MATHEMATICA SCANDINAVICA

https://www.mscand.dk/issue/feed MATHEMATICA SCANDINAVICA 2025-03-25T03:29:53+01:00 Arne Jensen mscand@math.au.dk Open Journal Systems https://www.mscand.dk/article/view/156272 Cover1 2025-03-25T03:09:39+01:00 Math Scand mscand@math.au.dk

2025-03-25T00:00:00+01:00 Copyright (c) 2025 MATHEMATICA SCANDINAVICA https://www.mscand.dk/article/view/156273 Announcement Board of associate editors 2025-03-25T03:16:08+01:00 Arne Jensen mscand@math.au.dk

Announcement of the board of associate editors for Mathematica Scandinavica

2025-03-25T00:00:00+01:00 Copyright (c) 2025 MATHEMATICA SCANDINAVICA https://www.mscand.dk/article/view/151573 On the Krull dimension of cofinite local cohomology modules 2024-11-27T10:42:50+01:00 Kamal Bahmanpour mscand@math.au.dk

Let $\mathfrak {a}$ be an ideal of a Noetherian ring $R$ such that the category of $\mathfrak {a}$-cofinite modules is an Abelian subcategory of the category of $R$-modules. Let $M$ be a finitely generated $R$-module such that the $R$-modules $H^i_{\mathfrak {a}}(M)$ are $\mathfrak {a}$-cofinite for all integers $i\in \mathbb{N}_0$. In this paper it is shown that $\dim H^i_{\mathfrak {a}}(M)\leq 1$ for all integers $i\geq 2$.

2025-03-25T00:00:00+01:00 Copyright (c) 2025 MATHEMATICA SCANDINAVICA https://www.mscand.dk/article/view/151576 On the distribution of sequences of the form $(q_ny)$ 2024-11-27T13:03:33+01:00 Simon Kristensen mscand@math.au.dk Tomas Persson mscand@math.au.dk

We study the distribution of sequences of the form $(q_ny)_{n=1}^\infty $, where $(q_n)_{n=1}^\infty $ is some increasing sequence of integers. In particular, we study the Lebesgue measure and find bounds on the Hausdorff dimension of the set of points $\gamma \in [0,1)$ which are well approximated by points in the sequence $(q_ny)_{n=1}^\infty $. The bounds on Hausdorff dimension are valid for almost every $y$ in the support of a measure of positive Fourier dimension. When the required rate of approximation is very good or if our sequence is sufficiently rapidly growing, our dimension bounds are sharp. If the measure of positive Fourier dimension is itself Lebesgue measure, our measure bounds are also sharp for a very large class of sequences.

2025-03-25T00:00:00+01:00 Copyright (c) 2025 MATHEMATICA SCANDINAVICA https://www.mscand.dk/article/view/153210 A study of H. Martens' Theorem on chains of cycles 2025-02-11T15:47:35+01:00 Marc Coppens mscand@math.au.dk

Let $C$ be a smooth curve of genus $g$ and let $d$,$r$ be integers with $1 \leq r \leq g-2$ and $2r\leq d \leq g-2+r$. H. Martens' Theorem states that $\dim (W^r_d(C))=d-2r$ implies $C$ is hyperelliptic. It is known that for a metric graph Γ of genus $g$ such statement using $\dim (W^r_d(\Gamma ))$ does not hold. However replacing $\dim (W^r_d(\Gamma ))$ by the so-called Brill-Noether rank $w^r_d(\Gamma )$ it was stated as a conjecture. Using a similar definition in the case of curves one has $\dim (W^r_d(C))=w^r_d(C)$. Let $\Gamma$ be a chain of cycles of genus $g$ and let $r,d$ be integers with $1 \leq r \leq g-2$ and $2r\leq d \leq g-3+r$. If $w^r_d(\Gamma )=d-2r$ then we prove Γ is hyperelliptic. In case $g \geq 2r+3$ then we prove there exist non-hyperelliptic chains of cycles satisfying $w^r_{g-2+r}(\Gamma )=g-2-r$, contradicting the conjecture. We give a complete description of all counterexamples within the set of chains of cycles to the statement of H. Martens' Theorem. Those counterexamples also give rise to chains of cycles such that $w^r_{g-2+r}(\Gamma ) \neq w^1_{g-r}(\Gamma )$. This shows that the Riemann-Roch duality does not hold for the Brill-Noether ranks of metric graphs.

2025-03-25T00:00:00+01:00 Copyright (c) 2025 MATHEMATICA SCANDINAVICA https://www.mscand.dk/article/view/150634 Real hypersurfaces with Reeb Jacobi operator of Codazzi type in the complex hyperbolic two-plane Grassmannians 2024-10-31T13:39:28+01:00 Young Jin Suh mscand@math.au.dk

Utilizing the concept of Reeb Jacobi operator of Codazzi type, we investigate Hopf real hypersurfaces in the complex hyperbolic two-plane Grassmannian $G^{*}_2(\mathbb {C}^{m+2})$ which admit a constant Reeb function α along the Reeb direction of ξ. If the Reeb function α is constant along the Reeb direction, then the Reeb vector field $\xi =-JN$ either belongs to the distribution ${\mathfrak D}$ or the distribution ${\mathfrak D}^{\bot }$. By virtue of this fact, we have proved a new result about Reeb Jacobi operator of Codazzi type according to the Reeb vector field $\xi \in \mathfrak {D}^{\bot }$ or $\xi \in \mathfrak {D}$, respectively.

2025-03-25T00:00:00+01:00 Copyright (c) 2025 MATHEMATICA SCANDINAVICA https://www.mscand.dk/article/view/152461 On the existence and uniqueness of the weak solution to spatial fractional nonlinear diffusion equation related to image processing 2024-12-28T10:06:02+01:00 Imane Boudrissa mscand@math.au.dk Noureddine Benhamidouche mscand@math.au.dk

This work discusses the existence and uniqueness of the weak solution to a spatial fractional diffusion equation, which can be applied in image processing. The proposed model combines the advantages of both second- and fourth-order diffusion equations along with Gaussian filtering by employing spatial fractional derivatives and a Gaussian filter. This approach enhance edges preservation and robustness to noise. The existence and uniqueness of the weak solution for the model are proved by applying Schauder's fixed-point theorem.

2025-03-25T00:00:00+01:00 Copyright (c) 2025 MATHEMATICA SCANDINAVICA https://www.mscand.dk/article/view/148979 Stable properties under weakly geometrically flat maps 2024-09-04T11:51:11+02:00 Daniel Barlet mscand@math.au.dk Jón Magnússon mscand@math.au.dk

In this note we show that a weakly geometrically flat map $\pi \colon M\rightarrow N$ between pure dimensional complex spaces has the local lifting property for cycles. From this result we also deduce that, under these hypotheses, several properties of $M$ are transferred to $N$.

2025-03-25T00:00:00+01:00 Copyright (c) 2025 MATHEMATICA SCANDINAVICA https://www.mscand.dk/article/view/151062 A unique continuation property for $\lvert\overline \partial u\rvert \leq V \lvert u\rvert$ 2024-11-15T06:52:40+01:00 Ziming Shi mscand@math.au.dk

Let $u\colon \Omega \subset \mathbb {C}^n \to \mathbb {C}^m$, for $n \geq 2$ and $m \geq 1$. Let $1 \leq p \leq 2$, and $2(2n)^2 -1 \leq q < \infty $ such that $\frac {1}{p} + \frac {1}{p'} = 1$ and $\frac {1}{p} - \frac {1}{p'} = \frac {1}{q}$. Suppose $\lvert \overline \partial u\rvert \leq V \lvert u\rvert $, where $V \in L^q_{\textrm {loc}}(\Omega )$. Then $u$ has a unique continuation property in the following sense: if $u \in W^{1,p}_{\textrm {loc}}(\Omega )$ and for some $z_0 \in \Omega $, $\lVert u \rVert _{L^{p'}(B(z_0,r))} $ decays faster than any powers of $r$ as $r \to 0$, then $u \equiv 0$. The same result holds for $q=\infty $ if $u$ is scalar-valued ($m=1$)