A general Bernoulli distribution in
\\n\\n
Released this past November, the list is based on data collected from the Web of Science and highlights some of the world’s most influential scientific minds by naming the researchers whose publications over the previous decade have included a high number of Highly Cited Papers placing them among the top 1% most-cited.
\\n\\nWe wish to congratulate all of the researchers named and especially our authors on this amazing accomplishment! We are happy and proud to share in their success!
Note: Edited in March 2021
\\n"}]',published:!0,mainMedia:{caption:"Highly Cited",originalUrl:"/media/original/117"}},components:[{type:"htmlEditorComponent",content:'IntechOpen is proud to announce that 191 of our authors have made the Clarivate™ Highly Cited Researchers List for 2020, ranking them among the top 1% most-cited.
\n\nThroughout the years, the list has named a total of 261 IntechOpen authors as Highly Cited. Of those researchers, 69 have been featured on the list multiple times.
\n\n\n\nReleased this past November, the list is based on data collected from the Web of Science and highlights some of the world’s most influential scientific minds by naming the researchers whose publications over the previous decade have included a high number of Highly Cited Papers placing them among the top 1% most-cited.
\n\nWe wish to congratulate all of the researchers named and especially our authors on this amazing accomplishment! We are happy and proud to share in their success!
Note: Edited in March 2021
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Firstly, classical mechanics is a theory in physics studying the macroscopic objects motion whether they are parts of machinery or projectiles or objects in astronomy like for example planets or spacecrafts or galaxies or stars. As it was established, classical mechanics is deterministic that means that we can predict the motion of objects in the future when we know their present state. It is also reversible and that means we can know the motion of objects in the past when we know their present state also [1].
Since classical mechanics was developed at the beginning by Sir Isaac Newton therefore it is usually referred to as Newtonian mechanics. It comprises the mathematical methods and the employed physical concepts developed, as we have mentioned, by Newton, Gottfried Wilhelm Leibniz and others in the seventeenth century to study the bodies motion under the effect of a set of forces. The theory was more developed later on to embody more abstract methods which have led to the reformulations of classical mechanics and hence to the establishment of Hamiltonian mechanics and Lagrangian mechanics. These developments which were done in the eighteenth and nineteenth centuries are substantial extensions beyond the work of Newton because they used more particularly analytical mechanics. After doing some modifications, modern physics makes use of them in all its areas [2].
Moreover, exceptionally precise results are provided by classical mechanics when considering objects with velocities far from the speed of light and when they do not possess extreme masses. It is mandatory to make use of quantum mechanics which is a sub-field of mechanics when studying objects which have an atom diameter size. Additionally, we need Albert Einstein’s special relativity when considering speeds near the velocity of light. Furthermore, Einstein’s general relativity is applied when objects have huge masses. It is important to note that many modern sources include in classical physics the relativistic mechanics which represents according to them the most precise, developed, and complete form of classical mechanics [3].
Furthermore, we now present classical mechanics fundamental concepts. The theory assumes that the objects of the real world are of negligible size that means that they are point particles. And it also characterizes the point particle motion by few parameters which are: its mass, its position, and the applied forces to it. We will discuss each of these parameters in turn [4].
In fact, and in reality, classical mechanics can describe always the kind of objects that have a non-zero size. Whereas, very small particles like electrons are described more accurately by the physics of quantum mechanics. Additionally, hypothetical point particles have more simplified behavior than non-zero size objects like for example a baseball that can spin when it is in motion. Moreover, such non-zero objects are considered as composite objects constituted of a large number of point particles acting collectively; hence, the point particles results can be used in such large objects study [5].
Common sense notions are used by classical mechanics of how matter and forces interact and exist. Its basic assumption is that energy and matter have knowable and definite attributes such as speed and location in space. Additionally, it is assumed by non-relativistic mechanics the instantaneous action of forces or instantaneous action at a distance [6].
The bodies motion study is very ancient, this makes classical mechanics one of the largest and oldest subjects in engineering, technology, and science [7].
Aristotle, one among antiquity Greek philosophers and who is the founder of Aristotelian physics, may have been the first to postulate that theoretical principles can assist nature understanding and to assume that “everything happens for a reason”. Many of these ideas preserved are considered as eminently reasonable by a modern reader but there is an obvious lack of controlled experiment and mathematical theory as we know it. In fact, modern science was formed by these later decisive factors and classical mechanics came to be known as their early application [8].
The medieval mathematician Jordanus de Nemore introduced in his Elementa demonstrationem ponderum the “positional gravity” concept and the component forces use [9].
Johannes Kepler published in 1609 Astronomia nova which was the first published causal explanation of the planets motion. Based on the observations made by Tycho Brahe on Mars orbit, he concluded that the orbits of the planet were ellipses. This epistemological revolution occurred at the same time when Galileo was proposing for objects motion abstract mathematical laws. Perhaps he may have performed the historical experiment of the two cannonballs of different weights dropping from Pisa tower. Hence, he showed that these two cannonballs hit the ground simultaneously. We doubt in fact the reality of that particular experiment, but Galileo conducted quantitative experiments which were to roll balls on an inclined plane. From such experiments results he derived his accelerated motion theory [10].
Sir Isaac Newton laid down classical mechanics foundations by founding his natural philosophy principles on three laws of motion proposed by him: the inertia first law, the acceleration second law, and the action and reaction third law. A proper mathematical and scientific treatment in Philosophiae Naturalis Principia Mathematica of Newton was given to his second and third laws. They are in fact different from the attempts laid earlier to explain similar phenomena and which were either incorrect, incomplete, or they lack a precise mathematical expression. Moreover, the principles of conservation of angular momentum and momentum were postulated by Newton. Additionally, the universal gravitational law of Newton was also provided by him to give the first accurate mathematical and scientific formulation of gravity. The most accurate and fullest description of classical mechanics was provided by the combination of the laws of motion and gravitation of Newton. Newton showed that his three laws can be applied to the objects of everyday as well to heavenly objects. Particularly, Newton derived a theoretical explanation of the planets’ laws of motion of Kepler [11].
Newton performed the mathematical calculation by inventing previously the mathematical calculus. In fact, calculus eclipsed his book, the Principia, which was formulated totally in terms of geometric methods which were long established and to gain hence acceptability. Moreover, the notation of the integral and of the derivative which are preferred today were developed by Leibniz however [12].
All phenomena, including light in the form of geometric optic, can be explained by classical mechanics as it was assumed by Newton and most of his contemporaries, with the notable exception of Christiaan Huygens. Newton maintained his own corpuscular light theory even when they discovered the wave interference phenomenon or the so-called Newton’s rings [13].
Classical mechanics became a major field of study in physics as well in mathematics and this after Newton. A far greater number to problems solutions were allowed by several progressive reformulations of his mechanics. Joseph Louis Lagrange was the first to reformulate in 1788 Newtons’ mechanics. William Rowan Hamilton in his turn reformulated Lagrangian mechanics in 1833 [14].
More modern physics resolved some difficulties that were discovered in the late nineteenth century. Compatibility with the theory of electromagnetism and the famous Michelson-Morley experiment were some of these difficulties. Often still considered as a part of classical mechanics, the special relativity theory was led by the resolution of these problems [15].
Explaining all thermodynamics, raised another set of difficulties and problems with classical mechanics. Gibbs paradox of classical statistical mechanics was the result of the combination of classical mechanics with thermodynamics. In this paradox, entropy is not a quantity which was well defined. We introduced quanta to explain the black-body radiation otherwise this was not possible. Classical mechanics was unable to explain, not even approximately, such basic things as the sizes of the atoms, the photo-electric effect, and the energy levels and this when experiments delved into the atomic world. Quantum mechanics was the result of the efforts to resolve these problems [16].
Classical mechanics has no longer been considered as an independent theory since the end of the twentieth century. We consider classical mechanics now as an approximate theory to quantum mechanics which is a more general theory. The desire to understand the fundamental forces of nature has shifted our emphasis in our research and investigation and has led to the Standard Model and also has directed the studies to a unified theory of everything. For the study of the motion of low-energy, of non-quantum mechanical particles in weak gravitational fields, it is useful to make use of classical mechanics. Additionally, we were successful to extend classical mechanics to the complex domain. In fact, this extended complex classical mechanics behaves very similarly to quantum mechanics [17].
At the end, and to conclude, this research work is organized as follows: After the introduction in section 1, Newton’s laws of classical mechanics are stated in section 2, then the purpose and the advantages of the present work are presented in section 3. Afterward, in section 4, the extended Kolmogorov’s axioms and hence the complex probability paradigm with their original parameters and interpretation will be explained and summarized. Moreover, in section 5, the complex probability paradigm axioms are applied to classical mechanics which will be hence extended to the imaginary and complex sets. Additionally, in section 6, the resultant complex random vector
The classical mechanics foundation was laid down by Isaac Newton’s three physical laws of motion. These laws define and describe the forces acting upon a body as well as the response of the body to those forces. Moreover, and more precisely, the first law defines the force qualitatively, the second law measures the force quantitively. The third law states that an isolated single force does not exist [18, 19, 20, 21]. Throughout nearly three centuries, these three laws have been stated in many different ways and we will summarize them as follows:
First law
In an inertial frame of reference, an object either remains at rest or continues to move at a constant velocity, unless acted upon by a force.
Second law
In an inertial frame of reference, the vector sum of the forces
Third law
When one body exerts a force on a second body, the second body simultaneously exerts a force equal in magnitude and opposite in direction on the first body.
Isaac Newton was the first to state in his Mathematical Principles of Natural Philosophy (Philosophiae Naturalis Principia Mathematica), first published in 1687, the three laws of motion. Many systems and physical objects were investigated and explained by the three laws of motion of Newton. As an example, the planetary motion laws of Johannes Kepler were proved and demonstrated by Newton’s laws when combined with the universal gravitational law, in the third volume of the text [22, 23, 24, 25].
Fourth law
Some also describe a Fourth law which states that forces add up like vectors, that is, that forces obey the principle of superposition.
A single point masses idealize the objects to which we apply the laws of Newton, that means that the object body shape and size are to be ignored in order to concentrate on the body’s motion more easily. This is achieved when the rotation and the deformation of the body are negligible and when the object is too small compared to the distances that the analysis involves. Hence, in the planet orbital motion around a star analysis, even a planet can be idealized as a particle [26, 27, 28, 29].
Moreover, deformable bodies and the rigid bodies motion are not characterized by the original form of the laws of motion of Newton which reveal to be inadequate. Additionally, a generalization of the laws of motion of Newton for rigid bodies was introduced and achieved by Leonhard Euler in 1750 and they were called accordingly Euler’s laws of motion. They were applied later on to deformable bodies which were postulated to be a continuum. Euler’s laws can be derived from the laws of Newton if we represent a body as an assemblage of discrete particles where every particle is governed by the motion laws of Newton. Independently of the structure of any particle, the laws of Euler can be considered, however, as axioms that describe the motion laws of extended bodies [30, 31, 32, 33].
Newtonian inertial reference frames are a certain set of frames that verify and confirm Newton’s laws. The first law defines what an inertial frame of reference is and this according to some authors interpretation. Therefore, the first law cannot be demonstrated as special case of the second law since the second law is only valid when an inertial frame of reference is used in the observation. The second law is considered as a corollary of the first law by other authors. It was long after Newton’s death that we have developed the inertial frame of reference explicit concept [34, 35, 36, 37].
Furthermore, we assume that, momentum, acceleration, and most importantly force to be quantities defined externally in the given interpretation. This is not the only interpretation, but the most common way one can consider the definition of these quantities by Newton’s laws [38, 39, 40, 41].
Additionally, when the speeds considered are much closer to the speed of light, then Albert Einstein’s special relativity replaces Newtonian mechanics which is still useful as an approximation of the studied phenomenon [42, 43, 44].
The crucial job of the theory of classical probability is to compute and to assess probabilities. A deterministic expression of probability theory can be attained by adding supplementary dimensions to nondeterministic and stochastic experiments. This original and novel idea is at the foundations of my new paradigm of complex probability. In its core, probability theory is a nondeterministic system of axioms that means that the phenomena and experiments outputs are the products of chance and randomness. In fact, a deterministic expression of the stochastic experiment will be realized and achieved by the addition of imaginary new dimensions to the stochastic phenomenon taking place in the real probability set
The advantages and the purpose of this current work are to:
Extend the theory of classical probability to encompass the complex numbers set, hence to bond the theory of probability to the field of complex variables and analysis in mathematics. This mission was elaborated and initiated in my earlier seventeen papers.
Apply the novel probability axioms and paradigm to Newton’s classical mechanics.
Show that all nondeterministic phenomena can be expressed deterministically in the complex probabilities set which is
Compute and quantify both the degree of our knowledge and the chaotic factor of all the forces acting on a body in classical mechanics and
Represent and show the graphs of the functions and parameters of the innovative paradigm related to Newton’s mechanics.
Demonstrate that the classical concept of probability is permanently equal to one in the set of complex probabilities; hence, no randomness, no chaos, no ignorance, no uncertainty, no nondeterminism, no unpredictability, and no disorder exist in:
Prove an important property at the foundation of statistical physics after applying
Prepare to implement this creative model to other topics in prognostics and to the field of stochastic processes. These will be the job to be accomplished in my future research publications.
Concerning some applications of the novel founded paradigm and as a future work, it can be applied to any nondeterministic phenomenon using classical mechanics whether in the continuous or in the discrete cases. Moreover, compared with existing literature, the major contribution of the current research work is to apply the innovative paradigm of complex probability to Newton’s classical mechanics and to statistical physics as well.
The next figure displays the major purposes and goals of the Complex Probability Paradigm (
The diagram of the complex probability paradigm major goals.
The simplicity of Kolmogorov’s system of axioms may be surprising. Let
hence, we say that
And we say also that:
Moreover, we can generalize and say that for
And we say also that for
Now, we can add to this system of axioms an imaginary part such that:
We can see that by taking into consideration the set of imaginary probabilities we added three new and original axioms and consequently the system of axioms defined by Kolmogorov was hence expanded to encompass the set of imaginary numbers [45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61].
As a summary of the new paradigm, we declare that in the universe
The
In this section we will relate and link Newton’s mechanics to the complex probability paradigm with all its parameters by using four novel concepts which are: the real stochastic force
The real stochastic force is defined by:
Here
Since
If
If
The imaginary stochastic force is defined by:
Here
Since
If
If
We have:
And since
And we can deduce that:
Therefore,
We define the resultant complex stochastic force by:
Here
Since
If
If
Since
where
Additionally, since
where
We define the deterministic real force by:
Since from
Here
Furthermore,
Since
Since
Since
Since
If
If
The second case shows and proves that if
Additionally, since
And
Since
And since
Furthermore, according to
which is a second-degree equation in terms of
Since
But according to
And if
And if
Consequently,
But
The graphs of the reduced real force
We have
The graphs of the reduced real force
We have
The graphs of the reduced real force
We can deduce also from
The graphs of the reduced real force
And we can infer using the fact that
The graphs of the reduced real force
Also, we can calculate (Figure 8):
The graphs of the reduced real force
But according to
As we have computed:
And since
We have
The graphs of the reduced imaginary force
And we can deduce that (Figure 10):
The graphs of the reduced imaginary force
And we can infer that (Figure 11):
The graphs of the reduced imaginary force
We can deduce also that (Figure 12):
The graphs of the reduced imaginary force
And we can compute (Figure 13):
The graphs of the reduced imaginary force
And we can calculate (Figure 14):
The graphs of the reduced imaginary force
But according to
Analogously, and since
And
And
We can deduce also that:
And
And
But according to
The graphs of the reduced real force
And since the deterministic force in
In this cube (Figure 15), we can notice the simulation of the complex resultant reduced force
We have:
But from
According to
Since also
Since we have:
And
Furthermore, since
Hence, we can conclude that no chaos, no ignorance, no disorder, no unpredictability, no chance, and no randomness exist in the probability universe
A powerful tool will be described in the current section which was developed in my personal previous research papers and which is founded on the concept of a complex random vector that is a vector combining the real and the imaginary probabilities of a random particle, defined in the three added axioms of
First, let us consider the following general Bernoulli distribution and let us define its complex random vectors and their resultant (Table 1):
Outcome | |||
---|---|---|---|
In | |||
In | |||
In |
A general Bernoulli distribution in
Where,
We have:
and
Where
The complex random vector corresponding to the random outcome
The complex random vector corresponding to the random outcome
The resultant complex random vector is defined as follows:
The probability
This is coherent with the three novel complementary axioms defined for the
Similarly,
The probability
Where
By analogy, for the case of one random vector
In general, for the vector
Where the degree of our knowledge of the whole distribution is equal to
Notice, if
which is coherent with the calculations already done.
To illustrate the concept of the resultant complex random vector
The resultant complex random vector
As a general case, let us consider then this discrete probability distribution with
Outcome | |||||
---|---|---|---|---|---|
In | |||||
In | |||||
In |
A discrete uniform distribution with
We have here in
Moreover, we can notice that:
And
Where
Therefore, the degree of our knowledge corresponding to the resultant complex vector
and its relative chaotic factor is:
Similarly, its relative magnitude of the chaotic factor is:
Thus, we can verify that we have always:
What is important here is that we can notice the following fact. Take for example:
We can deduce mathematically using calculus that:
From the above, we can also deduce this conclusion:
As much as
We have:
Where
= the mean of all the
Therefore,
Additionally, we have:
= the mean real random force acting on the whole macrosystem in
Moreover,
= the mean imaginary random force acting on the whole macrosystem in
Furthermore,
= the mean resultant complex random force acting on the whole macrosystem in
Also, we have:
And
Correspondingly, we can deduce the following result:
Therefore, this means that in the first case the mean real force acting on the macrosystem in the real set
That means that the mean norm of the resultant force acting on the whole macrosystem is totally deterministic in both cases in the probability set
Similarly, we can deduce also the following similar result:
Therefore, this means that in the first case the mean real force acting on the macrosystem in the real set
That means that the mean norm of the resultant force acting on the whole macrosystem is totally deterministic in both cases in the probability set
In addition, for
This means that we have a random experiment with only one outcome or vector, hence,
The law of large numbers states that:
“As
We can deduce now the following conclusion related to the law of large numbers:
We can see, as we have proved, that as much as
Hence, what we have done here is that we have proved the law of large numbers (already discussed in the published papers [46, 50, 57, 61]) as well as an important property of statistical mechanics using
The following flowchart summarizes all the procedures of the proposed complex probability prognostic model where
In this section, the simulation of the novel
The probability density function (
Note that in the simulation we have considered:
The cumulative distribution function (
Note that:
If
If
The mean or average or expectation is:
The variance is:
The standard deviation is:
The median
Since the distribution is uniform then it has no mode.
The real probability
The imaginary complementary probability
The real complementary probability
The complex probability or random vector and force are:
The Degree of Our Knowledge:
The Chaotic Factor:
The Magnitude of the Chaotic Factor
At any value of
then,
Hence, the prediction of all the probabilities and forces of the stochastic experiment in the universe
The graphs of
In the first cube (Figure 20), the simulation of
The graphs of
In the second cube (Figure 21), we can notice the simulation of the real reduced force
The graphs of
In the third cube (Figure 22), we can notice the simulation of the complex resultant reduced force
The graphs of the reduced forces
The probability density function (
I have taken the domain for the binomial random variable to be:
Taking in our simulation
The mean of this binomial discrete random distribution is:
The standard deviation is:
The median is
The mode for this symmetric distribution is = 6 =
The cumulative distribution function (
Note that:
If
If
The real probability
The imaginary complementary probability
The real complementary probability
The complex probability or random vector and force are:
The Degree of Our Knowledge:
The Chaotic Factor:
The Magnitude of the Chaotic Factor
At any value of
then,
Hence, the prediction of all the probabilities and forces of the stochastic experiment in the universe
The graphs of
In the first cube (Figure 24), the simulation of
The graphs of
In the second cube (Figure 25), we can notice the simulation of the real reduced force
The graphs of
In the third cube (Figure 26), we can notice the simulation of the complex resultant reduced force
The graphs of the reduced forces
The probability density function (
For the Poisson discrete random variable:
I have taken in the simulation the domain for the Poisson random variable to be equal to:
The mean of this Poisson discrete random distribution is:
The standard deviation is:
The median
The mode is =
Since
The cumulative distribution function (
Note that:
If
If
The real probability
The imaginary complementary probability
The real complementary probability
The complex probability or random vector and force are:
The Degree of Our Knowledge:
The Chaotic Factor:
The Magnitude of the Chaotic Factor
At any value of
then,
Hence, the prediction of all the probabilities and forces of the stochastic experiment in the universe
The graphs of
In the first cube (Figure 28), the simulation of
The graphs of
In the second cube (Figure 29), we can notice the simulation of the real reduced force
The graphs of
In the third cube (Figure 30), we can notice the simulation of the complex resultant reduced force
The graphs of the reduced forces
The probability density function (
and the cumulative distribution function (
I have taken the domain for the continuous uniform random variable to be equal to:
Note that:
If
If
The mean of this continuous uniform random distribution is:
The variance is:
The standard deviation is:
The median is
Since the distribution is uniform then it has no mode.
The real probability
The imaginary complementary probability
The real complementary probability
The complex probability or random vector and force are:
The Degree of Our Knowledge:
The Chaotic Factor:
The Magnitude of the Chaotic Factor
At any value of
then,
Hence, the prediction of all the probabilities and forces of the stochastic experiment in the universe
The graphs of
In the first cube (Figure 32), the simulation of
The graphs of
In the second cube (Figure 33), we can notice the simulation of the real reduced force
The graphs of
In the third cube (Figure 34), we can notice the simulation of the complex resultant reduced force
The graphs of the reduced forces
The probability density function (
and the cumulative distribution function (
The domain for this standard Gaussian normal variable is considered in the simulations to be equal to:
In the simulations, the mean of this standard normal random distribution is
The variance is
The standard deviation is
The median is
The mode for this symmetric distribution is = 0 =
The real probability
The imaginary complementary probability
The real complementary probability
The complex probability or random vector and force are:
The Degree of Our Knowledge:
The Chaotic Factor:
The Magnitude of the Chaotic Factor
At any value of
then,
Hence, the prediction of all the probabilities and forces of the stochastic experiment in the universe
The graphs of
In the first cube (Figure 36), the simulation of
The graphs of
In the second cube (Figure 37), we can notice the simulation of the real reduced force
The graphs of
In the third cube (Figure 38), we can notice the simulation of the complex resultant reduced force
The graphs of the reduced forces
In the current research work, the original extended model of eight axioms (
Additionally, as it was proved and verified in the novel model, before the beginning of the random phenomenon simulation and at its end we have the chaotic factor (
In addition, referring to all these obtained graphs and executed simulations throughout the whole research work, we are able to quantify and to visualize both the system chaos and stochastic effects and influences (expressed and materialized by
Moreover, it is important to mention here that one very well-known and important random distribution was considered in the current work which is the discrete and uniform random distribution that was used to prove an important and essential result at the foundation of statistical mechanics and physics, knowing that the novel
As a future and prospective research and challenges, we aim to more develop the novel prognostic paradigm conceived and to implement it to a large set of random and nondeterministic events like for other probabilistic phenomena as in stochastic processes and in the classical theory of probability. Additionally, we will apply
The author declares that there are no conflicts of interest regarding the publication of this paper.
The data used to support the findings of this study are available from the author upon request.
R | real set of events |
M | imaginary set of events |
C | complex set of events |
i | the imaginary number where i=−1 or i2=−1 |
EKA | Extended Kolmogorov’s Axioms |
CPP | Complex Probability Paradigm |
Prob | probability of any event |
Pr | probability in the real set R = probability of the real random force in R |
Pm | probability in the imaginary set M corresponding to the real probability in R = probability of the imaginary random force in M |
Pc | probability of an event in R with its associated complementary event in M = probability of the real deterministic force in the complex probability set C |
F→r | the real stochastic force in R |
F→m | the imaginary stochastic force in M |
F→ | the resultant complex stochastic force in C |
F→c | the real deterministic force in C |
z | complex probability number = sum of Pr and Pm = complex random vector = probability of the resultant complex stochastic force in C |
DOK =z2 | the degree of our knowledge of the random system or experiment, it is the square of the norm of z |
Chf | the chaotic factor of z |
MChf | magnitude of the chaotic factor of z |
N | number of random vectors = number of random atoms or particles or molecules |
Z | the resultant complex random vector = ∑j=1Nzj |
DOKZ=Z2N2 | the degree of our knowledge of the whole stochastic system |
ChfZ=ChfN2 | the chaotic factor of the whole stochastic system |
MChfZ | magnitude of the chaotic factor of the whole stochastic system |
ZU | the resultant complex random vector corresponding to a uniform random distribution |
DOKZU | the degree of our knowledge of the whole stochastic system corresponding to a uniform random distribution |
ChfZU | the chaotic factor of the whole stochastic system corresponding to a uniform random distribution |
MChfZU | the magnitude of the chaotic factor of the whole stochastic system corresponding to a uniform random distribution |
PcZU | probability in the complex probability set C of the whole stochastic system corresponding to a uniform random distribution |