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161  A New Tautological Relation in $\overline{\mathcal{M}}_{3,1}$ via the Invariance Constraint Arcara, D.; Lee, Y.P.
A new tautological relation of $\overline{\mathcal{M}}_{3,1}$ in codimension 3
is derived and proved, using an invariance constraint from
our previous work.


175  Connections on a Parabolic Principal Bundle, II Biswas, Indranil
In Connections on a parabolic principal bundle over a curve, I
we defined connections on a parabolic
principal bundle. While connections on usual principal bundles are
defined as splittings of the Atiyah exact sequence, it was noted in
the above article that the Atiyah exact sequence does not generalize to
the parabolic principal bundles.
Here we show that a twisted version
of the Atiyah exact sequence generalizes to the context of
parabolic principal bundles. For usual principal bundles, giving a
splitting of this twisted Atiyah exact sequence is equivalent
to giving a splitting of the Atiyah exact sequence. Connections on
a parabolic principal bundle can be defined using the
generalization of the twisted Atiyah exact sequence.


186  Extension of the Riemann $\xi$Function's Logarithmic Derivative Positivity Region to Near the Critical Strip Broughan, Kevin A.
If $K$ is a number field with $n_k=[k:\mathbb{Q}]$, and $\xi_k$
the symmetrized
Dedekind zeta function of the field, the inequality
$$\Re\,{\frac{ \xi_k'(\sigma + {\rm i} t)}{\xi_k(\sigma
+ {\rm i} t)}} > \frac{ \xi_k'(\sigma)}{\xi_k(\sigma)}$$ for $t\neq 0$ is
shown
to be true for $\sigma\ge 1+ 8/n_k^\frac{1}{3}$ improving the result of
Lagarias where the constant in the inequality was 9. In the case $k=\mathbb{Q}$
the
inequality is extended to $\si\ge 1$ for all $t$ sufficiently large or small
and to the region $\si\ge 1+1/(\log t 5)$ for all $t\neq 0$. This
answers positively a question posed by Lagarias.


195  The Waring Problem with the Ramanujan $\tau$Function, II Garaev, M. Z.; Garcia, V. C.; Konyagin, S. V.
Let $\tau(n)$ be the Ramanujan $\tau$function. We prove that for
any integer $N$ with $N\ge 2$ the diophantine equation
$$\sum_{i=1}^{148000}\tau(n_i)=N$$ has a solution in positive
integers $n_1, n_2,\ldots, n_{148000}$ satisfying the condition
$$\max_{1\le i\le 148000}n_i\ll N^{2/11}e^{c\log N/\log\log
N},$$ for some absolute constant $c>0.$


200  Schubert Calculus on a Grassmann Algebra Gatto, Letterio; Santiago, Ta\'\i se
The ({\em classical}, {\em small quantum}, {\em equivariant})
cohomology ring of the grassmannian $G(k,n)$ is generated by
certain derivations operating on an exterior algebra of a free
module of rank $n$ ( Schubert calculus on a Grassmann
algebra). Our main result gives, in a unified way, a presentation
of all such cohomology rings in terms of generators and
relations. Using results of Laksov and Thorup, it also provides
a presentation of the universal
factorization algebra of a monic polynomial of degree $n$ into the
product of two monic polynomials, one of degree $k$.


213  DunfordPettis Properties and Spaces of Operators Ghenciu, Ioana; Lewis, Paul
J. Elton used an application of Ramsey theory to show that
if $X$ is an infinite dimensional Banach space,
then $c_0$ embeds in $X$, $\ell_1$ embeds in $X$, or there
is a subspace of $X$ that fails to have the DunfordPettis property.
Bessaga and Pelczynski showed that if $c_0$ embeds in $X^*$,
then $\ell_\infty$ embeds in $X^*$. Emmanuele and John showed
that if $c_0$ embeds in $K(X,Y)$, then $K(X,Y)$ is not
complemented in $L(X,Y)$. Classical results from Schauder basis theory
are used in a study of DunfordPettis sets and strong
DunfordPettis sets to extend each of the preceding theorems. The space
$L_{w^*}(X^* , Y)$ of $w^*w$ continuous operators is also studied.


224  Equations and Complexity for the DuboisEfroymson Dimension Theorem Ghiloni, Riccardo
Let $\R$ be a real closed field, let $X \subset \R^n$ be an
irreducible real algebraic set and let $Z$ be an algebraic subset of
$X$ of codimension $\geq 2$. Dubois and Efroymson proved the existence
of an irreducible algebraic subset of $X$ of codimension $1$
containing~$Z$. We improve this dimension theorem as follows. Indicate
by $\mu$ the minimum integer such that the ideal of polynomials in
$\R[x_1,\ldots,x_n]$ vanishing on $Z$ can be generated by polynomials
of degree $\leq \mu$. We prove the following two results:
\begin{inparaenum}[\rm(1)]
\item There
exists a polynomial $P \in \R[x_1,\ldots,x_n]$ of degree~$\leq \mu+1$
such that $X \cap P^{1}(0)$ is an irreducible algebraic subset of $X$
of codimension $1$ containing~$Z$.
\item Let $F$ be a polynomial in
$\R[x_1,\ldots,x_n]$ of degree~$d$ vanishing on $Z$. Suppose there
exists a nonsingular point $x$ of $X$ such that $F(x)=0$ and the
differential at $x$ of the restriction of $F$ to $X$ is nonzero. Then
there exists a polynomial $G \in \R[x_1,\ldots,x_n]$ of degree $\leq
\max\{d,\mu+1\}$ such that, for each $t \in (1,1) \setminus \{0\}$,
the set $\{x \in X \mid F(x)+tG(x)=0\}$ is an irreducible algebraic
subset of $X$ of codimension $1$ containing~$Z$.
\end{inparaenum} Result (1) and a
slightly different version of result~(2) are valid over any
algebraically closed field also.


237  Points of Small Height on Varieties Defined over a Function Field Ghioca, Dragos
We obtain a Bogomolov type of result for the affine space defined
over the algebraic closure of a function field of transcendence
degree $1$ over a finite field.


245  Involutions of RA Loops Goodaire, Edgar G.; Milies, CĂ©sar Polcino
Let $L$ be an RA loop, that is, a loop whose loop ring
over any coefficient ring $R$
is an alternative, but not associative, ring. Let
$\ell\mapsto\ell^\theta$ denote an involution on $L$ and extend
it linearly to the loop ring $RL$. An element $\alpha\in RL$ is
symmetric if $\alpha^\theta=\alpha$ and skewsymmetric
if $\alpha^\theta=\alpha$. In this paper, we show that
there exists an involution making
the symmetric elements of $RL$ commute if and only if
the characteristic of $R$ is $2$ or $\theta$ is the
canonical involution on $L$,
and an involution making the skewsymmetric elements of $RL$
commute if and only if
the characteristic of $R$ is $2$ or $4$.


257  Essential Surfaces in Graph Link Exteriors Ikeda, Toru
An irreducible graph manifold $M$ contains an essential torus if
it is not a special Seifert manifold.
Whether $M$ contains a closed essential surface of
negative Euler characteristic or not
depends on the difference of Seifert fibrations from the two sides
of a torus system which splits $M$ into Seifert manifolds.
However,
it is not easy to characterize geometrically the class of
irreducible graph manifolds which contain such surfaces.
This article studies this problem in the case of graph link exteriors.


267  Extensions of Rings Having McCoy Condition Ko\c{s}an, Muhammet Tamer
Let $R$ be an associative ring with unity.
Then $R$ is said to be a {\it right McCoy ring} when the equation
$f(x)g(x)=0$ (over $R[x]$), where $0\neq f(x),g(x) \in R[x]$,
implies that there exists a nonzero element $c\in R$ such that
$f(x)c=0$. In this paper, we characterize some basic ring
extensions of right McCoy rings and we prove that if $R$ is a
right McCoy ring, then $R[x]/(x^n)$ is
a right McCoy ring for any positive integer $n\geq 2$ .


273  Amalgamations of Categories MacDonald, John; Scull, Laura
We consider the pushout of embedding functors in $\Cat$, the
category of small categories.
We show that if the embedding functors satisfy a 3for2
property, then the induced functors to the pushout category are
also embeddings. The result follows from the connectedness of
certain associated slice categories. The condition is motivated
by a similar result for maps of semigroups. We show that our
theorem can be applied to groupoids and to inclusions of full
subcategories. We also give an example to show that the theorem
does not hold when the
property only holds for one of the inclusion functors, or when it
is weakened to a onesided condition.


285  Global Geometrical Coordinates on Falbel's CrossRatio Variety Parker, John R.; Platis, Ioannis D.
Falbel has shown that four pairwise distinct points on the boundary
of a
complex hyperbolic $2$space are completely determined, up to conjugation
in ${\rm PU}(2,1)$, by three complex crossratios satisfying two real
equations. We give global geometrical coordinates on the resulting
variety.


295  On Functions Whose Graph is a Hamel Basis, II P{\l}otka, Krzysztof
We say that a function $h \from \real \to \real$ is a Hamel function
($h \in \ham$) if $h$, considered as a subset of $\real^2$, is a Hamel
basis for $\real^2$. We show that $\A(\ham)\geq\omega$, i.e., for
every finite $F \subseteq \real^\real$ there exists $f\in\real^\real$
such that $f+F \subseteq \ham$. From the previous work of the author
it then follows that $\A(\ham)=\omega$.


303  A Comment on ``$\mathfrak{p} < \mathfrak{t}$'' Shelah, Saharon
Dealing with the cardinal invariants ${\mathfrak p}$ and
${\mathfrak t}$ of the continuum, we prove that
${\mathfrak m}={\mathfrak p} = \aleph_2\ \Rightarrow\ {\mathfrak t} =\aleph_2$.
In other words, if ${\bf MA}_{\aleph_1}$ (or a weak version of
this) holds, then (of course $\aleph_2\le {\mathfrak p}\le
{\mathfrak t}$ and) ${\mathfrak p}=\aleph_2\ \Rightarrow\
{\mathfrak p}={\mathfrak t}$. The proof is based on a criterion
for ${\mathfrak p}<{\mathfrak t}$.


315  Generic QuasiConvergence for Essentially Strongly OrderPreserving Semiflows Yi, Taishan; Zou, Xingfu
By employing the limit set
dichotomy for essentially strongly orderpreserving semiflows and
the assumption that limit sets have infima and suprema in the
state space, we prove a generic quasiconvergence principle
implying the existence of an open and dense set of stable
quasiconvergent points. We also apply this generic quasiconvergence principle
to a model for biochemical feedback in protein
synthesis and obtain some results about the model which are of theoretical
and realistic significance.

