 真诚  勤勉    达观  自然

# 宇宙分维构造及其数学基础

(西安现代非线性科学应用研究所, 西安 710061)

探讨了宇宙分维构造的形式，给出了分维微积分及分形测度的数学基础，包括分维导数及分维微积分的表述形式、分维微分方程的规整空间积分解、分形测度的分维微积分定义及自相似分形的测度计算方程。作为诠释，探讨了原子核内中子与质子的趋势关系方程，以及其周期解和原子序数极限值。

Fractal dimension structure of Cosmos and its mathematical foundations

YAN Kun  (yankun@nature.ac.cn)

(Xi’an Modern Nonlinear Science Applying Institute, Xi’an 710061, China)

Abstract  Fractal dimension structure of the Cosmos are explored, and the mathematical foundation, which include the expressions of fractal dimension differential and calculus, regular space integral solutions of fractal dimension differential equations, the fractal calculus definitions of fractal measure as well as the measure computational equation of self-similar fractal, fractal dimension calculus and fractal measure are given. As annotation, an equation of the relation between neutrons and protons in nuclei and its periodical solutions as well as atomic number limit are discussed.

Keywords
Cosmos, fractal dimension structure, fractal dimension calculus, fractal differential equation, fractal measure, atomic number limit

Yan Kun. Fractal dimension structure of Cosmos and its mathematical foundations[J]. Progress in Geophysics(in Chinese), 2004, 19(3): 709
716. DOI:10.3969/j.issn.1004-2903.2004.03.036.
http://en.cnki.com.cn/Article_en/CJFDTOTAL-DQWJ200403036.htm

# 关于超光速与量子分形的能量交换描述方法

(西安现代非线性科学应用研究所, 西安 710061)

Energy-exchange descriptions on the superluminal velocity and quantum fractal

YAN Kun  (yankun@nature.ac.cn)

(Xi’an Modern Nonlinear Science Applying Institute, Xi’an 710061, China)

Abstract  In this paper, consider taking the vacuum as a form of medium, by exploring the constancy of light velocity in the vacuum, the energy-exchange equation and the vortex shedding wavelength equation in Karman vortex-street, the expressional forms of equation of the one-way velocity of light (also equation of the one-way speed of light, or equation of one-way variable speed of light) for the constancy of the two-way velocity of light (or constancy of the two-way speed of light), the experimental criterion equation for the closed strings mode of the photon (the general refraction equation of the photon with a same rotation plane), superluminal (also faster-than-light or FTL) velocity, medium action equation of particle fractal motion for the wave-particle duality, and quantum fractal are studied deeply. As a result, it shows that a tentative theoretical frame which includes not only the superluminal-velocity motion but consists with Einstein special relativity and quantum theory can be established.

Keywords
equation of one-way velocity of light, general refraction equation, medium action equation, superluminal velocity, quantum fractal, energy exchange, particle fractal motion.

Yan Kun. Energy-exchange descriptions on the superluminal velocity and quantum fractal[J]. Basic Science Journal of Textile Universities(in Chinese), 2004, 17(3):223227.  DOI:10.3969/j.issn.1006-8341.2004.03.010.

# 天体运行的介质层壳与离散轨道引论

(西安现代非线性科学应用研究所, 西安 710061)

采用介质层壳弯曲的唯象方法，在规整三维空间中给出了能量方程及物体间的能量引力形式表述，其引力方程的二个条件解分别与Newton引力理论及Einstein引力理论的有关结果相近。讨论了目前分维微积分在函数方面的局限性，给出了相似扩展方程，随后通过讨论天体运行轨道的基线扩展特征，给出了天体运行的离散轨道方程，并以太阳系行星及部分卫星为例，给出了这些天体运行离散轨道方程的具体表述形式。

Introductions on the medium shell and discrete orbits of celestial bodies motion

YAN Kun  (yankun@nature.ac.cn)

(Xi’an Modern Nonlinear Science Applying Institute, Xi’an 710061, China)

Abstract By using phenomenological method for the medium shell curve, an energy equation on three dimensions regular space and the energy-gravitation form about gravitational interaction between bodies are given. Further more, two condition solutions of the gravitational expression is close by with the results of Newton’s gravitational theory and Einstein’s general relativity respectively. The localizations in the functions of the fractal dimension calculus at present are discussed, and the similar expanded equation is given. Subsequentlyby discussing the expanded baseline property on the celestial motion orbit, the discrete orbital equation of the celestial bodies motion are given. And referring to the related orbital data of planets and some satellites in the solar system, the concrete expression on the discrete orbit of the celestial bodies motion are given.

Keywords
medium shell curve, discrete orbital equation of celestial bodies motion, energy equation, localization of fractal dimension calculus, similar expanded equation, fractal dimension expanded

Yan Kun. Introductions on the medium shell and discrete orbits of celestial bodies motion[J]. Progress in Geophysics(in Chinese), 2004, 19(4): 984995.  DOI:10.3969/j.issn.1004-2903.2004.04.046.

# 天体运行轨道的一般性Binet方程形式

(西安现代非线性科学应用研究所  西安 710061)

通过讨论已有质速关系的方程形式，给出质速关系的等效极坐标方程及其Binet方程，进而给出质速关系及质能关系的几个较为具体的方程形式，包括超光速运动形式。随后应用质能关系探讨介质层壳弯曲方法中能量方程解的一般形式，给出天体运行轨道的一般性Binet方程及其在弱场、强场时的近似解表述，给出行星近日点进动、光线弯曲的解析分析。

The general expression of Binet equation about celestial bodies motion orbits

YAN Kun  (yankun@nature.ac.cn)

(Xi’an Modern Nonlinear Science Applying Institute, Xi’an 710061, China)

Abstract By discussing the existent equations of mass-velocity relation, the equivalent polar coordinate equation and its Binet equation of the mass-velocity relation are given, and the expressions of the mass-velocity relation and mass-energy relation are given too, which include the forms of superluminal (also faster-than-light or FTL) motion. Subsequently, using the mass-energy relation, the general expression of the solution of the energy equation on the medium shell curve method is discussed, and the general expression of Binet equation and its approximate solutions about orbits of the celestial bodies motion in the weak and strong gravitational field are given. Further more, the analysis solutions of the advance of the perihelion of planets and bending of light for the gravitational force are given too.

Keywords
orbit of the celestial bodies motion, equations of mass-velocity relation, Binet equation, superluminal motion, advance of the perihelion of planets, bending of light, gravitational frequency shift

Yan Kun. The general expression of Binet equation about celestial bodies motion orbits[J]. Progress in Geophysics(in Chinese with abstract in English), 2005, 20(2): 534539.  DOI:10.3969/j.issn.1004-2903.2005.02.052.

(西安现代非线性科学应用研究所  西安 710061)

通过讨论地球空间已有稳定核素内质子数与中子数的分布趋势，介绍了稳定核素的趋势分析方法及其有关周期性分布方程形式，给出了理论方程曲线与地球空间稳定核素实验数据分布点的对比结果，进而给出了稳定核素极限值和元素周期表中化学元素极限，以及其与正负粒子对的可能对应关系方程，包括位于电子中微子层面附近的粒子质量量级初步估计。随后通过建立真空物质能量状态的二个假设，及基于等效Binet方程，给出了与Einstein狭义相对论有关结论相融合的物质粒子以光速及超光速运动的质量及能量方程；作为推论，对这些方程与暗物质及暗能量的可能对应关系予以了初步探讨。

The tendency analytical equations of stable nuclides and the superluminal
velocity motion laws of matter in geospace

YAN Kun  (yankun@nature.ac.cn)

(Xi’an Modern Nonlinear Science Applying Institute, Xi’an 710061, China)

Abstract   In this paper, by discussing the existent distribution trend of relation for the proton number and the neutron number to be included by the stable nuclides in geospace, the tendency analytical method and it’s periodic distribution equation forms of the stable nuclides are expressed at first. Then the comparison result between the curve of the theoretical equation analysis and the points of the experimental distribution data of the stable nuclides in geospace are given. Further more, the stable nuclide limit and the chemical element limit for the chemical element periodic table are given, and the possible corresponding relation equation with the positron-particle annihilation is expressed, which includes the estimation of the order of the static mass to be situated nearby at the electron neutrino structural dimension. Subsequently, by forming two hypotheses about the energy state of vacuum matter, and basing on the equivalent Binet’s equation, the mass equations and the energy equations of the partial moving with light-velocity or superluminal (also faster-than-light or FTL) velocity motion fusing with the results of Einstein special relativity are expressed. As inference, the possible corresponding relations between the mass equations and energy equations with the dark matter and dark energy are discussed tentatively.

Keywords
stable nuclide, tendency analytical equation, periodic law, chemical element limit, energy state of vacuum matter, equations of superluminal velocity motion

YAN Kun. The tendency analytical equations of stable nuclides and the superluminal velocity motion laws of matter in geospace[J]. Progress in Geophysics(in Chinese with abstract in English), 2006, 21(1): 3847.
DOI:10.3969/j.issn.1004-2903.2006.01.007.
http://en.cnki.com.cn/Article_en/CJFDTOTAL-DQWJ200601006.htm

( 西安现代非线性科学应用研究所, 西安 710061)

通过讨论天体运行背景介质理论的连续轨道及离散轨道这二个研究方向的基础假设，介绍了天体运行轨道的具体方程形式及理论框架概要；进一步地通过讨论天体运行轨道Binet方程的一般形式及其行星近日点进动角的解，给出了连续轨道理论与Newton理论及Einstein广义相对论的联系与区别；通过讨论天体运行轨道的分维扩展方程，给出了包括太阳系行星、天王星卫星、地球卫星、绕月航天器等在内的离散轨道（稳定性轨道）方程及其预言数据。特别地，作为对天体在较为广泛区域作用曲线的初步探讨推论，指出仅由天体引力难以形成质量密度趋于无穷大的理想黑洞。通过讨论函数的分维导数的位置假设及幂函数的分维导数的形式假设，进一步明晰了幂函数的分维导数、分维微分及分维积分的具体方程形式，给出分维导数与分数阶导数的区别，随后讨论了基于一般分形测度的非整数阶微积分定义导出的自相似分形测度趋势性微积分方程具体形式，给出了其与目前Hausdorff测度方法（覆盖方法）的区别，并对包括三分Cantor集合、Koch曲线、Sierpinski垫片及正交十字星形等自相似分形在内的测度进行了计算分析

Introduction on background medium theory about celestial body motion orbit

and foundation of fractional-dimension calculus about

self-similar fractal measure calculation

YAN Kun  (yankun@nature.ac.cn)

(Xi’an Modern Nonlinear Science Applying Institute, Xi’an 710061, China)

Abstract   In this paper, by discussing the basic hypotheses about the continuous orbit and discrete orbit in two research directions of the background medium theory for celestial body motion, the concrete equation forms and their summary of the theoretic frame of celestial body motion are introduced. Future more, by discussing the general form of Binet’s equation of celestial body motion orbit and it’s solution of the advance of the perihelion of planets, the relations and differences between the continuous orbit theory and Newton’s gravitational theory and Einstein’s general relativity are given. And by discussing the fractional-dimension expanded equation for the celestial body motion orbits, the concrete equations and the prophesy data of discrete orbit or stable orbits of celestial bodies which included the planets in the Solar system, satellites in the Uranian system, satellites in the Earth system and satellites obtaining the Moon obtaining from discrete orbit theory are given too. Especially, as the preliminary exploration and inference to the gravitational curve of celestial bodies in broadly range, the concept for the ideal black hole with trend to infinite in mass density difficult to be formed by gravitation only is explored. By discussing the position hypothesis of fractional-dimension derivative about  function and the formula form the hypothesis of fractional-dimension derivative about power function, the concrete equation formulas of fractional-dimension derivative, differential and integral are described distinctly further, and the difference between the fractional-dimension derivative and the fractional-order derivative are given too. Subsequently, the concrete forms of measure tendency calculus equations of self-similar fractal obtaining by based on the definition of form in non-integral order calculus about general fractal measure are discussed again, and the differences with Hausdorff measure method or the covering method at present are given. By applying the measure calculation equations, the measure of self-similar fractals which include middle-third Cantor set, Koch curve, Sierpinski gasket and orthogonal cross star are calculated and analyzed.

Keywords

orbit of celestial body motion, background medium theory, continuous orbit, discrete orbit, self-similar fractal measure, non-integral order calculus, fractional-order calculus, fractional-dimension calculus

YAN Kun. Introduction on background medium theory about celestial body motion orbit and foundation of fractional-dimension calculus about self-similar fractal measure calculation[J]. Progress in Geophysics(in Chinese with abstract in English), 2007, 22(2): 451
462.  DOI:10.3969/j.issn.1004-2903.2007.02.018.
http://en.cnki.com.cn/Article_en/CJFDTOTAL-DQWJ200702017.htm

( 西安现代非线性科学应用研究所  西安 710061)

给出对无穷予以虚数形式标记的初步讨论注释。

Primary annotation of symbol basing on imaginary form about infinity

YAN Kun  (yankun@nature.ac.cn)

(Xi’an Modern Nonlinear Science Applying Institute, Xi’an 710061, China)

Abstract   In this paper, the primary annotation of symbol basing on imaginary form about infinity is given.
Keywords
infinity, zero, imaginary form, symbol, annotation, Euler's formula

YAN Kun. Primary annotation of symbol basing on imaginary form about infinity[R]. Xi’an Modern Nonlinear Science Applying Institute, 18 March 2009.

60

( 西安现代非线性科学应用研究所  西安 710061)

通过讨论已有的60年来大气中CO2浓度数据的分布状态，采用趋势分析方法，给出了具体趋势方程形式。与冰芯分析或观测数据对比结果表明，趋势方程曲线与已有数据基本符合，随后初步给出了2010年至2016年间大气中CO2浓度预测值。

Research on tendency equation about the concentration data of carbon dioxide

in the atmosphere over the past 60 years

YAN Kun  (yankun@nature.ac.cn)

(Xi’an Modern Nonlinear Science Applying Institute, Xi’an 710061, China)

Abstract   In this paper, by discussing the existing distribution of the CO2 concentration data in the atmosphere over the past 60 years, and adopting the tendency analytical method, the concrete tendency equation forms of the CO2 concentration are presented at first. Further more, the comparison result between the curve of the theoretical equation and the data curve of CO2 concentration from the ice cores analysis or the observation is given. The result shows that the tendency equation curve agree well with the existing data. Subsequently, the predictive data of the CO2 concentration in the atmosphere during the year from 2010 to 2016 are suggested tentatively.

Keywords

concentration of carbon dioxide, atmosphere, tendency equation, curve fitting, predictive data, data analysis

YAN Kun.  Research on tendency equation about the concentration data of carbon dioxide in the atmosphere over the past 60 years[J]. Progress in Geophys. (in Chinese with abstract in English), 2009,24(5):16651670. DOI:10.3969/j.issn.1004 2903.2009.05.016.

( 西安现代非线性科学应用研究所  西安 710061 )

通过讨论非线性动力学方程的近似等效解析解及数据曲线间断区域的性质，给出间断区域的自适应连接方程构造形式及其参数确定的预置迭代方法，进而亦给出一般曲线间断区域的映射方程组连接方法，随后给出计算实例，其中包括位错间断区域与回转间断区域的映射方程组及光滑连接曲线。此自适应连接方程，可以作为经典S型曲线方程或Logistic函数的一般扩展形式（扩展型双曲正切函数）予以应用；其中对于缓变数据曲线阶跃间断区域，可自动计算生成连接方程形式。文中分析了非线性方程在经典Newton动力学方程及RLC串联电路电荷方程方面的兼容性；基于自适应连接方程探讨分析了数据曲线或部分连续函数的一种扩展型双曲正切级数近似表示形式、磁性材料磁滞回线方程、粒子统计分布的平均能量方程扩展形式及平均粒子数趋势性微分方程、稳定核素比结合能方程、势能函数曲线方程（诸如双原子分子势能函数曲线方程）、自然饱和过程方程（诸如钢材料断裂韧性方程）及金属或岩石蠕变过程方程。

Research on adaptive connection equation in discontinuous area of data curve

YAN Kun

( Xi’an Modern Nonlinear Science Applying Institute, Xi’an 710061, China )

Abstract  In this paper, by discussing the approximate equivalent analytical solution of the nonlinear dynamics equation and the properties in discontinuous area of data curve, a general constructing form of adaptive connection equation and the preset iteration method determining parameters in the discontinuous area are given. And then, the connecting method of the mapping equations for the discontinuous areas of general curves is also given. Subsequently, the computing examples which included the discontinuous areas with the dislocation and the turn-back are given too. This adaptive connection equation can be applied as a general form of expansion (extended hyperbolic tangent function) of the classical S-curve (sigmoid curve) equation or Logistic function, and for the step discontinuous area of slowly varying data curve, its form of the adaptive connection equation can be obtained by automatic calculating. In this paper, the compatibility of the nonlinear equation in classical Newton dynamic equation and the charge equation of the RLC series circuit is analyzed. Basing on the form of the adaptive connection equation, an approximate expression of extended hyperbolic tangent series for the data curve or the partial continuous functions, the equations of magnetic hysteresis loop for magnetic material, the extended form of average energy equation and the tendency differential equation of the average particle number for the statistical distributions of the particles, the equation of average binding energy per nucleon in stable nuclidethe curvilinear equation of potential energy function (such as curvilinear equation of potential energy function of diatomic molecule, etc), the equation of natural saturation process (such as tree growth and physical reaction or chemical reaction process, the equation of fracture toughness for steel material, etc) and the equation of typical creep process for metal or rock material are explored and analyzed tentatively.

Keywords
nonlinear dynamics equation, discontinuous area of data curve, adaptive connection equation, connecting method of the mapping equations, equations of magnetic hysteresis loop, equation of statistical distributions of the particles, equation of average binding energy per nucleon, curvilinear equation of potential energy function, equation of natural saturation process, equation of creep process

YAN Kun. Research on adaptive connection equation in discontinuous area of data curve[J]. Progress in Geophys(in Chinese with abstract in English), 2011,26(1): 162171.  DOI:10.3969/j.issn.1004-2903.2011.01.018.

( 西安现代非线性科学应用研究所  西安 710061 )

本文给出了连接方程的构造过程及其若干应用的简略注释，讨论了连接方程是一非线性动力学方程的近似等效解析解的分析方法，提出了现象在演化过程的状态转变方程，其在一般情况下遵循最优或最简洁路径光滑曲线方程形式这一自然最优原则或自然简洁原则，构建了数据曲线或部分连续函数的一种扩展型双曲正切级数近似表示形式；依据连接方程的非线性动力学方程近似形式，预言了二个基于非线性动力学方程的电路元件－电存器(nonlinstor)和电敏器(geomsentor)，其皆为深化型电容器的电路器件，分析了RLCNG串联电路微分方程的性质；给出了粒子统计分布趋势性方程及其若干条件解，探讨了具有近似线性变频解的非线性微分方程形式及在负频率情况下变频波动方程的曲线形态特征，讨论了数据库理论构架（此构架由基础数据库、趋势性方程、解析数据库构成）的一种简洁模式，建立了美国年度能源消费量与GDP关系方程及英国年度人口数量与GDP关系方程，并计算预测了美国年度能源消费极限值与英国年度人口极限值；随后探讨了岩石及单晶高温材料的蠕变过程曲线、半导体分立器件VA特性曲线、超导材料电阻R（或电阻率ρ绝对温度T曲线方程，双晶Josephson结直流IU特性曲线及Shapiro台阶电流阶跃幅值曲线的趋势拟合方程，机械系统或伺服系统摩擦力－速度特性曲线（包括Coulomb摩擦Stribeck摩擦、黏性摩擦、摩擦迟滞及反常摩擦迟滞效应）的趋势拟合方程，Newton冷却定律的扩展方程形式，高聚物熔体流动曲线剪切应力－剪切速率（包括第一Newton区、假塑性区、第二Newton区、胀流区、湍流及反常剪切效应）的趋势性方程等。在唯象及趋势层面，指出了材料蠕变过程曲线与高聚物熔体流动曲线具有类似性质，及材料断裂裂纹扩展结构与流体湍流漩涡嵌套结构这二种现象演化在分形测度表述上具有相通性的研究方向；后讨论了连接方程在数据拟合及长程预测方面的局限性。

Brief annotation of the connection equation

YAN Kun

( Xi’an Modern Nonlinear Science Applying Institute, Xi’an 710061, China )

Abstract  In this paper, the brief annotation of a constructing procedure and the applications of the connection equation are given. An analytical method of the connection equation as an approximate equivalent analytical solution of the nonlinear dynamics equations is discussed, the equation of the transformation of the phenomena between the states in evolutionary process is presented, that it follows a natural optimum principle or a natural conciseness principle in general, which having an equation form of the smooth curve with the optimal path or the most concise path. An approximate expression of extended hyperbolic tangent series for the data curve or the partial continuous functions is constructed. According to the approximate form of the nonlinear dynamics equation of the connection equation, two new electronic circuit elements (nonlinstor and geomsentor) with the deepening charge-controlled capacitor properties based on the form of the nonlinear differential equation is predicted, and the nonlinear differential equation for a RLCNG series circuit is also analyzed, the tendency equation and its conditional solutions of the statistical distributions of the particles are given, the nonlinear differential equation expressions with the solution of approximate linear frequency conversion and the characteristics of the curve shapes of frequency conversion wave equations at the negative frequency are discussed, a concise model of the database theoretical framework (this framework to be made up of the foundation database, the tendency equation, and the analytic database) is explored, the equation of relationship between the total annual energy consumption with the annual GDP in the United States, and the equation of relationship between the annual population with the annual GDP in the United Kingdom are established, and the limit values of the total annual energy consumption in the United States and the annual population in the United Kingdom are calculated and predicted. Subsequently, the tendency fitting equations of the curves are explored, which included the creep process curve of the rock and the single-crystal superalloy, the Volt-Ampere characteristic curve of the discrete semiconductor device, the resistance (or resistivity )-absolute temperature curve of the superconducting material, the direct current I-U characteristic curve of the bicrystal Josephson junction, and the current step amplitude of Shapiro steps, the friction-speed characteristic curve (that included the stages of Coulomb friction, Stribeck friction, viscous friction, friction hysteresis, and anomalous friction hysteresis effect) in mechanical system or servo system, the expanded equation form of the Newton's law of cooling, the shear stress-shear rate in the flow curve (that included the sections of first Newtonian fluid-flow, pseudoplastic flow, second Newtonian fluid-flow, dilatant flow, turbulent flow, and anomalous shearing effect) of the polymer melts, etc. On the phenomenological and tendency levels, the research directions of similar property of creep curves of materials with the flow curves of polymer melt, and the similarity in the fractal measure of the evolutions of fracture cracks growth structure and fluid turbulent eddies nested structure are pointed out. At the end, the limitations of the connection equation in the data fitting and long-range forecasting are discussed.

Keywords
connection equation, shape of curve, nonlinear dynamics equation, approximate equivalent analytical solution, states transforming equation, natural conciseness principle

YAN Kun. Brief annotation of the connection equation[R]. Xi’an Modern Nonlinear Science Applying Institute, 18 March 2011.

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 Email: yankun@nature.ac.cn