“About “bulky&rdquo, links generated by generalized Möbius-Listing bodies GML2n”. Gielis J, Tavkhelidze I, Ricci PE, Journal of mathematical sciences 193, 449 (2013). http://doi.org/10.1007/S10958-013-1474-7
Abstract: In this paper, we consider the bulky knots and bulky links, which appear after cutting of a Generalized MöbiusListing GMLn2 body (with the radial cross section a convex plane 2-symmetric figure with two vertices) along a different Generalized MöbiusListing surfaces GMLn2 situated in it. The aim of this report is to investigate the number and geometric structure of the independent objects that appear after such a cutting process of GMLn2 bodies. In most cases we are able to count the indices of the resulting mathematical objects according to the known classification for the standard knots and links.
Keywords: A2 Journal article; Sustainable Energy, Air and Water Technology (DuEL)
DOI: 10.1007/S10958-013-1474-7
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“About “bulky&rdquo, links, generated by generalized Möbius Listing's bodies GML3n”. Tavkhelidze I, Cassisa C, Gielis J, Ricci PE, Matematica e applicazioni : atti della Accademia nazionale dei Lincei 24, 11 (2013). http://doi.org/10.4171/RLM/643
Abstract: In the present paper we consider the “bulky knots'' and ”bulky links'', which appear after cutting a Generalized Möbius Listing's GMLn3 body (whose radial cross section is a plane 3-symmetric figure with three vertices) along different Generalized Möbius Listing's surfaces GMLn2 situated in it. This article is aimed to investigate the number and geometric structure of the independent objects appearing after such a cutting process of GMLn3 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.
Keywords: A1 Journal article; Engineering sciences. Technology; Sustainable Energy, Air and Water Technology (DuEL)
DOI: 10.4171/RLM/643
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“About some methods of analytic representation and classification of a wide set of geometric figures with “complex&rdquo, configuration”. Tavkhelidze I, Gielis J, Pinelas S, Sn –, 1512-0066 34, 81 (2020)
Keywords: A3 Journal article; Sustainable Energy, Air and Water Technology (DuEL)
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“About some methods of analytic representation and classification of a wide set of geometric figures with “complex” configuration”. Tavkhelidze I, Gielis J, Pinelas S page 347 (2020).
Keywords: H1 Book chapter; Sustainable Energy, Air and Water Technology (DuEL)
DOI: 10.1007/978-3-030-56323-3_27
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“The Dirichlet problem for the Laplace equation in supershaped annuli”. Caratelli D, Gielis J, Tavkhelidze I, Ricci PE, Boundary value problems , 113 (2013). http://doi.org/10.1186/1687-2770-2013-113
Abstract: The Dirichlet problem for the Laplace equation in normal-polar annuli is addressed by using a suitable Fourier-like technique. Attention is in particular focused on the wide class of domains whose boundaries are defined by the so-called superformula introduced by Gielis. A dedicated numerical procedure based on the computer algebra system Mathematica© is developed in order to validate the proposed methodology. In this way, highly accurate approximations of the solution, featuring properties similar to the classical ones, are obtained.
Keywords: A1 Journal article; Engineering sciences. Technology; Sustainable Energy, Air and Water Technology (DuEL)
DOI: 10.1186/1687-2770-2013-113
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“Fourier-Hankel solution of the Robin problem for the Helmholtz equation in supershaped annular domains”. Caratelli D, Gielis J, Tavkhelidze I, Ricci PE, Boundary value problems , 253 (2013). http://doi.org/10.1186/1687-2770-2013-253
Abstract: The Robin problem for the Helmholtz equation in normal-polar annuli is addressed by using a suitable Fourier-Hankel series technique. Attention is in particular focused on the wide class of domains whose boundaries are defined by the so-called superformula introduced by Gielis. A dedicated numerical procedure based on the computer algebra system Mathematica© is developed in order to validate the proposed methodology. In this way, highly accurate approximations of the solution, featuring properties similar to the classical ones, are obtained.
Keywords: A1 Journal article; Engineering sciences. Technology; Sustainable Energy, Air and Water Technology (DuEL)
DOI: 10.1186/1687-2770-2013-253
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“The general case of cutting GML bodies : the geometrical solution”. Gielis J, Caratelli D, Tavkhelidze I page 397 (2020).
Keywords: H1 Book chapter; Engineering sciences. Technology; Sustainable Energy, Air and Water Technology (DuEL)
DOI: 10.1007/978-3-030-56323-3_31
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“The general case of cutting of Generalized Möbius-Listing surfaces and bodies”. Gielis J, Tavkhelidze I, 4Open 3, 7 (2020). http://doi.org/10.1051/FOPEN/2020007
Abstract: 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.
Keywords: A1 Journal article; Engineering sciences. Technology; Sustainable Energy, Air and Water Technology (DuEL)
DOI: 10.1051/FOPEN/2020007
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“The Mӧbius phenomenon in Generalized Mӧbius-Listing bodies with cross sections of odd and even polygons”. Gielis J, Tavkhelidze I, Sn –, 1512-0066 34, 23 (2020)
Keywords: A3 Journal article; Sustainable Energy, Air and Water Technology (DuEL)
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“The Möbius phenomenon in Generalized Möbius-Listing surfaces and bodies, and Arnold's Cat phenomenon”. Gielis J, Ricci PE, Tavkhelidze I, Advanced Studies : Euro-Tbilisi Mathematical Journal 14, 17 (2021). http://doi.org/10.3251/ASETMJ/1932200812
Abstract: Möbius bands have been studied extensively, mainly in topology. Generalized Möbius-Listing surfaces and bodies providing a full geometrical generalization, is a quite new field, motivated originally by solutions of boundary value problems. Analogous to cutting of the original Möbius band, for this class of surfaces and bodies, results have been obtained when cutting such bodies or surfaces. In general, cutting leads to interlinked and intertwined different surfaces or bodies, resulting in very complex systems. However, under certain conditions, the result of cutting can be a single surface or body, which reduces complexity considerably. Our research is motivated by this reduction of complexity. In the study of cutting Generalized Möbius-Listing bodies with polygons as cross section, the conditions under which a single body results, displaying the Möbius phenomenon of a one-sided body, have been determined for even and odd polygons. These conditions are based on congruence and rotational symmetry of the resulting cross sections after cutting, and on the knife cutting the origin. The Möbius phenomenon is important, since the process of cutting (or separation of zones in a GML body in general) then results in a single body, not in different, intertwined domains. In all previous works it was assumed that the cross section of the GML bodies is constant, but the main result of this paper is that it is sufficient that only one cross section on the whole GML structure meets the conditions for the Möbius phenomenon to occur. Several examples are given to illustrate this.
Keywords: A1 Journal article; Sustainable Energy, Air and Water Technology (DuEL)
DOI: 10.3251/ASETMJ/1932200812
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“Modeling in mathematics : proceedings of the second Tbilisi-Salerno workshop on modeling in mathematics”. Gielis J, Ricci PE, Tavkhelidze I page 185 p. (2017).
Keywords: ME3 Book as editor; Engineering sciences. Technology; Sustainable Energy, Air and Water Technology (DuEL)
DOI: 10.2991/978-94-6239-261-8
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“On a geometric model of bodies with “complex” configuration and some movements”. Tavkhelidze I, Caratelli D, Gielis J, Ricci PE, Rogava M, Transirico M page 129 (2017).
Abstract: Aim of this chapter is analytical representation of one wide class of geometric figures (lines, surfaces and bodies) and their complicated displacements. 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. One central idea 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 of one class of bodies used of describe different human organs.
Keywords: H1 Book chapter; Engineering sciences. Technology; Sustainable Energy, Air and Water Technology (DuEL)
DOI: 10.2991/978-94-6239-261-8_10
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“On means, polynomials and special functions”. Gielis J, Verhulst R, Caratelli D, Ricci PE, Tavkhelidze I, The teaching of mathematics 17, 1 (2014)
Keywords: A1 Journal article; Educational sciences; Sustainable Energy, Air and Water Technology (DuEL)
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“The process of cutting GMLmn bodies with dm-knives”. Tavkhelidze I, Gielis J, Sn –, 1512-0066 32, 67 (2018)
Keywords: A3 Journal article; Sustainable Energy, Air and Water Technology (DuEL)
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“Some properties of “bulky&rdquo, links, generated by Generalized Möbius Listing's bodies GML4n”. Caratelli D, Gielis J, Ricci PE, Tavkhelidze I, Journal of mathematical sciences 216, 509 (2016). http://doi.org/10.1007/S10958-016-2907-X
Abstract: In the present paper, we consider the bulky knots and bulky links that appear after cutting of generalized MöbiusListing GML 4 n bodies (with corresponding radial cross sections square) along different generalized MöbiusListing surfaces GML 2 n 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 GML 4 n 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.
Keywords: A2 Journal article; Engineering sciences. Technology; Sustainable Energy, Air and Water Technology (DuEL)
DOI: 10.1007/S10958-016-2907-X
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“Some properties of “bulky&rdquo, links, generated by Generalized Möbius Listing's bodies GML4n”. Caratelli D, Gielis J, Ricci PE, Tavkhelidze I, (2013)
Keywords: P3 Proceeding; Engineering sciences. Technology; Sustainable Energy, Air and Water Technology (DuEL)
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“Spherical harmonic solution of the Robin problem for the Helmholtz equation in a supershaped shell”. Caratelli D, Gielis J, Tavkhelidze I, Ricci PE, Applied mathematics 4, 263 (2013). http://doi.org/10.4236/AM.2013.41A040
Abstract: The Robin problem for the Helmholtz equation in normal-polar shells is addressed by using a suitable spherical harmonic expansion technique. Attention is in particular focused on the wide class of domains whose boundaries are defined by a generalized version of the so-called superformula introduced by Gielis. A dedicated numerical procedure based on the computer algebra system Mathematica? is developed in order to validate the proposed methodology. In this way, highly accurate approximations of the solution, featuring properties similar to the classical ones, are obtained.
Keywords: A1 Journal article; Engineering sciences. Technology; Sustainable Energy, Air and Water Technology (DuEL)
DOI: 10.4236/AM.2013.41A040
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“Structure of the dm knives and process of cutting of GML(man) or GRT(man) bodies”. Tavkhelidze I, Gielis J, Sn –, 1512-0066 33 (2019)
Keywords: A3 Journal article; Engineering sciences. Technology; Sustainable Energy, Air and Water Technology (DuEL)
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“The Robin problem for the Helmholtz equation in a starlike planar domain”. Caratelli D, Gielis J, Natalini P, Ricci PE, Tavkhelidze I, Georgian mathematical journal 18, 465 (2011). http://doi.org/10.1515/GMJ.2011.0031
Abstract: The interior and exterior Robin problems for the Helmholtz equation in starlike planar domains are addressed by using a suitable Fourier-like technique. Attention is in particular focused on normal-polar domains whose boundaries are defined by the so-called superformula introduced by J. Gielis. A dedicated numerical procedure based on the computer algebra system Mathematica© is developed in order to validate the proposed approach. In this way, highly accurate approximations of the solution, featuring properties similar to the classical ones, are obtained. The computed results are found to be in good agreement with the theoretical findings on Fourier series expansion presented by L. Carleson.
Keywords: A1 Journal article; Sustainable Energy, Air and Water Technology (DuEL)
DOI: 10.1515/GMJ.2011.0031
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“Generalized Möbius-Listing bodies and the heart”. Gielis J, Tavkhelidze I, Ricci PE, Sn –, 2247-689x 13, 58 (2023)
Abstract: Generalized Möbius-Listing surfaces and bodies generalize Möbius bands, and this research was motivated originally by solutions of boundary value problems. Analogous to cutting of the original Möbius band, for this class of surfaces and bodies, results have been obtained when cutting such bodies or surfaces. The results can be applied in a wide range of fields in the natural science, and here we propose how they can serve as a model for the heart and the circulatory system.
Keywords: A3 Journal article; Sustainable Energy, Air and Water Technology (DuEL)
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“A note on Generalized Möbius-Listing Bodies”. Gielis J, Tavkhelidze I, , 31 (2023). http://doi.org/10.55060/s.atmps.231115.003
Abstract: Generalized Möbius-Listing surfaces and bodies generalize Möbius bands, and this research was motivated originally by solutions of boundary value problems. Analogous to cutting of the original Möbius band, for this class of surfaces and bodies, results have been obtained when cutting such bodies or surfaces. In general, cutting leads to interlinked and intertwined different surfaces or bodies, resulting in very complex systems. However, under certain conditions, the result of cutting can be a single surface or body, which reduces complexity considerably. These conditions are based on congruence and rotational symmetry of the resulting cross sections after cutting, and on the knife cutting the origin
Keywords: P1 Proceeding; Sustainable Energy, Air and Water Technology (DuEL)
DOI: 10.55060/s.atmps.231115.003
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