Topological conjugacy of 𝑛-multiple Cartesian products of circle rough transformations

It is known from the 1939 work of A. G. Mayer that rough transformations of the circle are limited to the diffeomorphisms of Morse – Smale. A topological conjugacy class of orientation-preserving diffeomorphism is entirely determined by its rotation number and the number of its periodic orbits, while for orientation-changing diffeomorphism the topological invariant will be only the number of periodic orbits. Thus, the purpose of this study is to find topological invariants of n-fold Cartesian products of diffeomorphisms of a circle. Methods. This paper explores the rough Morse – Smale diffeomorphisms on the n-torus surface. To prove the main result, additional constructions and formation of subsets of considered sets were used. Results. In this paper, a numerical topological invariant is introduced for n-fold Cartesian products of rough circle transformations. Conclusion.The criterion of topological conjugacy of n-fold Cartesian products of rough transformations of a circle is formulated.

A Primitive-Based Reverse Modeling Method for Obtaining the Editable CAD model of Reversed Parts

By the traditional inversion methods, the model of the part reversed is obtained in mesh feature , which is hard to edit and redesign. In the paper, for the revised part, a modeling method to obtain the editable CAD mode was proposed. Firstly, an approach to construct the primitive structure tree used to describe the geometric structure of a part was proposed, by which the all the basic geometry primitives and their Boolean operations are logically arranged in tree format. Then, for each basic geometry primitive, a method to calculate the characteristics parameters and position and posture was proposed, which could be used to guide reversion measurement and modeling. For the basic primitives made of quadratic surface, a method for modeling the torus surface, whose model little research has been carried on, was proposed in which the torus surface equation is deduced by coordinate transformation method, has 8 independent parameters. Last, based on the guidance of primitive structure tree, a process for the characteristics parameters, minimum measuring points, forward modeling, positioning and assembly process was proposed, which could be used to direct measurements and to forward modeling. The example showed that the method proposed could be used to reverse the parts in the editable CAD model and provide for subsequent optimization design services.

The study of marginal deformations of the leaf extracts with regard to plastic thinning and destruction of the material

The aim of work. The paper analyzes the process of sheet drawing and develops a method for determining the minimum curvature of the torus fillet in the formation of a cylindrical product (glass), taking into account the plastic thinning of the deformable material by the end edge of the pressing punch. Methods. The existing scheme of forming with the use of permissible coefficients of drawing and tensile strength of the material, representing the basis of deformation-force calculation, which often ignores the factor of deformation thinning and its impact on the strength of the material, which together can lead to excess of the allowable stress in the material, causing its rupture, is analyzed. A mathematical model of the volume stress state of the metal is developed, which allows to estimate the value of the thinning in the formation of the torus surface of different curvature, followed by the approximation of the values found by a polynomial function. Considering the criterion of limit loads the level of radial stresses during folding and stretching of the sheet material, taking into account its thinning, the minimum curvature of the torus surface of the glass is established. Results. The results obtained will allow to simulate the stress-strain state of the metal in the development of sheet drawing technology: to establish the value of thinning and to estimate the level of radial stresses in the formation of the torus surface rounding to prevent the destruction of the elongated glass (separation of the bottom), which guarantees quality products.

Effect of a rigid toroidal inhomogeneity on the elastic properties of a composite

An analytical solution is obtained for the problem of an infinite elastic medium containing a rigid toroidal inhomogeneity under remotely applied uniform strain. The traction on the torus surface is determined as a function of torus parameters and strain components applied at infinity. The results are utilized to calculate components of the stiffness contribution tensor of the rigid toroidal inhomogeneity that is required for calculation of the overall elastic properties of a material containing multiple toroidal inhomogeneities. The analytical results are verified by comparison with finite element model calculations.

Toroidal insulating inhomogeneity in an infinite space and related problems

An analytic solution for the steady-state temperature distribution in an infinite conductive medium containing an insulated toroidal inhomogeneity and subjected to remotely applied uniform heat flux is obtained. The temperature flux on the torus surface is then determined as a function of torus parameters. This result is used to calculate the resistivity contribution tensor for the toroidal inhomogeneity required to evaluate the effective conductive properties of a material containing multiple inhomogeneities of this shape.

PIT MEMBRANES OF EPHEDRA RESEMBLE GYMNOSPERMS MORE THAN ANGIOSPERMS

Bordered pit pairs of Ephedra species were characterized using different types of microscopy. Pit membranes contained tori that did not stain for lignin. SEM and AFM views of the torus surface showed no plasmodesmatal openings, but branched, secondary plasmodesmata were occasionally noted using TEM in conjunction with ultrathin sections. The margo consisted of radial microfibrils as well as finer diameter tangential fibrils. The former formed fascicles of fibrils that merged into even thicker buttresses during the act of pit membrane aspiration. AFM showed a discontinuous layer of non-microfibrillar material on the surface of both torus and margo. It is hypothesized that this material is responsible for adhesion of the pit membrane to the surface of the pit border during the process of aspiration. Taken as a whole, intervascular pit membranes of Ephedra more closely resemble those of conifers than those of torus-bearing pit membranes of angiosperms.

Green's function for the Laplace–Beltrami operator on a toroidal surface

Green's function for the Laplace–Beltrami operator on the surface of a three-dimensional ring torus is constructed. An integral ingredient of our approach is the stereographic projection of the torus surface onto a planar annulus. Our representation for Green's function is written in terms of the Schottky–Klein prime function associated with the annulus and the dilogarithm function. We also consider an application of our results to vortex dynamics on the surface of a torus.

Formation and evolution of black hole and accretion disk in collapse of massive stellar cores

We describe the results of our numerical simulations of the collapse of a massive stellar core to a BH, performed in the framework of full general relativity incorporating finite-temperature equation of state and neutrino cooling. We adopt a 100 M⊙ presupernova model calculated by Umeda & Nomoto (2008), which has a massive core with a high value of entropy per baryon. Changing the degree of rotation for the initial condition, we clarify the dependence of the outcome on this. When the rotation is rapid enough, the shock wave formed at the core bounce is deformed to be a torus-like shape. Then, the infalling matter is accumulated in the central region due to the oblique shock at the torus surface, hitting the hypermassive neutron star (HMNS) and dissipating the kinetic energy there. As a result, outflows can be launched. The HMNS eventually collapses to a BH and an accretion torus is formed around it. We also found that the evolution of the BH and torus depends strongly on the rotation initially given.