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3  Extensions of Positive Definite Functions on Amenable Groups Bakonyi, M.; Timotin, D.
Let $S$ be a subset of an amenable group $G$ such that $e\in S$ and
$S^{1}=S$. The main result of this paper states that if the Cayley
graph of $G$ with respect to $S$ has a certain combinatorial property,
then every positive definite operatorvalued function on $S$ can be
extended to a positive definite function on $G$. Several known
extension results are obtained as corollaries. New applications are
also presented.


12  Homotopy and the KestelmanBorweinDitor Theorem Bingham, N. H.; Ostaszewski, A. J.
The KestelmanBorweinDitor Theorem, on embedding a null sequence by
translation in (measure/category) ``large'' sets has two generalizations.
Miller replaces the translated sequence by a ``sequence homotopic
to the identity''. The authors, in a previous paper, replace points by functions:
a uniform functional null sequence replaces the null sequence, and
translation receives a functional form. We give a unified approach to
results of this kind. In particular, we show that (i) Miller's homotopy
version follows from the functional version, and (ii) the pointwise instance
of the functional version follows from Miller's homotopy version.


21  Generalized Dsymmetric Operators II Bouali, S.; Echchad, M.
Let $H$ be a separable,
infinitedimensional, complex Hilbert space and let $A, B\in{\mathcal L
}(H)$, where ${\mathcal L}(H)$ is the algebra of all bounded linear
operators on $H$. Let $\delta_{AB}\colon {\mathcal L}(H)\rightarrow {\mathcal
L}(H)$ denote the generalized derivation $\delta_{AB}(X)=AXXB$.
This note will initiate a study on the class of pairs $(A,B)$ such
that $\overline{{\mathcal R}(\delta_{AB})}= \overline{{\mathcal
R}(\delta_{A^{\ast}B^{\ast}})}$.


28  Generalized Solution of the Photon Transport Problem Chang, YuHsien; Hong, ChengHong
The purpose of this paper is to show the existence of a
generalized solution of the photon transport problem. By means of the theory of
equicontinuous $C_{0}$semigroup on a sequentially complete locally convex
topological vector space we show that the perturbed abstract Cauchy problem
has a unique solution when the perturbation operator and the forcing term
function satisfy certain conditions. A consequence of the abstract result is
that it can be directly applied to obtain a generalized solution of the photon
transport problem.


39  Elements in a Numerical Semigroup with Factorizations of the Same Length Chapman, S. T.; GarcíaSánchez, P. A.; Llena, D.; Marshall, J.
Questions concerning the lengths of factorizations into irreducible
elements in numerical monoids
have gained much attention in the recent literature. In this note,
we show that a numerical monoid has an element with two different
irreducible factorizations of the same length if and only if its
embedding dimension is greater than
two. We find formulas in embedding dimension three for the smallest
element with two different irreducible factorizations of the same
length and the largest element whose different irreducible
factorizations all have distinct lengths. We show that these
formulas do not naturally extend to higher embedding
dimensions.


44  StarShapedness and $K$Orbits in Complex Semisimple Lie Algebras Cheung, WaiShun; Tam, TinYau
Given a complex semisimple Lie algebra
$\mathfrak{g}=\mathfrak{k}+i\mathfrak{k}$ ($\mathfrak{k}$ is a compact
real form of $\mathfrak{g}$), let $\pi\colon\mathfrak{g}\to
\mathfrak{h}$ be the orthogonal projection (with respect to the
Killing form) onto the Cartan subalgebra
$\mathfrak{h}:=\mathfrak{t}+i\mathfrak{t}$, where $\mathfrak{t}$ is a
maximal abelian subalgebra of $\mathfrak{k}$. Given $x\in
\mathfrak{g}$, we consider $\pi(\mathop{\textrm{Ad}}(K) x)$, where $K$ is
the analytic subgroup $G$ corresponding to $\mathfrak{k}$, and show
that it is starshaped. The result extends a result of Tsing. We also
consider the generalized numerical range $f(\mathop{\textrm{Ad}}(K)x)$,
where $f$ is a linear functional on $\mathfrak{g}$. We establish the
starshapedness of $f(\mathop{\textrm{Ad}}(K)x)$ for simple Lie algebras
of type $B$.


56  Characteristic Varieties for a Class of Line Arrangements Dinh, Thi Anh Thu
Let $\mathcal{A}$ be a line arrangement in the complex projective plane
$\mathbb{P}^2$, having the points of multiplicity $\geq 3$ situated on two
lines in $\mathcal{A}$, say $H_0$ and $H_{\infty}$. Then we show that the
nonlocal irreducible components of the first resonance variety
$\mathcal{R}_1(\mathcal{A})$ are 2dimensional and correspond to parallelograms $\mathcal{P}$ in
$\mathbb{C}^2=\mathbb{P}^2 \setminus H_{\infty}$ whose sides are in $\mathcal{A}$ and for
which $H_0$ is a diagonal.


68  Nonsplitting in Kirchberg's Idealrelated $KK$Theory Eilers, Søren; Restorff, Gunnar; Ruiz, Efren
A. Bonkat obtained a universal coefficient theorem in the setting of Kirchberg's
idealrelated $KK$theory in the fundamental case of a
$C^*$algebra with one
specified ideal. The universal coefficient sequence was shown to split, unnaturally, under certain
conditions. Employing certain $K$theoretical information derivable
from the given operator algebras using a method introduced here, we shall
demonstrate that Bonkat's UCT does not split in general. Related
methods lead to information on the complexity of the $K$theory which
must be used to
classify $*$isomorphisms for purely infinite $C^*$algebras with
one nontrivial ideal.


82  Lefschetz Numbers for $C^*$Algebras Emerson, Heath
Using Poincar\'e duality, we obtain a formula of Lefschetz type
that computes the Lefschetz number of an endomorphism of a separable
nuclear $C^*$algebra satisfying Poincar\'e duality and the Kunneth
theorem. (The Lefschetz number of an endomorphism is the graded trace
of the induced map on $\textrm{K}$theory tensored with $\mathbb{C}$, as in the
classical case.) We then examine endomorphisms of CuntzKrieger
algebras $O_A$. An endomorphism has an invariant, which is a
permutation of an infinite set, and the contracting and expanding
behavior of this permutation describes the Lefschetz number of the
endomorphism. Using this description, we derive a closed polynomial
formula for the Lefschetz number depending on the matrix $A$ and the
presentation of the endomorphism.


100  On the Generalized Marcinkiewicz Integral Operators with Rough Kernels Fan, Dashan; Wu, Huoxiong
A class of generalized Marcinkiewicz
integral operators is introduced, and, under rather weak conditions
on the integral kernels, the boundedness of such operators on $L^p$
and TriebelLizorkin spaces is established.


113  On the Norm of the BeurlingAhlfors Operator in Several Dimensions Hytönen, Tuomas P.
The generalized BeurlingAhlfors operator $S$ on
$L^p(\mathbb{R}^n;\Lambda)$, where $\Lambda:=\Lambda(\mathbb{R}^n)$ is the
exterior algebra with its natural Hilbert space norm, satisfies the
estimate
$$\S\_{\mathcal{L}(L^p(\mathbb{R}^n;\Lambda))}\leq(n/2+1)(p^*1),\quad
p^*:=\max\{p,p'\}$$
This improves on earlier results in all dimensions $n\geq 3$. The
proof is based on the heat extension and relies at the bottom on
Burkholder's sharp inequality for martingale transforms.


126  Fundamental Solutions of Kohn SubLaplacians on Anisotropic Heisenberg Groups and Htype Groups Jin, Yongyang; Zhang, Genkai
We prove that the fundamental solutions
of Kohn subLaplacians $\Delta + i\alpha \partial_t$
on the anisotropic Heisenberg groups are tempered distributions and have
meromorphic continuation in $\alpha$ with simple poles. We compute the
residues and find the partial fundamental solutions
at the poles. We also find formulas for the
fundamental solutions for some matrixvalued
Kohn type subLaplacians
on Htype groups.


141  Linear Maps on $C^*$Algebras Preserving the Set of Operators that are Invertible in $\mathcal{A}/\mathcal{I}$ Kim, Sang Og; Park, Choonkil
For $C^*$algebras $\mathcal{A}$ of real rank zero, we describe
linear maps $\phi$ on $\mathcal{A}$ that are surjective up to ideals
$\mathcal{I}$, and $\pi(A)$ is invertible in $\mathcal{A}/\mathcal{I}$ if and only if
$\pi(\phi(A))$ is invertible in $\mathcal{A}/\mathcal{I}$, where $A\in\mathcal{A}$ and
$\pi:\mathcal{A}\to\mathcal{A}/\mathcal{I}$ is the quotient map. We also consider similar
linear maps preserving zero products on the Calkin algebra.


147  Generalized Quandle Polynomials Nelson, Sam
We define a family of generalizations of the twovariable quandle polynomial.
These polynomial invariants generalize in a natural way to eightvariable
polynomial invariants of finite biquandles. We use these polynomials to define
a family of link invariants that further generalize the quandle counting
invariant.


159  Hardy Inequalities on the Real Line Sababheh, Mohammad
We prove that some inequalities, which are considered to be
generalizations of Hardy's inequality on the circle,
can be modified and proved to be true for functions integrable on the real line.
In fact we would like to show that some constructions that were
used to prove the Littlewood conjecture can be used similarly to
produce real Hardytype inequalities.
This discussion will lead to many questions concerning the
relationship between Hardytype inequalities on the circle and
those on the real line.


172  Measures with Fourier Transforms in $L^2$ of a Halfspace Shayya, Bassam
We prove that if the Fourier transform of a compactly supported
measure is in $L^2$ of a halfspace, then the measure is
absolutely continuous to Lebesgue measure. We then show how this
result can be used to translate information about the
dimensionality of a measure and the decay of its Fourier
transform into geometric information about its support.


180  Additive Families of Low Borel Classes and Borel Measurable Selectors Spurný, J.; Zelený, M.
An important conjecture in the theory of Borel sets in nonseparable
metric spaces is whether any pointcountable Boreladditive family in
a complete metric space has a $\sigma$discrete refinement. We confirm the conjecture for
pointcountable $\mathbf\Pi_3^0$additive families, thus generalizing results of
R. W. Hansell and the first author. We apply this result to the
existence of Borel measurable selectors for multivalued mappings of
low Borel complexity, thus answering in the affirmative a particular
version of a question of J. Kaniewski and R. Pol.


193  Measurements and $G_\delta$Subsets of Domains Bennett, Harold; Lutzer, David
In this paper we study domains, Scott
domains, and the existence of measurements. We
use a space created by D.~K. Burke to show that
there is a Scott domain $P$ for which $\max(P)$ is
a $G_\delta$subset of $P$ and yet no measurement
$\mu$ on $P$ has $\ker(\mu) = \max(P)$. We also
correct a mistake in the literature asserting that
$[0, \omega_1)$ is a space of this type. We show
that if $P$ is a Scott domain and $X \subseteq
\max(P)$ is a $G_\delta$subset of $P$, then $X$
has a $G_\delta$diagonal and is weakly
developable. We show that if $X \subseteq
\max(P)$ is a $G_\delta$subset of $P$, where
$P$ is a domain but perhaps not a Scott domain,
then $X$ is domainrepresentable,
firstcountable, and is the union of dense,
completely metrizable subspaces. We also
show that there is a domain $P$ such that
$\max(P)$ is the usual space of countable
ordinals and is a $G_\delta$subset of $P$ in
the Scott topology. Finally we show that the
kernel of a measurement on a Scott domain can
consistently be a normal, separable,
nonmetrizable Moore space.


207  A Bilinear Fractional Integral on Compact Lie Groups Chen, Jiecheng; Fan, Dashan
As an analog of a wellknown theorem on the bilinear
fractional integral on $\mathbb{R}^{n}$ by Kenig and Stein,
we establish the similar boundedness
property for a bilinear fractional integral on a compact Lie group. Our
result is also a generalization of our recent theorem
about the
bilinear fractional integral on torus.


217  Recurrence Relations for Strongly $q$LogConvex Polynomials Chen, William Y. C.; Wang, Larry X. W.; Yang, Arthur L. B.
We consider a class of
strongly $q$logconvex polynomials based on a triangular recurrence
relation with linear coefficients, and we show that the Bell
polynomials, the Bessel polynomials, the Ramanujan polynomials and
the Dowling polynomials are strongly $q$logconvex. We also prove
that the Bessel transformation preserves logconvexity.


230  Universal Power Series in $\mathbb{C}^N$ Clouâtre, Raphaël
We establish the existence of power series in $\mathbb{C}^N$ with the property
that the subsequences of the sequence of partial sums uniformly
approach any holomorphic function on any well chosen compact subset
outside the set of convergence of the series. We also show that, in a
certain sense, most series enjoy this property.


237  The Structure of the Unit Group of the Group Algebra ${\mathbb{F}}_{2^k}D_{8}$ Creedon, Leo; Gildea, Joe
Let $RG$ denote the group ring of the group $G$ over
the ring $R$. Using an isomorphism between $RG$ and a
certain ring of $n \times n$ matrices in conjunction with other
techniques, the structure of the unit group of the group algebra
of the dihedral group of order $8$ over any
finite field of chracteristic $2$ is determined in
terms of split extensions of cyclic groups.


244  Homogeneous Suslinian Continua Daniel, D. ; Nikiel, J.; Treybig, L. B.; Tuncali, H. M.; Tymchatyn, E. D.
A continuum is said to be Suslinian if it does not
contain uncountably many
mutually exclusive nondegenerate subcontinua. Fitzpatrick and
Lelek have shown that a metric Suslinian continuum $X$ has the
property that the set of points at which $X$ is connected im
kleinen is dense in $X$. We extend their result to Hausdorff Suslinian continua
and obtain a number of corollaries. In particular, we prove that a homogeneous,
nondegenerate, Suslinian continuum is a simple closed curve and that each separable,
nondegenerate, homogenous, Suslinian continuum is metrizable.


249  A Note about Analytic Solvability of Complex Planar Vector Fields with Degeneracies Dattori da Silva, Paulo L.
This paper deals with the analytic solvability of a special class of
complex vector fields defined on the real plane, where they are
tangent to
a closed real curve, while off the real curve, they are elliptic.


255  On an Identity due to Bump and Diaconis, and Tracy and Widom Dehaye, PaulOlivier
A classical question for a Toeplitz matrix with given symbol is how to
compute asymptotics for the determinants of its reductions to finite
rank. One can also consider how those asymptotics are affected when
shifting an initial set of rows and columns (or, equivalently,
asymptotics of their minors). Bump and Diaconis
obtained a formula for such shifts involving Laguerre polynomials and
sums over symmetric groups. They also showed how the Heine identity
extends for such minors, which makes this question relevant to Random
Matrix Theory. Independently, Tracy and Widom
used the WienerHopf factorization to
express those shifts in terms of products of infinite matrices. We
show directly why those two expressions are equal and uncover some
structure in both formulas that was unknown to their authors. We
introduce a mysterious differential operator on symmetric functions
that is very similar to vertex operators. We show that the
BumpDiaconisTracyWidom identity is a differentiated version of the
classical JacobiTrudi identity.


270  Sequential Order Under PFA Dow, Alan
It is shown that it follows from PFA
that there is no
compact scattered space of height greater than $\omega$
in which the sequential order and the scattering heights coincide.


277  Maximal Sublattices of Finite Distributive Lattices. III: A Conjecture from the 1984 Banff Conference on Graphs and Order Farley, Jonathan David
Let $L$ be a finite distributive lattice. Let
$\operatorname{Sub}_0(L)$ be the lattice
$$
\{S\mid S\text{ is a sublattice of }L\}\cup\{\emptyset\}
$$
and let $\ell_*[\operatorname{Sub}_0(L)]$ be the length of the shortest maximal chain in $\operatorname{Sub}_0(L)$. It is proved that if $K$ and $L$ are nontrivial finite distributive lattices, then
$$
\ell_*[\operatorname{Sub}_0(K\times L)]=\ell_*[\operatorname{Sub}_0(K)]+\ell_*[\operatorname{Sub}_0(L)].
$$
A conjecture from the 1984 Banff Conference on Graphs and Order is thus proved.


283  Surgery on $\widetilde{\mathbb{SL}} \times \mathbb{E}^n$Manifolds Hillman, J. A.; Roushon, S. K.
We show that closed $\widetilde{\mathbb{SL}} \times \mathbb{E}^n$manifolds
are topologically rigid if $n\geq 2$, and are rigid up to
$s$cobordism, if $n=1$.


288  The Resultant of Chebyshev Polynomials Jacobs, David P.; Rayes, Mohamed O.; Trevisan, Vilmar
Let $T_{n}$ denote the $n$th
Chebyshev polynomial of the first kind,
and let $U_{n}$ denote the $n$th
Chebyshev polynomial of the second kind.
We give an explicit formula for the resultant
$\operatorname{res}( T_{m}, T_{n} )$.
Similarly, we give a formula for
$\operatorname{res}( U_{m}, U_{n} )$.


297  Lie Powers and PseudoIdempotents Johnson, Marianne; Stöhr, Ralph
We give a new factorisation of the classical Dynkin operator,
an element of the integral group ring of the symmetric group that
facilitates projections of tensor powers onto Lie powers.
As an application we show that the iterated Lie power $L_2(L_n)$ is
a module direct summand of the Lie power $L_{2n}$ whenever the
characteristic of the ground field does not divide $n$. An explicit
projection of the latter onto the former is exhibited in this case.


302  Structure of the Set of Normattaining Functionals on Strictly Convex Spaces Kurka, Ondřej
Let $X$ be a separable nonreflexive Banach space. We show that there
is no Borel class which contains the set of normattaining functionals
for every strictly convex renorming of $X$.


311  Some Remarks Concerning the Topological Characterization of Limit Sets for Surface Flows Marzougui, Habib
We give some extension to theorems of Jiménez López and Soler López concerning the topological characterization for limit sets of continuous flows on closed orientable surfaces.


316  The SaddlePoint Method and the Li Coefficients Mazhouda, Kamel
In this paper, we apply the saddlepoint method in conjunction with
the theory of the NörlundRice integrals to derive precise
asymptotic formula for the generalized Li coefficients established
by Omar and Mazhouda.
Actually, for any function $F$ in the Selberg class
$\mathcal{S}$ and under the Generalized Riemann Hypothesis, we have
$$
\lambda_{F}(n)=\frac{d_{F}}{2}n\log n+c_{F}n+O(\sqrt{n}\log n),
$$
with
$$
c_{F}=\frac{d_{F}}{2}(\gamma1)+\frac{1}{2}\log(\lambda
Q_{F}^{2}),\ \lambda=\prod_{j=1}^{r}\lambda_{j}^{2\lambda_{j}},
$$
where $\gamma$ is the Euler's constant and the notation is as below.


330  Sur la borne inférieure du rang du 2groupe de classes de certains corps multiquadratiques Mouhib, A.
Soient $p_1,p_2,p_3$ et $q$ des nombres premiers distincts tels que
$p_1\equiv p_2\equiv p_3\equiv q\equiv 1 \pmod{4}$, $k = \mathbf{Q}
(\sqrt{p_1}, \sqrt{p_2}, \sqrt{p_3}, \sqrt{q})$ et $\operatorname{Cl}_2(k)$ le
$2$groupe de classes de $k$. A. Fröhlich a
démontré que $\operatorname{Cl}_2(k)$ n'est jamais trivial. Dans cet article,
nous donnons une extension de ce résultat, en démontrant que le
rang de $\operatorname{Cl}_2(k)$ est toujours supérieur ou égal à $2$. Nous
démontrons aussi, que la valeur $2$ est optimale pour une famille
infinie de corps $k$.


338  Szegö's Theorem and Uniform Algebras Nakazi, Takahiko
We study Szegö's theorem for a uniform algebra.
In particular, we do it for the disc algebra or the bidisc algebra.


347  The Haar System in the Preduals of Hyperfinite Factors Potapov, D.; Sukochev, F.
We shall present examples of Schauder bases in the preduals to the
hyperfinite factors of types~$\hbox{II}_1$, $\hbox{II}_\infty$,
$\hbox{III}_\lambda$, $0 < \lambda \leq 1$. In the semifinite
(respectively, purely infinite) setting, these systems form Schauder bases
in any associated separable symmetric space of measurable operators
(respectively, in any noncommutative $L^p$space).


364  Lyapunov Theorems for the Asymptotic Behavior of Evolution Families on the HalfLine Preda, Ciprian; Preda, Petre
Two theorems regarding the
asymptotic behavior of evolution families are established in
terms of the solutions of a certain Lyapunov operator equation.


370  ManifoldValued Holomorphic Approximation Stout, Edgar Lee
This note considers the problem of
approximating continuous maps from sets in complex spaces into complex
manifolds by holomorphic maps.


381  A Short Note on the Higher Level Version of the KrullBaer Theorem Velušček, Dejan
Klep and Velu\v{s}\v{c}ek generalized the KrullBaer theorem for
higher level preorderings to the noncommutative setting. A $n$real valuation
$v$ on a skew field $D$ induces a group homomorphism $\overline{v}$. A section
of $\overline{v}$ is a crucial ingredient of the construction of a complete
preordering on the base field $D$ such that its projection on the residue skew
field $k_v$ equals the given level $1$ ordering on $k_v$. In the article we give
a proof of the existence of the section of $\overline{v}$, which was left as an
open problem by Klep and Velu\v{s}\v{c}ek, and thus
complete the generalization of the KrullBaer theorem for preorderings.


385  Irreducible Representations of Inner Quasidiagonal $C^*$Algebras Blackadar, Bruce; Kirchberg, Eberhard
It is shown that a separable $C^*$algebra is inner quasidiagonal if and
only if it has a separating family of quasidiagonal irreducible
representations. As a consequence, a separable $C^*$algebra is a strong
NF algebra if and only if it is nuclear and has a separating family of
quasidiagonal irreducible representations.
We also obtain some permanence properties of the class of inner
quasidiagonal $C^*$algebras.


396  Parabolic Geodesics in Sasakian $3$Manifolds Cho, Jong Taek; Inoguchi, Junichi; Lee, JiEun
We give explicit parametrizations for all
parabolic geodesics in 3dimensional Sasakian space forms.


411  Operator Algebras with Unique Preduals Davidson, Kenneth R.; Wright, Alex
We show that every free semigroup algebra has a (strongly) unique
Banach space predual. We also provide a new simpler proof that a
weak$*$ closed unital operator algebra containing a weak$*$
dense subalgebra of compact operators has a unique Banach space
predual.


422  Two Conditions on the Structure Jacobi Operator for Real Hypersurfaces in Complex Projective Space Pérez, Juan de Dios; Suh, Young Jin
We classify real hypersurfaces in complex projective space whose
structure Jacobi operator satisfies two conditions at the same time.


430  Complete Families of Linearly Nondegenerate Rational Curves DeLand, Matthew
We prove that every complete family of linearly nondegenerate
rational curves of degree $e > 2$ in $\mathbb{P}^{n}$ has at most $n1$
moduli. For $e = 2$ we prove that such a family has at most $n$
moduli. The general method involves exhibiting a map from the base of
a family $X$ to the Grassmannian of $e$planes in $\mathbb{P}^{n}$ and
analyzing the resulting map on cohomology.


442  Nondegeneracy for Lie Triple Systems and Kantor Pairs García, Esther; Lozano, Miguel Gómez; Neher, Erhard
We study the transfer of nondegeneracy
between Lie triple systems and their standard Lie algebra envelopes
as well as between Kantor pairs, their associated Lie triple systems,
and their Lie algebra envelopes. We also show that simple Kantor
pairs and Lie triple systems in characteristic $0$ are
nondegenerate.


456  On Operator Sum and Product Adjoints and Closures Gustafson, Karl
We comment on domain conditions that regulate when the adjoint of the
sum or product of two unbounded operators is the sum or product of their
adjoints, and related closure issues. The quantum mechanical problem PHP
essentially selfadjoint for unbounded Hamiltonians is addressed, with new
results.


464  A Characterization of the CompoundExponential Type Distributions Hwang, TeaYuan; Hu, ChinYuan
In this paper, a fixed point equation of the
compoundexponential type distributions is derived, and under some
regular conditions,
both the existence and uniqueness of
this fixed point equation are investigated.
A question posed by Pitman and Yor
can be partially answered by using our approach.


472  A Semiregularity Map Annihilating Obstructions to Deforming Holomorphic Maps Iacono, Donatella
We study infinitesimal deformations of holomorphic maps of
compact, complex, Kähler manifolds. In particular, we describe a
generalization of Bloch's semiregularity map that annihilates
obstructions to deform holomorphic maps with fixed codomain.


487  Some Properties Associated with Adequate Transversals Kong, Xiangjun
In this paper, another relationship between the quasiideal adequate transversals
of an abundant semigroup is given. We introduce the concept of a weakly multiplicative
adequate transversal and the classic result that an adequate transversal is multiplicative
if and only if it is weakly multiplicative and a quasiideal is obtained.
Also, we give two equivalent conditions for an adequate transversal to be weakly
multiplicative. We then consider the case when $I$ and $\Lambda$ (defined below) are
bands. This is analogous to the inverse transversal if the regularity condition is adjoined.


498  On the Adjoint and the Closure of the Sum of Two Unbounded Operators Mortad, Mohammed Hichem
We prove, under some conditions on the domains, that the adjoint of
the sum of two unbounded operators is the sum of their adjoints in
both Hilbert and Banach space settings. A similar result about the
closure of operators is also proved. Some interesting consequences
and examples "spice up" the paper.


506  On the Canonical Solution of the SturmLiouville Problem with Singularity and Turning Point of Even Order Neamaty, A.; Mosazadeh, S.
In this paper, we are going to investigate the canonical property of solutions of
systems of differential equations having a singularity and turning
point of even order. First, by a replacement, we transform the system
to the SturmLiouville equation with turning point. Using of the
asymptotic estimates provided by Eberhard, Freiling, and Schneider
for a special fundamental system of solutions of the SturmLiouville
equation, we study the infinite product representation of solutions of the systems. Then we
transform the SturmLiouville equation with
turning point to the
equation with singularity, then we study the asymptotic behavior of its solutions. Such
representations are relevant to the inverse spectral problem.


519  Erratum: Cartan Subalgebras of $\mathrm{gl}_\infty$ Neeb, K. H.; Penkov, I.
We correct an oversight in the the paper
Cartan Subalgebras of
$\mathrm{gl}_\infty$, Canad. Math. Bull. 46(2003), no. 4,
597616.
doi: 10.4153/CMB20030561


520  Simple Helices on Fano Threefolds Polishchuk, A.
Building on the work of Nogin,
we prove that the braid group $B_4$ acts transitively on full exceptional
collections of vector bundles on Fano threefolds with $b_2=1$ and
$b_3=0$. Equivalently,
this group acts transitively on the set of simple helices (considered
up to a shift in the derived category) on such a Fano threefold. We
also prove that on
threefolds with $b_2=1$ and very ample anticanonical class, every
exceptional coherent
sheaf is locally free.


527  On the Dichotomy of the Evolution Families: A DiscreteArgument Approach Preda, Ciprian; Sipos, Ciprian
We establish a discretetime criteria guaranteeing the existence of an
exponential dichotomy in the continuoustime
behavior of an abstract evolution family. We prove that an evolution
family ${\cal U}=\{U(t,s)\}_{t
\geq s\geq 0}$ acting on a Banach space $X$ is uniformly
exponentially dichotomic (with respect to its continuoustime
behavior) if and only if the
corresponding difference equation with the inhomogeneous term from
a vectorvalued Orlicz sequence space $l^\Phi(\mathbb{N}, X)$
admits
a solution in the same $l^\Phi(\mathbb{N},X)$. The technique of
proof effectively eliminates the continuity hypothesis on the
evolution family (i.e., we do not assume that $U(\,\cdot\,,s)x$
or $U(t,\,\cdot\,)x$ is continuous on $[s,\infty)$, and respectively
$[0,t]$). Thus, some known results given by
Coffman and Schaffer, Perron, and Ta Li are extended.


538  On the Horizontal Monotonicity of $\Gamma(s)$ Srinivasan, Gopala Krishna; Zvengrowski, P.
Writing $s = \sigma + it$ for a complex variable, it is proved
that the modulus of the gamma
function, $\Gamma(s)$, is strictly monotone increasing with
respect to $\sigma$ whenever
$t > 5/4$. It is also shown that this result is false for $t
\leq 1$.


544  Positive Definite Measures with Discrete Fourier Transform and Pure Point Diffraction Strungaru, Nicolae
In this paper we characterize the positive
definite measures with discrete Fourier transform. As an
application we provide a characterization of pure point
diffraction in locally compact Abelian groups.


556  Cyclic Surgery Between Toroidal Surgeries Teragaito, Masakazu
We show that there is an infinite family of hyperbolic knots such that
each knot admits a cyclic surgery $m$ whose adjacent surgeries $m1$
and $m+1$ are toroidal. This gives an affirmative answer to a
question asked by Boyer and Zhang.


561  A Note on Toric Varieties Associated with Moduli Spaces Uren, James J.
In this note we give a brief review of the construction of a toric
variety $\mathcal{V}$ coming from a genus $g \geq 2$ Riemann surface
$\Sigma^g$ equipped with a trinion, or pair of pants, decomposition.
This was outlined by J. Hurtubise and L.~C. Jeffrey.
A. Tyurin used this construction on a certain
collection of trinion decomposed surfaces to produce a variety
$DM_g$, the socalled Delzant model of moduli space, for
each genus $g.$ We conclude this note with some basic facts about
the moment polytopes of the varieties $\mathcal{V}.$ In particular,
we show that the varieties $DM_g$ constructed by Tyurin, and claimed
to be smooth, are in fact singular for $g \geq 3.$


566  Nonuniform Randomized Sampling for Multivariate Approximation by High Order Parzen Windows Zhou, XiangJun; Shi, Lei; Zhou, DingXuan
We consider approximation of multivariate functions in Sobolev
spaces by high order Parzen windows in a nonuniform sampling
setting. Sampling points are neither i.i.d. nor regular, but are
noised from regular grids by nonuniform shifts of a probability
density function. Sample function values at sampling points are
drawn according to probability measures with expected values being
values of the approximated function. The approximation orders are
estimated by means of regularity of the approximated function, the
density function, and the order of the Parzen windows, under
suitable choices of the scaling parameter.


577  Erratum: The Duality Problem For The Class of AMCompact Operators On Banach Lattices Aqzzouz, Belmesnaoui
It is proved that if a positive operator
$S: E \rightarrow F$ is AMcompact whenever its adjoint
$S': F' \rightarrow E'$ is AMcompact, then either the
norm of F is order continuous or $E'$ is discrete.


580  Kiguradzetype Oscillation Theorems for Second Order Superlinear Dynamic Equations on Time Scales Baoguo, Jia; Erbe, Lynn; Peterson, Allan
Consider the second order superlinear dynamic equation
\begin{equation*}
(*)\qquad
x^{\Delta\Delta}(t)+p(t)f(x(\sigma(t)))=0\tag{$*$}
\end{equation*}
where $p\in C(\mathbb{T},\mathbb{R})$, $\mathbb{T}$ is a time scale,
$f\colon\mathbb{R}\rightarrow\mathbb{R}$ is
continuously differentiable and satisfies $f'(x)>0$, and $xf(x)>0$ for
$x\neq 0$. Furthermore, $f(x)$ also satisfies a superlinear condition, which
includes the nonlinear function $f(x)=x^\alpha$ with $\alpha>1$, commonly
known as the EmdenFowler case. Here the coefficient function $p(t)$ is
allowed to be negative for arbitrarily large values of $t$. In addition to
extending the result of Kiguradze for \eqref{star1} in the real case $\mathbb{T}=\mathbb{R}$, we
obtain analogues in the difference equation and $q$difference equation cases.


593  Stability of Real $C^*$Algebras Boersema, Jeffrey L.; Ruiz, Efren
We will give a characterization of stable real $C^*$algebras
analogous to the one given for complex $C^*$algebras by Hjelmborg
and Rørdam. Using this result, we will prove
that any real $C^*$algebra satisfying the corona factorization
property is stable if and only if its complexification is stable.
Real $C^*$algebras satisfying the corona factorization property
include AFalgebras and purely infinite $C^*$algebras. We will also
provide an example of a simple unstable $C^*$algebra, the
complexification of which is stable.


607  Lightness of Induced Maps and Homeomorphisms Camargo, Javier
An example is given of a map $f$ defined between arcwise connected continua such that $C(f)$ is light and
$2^{f}$ is not light, giving a negative answer to a question of Charatonik and Charatonik. Furthermore, given a positive
integer $n$, we study when the lightness of the induced map $2^{f}$ or $C_n(f)$ implies that $f$ is a
homeomorphism. Finally, we show a result in relation with the lightness of $C(C(f))$.


619  Artinian and NonArtinian Local Cohomology Modules Dibaei, Mohammad T.; Vahidi, Alireza
Let $M$ be a finite module over a commutative noetherian ring $R$.
For ideals $\mathfrak{a}$ and $\mathfrak{b}$ of $R$, the relations between
cohomological dimensions of $M$ with respect to $\mathfrak{a},
\mathfrak{b}$,
$\mathfrak{a}\cap\mathfrak{b}$ and $\mathfrak{a}+ \mathfrak{b}$ are studied. When $R$ is local, it is
shown that $M$ is generalized CohenMacaulay if there exists an
ideal $\mathfrak{a}$ such that all local cohomology modules of $M$ with
respect to $\mathfrak{a}$ have finite lengths. Also, when $r$ is an integer
such that $0\leq r< \dim_R(M)$, any maximal element $\mathfrak{q}$ of the
nonempty set of ideals $\{\mathfrak{a} : \textrm{H}_\mathfrak{a}^i(M)
$ is not artinian for
some $ i, i\geq r \}$ is a prime ideal, and all Bass numbers
of $\textrm{H}_\mathfrak{q}^i(M)$ are finite for all $i\geq r$.


630  Mixed Norm Type Hardy Inequalities Fiorenza, Alberto; Gupta, Babita; Jain, Pankaj
Higher dimensional mixed norm type
inequalities involving certain integral operators are
characterized in terms of the corresponding lower dimensional
inequalities.


645  An Extension of Craig's Family of Lattices Flores, André Luiz; Interlando, J. Carmelo; Neto, Trajano Pires da Nóbrega
Let $p$ be a prime, and let $\zeta_p$ be a primitive $p$th root of
unity. The lattices in Craig's family are $(p1)$dimensional and
are geometrical representations of the integral $\mathbb
Z[\zeta_p]$ideals $\langle 1\zeta_p \rangle^i$, where $i$ is a
positive integer. This lattice construction technique is a powerful
one. Indeed, in dimensions $p1$ where $149 \leq p \leq 3001$,
Craig's lattices are the densest packings known. Motivated by this,
we construct $(p1)(q1)$dimensional lattices from the integral
$\mathbb Z[\zeta _{pq}]$ideals $\langle 1\zeta_p \rangle^i \langle
1\zeta_q \rangle^j$, where $p$ and $q$ are distinct primes and $i$
and $j$ are positive integers. In terms of spherepacking density,
the new lattices and those in Craig's family have the same
asymptotic behavior. In conclusion, Craig's family is greatly
extended while preserving its spherepacking properties.


654  Norm One Idempotent $cb$Multipliers with Applications to the Fourier Algebra in the $cb$Multiplier Norm Forrest, Brian E.; Runde, Volker
For a locally compact group $G$, let $A(G)$ be its Fourier algebra, let $M_{cb}A(G)$ denote the completely
bounded multipliers of $A(G)$, and let $A_{\mathit{Mcb}}(G)$ stand for the closure of $A(G)$ in $M_{cb}A(G)$. We
characterize the norm one idempotents in $M_{cb}A(G)$: the indicator function of a set $E \subset G$ is a norm
one idempotent in $M_{cb}A(G)$ if and only if $E$ is a coset of an open subgroup of $G$. As applications, we
describe the closed ideals of $A_{\mathit{Mcb}}(G)$ with an approximate identity bounded by $1$, and we characterize
those $G$ for which $A_{\mathit{Mcb}}(G)$ is $1$amenable in the sense of B. E. Johnson. (We can even slightly
relax the norm bounds.)


663  Admissible Sequences for Twisted Involutions in Weyl Groups Haas, Ruth; G. Helminck, Aloysius
Let $W$ be a Weyl group, $\Sigma$ a set of simple reflections in $W$
related to a basis $\Delta$ for the root system $\Phi$ associated with
$W$ and $\theta$ an involution such that $\theta(\Delta) = \Delta$. We
show that the set of $\theta$twisted involutions in $W$,
$\mathcal{I}_{\theta} = \{w\in W \mid \theta(w) = w^{1}\}$ is in one
to one correspondence with the set of regular involutions
$\mathcal{I}_{\operatorname{Id}}$. The elements of $\mathcal{I}_{\theta}$ are
characterized by sequences in $\Sigma$ which induce an ordering called
the RichardsonSpringer Poset. In particular, for $\Phi$ irreducible,
the ascending RichardsonSpringer Poset of $\mathcal{I}_{\theta}$,
for nontrivial $\theta$ is identical to the descending
RichardsonSpringer Poset of $\mathcal{I}_{\operatorname{Id}}$.


676  Quasiisometry and Plaque Expansiveness Hammerlindl, Andy
We show that a partially hyperbolic diffeomorphism is plaque
expansive (a form of structural stability for its center foliation) if the
strong stable and unstable foliations are quasiisometric in the universal
cover. In particular, all partially hyperbolic diffeomorphisms on the 3torus
are plaque expansive.


680  $2$Local Isometries on Spaces of Lipschitz Functions JiménezVargas, A.; VillegasVallecillos, Moisés
Let $(X,d)$ be a metric space, and let $\mathop{\textrm{Lip}}(X)$ denote the Banach
space of all scalarvalued bounded Lipschitz functions $f$ on $X$
endowed with one of the natural norms
$
\ f\ =\max \{\ f\ _\infty ,L(f)\}$ or $\f\ =\
f\ _\infty +L(f),
$
where $L(f)$ is the
Lipschitz constant of $f.$ It is said that the isometry
group of $\mathop{\textrm{Lip}}(X)$ is canonical if every
surjective linear isometry of
$\mathop{\textrm{Lip}}(X) $ is induced by a surjective isometry of $X$.
In this paper
we prove that if $X$ is bounded separable and the isometry group of
$\mathop{\textrm{Lip}}(X)$ is canonical, then every $2$local isometry
of $\mathop{\textrm{Lip}}(X)$ is
a surjective linear isometry. Furthermore, we give a complete
description of all $2$local isometries of $\mathop{\textrm{Lip}}(X)$ when $X$ is
bounded.


693  Stratified Subcartesian Spaces Lusala, Tsasa; Śniatycki, Jędrzej
We show that if the family $\mathcal{O}$ of orbits of all vector fields on
a subcartesian space $P$ is locally finite and each orbit in $\mathcal{O}$
is locally closed, then $\mathcal{O}$ defines a smooth Whitney A
stratification of $P$. We also show that the stratification by orbit type of
the space of orbits $M/G$ of a proper action of a Lie group $G$ on a smooth
manifold $M$ is given by orbits of the family of all vector fields on $M/G$.


706  Nonconstant Continuous Functions whose Tangential Derivative Vanishes along a Smooth Curve Moonens, Laurent
We provide a simple example showing that the tangential derivative of a
continuous function $\phi$
can vanish everywhere along a curve while the variation of $\phi$ along
this curve is nonzero. We give additional regularity conditions on the curve
and/or the function that prevent this from happening.


716  Symplectic LieRinehartJacobi Algebras and Contact Manifolds Okassa, Eugène
We give a characterization of contact manifolds in terms of symplectic
LieRinehartJacobi algebras. We also give a sufficient condition for a Jacobi
manifold to be a contact manifold.


726  Auerbach Bases and Minimal Volume Sufficient Enlargements Ostrovskii, M. I.
Let $B_Y$ denote the unit ball of a
normed linear space $Y$. A symmetric, bounded, closed, convex set
$A$ in a finite dimensional normed linear space $X$ is called a
sufficient enlargement for $X$ if, for an arbitrary
isometric embedding of $X$ into a Banach space $Y$, there exists a
linear projection $P\colon Y\to X$ such that $P(B_Y)\subset A$. Each
finite dimensional normed space has a minimalvolume sufficient
enlargement that is a parallelepiped; some spaces have ``exotic''
minimalvolume sufficient enlargements. The main result of the
paper is a characterization of spaces having ``exotic''
minimalvolume sufficient enlargements in terms of Auerbach
bases.


739  The Infimum in the Metric Mahler Measure Samuels, Charles L.
Dubickas and Smyth defined the metric Mahler measure on the
multiplicative group of nonzero algebraic numbers.
The definition involves taking an infimum over representations
of an algebraic number $\alpha$ by other
algebraic numbers. We verify their conjecture that the
infimum in its definition is always achieved, and we establish its
analog for the ultrametric Mahler measure.


748  On the Distribution of Irreducible Trinomials Shparlinski, Igor E.
We obtain new results about the number of trinomials $t^n + at + b$
with integer coefficients in a box $(a,b) \in [C, C+A] \times [D,
D+B]$ that are irreducible modulo a prime $p$. As a byproduct we
show that for any $p$ there are irreducible polynomials of height at
most $p^{1/2+o(1)}$, improving on the previous estimate of
$p^{2/3+o(1)}$ obtained by the author in 1989.


757  Cancellation of Cusp Forms Coefficients over Beatty Sequences on $\textrm{GL}(m)$ Sun, Qingfeng
Let $A(n_1,n_2,\dots,n_{m1})$
be the normalized Fourier coefficients of
a Maass cusp form on $\textrm{GL}(m)$.
In this paper, we study the cancellation of $A
(n_1,n_2,\dots,n_{m1})$ over Beatty sequences.

