Statistical Model of College Ideological and Political Learning Based on Fractional Differential Equations

Based on fractional differential equations, this paper focuses on the internal mechanism of college ideological and political learning. We also elaborate on its microstructure, channel characteristics, and competitive field. We put forward Gauss's theorem and the loop theorem of the related field of ideological and political learning in colleges and universities. At the same time, the convexity theorem of the information entropy of the competitive field and the principle of maximum entropy are proved. Research shows that college students can change the relationship between student learning and society by adapting, functioning, and coordinating. We need to help students develop more effective political learning strategies


Introduction
With the intensified trend of global economic integration and the integration of political economy, enterprises have entered the era of political competition.Some scholars have studied the issue of political connections from a global perspective.They found that corporate political ties are widespread in many countries.Political connections are especially prominent in countries and regions with weak property rights protection.Political success in many industries is as important as an economic success [1].The business often has a political strategy as part of its overall strategy.Political connections have become the foundation of business success.Many enterprises pay attention to formulating and implementing political strategies while formulating market competition strategies to defeat their opponents.They access various government resources through the decision-making process that influences government policy.This can help improve business performance.The more a company has a sense of long-term development, the more emphasis will be placed on formulating and implementing political strategies.This has become an important part of strategic enterprise management.In the context of China's transition and relationship-oriented social structure, the political connections of private enterprises are widespread.Political connections constitute an important part of the external environment of Chinese private enterprises [2].A good relationship with the government is an important prerequisite for the success of some businesses.Through different channels, private enterprises have government resources to establish close ties with government departments at different levels.This can bring political and economic benefits.However, political ties can also increase costs for private firms and thus damage firm performance.It can be seen that political connection is a neutral concept.It can play a positive role in promoting the survival of private enterprises and have a negative impact.However, the research on political connections in the existing literature still focuses on qualitative analysis and quantitative research, failing to form a theoretical breakthrough.The current research lacks dynamic analysis on the inter-firm relationship formed under the influence of corporate political competitiveness and the flow of resources in the competitive space.And these are the problems that need to be solved urgently in the study of corporate political connections.This paper focuses on the concept of the internal mechanism, microstructure, channel characteristics, and competitive field of the politically connected competitiveness of private enterprises under the Chinese institutional environment [3].A dynamic theoretical framework of political competitiveness is established for the first time.The evolution law of political competitiveness is analyzed.This provides a new perspective for the study of corporate political connections.

The political connection network and competitive field of Chinese private enterprises
Politically connected competitiveness is an active process for private enterprises to gather effective political resources.It is also a process of organically combining subjective and objective conditions of private enterprises [4].The degree of government control over resources, the degree of government intervention in business, and the degree of policy uncertainty directly affect the dependence of private enterprises on the government.This process takes the enterprise as the source of competition.It exerts a certain gravitational force on government agencies and their officials at different positions in social cyberspace.So we say that political competitiveness is defined in this social network space.If the competitiveness studied does not change over time, we call the field a static field.If it changes over time, we call it a dynamic field or a time-varying field.If the competitiveness understudy is scalar, we call the field a scalar field.If the competitiveness under study is a vector, we call the field a vector field.

Scalar playing field
A scalar politically connected arena can be denoted ( 1,2, , ) i x i n   denote the n dimensions of the competition space.In the scalar competition field, we make the points where the political relation competitiveness scalar achieves the same value constitute a space surface [5].This can be the isosurface of the scalar playing field.The isosurface equation can be expressed as 1 2 ( , , , ) . where C is a constant.There are differences in political connections between different enterprises and different competitiveness.This creates a politically connected arena.
( 1,2, , ) denotes the components of the competing field of different dimensions.The flux in the vector competition field can be expressed as Where S is the directed surface in the competition space, on its positive side, the normal unit vector is N.At any point M in the vector competition field F, draw an arbitrary closed surface S surrounding the point [6].When the volume V  defined by S approaches zero in any way, if the following limit exists, then we call the vector competition field F at Divergence at point M and denoted divF : divF represents the net flux through a unit volume at M in the competitive vector field.So divF describes the density of the flux source [7].The divergence of the vector competitive field F is a scalar field.Usually, we call divF the divergence field generated by the vector field F. The integral of the divergence F  of the vector competition field F over the volume V is equal to the area integral of the vector competition field F over the closed surface S bounding the volume: The curve integral of the political competitiveness vector field F along a space closed path C in the field is expressed as follows: We call this the circular volume of the vector playing field.dl is the path element vector with size dl .The direction is the tangent along path C. If the circulation of a vector competition field is not equal to zero, we usually think that a vortex source produces the vector competition field in the competition field.
The annular areal density is defined as . We take a point in the vector competition field as the face element s  .The curl of the vector competition field F at point M is defined as follows: the direction is along the normal direction of the surface element that maximizes the circular surface density, and the magnitude is equal to the vector of the maximum value of the circular surface density.Let's call it rotF : The curl of the vector field F is still a vector.So we call rotF the curl competition field generated by the vector competition field F. The curl of the vector competition field F at point M is the vortex source density.The circumferential areal density of the vector competition field F at point M along direction N. It is equal to the projection of rotF in that direction:

Elements and Distribution of Political Connections of Private Enterprises
We use the symbol q to represent the constituent elements of the politically connected competitiveness of private enterprises [8].According to the specific situation of the political connection distribution of private enterprises, the political connection density can be used to describe their spatial distribution.We set the total amount of politically related elements distributed in the spatial volume element V  as q  , and the density ρ of politically related elements is defined as 0 ( ) lim is a function of spatial location r. ( ) r  describes the density of politically relevant elements at any point in space and describes the spatial distribution of politically relevant elements as a whole.It constitutes a scalar field.We can use ( ) r  to find the total element ( ) of political connection in a certain space volume V .A very important concept must be raised in studying political competitiveness: the point element [9].When the total amount of politically related elements q is concentrated on a geometric point whose spatial volume is zero, this geometric point becomes a point element q.The volume density ( ) r  of the point element q located at the origin of the coordinates has the following properties: We can use the  function to represent the volume density of point features.We assume that the point element q is located at r′ , and the density ( ) r  of the politically related element body at any point r in the space can be expressed by the  function as: The total political connection factor Q in any volume V of the competition space can be given by: 0 ( ) ( ) When the point feature q is located at the origin of the spatial coordinate 0 r′ = , the volume density of the spatial point feature can be expressed as ( ) ( ) r q r    . If the space has N point features qi and they are located in ( 1, 2, , ) , the point feature density of the space can be expressed as 1 ( ) ( )

Gauss's theorem and loop theorem of the political competitiveness of private enterprises
Suppose two business entities have the basic characteristics of point elements.The interaction force between two stationary enterprise entities (point elements) 1 q and 2 q in this space can be defined as  ( , , , ) 4 Where 12 F represents the force of business entity 1 q acting on a business entity 2 q .
12 2 1 F r r   represents the distance vector of business entity 1 q to 2 q .12 12 is called the space constant.It represents the environmental parameters of the competition space.
When a point element 0 q is placed at a certain point in space, the competitiveness of the point element is related to the amount of the elements it brings and the competition intensity of the point [10].That is, there is a relation . In the formula, the vector E is the strength vector of the political-related competition field.The strength of the political competition field stimulated by point elements at any point in the competition space can be defined as: The intensity of the political competition field stimulated by any point in the competition space of the system composed of N point elements is is the position vector of the i point feature.The intensity of the political competition field generated by the continuously distributed element system (body element) is By assuming that the features are distributed in region V, we can obtain E G    .This is the differential form of Gauss's theorem, and it is also an important basic equation of the politically connected field of competition in the competitive space.It reflects the relationship between the divergence of the spatial competitiveness of political associations and the distribution of spatial elements [11].We integrate Gauss's theorem over the volume V of the competing space and use the Gaussian divergence theorem to have 0 E    .This is the second fundamental equation of the political arena, also called the loop theorem of the politically connected arena.Similarly, we arbitrarily take a surface S in the competition space, and its boundary is C.According to Stokes's formula we can integrate the loop theorem over S to get 0 . This is the integral form of the static playing field loop theorem.It shows that the integral of the competitive field along any closed loop is zero.

The polarization of the space medium of politically connected competition of private enterprises
Suppose there is an element system consisting of two-point elements q  with equal and opposite signs separated by l.We call it an element dipole.This is a simple and special element system.The element properties of element dipoles are represented by element dipole moments p ql  .This is a vector.For example, the relationship between small and medium- sized private enterprises and the government can be called a factor dipole.The consistency of the orientation of elements with extreme political relevance reflects the degree of medium polarization.We describe it by the polarization vector P , equal to the element dipole moment in a unit volume: . where i p is the element dipole moment of the ith element in the small volume 1 V  .After the medium is polarized, there will be a macroscopic distribution of elements related to competition on its surface and inside [12].The distribution of polarized elements is represented by the volumetric density P  of polarized elements or the surface density of polarized elements SP  . They are related to the degree of polarization of the medium.Take an arbitrary volume V in space.It has n element dipoles inside it.Its boundary is S. The positive element that passes through dS and exits dV is nql dS np dS P dS      .Then the positive element passing through S to V is . Negative elements should stay in V from the neutrality of the medium.These elements that are polarized due to the influence of the competitive field are the polarized elements qP , and are: Using the Gaussian divergence theorem, the above formula can be transformed into the differential form P P     .This will be a very important result.It reflects the relationship between the polarization intensity at any point in the competition space and the distribution of polarization elements.At this point, we can deduce . This formula is the boundary condition satisfied by the polarization vector P .When the elements in the medium are polarized, the political competitiveness effect produced by the polarized elements in the medium is the same as the competition field produced by the main economic object of the competition field.Polarization factors also act as sources of divergence to stimulate the playing field [13].The agglomeration effect of politically connected media can expand the intensity of the politically connected competitive field.Thus there is an , which is the differential form and integral form of Gauss's theorem of static competition field in a medium.It will be the basic equation of the static playing field.Where D is called the element displacement vector or competitive induction strength.It is an additional amount.The element displacement vector D forms an active competition field in the competition space.The free element is its divergence source.

Political connection competition potential of private enterprises and their differential equations
The static playing field is the active non-rotating playing field 0, E D       .Due to the rotation of the static playing field, we introduce a scalar function to describe the static playing field strength E    .The function  is called the competitive potential function.The competitive potential function  is not uniquely determined.The two can differ by an arbitrary constant.Therefore, the competitive potential of each point in the politically connected arena has a definite value.We need to determine a reference point and specify that the competitive potential for this point is zero.At this point we can get 2 G     .This is the Poisson equation satisfied by the competing potential quasi.In the competitive space without free element distribution, the competitive potential satisfies the Laplace equation 2 0    due to 0   .

The competitive field of political connection of private enterprises
Politically Connected Competitive Fields Field energies exist in any area of the competitive space.The ability of any business entity to utilize and integrate political resources requires the maintenance of energy.The energy density of a static competitive field in a medium can be expressed as .This formula is suitable for a static field of play but not for a time-varying field.The interaction energy of N point feature systems can be expressed as: Assume that the factor distribution density, the competition potential, and the competition field intensity of the two private enterprises are , , , 1, 2 , respectively.According to the superposition principle of the competition field, the intensity of the competition field of the politically related competition space is 1 2 , then the total energy of the competition field is The total energy of the competitive field is equal to the own energy of each private enterprise plus the interaction energy.Factors and competitive potential as can also express the interaction energy between private enterprises From the above formula, the energy of private enterprises in the external competition field can be obtained as . The competitiveness between competitors can be calculated by formula according to the energy change.

Information entropy of the politically connected competition field of private enterprises
Assuming that there are N private enterprises in the competitive field, the total energy is 1 1 2 represents the own energy of the i enterprise entity and the interaction energy with other private enterprises.If let is complete and non-negativity.It represents the private enterprise's energy share and interaction energy with other private enterprises in the total energy.
( 1, 2, , ) describes the distribution of the competitive field energy, which we call the density distribution function of the competitive field energy.
We define the information entropy of the political competition field as where h is a positive constant.

Convexity Theorem and Maximum Entropy Principle of Information Entropy in Politically Related Competition Field
(1) Convexity theorem of information entropy in the political connection competition.The set R corresponding to the density distribution function of the competition field energy of political connection is convex.The information entropy 1 ln of the competitive field is a convex function on the convex set R .
(2) The principle of maximum entropy.The information entropy of the political-related competition field is an upward convex function with a maximum value.
The information entropy E of the political-related competition field is an upward convex function with a single peak.The energy density distribution function 1 ( 1, 2, , ) presents a discrete uniform distribution if and only if the field energy densities of each private enterprise are equal to each other.At this time, the information entropy E of the competitive field reaches a maximum value.The concept of information entropy in the politically related competition field is beneficial to analyzing the political competitiveness and competitive behavior of private enterprises.Once the concept of entropy is established, it can directly use the currently known theories related to entropy to study the political competitiveness of private enterprises.

Conclusion
China's private enterprises are not only operating in a complex and changing political and economic environment but also in the process of transitioning from a traditional planned economy to a socialist market economy with Chinese characteristics.There is still a certain distance from the mature market economy environment regarding market development and regulation.The government is still interfering with private companies at many levels.So private enterprises face more dynamic political changes.In a relationship-dominated environment, private enterprises have a strong incentive to establish political connections to enhance their competitiveness in utilizing and integrating various resources.They use political power and government resources to promote business development.We have established a dynamic analysis framework for the political competitiveness of private enterprises.
T h i s p a g e i s i n t e n t i o n a l l y l e f t b l a n k this formula in the entire space of the politically related competition field.This formula is suitable for both static and time-varying politically connected fields.The total energy of the static competition field can be represented by the competition potential