The regularity of a quaternionic function is reinterpreted through a new canonical decomposition of the real differential, giving new insights into the algebraic properties of the regularity itself. The result comes from a somewhat unusual point of view on the automorphisms of the quaternionic field: a general notion of quaternionic linearity is associated to them, and some unnoticed metric properties of their inner representation are used to build up the theory.

For a nonzero character $\phi$ on a Banach algebra ${\frak A}$, we investigate some relations between $\phi$-amenability of ${\frak A}$ and ergodic theory. As the main result, we give a characterization for $\phi$-amenability of ${\frak A}$ in terms of antirepresentations of ${\frak A}$ on a Banach space.

The property $(\rm{gw})$ is a variant of generalized Weyl's theorem, for a bounded operator $T$ acting on a Banach space. In this note we consider the preservation of property $(\rm{gw})$ under a finite rank perturbation commuting with $T$, whenever $T$ is isoloid, polaroid, or $T$ has analytical core $K(\lamda_0 I -T ) = \set{0}$ for some $\lamda_0\in\mathbb{C}$. The preservation of property $(\rm{gw})$ is also studied under commuting nilpotent or under algebraic perturbations. The theory is exemplified in the case of some special classes of operators.

We consider almost Kenmotsu manifolds with conformal Reeb foliation. We prove that such a foliation produces harmonic morphisms, we study the $k$-nullity distributions and we discuss the isometrical immersion of such a manifold $M$ as hypersurface in a real space form $\widetilde{M}(c)$ of constant curvature $c$ proving that $c \leq -1$ and, if $c<-1$, $M$ is totally umbilical, Kenmotsu and locally isometric to the hyperbolic space of constant curvature $-1$. Finally, the Einstein and $\eta$-Einstein conditions are discussed.

Let $K$ be an algebraically closed field of characteristic $0$, complete with respect to an ultrametric absolute value. Let $f$ be a transcendental meromorphic function in $K$. We prove that if all zeroes and poles are of order $\geq 2$, then $f$ has no Picard exceptional value different from zero. More generally, if all zeroes and poles are of order $\geq k\geq 3$, then $f^{(k-2)}$ has no exceptional value different from zero. Similarly, a result of this kind is obtained for the $k-th$ derivative when the zeroes of $f $ are at least of order $m$ and the poles of order $n$, such that $mn>m+n+kn$. If $f$ admits a sequence of zeroes $a_n$ such that the open disk containing $a_n$, of diameter $|a_n|$ contains no pole, then $f$ and all its derivatives assume each non-zero value infinitely often. Several corollaries apply to the Hayman conjecture in the non-solved cases. Similar results are obtained concerning ''unbounded '' meromorphic functions inside an ''open'' disk.

We present a topological classification of some trichotomic evolution families, in the sense of Sacker-Sell, Elaydi-Hajek, and the one introduced in ([7]). We hope our work will contribute to the general theory of exponential dichotomies and trichotomies.

Topologically $Q$-algebras, a very convenient generalization of (topological) $Q$-algebras, have been recently considered by {\em A. Najmi} and independently by {\em H. Arizmendi et al.}. Here we extend certain basic results of {\em A. Najmi}. Within the same vein of ideas, we introduce a new class of topological algebras, the {\em ``Cauchy topologically $Q$-algebras"} (Cauchy $tQ$-algebras), that yields a new characterization of the standard advertibly complete algebras, as defined by {\em S. Warner}.

Following the concept of normal to a convex body, which is due to Eggleston, we introduce the notion of affine orthogonality with respect to a convex body. In contrast to Eggleston's considerations, we do not need a metric and via this notion we are able to characterize convex bodies of constant width and further interesting classes of convex bodies.

Let $\Omega$ be an $n$-dimensional convex domain with finite inradius \linebreak$\delta _0=\sup_{x\in \Omega} \,\,\delta$, where $\delta = \,\,dist (x, \partial \Omega)$, and let $(p,q)$ be a pair of positive numbers. For functions vanishing at the boundary of the domain and any $\nu \in [0, p/q]$ we prove the following Hardy-type inequality $$ \int _{\Omega}\frac{|\nabla f|^2}{\delta ^{p-1}} \,dx \,\,\geq \,\,h \int_{\Omega}\frac{|f|^2}{\delta ^{p+1}} \,dx + \frac{ \lambda^2}{\delta_0^q} \int_{\Omega}\frac{|f|^2}{\delta ^{p-q+1}} \,dx $$ with two sharp constants $$ h=\frac{p^2-\nu^2 q^2}{4}\geq 0 \qquad\,\,\,\,\,\,\mbox{and} \qquad\,\,\,\,\,\,\lambda = \frac{q}{2} \lambda_{\nu}(2p/q) > 0, $$ where $z = \lambda_{\nu}(p)$ is the Lamb constant defined as the first positive root of the equation $ p J_{\nu}(z)+ 2 z J_{\nu}'(z)=0$ for the Bessel function $J_{\nu}$. We prove that $z = \lambda_{\nu}(p)$ as a function in $p$ can be found as the solution of an initial value problem for the differential equation $$ \frac{d z}{dp}=\frac{2z}{p^2-4\nu^2 +4z^2}. $$ For $n=1$ our inequality is an improvement of the original Hardy inequality for finite intervals. For $n\geq 1$ and $p=q/2=1$ it gives a new sharp form of the Hardy-type inequality due to H. Brezis and M. Marcus. The case $h=0$, $\nu =1/2$, $p=1$ and $q=2$ coincides with sharp eigenvalue estimates due to J. Hersch for $n=2$, and L. E. Payne and I. Stakgold for $n\geq 3$.

In this paper we give Weierstrass type representation formulas for spacelike surfaces and maximal spacelike surfaces in 3-dimensional lightlike cone $\mathbb{Q}^3$. Then we discuss some properties and structures of spacelike surface and its associated surface.

We investigate Banach lattices for which the class of positive semi-compact operators coincides with that of L-weakly compact (resp. M-weakly compact) operators, and we give some consequences.