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
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.
2. Hypersphere Function with Two Degrees of Freedom
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.
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 .
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
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.
- (1)for the matrix(4.7)
- (2)and for the block-matrix (Figure 3)(4.8)
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 .
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