1. The liberation set in the inverse eigenvalue problem of a graphJephian C.-H. Lin, Polona Oblak, Helena Šmigoc, 2023, original scientific article Abstract: The inverse eigenvalue problem of a graph $G$ is the problem of characterizing all lists of eigenvalues of real symmetric matrices whose off-diagonal pattern is prescribed by the adjacencies of $G$. The strong spectral property is a powerful tool in this problem, which identifies matrices whose entries can be perturbed while controlling the pattern and preserving the eigenvalues. The Matrix Liberation Lemma introduced by Barrett et al. in 2020 advances the notion to a more general setting. In this paper we revisit the Matrix Liberation Lemma and prove an equivalent statement, that reduces some of the technical difficulties in applying the result. We test our method on matrices of the form $M=A \oplus B$ and show how this new approach supplements the results that can be obtained from the strong spectral property only. While extending this notion to the direct sums of graphs, we discover a surprising connection with the zero forcing game on Cartesian products of graphs. Throughout the paper we apply our results to resolve a selection of open cases for the inverse eigenvalue problem of a graph on six vertices.
Keywords: symmetric matrix, inverse eigenvalue problem, strong spectral property, Matrix Liberation Lemma, zero forcing
Published in DiRROS: 14.03.2024; Views: 115; Downloads: 57
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2. Recent developments on Oka manifoldsFranc Forstnerič, 2023, review article Abstract: In this paper we present the main developments in Oka theory since the publication of my book "Stein Manifolds and Holomorphic Mappings (The Homotopy Principle in Complex Analysis)", 2nd ed., Springer, 2017. We also give several new results, examples and constructions of Oka domains in Euclidean and projective spaces. Furthermore, we show that for $n > 1$ the fibre $\mathbb C^n$ in a Stein family can degenerate to a non-Oka fibre, thereby answering a question of Takeo Ohsawa. Several open problems are discussed.
Keywords: Oka manifold, Oka map, Stein manifold, elliptic manifold, algebraically subelliptic manifold, Calabi–Yau manifold, density property
Published in DiRROS: 14.03.2024; Views: 426; Downloads: 456
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3. Oka domains in Euclidean spacesFranc Forstnerič, Erlend Fornæss Wold, 2024, original scientific article Abstract: In this paper, we find surprisingly small Oka domains in Euclidean spaces $\mathbb C^n$ of dimension $n>1$ at the very limit of what is possible. Under a mild geometric assumption on a closed unbounded convex set $E$ in $\mathbb C^n$, we show that $\mathbb C^n\setminus E$ is an Oka domain. In particular, there are Oka domains only slightly bigger than a halfspace, the latter being neither Oka nor hyperbolic. This gives smooth families of real hypersurfaces $\Sigma_t \subset \mathbb C^n$ for $t \in \mathbb R$ dividing $\mathbb C^n$ in an unbounded hyperbolic domain and an Oka domain such that at $t=0$, $\Sigma_0$ is a hyperplane and the character of the two sides gets reversed. More generally, we show that if $E$ is a closed set in $\mathbb C^n$ for $n>1$ whose projective closure $\overline E \subset \mathbb{CP}^n$ avoids a hyperplane $\Lambda \subset \mathbb{CP}^n$ and is polynomially convex in $\mathbb{CP}^n\setminus \Lambda\cong\mathbb C^n$, then $\mathbb C^n\setminus E$ is an Oka domain.
Keywords: Oka manifold, hyperbolic manifolds, density property, projectively convex sets
Published in DiRROS: 19.02.2024; Views: 230; Downloads: 75
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4. Complete nonsingular holomorphic foliations on Stein manifoldsAntonio Alarcón, Franc Forstnerič, 2024, original scientific article Abstract: Let $X$ be a Stein manifold of complex dimension $n \ge 1$ endowed with a Riemannian metric ${\mathfrak g}$. We show that for every integer $k$ with $\left[\frac{n}{2}\right] \le k \le n-1$ there is a nonsingular holomorphic foliation of dimension $k$ on $X$ all of whose leaves are topologically closed and ${\mathfrak g}$-complete. The same is true if $1\le k \left[\frac{n}{2}\right]$ provided that there is a complex vector bundle epimorphism $TX\to X \times \mathbb{C}^{n-k}$. We also show that if $\mathcal{F}$ is a proper holomorphic foliation on $\mathbb{C}^n$ $(n > 1)$ then for any Riemannian metric ${\mathfrak g}$ on $\mathbb{C}^n$ there is a holomorphic automorphism $\Phi$ of $\mathbb{C}^n$ such that the image foliation $\Phi_*\mathcal{F}$ is ${\mathfrak g}$-complete. The analogous result is obtained on every Stein manifold with Varolin's density property.
Keywords: Stein manifolds, complete holomorphic foliations, density property
Published in DiRROS: 19.02.2024; Views: 189; Downloads: 66
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5. Experimental study of the fire dynamics in a semi-enclosure formed by photovoltaic (PV) installations on flat roof constructionsJens Steemann Kristensen, Benjamin Jacobs, Grunde Jomaas, 2022, original scientific article Abstract: Flame spread experiments upon a BROOF(t4) compliant flat roof mock-up located below a vertical barrier were carried out for variations in gap height, inclination, subjacent insulation material, and the barrier type (stainless-steel board or photovoltaic (PV) module). A binary flame spread scenario was identified, where re-radiation from the flame facilitated self-sustained flame spread if the gap height to the horizontal panel was below 10 cm for the stainless-steel board and 11 cm for PV modules. These were defined as the critical gap heights. Inclination of the PV modules increased the critical gap height and caused a 25% faster flame spread rate (FSR) than the FSR below horizontal modules with the same gap height at the location of ignition. The faster FSR for inclined modules caused a 40% reduction of the maximum temperature measured at a depth of 70 mm in the insulation materials (242°C). Based on temperatures measured in the insulation materials, the 60 mm polyisocyanurate (PIR) insulation performed slightly better than the 50 mm mineral wool insulation. However, it is expected that the mineral wool would outperform the PIR insulation if tested with the same thickness, as it insulates significantly better at high temperatures. Finally, no sustained flame spread was observed on the back side polymer sheet of the PV modules, but one of the three PV module brands produced burning droplets. Based on the experiments, it can be concluded that the current standards are inadequate as the introduction of a PV system on a compliant roof construction enables flame spread.
Keywords: photovoltaic (PV) installations, flame spread, fire dynamics, property protection, open access
Published in DiRROS: 31.05.2023; Views: 325; Downloads: 246
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