51. CMB 2016 (vol 60 pp. 12)
 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 

52. CMB 2016 (vol 60 pp. 43)
53. CMB 2016 (vol 60 pp. 95)
 Choi, ChangKwon; 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 

54. CMB 2016 (vol 60 pp. 104)
55. CMB 2016 (vol 60 pp. 184)
 Pathak, Siddhi

On a Conjecture of Livingston
In an attempt to resolve a folklore conjecture of ErdÃ¶s regarding
the nonvanishing at $s=1$ of the $L$series
attached to a periodic arithmetical function with period $q$
and values in $\{ 1, 1\} $, Livingston conjectured the $\bar{\mathbb{Q}}$
 linear independence of logarithms of certain algebraic numbers.
In this paper, we disprove Livingston's conjecture for composite
$q \geq 4$, highlighting that a new approach is required to settle
ErdÃ¶s's conjecture. We also prove that the conjecture is
true for prime $q \geq 3$, and indicate that more ingredients
will be needed to settle ErdÃ¶s's conjecture for prime $q$.
Keywords:nonvanishing of Lseries, linear independence of logarithms of algebraic numbers Categories:11J86, 11J72 

56. CMB 2016 (vol 60 pp. 197)
 Tang, Zikai; Deng, Hanyuan

Degree Kirchhoff Index of Bicyclic Graphs
Let $G$ be a connected graph with vertex set $V(G)$. The degree
Kirchhoff index of $G$ is defined as $S'(G) =\sum_{\{u,v\}\subseteq
V(G)}d(u)d(v)R(u,v)$, where $d(u)$ is the degree of vertex $u$,
and
$R(u, v)$ denotes the resistance distance between vertices $u$
and
$v$. In this paper, we characterize the graphs having maximum
and
minimum degree Kirchhoff index among all $n$vertex bicyclic
graphs
with exactly two cycles.
Keywords:degree Kirchhoff index, resistance distance, bicyclic graph, extremal graph Categories:05C12, 05C35 

57. CMB Online first
 Xu, Xu; Zhu, Laiyi

Rational function operators from Poisson integrals
In this paper, we construct two classes of rational function
operators by using the Poisson integrals of the function on the
whole real
axis. The convergence rates of the uniform and mean approximation
of such rational function operators on the whole real axis are
studied.
Keywords:rational function operators, Poisson integrals, convergence rate, uniform approximation, mean approximation Categories:41A20, 41A25, 41A35 

58. CMB 2016 (vol 60 pp. 173)
 Oubbi, Lahbib

On Ulam Stability of a Functional Equation in Banach Modules
Let $X$ and $Y$ be Banach spaces and $f : X \to Y$ an odd mapping.
For any rational number $r \ne 2$, C. Baak, D. H.
Boo, and Th. M. Rassias have proved the HyersUlam stability
of the following functional equation:
\begin{align*}
r f
\left(\frac{\sum_{j=1}^d x_j}{r}
\right)
& + \sum_{\substack{i(j) \in \{0,1\}
\\ \sum_{j=1}^d i(j)=\ell}} r f
\left(
\frac{\sum_{j=1}^d (1)^{i(j)}x_j}{r}
\right)
= (C^\ell_{d1}  C^{\ell 1}_{d1} + 1) \sum_{j=1}^d
f(x_j)
\end{align*}
where $d$ and $\ell$ are positive integers so that $1 \lt \ell
\lt \frac{d}{2}$, and $C^p_q := \frac{q!}{(qp)!p!}$,
$p, q \in \mathbb{N}$ with $p \le q$.
In this note we solve this equation for arbitrary nonzero scalar
$r$ and show that it is actually HyersUlam stable.
We thus extend and generalize Baak et al.'s result.
Different questions concerning the *homomorphisms and the
multipliers between C*algebras are also
considered.
Keywords:linear functional equation, HyersUlam stability, Banach modules, C*algebra homomorphisms. Categories:39A30, 39B10, 39A06, 46Hxx 

59. CMB 2016 (vol 59 pp. 806)
 Izumiya, Shyuichi

Geometric Interpretation of Lagrangian Equivalence
As an application of the theory of
graphlike Legendrian unfoldings, relations of the hidden structures
of caustics and wave front propagations are revealed.
Keywords:wave front propagations, big wave fronts, graphlike Legendrian unfoldings, caustics Categories:58K05, 57R45, 58K60 

60. CMB 2016 (vol 59 pp. 776)
61. CMB 2016 (vol 59 pp. 849)
 Nah, Kyeongah; Röst, Gergely

Stability Threshold for Scalar Linear Periodic Delay Differential Equations
We prove that for the linear scalar delay differential
equation
$$ \dot{x}(t) =  a(t)x(t) + b(t)x(t1) $$
with nonnegative periodic coefficients of period $P\gt 0$, the
stability threshold for the trivial solution is
$r:=\int_{0}^{P}
\left(b(t)a(t)
\right)\mathrm{d}t=0,$
assuming that $b(t+1)a(t)$ does not change its sign. By constructing
a class of explicit examples, we show the counterintuitive result
that in general, $r=0$ is not a stability threshold.
Keywords:delay differential equation, stability, periodic system Categories:34K20, 34K06 

62. CMB 2016 (vol 60 pp. 300)
 Gauthier, Paul M; Sharifi, Fatemeh

Luzintype Holomorphic Approximation on Closed Subsets of Open Riemann Surfaces
It is known that if $E$ is a closed subset of an open Riemann
surface $R$ and $f$ is a holomorphic function on a neighbourhood
of $E,$ then it is ``usually" not possible to approximate $f$
uniformly by functions holomorphic on all of $R.$ We show, however,
that for every open Riemann surface $R$ and every closed subset
$E\subset R,$ there is closed subset $F\subset E,$ which approximates
$E$ extremely well, such that every function holomorphic on $F$
can be approximated much better than uniformly by functions holomorphic
on $R$.
Keywords:Carleman approximation, tangential approximation, Myrberg surface Categories:30E15, 30F99 

63. CMB Online first
 Werner, Elisabeth; Ye, Deping

Mixed $f$divergence for multiple pairs of measures
In this paper, the concept of the classical $f$divergence for
a pair of measures is extended to the mixed $f$divergence for
multiple pairs of measures. The mixed $f$divergence provides
a way to measure the difference between multiple pairs of (probability)
measures. Properties for the mixed $f$divergence are established,
such as permutation invariance and symmetry in distributions.
An
AlexandrovFenchel type inequality and an isoperimetric inequality
for the
mixed $f$divergence are proved.
Keywords:AlexandrovFenchel inequality, $f$dissimilarity, $f$divergence, isoperimetric inequality Categories:28XX, 52XX, 60XX 

64. CMB Online first
 Liu, Feng; Wu, Huoxiong

Endpoint Regularity of Multisublinear Fractional Maximal Functions
In this paper we investigate
the endpoint regularity properties of the multisublinear
fractional maximal operators, which include the multisublinear
HardyLittlewood maximal operator. We obtain some new bounds
for the derivative of the onedimensional multisublinear
fractional maximal operators acting on vectorvalued function
$\vec{f}=(f_1,\dots,f_m)$ with all $f_j$ being $BV$functions.
Keywords:multisublinear fractional maximal operators, Sobolev spaces, bounded variation Categories:42B25, 46E35 

65. CMB 2016 (vol 60 pp. 154)
 Liu, Ye

On Chromatic Functors and Stable Partitions of Graphs
The chromatic functor of a simple graph is a functorization of
the chromatic polynomial. M. Yoshinaga showed
that two finite graphs have isomorphic chromatic functors if
and only if they have the same chromatic polynomial. The key
ingredient in the proof is the use of stable partitions of graphs.
The latter is shown to be closely related to chromatic functors.
In this note, we further investigate some interesting properties
of chromatic functors associated to simple graphs using stable
partitions. Our first result is the determination of the group
of natural automorphisms of the chromatic functor, which is in
general a larger group than the automorphism group of the graph.
The second result is that the composition of the chromatic functor
associated to a finite graph restricted to the category $\mathrm{FI}$
of finite sets and injections with the free functor into the
category of complex vector spaces yields a consistent sequence
of representations of symmetric groups which is representation
stable in the sense of ChurchFarb.
Keywords:chromatic functor, stable partition, representation stability Categories:05C15, 20C30 

66. CMB 2016 (vol 59 pp. 813)
67. CMB Online first
68. CMB 2016 (vol 60 pp. 63)
 Chang, Gyu Whan

Power Series Rings Over PrÃ¼fer $v$multiplication Domains, II
Let $D$ be an integral domain, $X^1(D)$ be the set of heightone
prime ideals of $D$,
$\{X_{\beta}\}$ and $\{X_{\alpha}\}$ be
two disjoint nonempty sets of indeterminates over $D$,
$D[\{X_{\beta}\}]$ be the polynomial ring over $D$, and
$D[\{X_{\beta}\}][\![\{X_{\alpha}\}]\!]_1$ be the first type
power series ring over $D[\{X_{\beta}\}]$.
Assume that $D$ is a PrÃ¼fer $v$multiplication domain (P$v$MD)
in which each proper integral $t$ideal has only finitely many
minimal prime ideals
(e.g., $t$SFT P$v$MDs, valuation domains, rings of Krull type).
Among other things, we show that if
$X^1(D) = \emptyset$ or $D_P$ is a DVR for all $P \in X^1(D)$,
then
${D[\{X_{\beta}\}][\![\{X_{\alpha}\}]\!]_1}_{D  \{0\}}$ is a
Krull domain.
We also prove that if $D$ is a $t$SFT P$v$MD, then the complete
integral closure of $D$ is a Krull domain and
ht$(M[\{X_{\beta}\}][\![\{X_{\alpha}\}]\!]_1)$ = $1$ for every
heightone maximal $t$ideal $M$ of $D$.
Keywords:Krull domain, P$v$MD, multiplicatively closed set of ideals, power series ring Categories:13A15, 13F05, 13F25 

69. CMB 2016 (vol 60 pp. 269)
 Chen, Jianlong; Zhu, Huihui; Patricio, Pedro; Zhang, Yulin

Characterizations and Representations of Core and Dual Core Inverses
In this
paper,
double commutativity and the reverse order law for the core inverse
are considered. Then, new characterizations of the MoorePenrose
inverse of a regular element are given by onesided invertibilities
in a ring. Furthermore, the characterizations and representations
of
the core and dual core inverses of a regular element are considered.
Keywords:regularities, group inverses, MoorePenrose inverses, core inverses, dual core inverses, Dedekindfinite rings Categories:15A09, 15A23 

70. CMB 2016 (vol 59 pp. 834)
 Liao, Fanghui; Liu, Zongguang

Some Properties of TriebelLizorkin and Besov Spaces Associated with Zygmund Dilations
In this paper, using CalderÃ³n's
reproducing formula and almost orthogonality estimates, we
prove the lifting property and the embedding theorem of the TriebelLizorkin
and Besov spaces associated with Zygmund dilations.
Keywords:TriebelLizorkin and Besov spaces, Riesz potential, CalderÃ³n's reproducing formula, almost orthogonality estimate, Zygmund dilation, embedding theorem Categories:42B20, 42B35 

71. CMB 2016 (vol 60 pp. 111)
 Ghaani Farashahi, Arash

Abstract Plancherel (Trace) Formulas over Homogeneous Spaces of Compact Groups
This paper introduces a unified operator theory approach to the
abstract Plancherel (trace) formulas over
homogeneous spaces of compact groups. Let $G$ be a compact group
and $H$ be a closed subgroup of $G$.
Let $G/H$ be the left coset space of $H$ in $G$ and $\mu$ be
the normalized $G$invariant measure on $G/H$ associated to the
Weil's formula.
Then, we present a generalized abstract notion of Plancherel
(trace) formula for the Hilbert space $L^2(G/H,\mu)$.
Keywords:compact group, homogeneous space, dual space, Plancherel (trace) formula Categories:20G05, 43A85, 43A32, 43A40 

72. CMB 2016 (vol 59 pp. 760)
 Fichou, Goulwen; Quarez, Ronan; Shiota, Masahiro

Artin Approximation Compatible with a Change of Variables
We propose a version of the classical Artin
approximation
which allows to perturb the variables of the approximated solution. Namely, it is possible to approximate a formal solution of a
Nash equation by a Nash solution in a
compatible way with a given Nash change of variables.
This result is closely related to the socalled nested Artin
approximation and becomes false in the analytic setting. We provide
local and global versions of this approximation in real and complex
geometry together with an application to the RightLeft equivalence
of Nash maps.
Keywords:Artin approximation, global case, Nash functions Categories:14P20, 58A07 

73. 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 

74. 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 

75. 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 
