#### Archivum Mathematicum

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ISSN / EISSN
:
0044-8753 / 1212-5059

Published by: Masaryk University Press
(10.5817)

Total articles ≅ 252

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#### Latest articles in this journal

Archivum Mathematicum, Volume 57, pp 1-11; https://doi.org/10.5817/am2021-1-1

**Abstract:**

In the present paper we study naturally reductive homogeneous $(\alpha ,\beta )$-metric spaces. We show that for homogeneous $(\alpha ,\beta )$-metric spaces, under a mild condition, the two definitions of naturally reductive homogeneous Finsler space, given in the literature, are equivalent. Then, we compute the flag curvature of naturally reductive homogeneous $(\alpha ,\beta )$-metric spaces.

Archivum Mathematicum, Volume 57, pp 13-26; https://doi.org/10.5817/am2021-1-13

**Abstract:**

Let $d$ be an odd square-free integer, $m\ge 3$ any integer and $L_{m, d}:=\mathbb{Q}(\zeta _{2^m},\sqrt{d})$. In this paper, we shall determine all the fields $L_{m, d}$ having an odd class number. Furthermore, using the cyclotomic $\mathbb{Z}_2$-extensions of some number fields, we compute the rank of the $2$-class group of $L_{m, d}$ whenever the prime divisors of $d$ are congruent to $3$ or $5\pmod 8$.

Archivum Mathematicum, Volume 57, pp 285-297; https://doi.org/10.5817/am2021-5-285

**Abstract:**

In this paper, the authors establish sufficient conditions for the existence of solutions to implicit fractional differential inclusions with nonlocal conditions. Both of the cases of convex and nonconvex valued right hand sides are considered.

Archivum Mathematicum, Volume 57, pp 41-60; https://doi.org/10.5817/am2021-1-41

**Abstract:**

We consider the half-linear differential equation of the form \[ (p(t)|x^{\prime }|^{\alpha }\mathrm{sgn}\,x^{\prime })^{\prime } + q(t)|x|^{\alpha }\mathrm{sgn}\,x = 0\,, \quad t \ge t_{0} \,, \] under the assumption that $p(t)^{-1/\alpha }$ is integrable on $[t_{0}, \infty )$. It is shown that if a certain condition is satisfied, then the above equation has a pair of nonoscillatory solutions with specific asymptotic behavior as $t \rightarrow \infty $.

Archivum Mathematicum, Volume 57, pp 27-39; https://doi.org/10.5817/am2021-1-27

**Abstract:**

Asymptotic forms of solutions of half-linear ordinary differential equation $\big (|u^{\prime }|^{\alpha -1}u^{\prime }\big )^{\prime }= \alpha \big (1+b(t)\big ) |u|^{\alpha -1}u$ are investigated under a smallness condition and some signum conditions on $b(t)$. When $\alpha =1$, our results reduce to well-known ones for linear ordinary differential equations.

Archivum Mathematicum, Volume 57, pp 83-99; https://doi.org/10.5817/am2021-2-83

**Abstract:**

Some global existence criteria for quaternionic Riccati equations are established. Two of them are used to prove a completely non conjugation theorem for solutions of linear systems of ordinary differential equations.

Archivum Mathematicum, Volume 57, pp 101-111; https://doi.org/10.5817/am2021-2-101

**Abstract:**

We first classify left invariant Douglas $(\alpha , \beta )$-metrics on the Heisenberg group $H_{2n+1}$ of dimension $2n + 1$ and its extension i.e., oscillator group. Then we explicitly give the flag curvature formulas and geodesic vectors for these spaces, when equipped with these metrics. We also explicitly obtain $S$-curvature formulas of left invariant Randers metrics of Douglas type on these spaces and obtain a comparison on geometry of these spaces, when equipped with left invariant Douglas $(\alpha , \beta )$-metrics. More exactly, we show that although the results concerning bi-invariant Douglas $(\alpha ,\beta )$-metrics on these spaces are similar, several results concerning left invariant Douglas $(\alpha ,\beta )$-metrics on these spaces are different. For example we prove that the existence of left-invariant Matsumoto, Kropina and Randers metrics of Berwald type on oscillator groups can not extend to Heisenberg groups. Also we prove that oscillator groups have always vanishing $S$-curvature, whereas this can not occur on Heisenberg groups. Moreover, we prove that there exist new geodesic vectors on oscillator groups which can not extend to the Heisenberg groups.

Archivum Mathematicum, Volume 57, pp 157-174; https://doi.org/10.5817/am2021-3-157

**Abstract:**

The concept of generalized prime $D$-filters is introduced in distributive lattices. Generalized prime $D$-filters are characterized in terms of principal filters and ideals. The notion of generalized minimal prime $D$-filters is introduced in distributive lattices and properties of minimal prime $D$-filters are then studied with respect to congruences. Some topological properties of the space of all prime $D$-filters of a distributive lattice are also studied.

Archivum Mathematicum, Volume 57, pp 131-150; https://doi.org/10.5817/am2021-3-131

**Abstract:**

In this paper, we clarify the relationship among the Vietoris-type homology theories and the microsimplicial homology theories, where the latter are nonstandard homology theories defined by M.C. McCord (for topological spaces), T. Korppi (for completely regular topological spaces) and the author (for uniform spaces). We show that McCord’s and our homology are isomorphic for all compact uniform spaces and that Korppi’s and our homology are isomorphic for all fine uniform spaces. Our homology shares many good properties with Korppi’s homology. As an example, we outline a proof of the continuity of our homology with respect to uniform resolutions. S. Garavaglia proved that McCord’s homology is isomorphic to Vietoris homology for all compact topological spaces. Inspired by this result, we prove that our homology is isomorphic to uniform Vietoris homology for all precompact uniform spaces and that Korppi’s homology is isomorphic to normal Vietoris homology for all pseudocompact completely regular topological spaces.

Archivum Mathematicum, Volume 57, pp 195-219; https://doi.org/10.5817/am2021-4-195

**Abstract:**

Let $\Phi _1,\ldots ,\Phi _{k+1}$ (with $k\ge 1$) be vector fields of class $C^k$ in an open set $U\subset ^{N+m}$, let $\mathbb{M}$ be a $N$-dimensional $C^k$ submanifold of $U$ and define \[ \mathbb{T}:=\lbrace z\in \mathbb{M}: \Phi _1(z), \ldots , \Phi _{k+1}(z) \in T_z \mathbb{M}\rbrace \] where $T_z \mathbb{M}$ is the tangent space to $\mathbb{M}$ at $z$. Then we expect the following property, which is obvious in the special case when $z_0$ is an interior point (relative to $\mathbb{M}$) of $\mathbb{T}$: If $z_0\in \mathbb{M}$ is a $(N+k)$-density point (relative to $\mathbb{M}$) of $\mathbb{T}$ then all the iterated Lie brackets of order less or equal to $k$ \[ \Phi _{i_1}(z_0),\, [\Phi _{i_1}, \Phi _{i_2}](z_0), \, [[\Phi _{i_1}, \Phi _{i_2}], \Phi _{i_3}](z_0),\, \ldots \qquad (h, i_h\le k+1) \] belong to $T_{z_0}\mathbb{M}$. Such a property has been proved in [9] for $k=1$ and its proof in the case $k=2$ is the main purpose of the present paper. The following corollary follows at once: Let $\mathbb{D}$ be a $C^2$ distribution of rank $N$ on an open set $U\subset ^{N+m}$ and $\mathbb{M}$ be a $N$-dimensional $C^2$ submanifold of $U$. Moreover let $z_0\in \mathbb{M}$ be a $(N+2)$-density point of the tangency set $\lbrace z\in \mathbb{M}\,\vert \, T_z\mathbb{M}=\mathbb{D}(z)\rbrace $. Then $\mathbb{D}$ must be $2$-involutive at $z_0$, i.e., for every family $\lbrace X_j\rbrace _{j=1}^N$ of class $C^2$ in a neighborhood $V\subset U$ of $z_0$ which generates $\mathbb{D}$ one has \[ X_{i_1} (z_0), [X_{i_1},X_{i_2}](z_0), [[X_{i_1},X_{i_2}],X_{i_3}](z_0)\in T_{z_0}\mathbb{M}\] for all $1\le i_1, i_2, i_3\le N$.