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3  The Coannihilatingideal Graphs of Commutative Rings Akbari, Saeeid; Alilou, Abbas; Amjadi, Jafar; Sheikholeslami, Seyed Mahmoud
Let $R$ be a commutative ring with identity. The
coannihilatingideal graph of $R$, denoted by $\mathcal{A}_R$,
is
a graph whose vertex set is the set of all nonzero proper ideals
of $R$ and two distinct vertices $I$ and $J$ are adjacent
whenever ${\operatorname {Ann}}(I)\cap {\operatorname {Ann}}(J)=\{0\}$. In this paper we
initiate the study of the coannihilating ideal graph of a
commutative ring and we investigate its properties.


12  Comaximal Graphs of Subgroups of Groups Akbari, Saieed; Miraftab, Babak; Nikandish, Reza
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.


26  Self $2$distance Graphs Azimi, Ali; Farrokhi Derakhshandeh Ghouchan, Mohammad
All finite simple self $2$distance graphs with no square, diamond,
or triangles with a common vertex as subgraph are determined.
Utilizing these results, it is shown that there is no cubic self
$2$distance graph.


43  On the Dual König Property of the Orderinterval Hypergraph of Two Classes of Nfree Posets Bouchemakh, Isma; Fatma, Kaci
Let $P$ be a finite Nfree poset. We consider the hypergraph
$\mathcal{H}(P)$ whose vertices are the elements of $P$ and whose
edges are the maximal intervals of $P$. We study the dual
König property of $\mathcal{H}(P)$ in two subclasses of Nfree class.


54  Tubular Free by Cyclic Groups Act Freely on CAT(0) Cube Complexes Button, Jack
We identify when a tubular group (the fundamental group of a
finite
graph of groups with $\mathbb{Z}^2$ vertex and $\mathbb{Z}$ edge groups) is free
by
cyclic and show, using Wise's equitable sets criterion, that
every
tubular free by
cyclic group acts freely on a CAT(0) cube complex.


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


77  Nilpotent Group C*algebras as Compact Quantum Metric Spaces Christ, Michael; Rieffel, Marc A.
Let $\mathbb{L}$ be a length function on a group $G$, and let $M_\mathbb{L}$
denote the
operator of pointwise multiplication by $\mathbb{L}$ on $\lt(G)$.
Following Connes,
$M_\mathbb{L}$ can be used as a ``Dirac'' operator for the reduced
group C*algebra $C_r^*(G)$. It defines a
Lipschitz seminorm on $C_r^*(G)$, which defines a metric on the
state space of
$C_r^*(G)$. We show that
for any length function satisfying a strong form of polynomial
growth on a discrete group,
the topology from this metric
coincides with the
weak$*$ topology (a key property for the
definition of a ``compact quantum metric
space''). In particular, this holds for all wordlength functions
on finitely generated nilpotentbyfinite groups.


95  Cubic Functional Equations on Restricted Domains of Lebesgue Measure Zero Choi, ChangKwon; Chung, Jaeyoung; Ju, Yumin; Rassias, John
Let $X$ be a real normed space, $Y$ a Bancch space and $f:X \to
Y$.
We prove the UlamHyers stability theorem
for the cubic functional equation
\begin{align*}
f(2x+y)+f(2xy)2f(x+y)2f(xy)12f(x)=0
\end{align*}
in restricted domains. As an application we consider a measure
zero stability problem
of the inequality
\begin{align*}
\f(2x+y)+f(2xy)2f(x+y)2f(xy)12f(x)\\le \epsilon
\end{align*}
for all $(x, y)$ in $\Gamma\subset\mathbb R^2$ of Lebesgue measure
0.


104  An Extension of Nikishin's Factorization Theorem Diestel, Geoff
A NikishinMaurey characterization is given for bounded subsets
of weaktype Lebesgue spaces. New factorizations for linear and
multilinear operators are shown to follow.


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


122  A Homological Property and Arens Regularity of Locally Compact Quantum Groups Ghanei, Mohammad Reza; NasrIsfahani, Rasoul; Nemati, Mehdi
We characterize two important notions of amenability and compactness
of
a locally compact quantum group ${\mathbb G}$ in terms of certain
homological
properties. For this, we show that ${\mathbb G}$ is character
amenable if and only if it is both amenable and coamenable.
We finally apply our results to
Arens regularity problems of the quantum group algebra
$L^1({\mathbb G})$; in particular, we improve an interesting result
by Hu, Neufang and Ruan.


131  Some Estimates for Generalized Commutators of Rough Fractional Maximal and Integral Operators on Generalized Weighted Morrey Spaces Gürbüz, Ferit
In this paper, we establish $BMO$ estimates for generalized commutators
of
rough fractional maximal and integral operators on generalized
weighted
Morrey spaces, respectively.


146  The Dirichlet Problem for the Slab with Entire Data and a Difference Equation for Harmonic Functions Khavinson, Dmitry; Lundberg, Erik; Render, Hermann
It is shown that the Dirichlet problem for the slab $(a,b) \times
\mathbb{R}^{d}$ with entire boundary data has an entire solution. The proof
is based
on a generalized Schwarz reflection principle. Moreover, it is
shown that
for a given entire harmonic function $g$
the inhomogeneous difference equation $h
( t+1,y) h (t,y) =g ( t,y)$
has an entire harmonic solution $h$.


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


165  Cokernels of Homomorphisms from Burnside Rings to Inverse Limits Morimoto, Masaharu
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 $G$.


173  On Ulam Stability of a Functional Equation in Banach Modules Oubbi, Lahbib
Let $X$ and $Y$ be Banach spaces and $f : X \to Y$ an odd mapping.
For any rational number $r \ne 2$, C. Baak, D. H.
Boo, and Th. M. Rassias have proved the HyersUlam stability
of the following functional equation:
\begin{align*}
r f
\left(\frac{\sum_{j=1}^d x_j}{r}
\right)
& + \sum_{\substack{i(j) \in \{0,1\}
\\ \sum_{j=1}^d i(j)=\ell}} r f
\left(
\frac{\sum_{j=1}^d (1)^{i(j)}x_j}{r}
\right)
= (C^\ell_{d1}  C^{\ell 1}_{d1} + 1) \sum_{j=1}^d
f(x_j)
\end{align*}
where $d$ and $\ell$ are positive integers so that $1 \lt \ell
\lt \frac{d}{2}$, and $C^p_q := \frac{q!}{(qp)!p!}$,
$p, q \in \mathbb{N}$ with $p \le q$.


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


196  Corrigendum to "Generalized Cesàro Matrices" Rhaly, H. C. Jr.
This note corrects an error in Theorem 1 of
"Generalized Cesàro matrices"
Canad. Math. Bull. 27 (1984), no. 4, 417422.


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


206  The Metric Dimension of Circulant Graphs Vetrik, Tomáš
A subset $W$ of the vertex set of a graph $G$ is called a resolving
set of $G$ if for every pair of distinct vertices $u, v$ of $G$,
there is $w \in W$ such that the~distance of $w$ and $u$ is different
from the distance of $w$ and $v$. The~cardinality of a~smallest
resolving set is called the metric dimension of $G$, denoted
by $dim(G)$. The circulant graph $C_n (1, 2, \dots , t)$ consists
of the vertices $v_0, v_1, \dots , v_{n1}$ and the~edges $v_i
v_{i+j}$, where $0 \le i \le n1$, $1 \le j \le t$ $(2 \le t
\le \lfloor \frac{n}{2} \rfloor)$, the indices are taken modulo
$n$. Grigorious et al. [On the metric dimension of circulant
and Harary graphs, Applied Mathematics and Computation 248 (2014),
4754] proved that $dim(C_n (1,2, \dots , t))
\ge t+1$ for $t \lt \lfloor \frac{n}{2} \rfloor$, $n \ge 3$, and they
presented a~conjecture saying that $dim(C_n (1,2, \dots , t))
= t+p1$ for $n=2tk+t+p$, where $3 \le p \le t+1$. We disprove
both statements. We show that if $t \ge 4$ is even, there exists
an infinite set of values of $n$ such that $dim(C_n (1,2, \dots
, t)) = t$. We also prove that $dim(C_n (1,2, \dots , t)) \le
t + \frac{p}{2}$ for $n=2tk+t+p$, where $t$ and $p$ are even,
$t \ge 4$, $2 \le p \le t$ and $k \ge 1$.


217  Condition $C'_{\wedge}$ of Operator Spaces Wang, Yuanyi
In this paper, we study condition $C'_{\wedge}$ which is a
projective tensor product analogue of condition $C'$. We show
that
the finitedimensional OLLP operator spaces have condition
$C'_{\wedge}$ and $M_{n}$ $(n\gt 2)$ does not have that property.


225  Faltings' Finiteness Dimension of Local Cohomology Modules Over Local CohenMacaulay Rings Bahmanpour, Kamal; Naghipour, Reza
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.


235  Topology of Certain Quotient Spaces of Stiefel Manifolds Basu, Samik; Subhash, B
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.


246  On Radicals of Green's Relations in Ordered Semigroups Bhuniya, Anjan Kumar; Hansda, Kalyan
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.


253  On a Yamabe Type Problem in Finsler Geometry Chen, Bin; Zhao, Lili
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.


269  Characterizations and Representations of Core and Dual Core Inverses Chen, Jianlong; Zhu, Huihui; Patricio, Pedro; Zhang, Yulin
In this
paper,
double commutativity and the reverse order law for the core inverse
are considered. Then, new characterizations of the MoorePenrose
inverse of a regular element are given by onesided invertibilities
in a ring. Furthermore, the characterizations and representations
of
the core and dual core inverses of a regular element are considered.


283  Twisted Alexander Invariants Detect Trivial Links Friedl, Stefan; Vidussi, Stefano
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.


300  Luzintype Holomorphic Approximation on Closed Subsets of Open Riemann Surfaces Gauthier, Paul M; Sharifi, Fatemeh
It is known that if $E$ is a closed subset of an open Riemann
surface $R$ and $f$ is a holomorphic function on a neighbourhood
of $E,$ then it is ``usually" not possible to approximate $f$
uniformly by functions holomorphic on all of $R.$ We show, however,
that for every open Riemann surface $R$ and every closed subset
$E\subset R,$ there is closed subset $F\subset E,$ which approximates
$E$ extremely well, such that every function holomorphic on $F$
can be approximated much better than uniformly by functions holomorphic
on $R$.


309  A Congruence Modulo Four for Real Schubert Calculus with Isotropic Flags Hein, Nickolas; Sottile, Frank; Zelenko, Igor
We previously obtained a congruence modulo four for the number
of real solutions to many Schubert problems on
a square Grassmannian given by osculating flags.
Here, we consider Schubert problems given by more general isotropic
flags, and prove this
congruence modulo four for the largest class of Schubert problems
that could be expected to exhibit this
congruence.


319  The Weakly Nilpotent Graph of a Commutative Ring Khojasteh, Sohiela; Nikmehr, Mohammad Javad
Let $R$ be a commutative ring with nonzero identity. In this
paper, we introduced the weakly nilpotent graph of a commutative
ring. The weakly nilpotent graph of $R$ is denoted by $\Gamma_w(R)$
is a graph with the vertex set $R^{*}$ and two vertices $x$ and
$y$ are adjacent if and only if $xy\in N(R)^{*}$, where $R^{*}=R\setminus\{0\}$
and $N(R)^{*}$ is the set of all nonzero nilpotent elements
of $R$. In this article, we determine the diameter of weakly
nilpotent graph of an Artinian ring. We prove that if $\Gamma_w(R)$
is a forest, then $\Gamma_w(R)$ is a union of a star and some
isolated vertices. We study the clique number, the chromatic
number and the independence number of $\Gamma_w(R)$. Among other
results, we show that for an Artinian ring $R$, $\Gamma_w(R)$
is not a disjoint union of cycles or a unicyclic graph. For Artinan
ring, we determine $\operatorname{diam}(\overline{\Gamma_w(R)})$. Finally, we
characterize all commutative rings $R$ for which $\overline{\Gamma_w(R)}$
is a cycle, where $\overline{\Gamma_w(R)}$ is the complement
of the weakly nilpotent graph of $R$.


329  Nonvanishing of Central Values of $L$functions of Newforms in $S_2 (\Gamma_0 (dp^2))$ Twisted by Quadratic Characters Le Fourn, Samuel
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.


350  Isometry on Linear $n$Gquasi Normed Spaces Ma, Yumei
This paper generalizes the Aleksandrov problem: the MazurUlam
theorem on $n$Gquasi normed spaces. It proves that a one$n$distance
preserving mapping is an $n$isometry if and only if it has the
zero$n$Gquasi preserving property, and two kinds of $n$isometries
on $n$Gquasi normed space are equivalent; we generalize the
Benz theorem to nnormed spaces with no restrictions on the dimension
of spaces.


364  On the Roughness of Quasinilpotency Property of One–parameter Semigroups Preda, Ciprian
Let $\mathbf{S}:=\{S(t)\}_{t\geq0}$ be a C$_0$semigroup of quasinilpotent
operators
(i.e. $\sigma(S(t))=\{0\}$ for each $t\gt 0$).
In the dynamical systems theory the above quasinilpotency property
is equivalent
to a very strong concept of stability for the solutions of autonomous
systems.
This concept is frequently called superstability and weakens
the classical finite time extinction property
(roughly speaking, disappearing solutions).
We show that under some assumptions, the quasinilpotency, or
equivalently, the superstability property
of a C$_0$semigroup is preserved under the perturbations of
its infinitesimal generator.


372  Coaxer Lattices Rao, M. Sambasiva
The notion of coaxers is introduced in a pseudocomplemented
distributive lattice. Boolean algebras are characterized in terms
of coaxer ideals and congruences. The concept of coaxer lattices
is introduced in pseudocomplemented distributive lattices and
characterized in terms of coaxer ideals and maximal ideals. Finally,
the coaxer lattices are also characterized in topological terms.


381  The Bifurcation Diagram of Cubic Polynomial Vector Fields on $\mathbb{C}\mathbb{P}^1$ Rousseau, C.
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.


402  Invariant Means on a Class of von Neumann Algebras Related to Ultraspherical Hypergroups II Shravan Kumar, N.
Let $K$ be an ultraspherical hypergroup associated to a locally
compact group $G$ and a spherical projector $\pi$ and let $VN(K)$
denote the dual of the Fourier algebra $A(K)$ corresponding to
$K.$ In this note, we show that the set of invariant means on
$VN(K)$ is singleton if and only if $K$ is discrete. Here $K$
need not be second countable. We also study invariant means on
the dual of the Fourier algebra $A_0(K),$ the closure of $A(K)$
in the $cb$multiplier norm. Finally, we consider generalized
translations and generalized invariant means.


411  On Gibbs Measures and Spectra of Ruelle Transfer Operators Stoyanov, Luchezar
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 Hölder
constant of the function generating the operator appears only
polynomially, not exponentially as
in previous known estimates.


422  New Superquadratic Conditions for Asymptotically Periodic Schrödinger Equations Tang, Xianhua
This paper is dedicated to studying the
semilinear Schrödinger 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.


436  Globally Asymptotic Stability of a Delayed IntegroDifferential Equation with Nonlocal Diffusion Weng, Peixuan; Liu, Li
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.


449  Character Density in Central Subalgebras of Compact Quantum Groups Alaghmandan, Mahmood; Crann, Jason
We investigate quantum group generalizations
of various density results from Fourier analysis on compact groups.
In particular, we establish the density of characters in the
space of fixed points of the conjugation action on $L^2(\mathbb{G})$, and
use this result to show the weak* density and norm density of
characters in $ZL^\infty(\mathbb{G})$ and $ZC(\mathbb{G})$, respectively. As a corollary,
we partially answer an open question of Woronowicz.
At the level of $L^1(\mathbb{G})$, we show that the center
$\mathcal{Z}(L^1(\mathbb{G}))$
is precisely the closed linear span of the quantum characters
for a large class of compact quantum groups, including arbitrary
compact Kac algebras. In the latter setting, we show, in addition,
that $\mathcal{Z}(L^1(\mathbb{G}))$ is a completely complemented
$\mathcal{Z}(L^1(\mathbb{G}))$submodule
of $L^1(\mathbb{G})$.


462  Functions Universal for All Translation Operators in Several Complex Variables Bayart, Frédéric; Gauthier, Paul M
We prove the existence of
a (in fact many)
holomorphic function $f$ in $\mathbb{C}^d$ such that, for any $a\neq
0$, its translations $f(\cdot+na)$ are dense in $H(\mathbb{C}^d)$.


470  MaurerCartan Elements in the Lie Models of Finite Simplicial Complexes Buijs, Urtzi; Félix, Yves; Murillo, Aniceto; Tanré, Daniel
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.


478  Springer's Weyl Group Representation via Localization Carrell, Jim; Kaveh, Kiumars
Let $G$ denote a reductive algebraic group over
$\mathbb{C}$
and $x$ a nilpotent element of its Lie algebra $\mathfrak{g}$. The Springer
variety $\mathcal{B}_x$
is the closed subvariety of the flag variety $\mathcal{B}$ of $G$ parameterizing
the
Borel subalgebras of $\mathfrak{g}$ containing $x$. It has the remarkable
property that
the Weyl group $W$ of $G$ admits a representation on the cohomology
of $\mathcal{B}_x$
even though $W$ rarely acts on $\mathcal{B}_x$ itself. Wellknown constructions
of this action
due to Springer et al use technical machinery from algebraic
geometry.
The purpose of this note is to describe an elementary approach
that gives this action
when $x$ is what we call parabolicsurjective. The idea is to
use localization to construct an action of $W$ on
the equivariant cohomology algebra $H_S^*(\mathcal{B}_x)$, where $S$ is a certain algebraic
subtorus of $G$.
This action descends to $H^*(\mathcal{B}_x)$ via the forgetful map and
gives the desired representation. The parabolicsurjective case
includes all nilpotents of type $A$ and,
more generally, all nilpotents for which it is known that $W$
acts on
$H_S^*(\mathcal{B}_x)$ for some torus $S$.
Our result is deduced from a general theorem describing
when a group action on the cohomology of the fixed point set of a
torus action
on a space lifts to the full cohomology algebra of the space.


484  A Note on Lawton's Theorem Dobrowolski, Edward
We prove Lawton's conjecture about the upper bound on the measure
of the set on the unit circle on which a complex polynomial with
a bounded number of coefficients takes small values. Namely,
we prove that Lawton's bound holds for polynomials that are not
necessarily monic. We also provide an analogous bound for polynomials
in several variables. Finally, we investigate the dependence
of the bound on the multiplicity of zeros for polynomials in
one variable.


490  A RiemannHurwitz Theorem for the Algebraic Euler Characteristic Fiori, Andrew
We prove an analogue of the RiemannHurwitz theorem for computing
Euler characteristics of pullbacks of coherent sheaves through
finite maps of smooth projective varieties in arbitrary dimensions,
subject only to the condition that the irreducible components
of the branch and ramification locus have simple normal crossings.


510  Convexnormal (Pairs of) Polytopes Haase, Christian; Hofmann, Jan
In 2012 Gubeladze (Adv. Math. 2012)
introduced the notion of $k$convexnormal polytopes to show
that
integral polytopes all of whose edges are longer than $4d(d+1)$
have
the integer decomposition property.
In the first part of this paper we show that for lattice polytopes
there is no difference between $k$ and $(k+1)$convexnormality
(for
$k\geq 3 $) and improve the bound to $2d(d+1)$. In the second
part we
extend the definition to pairs of polytopes. Given two rational
polytopes $P$ and $Q$, where the normal fan of $P$ is a refinement
of
the normal fan of $Q$.
If every edge $e_P$ of $P$ is at least $d$ times as long as the
corresponding face (edge or vertex) $e_Q$ of $Q$, then $(P+Q)\cap
\mathbb{Z}^d
= (P\cap \mathbb{Z}^d ) + (Q \cap \mathbb{Z}^d)$.


522  On the Singular Sheaves in the Fine Simpson Moduli Spaces of $1$dimensional Sheaves Iena, Oleksandr; Leytem, Alain
In the Simpson moduli space $M$ of semistable sheaves with
Hilbert polynomial $dm1$ on a projective plane we study the
closed subvariety $M'$ of sheaves that are not locally free on
their support. We show that for $d\ge 4$ it is a singular subvariety
of codimension $2$ in $M$. The blow up of $M$ along $M'$ is interpreted
as a (partial) modification of $M\setminus M'$ by line bundles
(on support).


536  The Gradient of a Solution of the Poisson Equation in the Unit Ball and Related Operators Kalaj, David; Vujadinović, Djordjije
In this paper we determine the $L^1\to L^1$ and $L^{\infty}\to
L^\infty$ norms of an integral operator $\mathcal{N}$ related
to the gradient of the solution of Poisson equation in the unit
ball with vanishing boundary data in sense of distributions.


546  On Polarized K3 Surfaces of Genus 33 Karzhemanov, Ilya
We prove that the moduli space of smooth primitively polarized
$\mathrm{K3}$ surfaces of genus $33$ is unirational.


561  Nuij Type Pencils of Hyperbolic Polynomials Kurdyka, Krzysztof; Paunescu, Laurentiu
Nuij's theorem states that if a polynomial $p\in \mathbb{R}[z]$ is hyperbolic
(i.e. has only real roots) then $p+sp'$ is also hyperbolic for
any
$s\in \mathbb{R}$. We study other perturbations of hyperbolic polynomials
of the form $p_a(z,s): =p(z) +\sum_{k=1}^d a_ks^kp^{(k)}(z)$.
We give a full characterization of those $a= (a_1, \dots,
a_d) \in \mathbb{R}^d$ for which $p_a(z,s)$ is a pencil of hyperbolic
polynomials.
We give also a full characterization of those $a= (a_1, \dots,
a_d) \in \mathbb{R}^d$ for which the associated families $p_a(z,s)$
admit universal determinantal representations. In fact we show
that all these sequences come from special symmetric Toeplitz
matrices.


571  Weak Factorizations of the Hardy space $H^1(\mathbb{R}^n)$ in terms of Multilinear Riesz Transforms Li, Ji; Wick, Brett D.
This paper provides a constructive proof of the weak factorization
of the classical Hardy space $H^1(\mathbb{R}^n)$ in terms of
multilinear Riesz transforms. As a direct application, we obtain
a new proof of the characterization of ${\rm BMO}(\mathbb{R}^n)$
(the dual of $H^1(\mathbb{R}^n)$) via commutators of the multilinear
Riesz transforms.


586  Endpoint Regularity of Multisublinear Fractional Maximal Functions Liu, Feng; Wu, Huoxiong
In this paper we investigate
the endpoint regularity properties of the multisublinear
fractional maximal operators, which include the multisublinear
HardyLittlewood maximal operator. We obtain some new bounds
for the derivative of the onedimensional multisublinear
fractional maximal operators acting on vectorvalued function
$\vec{f}=(f_1,\dots,f_m)$ with all $f_j$ being $BV$functions.


604  Stackings and the $W$cycles Conjecture Louder, Larsen; Wilton, Henry
We prove Wise's $W$cycles conjecture: Consider a compact graph
$\Gamma'$ immersing into another graph $\Gamma$. For any immersed
cycle $\Lambda:S^1\to \Gamma$, we consider the map $\Lambda'$
from
the circular components $\mathbb{S}$ of the pullback to $\Gamma'$.
Unless
$\Lambda'$ is reducible, the degree of the covering map $\mathbb{S}\to
S^1$ is bounded above by minus the Euler characteristic of
$\Gamma'$. As a corollary, any finitely generated subgroup
of a
onerelator group has finitely generated Schur multiplier.


613  On the Dimension of the Locus of Determinantal Hypersurfaces Reichstein, Zinovy; Vistoli, Angelo
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 $(r1)n^2 + 1$.


631  Traceless Maps as the Singular Minimizers in the Multidimensional Calculus of Variations ShahrokhiDehkordi, M. S.
Let $\Omega \subset \mathbb{R}^n$ be a bounded Lipschitz domain and
consider
the energy functional
\begin{equation*}
{\mathcal F}[u, \Omega] := \int_{\Omega} {\rm F}(\nabla {\bf u}(\bf x))\, d{\bf x},
\end{equation*}
over the space of $W^{1,2}(\Omega, \mathbb{R}^m)$ where the integrand
${\rm F}: \mathbb M_{m\times n}\to \mathbb{R}$ is a smooth uniformly
convex function
with bounded second derivatives. In this paper we address the
question of
regularity for solutions of the corresponding system of
EulerLagrange equations.
In particular we introduce a class of singular maps referred
to as traceless and
examine them as a new counterexample to the regularity of minimizers
of the energy
functional $\mathcal F[\cdot,\Omega]$ using a method based on
null Lagrangians.


641  Mixed $f$divergence for Multiple Pairs of Measures Werner, Elisabeth; Ye, Deping
In this paper, the concept of the classical $f$divergence for
a pair of measures is extended to the mixed $f$divergence for
multiple pairs of measures. The mixed $f$divergence provides
a way to measure the difference between multiple pairs of (probability)
measures. Properties for the mixed $f$divergence are established,
such as permutation invariance and symmetry in distributions.
An
AlexandrovFenchel type inequality and an isoperimetric inequality
for the
mixed $f$divergence are proved.


655  Characterizations of BesovType and TriebelLizorkinType Spaces via Averages on Balls Zhuo, Ciqiang; Sickel, Winfried; Yang, Dachun; Yuan, Wen
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.

