 Research Article
 Open Access
The Relational Translators of the Hyperspherical Functional Matrix
 Dusko Letic^{1},
 Nenad Cakic^{2}Email author and
 Branko Davidovic^{3}
https://doi.org/10.1155/2010/261290
© Dusko Letic et al. 2010
 Received: 18 March 2010
 Accepted: 6 July 2010
 Published: 26 July 2010
Abstract
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
Keywords
 Spherical Function
 Generalize Translator
 Translator Function
 Geometrical Entity
 Matrix Conversion
1. Introduction
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 [1], Collins [2], Conway and Sloane [3], Dodd and Coll [4], Hinton [5], Hocking and Young [6], Manning [7], Maunder [8], Neville [9], Rohrmann and Santos [10], Rucker [11], Maeda et al. [12], Sloane [13], Sommerville [14], Wells et al. [15] 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 [4–30]) as it is known present twodimensional (surfacesurfs ), respectively, threedimensional (volumesolids) geometrical entities. In addition to certain generalizations [27], 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.
Definition 2.1.
where is the gamma function.
3. Translators in the Matrix Conversion of Functions
Note.
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 .
Table 1
Translators  Formula  Destination function  

1 



2 



3 



4 



5 



6 



7 



8 



9 



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 will get the extended form
Definition 4.1.
Table 2




























The schematic presentation of "3D motions'' through the space of blocksubmatrix and locating the assigned HS function on the basis of translators and the starting hyperspherical function is given in Figure 4.
Examples 4.2.
 (1)
 (2)and for the blockmatrix (Figure 3)
5. Conversion of the Basic Spheric Entities
Example 5.1.
6. The Relation of a Point and Real Spherical Entities
Table 3































Table 4
Relations  Referent and assigned coordinates  Type of translator  Conversion 









































7. Conclusion
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 [27].
Authors’ Affiliations
References
 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):94103. 10.1038/scientificamerican070494MathSciNetView 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(12):4755. 10.1007/s0002200820152MATHMathSciNetView 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 hardhypersphere 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):265270. 10.1016/j.cplett.2005.08.063View ArticleGoogle Scholar
 Sloane NJA: Sequences, A072478, A072479, A072345, A072346, A087299, A087300 and A074457. The OnLine 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):882883. 10.2307/2324678MathSciNetView ArticleGoogle Scholar
 Joshi JMC: Random walk over a hypersphere. International Journal of Mathematics and Mathematical Sciences 1985,8(4):683688. 10.1155/S0161171285000758MATHMathSciNetView ArticleGoogle Scholar
 Kabatiansky AG, Levenshtein IV: Bounds for packings on a sphere and in space. Problemy Peredachi Informatsii 1978,14(1):325.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(12):9698. 10.1007/s0002200718911MATHMathSciNetView 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: 107123.MATHMathSciNetGoogle Scholar
 Tu SJ, Fischbach E: A new geometric probability technique for an Ndimensional sphere and its applications to physics. Mathematical Physics, http://arxiv.org/abs/mathph/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
 http://mathworld.wolfram.com/Ball.html
 http://mathworld.wolfram.com/FourDimensionalGeometry.html
 http://mathworld.wolfram.com/Hypersphere.html
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