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We consider endomorphism algebras of $n$-term silting complexes in derived categories of hereditary algebras, and we show that the module category of such an endomorphism algebra has a separated $n$-section. For $n=3$ we obtain a trisection in the sense of [2]., Comment: 17 pages

We establish a bijection between torsion pairs in the category of finite-dimensional modules over a finite-dimensional algebra A and pairs (Z, I) formed by a closed rigid set Z in the Ziegler spectrum of A and a set I of indecomposable injective A-modules. This can be regarded as an extension of a result from $\tau$-tilting theory which parametrises the functorially finite torsion pairs over A. We also obtain a one-one-correspondence between finite-dimensional bricks and certain (possibly infinite-dimensional) indecomposable modules satisfying a rigidity condition. Our results also hold when A is an artinian ring.

Given a presilting object in a triangulated category, we find necessary and sufficient conditions for the existence of a complement. This is done both for classic (pre)silting objects and for large (pre)silting objects. The key technique is the study of associated co-t-structures. As a consequence of our techniques we recover some known cases of the existence of complements, including for derived categories of some hereditary abelian categories and for silting-discrete algebras. Moreover, we also show that a finite-dimensional algebra is silting discrete if and only if every bounded large silting complex is equivalent to a compact one., Comment: 25 pages, 1 figure. Comments are welcome!

The increasing availability of large clinical datasets collected from patients can enable new avenues for computational characterization of complex diseases using different analytic algorithms. One of the promising new methods for extracting knowledge from large clinical datasets involves temporal pattern mining integrated with machine learning workflows. However, mining these temporal patterns is a computational intensive task and has memory repercussions. Current algorithms, such as the temporal sequence pattern mining (tSPM) algorithm, are already providing promising outcomes, but still leave room for optimization. In this paper, we present the tSPM+ algorithm, a high-performance implementation of the tSPM algorithm, which adds a new dimension by adding the duration to the temporal patterns. We show that the tSPM+ algorithm provides a speed up to factor 980 and a up to 48 fold improvement in memory consumption. Moreover, we present a docker container with an R-package, We also provide vignettes for an easy integration into already existing machine learning workflows and use the mined temporal sequences to identify Post COVID-19 patients and their symptoms according to the WHO definition., Comment: Supplementary data: https://doi.org/10.5281/zenodo.8329519

Finding similar patients is a common objective in precision medicine, facilitating treatment outcome assessment and clinical decision support. Choosing widely-available patient features and appropriate mathematical methods for similarity calculations is crucial. International Statistical Classification of Diseases and Related Health Problems (ICD) codes are used worldwide to encode diseases and are available for nearly all patients. Aggregated as sets consisting of primary and secondary diagnoses they can display a degree of comorbidity and reveal comorbidity patterns. It is possible to compute the similarity of patients based on their ICD codes by using semantic similarity algorithms. These algorithms have been traditionally evaluated using a single-term expert rated data set. However, real-word patient data often display varying degrees of documented comorbidities that might impair algorithm performance. To account for this, we present a scale term that considers documented comorbidity-variance. In this work, we compared the performance of 80 combinations of established algorithms in terms of semantic similarity based on ICD-code sets. The sets have been extracted from patients with a C25.X (pancreatic cancer) primary diagnosis and provide a variety of different combinations of ICD-codes. Using our scale term we yielded the best results with a combination of level-based information content, Leacock & Chodorow concept similarity and bipartite graph matching for the set similarities reaching a correlation of 0.75 with our expert's ground truth. Our results highlight the importance of accounting for comorbidity variance while demonstrating how well current semantic similarity algorithms perform., Comment: 11 pages, 6 figures, 1 table

We revisit a construction of wide subcategories going back to work of Ingalls and Thomas. To a torsion pair in the category $ R\operatorname{-}\operatorname{mod}$ of finitely presented modules over a left artinian ring $R$, we assign two wide subcategories in the category $ R\operatorname{-}\operatorname{Mod}$ of all $R$-modules and describe them explicitly in terms of an associated cosilting module. It turns out that these subcategories are coreflective, and we address the question of which wide coreflective subcategories can be obtained in this way. Over a tame hereditary algebra, they are precisely the categories which are perpendicular to collections of pure-injective modules., Comment: 28 pages, 1 figure

Every cotilting module over a ring R induces a t-structure with a Grothendieck heart in the derived category D(Mod R). We determine the simple objects in this heart and their injective envelopes, combining torsion-theoretic aspects with methods from the model theory of modules and Auslander-Reiten theory., Comment: The paper will appear in Mathematische Zeitschrift

Mutation of compact silting objects is a fundamental operation in the representation theory of finite-dimensional algebras due to its connections to cluster theory and to the lattice of torsion pairs in module or derived categories. In this paper we develop a theory of mutation in the broader framework of silting or cosilting t-structures in triangulated categories. We show that mutation of pure-injective cosilting objects encompasses the classical concept of mutation for compact silting complexes. As an application we prove that any minimal inclusion of torsion classes in the category of finitely generated modules over an artinian ring corresponds to an irreducible mutation. This generalises a well-known result for functorially finite torsion classes., Comment: 37 pages

Ring epimorphisms often induce silting modules and cosilting modules, termed minimal silting or minimal cosilting. The aim of this paper is twofold. Firstly, we determine the minimal tilting and minimal cotilting modules over a tame hereditary algebra. In particular, we show that a large cotilting module is minimal if and only if it has an adic module as a direct summand. Secondly, we discuss the behaviour of minimality under ring extensions. We show that minimal cosilting modules over a commutative noetherian ring extend to minimal cosilting modules along any flat ring epimorphism. Similar results are obtained for commutative rings of small homological dimension.