First some properties are recalled of the Hilbert transform introduced in the 1990's within the framework of Clifford analysis. Furthermore algebraic and geometric characterizations are given for this operator to be unitary. Special attention is paid to the Hilbert transform on the unit sphere $S^m$ and the hyperplane $\mathbb R^m$ in $\mathbb R^{m+1}$ and classical results in the plane are revisited.

In this paper,we will prove the prime-to-p-part of a cohomological Hasse principle for the ring $A=\mathbb{F}_{p}((t))[[X,Y]].$ The proof is based on recent results by Fujiwara and Panin.

It is shown that the center of a unital primitive locally $A$-pseudoconvex Hausdorff algebra over $\mathbb{C}$ and of a unital topologically primitive locally pseudoconvex Fréchet algebra over $\mathbb{C}$ are topologically isomorphic to $\mathbb{C}$.

Higher dimensional wavelets are constructed in the framework of Clifford analysis by taking the Clifford-monogenic extension of specific functions. Clifford-monogenic functions are direct higher dimensional generalizations of holomorphic functions in the complex plane. In this way also new generalized polynomials, the so-called Clifford-Laguerre polynomials, are obtained.

In this paper, we establish the existence and non-existence results of positive solutions for the (n-1,1) three-point boundary value problems consisting of the equation $$ u^{(n)}+\lambda a(t)f(u(t))=0,\;\;\;t\in (0,1) $$ and one of the following boundary value conditions $$ u(1)=\beta u(\eta ),\;\;u^{(i)}(0)=0\;\hbox{ for }\;i=1,2,\cdots,n-1 $$ and $$ u^{(n-1)}(1)=\beta u^{(n-1)}(\eta ),\;u^{(i)}(0)=0\;\hbox{ for }\;i=0,1,\cdot,n-2, $$ where $\eta \in [0,1)$, $\beta\in [0,1)$ and $a:\;(0,1)\rightarrow R$ may change sign. $f(0)>0$, $\lambda >0$ is a parameter. Our approach is based on the Leray-Schauder fixed point Theorem. This paper is motivated by Eloe and Henderson [6].

We consider the principal configurations associated to smooth vector fields $\nu$ normal to a manifold $M$ immersed into a euclidean space and give conditions on the number of principal directions shared by a set of $k$ normal vector fields in order to guaranty the umbilicity of $M$ with respect to some normal field $\nu$. Provided that the umbilic curvature is constant, this will imply that $M$ is hyperspherical. We deduce some results concerning binormal fields and asymptotic directions for manifolds of codimension 2. Moreover, in the case of a surface $M$ in $\mathbb R^N$, we conclude that if $N>4$, it is always possible to find some normal field with respect to which $M$ is umbilic and provide a geometrical characterization of such fields.

For realcompact spaces $X$ and $Y$ we give a complete description of the linear biseparating maps between spaces of vector-valued continuous functions on $X$ and $Y$ in two cases: the spaces of all continuous functions and the spaces of {\em bounded} continuous functions. With similar techniques we also describe the linear biseparating maps defined between some other families of spaces, in particular spaces of vector-valued uniformly continuous bounded functions.

A non-abelian tensor product for Leibniz $n$-algebras is introduced as a generalization of the non-abelian tensor product for Leibniz algebras introduced by Kurdiani and Pirashvili. We use it to construct the universal central extension of a perfect Leibniz $n$-algebra.

In this paper we present a simple and self contained proof of the great Picard Theorem based on certain Harnack-type inequalities due to J. Lewis.

In this paper quadratic sets of a $3$-dimensional locally projective regular planar space $(\cal S,\cal L,\cal P)$ of order $n$ are studied and classified. It is proved that if in $(\cal S,\cal L,\cal P)$ there is a non-degenerate quadratic set $\bf H$, then the planar space is either $\mathop{\rm{PG}}(3,n)$ or $\mathop{\rm{AG}}(3,n)$. Moreover in the first case $\bf H$ is either an ovoid or an hyperbolic quadric, in the latter case $\bf H$ is either a cylinder with base an oval or a pair of parallel planes.

A fixed point and homotopic invariance result is presented for set valued contractive type maps in complete gauge spaces.

We consider a nonlinear coupled system of two wave equations with memory condition at the boundary and we study the asymptotic behavior of the corresponding solutions. We proved that the energy decay with the same rate of decay of the relaxation functions, that is, the energy decays exponentially when the relaxation functions decay exponentially and decay polynomially when the relaxation functions decay polynomially.