module at a localizable set are introduced and studied. Localizable sets are generalization of Ore sets and denominator sets, and the localization of a ring/module at a…
Characterizations of ultrafilters belong to the smallest ideal of Stone-\v{C}ech compactification of a discrete semigroup are exhibited using syndetic sets, strongly central sets and very strongly central sets respectively. These lead to…
In the first part, after showing that the most natural approach to define an order on sets of conformal classes fails, we define a nontrivial order $\leq_2$ on the set of conformal classes of compact Cauchy slabs with fixed past boundary…
The dimension of a partially-ordered set $P$ is the smallest integer $d$ such that one can embed $P$ into a product of $d$ linear orders. We prove that the dimension of the divisibility order on the interval $\{1, \dotsc, n\}$ is bounded…
We study three kinds of compactness in some variants of G\"odel logic: compactness, entailment compactness, and approximate entailment compactness. For countable first-order underlying language we use the Henkin construction to prove the…
We classify indecomposable racks of order p^2 (p a prime). There are 2p^2 - 2p - 2 isomorphism classes, among which 2p^2 - 3p - 1 correspond to quandles. In particular, we prove that an indecomposable quandle of order p^2 is affine…