The paper suggests that E = mc<sup>2</sup> may be open to misinterpretation and that in this form it is not what Einstein advanced first. It is further suggested to return to the slightly less compact form...The paper suggests that E = mc<sup>2</sup> may be open to misinterpretation and that in this form it is not what Einstein advanced first. It is further suggested to return to the slightly less compact formula E = Kmc<sup>2</sup> where a < K < 1 which has the merit of accounting for the measured ordinary energy density of the cosmos (K = 1/22) and the conjectured missing dark energy density of the universe (K = 21/22) from the view point of economical notation.展开更多
Intensive use of geometrical modelling started with the development of computer graphics and CAD/CAM technologies, comprising a wide area of mathematical disciplines. Geometrical modelling is a synthesis of the geomet...Intensive use of geometrical modelling started with the development of computer graphics and CAD/CAM technologies, comprising a wide area of mathematical disciplines. Geometrical modelling is a synthesis of the geometry and computer graphics, which enable us to develop complex mathematical models that would be rather difficult to display without using a computer. The author recognizes two basic methods of research of geometrical models, synthetic and analytic. The synthetic method makes use of geometrical constructions and in the analytic method geometrical objects are characterized by numerical data. The author uses these methods in this contribution. The aim of this research is to show central cyclides （Dupin＇s cyclides） as a special kind of the cyclical elliptical pedal surfaces. Firstly, applying the method of synthetic geometry, a new class of such surfaces in the 3-dimensional Euclidean space （model of the projective space） is defined. The geometrical construction of these surfaces is dependent on the given ellipse and the position of the pole P. It is the point at which the pencil of perpendicular planes passes to the plane of the ellipse, wherein the generating circles of the surface are lying. The parametric equations of the cyclical elliptical surface are derived applying the method of analytic geometry. The author classifies the surfaces according to the number of generating circles with a zero radius. The evolute of the ellipse divides the plane of the ellipse into two areas, Ω1 and Ω2. The shape of the surface depends on the position of the pole P in the areas of Ω1 and Ω2. The surface can have 4, 3 or 2 circles with a zero radius. The author then describes the relationship between these surfaces and the Dupin＇s cyclides, obtained in the case when the pole P is the point on the major axis of the ellipse. Finally, the transformation of these surfaces is shown by changing of the orthonormal base in the parameterization of the generating circles of the surface. The author visualizes the resulting surfaces obtained by the parametric approach in the MAPLE program environment.展开更多
When initial radius if Stoica actually derived Einstein equations in a formalism which removes the big bang singularity pathology, then the reason for Planck length no longer holds. We follow what Ng derived as limit ...When initial radius if Stoica actually derived Einstein equations in a formalism which removes the big bang singularity pathology, then the reason for Planck length no longer holds. We follow what Ng derived as limit calculations as to a space time length factor Without the drop off of the vacuum energy as given by is at least the value of . We review the work by Ng as to quantum foam as to how that affects a general expression as to energy when , with determined at least approximately by arguments he presented in 2008 in the Dark side of the universe conference. Well before certain effects make themselves apparent, in ways which are illustrated in the manuscript. Having at a point singularity would remove expansion by the scale factor, so that the extreme version of Stoica’s treatment in an isolated 4-dimensional universe would be no expansion at all.展开更多
文摘The paper suggests that E = mc<sup>2</sup> may be open to misinterpretation and that in this form it is not what Einstein advanced first. It is further suggested to return to the slightly less compact formula E = Kmc<sup>2</sup> where a < K < 1 which has the merit of accounting for the measured ordinary energy density of the cosmos (K = 1/22) and the conjectured missing dark energy density of the universe (K = 21/22) from the view point of economical notation.
文摘Intensive use of geometrical modelling started with the development of computer graphics and CAD/CAM technologies, comprising a wide area of mathematical disciplines. Geometrical modelling is a synthesis of the geometry and computer graphics, which enable us to develop complex mathematical models that would be rather difficult to display without using a computer. The author recognizes two basic methods of research of geometrical models, synthetic and analytic. The synthetic method makes use of geometrical constructions and in the analytic method geometrical objects are characterized by numerical data. The author uses these methods in this contribution. The aim of this research is to show central cyclides （Dupin＇s cyclides） as a special kind of the cyclical elliptical pedal surfaces. Firstly, applying the method of synthetic geometry, a new class of such surfaces in the 3-dimensional Euclidean space （model of the projective space） is defined. The geometrical construction of these surfaces is dependent on the given ellipse and the position of the pole P. It is the point at which the pencil of perpendicular planes passes to the plane of the ellipse, wherein the generating circles of the surface are lying. The parametric equations of the cyclical elliptical surface are derived applying the method of analytic geometry. The author classifies the surfaces according to the number of generating circles with a zero radius. The evolute of the ellipse divides the plane of the ellipse into two areas, Ω1 and Ω2. The shape of the surface depends on the position of the pole P in the areas of Ω1 and Ω2. The surface can have 4, 3 or 2 circles with a zero radius. The author then describes the relationship between these surfaces and the Dupin＇s cyclides, obtained in the case when the pole P is the point on the major axis of the ellipse. Finally, the transformation of these surfaces is shown by changing of the orthonormal base in the parameterization of the generating circles of the surface. The author visualizes the resulting surfaces obtained by the parametric approach in the MAPLE program environment.
文摘When initial radius if Stoica actually derived Einstein equations in a formalism which removes the big bang singularity pathology, then the reason for Planck length no longer holds. We follow what Ng derived as limit calculations as to a space time length factor Without the drop off of the vacuum energy as given by is at least the value of . We review the work by Ng as to quantum foam as to how that affects a general expression as to energy when , with determined at least approximately by arguments he presented in 2008 in the Dark side of the universe conference. Well before certain effects make themselves apparent, in ways which are illustrated in the manuscript. Having at a point singularity would remove expansion by the scale factor, so that the extreme version of Stoica’s treatment in an isolated 4-dimensional universe would be no expansion at all.
