Curves and surfaces making a constant angle with a parallel transported direction in Riemannian spaces

In the last two decades, much effort has been dedicated to studying curves and surfaces according to their angle with a given direction. How- ever, most findings were obtained using a case-by-case approach, and it is often unclear what is a consequence of specificities of the ambient manifold and what could be generic. In this work, we propose a theo- retical framework to unify parts of these findings. We study curves and surfaces by prescribing the angle they make with a parallel transported vector field. We show that the characterization of Euclidean helices in terms of their curvature and torsion is also valid in any Riemannian manifold. Among other properties, we prove that surfaces making a con- stant angle with a parallel transported direction are extrinsically flat ruled surfaces. We also investigate the relation between their geodesics and the so-called slant helices. We prove that surfaces of constant angle are the rectifying surface of a slant helix, i.e., the ruled surface with rulings given by the Darboux field of the directrix. We characterize recti- fying surfaces of constant angle or, equivalently, when their geodesics are slant helices. As a corollary, we show that if every geodesic of a surface of constant angle is a slant helix, the ambient manifold is flat. Finally, we characterize surfaces in the product of a Riemannian surface with the real line making a constant angle with the vertical real direction

Curves and surfaces making a constant angle with a parallel transported direction in Riemannian spaces

In the last two decades, much effort has been dedicated to studying curves and surfaces according to their angle with a given direction. How- ever, most findings were obtained using a case-by-case approach, and it is often unclear what is a consequence of specificities of the ambient manifold and what could be generic. In this work, we propose a theo- retical framework to unify parts of these findings. We study curves and surfaces by prescribing the angle they make with a parallel transported vector field. We show that the characterization of Euclidean helices in terms of their curvature and torsion is also valid in any Riemannian manifold. Among other properties, we prove that surfaces making a con- stant angle with a parallel transported direction are extrinsically flat ruled surfaces. We also investigate the relation between their geodesics and the so-called slant helices. We prove that surfaces of constant angle are the rectifying surface of a slant helix, i.e., the ruled surface with rulings given by the Darboux field of the directrix. We characterize recti- fying surfaces of constant angle or, equivalently, when their geodesics are slant helices. As a corollary, we show that if every geodesic of a surface of constant angle is a slant helix, the ambient manifold is flat. Finally, we characterize surfaces in the product of a Riemannian surface with the real line making a constant angle with the vertical real direction

Curves orthogonal to a vector field in Euclidean spaces

A curve is rectifying if it lies on a moving hyperplane orthogonal to its curvature vector. In this work, we extend the main result of [B.-Y. Chen, Tamkang J. Math. \textbf{48} (2017) 209--214] to any space dimension: we prove that rectifying curves are geodesics on the hypersurface of higher dimensional cones. We later use this association to characterize rectifying curves that are also slant helices in three-dimensional space as geodesics of circular cones. In addition, we consider curves that lie on a moving hyperplane normal to (i) one of the normal vector fields of the Frenet frame and to (ii) a rotation minimizing vector field along the curve. The former class is characterized in terms of the constancy of a certain vector field normal to the curve, while the latter contains spherical and plane curves. Finally, we establish a formal mapping between rectifying and spherical curves in any dimension.Comment: 12 pages; keywords: Rectifying curve, geodesic, cone, spherical curve, plane curve, slant heli

A EXPANSÃO URBANA DA CIDADE DE EUNÁPOLIS E SUA INTERFACE COM A OFERTA DE SANEAMENTO BÁSICO

Este estudo buscou analisar a interface entre a expansão da cidade de Eunápolis e a oferta de saneamento básico. Impulsionada pelo plantio de eucalipto na região e pela implantação da indústria de celulose, a dinâmica econômica da cidade se altera com crescimento econômico e aumento populacional. Para atender à demanda do contingente de migrantes em busca de oportunidades, novos bairros foram surgindo, porém, sem a infraestrutura adequada de saneamento básico, que é um importante elemento na qualidade de vida das pessoas. Como metodologia para atingir o objetivo, foi realizado estudo bibliográfico para maior aproximação dos estudiosos sobre o tema e a análise de dados secundários do Sistema de Informação da Atenção Básica (SIAB) no período de 1998 – 2013; Atlas de Saneamento Básico (2011); Instituto Trata Brasil (2012); e, da Empresa Baiana de Água e Saneamento (2005 – 2013). Evidenciou-se no estudo que não há uma universalização da cobertura pela rede pública de esgoto, na qual apenas 20,4% das famílias cadastradas no SIAB declararam ter acesso a esse serviço. A fossa como destino do esgoto foi utilizada por 68,9% das famílias no ano 2013. Da mesma forma, a disposição do esgoto a céu aberto  foi referida como prática por 10,3% das famílias. O fornecimento de água por intermédio da rede pública teve cobertura de 82,2% em 2013, porém 15,7% das famílias declaram fazer uso de água de poço ou nascente. A coleta de lixo também teve uma boa cobertura pelo serviço público de coleta, mas persistiu a prática de enterro, queima e disposição do lixo a céu aberto. O destino do lixo ainda ocorreu nos chamados “lixões”. Em 2013, a cidade não contava com rede pluvial, acarretando ruas alagadas em vários pontos da cidade. Considera-se que a cidade precisa ser (re)pensada para que o crescimento econômico e o desenvolvimento humano caminhem próximos para que a população tenha acesso aos serviços públicos, incluindo o saneamento básico que é um elemento importante para a saúde das pessoas e preservação do meio ambiente. Para isso, o Plano Diretor Municipal e o Plano Municipal de saneamento básico deverão ser mecanismos tanto para a regulação do uso do solo como para o destino correto do esgoto sanitário, do lixo e das águas pluviais

EFFECT OF EXTRACTIVE CONTENTS ON NATURAL RESISTANCE OF FIVE DIFFERENT WOODS TO XILOPHAGAUS TERMITES ATTACK

Quando o homem deixou de ser n\uf4made, teve de se adaptar \ue0s novas condi\ue7\uf5es, passando a empregar a madeira para energia, instrumentos de ca\ue7a e defesa e elementos construtivos diversos. Desde ent\ue3o, em fun\ue7\ue3o dos usos e do r\ue1pido crescimento, a import\ue2ncia da madeira vem se acentuando, principalmente porque sua explora\ue7\ue3o agride menos o ambiente quando comparada a outros materiais n\ue3o renov\ue1veis. Embora existam v\ue1rias pesquisas sobre Eucalyptus e Pinus , estudos com outras esp\ue9cies s\ue3o incipientes, principalmente quanto ao efeito dos extrativos na resist\ueancia natural das madeiras. Assim, o objetivo da pesquisa foi avaliar a influ\ueancia do teor de extrativos na resist\ueancia natural das madeiras de Acacia mangium , Casuarina sp., Eucalyptus cloeziana , Corymbia torelliana e Tectona grandis ao t\ue9rmita xil\uf3fago Nasutitermes corniger , esp\ue9cie de ocorr\ueancia frequente em v\ue1rias regi\uf5es no Brasil. De cada esp\ue9cie foram retirados corpos de prova, com dimens\uf5es de 2,00 x 10,16 x 0,64 cm (radial x longitudinal x tangencial) em quatro posi\ue7\uf5es no sentido medula-alburno (cerne interno, cerne intermedi\ue1rio, cerne externo e alburno). As madeiras foram expostas \ue0 a\ue7\ue3o dos cupins durante 45 dias em ensaio de prefer\ueancia alimentar. Amostras n\ue3o selecionadas para o ensaio com cupins foram transformadas em serragem e o teor de extrativos obtido ao empregar a fra\ue7\ue3o que passou pela peneira de 40 e ficou retida na de 60 mesh. A resist\ueancia natural n\ue3o esteve associada ao teor de extrativos presentes na madeira. A resist\ueancia das madeiras variou no sentido medula-alburno, com padr\ue3o vari\ue1vel para cada esp\ue9cie. As madeiras mais resistentes foram Tectona grandis e Corymbia torelliana e a mais deteriorada a Acacia mangium.When man stopped being nomadic, he had to adapt to new conditions, and started employing wood for energy, hunting and defense instruments and miscellaneous constructive elements. Since then, in the light of usages and rapid growth, the importance of wood has been increasing, primarily by their exploitation assaults less the environment when compared to other non-renewable materials. Although there are several researches about Eucalyptus and Pinus, but studies with other species are incipient mainly about the extractive effects on the wood natural resistance. Thus, this research aimed to assess the influence of extractive contents with the natural resistance of Acacia mangium, Casuarina sp. Eucalyptus cloeziana, Corymbia torelliana and Tectona grandis woods to Nasutitermes corniger xylophagous termite, species of frequent occurrence in several regions in Brazil. From each species, test samples were taken, with dimensions of 2.00 x 10.16 x 0.64 (radial x longitudinal x tangential) in four positions in the medulla-sapwood direction (internal core, intermediate core, outer core and sapwood). The woods were exposed to the action of termites for 45 days in a food preference assay. The samples not selected for the test with termites were turned into sawdust and the extractive contents were obtained by using the fraction which passed through the sieve 40 and was retained in 60 mesh. The natural resistance has not been linked to extractive levels present in the wood. The resistance of wood varied in medulla-sapwood direction, with default variable for each species. The more resistant timbers were Tectona grandis and Corymbia torelliana and the most deteriorated was Acacia mangium

Correlação de Spearman aplicada ao estudo de adaptabilidade e estabilidade em genótipos de alfafa

Este trabalho teve por objetivo propor uma nova metodologia, baseada no coeficiente de correlação de Spearman para o estudo da adaptabilidade e estabilidade fenotípica de genótipos de alfafa. Além disso, os resultados foram comparados com os obtidos pela metodologia baseada no teste dos sinais. Para tanto, foram utilizados dados provenientes de um experimento em blocos casualizados com 2 repetições, que constituiu-se da avaliação da produção de matéria seca de 92 cultivares de alfafa em 20 cortes, realizados no período de novembro de 2004 a junho de 2006 no Campo Experimental da Embrapa Pecuária Sudeste - São Carlos/SP. Os resultados encontrados mostram que o coeficiente de correlação de Spearman é eficiente para o estudo de adaptabilidade e estabilidade fenotípica, sendo possível classificar os cultivares conforme o interesse do estudo. Ademais, quando comparado com a metodologia baseado no teste dos sinais, o mesmo se mostra mais eficiente na discriminação de genótipos

Individual tree growth models for eucalyptus in northern Brazil

The diameter and height growth model is one of three submodels used for simulating individual tree growth. In Brazil, there are few studies on the dimensional growth of individual trees be they native or exotic species, despite their potential. This study aimed to evaluate diameter and height growth models for individual trees for eucalyptus stands and to validate the best fitting model. Tree diameter and height data were obtained from 48 permanent plots of unthinned stands of Eucalyptus grandis × Eucalyptus urophylla hybrid located in northern Brazil. The evaluation of the diameter and height growth models was based on adjusted coefficient of determination, standard error of estimate as a percentage, trend, root mean square error and Akaike Information Criterion. Analysis also included distribution of residual percentage, statistical significance and signs of the coefficients. The Lundqvist-Korf model provided the most accurate estimates for diameter and height growth, in comparison with the other models, providing better statistical values, greater proximity to observed values and better distribution of residual percentages. The use of this type of model is feasible and can result in significant improvements in the accuracy of yield estimates