MATHEMATICA SCANDINAVICA

Critical configurations for three projective views

2023-07-30T14:16:49+02:00

The problem of structure from motion is concerned with recovering the 3-dimensional structure of an object from a set of 2-dimensional images taken by unknown cameras. Generally, all information can be uniquely recovered if enough images and point correspondences are provided, yet there are certain cases where unique recovery is impossible; these are called \emph{critical configurations}. We use an algebraic approach to study the critical configurations for three projective cameras. We show that all critical configurations lie on the intersection of quadric surfaces, and classify exactly which intersections constitute a critical configuration.

A note on $\mathcal{M}$-normal embedded subgroups of finite groups

2023-05-16T05:47:18+02:00

In this note, we obtain a criterion for $p$-nilpotency of a finite group. We not only simplify the proofs of the main theorems of Chen et al. [On $\mathcal{M}$-normal embedded subgroups and the structure of finite groups, Math. Scand. 127(2021), no. 2, 243–251] but also generalize these results.

Attainable measures for certain types of $p$-adic Duffin-Schaeffer sets

2023-08-02T09:10:31+02:00

This paper settles recent conjectures concerning the $p$-adic Haar measure applied to a family of sets defined in terms of Diophantine approximation. This is done by determining the spectrum of measure values for each family and seeing that this contradicts the corresponding conjectures.

Harmonic analogue of Bohr phenomenon of certain classes of univalent and analytic functions

2023-07-24T09:54:34+02:00

A class $ \mathcal {F} $ consisting of analytic functions $ f(z)=\sum _{n=0}^{\infty }a_nz^n $ in the unit disk $ \mathbb {D}=\{z\in \mathbb {C}:\lvert z\rvert <1\} $ is said to satisfy Bohr phenomenon if there exists an $ r_f>0 $ such that $$ \sum _{n=1}^{\infty }\lvert a_n\rvert r^n\leq d(f(0),\partial \mathbb {D}) $$ for every function $ f\in \mathcal {F} $, and $\lvert z\rvert =r\leq r_f $. The largest radius $ r_f $ is known as the Bohr radius and the inequality $ \sum _{n=1}^{\infty }\lvert a_n\rvert r^n\leq d(f(0),\partial f(\mathbb {D})) $ is known as the Bohr inequality for the class $ \mathcal {F} $, where $d$ is the Euclidean distance. In this paper, we prove several sharp improved and refined versions of Bohr-type inequalities in terms of area measure of functions in a certain subclass of analytic and univalent (i.e. one-to-one) functions. As a consequence, we obtain several interesting corollaries on the Bohr-type inequality for the class which are the harmonic analogue of some Bohr-type inequality for the class of analytic functions.

On scaling limits of random Halin-like maps

2023-08-08T14:18:19+02:00

We consider maps which are constructed from plane trees by assigning marks to the corners of each vertex and then connecting each pair of consecutive marks on their contour by a single edge. A measure is defined on the set of such maps by assigning Boltzmann weights to the faces. When every vertex has exactly one marked corner, these maps are dissections of a polygon which are bijectively related to non-crossing trees. When every vertex has at least one marked corner, the maps are outerplanar and each of its two-connected component is bijectively related to a non-crossing tree. We study the scaling limits of the maps under these conditions and establish that for certain choices of the weights the scaling limits are either the Brownian CRT or the α-stable looptrees of Curien and Kortchemski.

Bilinear forms, Schur multipliers, complete boundedness and duality

2023-08-26T10:31:56+02:00

Grothendieck's inequalities for operators and bilinear forms imply some factorization results for complex $m \times n$ matrices. Based on the theory of operator spaces and completely bounded maps we present norm optimal versions of these results and two norm optimal factorization results related to the Schur product. We show that the spaces of bilinear forms and of Schur multipliers are conjugate duals to each other with respect to their completely bounded norms.

On shifting the principal eigenvalue of Dirichlet problem to infinity with non-transversal incompressible drift

2023-07-24T14:16:35+02:00

We prove that it is always possible to add some divergence free drift vector field to some two dimensional spherical Dirichlet problem, such that the resulting principal eigenvalue lies above a prescribed bound. By construction those drift vector fields vanish on the boundary and their flow lines individually stay away from the boundary. The capacity of those drift vector fields to accelerate diffusivity originates from high frequency oscillation of the associated flow lines. The lower bounds for the spectrum are obtained through isoperimetric inequalities for flow invariant functions.

Weyl asymptotics for fractional-order Dirichlet realizations in nonsmooth cases

2023-06-12T13:11:03+02:00

Let $P$ be a symmetric $2a$-order classical strongly elliptic pseudodifferential operator with \emph{even} symbol $p(x,\xi)$ on $\mathbb{R}^n $ ($0<a<1$), for example a perturbation of $(-\Delta )^a$. Let $\Omega \subset \mathbb{R}^n$ be bounded, and let $P_D$ be the Dirichlet realization in $L_2(\Omega)$ defined under the exterior condition $u=0$ in $\mathbb{R}^n\setminus\Omega$. When $p(x,\xi)$ and $\Omega$ are $C^\infty $, it is known that the eigenvalues $\lambda_j$ (ordered in a nondecreasing sequence for $j\to \infty$) satisfy a Weyl asymptotic formula \begin{equation*} \lambda _j(P_{D})=C(P,\Omega )j^{2a/n}+o(j^{2a/n}) \text {for $j\to \infty $}, \end{equation*} with $C(P,\Omega)$ determined from the principal symbol of $P$. We now show that this result is valid for more general operators with a possibly nonsmooth $x$-dependence, over Lipschitz domains, and that it extends to $\tilde P=P+P'+P”$, where $P'$ is an operator of order $<\min\{2a, a+\frac 12\}$ with certain mapping properties, and $P”$ is bounded in $L_2(\Omega )$ (e.g. $P”=V(x)\in L_\infty(\Omega)$). Also the regularity of eigenfunctions of $P_D$ is discussed.

Finitely presented isomorphisms of Cuntz-Krieger algebras and continuous orbit equivalence of one-sided topological Markov shifts

2023-08-01T08:40:30+02:00

We introduce the notion of finitely presented isomorphism between Cuntz–Krieger algebras, and of finitely presented isomorphic Cuntz–Krieger algebras. We prove that there exists a finitely presented isomorphism between Cuntz–Krieger algebras $\mathcal{O}_A$ and $\mathcal{O}_B$ if and only if their one-sided topological Markov shifts $(X_A,\sigma_A)$ and $(X_B,\sigma_B)$ are continuously orbit equivalent. Hence the value $\det (I-A)$ is a complete invariant for the existence of a finitely presented isomorphism between isomorphic Cuntz–Krieger algebras, so that there exists a pair of Cuntz–Krieger algebras which are isomorphic but not finitely presented isomorphic.

Corrections to "Vanishing Morrey integrability for Riesz potentials in Morrey-Orlicz spaces"

2023-08-20T14:51:56+02:00

In this note some flaws in our recent paper by Mizuta and Shimomura (Math. Scand. 129 (2023), no.~2, 374--400) are corrected.

Index

2023-10-23T11:15:20+02:00

Index Vol. 129

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2023-10-23T11:30:36+02:00

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