51. CMB 2017 (vol 60 pp. 422)
 Tang, Xianhua

New Superquadratic Conditions for Asymptotically Periodic SchrÃ¶dinger Equations
This paper is dedicated to studying the
semilinear SchrÃ¶dinger equation
$$
\left\{
\begin{array}{ll}
\triangle u+V(x)u=f(x, u), \ \ \ \ x\in {\mathbf{R}}^{N},
\\
u\in H^{1}({\mathbf{R}}^{N}),
\end{array}
\right.
$$
where $f$ is a superlinear, subcritical nonlinearity. It focuses
on the case where
$V(x)=V_0(x)+V_1(x)$, $V_0\in C(\mathbf{R}^N)$, $V_0(x)$ is 1periodic
in each of $x_1, x_2, \ldots, x_N$ and
$\sup[\sigma(\triangle +V_0)\cap (\infty, 0)]\lt 0\lt \inf[\sigma(\triangle
+V_0)\cap (0, \infty)]$,
$V_1\in C(\mathbf{R}^N)$ and $\lim_{x\to\infty}V_1(x)=0$. A new superquadratic
condition is obtained,
which is weaker than some well known results.
Keywords:SchrÃ¶dinger equation, superlinear, asymptotically periodic, ground state solutions of NehariPankov type Categories:35J20, 35J60 

52. CMB 2017 (vol 60 pp. 690)
 Bao, Guanlong; Göğüş, Nıhat Gökhan; Pouliasis, Stamatis

$\mathcal{Q}_p$ Spaces and Dirichlet Type Spaces
In this paper, we show that the MÃ¶bius invariant
function space $\mathcal {Q}_p$ can be generated by variant
Dirichlet type spaces
$\mathcal{D}_{\mu, p}$ induced by finite positive Borel measures
$\mu$ on the open unit disk. A criterion for the equality between
the space $\mathcal{D}_{\mu, p}$ and the usual Dirichlet type
space $\mathcal {D}_p$ is given. We obtain a sufficient condition
to construct different $\mathcal{D}_{\mu, p}$ spaces
and we provide examples.
We establish decomposition theorems for $\mathcal{D}_{\mu,
p}$ spaces, and prove that the nonHilbert space $\mathcal
{Q}_p$ is equal to the intersection of Hilbert spaces $\mathcal{D}_{\mu,
p}$. As an application of the relation between $\mathcal {Q}_p$
and $\mathcal{D}_{\mu, p}$ spaces, we also obtain that there
exist different $\mathcal{D}_{\mu, p}$ spaces; this is a trick
to prove the existence without constructing examples.
Keywords:$\mathcal {Q}_p$ space, Dirichlet type space, MÃ¶bius invariant function space Categories:30H25, 31C25, 46E15 

53. CMB 2017 (vol 60 pp. 830)
 Motegi, Kimihiko; Teragaito, Masakazu

Generalized Torsion Elements and Biorderability of 3manifold Groups
It is known that a biorderable group has no generalized torsion
element,
but the converse does not hold in general.
We conjecture that the converse holds for the fundamental groups
of $3$manifolds,
and verify the conjecture for nonhyperbolic, geometric $3$manifolds.
We also confirm the conjecture for some infinite families of
closed hyperbolic $3$manifolds.
In the course of the proof,
we prove that each standard generator of the Fibonacci group
$F(2, m)$ ($m \gt 2$) is a generalized torsion element.
Keywords:generalized torsion element, biordering, 3manifold group Categories:57M25, 57M05, 06F15, 20F05 

54. CMB 2017 (vol 60 pp. 816)
 Moslehian, Mohammad Sal; Zamani, Ali

Characterizations of Operator BirkhoffJames Orthogonality
In this paper, we obtain some characterizations of the (strong)
BirkhoffJames orthogonality for elements of Hilbert $C^*$modules
and certain elements of $\mathbb{B}(\mathscr{H})$.
Moreover, we obtain a kind of Pythagorean relation for bounded
linear operators.
In addition, for $T\in \mathbb{B}(\mathscr{H})$ we prove that if the
norm attaining
set $\mathbb{M}_T$ is a unit sphere of some finite dimensional
subspace $\mathscr{H}_0$ of $\mathscr{H}$ and $\T\_{{{\mathscr{H}}_0}^\perp}
\lt \T\$, then for every $S\in\mathbb{B}(\mathscr{H})$, $T$ is the strong
BirkhoffJames orthogonal to $S$ if and only if there exists
a unit vector $\xi\in {\mathscr{H}}_0$ such that $\T\\xi =
T\xi$ and $S^*T\xi = 0$.
Finally, we introduce a new type of approximate orthogonality
and investigate this notion in the setting of inner product $C^*$modules.
Keywords:Hilbert $C^*$module, BirkhoffJames orthogonality, strong BirkhoffJames orthogonality, approximate orthogonality Categories:46L05, 46L08, 46B20 

55. CMB 2017 (vol 60 pp. 253)
 Chen, Bin; Zhao, Lili

On a Yamabe Type Problem in Finsler Geometry
In this paper, a new notion of scalar curvature for a Finsler
metric $F$ is introduced, and two conformal invariants $Y(M,F)$
and $C(M,F)$ are defined. We prove that there exists a Finsler
metric with constant scalar curvature in the conformal class
of $F$ if the Cartan torsion of $F$ is sufficiently small and
$Y(M,F)C(M,F)\lt Y(\mathbb{S}^n)$ where $Y(\mathbb{S}^n)$ is the
Yamabe constant of the standard sphere.
Keywords:Finsler metric, scalar curvature, Yamabe problem Categories:53C60, 58B20 

56. CMB 2017 (vol 60 pp. 436)
 Weng, Peixuan; Liu, Li

Globally Asymptotic Stability of a Delayed IntegroDifferential Equation with Nonlocal Diffusion
We study a population model with nonlocal diffusion, which
is a delayed integrodifferential equation with double nonlinearity
and two integrable kernels. By comparison method and analytical
technique, we obtain globally asymptotic stability of the zero
solution and the positive equilibrium. The results obtained
reveal that the globally asymptotic stability only depends on
the property of nonlinearity. As application, an example for
a population model with age structure is discussed at the end
of the article.
Keywords:integrodifferential equation, nonlocal diffusion, equilibrium, globally asymptotic stability, population model with age structure Categories:45J05, 35K57, 92D25 

57. CMB 2017 (vol 60 pp. 381)
 Rousseau, C.

The Bifurcation Diagram of Cubic Polynomial Vector Fields on $\mathbb{C}\mathbb{P}^1$
In this paper we give the bifurcation diagram
of the family of cubic vector fields $\dot z=z^3+ \epsilon_1z+\epsilon_0$
for $z\in \mathbb{C}\mathbb{P}^1$, depending on the values of
$\epsilon_1,\epsilon_0\in\mathbb{C}$.
The bifurcation diagram is in $\mathbb{R}^4$, but its conic structure
allows describing it for parameter values in $\mathbb{S}^3$. There are
two open simply connected regions of structurally stable vector
fields separated by surfaces corresponding to bifurcations of
homoclinic connections between two separatrices of the pole at
infinity. These branch from the codimension 2 curve of double
singular points. We also explain the bifurcation of homoclinic
connection in terms of the description of Douady and Sentenac
of polynomial vector fields.
Keywords:complex polynomial vector field, bifurcation diagram, DouadySentenac invariant Categories:34M45, 32G34 

58. CMB 2017 (vol 60 pp. 246)
 Bhuniya, Anjan Kumar; Hansda, Kalyan

On Radicals of Green's Relations in Ordered Semigroups
In this paper, we give a new definition of radicals of Green's
relations in an ordered semigroup and characterize left regular
(right regular), intra regular ordered semigroups by radicals
of Green's relations. Also we characterize the ordered semigroups
which are unions and complete semilattices of tsimple ordered
semigroups.
Keywords:radical of Green's relation, intra regular ordered semigroup, left regular, tsimple ordered semigroup Category:06F05 

59. CMB 2017 (vol 60 pp. 470)
 Buijs, Urtzi; Félix, Yves; Murillo, Aniceto; Tanré, Daniel

MaurerCartan Elements in the Lie Models of Finite Simplicial Complexes
In a previous work, we have associated a complete differential
graded Lie algebra
to any finite simplicial complex in a functorial way.
Similarly, we have also a realization functor from the category
of complete differential graded Lie algebras
to the category of simplicial sets.
We have already interpreted the homology of a Lie algebra
in terms of homotopy groups of its realization.
In this paper, we begin a dictionary between models
and simplicial complexes by establishing a correspondence
between the Deligne groupoid of the model and the connected components
of the finite simplicial complex.
Keywords:complete differential graded Lie algebra, MaurerCartan element, rational homotopy theory Category:16E45 

60. CMB 2017 (vol 60 pp. 283)
 Friedl, Stefan; Vidussi, Stefano

Twisted Alexander Invariants Detect Trivial Links
It follows from earlier work of SilverWilliams and the authors
that twisted Alexander polynomials detect the unknot and the
Hopf link.
We now show that twisted Alexander polynomials also detect the
trefoil and the figure8 knot,
that twisted Alexander polynomials detect whether a link is split
and that twisted Alexander modules detect trivial links. We use
this result to provide algorithms for detecting whether a link
is the unlink, whether it is split and whether it is totally
split.
Keywords:twisted Alexander polynomial, virtual fibering theorem, unlink detection Category:57M27 

61. CMB 2017 (vol 60 pp. 225)
 Bahmanpour, Kamal; Naghipour, Reza

Faltings' Finiteness Dimension of Local Cohomology Modules Over Local CohenMacaulay Rings
Let $(R, \frak m)$ denote a local CohenMacaulay ring and $I$
a nonnilpotent ideal of $R$. The purpose of this article is
to investigate Faltings' finiteness
dimension $f_I(R)$ and equidimensionalness of certain homomorphic
image of $R$. As a consequence
we deduce that $f_I(R)=\operatorname{max}\{1, \operatorname{ht} I\}$
and if $\operatorname{mAss}_R(R/I)$
is contained in $\operatorname{Ass}_R(R)$, then the ring $R/ I+\cup_{n\geq
1}(0:_RI^n)$ is equidimensional of dimension $\dim R1$.
Moreover, we will obtain a lower bound for injective dimension
of the local cohomology module $H^{\operatorname{ht} I}_I(R)$, in the case
$(R, \frak m)$ is a complete equidimensional local ring.
Keywords:Cohen Macaulay ring, equidimensional ring, finiteness dimension, local cohomology Categories:13D45, 14B15 

62. CMB 2017 (vol 60 pp. 235)
 Basu, Samik; Subhash, B

Topology of Certain Quotient Spaces of Stiefel Manifolds
We compute the cohomology of the right generalised projective
Stiefel manifolds. Following this, we discuss some easy applications
of the computations to the ranks of complementary bundles, and
bounds on the span and immersibility.
Keywords:projective Stiefel manifold, span, spectral sequence Categories:55R20, 55R25, 57R20 

63. CMB Online first
 MirandaNeto, Cleto Brasileiro

A moduletheoretic characterization of algebraic hypersurfaces
In this note we prove the following surprising characterization:
if
$X\subset {\mathbb A}^n$ is an (embedded, nonempty, proper)
algebraic variety defined over a
field $k$ of characteristic zero, then $X$ is a hypersurface
if and only if the module $T_{{\mathcal O}_{{\mathbb
A}^n}/k}(X)$ of logarithmic vector fields of
$X$ is a reflexive ${\mathcal
O}_{{\mathbb A}^n}$module. As a consequence of this result,
we derive that if $T_{{\mathcal O}_{{\mathbb A}^n}/k}(X)$ is a
free ${\mathcal
O}_{{\mathbb A}^n}$module, which is shown to be equivalent
to the freeness of the $t$th exterior power of $T_{{\mathcal O}_{{\mathbb
A}^n}/k}(X)$ for some (in fact, any) $t\leq n$, then necessarily
$X$ is a Saito free divisor.
Keywords:hypersurface, logarithmic vector field, logarithmic derivation, free divisor Categories:14J70, 13N15, 32S22, 13C05, 13C10, 14N20, , , , , 14C20, 32M25 

64. CMB Online first
 Lee, TsiuKwen

Adnilpotent elements of semiprime rings with involution
Let $R$ be an $n!$torsion free semiprime ring with
involution $*$ and with extended centroid $C$, where $n\gt 1$ is
a positive integer. We characterize $a\in K$, the Lie algebra
of skew elements in $R$, satisfying $(\operatorname{ad}_a)^n=0$ on $K$. This
generalizes both Martindale and Miers' theorem
and the theorem of Brox et al.
To prove it we
first prove that if $a, b\in R$ satisfy
$(\operatorname{ad}_a)^n=\operatorname{ad}_b$ on
$R$, where either $n$ is even or $b=0$, then
$\big(a\lambda\big)^{[\frac{n+1}{2}]}=0$
for some $\lambda\in C$.
Keywords:Semiprime ring, Lie algebra, Jordan algebra, faithful $f$free, involution, skew (symmetric) element, adnilpotent element, Jordan element Categories:16N60, 16W10, 17B60 

65. CMB 2017 (vol 60 pp. 329)
 Le Fourn, Samuel

Nonvanishing of Central Values of $L$functions of Newforms in $S_2 (\Gamma_0 (dp^2))$ Twisted by Quadratic Characters
We prove that for $d \in \{ 2,3,5,7,13 \}$ and $K$ a quadratic
(or rational) field of discriminant $D$ and Dirichlet character
$\chi$, if a prime $p$ is large enough compared to $D$, there
is a newform $f \in S_2(\Gamma_0(dp^2))$ with sign $(+1)$ with
respect to the AtkinLehner involution $w_{p^2}$ such that $L(f
\otimes \chi,1) \neq 0$. This result is obtained through an estimate
of a weighted sum of twists of $L$functions which generalises
a result of Ellenberg. It relies on the approximate functional
equation for the $L$functions $L(f \otimes \chi, \cdot)$ and
a Petersson trace formula restricted to AtkinLehner eigenspaces.
An application of this nonvanishing theorem will be given in
terms of existence of rank zero quotients of some twisted jacobians,
which generalises a result of Darmon and Merel.
Keywords:nonvanishing of $L$functions of modular forms, Petersson trace formula, rank zero quotients of jacobians Categories:14J15, 11F67 

66. CMB 2017 (vol 60 pp. 807)
 Liu, Zhongyun; Qin, Xiaorong; Wu, Nianci; Zhang, Yulin

The Shifted Classical Circulant and Skew Circulant Splitting Iterative Methods for Toeplitz Matrices
It is known that every Toeplitz matrix $T$ enjoys a circulant
and skew circulant splitting (denoted by CSCS)
i.e., $T=CS$ with $C$ a circulant matrix and $S$ a skew circulant
matrix. Based on the variant of such a splitting (also referred
to as CSCS), we first develop classical CSCS iterative methods
and then introduce shifted CSCS iterative methods for solving
hermitian positive definite Toeplitz systems in this paper. The
convergence of each method is analyzed. Numerical experiments
show that the classical CSCS iterative methods work slightly
better than the GaussSeidel (GS) iterative methods if the CSCS
is convergent, and that there is always a constant $\alpha$ such
that the shifted CSCS iteration converges much faster than the
GaussSeidel iteration, no matter whether the CSCS itself is
convergent or not.
Keywords:Hermitian positive definite, CSCS splitting, GaussSeidel splitting, iterative method, Toeplitz matrix Categories:15A23, 65F10, 65F15 

67. CMB 2016 (vol 60 pp. 655)
 Zhuo, Ciqiang; Sickel, Winfried; Yang, Dachun; Yuan, Wen

Characterizations of BesovType and TriebelLizorkinType Spaces via Averages on Balls
Let $\ell\in\mathbb N$ and $\alpha\in (0,2\ell)$. In this article,
the authors establish
equivalent characterizations
of Besovtype spaces, TriebelLizorkintype
spaces and BesovMorrey spaces via the sequence
$\{fB_{\ell,2^{k}}f\}_{k}$ consisting of the difference between
$f$ and
the ball average $B_{\ell,2^{k}}f$. These results give a way
to introduce Besovtype spaces,
TriebelLizorkintype spaces and BesovMorrey spaces with any
smoothness order
on metric measure spaces. As special cases, the authors obtain
a new characterization of MorreySobolev spaces
and $Q_\alpha$ spaces with $\alpha\in(0,1)$, which are of independent
interest.
Keywords:Besov space, TriebelLizorkin space, ball average, CalderÃ³n reproducing formula Categories:42B25, 46E35, 42B35 

68. CMB 2016 (vol 60 pp. 411)
 Stoyanov, Luchezar

On Gibbs Measures and Spectra of Ruelle Transfer Operators
We prove a comprehensive version of the RuellePerronFrobenius
Theorem
with explicit estimates of the spectral radius of the Ruelle
transfer operator and various other
quantities related to spectral properties of this operator. The
novelty here is that the HÃ¶lder
constant of the function generating the operator appears only
polynomially, not exponentially as
in previous known estimates.
Keywords:subshift of finite type, Ruelle transfer operator, Gibbs measure Categories:37A05, 37B10 

69. CMB 2016 (vol 60 pp. 165)
 Morimoto, Masaharu

Cokernels of Homomorphisms from Burnside Rings to Inverse Limits
Let $G$ be a finite group and
let $A(G)$ denote the Burnside ring of $G$.
Then an inverse limit $L(G)$ of the groups $A(H)$ for
proper subgroups $H$ of $G$ and a homomorphism
${\operatorname{res}}$ from $A(G)$ to $L(G)$ are obtained in a natural
way.
Let $Q(G)$ denote the cokernel of ${\operatorname{res}}$.
For a prime $p$,
let $N(p)$ be the minimal
normal subgroup of $G$ such that the order of $G/N(p)$ is
a power of $p$, possibly $1$.
In this paper we prove that $Q(G)$ is isomorphic to
the cartesian product of the groups $Q(G/N(p))$, where $p$
ranges over the primes dividing the order of $G$.
Keywords:Burnside ring, inverse limit, finite group Categories:19A22, 57S17 

70. CMB 2016 (vol 60 pp. 26)
71. CMB 2016 (vol 60 pp. 721)
 Eroǧlu, Münevver Pınar; Argaç, Nurcan

On Identities with Composition of Generalized Derivations
Let $R$ be a prime ring with extended
centroid $C$, $Q$ maximal right ring of quotients of $R$, $RC$
central closure of $R$ such that $dim_{C}(RC)
\gt 4$, $f(X_{1},\dots,X_{n})$
a multilinear polynomial over $C$ which is not centralvalued
on $R$ and $f(R)$ the set of all evaluations of the multilinear
polynomial $f\big(X_{1},\dots,X_{n}\big)$ in $R$. Suppose that
$G$ is a nonzero generalized derivation of $R$ such that $G^2\big(u\big)u
\in C$ for all $u\in f(R)$ then one of the following conditions
holds:
(I) there exists $a\in Q$ such that $a^2=0$ and
either $G(x)=ax$ for all $x\in R$ or $G(x)=xa$ for all $x\in
R$;
(II) there exists $a\in Q$ such that $0\neq a^2\in
C$ and either $G(x)=ax$ for all $x\in R$ or $G(x)=xa$ for all
$x\in R$ and $f(X_{1},\dots,X_{n})^{2}$ is centralvalued on
$R$;
(III) $char(R)=2$ and one of the following holds:
(i) there exist $a, b\in Q$ such that $G(x)=ax+xb$ for all
$x\in R$ and $a^{2}=b^{2}\in C$;
(ii) there exist $a, b\in Q$ such that $G(x)=ax+xb$ for all
$x\in R$, $a^{2}, b^{2}\in C$ and $f(X_{1},\ldots,X_{n})^{2}$
is centralvalued on $R$;
(iii) there exist $a \in Q$ and an $X$outer derivation $d$
of $R$ such that $G(x)=ax+d(x)$ for all $x\in R$, $d^2=0$ and
$a^2+d(a)=0$;
(iv) there exist $a \in Q$ and an $X$outer derivation $d$
of $R$ such that $G(x)=ax+d(x)$ for all $x\in R$, $d^2=0$,
$a^2+d(a)\in C$ and $f(X_{1},\dots,X_{n})^{2}$ is centralvalued
on $R$.
Moreover, we characterize the form of nonzero generalized derivations
$G$ of $R$ satisfying $G^2(x)=\lambda x$ for all $x\in R$, where
$\lambda \in C$.
Keywords:prime ring, generalized derivation, composition, extended centroid, multilinear polynomial, maximal right ring of quotients Categories:16N60, 16N25 

72. CMB 2016 (vol 60 pp. 613)
 Reichstein, Zinovy; Vistoli, Angelo

On the Dimension of the Locus of Determinantal Hypersurfaces
The characteristic polynomial $P_A(x_0, \dots,
x_r)$
of an $r$tuple $A := (A_1, \dots, A_r)$ of $n \times n$matrices
is
defined as
\[ P_A(x_0, \dots, x_r) := \det(x_0 I + x_1 A_1 + \dots + x_r
A_r) \, . \]
We show that if $r \geqslant 3$
and $A := (A_1, \dots, A_r)$ is an $r$tuple of $n \times n$matrices in general position,
then up to conjugacy, there are only finitely many $r$tuples
$A' := (A_1', \dots, A_r')$ such that $p_A = p_{A'}$. Equivalently,
the locus of determinantal hypersurfaces of degree $n$ in $\mathbf{P}^r$
is irreducible of dimension $(r1)n^2 + 1$.
Keywords:determinantal hypersurface, matrix invariant, $q$binomial coefficient Categories:14M12, 15A22, 05A10 

73. CMB 2016 (vol 60 pp. 131)
74. 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 

75. CMB 2016 (vol 60 pp. 43)