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Some certain properties of the generalized hypercubical functions
© Letić et al; licensee Springer. 2011
Received: 4 May 2011
Accepted: 6 December 2011
Published: 6 December 2011
In this article, the results of theoretical research of the generalized hypercube function by generalizing two known functions referring to the cube hypervolume and hypersurface and the recurrent relation between them have been presented. By introducing two degrees of freedom k and n (and the third half-edge r), we are able to develop the derivative functions for all three arguments and discuss the possibilities of their use. The symbolic evaluation, numerical experiment, and graphic presentation of the functions are realized using Mathcad Professional and Mathematica.
MSC 2010: 33E30; 33E50; 33E99; 52B11.
The hypercube function (HC) is a hypothetical function connected with multidimensional space. It belongs to the group of special functions, so its testing is being performed on the basis of known functions of the type: Γ--gamma, ψ--psi, ln-- logarithm, exp-- exponential function, and so on. By introducing two degrees of freedom k and n, we generalize it from discrete to continual [1, 2]. In addition, we can advance from the field of the natural integers of the dimensions--degrees of freedom of cube geometry, to the field of real and non-integer values, where all the conditions concur for a more condense mathematical analysis of the function HC(k, n, r). In this article, the analysis is focused on the infinitesimal calculus application of the HC which is given in the generalized form. For research papers on the development of multidimensional function theory, see Bowen , Conway , Coxeter , Dewdney , Hinton , Hocking and Young , Gardner , Manning , Maunder , Neville , Rucker , Skiena , Sloane , Sommerville , Wilker , and others and for its testing, see Letić et al. . Today the results of the HC research are represented both in geometry and topology and in other branches of mathematics and physics, such as Boole's algebra, operational researches, theory of algorithms and graphs, combinatorial analysis, nuclear and astrophysics, molecular dynamics, and so on.
2. The derivative HCs
2.1. The hypercube functional matrix
On the basis of the above recurrent relations, we formulated the general form of the HC .
The matrix M[HC] kxn is based on the characteristic that each of its vectors of the (n - 1)-column (also marked as <n - 1>) is equal to the derivative with respect to the half-edge r of the following vector (<n>) and in the order according to Figure 2. This recursive characteristic ordinates among the initial assumptions (2.2).
This characteristic is very significant, because we can obtain the same result in view of the two special formulae, or using only one, the general.
2.2. The analysis if the recurrent potential function of the type z υ
After applying some transformations on the above expression, we obtain the function form.
originating from the known characteristic Γ(z+1) = z Γ(z). Equation 2.5 is recurrent by nature, and with it we find every degree of the expression (for +m) and integral (for -m), depending on the position of (n) for which we do these operations. In that sense we define and, where appropriate, use a unique operator with which we merge the operations of differentiating and integrating (using the unique symbol D ± m ) on the radius of the HC function. These operations are generalized as well on non-integer (fractional) degrees of derivative/integral.
Having in mind the known characteristics of the gamma function, the value of differential and integral degree m need not be integer, as e.g., with classical differentiating (integrating).
2.3. Fractional differentials/integrals of the HC function
The degree of derivation (or integration) m may be integer or non-integer, consequently, out of the field of real numbers. So, for example, for the integer derivatives the following values are representative and each of them gives the same result.
Evidently that the results (2.6), (2.7), (2.8), and (2.9) are identical.
2.4. The HC gradient
This function is particularly noteworthy with fixing the extreme of the contour HC functions.
2.5. Contour graphics of HCs
2.6. The extreme values of the HC function from the viewpoint of the freedom degree k
Minimum cube surface for optimal dimension and various degrees of freedom
Optimal dimension k 0
4.293 × 10-12
1.940 × 10-11
-1.836 × 10-12
5.191 × 10-13
. . .
. . .
. . .
. . .
. . .
. . .
. . .
. . .
2.7. The first derivatives of the ch 1 function of HC in relation to the degree of freedom n
Since the function HC(k, n, r) is the function of two variables (not taking into consideration the radius r), we can determine its partial derivatives on the degree of freedom n, so the derivatives are set, respectively.
Extreme values of contour functions where k is known
Optimal dimension n 0
6.619 × 10-15
3.818 × 10-14
-3.179 × 10-12
-2.964 × 10-11
. . .
. . .
. . .
. . .
. . .
. . .
. . .
. . .
2.8. The higher-order derivatives of the HC function on the k argument
Here, we include a degree of freedom as υ = n + k - 3.
where ζ(z) is the Riemann's zeta function, and ζ(3) ≈ 1,202057 the Apery's constant .
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