A map $f$ on a ring $\cal A$ is said to be commuting if $f(x)$ commutes with $x$ for every $x\in \cal A$. The paper surveys the development of the theory of commuting maps and their applications. The following topics are discussed: commuting derivations, commuting additive maps, commuting traces of multiadditive maps, various generalizations of the notion of a commuting map, and applications of results on commuting maps to different areas, in particular to Lie theory.

In this paper, we extend the existence of traveling wave result in [1] to the case where the rate of disassembly of free polymers is non-zero. Mathematical methods used include dynamical system techniques and global bifurcation theory.

Multidimensional extensions of the Bernoulli and Appell polynomials are defined generalizing the corresponding generating functions, and using the Hermite-Kamp´e de F´eriet (or Gould-Hopper) polynomials. Furthermore the differential equations satisfied by the corresponding 2D polynomials are derived exploiting the factorization method, introduced in [15].

Let $\{\beta(n)\}$ be a sequence of positive numbers with $\beta(0) = 1$ and let $p\gt 0$. By the space $H^{p}(\beta)$, we mean the set of all formal power series $\sum^{\infty}_{n=0} \hat{f}(n) z^{n}$ for which $\sum^{\infty}_{n=0} |\hat{f}(n)|^{p} \beta(n)^{p} \lt \infty$. In this paper, we study cyclic vectors for the forward shift operator and supercyclic vectors for the backward shift operator on the space $H^{p} (\beta)$.

In the present paper, by making use of certain operators of generalized fractional calculus, we introduce a novel class ${\cal T}_{\lambda}^{\mu , \varphi , \eta} (n; \alpha )$ of functions which are analytic and univalent in the open unit disk ${\cal U}.$ A necessary and sufficient condition for a function to be in the class ${\cal T}_{\lambda}^{\mu , \varphi , \eta}(n; \alpha )$ is obtained. In addition, this paper includes distortion theorems involving generalized fractional integrals (and generalized fractional derivatives), radii of close-to-convexity, starlikeness, and convexity. Relevance with some new (or known) special cases are also pointed out.

Let $A$ and $B$ be two summability methods. We shall use the notation $\lambda \in(A, B)$ to denote the set of all sequences $\lambda$ such that $\sum\nolimits a_{n}\lambda_{n}$ is summable $B$, whenever $\sum\nolimits a_{n}$ is summable $A$. In the present paper we characterize the sets $\lambda \in (|\overline{N}, p_{n}|, |T|_{k})$ and $\lambda \in (|\overline{N}, p_{n}|_{k}, |T|)$, where $T$ is a lower triangular matrix with positive entries and row sums $1$. As special cases we obtain summability factor theorems and inclusion theorems for pairs of weighted mean matrices.

Let $M^2$ be a compact Willmore surface in the $n$-dimensional unit sphere. Denote by $\phi_{ij}^{\alpha}$ the tracefree part of the second fundamental form $h_{ij}^{\alpha}$ of $M^2,$ and by $\mathbb{H}$ the mean curvature vector of $M^2.$ Let $\Phi$ be the square of the length of $\phi_{ij}^{\alpha}$ and $H=|\mathbb{H}|$. We prove that if $0\leq\Phi\leq C(1+\frac{H^2}{8})$, where $C=2$ when $n=3$ and $C=\frac{4}{3}$ when $n \geq 4$, then either $\Phi=0$ and $M^2$ is totally umbilic or $\Phi=C(1+\frac{H^2}{8})$. In the latter case, either $n=3$ and $M^2$ is the Clifford torus or $n=4$ and $M^2$ is the Veronese surface.

Let $(\cal Z \, , \, C)$ be an ordered Hausdorff real topological vector space. Some conditions for assuring that a nonempty set $K \subset {\cal Z}$ has a nonempty superior or inferior are established. Ordering-conically compact ordered Hausdorff real topological vector spaces are introduced so that in such a space every nonempty bounded below (respectively, bounded above) set has a nonempty inferior (respectively, superior).

\noindent We consider eigenvalue problems for the vibrating string \vspace{-0.1cm} \[ u^{\prime \prime }(x)+\lambda \rho (x)u(x)=0,\;\;\;\;u(0)=u(a)=0 \] \vspace{-0.6cm} \noindent where the density $\rho (x)\;$is a positive continuous function on $% [0,a]$. Let $\lambda _{n}(t)$ be the $n$th eigenvalue of the string with $\rho =\rho (x,t)$. A classical convexity theorem of P\'{o}lya and Schiffer states that for any $k\geq 1$, the sum $% \sum_{n=1}^{k}\frac{1}{\lambda _{n}(t)}\;$is a convex function of $t$ if $\rho (x,t)$ is convex with respect to $t$. In this paper, we shall give a different approach to this result based on variational analysis. The ideas used also lead to applications in the case of symmetric densities and in the case of concave densities.

The MacWilliams-Delsarte identity is very important in coding theory. We will give a new proof of the identity using elementary method in this paper, which is much simpler than the original one [1].

In this paper we investigate the existence of nontrivial solutions of a two-point boundary value problem. Under the condition that the associated linear boundary value problem has no nontrivial solutions or no sign-changing solutions and some other additional conditions, we prove some existence theorems of (nontrivial) solutions.

This paper is concerned with local C-cosine families and local ntimes integrated cosine families. The characterization of each cosine families is mentioned through their infinitesimal generators. The relationship among local C-cosine families, local integrated cosine families and the abstract Cauchy problem is also mentioned.

We consider the singular integral operators which are more singular than Calder´on-Zygmund operator and include pseudo-differential operators. We obtain the boundedness of these operators on inhomogeneous Herz spaces and Herz-type Hardy spaces.

It has come to our attention that there is a positive answer to the following question mentioned in [2, p. 734] : ``We may conjecture whether a $(WC_{11})$-module with essential socle is a $(C_{11})$-module? ". Unfortunately in [2, Example 9] we mistakenly stated that \begin{eqnarray*} R=\Big[\begin{array}{ll} \mathbb{Z}/\mathbb{Z}p^{2} & \mathbb{Z}/\mathbb{Z}p^{2} \\ 0 & \mathbb{Z}/\mathbb{Z}p^{2} \end{array}\Big] \end{eqnarray*} is not right $(C_{11})$. However this $R$ is, in fact, right $(C_{11})$ (see, [1, Theorem 2.4]). Moreover the following proposition provides the answer to the question which do not affect the other results in the paper.