Absolutely kk-convex domains and holomorphic foliations on homogeneous manifolds

We consider a holomorphic foliation $\mathcal{F}$ of codimension $k\geq 1$ on a homogeneous compact K\"ahler manifold $X$ of dimension $n>k$. Assuming that the singular set $Sing(\mathcal{F})$ of $\mathcal{F}$ is contained in an absolutely $k$-convex domain $U\subset X$, we prove that the determinant of normal bundle $\det(N_{\mathcal{F}})$ of $\mathcal{F}$ cannot be an ample line bundle, provided $[n/k]\geq 2k+3$. Here $[n/k]$ denotes the largest integer $\leq n/k.

Higher codimensional foliations and Kupka singularities

We consider holomorphic foliations of dimension $k>1$ and codimension $\geq 1$ in the projective space $\mathbb{P}^n$, with a compact connected component of the Kupka set. We prove that, if the transversal type is linear with positive integers eigenvalues, then the foliation consist on the fibers of a rational fibration. As a corollary, if $\mathcal{F}$ is a foliation such that $dim(\mathcal{F})\geq cod(\mathcal{F})+2$ and has transversal type diagonal with different eigenvalues, then the Kupka component $K$ is a complete intersection and we get the same conclusion. The same conclusion holds if the Kupka set is a complete intersection and has radial transversal type. Finally, as an application, we find a normal form for non integrable codimension one distributions on $\mathbb{P}^{n}$

Lagrangian Formalism for Multiform Fields on Minkowski Spacetime

We present an introduction to the mathematical theory of the Lagrangian formalism for multiform fields on Minkowski spacetime based on the multiform and extensor calculus. Our formulation gives a unified mathematical description for the main relativistic field theories including the gravitational field (which however will be discussed in a separate paper). We worked out several examples (including tricks of the trade), from simple to very sophisticated ones (like, e.g., the Dirac-Hestenes field on the more general gravitational background) which show the power and beauty of the formalism

Metric Clifford Algebra

In this paper we introduce the concept of metric Clifford algebra $\mathcal{C\ell}(V,g)$ for a $n$-dimensional real vector space $V$ endowed with a metric extensor $g$ whose signature is $(p,q)$, with $p+q=n$. The metric Clifford product on $\mathcal{C\ell}(V,g)$ appears as a well-defined \emph{deformation}(induced by $g$) of an euclidean Clifford product on $\mathcal{C\ell}(V)$. Associated with the metric extensor $g,$ there is a gauge metric extensor $h$ which codifies all the geometric information just contained in $g.$ The precise form of such $h$ is here determined. Moreover, we present and give a proof of the so-called \emph{golden formula,} which is important in many applications that naturally appear in ours studies of multivector functions, and differential geometry and theoretical physics

Multivector Functions of a Real Variable

This paper is an introduction to the theory of multivector functions of a real variable. The notions of limit, continuity and derivative for these objects are given. The theory of multivector functions of a real variable, even being similar to the usual theory of vector functions of a real variable, has some subtle issues which make its presentation worhtwhile.We refer in particular to the derivative rules involving exterior and Clifford products, and also to the rule for derivation of a composition of an ordinary scalar function with a multivector function of a real variable

Metric Tensor Vs. Metric Extensor

In this paper we give a comparison between the formulation of the concept of metric for a real vector space of finite dimension in terms of \emph{tensors} and \emph{extensors}. A nice property of metric extensors is that they have inverses which are also themselves metric extensors. This property is not shared by metric tensors because tensors do \emph{not} have inverses. We relate the definition of determinant of a metric extensor with the classical determinant of the corresponding matrix associated to the metric tensor in a given vector basis. Previous identifications of these concepts are equivocated. The use of metric extensor permits sophisticated calculations without the introduction of matrix representations

Extensors

In this paper we introduce a class of mathematical objects called \emph{extensors} and develop some aspects of their theory with considerable detail. We give special names to several particular but important cases of extensors. The \emph{extension,} \emph{adjoint} and \emph{generalization} operators are introduced and their properties studied. For the so-called $(1,1)$-extensors we define the concept of \emph{determinant}, and their properties are investigated. Some preliminary applications of the theory of extensors are presented in order to show the power of the new concept in action. An useful formula for the inversion of $(1,1)$-extensors is obtained

Euclidean Clifford Algebra

Let $V$ be a $n$-dimensional real vector space. In this paper we introduce the concept of \emph{euclidean} Clifford algebra $\mathcal{C\ell}(V,G_{E})$ for a given euclidean structure on $V,$ i.e., a pair $(V,G_{E})$ where $G_{E}$ is a euclidean metric for $V$ (also called an euclidean scalar product). Our construction of $\mathcal{C\ell}(V,G_{E})$ has been designed to produce a powerful computational tool. We start introducing the concept of \emph{multivectors} over $V.$ These objects are elements of a linear space over the real field, denoted by $\bigwedge V.$ We introduce moreover, the concepts of exterior and euclidean scalar product of multivectors. This permits the introduction of two \emph{contraction operators} on $\bigwedge V,$ and the concept of euclidean \emph{interior} algebras. Equipped with these notions an euclidean Clifford product is easily introduced. We worked out with considerable details several important identities and useful formulas, to help the reader to develope a skill on the subject, preparing himself for the reading of the following papers in this series.Comment: Latex accent in author(s) was introduced Latex commands in abstract were correcte

Joint Data-Aided Carrier Frequency Offset, Phase Offset, Amplitude and SNR Estimation for Millimeter-Wave MIMO Systems

This work is devoted to solve the problem of estimating the carrier frequency offset, phase offset, amplitude, and SNR between two mmWave transceivers. The Cram\'{e}r-Rao Lower Bound (CRLB) for the different parameters is provided first, as well as the condition for the CRLB to exist, known as Regularity Condition. Thereafter, the problem of finding suitable estimators for the parameters is adressed, for which the proposed solution is the Maximum Likelihood estimator (ML)

Multivector Functionals

In this paper we introduce the concept of \emph{multivector functionals.} We study some possible kinds of derivative operators that can act in interesting ways on these objects such as, e.g., the $A$-directional derivative and the generalized concepts of curl, divergence and gradient. The derivation rules are rigorously proved. Since the subject of this paper has not been developed in previous literature, we work out in details several examples of derivation of multivector functionals.Comment: Some references change