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225  Convex Functions on Discrete Time Domains Atıcı, Ferhan M.; Yaldız, Hatice
In this paper, we introduce the definition of a convex real
valued function $f$ defined on the set of integers, ${\mathbb{Z}}$. We
prove that $f$ is convex on ${\mathbb{Z}}$ if and only if $\Delta^{2}f
\geq 0$ on ${\mathbb{Z}}$. As a first application of this new concept,
we state and prove discrete HermiteHadamard inequality using
the basics of discrete calculus (i.e. the calculus on ${\mathbb{Z}}$).
Second, we state and prove the discrete fractional HermiteHadamard
inequality using the basics of discrete fractional calculus.
We close the paper by defining the convexity of a real valued
function on any time scale.


234  Nondiscrete Frieze Groups Beardon, Alan F.
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.


244  A Note on Quaternionic Hyperbolic Ideal Triangle Groups Cao, Wensheng; Huang, Xiaolin
In this paper, the quaternionic hyperbolic
ideal triangle groups are parameterized by a real oneparameter
family $\{\phi_s: s\in \mathbb{R}\}$. The indexing parameter $s$ is
the tangent of the quaternionic angular invariant of a triple
of points in $\partial \mathbf{H}_{\mathbb{h}}^2 $ forming this ideal
triangle. We show that if $s \gt \sqrt{125/3}$ then $\phi_s$ is
not a discrete embedding, and if $s \leq \sqrt{35}$
then $\phi_s$ is a discrete embedding.


258  Annihilators and Power Values of Generalized Skew Derivations on Lie Ideals De Filippis, Vincenzo
Let $R$ be a prime ring of characteristic different from
$2$, $Q_r$ be its right Martindale quotient ring and
$C$ be its extended centroid. Suppose that $F$ is
a generalized skew derivation of $R$, $L$ a noncentral Lie ideal
of $R$, $0 \neq a\in R$,
$m\geq 0$ and $n,s\geq 1$ fixed integers. If
\[
a\biggl(u^mF(u)u^n\biggr)^s=0
\]
for all $u\in L$, then either $R\subseteq M_2(C)$, the ring of
$2\times 2$ matrices over $C$, or $m=0$ and there exists $b\in
Q_r$ such that
$F(x)=bx$, for any $x\in R$, with $ab=0$.


271  Artinianness of Composed Graded Local Cohomology Modules DehghaniZadeh, Fatemeh
Let $R=\bigoplus_{n\geq0}R_{n}$ be a graded Noetherian ring with
local base ring $(R_{0}, \mathfrak{m}_{0})$ and let
$R_{+}=\bigoplus_{n\gt 0}R_{n}$, $M$ and $N$ be finitely generated
graded $R$modules and $\mathfrak{a}=\mathfrak{a}_{0}+R_{+}$ an ideal of $R$. We
show that $H^{j}_{\mathfrak{b}_{0}}(H^{i}_{\mathfrak{a}}(M,N))$ and $H^{i}_{\mathfrak{a}}(M,
N)/\mathfrak{b}_{0}H^{i}_{\mathfrak{a}}(M,N)$ are Artinian for some $i^{,}s$ and
$j^{,}s$ with a specified property, where $\mathfrak{b}_{o}$ is an ideal
of
$R_{0}$ such that $\mathfrak{a}_{0}+\mathfrak{b}_{0}$ is an $\mathfrak{m}_{0}$primary ideal.


279  The PoincaréDeligne Polynomial of Milnor Fibers of Triple Point Line Arrangements is Combinatorially Determined Dimca, Alexandru
Using a recent result by S. Papadima and A.
Suciu, we show that
the equivariant PoincaréDeligne polynomial of the Milnor
fiber of a projective line arrangement having only double and
triple points is combinatorially determined.


287  An Existence Theory for Incomplete Designs Dukes, Peter; Lamken, E.R.; Ling, Alan C.H.
An incomplete pairwise balanced design is equivalent to a pairwise
balanced design with a distinguished block, viewed as a `hole'.
If there are $v$ points, a hole of size $w$, and all (other)
block sizes equal $k$, this is denoted IPBD$((v;w),k)$. In addition
to congruence restrictions on $v$ and $w$, there is also a necessary
inequality: $v \gt (k1)w$. This article establishes two main existence
results for IPBD$((v;w),k)$: one in which $w$ is fixed and $v$
is large, and the other in the case $v \gt (k1+\epsilon) w$ when
$w$ is large (depending on $\epsilon$). Several possible generalizations
of the problem are also discussed.


303  Nonextendable Zero Sets of Harmonic and Holomorphic Functions Gauthier, P. M.
In this paper we study the zero sets of harmonic functions on
open sets in $\mathbb{R}^N$ and holomorphic functions on open sets in
$\mathbb{C}^N.$
We show that the nonextendability of such zero sets is a generic
phenomenon.


311  Product Ranks of the $3\times 3$ Determinant and Permanent Ilten, Nathan; Teitler, Zach
We show that the product rank of the $3 \times 3$ determinant
$\det_3$ is $5$,
and the product rank of the $3 \times 3$ permanent
$\operatorname{perm}_3$
is $4$.
As a corollary, we obtain that the tensor rank of $\det_3$ is
$5$ and the tensor rank of $\operatorname{perm}_3$ is $4$.
We show moreover that the border product rank of $\operatorname{perm}_n$ is
larger than $n$ for any $n\geq 3$.


320  Perturbations of Von Neumann Subalgebras with Finite Index Ino, Shoji
In this paper, we study uniform perturbations of von Neumann
subalgebras of a von Neumann algebra.
Let $M$ and $N$ be von Neumann subalgebras of a von Neumann algebra
with finite probabilistic index in the sense of PimsnerPopa.
If $M$ and $N$ are sufficiently close,
then $M$ and $N$ are unitarily equivalent.
The implementing unitary can be chosen as being close to the
identity.


326  On the Uniqueness of Jordan Canonical Form Decompositions of Operators by $K$theoretical Data Jiang, Chunlan; Shi, Rui
In this paper, we develop a generalized Jordan canonical form
theorem for a certain class of operators in $\mathcal
{L}(\mathcal {H})$. A complete criterion for similarity for this
class of operators in terms of $K$theory for Banach
algebras is given.


340  A Note on Algebras that are Sums of Two Subalgebras Kȩpczyk, Marek
We study an associative algebra $A$ over an arbitrary field,
that is
a sum of two subalgebras $B$ and $C$ (i.e. $A=B+C$). We show
that if $B$ is a right or left Artinian $PI$ algebra and $C$
is a $PI$ algebra, then $A$ is a $PI$ algebra. Additionally we
generalize this result for semiprime algebras $A$.
Consider the class of
all semisimple finite dimensional algebras $A=B+C$ for some
subalgebras $B$ and $C$ which satisfy given polynomial identities
$f=0$ and $g=0$, respectively.
We prove that all algebras in this class satisfy a common polynomial
identity.


346  On a Theorem of Bers, with Applications to the Study of Automorphism Groups of Domains Krantz, Steven
We study and generalize a classical theorem of L. Bers that classifies
domains up to biholomorphic equivalence in terms of the algebras
of
holomorphic functions on those domains. Then we develop applications
of these results to the study of domains with noncompact automorphism
group.


354  Factoring a Quadratic Operator as a Product of Two Positive Contractions Li, ChiKwong; Tsai, MingCheng
Let $T$ be a quadratic operator on a complex Hilbert space $H$.
We show that $T$ can be written as a product of two positive
contractions if and only if $T$ is of the form
\begin{equation*}
aI \oplus bI \oplus
\begin{pmatrix} aI & P \cr 0 & bI \cr
\end{pmatrix} \quad \text{on} \quad H_1\oplus H_2\oplus (H_3\oplus
H_3)
\end{equation*}
for some $a, b\in [0,1]$ and strictly positive operator $P$ with
$\P\ \le \sqrt{a}  \sqrt{b}\sqrt{(1a)(1b)}.$ Also, we
give a necessary condition for a bounded linear operator $T$
with operator matrix
$
\big(
\begin{smallmatrix} T_1 & T_3
\\ 0 & T_2\cr
\end{smallmatrix}
\big)
$ on $H\oplus K$ that can be written as a product
of two positive contractions.


363  Dynamical Analysis of a StageStructured Model for Lyme Disease with Two Delays Li, Dan; Ma, Wanbiao
In this paper, a
nonlinear stagestructured model for Lyme disease is considered.
The model is a system of differential equations with two time
delays. The basic reproductive rate, $R_0(\tau_1,\tau_2)$, is
derived. If $R_0(\tau_1,\tau_2)\lt 1$, then the boundary equilibrium
is globally asymptotically stable. If $R_0(\tau_1,\tau_2)\gt 1$,
then there exists
a unique positive equilibrium whose local asymptotical stability
and the existence of
Hopf bifurcations are established by analyzing the distribution
of the characteristic values.
An explicit algorithm for determining the direction of Hopf bifurcations
and the
stability of the bifurcating periodic solutions is derived by
using the normal form and
the center manifold theory. Some numerical simulations are performed
to confirm the correctness
of theoretical analysis. At last, some conclusions are given.


381  Supports of Extremal Doubly Stochastic Measures Moameni, Abbas
A doubly stochastic measure on the unit square is a Borel probability
measure whose horizontal and vertical marginals both coincide
with the Lebesgue measure. The set of doubly stochastic measures
is convex and compact so its
extremal points are of particular interest. The problem number 111
of
Birkhoff (Lattice Theory 1948) is to provide a necessary and
sufficient condition on the support of a doubly stochastic measure
to guarantee extremality. It was proved by
Beneš and Štėpán that an extremal doubly stochastic measure is concentrated
on a set which admits an aperiodic decomposition.
Hestir and Williams later found a necessary condition which
is nearly sufficient by
further refining the aperiodic structure of the support of extremal
doubly stochastic measures.
Our objective in this work is to
provide a more practical necessary and nearly sufficient
condition for a set to support an extremal doubly stochastic
measure.


392  Total Character of a Group $G$ with $(G,Z(G))$ as a Generalized Camina Pair Prajapati, S. K.; Sarma, R.
We investigate whether the total character of a finite group $G$
is a polynomial in a suitable irreducible character of $G$. When
$(G,Z(G))$ is a generalized Camina
pair, we show that the total character is a polynomial in a faithful
irreducible character of $G$
if and only if $Z(G)$ is cyclic.


403  On Flat and Gorenstein Flat Dimensions of Local Cohomology Modules Zargar, Majid Rahro; Zakeri, Hossein
Let $\mathfrak{a}$ be an ideal of a Noetherian local
ring $R$ and let $C$ be a semidualizing $R$module. For an $R$module
$X$, we denote any of the quantities $\mathfrak{d}_R X$,
$\operatorname{\mathsf{Gfd}}_R X$ and
$\operatorname{\mathsf{G_Cfd}}_RX$ by $\operatorname{\mathsf{T}}(X)$. Let $M$ be an $R$module such that
$\operatorname{H}_{\mathfrak{a}}^i(M)=0$
for all $i\neq n$. It is proved that if $\operatorname{\mathsf{T}}(X)\lt \infty$, then
$\operatorname{\mathsf{T}}(\operatorname{H}_{\mathfrak{a}}^n(M))\leq\operatorname{\mathsf{T}}(M)+n$ and the equality holds whenever
$M$ is finitely generated. With the aid of these results, among
other things, we characterize CohenMacaulay modules, dualizing
modules and Gorenstein rings.


417  Existence of Multiple Solutions for a $p$Laplacian System in $\textbf{R}^{N}$ with Signchanging Weight Functions Song, Hongxue; Chen, Caisheng; Yan, Qinglun
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.


435  On Extensions of Stably Finite C*algebras (II) Yao, Hongliang
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.


440  A Note on 3choosability of Planar Graphs Related to Montanssier's Conjecture Zhang, Haihui
A graph $G=(V,E)$ is $L$colorable if for a given list
assignment $L=\{L(v):v\in V(G)\}$, there exists a proper coloring
$c$ of $G$ such that $c(v)\in L(v)$ for all $v\in V$. If $G$ is
$L$colorable for every list assignment $L$ with $L(v)\geq
k$ for
all $v\in V$, then $G$ is said to be $k$choosable. Montassier
(Inform. Process. Lett. 99 (2006) 6871) conjectured that every
planar
graph without cycles of length 4, 5, 6, is 3choosable. In this
paper,
we prove that every planar graph without 5, 6 and 10cycles,
and
without two triangles at distance less than 3 is 3choosable.

