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A partial group with $n+1$ elements is, when regarded as a symmetric simplicial set, of dimension at most $n$. This dimension is $n$ if and only if the partial group is a group. As a consequence of the first statement, finite partial groups are genuinely finite, despite being seemingly specified by infinitely much data. In particular, finite partial groups have only finitely many im-partial subgroups. We also consider dimension of partial groupoids., Comment: 7 pages

Stability, akin to reproducibility, is crucial in statistical analysis. This paper examines the stability of sparse network inference in high-dimensional graphical models, where selected edges should remain consistent across different samples. Our study focuses on the Graphical Lasso and its decomposition into two steps, with the first step involving hierarchical clustering using single linkage.We provide theoretical proof that single linkage is stable, evidenced by controlled distances between two dendrograms inferred from two samples. Practical experiments further illustrate the stability of the Graphical Lasso's various steps, including dendrograms, variable clusters, and final networks. Our results, validated through both theoretical analysis and practical experiments using simulated and real datasets, demonstrate that single linkage is more stable than other methods when a modular structure is present.

Machine learning methods usually rely on large sample size to have good performance, while it is difficult to provide labeled set in many applications. Pool-based active learning methods are there to detect, among a set of unlabeled data, the ones that are the most relevant for the training. We propose in this paper a meta-approach for pool-based active learning strategies in the context of multi-class classification tasks based on Proper Topological Regions. PTR, based on topological data analysis (TDA), are relevant regions used to sample cold-start points or within the active learning scheme. The proposed method is illustrated empirically on various benchmark datasets, being competitive to the classical methods from the literature.

In this article, we compare two different notions of partially defined group strutures, namely partial groups and pregroups, as introduced by Chermak and Stallings respectively. In particular we prove that the category of pregroups can be seen as a full subcategory of the category of partial groups. We also bring out some conjugation properties about elements and subgroups of finite order in pregroups and their universal groups. We then use these to investigate the question of realisability of fusion systems in finite pregroups.

Crystallization represents a fundamental process engendering solidification of a material and determines its microstructure. Driven by complex phenomena at the atomic scale, its understanding for alloys still remains elusive. The present work proposes a large scale molecular dynamics simulation study of the homogeneous crystal nucleation pathways of prototypical undercooled Al-Ni binary alloys. An unsupervised topological learning analysis shows that the nucleation sets in first from a chemical ordering, followed by a bond-orientational ordering of the underlying crystal phase. Our results indicate also a different polymorph selection that depends on composition. While the nucleation pathway of Al50 Ni50 displays a single step with the emergence of B2 short-range order, a step-wise nucleation toward the L12 phase is seen for Al25 Ni75 . The influence of the nucleation of pure Al and Ni counterparts is further discussed.

Nucleation phenomena commonly observed in our every day life are of fundamental, technological and societal importance in many areas, but some of their most intimate mechanisms remain however to be unraveled. Crystal nucleation, the early stages where the liquid-to-solid transition occurs upon undercooling, initiates at the atomic level on nanometer length and sub-picoseconds time scales and involves complex multidimensional mechanisms with local symmetry breaking that can hardly be observed experimentally in the very details. To reveal their structural features in simulations without a priori, an unsupervised learning approach founded on topological descriptors loaned from persistent homology concepts is proposed. Applied here to a monatomic metal, namely Tantalum (Ta), it shows that both translational and orientational ordering always come into play simultaneously when homogeneous nucleation starts in regions with low five-fold symmetry., Comment: 6 pages, 3 figures, Conference Neurip 2021. arXiv admin note: substantial text overlap with arXiv:2109.06797

We propose an unsupervised learning methodology with descriptors based on Topological Data Analysis (TDA) concepts to describe the local structural properties of materials at the atomic scale. Based only on atomic positions and without a priori knowledge, our method allows for an autonomous identification of clusters of atomic structures through a Gaussian mixture model. We apply successfully this approach to the analysis of elemental Zr in the crystalline and liquid states as well as homogeneous nucleation events under deep undercooling conditions. This opens the way to deeper and autonomous study of complex phenomena in materials at the atomic scale.

Nucleation phenomena commonly observed in our every day life are of fundamental, technological and societal importance in many areas, but some of their most intimate mechanisms remain however to be unravelled. Crystal nucleation, the early stages where the liquid-to-solid transition occurs upon undercooling, initiates at the atomic level on nanometre length and sub-picoseconds time scales and involves complex multidimensional mechanisms with local symmetry breaking that can hardly be observed experimentally in the very details. To reveal their structural features in simulations without a priori, an unsupervised learning approach founded on topological descriptors loaned from persistent homology concepts is proposed. Applied here to monatomic metals, it shows that both translational and orientational ordering always come into play simultaneously when homogeneous nucleation starts in regions with low five-fold symmetry. It also reveals the specificity of the nucleation pathways depending on the element considered, with features beyond the hypothesis of Classical Nucleation Theory.

It is well known that not every finite group arises as the full automorphism group of some group. Here we show that the situation is dramatically different when considering the category of partial groups, ${{\mathcal P}art}$, as defined by Chermak: given any group $H$ there exists infinitely many non isomorphic partial groups ${\mathbb M}$ such that $\operatorname{Aut}_{{\mathcal P}art}({\mathbb M})\cong H$. To prove this result, given any simple undirected graph $G$ we construct a partial group ${\mathbb P}(G)$, called the path partial group associated to $G$, such that $\operatorname{Aut}_{{\mathcal P}art}\big({\mathbb P}(G)\big)\cong \operatorname{Aut}_{{\mathcal G}raphs}(G)$. As a consequence, we solve the Kahn realizability problem in the category of simplicial sets.

These notes describe some results on dice comparisons when changing the numbers on the faces while the sum of all the face stay the same., Comment: 7 pages. there are some minor changes, in particular in the last section. Also, it is in the DMTCS format

These notes are defining the notion of centric linking system for a locally finite group If a locally finite group $G$ has countable Sylow $p$-subgroups, we prove that, with a countable condition on the set of intersections, the $p$-completion of its classifying space is homotopy equivalent to the $p$-completion of the nerve of its centric linking system., Comment: 9 pages, any comments is welcome

In these notes we give an Alperin's Fusion Theorem for localities., Comment: 6 pages. Published in Communication in Algebra

The existence and uniqueness of linking systems associated to saturated fusion systems over discrete $p$-toral groups were proved by Levi and Libman. Their proof make indirectly use of the classification of the finite simple groups. Here we extend some results and arguments of Glauberman and Lynd to show how to remove this assumption., Comment: 26 pages. Published in Journal of Algebra. arXiv admin note: substantial text overlap with arXiv:1506.01307 by other authors

In this paper we study the cohomology of the geometric realization of linking systems with twisted coefficients. More precisely, given a prime $p$ and a $p$-local finite group $(S,\mathcal{F},\mathcal{L})$, we compare the cohomology of $\mathcal{L}$ with twisted coefficients with the submodule of $\mathcal{F}^c$-stable elements in the cohomology of $S$. We start with the particular case of constrained fusion systems. Then, we study the case of $p$-solvable actions on the coefficients., Comment: 20 pages. a few typos corrected. Published in Homology Homotopy and Applications

We study the cohomology with twisted coefficients of the geometric realization of a linking system associated to a saturated fusion system $\mathcal{F}$. More precisely, we extend a result due to Broto, Levi and Oliver to twisted coefficients. We generalize the notion of $\mathcal{F}$-stable elements to $\mathcal{F}^c$-stable elements in a setting of cohomology with twisted coefficients by an action of the fundamental group.% or, in other word, with locally constant coefficients. We then study the problem of inducing an idempotent from an $\mathcal{F}$-characteristic $(S,S)$-biset and we show that, if the coefficient module is nilpotent, then the cohomology of the geometric realization of a linking system can be computed by $\mathcal{F}^c$-stable elements. As a corollary, we show that for any coefficient module, the cohomology of the classifying space of a $p$-local finite group can be computed by these $\mathcal{F}^c$-stable elements., Comment: 18 pages. Published in Topology and its Applications