On microscopic sets and Fubini Property in all directions

2018 ◽  
Vol 68 (5) ◽  
pp. 1041-1048
Author(s):  
Adam Paszkiewicz

AbstractFor theσ-ideal$\mathcal{N} $of nullsets andσ-ideal$\mathcal{M} $of microscopic sets, it was recently obtained that there exists a Borel set$E\subset\mathbb{R}^{2} $with the following property:$E_{x}\in\mathcal{M} $for any$x\in\mathbb{R} $and$\{y;E^{y}\notin\mathcal{N}\}\notin\mathcal{M} $, for vertical sections$E_{x}=\{y;(x,y)\in E\} $and horizontal sections$E^{y}=\{x;(x,y)\in E\} $for$E\subset\mathbb{R}^{2} $. Thus$(\mathcal{N},\mathcal{M}) $does not satisfy Fubini Property. In this paper we obtain such Borel setE, that$\{y;E^{y}\notin\mathcal{N}\}\notin\mathcal{M} $and all non-horizontal (in a natural sense) sections ofEare in$\mathcal{M} $. Other Fubini type properties, with conditions written for all directions are also discussed.

Bone ◽  
1987 ◽  
Vol 8 (1) ◽  
pp. 13-17 ◽  
Author(s):  
A. Vesterby ◽  
J. Kragstrup ◽  
H.J.G. Gundersen ◽  
F. Melsen

2015 ◽  
Vol 33 (11) ◽  
pp. 1170-1174
Author(s):  
Bowen WANG ◽  
Wenmei HUANG ◽  
Ling WENG ◽  
Ying SUN ◽  
Shuying CAO ◽  
...  
Keyword(s):  

2008 ◽  
Vol 40 (01) ◽  
pp. 31-48
Author(s):  
Markus Kiderlen

We discuss the determination of the mean normal measure of a stationary random set Z ⊂ ℝ d by taking measurements at the intersections of Z with k-dimensional planes. We show that mean normal measures of sections with vertical planes determine the mean normal measure of Z if k ≥ 3 or if k = 2 and an additional mild assumption holds. The mean normal measures of finitely many flat sections are not sufficient for this purpose. On the other hand, a discrete mean normal measure can be verified (i.e. an a priori guess can be confirmed or discarded) using mean normal measures of intersections with m suitably chosen planes when m ≥ ⌊d / k⌋ + 1. This even holds for almost all m-tuples of k-dimensional planes are viable for verification. A consistent estimator for the mean normal measure of Z, based on stereological measurements in vertical sections, is also presented.


1963 ◽  
Vol 20 (4) ◽  
pp. 939-967 ◽  
Author(s):  
Robert K. Lane

Oceanographic data collected in a line of stations extending seaward of the west coast of Vancouver Island, British Columbia, were reviewed and analyzed. On the basis of these data and the large-scale meteorological processes of wind, insolation, and precipitation, the characteristic structure of temperature and salinity in the coastal region was denned in five temporal stages throughout the year. These stages are presented as vertical sections along the line with characteristic ranges of values to be found in each of the structural elements.


Author(s):  
Titi Christiana Falana

<p>Breast ironing also known as breast flattening, has been considered as one of the most widespread and systematic violations of the universal human rights to personal integrity and sexual autonomy perpetuated against the girl child. This exhilarating practice which is typically carried out by the girl’s mother on the pretext of protecting the female child from teenage sexual harassment and rape, early pregnancy and dropping out of school is unfortunately done to protect family name. Employing the qualitative research approach and the expository analytic method, this research reveals that this practice has ensuring clinical, psychological and social consequences on the female victim. The paper considers this practice a rape based on the fact that the female victim’s consent is never sought. Natural sense of justice demands that the victim whose personality integrity and right to sexual autonomy was to be defiled and destroyed deserves to give an informed consent. Therefore, the paper concludes that this harmful practice is an abuse because it violently violates the girl child right full sexual autonomy and right to possess natural physiological endowment that adorns a woman. This paper therefore recommends that strict laws and penalties should be promulgated to totally abolish and eradicate this barbaric and horrific mutilation.</p>


Author(s):  
K.A. Afonin ◽  
◽  

One of the main forms of the measurable selection theorem is connected with the existence of the graph of a measurable mapping in a given measurable set 𝑆 in the product of two measurable spaces 𝑋 and 𝑌 . Such a graph enables one to pick a point in the section 𝑆𝑥 for each 𝑥 in 𝑋 such that the obtained mapping will be measurable. The indicated selection is called a measurable selection of the multi-valued mapping associating to the point 𝑥 the section 𝑆𝑥 , which is a set in 𝑌 . The classical theorem of Blackwell and Ryll-Nardzewski states that a Borel set 𝑆 in the product of two complete separable metric spaces contains the graph of a Borel mapping (hence admits a Borel selection) provided that there is a transition probability on this product with positive measures for all sections of 𝑆 . The main result of this paper gives a generalization to the case where only one of the two spaces is complete separable and the other one is a general measurable space whose points parameterize a family of Borel probability measures on the first space such that the sections of the given set 𝑆 in the product have positive measures.


2019 ◽  
Vol 8 (4) ◽  
pp. 12274-12278

Rubber seed though not common known is very rich in its oil yield and over time its importance is becoming more prominent. Bleaching in the natural sense is relative to the removal of impurities from the oil or material. In this research work, Rubber seed oil was bleached using bentonite clay and Ibeshe clay at 0.5M, 1M and 2M concentrations of Hydrochloric acid. Physiochemical properties as well as spectroscopic analysis such as FT-IR and XRD analysis were carried out and aided in obtaining the bleaching efficiency of both clays. The FT-IR results displayed a visible change in the oil after it was bleached with Ibeshe clay but still retained most its functional group when bleached with bentonite clay. At 2M concentration of the acid, the oil bleached with bentonite showed 53% while with Ibeshe clay it remained at 16%. This summarizes that Ibeshe clay has little to no effect on bleaching performance.


Ars Adriatica ◽  
2015 ◽  
pp. 115
Author(s):  
Bojan Goja

The article analyses the illuminations of two incunables which are housed in the monastery of St Francis at Šibenik. The front page of the incunable of John Duns Scotus’ Scriptum in quattuor libros Sententiarum Petri Lombardi (Johannes de Colonia et Johannes Manthen, Venice, 1477) is decorated with high-quality figural and phytomorphic illuminations. In the corners of the decorative frame in the upper margin are the figures of a heronand a monkey. Vertical sections of the frame are filled with flowers, leaves and berries in the colour blue, green and cyclamen pink and with numerous stylized golden burdock flowers (Arctium). The central part of the frame in the upper and lower margin is filled with dense, symmetrically placed thick leaves in the colour blue, green, purple and cyclamen pink with a stylized golden burdockflower (Arctium) appearing here and there. In the centre of the page is a crest composed of two fields separated by a horizontal line; the upper on is red and the lower one white. Two winged putti are set in the corners andthey hold red ribbons. Each wears a necklace made of red corals and classical sandals on their feet. They landscape around them is arid and there is only one tree, its bark dry, standing in it. The rocky ground with jagged edges is covered in small stones. The distinctly painted winged putti, the depiction of the landscape and the dense vegetal decoration filling the frame in the upper and lower margin demonstrate noticeable similarities with the works of Giovanni Vendramin, a prominent representative of Paduan Renaissance miniature. Thefront page of the aforementioned incunable at Šibenik can be attributed to him; he may well have been helped by his workshop and collaborators. First and foremost, it ought to be mentioned that the decorative frame on one of the opening pages (c. 4v) in an Antiphonary at Ferrara features identical type of leaf decoration as the one that fills the upper and lower margin in the incunable at Šibenik. Here too, the playful putti wear classical sandals and necklaces made of red coral. Furthermore, putti with identical physiohnomies – wearing coral necklaces and classical sandals while holding ribbons in their hands – can be found on fol. 2r in the incunable of Marcus Tullius Cicero’s Orationes (Venice, Christophorus Valdarfer, 1471, Philadelphia, The Rosenbach Museum and Library, Inc 471ci). The landscape in which the putti are depictedis also arid and marked by a single dry tree rising from the ground covered with small stones. Identical putti can be seen on the cover of the incunable of Marcus Tullius Cicero’s, Tusculanae Quaestiones (Venice, N. Jenson, 1472, London, British Library, C.1c.10, fol. 1). The landscape is also depicted in the same way. An excellent comparative example can be found in the winged putti standing on an all’antica structure on the cover of the manuscript of Jacopo Camphora’s, De immortalitate animae (London, Brittish Library, MS Add. 22325) which is decorated with architectural forms. The left and the upper margins of the opening pages of Book I and Book III of Gaius Julius Caesar’s Commentariorvm de bello Gallico (Milan, Antonius Zarotus, 1477) are decorated with frames filled with white vine scrolls on red, green and blue background with white dots. The decoration extends beyond the ornamental frames and reaches into the gold initials G and C. Although the decorative frames were not completely finished, it can be ascertained that they were made with great skill and are of high quality. This frame type was frequently used by Giovanni Vendramin and the examples from Šibenik are very close to some of his works, especially those made for Jacopo Zeno, the Bishop of Padua (Padua, Biblioteca Capitolare).


2018 ◽  
Vol 78 (6) ◽  
pp. 1178-1184 ◽  
Author(s):  
Julio Jasso-Olivares ◽  
José Manuel Diaz-Gonzalez ◽  
Mariya Miteva

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