Cyclic homogeneous Riemannian manifolds

In spin geometry, traceless cyclic homogeneous Riemannian manifolds equipped with a homogeneous spin structure can be viewed as the simplest manifolds after Riemannian symmetric spin spaces. In this paper, we give some characterizations and properties of cyclic and traceless cyclic homogeneous Riemannian manifolds and we obtain the classification of simply-connected cyclic homogeneous Riemannian manifolds of dimension less than or equal to four. We also present a wide list of examples of non-compact irreducible Riemannian $3$-symmetric spaces admitting cyclic metrics and give the expression of these metrics

Cyclic metric Lie groups

Cyclic metric Lie groups are Lie groups equipped with a left-invariant metric which is in some way far from being biinvariant, in a sense made explicit in terms of Tricerri and Vanhecke's homogeneous structures. The semisimple and solvable cases are studied. We extend to the general case, Kowalski-Tricerri's and Bieszk's classifications of connected and simply-connected unimodular cyclic metric Lie groups for dimensions less than or equal to five

Homogeneous spin Riemannian manifolds with the simplest Dirac operator

We show the existence of nonsymmetric homogeneous spin Riemannian manifolds whose Dirac operator is like that on a Riemannian symmetric spin space. Such manifolds are exactly the homogeneous spin Riemannian manifolds $(M,g)$ which are traceless cyclic with respect to some quotient expression $M=G/K$ and reductive decomposition $\mathfrak{g} = \mathfrak{k} \oplus \mathfrak{m}$. Using transversally symmetric fibrations of noncompact type, we give a list of them

Cycles inside cycles: Spanish regional aggregation

This paper sets out a comprehensive framework to identify regional business cycles within Spain and analyses their stylised features and the degree of synchronization present among them and the Spanish economy. We show that the regional cycles are quite heterogeneous although they display some degree of synchronization that can be partially explained using macroeconomic variables. We also propose a dynamic factor model to cluster the regional comovements and Önd out if the country cycle is simply the aggregation of the regional ones. We Önd that the Spanish business cycle is not shared by the seventeen regions, but is the sum of the di§erent regional behaviours. The implications derived from our results are useful both for policy makers and analysts.

Editorial introduction: How has the pandemic hit the Spanish economy and what should governments do?

In 2020, the world was suddenly hit by the most severe global pandemic in a century due to the coronavirus disease (COVID-19), first detected in late 2019. Beyond the health and human tragedy of the disease, the pandemic triggered the most serious global economic crisis since Second World War in terms of speed and intensity. To mitigate the propagation of the virus, governments around the world implemented unprecedented protective strategies, such as restrictions on movement and travel, recurrent population lockdowns and social distancing imposed in shops, eateries, public transport, schools and cultural activities. As a result, hospitality and travel industry has collapsed; businesses have closed, unemployment has skyrocketed, the inequalities in our societies have been exposed; and social isolation has been widely imposed worldwide, which plunged most major economies into recession..

On the classification theorems of almost-Hermitian or homogeneous Kähler structures

A proof by Young tableaux and symmetrizers is given of the classification theorems by Gray and Hervella of almost-Hermitian structures and by Abbena and Garbiero of homogeneous Kähler structures.Partially supported by DGICYT, Spain, under Grant no. BFM2002–00141.Peer reviewe

The canonical 8-form on manifolds with holonomy group Spin(9)

An explicit expression of the canonical 8-form on a Riemannian manifold with a Spin(9)-structure, in terms of the nine local symmetric involutions involved, is given. The list of explicit expressions of all the canonical forms related to Berger's list of holonomy groups is thus completed. Moreover, some results on Spin(9)-structures as G-structures defined by a tensor and on the curvature tensor of the Cayley planes, are obtained

Econometric methods for business cycle dating

© 2023. This document is made available under the CC-BY-NC-ND 4.0 license http://creativecommons.org/licenses/by-nc-nd/4.0/ This document is the submitted version of a published work that appeared in final form in Oxford Research Encyclopedia of Economics and Finance.Business cycle dating helps in developing economic analysis and is useful for economic agents whether they be policy makers, investors or academics. This paper reviews old and recent research on dating the reference cycle turning points and is intended as a guide to the applied researcher. All these methods provide a statistical alternative to cycle dating committees, although full automatism and researcher’s art could be complements rather than substitutes in some dating scenarios. Our survey divides the dating literature into two groups with different approaches to dating the business cycle from a set of coincident economic indicators: averagethen- date or date-then average. In both cases, the dating techniques can be divided into nonparametric and parametric. The paper shows the theoretical foundations of both types of techniques and describes in detail the algorithms or estimation methods necessary for their implementation. Finally, the paper describes empirical applications of the different methods with data of different frequencies, trying to show how they work in practice and pointing out their advantages and disadvantages. This empirical illustrations include a compilation of the codes in different languages (R, Matlab or Gauss). In our opinion, future research should focus on developing methods that are robust to changes in volatility or large outliers and on exploring the usefulness of big data sources and the classification ability offered by machine learning methods