- Research Article
- Open Access
The Relational Translators of the Hyperspherical Functional Matrix
© Dusko Letic et al. 2010
- Received: 18 March 2010
- Accepted: 6 July 2010
- Published: 26 July 2010
We present the results of theoretical researches of the developed hyperspherical function for the appropriate functional matrix, generalized on the basis of two degrees of freedom, and , and the radius . 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
- Spherical 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 -gamma, -psi, - 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.
The former results (see [4–30]) 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 , with two degrees of freedom and ( ), 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 is the gamma function.
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 .
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 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.
- (2)and for the block-matrix (Figure 3)
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 .
- Bishop M, Whitlock PA: The equation of state of hard hyperspheres in four and five dimensions. Journal of Chemical Physics 2005,123(1):-3.View ArticleGoogle Scholar
- Collins GP: The shapes of space. Scientific American 2004,291(1):94-103. 10.1038/scientificamerican0704-94MathSciNetView ArticleGoogle Scholar
- Conway JH, Sloane NJA: Sphere Packings, Lattices and Groups, Grundlehren der Mathematischen Wissenschaften. Volume 290. 2nd edition. Springer, New York, NY, USA; 1993:xliv+679.View ArticleGoogle Scholar
- Dodd J, Coll V: Generalizing the equal area zones property of the sphere. Journal of Geometry 2008,90(1-2):47-55. 10.1007/s00022-008-2015-2MATHMathSciNetView ArticleGoogle Scholar
- Hinton CH: The Fourth Dimension. Health Research, Pomeroy, Wash, USA; 1993.Google Scholar
- Hocking JG, Young GS: Topology. Dover, New York, NY, USA; 1988.MATHGoogle Scholar
- Manning PH: Geometry of Four Dimensions. Phillips Press; 2010.Google Scholar
- Maunder CRF: Algebraic Topology. Dover, New York, NY, USA; 1997.Google Scholar
- Neville EH: The Fourth Dimension. Cambridge University Press, Cambridge, UK; 1921.MATHGoogle Scholar
- Rohrmann RD, Santos A: Structure of hard-hypersphere fluids in odd dimensions. Physical Review E 2007.,76(5):Google Scholar
- Rucker R: The Fourth Dimension: A Guided Tour of the Higher Universes. Houghton Mifflin, Boston, Mass, USA; 1985:xii+228.Google Scholar
- Maeda S, Watanabe Y, Ohno K: A scaled hypersphere interpolation technique for efficient construction of multidimensional potential energy surfaces. Chemical Physics Letters 2005,414(4–6):265-270. 10.1016/j.cplett.2005.08.063View ArticleGoogle Scholar
- Sloane NJA: Sequences, A072478, A072479, A072345, A072346, A087299, A087300 and A074457. The On-Line Encyclopedia of Integer SequencesGoogle Scholar
- Sommerville DMY: An Introduction to the Geometry of n Dimensions. Dover, New York, NY, USA; 1958:xviii+196.MATHGoogle Scholar
- Wells D: The Penguin Dictionary of Curious and Interesting Numbers. Penguin Books, Middlesex, UK; 1986.Google Scholar
- Freden E: Problems and solutions: solutions: 10207. The American Mathematical Monthly 1993,100(9):882-883. 10.2307/2324678MathSciNetView ArticleGoogle Scholar
- Joshi JMC: Random walk over a hypersphere. International Journal of Mathematics and Mathematical Sciences 1985,8(4):683-688. 10.1155/S0161171285000758MATHMathSciNetView ArticleGoogle Scholar
- Kabatiansky AG, Levenshtein IV: Bounds for packings on a sphere and in space. Problemy Peredachi Informatsii 1978,14(1):3-25.Google Scholar
- Letić D, Cakić N: Srinivasa Ramanujan, The Prince of Numbers. Computer Library, Belgrade, Serbia; 2010.Google Scholar
- Letić D, Cakić N, Davidović B: Mathematical Constants—Exposition in Mathcad. , Belgrade, Serbia; 2010.Google Scholar
- Letić D, Davidović B, Berković I, Petrov T: Mathcad 13 in Mathematics and Visualization. Computer Library, Belgrade, Serbia; 2007.Google Scholar
- Loskot P, Beaulieu NC: On monotonicity of the hypersphere volume and area. Journal of Geometry 2007,87(1-2):96-98. 10.1007/s00022-007-1891-1MATHMathSciNetView ArticleGoogle Scholar
- Mitrinović SD: An Introduction into Special Functions. Scientific Book, Belgrade, Serbia; 1991.Google Scholar
- Sasaki T: Hyperbolic affine hyperspheres. Nagoya Mathematical Journal 1980, 77: 107-123.MATHMathSciNetGoogle Scholar
- 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,
- Woonchul H, Zhou K: A Short Note on the Volume of Hypersphere. http://arxiv.org/abs/cs/0604056v1
- 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.