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 reaction-diffusion equations.

We give the sufficient and necessary conditions
of Browder's convergence theorem
for one-parameter nonexpansive semigroups
which was proved by Suzuki.
We also discuss the perfect kernels of topological spaces.

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.

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 ${\textrm{End}}_{\overline{\mathbb{Q}}}(J)$.

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 finite-dimensional
$kG$-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.

In this paper, we prove that a non--zero power series $F(z)\in\mathbb{C}
[\mkern-3mu[ z]\mkern-3mu]
$
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 $G_k(z):= \sum_{n=0}^\infty z^{k^n}(1-z^{k^n})^{-1}$ 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.

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.

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.

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:
$$\mu_{k+1}\leq\frac{1}{k}\sum_{i=1}^k\mu_i+\Bigl[\frac{8(n+2)}{n^2}\Bigl(\frac{\rho_{\max}}
{\rho_{\min}}\Bigr)^2\Bigr]^{1/2}\times
\frac{1}{k}\sum_{i=1}^k[\mu_i(\mu_{k+1}-\mu_i)]^{1/2},
$$
$$\frac{n^2k^2}{8(n+2)}\leq
\Bigl(\frac{\rho_{\max}}{\rho_{\min}}\Bigr)^2\Bigl[\sum_{i=1}^k\frac{\mu_i^{1/2}}{\mu_{k+1}-\mu_i}\Bigr]
\times\sum_{i=1}^k\mu_i^{1/2}.
$$

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.

In this paper we study real hypersurfaces in a non-flat complex space form with $\eta$-parallel shape operator. Several partial characterizations of these real hypersurfaces are obtained.

The zero-divisor graph $\Gamma(R)$ of a commutative ring $R$ is the graph whose vertices consist of
the nonzero zero-divisors 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 zero-divisor graphs of Boolean
rings. Also, commutative rings $R$ such that
$\Gamma(R)$ is isomorphic to the zero-divisor graph of a direct product of integral domains are classified, as well as
those whose zero-divisor graphs are central vertex complete.

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.

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 $\beta=(p+\alpha-n-1)/2$ and $dv_\beta(z)=
(1-|z|^2)^\beta\,dv(z)$. 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 $|f(z)-f(w)|/|z-w|$ 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.

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.

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.

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 $n-j=2^pq$, where $q \ge 1$ is odd and $p \geq 0$, and set $m(n-j) = 2n+p-q+1$ if $p \leq q + 1$
and $m(n-j)= 2n + 2^{p-q}$ if $p \geq q$. In this paper we show that $m \le m(n-j) + 2j+1$. Further, we show that this bound is almost best possible, by exhibiting examples $(M^{m(n-j) +2j},T)$ 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 $2^t-1$ (for $j=0$ and $2$, it has been previously shown that $m(n-j) +2j$ 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 $m \le \frac{5} {2}n$.

We consider the linearization of the three-dimensional 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 so-called "slowest
wave". Additionally, we combine our dispersive estimates with $L^2$
type energy bounds to prove a family of Strichartz estimates.

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.

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.

In this paper we show that there exists a polynomial $k\in\mathbb{Z}[t]$
with the property that on the elliptic curve $\mathcal{E}':
y^2=x^3+k(t)$ (defined over the field $\mathbb{Q}(t)$) we can find four
points in arithmetic progression that are independent in the group
of all $\mathbb{Q}(t)$-rational points on the curve $\mathcal{E}'$. In
particular this result generalizes earlier results of Lee and
V\'{e}lez. We also show that if $n\in\mathbb{N}$ is odd,
then there are infinitely many $k$'s with the property that on
curves $y^2=x^n+k$ there are four rational points in arithmetic
progressions. In the case when $n$ is even we can find infinitely
many $k$'s such that on curves $y^2=x^n+k$ there are six rational
points in arithmetic progression.

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.

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.