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.
Keywords
special functionshypercube functionderivate
1. Introduction
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 [3], Conway [4], Coxeter [5], Dewdney [6], Hinton [7], Hocking and Young [8], Gardner [9], Manning [10], Maunder [11], Neville [12], Rucker [13], Skiena [14], Sloane [15], Sommerville [16], Wilker [17], and others and for its testing, see Letić et al. [18]. 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
The former results [2], as it is known, give the functions of the hypercube surface (n = 2), i.e., volume (n = 3), therefore, we have, respectively
where r is the half-edge of the hypercube. These functions need to be the functions of three variables with two degrees of freedom k and n and the hypercube radius r. With real cubic entities there exist, for example, square edge, length, size, or surface, then the cube surface and volume, where there exists only the variable--the half-edge r (Figure 1).
Having in mind the characteristic that the derivatives with respect to the half-edge r generate new functions (the HC matrix columns), we perform "movements" to lower or higher degrees of freedom. We start from the n th degree of freedom, on the basis of the following recurrent relations:
The previous characteristics are essential and hypothetical. They also hold for elements outside of this submatrix of six elements (see Figures 1 and 2). For example, the derivations (2.3, left) show that we have obtained the zeroth (n = 0) degree of freedom, if we perform the n th degree derivation. Using the defined derivative degree, the HC function is being calculated as well for the complex field of the hypercube matrix, namely, we obtain an expression that is equal to (2.3, right)
Due to the known characteristics of the gamma function and on the basis of relations generalizing (2.2) and (2.3), we obtain the following differential equation:
So, on the basis of the recurrent relation, we establish a connection with the positive degree of freedom n and its symmetrical negative degree (-n), through the differential equation which describes the relation among the columns of the hypercube matrix for k, n∈ℜ (2.4). In view of the general HC (2.1), we develop an adequate matrix M[HC]_{
kxn
} (k, n∈ℜ), where concrete values for the selected submatrix 9 × 9 follow as well.
$M{\left[HC\right]}_{kxn}=$
(2.4)
For example when n = 3 and $k=\overline{1,4}$ we obtain the following relation
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).
Theorem 2.2. From the columns of the matrix [M]_{
kxn
}, the following equality holds:
Interesting results can be obtained on the basis of horizontal (n) or vertical (k) degrees of freedom (arranged in 2.4) and the generalized HC function, e.g., for the level surface, as follows:
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
}
The HC, besides the gamma function, also has the potential component r^{
k+n-3}. The generalized equation of the h th derivation of the fractional power function z^{
υ
} amounts to [19]
Namely, here we know that the exponent with the basis of the half-edge r of the HC function is υ = k+n-3. Now, the m th derivation of the HC on the radius is defined by the following equation
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.
Definition 2.4. The unique operator that merges the operations of differentiating and integrating with respect to the radius of the hypercubical function is given by
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.
Example 2.5. In the first case, there exists a second derivative of the HC function connected to the degree of freedom n = 5 and derivative degree m = 2, as follows
Both operations give the same result. Applying non-integer (fractional) derivative degrees, e.g., m = ± 1/2, starting with the fixed degrees of freedom n = 7/2 and n = 5/2, we obtain the same results as the previous ones, with the procedure of integer differentiating/integrating (2.6) and (2.7). Thus, the results follow
Evidently that the results (2.6), (2.7), (2.8), and (2.9) are identical.
2.4. The HC gradient
Gradient may be applied on the hypercubical function, taking into consideration its differentiability and multidimensionality. As this function has three variables, k, n, and r, the solution of gradient functions ∇_{
k, n, r
} is given as follows
This function is particularly noteworthy with fixing the extreme of the contour HC functions.
2.5. Contour graphics of HCs
The graphics of the function and their derivatives may be simply obtained by computer analysis. Each checking in sense of the analytical expanding of the HCs (Figure 3) and in the left half-plane (k ≤ 0), it relies on the characteristics of the gamma (Γ) and digamma (ψ_{0}) functions, which make the HC.
The derivation functions hc 1 = dHC/dk = ∇_{
k
}{HC} and ch 1 = dHC/dn = ∇_{
n
}{HC} also belong to the family of HCs. Using them, we separate two groups of functions, according to the degrees of freedom k and n. The first derivate function on the degree of freedom k is as follows
and has a special meaning when we determine the maximum of HCs, e.g., for the common half-edge and the domain k∈N (Figure 4). Also, on the basis of the known criteria, for each function in the family of derivations functions, we define the maximum value of the HC function by equating its derivative with zero. Other than the maximum, we give the following, where the value of the degree of freedom k_{0} is named as "optimal". Consequently
2.6. The extreme values of the HC function from the viewpoint of the freedom degree k
The function hc 1(k, n, r) has a special meaning (Figure 5) in defining the extreme values of the HCs, e.g., for the common radius and the domain k∈ℜ. Also, in view of the known criteria, for each function in the family of derivative functions, we obtain minimum values of the HC functions by equating its derivative with zero. Other than the minimum, we give the following, where the value of the degree of freedom k_{0} is named as "optimal".
On the basis of (2.10)-(2.11), we define the minimum cube surface for the "optimal" dimension k_{0}. Also, we obtain k_{0} ≈ -1,44269, which gives the lateral surface of min HC(k_{0}, n ,1) ≈ -0,53074 as in (Table 1). For other HCs, we also define minimum values, but clearly on numeric bases. Meanwhile, for degrees of freedom n ≥ 3, the minimum comes down to zero. The greatest values of the considered functions are infinite, when k_{0} = ∞.
Table 1
Minimum cube surface for optimal dimension and various degrees of freedom
Dimension n
Optimal dimension k_{0}
minHC
Error
-1
2.680571494
-6.2786205562
4.293 × 10^{-12}
0
1.648207317
-1.1780735642
1.940 × 10^{-11}
1
0.584187070
-0.3641730147
-1.836 × 10^{-12}
2
-1.442695041
-0.5307378454
5.191 × 10^{-13}
3
-∞
0
0
. . .
. . .
. . .
. . .
i
-∞
0
0
. . .
. . .
. . .
. . .
n
-∞
0
0
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.
These functions can be important in further research of the HC. The extreme values of these contour functions HC(k, n, r) (where k is known) can be presented in Table 2, and they are visually shown in Figures 7 and 8,
Table 2
Extreme values of contour functions where k is known
2.8. The higher-order derivatives of the HC function on the k argument
The derivatives of the HC for any argument retain the characteristics similar to the original HC function. This is especially noticeable in graphical representation. In addition, it would be interesting to examine the regularities in the structure of this function of the higher degree derivation. In that sense, later in this article, we present the derivative functions on the k argument (the more complex case) i.e. n (the simpler case) analytically and graphically. The second derivative of the HC on the argument k is
where ${\psi}_{0}\left(z\right)=\frac{d}{dz}ln\Gamma \left(z\right)$ is the digamma, ${\psi}_{1}\left(z\right)=\frac{{d}^{2}}{d{z}^{2}}ln\Gamma \left(z\right)$ the trigamma function, and ζ(z) Riemann's zeta function. The concrete value of this derivative for the fixed parameters k = 3 and n = 3 is obtained later. The mixed derivation of the HC is
where ζ(z) is the Riemann's zeta function, and ζ(3) ≈ 1,202057 the Apery's constant [20].
3. Conclusion
When we obtain the HC functions of the m th derivative, we notice a certain regularity in the disposal of coefficients in addition to the polygamma functions (-1, 10, -10, -15, 5, 10, -1; see (3.1)). Partial derivatives with respect to the argument k are relatively complicated and they have been obtained only for the first and second derivatives. The previous partial derivations of the HC function have been derived to the fifth degree of freedom n, so in that case, for r = 1, we obtain the following coefficients along with psi (polygamma) function (3.1). As it is known, this derivative can be obtained as well on the basis of integral equation, where we consider m = 5.
Such integral presentation of the derivative of reciprocal gamma function, and also HC is known, but the equivalent solution (3.1) is not explained enough with the alternative members of the psi functions.
In general, the HC of the m th derivative of n can be defined on the basis of the product of polygamma (psi) polynomial of the m th degree and HC. Its general form is the following
The analysis of the multidimensional object-cube and the formulas of this geometry--suggests its complexity and connection with special functions and the other fields of geometry. Certain issues (for which we do not give solutions in this article) relate to the rational and general forms of the derivative on the two degrees of freedom, while the general derivative/integral on the argument r is obtained in (2.1) and it amounts to
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