Characterization of manifolds of constant curvature by spherical curves

It is known that the so-called rotation minimizing (RM) frames allow for a simple and elegant characterization of geodesic spherical curves in Euclidean, hyperbolic, and spherical spaces through a certain linear equation involving the coefficients that dictate the RM frame motion (da Silva, da Silva in Mediterr J Math 15:70, 2018). Here, we shall prove the converse, i.e., we show that if all geodesic spherical curves on a Riemannian manifold are characterized by a certain linear equation, then all the geodesic spheres with a sufficiently small radius are totally umbilical and, consequently, the given manifold has constant sectional curvature. We also furnish two other characterizations in terms of (i) an inequality involving the mean curvature of a geodesic sphere and the curvature function of their curves and (ii) the vanishing of the total torsion of closed spherical curves in the case of three-dimensional manifolds. Finally, we also show that the same results are valid for semi-Riemannian manifolds of constant sectional curvature.Comment: To appear in Annali di Matematica Pura ed Applicat

The core-periphery model with three regions

We study a 3-region core-periphery model à la Krugman and compare our results with those of the standard 2-region model. The conditions for the stability of the dispersion and concentration configurations are established. Like in the 2-region model, dispersion and concentration can be simultaneously stable. We show that the 2- region (3-region) model favors the dispersion (concentration) of economic activity. Finally, we extend the core-periphery model to the case of n regions and show that stability of concentration with 2 regions implies stability of concentration with any even number of regions.new economic geography, core-periphery

Topological Approach to Microcanonical Thermodynamics and Phase Transition of Interacting Classical Spins

We propose a topological approach suitable to establish a connection between thermodynamics and topology in the microcanonical ensemble. Indeed, we report on results that point to the possibility of describing {\it interacting classical spin systems} in the thermodynamic limit, including the occurrence of a phase transition, using topology arguments only. Our approach relies on Morse theory, through the determination of the critical points of the potential energy, which is the proper Morse function. Our main finding is to show that, in the context of the studied classical models, the Euler characteristic $\chi(E)$ embeds the necessary features for a correct description of several magnetic thermodynamic quantities of the systems, such as the magnetization, correlation function, susceptibility, and critical temperature. Despite the classical nature of the studied models, such quantities are those that do not violate the laws of thermodynamics [with the proviso that Van der Waals loop states are mean field (MF) artifacts]. We also discuss the subtle connection between our approach using the Euler entropy, defined by the logarithm of the modulus of $\chi(E)$ per site, and that using the {\it Boltzmann} microcanonical entropy. Moreover, the results suggest that the loss of regularity in the Morse function is associated with the occurrence of unstable and metastable thermodynamic solutions in the MF case. The reliability of our approach is tested in two exactly soluble systems: the infinite-range and the short-range $XY$ models in the presence of a magnetic field. In particular, we confirm that the topological hypothesis holds for both the infinite-range ($T_c \neq 0$) and the short-range ($T_c = 0$) $XY$ models. Further studies are very desirable in order to clarify the extension of the validity of our proposal

Sensoriamento remoto e geoprocessamento como ferramentas para o estudo da sedimentação do rio São Francisco.

O processo de sedimentação no curso do rio São Francisco está sendo estudado na sua porção média e sub média, que compreende o trecho entre Três Marias em Minas Gerais até a localidade de Paulo Afonso no Estado da Bahia. O objetivo da presente pesquisa é verificar os possíveis agentes causais do referido processo e as variações temporais e espaciais da sedimentação. Até o presente momento verificou-se que as variações temporais e espaciais da sedimentação estão intimamente ligadas às alterações da cobertura vegetal que margeia o curso principal e em menor medida os afluentes, tendo o desmatamento e retirada da mata ciliar como principais vetores para desencadear o processo de desbarrancamento e o conseqüente assoreamento da bacia hidráulica. A atuação conjunta desses fatores interagindo com as variáveis climáticas, de relevo, da vegetação circundante, dos tipos de solos e as ações antrópicas promove em maior ou menor grau variações na taxa de sedimentação. Diante destas constatações objetivou-se detectar a origem e os principais agentes causais do elevado carreamento dos sedimentos superficiais ao leito do rio e conseqüentemente das obras de engenharia ali instaladas. Para tanto, foram utilizadas conjuntamente, técnicas de sensoriamento remoto, utilizando imagens de satélites de última geração, sistema de informação geográfica para análise, interpretação e apresentação gráfica dos resultados e verificação em campo, com apoio de equipamentos modernos de posicionamento no terreno (GPS)

Tendências fluviométricas nas áreas estuarinas de Goiana-Megaó e Pirapama/Jaboatão e das tabuas de maré no Porto de Suape-PE.

Os estuários são ambientes de transição entre o oceano e o continente, ocorrendo na desmbocadura dos rios, resultando na diluição da água salgada