Table of Contents Author Guidelines Submit a Manuscript
Abstract and Applied Analysis
Volume 2011 (2011), Article ID 245326, 18 pages
http://dx.doi.org/10.1155/2011/245326
Research Article

The Dimensional Fluxes of the Hypercylindrical Function

1Technical Faculty M. Pupin, University of Novi Sad, Zrenjanin, Serbia
2Technical High School, 34000 Kragujevac, Serbia

Received 27 January 2011; Accepted 29 March 2011

Academic Editor: Narcisa C. Apreutesei

Copyright © 2011 Dusko Letic and Branko Davidovic. 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.

Abstract

We present the results of the theoretical researches of some characteristics of the generalized hyperspherical function with two degrees as independent dimensions. Primarily are given the answers on the quantification of the dimensional potentials (fluxes) of this function in the domain of the integer natural numbers. Beside them, we have got the solutions for some continual fluxes of the contour hypercylindrical (HY) functions on the basis. The symbolical evaluation and numerical verification of the series values and integrals are realized through the program packages Mathcad Professional and Mathematica.

1. Introduction

Hypercylinder function is the hypothetical function connected to multidimensional space. The most significant value of this function is in the fact that it originated [1] on the properties of the cylindrical entities: point, diameter, square, surface, and volume of cylinder. Another property is generalizing of these functions from discretion to continuum. It belongs to the group of special functions, so its testing is performed on the basis of known functions of these types: gamma (), psi (), error function (erf), and the like.

Definition 1.1. The generalized hypercylindrical function is defined by equality [2] where is the gamma function.

Thanks to the interpolation properties of the gamma function, we can analytically pass from the field of the natural integer values on the set of real and noninteger values with which there is concurrence of events both for its graphic interpretation and more concise mathematical analysis. It is developed on the basis of the two freedom degrees and , as special (vectors) dimensions, besides radius, as implicitly included freedom degree for every hypercylinder (Figures 1 and 2). The dominant theorem that is set is the one that refers to the recurrent property of this function (when the height is ). It implicitly includes that the left vectors () of the matrix columns in Figure 1 we get on the basis of the reverent vector () deduction, and the right vectors () on the basis of integrals on radius [2]

245326.fig.001
Figure 1: The functional submatrix for the freedom degrees and and with six characteristics hypercylindrical functions (undef. are nondefined, most often singular values of this function).
245326.fig.002
Figure 2: The surface graphic of the hypercylindrical function with the constant radius and the selected fields of freedom degrees () and () (using the software Mathematica).

To the development of the theory of the multidimensional objects, especially have contributed: Conway and Sloane [3], Gwak et al. [4], Hinton [5], Hocking and Young [6], Manning [7], Maunder [8], Neville [9], Von R. Rucker [10], Sommerville [11], Sun and Bowman [12], and the others, and to its testing, Ramanujan and Hardy [16, 17]. Today the researches of the hypercylindrical function are represented both in Euclid's and Riemann's geometry (molecular dynamics, neural networks, hypercylindrical black holes and the like).

2. Dimensional Potentials: The Fluxes of the HY Function

2.1. Vertical Dimensional Flux of the Hypercylindrical Function

The discrete dimensional potential or flux of the hypercylindrical function presents the total of all single functions in the (sub)matrix of this function that develops for the integer natural freedom degrees. Formally, flux can be quantification by twofold series that covers this area of function. The first step is to define the value of the infinite succession of functions ordered in columns (vectors) of the submatrix . This is at the same time as well the definition of the vertical dimensional fluxes of function. The first value that is being calculated, refers to the fourth column () of the submatrix in Figure 1. In this case, it follows that the flux is equal The analytical value of this series is In the paper are used three known error functions as follows: -basic, -cumulative and -imaginary. When values are even (), in other words odd ones (), the series can be divided as dichotomous, so we can now obtain two complementary series Well, the result (2.3) can be presented in the from of series with even () and odd members () that complement (one another). In that sense follows On the basis of the solution (2.3), as the starting and reference (one), and applying the reference relations (1.2), we can obtain the series values for lower freedom degrees (). So, we establish the connection of cylinder hypervolume () with its hypersurface (). In that sense follows a new vector flux for : so that Further, for the hypercylinder (), we obtain a series on the basis of the derivative of the previous series, so that For the zero dimension , the series value is as well located on the derivative basis, so it follows that For greater freedom degree that , series are found through an inverse operation, that is, by recurrent relation on the basis of integrating on the radius . Well, and for the fourth dimension is valid the next integral form For greater freedom degree than , series are found through a recurrent relation (1.2), by integrating on radius So, we come to the expression Solving the integrals and using the relation on dichotomous series [2], we obtain the dimension flux of the fourth dimension For the freedom degree of , series can be found through the integrating of the obtained solution (2.10) The general solution is known, and it is After settling the expression, we get the concrete solution For the same freedom degree, the series can be found with the integrating of the dichotomous expression So that The same result is obtained as well on the basis of the complementary dichotomous series Here, we use the imaginary error function that is equal to [14] For the last analysed vector flux, through similar procedures follow: It can be presented that the first subintegral member of the dichotomous series (2.19) is equal to So, its integral is The dimensional flux for the sixth freedom degree is now For each freedom degree the recurrent relation would be formulated as The similar formulation would be related as well to the recurrent relation of the type It can be supposed that the values of the vector fluxes are less and less with the increasing of the freedom degree , so that the limit values are equal The systematized numerical values of the discrete and continual fluxes, (for ), are given in Table 1.

tab1
Table 1

The dimensional fluxes can be studied as well for the complex part. So, for example with recurrence we get the series value for negative freedom degrees, and so for , follows [2] For more lower freedom degree, that is , the flux is more complex and is defined by the next analytical value So, for example, the flux for radius is . The other values, also can be established on the basis of the recurrent relations (2.25), in other words (2.26).

2.2. The Fluxes on the Basis of the Series of the Hypercylindrical Functional Matrix

The discrete dimensional fluxes can be calculated as well on the “horizontal line”, that is, adding functions values on the submatrix series (Figure 1). For example, through the series development for the flux would contain the next members: Some values of discrete and continual fluxes (for ) are given in Table 2.

tab2
Table 2
2.3. Some Continual Fluxes of the Hypercylindrical Function

The trend of the distribution of the vector fluxes is increasing, and then asymptotically falling, with the linear growth of the freedom degree . From the standpoint of the functional analysis the most interesting series of the matrix are the ones referring to the freedom degrees and . The first series () covers the known functions for the square size () and surface (). The members of the following series are, among the others, the cylinder functions of the surface () and volume () (Figure 1). The same series are interesting as well for calculatiing continual fluxes. So the continual natural flux for the hypercylinder surface is analysed in view of integrals, instead of series. This integral is specific, because its subintegral function is the reciprocal gamma function. Its value, as it is known, is equal to the value of Fransen-Robinson's constant [15] The integral values of the wanted flux is now , what concretely for the unit radius is In respect to the continual dimension , the more generalized cylinder volume flux follows in view of Ramanujan-Hardy's integral [16, 17]: Ramanujan defined this integral, and it was analytically intensified by Hardy. In that sense, the previous expression can be applied in calculating the hypercylindrical function flux, when , as The integral can be calculated as well as the difference between the series and the integral with the value (for )

2.4. The Progressions of the Vector Fluxes

The whole dimension flux in the freedom degree domain with the natural numbers, is obtained in the result of double amount with which are considered the integer values of the hypercylindrical function , for all . This double series must be convergent, and this characteristic is in the function of hypercylinder radius. The flux can be watched as well for every column of the matrix individually. So, there is for the column (in the mark ), the flux in the form of series

2.5. The Orthogonal Dimensional Fluxes

These fluxes are the fluxes of the all columns or of all series of the matrix . As the number of columns, that is of the series infinite, the total flux is as follows.

Definition 2.1. The dimensional flux (potentional) of the functional matrix with two freedom degrees and is defined as a double series

When the certain number of members is calculated, the flux has the next form: The flux on the matrix series in the domain of the natural number is defined as a double series, but with the changed sequence of summing. This dimensional flux is, thus, defined as In view of previously established members, the matrix flux has the following form: The equivalence of the orthogonal dimensional fluxes means the equality of the double series So, for example, for the dimensional fluxes have the unambiguous numerical value that is:

2.6. The Application of the Recurrent Operators with Defining the Diagonal Dimensional Fluxes of the Function

In the previous analyses the defining of the dimensional fluxes of the matrix is performed in view of the adding up of the function value for (on) the columns, in other words, the series of this matrix. The more detailed analysis would be very ample including the exponential function, then error functions , , , the incomplete gamma function and the like. When we use the ideas of transition operators from the reference function into the destined function, in the functional hypercylindrical matrix we can also establish the dimensional fluxes values on the diagonals (2.44), whose total would present the whole flux for the matrix in the field of freedom degrees for natural numbers, that is, . Such matrix contains infinite number of elements. For reference functions we take functions on the positions of the first series of the matrix, and they are so-called zero functions: The destined functions are arranged on the “gradual” growth law () and decrease () of increments.

Definition 2.2. The flux series operator is defined by the relation 245326.fig.eq(2.44)

Meanwhile, as increments for the absolute value are mutually equal and unit ones, that is, , to them is assigned a new joint argument (). In addition, the starting value of the -og freedom degree states , so the theta operator gets the form The destined function is now calculated as The dimensional flux on the diagonal presents the total of ist particular nembers. So, for the first diagonal (with the mark ) flux is equal then for the second, For the third, Or for the fourth, The flux in the diagonal would be reckoned in the sum form The flux for the value is calculated on the basis of the highest function value. Considering that the equivalence of the double factorial and trigonometric functions [18] is known The expression (2.19) can after settling be written as well in the equivalent form So, we get for the fifth diagonal (), using the expressions (2.51) or (2.53) For the sixth diagonal () follows: As the number of diagonals is infinite, the total flux is formed as the series of all diagonal fluxes For example, approximately, the flux for and it concretely is In the developed, form the total flux has the polynominal structure of members The diagonal flux of the hypercylindrical function can be expressed by the series of general form Here, is the summation index with which we take into consideration the order of elements from left to right and from top to bottom on (along) the diagonal (Figure 3). The polynomial coefficients contain the rational numbers and the graded constant (Table 3). The first three coefficients are equal to zero, so they are not put in the summation sequence. Its other values are () given in Table 3.

tab3
Table 3
245326.fig.003
Figure 3: The representative submatrix (above) and the addition principle of the matrix members on the diagonal.

The series approximation with 13 coefficients in the decimal notation is in the form Approximately, the double series leads us to the solution that is very near to the correct one. Namely, for the unit radius and reducing on the double series of the diagonal flux develops the following structure: The diagonal dimension flux is characteristic with its coefficients that contain the constant in the graded series numbers, as distinguished from the vertical fluxes with the destination of the errors functions and and constants. The horizontal fluxes, as it is presented (2.40) contain the exponential functions. Meanwhile, the total flux for the unit radius is convergent and it can be calculated with considerably greater value. The value aberration (2.61) of the accurate value is only (or 0,00467%) The whole dimensional continual flux () of the unit hypercylindrical function is equal to the value of the twofold integral and we find its solution on the analytical and numerical bases.

3. Conclusion

On view of the supposition of recurrent relations (1.2), namely, (2.25) and (2.26) that exist in the scope of the hypercylindrical function, we can calculate the discrete dimension flux of this function in the domain of integer freedom degrees. The quantitative flux value at the most depends of the formulated value of the hypercylinder radius. Meanwhile, as the function is the three variables function, its dependence is by all means restricted by the values of variables, namely . In the paper are calculated several continual fluxes for the contour hypercylindrical functions, in view of Ramanujan-Hardy's integrals. With the continual flux in the domain the problem is more complex, because we must, for its defining, to perform the twofold integration (2.63). Although it is not yet calculated, we suppose that its value is very close to the discrete flux, obtained in view of the double series.

Calculating the dimensional flux by the diagonal algorithm is much simpler and faster by computer, because the total flux is now defined as the convergent graded series and does not contain as components the special functions. In any case its value is identical with the fluxes that are calculated on the basis of the series, relatively to -matrix columns, so there is valid the numerically verified statement that In any case, this solution is initial for solving the other dimensional fluxes, both for hypercylindrical and hyperspherical [3, 19], in other words hypercube function [20].

References

  1. http://functions.wolfram.com/Constants/Khinchin/introductions/ClassicalConstants/01/.html.
  2. D. Letić and N. Cakić, Generalization of the Hypercylindrical Function, (Monograph to Prepare), Technical Faculty, Zrenjanin, Serbia, 2010.
  3. J. H. Conway and N. J. A. Sloane, Sphere Packing: Lattices and Groups, Springer, New York, NY, USA, 2nd edition, 1993.
  4. B. Gwak, B. H. Lee, and W. Lee, “Geodesic properties and orbits in 5-dimensional hypercylindrical spacetime,” Journal of Korean Physical Society, vol. 54, pp. 2202–2211, 2009. View at Publisher · View at Google Scholar
  5. H. C. Hinton, The Fourth Dimenzion, Health Research, Pomeroy, Wash, USA, 1993.
  6. G. J. Hocking and S. G. Young, Topology, Dover, New York, NY, USA, 1988.
  7. H. P. Manning, Geometry of Four Dimensions, Phillips Press, 2010.
  8. F. R. C. Maunder, Algebraic Topology, Dover, New York, NY, USA, 1997.
  9. H. E. Neville, The Fourth Dimension, Cambridge University Press, Cambridge, UK, 1921.
  10. B. Von R. Rucker, The Fourth Dimension: A Guided Tour of the Higher Universes, Houghton Mifflin Harcourt, Boston, Mass, USA, 1984.
  11. Y. M. D. Sommerville, An Introduction To The Geometry of n Dimensions, Dover, New York, NY, USA, 1958.
  12. Q. Sun and J. M. Bowman, “Diatom-diatom reactive scattering in hypercylindrical,” International Journal of Quantum Chenistry, vol. 36, no. S23, pp. 115–126, 2009. View at Google Scholar
  13. K. Gohara and J. Nishikawa, “From hybrid dynamical systems point of view,” Artificial Life and Robotics, vol. 7, no. 4, pp. 189–192, 2004. View at Google Scholar
  14. M. Abramowitz and I. A. Stegun, Eds., Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, Dover, New York, NY, USA, 9th edition, 1972.
  15. D. Letić, N. Cakić, and B. Davidović, Mathematical Constants—Exposition in Mathcad, Belgrade, Serbia, 2010.
  16. H. G. Hardy, Ramanujan: Twelve Lectures on Subjects Suggested by his Life and Work, Cambridge. University Press, 1st edition, 1940.
  17. D. Letić and N. Cakić, Srinivasa Ramanujan: The Prince of Numbers, Computer Library, Belgrade, Serbia, 2010.
  18. http://mathworld.wolfram.com/DoubleFactorial.html.
  19. http://mathworld.wolfram.com/Hypersphere.html.
  20. http://mathworld.wolfram.com/Hypercube.html.