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Register a new userGeneralized Potential and Mathematical Principles of Nonlinear Analysis
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Authors
Peng Yue
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In the past hundred years, chaos has always been a mystery to human beings, including the butterfly effect discovered in 1963 and the dissipative structure theory which won the chemistry Nobel Prize in 1977. So far, there is no quantitative mathematical-physical method to solve and analyze these problems. In this paper, the idea of using field theory to study nonlinear systems is put forward, and the concept of generalized potential is established mathematically. The physical essence of generalized potential promoting the development of nonlinear field is extended and the spatiotemporal evolution law of generalized potential is clarified. Then the spatiotemporal evolution law of conservative system and pure dissipative system is clarified. Acceleration field, conservative vector field and dissipation vector field are established to evaluate the degree of conservation and dissipation of physical field. Finally, the development route of new field research and the precondition of promoting engineering application in the future are discussed.
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In this paper, we consider the generalized stationary Stokes system with $p$-growth and Dini-$operatorname{BMO}$ regular coefficients. The main purpose is to establish pointwise estimates for the shear rate and the associated pressure to such Stokes system in terms of an unconventional nonlinear Havin-Mazya-Wolff type potential of the nonhomogeneous term in the plane. As a consequence, a symmetric gradient $L^{infty}$ estimate is obtained. Moreover, we derive potential estimates for the weak solution to the Stokes system without additional regularity assumptions on the coefficients in higher dimensional space.
A potential representation for the subset of traveling solutions of nonlinear dispersive evolution equations is introduced. The procedure involves a reduction of a third order partial differential equation to a first order ordinary differential equation. In this representation it can be shown that solitons and solutions with compact support only exist in systems with linear or quadratic dispersion, respectively. In particular, this article deals with so the called K(n,m) equations. It is shown that these equations can be classified according to a simple point transformation. As a result, all equations that allow for soliton solutions join the same equivalence class with the Korteweg-deVries equation being its representative.
We use the data of tenured and tenure-track faculty at ten public and private math departments of various tiered rankings in the United States, as a case study to demonstrate the statistical and mathematical relationships among several variables, e.g., the number of publications and citations, the rank of professorship and AMS fellow status. At first we do an exploratory data analysis of the math departments. Then various statistical tools, including regression, artificial neural network, and unsupervised learning, are applied and the results obtained from different methods are compared. We conclude that with more advanced models, it may be possible to design an automatic promotion algorithm that has the potential to be fairer, more efficient and more consistent than human approach.
This paper has been withdrawn by the authors because the paper is largely revised and improved, and to appear in Mechanics Research Communications.
In terms of the quantitative causal principle, this paper obtains a general variational principle, gives unified expressions of the general, Hamilton, Voss, H{o}lder, Maupertuis-Lagrange variational principles of integral style, the invariant quantities of the general, Voss, H{o}lder, Maupertuis-Lagrange variational principles are given, finally the Noether conservation charges of the general, Voss, H{o}lder, Maupertuis-Lagrange variational principles are deduced, and the intrinsic relations among the invariant quantities and the Noether conservation charges of the all integral variational principles are achieved.
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