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3  On a Local Theory of Asymptotic Integration for Nonlinear Differential Equations Agarwal, Ravi P.; Mustafa, Octavian G.
We improve several recent results in the
asymptotic integration theory of nonlinear ordinary differential
equations via a variant of the method devised by J. K. Hale and
N. Onuchic The results
are used for investigating the existence of positive solutions to
certain reactiondiffusion equations.


15  Browder's Convergence for OneParameter Nonexpansive Semigroups Akiyama, Shigeki; Suzuki, Tomonari
We give the sufficient and necessary conditions
of Browder's convergence theorem
for oneparameter nonexpansive semigroups
which was proved by Suzuki.
We also discuss the perfect kernels of topological spaces.


26  A Mahler Measure of a $K3$ Surface Expressed as a Dirichlet $L$Series Bertin, Marie José
We present another example of a $3$variable polynomial defining a $K3$hypersurface
and having a logarithmic Mahler measure expressed in terms of a Dirichlet
$L$series.


38  Endomorphisms of Two Dimensional Jacobians and Related Finite Algebras Butske, William
Zarhin proves that if $C$ is the curve $y^2=f(x)$ where
$\textrm{Gal}_{\mathbb{Q}}(f(x))=S_n$ or $A_n$, then
${\textrm{End}}_{\overline{\mathbb{Q}}}(J)=\mathbb{Z}$. In seeking to examine his
result in the genus $g=2$ case supposing other Galois groups, we
calculate
$\textrm{End}_{\overline{\mathbb{Q}}}(J)\otimes_{\mathbb{Z}} \mathbb{F}_2$
for a genus $2$ curve where $f(x)$ is irreducible.
In particular, we show that unless the Galois group is $S_5$ or
$A_5$, the Galois group does not determine ${\textrm{End}}_{\overline{\mathbb{Q}}}(J)$.


48  Freyd's Generating Hypothesis for Groups with Periodic Cohomology Chebolu, Sunil K.; Christensen, J. Daniel; Mináč, Ján
Let $G$ be a finite group, and let $k$ be a field whose characteristic $p$
divides
the order of $G$.
Freyd's generating hypothesis for the stable module category of
$G$ is the statement that a map between finitedimensional
$kG$modules in the thick subcategory generated by $k$ factors through a
projective if the induced map on Tate cohomology is trivial. We show that if
$G$
has periodic cohomology, then the generating hypothesis holds if and only if
the Sylow
$p$subgroup of $G$ is $C_2$ or $C_3$. We also give some other conditions
that are equivalent to the GH
for groups with periodic cohomology.


60  Extension of Some Theorems of W. Schwarz Coons, Michael
In this paper, we prove that a nonzero power series $F(z)\in\mathbb{C}
[\mkern3mu[ z]\mkern3mu]
$
satisfying $$F(z^d)=F(z)+\frac{A(z)}{B(z)},$$ where $d\geq 2$, $A(z),B(z)\in\mathbb{C}[z]$
with $A(z)\neq 0$ and $\deg A(z),\deg B(z)<d$ is transcendental over $\mathbb{C}(z)$. Using
this result and a theorem of Mahler's, we extend results of Golomb and Schwarz on
transcendental values of certain power series. In particular, we prove that for all $k\geq 2$ the
series $G_k(z):= \sum_{n=0}^\infty z^{k^n}(1z^{k^n})^{1}$ is transcendental for all algebraic
numbers $z$ with $z<1$. We give a similar result for $F_k(z):= \sum_{n=0}^\infty z^{k^n}
(1+z^{k^n})^{1}$. These results were known to Mahler, though our proofs of the function
transcendence are new and elementary; no linear algebra or differential calculus is used.


67  An $E_8$ Correspondence for Multiplicative EtaProducts Cummins, C. J.; Duncan, J. F.
We describe an $E_8$ correspondence for the multiplicative
etaproducts of weight at least $4$.


73  Classification of Inductive Limits of Outer Actions of ${\mathbb R}$ on Approximate Circle Algebras Dean, Andrew J.
In this paper we present a classification,
up to equivariant isomorphism, of $C^*$dynamical systems $(A,{\mathbb R},\alpha )$
arising as inductive limits of directed systems
$\{ (A_n,{\mathbb R},\alpha_n),\varphi_{nm}\}$, where each $A_n$
is a finite direct sum of matrix algebras over the continuous
functions on the unit circle, and the $\alpha_n$s are outer actions
generated by rotation of the spectrum.


81  Cofiniteness of Generalized Local Cohomology Modules for OneDimensional Ideals DivaaniAazar, Kamran; Hajikarimi, Alireza
Let $\mathfrak a$ be an ideal of a commutative Noetherian
ring $R$ and $M$ and $N$ two finitely generated $R$modules. Our
main result asserts that if $\dim R/\mathfrak a\leq 1$, then all generalized
local cohomology modules $H^i_{\mathfrak a}(M,N)$ are $\mathfrak a$cofinite.


88  Inequalities for Eigenvalues of a General Clamped Plate Problem Ghanbari, K.; Shekarbeigi, B.
Let $D$ be a
connected bounded domain in $\mathbb{R}^n$. Let
$0<\mu_1\leq\mu_2\leq\dots\leq\mu_k\leq\cdots$ be the eigenvalues
of the following Dirichlet
problem:
$$
\begin{cases}\Delta^2u(x)+V(x)u(x)=\mu\rho(x)u(x),\quad x\in
D
u_{\partial D}=\frac{\partial u}{\partial n}_{\partial
D}=0,
\end{cases}
$$
where $V(x)$ is a nonnegative potential,
and $\rho(x)\in C(\bar{D})$ is positive.
We prove the following inequalities:
$$\mu_{k+1}\leq\frac{1}{k}\sum_{i=1}^k\mu_i+\Bigl[\frac{8(n+2)}{n^2}\Bigl(\frac{\rho_{\max}}
{\rho_{\min}}\Bigr)^2\Bigr]^{1/2}\times
\frac{1}{k}\sum_{i=1}^k[\mu_i(\mu_{k+1}\mu_i)]^{1/2},
$$
$$\frac{n^2k^2}{8(n+2)}\leq
\Bigl(\frac{\rho_{\max}}{\rho_{\min}}\Bigr)^2\Bigl[\sum_{i=1}^k\frac{\mu_i^{1/2}}{\mu_{k+1}\mu_i}\Bigr]
\times\sum_{i=1}^k\mu_i^{1/2}.
$$


98  Similarity and Coincidence Isometries for Modules Glied, Svenja
The groups of (linear) similarity and coincidence isometries of
certain modules $\varGamma$ in $d$dimensional Euclidean space, which
naturally occur in quasicrystallography, are considered. It is shown
that the structure of the factor group of similarity modulo
coincidence isometries is the direct sum of cyclic groups of prime
power orders that divide $d$. In particular, if the dimension $d$ is a
prime number $p$, the factor group is an elementary abelian
$p$group. This generalizes previous results obtained for lattices to
situations relevant in quasicrystallography.


108  On Segre Forms of Positive Vector Bundles Guler, Dincer
The goal of this note is to prove that the signed Segre forms of Griffiths' positive vector bundles are
positive.


114  On Characterizations of Real Hypersurfaces in a Complex Space Form with $\eta$Parallel Shape Operator Kon, S. H.; Loo, TeeHow
In this paper we study real hypersurfaces in a nonflat complex space form with $\eta$parallel shape operator. Several partial characterizations of these real hypersurfaces are obtained.


127  Characterizations of Three Classes of ZeroDivisor Graphs LaGrange, John D.
The zerodivisor graph $\Gamma(R)$ of a commutative ring $R$ is the graph whose vertices consist of
the nonzero zerodivisors of $R$ such that distinct vertices $x$ and
$y$ are adjacent if and only if $xy=0$. In this paper,
a characterization is provided for zerodivisor graphs of Boolean
rings. Also, commutative rings $R$ such that
$\Gamma(R)$ is isomorphic to the zerodivisor graph of a direct product of integral domains are classified, as well as
those whose zerodivisor graphs are central vertex complete.


138  Projectively Flat Fourth Root Finsler Metrics Li, Benling; Shen, Zhongmin
In this paper, we study locally projectively flat fourth root
Finsler metrics and their generalized metrics. We prove that if they
are irreducible, then they must be locally Minkowskian.


146  A Characterization of Bergman Spaces on the Unit Ball of ${\mathbb C}^n$. II Li, Songxiao; Wulan, Hasi; Zhu, Kehe
It has been shown that a holomorphic function $f$ in the unit ball
$\mathbb{B}_n$ of ${\mathbb C}_n$ belongs to the weighted Bergman space $A^p_\alpha$,
$p>n+1+\alpha$, if and only if the function
$f(z)f(w)/1\langle z,w\rangle$ is in $L^p(\mathbb{B}_n\times\mathbb{B}_n,dv_\beta
\times dv_\beta)$, where $\beta=(p+\alphan1)/2$ and $dv_\beta(z)=
(1z^2)^\beta\,dv(z)$. In this paper
we consider the range $0<p<n+1+\alpha$ and show that in this case,
$f\in A^p_\alpha$ (i)~if and only if the function $f(z)f(w)/1\langle z,
w\rangle$ is in $L^p(\mathbb{B}_n\times\mathbb{B}_n,dv_\alpha\times
dv_\alpha)$,
(ii)~if and only
if the function $f(z)f(w)/zw$ is in $L^p(\mathbb{B}_n\times\mathbb{B}_n,dv_\alpha\times
dv_\alpha)$. We think the revealed difference in the weights for the double
integrals between the cases $0<p<n+1+\alpha$ and $p>n+1+\alpha$ is
particularly interesting.


153  Artinianness of Certain Graded Local Cohomology Modules Mafi, Amir; Saremi, Hero
We show that if
$R=\bigoplus_{n\in\mathbb{N}_0}R_n$ is a Noetherian homogeneous ring
with local base ring $(R_0,\mathfrak{m}_0)$, irrelevant ideal $R_+$, and
$M$ a finitely generated graded $R$module, then
$H_{\mathfrak{m}_0R}^j(H_{R_+}^t(M))$ is Artinian for $j=0,1$ where
$t=\inf\{i\in{\mathbb{N}_0}: H_{R_+}^i(M)$ is not finitely
generated $\}$. Also, we prove that if $\operatorname{cd}(R_+,M)=2$, then for
each $i\in\mathbb{N}_0$, $H_{\mathfrak{m}_0R}^i(H_{R_+}^2(M))$ is
Artinian if and only if $H_{\mathfrak{m}_0R}^{i+2}(H_{R_+}^1(M))$ is
Artinian, where $\operatorname{cd}(R_+,M)$ is the cohomological dimension of $M$
with respect to $R_+$. This improves some results of R. Sazeedeh.


157  Subdivisions of Simplicial Complexes Preserving the Metric Topology Mine, Kotaro; Sakai, Katsuro
Let $K$ be the metric polyhedron of a simplicial complex $K$.
In this paper,
we characterize a simplicial subdivision $K'$ of $K$
preserving the metric topology for $K$ as the one such that
the set $K'{}^{(0)}$ of vertices of $K'$ is discrete in $K$.
We also prove that two such subdivisions of $K$
have such a common subdivision.


164  Involutions Fixing $F^n \cup \{\text{Indecomposable}\}$ Pergher, Pedro L. Q.
Let $M^m$ be an $m$dimensional, closed and smooth manifold, equipped with a smooth involution $T\colon M^m \to M^m$ whose fixed point set has the form $F^n \cup F^j$, where $F^n$ and $F^j$ are submanifolds with dimensions $n$ and $j$, $F^j$ is indecomposable and $ n >j$. Write $nj=2^pq$, where $q \ge 1$ is odd and $p \geq 0$, and set $m(nj) = 2n+pq+1$ if $p \leq q + 1$
and $m(nj)= 2n + 2^{pq}$ if $p \geq q$. In this paper we show that $m \le m(nj) + 2j+1$. Further, we show that this bound is almost best possible, by exhibiting examples $(M^{m(nj) +2j},T)$ where the fixed point set of
$T$ has the form $F^n \cup F^j$ described above, for every $2 \le j <n$ and $j$ not of the form $2^t1$ (for $j=0$ and $2$, it has been previously shown that $m(nj) +2j$ is the best possible bound). The existence of these bounds is guaranteed by the famous $5/2$theorem of J. Boardman, which establishes that under the above hypotheses $m \le \frac{5} {2}n$.


172  Hausdorff Prime Matrices Rhoades, B. E.
In this paper we give the form of every multiplicative Hausdorff
prime matrix, thus answering a longstanding open question.


176  Linear Dispersive Decay Estimates for the 3+1 Dimensional Water Wave Equation with Surface Tension Spirn, Daniel; Wright, J. Douglas
We consider the linearization of the threedimensional water waves
equation with surface tension about a flat interface. Using
oscillatory integral methods, we prove that solutions of this equation
demonstrate dispersive decay at the somewhat surprising rate of
$t^{5/6}$. This rate is due to competition between surface tension
and gravitation at $O(1)$ wave numbers and is connected to the fact
that, in the presence of surface tension, there is a socalled "slowest
wave". Additionally, we combine our dispersive estimates with $L^2$
type energy bounds to prove a family of Strichartz estimates.


188  Yet Another Solution to the Burnside Problem for Matrix Semigroups Steinberg, Benjamin
We use the kernel category to give a finiteness condition for semigroups. As a consequence we provide yet another proof that finitely generated periodic semigroups of matrices are finite.


193  Rational Points in Arithmetic Progressions on $y^2=x^n+k$ Ulas, Maciej
Let $C$ be a hyperelliptic curve given by the equation $y^2=f(x)$
for $f\in\mathbb{Z}[x]$ without multiple roots. We say that points
$P_{i}=(x_{i}, y_{i})\in C(\mathbb{Q})$ for $i=1,2,\dots, m$ are in
arithmetic progression if the numbers $x_{i}$ for $i=1,2,\dots, m$
are in arithmetic progression.


208  Abelian Gradings on Upper Block Triangular Matrices Valenti, Angela; Zaicev, Mikhail
Let $G$ be an arbitrary finite abelian group. We describe all
possible $G$gradings on upper block triangular matrix algebras
over an algebraically closed field of characteristic zero.


214  Positive Solutions of Impulsive Dynamic System on Time Scales Wang, DaBin
In this paper, some criteria for the existence of positive solutions of a class
of systems of impulsive dynamic equations on time scales are obtained by
using a fixed point theorem in cones.


225  Limit Sets of Typical Homeomorphisms Bernardes, Nilson C.
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
$f \colon X \to X$, it is true that for $\mu$almost every point $x$ in $X$
the limit set $\omega(f,x)$ is a Cantor set of Hausdorff dimension zero,
each point of $\omega(f,x)$ has a dense orbit in $\omega(f,x)$, $f$ is
nonsensitive at each point of $\omega(f,x)$, and the function
$a \to \omega(f,a)$ is continuous at $x$.


233  On Algebraically Maximal Valued Fields and Defectless Extensions Bishnoi, Anuj; Khanduja, Sudesh K.
Let $v$ be a Henselian Krull valuation of a field $K$. In this paper,
the authors give some necessary and sufficient conditions for a
finite simple extension of $(K,v)$ to be defectless. Various
characterizations of algebraically maximal valued fields are also
given which lead to a new proof of a result proved by Yu. L. Ershov.


242  Convergence in Capacity Cegrell, Urban
In this note we study the convergence of sequences of MongeAmpère measures $\{(dd^cu_s)^n\}$,
where $\{u_s\}$ is a given sequence of plurisubharmonic functions, converging in capacity.


249  Description of Entire Solutions of Eiconal Type Equations Chang, DerChen; Li, Bao Qin
The paper describes entire solutions to the eiconal type nonlinear partial differential
equations, which include the eiconal equations $(X_1(u))^2+(X_2(u))^2=1$ as special cases,
where
$X_1=p_1{\partial}/{\partial z_1}+p_2{\partial}/{\partial z_2}$,
$X_2=p_3{\partial}/{\partial z_1}+p_4{\partial}/{\partial z_2}$
are linearly independent operators with $p_j$ being arbitrary
polynomials in $\mathbf{C}^2$.


260  A Note on the Antipode for Algebraic Quantum Groups Delvaux, L.; Van Daele, A.; Wang, Shuanhong
Recently, Beattie, Bulacu ,and Torrecillas proved Radford's formula for the fourth power of the antipode for a coFrobenius Hopf algebra.


271  On the Existence of the Graded Exponent for Finite Dimensional $\mathbb{Z}_p$graded Algebras Di Vincenzo, M. Onofrio; Nardozza, Vincenzo
Let $F$ be an algebraically closed field of characteristic zero, and
let $A$ be an associative unitary $F$algebra graded by a group of
prime order. We prove that if $A$ is finite dimensional then the
graded exponent of $A$ exists and is an integer.


285  Uniqueness Implies Existence and Uniqueness Conditions for a Class of $(k+j)$Point Boundary Value Problems for $n$th Order Differential Equations Eloe, Paul W.; Henderson, Johnny; Khan, Rahmat Ali
For the $n$th order nonlinear differential equation, $y^{(n)} = f(x, y, y',
\dots, y^{(n1)})$, we consider uniqueness implies uniqueness and existence
results for solutions satisfying certain $(k+j)$point
boundary conditions for $1\le j \le n1$ and $1\leq k \leq nj$. We
define $(k;j)$point unique solvability in analogy to $k$point
disconjugacy and we show that $(nj_{0};j_{0})$point
unique solvability implies $(k;j)$point unique solvability for $1\le j \le
j_{0}$, and $1\leq k \leq nj$. This result is
analogous to
$n$point disconjugacy implies $k$point disconjugacy for $2\le k\le
n1$.


297  The Group $\operatorname{Aut}(\mu)$ is Roelcke Precompact Glasner, Eli
Following a similar result of Uspenskij on the unitary group of a
separable Hilbert space, we show that, with respect to the lower (or
Roelcke) uniform structure, the Polish group $G=
\operatorname{Aut}(\mu)$ of automorphisms of an atomless standard
Borel probability space $(X,\mu)$ is precompact. We identify the
corresponding compactification as the space of Markov operators on
$L_2(\mu)$ and deduce that the algebra of right and left uniformly
continuous functions, the algebra of weakly almost periodic functions,
and the algebra of Hilbert functions on $G$, i.e., functions on
$G$ arising from unitary representations, all coincide. Again
following Uspenskij, we also conclude that $G$ is totally minimal.


303  Atomic Decomposition and Boundedness of Operators on Weighted Hardy Spaces Han, Yongsheng; Lee, MingYi; Lin, ChinCheng
In this article, we establish a new atomic decomposition for $f\in L^2_w\cap H^p_w$,
where the decomposition converges in $L^2_w$norm rather than in the distribution sense.
As applications of this decomposition, assuming that $T$ is a linear
operator bounded on $L^2_w$ and $0<p\le 1$, we obtain
(i) if $T$ is uniformly bounded in $L^p_w$norm for all $w$$p$atoms,
then $T$ can be extended to be bounded from $H^p_w$ to $L^p_w$;
(ii) if $T$ is uniformly bounded in $H^p_w$norm for all $w$$p$atoms,
then $T$ can be extended to be bounded on $H^p_w$;
(iii) if $T$ is bounded on $H^p_w$,
then $T$ can be extended to be bounded from $H^p_w$ to $L^p_w$.


315  A Note on the Vanishing of Certain Local Cohomology Modules Hellus, M.
For a finite module $M$ over a local, equicharacteristic ring $(R,m)$,
we show that the wellknown formula $\textrm{cd}(m,M)=\dim M$ becomes trivial
if ones uses Matlis duals of local cohomology modules together with spectral sequences.
We also prove a new ringtheoretic vanishing criterion for local cohomology modules.


319  The Verdier Hypercovering Theorem Jardine, J. F.
This note gives a simple cocycletheoretic proof of the Verdier
hypercovering theorem. This theorem approximates morphisms $[X,Y]$ in the
homotopy category of simplicial sheaves or presheaves by simplicial
homotopy classes of maps, in the case where $Y$ is locally fibrant. The
statement proved in this paper is a generalization of the standard
Verdier hypercovering result in that it is pointed (in a very broad
sense) and there is no requirement for the source object $X$ to be
locally fibrant.


329  NonDiscrete Complex Hyperbolic Triangle Groups of Type $(n,n, \infty;k)$ Kamiya, Shigeyasu; Parker, John R.; Thompson, James M.
A complex hyperbolic triangle group is a group
generated by three involutions fixing complex lines in complex
hyperbolic space. Our purpose in this paper is to improve a previous result
and to discuss discreteness of complex hyperbolic
triangle groups of type $(n,n,\infty;k)$.


339  From Matrix to Operator Inequalities Loring, Terry A.
We generalize Löwner's method for proving that matrix monotone
functions are operator monotone. The relation $x\leq y$ on bounded
operators is our model for a definition of $C^{*}$relations
being residually finite dimensional.


351  Rational Homogeneous Algebras MacDougall, J. A.; Sweet, L. G.
An algebra $A$ is <em>homogeneous</em> if the automorphism group of $A$
acts transitively on the onedimensional subspaces of $A$. The existence of homogeneous algebras depends critically on the choice of the scalar field. We examine the case where the scalar field is the rationals. We prove that if $A$ is a rational homogeneous algebra with $\operatorname{dim} A>1$, then $A^{2}=0$.


355  Convolution Inequalities in $l_p$ Weighted Spaces Nhan, Nguyen Du Vi; Duc, Dinh Thanh
Various weighted $l_p$norm inequalities in convolutions are derived
by a simple and general principle whose $l_2$ version was obtained by
using the theory of reproducing kernels. Applications to the Riemann zeta
function and a difference equation are also considered.


368  The Secondary ChernEuler Class for a General Submanifold Nie, Zhaohu
We define and study the secondary ChernEuler class for a general submanifold of a Riemannian manifold. Using this class, we define and study the index for a vector field with nonisolated singularities on a submanifold. As an application, we give conceptual proofs of a result of Chern.


378  On Modules Whose Proper Homomorphic Images Are of Smaller Cardinality Oman, Greg; Salminen, Adam
Let $R$ be a commutative ring with identity, and let $M$ be a
unitary module over $R$. We call $M$ Hsmaller (HS for short) if and only if
$M$ is infinite and $M/N<M$ for every nonzero submodule $N$ of
$M$. After a brief introduction, we show that there exist nontrivial
examples of HS modules of arbitrarily large cardinality over
Noetherian and nonNoetherian domains. We then prove the following
result: suppose $M$ is faithful over $R$, $R$ is a domain (we will
show that we can restrict to this case without loss of generality),
and $K$ is the quotient field of $R$. If $M$ is HS over $R$, then
$R$ is HS as a module over itself, $R\subseteq M\subseteq K$, and
there exists a generating set $S$ for $M$ over $R$ with $S<R$.
We use this result to generalize a problem posed by Kaplansky and
conclude the paper by answering an open question on Jónsson
modules.


390  Automorphisms of Iterated Wreath Product $p$Groups Riedl, Jeffrey M.
We determine the order of
the automorphism group
$\operatorname{Aut}(W)$ for each member
$W$ of an important family
of finite $p$groups that
may be constructed as
iterated regular wreath
products of cyclic groups.
We use a method based on
representation theory.


400  Eisenstein Series and Modular Differential Equations Sebbar, Abdellah; Sebbar, Ahmed
The purpose of this paper is to solve various differential
equations having Eisenstein series as coefficients using various tools and techniques. The solutions
are given in terms of modular forms, modular functions, and
equivariant forms.


410  A Ramsey Theorem with an Application to Sequences in Banach Spaces Service, Robert
The notion of a maximally conditional sequence is introduced for sequences in a Banach space. It is then proved using
Ramsey theory that every basic sequence in a Banach space has a subsequence which is either an unconditional
basic sequence or a maximally conditional sequence. An apparently novel, purely combinatorial lemma in the spirit of
Galvin's theorem is used in the proof. An alternative proof
of the dichotomy result for sequences in Banach spaces is
also sketched,
using the GalvinPrikry theorem.


418  Maximal Sets of Pairwise Orthogonal Vectors in Finite Fields Vinh, Le Anh
Given a positive integer $n$, a finite field $\mathbb{F}_q$ of $q$ elements
($q$ odd), and a nondegenerate symmetric bilinear form $B$ on
$\mathbb{F}_q^n$, we determine the largest possible cardinality of pairwise
$B$orthogonal subsets $\mathcal{E} \subseteq \mathbb{F}_q^n$, that is, for
any two vectors $\mathbf{x}, \mathbf{y} \in \mathcal{E}$, one has $B
(\mathbf{x}, \mathbf{y}) = 0$.


424  Convergence Rates of Cascade Algorithms with Infinitely Supported Masks Yang, Jianbin; Li, Song
We investigate the solutions of refinement equations of the form
$$
\phi(x)=\sum_{\alpha\in\mathbb
Z^s}a(\alpha)\:\phi(Mx\alpha),
$$ where the function $\phi$
is in $L_p(\mathbb R^s)$$(1\le p\le\infty)$, $a$ is an infinitely
supported sequence on $\mathbb Z^s$ called a refinement mask, and
$M$ is an $s\times s$ integer matrix such that
$\lim_{n\to\infty}M^{n}=0$. Associated with the mask $a$ and $M$ is
a linear operator $Q_{a,M}$ defined on $L_p(\mathbb R^s)$ by
$Q_{a,M} \phi_0:=\sum_{\alpha\in\mathbb
Z^s}a(\alpha)\phi_0(M\cdot\alpha)$. Main results of this paper are
related to the convergence rates of $(Q_{a,M}^n
\phi_0)_{n=1,2,\dots}$ in $L_p(\mathbb R^s)$ with mask $a$ being
infinitely supported. It is proved that under some appropriate
conditions on the initial function $\phi_0$, $Q_{a,M}^n \phi_0$
converges in $L_p(\mathbb R^s)$ with an exponential rate.


435  A Note on the Diophantine Equation $x^2 + y^6 = z^e$, $e \geq 4$ Zelator, Konstantine
We consider the diophantine equation $x^2 + y^6 = z^e$, $e \geq 4$.
We show that, when $e$ is a multiple of $4$ or $6$, this equation
has no solutions in positive integers with $x$ and $y$ relatively prime.
As a corollary, we show
that there exists no primitive Pythagorean triangle one of whose
leglengths is a perfect cube, while the hypotenuse length is an
integer square.


441  Univalently Induced, Closed Range, Composition Operators on the Blochtype Spaces Zorboska, Nina
While there is a large variety of univalently induced closed range
composition operators on the Bloch space,
we show that the only univalently induced, closed range, composition
operators on the Blochtype spaces $B^{\alpha}$ with $\alpha \ne 1$
are the ones induced by a disc automorphism.


449  Complemented Subspaces of Linear Bounded Operators Bahreini, Manijeh; Bator, Elizabeth; Ghenciu, Ioana
We study the complementation of the space $W(X,Y)$ of weakly compact operators, the space $K(X,Y)$ of compact operators, the space $U(X,Y)$ of unconditionally converging operators, and the space $CC(X,Y)$ of completely continuous operators in the space $L(X,Y)$ of bounded linear operators from $X$ to $Y$.
Feder proved that if $X$ is infinitedimensional and $c_0
\hookrightarrow Y$, then $K(X,Y)$ is uncomplemented in
$L(X,Y)$. Emmanuele and John showed that if $c_0 \hookrightarrow
K(X,Y)$, then $K(X,Y)$ is uncomplemented in $L(X,Y)$.
Bator and Lewis showed that if $X$ is not a Grothendieck space and
$c_0 \hookrightarrow Y$, then $W(X,Y)$ is uncomplemented in
$L(X,Y)$. In this paper, classical results of Kalton and separably
determined operator ideals with property $(*)$ are used to obtain
complementation results that yield these theorems as corollaries.


462  Hookcontent Formulae for Symplectic and Orthogonal Tableaux Campbell, Peter S.; Stokke, Anna
By considering the specialisation
$s_{\lambda}(1,q,q^2,\dots,q^{n1})$ of
the Schur function, Stanley was able to describe a formula for the
number of semistandard Young tableaux of shape $\lambda$ in terms of
the contents and hook lengths of the boxes in the Young diagram.
Using specialisations of symplectic and orthogonal Schur functions,
we derive corresponding formulae,
first given by El Samra and King, for the number of semistandard
symplectic and orthogonal $\lambda$tableaux.


474  A Note on Randers Metrics of Scalar Flag Curvature Chen, Bin; Zhao, Lili
Some families of Randers metrics of scalar flag curvature are
studied in this paper. Explicit examples that are neither locally
projectively flat nor of isotropic $S$curvature are given. Certain
Randers metrics with Einstein $\alpha$ are considered and proved to
be complex. Three dimensional Randers manifolds, with $\alpha$
having constant scalar curvature, are studied.


487  Weighted Model Sets and their Higher PointCorrelations Deng, Xinghua; Moody, Robert V.
Examples of distinct weighted model sets with equal $2,3,4, 5$point
correlations are given.


498  Simplices in the Euclidean Ball Fradelizi, Matthieu; Paouris, Grigoris; Schütt, Carsten
We establish some inequalities for the second moment
$$
\frac{1}{K} \int_{K}x_2^2 \,dx
$$
of a convex body $K$ under various assumptions on the position of $K$.


509  Domains of Injective Holomorphy Gauthier, P. M.; Nestoridis, V.
A domain $\Omega$ is called a domain of injective holomorphy if
there exists an injective holomorphic function
$f\colon \Omega\rightarrow\mathbb{C}$ that is nonextendable. We give examples of
domains that are domains of injective holomorphy and others that
are not. In particular, every regular domain
$(\overline\Omega^o=\Omega)$ is a domain of injective holomorphy, and
every simply connected domain is a domain of injective holomorphy
as well.


523  The MilnorStasheff Filtration on Spaces and Generalized Cyclic Maps Iwase, Norio; Mimura, Mamoru; Oda, Nobuyuki; Yoon, Yeon Soo
The concept of $C_{k}$spaces is introduced, situated at an
intermediate stage between $H$spaces and $T$spaces. The
$C_{k}$space corresponds to the $k$th MilnorStasheff filtration on
spaces. It is proved that a space $X$ is a $C_{k}$space if and only
if the Gottlieb set $G(Z,X)=[Z,X]$ for any space $Z$ with ${\rm cat}\,
Z\le k$, which generalizes the fact that $X$ is a $T$space if and
only if $G(\Sigma B,X)=[\Sigma B,X]$ for any space $B$. Some results
on the $C_{k}$space are generalized to the $C_{k}^{f}$space for a
map $f\colon A \to X$. Projective spaces, lens spaces and spaces with
a few cells are studied as examples of $C_{k}$spaces, and
non$C_{k}$spaces.


537  Asymptotic Properties of Solutions to Semilinear Equations Involving Multiple Critical Exponents Kang, Dongsheng
In this paper, we investigate
a semilinear elliptic equation that involves multiple
Hardytype terms and critical HardySobolev exponents. By the
Moser iteration method and analytic techniques, the asymptotic
properties of its nontrivial solutions at the singular points are
investigated.


548  Noncomplemented Spaces of Operators, Vector Measures, and $c_o$ Lewis, Paul; Schulle, Polly
The Banach spaces $L(X, Y)$, $K(X, Y)$, $L_{w^*}(X^*, Y)$, and
$K_{w^*}(X^*, Y)$ are studied to determine when they contain the
classical Banach spaces $c_o$ or $\ell_\infty$. The complementation of
the Banach space $K(X, Y)$ in $L(X, Y)$ is discussed as well as what
impact this complementation has on the embedding of $c_o$ or
$\ell_\infty$ in $K(X, Y)$ or $L(X, Y)$. Results of Kalton, Feder, and
Emmanuele concerning the complementation of $K(X, Y)$ in $L(X, Y)$ are
generalized. Results concerning the complementation of the Banach
space $K_{w^*}(X^*, Y)$ in $L_{w^*}(X^*, Y)$ are also explored as well
as how that complementation affects the embedding of $c_o$ or
$\ell_\infty$ in $K_{w^*}(X^*, Y)$ or $L_{w^*}(X^*, Y)$. The $\ell_p$
spaces for $1 = p < \infty$ are studied to determine when the space of
compact operators from one $\ell_p$ space to another contains
$c_o$. The paper contains a new result which classifies these spaces
of operators. A new result using vector measures is given to
provide more efficient proofs of theorems by Kalton, Feder, Emmanuele,
Emmanuele and John, and Bator and Lewis.


555  Weighted $L^p$ Boundedness of Pseudodifferential Operators and Applications Michalowski, Nicholas; Rule, David J.; Staubach, Wolfgang
In this paper we prove weighted norm inequalities with weights in
the $A_p$ classes, for pseudodifferential operators with symbols in
the class ${S^{n(\rho 1)}_{\rho, \delta}}$ that fall outside the
scope of CalderónZygmund theory. This is accomplished by
controlling the sharp function of the pseudodifferential operator by
HardyLittlewood type maximal functions. Our weighted norm
inequalities also yield $L^{p}$ boundedness of commutators of
functions of bounded mean oscillation with a wide class of operators
in $\mathrm{OP}S^{m}_{\rho, \delta}$.


571  A Generalised KummerType Transformation for the ${}_pF_p(x)$ Hypergeometric Function Miller, A. R.; Paris, R. B.
In a recent paper, Miller derived a Kummertype
transformation for the generalised hypergeometric function ${}_pF_p(x)$ when pairs of
parameters differ by unity, by means of a reduction
formula for a certain Kampé de Fériet function. An alternative and simpler derivation of this
transformation is obtained here by application of the wellknown Kummer transformation for the
confluent hypergeometric function corresponding to $p=1$.


579  Casimir Operators and Nilpotent Radicals Ndogmo, J. C.
It is shown that a Lie algebra having a nilpotent radical has a
fundamental set of invariants consisting of Casimir operators. A
different proof is given in the well known special case of an
abelian radical. A result relating the number of invariants to the
dimension of the Cartan subalgebra is also established.


586  On Sha's Secondary ChernEuler Class Nie, Zhaohu
For a manifold with boundary, the restriction of Chern's transgression
form of the Euler curvature form over the boundary is closed. Its
cohomology class is called the secondary ChernEuler class and was
used by Sha to formulate a relative PoincaréHopf theorem under the
condition that the metric on the manifold is locally product near the
boundary. We show that the secondary ChernEuler form is exact away
from the outward and inward unit normal vectors of the boundary by
explicitly constructing a transgression form. Using Stokes' theorem,
this evaluates the boundary term in Sha's relative PoincaréHopf
theorem in terms of more classical indices of the tangential
projection of a vector field. This evaluation in particular shows
that Sha's relative PoincaréHopf theorem is equivalent to the more
classical law of vector fields.


597  Sharp Inequalities for Differentially Subordinate Harmonic Functions and Martingales Osękowski, Adam
We determine the best constants $C_{p,\infty}$ and $C_{1,p}$,
$1 < p < \infty$, for which the following holds. If $u$, $v$ are
orthogonal harmonic functions on a Euclidean domain such that $v$ is
differentially subordinate to $u$, then
$$ \v\_p \leq C_{p,\infty}
\u\_\infty,\quad
\v\_1 \leq C_{1,p} \u\_p.
$$
In particular, the inequalities are still sharp for the conjugate
harmonic functions on the unit disc of $\mathbb R^2$.
Sharp probabilistic versions of these estimates are also studied.
As an application, we establish a sharp version of the classical logarithmic inequality of Zygmund.


611  Chen Inequalities for Submanifolds of Real Space Forms with a SemiSymmetric NonMetric Connection Özgür, Cihan; Mihai, Adela
In this paper we prove Chen inequalities for submanifolds of real space
forms endowed with a semisymmetric nonmetric connection, i.e., relations
between the mean curvature associated with a semisymmetric nonmetric
connection, scalar and sectional curvatures, Ricci curvatures and the
sectional curvature of the ambient space. The equality cases are considered.


623  The Continuous Dependence on the Nonlinearities of Solutions of Fast Diffusion Equations Pan, Jiaqing
In this paper, we consider the Cauchy problem
$$
\begin{cases}
u_{t}=\Delta(u^{m}), &x\in{}\mathbb{R}^{N}, t>0, N\geq3,
\\
% ^^ here
u(x,0)=u_{0}(x), &x\in{}\mathbb{R}^{N}.
\end{cases}
$$
We will prove that:


632  Characterizations of Model Manifolds by Means of Certain Differential Systems Pigola, S.; Rimoldi, M.
We prove metric rigidity for complete manifolds supporting solutions of
certain second order differential systems, thus extending classical works on a
characterization of spaceforms. Along the way, we also discover
new characterizations of spaceforms. We next generalize results concerning metric
rigidity via equations involving vector fields.


646  Marcinkiewicz Commutators with Lipschitz Functions in Nonhomogeneous Spaces Zhou, Jiang; Ma, Bolin
Under the assumption that $\mu$ is a nondoubling
measure, we study certain commutators generated by the
Lipschitz function and the Marcinkiewicz integral whose kernel
satisfies a Hörmandertype condition. We establish the boundedness
of these commutators on the Lebesgue spaces, Lipschitz spaces, and
Hardy spaces. Our results are extensions of known theorems in the
doubling case.


663  An Onofritype Inequality on the Sphere with Two Conical Singularities Zhou, Chunqin
In this paper, we give a new proof of the Onofritype inequality
\begin{equation*}
\int_S e^{2u} \,ds^2 \leq 4\pi(\beta+1) \exp \biggl\{
\frac{1}{4\pi(\beta+1)} \int_S \nabla u^2 \,ds^2 +
\frac{1}{2\pi(\beta+1)} \int_S u \,ds^2 \biggr\}
\end{equation*}
on the sphere $S$ with Gaussian curvature $1$ and with conical
singularities divisor $\mathcal A = \beta\cdot p_1 + \beta \cdot p_2$ for
$\beta\in (1,0)$; here $p_1$ and $p_2$ are antipodal.


673  Multiplicity Free Jacquet Modules Aizenbud, Avraham; Gourevitch, Dmitry
Let $F$ be a nonArchimedean local field or a finite field.
Let $n$ be a natural number and $k$ be $1$ or $2$.
Consider $G:=\operatorname{GL}_{n+k}(F)$ and let
$M:=\operatorname{GL}_n(F) \times \operatorname{GL}_k(F)\lt G$ be a maximal Levi subgroup.
Let $U\lt G$ be the corresponding unipotent subgroup and let $P=MU$ be the corresponding parabolic subgroup.
Let $J:=J_M^G: \mathcal{M}(G) \to \mathcal{M}(M)$ be the Jacquet functor, i.e., the functor of coinvariants with respect to $U$.
In this paper we prove that $J$ is a multiplicity free functor, i.e.,
$\dim \operatorname{Hom}_M(J(\pi),\rho)\leq 1$,
for any irreducible representations $\pi$ of $G$ and $\rho$ of $M$.
We adapt the classical method of Gelfand and Kazhdan, which proves the ``multiplicity free" property of certain representations to prove the ``multiplicity free" property of certain functors.
At the end we discuss whether other Jacquet functors are multiplicity free.


689  A Pointwise Estimate for the Fourier Transform and Maxima of a Function Berndt, Ryan
We show a pointwise estimate for the Fourier
transform on the line involving the number of times the function
changes monotonicity. The contrapositive of the theorem may be used to
find a lower bound to the number of local maxima of a function. We
also show two applications of the theorem. The first is the two weight
problem for the Fourier transform, and the second is estimating the
number of roots of the derivative of a function.


697  Constructions of Uniformly Convex Functions Borwein, Jonathan M.; Vanderwerff, Jon
We give precise conditions under which the composition
of a norm with a convex function yields a
uniformly convex function on a Banach space.
Various applications are given to functions of power type.
The results are dualized to study uniform smoothness
and several examples are provided.


708  Improved Range in the Return Times Theorem Demeter, Ciprian
We prove that the Return Times Theorem holds true for pairs of $L^pL^q$ functions,
whenever $\frac{1}{p}+\frac{1}{q}<\frac{3}{2}$.


723  First Variation Formula in Wasserstein Spaces over Compact Alexandrov Spaces Gigli, Nicola; Ohta, ShinIchi
We extend results proved by the second author (Amer. J. Math., 2009)
for nonnegatively curved Alexandrov spaces
to general compact Alexandrov spaces $X$ with curvature bounded
below.
The gradient flow of a geodesically convex functional on the quadratic Wasserstein
space $(\mathcal P(X),W_2)$ satisfies the evolution variational inequality.
Moreover, the gradient flow enjoys uniqueness and contractivity.
These results are obtained by proving a first variation formula for
the Wasserstein distance.


736  Existence of Solutions for Abstract NonAutonomous Neutral Differential Equations Hernández, Eduardo; O'Regan, Donal
In this paper we discuss the existence of mild and classical solutions for a class of abstract nonautonomous
neutral functional differential equations. An application to partial neutral differential equations is considered.


752  Approximation of Holomorphic Solutions of a System of Real Analytic Equations Hickel, M.; Rond, G.
We prove the existence of an approximation function for holomorphic
solutions of a system of real analytic equations. For this we use
ultraproducts and Weierstrass systems introduced by J. Denef and L.
Lipshitz. We also prove a version of the Płoski smoothing theorem in
this case.


762  Smooth Approximation of Lipschitz Projections Li, Hanfeng
We show that any Lipschitz projectionvalued function
$p$ on a connected closed Riemannian manifold
can be approximated uniformly by smooth
projectionvalued functions $q$ with Lipschitz constant
close to that of $p$.
This answers a question of Rieffel.


767  On Zindler Curves in Normed Planes Martini, Horst; Wu, Senlin
We extend the notion of Zindler curve from the Euclidean plane to
normed planes. A characterization of Zindler curves for general
normed planes is given, and the relation between Zindler curves and
curves of constant areahalving distances in such planes is
discussed.


774  Pell Equations: NonPrincipal Lagrange Criteria and Central Norms Mollin, R. A.; Srinivasan, A.
We provide a criterion for the central norm to be
any value in the simple continued fraction expansion of $\sqrt{D}$
for any nonsquare integer $D>1$. We also provide a simple criterion
for the solvability of the Pell equation $x^2Dy^2=1$ in terms of
congruence conditions modulo $D$.


783  Products and Direct Sums in Locally Convex Cones Motallebi, M. R.; Saiflu, H.
In this paper we define lower, upper, and symmetric completeness and
discuss closure of the sets in product and direct sums. In particular,
we introduce suitable bases for these topologies, which leads us to
investigate completeness of the direct sum and its components. Some
results obtained about $X$topologies and polars of the neighborhoods.


799  Manifolds Covered by Lines and Extremal Rays Novelli, Carla; Occhetta, Gianluca
Let $X$ be a smooth complex projective variety, and let $H \in
\operatorname{Pic}(X)$
be an ample line bundle. Assume that $X$ is covered by rational
curves with degree one with respect to $H$ and with anticanonical
degree greater than or equal to $(\dim X 1)/2$. We prove that there
is a covering family of such curves whose numerical class spans an
extremal ray in the cone of curves $\operatorname{NE}(X)$.


815  Restricted Radon Transforms and Projections of Planar Sets Oberlin, Daniel M.
We establish a mixed norm estimate for the Radon transform in
$\mathbb{R}^2$ when the set of directions has fractional dimension.
This estimate is used to prove a result about an exceptional set of directions connected with projections of planar sets. That leads to
a conjecture analogous to a wellknown conjecture of Furstenberg.


821  New Examples of NonArchimedean Banach Spaces and Applications PerezGarcia, C.; Schikhof, W. H.
The study carried out in this paper about some new examples of
Banach spaces, consisting of certain valued fields extensions, is
a typical nonarchimedean feature. We determine whether these
extensions are of countable type, have $t$orthogonal bases, or are
reflexive.
As an application we construct, for a class of base fields, a norm
$\\cdot\$ on $c_0$, equivalent to the canonical supremum norm,
without nonzero vectors that are $\\cdot\$orthogonal and such
that there is a multiplication on $c_0$ making $(c_0,\\cdot\)$
into a valued field.


830  Almost Everywhere Convergence of Convolution Measures Reinhold, Karin; Savvopoulou, Anna K.; Wedrychowicz, Christopher M.
Let $(X,\mathcal{B},m,\tau)$ be a dynamical system with $(X,\mathcal{B},m)$ a probability
space and $\tau$ an invertible, measure preserving transformation.
This paper deals with the almost everywhere convergence in $\textrm{L}^1(X)$ of a
sequence of operators of weighted averages. Almost everywhere convergence follows
once we obtain an appropriate maximal estimate and once we provide
a dense class where convergence holds almost everywhere.
The weights are given by convolution products of members of a sequence of probability
measures $\{\nu_i\}$ defined on $\mathbb{Z}$.
We then exhibit cases of such averages where convergence fails.


842  The Rank of Jacobian Varieties over the Maximal Abelian Extensions of Number Fields: Towards the FreyJarden Conjecture Sairaiji, Fumio; Yamauchi, Takuya
Frey and Jarden asked if
any abelian variety over a number field $K$
has the infinite MordellWeil rank over
the maximal abelian extension $K^{\operatorname{ab}}$.
In this paper,
we give an affirmative answer to their conjecture
for the Jacobian variety
of any smooth projective curve $C$
over $K$
such that $\sharp C(K^{\operatorname{ab}})=\infty$
and for any abelian variety of $\operatorname{GL}_2$type with trivial character.


850  Character Sums with Division Polynomials Shparlinski, Igor E.; Stange, Katherine E.
We obtain nontrivial estimates of quadratic character sums of division polynomials $\Psi_n(P)$, $n=1,2, \dots$, evaluated at a given point $P$ on an elliptic curve over a finite field of $q$ elements. Our bounds are nontrivial if the order of $P$ is at least $q^{1/2 + \varepsilon}$ for some fixed $\varepsilon > 0$. This work is motivated by an open question about statistical indistinguishability of some cryptographically relevant sequences that was recently brought up by K. Lauter and the second author.


858  An Optimal Transport View of Schrödinger's Equation von Renesse, MaxK.
We show that the Schrödinger equation is a lift of Newton's third law
of motion $\nabla^\mathcal W_{\dot \mu} \dot \mu = \nabla^\mathcal W F(\mu)$ on
the space of probability measures, where derivatives are taken
with respect to the Wasserstein Riemannian metric. Here the potential
$\mu \to F(\mu)$ is the sum of the total classical potential energy $\langle V,\mu\rangle$
of the extended system
and its Fisher information
$ \frac {\hbar^2} 8 \int \nabla \ln \mu ^2
\,d\mu$. The precise relation is established via a wellknown
(Madelung) transform which is shown to be a symplectic submersion
of the standard symplectic
structure of complex valued functions into the
canonical symplectic space over the Wasserstein space.
All computations are conducted in the framework of Otto's formal
Riemannian calculus for optimal transportation of probability
measures.


870  Left Invariant EinsteinRanders Metrics on Compact Lie Groups Wang, Hui; Deng, Shaoqiang
In this paper we study left invariant EinsteinRanders metrics on compact Lie
groups. First, we give a method to construct left invariant nonRiemannian EinsteinRanders metrics
on a compact Lie group, using the Zermelo navigation data.
Then we prove that this gives a complete classification of left invariant EinsteinRanders metrics on compact simple
Lie groups with the underlying Riemannian metric naturally reductive.
Further, we completely determine the identity component of the group of
isometries for this type of metrics on simple groups. Finally, we study some
geometric properties of such metrics. In particular, we give the formulae of geodesics and flag curvature
of such metrics.


882  Equivalence of $L_p$ Stability and Exponential Stability of Nonlinear Lipschitzian Semigroups Xueli, Song; Jigen, Peng
$L_p$ stability and exponential stability are two important concepts
for nonlinear dynamic systems. In this paper, we prove that a
nonlinear exponentially bounded Lipschitzian semigroup is
exponentially stable if and only if the semigroup is $L_p$ stable
for some $p>0$. Based on the equivalence, we derive two sufficient
conditions for exponential stability of the nonlinear semigroup. The
results obtained extend and improve some existing ones.
