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 $\text{Ga}{{\text{l}}_{\mathbb{Q}}}(f(x))\,=\,{{S}_{n}}$ or ${{A}_{n}}$, then $\text{En}{{\text{d}}_{\overline{\mathbb{Q}}}}(J)\,=\,\mathbb{Z}$. In seeking to examine his result in the genus $g\,=\,2$ case supposing other Galois groups, we calculate $\text{En}{{\text{d}}_{\overline{\mathbb{Q}}}}(J)\,{{\otimes }_{\mathbb{Z}}}\,{{\mathbb{F}}_{2}}$ 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 $\text{En}{{\text{d}}_{\overline{\mathbb{Q}}}}(J)$.

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 $\text{GH}$ for groups with periodic cohomology.

In this paper, we prove that a non–zero power series $F(z\text{)}\in \mathbb{C}\text{ }[[z]]$ satisfying

$$F({{z}^{d}})\,=\,F(z)\,+\,\frac{A(z)}{B(z)},$$

where $d\,\ge \,2,\,A(z),\,B(z)\,\in \,\mathbb{C}[z]$, with $A(z)\,\ne \,0$ 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 $k\,\ge \,2$ the series ${{G}_{k}}(z):=\mathop{\sum }_{n=0}^{\infty }{{z}^{{{k}^{n}}}}{{(1-{{z}^{{{k}^{n}}}})}^{-1}}$ is transcendental for all algebraic numbers $z$ with $\left| z \right|\,<\,1$. We give a similar result for ${{F}_{k}}(z):=\mathop{\sum }_{n=0}^{\infty }{{z}^{{{k}^{n}}}}{{(1+{{z}^{{{k}^{n}}}})}^{-1}}$. 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.

We describe an ${{E}_{8}}$ correspondence for the multiplicative eta-products of weight at least 4.

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 ${{\alpha }_{n}}\text{s}$ 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 $R/\mathfrak{a}\,\le \,1$, then all generalized local cohomology modules $H_{\mathfrak{a}}^{i}(M,\,N)$ are $\mathfrak{a}$-cofinite.

Let $D$ be a connected bounded domain in ${{\mathbb{R}}^{n}}$. Let $0\,<\,{{\mu }_{1}}\,\le \,{{\mu }_{2}}\,\le \,\cdots \,\le \,{{\mu }_{k}}\,\le \,\cdots $ be the eigenvalues of the following Dirichlet problem:

$$\left\{ \begin{align} & {{\Delta }^{2}}u(x)\,+\,V(x)u(x)\,=\,\mu \rho (x)u(x),x\in \,D \\ & u{{|}_{\partial D}}\,=\,\frac{\partial u}{\partial n}\,{{|}_{\partial D}}\,=\,0, \\ \end{align} \right.$$

where $V(x)$ is a nonnegative potential, and $\rho (x)\,\in \,C(\overset{-}{\mathop{D}}\,)$ is positive. We prove the following inequalities:

$$\begin{align} & {{\mu }_{k+1}}\le \frac{1}{k}\sum\limits_{i=1}^{k}{\mu i+}{{[\frac{8(n+2)}{{{n}^{2}}}{{\left( \frac{\rho \max }{\rho \min } \right)}^{2}}]}^{1/2}}\times \frac{1}{k}{{\sum\limits_{i=1}^{k}{[{{\mu }_{i}}({{\mu }_{k+1}}-{{\mu }_{i}})]}}^{1/2}}, \\ & \frac{{{n}^{2}}{{k}^{2}}}{8(n+2)}\le {{\left( \frac{\rho \max }{\rho \min } \right)}^{2}}[\sum\limits_{i=1}^{k}{\frac{\mu _{i}^{1/2}}{{{\mu }_{k+1}}-{{\mu }_{i}}}}]\times \sum\limits_{i=1}^{k}{\mu _{i}^{1/2}}. \\ \end{align}$$

The groups of (linear) similarity and coincidence isometries of certain modules $\Gamma $ 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.

The goal of this note is to prove that the signed Segre forms of Griffiths’ positive vector bundles are positive.

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_{\alpha }^{p},\,p\,>\,n\,+\,1\,+\alpha $, if and only if the function $\left| f(z)\,-\,f(w) \right|/\left| 1\,-\,\left\langle z,\,w \right\rangle \right|$ is in ${{L}^{p}}({{\mathbb{B}}_{n}}\,\times \,{{\mathbb{B}}_{n}},\,d{{v}_{\beta }}\,\times \,d{{v}_{\beta }})$, where $\beta \,=\,(p\,+\,\alpha \,-\,n\,-\,1)/2$ and $d{{v}_{\beta }}(z)\,=\,{{(1\,-\,{{\left| z \right|}^{2}})}^{\beta }}\,dv(z)$. In this paper we consider the range $0\,<\,p\,<\,n\,+\,1\,+\,\alpha $ and show that in this case, $f\,\in \,A_{\alpha }^{p}\,(\text{i})$ (i) if and only if the function $\left| f(z)\,-\,f(w) \right|/\left| 1\,-\,\left\langle z,\,w \right\rangle \right|$ is in ${{L}^{p}}({{\mathbb{B}}_{n}}\,\times \,{{\mathbb{B}}_{n}},\,d{{v}_{\alpha }}\,\times \,d{{v}_{\alpha }})$, (ii) if and only if the function $\left| f(z)\,-\,f(w) \right|/\left| z\,-\,w \right|$ is in ${{L}^{p}}({{\mathbb{B}}_{n}}\,\times \,{{\mathbb{B}}_{n}},\,d{{v}_{\alpha }}\,\times \,d{{v}_{\alpha }})$. 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 $R\,=\,{{\oplus }_{n\in \mathbb{N}0}}\,{{R}_{n}}$ is a Noetherian homogeneous ring with local base ring $({{R}_{0}},\,{{m}_{0}})$, irrelevant ideal ${{R}_{+}}$, and $M$ a finitely generated graded $R$-module, then $H_{{{m}_{0}}R}^{j}\,(H_{R+}^{t}\,(M))$ 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 $\text{cd(}{{R}_{+}},M)\,=\,2$, then for each $i\,\in \,{{\mathbb{N}}_{0}},\,H_{{{m}_{0}}R}^{i}\,(H_{R+}^{2}\,(M))$ is Artinian if and only if $H_{{{m}_{0}}R}^{i+2}(H_{R+}^{1}(M))$ is Artinian, where $ \text{cd(}{{R}_{+}},\,M)$ is the cohomological dimension of $M$ with respect to ${{R}_{+}}$. This improves some results of R. Sazeedeh.

Let $\left| K \right|$ be the metric polyhedron of a simplicial complex $K$. In this paper, we characterize a simplicial subdivision ${{K}^{\prime }}$ of $K$ preserving the metric topology for $\left| K \right|$ as the one such that the set ${{K}^{\prime }}(0)$ of vertices of ${{K}^{\prime }}$ is discrete in $\left| K \right|$. 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 $T\,:\,{{M}^{m}}\,\to \,{{M}^{m}}$ whose fixed point set has the form ${{F}^{n}}\,\bigcup \,{{F}^{j}}$, where ${{F}^{n}}$ and ${{F}^{j}}$ are submanifolds with dimensions $n$ and $j$, ${{F}^{j}}$ is indecomposable and $n\,>\,j$. Write $n\,-\,j\,={{2}^{p}}q$, where $q\,\ge \,1$ is odd and $p\,\ge \,0$, and set $m(n\,-\,j)\,=\,2n\,+\,p\,-q\,+\,1$ if $p\,\le \,q\,+\,1$ and $m(n\,-\,j)\,=\,2n\,+\,{{2}^{p-q}}$ if $p\,\ge \,q$. 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 ${{F}^{n}}\,\bigcup \,{{F}^{j}}$ 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 $m\,\le \,\frac{5}{2}\,n$.

In this paper we give the form of every multiplicative Hausdorff prime matrix, thus answering a long-standing open question.