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3  The Contraction Principle for Multivalued Mappings on a Modular Metric Space with a Graph Alfuraidan, Monther Rashed
We study the existence of fixed points for contraction multivalued
mappings in modular metric spaces endowed with a graph. The
notion of a modular metric on an arbitrary set and the corresponding
modular spaces, generalizing classical modulars over linear spaces
like Orlicz spaces, were recently introduced. This paper can
be seen as a generalization of Nadler's and Edelstein's fixed
point theorems to modular metric spaces endowed with a graph.


13  On classes $Q_p^\#$ for Hyperbolic Riemann surfaces Aulaskari, Rauno; Chen, Huaihui
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.


30  A Geometric Extension of Schwarz's Lemma and Applications Cleanthous, Galatia
Let $f$ be a holomorphic function of the unit
disc $\mathbb{D},$ preserving the origin. According to Schwarz's
Lemma, $f'(0)\leq1,$ provided that $f(\mathbb{D})\subset\mathbb{D}.$
We prove that this bound still holds, assuming only that $f(\mathbb{D})$
does not contain any closed rectilinear segment
$[0,e^{i\phi}],\;\phi\in[0,2\pi],$ i.e. does not contain any
entire radius of the closed unit disc. Furthermore, we apply
this result to the hyperbolic density and we give a covering
theorem.


36  Distributive and Antidistributive Mendelsohn Triple Systems Donovan, Diane M.; Griggs, Terry S.; McCourt, Thomas A.; Opršal, Jakub; Stanovský, David
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.


50  On the Bernstein Problem in the Threedimensional Heisenberg Group Dorfmeister, Josef F.; Inoguchi, Junichi; Kobayashi, Shimpei
In this note we present a simple alternative proof
for the Bernstein problem in the threedimensional Heisenberg
group $\operatorname{Nil}_3$
by using the loop group technique. We clarify the geometric
meaning of the twoparameter ambiguity of entire minimal graphs
with prescribed AbreschRosenberg differential.


62  Uncertainty Principles on Weighted Spheres, Balls and Simplexes Feng, Han
This paper studies the uncertainty principle for spherical
$h$harmonic expansions on the unit sphere of $\mathbb{R}^d$ associated
with a weight function invariant under a general finite reflection
group, which
is in full analogy with the classical Heisenberg inequality.
Our proof is motivated by a new decomposition of the DunklLaplaceBeltrami
operator on the weighted sphere.


73  Positive Solutions for the Generalized Nonlinear Logistic Equations Gasiński, Leszek; Papageorgiou, Nikolaos S.
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.


87  Approximation of a Function and its Derivatives by Entire Functions Gauthier, Paul M.; Kienzle, Julie
A simple proof is given for the fact that, for $m$ a nonnegative
integer, a function $f\in C^{(m)}(\mathbb{R}),$ and an arbitrary positive
continuous function $\epsilon,$ there is an entire function $g,$
such that $g^{(i)}(x)f^{(i)}(x)\lt \epsilon(x),$ for all $x\in\mathbb{R}$
and for each $i=0,1\dots,m.$ We also consider the situation,
where $\mathbb{R}$ is replaced by an open interval.


95  Faithful Representations of Graph Algebras via Branching Systems Gonçalves, Daniel; Li, Hui; Royer, Danilo
We continue to investigate branching systems of directed graphs
and their connections with graph algebras. We give a sufficient
condition under which the representation induced from a branching
system of a directed graph is faithful and construct a large
class of branching systems that satisfy this condition. We finish
the paper by providing a proof of the converse of the CuntzKrieger
uniqueness theorem for graph algebras by means of branching systems.


104  LittlewoodPaley Characterizations of SecondOrder Sobolev Spaces via Averages on Balls He, Ziyi; Yang, Dachun; Yuan, Wen
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.


119  A Simple Proof and Strengthening of a Uniqueness Theorem for Lfunctions Hu, PeiChu; Li, Bao Qin
We give a simple proof and strengthening of a uniqueness theorem
for functions in the extended Selberg class.


123  Discrete Spacetime and Lorentz Transformations Jensen, Gerd; Pommerenke, Christian
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.


136  Transformation Formulas for Bilinear Sums of Basic Hypergeometric Series Kajihara, Yasushi
A master formula of transformation formulas for bilinear sums
of basic hypergeometric series
is proposed.
It is obtained from the author's previous results on
a transformation formula for Milne's multivariate generalization
of basic hypergeometric
series of type $A$ with different dimensions and it can be considered
as a
generalization of the WhippleSears transformation formula for
terminating balanced ${}_4 \phi_3$
series.
As an application of the master formula, the one variable cases
of some transformation formulas
for bilinear sums of basic hypergeometric series are given as
examples.
The bilinear transformation formulas seem to be new in the literature,
even in one variable case.


144  A Brief Note Concerning Hard Lefschetz for Chow Groups Laterveer, Robert
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.


159  Rotors in Khovanov Homology MacColl, Joseph
Anstee, Przytycki, and Rolfsen introduced the idea of rotants,
pairs of links related by a generalised form of link mutation.
We exhibit infinitely many pairs of rotants which can be distinguished
by Khovanov homology, but not by the Jones polynomial.


170  A Note on Fine Graphs and Homological Isoperimetric Inequalities MartínezPedroza, Eduardo
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.


182  Generalized Torsion in Knot Groups Naylor, Geoff; Rolfsen, Dale
In a group, a nonidentity element is called
a generalized torsion element if some product of its conjugates
equals the identity. We show that for many classical knots one
can find generalized torsion in the fundamental group of its
complement, commonly called the knot group. It follows that
such a group is not biorderable. Examples include all torus
knots, the (hyperbolic) knot $5_2$ and algebraic knots in the
sense of Milnor.


190  Ramsey Number of Wheels Versus Cycles and Trees Raeisi, Ghaffar; Zaghian, Ali
Let $G_1, G_2, \dots , G_t$ be arbitrary graphs. The
Ramsey number $R(G_1, G_2, \dots, G_t)$ is the smallest positive
integer $n$ such that if the edges of the complete graph $K_n$
are
partitioned into $t$ disjoint color classes giving $t$ graphs
$H_1,H_2,\dots,H_t$, then at least one $H_i$ has a subgraph
isomorphic to $G_i$. In this paper, we provide the exact value
of
the $R(T_n,W_m)$ for odd $m$, $n\geq m1$, where $T_n$ is
either a caterpillar, a tree with diameter at most four or a
tree
with a vertex adjacent to at least $\lceil
\frac{n}{2}\rceil2$ leaves. Also, we
determine $R(C_n,W_m)$ for even integers $n$ and $m$, $n\geq
m+500$, which improves a result of Shi and confirms a
conjecture of Surahmat et al. In addition, the multicolor Ramsey
number of trees
versus an odd wheel is discussed in this paper.


197  Quasicopure Submodules Rajaee, Saeed
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.


204  Restricted Khinchine Inequality Spektor, Susanna
We prove a Khintchine type inequality under the assumption that
the sum of
Rademacher random variables equals zero. We also show a new
tailbound for a hypergeometric random variable.


211  Universality Under Szegő's Condition Totik, Vilmos
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 $\log w$.


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.


449  On Hodge Theory of Singular Plane Curves Abdallah, Nancy
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 $H^2(U,\mathbb C)$.


461  The Nilpotent Regular Element Problem Ara, Pere; O'Meara, Kevin C.
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.


472  Testing Biorderability of Knot Groups Clay, Adam; Desmarais, Colin; Naylor, Patrick
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.


483  Generalized Equivariant Cohomology and Stratifications Crooks, Peter; Holden, Tyler
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.


497  Oneparameter Groups of Operators and Discrete Hilbert Transforms De Carli, Laura; Samad, Gohin Shaikh
We show that the discrete Hilbert transform
and the discrete KakHilbert transform
are infinitesimal generator of oneparameter groups of
operators in $\ell^2$.


508  Generalized Goldberg Formula De Nicola, Antonio; Yudin, Ivan
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 Kähler
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.


521  The Relationship Between $\epsilon$Kronecker Sets and Sidon Sets Hare, Kathryn; Ramsey, L. Thomas
A subset $E$ of a discrete abelian group is called $\epsilon
$Kronecker if
all $E$functions of modulus one can be approximated to within
$\epsilon $
by characters. $E$ is called a Sidon set if all bounded $E$functions
can be
interpolated by the Fourier transform of measures on the dual
group. As $%
\epsilon $Kronecker sets with $\epsilon \lt 2$ possess the same
arithmetic
properties as Sidon sets, it is natural to ask if they are Sidon.
We use the
Pisier net characterization of Sidonicity to prove this is true.


528  Characterization of Lowpass Filters on Local Fields of Positive Characteristic Jahan, Qaiser
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.


542  Spatial Homogenization of Stochastic Wave Equation with Large Interaction Jiang, Yongxin; Wang, Wei; Feng, Zhaosheng
A dynamical approximation of a stochastic wave
equation with large interaction is derived.
A random invariant manifold is discussed. By a key linear transformation,
the random invariant manifold is shown to be close to the random
invariant manifold
of a secondorder stochastic ordinary differential equation.


553  A New Formula for the Energy of Bulk Superconductivity Kachmar, Ayman
The energy of a type II superconductor submitted to an external
magnetic field of intensity close to the second critical field
is given by the celebrated Abrikosov energy. If the external
magnetic field is comparable to and below the second critical
field, the energy is given by a reference function obtained as
a special (thermodynamic) limit of a nonlinear energy. In this
note, we give a new formula for this reference energy. In particular,
we obtain it as a special limit of a linear energy defined
over configurations normalized in the $L^4$norm.


564  Normal Extensions of Representations of Abelian Semigroups Li, Boyu
A commuting family of subnormal operators need
not have a commuting normal extension. We study when a representation
on an abelian semigroup can be extended to a normal representation,
and show that it suffices to extend the set of generators to
commuting normals. We also extend a result due to Athavale to
representations on abelian lattice ordered semigroups.


575  Cohomogeneity One Randers Metrics Li, Jifu; Hu, Zhiguang; Deng, Shaoqiang
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.


585  A Determinantal Inequality Involving Partial Traces Lin, Minghua
Let $\mathbf{A}$ be a density matrix in $\mathbb{M}_m\otimes
\mathbb{M}_n$. Audenaert [J. Math. Phys. 48 (2007) 083507] proved
an inequality for Schatten $p$norms:
\[
1+\\mathbf{A}\_p\ge \\tr_1 \mathbf{A}\_p+\\tr_2 \mathbf{A}\_p,
\]
where $\tr_1, \tr_2$ stand for the first and second partial
trace, respectively. As an analogue of his result, we prove a
determinantal inequality
\[
1+\det \mathbf{A}\ge \det(\tr_1 \mathbf{A})^m+\det(\tr_2 \mathbf{A})^n.
\]


592  The Dirichlet Divisor Problem of Arithmetic Progressions Liu, H. Q.
We design an elementary method to study the problem, getting
an asymptotic formula which is better than Hooley's and HeathBrown's
results for certain cases.


599  Small Prime Solutions to Cubic Diophantine Equations II Liu, Zhixin
Let $a_1, \cdots, a_9$ be nonzero integers and $n$ any integer.
Suppose
that $a_1+\cdots+a_9 \equiv n( \textrm{mod}\,2)$ and $(a_i, a_j)=1$
for $1 \leq i \lt j \leq 9$.
In this paper we prove that


606  Eigenvalues of $ \Delta_p \Delta_q $ Under Neumann Boundary Condition Mihăilescu, Mihai; Moroşanu, Gheorghe
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.


617  Canonical Systems of Basic Invariants for Unitary Reflection Groups Nakashima, Norihiro; Terao, Hiroaki; Tsujie, Shuhei
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.


624  Homology of the Fermat Tower and Universal Measures for Jacobi Sums Otsubo, Noriyuki
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.


641  Some Results on the Annihilatingideal Graphs Shaveisi, Farzad
The annihilatingideal graph
of a commutative ring $R$, denoted by $\mathbb{AG}(R)$, is a
graph whose vertex set consists of all nonzero annihilating
ideals and two distinct
vertices $I$ and $J$ are adjacent if and only if $IJ=(0)$. Here,
we show that if $R$ is a reduced ring and the independence
number of $\mathbb{AG}(R)$ is finite, then the edge chromatic
number of $\mathbb{AG}(R)$ equals its maximum degree
and this number equals $2^{{\rm Min}(R)1}1$; also, it is
proved that the independence number of $\mathbb{AG}(R)$ equals
$2^{{\rm Min}(R)1}$, where ${\rm Min}(R)$ denotes the set
of minimal prime ideals of $R$.
Then we give some criteria for a graph to be isomorphic with
an annihilatingideal graph of a ring.
For example, it is shown that every bipartite annihilatingideal
graph is a complete bipartite graph with at most two horns. Among
other results, it is shown that a finite graph $\mathbb{AG}(R)$
is not Eulerian, and it is Hamiltonian if and only if $R$ contains
no Gorenstain ring as its direct summand.


652  On the Diameter of Unitary Cayley Graphs of Rings Su, Huadong
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.


661  Rings in Which Every Element is a Sum of Two Tripotents Ying, Zhiling; Koşan, Tamer; Zhou, Yiqiang
Let $R$ be a ring. The following results are proved: $(1)$ every
element of $R$ is a sum of an idempotent and a tripotent that
commute iff $R$ has the identity $x^6=x^4$ iff $R\cong R_1\times
R_2$, where $R_1/J(R_1)$ is Boolean with $U(R_1)$ a group of
exponent $2$ and $R_2$ is zero or a subdirect product of $\mathbb
Z_3$'s; $(2)$ every element of $R$ is either a sum or a difference
of two commuting idempotents iff $R\cong R_1\times R_2$, where
$R_1/J(R_1)$ is Boolean with $J(R_1)=0$ or $J(R_1)=\{0,2\}$,
and $R_2$ is zero or a subdirect product of $\mathbb Z_3$'s;
$(3)$ every element of $R$ is a sum of two commuting tripotents
iff $R\cong R_1\times R_2\times R_3$, where $R_1/J(R_1)$ is Boolean
with $U(R_1)$ a group of exponent $2$, $R_2$ is zero or a subdirect
product of $\mathbb Z_3$'s, and $R_3$ is zero or a subdirect
product of $\mathbb Z_5$'s.


673  Lipschitz Retractions in Hadamard Spaces Via Gradient Flow Semigroups Bačák, Miroslav; Kovalev, Leonid V.
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.


682  Ghosts and Strong Ghosts in the Stable Category Carlson, Jon F.; Chebolu, Sunil K.; Mináč, Ján
Suppose that $G$ is a finite group and $k$ is a field of characteristic
$p\gt 0$. A ghost map is a map in the stable category of
finitely generated $kG$modules which induces the zero map
in Tate cohomology in all degrees. In an earlier paper we showed
that the
thick subcategory generated by the trivial module
has no nonzero ghost maps if and only if
the Sylow $p$subgroup of $G$ is cyclic of order 2 or 3.
In this paper we introduce and study variations of ghost
maps.
In particular, we consider the behavior of ghost maps under
restriction
and induction functors. We find all groups satisfying a strong
form
of Freyd's generating hypothesis and show that ghosts can
be detected on a finite range of degrees of Tate cohomology.
We also
consider maps which mimic ghosts in high degrees.


693  Recurrence of Cosine Operator Functions on Groups Chen, ChungChuan
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.


705  The Thickness of the Cartesian Product of Two Graphs Chen, Yichao; Yin, Xuluo
The thickness of a graph $G$ is the minimum number
of planar subgraphs whose union is $G.$ A
$t$minimal graph is a graph of thickness $t$ which contains
no proper subgraph of thickness $t.$ In this paper, upper and
lower bounds are obtained for the thickness, $t(G\Box H)$, of
the Cartesian
product of two graphs $G$ and $H$, in terms of the thickness
$t(G)$ and $t(H)$.
Furthermore, the thickness of the Cartesian product of two planar
graphs and of a $t$minimal graph and a planar graph are determined.
By using a new planar decomposition of the complete bipartite
graph $K_{4k,4k},$ the thickness of the Cartesian product of
two complete bipartite graphs $K_{n,n}$ and $K_{n,n}$ is also
given, for $n\neq 4k+1$.


721  Real Hypersurfaces in Complex Twoplane Grassmannians with Reeb Parallel Ricci Tensor in the GTW Connection Pérez, Juan de Dios; Lee, Hyunjin; Suh, Young Jin; Woo, Changhwa
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.


734  Semiclassical Asymptotics for Schrödinger Operator with Oscillating Decaying Potential Dimassi, Mouez
We study the distribution of the discrete spectrum of the Schrödinger
operator perturbed by a fast oscillating decaying potential depending
on a small parameter $h$.


748  The Metric Dimension of the Total Graph of a Finite Commutative Ring Dolžan, David
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.


760  Artin Approximation Compatible with a Change of Variables Fichou, Goulwen; Quarez, Ronan; Shiota, Masahiro
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.


769  Geometric Characterizations of Hilbert Spaces GarcíaPacheco, Francisco Javier; Hill, Justin R.
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.


776  The Carathéodory Reflection Principle and OsgoodCarathéodory Theorem on Riemann Surfaces Gauthier, Paul M; Sharifi, Fatemeh
The OsgoodCarathéodory theorem asserts that
conformal mappings between Jordan domains extend to homeomorphisms
between their closures.
For multiplyconnected domains on Riemann surfaces, similar results
can be reduced to the simplyconnected case, but we find it simpler
to deduce such results using a direct analogue of the Carathéodory
reflection principle.


794  Zerodivisor Graphs of Ore Extensions over Reversible Rings Hashemi, Ebrahim; Amirjan, R.
Let $R$ be an associative ring with identity.
First we prove some results about zerodivisor graphs of reversible
rings. Then we study the zerodivisors of the skew power series
ring $R[[x;\alpha]]$, whenever $R$ is reversible and $\alpha$compatible. Moreover, we compare the diameter and girth of the zerodivisor
graphs of $\Gamma(R)$, $\Gamma(R[x;\alpha,\delta])$ and $\Gamma(R[[x;\alpha]])$,
when
$R$ is reversible and $(\alpha,\delta)$compatible.


806  Geometric Interpretation of Lagrangian Equivalence Izumiya, Shyuichi
As an application of the theory of
graphlike Legendrian unfoldings, relations of the hidden structures
of caustics and wave front propagations are revealed.


813  A Classification of Threedimensional Real Hypersurfaces in Nonflat Complex Space Forms in Terms of Their generalized TanakaWebster Lie Derivative Kaimakamis, George; Panagiotidou, Konstantina; Pérez, Juan de Dios
On a real hypersurface $M$ in a nonflat complex
space form there exist the LeviCivita and the kth generalized
TanakaWebster connections. The aim of the present paper is to
study three dimensional real hypersurfaces in nonflat complex
space forms, whose Lie derivative of the structure Jacobi operator
with respect to the LeviCivita connections coincides with the
Lie derivative of it with respect to the kth generalized TanakaWebster
connection. The Lie derivatives are considered in direction of
the structure vector field and in directions of any vecro field
orthogonal to the structure vector field.


824  Incompressibility of Products of Pseudohomogeneous Varieties Karpenko, Nikita A.
We show that the conjectural criterion of $p$incompressibility
for products of projective homogeneous varieties in terms of
the factors, previously known in a few special cases only, holds
in general.
Actually, the proof goes through for a wider class of varieties
which includes the norm varieties associated to symbols in Galois
cohomology of arbitrary degree.


834  Some Properties of TriebelLizorkin and Besov Spaces Associated with Zygmund Dilations Liao, Fanghui; Liu, Zongguang
In this paper, using Calderón'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.


849  Stability Threshold for Scalar Linear Periodic Delay Differential Equations Nah, Kyeongah; Röst, Gergely
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.


858  Stability of Vector Bundles on Curves and Degenerations Osserman, Brian
We introduce a weaker notion of (semi)stability for vector bundles
on
reducible curves which does not depend on a choice of polarization,
and
which suffices for many applications of degeneration techniques.
We explore the basic
properties of this alternate notion of (semi)stability. In a
complementary
direction, we record a proof of the existence of semistable extensions
of vector bundles in suitable degenerations.


865  Moduli of Rank 2 Stable Bundles and Hecke Curves Pal, Sarbeswar
Let $X$ be smooth projective curve of arbitrary genus $g \gt 3$
over complex numbers. In this short note we will show that the
moduli
space of rank $2$ stable vector bundles with determinant isomorphic
to $L_x$, where $L_x$ denote the line bundle corresponding to
a point $x \in X$ is isomorphic to certain lines in the moduli
space of Sequivalence classes of semistable bundles of rank
2 with
trivial determinant.


878  The Carleson Measure Problem Between Analytic Morrey Spaces Wang, Jianfei
The purpose of this paper is to characterize positive measure
$\mu$ on the unit disk such that the analytic
Morrey space $\mathcal{AL}_{p,\eta}$ is boundedly and compactly
embedded to the tent space
$\mathcal{T}_{q,1\frac{q}{p}(1\eta)}^{\infty}(\mu)$ for the
case $1\leq q\leq p\lt \infty$
respectively. As an application, these results are used to
establish the boundedness and compactness of integral operators
and multipliers between analytic Morrey spaces.

