
« 2012 (v55)  2014 (v57) » 
Page 


3  Semiclassical Limits of Eigenfunctions on Flat $n$Dimensional Tori Aïssiou, Tayeb
We provide a proof of a conjecture by Jakobson, Nadirashvili, and
Toth stating
that on an $n$dimensional flat torus $\mathbb T^{n}$, and the Fourier transform
of squares of the eigenfunctions $\varphi_\lambda^2$ of the Laplacian have
uniform $l^n$ bounds that do not depend on the eigenvalue $\lambda$. The proof
is a generalization of an argument by Jakobson, et al. for the
lower dimensional cases. These results imply uniform bounds for semiclassical
limits on $\mathbb T^{n+2}$. We also prove a geometric lemma that bounds the number of
codimensionone simplices satisfying a certain restriction on an
$n$dimensional sphere $S^n(\lambda)$ of radius $\sqrt{\lambda}$, and we use it in
the proof.


13  Ordering the Representations of $S_n$ Using the Interchange Process Alon, Gil; Kozma, Gady
Inspired by Aldous' conjecture for
the spectral gap of the interchange process and its recent
resolution by Caputo, Liggett, and Richthammer, we define
an associated order $\prec$ on the irreducible representations of $S_n$. Aldous'
conjecture is equivalent to certain representations being comparable
in this order, and hence determining the ``Aldous order'' completely is a
generalized question. We show a few additional entries for this order.


31  Derivations and Valuation Rings Ayuso, Fortuny P.
A complete characterization of valuation rings closed for a
holomorphic derivation is given, following an idea of Seidenberg,
in dimension $2$.


39  Comparison Theorem for Conjugate Points of a Fourthorder Linear Differential Equation Ben Amara, Jamel
In 1961, J. Barrett showed that if the first conjugate point
$\eta_1(a)$ exists for the differential equation $(r(x)y'')''=
p(x)y,$ where $r(x)\gt 0$ and $p(x)\gt 0$, then so does the first
systemsconjugate point $\widehat\eta_1(a)$. The aim of this note is to
extend this result to the general equation with middle term
$(q(x)y')'$ without further restriction on $q(x)$, other than
continuity.


44  Polystable Parabolic Principal $G$Bundles and HermitianEinstein Connections Biswas, Indranil; Dey, Arijit
We show that there
is a bijective correspondence between the polystable parabolic
principal $G$bundles and solutions of the HermitianEinstein
equation.


55  Cliquishness and Quasicontinuity of TwoVariable Maps Bouziad, A.
We study the existence of continuity points for mappings
$f\colon X\times Y\to Z$ whose $x$sections $Y\ni y\to f(x,y)\in Z$ are
fragmentable and $y$sections $X\ni x\to f(x,y)\in Z$ are
quasicontinuous, where $X$ is a Baire space and $Z$
is a metric space. For the factor $Y$, we consider two
infinite ``pointpicking'' games $G_1(y)$ and $G_2(y)$ defined respectively
for each $y\in Y$ as follows: in the $n$th
inning, Player I gives a dense set $D_n\subset Y$, respectively, a dense open set $D_n\subset Y$. Then
Player II picks a point $y_n\in D_n$;
II wins if $y$ is in the closure of ${\{y_n:n\in\mathbb N\}}$, otherwise
I wins. It is shown that
(i) $f$ is
cliquish
if II has a winning strategy in $G_1(y)$ for every $y\in Y$, and (ii) $
f$ is quasicontinuous if
the $x$sections of $f$ are continuous and the set of $y\in Y$
such that II has a winning strategy in $G_2(y)$ is dense in $Y$. Item (i) extends substantially
a result of Debs and item (ii) indicates that
the problem of Talagrand on separately continuous maps has a positive answer for a wide
class of ``small'' compact spaces.


65  The Uncomplemented Subspace $\mathbf K(X,Y) $ Ghenciu, Ioana
A vector measure result is used to study the complementation of the
space $K(X,Y)$ of compact operators in the spaces $W(X,Y)$ of weakly
compact operators, $CC(X,Y)$ of completely continuous operators, and
$U(X,Y)$ of unconditionally converging operators.
Results of Kalton and Emmanuele concerning the complementation of
$K(X,Y)$ in $L(X,Y)$ and in $W(X,Y)$ are generalized. The containment
of $c_0$ and $\ell_\infty$ in spaces of operators is also studied.


70  An Asymptotic Bound on the Composition Number of Integer Sums of Squares Formulas Hrubeš, P.; Wigderson, A.; Yehudayoff, A.
Let $\sigma_{\mathbb Z}(k)$ be the smallest $n$ such that there exists an
identity
\[
(x_1^2 + x_2^2 + \cdots + x_k^2) \cdot (y_1^2 + y_2^2 + \cdots + y_k^2)
= f_1^2 + f_2^2 + \cdots + f_n^2,
\]
with $f_1,\dots,f_n$ being polynomials with integer coefficients in
the variables $x_1,\dots,x_k$ and $y_1,\dots,y_k$. We prove that
$\sigma_{\mathbb Z}(k) \geq \Omega(k^{6/5})$.


80  Three Fixed Point Theorems: Periodic Solutions of a Volterra Type Integral Equation with Infinite Heredity Islam, Muhammad N.
In this paper we study the existence of periodic solutions of a Volterra type integral equation with infinite heredity. Banach fixed point theorem, Krasnosel'skii's fixed point theorem, and a combination of Krasnosel'skii's
and Schaefer's fixed point theorems are employed in the analysis.
The combination theorem of Krasnosel'skii and Schaefer requires an a priori bound on all solutions.
We employ Liapunov's direct method to obtain such an a priori bound.
In the process, we compare these theorems in terms of assumptions and outcomes.


92  On Perturbations of Continuous Maps Jacob, Benoît
We give sufficient conditions for the following problem: given a
topological space $X$, a metric space $Y$, a subspace $Z$ of $Y$, and
a continuous map $f$ from $X$ to $Y$, is it possible, by applying to
$f$ an arbitrarily small perturbation, to ensure that $f(X)$ does not
meet $Z$? We also give a relative variant: if $f(X')$ does not meet
$Z$ for a certain subset $X'\subset X$, then we may keep $f$ unchanged
on $X'$. We also develop a variant for continuous sections of
fibrations and discuss some applications to matrix perturbation
theory.


102  Eigenvalue Approach to Even Order System Periodic Boundary Value Problems Kong, Qingkai; Wang, Min
We study an even order system boundary value problem with
periodic boundary conditions. By establishing
the existence of a positive eigenvalue of an associated linear system
SturmLiouville problem, we obtain new conditions for the boundary
value problem to have a positive solution. Our major tools are the
KreinRutman theorem for linear spectra and the fixed point index theory
for compact operators.


116  Central Extensions of Loop Groups and Obstruction to PreQuantization Krepski, Derek
An explicit construction of a prequantum line bundle for the moduli
space of flat $G$bundles over a Riemann surface is given, where $G$
is any nonsimply connected compact simple Lie group. This work helps
to explain a curious coincidence previously observed between
Toledano Laredo's work classifying central extensions of loop groups
$LG$ and the author's previous work on the obstruction to
prequantization of the moduli space of flat $G$bundles.


127  Evolution of Eigenvalues along Rescaled Ricci Flow Li, Junfang
In this paper, we discuss monotonicity formulae of various entropy functionals under various
rescaled versions of Ricci flow. As an application, we prove that the lowest eigenvalue
of a family of geometric operators $4\Delta + kR$ is monotonic along the
normalized Ricci flow for all $k\ge 1$ provided the initial manifold has
nonpositive total scalar curvature.


136  On Constructing Ergodic Hyperfinite Equivalence Relations of NonProduct Type Munteanu, RaduBogdan
Product type equivalence relations are hyperfinite measured
equivalence relations, which, up to orbit equivalence, are generated
by product type odometer actions. We give a concrete example of a
hyperfinite equivalence relation of nonproduct type, which is the
tail equivalence on a Bratteli diagram.
In order to show that the equivalence relation constructed is not of
product type we will use a criterion called property A. This
property, introduced by Krieger for nonsingular transformations, is
defined directly for hyperfinite equivalence relations in this paper.


148  On the Gras Conjecture for Imaginary Quadratic Fields Oukhaba, Hassan; Viguié, Stéphane
In this paper we extend K. Rubin's methods to prove the Gras conjecture
for abelian extensions of a given imaginary quadratic field $k$ and
prime numbers $p$ that divide the number of roots of unity in $k$.


161  An Extension of the Dirichlet Density for Sets of Gaussian Integers Rêgo, L. C.; Cintra, R. J.
Several measures for the density of sets of integers have been proposed,
such as the asymptotic density, the Schnirelmann density, and the Dirichlet density. There has been some work in the literature on extending some of these concepts of density to higher dimensional sets of integers. In this work, we propose an extension of the Dirichlet density for sets of Gaussian integers and
investigate some of its properties.


173  Semiinvariant Submersions from Almost Hermitian Manifolds Sahin, Bayram
We introduce semiinvariant Riemannian submersions from almost
Hermitian manifolds onto Riemannian manifolds. We give examples,
investigate the geometry of foliations that arise from the
definition of a Riemannian submersion, and find necessary sufficient
conditions for total manifold to be a locally product Riemannian
manifold. We also find necessary and sufficient conditions for a
semiinvariant submersion to be totally geodesic. Moreover, we
obtain a classification for semiinvariant submersions with totally
umbilical fibers and show that such submersions put some
restrictions on total manifolds.


184  On Some NonRiemannian Quantities in Finsler Geometry Shen, Zhongmin
In this paper we study several nonRiemannian quantities in Finsler
geometry. These nonRiemannian quantities play an important role in
understanding the geometric properties of Finsler metrics. In
particular, we study a new nonRiemannian quantity defined by the
Scurvature. We show some relationships among the flag curvature,
the Scurvature, and the new nonRiemannian quantity.


194  On the Smallest and Largest Zeros of MüntzLegendre Polynomials Stefánsson, Úlfar F.
MüntzLegendre
polynomials $L_n(\Lambda;x)$ associated with a
sequence $\Lambda=\{\lambda_k\}$ are obtained by orthogonalizing the
system $(x^{\lambda_0}, x^{\lambda_1}, x^{\lambda_2}, \dots)$ in
$L_2[0,1]$ with respect to the Legendre weight. If the $\lambda_k$'s
are distinct, it is well known that $L_n(\Lambda;x)$ has exactly $n$
zeros $l_{n,n}\lt l_{n1,n}\lt \cdots \lt l_{2,n}\lt l_{1,n}$ on $(0,1)$.
First we prove the following global bound for the smallest zero,
$$
\exp\biggl(4\sum_{j=0}^n \frac{1}{2\lambda_j+1}\biggr) \lt l_{n,n}.
$$
An important consequence is that if the associated Müntz space is
nondense in $L_2[0,1]$, then
$$
\inf_{n}x_{n,n}\geq
\exp\biggl({4\sum_{j=0}^{\infty} \frac{1}{2\lambda_j+1}}\biggr)\gt 0,
$$
so
the elements $L_n(\Lambda;x)$ have no zeros close to 0.


203  Productively Lindelöf Spaces May All Be $D$ Tall, Franklin D.
We give easy proofs that (a) the Continuum Hypothesis implies that if
the product of $X$ with every Lindelöf space is Lindelöf, then $X$ is
a $D$space, and (b) Borel's Conjecture implies every Rothberger space
is Hurewicz.


213  A Locally Compact Non Divisible Abelian Group Whose Character Group Is Torsion Free and Divisible Tausk, Daniel V.
It was claimed by Halmos in 1944 that if $G$ is a
Hausdorff locally compact topological abelian
group and if the character group of $G$ is torsion
free, then $G$ is divisible.
We prove that such a claim is false by
presenting a family of counterexamples.
While other counterexamples are known,
we also present a family of stronger counterexamples,
showing that even if one assumes that the character
group of $G$ is both torsion free and divisible,
it does not follow that $G$ is divisible.


218  Functional Equations and Fourier Analysis Yang, Dilian
By exploring the relations among functional equations, harmonic analysis and representation theory,
we give a unified and very accessible approach to solve three important functional equations 
the d'Alembert equation, the Wilson equation, and the d'Alembert long equation 
on compact groups.


225  On the Notion of Visibility of Torsors Agashe, Amod
Let $J$ be an abelian variety and
$A$ be an abelian subvariety of $J$, both defined over $\mathbf{Q}$.
Let $x$ be an element of $H^1(\mathbf{Q},A)$.
Then there are at least two definitions of $x$ being visible in $J$:
one asks that the torsor corresponding to $x$ be isomorphic over $\mathbf{Q}$
to a subvariety of $J$, and the other asks that $x$ be in the kernel
of the natural map $H^1(\mathbf{Q},A) \to H^1(\mathbf{Q},J)$. In this article, we
clarify the relation between the two definitions.


229  Cesàro Operators on the Hardy Spaces of the HalfPlane Arvanitidis, Athanasios G.; Siskakis, Aristomenis G.
In this article we study the Cesàro
operator
$$
\mathcal{C}(f)(z)=\frac{1}{z}\int_{0}^{z}f(\zeta)\,d\zeta,
$$
and its companion operator $\mathcal{T}$ on Hardy spaces of the
upper half plane. We identify $\mathcal{C}$ and $\mathcal{T}$ as
resolvents for appropriate semigroups of composition operators and we
find the norm and the spectrum in each case. The relation of
$\mathcal{C}$ and $\mathcal{T}$ with the corresponding Ces\`{a}ro
operators on Lebesgue spaces $L^p(\mathbb R)$ of the boundary line is also
discussed.


241  Versions of Schwarz's Lemma for Condenser Capacity and Inner Radius Betsakos, Dimitrios; Pouliasis, Stamatis
We prove variants of Schwarz's lemma involving monotonicity
properties of condenser capacity and inner radius. Also, we
examine when a similar monotonicity property holds for the
hyperbolic metric.


251  Sign Changes of the Liouville Function on Quadratics Borwein, Peter; Choi, Stephen K. K.; Ganguli, Himadri
Let $\lambda (n)$ denote the Liouville function. Complementary to the prime number theorem, Chowla conjectured
that
\begin{equation*}
\label{a.1}
\sum_{n\le x} \lambda (f(n)) =o(x)\tag{$*$}
\end{equation*}
for any polynomial $f(x)$ with integer coefficients which is not of
form $bg(x)^2$.


258  The Smallest Pisot Element in the Field of Formal Power Series Over a Finite Field Chandoul, A.; Jellali, M.; Mkaouar, M.
Dufresnoy and Pisot characterized the smallest
Pisot number of degree $n \geq 3$ by giving explicitly its minimal
polynomial. In this paper, we translate Dufresnoy and Pisot's
result to the Laurent series case.


265  Embedding Distributions of Generalized Fan Graphs Chen, Yichao; Mansour, Toufik; Zou, Qian
Total embedding distributions have been known for a few classes of graphs.
Chen, Gross, and Rieper
computed it for necklaces, closeend ladders and cobblestone
paths. Kwak and Shim computed it for bouquets of circles and
dipoles. In this paper, a splitting theorem is generalized
and the embedding distributions of
generalized fan graphs are obtained.


272  On Super Weakly Compact Convex Sets and Representation of the Dual of the Normed Semigroup They Generate Cheng, Lixin; Luo, Zhenghua; Zhou, Yu
In this note, we first give a characterization of super weakly
compact convex sets of a Banach space $X$:
a closed bounded convex set $K\subset X$ is
super weakly compact if and only if there exists a $w^*$ lower
semicontinuous seminorm $p$ with $p\geq\sigma_K\equiv\sup_{x\in
K}\langle\,\cdot\,,x\rangle$ such that $p^2$ is uniformly Fréchet
differentiable on each bounded set of $X^*$. Then we present a
representation theorem for the dual of the semigroup $\textrm{swcc}(X)$
consisting of all the nonempty super weakly compact convex sets of the
space $X$.


283  Transcendental Solutions of a Class of Minimal Functional Equations Coons, Michael
We prove a result concerning power series
$f(z)\in\mathbb{C}[\mkern3mu[z]\mkern3mu]$
satisfying a functional equation of the form
$$
f(z^d)=\sum_{k=1}^n
\frac{A_k(z)}{B_k(z)}f(z)^k,
$$
where $A_k(z),B_k(z)\in
\mathbb{C}[z]$. In particular, we show that if $f(z)$ satisfies a
minimal functional equation of the above form with $n\geqslant 2$,
then $f(z)$ is necessarily transcendental. Towards a more complete
classification, the case $n=1$ is also considered.


292  Quasisymmetrically Minimal Moran Sets Dai, MeiFeng
M. Hu and S. Wen considered quasisymmetrically minimal uniform Cantor
sets of Hausdorff dimension $1$, where at the $k$th set one removes
from each interval $I$ a certain number $n_{k}$ of open subintervals
of length $c_{k}I$, leaving $(n_{k}+1)$ closed subintervals of
equal length. Quasisymmetrically Moran sets of Hausdorff dimension $1$
considered in the paper are more general than uniform Cantor sets in
that neither the open subintervals nor the closed subintervals are
required to be of equal length.


306  Real Hypersurfaces in Complex Projective Space Whose Structure Jacobi Operator is Lie $\mathbb{D}$parallel Pérez, Juan de Dios; Suh, Young Jin
We prove the nonexistence of real hypersurfaces in complex projective
space whose structure Jacobi operator is Lie $\mathbb{D}$parallel and
satisfies a further condition.


317  A Note on Conjectures of F. Galvin and R. Rado Dorais, François G.
In 1968, Galvin conjectured that an uncountable poset $P$ is the
union of countably many chains if and only if this is true for every
subposet $Q \subseteq P$ with size $\aleph_1$. In 1981, Rado
formulated a similar conjecture that an uncountable interval graph $G$ is countably
chromatic if and only if this is true for every induced subgraph $H
\subseteq G$ with size $\aleph_1$. Todorčević has shown
that Rado's Conjecture is consistent relative to the existence of a
supercompact cardinal, while the consistency of Galvin's Conjecture
remains open. In this paper, we survey and collect a variety of
results related to these two conjectures. We also show that the
extension of Rado's conjecture to the class of all chordal graphs is
relatively consistent with the existence of a supercompact cardinal.


326  Restricting Fourier Transforms of Measures to Curves in $\mathbb R^2$ Erdoğan, M. Burak; Oberlin, Daniel M.
We establish estimates for restrictions to certain curves in $\mathbb R^2$ of the Fourier transforms
of some fractal measures.


337  Certain Properties of $K_0$monoids Preserved by Tracial Approximation Fan, Qingzhai
We show that the following $K_0$monoid properties of $C^*$algebras
in the class $\Omega$ are inherited by simple unital $C^*$algebras in
the class $TA\Omega$: (1) weak comparability, (2) strictly
unperforated, (3) strictly cancellative.


344  Involutions and Anticommutativity in Group Rings Goodaire, Edgar G.; Milies, César Polcino
Let $g\mapsto g^*$ denote an involution on a
group $G$. For any (commutative, associative) ring
$R$ (with $1$), $*$ extends linearly to an involution
of the group ring $RG$. An element $\alpha\in RG$
is symmetric if $\alpha^*=\alpha$ and
skewsymmetric if $\alpha^*=\alpha$.
The skewsymmetric elements are closed under
the Lie bracket, $[\alpha,\beta]=\alpha\beta\beta\alpha$.
In this paper, we investigate when this set is also closed
under the ring product in $RG$.
The symmetric elements are closed under the Jordan
product, $\alpha\circ\beta=\alpha\beta+\beta\alpha$.
Here, we determine when this product is trivial.
These two problems
are analogues of problems about the skewsymmetric and
symmetric elements in group rings that have received a
lot of attention.


354  The Sizes of Rearrangements of Cantor Sets Hare, Kathryn E.; Mendivil, Franklin; Zuberman, Leandro
A linear Cantor set $C$ with zero Lebesgue measure is associated with
the countable collection of the bounded complementary open intervals. A
rearrangment of $C$ has the same lengths of its complementary
intervals, but with different locations. We study the Hausdorff and packing
$h$measures and dimensional properties of the set of all rearrangments of
some given $C$ for general dimension functions $h$. For each set of
complementary lengths, we construct a Cantor set rearrangement which has the
maximal Hausdorff and the minimal packing $h$premeasure, up to a constant.
We also show that if the packing measure of this Cantor set is positive,
then there is a rearrangement which has infinite packing measure.


366  Multiple Solutions for Nonlinear Periodic Problems Kyritsi, Sophia Th.; Papageorgiou, Nikolaos S.
We consider a nonlinear periodic problem driven by a
nonlinear nonhomogeneous differential operator and a
Carathéodory reaction term $f(t,x)$ that exhibits a
$(p1)$superlinear growth in $x \in \mathbb{R}$
near $\pm\infty$ and near zero.
A special case of the differential operator is the scalar
$p$Laplacian. Using a combination of variational methods based on
the critical point theory with Morse theory (critical groups), we
show that the problem has three nontrivial solutions, two of which
have constant sign (one positive, the other negative).


378  Sharp Threshold of the GrossPitaevskii Equation with Trapped Dipolar Quantum Gases Ma, Li; Wang, Jing
In this paper, we consider the GrossPitaevskii equation for the
trapped dipolar quantum gases. We obtain the sharp criterion for the
global existence and finite time blow up in the unstable regime by
constructing a variational problem and the socalled invariant
manifold of the evolution flow.


388  Application of Measure of Noncompactness to Infinite Systems of Differential Equations Mursaleen, M.
In this paper we determine the Hausdorff measure of noncompactness on
the sequence space $n(\phi)$ of W. L. C. Sargent.
Further we apply
the technique of measures of noncompactness to the theory of infinite
systems of differential equations in the Banach sequence spaces
$n(\phi)$ and $m(\phi)$. Our aim is to present some existence results
for infinite systems of differential equations formulated with the help
of measures of noncompactness.


395  Coessential Abelianization Morphisms in the Category of Groups Oancea, D.
An epimorphism $\phi\colon G\to H$ of groups, where $G$ has rank $n$, is called
coessential if every (ordered) generating $n$tuple of $H$ can be
lifted along $\phi$ to a generating $n$tuple for $G$. We discuss this
property in the context of the category of groups, and establish a criterion
for such a group $G$ to have the property that its abelianization
epimorphism $G\to G/[G,G]$, where $[G,G]$ is the commutator subgroup, is
coessential. We give an example of a family of 2generator groups whose
abelianization epimorphism is not coessential.
This family also provides counterexamples to the generalized AndrewsCurtis conjecture.


400  A Factorization Theorem for Multiplier Algebras of Reproducing Kernel Hilbert Spaces Prunaru, Bebe
Let $(X,\mathcal B,\mu)$ be a $\sigma$finite
measure space and let $H\subset L^2(X,\mu)$
be a separable reproducing kernel Hilbert
space on $X$. We show that the multiplier
algebra of $H$ has property $(A_1(1))$.


407  On Domination in ZeroDivisor Graphs Rad, Nader Jafari; Jafari, Sayyed Heidar; Mojdeh, Doost Ali
We first determine the domination number for the zerodivisor
graph of the product of two commutative rings with $1$. We then
calculate the domination number for the zerodivisor graph of any
commutative artinian ring. Finally, we extend some of the results
to noncommutative rings in which an element is a left
zerodivisor if and only if it is a right zerodivisor.


412  Structure in Sets with Logarithmic Doubling Sanders, T.
Suppose that $G$ is an abelian group, $A \subset G$ is finite with $A+A \leq KA$ and $\eta \in (0,1]$ is a parameter.
Our main result is that there is a set $\mathcal{L}$ such that
\begin{equation*}
A \cap \operatorname{Span}(\mathcal{L}) \geq K^{O_\eta(1)}A \quad\text{and}\quad \mathcal{L} = O(K^\eta\log A).
\end{equation*}
We include an application of this result to a generalisation of the RothMeshulam theorem due to Liu and Spencer.


424  Convergent Sequences in Discrete Groups Thom, Andreas
We prove that a finitely generated group contains a
sequence of nontrivial elements that converge to the identity in
every compact homomorphic image if and only if the group is not
virtually abelian. As a consequence of the methods used, we show that a finitely generated
group satisfies Chu duality if and only if it is virtually abelian.


434  Some Remarks on the Algebraic Sum of Ideals and Riesz Subspaces Wnuk, Witold
Following ideas used by Drewnowski and Wilansky we prove that if $I$
is an infinite dimensional and
infinite codimensional closed ideal in a complete metrizable locally
solid Riesz space and $I$ does
not contain any order copy of $\mathbb R^{\mathbb N}$ then there exists a
closed, separable, discrete Riesz subspace
$G$ such that the topology induced on $G$ is Lebesgue, $I \cap G =
\{0\}$, and $I + G$ is not closed.


442  Closed Left Ideal Decompositions of $U(G)$ Zelenyuk, Yevhen
Let $G$ be an infinite discrete group and let $\beta G$ be the
StoneČech compactification of $G$. We take the points of $ėta
G$ to be the ultrafilters on $G$, identifying the principal
ultrafilters with the points of $G$. The set $U(G)$ of uniform
ultrafilters on $G$ is a closed twosided ideal of $\beta G$. For
every $p\in U(G)$, define $I_p\subseteq\beta G$ by $I_p=\bigcap_{A\in
p}\operatorname{cl} (GU(A))$, where $U(A)=\{p\in U(G):A\in p\}$. We show
that if $G$ is a regular cardinal, then $\{I_p:p\in U(G)\}$ is the
finest decomposition of $U(G)$ into closed left ideals of $\beta G$
such that the corresponding quotient space of $U(G)$ is Hausdorff.


449  The $f$Chromatic Index of a Graph Whose $f$Core has Maximum Degree $2$ Akbari, S.; Chavooshi, M.; Ghanbari, M.; Zare, S.
Let $G$ be a graph. The minimum number of colors needed to color the edges of
$G$ is called the chromatic index of $G$ and is denoted by $\chi'(G)$.
It is wellknown that $\Delta(G) \leq \chi'(G) \leq \Delta(G)+1$, for any
graph $G$, where $\Delta(G)$ denotes the maximum degree of $G$. A graph $G$ is said to be
Class $1$ if $\chi'(G) = \Delta(G)$ and Class $2$ if
$\chi'(G) = \Delta(G) + 1$. Also, $G_\Delta$ is the induced subgraph on all vertices of degree $\Delta(G)$.
Let $f:V(G)\rightarrow \mathbb{N}$ be a function.
An $f$coloring of a graph $G$ is a coloring of the edges
of $E(G)$ such that each color appears at each vertex $v\in V(G)$ at
most $f (v)$ times. The minimum number of colors needed
to $f$color $G$ is called the $f$chromatic index of $G$ and
is denoted by $\chi'_{f}(G)$. It was shown that for every graph $G$, $\Delta_{f}(G)\le \chi'_{f}(G)\le \Delta_{f}(G)+1$, where $\Delta_{f}(G)=\max_{v\in V(G)} \big\lceil \frac{d_G(v)}{f(v)}\big\rceil$. A graph $G$ is said to be $f$Class $1$ if $\chi'_{f}(G)=\Delta_{f}(G)$, and $f$Class $2$, otherwise. Also, $G_{\Delta_f}$ is the induced subgraph of $G$ on $\{v\in V(G):\,\frac{d_G(v)}{f(v)}=\Delta_{f}(G)\}$.
Hilton and Zhao showed that if $G_{\Delta}$ has maximum degree two and $G$ is Class $2$, then $G$ is critical, $G_{\Delta}$ is a disjoint union of cycles and $\delta(G)=\Delta(G)1$, where $\delta(G)$ denotes the minimum degree of $G$, respectively. In this paper, we generalize this theorem to $f$coloring of graphs. Also, we determine the $f$chromatic index of a connected graph $G$ with $G_{\Delta_f}\le 4$.


459  On Certain Multivariable Subnormal Weighted Shifts and their Duals Athavale, Ameer; Patil, Pramod
To every subnormal $m$variable weighted shift $S$ (with bounded
positive weights) corresponds a positive Reinhardt measure $\mu$
supported on a compact Reinhardt subset of $\mathbb C^m$. We show that, for
$m \geq 2$, the dimensions of the $1$st cohomology vector spaces
associated with the Koszul complexes of $S$ and its dual ${\tilde S}$
are different if a certain radial function happens to be integrable
with respect to $\mu$ (which is indeed the case with many classical
examples). In particular, $S$ cannot in that case be similar to
${\tilde S}$. We next prove that, for $m \geq 2$, a Fredholm subnormal
$m$variable weighted shift $S$ cannot be similar to its dual.


466  Inclusion Relations for New Function Spaces on Riemann Surfaces Aulaskari, Rauno; Rättyä, Jouni
We introduce and study some new function spaces on Riemann
surfaces. For certain parameter values these spaces coincide with
the classical Dirichlet space, BMOA or the recently
defined $Q_p$ space. We establish inclusion relations that
generalize earlier known inclusions between the abovementioned
spaces.


477  Hypercyclic Abelian Groups of Affine Maps on $\mathbb{C}^{n}$ Ayadi, Adlene
We give a characterization of hypercyclic abelian group
$\mathcal{G}$ of affine maps on $\mathbb{C}^{n}$. If $\mathcal{G}$
is finitely generated, this characterization is explicit. We prove
in particular
that no abelian group generated by $n$ affine maps on $\mathbb{C}^{n}$ has a dense orbit.


491  A Note on Homological Dimensions of Artinian Local Cohomology Modules Bahmanpour, Kamal
Let $(R,{\frak m})$ be a nonzero commutative Noetherian local ring
(with identity), $M$ be a nonzero finitely generated $R$module. In
this paper for any ${\frak p}\in {\rm Spec}(R)$ we show that
$
\operatorname{{\rm injdim_{_{R_{\frak p}}}}}
H^{i\dim(R/{\frak p})}_{{\frak p}R_{\frak p}}(M_{\frak p})$ and
${\rm fd}_{R_{\p}} H^{i\dim(R/{\frak p})}_{{\frak p}R_{\frak
p}}(M_{\frak p})$ are bounded from above by $
\operatorname{{\rm injdim_{_{R}}}}
H^i_{\frak
m}(M)$ and
$ {\rm fd}_R H^i_{\frak m}(M)$ respectively, for all integers $i\geq \dim(R/{\frak p})$.


500  The LangWeil Estimate for Cubic Hypersurfaces Browning, T. D.
An improved estimate is provided for the number of $\mathbb{F}_q$rational points
on a geometrically irreducible, projective, cubic hypersurface that is
not equal to a cone.


503  Weak Sequential Completeness of $\mathcal K(X,Y)$ Bu, Qingying
For Banach spaces $X$ and $Y$, we show that if $X^\ast$ and $Y$ are
weakly sequentially complete and every weakly compact operator from
$X$ to $Y$ is compact then the space of all compact operators from $X$
to $Y$ is weakly sequentially complete. The converse is also true if,
in addition, either $X^\ast$ or $Y$ has the bounded compact
approximation property.


510  Linear Forms in Monic Integer Polynomials Dubickas, Artūras
We prove a necessary and sufficient condition on the list of
nonzero integers $u_1,\dots,u_k$, $k \geq 2$, under which a monic
polynomial $f \in \mathbb{Z}[x]$ is expressible by a linear form
$u_1f_1+\dots+u_kf_k$ in monic polynomials $f_1,\dots,f_k \in
\mathbb{Z}[x]$. This condition is independent of $f$. We also show that if
this condition holds, then the monic polynomials $f_1,\dots,f_k$
can be chosen to be irreducible in $\mathbb{Z}[x]$.


520  Equivariant Forms: Structure and Geometry Elbasraoui, Abdelkrim; Sebbar, Abdellah
In this paper we study the notion of equivariant forms introduced in
the authors' previous works. In particular, we completely classify all the
equivariant forms for a subgroup of
$\operatorname{SL}_2(\mathbb{Z})$
by means of the crossratio, the weight
2 modular forms, the quasimodular forms, as well as differential forms
of a Riemann surface and sections of a canonical line bundle.


534  A Cohomological Property of $\pi$invariant Elements Filali, M.; Monfared, M. Sangani
Let $A$ be a Banach algebra and $\pi \colon A \longrightarrow \mathscr L(H)$
be a continuous representation of $A$ on a separable Hilbert space $H$
with $\dim H =\frak m$. Let $\pi_{ij}$ be the coordinate functions of
$\pi$ with respect to an orthonormal basis and suppose that for each
$1\le j \le \frak m$, $C_j=\sum_{i=1}^{\frak m}
\\pi_{ij}\_{A^*}\lt \infty$ and $\sup_j C_j\lt \infty$. Under these
conditions, we call an element $\overline\Phi \in l^\infty (\frak m , A^{**})$
left $\pi$invariant if $a\cdot \overline\Phi ={}^t\pi (a) \overline\Phi$ for all
$a\in A$. In this paper we prove a link between the existence
of left $\pi$invariant elements and the vanishing of certain
Hochschild cohomology groups of $A$. Our results extend an earlier
result by Lau on $F$algebras and recent results of KaniuthLauPym
and the second named author in the special case that $\pi \colon A
\longrightarrow \mathbf C$ is a nonzero character on $A$.


544  Universally Overconvergent Power Series via the Riemann Zetafunction Gauthier, P. M.
The Riemann zetafunction is employed to generate universally overconvergent power series.


551  Real Dimension Groups Handelman, David
Dimension groups (not countable) that are also real ordered vector
spaces can be obtained as direct limits (over directed sets) of
simplicial real vector spaces (finite dimensional vector spaces with
the coordinatewise ordering), but the directed set is not as
interesting as one would like, i.e., it is not true that a
countabledimensional real vector space that has interpolation can be
represented as such a direct limit over the a countable directed
set. It turns out this is the case when the group is additionally
simple, and it is shown that the latter have an ordered tensor product
decomposition. In the Appendix, we provide a huge class of polynomial
rings that, with a pointwise ordering, are shown to satisfy
interpolation, extending a result outlined by Fuchs.


564  Ziegler's Indecomposability Criterion Herzog, Ivo
Ziegler's Indecomposability Criterion is used to prove that a totally
transcendental, i.e., $\Sigma$pure injective, indecomposable left
module over a left noetherian ring is a directed union of finitely
generated indecomposable modules. The same criterion is also used to
give a sufficient condition for a pure injective indecomposable module
${_R}U$ to have an indecomposable local dual $U_R^{\sharp}.$


570  Conjugacy Classes and Binary Quadratic Forms for the Hecke Groups Hoang, Giabao; Ressler, Wendell
In this paper we give a lower bound
with respect to block length
for the trace of nonelliptic conjugacy classes
of the Hecke groups.
One consequence of our bound
is that there are finitely many
conjugacy classes of a given trace in any Hecke group.
We show that another consequence of our bound
is that
class numbers are finite for
related hyperbolic \( \mathbb{Z}[\lambda] \)binary quadratic forms.
We give canonical class representatives
and calculate class numbers
for some classes of hyperbolic \( \mathbb{Z}[\lambda] \)binary quadratic forms.


584  On Automorphisms and Commutativity in Semiprime Rings Liau, PaoKuei; Liu, ChengKai
Let $R$ be a semiprime ring with center
$Z(R)$. For $x,y\in R$, we denote by $[x,y]=xyyx$ the commutator of
$x$ and $y$. If $\sigma$ is a nonidentity automorphism of $R$ such
that
$$
\Big[\big[\dots\big[[\sigma(x^{n_0}),x^{n_1}],x^{n_2}\big],\dots\big],x^{n_k}\Big]=0
$$
for all $x \in R$, where $n_{0},n_{1},n_{2},\dots,n_{k}$ are fixed
positive integers, then there exists a map $\mu\colon R\rightarrow Z(R)$
such that $\sigma(x)=x+\mu(x)$ for all $x\in R$. In particular, when
$R$ is a prime ring, $R$ is commutative.


593  On the $p$norm of an Integral Operator in the Half Plane Liu, Congwen; Zhou, Lifang
We give a partial answer to a conjecture of Dostanić on the
determination of the norm of a class of integral operators induced
by the weighted Bergman projection in the upper half plane.


602  Resultants of Chebyshev Polynomials: A Short Proof Louboutin, Stéphane R.
We give a simple proof of the value of the resultant of two Chebyshev polynomials
(of the first or the second kind),
values lately obtained by D. P. Jacobs, M. O. Rayes and V. Trevisan.


606  Characterization of Simple Highest Weight Modules Mazorchuk, Volodymyr; Zhao, Kaiming
We prove that for simple complex finite dimensional
Lie algebras, affine KacMoody Lie algebras, the
Virasoro algebra and the HeisenbergVirasoro algebra,
simple highest weight modules are characterized
by the property that all positive root elements
act on these modules locally nilpotently. We
also show that this is not the case for higher rank
Virasoro and for Heisenberg algebras.


615  Randers Metrics of Constant Scalar Curvature Sevim, Esra Sengelen; Shen, Zhongmin
Randers metrics are a special class of Finsler metrics. Every Randers
metric can be expressed in terms of a Riemannian metric and a vector
field via Zermelo navigation.
In this paper, we show that a Randers metric has constant scalar
curvature if the Riemannian metric has constant scalar curvature and
the vector field is homothetic.


621  Optimal Control Strategies for Virus Spreading in Inhomogeneous Epidemic Dynamics Shang, Yilun
In this paper, we study the spread of virus/worm in computer
networks with a view to addressing cyber security problems. Epidemic
models have been applied extensively to model the propagation of
computer viruses, which characterize the fact that infected machines
may spread malware to other hosts connected to the network. In our
framework, the dynamics of hosts evolves according to a modified
inhomogeneous SusceptibleInfectiousSusceptible (SIS) epidemic
model with timevarying transmission rate and recovery rate. The
infection of computers is subject to direct attack as well as
propagation among hosts. Based on optimal control theory, optimal
attack strategies are provided by minimizing the cost (equivalently
maximizing the profit) of the attacker. We present a threshold
function of the fraction of infectious hosts, which captures the
dynamically evolving strategies of the attacker and reflects the
persistence of virus spreading. Moreover, our results indicate that
if the infectivity of a computer worm is low and the computers are
installed with antivirus software with high reliability, the
intensity of attacks incurred will likely be low. This agrees with
our intuition.


630  Inverse Semigroups and Sheu's Groupoid for the Odd Dimensional Quantum Spheres Sundar, S.
In this paper, we give a different proof of the fact that the odd dimensional
quantum spheres are groupoid $C^{*}$algebras. We show that the $C^{*}$algebra
$C(S_{q}^{2\ell+1})$ is generated by an inverse semigroup $T$ of partial
isometries. We show that the groupoid $\mathcal{G}_{tight}$ associated with the
inverse semigroup $T$ by Exel is exactly the same as the groupoid
considered by Sheu.


640  Regulator Indecomposable Cycles on a Product of Elliptic Curves Türkmen, İnan Utku
We provide a novel proof of the existence
of regulator indecomposables in the cycle group $CH^2(X,1)$,
where $X$ is a sufficiently general product of two elliptic
curves. In particular, the nature of our proof provides an illustration of
Beilinson rigidity.


647  On Induced Representations Distinguished by Orthogonal Groups Valverde, Cesar
Let $F$ be a local nonarchimedean field of characteristic zero. We
prove that a representation of $GL(n,F)$ obtained from irreducible
parabolic induction of supercuspidal representations is distinguished
by an orthogonal group only if the inducing data is distinguished by
appropriate orthogonal groups. As a corollary, we get that an
irreducible representation induced from supercuspidals that is
distinguished by an orthogonal group is metic.


659  Asymptotics and Uniqueness of Travelling Waves for NonMonotone Delayed Systems on 2D Lattices Yu, ZhiXian; Mei, Ming
We establish asymptotics and uniqueness (up
to translation) of travelling waves for delayed 2D lattice equations
with nonmonotone birth functions. First, with the help of
Ikehara's Theorem, the a priori asymptotic behavior of
travelling wave is exactly derived. Then, based on the obtained
asymptotic behavior, the uniqueness of the traveling waves is
proved. These results complement earlier results in the literature.


673  Diophantine Approximation for Certain Algebraic Formal Power Series in Positive Characteristic Ayadi, K.; Hbaib, M.; Mahjoub, F.
In this paper, we study rational approximations for certain algebraic power series over a finite field.
We obtain results for irrational elements of strictly positive degree
satisfying an equation of the type
\begin{equation}
\alpha=\displaystyle\frac{A\alpha^{q}+B}{C\alpha^{q}}
\end{equation}
where $(A, B, C)\in
(\mathbb{F}_{q}[X])^{2}\times\mathbb{F}_{q}^{\star}[X]$.
In particular,
we will give, under some conditions on the polynomials $A$, $B$
and $C$, well approximated elements satisfying this equation.


683  Envelope Dimension of Modules and the Simplified Radical Formula Nikseresht, A.; Azizi, A.
We introduce and investigate the notion of envelope dimension of
commutative rings and modules over them. In particular, we show that
the envelope dimension of a ring, $R$, is equal to that of the
$R$module $R^{(\mathbb{N})}$. Also we prove that the Krull dimension of a
ring is no more than its envelope dimension and characterize
Noetherian rings for which these two dimensions are equal. Moreover we
generalize and study the concept of simplified radical formula for
modules, which
we defined in an earlier paper.


695  Carmichael meets Chebotarev Banks, William D.; Güloğlu, Ahmet M.; Yeager, Aaron M.
For any finite Galois extension $K$ of $\mathbb Q$
and any conjugacy class $C$ in $\operatorname {Gal}(K/\mathbb Q)$,
we show that there exist infinitely many Carmichael numbers
composed solely of primes for which the associated class of Frobenius
automorphisms is $C$. This result implies that for every natural
number $n$ there are infinitely many Carmichael numbers of the form
$a^2+nb^2$ with $a,b\in\mathbb Z $.


709  Universal Minimal Flows of Groups of Automorphisms of Uncountable Structures Bartošová, Dana
It is a wellknown fact, that the greatest ambit for
a topological group $G$ is the Samuel compactification of $G$ with
respect to the right uniformity on $G.$ We apply the original
description by Samuel from 1948 to give a simple computation of the
universal minimal flow for groups of automorphisms of uncountable
structures using Fraïssé theory and Ramsey theory. This work
generalizes some of the known results about countable structures.


723  On the Sum of Digits of Numerators of Bernoulli Numbers Bérczes, Attila; Luca, Florian
Let $b\gt 1$ be an integer. We prove that for almost all $n$, the sum of the
digits in base $b$ of the numerator of the Bernoulli number $B_{2n}$
exceeds $c\log n$, where $c:=c(b)\gt 0$ is some constant depending on
$b$.


729  The Orthonormal Dilation Property for Abstract Parseval Wavelet Frames Currey, B.; Mayeli, A.
In this work we introduce a class of discrete groups containing
subgroups of abstract translations and dilations, respectively. A
variety of wavelet systems can appear as $\pi(\Gamma)\psi$, where $\pi$ is
a unitary representation of a wavelet group and $\Gamma$ is the abstract
pseudolattice $\Gamma$. We prove a condition in order that a Parseval
frame $\pi(\Gamma)\psi$ can be dilated to an orthonormal basis of the
form $\tau(\Gamma)\Psi$ where $\tau$ is a superrepresentation of
$\pi$. For a subclass of groups that includes the case where the
translation subgroup is Heisenberg, we show that this condition
always holds, and we cite familiar examples as applications.


737  On the Radius of Comparison of a Commutative C*algebra Elliott, George A.; Niu, Zhuang
Let $X$ be a compact metric space. A lower bound for the radius of
comparison of the C*algebra $\operatorname{C}(X)$ is given in terms of
$\operatorname{dim}_{\mathbb{Q}} X$, where $\operatorname{dim}_{\mathbb{Q}} X $ is
the cohomological dimension with rational coefficients. If
$\operatorname{dim}_{\mathbb{Q}} X =\operatorname{dim} X=d$, then the
radius of comparison of the C*algebra $\operatorname{C}(X)$ is $\max\{0, (d1)/21\}$ if $d$ is odd, and must be either $d/21$ or $d/22$ if $d$ is even (the possibility of $d/21$ does occur, but we do not know if the possibility of $d/22$ also can occur).


745  Dimension Functions of SelfAffine Scaling Sets Fu, Xiaoye; Gabardo, JeanPierre
In this paper, the dimension function of a selfaffine generalized scaling set associated with an $n\times n$ integral expansive dilation $A$ is studied. More specifically, we consider the dimension function of an $A$dilation generalized scaling set $K$ assuming that $K$ is a selfaffine tile satisfying $BK = (K+d_1) \cup (K+d_2)$, where $B=A^t$, $A$ is an $n\times n$ integral expansive matrix with $\lvert \det A\rvert=2$, and $d_1,d_2\in\mathbb{R}^n$. We show that the dimension function of $K$ must be constant if either $n=1$ or $2$ or one of the digits is $0$, and that it is bounded by $2\lvert K\rvert$ for any $n$.


759  A Generalization of a Theorem of Boyd and Lawton Issa, Zahraa; Lalín, Matilde
The Mahler measure of a nonzero $n$variable polynomial $P$ is the integral of
$\logP$ on the unit $n$torus. A result of Boyd and Lawton says that
the Mahler measure of a multivariate polynomial is the limit of Mahler
measures of univariate polynomials. We prove the analogous
result for different extensions of Mahler measure such as generalized
Mahler measure (integrating the maximum of $\logP$ for possibly
different $P$'s),
multiple Mahler measure (involving products of $\logP$ for possibly
different $P$'s), and higher Mahler measure (involving $\log^kP$).


769  A Nonzero Value Shared by an Entire Function and its Linear Differential Polynomials Lahiri, Indrajit; Kaish, Imrul
In this paper we study uniqueness of entire functions
sharing a nonzero finite value with linear differential polynomials
and address a result of W. Wang and P. Li.


785  Small Prime Solutions to Cubic Diophantine Equations 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 (i) if $a_j$ are not all of the same sign, then the above cubic
equation has prime solutions satisfying
$p_j \ll n^{1/3}+\textrm{max}\{a_j\}^{14+\varepsilon};$
and (ii) if all $a_j$ are positive and $n \gg \textrm{max}\{a_j\}^{43+\varepsilon}$, then the cubic
equation $a_1p_1^3+\cdots +a_9p_9^3=n$ is soluble in primes $p_j$.
This result is the extension of the linear and quadratic relative problems.


795  Upper Bounds for the Essential Dimension of $E_7$ MacDonald, Mark L.
This paper gives a new upper bound for the essential dimension and the
essential 2dimension of the split simply connected group of type
$E_7$ over a field of characteristic not 2 or 3. In particular,
$\operatorname{ed}(E_7) \leq 29$, and $\operatorname{ed}(E_7;2) \leq 27$.


801  Estimates for Compositions of Maximal Operators with Singular Integrals Oberlin, Richard
We prove weaktype $(1,1)$ estimates for compositions of maximal
operators with singular integrals. Our main object of interest is the
operator $\Delta^*\Psi$ where $\Delta^*$ is Bourgain's maximal
multiplier operator and $\Psi$ is the sum of several modulated
singular integrals; here our method yields a significantly improved
bound for the $L^q$ operator norm when $1 \lt q \lt 2.$ We also consider
associated variationnorm estimates.


814  Quantum Limits of Eisenstein Series and Scattering States Petridis, Yiannis N.; Raulf, Nicole; Risager, Morten S.
We identify the quantum limits of scattering states
for the modular surface. This is obtained through the study of quantum
measures of nonholomorphic Eisenstein series away from the critical
line. We provide a range of stability for the quantum unique
ergodicity theorem of Luo and Sarnak.


827  Erratum to ``Quantum Limits of Eisenstein Series and Scattering States'' Petridis, Yiannis N.; Raulf, Nicole; Risager, Morten S.
This paper provides an erratum to Y. N. Petridis,
N. Raulf, and M. S. Risager, ``Quantum Limits
of Eisenstein Series and Scattering States.'' Canad. Math. Bull., published
online 20120203, http://dx.doi.org/10.4153/CMB20112002.


829  On Mertens' Theorem for Beurling Primes Pollack, Paul
Let $1 \lt p_1 \leq p_2 \leq p_3 \leq \dots$ be an infinite sequence
$\mathcal{P}$ of real numbers for which $p_i \to \infty$, and associate to
this sequence the Beurling zeta function $\zeta_{\mathcal{P}}(s):=
\prod_{i=1}^{\infty}(1p_i^{s})^{1}$. Suppose that for some constant
$A\gt 0$, we have
$\zeta_{\mathcal{P}}(s) \sim A/(s1)$, as $s\downarrow 1$. We prove that
$\mathcal{P}$ satisfies an analogue of a classical theorem of Mertens:
$\prod_{p_i \leq x}(11/p_i)^{1} \sim A \e^{\gamma} \log{x}$, as
$x\to\infty$.
Here $\e = 2.71828\ldots$ is the base of the natural logarithm and
$\gamma = 0.57721\ldots$ is the usual EulerMascheroni constant. This
strengthens a recent theorem of Olofsson.


844  On the Average Number of SquareFree Values of Polynomials Shparlinski, Igor E.
We obtain an asymptotic formula for the number
of squarefree integers in $N$ consecutive values
of polynomials on average over integral
polynomials of degree at most $k$ and of
height at most $H$, where $H \ge N^{k1+\varepsilon}$
for some fixed $\varepsilon\gt 0$.
Individual results of this kind for polynomials of degree $k \gt 3$,
due to A. Granville (1998),
are only known under the $ABC$conjecture.


850  Leftorderability and Exceptional Dehn Surgery on Twist Knots Teragaito, Masakazu
We show that any exceptional nontrivial Dehn surgery on a twist knot, except the trefoil,
yields a $3$manifold whose fundamental group is leftorderable.
This is a generalization of a result of Clay, Lidman and Watson, and
also gives a new supporting evidence for a conjecture of Boyer, Gordon and Watson.


860  On Countable Dense and $n$homogeneity van Mill, Jan
We prove that a connected, countable dense homogeneous space is
$n$homogeneous for every $n$, and strongly 2homogeneous provided it
is locally connected. We also present an example of a connected and
countable dense homogeneous space which is not strongly
2homogeneous. This answers Problem 136 of Watson in the Open Problems
in Topology Book in the negative.


870  Note on Kasparov Product of $C^*$algebra Extensions Wei, Changguo
Using the Dadarlat isomorphism, we give a characterization for the
Kasparov product of $C^*$algebra extensions. A certain relation
between $KK(A, \mathcal q(B))$ and $KK(A, \mathcal q(\mathcal k B))$ is also considered when
$B$ is not stable and it is proved that $KK(A, \mathcal q(B))$ and
$KK(A, \mathcal q(\mathcal k B))$ are not isomorphic in general.


881  Free Groups Generated by Two Heisenberg Translations Xie, BaoHua; Wang, JieYan; Jiang, YuePing
In this paper, we will discuss the groups generated by two
Heisenberg translations of $\mathbf{PU}(2,1)$ and determine when they are free.

