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Calculus of variations and partial differential equations 60(1), 32 (2021). doi:10.1007/s00526-020-01875-6, Published by Springer, Heidelberg

Liquid storage structures represent an important component of modern infrastructure. They can take variety of shapes of which the conical shape is one of the most common configurations. Conical tanks are preferred by both architects and structural engineers because of their appealing look and structural efficiency in addition to their large capacities with relatively small footprint area. The state of stresses in these tanks is rather complicated and needs powerful computational tools. However, in the preliminary design phase, it is important to have a simplified analysis method for selection of economic design parameters including tank height, inclination angle, and footprint radius needed to achieve the desired tank capacity. It is also of importance that the structural engineer has an insight and understanding of the effect of these various parameters on the resulting internal forces acting on the tank. This paper presents a simplified analysis of conical tanks under hydrostatic loading based on the application of the membrane theory. The equations governing the behavior of these structures are first derived. Then, they are applied on several vessels of practical dimensions and the resulting of stresses are presented to give a deeper understanding of the resulting internal actions. Moreover, a simple guide to achieve efficient structural preliminary design parameters for a wide range of tank capacities is introduced.

In this study, we examine the condition of the conchoidal surface to be a Bonnet surface in Euclidean 3-space. Especially, we consider the Bonnet conchoidal surfaces which admit an infnite number of isometries. In addition, we study the necessary conditions which have to be fulflled by the surface of revolution with the rotating curve c(t) and its conchoid curve cd(t) to be the Bonnet surface in Euclidean 3-space.

For constructions bounded by the surfaces of revolution, structured grid generation technique is presented. Its technology has been elaborated within the variational approach for constructing optimal grids satisfying optimality criteria: Closeness of grids to uniform ones, closeness of grids to orthogonal ones and adaptation to a given function. Grid generation has been designed for numerical solution of the differential equations modeling the vortex processes of multi-component hydrodynamics. In the technology, the three-dimensional construction in which it is required to construct a grid is represented in the form of the curvilinear hexahedron defining its configuration. The specific feature of the required configurations is that some faces of a curvilinear hexahedron lie in one plane and along edges of adjoining faces grid cells degenerate into prisms. Grid generation in the considered constructions has started to be developed by the elaboration of algorithms for the volume of revolution which has become the basic construction. The basic construction is obtained by the rotation through 180° around the axis of a generatrix consisting of straight line segments, arcs of circles and ellipses. Then the deformed volumes of revolutions are considered along with the generalizations of the volume of revolution which represent constructions obtained by the surfaces of revolution with parallel axis of rotation. The aim of the further development of the technology is to consider more and more complicated constructions and elaborate the technology for them. In the presentation, the current state of the development of the technology is given. Examples of generated grids are supplied. © 2021 Institute of Physics Publishing. All rights reserved.

Anadolu Üniversitesi ve Bilecik Şeyh Edebali Üniversitesi tarafından ortak yürütülen program. Bu çalışma dört bölümden oluşmuştur. Birinci bölümde giriş kısmına yer verilmiştir. İkinci bölümde 3-boyutlu Öklid uzayı ve Minkowski uzaylarında temel kavramlar verilmiş ve yüzeyler tanıtılmıştır. Üçüncü bölümde 3-boyutlu Öklid ve Minkowski uzaylarında dönel yüzeyler ve yoğunluklu yüzeyler tanıtıldı. Ayrıca 3- boyutlu Minkowski uzayında dönel yüzeyler eksen tiplerine göre incelenmiştir. Dördüncü bölüm ise bu çalışmanın orijinal kısmını oluşturmaktadır. Bu bölümde yoğunluklu 3-boyutlu Minkowski uzayında lightlike dönme eksenli dönel yüzeyler incelenmiştir. Dönel yüzeylerin ağırlıklı ortalama eğriliği ve ağırlıklı Gauss eğriliği hesaplanmıştır. Ağırlıklı Gauss eğriliğinden hareketle dönel yüzey elde edilmiştir. Son olarak ağırlıklı Gauss eğriliği belli olan dönel yüzeylere ait örnekler verilmiştir. This study consists of four chapters. The first chapter is included the introduction part. In the second chapter, basic concepts in 3-dimensional Euclidean and Minkowski are given and surfaces are introduced. In the third chapter, surfaces of revolution and revolution of surfaces with density are introduced in 3-dimensional Euclidean and Minkowski space. Moreover, surfaces of revolution are examined according to axis types in 3-dimensional Minkowski space. The fourth chapter is the original part of this study. In this chapter, surfaces of revolution with lightlike axis are examined in Minkowski space with density. The surfaces of revolution weighted mean curvature and weighted Gaussian curvature are calculated. By using weighted Gauss curvature, surface of revolution is obtained. Finally, examples surfaces of revolution with weighted Gaussian curvature are given.

The problem of rolling without sliding of a homogeneous ball on a fixed surface under the action of gravity is a classical problem of nonholonomic system dynamics. Usually, when considering this problem, following the E. J. Routh approach [1] it is convenient to define explicitly the equation of the surface, on which the ball’s centre is moving. This surface is equidistant to the surface, over which the contact point is moving. From the classical works of E. J. Routh [1] and F. Noether [2] it was known that if the ball rolls on a surface such that its centre moves along a surface of revolution, then the problem is reduced to solving the second order linear differential equation. However it is impossible to find the general solution of this differential equation for an arbitrary surface of revolution. Therefore it is interesting to study for which surface of revolution the corresponding second order linear differential equation admits the explicit solution, for example, Liouvillian solution. To solve this problem it is possible to apply the Kovacic algorithm [3] to the corresponding second order linear differential equation. In this paper we present our own method to derive the corresponding second order linear differential equation. In the case when the centre of the ball moves along the paraboloid of revolution we prove that the corresponding second order linear differential equation admits a Liouvillian solution.

In the present paper, we study the global behaviour of geodesics of a Randers metric, defined on Finsler surfaces of revolution, obtained as the solution of the Zermelo&rsquo, s navigation problem. Our wind is not necessarily a Killing field. We apply our findings to the case of the topological cylinder R&times, S1 and describe in detail the geodesics behaviour, the conjugate and cut loci.

In the present work, the notion of generalized Cheng&ndash, Yau 1-type Gauss map is proposed, which is similar to the idea of generalized 1-type Gauss maps. Based on this concept, the surfaces of revolution and the canal surfaces in the Euclidean three-space are classified. First of all, we show that the Gauss map of any surfaces of revolution with a unit speed profile curve is of generalized Cheng&ndash, Yau 1-type. At the same time, an oriented canal surface has a generalized Cheng&ndash, Yau 1-type Gauss map if, and only if, it is an open part of a surface of revolution or a torus.

In this article, we establish the Lp estimates for a certain class of rough parabolic maximal functions related to surfaces of revolution. The obtained estimates allow us to apply an extrapolation argument to extend and improve some previously known results., Mathematics; Lp boundedness; Rough kernels; Surfaces of revolution; Parabolic maximal operators; Extrapolation

Tanks containing liquid propellants for rocket engines can be considered as shells whose thin walls are surfaces of revolution. Such surfaces result from a plane curve which rotates about some straight line lying in the plane which contains the curve. The structural analysis of tanks is presented here by using Roark’s formulas for stress and strain. Loads due to propellant sloshing and slosh-suppression devices are also discussed.

For a specific class of surfaces of revolution S, the existence of a smooth map {\Phi} from a neighbourhood U of S to the Euclidean plane E2 preserving distances infinitesimally along the meridians and the parallels of S and sending the meridional arcs of U \ S to straight lines of E2; is proven., Comment: 1figure

In this paper, we classify conformal surfaces of revolution in hyperbolic3-space $\mathbb{H}^{3}(-c^{2})$ satisfying an equation in terms of the position vector field and the Laplace operators with respect to the first,the second and the third fundamental forms of the surface.

Here we introduce a new forward model and imaging modality for Bragg Scattering Tomography (BST). The model we propose is based on an X-ray portal scanner with linear detector collimation, currently being developed for use in airport baggage screening. The geometry under consideration leads us to a novel two-dimensional inverse problem, where we aim to reconstruct the Bragg scattering differential cross section function from its integrals over a set of symmetric $C^2$ curves in the plane. The integral transform which describes the forward problem in BST is a new type of Radon transform, which we introduce and denote as the Bragg transform. We provide new injectivity results for the Bragg transform here, and describe how the conditions of our theorems can be applied to assist in the machine design of the portal scanner. Further we provide an extension of our results to $n$-dimensions, where a generalization of the Bragg transform is introduced. Here we aim to reconstruct a real valued function on $\mathbb{R}^{n+1}$ from its integrals over $n$-dimensional surfaces of revolution of $C^2$ curves embedded in $\mathbb{R}^{n+1}$. Injectivity proofs are provided also for the generalized Bragg transform., 39 pages, 13 figures

In the ideal case, surfaces can be defined by motion. Surfaces of revolution are produced by a rotation of a curve about an axis. In the case of helical surfaces, the generators additionally experience a translation along the axis of rotation.

The Riemann–Darboux integral is defined and integrable functions are studied. Principal results include the fundamental theorem of calculus. Nuggets include Riemann sums, volumes and surfaces of revolution.

Bu çalışmada, ... yoğunluklu ... 1 3 Lorentz-Minkowski uzayında, ikisi aynı anda sıfır olmayan ... ve ... sabitlerinin durumlarına göre, ağırlıklı eğrilikleri sıfır olan spacelike ve timelike düzlemsel eğriler yardımıyla oluşturulan dönel yüzeyler ve regle yüzeyler çalışılmıştır. In the present paper, the surfaces of revolution and ruled surfaces which are constructed with the aid of spacelike and timelike planar curves with vanishing weighted curvatures in Lorentz-Minkowski space ... 1 3 with density .... according to the cases of not all zero constants ... and .... are studied.

Bu çalışmada iki konveks objenin Minkowski toplamlarından haraket ederek ℝ3teki iki yüzeyin konvolüsyonu incelenmiştir. Bu tez 5 bölümden oluşmaktadır. İlk bölüm giriş bölümüdür. İkinci bölümde sonraki bölüm için gerekli olan kuramsal temeller verilmiştir. Üçüncü bölümde ℝ3 teki paraboloid yüzeyi ile bir keyfi parametreli yüzeyin konvolüsyonu ile ilgili yapılan hesaplamalar verilmiştir. Ayrıca bu yüzeylerin Gauss eğrilikleri hesaplanması gösterilmiştir. Bununla birlikte konvolüsyon yüzeyinin Gauss eğriliği karakterize edilmiştir. Dördüncü bölümde ℝ3 teki paraboloid yüzeyi sırasıyla Monge yaması ile verilen graf yüzeyi ve dönel yüzey ile konvolüsyonları incelenmiştir. Orijinal sonuç olarak bu konvolüsyon yüzeylerinin Gauss eğrilikleri hesaplanmıştır. Ayrıca bu sonuçları destekleyici bazı örnekler verilmiştir. Beşinci bölümde diğer bölümlerde elde edilen sonuçlar tartışılmış ve sonuç ve öneriler dile getirilmiştir. In this study, the convolution of two surfaces in R^3 is studied by the use of two Minkowski sums of two convex objects. This thesis consists of 5 chapters. The first chapter is the introduction. In the second chapter, theoretical foundations for the next part are given. In the third chapter, the calculations related to the convolution of the surface with an arbitrary parameter and the paraboloid surface in R^3 are given. In addition, the calculation of Gaussian curvature of these surfaces is shown. However, Gaussian curvature of the convolution surface was characterized. In the fourth chapter, the paraboloid surface in R^3 is examined by the graf surface and the rotational surface given by the monge patch. As a result, Gaussian curvatures of these convolution surfaces were calculated. In addition, some examples are given to support these results. In the fifth chapter, the results obtained in the other chapters are discussed and the results and suggestions are expressed.

Further development of the technology for generation of three-dimensional structured grids in constructions bounded by the surfaces of revolutions is presented. The technology has started to be developed first for the volumes of revolution. Then algorithms for deformed volumes of revolution and for the volumes bonded by the surfaces of revolution with parallel axis of rotation have been elaborated. The technology is designed for numerical modeling the processes of multicomponent hydrodynam- ics. It is being developed within the variational approach for constructing optimal grids satisfying optimality criteria: closeness of grids to uniform ones in the distance between nodes (criterion of uniformity), closeness of grids to orthogonal ones (criterion of orthogonality) and adaptation to a given function (criterion of adaptation). At present time the technology is enlarged by the algorithms and programs for grid generation in volumes of revolution deformed by the volumes of revolution formed by the several surfaces of revolution. Besides this, in the technology the criterion of adaptation is realized in the three-dimensional case. Earlier, the algorithms were developed for the cases of deformations by a conus, a cylinder and a sphere and the criterion of adaptation was realized in the two-dimensional case. Examples of grids are given.

We extend the decoupling results of the first two authors to the case of real analytic surfaces of revolution in \({\mathbb R}^3\). New examples of interest include the torus and the perturbed cone.

Here we present a novel microlocal analysis of generalized Radon transforms which describe the integrals of $L^2$ functions of compact support over surfaces of revolution of $C^{\infty}$ curves $q$. We show that the Radon transforms are elliptic Fourier Integral Operators (FIO) and provide an analysis of the left projections $\Pi_L$. Our main theorem shows that $\Pi_L$ satisfies the semi-global Bolker assumption if and only if $g=q'/q$ is an immersion. An analysis of the visible singularities is presented, after which we derive novel Sobolev smoothness estimates for the Radon FIO. Our theory has specific applications of interest in Compton Scattering Tomography (CST) and Bragg Scattering Tomography (BST). We show that the CST and BST integration curves satisfy the Bolker assumption and provide simulated reconstructions from CST and BST data. Additionally we give example "sinusoidal" integration curves which do not satisfy Bolker and provide simulations of the image artefacts. The observed artefacts in reconstruction are shown to align exactly with our predictions., Comment: 24 pages, 9 figures

Solutions of the planar Kepler problem with fixed energy h determine geodesics of the corresponding Jacobi–Maupertuis metric. This is a Riemannian metric on ℝ2 if h ⩾ 0 or on a disk D ⊂ ℝ2 if h < 0. The metric is singular at the origin (the collision singularity) and also on the boundary of the disk when h < 0. The Kepler problem and the corresponding metric are invariant under rotations of the plane and it is natural to wonder whether the metric can be realized as a surface of revolution in ℝ3 or some other simple space. In this note, we use elementary methods to study the geometry of the Kepler metric and the embedding problem. Embeddings of the metrics with h ⩾ 0 as surfaces of revolution in ℝ3 are constructed explicitly but no such embedding exists for h < 0 due to a problem near the boundary of the disk. We prove a theorem showing that the same problem occurs for every analytic central force potential. Returning to the Kepler metric, we rule out embeddings in the three-sphere or hyperbolic space, but succeed in constructing an embedding in four-dimensional Minkowski spacetime. Indeed, there are many such embeddings.

There is a large number of complex surfaces of the facades of houses of the 2nd and higher orders. Each surface has its own reflection of sun rays. There is a need for an analysis of these reflections for the detection of zones of concentration of the stream of sun rays, which create zones of overheating, and zones with the diffuse reflection of sun rays. Detection of these zones makes it possible to correct facades at the design stage of a complex form to create a comfortable environment for existing homes and territories and allows to analyze the illumination of opposite buildings. The existing theory of congruence allows us to analyze the reflection of solar rays synthetically from a geometric point of view and to present them analytically. The two-parameter set of normals of the facade surface is layered on the corresponding surfaces of the normals to the main generatrices along the plane section. Relatively to the same generatrices congruence reflected by the surface sun rays is layered on the surface of reflected rays. It is advisable to use the systematization of reflecting surfaces on the principle of surfaces of normals to their generations, which makes it possible to use the surfaces of reflected rays with typical characteristics for certain types of surface of normals. Thanks to this method, quasi-focal lines of the surfaces of reflected rays can be obtained as reflection concentration and, in response, the overheating zone. From the indicated method directly, it is possible to select the shape of the facade surfaces according to the given conditions and to model them on the basis of the known surfaces of the reflected rays. On the basis of the specified analytic algorithm, it is possible to use reflecting surfaces of any complexity, in particular, surfaces of the 2nd order, developable surfaces, channel, transfer surfaces, surfaces of revolution and others, as well as joining surfaces along the lines of main sections. The article considers the modeling of reflected sun rays by means of analytical equations of surfaces of reflected rays. These calculations are necessary for designing facades of new homes, depending on the tasks. Further research is possible for computer visualization of surfaces of reflected sun rays.

In this article we consider a one-dimensional reaction-diffusion problem with mixed boundary conditions. We provide conditions for the existence or nonexistence of stable nonconstant solutions whose derivative vanishes at some point. As an application, we obtain similar results for problems with Dirichlet boundary conditions posed in some symmetric domains: an n-dimensional ball, surfaces of revolution, and model manifolds.

The paper studies surfaces with constant mean curvature (CMC) H. If H = 0 then the surfaces are minimal. CMC tori were studied by H. Wente. U. Abresz proved that Wente tori have one family of planar lines of curvature and characterized them with elliptic integrals.A.I. Bobenko in his studies considered the problem of constructing CMC tori E3, S3, H3. In this paper, CMC surfaces of revolution are investigated. For a surface in E3 the Bonnet’s theorem states that for any surface having constant positive Gaussian curvature, there exists a surface parallel to it with a constant mean curvature.According to this statement, for surfaces of revolution with constant positive Gaussian curvature, CMC surfaces are constructed using the Bonnet’s theorem. It is proved that constructed surfaces are also surfaces of revolution. A family of plane curvature lines (meridians) is described by elliptic integrals, and surfaces with Gaussian curvature are also described by elliptic integrals. These surfaces are constructed using the mathematical software package. DOI 10.14258/izvasu(2018)4-22

In this work, the generalized parametric Marcinkiewicz integral operators with mixed homogeneity related to surfaces of revolution are studied. Under some weak conditions on the kernels, the boundedness of such operators from Triebel&ndash, Lizorkin spaces to L p spaces are established. Our results, with the help of an extrapolation argument, improve and extend some previous known results.

In this article, we define and study three types of surfaces of revolution in Galilean 3-space. The construction of the well-known surface of revolution, being the trace of a planar curve that is rotated about an axis in the supporting plane of the curve, is carried over to Galilean 3-space. Then, we classify the surfaces of revolution with vanishing Gaussian curvature or vanishing mean curvature in Galilean 3-space. Depending on the type of surface of revolution, either the vanishing of the Gaussian curvature turns out to be equivalent to the vanishing of the mean curvature or one concludes immediately that there do not exist surfaces of revolution of that type with vanishing mean curvature. ispartof: Journal of Mathematical Physics, Analysis, Geometry vol:14 issue:2 pages:141-152 status: Published online

An invariant submodel of gas dynamics equations constructed on a three-dimensional subalgebra with a projective operator for the case of monatomic gas is under consideration. The submodel is reduced to an Abel equation, with integral curves constructed for it. For a separatrix of a saddle, an approximate solution is studied. Such solutions describe the vortex scattering of gas along plane curves placed on the surface of revolution.

The centroid theorems of Pappus (or the Pappus–Guldin theorems, or the Guldin theorems) show deep connections between areas, arc lengths, volumes of revolution, surfaces of revolution, and centers ...

We first show that trajectories traced by charges moving in rotational magnetic fields are, basically, the non-parallel geodesics of surfaces of revolution with coincident axis. Thus, people living in a surface of revolution are not able to sense the magnetic Hall effect induced by the surrounding magnetic field and perceive charges as influenced, exclusively, by the gravity action on the surface of revolution. Secondly, the extended Hasimoto transformations are introduced and then used to identify trajectories of charges moving through a Killing rotational magnetic field in terms of non-circular elastic curves. As a consequence, we see that in this case charges evolve along trajectories which are obtained as extended Hasimoto transforms of solitons of the filament equation.

We consider a class of linear Weingarten surfaces of revolution whose principal curvatures, meridional $k_{\mu}$ and parallel $k_{\pi}$, satisfy the relation $k_{\mu}=(n+1)k_{\pi}$, $n=0,\,1,\,2,\ldots\, .$ The first two members of this class of surfaces are the sphere $(n=0)$ and the Mylar balloon $(n=1)$. Elsewhere the Mylar balloon has been parameterized via the Jacobian and Weierstrassian elliptic functions and elliptic integrals. Here we derive six alternative parameterizations describing the third type of surfaces when $n=2$. The so obtained explicit formulas are applied for the derivation of the basic geometrical characteristics of this surface.

The basing of parts with surfaces of revolution in a self-centering vise and a motionless prism is considered, for the example of a disk and a shaft. In the case of an immobile prism, violation of basing principles produces indeterminacy because of redundant reference points. That gives rise to a basing error.

Given an implicit polynomial equation or a rational parametrization, we develop algorithms to determine whether the set of real and complex points defined by the equation, i.e. the surface defined by the equation, in the sense of Algebraic Geometry, is a cylindrical surface, a conical surface, or a surface of revolution. The algorithms are directly applicable to, and formulated in terms of, the implicit equation or the rational parametrization. When the surface is cylindrical, we show how to compute the direction of its rulings; when the surface is conical, we show how to compute its vertex; and when the surface is a surface of revolution, we show how to compute its axis of rotation directly from the defining equations, Ministerio de Economía y Competitividad

It is shown that the surface with a constant mean curvature encloses the extremal volume among all toroidal surfaces of given area. The exact solution for the corresponding variational problem is derived, and its parametric analysis is performed in the limits of high and small mean curvatures. An absence of smooth torus with constant mean curvature is proved, and the extremal surface is demonstrated to have at least one edge located on the outer side of the torus.

Origami research is always devoted to exploring the transformation from a paper-like material into a three-dimensional (3D) object, which is widely used in the storage of planar structures. However, origami only has a two-dimensional crease map, so it performs limited effect on the storage of 3D surface. Inspired by waterbomb origami, this paper presents a scheme of folding discretized surfaces of revolution to create foldable 3D geometries. The geometries, which are the approximation of target surfaces of revolution, consist of discretizing surfaces of revolution and have 3D crease maps to guide folding. The discretized surfaces constructed by the approaching scheme can satisfy the error requirements and retain the features of original surfaces. The paper verifies the feasibility of the folding scheme by showing the formula of four typical surfaces, including convex shape, concave shape, concave-convex shape and capsule shape. This scheme of folding discretized surfaces of revolution shows the potential of origami to 3D deployable structures in the field of aerospace.