Integrating the geodesic equations in the Schwarzschild and Kerr space-times using Beltrami's "geometrical" method

We revisit a little known theorem due to Beltrami, through which the integration of the geodesic equations of a curved manifold is accomplished by a method which, even if inspired by the Hamilton-Jacobi method, is purely geometric. The application of this theorem to the Schwarzschild and Kerr metrics leads straightforwardly to the general solution of their geodesic equations. This way of dealing with the problem is, in our opinion, very much in keeping with the geometric spirit of general relativity. In fact, thanks to this theorem we can integrate the geodesic equations by a geometrical method and then verify that the classical conservation laws follow from these equations.Comment: 12 pages; corrected typos, journal-ref adde

Dimensional stability of lightweight foamed concrete

Thesis (MEng)--Stellenbosch University, 2019.ENGLISH ABSTRACT: Lightweight Foamed Concrete (LWFC) is a variation of low density cellular concrete produced by entraining a system of regular and stable air voids into a mortar base mix. The air void structure in LWFC is created by incorporating a pre-formed aqueous foam into a regular mortar base mix which typically consists of ordinary Portland cement, water, fine aggregates and a variety of supplementary cementitious materials. The result of incorporating this pre-formed aqueous foam is a hardened concrete material which varies in density from 450 kg/m3 – 1600 kg/m3, possess superior fresh state and insulative properties and drastically reduces dead loads on structures. Although significant strides have been made when concerning the mechanical characteristics of LWFC, the durability and long term behaviour properties of this material such as creep and shrinkage remain relatively undocumented. As the long-term dimensional stability of concrete is an essential factor to consider when designing any concrete structure, it is imperative that the creep and shrinkage behaviour of LWFC be investigated, quantified and recorded to aid in commercialising this innovative construction material. This study investigates and quantifies the long-term creep and shrinkage behaviour of LWFC by means of three separate methods. The first of these is experimental testing on a variety of physical LWFC samples. The effects of density, foam volume, filler type and filler quantity as well as the effects of other additives have all been noted as important parameters to investigate. For this reason, LWFC mixes which vary in density from 1200 kg/m3 – 1600 kg/m3 and contain only cement have been tested to investigate the effects of density and foam volume whilst other mixes all with a density of 1400 kg/m3 containing sand, fly ash and propylene glycol have been tested to investigate the effects of various fillers and additives in different quantities. The second investigation method is the study of three prominent theoretical prediction models which have been examined and their applicability to LWFC assessed. These prediction models are that of the Fib Model Code 2010, EN 1992-1 (2004) and Rilem Model B3. All three models have been calibrated with typical LWFC properties and adjustment factors proposed by comparing their results to experimentally obtained values from physical samples. The last research method utilised is that of developing a nonlinear viscoelastic Kelvin-Chain finite element model in Diana Finite Element Analysis Software. The finite element model utilises direct input of experimental creep and shrinkage curves to accurately replicate the results of physical tests. The results obtained from physical creep and shrinkage tests indicate that LWFC exhibits drying creep and drying shrinkage strains in excess of 6000 μm which is more than six times what is generally seen from ordinary concrete. Additionally, under sealed conditions LWFC exhibits autogenous shrinkage and basic creep strain values which may be as high as 3000 μm. It has been noted that for drying shrinkage, an inverse relationship appears to be present between density and drying shrinkage strain. For both creep and shrinkage testing it has been found that the inclusion of both fillers in all ratios results in a decrease in strain values compared to mixes containing only cement. The inclusion of propylene glycol into LWFC is seen to significantly reduce the total amount of drying shrinkage and drying creep strain exhibited. Results from theoretical prediction models show good correlation to ordinary concrete results, however, adjustments factors ranging between 2.7 – 6.3 for shrinkage strains and 0.9 – 1.4 for creep strains have been applied to account for the increased shrinkage and creep behaviour of LWFC. The results obtained from Diana FEA are able to replicate LWFC shrinkage behaviour exactly whilst creep results may be replicated with an average accuracy of 0.46%. The findings presented in this thesis indicate that the importance of quantifying the long-term deformation behaviour of LWFC cannot be understated. Results indicate that the inclusion of fillers such as sand and fly ash as well as the addition of additives such as propylene glycol are capable of reducing shrinkage and creep strains significantly, however, these strains are still far above what is typically accepted for ordinary concretes.AFRIKAANSE OPSOMMING: Liggewig skuimbeton is 'n variasie van sellulêre beton met 'n lae digtheid wat geproduseer word deur 'n stelsel van gereelde en stabiele lugruimtes, wat in 'n mortelbasis mengsel betrek word. Die lugvloeistofstruktuur in liggewig skuimbeton word geskep deur 'n vooraf gevormde waterige skuim in 'n gewone basismengsel in te sluit, wat gewoonlik bestaan uit gewone Portland-sement, water, fyn aggregate en 'n verskeidenheid aanvullende sementhoudende materiale. Die resultaat van die voorgevormde waterige skuim is 'n geharde betonmateriaal wat in digtheid van 450 kg / m3 - 1600 kg / m3 wissel, wat 'n uitstekende vars toestand en isolerende eienskappe bevat, in gevolg verminder dit drasties die dooie las op strukture. Alhoewel daar belangrike vordering gemaak is met betrekking tot die meganiese eienskappe van liggewig skuimbeton , bly die duursaamheid en langdurige gedragseienskappe van hierdie materiaal soos die kruip en krimp gedrag relatief ongedokumenteerd. Die langtermyn dimensionele stabiliteit van beton is ‘n wesenlike faktor wat in ag geneem moet word by die ontwerp van enige betonstruktuur, dit noodsaak dat die kruip en krimp gedrag van liggewig skuimbeton ondersoek, gekwantifiseer en aangeteken moet word om hierdie innoverende konstruksiemateriaal te kommersialiseer. Hierdie studie ondersoek en kwantifiseer die langtermyn -kruip- en krimpgedrag van liggewig skuimbeton met behulp van drie afsonderlike metodes. Die eerste hiervan is eksperimentele toetsing op 'n verskeidenheid fisiese liggewig skuimbeton-monsters. Die gevolge van digtheid, skuimvolume, tipe en hoeveelheid opvulmateriaal sowel as die gevolge van ander bymiddels is al opgemerk as belangrike parameters om te ondersoek. Om hierdie rede is liggewig skuimbeton-mengsels wat in digtheid van 1200 kg / m3 - 1600 kg / m3 verskil en slegs sement bevat, getoets om die gevolge van digtheid en skuimvolume te ondersoek, terwyl ander mengsels met 'n digtheid van 1400 kg / m3 sand, vliegas en propyleenglikol is getoets om die gevolge van verskillende vullers en bymiddels in verskillende hoeveelhede te ondersoek. Die tweede ondersoekmetode is die studie van drie prominente teoretiese voorspellingsmodelle, en die toepaslikheid daarvan op liggewig skuimbeton. Hierdie voorspellingsmodelle is dié van die Fib Model Kode 2010, EN 1992-1 (2004) en Rilem Model B3. Al drie modelle is gekalibreer met tipiese liggewig skuimbeton-eienskappe en aanpassingsfaktore wat voorgestel is deur hul resultate te vergelyk met eksperimenteel verkryde waardes uit fisiese monsters. Die laaste navorsingsmetode wat gebruik is, is die ontwikkeling van 'n nie-lineêre visco-elastiese Kelvin ketting eindige elementmodel in Diana Eindige Element Analiserende Sagteware. Die eindige elementmodel maak gebruik van direkte invoer van eksperimentele kruip- en krimp kurwes om die resultate van fisiese toetse akkuraat te repliseer. Die resultate verkry uit fisiese kruip- en krimptoetse dui aan dat liggewig skuimbeton droogkruip en droë krimp vervormings van meer as 6000 μm vertoon, wat dus ses keer meer is as wat algemeen gesien word uit gewone beton. Boonop het liggewig skuimbeton onder verseëlde toestande outogene krimping en basiese kruipvervormingwaardes wat tot 3000μm kan wees. Daar is opgemerk dat vir die droging van krimping 'n omgekeerde verband blyk te wees tussen digtheid en drogingskrimpvervorming. Vir beide die kruip- en krimpingstoetse is daar gevind dat die insluiting van beide vullers in alle verhoudings lei tot 'n afname in vervormingwaardes in vergelyking met mengsels wat slegs sement bevat. Daar word gesien dat die insluiting van propyleenglikol in liggewig skuimbeton die totale hoeveelheid drogingskrimping en droë kruipvervorming aansienlik verminder. Resultate van teoretiese voorspellingsmodelle toon goeie korrelasie met gewone beton resultate, maar aanpassingsfaktore wat wissel tussen 2,7 - 6,3 vir krimpvervormings en 0,9 - 1,4 vir kruipvervormings is toegepas om rekening te hou met die verhoogde krimping en kruipgedrag van liggewig skuimbeton. Die resultate van Diana FEA kan liggewig skuimbeton se krimpgedrag presies naboots, terwyl kruipresultate met 'n gemiddelde akkuraatheid van 0.46% naboots kan word.Master

Grupos de interaprendizaje docente y trabajo colegiado en instituciones educativas con jornada escolar completa en la UGEL Caravelí. Arequipa, 2022

En el presente trabajo de investigación se planteó como objetivo general: Determinar la relación entre los grupos de interaprendizaje docente y el trabajo colegiado en instituciones educativas con jornada escolar completa en la UGEL Caraveli. Arequipa, 2022. El diseño de investigación fue de tipo aplicada con un diseño no experimental, correlacional, transversal, observacional y descriptivo con un enfoque cuantitativo. Se consideró una muestra censal compuesta por 80 colaboradores, entre directivos y docentes de las instituciones educativas con jornada escolar completa de la UGEL Caravelí. Arequipa, 2022. Para el acopio de los datos se utilizaron dos cuestionarios elaborados por el propio investigador según la escala de Likert para cada variable. Para el análisis descriptivo e inferencial se procesó una base de datos con el programa Microsoft Excel, para poder presentar los resultados se utilizó el programa estadístico SPSS versión 25. Por otro lado, para determinar la correlación entre las variables se aplicó el coeficiente de Rho Spearman. Los resultados de este trabajo de investigación demostraron que existe una correlación positiva fuerte entre las variables grupos de interaprendizaje docente y el trabajo colegiado, con un valor del coeficiente de correlación de 0.779, con lo que se concluye que existe una correlación significativ

FairSwiRL: fair semi-supervised classification with representation learning

International audienceSemi-supervised learning has shown its potential in many real-world applications where only few labeled examples are available. However, when some fairness constraints need to be satisfied, semi-supervised classification models often struggle as they are required to cope with the lack of sufficient information for predicting the target variable while forgetting its relationships with any sensitive and potentially discriminatory attribute. To address this issue, we propose a fair semi-supervised representation learning architecture that leads to fair and accurate classification results even in very challenging scenarios with few labeled (but biased) instances. We show experimentally that our model can be easily adopted in very general settings, as the learned representations may be employed to train any supervised classifier. Moreover, when applied to several synthetic and real-world datasets, our method is competitive with state-of-the-art fair semi-supervised approaches