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703. Reflexivity of the space of transversal distributionsJure Kališnik, 2023, izvirni znanstveni članek Povzetek: For any smooth, Hausdorff and second-countable manifold $N$ one can define the Fréchet space ${\mathcal C}^{\infty}(N)$ of smooth functions on $N$ and its strong dual ${\cal E}'(N)$ of compactly supported distributions on $N$. It is a standard result that the strong dual of ${\cal E}'(N)$ is naturally isomorphic to ${\mathcal C}^{\infty}(N)$, which implies that both ${\mathcal C}^{\infty}(N)$ and ${\cal E}'(N)$ are reflexive locally convex spaces. In this paper we generalise that result to the setting of transversal distributions on the total space of a surjective submersion $\pi : P\to M$. We show that the strong ${\mathcal C}^{\infty}_c(M)$-dual of the space ${\cal E}'_{\pi} (P)$ of $\pi$-transversal distributions is naturally isomorphic to the ${\mathcal C}^{\infty}_c(M)$-module ${\mathcal C}^{\infty}(P)$.
Ključne besede: distributions with compact support, Fréchet spaces, transversal distributions, homomorphisms of modules, reflexive modules
Objavljeno v DiRROS: 15.03.2024; Ogledov: 261; Prenosov: 138
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704. Rigidity of terminal simplices in persistent homologyAleksandra Franc, Žiga Virk, 2023, izvirni znanstveni članek Povzetek: Given a filtration function on a finite simplicial complex, stability theorem of persistent homology states that the corresponding barcode is continuous with respect to changes in the filtration function. However, due to the discrete setting of simplicial complexes, the simplices terminating matched bars cannot change continuously for arbitrary perturbations of filtration functions. In this paper we provide a sufficient condition for rigidity of a terminal simplex, i.e., a condition on $\varepsilon > 0$ implying that the terminal simplex of a homology class or a bar in persistent homology remains constant through $\varepsilon$-perturbations of filtration function. The condition for a homology class or a bar in dimension $n$ depends only on the barcodes in dimensions $n$ and $n+1$.
Ključne besede: persistent homology, stability theorem, terminal simplex, rigidity
Objavljeno v DiRROS: 15.03.2024; Ogledov: 203; Prenosov: 93
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710. Trilinear embedding for divergence-form operators with complex coefficientsAndrea Carbonaro, Oliver Dragičević, Vjekoslav Kovač, Kristina Ana Škreb, 2023, izvirni znanstveni članek Povzetek: We prove a dimension-free $L^p(\Omega)\times L^q(\Omega)\times L^r(\Omega)\rightarrow L^1(\Omega\times (0,\infty))$ embedding for triples of elliptic operators in divergence form with complex coefficients and subject to mixed boundary conditions on $\Omega$, and for triples of exponents $p,q,r \in (1,\infty)$ mutually related by the identity $1/p+1/q+1/r=1$. Here $\Omega$ is allowed to be an arbitrary open subset of $\mathbb{R}^d$. Our assumptions involving the exponents and coefficient matrices are expressed in terms of a condition known as $p$-ellipticity. The proof utilizes the method of Bellman functions and heat flows. As a corollary, we give applications to (i) paraproducts and (ii) square functions associated with the corresponding operator semigroups, moreover, we prove (iii) inequalities of Kato-Ponce type for elliptic operators with complex coefficients. All the above results are the first of their kind for elliptic divergence-form operators with complex coefficients on arbitrary open sets. Furthermore, the approach to (ii),(iii) through trilinear embeddings seems to be new.
Ključne besede: elliptic differential operator, p-ellipticity, operator semigroup, multilinear estimate
Objavljeno v DiRROS: 15.03.2024; Ogledov: 202; Prenosov: 113
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