In the real world, we are enlightened from some concepts and phenomena such as “biosphere”, “food chain”, “ecological balance” etc. With research and practice, by using the theory of multilateral matrices [
Let V be a non-empty set and 
. The non-empty subset 
is called a relation of V. Image mathematics mainly researches general relation rules for general V rather than for special V. So the general V cannot be supposed. We can only do matter is to research the structure of the set of relations
.
For any relation set
, we can define both an inverse relationship of 
and a relation multiplication between 
and 
as follows:
and 
there is at least a 
such that 
and
.
The relation 
is called reasonable if
. A generalized reasoning of general V is defined as for 
there is a relation 
such that 
.
The generalized reasoning satisfies the associative law of reasoning, i.e.,
. This is the basic requirement of reasoning in TCMath. But there are a lot of reasoning forms which do not satisfy the associative law of reasoning in Western Science. For example, in true and false binary of proposition logic, the associative law does not hold on its reasoning because
Let V be a non empty set and 
be its a relation. We call it an equivalence relation, denoted by ~, if the following three conditions are all true:
1) Reflexive: 
for all
, i.e.,
;
2) Symmetric: if
, then
, i.e., if
, then
;
3) Conveyable (Transitivity): if
, 
, then
, i.e., if
, 
, then 
Furthermore, the relation R is called a compatibility relation if there is a non-empty subset 
such that 
satisfies at least one of the conditions above. And the relation R is called a non-compatibility relation if there doesn’t exist any non-empty subset 
such that 
satisfies any one of the conditions above. Any one of compatibility relations can be expanded into an equivalent relation to some extent [
Western Science only considers the reasoning under one Axiom system such that only compatibility relation reasoning is researched. However there are many Axiom systems in Nature. Traditional Chinese Science mainly researches the reasoning among many Axiom systems in Nature. Of course, she also considers the reasoning under one Axiom system but she only expands the reasoning as the equivalence relation reasoning.
Equivalence relations, even compatibility relations, can not portray the structure of the complex systems clearly. In the following, we consider two non-compatibility relations.
In image mathematics, any Axiom system is not considered, but should first consider to use a logic system. Believe that the rules of Heaven and the behavior of Human can follow the same logic system (天人合一). This logic system is equivalent to a group of computation. The method is to abide by the selected logic system to the research object classification, without considering the specific content of the research object, namely classification taking images (比类取象). Analysis of the relationship between research object, make relationships with computational reasoning comply with the selected logic system operation. And then in considering the research object of the specific content of the conditions, according to the logic of the selected system operation to solve specific problems. In mathematics, the method of classification taking images is explained in the following Definition 2.1.
Definition 2.1. Suppose that there exists a finite group 
of order m where g0 is identity. Let V be a none empty set satisfying that 
where the notation means that
, 
(the following the same).
In image mathematics, the 
is first called a factor image of group element 
for any j, and 
is called a factor space (all “Gua (卦)”). We do not consider the factor size (class variables) and only consider it as mathematical symbols (“Gua (卦)”), such as, 0 or 1, because the size is defined by a human behavior for V, but we have no assumption of V.
A mathematical index of the unknown multivariate function
, 
is called a function image of V. All mathematical indexes of the unknown multivariate function f compose of the formation of a new set, namely image space
where 
is also a finite group of order g. The 
is also called an Axiom system for any j if 
because any Axiom system is the assumption of
(or, equivalent, V) in which there is only the compatibility relations, i.e., pursuing the same mathematics index
. We do not consider the special multivariate function f (i.e., special function image) and only consider the calculation way of the general mathematical indexes of f from the factor space 
in order to know some causal relations, because we have no assumption of f. But the size of the data image should be considered if we study specific issues by the general rules of data images.
Say simply, a study of the hexagrams (“Gua (卦)”) in image mathematics is to learn the generalized properties of the inputs 
of any multivariate function f for the given factor space
, such as there are non-size, non-order relation, orthogonal relations and symmetrical relations, and so on. A study of the image (“Xiang (象)”) in image mathematics is to learn the generalized properties of all outputs f for the image space
, such as there are size specific meaning, a sequence of relationship, killing relations, loving relations, equivalent relations, and so on.
Without loss of generality, we put the function image space 
and the factor image space
, still keep for V because of no assumption of V. In order to study the generalized relations and generalized reasoning, image mathematics researches the following relations.
Denoted
, where the note × is the usual Cartesian product or cross join. Define relations
where 
is called an equivalence relation of V if g0 is identity; denoted by ~; 
is called a symmetrical relation of V if
,
; denoted by 
or
; 
is called a neighboring relation of V if
; denoted by 
or
; 
is called an alternate (or atavism) relation of V if
, 
, 
,
; denoted by 
or
. #
In this case, the equivalence relations and symmetrical relations are compatibility relations but both neighboring relations and alternate relations are non-compatibility relations. For the given relation set
, these relations 
are all reasoning relations since the relation 
if
.
The equivalence relation
, symmetrical relations
, neighboring relation 
and alternate relations 
are all the possible relations for the method of classification taking images. In this paper, we mainly consider the equivalence relation
, neighboring relation 
and alternate relations
.
There is an unique generalized reasoning between the two kinds of opposite non-compatibility relations for case m = 5. For example, let V be a none empty set, there are two kinds of opposite relations: the neighboring relation
, denoted ® and the alternate (or atavism) relation
, denoted
. The logic reasoning architecture [2-19] of “Yin Yang Wu Xing” Theory in Ancient China is equivalent to the following reasoning:
1) If x ® y, y ® z, then
; i.e., if
, 
, then
; or,
; 
if x ® y, 
, then y ® z; i.e., if
, 
, then
; or,
; 
if
, y ® z, then x ® y; i.e., if
, 
, then
; or,
.
2) If
, 
then z ® x; i.e., if
, 
, then
; or,
; 
if z ® x, 
, then
; i.e., if
, 
then
; or,
; 
if
, z ® x then
; i.e., if
, 
, then
; or,
.
Let 
and
. Then above reasoning is equivalent to the calculating as follows:
where the 
is the addition of module 5.
Two kinds of opposite relations can not be exist separately. Such reasoning can be expressed in
Let V be a none empty set with the equivalent relation
, the neighboring relation 
and the alternate relations
,
. Then a genetic reasoning is defined as follows:
1) If x ~ y, y ® z, then x ® z; i.e., if
, 
, then
; or,
;
2) If x ~ y, 
, then
; i.e., if
, 
, then
; or,
;
3) If x ® y, y ~ z, then x ® z; i.e., if
, 
, then
; or,
;
4) If
, y ~ z, then
; i.e., if
, 
, then
; or,
.
The genetic reasoning is equivalent to that
since
,
. The genetic reasoning is equivalent to that there is a group 
with the operation * such that V can be cut into 
where Vi may be an empty set and the corresponding relations of reasoning can be written as the forms as follows
satisfying
.
For a none empty set V and its a relation set 
, the form 
(or simply, V) is called a multilateral system [2-19], if 
satisfies the following properties:
1) 
i.e.
.
2) 
3) The relation
if
.
4) For
, there is a relation 
such that
.
The 4) is called the reasoning, the 1) the uniqueness of reasoning, the 2) the hereditary of reasoning (or genetic reasoning) and the 3) the equivalent property of reasoning of both relations 
and
, i.e., the reasoning of 
is equivalent to the reasoning of
. In this case, the two-relation set 
is a lateral relation of
. The 
is called an equivalence relation. The multilateral system 
can be written as
. Furthermore, the V and 
are called the state space and relation set considered of
, respectively. For a multilateral system
, it is called complete (or, perfect) if “
” changes into “=”. And it is called complex if there exists at least a non-compatibility relation
. In this case, the multilateral system is also called a logic analysis model of complex systems.
Let 
be a non-compatibility relation. A complex multilateral system
is said as a steady multilateral system (or, a stable multilateral system) if there exists a number n such that 
where
. The condition is equivalent to there is a the chain 
such that
, or
. The steady multilateral system is equivalent to the complete multilateral system. The stability definition given above, for a relatively stable system, is most essential. If there is not the chain or circle, then there will be some elements without causes or some elements without results in a system. Thus, this system is to be in the state of finding its results or causes, i.e., this system will fall into an unstable state, and there is not any stability to say.
Theorem 2.1. The system 
is a multilateral system if and only if there exists a finite group
of order m where g0 is identity such that the relation set 
satisfying 
.
In this case, the multilateral system 
can be written as 
satisfying 
where 
may be an empty set.
Theorem 2.2. If the multilateral system
is a steady multilateral system, then 
and 
is a finite group of order m about the relation multiplication 
where 
must be a non empty set.
Definition 2.2. Suppose that a multilateral system 
can be written as 
satisfying 
and 
.
The group 
of order m where g0 is identity is called the representation group of the multilateral system
. The representing function of 
is defined as follows
.
Let multilateral systems 
be with two representation groups
, 
, respectively. Both multilateral systems 
are called isomorphic if the two representation groups
, 
are isomorphic.
Theorems 2.1 and 2.2 and Definitions 2.1 and 2.2 are key concepts in multilateral system theory because they show the classification taking images as the basic method. In the following, introduce two basic models to illustrate the method.
Theorem 2.3 Suppose that 
with multiplication table
i.e., the multiplication of 
is the addition of module 2. In other words,
.
And assume that 
satisfying
.
Then 
is a steady multilateral system with one equivalent relation 
and one symmetrical relation 
which is a simple system since there is not any noncompatibility relation. In other words, the relations
’s are the simple forms as follows:
, 
where (i, j) is corresponding to
.
It will be proved that the steady multilateral system in Theorem 2.3 is the reasoning model of “Tao” (道) in TCMath if there are two energy functions 
and 
satisfying
, called Dao model, denoted by
.
Theorem 2.4. For each element x in a steady multilateral system V with two non-compatibility relations, there exist five equivalence classes below:
which the five equivalence classes have relations in
It can be proved that the steady multilateral system in Theorem 2.4 is the reasoning model of “Yin Yang Wu Xing” in TCMath if there are five energy functions (Defined in Section 3) 
and 
satisfying 
called Wu-Xing model, denoted by
.
By Definition 2.2, the Wu-Xing model 
can be written as follows:
Define
, 
, 
, 
, 
, corresponding to wood, fire, soil, metal, water, and assume 
and 
satisfying
i.e., the relation multiplication of 
is isomorphic to the addition of module 5. Then 
is a steady multilateral system with one equivalent relation 
and two non-compatibility relations 
and
.
These Theorems can been found in [1-6,11-16].
which makes every subsystem meaningful of the multilateral system.
(or
, or
, or
) means
, 
, 
(or
, 
, 
, or
, 
, 
, or
, 
,
).
,
. Let
, 
and 
be the energy function of Vi, the energy function of Vj and the total energy of both Vi and Vj, respectively.
, if 
(the normal state of the energy of
), then the neighboring relation 
is called that Vi likes Vj which means that Vi is similar to Vj. In this case, the Vi is also called the brother of Vj while the Vj is also called the brother of Vi. In the causal model, the Vi is called the similar family member of Vj while the Vj is also called the similar family member of Vi. There are not any causal relation considered between Vi and Vj.
, if 
) where the 
is an interaction of 
and 
satisfying 
, then the symmetrical relation 
is called that Vi is corresponding to Vj which means that Vi is positively (or non-negatively) corresponding to Vj if 
(or
) and that Vi is negatively corresponding to Vj if
. In this case, the Vi is also called the counterpart of Vj while the Vj is also called the counterpart of Vi. In the causal model, the Vi is called the reciprocal causation of Vj while the Vj is also called the reciprocal causation of Vi. There is a reciprocal causation relation considered between Vi and Vj.
, if 
), then the neighboring relation 
is called that Vi bears (or loves) Vj [or that Vj is born by (or is loved by) Vj] which means that Vi is beneficial on Vj each other. In this case, the Vi is called the mother of Vj while the Vj is called the son of Vi. In the causal model, the Vi is called the beneficial cause of Vj while the Vj is called the beneficial effect of Vi.
, if 
), then the alternate relation 
is called as that Vi kills (or hates) Vj [or that Vj is killed by (or is hated by) Vi] which means that Vi is harmful on Vj each other. In this case, the Vi is called the bane of Vj while the Vj is called the prisoner of Vi. In the causal model, the Vi is called the harmful cause of Vj while the Vj is called the harmful effect of Vi.
change increment
, then the energies 
of other subsystems XS, XK, KX, SX (defined in Theorem 2.4) of V will be changed by the increments
, 
, 
and
, respectively.
of the son XS of X is greater than the energy increment 
of the mother SX of X, i.e., the best beneficiary is the son XS of X. But the energy decrease degree 
of the prisoner XK of X is greater than the energy decrease degree 
of the bane of X, i.e., the worst victim is the prisoner XK of X.
of the mother SX of X is greater than the energy decrease degree 
of the son of X, i.e., the best beneficiary is the mother SX of X. But the energy increment 
of the bane KX of X is greater than the energy increment 
of the prisoner XK of X, i.e., the worst victim is the bane KX of X.
, then
, 
, 
,
;
, then
, 
, 
,
;
. Both 
and 
are called intervention reaction coefficients, which are used to represent the capability of intervention reaction. The larger 
and
, the better the capability of intervention reaction. The state 
is the best state but the state 
is the worst state.
and 
are decreasing functions of the intervention force Δ since the intervention force 
is easily to transfer all if Δ is small but the intervention force is not easily to transfer all if Δ is large. The energy function of complex system, the stronger the more you use. In order to magnify 
and
, should set up a mathematical complex system of the intervention reaction system, and often use it.
and 
is too small. In the state
, any mathematical intervening resistance problem is non-existence but in the state
, mathematical intervening resistance problem is always existence. At this point, the paper advocates the essential principle of intervening to avoid mathematical intervening resistance problems.
changed by increment 
such that the energies 
of other subsystems XS, XK, KX, SX (defined in Theorem 2.4) of V will be changed by the increments
, 
, 
, 
, respectively, then can the multilateral system V has capability to protect the worst victim to restore?
, the worst victim is the prisoner XK of X which has the increment
. Thus the intervening principle of self-protection is to restore the prisoner XK of X and the restoring method of self-protection is to increase the energy 
of the prisoner XK of X by using the intervention force on X according to the intervention rule. In general, the increase degree is 
where
.
, the worst victim is the bane KX of X which has the increment
. Thus the intervening principle of self-protection is to restore the bane KX of X and the restoring method of self-protection is to decrease the energy 
of the bane KX of X by using the same intervention force on X according to the intervention rule. In general, the decrease degree is 
where
.
, then the energy of subsystem XK will decrease the increment
, which is the worst victim. So the capability of self-protection increases the energy of subsystem XK by increment 
where
, in order to restore the worst victim by according to the intervention rule.
, then the energy of subsystem KX will increase the increment
, which is the worst victim. So the capability of self-protection decreases the energy of subsystem KX by increment 
where
, in order to restore the worst victim by according to the intervention rule.
The 
is the intervenetion reaction coefficient. The 
is called an self-protection coefficient, which is used to represent the capability of self-protection. The larger
, the better the capability of self-protection. The state 
is the best state but the state 
is the worst state of self-protection. According to the general economy of the protection principle, 
should be not greater than 
since the purpose of protection is to restore the victims and not reward the victims.
is also a decreasing function of the intervention force Δ since the worst victim is easily to restore all if Δ is small but the worst victim is not easily to restore all if Δ is large. The energy function of complex system, the stronger the more you use. In order to magnify
, should set up an economic society of the selfprotection system, and often use it.
and capabilities of intervention reaction and self-protection is with intervention reaction coefficients 
and
, and with self-protection coefficient
. If the capability of self-protection wants to restore both subsystems XK and KX, then the following statements are true.
has been changed by increment
, then all five subsystems will be changed finally by the increments as follows:
has been changed by increment
, then all five subsystems will be changed by the increments as follows:
’s and
’s are the increments under the capability of self-protection.
and capabilities of intervention reaction and self-protection is with intervention reaction coefficients 
and
, and with self-protection coefficient
. Then the capability of self-protection can make both subsystems XK and KX to be restored at the same time, i.e., the capability of self-protection is better, if and only if 
and
.
and
. General for a complex system of mathematical complex system, the condition 
is easy to meet since it can restore two subsystems by Theorem 3.1, the condition 
is difficult to meet it only can restore one subsystem by Theorem 3.1. At this point, the paper advocates the principle to avoid any side effects of intervening.
and
, and with self-protection coefficient 
satisfying 
and
. Then the following statements are true.
increases the increment
, then the subsystems SX, XK and KX can be restored at the same time, but the subsystems X and XS will increase their energies by the increments
decreases, i.e., by the increment
, the subsystems XS, XK and KX can also be restored at the same time, and the subsystems X and XS will decrease their energies, i.e., by the increments
and capabilities of interveneing reaction and self-protection, assume intervention reaction coefficients are 
and
, and let the selfprotection coefficientbe
, which satisfy
, 
and 
where 
(the following the same) is the solution of
. Then the following statements are true.
has been changed by increment
, then the final increment 
of the energy 
of the subsystem XS changed is greater than or equal to the final increment 
of the energy 
of the subsystem X changed based on the capability of self-protection.
has been changed by increment
, then the final increment 
of the energy 
of the subsystem SX changed is less than or equal to the final increment 
of the energy 
of the subsystem X changed based on the capability of self-protection.
and capabilities of intervening reaction and self-protection, assume intervention reaction coefficients are 
and
, and let the self-protection coefficient be
, which satisfy
, 
and
. Then the following statements are true.
has been changed by increment
, then the final increment 
of the energy 
of the subsystem XS changed is less than the final increment 
of the energy 
of the subsystem X changed based on the capability of self-protection.
has been changed by increment
, then the final increment 
of the energy 
of the subsystem SX changed is greater than the final increment 
of the energy 
of the subsystem X changed based on the capability of self-protection. #
, 
and 
since in this time,
.
and capabilities of intervention reaction and self-protection is with intervention reaction coefficients 
and
, and with self-protection coefficient
. Then the following statements are true.
has been changed by increment
, and at the same time, another intervention force on the subsystem XK of steady multilateral system V is also implemented such that its energy 
has been changed by increment
, then all other subsystems: SX, KX and XS can be restored at the same time, and the subsystems X and XK will increase and decrease their energies by the same size but the direction opposite, i.e., by the increments
has been changed by increment
, and at the same time, another intervention force on the subsystem KX of steady multilateral system V is also implemented such that its energy 
has been changed by increment
, then all other subsystems: SX, XK and XS can be restored at the same time, and the subsystems X and KX will decrease and increase their energies by the same size but the direction opposite, i.e., by the increments
, 
and 
since
.
as harmful, conflict, ruinous, killing, etc., then the above defined stable logic analysis model is similar to the logic architecture of reasoning of “Yin Yang Wu Xing”. Both “Yin” and “Yang” mean that there are two opposite relations in the world: harmony or loving ® and conflict or killing
, as well as a general equivalent category ~. There is only one of three relations ~, ® and 
between every two objects. Everything X makes something (
), and is made by something (
); Everything restrains something (
), and is restrained by something (
); i.e., one thing overcomes another thing and one thing is overcome by another thing. The ever changing world V, following the relations: ~, ® and
, must be divided into five categories by the equivalent relation ~, being called “Wu Xing”: wood (X), fire (XS), soil (XK), metal (KX), water (SX). The “Wu Xing” is to be “neighbor is friend”: wood (X) ® fire (XS) ® soil (XK) ® metal (KX) ® water (SX) ® wood (X), and “alternate is foe”: wood (X) 
soil (XK) 
water (SX) 
fire (XS) 
metal (KX) 
wood (X). In other words, the ever changing world must be divided into five categories:
and output y can be written as
, 
and 
are not known, but also the input variables 
are not known. The problem is called model-free.
are called controllable if they are observed and controlled by human. So
can be choose.
is called uncontrolled if it is not observed or controlled by human. So the freedom model error
as independent random variables with continuous distributions
, respectively, since the inputs 
are controllable which can be selected independently by human under some similar conditions. If not independent, by factor analysis can select orthogonal factor. This operation of experiment under some similar conditions is equivalent to the uncontrolled input 
is independent of the controllable input 
since 
This operation of experiment that the inputs 
can be selected independently by human is equivalent to that the inputs 
are independent random variables one another. For example, take 
based on an orthogonal array since the orthogonality is equivalent to independence for discrete random variables and continuous random variables can be in a discrete random variable approximation [
Assume
. Then the inputs 
are independent random variables with continuous distributions
. In this case, the function
as independent random variables with continuous distributions
.
and testing or experimental tolerance 
for the observed function
are independent random variables and
.
#
is called Distortion Degree. It belongs to the wood (X) subsystem of the complex system f since it cognizes the structure function g of the complex system f which is the beginning or birth stage of all things, just like in the Spring of a year. In mathematics,
It is thought as the wood (X) image of generally complex system f since it is an objective constant independent of human observations, expressing birth, although the maximum and controllable experimental tolerance 
can be choose by human.
can be taken as the energy function of the subsystem wood (X) since the smaller the distortion degree
, the better the stability of the complex system f.
is unknown for freedom model speaking, it can be easily estimated by using orthogonal arrays
,
(see [
then 
where
.
, 
, we have
,
. Thus, we can obtained the estimation of the distortion degree 
as
.
smaller, more exact estimate.
is called Disturb Degree. It belongs to the fire (XS) subsystem of the complex system f since it controls the fluctuations of the complex system f which is the development and growth stage of all things, just like in the Summer of a year. In mathematics,
It is thought as the fire (XS) image of generally complex system f since it is also an objective constant independent of human observations, expressing growth, although the maximum and controllable experimental tolerance 
can be choose by human. The index 
can be taken as the energy function of the subsystem fire (XS) since the smaller the disturb degree
, the better the stability of the complex system f.
is unknown for freedom model speaking, it can be easily estimated by using experiment data 
of orthogonal arrays (see [
is the data standard variance
where
.
is called Information Decomposition Ratio (or Signal to Noise Ratio). It belongs to the soil (XK) subsystem of the complex system f since it makes the coordination of the center and fluctuation in the complex system f which is the continuous development and combined stage of all things, just like in the Long-Summer of a year. In mathematics,
It is thought as the soil (XK) image of generally complex system f since it is an objective constant independent of human observations, expressing combined, although the maximum and controllable experimental tolerance 
can be choose by human. The index 
can be taken as the energy function of the subsystem soil (XK) since the bigger the information decomposition ratio
, the better the stability of the complex system f.
is unknown for freedom model speaking, it can be easily estimated by using the experimental data 
of orthogonal arrays (see [
is
instead of
. Limited to data observation point of view, they are in the statistical meaning equivalent, but the former is more advantageous to the statistical analysis.
is called Deviation Degree. It belongs to the metal (KX) subsystem of the complex system f since it makes function characteristics (or data center and the expected goal deviation) in the complex system f which is the getting-results and accepted stage of all things, just like in the Autumn of a year. In mathematics,
It is thought as the metal (KX) image of a generally complex system f since it is an objective constant independent of human observations expressing accepted, although the controllable inputs 
of the observed function
can be taken as the energy function of the subsystem metal (KX) since the smaller the deviation degree
, the better the stability of the complex system f.
is unknown for freedom model speaking, it can be easily estimated by using the experimental data 
of orthogonal arrays (see [
is
is called Risk Function (or Risk, or Loss Function). It belongs to the water (SX) subsystem of the complex system f since it makes the expected value of loss 
between each data y and the goal t in the complex system f which is the risk and hiding stage of all things, just like in the winter of a year. The best condition is
, but generally this is very hard to achieve, because right now all the data are equal to the target t and the system f is a simple system
. In mathematics,
It is thought as the water (SX) image of generally complex system f since it is an objective constant independent of human observations, expressing hiding, although the observed function
can be taken as the energy function of the subsystem water (SX) since the smaller the risk
, the better the stability of the complex system f.
is unknown for freedom model speaking, it can be easily estimated by using the experimental data 
of orthogonal arrays (see [
is
based the unknown system function f is the set of mathematical indexes of the unknown system function f. By Theorem 2.4, the set V can be divided into five categories. Corresponding to every kind of five categories, each of mathematical indexes of the unknown system function f is called an image. Each of images must shows the complex system certain characteristics, such as, wood, fire, soil, metal, water.
becomes larger, then the loving force is large and the force of similar material accumulation of each other is also large. They can make the complex system more stable. If the killing force is small, i.e., 
becomes smaller, then the loving force is small and the force of similar material accumulation of each other is also small. They can make the complex system becoming unstable.
. In this time, the energy of the system will be the sum of energy of each part. Thus the mathematical system will be a simple mathematical system which is not what we consider range.
and 
near to 0. The state 
is the worst state of the mathematical complex system, namely mathematical intervenetion failure. On the way, the mathematical intervening resistance problem will be occurred since any mathematical intervening method is possible too little for some small 
and
.
, 
and
. In this case:
. It can be explained that if a multilateral system which has a poor capability of intervention reaction, then it is better to intervene the subsystem itself directly than indirectly. But similar to above, the intervention method is always to destroy the balance of multilateral systems such that there is at least one of side effects occurred. And the intervention method also have harmful to the capability of intervention reaction making the mathematical interveneing resistance problem also occurred by Theorem 3.2. Therefore the intervention method directly can be used in case 
but should be used as little as possible.
and 
near to 1. The state 
is the best state of the mathematical complex system. On the way, it is almost none mathematical intervening resistance problem since any mathematical intervening method is possible good for some large 
and
.
(or, income, water (SX)) and interfere 
(or, disturb, fire (XS)) as two mathematical indexes which have the killing relations for a complicated engineering problem based on the freedom model (3). They frequently used method is: If only the risk 
(i.e., water (SX) is not normal (or, the bigger, virtual disease), then they reduce deviation degree 
(i.e., increase the energy 
of metal (KX)) with linear optimization method since metal (KX) is the mother of water (SX); If only the interfere degree 
(i.e., fire (XS)) is not normal (or, the bigger, virtual disease), then they reduce distortion degree 
(i.e., increase the energy 
of wood (X)) of system function f with the method of adjusting stable center 
of the system function since wood (X) is the mother of fire (XS) (see [
satisfies
. In logic and practice, it’s reasonable 
near to 1 if the input and output in a complex system is balanced, since a mathematical output subsystem is absolutely necessary social other subsystems of all consumption. In case:
, all the energy for intervening mathematical complex subsystem can transmit to other mathematical complex subsystems which have neighboring relations or alternate relations with the intervening mathematical complex subsystem. The condition 
can be satisfied when 
and 
for a mathematical complex system since 
implies 
and
. If this assumptions is set upthen the intervening principle: “Real disease is to rush down his son and virtual disease is to fill his mother” based on “Yin Yang Wu Xing” Theory in image mathematics, is quite reasonable. But, in general, the ability of self-protection often is insufficient for an usual mathematical complex system, i.e., 
is small. A common standard is
, i.e., there is a principle which all losses are bear in mathematical complex system. Thus the general condition often is 
. Interestingly, they near to the golden numbers.
, 
and
, if one wants to treat the abnormal subsystems X and XS, then 1) For mathematical virtual disease, the one should intervene subsystem X directly by increasing its energy. It means “mathematical virtual disease is to fill his mother” because the mathematical virtual disease causal is X;
(or, risk, water (SX)) and quality 
(or, deviation degree, metal (KX)) as two mathematical indexes which have the loving relation for a complicated economical problem based on the freedom model (3). They frequently used method is: If both cost and quality are all not normal (or, virtual diseases, i.e., high cost and poor quality), then they reduce deviation degree 
(i.e., improve quality, increase the energy 
of metal (KX)) with linear optimization method since metal (KX) is the mother of water (SX) (see [
and 
greater and near to 1. The state 
is the best state of the mathematical complex system. On the way, it almost have none mathematical intervening resistance problem since any mathematical intervening method is possible good for some large 
and
.
, 
and
, if one wants to treat the abnormal subsystems X and XK, the one should intervene subsystem X directly by increasing its energy, and at the same, intervene subsystem XK directly by decreasing its energy. It means that “Strong inhibition of the same time, support the weak”.
(or social benefits, water (SX)) and social equity extent 
(or, information decomposition ratio, soil (XK)) with two mathematical killing indexes for a complicated social system based on the freedom model (3). If both social benefit 
and social equity extent 
are not normal, often appear serious problem is that social benefit 
is too good (or, 
too low, real disease) but social equity extent 
is too bad (or, 
too low, i.e., virtual disease).The disease is serious because the XK has been harmed by the SX with the method of incest such that the soil (XK) cannot kill water (SX). For the serious disease, many mathematical sociologists of China now frequently used method is: to increase social equity extent 
(or, to increase the information decomposition ratio 
which is the energy of the soil (XK)) but to decrease social and economic benefits 
(i.e., to decrease the energy of water (SX)) at the same time since soil (XK) is the bane of water (SX). The idea is “Strong inhibition of the same time, support the weak” if XK falls virtual disease and at the same time, SX befalls real disease.
is very good, the complex system is rife with average socialist, but both social benefit 
and social structure 
are all very poor. In addition, the mathematical complex system is not rich. In other words, both SX and X fall virtual diseases and at the same time, XK befalls real disease based on the freedom model (3). The disease is very serious because not only the wood (X) has been harmed by the soil (XK) with the method of incest such that wood (X) cannot kill soil (XK), but also there are 3 subsystems falling ill. Generally speaking, there are three or more than three of disease of the subsystem of a complex system, it is very difficult to cure. In order to cure the very serious disease, Deng Xiao-Ping’s taking method is to break the “iron bowl” and to develop the economy (to fill up the energies 
and 
of both wood (X) and water (SX) at the same time, i.e., strengthen social structure and social benefit), and to allow a few people to get rich (to rush down the energy 
of soil (XK), abate the coordinated ability). The idea is, at the same time, to use both “Strong inhibition of the same time, support the weak” and “Virtual disease is to fill his mother”, if both SX and X fall virtual diseases and at the same time, XK befalls real disease.
and 
greater and near to 1. The state 
is the best state of the steady multilateral system. On the way, it almost have none mathematical intervening resistance problem since any mathematical intervening method is possible good for some large 
and
.
, 
and
, according to the intervening principle of “Real disease is to rush down his son but virtual disease is to fill his mother”. The intervention method directly can be used in case
, 
and 
but should be used as little as possible.