1. 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 Category:05C10 

2. CMB 2015 (vol 59 pp. 50)
3. 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 nonself dual if every
proper subgroup of $G$
is self dual but $G$ is not self dual. In this paper, the structure
of minimal nonself dual groups is determined.
Keywords:minimal nonself dual group, finite group, metacyclic group, metabelian group Category:20D15 

4. CMB 2015 (vol 58 pp. 285)
 Karpukhin, Mikhail

Spectral Properties of a Family of Minimal Tori of Revolution in Fivedimensional Sphere
The normalized eigenvalues $\Lambda_i(M,g)$ of the LaplaceBeltrami
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 Category:58J50 

5. 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 BusemannHausdorff 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 

6. 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 computeraided.
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, 5504, 0504 

7. CMB 2012 (vol 56 pp. 709)
 Bartošová, Dana

Universal Minimal Flows of Groups of Automorphisms of Uncountable Structures
It is a wellknown 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 FraÃ¯ssÃ© 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 

8. 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 

9. CMB 2011 (vol 54 pp. 311)
10. 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 

11. 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 

12. CMB 1999 (vol 42 pp. 104)
 Nikolskaia, Ludmila

InstabilitÃ© de vecteurs propres d'opÃ©rateurs linÃ©aires
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 
