transitional processes as a tool for revealing universe's hidden

Int. J. Emerg. Sci., 1(2), 83-94, June 2011ISSN: 2222-4254© IJES

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Transitional Processes as a Tool for RevealingUniverse’s Hidden Dimensions

Alexander A. Antonov

Research Center of Information Technologies, P.O.Box 73, Kiev, 03142, Ukraine

[email protected]

Abstract. Contemporary algebra is self-contradictory. Indeed, it offers the algorithms of solving algebraic equations using either the real or the complex numbers, which are, however, mutually exclusive. For instance, even the

simplest quadratic equation cbxaxy 2 in the case of a negative

discriminant acb 42 has no solution on the set of real numbers, but does have a solution on the set of complex numbers. Nevertheless, there can naturally be only one truth. So, which of the two solutions is true? The answer to this question can be given by the physics of transitional processes in oscillating systems. We present the solution of the problem, which eventually leads to the discovery of hidden extra dimensions in the physical macrocosm.

Keywords: Algebraic Equations, Complex Numbers, Oscillating Systems, Extra Dimensions, Hidden Dimensions

1 INTRODUCTION

In 1545, Girolamo Cardano (1501 – 1576) discovered complex numbers when solving algebraic equations. In 1777, Leonhard Euler (1707 – 1783) introduced the symbol 1-i . The term ‘complex numbers’ was introduced by Johann CarlFriedrich Gauss (1777 – 1855) in 1831 [1]. Several centuries have passed. However, it turns out that the search for the true solutions of algebraic equations and their understanding is not over yet, since the physical meaning of complex numbers has not yet been revealed. Therefore, until now algebraic equations are solved using both the real and the complex numbers, although these solutions are mutually exclusive.


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2 MATHEMATICAL ASPECT OF SOLVING ALGEBRAICEQUATIONS

In the course of algebra, two algorithms are used to solve algebraic equations of any degree, the first one – on the set of the generally understood real numbers, the second one – on the set of the generally incomprehensible complex numbers. The first algorithm has it that a different number of solutions may exist in different situations. According to the second algorithm, the number of solutions always (at any magnitude of coefficients) equals the degree of the algebraic equation.

Figure 1. Graphical Solutions of Algebraic Equations on the Set of Real Numbers

For example, a quadratic equation is claimed to have either two, or one, or none solutions on the set of real numbers. For a cubic equation, it is stated that on the set of real numbers it can have either three, or two, or one solution.

These conclusions are often illustrated by finding the intersection of the parabola c+bx+ax=y 2 and the line 0y (Figure 1a), which corresponds to the graphical solution of the set of equations:

0y

cbxaxy 2

or the intersection of the curve dcхbxaxy 23 +++= and the line0y (Figure 1b), which corresponds to the graphical solution of the set of

equations

0y

dcxbxaxy 23

It is easy to give similar graphic illustrations for the solutions of equations of any higher degree.

The use of the second algorithm allows claiming that the solutions always exist. The number of the solutions always equals the degree of the algebraic equation. Moreover, at least some of the solutions may be complex numbers.

International Journal of Emerging. Sciences, 1(2), 83-94, June 2011

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However, it is easy to notice that in the cases when, according to the second algorithm, the solutions exist as complex numbers, they do not exist in accordance with the first algorithm. Obviously, these two statements are mutually exclusive.

Therefore, naturally, we should ask which of these solutions is eventually the true one and, consequently, the one that exists in nature.

3 PHYSICAL ASPECT OF SOLVING ALGEBRAIC EQUATIONS

Since there is still no strictly mathematical answer to the question above, let usresort to the help of a physical experiment. To this effect, let us consider the so-called transitional processes, which correspond to the transition of any energy-intensive system from one energy state into another. The transitional processes are accounted for by the very important physical circumstance that in nature energy cannot change instantly, simply because the time derivative of energy is power. Consequently, instant change of energy should correspond to the infinitely large power. This is why a transitional process always takes certain time.

In order to determine the parameters of transitional processes, algebraic, the so-called characteristic equations, are solved, which are in a certain way connected with the differential equations describing the behaviour of these systems [2], [3], [4].

Transitional processes are everywhere around us. Here belong shock oscillations in electric oscillating LCR-circuits, in hydraulic systems (tsunami), acoustic systems (percussion instruments), swinging of a pendulum after it is pushed, and other processes. Thus, anyone can perform a simple physical experiment (e.g., push a pendulum) and make sure that:

oscillating transitional processes exist; transitional processes are damped; damping of oscillations takes certain time.

This simple experimental evidence turns out to be sufficient to answer the question above.

Indeed, the solution of the algebraic characteristic equation 0cbxax2 at the positive value of the discriminant 0ac4b2 corresponds to the aperiodic transitional process. The solution of the quadratic characteristic equationat 0ac4b2 corresponds to the critical transitional process. The solution of the quadratic characteristic equation at 0ac4b2 corresponds to the oscillation (in the form of damped oscillations – remember the swinging of the pendulum after it is pushed) transitional process.

The solutions of the algebraic characteristic equation 0dcхbxax 23 =+++ correspond to the same processes with the additional

exponential component.Moreover, for the cases when the solutions of the characteristic equation of the

second degree 0cbxax2 , the third degree 0dcхbxax 23 =+++ or


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any higher degree contain pairs of complex-conjugate numbers, the transitional processes possess the oscillating component(s).

And they always exist.If the characteristic equations were solved using the above algorithms only on

the set of real numbers, oscillation transitional processes in nature would have been non-existent.

Therefore, we have to conclude that the only correct solutions of algebraic equations are the solutions on the set of complex numbers [5], [6].

4 GRAPHICAL INTERPRETATION OF THE SOLUTIONS OF ALGEBRAIC EQUATIONS

The results obtained turned out to be so unusual that in order to comprehend them some additional explanations are necessary.

First, the conclusion made above has to be illustrated with the corresponding charts. For instance, the graphs in Figure 1 illustrate the solutions of the quadratic and cubic equations on the set of real numbers in a very convincing way. Even though no one claims based on these graphs that the solutions of algebraic equations on the set of complex numbers actually do not exist, that these solutions are something unreal, in fact this is exactly the way many people think.

Figure 2. Graphical Solution of the Quadratic Equation on the Set of Complex Numbers

This is why the solutions of algebraic equations using complex numbers also require to be graphically illustrated. This is obviously a complicated task, because, if x and y are complex numbers, then the graph of the function xfy requires four-dimensional space to be illustrated. However, psychologists found that people think with visual, i.e., not more than three-dimensional, images. In order to make sure this is so, try to imagine, for instance, a four-dimensional cube (or, in other words, the hypercube, tesseract, octachoron). Therefore, we cannot, unfortunately,


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either plot or imagine the function graph of the complex variable xfy .Nevertheless, the above complication can be overcome if we remember that

apart from the algebraic representation of complex numbers, there are also the trigonometric and exponential forms of their representation. They make it possible to understand that the complex number takes zero value only at zero absolute value irrespective of the phase value. Therefore, the task may be simplified by plotting the three-dimensional graph of the function ivufy instead of the four-

dimensional graph of the function of complex variable xfy .

Figure 3. Graphical Solution of the Cubic Equation on the Set of Complex Numbers

In this case, the solution of the quadratic equation on the set of complex numbers can be illustrated by graphical solution of the equations set:

0y

civubivuacbxaxy 22

(1)

and the solution of the cubic equation on the set of complex numbers can be illustrated by graphical solution of the set of equations:


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0y

divucivubivua

dcxbxaxy

23

23

(2)

where ivux is the complex argument;

y is the absolute value of the complex variabley .

Graphical solutions of the equations set (1) for different values of parametersc,b,a are given in Figures 2a, 2b and 2c. Graphical solutions of the equations set

(2) for different values of d,c,b,a are presented in Figures 3a, 3b, 3c, and 3d. These three-dimensional graphs clearly illustrate that at any parameters of the algebraic equations the number of their solutions (i.e. the intersections of the surface and the plane) is indeed equal to the degree of the equation.

However, upon closer examination of Figure 2 and Figure 3 it turns out that in Figure 2b and Figure 3c there is one less point of contact of the surface and the plane, compared to other graphs. Thus, these two plots seem to contradict the statement made above. However, this is not the case, because in Figure 2b the point of contact corresponds to the once repeated root of the quadratic equation, e.g. 02x 2 , and in Figure 3c one of the points of contact corresponds to the once

repeated root of the cubic equation, e.g. 01x2x 2 .Then how are we supposed to understand the solutions of the quadratic and the

cubic equations on the set of real numbers, and the corresponding graphs given above (Figure 1)? Why were the solutions found earlier for these equations represented only by real numbers?

The answer is very simple. The matter is that when solving the quadratic and the cubic equations in accordance with the first algorithm, the solution was searched only in the form of real numbers. Therefore, the solution corresponding to this very condition was found. You only get what you are looking for.

Thus, since all, without any exception, solutions of characteristic algebraic equations on the set of complex numbers correspond to the physically real transitional processes, complex numbers are the only true solutions of any algebraic equations.

Therefore, complex numbers, or, to be more precise, concrete (i.e., having a dimension) complex numbers are physically real. Thus, not only the real, but the imaginary components of complex numbers, as well, are physically real.

5 OTHER EVIDENCE OF PHYSICAL REALITY OF COMPLEX NUMBERS

There is other evidence of physical reality of concrete complex numbers. For example, it can be proven with the help of the new theory of resonance [7], [8].


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The current theory of resonance in linear electric circuits based on the concept of real frequency (i.e., the frequency measured with real numbers) is imperfect and suitable only for simplified engineering calculations. Let us illustrate this.

Figure 4. Electric Circuits under Investigation

The current theory of resonance usually starts with the analysis of processes in the simplest oscillating LC-circuits, e.g., represented by the parallel oscillating LC-circuit (Fig. 4a), the complex admittance of which is

20

2201

LjLj

CjjY (3a)

whereLC

10 .

Thus, the frequency of free oscillations in such an oscillating circuit is

0free .The imaginary component of this complex admittance:

20

20

2

L

jYIm (3b)

as can be seen, takes zero value at the frequency 01 res . Therefore, it isreferred to as resonance frequency.

The absolute value of this complex admittance is

20

20

2

LY (3c)

It takes zero value at the frequency 02 res , which is also referred to as resonance frequency.

As can be seen, both resonance frequencies and the frequency of free oscillations are equal: 0free2res1res . It is this circumstance that in the current theory is considered an attribute of resonance in any oscillating circuit.

However, in fact, the relation 021 freeresres in LCR-circuits is never fulfilled. For instance, even in the simplest oscillating LCR-circuit (Fig. 4b) yields different results. Its complex admittance is


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0

20

022

0

2

211

1

jLj

j

LjRCj

jY (4а)

whereLC

10 ; L

R0 .

Thus, the frequency of free oscillations in this LCR-circuit is (the corresponding mathematical calculations are omitted for simplicity):

020

20 free .

The imaginary component of this complex admittance:

22

040

40

20

20

2

4

41

LjYIm (4b)

takes zero value already at the resonance frequency:

020

20

20

14

res .

The absolute value of this complex admittance is

22

040

2

20

20

2220

4

41

LjY (4c)

Figure 5. Amplitude-frequency curve of the LCR-circuit under investigation

As the function (4c) does not take zero value at any values of its argument (see Fig. 5), we have to find out which values of the argument correspond to its


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minimum value. To this end, let us find out the derivative of the radicand and set it to zero. By solving the equation obtained

0

4

]8)168([2

4

4

dd

222

0222

020

80

40

20

240

20

20

40

4

220

40

2

220

2220

We find the set of two resonance frequencies (not one frequency!) which correspond to the second of the above attributes of resonance.

040

20

20

40

20

20

20

20

30

02res

2res

168

48

0

Thus, in the LCR-circuit under investigation, there are three resonance frequencies, which are not equal to each other and to the frequency of free oscillations. Other LCR-circuits also have certain peculiarities.

Only small values of 0 provide for the approximate parity

021 freeresres . At small values of 0 , the formulae (4a), (4b), and (4c) also take the form (3a), (3b), and (3c). Therefore, the processes taking place in this case, according to the current theory, are regarded as resonance, whereas, in fact, they are just quasi-resonance processes. However, the current theory of resonance does not explain why in LCR-circuits, contrary to LC-circuits, the precise equality 021 freeresres is not fulfilled.

Therefore, the current theory of resonance in linear electric circuits, based on the concept of real frequency (i.e., the frequency measured with real numbers), is, indeed, imperfect.

At the same time, there is a different theory of electric circuits [2], [3], [4]based on the concept of complex frequency (i.e., the frequency measured withcomplex numbers). In accordance with this theory, the immittance functions of complex frequency are fractional rational. Thus, at certain complex frequencies (which in a particular case can be real), referred to as the terminals, the denominator of the immittance function takes zero values. This theory is consistent and allows solving many important problems. However, it does not mention resonance. It also does not explain the physical meaning of the concept of complex frequency.

Obviously, this was accounted for by the lack of theoretical and experimental evidence of the physical reality of complex frequencies and resonance at complex frequencies. However, [9], [10], [11] offer such theoretical and experimental evidence. Moreover, the new theory explains all the paradoxes of the current theory of resonance, which uses the concept of the real frequency.

In particular, it follows from the new theory, that complex resistances and


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admittances, voltages and currents, capacities and energies (as well as their imaginary components) are also physically real. Since oscillations and resonance can exist not only in electric circuits, but also in mechanical, acoustic, hydraulic and other systems, the corresponding parameters measured with complex numbers are also physically real.

5 UNIVERSE’S HIDDEN EXTRA DIMENSIONS

The latter statement provokes new questions: What is the physical reality of concrete imaginary numbers if they cannot

bee seen or felt in any other way? If concrete imaginary numbers are physically real, do they measure

anything? Are they the extra dimensions?As for the first question, we should not forget that we do not see or feel many

other things and phenomena, such as the magnetic field, the ultraviolet, the X-rays, the radioactive emanation, the elementary particles, the atomic structure, the black holes, etc. Nevertheless, we know they exist based on the existing theoretical and experimental data. Therefore, we also have to admit the physical reality of concrete complex and imaginary numbers, taking into account the above theoretical and experimental evidence. Oscillograph recordings of damped oscillations, where, according to the Euler formula, one of the components of the complex frequency characterizes the damping rate, and the other one – the frequency of intersecting the time axis, can help overcome the psychological barrier of distrusting this new knowledge.

As for the second question, it is clear that if the physically real concrete imaginary numbers can have different values, they naturally measure something. Moreover, as has been proven above, they measure something that really exists. For instance, the available equipment allows measuring complex, real and imaginary voltage drop and power. In this respect, they are certainly physically real dimensions, or, to be more precise, the extra dimensions in addition to the physically real dimensions measured with the concrete real numbers. Since we do not see or feel them, they can be named hidden dimensions, similar to the hidden dimensions which are described in [12] (Randall 2005) and which are expected to be discovered at the Large Hadron Collider. However, the extra dimensions described in this paper are different, since they belong to the macrocosm, whereas the Large Hadron Collider deals with the microcosm. We do not rule out that the comprehension of the physical reality of concrete complex numbers will allow revealing other extra dimensions in the microcosm, as well.

6 CONCLUSION

Universe’s hidden extra dimensions have been revealed. This was made in an unusual way, namely, by employing a physical experiment to solve the centuries-old mathematical problem. In other words, the problem solved was neither purely


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mathematical nor purely physical, but a complex physical and mathematical one. This approach yielded fruitful results in solving the problem unsolvable by

purely mathematical methods, namely, which of the two mutually exclusive solutions of algebraic equations (using only the real or only the complex numbers) is the only true one. The convincing answer to this question was obtained only after specifying the selection criterion. The correct decision was recognized as the one corresponding to the physical processes actually taking place in nature. Since only the solutions of characteristic algebraic equations on the set of complex numbers correspond to the physically real transitional processes, these solutions have been recognized as being the only correct ones.

For the same reason, concrete complex numbers have also been recognized as physically real, including their imaginary components. However, these physically real numbers are complex only for the period of damped oscillation processes. In other words, it is only for this period that extra dimensions open up in out physical world. Since we cannot see or feel concrete imaginary numbers, they can be justly referred to as the hidden dimensions, similar to the hidden dimensions, which are expected to be discovered during the experiments at the Large Hadron Collider.

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1. Vinogradov. IМ, ed, “Mathematical Encyclopaedia”, vol. 2, Moscow, Soviet Encyclopaedia, 1979.

2. Bode. HW, “Network Analysis and Feedback Amplifier Design”, Van Nostrand, Princeton, UK, 1945.

3. Steinmetz. CP, “Theory and Calculation of Electric Circuits”, Wexford College Press, 2006.

4. Bell. DA, “Fundamentals of Electric Circuits”, Oxford University Press, Oxford, UK, 2009.

5. Antonov. AA, “Solution of Algebraic Quadratic Equations Taking into Account Transitional Processes in Oscillation Systems”, General Mathematical Notes 2010a, 1.2: 11-16. http://geman.in/yahoo_site_admin/assets/docs/2_Solution_of_algebraic_quadraticAntonov.31223248.pdf

6. Antonov. AA, “Oscillation Processes as a Tool of Physics Cognition”, American Journal Scientific and Industrial Research 2010b, 1.2: 342 – 349. doi:10.5251/ajsir.2010.1.2.342.349

7. Antonov. AA, “Investigation of Resonance”, Preprint № 67 of the Institute for Modelling in Energy Engineering of AS of the UkrSSR, Kiev, 1987.

8. Antonov. AA, “New Interpretation of Resonance”, International Journal of Pure and Applied Sciences and Technology 2010c, 1.2: 1-12.http://ijopaasat.in/yahoo_site_admin/assets/docs/Antonov_Paper-1.18191424.pdf

9. Antonov. AA, Bazhev. VM, Patent of the USSR № 433650, 1970.


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10. Antonov. AA, “Physical Reality of Resonance on Complex Frequencies” European Journal of Scientific Research 2008, 21.4: 627-641. http://www.eurojournals.com/ejsr_21_4_06Alexander.pdf

11. Antonov. AA, “Resonance on Real and Complex Frequencies”, European Journal of Scientific Research 2009, 28.2: 193-204. http://www.eurojournals.com/ejsr_28_2_03.pdf

12. Randall. L, “Warped Passages: Unraveling The Mysteries Of The Universe's Hidden Dimensions”, Ecco, NY, 2005.

transitional processes as a tool for revealing universe's hidden

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