Development of singularities for the compressible Euler equations with external force in several dimensions

We consider solutions to the Euler equations in the whole space from a certain class, which can be characterized, in particular, by finiteness of mass, total energy and momentum. We prove that for a large class of right-hand sides, including the viscous term, such solutions, no matter how smooth initially, develop a singularity within a finite time. We find a sufficient condition for the singularity formation, "the best sufficient condition", in the sense that one can explicitly construct a global in time smooth solution for which this condition is not satisfied "arbitrary little". Also compactly supported perturbation of nontrivial constant state is considered. We generalize the known theorem by Sideris on initial data resulting in singularities. Finally, we investigate the influence of frictional damping and rotation on the singularity formation.Comment: 23 page

Computer-aided research of ESP class materials: vocabulary potential and learning opportunities

The present study focuses on the assessment of vocabulary potential of teacher-created pedagogical materials to use in the class of English for Specific Purposes (ESP) taught to Russian undergraduates, majoring in Chemical Engineering, in a large public technical university. Specifically, the aim of the research is to examine the vocabulary distribution in the collection of course materials in terms of its range, frequency and variety. The motivation behind the study is that ESP instructors often have to rely on their own intuition during the text selection and material development due to the fact that there is the lack of pedagogical materials covering discipline-specific vocabulary of every specialization. ESP instructors teach English first, integrating the presentation of topics from subject matter classes and the major challenge consists in the lack of content-based knowledge of ESP instructors to select the domain vocabulary to L2 learners. Thus, to facilitate customized language acquisition (basic engineering, academic and specialized vocabulary) and to determine the extent to which the vocabulary contained in the texts is specialized and relevant to L2 learners’ domain, corpus software “Range” is used to run the analysis. The results suggest that the ESP corpus promotes the acquisition of high frequency English words (the first and the second 1,000), basic engineering and academic vocabulary. The results also demonstrate insufficient level of teacher-created materials for specialized vocabulary development and highlight the need to include texts of varying vocabulary types, and to optimize specialized wordlist

Regularities of changes of irido-ciliary-lenticular interrelations during presbyopia evelop-ment in patients with different types of refraction

The presbyopia formation in patients with different types of refractive errors has its own characteristics, which are unexplored. The purpose of this study was to identify the regularities of changes of irido-ciliary-lenticular relationships in presbyopia formation in different refractive conditions. 170 patients with various types of refraction (emmetropia, myopia, hyperopia) with or without presbyopia were examined. The infrared pupillometry, the ultrasound biomicroscopy and examination of the eye anterior segment under rotary Scheimpflug camera («Pentacam») were performed besides standard ophthalmic examination. In patients with emmetropic refraction the increase in the size of the lens, the reduction of the thickness of the ciliary body, the narrowing the diameter of the pupil in both photopic and scotopic conditions, the increase in the largest range of pupillary excursion and the first signs of intraocular asymmetry were noted during the formation of presbyopia. Presbyopes with axial myopia were distinguished from presbyopes with emmetropia by more pronounced constriction of the pupil in scotopic lighting conditions and increased intraocular initial asymmetry. The degree of increase in the anterior-posterior size of the lens in presbyopia formation was the most pronounced in the hyperopic eyes. The loss of accommodation in hyperopeswas accompanied by hypertrophy of anterior part of the ciliary muscle and an increase in pupil diameter. The development of presbyopia in patients with different types of refraction increases the initial differences of irido-ciliary-lenticular relationships

Analysis of Optical Properties of Posterior Surface of Cornea in Patients after Anterior Radial Keratotomy

Taking into account the constant increase in patients with age-related cataracts after radial keratotomy, a careful analysis of both the optical and anatomical properties of the cornea with the examination of the posterior surface is of particular importance.Aim. To analyze the optical properties of the posterior surface of the cornea in patients after anterior radial keratotomy. Materials and methods. An examination of 24 patients (48 eyes) with age-related cataracts of varying degrees of density, myopia and the presence of a previous anterior radial keratotomy or radial-tangential keratotomy in history. The average age of patients was 59.5 years (from 47 to 68), there were 19 women and 5 men.Results. The radius of curvature of the anterior surface in patients after anterior radial keratotomy was 9.45 ± 0.91 mm on average along the meridians, which is significantly more in comparison with these indices in control patients – 7.70 ± 0.19 mm (p = 0.0001). The ratio of the radii of the posterior cornea curvature to the anterior radius on average along the meridians in patients after anterior radial keratotomy was 1.07 ± 0.70, and in control patients – 1.20 ± 0.02 (p = 0.0001). The keratometric index in patients after anterior radial keratotomy was 1.3538 ± 0.0239, and in the control group – 1.3372 ± 0.0003 (p = 0.23).Conclusion. In patients after anterior radial keratotomy, keratometry of the posterior surface of the cornea is significantly higher than in the control. The ratio of the radius of curvature of the posterior cornea to the radius of curvature of the anterior cornea varies significantly after anterior radial keratotomy, which is due to a more pronounced flattening of the posterior cornea. The standard keratometric index (1.3375) is invalid for patients after anterior radial keratotomy and must be calculated individually for each patient when deciding on the operative treatment of cataracts

Relief-topography of anterior and posterior corneal surfaces in myopic patients in the long-term follow-up period after anterior radial keratotomy

Purpose. To analyze the topographical features of the relief of anterior and posterior corneal surfaces in patients with myopia in the long-term period after anterior r adial keratotomy (ARK).Material and methods. We examined 132 patients with age-related cataracts of varying degrees of density and a history of myopia with previously performed ARK. Mean age of patients was 59.67±6.09 (from 47 to 76). In the control group, 30 patients of the same age group with myopia were examined. A comprehensive examination was performed, including optical biometry, keratotopography on the Pentacam HR device with the determination of the elevation points of the anterior and posterior corneal surfaces.Results. The corneal profile differed from the correct spherical configuration in the control group. The average deviation of the profile of the anterior part of the cornea from the BFS was (–)1.41±5.10 μm, the posterior part was (–)5.12±12.25 μm (p=0.0001). The profile of the posterior corneal surface had a pronounced prolapse in the paracentral area with positive elevation values in the lower-outer segment. Negative values were found in the upper and lower sectors in the peripheral area and positive values were found in the inner and outer sectors. The corneal profile was deformed in patients after ARK. Negative elevation values in the central and paracentral area and positive elevation values in the peripheral segments corresponding to its protrusion were noted. At the same time, the relief of anterior and posterior surfaces of the cornea was not regular. Prolapse from the posterior surface of the cornea in the peripheral area was more pronounced than its anterior surface (p=0.0001).Conclusion. The relief of anterior and posterior corneal surfaces in patients of the control group differs in the degree and topography of elevation. In patients after ARK, obvious topographic deformities of the anterior and posterior corneal pr ofiles were detected