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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.

