Экономика

Pil E.A.

Russia, Saint-Petersburg

Academic of the RANH, Dr.S. (eng.). Professor,

Options OF CALCULATING AREA OF SPHERICAL ECONOMIC SHELL

This article describes the economic shell that can be presented in the form of area of sphere surface. This area can be presented as the gross domestic product (GDP). The types of sphere deformation are shown under the effect of external and internal pressure.

Earlier the author showed in his articles that GDP of any country can be presented as a spherical shell or more precisely as the area of its surface [1, 2, 3].

Gross domestic product is a macroeconomic indicator that reflects the market value of all final goods and services produced for consumption, export and accumulation in all economic sectors within the borders of a country during a year, irrespective of national identity of the used production factors. That is, it can be presented as a package of final goods and services of enterprises (companies) and, consequently, each enterprise (company) constitutes its separate part of the area of spherical economic shell.

For this purpose, the area of spherical surface can be categorized into two classes:

1. the first class - the surface area of the economic shell is a closed sphere without cracks S_sc;

2. the second class – the surface area of the economic shell is an open sphere S_so where at least two sides of two polygons are out of contact with each other, i.e, there is a crack.

Each class is further divided into the following ten equal subclasses:

1. The plotted surface area of the economic shell S_s is an aggregate of separate identical polygons of the same surface area S_si and radius R_si;

2. The plotted surface area of the economic shell S_s is an aggregate of separate identical polygons with different surface areas S_si and radius Rsi;

3. The plotted surface area of the economic shell S_s is a convex and concave aggregate of separate identical polygons of the same surface areas S_si and with constant radiuses R_si for convex R_s₁ and concave R_s₂ surfaces;

5. The plotted surface area of the economic shell S_s is a convex and concave aggregate of separate identical polygons with the same surface areas S_si and with constant and varying radiuses R_si (varying radiuses R_s₁ are for convex surfaces and constant radius R_s₂ is for concave surfaces).

6. The plotted surface area of the economic shell S_s is a convex and concave aggregate of separate identical polygons of the same surface areas S_si and with varying radiuses R_si (radius R_s₁ is for convex surfaces and radius R_s₂ is for concave surfaces).

8. The plotted surface area of the economic shell S_s is a convex and concave aggregate of separate identical polygons of varying surface areas S_si and with different radiuses R_si (constant radius R_s₁ is for convex surfaces and varying radius R_s₂ is for concave surfaces).

9. The plotted surface area of the economic shell S_s is a convex and concave aggregate of separate identical polygons of varying surface areas S_siand with different radiuses R_si (varying radius R_s₁is for convex surfaces and constant radius R_s₂is for concave surfaces).

Let us consider the above stated subclasses individually:

Subclasses without cracks:

1. The plotted surface area of the economic shell S_s is an aggregate of separate identical polygons of the same surface area S_si and radius R_si.

This subclass represents an ordinary sphere plotted from identical polygons. The minimum number of sides of a polygon is equal to three, i.e. . n_p_min = 3, and the maximum number is equal to n_p_max, i.e. n_p = n_p_max. As we calculate the area of a polygon, then the number of its sides shall correspond to the number of the variables n_GDP used when calculating GDP of a country. Thus, it is possible to record the following boundaries for the number of sides of a polygon within which they can exist 3 £ n_p £ n_GDP. It should be noted that the areas of polygons are equal among themselves S_s₁ = S_s₂ … = S_si (Fig. 1).

2. The plotted surface area of the economic shell S_s is an aggregate of separate identical polygons with different surface areas S_si and radius R_si;

Figure 2 presents this subclass with two contact equilateral triangles of areas S_s₁ and S_s₁ where the radius for the convex part of the sphere R_s₁ is equal to the radius for the concave surface of the sphere R_s₂, that is R_s₁ = R_s₂.

Figure 3 represents this subclass with two contact equilateral triangles of areas S_s₁ and S_s₂ where S_s₁= S_s₂. Here the radius for the convex part of the sphere R_s₁ is equal to the radius for the concave part of the sphere R_s₂, that is R_s₁ = R_s_2.

4. The plotted surface area of the economic shell S_s is a convex and concave aggregate of separate identical polygons of the same surface areas S_si and with constant and varying radiuses R_si (constant radius R_s₁ is for convex surfaces and varying radiuses R_s₂ are for concave surfaces) and it is shown on Figure 4. In this subclass S_s₁ = S_s₂, R_s₁ = R_s₂ =…R_si = const, and R_s₂₁ ¹ R_s₂₂.

6. The plotted surface area of the economic shell S_s is a convex and concave aggregate of separate identical polygons of the same surface areas S_si and with varying radiuses R_si: radius R_s₁ is for convex surfaces and radius R_s₂ is for concave surfaces (Figure 6). In this subclass S_s₁ = S_s₂, R_s₁₁ ¹ R_s₁₂, R_s₂₁ ¹ R_s₂₂.

7. The plotted surface area of the economic shell S_s is a convex and concave aggregate of separate identical polygons of varying surface areas S_si and with the same radiuses R_si (radius R_s₁ is for convex surfaces and radius R_s₂ is for concave surfaces). Figure 7 represents this subclass with two contact equilateral triangles of areas S_s₁ and S_s₂ where S_s₁ ¹ S_s₂.

Here the radiuses for a convex part R_s₁ and for a concave part R_s₂ of spheres can be divided into the following two groups:

· all radiuses of convex R_s₁ and concave R_s₂ spheres are equal among themselves, that is R_s₁₁ = R_s₁₂ =…R_s_1i = R_s₂₁= R_s₂₂ =…R_s_2j;

· all radiuses of convex R_s₁ and concave R_s₂ spheres are equal among themselves only in their groups, that is R_s₁₁ = R_s₁₂ =…R_s_1i и R_s₂₁= R_s₂₂ =…R_s_2j and in this way R_s_1i ¹ R_s_2j.

9. The plotted surface area of the economic shell S_s is a convex and concave aggregate of separate identical polygons of varying surface areas S_siand different radiuses R_si (varying radius R_s₁is for convex surfaces and constant radius R_s₂ is for concave surfaces) (See Figure 9). In this subclass S_s₁ ¹ S_s₂, R_s₁₁ ¹ R_s₁₂, and R_s₂₁ = R_s₂₁ =…R_s_2j = const.

10. The plotted surface area of the economic shell S_s is a convex and concave aggregate of separate identical polygons of varying surface areas S_siand with varying radiuses R_si (varying radiuses R_s₁ and R_s₂are for convex and concave surfaces) (Figure 10). In this subclass S_s₁ ¹ S_s₂, R_s₁₁ ¹ R_s₁₂, R_s₂₁ ¹ R_s₂₂.

Two formulae are acceptable for surface area calculation for the above described 10 variants of close economic sphere

1. Surface area calculations are made using a corresponding formula for the first two variants when the economic sphere is a sphere in radius R_si

2. Surface area calculations are made for convex and concave sphere as an aggregate of separate areas of polygons S_si.

Similar figures and descriptions are also applied to the subclass of open spheres having even one crack, that is when two sides of two polygons are out of contact against each other and it is shown in Figure 11.

If it is necessary to change any economic surface from an open one, that is with a crack (cracks), into a close one, then it can be done in the following way. It is necessary to base on an equilateral single economic polygon where its side length l_sp is equal to one (l_sp₁ = 1) and which area we indicate as S_pht₁.

On this basis, here, we will introduce a concept of “unit theoretical are” of an economic polygon S_phtwhere the length of each side of this polygon is equal to one, squared.

Now we will perform the following transformation, we will increase (decrease) all sides of polygons depending on their sizes in such a way so that the lengths of all their sides are to be equal to one. Thus, we reduce all polygons of economic shell to the unit theoretical area S_pht₁. In this case, we will receive a polyhedral theoretical economic shell V_pht consisting of a set of unit theoretical areas of economic polyhedrons S_pht_1n. From this perspective, the area of polyhedral theoretical economic shell S_pht can be calculated under formula (2)

After that, we will introduce a coefficient of increase (decrease) for a side of a polyhedron l_sp and indicate it as K_sp. The application of this coefficient gives us a possibility to reduce any side of a polyhedron l_sp to be equal to unity, i.e. l_sp₁ = 1, that is to equilateral polyhedron and it is calculated under formula (3)

The value of the coefficient is always greater than zero, that is K_sp > 0. In this case, if the values of coefficient K_sp are within the following boundaries 0 < K_sp < 1, then, consequently, the considerd side of a polygon l_sp is decreased, that is its value is more than one. If the value of coefficient K_sp is more than one, that is K_sp > 1, it means that a side of a polygon l_sp is within the boundaries 0 < l_sp < 1. When K_sp = 1, it means that a side of a polygon is l_sp = 1.

It follows that in order to transform the open spherical economic shell S_so into a closed one S_sc, it is necessary to recalculate the lengths of all sides of polygons l_sp that are different from one under formula (3).

Since there can be a great number of such sides and polygons in a spherical economic shell if the GDP of a country is calculated, then it is necessary to use the corresponding table which example is represented below.

Table 1. Reduction the values of all lengths of triangle sides l_sp to unity
№ п/п	l_sp	K_sp	l_sp	K_sp	l_sp	K_sp
1	l_sp₁₁	K_sp₁₁	l_sp₁₂	K_sp₁₂	l_sp₁₃	K_sp₁₃
2	l_sp₂₁	K_sp₂₁	l_sp₂₂	K_sp₂₂	l_sp₂₃	K_sp₂₃
	…	…	…	…	…	…
i	l_spi₁	K_spi₁	l_spi₂	K_spi₂	l_spi₃	K_spi₃

Table 1 represents an example of values reduction of all lengths of sides of triangles l_sp to unity except when their values are equal to unity (K_sp = 1). Here a line number (serial number) (item No.) corresponds to the number of a considered triangle.

While calculating the area of spherical economic shell, it is expedient to represent it in the form of a set of polyhedrons as it was done in Figure 12, than in the form of convex or concave surfaces like it is represented in Figures above.

REFERENCES

1. Pil E.A. Types of deformation of economic shell under influence of various forces // Bulletin of St. Petersburg State University of engineering and economics. Series “Economy” №. 14(17), 2007 – P.226-231.

2. Pil E.A. Application of theory and shells for the purpose of description of processes taking place in economy // Almanac of modern science and education. 2009. No.3. P. 137-139.

3. Pil E.A. Influence of different variables onto economic shell of a country // Almanac of modern science and education. 2012. №. 12(67). P. 123-126.