The Relational Translators of the Hyperspherical Functional Matrix
- 723 Downloads
We present the results of theoretical researches of the developed hyperspherical function Open image in new window for the appropriate functional matrix, generalized on the basis of two degrees of freedom, Open image in new window and Open image in new window , and the radius Open image in new window . The precise analysis of the hyperspherical matrix for the field of natural numbers, more specifically the degrees of freedom, leads to forming special translators that connect functions of some hyperspherical and spherical entities, such as point, diameter, circle, cycle, sphere, and solid sphere
KeywordsSpherical Function Generalize Translator Translator Function Geometrical Entity Matrix Conversion
The hypersphere function is a hypothetical function connected to multidimensional space. It belongs to the group of special functions, so its testing is performed on the basis of known functions such as the Open image in new window -gamma, Open image in new window -psi, Open image in new window - beta, anderf- error function. The most significant value is in its generalization from discrete to continuous. In addition, we can move from the scope of natural integers to the set of real and noninteger values. Therefore, there exist conditions both for its graphical interpretation and a more concise analysis. For the development of the hypersphere function theory see Bishop and Whitlock , Collins , Conway and Sloane , Dodd and Coll , Hinton , Hocking and Young , Manning , Maunder , Neville , Rohrmann and Santos , Rucker , Maeda et al. , Sloane , Sommerville , Wells et al.  Nowadays, the research of hyperspherical functions is given both in Euclid's and Riemann's geometry and topology (Riemann's and Poincare's sphere) multidimensional potentials, theory of fluids, nuclear physics, hyperspherical black holes, and so forth.
2. Hypersphere Function with Two Degrees of Freedom
The former results (see [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30]) as it is known present two-dimensional (surface-surfs ), respectively, three-dimensional (volume-solids) geometrical entities. In addition to certain generalizations , there exists a family of hyperspherical functions that can be presented in the simplest way through the hyperspherical matrix Open image in new window , with two degrees of freedom Open image in new window and Open image in new window ( Open image in new window ), instead of the former presentation based only on vector approach (on the degree of freedom k). This function is based on the general value of integrals, and so we obtain it's generalized form.
where Open image in new window is the gamma function.
3. Translators in the Matrix Conversion of Functions
Every matrix element as a referring one can have in total eight elements in its neighbourhood, and it makes nine types of connections (one with itself) in the matrix plane (Figure 3). Considering that two degrees of freedom have a positive or negative increment (in this case integer), the selected submatrix is representative enough from the aspect of the functions conversion in plane with the help of the translator Open image in new window .
4. Generalized Translators of the Hyperspherical Matrix
In this section the extended recurrent operators include one more dimension as a degree of freedom, which is the radius r. If the increment and/or reduction is applied on this argument as well, the translator Open image in new window will get the extended form
The schematic presentation of "3D motions'' through the space of block-submatrix and locating the assigned HS function on the basis of translators and the starting hyperspherical function is given in Figure 4.
5. Conversion of the Basic Spheric Entities
6. The Relation of a Point and Real Spherical Entities
Referent and assigned coordinates
Type of translator
Here, the translators are applied taking into consideration that every defining function can be presented on the basis of the reference HS function, if we correctly define the recurrent relations both for the series and for the columns of the hyperspherical matrix .
- 5.Hinton CH: The Fourth Dimension. Health Research, Pomeroy, Wash, USA; 1993.Google Scholar
- 7.Manning PH: Geometry of Four Dimensions. Phillips Press; 2010.Google Scholar
- 8.Maunder CRF: Algebraic Topology. Dover, New York, NY, USA; 1997.Google Scholar
- 10.Rohrmann RD, Santos A: Structure of hard-hypersphere fluids in odd dimensions. Physical Review E 2007.,76(5):Google Scholar
- 11.Rucker R: The Fourth Dimension: A Guided Tour of the Higher Universes. Houghton Mifflin, Boston, Mass, USA; 1985:xii+228.Google Scholar
- 13.Sloane NJA: Sequences, A072478, A072479, A072345, A072346, A087299, A087300 and A074457. The On-Line Encyclopedia of Integer SequencesGoogle Scholar
- 15.Wells D: The Penguin Dictionary of Curious and Interesting Numbers. Penguin Books, Middlesex, UK; 1986.Google Scholar
- 18.Kabatiansky AG, Levenshtein IV: Bounds for packings on a sphere and in space. Problemy Peredachi Informatsii 1978,14(1):3-25.Google Scholar
- 19.Letić D, Cakić N: Srinivasa Ramanujan, The Prince of Numbers. Computer Library, Belgrade, Serbia; 2010.Google Scholar
- 20.Letić D, Cakić N, Davidović B: Mathematical Constants—Exposition in Mathcad. , Belgrade, Serbia; 2010.Google Scholar
- 21.Letić D, Davidović B, Berković I, Petrov T: Mathcad 13 in Mathematics and Visualization. Computer Library, Belgrade, Serbia; 2007.Google Scholar
- 23.Mitrinović SD: An Introduction into Special Functions. Scientific Book, Belgrade, Serbia; 1991.Google Scholar
- 25.Tu S-J, Fischbach E: A new geometric probability technique for an N-dimensional sphere and its applications to physics. Mathematical Physics, http://arxiv.org/abs/math-ph/0004021v3 Mathematical Physics,
- 26.Woonchul H, Zhou K: A Short Note on the Volume of Hypersphere. http://arxiv.org/abs/cs/0604056v1
- 27.Group of authors : Three Archimedes' Bodies, edited by D. Letic. Electrotechnical Faculty, Belgrade, Serbia; 2010. Technical Faculty M. Pupin, ZrenjaninGoogle Scholar
This article is published under license to BioMed Central Ltd. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.