The Elser nuclei sum revisited

Fix a finite undirected graph $\Gamma$ and a vertex $v$ of $\Gamma$. Let $E$ be the set of edges of $\Gamma$. We call a subset $F$ of $E$ pandemic if each edge of $\Gamma$ has at least one endpoint that can be connected to $v$ by an $F$-path (i.e., a path using edges from $F$ only). In 1984, Elser showed that the sum of $\left(-1\right)^{\left| F\right|}$ over all pandemic subsets $F$ of $E$ is $0$ if $E\neq \varnothing$. We give a simple proof of this result via a sign-reversing involution, and discuss variants, generalizations and refinements, revealing connections to abstract convexity (the notion of an antimatroid) and discrete Morse theory. Comment: 25 pages. Final version (published in DMTCS, 2021). More detailed variants of the text can be found in version 8 (arXiv:2009.11527v8)

The action of Young subgroups on the partition complex

We study the restrictions, the strict fixed points, and the strict quotients of the partition complex $|\Pi_{n}|$ | Π n | , which is the $\Sigma_{n}$ Σ n -space attached to the poset of proper nontrivial partitions of the set $\{1,\ldots,n\}$ { 1 , … , n } .We express the space of fixed points $|\Pi_{n}|^{G}$ | Π n | G in terms of subgroup posets for general $G\subset \Sigma_{n}$ G ⊂ Σ n and prove a formula for the restriction of $|\Pi_{n}|$ | Π n | to Young subgroups $\Sigma_{n_{1}}\times \cdots\times \Sigma_{n_{k}}$ Σ n 1 × ⋯ × Σ n k . Both results follow by applying a general method, proven with discrete Morse theory, for producing equivariant branching rules on lattices with group actions.We uncover surprising links between strict Young quotients of $|\Pi_{n}|$ | Π n | , commutative monoid spaces, and the cotangent fibre in derived algebraic geometry. These connections allow us to construct a cofibre sequence relating various strict quotients $|\Pi_{n}|^{\diamond} \mathbin {\operatorname* {\wedge }_{\Sigma_{n}}^{}} (S^{\ell})^{\wedge n}$ | Π n | ⋄ ∧ Σ n ( S ℓ ) ∧ n and give a combinatorial proof of a splitting in derived algebraic geometry.Combining all our results, we decompose strict Young quotients of $|\Pi_{n}|$ | Π n | in terms of “atoms” $|\Pi_{d}|^{\diamond} \mathbin {\operatorname* {\wedge }_{\Sigma_{d}}^{}} (S^{\ell})^{\wedge d}$ | Π d | ⋄ ∧ Σ d ( S ℓ ) ∧ d for $\ell$ ℓ odd and compute their homology. We thereby also generalise Goerss’ computation of the algebraic André-Quillen homology of trivial square-zero extensions from $\mathbf {F}_{2}$ F 2 to $\mathbf {F}_{p}$ F p for $p$ p an odd prime.

Outlier data mining model for sports data analysis

The results of data mining can be used to predict the physical health status of sports athletes and college sports students and provide physical fitness warnings, so that students can pay attention to physical health status and adjust their physical exercise status. Discrete Morse theory, as a powerful optimization theory, plays a big role in algorithm optimization. This paper combines data mining and discrete Morse theory to propose a grid clustering algorithm based on discrete Morse theory. Moreover, according to the theorem that the cell complex reaches the optimum when it has the smallest possible critical point, this study applies the concept of critical points in the discrete Morse theory to optimize the grid clustering process to obtain clustering results. In addition, this study uses the improved C4.5 algorithm to analyze the physical fitness assessment results and obtains a valuable analysis of the physical fitness assessment results.