26. CMB 2016 (vol 59 pp. 673)
 Bačák, Miroslav; Kovalev, Leonid V.

Lipschitz Retractions in Hadamard Spaces Via Gradient Flow Semigroups
Let $X(n),$ for $n\in\mathbb{N},$ be the set of all subsets of a metric
space $(X,d)$ of cardinality at most $n.$ The set $X(n)$ equipped
with the Hausdorff metric is called a finite subset space. In
this paper we are concerned with the existence of Lipschitz retractions
$r\colon X(n)\to X(n1)$ for $n\ge2.$ It is known that such retractions
do not exist if $X$ is the onedimensional sphere. On the other
hand L. Kovalev has recently established their existence in case $X$
is a Hilbert space and he also posed a question as to whether
or not such Lipschitz retractions exist for $X$ being a Hadamard
space. In the present paper we answer this question in the positive.
Keywords:finite subset space, gradient flow, Hadamard space, LieTrotterKato formula, Lipschitz retraction Categories:53C23, 47H20, 54E40, 58D07 

27. CMB Online first
 Chung, Jaeyoung; Ju, Yumin; Rassias, John

Cubic functional equations on restricted domains of Lebesgue measure zero
Let $X$ be a real normed space, $Y$ a Bancch space and $f:X \to
Y$.
We prove the UlamHyers stability theorem
for the cubic functional equation
\begin{align*}
f(2x+y)+f(2xy)2f(x+y)2f(xy)12f(x)=0
\end{align*}
in restricted domains. As an application we consider a measure
zero stability problem
of the inequality
\begin{align*}
\f(2x+y)+f(2xy)2f(x+y)2f(xy)12f(x)\\le \epsilon
\end{align*}
for all $(x, y)$ in $\Gamma\subset\mathbb R^2$ of Lebesgue measure
0.
Keywords:Baire category theorem, cubic functional equation, first category, Lebesgue measure, UlamHyers stability Category:39B82 

28. CMB 2016 (vol 59 pp. 721)
 Pérez, Juan de Dios; Lee, Hyunjin; Suh, Young Jin; Woo, Changhwa

Real Hypersurfaces in Complex Twoplane Grassmannians with Reeb Parallel Ricci Tensor in the GTW Connection
There are several kinds of classification problems for real hypersurfaces
in complex twoplane Grassmannians $G_2({\mathbb C}^{m+2})$.
Among them, Suh classified Hopf hypersurfaces $M$ in $G_2({\mathbb
C}^{m+2})$ with Reeb parallel Ricci tensor in LeviCivita connection.
In this paper, we introduce the notion of generalized TanakaWebster
(in shortly, GTW) Reeb parallel Ricci tensor for Hopf hypersurface
$M$ in $G_2({\mathbb C}^{m+2})$. Next, we give a complete classification
of Hopf hypersurfaces in $G_2({\mathbb C}^{m+2})$ with GTW Reeb
parallel Ricci tensor.
Keywords:Complex twoplane Grassmannian, real hypersurface, Hopf hypersurface, generalized TanakaWebster connection, parallelism, Reeb parallelism, Ricci tensor Categories:53C40, 53C15 

29. CMB 2016 (vol 59 pp. 472)
 Clay, Adam; Desmarais, Colin; Naylor, Patrick

Testing Biorderability of Knot Groups
We investigate the biorderability of twobridge knot groups
and the groups of knots with 12 or fewer crossings by applying
recent theorems of Chiswell, Glass and Wilson.
Amongst all knots with 12 or fewer crossings (of which there
are 2977), previous theorems were only able to determine biorderability
of 499 of the corresponding knot groups. With our methods we
are able to deal with 191 more.
Keywords:knots, fundamental groups, orderable groups Categories:57M25, 57M27, 06F15 

30. CMB 2016 (vol 59 pp. 483)
 Crooks, Peter; Holden, Tyler

Generalized Equivariant Cohomology and Stratifications
For $T$ a compact torus and $E_T^*$ a generalized $T$equivariant
cohomology theory, we provide a systematic framework for computing
$E_T^*$ in the context of equivariantly stratified smooth complex
projective varieties. This allows us to explicitly compute $E_T^*(X)$
as an $E_T^*(\text{pt})$module when $X$ is a direct limit of
smooth complex projective $T_{\mathbb{C}}$varieties with finitely
many $T$fixed points and $E_T^*$ is one of $H_T^*(\cdot;\mathbb{Z})$,
$K_T^*$, and $MU_T^*$. We perform this computation on the affine
Grassmannian of a complex semisimple group.
Keywords:equivariant cohomology theory, stratification, affine Grassmannian Categories:55N91, 19L47 

31. CMB 2016 (vol 59 pp. 769)
 GarcíaPacheco, Francisco Javier; Hill, Justin R.

Geometric Characterizations of Hilbert Spaces
We study some geometric properties related to the set $\Pi_X:=
\{
(x,x^*
)\in\mathsf{S}_X\times \mathsf{S}_{X^*}:x^*
(x
)=1
\}$ obtaining two characterizations of Hilbert spaces
in the category of Banach spaces. We also compute the distance
of a generic element $
(h,k
)\in H\oplus_2 H$ to $\Pi_H$ for $H$ a Hilbert space.
Keywords:Hilbert space, extreme point, smooth, $\mathsf{L}^2$summands Categories:46B20, 46C05 

32. CMB 2016 (vol 59 pp. 652)
 Su, Huadong

On the Diameter of Unitary Cayley Graphs of Rings
The unitary Cayley graph of a ring $R$, denoted
$\Gamma(R)$, is the simple graph
defined on all elements of $R$, and where two vertices $x$ and
$y$
are adjacent if and only if $xy$ is a unit in $R$. The largest
distance between all pairs of vertices of a graph $G$ is called
the
diameter of $G$, and is denoted by ${\rm diam}(G)$. It is proved
that for each integer $n\geq1$, there exists a ring $R$ such
that
${\rm diam}(\Gamma(R))=n$. We also show that ${\rm
diam}(\Gamma(R))\in \{1,2,3,\infty\}$ for a ring $R$ with $R/J(R)$
selfinjective and classify all those rings with ${\rm
diam}(\Gamma(R))=1$, 2, 3 and $\infty$, respectively.
Keywords:unitary Cayley graph, diameter, $k$good, unit sum number, selfinjective ring Categories:05C25, 16U60, 05C12 

33. CMB 2016 (vol 59 pp. 606)
 Mihăilescu, Mihai; Moroşanu, Gheorghe

Eigenvalues of $ \Delta_p \Delta_q $ Under Neumann Boundary Condition
The
eigenvalue problem $\Delta_p u\Delta_q u=\lambdau^{q2}u$
with $p\in(1,\infty)$, $q\in(2,\infty)$, $p\neq q$ subject to
the
corresponding homogeneous Neumann boundary condition is
investigated on a bounded open set with smooth boundary from
$\mathbb{R}^N$ with $N\geq 2$. A careful analysis of this problem leads
us to a complete description of the set of eigenvalues as being
a
precise interval $(\lambda_1, +\infty )$ plus an isolated point
$\lambda =0$. This comprehensive result is strongly related to
our
framework which is complementary to the wellknown case $p=q\neq
2$ for which a full description of the set of eigenvalues is
still
unavailable.
Keywords:eigenvalue problem, Sobolev space, Nehari manifold, variational methods Categories:35J60, 35J92, 46E30, 49R05 

34. CMB 2016 (vol 59 pp. 617)
 Nakashima, Norihiro; Terao, Hiroaki; Tsujie, Shuhei

Canonical Systems of Basic Invariants for Unitary Reflection Groups
It has been known that there exists a canonical system for every
finite real reflection group. The first and the third authors
obtained
an explicit formula for a canonical system in the previous paper.
In this article, we first define canonical systems for the finite
unitary reflection groups, and then prove their existence.
Our proof does not depend on the classification of unitary reflection
groups.
Furthermore, we give an explicit formula for a canonical system
for every unitary reflection group.
Keywords:basic invariant, invariant theory, finite unitary reflection group Categories:13A50, 20F55 

35. CMB Online first
36. CMB 2016 (vol 59 pp. 449)
 Abdallah, Nancy

On Hodge Theory of Singular Plane Curves
The dimensions of the graded quotients of the
cohomology of a plane curve complement $U=\mathbb P^2 \setminus C$
with respect to the Hodge filtration are described in terms of
simple geometrical invariants. The case of curves with ordinary
singularities is discussed in detail. We also give a precise
numerical estimate for the difference between the Hodge filtration
and the pole order filtration on $H^2(U,\mathbb C)$.
Keywords:plane curves, Hodge and pole order filtrations Categories:32S35, 32S22, 14H50 

37. CMB 2016 (vol 59 pp. 528)
 Jahan, Qaiser

Characterization of Lowpass Filters on Local Fields of Positive Characteristic
In this article, we give necessary and sufficient conditions
on a function to be a lowpass filter on a local field $K$ of
positive characteristic associated to the scaling function for
multiresolution analysis of $L^2(K)$. We use probability and
martingale methods to provide such a characterization.
Keywords:multiresolution analysis, local field, lowpass filter, scaling function, probability, conditional probability and martingales Categories:42C40, 42C15, 43A70, 11S85 

38. CMB 2016 (vol 59 pp. 734)
39. CMB 2016 (vol 59 pp. 497)
40. CMB 2016 (vol 59 pp. 461)
 Ara, Pere; O'Meara, Kevin C.

The Nilpotent Regular Element Problem
We use George Bergman's recent normal form for universally adjoining
an inner inverse to show that, for general rings, a nilpotent
regular element $x$ need not be unitregular.
This contrasts sharply with the situation for nilpotent regular
elements in exchange rings (a large class of rings), and for
general rings when all powers of the nilpotent element $x$ are
regular.
Keywords:nilpotent element, von Neumann regular element, unitregular, Bergman's normal form Categories:16E50, 16U99, 16S10, 16S15 

41. CMB Online first
 Akbari, Saieed; Miraftab, Babak; Nikandish, Reza

CoMaximal Graphs of Subgroups of Groups
Let $H$ be a group. The comaximal graph of subgroups
of $H$, denoted by $\Gamma(H)$, is a
graph whose vertices are nontrivial and proper subgroups of
$H$ and two distinct vertices $L$
and $K$ are adjacent in $\Gamma(H)$ if and only if $H=LK$. In
this paper, we study the connectivity, diameter, clique number
and vertex
chromatic number of $\Gamma(H)$. For instance, we show that
if $\Gamma(H)$ has no isolated vertex, then $\Gamma(H)$
is connected with diameter at most $3$. Also, we characterize
all finite groups whose comaximal graphs are connected.
Among other results, we show that if $H$ is a finitely generated
solvable group and $\Gamma(H)$ is connected and moreover the
degree of a maximal subgroup is finite, then $H$ is finite.
Furthermore, we show that the degree of each vertex in the
comaximal graph of a general linear group over an algebraically
closed field is zero or infinite.
Keywords:comaximal graphs of subgroups of groups, diameter, nilpotent group, solvable group Categories:05C25, 05E15, 20D10, 20D15 

42. CMB Online first
 Khavinson, Dmitry; Lundberg, Erik; Render, Hermann

The Dirichlet problem for the slab with entire data and a difference equation for harmonic functions
It is shown that the Dirichlet problem for the slab $(a,b) \times
\mathbb{R}^{d}$ with entire boundary data has an entire solution. The proof
is based
on a generalized Schwarz reflection principle. Moreover, it is
shown that
for a given entire harmonic function $g$
the inhomogeneous difference equation $h
( t+1,y) h (t,y) =g ( t,y)$
has an entire harmonic solution $h$.
Keywords:reflection principle, entire harmonic function, analytic continuation Categories:31B20, 31B05 

43. CMB 2016 (vol 59 pp. 575)
 Li, Jifu; Hu, Zhiguang; Deng, Shaoqiang

Cohomogeneity One Randers Metrics
An action of a Lie group $G$ on a smooth manifold $M$ is called
cohomogeneity one if the orbit space $M/G$ is of dimension $1$.
A Finsler metric $F$ on $M$ is called invariant if $F$ is
invariant under the action of $G$. In this paper,
we study invariant
Randers metrics on cohomogeneity one manifolds. We first give a
sufficient and necessary condition for the existence of invariant
Randers metrics on cohomogeneity one manifolds. Then we obtain
some results on invariant Killing vector fields on the
cohomogeneity one manifolds and use that to deduce some
sufficient and necessary condition for a cohomogeneity one
Randers metric to be Einstein.
Keywords:cohomogeneity one actions, normal geodesics, invariant vector fields, Randers metrics Categories:53C30, 53C60 

44. CMB 2016 (vol 59 pp. 748)
 Dolžan, David

The Metric Dimension of the Total Graph of a Finite Commutative Ring
We study the total graph of a finite commutative ring. We calculate
its metric dimension in the case when the Jacobson radical of
the ring is nontrivial and we examine the metric dimension of
the total graph of a product of at most two fields, obtaining
either exact values in some cases or bounds in other, depending
on the number of elements in the respective fields.
Keywords:total graph, finite ring, metric dimension Categories:13M99, 05E40 

45. CMB 2016 (vol 59 pp. 624)
 Otsubo, Noriyuki

Homology of the Fermat Tower and Universal Measures for Jacobi Sums
We give a precise description of the homology group of the Fermat
curve as a cyclic module over a group ring.
As an application, we prove the freeness of the profinite homology
of the Fermat tower.
This allows us to define measures, an equivalent of Anderson's
adelic beta functions,
in a similar manner to Ihara's definition of $\ell$adic universal
power series for Jacobi sums.
We give a simple proof of the interpolation property using a
motivic decomposition of the Fermat curve.
Keywords:Fermat curves, IharaAnderson theory, Jacobi sums Categories:11S80, 11G15, 11R18 

46. CMB 2016 (vol 59 pp. 693)
 Chen, ChungChuan

Recurrence of Cosine Operator Functions on Groups
In this note, we study the recurrence and topologically multiple
recurrence of a sequence of operators on Banach spaces.
In particular, we give a sufficient and necessary condition for
a cosine operator function,
induced by a sequence of operators on the Lebesgue space of a
locally compact group, to be topologically multiply recurrent.
Keywords:topologically multiple recurrence, recurrence, topological transitivity, hypercyclicity, cosine operator function Categories:47A16, 54B20, 43A15 

47. CMB 2016 (vol 59 pp. 417)
 Song, Hongxue; Chen, Caisheng; Yan, Qinglun

Existence of Multiple Solutions for a $p$Laplacian System in $\textbf{R}^{N}$ with Signchanging Weight Functions
In this paper, we consider the quasilinear elliptic
problem
\[
\left\{
\begin{aligned}
&
M
\left(\int_{\mathbb{R}^{N}}x^{ap}\nabla u^{p}dx
\right){\rm
div}
\left(x^{ap}\nabla u^{p2}\nabla u
\right)
\\
&
\qquad=\frac{\alpha}{\alpha+\beta}H(x)u^{\alpha2}uv^{\beta}+\lambda
h_{1}(x)u^{q2}u,
\\
&
M
\left(\int_{\mathbb{R}^{N}}x^{ap}\nabla v^{p}dx
\right){\rm
div}
\left(x^{ap}\nabla v^{p2}\nabla v
\right)
\\
&
\qquad=\frac{\beta}{\alpha+\beta}H(x)v^{\beta2}vu^{\alpha}+\mu
h_{2}(x)v^{q2}v,
\\
&u(x)\gt 0,\quad v(x)\gt 0, \quad x\in \mathbb{R}^{N}
\end{aligned}
\right.
\]
where $\lambda, \mu\gt 0$, $1\lt p\lt N$,
$1\lt q\lt p\lt p(\tau+1)\lt \alpha+\beta\lt p^{*}=\frac{Np}{Np}$, $0\leq
a\lt \frac{Np}{p}$, $a\leq b\lt a+1$, $d=a+1b\gt 0$, $M(s)=k+l s^{\tau}$,
$k\gt 0$, $l, \tau\geq0$ and the weight $H(x), h_{1}(x), h_{2}(x)$
are
continuous functions which change sign in $\mathbb{R}^{N}$. We
will prove that the problem has at least two positive solutions
by
using the Nehari manifold and the fibering maps associated with
the Euler functional for this problem.
Keywords:Nehari manifold, quasilinear elliptic system, $p$Laplacian operator, concave and convex nonlinearities Category:35J66 

48. CMB 2016 (vol 59 pp. 234)
 Beardon, Alan F.

Nondiscrete Frieze Groups
The classification of Euclidean frieze groups into seven conjugacy
classes is well known, and many articles on recreational mathematics
contain frieze patterns that illustrate these classes. However,
it is
only possible to draw these patterns because the subgroup of
translations that leave the pattern invariant is (by definition)
cyclic, and hence discrete. In this paper we classify the conjugacy
classes of frieze groups that contain a nondiscrete subgroup of
translations, and clearly these groups cannot be represented
pictorially in any practical way. In addition, this discussion
sheds
light on why there are only seven conjugacy classes in the classical
case.
Keywords:frieze groups, isometry groups Categories:51M04, 51N30, 20E45 

49. CMB 2016 (vol 59 pp. 326)
50. CMB 2016 (vol 59 pp. 508)
 De Nicola, Antonio; Yudin, Ivan

Generalized Goldberg Formula
In this paper we prove a useful formula for the graded commutator
of the Hodge
codifferential with the left wedge multiplication by a fixed
$p$form acting on
the de Rham algebra of a Riemannian manifold. Our formula generalizes
a formula
stated by Samuel I. Goldberg for the case of 1forms. As first
examples of
application we obtain new identities on locally conformally KÃ¤hler
manifolds
and quasiSasakian manifolds. Moreover, we prove that under suitable
conditions
a certain subalgebra of differential forms in a compact manifold
is quasiisomorphic as a CDGA to the full de Rham algebra.
Keywords:graded commutator, Hodge codifferential, Hodge laplacian, de Rham cohomology, locally conformal Kaehler manifold, quasiSasakian manifold Categories:53C25, 53D35 
