51. CMB 2015 (vol 59 pp. 73)
 Gasiński, Leszek; Papageorgiou, Nikolaos S.

Positive Solutions for the Generalized Nonlinear Logistic Equations
We consider a nonlinear parametric elliptic equation driven
by a nonhomogeneous differential
operator with a logistic reaction of the superdiffusive type.
Using variational methods coupled with suitable truncation
and comparison techniques,
we prove a bifurcation type result describing the set of positive
solutions
as the parameter varies.
Keywords:positive solution, bifurcation type result, strong comparison principle, nonlinear regularity, nonlinear maximum principle Categories:35J25, 35J92 

52. CMB 2015 (vol 59 pp. 435)
 Yao, Hongliang

On Extensions of Stably Finite C*algebras (II)
For any $C^*$algebra $A$ with an approximate
unit of projections, there is a smallest ideal $I$ of $A$ such
that the quotient $A/I$ is stably finite.
In this paper, a sufficient and necessary condition is obtained
for an ideal of a $C^*$algebra with real rank zero is this smallest
ideal by $K$theory.
Keywords:extension, stably finite C*algebra, index map Categories:46L05, 46L80 

53. CMB 2015 (vol 59 pp. 170)
 MartínezPedroza, Eduardo

A Note on Fine Graphs and Homological Isoperimetric Inequalities
In the framework of homological characterizations of relative
hyperbolicity, Groves and Manning posed the question of whether
a simply connected $2$complex $X$ with a linear homological
isoperimetric inequality, a bound on the length of attaching
maps of $2$cells and finitely many $2$cells adjacent to any
edge must have a fine $1$skeleton. We provide a positive answer
to this question. We revisit a homological characterization
of relative hyperbolicity, and show that a group $G$ is hyperbolic
relative to a collection of subgroups $\mathcal P$ if and only if
$G$ acts cocompactly with finite edge stabilizers on an connected
$2$dimensional cell complex with a linear homological isoperimetric
inequality and $\mathcal P$ is a collection of representatives of
conjugacy classes of vertex stabilizers.
Keywords:isoperimetric functions, Dehn functions, hyperbolic groups Categories:20F67, 05C10, 20J05, 57M60 

54. CMB 2015 (vol 59 pp. 197)
 Rajaee, Saeed

Quasicopure Submodules
All rings are commutative with identity and all modules are unital.
In this paper we introduce the concept of quasicopure submodule
of
a multiplication $R$module $M$ and will give some results of
them.
We give some properties of tensor product of finitely generated
faithful multiplication modules.
Keywords:multiplication module, arithmetical ring, copure submodule, radical of submodules Categories:13A15, 13C05, 13C13, , 13C99 

55. CMB 2015 (vol 59 pp. 123)
 Jensen, Gerd; Pommerenke, Christian

Discrete Spacetime and Lorentz Transformations
Alfred Schild has established conditions
that Lorentz transformations map worldvectors $(ct,x,y,z)$ with
integer coordinates onto vectors of the same kind. The problem
was dealt with in the context of tensor and spinor calculus.
Due to Schild's numbertheoretic arguments, the subject is also
interesting when isolated from its physical background.
The paper of Schild is not easy to understand. Therefore we first
present a streamlined version of his proof which is based on
the use of null vectors. Then we present a purely algebraic proof
that is somewhat shorter. Both proofs rely on the properties
of Gaussian integers.
Keywords:Lorentz transformation, integer lattice, Gaussian integers Categories:22E43, 20H99, 83A05 

56. CMB 2015 (vol 58 pp. 818)
 Llibre, Jaume; Zhang, Xiang

On the Limit Cycles of Linear Differential Systems with Homogeneous Nonlinearities
We consider the class of polynomial differential systems of the
form
$\dot x= \lambda xy+P_n(x,y)$, $\dot y=x+\lambda y+ Q_n(x,y),$ where
$P_n$ and $Q_n$ are homogeneous polynomials of degree $n$. For
this
class of differential systems we summarize the known results
for the
existence of limit cycles, and we provide new results for their
nonexistence and existence.
Keywords:polynomial differential system, limit cycles, differential equations on the cylinder Categories:34C35, 34D30 

57. CMB 2015 (vol 58 pp. 704)
 Benamar, H.; Chandoul, A.; Mkaouar, M.

On the Continued Fraction Expansion of Fixed Period in Finite Fields
The Chowla conjecture
states that,
if $t$ is any given
positive integer, there are infinitely many prime positive
integers $N$ such that $\operatorname{Per} (\sqrt{N})=t$, where
$\operatorname{Per} (\sqrt{N})$
is the period length of the continued fraction expansion for
$\sqrt{N}$.
C. Friesen proved
that, for any $k\in \mathbb{N}$, there are infinitely many
squarefree integers $N$, where the continued fraction expansion
of $\sqrt{N}$ has a fixed period. In this paper, we describe all
polynomials $Q\in \mathbb{F}_q[X] $ for which the continued fraction
expansion of $\sqrt {Q}$ has a fixed period, also we give a
lower
bound of the number of monic, nonsquares polynomials $Q$ such
that $\deg Q= 2d$ and $ Per \sqrt {Q}=t$.
Keywords:continued fractions, polynomials, formal power series Categories:11A55, 13J05 

58. CMB 2015 (vol 58 pp. 799)
 Kong, Qingjun; Guo, Xiuyun

On $s$semipermutable or $s$quasinormally Embedded Subgroups of Finite Groups
Suppose that $G$ is a
finite group and $H$ is a subgroup of $G$. $H$ is said to be
$s$semipermutable in $G$ if $HG_{p}=G_{p}H$ for any Sylow
$p$subgroup $G_{p}$ of $G$ with $(p,H)=1$; $H$ is said to be
$s$quasinormally embedded in $G$ if for each prime $p$ dividing the
order of $H$, a Sylow $p$subgroup of $H$ is also a Sylow
$p$subgroup of some $s$quasinormal subgroup of $G$. We fix in
every noncyclic Sylow subgroup $P$ of $G$ some subgroup $D$
satisfying $1\lt D\lt P$ and study the structure of $G$ under the
assumption that every subgroup $H$ of $P$ with $H=D$ is either
$s$semipermutable or $s$quasinormally embedded in $G$.
Some recent results are generalized and unified.
Keywords:$s$semipermutable subgroup, $s$quasinormally embedded subgroup, saturated formation. Categories:20D10, 20D20 

59. CMB 2015 (vol 58 pp. 741)
 Gao, Zenghui

Homological Properties Relative to Injectively Resolving Subcategories
Let $\mathcal{E}$ be an injectively resolving subcategory of
left $R$modules. A left $R$module $M$
(resp. right $R$module $N$) is called $\mathcal{E}$injective
(resp. $\mathcal{E}$flat)
if $\operatorname{Ext}_R^1(G,M)=0$ (resp. $\operatorname{Tor}_1^R(N,G)=0$)
for any $G\in\mathcal{E}$.
Let $\mathcal{E}$ be a covering subcategory.
We prove that a left $R$module $M$ is $\mathcal{E}$injective
if and only if $M$ is a direct sum
of an injective left $R$module and a reduced $\mathcal{E}$injective
left $R$module.
Suppose $\mathcal{F}$ is a preenveloping subcategory of right
$R$modules such that
$\mathcal{E}^+\subseteq\mathcal{F}$ and $\mathcal{F}^+\subseteq\mathcal{E}$.
It is shown that a finitely presented right $R$module $M$ is
$\mathcal{E}$flat if and only if
$M$ is a cokernel of an $\mathcal{F}$preenvelope of a right
$R$module.
In addition, we introduce and investigate the
$\mathcal{E}$injective and $\mathcal{E}$flat dimensions of
modules and rings. We also introduce $\mathcal{E}$(semi)hereditary
rings and $\mathcal{E}$von Neumann regular rings and characterize
them in terms of $\mathcal{E}$injective and $\mathcal{E}$flat
modules.
Keywords:injectively resolving subcategory, \mathcal{E}injective module (dimension), \mathcal{E}flat module (dimension), cover, preenvelope, \mathcal{E}(semi)hereditary ring Categories:16E30, 16E10, 16E60 

60. CMB 2015 (vol 59 pp. 104)
 He, Ziyi; Yang, Dachun; Yuan, Wen

LittlewoodPaley Characterizations of SecondOrder Sobolev Spaces via Averages on Balls
In this paper, the authors characterize secondorder Sobolev
spaces $W^{2,p}({\mathbb R}^n)$,
with $p\in [2,\infty)$ and $n\in\mathbb N$ or $p\in (1,2)$ and
$n\in\{1,2,3\}$, via the Lusin area
function and the LittlewoodPaley $g_\lambda^\ast$function in
terms of ball means.
Keywords:Sobolev space, ball means, Lusinarea function, $g_\lambda^*$function Categories:46E35, 42B25, 42B20, 42B35 

61. CMB 2015 (vol 58 pp. 824)
 Luo, XiuHua

Exact Morphism Category and Gorensteinprojective Representations
Let $Q$ be a finite acyclic quiver, $J$ be an ideal of $kQ$ generated
by all arrows in $Q$, $A$ be a finitedimensional $k$algebra. The
category of all finitedimensional representations of $(Q, J^2)$ over
$A$ is denoted by $\operatorname{rep}(Q, J^2, A)$. In this paper, we
introduce the category $\operatorname{exa}(Q,J^2,A)$, which is a
subcategory of
$\operatorname{rep}{}(Q,J^2,A)$ of all exact representations.
The main result of this paper explicitly describes the Gorensteinprojective representations in $\operatorname{rep}{}(Q,J^2,A)$,
via the exact representations plus an extra condition.
As a corollary, $A$ is a selfinjective algebra, if
and only if the Gorensteinprojective representations are exactly the
exact representations of $(Q, J^2)$ over $A$.
Keywords:representations of a quiver over an algebra, exact representations, Gorensteinprojective modules Category:18G25 

62. CMB 2015 (vol 58 pp. 774)
 Hanson, Brandon

Character Sums over Bohr Sets
We prove character sum estimates for additive Bohr subsets modulo
a prime.
These estimates are analogous to classical character sum bounds
of
PÃ³lyaVinogradov and Burgess. These estimates are applied to
obtain results on
recurrence mod $p$ by special elements.
Keywords:character sums, Bohr sets, finite fields Categories:11L40, 11T24, 11T23 

63. CMB 2015 (vol 58 pp. 673)
 Achter, Jeffrey; Williams, Cassandra

Local Heuristics and an Exact Formula for Abelian Surfaces Over Finite Fields
Consider a quartic $q$Weil polynomial $f$. Motivated by equidistribution
considerations, we define, for each prime $\ell$, a local factor
that
measures the relative frequency with which $f\bmod \ell$ occurs
as the
characteristic polynomial of a symplectic similitude over $\mathbb{F}_\ell$.
For a certain
class of polynomials, we show that the resulting infinite product
calculates the number of principally polarized abelian surfaces
over $\mathbb{F}_q$
with Weil polynomial $f$.
Keywords:abelian surfaces, finite fields, random matrices Category:14K02 

64. CMB 2015 (vol 58 pp. 877)
 Zaatra, Mohamed

Generating Some Symmetric Semiclassical Orthogonal Polynomials
We show that if $v$ is a regular semiclassical form
(linear functional), then the symmetric form $u$ defined by the
relation
$x^{2}\sigma u = \lambda v$,
where $(\sigma f)(x)=f(x^{2})$ and the odd
moments of $u$ are $0$, is also
regular and semiclassical form for every
complex $\lambda $ except for a discrete set of numbers depending
on $v$. We give explicitly the threeterm recurrence relation
and the
structure relation coefficients of the orthogonal polynomials
sequence associated with $u$ and the class of the form $u$ knowing
that of $v$. We conclude with an illustrative example.
Keywords:orthogonal polynomials, quadratic decomposition, semiclassical forms, structure relation Categories:33C45, 42C05 

65. CMB 2015 (vol 58 pp. 835)
66. CMB 2015 (vol 59 pp. 144)
 Laterveer, Robert

A Brief Note Concerning Hard Lefschetz for Chow Groups
We formulate a conjectural hard Lefschetz property
for Chow groups, and prove this in some special cases: roughly
speaking, for varieties with finitedimensional motive, and
for varieties whose selfproduct has vanishing middledimensional
Griffiths group. An appendix includes related statements that
follow from results of Vial.
Keywords:algebraic cycles, Chow groups, finitedimensional motives Categories:14C15, 14C25, 14C30 

67. CMB 2015 (vol 59 pp. 211)
 Totik, Vilmos

Universality Under SzegÅ's Condition
This paper presents a
theorem on universality on orthogonal polynomials/random matrices
under a weak local condition on the weight function $w$.
With a new inequality for
polynomials and with the use of fast decreasing polynomials,
it is shown that an approach of
D. S. Lubinsky is applicable. The proof works
at all points which are Lebesguepoints both
for the weight function $w$ and for $\log w$.
Keywords:universality, random matrices, Christoffel functions, asymptotics, potential theory Categories:42C05, 60B20, 30C85, 31A15 

68. CMB 2015 (vol 58 pp. 858)
 Williams, Kenneth S.

Ternary Quadratic Forms and Eta Quotients
Let $\eta(z)$ $(z \in \mathbb{C},\;\operatorname{Im}(z)\gt 0)$
denote the Dedekind eta function. We use a recent producttosum
formula in conjunction with conditions for the nonrepresentability
of integers by certain ternary quadratic forms to give explicitly
10 eta quotients
\[
f(z):=\eta^{a(m_1)}(m_1 z)\cdots \eta^{{a(m_r)}}(m_r z)=\sum_{n=1}^{\infty}c(n)e^{2\pi
i nz},\quad z \in \mathbb{C},\;\operatorname{Im}(z)\gt 0,
\]
such that the Fourier coefficients $c(n)$ vanish for all positive
integers $n$ in each of infinitely many nonoverlapping arithmetic
progressions. For example, it is shown that for $f(z)=\eta^4(z)\eta^{9}(4z)\eta^{2}(8z)$
we have $c(n)=0$ for all $n$ in each of the arithmetic progressions
$\{16k+14\}_{k \geq 0}$, $\{64k+56\}_{k \geq 0}$, $\{256k+224\}_{k
\geq 0}$, $\{1024k+896\}_{k \geq 0}$, $\ldots$.
Keywords:Dedekind eta function, eta quotient, ternary quadratic forms, vanishing of Fourier coefficients, producttosum formula Categories:11F20, 11E20, 11E25 

69. CMB 2015 (vol 58 pp. 787)
 Kitabeppu, Yu; Lakzian, Sajjad

Nonbranching RCD$(0,N)$ Geodesic Spaces with Small Linear Diameter Growth have Finitely Generated Fundamental Groups
In this paper, we generalize the finite generation result of
Sormani
to nonbranching $RCD(0,N)$
geodesic spaces (and in particular, Alexandrov spaces) with full
support measures. This is a special case of the Milnor's Conjecture
for complete noncompact $RCD(0,N)$ spaces. One of the key tools
we use is the AbreschGromoll type excess estimates for nonsmooth
spaces obtained by GigliMosconi.
Keywords:Milnor conjecture, non negative Ricci curvature, curvature dimension condition, finitely generated, fundamental group, infinitesimally Hilbertian Categories:53C23, 30L99 

70. CMB 2015 (vol 59 pp. 13)
 Aulaskari, Rauno; Chen, Huaihui

On classes $Q_p^\#$ for Hyperbolic Riemann surfaces
The $Q_p$ spaces of holomorphic functions on
the disk, hyperbolic Riemann surfaces or complex unit ball have
been studied deeply.
Meanwhile, there are a lot of papers devoted to the $Q^\#_p$
classes of meromorphic functions on the disk or hyperbolic Riemann
surfaces. In this paper, we prove the nesting property (inclusion
relations) of $Q^\#_p$ classes on hyperbolic Riemann surfaces.
The same property for $Q_p$ spaces was also established systematically
and precisely in
earlier work
by the authors of this paper.
Keywords:$Q_p^\#$ class, hyperbolic Riemann surface, spherical Dirichlet function, Categories:30D50, 30F35 

71. CMB 2015 (vol 59 pp. 119)
72. CMB 2015 (vol 59 pp. 36)
 Donovan, Diane M.; Griggs, Terry S.; McCourt, Thomas A.; Opršal, Jakub; Stanovský, David

Distributive and Antidistributive Mendelsohn Triple Systems
We prove that the existence spectrum of Mendelsohn triple systems
whose associated quasigroups satisfy distributivity corresponds
to the Loeschian numbers, and provide some enumeration results.
We do this by considering a description of the quasigroups in
terms of commutative Moufang loops.
In addition we provide constructions of Mendelsohn quasigroups
that fail distributivity for as many combinations of elements
as possible.
These systems are analogues of Hall triple systems and antimitre
Steiner triple systems respectively.
Keywords:Mendelsohn triple system, quasigroup, distributive, Moufang loop, Loeschian numbers Categories:20N05, 05B07 

73. CMB 2015 (vol 58 pp. 664)
 Vahidi, Alireza

Betti Numbers and Flat Dimensions of Local Cohomology Modules
Assume that $R$ is a commutative Noetherian ring with nonzero
identity, $\mathfrak{a}$ is an ideal of $R$ and $X$ is an $R$module.
In this paper, we first study the finiteness of Betti numbers
of local cohomology modules $\operatorname{H}_\mathfrak{a}^i(X)$. Then we give some
inequalities between the Betti numbers of $X$ and those of its
local cohomology modules. Finally, we present many upper bounds
for the flat dimension of $X$ in terms of the flat dimensions
of its local cohomology modules and an upper bound for the flat
dimension of $\operatorname{H}_\mathfrak{a}^i(X)$ in terms of the flat dimensions of
the modules $\operatorname{H}_\mathfrak{a}^j(X)$, $j\not= i$, and that of $X$.
Keywords:Betti numbers, flat dimensions, local cohomology modules Categories:13D45, 13D05 

74. 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 

75. CMB 2015 (vol 58 pp. 651)
 Tang, Xianhua

Ground State Solutions of NehariPankov Type for a Superlinear Hamiltonian Elliptic System on ${\mathbb{R}}^{N}$
This paper is concerned with the following
elliptic system of Hamiltonian type
\[
\left\{
\begin{array}{ll}
\triangle u+V(x)u=W_{v}(x, u, v), \ \ \ \ x\in {\mathbb{R}}^{N},
\\
\triangle v+V(x)v=W_{u}(x, u, v), \ \ \ \ x\in {\mathbb{R}}^{N},
\\
u, v\in H^{1}({\mathbb{R}}^{N}),
\end{array}
\right.
\]
where the potential $V$ is periodic and $0$ lies in a gap of
the spectrum of $\Delta+V$, $W(x, s, t)$ is
periodic in $x$ and superlinear in $s$ and $t$ at infinity.
We develop a direct approach to find ground
state solutions of NehariPankov type for the above system.
Especially, our method is applicable for the
case when
\[
W(x, u, v)=\sum_{i=1}^{k}\int_{0}^{\alpha_iu+\beta_iv}g_i(x,
t)t\mathrm{d}t
+\sum_{j=1}^{l}\int_{0}^{\sqrt{u^2+2b_juv+a_jv^2}}h_j(x,
t)t\mathrm{d}t,
\]
where $\alpha_i, \beta_i, a_j, b_j\in \mathbb{R}$ with $\alpha_i^2+\beta_i^2\ne
0$ and $a_j\gt b_j^2$, $g_i(x, t)$
and $h_j(x, t)$ are nondecreasing in $t\in \mathbb{R}^{+}$ for every
$x\in \mathbb{R}^N$ and $g_i(x, 0)=h_j(x, 0)=0$.
Keywords:Hamiltonian elliptic system, superlinear, ground state solutions of NehariPankov type, strongly indefinite functionals Categories:35J50, 35J55 
