Opuscula Mathematica

In this paper, we investigate a free boundary problem relevant in several applications, such as tumor growth models. Our problem is expressed as an elliptic equation involving discontinuous nonlinearities in a specified domain with a moving boundary. We establish the existence and uniqueness of solutions and provide a qualitative analysis of the free boundaries generated by the nonlinear term (inner boundaries). Furthermore, we analyze the dynamics of the outer region boundary. The final result demonstrates that under certain conditions, our problem is solvable in the neighborhood of a radial solution.

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Unitarily equivalent bilateral weighted shifts with operator weights

(Wydawnictwa AGH, 2024) Buchała, Michał

The aim of this paper is to study unitarily equivalent bilateral weighted shifts with operator weights. Our purpose is to establish a general characterization of unitary equivalence of such shifts under the assumption that the weights are quasi-invertible. Under further assumptions on weights it was proved that unitary equivalence of bilateral weigthed shifts with operator weights defined on $\mathbb{C}^{2}$ can always be given by a unitary operator with at most two non-zero diagonals. The paper contains also examples of unitarily equivalent shifts with weights defined on $\mathbb{C}^{k}$ such that every unitary operator, which intertwines them has at least $k$ non-zero diagonals

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Open Access

Seven largest trees pack

(Wydawnictwa AGH, 2024) Cisiński, Maciej; Żak, Andrzej

The Tree Packing Conjecture (TPC) by Gyárfás states that any set of trees $T_2,\dots,T_{n-1}, T_n$ such that $T_i$ has $i$ vertices pack into $K_n$. The conjecture is true for bounded degree trees, but in general, it is widely open. Bollobás proposed a weakening of TPC which states that $k$ largest trees pack. The latter is true if none tree is a star, but in general, it is known only for $k=5$. In this paper we prove, among other results, that seven largest trees packThe Tree Packing Conjecture (TPC) by Gyárfás states that any set of trees $T_2,\dots,T_{n-1}, T_n$ such that $T_i$ has $i$ vertices pack into $K_n$. The conjecture is true for bounded degree trees, but in general, it is widely open. Bollobás proposed a weakening of TPC which states that $k$ largest trees pack. The latter is true if none tree is a star, but in general, it is known only for $k=5$. In this paper we prove, among other results, that seven largest trees pack

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Open Access

Recovering the shape of an equilateral quantum tree with the Dirichlet conditions at the pendant vertices

(Wydawnictwa AGH, 2024) Dudko, Anastasia; Lesechko, Oleksandr; Pivovarchik, Vyacheslav

We consider two spectral problems on an equilateral rooted tree with the standard (continuity and Kirchhoff's type) conditions at the interior vertices (except of the root if it is interior) and Dirichlet conditions at the pendant vertices (except of the root if it is pendant). For the first (Neumann) problem we impose the standard conditions (if the root is an interior vertex) or Neumann condition (if the root is a pendant vertex) at the root, while for the second (Dirichlet) problem we impose the Dirichlet condition at the root. We show that for caterpillar trees the spectra of the Neumann problem and of the Dirichlet problem uniquely determine the shape of the tree. Also, we present an example of co-spectral snowflake graphs

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Open Access

Isoperimetric inequalities in nonlocal diffusion problems with integrable kernel

(Wydawnictwa AGH, 2024) Galiano, Gonzalo

We deduce isoperimetric estimates for solutions of linear stationary and evolution problems. Our main result establishes the comparison in norm between the solution of a problem and its symmetric version when nonlocal diffusion defined through integrable kernels is replacing the usual local diffusion defined by a second order differential operator. Since an appropriate kernel rescaling allows to define a sequence of solutions of the nonlocal diffusion problems converging to their local diffusion counterparts, we also find the corresponding isoperimetric inequalities for the latter, i.e. we prove the classical Talenti's theorem. The novelty of our approach is that we replace the measure geometric tools employed in Talenti's proof, such as the geometric isoperimetric inequality or the coarea formula, by the Riesz's rearrangement inequality. Thus, in addition to providing a proof for the nonlocal diffusion case, our technique also introduces an alternative proof to Talenti's theorem

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