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5. On the diagonal of Riesz operators on Banach latticesRoman Drnovšek, Marko Kandić, 2024, original scientific article Abstract: This paper extends the well-known Ringrose theory for compact operators to polynomially Riesz operators on Banach spaces. The particular case of an ideal-triangularizable Riesz operator on an order continuous Banach lattice yields that the spectrum of such operator lies on its diagonal, which motivates the systematic study of an abstract diagonal of a regular operator on an order complete vector lattice $E$. We prove that the class $\mathscr D$ of regular operators for which the diagonal coincides with the atomic diagonal is always a band in $\mathcal L_r(E)$, which contains the band of abstract integral operators. If $E$ is also a Banach lattice, then $\mathscr D$ contains positive Riesz and positive AM-compact operators.
Keywords: vector lattices, Banach lattices, Riesz operators, diagonal of an operator
Published in DiRROS: 24.01.2025; Views: 28; Downloads: 7
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8. Optimal strategies in fractional games: vertex cover and dominationCsilla Bujtás, Günter Rote, Zsolt Tuza, 2024, original scientific article Abstract: In a hypergraph ${\cal H}=(V,{\cal E})$ with vertex set $V$ and edge set ${\cal E}$, a real-valued function $f: V \to [0, 1]$ is a fractional transversal if $\sum_{v\in E} f(v) \ge 1$ for every edge $E \in {\cal E}$. Its size is $|f| := \sum_{v \in V} f(v)$, and the fractional transversal number $\tau^\ast({\cal H})$ is the smallest possible $|f|$. We consider a game scenario where two players have opposite goals, one of them trying to minimize and the other to maximize the size of a fractional transversal constructed incrementally. We prove that both players have strategies to achieve their common optimum, and they can reach their goals using rational weights.
Keywords: fractional vertex cover, fractional transversal game, fractional domination game
Published in DiRROS: 24.01.2025; Views: 28; Downloads: 9
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9. Classification of cubic tricirculant nut graphsIvan Damnjanović, Nino Bašić, Tomaž Pisanski, Arjana Žitnik, 2024, original scientific article Abstract: A nut graph is a simple graph whose adjacency matrix has the eigenvalue zero with multiplicity one such that its corresponding eigenvector has no zero entries. It is known that there exist no cubic circulant nut graphs. A bicirculant (resp. tricirculant) graph is defined as a graph that admits a cyclic group of automorphisms having two (resp. three) orbits of vertices of equal size. We show that there exist no cubic bicirculant nut graphs and we provide a full classification of cubic tricirculant nut graphs.
Keywords: bicirculant, tricirculant, eigenvalue
Published in DiRROS: 24.01.2025; Views: 25; Downloads: 7
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10. Genomic insights into genetic diversity and seed coat color change in common bean composite populationsEva Plestenjak, Mohamed Neji, Lovro Sinkovič, Vladimir Meglič, Barbara Pipan, 2024, original scientific article Keywords: seed coat color, phenotypic variation, composite populations, whole genome sequencing
Published in DiRROS: 24.01.2025; Views: 33; Downloads: 13
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