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449  On the Graph of Divisibility of an Integral Domain Boynton, Jason Greene; Coykendall, Jim
It is well known that the factorization properties of a domain are reflected
in the structure of its group of divisibility. The main theme of this paper
is to introduce a topological/graphtheoretic point of view to the current
understanding of factorization in integral domains. We also show that
connectedness properties in the graph and topological space give rise to a
generalization of atomicity.


459  Hyperplanes in the Space of Convergent Sequences and Preduals of $\ell_1$ Casini, Emanuele; Miglierina, Enrico; Piasecki, Lukasz
The main aim of the present paper is to investigate various structural
properties
of hyperplanes of $c$, the Banach space of the convergent sequences.
In particular, we give an explicit formula for the projection
constants and we prove that an hyperplane of $c$ is isometric
to the whole space if and only if it is $1$complemented. Moreover,
we obtain the classification
of those hyperplanes for which their duals are isometric to
$\ell_{1}$ and we give a complete description of the preduals
of $\ell_{1}$ under the assumption that the standard basis of
$\ell_{1}$
is weak$^{*}$convergent.


471  Almost Sure Global Wellposedness for the Fractional Cubic Schrödinger Equation on Torus Demirbas, Seckin
In a previous paper, we proved that $1$d periodic fractional
Schrödinger equation with cubic nonlinearity is locally wellposed
in $H^s$ for $s\gt \frac{1\alpha}{2}$ and globally wellposed for
$s\gt \frac{10\alpha1}{12}$. In this paper we define an invariant
probability measure $\mu$ on $H^s$ for $s\lt \alpha\frac{1}{2}$,
so that for any $\epsilon\gt 0$ there is a set $\Omega\subset H^s$
such that $\mu(\Omega^c)\lt \epsilon$ and the equation is globally
wellposed for initial data in $\Omega$. We see that this fills
the gap between the local wellposedness and the global wellposedness
range in almost sure sense for $\frac{1\alpha}{2}\lt \alpha\frac{1}{2}$,
i.e. $\alpha\gt \frac{2}{3}$ in almost sure sense.


486  Inequalities for Partial Derivatives and their Applications Duc, Dinh Thanh; Nhan, Nguyen Du Vi; Xuan, Nguyen Tong
We present various weighted integral inequalities for partial
derivatives acting on products and compositions of functions
which are applied to establish some new Opialtype inequalities
involving functions of several independent variables. We also
demonstrate the usefulness of our results in the field of partial
differential equations.


497  Constructing Double Magma on Groups Using Commutation Operations Edmunds, Charles C.
A magma $(M,\star)$ is a nonempty set with a binary
operation. A double magma $(M, \star, \bullet)$ is a
nonempty set with two binary operations satisfying the
interchange law,
$(w \star x) \bullet (y\star z)=(w\bullet y)\star(x \bullet
z)$. We call a double magma proper if the two operations
are distinct and commutative if the operations are commutative.
A double semigroup, first introduced by Kock,
is a double magma for which both operations are associative.
Given a nontrivial group $G$ we define a system of two magma
$(G,\star,\bullet)$ using the commutator operations $x \star
y = [x,y](=x^{1}y^{1}xy)$ and $x\bullet y = [y,x]$. We show
that $(G,\star,\bullet)$ is a double magma if and only if $G$
satisfies the commutator laws $[x,y;x,z]=1$ and $[w,x;y,z]^{2}=1$.
We note that the first law defines the class of 3metabelian
groups. If both these laws hold in $G$, the double magma is proper
if and only if there exist $x_0,y_0 \in G$ for which $[x_0,y_0]^2
\not= 1$. This double magma is a double semigroup if and only
if $G$ is nilpotent of class two. We construct a specific example
of a proper double semigroup based on the dihedral group of order
16. In addition we comment on a similar construction for rings
using Lie commutators.


507  VMO Space Associated with Parabolic Sections and its Application Hsu, MingHsiu; Lee, MingYi
In this paper we define $VMO_\mathcal{P}$ space associated with
a family $\mathcal{P}$ of parabolic sections and show that the
dual of $VMO_\mathcal{P}$ is the Hardy space $H^1_\mathcal{P}$.
As an application, we prove that almost everywhere convergence
of a bounded sequence in $H^1_\mathcal{P}$ implies weak* convergence.


519  Refined Motivic Dimension Kang, SuJeong
We define a refined motivic dimension for an algebraic variety
by modifying the definition of motivic dimension by Arapura.
We apply this to check and recheck the generalized Hodge conjecture
for certain varieties, such as uniruled, rationally connected
varieties and a rational surface fibration.


530  Ricci Curvature Tensor and NonRiemannian Quantities Li, Benling; Shen, Zhongmin
There are several notions of Ricci curvature tensor
in Finsler geometry and spray geometry. One of them is defined by the
Hessian of the wellknown Ricci curvature.
In this paper we will introduce a new notion of Ricci curvature
tensor and discuss its relationship with the Ricci curvature and some
nonRiemannian quantities. By this Ricci curvature tensor, we shall
have a better understanding on these nonRiemannian quantities.


538  Minimal Non Self Dual Groups Li, Lili; Chen, Guiyun
A group $G$ is self dual if every
subgroup
of $G$ is isomorphic to a quotient of $G$ and every quotient
of $G$ is isomorphic to
a subgroup of $G$. It is minimal nonself dual if every
proper subgroup of $G$
is self dual but $G$ is not self dual. In this paper, the structure
of minimal nonself dual groups is determined.


548  Higher Moments of Fourier Coefficients of Cusp Forms Lü, Guangshi; Sankaranarayanan, Ayyadurai
Let $S_{k}(\Gamma)$ be the space of holomorphic cusp
forms of even integral weight $k$ for the full modular group
$SL(2, \mathbb{Z})$. Let
$\lambda_f(n)$, $\lambda_g(n)$, $\lambda_h(n)$ be the $n$th normalized
Fourier
coefficients of three distinct holomorphic primitive cusp forms
$f(z) \in S_{k_1}(\Gamma), g(z) \in S_{k_2}(\Gamma), h(z) \in
S_{k_3}(\Gamma)$ respectively.
In this paper we study the cancellations of sums related to arithmetic
functions, such as $\lambda_f(n)^4\lambda_g(n)^2$, $\lambda_g(n)^6$,
$\lambda_g(n)^2\lambda_h(n)^4$, and $\lambda_g(n^3)^2$ twisted
by
the arithmetic function $\lambda_f(n)$.


561  Plane Lorentzian and Fuchsian Hedgehogs MartinezMaure, Yves
Parts of the BrunnMinkowski theory can be extended to hedgehogs, which are
envelopes of families of affine hyperplanes parametrized by their Gauss map.
F. Fillastre introduced Fuchsian convex bodies, which are the
closed convex sets of LorentzMinkowski space that are globally invariant
under the action of a Fuchsian group. In this paper, we undertake a study of
plane Lorentzian and Fuchsian hedgehogs. In particular, we prove the
Fuchsian analogues of classical geometrical inequalities (analogues which
are reversed as compared to classical ones).


575  The Diffeomorphism Type of Canonical Integrations Of Poisson Tensors on Surfaces MartinezTorres, David
A surface $\Sigma$ endowed with a Poisson tensor
$\pi$ is known to admit
canonical integration, $\mathcal{G}(\pi)$,
which is a 4dimensional manifold with a (symplectic) Lie groupoid
structure.
In this short note we show that if $\pi$ is not an area
form on the 2sphere, then $\mathcal{G}(\pi)$ is diffeomorphic
to the cotangent bundle $T^*\Sigma$. This extends
results by the author and by Bonechi, Ciccoli, Staffolani, and Tarlini.


580  A Specialisation of the BumpFriedberg $L$function Matringe, Nadir
We study the restriction of the BumpFriedberg integrals to affine
lines $\{(s+\alpha,2s),s\in\mathbb{C}\}$.
It has a simple theory, very close to that of the Asai $L$function.
It is an integral representation of the product
$L(s+\alpha,\pi)L(2s,\Lambda^2,\pi)$ which we denote by $L^{lin}(s,\pi,\alpha)$
for this abstract, when $\pi$ is a cuspidal automorphic
representation of $GL(k,\mathbb{A})$ for
$\mathbb{A}$ the adeles of a number field. When $k$ is even, we show
that for a cuspidal automorphic representation $\pi$,
the partial $L$function $L^{lin,S}(s,\pi,\alpha)$ has a pole
at $1/2$, if and only if $\pi$ admits a (twisted) global
period, this gives a more direct proof of a
theorem of Jacquet and Friedberg, asserting
that $\pi$ has a twisted global period if and only if $L(\alpha+1/2,\pi)\neq
0$ and $L(1,\Lambda^2,\pi)=\infty$.
When $k$ is odd, the partial $L$function is holmorphic in a
neighbourhood of $Re(s)\geq 1/2$ when $Re(\alpha)$ is
$\geq 0$.


596  A Note on Planarity Stratification of Hurwitz Spaces Ongaro, Jared; Shapiro, Boris
One can easily show that any meromorphic function
on a complex closed Riemann surface can be represented as a
composition of a birational map of this surface to $\mathbb{CP}^2$ and
a projection of the image curve from an appropriate point
$p\in \mathbb{CP}^2$ to the pencil of lines through $p$. We introduce
a natural stratification of Hurwitz spaces according to the
minimal degree of a plane curve such that a given meromorphic
function can be represented in the above way and calculate the
dimensions of these strata. We observe that they are closely
related to a family of Severi varieties studied earlier by J. Harris,
Z. Ran and I. Tyomkin.


610  Path Decompositions of Kneser and Generalized Kneser Graphs Rodger, C. A.; Whitt, Thomas Richard III
Necessary and sufficient conditions are given for the existence
of a graph decomposition of the Kneser Graph $KG_{n,2}$ and of
the Generalized Kneser Graph $GKG_{n,3,1}$ into paths of length
three.


620  $L$functions for Quadratic Characters and Annihilation of Motivic Cohomology Groups Sands, Jonathan W.
Let $n$ be a positive even integer, and let $F$ be a totally real
number field and $L$ be an abelian Galois extension which is totally
real or CM.
Fix a finite set $S$ of primes of $F$ containing the infinite primes
and all those which ramify in
$L$, and let $S_L$ denote the primes of $L$ lying above those in
$S$. Then $\mathcal{O}_L^S$ denotes the ring of $S_L$integers of $L$.
Suppose that $\psi$ is a quadratic character of the Galois group of
$L$ over $F$. Under the assumption of the motivic Lichtenbaum
conjecture, we obtain a nontrivial annihilator of the motivic
cohomology group
$H_\mathcal{M}^2(\mathcal{O}_L^S,\mathbb{Z}(n))$ from the lead term of the Taylor series for the
$S$modified Artin $L$function $L_{L/F}^S(s,\psi)$ at $s=1n$.


632  Quantum Unique Ergodicity on Locally Symmetric Spaces: the Degenerate Lift Silberman, Lior
Given a measure $\bar\mu_\infty$ on a locally symmetric space $Y=\Gamma\backslash
G/K$,
obtained as a weak{*} limit of probability measures associated
to
eigenfunctions of the ring of invariant differential operators,
we
construct a measure $\bar\mu_\infty$ on the homogeneous space $X=\Gamma\backslash
G$
which lifts $\bar\mu_\infty$ and which is invariant by a connected subgroup
$A_{1}\subset A$ of positive dimension, where $G=NAK$ is an Iwasawa
decomposition. If the functions are, in addition, eigenfunctions
of
the Hecke operators, then $\bar\mu_\infty$ is also the limit of measures
associated
to Hecke eigenfunctions on $X$. This generalizes results of the
author
with A. Venkatesh in the case where the spectral parameters
stay
away from the walls of the Weyl chamber.


651  Ground State Solutions of NehariPankov Type for a Superlinear Hamiltonian Elliptic System on ${\mathbb{R}}^{N}$ Tang, Xianhua
This paper is concerned with the following
elliptic system of Hamiltonian type
\[
\left\{
\begin{array}{ll}
\triangle u+V(x)u=W_{v}(x, u, v), \ \ \ \ x\in {\mathbb{R}}^{N},
\\
\triangle v+V(x)v=W_{u}(x, u, v), \ \ \ \ x\in {\mathbb{R}}^{N},
\\
u, v\in H^{1}({\mathbb{R}}^{N}),
\end{array}
\right.
\]
where the potential $V$ is periodic and $0$ lies in a gap of
the spectrum of $\Delta+V$, $W(x, s, t)$ is
periodic in $x$ and superlinear in $s$ and $t$ at infinity.
We develop a direct approach to find ground
state solutions of NehariPankov type for the above system.
Especially, our method is applicable for the
case when
\[
W(x, u, v)=\sum_{i=1}^{k}\int_{0}^{\alpha_iu+\beta_iv}g_i(x,
t)t\mathrm{d}t
+\sum_{j=1}^{l}\int_{0}^{\sqrt{u^2+2b_juv+a_jv^2}}h_j(x,
t)t\mathrm{d}t,
\]
where $\alpha_i, \beta_i, a_j, b_j\in \mathbb{R}$ with $\alpha_i^2+\beta_i^2\ne
0$ and $a_j\gt b_j^2$, $g_i(x, t)$
and $h_j(x, t)$ are nondecreasing in $t\in \mathbb{R}^{+}$ for every
$x\in \mathbb{R}^N$ and $g_i(x, 0)=h_j(x, 0)=0$.


664  Betti Numbers and Flat Dimensions of Local Cohomology Modules Vahidi, Alireza
Assume that $R$ is a commutative Noetherian ring with nonzero
identity, $\mathfrak{a}$ is an ideal of $R$ and $X$ is an $R$module.
In this paper, we first study the finiteness of Betti numbers
of local cohomology modules $\operatorname{H}_\mathfrak{a}^i(X)$. Then we give some
inequalities between the Betti numbers of $X$ and those of its
local cohomology modules. Finally, we present many upper bounds
for the flat dimension of $X$ in terms of the flat dimensions
of its local cohomology modules and an upper bound for the flat
dimension of $\operatorname{H}_\mathfrak{a}^i(X)$ in terms of the flat dimensions of
the modules $\operatorname{H}_\mathfrak{a}^j(X)$, $j\not= i$, and that of $X$.

