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Search: All articles in the CMB digital archive with keyword minimal

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1. CMB Online first

Bavula, V. V.; Lu, T.
Classification of simple weight modules over the Schrödinger algebra
A classification of simple weight modules over the Schrödinger algebra is given. The Krull and the global dimensions are found for the centralizer $C_{\mathcal{S}}(H)$ (and some of its prime factor algebras) of the Cartan element $H$ in the universal enveloping algebra $\mathcal{S}$ of the Schrödinger (Lie) algebra. The simple $C_{\mathcal{S}}(H)$-modules are classified. The Krull and the global dimensions are found for some (prime) factor algebras of the algebra $\mathcal{S}$ (over the centre). It is proved that some (prime) factor algebras of $\mathcal{S}$ and $C_{\mathcal{S}}(H)$ are tensor homological/Krull minimal.

Keywords:weight module, simple module, centralizer, Krull dimension, global dimension, tensor homological minimal algebra, tensor Krull minimal algebra
Categories:17B10, 17B20, 17B35, 16E10, 16P90, 16P40, 16P50

2. CMB Online first

Ha, Pham Hoang; Kawakami, Yu
A note on a unicity theorem for the Gauss maps of complete minimal surfaces in Euclidean four-space
The classical result of Nevanlinna states that two nonconstant meromorphic functions on the complex plane having the same images for five distinct values must be identically equal to each other. In this paper, we give a similar uniqueness theorem for the Gauss maps of complete minimal surfaces in Euclidean four-space.

Keywords:minimal surface, Gauss map, unicity theorem
Categories:53A10, 30D35, 53C42

3. CMB 2016 (vol 59 pp. 705)

Chen, Yichao; Yin, Xuluo
The Thickness of the Cartesian Product of Two Graphs
The thickness of a graph $G$ is the minimum number of planar subgraphs whose union is $G.$ A $t$-minimal graph is a graph of thickness $t$ which contains no proper subgraph of thickness $t.$ In this paper, upper and lower bounds are obtained for the thickness, $t(G\Box H)$, of the Cartesian product of two graphs $G$ and $H$, in terms of the thickness $t(G)$ and $t(H)$. Furthermore, the thickness of the Cartesian product of two planar graphs and of a $t$-minimal graph and a planar graph are determined. By using a new planar decomposition of the complete bipartite graph $K_{4k,4k},$ the thickness of the Cartesian product of two complete bipartite graphs $K_{n,n}$ and $K_{n,n}$ is also given, for $n\neq 4k+1$.

Keywords:planar graph, thickness, Cartesian product, $t$-minimal graph, complete bipartite graph

4. CMB 2015 (vol 59 pp. 50)

Dorfmeister, Josef F.; Inoguchi, Jun-ichi; Kobayashi, Shimpei
On the Bernstein Problem in the Three-dimensional Heisenberg Group
In this note we present a simple alternative proof for the Bernstein problem in the three-dimensional Heisenberg group $\operatorname{Nil}_3$ by using the loop group technique. We clarify the geometric meaning of the two-parameter ambiguity of entire minimal graphs with prescribed Abresch-Rosenberg differential.

Keywords:Bernstein problem, minimal graphs, Heisenberg group, loop groups, spinors
Categories:53A10, 53C42

5. CMB 2015 (vol 58 pp. 538)

Li, Lili; Chen, Guiyun
Minimal Non Self Dual Groups
A group $G$ is self dual if every subgroup of $G$ is isomorphic to a quotient of $G$ and every quotient of $G$ is isomorphic to a subgroup of $G$. It is minimal non-self dual if every proper subgroup of $G$ is self dual but $G$ is not self dual. In this paper, the structure of minimal non-self dual groups is determined.

Keywords:minimal non-self dual group, finite group, metacyclic group, metabelian group

6. CMB 2015 (vol 58 pp. 285)

Karpukhin, Mikhail
Spectral Properties of a Family of Minimal Tori of Revolution in Five-dimensional Sphere
The normalized eigenvalues $\Lambda_i(M,g)$ of the Laplace-Beltrami operator can be considered as functionals on the space of all Riemannian metrics $g$ on a fixed surface $M$. In recent papers several explicit examples of extremal metrics were provided. These metrics are induced by minimal immersions of surfaces in $\mathbb{S}^3$ or $\mathbb{S}^4$. In the present paper a family of extremal metrics induced by minimal immersions in $\mathbb{S}^5$ is investigated.

Keywords:extremal metric, minimal surface

7. CMB 2014 (vol 57 pp. 765)

da Silva, Rosângela Maria; Tenenblat, Keti
Helicoidal Minimal Surfaces in a Finsler Space of Randers Type
We consider the Finsler space $(\bar{M}^3, \bar{F})$ obtained by perturbing the Euclidean metric of $\mathbb{R}^3$ by a rotation. It is the open region of $\mathbb{R}^3$ bounded by a cylinder with a Randers metric. Using the Busemann-Hausdorff volume form, we obtain the differential equation that characterizes the helicoidal minimal surfaces in $\bar{M}^3$. We prove that the helicoid is a minimal surface in $\bar{M}^3$, only if the axis of the helicoid is the axis of the cylinder. Moreover, we prove that, in the Randers space $(\bar{M}^3, \bar{F})$, the only minimal surfaces in the Bonnet family, with fixed axis $O\bar{x}^3$, are the catenoids and the helicoids.

Keywords:minimal surfaces, helicoidal surfaces, Finsler space, Randers space
Categories:53A10, 53B40

8. CMB 2013 (vol 57 pp. 225)

Adamaszek, Michał
Small Flag Complexes with Torsion
We classify flag complexes on at most $12$ vertices with torsion in the first homology group. The result is moderately computer-aided. As a consequence we confirm a folklore conjecture that the smallest poset whose order complex is homotopy equivalent to the real projective plane (and also the smallest poset with torsion in the first homology group) has exactly $13$ elements.

Keywords:clique complex, order complex, homology, torsion, minimal model
Categories:55U10, 06A11, 55P40, 55-04, 05-04

9. CMB 2012 (vol 56 pp. 709)

Bartošová, Dana
Universal Minimal Flows of Groups of Automorphisms of Uncountable Structures
It is a well-known fact, that the greatest ambit for a topological group $G$ is the Samuel compactification of $G$ with respect to the right uniformity on $G.$ We apply the original description by Samuel from 1948 to give a simple computation of the universal minimal flow for groups of automorphisms of uncountable structures using Fraïssé theory and Ramsey theory. This work generalizes some of the known results about countable structures.

Keywords:universal minimal flows, ultrafilter flows, Ramsey theory
Categories:37B05, 03E02, 05D10, 22F50, 54H20

10. CMB 2011 (vol 56 pp. 434)

Wnuk, Witold
Some Remarks on the Algebraic Sum of Ideals and Riesz Subspaces
Following ideas used by Drewnowski and Wilansky we prove that if $I$ is an infinite dimensional and infinite codimensional closed ideal in a complete metrizable locally solid Riesz space and $I$ does not contain any order copy of $\mathbb R^{\mathbb N}$ then there exists a closed, separable, discrete Riesz subspace $G$ such that the topology induced on $G$ is Lebesgue, $I \cap G = \{0\}$, and $I + G$ is not closed.

Keywords:locally solid Riesz space, Riesz subspace, ideal, minimal topological vector space, Lebesgue property
Categories:46A40, 46B42, 46B45

11. CMB 2011 (vol 54 pp. 311)

Marzougui, Habib
Some Remarks Concerning the Topological Characterization of Limit Sets for Surface Flows
We give some extension to theorems of Jiménez López and Soler López concerning the topological characterization for limit sets of continuous flows on closed orientable surfaces.

Keywords:flows on surfaces, orbits, class of an orbit, singularities, minimal set, limit set, regular cylinder
Categories:37B20, 37E35

12. CMB 2003 (vol 46 pp. 632)

Runde, Volker
The Operator Amenability of Uniform Algebras
We prove a quantized version of a theorem by M.~V.~She\u{\i}nberg: A uniform algebra equipped with its canonical, {\it i.e.}, minimal, operator space structure is operator amenable if and only if it is a commutative $C^\ast$-algebra.

Keywords:uniform algebras, amenable Banach algebras, operator amenability, minimal, operator space
Categories:46H20, 46H25, 46J10, 46J40, 47L25

13. CMB 2002 (vol 45 pp. 154)

Weitsman, Allen
On the Poisson Integral of Step Functions and Minimal Surfaces
Applications of minimal surface methods are made to obtain information about univalent harmonic mappings. In the case where the mapping arises as the Poisson integral of a step function, lower bounds for the number of zeros of the dilatation are obtained in terms of the geometry of the image.

Keywords:harmonic mappings, dilatation, minimal surfaces
Categories:30C62, 31A05, 31A20, 49Q05

14. CMB 1999 (vol 42 pp. 104)

Nikolskaia, Ludmila
Instabilité de vecteurs propres d'opérateurs linéaires
We consider some geometric properties of eigenvectors of linear operators on infinite dimensional Hilbert space. It is proved that the property of a family of vectors $(x_n)$ to be eigenvectors $Tx_n= \lambda_n x_n$ ($\lambda_n \noteq \lambda_k$ for $n\noteq k$) of a bounded operator $T$ (admissibility property) is very instable with respect to additive and linear perturbations. For instance, (1)~for the sequence $(x_n+\epsilon_n v_n)_{n\geq k(\epsilon)}$ to be admissible for every admissible $(x_n)$ and for a suitable choice of small numbers $\epsilon_n\noteq 0$ it is necessary and sufficient that the perturbation sequence be eventually scalar: there exist $\gamma_n\in \C$ such that $v_n= \gamma_n v_{k}$ for $n\geq k$ (Theorem~2); (2)~for a bounded operator $A$ to transform admissible families $(x_n)$ into admissible families $(Ax_n)$ it is necessary and sufficient that $A$ be left invertible (Theorem~4).

Keywords:eigenvectors, minimal families, reproducing kernels
Categories:47A10, 46B15

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