Ilia Tavkhelidze

Meromorphic solutions of algebraic differential equations by A. E. Eremenko Some questions in spectral theory for the operator Sturm-Liouville equation on the half-line by M. L. Gorbachuk and V. A. Kutovoi On the connection between spectral and oscillatory properties of the matrix Jacobi problem by V. S. Bondarchuk On the numerical solution of the Cauchy problem for ordinary linear homogeneous differential equations on large intervals of integration by S. F. Zaletkin On the solutions of a uniformly elliptic complex equation of first order connected with the convergence of analytic functions by S. A. Akhmedov On the behavior of solutions of the equations of plane elasticity theory in the neighborhood of irregular boundary points and at infinity by O. A. Oleinik, G. A. Iosifyan, and I. N. Tavkhelidze The Dirichlet problem for the Laplace operator in the exterior of a thin body of revolution by M. V. Fedoryuk Variational methods in a mixed problem of thermal equilibrium with a free boundary by B. V. Bazalii and V. Yu. Shelepov On an estimate of $N^\ast(\lambda)$ for the series of quasimodes of the Laplace operator by V. F. Lazutkin On the selfadjointness and maximal dissipativity of differential operators for functions of an infinite-dimensional argument by Yu. L. Daletskii Selfadjoint differential operators acting in spaces of functions of infinitely many variables by Yu. M. Berezanskii.

In previous articles [1-7] a wide class of geometric figures-"Generalized Twisting and Rotated" bodies (sometimes called "surface of Revolution" see [11])-shortly n m GTR-was defined through their analytic representation. In particular cases, this analytic representation gives back many classical objects (torus, helicoid, helix, Möbius strip ... etc.). Aim of this article is to consider some geometric properties of a wide subclass of the already defined surfaces, by using their analytical representation. In previous articles [1-7] a set of the Generalized Möbius Listing's bodies-shortly n m GML , which are a particular case of the n m GTR bodies, have been defined. In the present paper we show some geometric properties of Generalized Twisting and Rotated-surfaces and relationships between the set n GML 2 and the sets of Knots and Links.

Natural forms affect all of us, not only for their beauty, but also for their diversity (see e.g. Fig. 1). It is still not known whether forms define the essence of the phenomena associated with them, or vice versa - that is, forms are natural consequences of the phenomena. The essence of one “unexpected” phenomenon is as follows: Usually after one “full cutting”, an object is split into two parts. The Mobius strip is a well-known exception, however, which still remains whole after cutting. The first author discovered a class of surfaces, which have following properties - after full cutting more than two surfaces appear, but this is a result for specific class of pure mathematical surfaces [1, 2]. It turns out that three-dimensional Mobius Listing bodies, \(GML_m^n\), which is a wide subclass of the Generalized Twisting and Rotated figures - shortly \(GTR_m^n\) - which, through their analytic representation, could yield more than two objects after only single cutting ([3] or [2]). These are not only theoretical results, as can be proved by real-life examples. Many classical objects (torus with different forms of radial cross sections, helicoid, helix, Mobius strip,... etc.) are elements of this wide class of \(GTR_m^n\) figures, so it is important to study the similarity and difference between these figures and surfaces. In this chapter we study some questions of similarity and difference in the cases of the “cut” of Generalized Mobius–Listing’s figures.

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In the present paper, we consider the “bulky knots” and “bulky links” that appear after cutting of generalized Möbius–Listing GML4n bodies (with corresponding radial cross sections square) along different generalized Möbius–Listing surfaces GML2n situated in it. The aim of this article is to examine the number and geometric structure of independent objects that appear after such a cutting process of GML4n bodies. In most cases, we are able to count the indices of the resulting mathematical objects according to the known tabulation for knots and links of small complexity.

Abstract. Aim of this article is the analytical representation of a class of geometric figures, surfaces and lines. This class of surfaces includes the sur-faces appearing in some problems of Shell Theory or problems of spreading of smoke-rings; furthermore, the lines of this class can be used for describ-ing the complicated orbit of some celestial objects. In previous articles [1-5] sets of Generalized Möbius Listing’s bodies, which are a particular case of this class in static case, have been already defined. In particular cases, this analytic representation gives back many classical objects (torus, helicoid, helix, Möbius strip... etc.). In present paper was studied some relations between set of GMLn2 (Generalized Möbius-Listing’s surfases) and sets of Knots and Links. Also, here was defined classes of DMLn2 (Degenerated

We will present 2 different analytical representations of only one general idea—this is the representation of complex movements using the superposition of certain elementary displacements! Despite of the analytical and structural similarity of these representations, they describe fundamentally different geometric figures (in statics) and trajectories of motion (in dynamics). In previous articles [1, 2, 3, 4, 5, 6, 7, 8, 9] a wide class of geometric figures—“Generalized Twisting and Rotated” bodies \(GRT^n_m\) in short—was defined through their analytic representation. In particular cases, this analytic representation gives back many classical objects (torus, helicoid, helix, Mobius strip ... etc.). The aim of this article is to consider some geometric properties of a wide subclass of the generally defined surfaces. We show some geometric properties of GRT and GML—surfaces.

For more than almost 200 years the Mobius strip and its “mysterious” property attracts the attention of mathematicians. After a “complete cut” of this surface, one object appears, but already with a fourfold twist. The generalization of this phenomenon to figures of a more complex configuration led to an “unexpected” result: after the cut of the generalized Mobius-Listing body, more than two geometric shapes may appear. In this paper, we consider all possible cases of a complete cut of the generalized Mobius-Listing body with a regular hexagon as radial section. In early works, together with different colleagues, on the basis of importance, they separately examined the case of Mobius-Listing’s bodies with a radial section of regular 3, 4 and 5 angular figures. Also, cases of similar bodies with a radial section of convex regular two and three angular figures were considered separately. One possible application of these results is assumed in the description of the properties of the m...

The original motivation to study this class of geometrical objects of Generalized Mobius-Listing GML surfaces and bodies was the observation that the solution of boundary value problems greatly depends on the structure of the boundary of domains. Since around 2010 GML’s were merged with (continuous) Gielis Transformations, which provide a unifying description of geometrical shapes, as a generalization of the Pythagorean Theorem. The resulting geometrical objects can be used for modeling a wide range of natural shapes and phenomena.

Generalized Mobius-Listing bodies and surfaces are generalizations of the classic Mobius band. The original motivation is that for solutions of boundary value problems the knowledge of the domain is essential. In previous papers cutting of GML bodies with cross section symmetrical disks with symmetry 2, 3, 4, 5 and 6 have been classified. In this paper we solve the general case, using regular m-gons as cross section. The 3D problem is reduced to the problem of cutting regular m-polygons with d-knives, related to the number of divisors of m. The problem has both a geometrical and topological solution, and has many connections to other fields of mathematics.

The original motivation to study Generalized Möbius-Listing GML surfaces and bodies was the observation that the solution of boundary value problems greatly depends on the domains. Since around 2010 GML’s were merged with (continuous) Gielis Transformations, which provide a unifying description of geometrical shapes, as a generalization of the Pythagorean Theorem. The resulting geometrical objects can be used for modeling a wide range of natural shapes and phenomena. The cutting of GML bodies and surfaces, with the Möbius strip as one special case, is related to the field of knots and links, and classifications were obtained for GML with cross sectional symmetry of 2, 3, 4, 5 and 6. The general case of cutting GML bodies and surfaces, in particular the number of ways of cutting, could be solved by reducing the 3D problem to planar geometry. This also unveiled a range of connections with topology, combinatorics, elasticity theory and theoretical physics.

Starting from the exponential, some classes of analytic functions of the derivative operator are studied, including pseudo-hyperbolic and pseudo-circular functions. Some formulas related to operational calculus are deduced, and the important role played in such a context by Hermite–Kampé de Fériet polynomials is underlined.

We prove uniqueness theorems for the solutions of the Dirichlet and Riquier boundary value problems for the biharmonic equation in a half-space. We construct Green functions, based on which we study the behavior of the solutions in a neighborhood of infinity.

We use the multi-dimensional polynomials considered by Hermite, and subsequently studied by P. Appell and J. Kampé de Fériet, in order to obtain explicit solutions of pseudo-classical PDE problems in the half-plane y>0. We consider systems of PDE, including some problems with degeneration on the x-axis.

In this paper, the authors use the multidimensional polynomials considered by Kampé de Fériet to define the hyperbolic and circular functions of the derivative operator. Then they apply this tool to solve classical and generalized PDE problems in the half-plane y>0, connecting systems with equations, and including some problems with degeneration on the x-axis.

We consider the apriory energetic estimates Saint-Venant’s type. We study problem of dependence of coefficients in this estimates with geometric structure of considered domain, with dimension of space and with order of polyharmonic equation.

The aim of this article is the analytical representation of one class of geometrical figures, surfaces and lines. This class of surfaces appear, when we study the problems of spreading of smoke-rings, also this class of lines describe the complicated orbit of some celestial objects. In particularly cases of this analytic representation give as classical objects (torus, helicoid, helix and ...).

The accurate estimation of physical characteristics (such as volume, surface area, length, or other specific parameters) relevant to human organs is of fundamental importance in medicine. The aim of this article is, in this respect, to provide a general methodology for the evaluation, as a function of time, of the volume and center of gravity featured by moving generalized Möbius listing’s bodies used to describe different human organs.

We consider the cutting process of the generalized Möbius-Listing surface GML 2 n along some closed line which is parallel to its border line. We show connections of the resulting mathematical objects with the set of knots and links. In some cases we count the indices of the corresponding objects according to the known classification.

For a higher order dierential equation with the polyharmonic operator, the Dirichlet and Riquier boundary value problems are studied in some polyhedral angles. Uniqueness theorems for solutions with a bounded "energy integral" of the corresponding BVPs are proved. Recurrent formu- las are constructed for representation of fundamental solutions and Green's functions. The asymptotic behavior of solutions at infinity is studied.