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A $k$-Stirling permutation of order $n$ is said to be "flattened" if the leading terms of its increasing runs are in ascending order. We show that flattened $k$-Stirling permutations of order $n+1$ are in bijection correspondence with a…
In this note, we present a characterization of sets definable in Skolem arithmetic, i.e., the first-order theory of natural numbers with multiplication. This characterization allows us to prove the decidability of the theory. The idea is…
We show that the first order theory of the lattice of open sets in some natural topological spaces is $m$-equivalent to second order arithmetic. We also show that for many natural computable metric spaces and computable domains the first…
We develop the theory of set-indexed families of sets and subsets within the informal Bishop Set Theory BST, a reconstruction of Bishop's theory of sets.
Young asteroid families are unique sources of information about fragmentation physics and the structure of their parent bodies, since their physical properties have not changed much since their birth. Families have different properties such…
We introduce a first-order theory $\mathsf{Seq}$ which is mutually interpretable with Robinson's $\mathsf{Q}$. The universe of a standard model for $\mathsf{Seq}$ consists of sequences. We prove that $\mathsf{Seq}$ directly interprets the…
If phi is a scattered order type, mu a cardinal, then there exists a scattered order type psi such that psi->[phi]^1_{mu,aleph_0} holds.
Diestel, et al. (see Order 35 (2017), JCT-A 167 (2019), arXiv:1805.01439) introduced the notion of abstract separation systems that satisfy a submodularity property, and they call this structural submodularity. Williamson, Goemans, Mihail,…
We answer a question from Raghavan and Stepr{\=a}ns' paper on weakly tight families by showing that $\mathfrak{s} = {\mathfrak{s}}_{\omega, \omega}$. Then we use this to construct a completely separable maximal almost disjoint family under…
A classical theorem of Baranyai states that, given integers $2\leq k < n$ such that $k$ divides $n$, one can find a family of ${n-1\choose k-1}$ partitions of $[n]$ into $k$-element subsets such that every subset appears in exactly one…
H. Furstenberg introduced the notion of central sets in terms of topological dynamics and established the famous Central Sets Theorem. Later in [A new and stronger Central Sets Theorem, Fund. Math. 199 (2008), 155-175], D. De, N. Hindman,…
If $G$ is a finite group and $x\in G$ then the set of all elements of $G$ having the same order as $x$ is called {\em an order subset of $G$ determined by $x$} (see [2]). We say that $G$ is a {\em group with perfect order subsets} or…
Fracton order is an intriguing new type of order which shares many common features with topological order, such as topology-dependent ground state degeneracies, and excitations with mutual statistics. However, it also has several…
We propose the representation principle to study physical systems with a given symmetry. In the context of symmetry enriched topological orders, we give the appropriate representation category, the category of SET orders, which include SPT…
This dissertation is a contribution to the project of second-order set theory, which has seen a revival in recent years. The approach is to understand second-order set theory by studying the structure of models of second-order set theories.…
The Springer numbers, introduced by Arnold, are generalizations of Euler numbers in the sense of Coxeter groups. They appear as the row sums of a double triangular array $(v_{n,k})$ of integers, $1\leq|k|\leq n$, defined recursively by a…
A system of O(N)-matrix difference equations is solved by means of the off-shell version of the nested algebraic Bethe ansatz. In the nesting process a new object, the $\Pi$-matrix, is introduced to overcome the complexities of the O(N)…
The Union Closed Sets Conjecture states that in every finite, nontrivial set family closed under taking unions there is an element contained in at least half of all the sets of the family. We investigate two new directions with respect to…
New Foundations ($\mathrm{NF}$) is a set theory obtained from naive set theory by putting a stratification constraint on the comprehension schema; for example, it proves that there is a universal set $V$. $\mathrm{NFU}$ ($\mathrm{NF}$ with…
We establish a cutting lemma for definable families of sets in distal structures, as well as the optimality of the distal cell decomposition for definable families of sets on the plane in $o$-minimal expansions of fields. Using it, we…