On one-sided densities of arcs of positive two-dimensional measure

1964 ◽  
Vol 60 (3) ◽  
pp. 517-524
Author(s):  
A. S. Besicovitch

1. Let AB be a plane simple arc, which, considered as a point-set, has a positive two-dimensional measure. If x be a point of AB, a point x′ ≠ x is said to be to the left or to the right from x according as it belongs to the arc Ax or xB. The upper and the lower limits of the ratio [m2{Ax C(x, r)}]/πr2, where C(x, r) denotes the closed disc with centre x and radius r, as r → 0 define the upper and the lower left densities at the point x. When the two densities are equal their common value defines the left density. The right densities are defined similarly.

Author(s):  
D. G. Larman

We use J(a, b) to denote a Jordan curve of positive two-dimensional measure in the plane, with end-points a and b. If υ is a point of J(a, b), we define the right lower arc density at υ bywhere J( υ, υ′) is the largest arc, whose left-end point is υ, which is contained in the disc c(υ, r).


2021 ◽  
Vol 2021 (7) ◽  
Author(s):  
Mert Besken ◽  
Jan de Boer ◽  
Grégoire Mathys

Abstract We discuss some general aspects of commutators of local operators in Lorentzian CFTs, which can be obtained from a suitable analytic continuation of the Euclidean operator product expansion (OPE). Commutators only make sense as distributions, and care has to be taken to extract the right distribution from the OPE. We provide explicit computations in two and four-dimensional CFTs, focusing mainly on commutators of components of the stress-tensor. We rederive several familiar results, such as the canonical commutation relations of free field theory, the local form of the Poincaré algebra, and the Virasoro algebra of two-dimensional CFT. We then consider commutators of light-ray operators built from the stress-tensor. Using simplifying features of the light sheet limit in four-dimensional CFT we provide a direct computation of the BMS algebra formed by a specific set of light-ray operators in theories with no light scalar conformal primaries. In four-dimensional CFT we define a new infinite set of light-ray operators constructed from the stress-tensor, which all have well-defined matrix elements. These are a direct generalization of the two-dimensional Virasoro light-ray operators that are obtained from a conformal embedding of Minkowski space in the Lorentzian cylinder. They obey Hermiticity conditions similar to their two-dimensional analogues, and also share the property that a semi-infinite subset annihilates the vacuum.


2012 ◽  
Vol 56 (6) ◽  
pp. 2159-2181 ◽  
Author(s):  
Meng Sang Ong ◽  
Ye Chow Kuang ◽  
Melanie Po-Leen Ooi

2021 ◽  
Vol 49 ◽  
Author(s):  
Sabrina Barros Araújo ◽  
Flávio Ribeiro Alves ◽  
Gerson Tavares Pessoa ◽  
Renan Paraguassu De Sá Rodrigues ◽  
Laecio Da Silva Moura ◽  
...  

Background: Veterinary Ophthalmology provides complementary information for the diagnosis of ocular pathologies. Studies in wild species are essential. Among the diagnostic techniques in ophthalmology, two-dimensional ultrasonography stands out. The agouti is a rodent belonging to the Dasyproctidae family that has been widely used as an experimental model. For these animals, sight is one of the crucial senses for their survival. The aim of this study was to evaluate the effectiveness of the two-dimensional ocular ultrasound technique to obtain anatomical measurements and the external ophthalmic artery resistivity index, presumably normal in the species Dasyprocta prymnolopha.Materials, Methods & Results: Forty eye bulbs of 20 adult rodents of the species were evaluated by ultrasonography. In these animals, B-mode echobiometry was performed using the transpalpebral approach and the hemodynamic study of the external ophthalmic artery using the color Doppler technique. All examinations were initiated by the left eye bulb and all measurements were performed by only one examiner. The collected data related to echobiometry were analyzed using Bioest 5.0 for Windows. Initially, normality was tested using the Shapiro-Wilk test for each parameter, then the paired t-test was performed, comparing right and left eyes, and a significance level of 5% (P < 0.05) was adopted. Based on the methodology used, the following values were obtained for the right and left eyeballs, respectively: anterior chamber thickness - mean of 1.28 ± 0.3 mm and 1.22 ± 0.1 mm; lens thickness - 8.27 ± 0.9 mm and 8.11 ± 0.9 mm; vitreous chamber thickness - 5.35 ± 0.48 mm and 5.30 ± 0.47 mm and axial length - 12.7 ± 0.9 mm and 13 ± 0.68 mm. The mean external ophthalmic artery resistivity values were 0.4305 ± 0.0390 and 0.4258 ± 0.0387 (right and left eye, respectively), characterizing a low resistance. There was no statistical difference between the right and left eyeballs in any of the studied parameters.Discussion: The use of the convex transducer was feasible, promoting adequate contact with the ocular surface and images of satisfactory quality for obtaining measurements, similar to what was observed in studies evaluating the ocular biometry of primates and dogs. The anterior chamber thickness values in this experiment did not differ statistically between the antimers, as well as observed for dogs. The data obtained for lens thickness did not differ statistically for antimers, like those obtained for other rodent species evaluated with the same methodology. The mean values of vitreous chamber thickness were like those observed in chinchillas but correspond to about half of that obtained for capybaras. In this study, the external ophthalmic artery was characterized in all animals, but obtaining the spectral tracing was difficult due to its fine caliber. In wild animals, and especially in wild rodents, there are few data reporting the resistivity of the ophthalmic artery, and there is a lack of studies, which can be explained by the behavioral characteristics of defense and by the high susceptibility to stress in capture, since the performing the technique requires, as in other procedures, the use of chemical containment.Keywords: Doppler flow, ultrasound, eye, agouti.


Author(s):  
Victor Revenko ◽  
Andrian Revenko

The three-dimensional stress-strain state of an isotropic plate loaded on all its surfaces is considered in the article. The initial problem is divided into two ones: symmetrical bending of the plate and a symmetrical compression of the plate, by specified loads. It is shown that the plane problem of the theory of elasticity is a special case of the second task. To solve the second task, the symmetry of normal stresses is used. Boundary conditions on plane surfaces are satisfied and harmonic conditions are obtained for some functions. Expressions of effort were found after integrating three-dimensional stresses that satisfy three equilibrium equations. For a thin plate, a closed system of equations was obtained to determine the harmonic functions. Displacements and stresses in the plate were expressed in two two-dimensional harmonic functions and a partial solution of the Laplace equation with the right-hand side, which is determined by the end loads. Three-dimensional boundary conditions were reduced to two-dimensional ones. The formula was found for experimental determination of the sum of normal stresses via the displacements of the surface of the plate.


2019 ◽  
Vol 15 (1) ◽  
Author(s):  
Natalia Siwinska ◽  
Marcin Michalek ◽  
Agnieszka Zak ◽  
Malwina Slowikowska ◽  
Agnieszka Noszczyk-Nowak ◽  
...  

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