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Abstract: A factorization of an $n \times n$ nonnegative symmetric matrix $A$ of the form $BCB^T$, where $C$ is a $k \times k$ symmetric matrix, and both $B$ and $C$ are required to be nonnegative, is called the Symmetric Nonnegative Matrix Trifactorization (SN-Trifactorization). The SNT-rank of $A$ is the minimal $k$ for which such factorization exists. The SNT-rank of a simple graph $G$ that allows loops is defined to be the minimal possible SNT-rank of all symmetric nonnegative matrices whose zero-nonzero pattern is prescribed by the graph $G$. We define set-join covers of graphs, and show that finding the SNT-rank of $G$ is equivalent to finding the minimal order of a set-join cover of $G$. Using this insight we develop basic properties of the SNT-rank for graphs and compute it for trees and cycles without loops. We show the equivalence between the SNT-rank for complete graphs and the Katona problem, and discuss uniqueness of patterns of matrices in the factorization.
Keywords: mathematics, mathematical economy, matrix algebra, nonnegative matrix factorization, nonnegative symmetric matrices, symmetric nonnegative trifactorization, pattern matrices
Published in DiRROS: 03.11.2025; Views: 139; Downloads: 76
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Abstract: Motivated by recent interest to $F$-inverse monoids, on the one hand, and to restriction and birestriction monoids, on the other hand, we initiate the study of $F$-birestriction monoids as algebraic structures in the enriched signature $(\cdot, \, ^*, \,^+, \, ^\mathfrak{m},1)$ where the unary operation $(\cdot)^\mathfrak{m}$ maps each element to the maximum element of its $\sigma$-class. We find a presentation of the free $F$-birestriction monoid ${\mathsf{FFBR}}(X)$ as a birestriction monoid ${\mathcal F}$ over the extended set of generators $X\cup\overline{X^+}$ where $\overline{X^+}$ is a set in a bijection with the free semigroup $X^+$ and encodes the maximum elements of (non-projection) $\sigma$-classes. This enables us to show that ${\mathsf{FFBR}}(X)$ decomposes as the partial action product $E({\mathcal I})\rtimes X^*$ of the idempotent semilattice of the universal inverse monoid ${\mathcal I}$ of ${\mathcal F}$ partially acted upon by the free monoid $X^*$. Invoking Schützenberger graphs, we prove that the word problem for ${\mathsf{FFBR}}(X)$ and its strong and perfect analogues is decidable. Furthermore, we show that ${\mathsf{FFBR}}(X)$ does not admit a geometric model based on a quotient of the Margolis-Meakin expansion $M({\mathsf{FG}}(X), X\cup \overline{X^+})$ over the free group ${\mathsf{FG}}(X)$, but the free perfect $X$-generated $F$-birestriction monoid admits such a model.
Keywords: birestriction monoid, F-birestriction monoid, free F-birestriction monoid, inverse monoid, F-inverse monoid, Margolis-Meakin expansion, Schützenberger graph, partial action, partial action product
Published in DiRROS: 03.11.2025; Views: 201; Downloads: 105
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Keywords: common buckwheat, grain yield stability, buckwheat, nutritional quality, temporal genetic structure, crop protein
Published in DiRROS: 30.10.2025; Views: 194; Downloads: 103
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Abstract: In the framework of the EURAD-2 partnership, the ICARUS (Innovative ChARacterization techniques for large volUmeS) work package aims at further developing, optimizing and harmonizing innovative techniques for the radiological, physical and chemical characterization of large volume of low/intermediate-level mixed waste, as it could be critical for the safe implementation of radioactive waste management programmes of the member states. Destructive Techniques (DT), Non-Destructive Techniques (NDT), and Scaling Factors (SF) will be investigated. Four use cases will be considered according to the end users’ needs: (i) enabling fast and sufficiently accurate characterization of gamma activity distribution by NDT in complex large packages, including mixed wastes as heterogeneous legacy waste; (ii) improving and simplifying NDT for determining physico-chemical properties; (iii) improving sensitivity, accuracy, and uncertainty and replacing the expensive and time-consuming radiochemical analysis of long-lived Difficult To Measure (DTM) radionuclides by DT; (iv) improving accuracy, uncertainty, and reliability of the SF approach to estimate DTM radionuclides in raw mixed waste. Specific education and training materials and opportunities will be provided to foster career advancement of young professionals and researchers. The research activities will be carried out by 29 organizations from 17 countries.
Keywords: nuclear physics, nuclear waste, nuclear fuel, nuclear decay
Published in DiRROS: 30.10.2025; Views: 150; Downloads: 80
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Keywords: climate adaptation, coastal area, cultural heritage, nature-based solutions
Published in DiRROS: 30.10.2025; Views: 168; Downloads: 90
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