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3  A Short Proof of Paouris' Inequality Adamczak, Radosław; Latała, Rafał; Litvak, Alexander E.; Oleszkiewicz, Krzysztof; Pajor, Alain; TomczakJaegermann, Nicole
We give a short proof of a result of G.~Paouris on
the tail behaviour of the Euclidean norm $X$ of an isotropic
logconcave random vector $X\in\mathbb{R}^n,$
stating that for every $t\geq 1$,
\[\mathbb{P} \big( X\geq ct\sqrt n\big)\leq \exp(t\sqrt n).\]
More precisely we show that for any logconcave random vector $X$
and any $p\geq 1$,
\[(\mathbb{E}X^p)^{1/p}\sim \mathbb{E} X+\sup_{z\in
S^{n1}}(\mathbb{E} \langle
z,X\rangle^p)^{1/p}.\]


9  Integral Sets and the Center of a Finite Group Alperin, Roger C.; Peterson, Brian L.
We give a description of the atoms in the Boolean algebra generated by the integral subsets of a finite group.


12  On the Continuity of the Eigenvalues of a Sublaplacian Aribi, Amine; Dragomir, Sorin; El Soufi, Ahmad
We study the behavior of the eigenvalues of a sublaplacian $\Delta_b$ on a compact strictly pseudoconvex CR manifold $M$, as functions on the set
${\mathcal P}_+$ of positively oriented contact forms on $M$ by endowing ${\mathcal P}_+$ with a natural metric topology.


25  Subadditivity Inequalities for Compact Operators Bourin, JeanChristophe; Harada, Tetsuo; Lee, EunYoung
Some subadditivity inequalities for matrices and concave functions also hold for Hilbert space operators, but (unfortunately!) with an additional $\varepsilon$ term. It seems not possible to erase this residual term. However, in case of compact operators we show that the $\varepsilon$ term is unnecessary. Further, these inequalities are strict in a certain sense when some natural assumptions are satisfied. The discussion also stresses on matrices and their compressions and several open questions or conjectures are considered, both in the matrix and operator settings.


37  Character Amenability of Lipschitz Algebras Dashti, Mahshid; NasrIsfahani, Rasoul; Renani, Sima Soltani
Let ${\mathcal X}$ be a locally compact metric space and let
${\mathcal A}$ be any of the Lipschitz algebras
${\operatorname{Lip}_{\alpha}{\mathcal X}}$, ${\operatorname{lip}_{\alpha}{\mathcal X}}$ or
${\operatorname{lip}_{\alpha}^0{\mathcal X}}$. In this paper, we show, as a
consequence of rather more general results on Banach algebras,
that ${\mathcal A}$ is $C$character amenable if and only if
${\mathcal X}$ is uniformly discrete.


42  Covering the Unit Sphere of Certain Banach Spaces by Sequences of Slices and Balls Fonf, Vladimir P.; Zanco, Clemente
e prove that, given any covering of any infinitedimensional Hilbert space $H$ by countably many closed balls, some point exists in $H$ which belongs to infinitely many balls. We do that by characterizing isomorphically polyhedral separable Banach spaces as those whose unit sphere admits a pointfinite covering by the union of countably many slices of the unit ball.


51  Jordan $*$Derivations of FiniteDimensional Semiprime Algebras Fošner, Ajda; Lee, TsiuKwen
In the paper, we characterize Jordan $*$derivations of a $2$torsion
free, finitedimensional semiprime algebra $R$ with involution $*$. To
be precise, we prove the theorem: Let $deltacolon R o R$ be a Jordan
$*$derivation. Then there exists a $*$algebra decomposition
$R=Uoplus V$ such that both $U$ and $V$ are invariant under
$delta$. Moreover, $*$ is the identity map of $U$ and $delta,_U$ is a
derivation, and the Jordan $*$derivation $delta,_V$ is inner.
We also prove the theorem: Let $R$ be a noncommutative, centrally
closed prime algebra with involution $*$, $operatorname{char},R
e 2$,
and let $delta$ be a nonzero Jordan $*$derivation of $R$. If $delta$ is
an elementary operator of $R$, then $operatorname{dim}_CRlt infty$ and
$delta$ is inner.


61  2dimensional Convexity Numbers and $P_4$free Graphs Geschke, Stefan
For $S\subseteq\mathbb R^n$ a set
$C\subseteq S$ is an $m$clique if the convex hull of no $m$element subset of
$C$ is contained in $S$.
We show that there is essentially just one way to construct
a closed set $S\subseteq\mathbb R^2$ without an uncountable
$3$clique that is not the union of countably many convex sets.
In particular, all such sets have the same convexity number;
that is, they
require the same number of convex subsets to cover them.
The main result follows from an analysis of the convex structure of closed
sets in $\mathbb R^2$ without uncountable 3cliques in terms of
clopen, $P_4$free graphs on Polish spaces.


72  Un Anneau Commutatif associé à un design symétrique Grari, A.
Dans les articles \cite{1}, \cite{2} et \cite{3}; l'auteur développe une représentation
d'un plan projectif fini par un
anneau commutatif unitaire dont les propriétés algébriques dépendent
de la structure géométrique du plan. Dans l'article \cite{4}; il étend cette représentation aux designs symétriques. Cependant l'auteur de l'article \cite{7} fait remarquer que la multiplication définie dans ce cas ne peut être associative que si le design est un plan projectif.
Dans ce papier on mènera
une étude de cette représentation dans le cas des designs
symétriques. On y montrera comment on peut faire associer un
anneau commutatif unitaire à
tout design symétrique , on y précisera certaines de ses propriétés, en
particulier, celles qui relèvent de son invariance. On caractérisera aussi les géométries projectives finies de dimension supérieure moyennant cette représentation.


80  Semicrossed Products of the Disk Algebra and the Jacobson Radical Khemphet, Anchalee; Peters, Justin R.
We consider semicrossed products of the disk algebra with respect to
endomorphisms defined by finite Blaschke products. We characterize the Jacobson radical
of these operator algebras. Furthermore, in the case the finite Blaschke product is elliptic,
we show that the semicrossed product contains no nonzero quasinilpotent
elements. However, if the finite Blaschke product is hyperbolic or parabolic with positive hyperbolic step,
the Jacobson radical is nonzero and a proper subset of the set of quasinilpotent elements.


90  Compact Subsets of the Glimm Space of a $C^*$algebra Lazar, Aldo J.
If $A$ is a $\sigma$unital $C^*$algebra and $a$ is a strictly positive element of $A$ then for every compact subset $K$ of the complete
regularization $\mathrm{Glimm}(A)$ of $\mathrm{Prim}(A)$ there exists
$\alpha \gt 0$ such that $K\subset \{G\in \mathrm{Glimm}(A) \mid \Vert a + G\Vert \geq
\alpha\}$. This extends
a result of J. Dauns
to all $\sigma$unital $C^*$algebras. However, there are a $C^*$algebra $A$
and a compact subset of $\mathrm{Glimm}(A)$ that is not contained in any set of the form $\{G\in \mathrm{Glimm}(A) \mid \Vert a + G\Vert \geq
\alpha\}$, $a\in A$ and $\alpha \gt 0$.


97  Rationality and the JordanGattiViniberghi decomposition Levy, Jason
We verify
our earlier conjecture
and use it to prove that the
semisimple parts of the rational JordanKacVinberg decompositions of
a rational vector all lie in a single rational orbit.


105  On the Counting Function of Elliptic Carmichael Numbers Luca, Florian; Shparlinski, Igor E.
We give an upper bound for the number elliptic Carmichael numbers $n \le x$
that have recently been introduced by J. H. Silverman in the case of an elliptic curve without complex multiplication (non CM). We also discuss
several possible ways for further improvements.


113  A Lower Bound for the EndtoEnd Distance of SelfAvoiding Walk Madras, Neal
For an $N$step selfavoiding walk on the hypercubic lattice ${\bf Z}^d$,
we prove that the meansquare endtoend distance is at least
$N^{4/(3d)}$ times a constant.
This implies that the associated critical exponent $\nu$ is
at least $2/(3d)$, assuming that $\nu$ exists.


119  Splitting Families and Complete Separability Mildenberger, Heike; Raghavan, Dilip; Steprans, Juris
We answer a question from Raghavan and Steprāns
by showing that $\mathfrak{s} = {\mathfrak{s}}_{\omega, \omega}$. Then we use this to construct a completely separable maximal almost disjoint family under $\mathfrak{s} \leq \mathfrak{a}$, partially answering a question of Shelah.


125  Camina Triples Mlaiki, Nabil M.
In this paper, we study Camina triples. Camina triples are a
generalization of Camina pairs. Camina pairs were first introduced
in 1978 by A .R. Camina.
Camina's work
was inspired by the study of Frobenius groups. We
show that if $(G,N,M)$ is a Camina triple, then either $G/N$ is a
$p$group, or $M$ is abelian, or $M$ has a nontrivial nilpotent or
Frobenius quotient.


132  Twisted Conjugacy Classes in Abelian Extensions of Certain Linear Groups Mubeena, T.; Sankaran, P.
Given a group automorphism $\phi:\Gamma\longrightarrow \Gamma$, one has
an action of $\Gamma$ on itself by $\phi$twisted conjugacy, namely, $g.x=gx\phi(g^{1})$.
The orbits of this action are called $\phi$twisted conjugacy classes. One says
that $\Gamma$ has the $R_\infty$property if there are infinitely many $\phi$twisted conjugacy
classes for every automorphism $\phi$ of $\Gamma$. In this paper we
show that $\operatorname{SL}(n,\mathbb{Z})$ and its
congruence subgroups have the $R_\infty$property. Further we show that
any (countable) abelian extension of $\Gamma$ has the $R_\infty$property where $\Gamma$ is a torsion
free nonelementary hyperbolic group, or $\operatorname{SL}(n,\mathbb{Z}),
\operatorname{Sp}(2n,\mathbb{Z})$ or a principal congruence
subgroup of $\operatorname{SL}(n,\mathbb{Z})$ or the fundamental group of a complete Riemannian
manifold of constant negative curvature.


141  Size, Order, and Connected Domination Mukwembi, Simon
We give a sharp upper bound on the size of a
trianglefree graph of a given order and connected domination. Our
bound, apart from
strengthening an old classical theorem of Mantel and of
Turán , improves on a theorem of Sanchis.
Further, as corollaries, we settle a long standing
conjecture of Graffiti on the leaf number and local independence for
trianglefree graphs and answer a question of Griggs, Kleitman and
Shastri on a lower bound of the leaf number in
trianglefree graphs.


145  The Essential Spectrum of the Essentially Isometric Operator Mustafayev, H. S.
Let $T$ be a contraction on a complex, separable, infinite dimensional
Hilbert space and let $\sigma \left( T\right) $ (resp. $\sigma _{e}\left(
T\right) )$ be its spectrum (resp. essential spectrum). We assume that $T$
is an essentially isometric operator, that is $I_{H}T^{\ast }T$ is compact.
We show that if $D\diagdown \sigma \left( T\right) \neq \emptyset ,$ then
for every $f$ from the discalgebra,
\begin{equation*}
\sigma _{e}\left( f\left( T\right) \right) =f\left( \sigma _{e}\left(
T\right) \right) ,
\end{equation*}
where $D$ is the open unit disc. In addition, if $T$ lies in the class
$ C_{0\cdot }\cup C_{\cdot 0},$ then
\begin{equation*}
\sigma _{e}\left( f\left( T\right) \right) =f\left( \sigma \left( T\right)
\cap \Gamma \right) ,
\end{equation*}
where $\Gamma $ is the unit circle. Some related problems are also discussed.


159  Strongly $0$dimensional Modules Oral, Kürşat Hakan; Özkirişci, Neslihan Ayşen; Tekir, Ünsal
In a multiplication module, prime submodules have the property, if a prime
submodule contains a finite intersection of submodules then one of the
submodules is contained in the prime submodule. In this paper, we generalize
this property to infinite intersection of submodules and call such prime
submodules strongly prime submodule. A multiplication module in which every
prime submodule is strongly prime will be called strongly 0dimensional
module. It is also an extension of strongly 0dimensional rings. After
this we investigate properties of strongly 0dimensional modules and give
relations of von Neumann regular modules, Qmodules and strongly
0dimensional modules.


166  On Minimal and Maximal $p$operator Space Structures Öztop, Serap; Spronk, Nico
We show that for $p$operator spaces, there are natural notions of minimal and maximal
structures. These are useful for dealing with tensor products.


178  Quasiconvexity and Density Topology Rabier, Patrick J.
We prove that if $f:\mathbb{R}^{N}\rightarrow \overline{\mathbb{R}}$ is
quasiconvex and $U\subset \mathbb{R}^{N}$ is open in the density topology, then
$\sup_{U}f=\operatorname{ess\,sup}_{U}f,$ while
$\inf_{U}f=\operatorname{ess\,inf}_{U}f$
if and only if the equality holds when $U=\mathbb{R}^{N}.$ The first (second)
property is typical of lsc (usc) functions and, even when $U$ is an ordinary
open subset, there seems to be no record that they both hold for all
quasiconvex functions.


188  A Characterization of Bipartite Zerodivisor Graphs Rad, Nader Jafari; Jafari, Sayyed Heidar
In this paper we obtain a characterization for all bipartite
zerodivisor graphs of commutative rings $R$ with $1$, such that
$R$ is finite or $Nil(R)\neq2$.


194  A Lower Bound for the Length of Closed Geodesics on a Finsler Manifold Zhao, Wei
In this paper, we obtain a lower bound for the length of closed geodesics on an arbitrary closed Finsler manifold.


209  Erratum to the Paper "A Lower Bound for the Length of Closed Geodesics on a Finsler Manifold" Zhao, Wei
We correct two clerical errors made in the paper "A Lower Bound for
the Length of Closed Geodesics on a Finsler Manifold".


210  An Explicit Formula for the Generalized Cyclic Shuffle Map Zhang, Jiao; Wang, QingWen
We provide an explicit formula for the generalized cyclic shuffle map for cylindrical modules.
Using this formula we give a combinatorial proof of the generalized
cyclic EilenbergZilber theorem.


225  Small Flag Complexes with Torsion Adamaszek, Michał
We classify flag complexes on at most $12$ vertices with torsion in
the first homology group. The result is moderately computeraided.


231  On the Multiplicities of Characters in Table Algebras Bagherian, J.
In this paper we show that every module of a table algebra
can be considered as a faithful module of some quotient table
algebra.
Also we prove that every faithful module of a table algebra
determines a closed subset which is a cyclic group.
As a main result we give some information about multiplicities
of characters in table algebras.


240  Addendum to ``Limit Sets of Typical Homeomorphisms'' Bernardes, Nilson C. Jr.
Given an integer $n \geq 3$,
a metrizable compact topological $n$manifold $X$ with boundary,
and a finite positive Borel measure $\mu$ on $X$,
we prove that for the typical homeomorphism $f : X \to X$,
it is true that for $\mu$almost every point $x$ in $X$ the restriction of
$f$ (respectively of $f^{1}$) to the omega limit set $\omega(f,x)$
(respectively to the alpha limit set $\alpha(f,x)$) is topologically
conjugate to the universal odometer.


245  AssouadNagata Dimension of Wreath Products of Groups Brodskiy, N.; Dydak, J.; Lang, U.
Consider the wreath product $H\wr G$, where $H\ne 1$ is finite and $G$ is finitely generated.
We show that the AssouadNagata dimension $\dim_{AN}(H\wr G)$ of $H\wr G$
depends on the growth of $G$ as follows:
\par If the growth of $G$ is not bounded by a linear function, then $\dim_{AN}(H\wr G)=\infty$,
otherwise $\dim_{AN}(H\wr G)=\dim_{AN}(G)\leq 1$.


254  On Parseval Wavelet Frames with Two or Three Generators via the Unitary Extension Principle Christensen, Ole; Kim, Hong Oh; Kim, Rae Young
The unitary extension principle (UEP) by Ron and Shen yields a
sufficient condition for the construction of Parseval wavelet frames with
multiple generators. In this paper we characterize the UEPtype wavelet systems that
can be extended to a Parseval wavelet frame by adding just one UEPtype wavelet
system. We derive a condition that is necessary for the extension of a UEPtype
wavelet system to any Parseval wavelet frame with any number of generators, and
prove that this condition is also sufficient to ensure that an extension
with just two generators is possible.


264  On Semisimple Hopf Algebras of Dimension $pq^n$ Dai, Li; Dong, Jingcheng
Let $p,q$ be prime numbers with $p^2\lt q$, $n\in \mathbb{N}$, and $H$ a
semisimple Hopf algebra of dimension $pq^n$ over an algebraically
closed field of characteristic $0$. This paper proves that $H$ must
possess one of the following structures: (1) $H$ is semisolvable;
(2) $H$ is a Radford biproduct $R\# kG$, where $kG$ is the group
algebra of group $G$ of order $p$, and $R$ is a semisimple YetterDrinfeld
Hopf algebra in ${}^{kG}_{kG}\mathcal{YD}$ of dimension $q^n$.


270  Derivations on Toeplitz Algebras Didas, Michael; Eschmeier, Jörg
Let $H^2(\Omega)$ be the Hardy space on a strictly pseudoconvex domain $\Omega \subset
\mathbb{C}^n$,
and let $A \subset L^\infty(\partial \Omega)$ denote the subalgebra of all $L^\infty$functions $f$
with compact Hankel operator $H_f$. Given any closed subalgebra $B \subset A$ containing $C(\partial \Omega)$,
we describe the first Hochschild cohomology group of the
corresponding Toeplitz algebra $\mathcal(B) \subset B(H^2(\Omega))$.
In particular, we show that every derivation on $\mathcal{T}(A)$ is inner. These results are new even for $n=1$,
where it follows that every derivation on $\mathcal{T}(H^\infty+C)$ is inner, while there are noninner
derivations on $\mathcal{T}(H^\infty+C(\partial \mathbb{B}_n))$ over
the unit ball $\mathbb{B}_n$ in dimension $n\gt 1$.


277  On Mutually $m$permutable Product of Smooth Groups Elkholy, A. M.; ElLatif, M. H. Abd
Let $G$ be a
finite group and $H$, $K$ two subgroups of G. A group $G$ is said to
be a mutually mpermutable product of $H$ and $K$ if $G=HK$ and
every maximal subgroup of $H$ permutes with $K$ and every maximal
subgroup of $K$ permutes with $H$. In this paper, we investigate the
structure of a finite group which is a mutually mpermutable product
of two subgroups under the assumption that its maximal subgroups are
totally smooth.


283  Infinite Dimensional DeWitt Supergroups and their Bodies Fulp, Ronald
For Dewitt super groups $G$ modeled via an underlying finitely
generated Grassmann algebra it is wellknown that when there exists a
body group $BG$ compatible with the group operation on $G,$ then
generically the kernel $K$ of the body homomorphism is nilpotent. This
is not true when the underlying Grassmann algebra is infinitely
generated. We show that it is quasinilpotent in the sense that as a
Banach Lie group its Lie algebra $\kappa$ has the property that for
each $a\in \kappa$, $ad_a$ has a zero spectrum. We also show that
the exponential mapping from $\kappa$ to $K$ is surjective and that
$K$ is a quotient manifold of the Banach space $\kappa$ via a lattice
in $\kappa.$


289  Closure of the Cone of Sums of $2d$powers in Certain Weighted $\ell_1$seminorm Topologies Ghasemi, Mehdi; Marshall, Murray; Wagner, Sven
In a paper from 1976, Berg, Christensen and Ressel prove that the
closure of the cone of sums of squares $\sum
\mathbb{R}[\underline{X}]^2$ in the polynomial ring
$\mathbb{R}[\underline{X}] := \mathbb{R}[X_1,\dots,X_n]$ in the
topology induced by the $\ell_1$norm is equal to
$\operatorname{Pos}([1,1]^n)$, the cone consisting of all polynomials
which are nonnegative on the hypercube $[1,1]^n$. The result is
deduced as a corollary of a general result, established in the same
paper, which is valid for any commutative semigroup.
In later work, Berg and Maserick and Berg, Christensen and Ressel
establish an even more general result, for a commutative semigroup
with involution, for the closure of the cone of sums of squares of
symmetric elements in the weighted $\ell_1$seminorm topology
associated to an absolute value.
In the present paper we give a new proof of these results which is
based on Jacobi's representation theorem from 2001. At the same time,
we use Jacobi's representation theorem to extend these results from
sums of squares to sums of $2d$powers, proving, in particular, that
for any integer $d\ge 1$, the closure of the cone of sums of
$2d$powers $\sum \mathbb{R}[\underline{X}]^{2d}$ in
$\mathbb{R}[\underline{X}]$ in the topology induced by the
$\ell_1$norm is equal to $\operatorname{Pos}([1,1]^n)$.


303  Octonion Algebras over Rings are not Determined by their Norms Gille, Philippe
Answering a question of H. Petersson, we provide
a class of examples of pair of octonion algebras over a ring having isometric
norms.


310  Leftorderable Fundamental Group and Dehn Surgery on the Knot $5_2$ Hakamata, Ryoto; Teragaito, Masakazu
We show that the resulting manifold by $r$surgery on the knot $5_2$, which is
the twobridge knot corresponding to the rational number $3/7$, has leftorderable
fundamental group if the slope $r$ satisfies $0\le r \le 4$.


318  Duality of Preenvelopes and Pure Injective Modules Huang, Zhaoyong
Let $R$ be an arbitrary ring and $()^+=\operatorname{Hom}_{\mathbb{Z}}(,
\mathbb{Q}/\mathbb{Z})$ where $\mathbb{Z}$ is the ring of integers
and $\mathbb{Q}$ is the ring of rational numbers, and let
$\mathcal{C}$ be a subcategory of left $R$modules and $\mathcal{D}$
a subcategory of right $R$modules such that $X^+\in \mathcal{D}$
for any $X\in \mathcal{C}$ and all modules in $\mathcal{C}$ are pure
injective. Then a homomorphism $f: A\to C$ of left $R$modules with
$C\in \mathcal{C}$ is a $\mathcal{C}$(pre)envelope of $A$ provided
$f^+: C^+\to A^+$ is a $\mathcal{D}$(pre)cover of $A^+$. Some
applications of this result are given.


326  On Zerodivisors in Group Rings of Groups with Torsion Ivanov, S. V.; Mikhailov, Roman
Nontrivial pairs of zerodivisors in group rings are
introduced and discussed. A problem on the existence of nontrivial
pairs of zerodivisors in group rings of free Burnside groups of odd
exponent $n \gg 1$ is solved in the affirmative. Nontrivial pairs of
zerodivisors are also found in group rings of free products of groups
with torsion.


335  Alexandroff Manifolds and Homogeneous Continua Karassev, A.; Todorov, V.; Valov, V.
ny homogeneous,
metric $ANR$continuum is a $V^n_G$continuum provided $\dim_GX=n\geq
1$ and $\check{H}^n(X;G)\neq 0$, where $G$ is a principal ideal
domain.
This implies that any homogeneous $n$dimensional metric $ANR$continuum is a $V^n$continuum in the sense of Alexandroff.
We also prove that any finitedimensional homogeneous metric continuum
$X$, satisfying $\check{H}^n(X;G)\neq 0$ for some group $G$ and $n\geq
1$, cannot be separated by
a compactum $K$ with $\check{H}^{n1}(K;G)=0$ and $\dim_G K\leq
n1$. This provides a partial answer to a question of
KallipolitiPapasoglu
whether any twodimensional homogeneous Peano continuum cannot be separated by arcs.


344  On Localized Unstable $K^1$groups and Applications to Selfhomotopy Groups Kishimoto, Daisuke; Kono, Akira; Tsutaya, Mitsunobu
The computing method for the $p$localization of the group
$[X,\mathrm{U}(n)]$
by Hamanaka in 2004
is revised. As an application, an explicit description of the
selfhomotopy group of $\mathrm{Sp}(3)$ localized at $p\ge 5$ is given and
the homotopy nilpotency of $\mathrm{Sp}(3)$ localized at $p\ge 5$ is determined.


357  Representation Equivalent Bieberbach Groups and Strongly Isospectral Flat Manifolds Lauret, Emilio A.
Let $\Gamma_1$ and $\Gamma_2$ be Bieberbach groups contained in the
full isometry group $G$ of $\mathbb{R}^n$.
We prove that if the compact flat manifolds $\Gamma_1\backslash\mathbb{R}^n$ and
$\Gamma_2\backslash\mathbb{R}^n$ are strongly isospectral then the Bieberbach groups
$\Gamma_1$ and $\Gamma_2$ are representation equivalent, that is, the
right regular representations $L^2(\Gamma_1\backslash G)$ and
$L^2(\Gamma_2\backslash G)$ are unitarily equivalent.


364  How Lipschitz Functions Characterize the Underlying Metric Spaces Li, Lei; Wang, YaShu
Let $X, Y$ be metric spaces and $E, F$ be Banach spaces. Suppose that
both $X,Y$ are realcompact, or both $E,F$ are realcompact.
The zero set of a vectorvalued function $f$ is denoted by $z(f)$.
A linear bijection $T$ between local or generalized Lipschitz vectorvalued function spaces
is said to preserve zeroset containments or nonvanishing functions
if
\[z(f)\subseteq z(g)\quad\Longleftrightarrow\quad z(Tf)\subseteq z(Tg),\]
or
\[z(f) = \emptyset\quad \Longleftrightarrow\quad z(Tf)=\emptyset,\]
respectively.
Every zeroset containment preserver, and every nonvanishing function preserver when
$\dim E =\dim F\lt +\infty$, is a weighted composition operator
$(Tf)(y)=J_y(f(\tau(y)))$.
We show that the map $\tau\colon Y\to X$ is a locally (little) Lipschitz homeomorphism.


375  A Problem on Edgemagic Labelings of Cycles López, S. C.; MuntanerBatle; RiusFont
Kotzig and Rosa defined in 1970 the concept of edgemagic labelings as
follows: let $G$ be a simple $(p,q)$graph (that is, a graph of order $p$
and size $q$ without loops or multiple edges). A bijective function $f:V(G)\cup
E(G)\rightarrow \{1,2,\ldots,p+q\}$ is an edgemagic labeling of $G$ if
$f(u)+f(uv)+f(v)=k$, for all $uv\in E(G)$. A graph that admits an edgemagic
labeling is called an edgemagic graph, and $k$ is called the magic sum
of the labeling. An old conjecture of Godbold and Slater sets that all
possible theoretical magic sums are attained for each cycle of order $n\ge
7$. Motivated by this conjecture, we prove that for all $n_0\in \mathbb{N}$,
there exists $n\in \mathbb{N}$, such that the cycle $C_n$ admits at least
$n_0$ edgemagic labelings with at least $n_0$ mutually distinct magic
sums. We do this by providing a lower bound for the number of magic sums
of the cycle $C_n$, depending on the sum of the exponents of the odd primes
appearing in the prime factorization of $n$.


381  On Complex Explicit Formulae Connected with the Möbius Function of an Elliptic Curve Łydka, Adrian
We study analytic properties function $m(z, E)$, which is defined on the upper halfplane as an integral from the shifted $L$function of an elliptic curve. We show that $m(z, E)$ analytically continues to a meromorphic function on the whole complex plane and satisfies certain functional equation. Moreover, we give explicit formula for $m(z, E)$ in the strip $\Im{z}\lt 2\pi$.


390  Simplicity of Some Twin Tree Automorphism Groups with Trivial Commutation Relations Morita, Jun; Rémy, Bertrand
We prove simplicity for incomplete rank 2 KacMoody groups over algebraic closures of finite fields with trivial commutation relations between root groups corresponding to prenilpotent pairs.
We don't use the (yet unknown) simplicity of the corresponding finitely generated groups (i.e., when the ground field is finite).
Nevertheless we use the fact that the latter groups are just infinite
(modulo center).


401  Curvature of $K$contact SemiRiemannian Manifolds Perrone, Domenico
In this paper we characterize $K$contact semiRiemannian manifolds
and Sasakian semiRiemannian manifolds in terms of
curvature. Moreover, we show that any conformally flat $K$contact
semiRiemannian manifold is Sasakian and of constant sectional
curvature $\kappa=\varepsilon$, where $\varepsilon =\pm 1$ denotes
the causal character of the Reeb vector field. Finally, we give some
results about the curvature of a $K$contact Lorentzian manifold.


413  On the Comaximal Graph of a Commutative Ring Samei, Karim
Let $R$ be a commutative ring with $1$. In [P. K. Sharma, S. M.
Bhatwadekar, A note on graphical representation of rings, J.
Algebra 176(1995) 124127], Sharma and Bhatwadekar defined a
graph on $R$, $\Gamma(R)$, with vertices as elements of $R$, where
two distinct vertices $a$ and $b$ are adjacent if and only if $Ra
+ Rb = R$. In this paper, we consider a subgraph $\Gamma_2(R)$ of
$\Gamma(R)$ which consists of nonunit elements. We investigate
the behavior of $\Gamma_2(R)$ and $\Gamma_2(R) \setminus \operatorname{J}(R)$,
where $\operatorname{J}(R)$ is the Jacobson radical of $R$. We associate the
ring properties of $R$, the graph properties of $\Gamma_2(R)$ and
the topological properties of $\operatorname{Max}(R)$. Diameter, girth, cycles
and dominating sets are investigated and the algebraic and the
topological characterizations are given for graphical properties
of these graphs.


424  A Note on Amenability of Locally Compact Quantum Groups Sołtan, Piotr M.; Viselter, Ami
In this short note we introduce a notion called ``quantum injectivity''
of locally compact quantum groups, and prove that it is equivalent
to amenability of the dual. Particularly, this provides a new characterization
of amenability of locally compact groups.


431  The Rasmussen Invariant, Fourgenus and Threegenus of an Almost Positive Knot Are Equal Tagami, Keiji
An oriented link is positive if it has a link diagram whose crossings are all positive.
An oriented link is almost positive if it is not positive and has a link diagram with exactly one negative crossing.
It is known that the Rasmussen invariant, $4$genus and $3$genus of a positive knot are equal.
In this paper, we prove that the Rasmussen invariant, $4$genus and $3$genus of an almost positive knot are equal.
Moreover, we determine the Rasmussen invariant of an almost positive knot in terms of its almost positive knot diagram.
As corollaries, we prove that all almost positive knots are not homogeneous, and there is no almost positive knot of $4$genus one.


439  The Fixed Point Locus of the Verschiebung on $\mathcal{M}_X(2, 0)$ for Genus2 Curves $X$ in Charateristic $2$ Yang, YanHong
We prove that for every ordinary genus$2$ curve $X$ over a finite
field $\kappa$ of characteristic $2$ with
$\textrm{Aut}(X/\kappa)=\mathbb{Z}/2\mathbb{Z} \times S_3$, there exist
$\textrm{SL}(2,\kappa[\![s]\!])$representations of $\pi_1(X)$ such
that the image of $\pi_1(\overline{X})$ is infinite. This result
produces a family of examples similar to Laszlo's counterexample
to de Jong's question regarding the finiteness of the geometric
monodromy of representations of the fundamental group.
