## 抑制不住创作的欲望——cosea

☼｡◕‿◕｡♪≧◠◡◠≦✌♫(͡° ͜ʖ ͡°)♬≧◔◡◔≦⚽
╔＊═╗ ╔╗
╚╗╔╝ ║║★═╦╦╦═╗ ║☆╝╠═╦╦
╔╝╚╗ ＊╚╣║║║║╠╣ ╚╗╔╣║♀║☆║
╚═♂╝ ╚═╩═╩＊╩═╝♪ ╚╝╚╩╝╚╩╝
~@^_^@~ ღ…(⊙_⊙;)… ☠(⊙o⊙)❣ ｡◕‿-｡⚽
• 前言说明：首次使用LaTeX，此文写得满满的兴奋，除少数引用外的所有内容，全凭个人人工输入完成。多图预警，建议WiFi下查看。

## ——小海cosea

reference：Golden ratio

• 黄金分割点：

reference：Heptadecagon 知乎上传gif动态图无法动起来······作图过程的gif可以在Heptadecagon上查看

（LaTeX，音译“拉泰赫”）是一种基于ΤΕΧ的排版系统，由美国计算机学家莱斯利·兰伯特（Leslie Lamport）在20世纪80年代初期开发，利用这种格式，即使使用者没有排版和程序设计的知识也可以充分发挥由TeX所提供的强大功能，能在几天，甚至几小时内生成很多具有书籍质量的印刷品。对于生成复杂表格和数学公式，这一点表现得尤为突出。因此它非常适用于生成高印刷质量的科技和数学类文档。这个系统同样适用于生成从简单的信件到完整书籍的所有其它种类的文档。

reference：LaTeX

polar equation:

parametric equations:

reference：Butterfly curve (transcendental)

reference：Cardioid

reference：Folium of Descartes

reference：Fibonacci number 发现什么了吗？······Golden spiral

In parametric form, the curve is:

reference：Logarithmic spiral

Spiral

or , it can be described by the equation:

reference：Archimedean spiral

The involute of a circle (black) is not identical to the Archimedean spiral (red).

reference：Spiral

reference：Hyperbolic spiral

reference：Fermat's spiral

Euler spiral

reference：Euler spiral

lituus

reference：Lituus (mathematics)

spiral of Theodorus

reference：Spiral of Theodorus

Spiral

As a parametric equation:

reference：Lemniscate of Bernoulli

reference related articles：

## A quartic plane curve

A quartic plane curve is a plane algebraic curve of the fourth degree. It can be defined by a bivariate quartic equation:

reference：Quartic plane curve

Ampersand curve

reference：Quartic plane curve

Bicuspid curve

The biscuspid is a quartic plane curve with the equation:

reference：Quartic plane curve

The three-leaved clover is the quartic plane curve.

reference：Quartic plane curve

quadrifolium,also known as four-leaved clover.

polar equation:

with corresponding algebraic equation:

polar equation:

with corresponding algebraic equation:

reference：Quadrifolium

rose curve

these curves can all be expressed by a polar equation of the form:

and

reference related articles：Rose (mathematics)

Maurer rose

Let be a rose in the polar coordinate system, where is a positive integer. The rose has petals if is odd, and petals if is even.

We then take 361 points on the rose:

where is a positive integer and the angles are in degrees, not radians.

A Maurer rose of the rose consists of the 360 lines successively connecting the above 361 points. Thus a Maurer rose is a polygonal curve with vertices on a rose.

Explanation:A Maurer rose can be described as a closed route in the polar plane. A walker starts a journey from the origin, (0, 0), and walks along a line to the point . Then, in the second leg of the journey, the walker walks along a line to the next point, (sin(n·2d), 2d), and so on. Finally, in the final leg of the journey, the walker walks along a line, from (sin(n·359d), 359d) to the ending point, (sin(n·360d), 360d). The whole route is the Maurer rose of the rose . A Maurer rose is a closed curve since the starting point, (0, 0) and the ending point, (sin(n·360d), 360d), coincide.

The following figure shows the evolution of a Maurer rose (n = 2, d = 29° ).

The following are some Maurer roses drawn with some values for n and d:

reference：Maurer rose

reference related articles：

Rose (mathematics)

Dual curve

Quatrefoil

The Koch snowflake (also known as the Koch curve, Koch star, or Koch island).

After each iteration, the number of sides of the Koch snowflake increases by a factor of 4, so the number of sides after iterations is given by:

If the original equilateral triangle has sides of length , the length of each side of the snowflake after iterations is:

the perimeter of the snowflake after iterations is:

reference：Koch snowflake

Variants of the Koch curve , for example:

Quadratic type 2 curve

1D, 90° angle

The first 2 iterations. Its fractal dimension equals 3/2 and is exactly half-way between dimension 1 and 2. It is therefore often chosen when studying the physical properties of non-integer fractal objects.

reference：Ellipsoid

reference：Hyperboloid

Paraboloid of revolution

reference：Paraboloid

reference：Paraboloid

reference：Pythagoras tree (fractal)

reference：Mass–energy equivalence

In quantum mechanics, the Schrödinger equation is a partial differential equation that describes how the quantum state of a quantum system changes with time.

The concept of a wavefunction is a fundamental postulate of quantum mechanics.

Time-dependent Schrödinger equation (general)

The form of the Schrödinger equation depends on the physical situation (see below for special cases). The most general form is the time-dependent Schrödinger equation, which gives a description of a system evolving with time.

is the Hamiltonian operator (which characterizes the total energy of any given wave function and takes different forms depending on the situation).

reference：Hamiltonian (quantum mechanics)

Time-dependent Schrödinger equation (single non-relativistic particle)

The most famous example is the non-relativistic Schrödinger equation for a single particle moving in an electric field (but not a magnetic field; see the Pauli equation)——Pauli equation

The term "Schrödinger equation" can refer to both the general equation (first box above), or the specific nonrelativistic version (second box above and variations thereof). The general equation is indeed quite general, used throughout quantum mechanics, for everything from the Dirac equation to quantum field theory, by plugging in various complicated expressions for the Hamiltonian. The specific nonrelativistic version is a simplified approximation to reality, which is quite accurate in many situations, but very inaccurate in others (see relativistic quantum mechanics and relativistic quantum field theory).

The time-dependent Schrödinger equation described above predicts that wave functions can form standing waves, called stationary states (also called "orbitals", as in atomic orbitals or molecular orbitals). These states are important in their own right, and if the stationary states are classified and understood, then it becomes easier to solve the time-dependent Schrödinger equation for any state. Stationary states can also be described by a simpler form of the Schrödinger equation, the time-independent Schrödinger equation. (This is only used when the Hamiltonian itself is not dependent on time explicitly. However, even in this case the total wave function still has a time dependency.)

Time-independent Schrödinger equation (general)

Time-independent Schrödinger equation (single non-relativistic particle)

As before, the most famous manifestation is the non-relativistic Schrödinger equation for a single particle moving in an electric field (but not a magnetic field):

reference：Schrödinger equation

reference：Wave function

is the reduced Planck constant,.

The most common general form of the uncertainty principle is the Robertson uncertainty relation:

The Robertson uncertainty relation immediately follows from a slightly stronger inequality, the Schrödinger uncertainty relation:

reference：Uncertainty principle

As a solution for a certain partial differential equation, the quantized angular momentum can be written as:

( is the norm of the spin vector)

reference：Spin quantum number

reference：Pauli exclusion principle

reference：Fermion

Schrödinger's cat is a thought experiment, sometimes described as a paradox, devised by Austrian physicist Erwin Schrödinger in 1935.It illustrates what he saw as the problem of the Copenhagen interpretation of quantum mechanics applied to everyday objects. The scenario presents a cat that may be simultaneously both alive and dead,a state known as a quantum superposition, as a result of being linked to a random subatomic event that may or may not occur. The thought experiment is also often featured in theoretical discussions of the interpretations of quantum mechanics. Schrödinger coined the term Verschränkung (entanglement) in the course of developing the thought experiment.

reference：Schrödinger's cat

reference：Matter wave

Wave–particle duality

Louis de Broglie

reference：Fourier transform

Joseph Fourier

reference：Gamma function

(P>0，Q>0)

reference：Beta function

reference：Normal distribution

reference：Fundamental theorem of calculus

reference：Newton's laws of motion

reference：Isaac Newton

（高阶无穷小）时，

reference：Taylor's theorem

Brook Taylor

reference：Brook Taylor

is delta-v - the maximum change of velocity of the vehicle (with no external forces acting).

is the initial total mass, including propellant.

is the final total mass without propellant, also known as dry mass.

is the effective exhaust velocity.

refers to the natural logarithm function.

The Tsiolkovsky rocket equation, or ideal rocket equation, describes the motion of vehicles that follow the basic principle of a rocket: a device that can apply acceleration to itself (a thrust) by expelling part of its mass with high speed and thereby move due to the conservation of momentum. The equation relates the delta-v (the maximum change of velocity of the rocket if no other external forces act) with the effective exhaust velocity and the initial and final mass of a rocket (or other reaction engine).

reference：Tsiolkovsky rocket equation

reference：Lever

Archimedes

reference：Boltzmann's entropy formula

Ludwig Boltzmann

reference：Boltzmann's entropy formula

Clausius theorem

The Riemann zeta function ζ(s) is a function of a complex variable . (The notation , , and is used traditionally in the study of the ζ-function, following Riemann.)

The following infinite series converges for all complex numbers s with real part greater than 1, and defines ζ(s) in this case:

It can also be defined by the integral:

The Riemann zeta function is defined as the analytic continuation of the function defined for by the sum of the preceding series.

Leonhard Euler considered the above series in 1740 for positive integer values of , and later Chebyshev extended the definition to

The above series is a prototypical Dirichlet series that converges absolutely to an analytic function for such that and diverges for all other values of . Riemann showed that the function defined by the series on the half-plane of convergence can be continued analytically to all complex values . For the series is the harmonic series which diverges to , and

Thus the Riemann zeta function is a meromorphic function on the whole complex , which is holomorphic everywhere except for a simple pole at with residue 1.

In mathematics, a holomorphic function is a complex-valued function of one or more complex variables that is complex differentiable in a neighborhood of every point in its domain. The existence of a complex derivative in a neighborhood is a very strong condition, for it implies that any holomorphic function is actually infinitely differentiable and equal to its own Taylor series (analytic). Holomorphic functions are the central objects of study in complex analysis.

Given a complex-valued function f of a single complex variable, the derivative of f at a point in its domain is defined by the limit:

This is the same as the definition of the derivative for real functions, except that all of the quantities are complex. In particular, the limit is taken as the complex number approaches , and must have the same value for any sequence of complex values for that approach on the complex plane. If the limit exists, we say that is complex-differentiable at the point . This concept of complex differentiability shares several properties with real differentiability: it is linear and obeys the product rule, quotient rule, and chain rule.
If is complex differentiable at every point in an open set , we say that is holomorphic on . We say that is holomorphic at the point if it is holomorphic on some neighborhood of .We say that is holomorphic on some non-open set if it is holomorphic in an open set containing .

reference：Riemann zeta function

reference：Holomorphic function

## 黎曼猜想——Riemann hypothesis

In mathematics, the Riemann hypothesis is a conjecture that the Riemann zeta function has its zeros only at the negative even integers and the complex numbers with real part 1/2. It was proposed by Bernhard Riemann (1859), after whom it is named. The name is also used for some closely related analogues, such as the Riemann hypothesis for curves over finite fields.

The Riemann hypothesis implies results about the distribution of prime numbers. Along with suitable generalizations, some mathematicians consider it the most important unresolved problem in pure mathematics (Bombieri 2000). The Riemann hypothesis, along with Goldbach's conjecture, is part of Hilbert's eighth problem in David Hilbert's list of 23 unsolved problems; it is also one of the Clay Mathematics Institute's Millennium Prize Problems.

It can also be defined by the integral:

The Riemann zeta function satisfies the functional equation (known as the Riemann functional equation or Riemann's functional equation):

reference：Riemann hypothesis

reference related articles：Riemann Xi function

Theta function

reference related articles：Theta function

In mathematics, particularly in complex analysis, a Riemann surface, first studied by and named after Bernhard Riemann, is a one-dimensional complex manifold. Riemann surfaces can be thought of as deformed versions of the complex plane: locally near every point they look like patches of the complex plane, but the global topology can be quite different. For example, they can look like a sphere or a torus or several sheets glued together.

The main point of Riemann surfaces is that holomorphic functions may be defined between them. Riemann surfaces are nowadays considered the natural setting for studying the global behavior of these functions, especially multi-valued functions such as the square root and other algebraic functions, or the logarithm.

Every Riemann surface is a two-dimensional real analytic manifold (i.e. a surface), but it contains more structure (specifically a complex structure) which is needed for the unambiguous definition of holomorphic functions. A two-dimensional real manifold can be turned into a Riemann surface (usually in several inequivalent ways) if and only if it is orientable and metrizable. So the sphere and torus admit complex structures, but the Möbius strip, Klein bottle and projective plane do not.

There are several equivalent definitions of a Riemann surface.

1.A Riemann surface X is a complex manifold of complex dimension one. This means that X is a Hausdorff topological space endowed with an atlas: for every point x ∈ X there is a neighbourhood containing x homeomorphic to the unit disk of the complex plane. The map carrying the structure of the complex plane to the Riemann surface is called a chart. Additionally, the transition maps between two overlapping charts are required to be holomorphic.

2.A Riemann surface is an oriented manifold of (real) dimension two – a two-sided surface – together with a conformal structure. Again, manifold means that locally at any point x of X, the space is homeomorphic to a subset of the real plane. The supplement "Riemann" signifies that X is endowed with an additional structure which allows angle measurement on the manifold, namely an equivalence class of so-called Riemannian metrics. Two such metrics are considered equivalent if the angles they measure are the same. Choosing an equivalence class of metrics on X is the additional datum of the conformal structure.

1857年，他初次登台作了题为“论作为几何基础的假设”的演讲，开创了黎曼几何，并为爱因斯坦的广义相对论提供了数学基础。他在1857年升为哥廷根大学的编外教授，并在1859年狄利克雷去世后成为正教授。

reference：Riemann surface

Riemann Surfaces-PDF

（k，j为整数）也可以简单地表示分段函数的形式，D(x)= 0（x是无理数）或1（x是有理数）。

reference：Nowhere continuous function

In mathematics, the Poincaré conjecture is a theorem about the characterization of the 3-sphere, which is the hypersphere that bounds the unit ball in four-dimensional space. The conjecture states:

Every simply connected, closed 3-manifold is homeomorphic to the 3-sphere.

reference：Poincaré conjecture

Maxwell's equations are a set of partial differential equations that, together with the Lorentz force law, form the foundation of classical electrodynamics, classical optics, and electric circuits. These areas of physics are the basis for all electric, optical and radio technologies like power generation, electric motors, wireless communication, cameras, televisions, computers etc. Maxwell's equations describe how electric and magnetic fields are generated by charges, currents and changes of each other. One important consequence of the equations is that fluctuating electric and magnetic fields can propagate at the speed of light, and this electromagnetic radiation manifests itself in manifold ways from radio waves to light and X- or γ-rays. The equations are named after the physicist and mathematician James Clerk Maxwell, who published an early form of the equations between 1861 and 1862, and first proposed that light is an electromagnetic phenomenon.

In many situations, though, deviations from Maxwell's equations are immeasurably small. Exceptions include nonclassical light, photon-photon scattering, quantum optics, and many other phenomena related to photons or virtual photons.

The CGS system uses a unit of charge defined in such a way that the permittivity of the vacuum , hence .These units are sometimes preferred over SI units in the context of special relativity,since when using them, the components of the electromagnetic tensor, the Lorentz covariant object describing the electromagnetic field, have the same unit without constant factors.

## 高斯单位制——Formulation in Gaussian units convention

reference：Gauss's law

reference：Gauss's law for magnetism

reference：Faraday's law of induction

reference：Ampère's circuital law

## Vacuum equations , electromagnetic waves and speed of light

In a region with no charges (ρ = 0) and no currents (J = 0), such as in a vacuum, Maxwell's equations reduce to:

Taking the curl of the curl equations, and using the curl of the curl identitywe obtain the wave equations:

which identify

with the speed of light in free space. In materials with relative permittivity , and relative permeability , the phase velocity of light becomes:

which is usually less than .

In addition, and are mutually perpendicular to each other and the direction of wave propagation, and are in phase with each other. A sinusoidal plane wave is one special solution of these equations. Maxwell's equations explain how these waves can physically propagate through space. The changing magnetic field creates a changing electric field through Faraday's law. In turn, that electric field creates a changing magnetic field through Maxwell's addition to Ampère's law. This perpetual cycle allows these waves, now known as electromagnetic radiation, to move through space at velocity, .

reference：Electromagnetic radiation

is the Hamiltonian operator (which characterizes the total energy of any given wave function and takes different forms depending on the situation).

reference：Hamiltonian (quantum mechanics)

reference：Curl (mathematics)

## 宏观麦克斯韦方程组

The microscopic variant of Maxwell's equation is the version given above. It expresses the electric field and the magnetic field in terms of the total charge and total current present, including the charges and currents at the atomic level. The "microscopic" form is sometimes called the "general" form of Maxwell's equations. The macroscopic variant of Maxwell's equation is equally general, however, with the difference being one of bookkeeping.

The "microscopic" variant is sometimes called "Maxwell's equations in a vacuum". This refers to the fact that the material medium is not built into the structure of the equation; it does not mean that space is empty of charge or current. They are also known as the "Maxwell-Lorentz equations". Lorentz tried to use these equations to predict the macroscopic properties of bulk matter from the physical behavior of its microscopic constituents.

"Maxwell's macroscopic equations", also known as Maxwell's equations in matter, are more similar to those that Maxwell introduced himself.

Following is a summary of some of the numerous other ways to write the microscopic Maxwell's equations, showing they can be formulated using different points of view and mathematical formalisms that describe the same physics. Often, they are also called the Maxwell equations.

The direct space–time formulations make manifest that the Maxwell equations are relativistically invariant (in fact studying the hidden symmetry of the vector calculus formulation was a major source of inspiration for relativity theory). In addition, the formulation using potentials was originally introduced as a convenient way to solve the equations but with all the observable physics contained in the fields. The potentials play a central role in quantum mechanics, however, and act quantum mechanically with observable consequences even when the fields vanish (Aharonov–Bohm effect).

reference：Spacetime

AB效应——阿哈罗诺夫-玻姆效应——Aharonov–Bohm effect

1959年，阿哈罗诺夫（Y.Aharonov）和玻姆（D.Bohm）发表了一篇论文，该论文认为，在电子运动的空间中，无论是否存在电磁场，电子波函数的位相都会受到空间中电磁势的影响。由此他们做出结论，在量子理论中，电磁势要比经典电磁理论中的电场与磁场强度更有意义。

AB效应实验：两束同相位的电子，通过一个磁线圈，到屏上成像。磁场不改变，而磁矢势变化时，屏上的成像有变化。

AB效应主要就是证明，电磁场的矢势有直接的可观测的物理效应。

The electromagnetic four-potential can be defined as:

SI国际单位制下：.

in which is the electric potential, and A is the magnetic potential (a vector potential). The units of are in SI, and in Gaussian-cgs.

The electric and magnetic fields associated with these four-potentials are:

SI国际单位制下： ,

In special relativity, the electric and magnetic fields must be written in the form of a tensor so they transform correctly under Lorentz transformations - achieved by the electromagnetic tensor. This is written in terms of the electromagnetic four-potential and the four-gradient as:

This essentially defines the four-potential in terms of physically observable quantities, as well as reducing to the above definition.

reference：Electromagnetic four-potential

Aharonov–Bohm effect

Quantum mechanics

Fields -3D Euclidean space + time

Potentials (any gauge)-3D Euclidean space + time

Potentials (Lorenz gauge)-3D Euclidean space + time

Fields -3D Euclidean space + time

Potentials (any gauge)-3D Euclidean space + time

Potentials (Lorenz gauge)-3D Euclidean space + time

Fields -Minkowski space

Potentials (any gauge) -Minkowski space

Potentials (Lorenz gauge) -Minkowski space

Fields -Any space–time

Potentials (any gauge) -Any space–time (with topological restrictions)

Potentials (Lorenz gauge) -Any space–time (with topological restrictions

Fields -Minkowski space

Potentials (any gauge) -Minkowski space

Potentials (Lorenz gauge) -Minkowski space

Fields -Any space–time

Potentials (any gauge) -Any space–time (with topological restrictions)

Potentials (Lorenz gauge) -Any space–time (with topological restrictions

Fields -Any space–time

Potentials (any gauge) -Any space–time (with topological restrictions)

Potentials (Lorenz gauge) -Any space–time (with topological restrictions)

Fields -Any space–time

Potentials (any gauge) -Any space–time (with topological restrictions)

Potentials (Lorenz gauge) -Any space–time (with topological restrictions)

reference：Maxwell's equations

Maxwell's equations in curved spacetime

## 天长地久，生生长流

——小海cosea

——2016年9月28日晚

The Seven Bridges of Königsberg is a historically notable problem in mathematics. Its negative resolution by Leonhard Euler in 1736 laid the foundations of graph theory and prefigured the idea of topology.

The city of Königsberg in Prussia (now Kaliningrad, Russia) was set on both sides of the Pregel River, and included two large islands which were connected to each other and the mainland by seven bridges. The problem was to devise a walk through the city that would cross each bridge once and only once, with the provisos that: the islands could only be reached by the bridges and every bridge once accessed must be crossed to its other end. The starting and ending points of the walk need not be the same.

Euler proved that the problem has no solution. The difficulty was the development of a technique of analysis and of subsequent tests that established this assertion with mathematical rigor.

1736年29岁的欧拉向圣彼得堡科学院递交了《哥尼斯堡的七座桥》的论文，在解答问题的同时，开创了数学的一个新的分支——图论与几何拓扑，也由此展开了数学史上的新历程。七桥问题提出后，很多人对此很感兴趣，纷纷进行试验，但在相当长的时间里，始终未能解决。欧拉通过对七桥问题的研究，不仅圆满地回答了哥尼斯堡居民提出的问题，而且得到并证明了更为广泛的有关一笔画的三条结论，人们通常称之为“欧拉定理F”。

reference：Seven Bridges of Königsberg

——2016年国庆节期间

Quora:What is the most beautiful equation?

## Laplace's Equation, the homogeneous analogue of Poisson's Equation, produces functions, where every point is the average of some epsilon region around it. This means that there can be no maxima or minima on the interior of the region. Functions that satisfy this differential equation are called "harmonic", and I find them to be very peaceful functions.Here are some examples of functions over a finite domain which satisfy Laplace's equation.

• 后记：楼主高中开始推数学公式，现在已经有不少原创数学公式和研究成果，其中泰勒级数我高中就推过，那个时候有点苗头，只可惜，上大学后明白了什么是“井底之蛙”。数学发展到今天，路几乎被数学家们踩烂了，当然，每个时代都有每个时代的命运，无论什么时代，都会有数学难题困扰着数学家，路漫漫其修远，每个人都有机会。
• 4年，一度荒废后，2016年重拾数学，经过大半年的学习，现在已经找回巅峰时期的状态，大学虽然几乎荒废了数学，但也有几个创造性的数学研究，总体来说都不是很满意。接下来的目标就是将所有数学专业的专业课程从低到高，一直研究到专业学术层，在闲暇时间的专业数学研究道路上朝着想去的方向越走越远······毫无疑问，我将不断进行数学研究，我将一直行走在这条路上，直至生命的尽头。

NP完全问题、霍奇猜想、庞加莱猜想、黎曼假设、杨·米尔斯理论、纳卫尔-斯托可方程、BSD猜想。

1.NP完全问题

2.霍奇猜想

3.庞加莱猜想

2006年8月，第25届国际数学家大会授予佩雷尔曼菲尔兹奖。数学界最终确认佩雷尔曼的证明解决了庞加莱猜想。

4.黎曼假设

5.杨－米尔斯存在性和质量缺口

6.纳卫尔-斯托可方程的存在性与光滑性

7.BSD猜想

Hilbert's problems are a list of twenty-three problems in mathematics published by German mathematician David Hilbert in 1900. The problems were all unsolved at the time, and several of them were very influential for 20th century mathematics. Hilbert presented ten of the problems (1, 2, 6, 7, 8, 13, 16, 19, 21 and 22) at the Paris conference of the International Congress of Mathematicians, speaking on August 8 in the Sorbonne. The complete list of 23 problems was published later, most notably in English translation in 1902 by Mary Frances Winston Newson in the Bulletin of the American Mathematical Society.

reference：Hilbert's problems

2：德国

3：法国

4：美国

5：英国

6：瑞士

7：匈牙利

8：挪威

9：澳大利亚

10：苏联

11：意大利

12：印度

13：爱尔兰

14：瑞典

15：丹麦

16：捷克

17：日本

18：比利时

19：波兰

20：墨西哥

21：奥地利

22：阿拉伯

23：罗马尼亚

## 祖冲之

1983年王见定教授在世界上首次提出半解析函数理论，1988年又首次提出并系统建立了共轭解析函数理论；并将这两项理论成功地应用于电场、磁场、流体力学、弹性力学等领域。此两项理论受到众多专家、学者的引用和发展，并由此引发双解析函数、复调和函数、多解析函数（k阶解析函数）、半双解析函数、半共轭解析函数以及相应的边值问题、微分方程、积分方程等一系列新的数学分支的产生，而且这种发展势头强劲有力，不可阻挡。这是中国学者对发展世界数学作出的前所未有的大范围的原创工作。

1911年10月28日生于浙江嘉兴秀水县，美籍华人，20世纪世界级的几何学家，他开创并领导着整体微分几何、纤维丛微分几何、“陈示性类”等领域的研究，在国际上享有“微分几何之父”的美誉，曾获得美国国家科学奖、“沃尔夫奖”和“邵逸夫奖”等殊荣。

## 伽罗瓦

1933年，匈牙利数学家乔治·塞凯赖什（George Szekeres）还只有22岁。那时，他常常和朋友们在匈牙利的首都布达佩斯讨论数学。这群人里面还有同样生于匈牙利的数学怪才——保罗·埃尔德什（PAUL ERDŐS）大神。不过当时，埃尔德什只有20岁。

1 、数值计算软件，如matlab（商业软件），scilab(开源自由软件）等等；
2 、统计软件，如SAS（商业软件）、minitab（商业软件）、SPSS（商业软件），R（开源自由软件）等；
3 、符号运算软件，这类软件不同于前两种，它不仅能计算出数值，还可以把符号表达的公式、方程进行推导和化简，可以求出微分积分的表达式，这类软件的代表有：MathType、maple（商业软件）、mathematica（商业软件）、maxima（开源自由软件）、mathcad（商业软件）、Microsoft Mathematics（商业软件，可以通过DreamSpark免费下载）等等。

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## 数学篇

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《自然哲学之数学原理》·牛顿 (Isaac Newton) , 王克迪译·北京大学出版社

《数学分析》第四版（上下册）·华东师范大学数学系·高等教育出版社

《线性代数》第六版·同济大学数学系·高等教育出版社

《复变函数》第四版·西安交通大学高等数学教研室·高等教育出版社

《概率论与数理统计》浙大第四版·高等教育出版社

《高等代数》第四版·北京大学数学系前代数小组·王萼芳·高等教育出版社

《陶哲轩实分析》·陶哲轩·人民邮电出版社

Visual Complex Analysis, by Tristan Needham

Game Theory, by Drew Fudenberg and Jean Tirole

Game Theory: Analysis of Conflict, by Roger B. Myerson

Game Theory, by Michael Maschler, Eilon Solan and Shmuel Zamir

A Course in Game Theory, by Martin Osborne and Ariel Rubinstein

《组合数学》英文第5版·Richard A.Brualdi·机械工业出版社

《微分几何》第四版·梅向明·高等教育出版社

A Course in Arithmetic, by Jean-Pierre Serre

Topology and geometry, by Glen E.Bredon

Introduction to Topological Manifolds, by John M.Lee

Topology from the Differentiable Viewpoint, by John Willard Milnor

Differential Topology, by Morris W. Hirsch

Algebraic Topology, by Allen Hatcher

Algebraic Topology, by Tammo tom Dieck

Algebraic Topology, Corr. 3rd Edition by Edwin H. Spanier

Three-Dimensional Geometry and Topology, Volume 1, William P. Thurston, Edited by Silvio Levy

Mathematical Physics ,2nd Edition by Sadri Hassani

Mathematica Methods for Physicists:A Comprehensive Guide, 7th Edition by GeorgeB.Arfken HansJ.Weber

Functional Analysis, 2nd Edition by Walter Rudin

A Course in Functional Analysis, by John B. Conway

Real and Functional Analysis, by Serge Lang

Real Analysis, Fourth Edition by H.L. Royden P.MP.M. Fitzpatrick

Real and complex analysis, by Walter Rudin

Functions of One Complex Variable, by John B.Conway

Algebra, by Serge Lang

Advanced Linear Algebra, by Steven Roman

Complex Analysis, by Lars V.Ahlfors

《简明复分析》·龚昇·中国科学技术大学出版社

Complex Analysis, by Elias M. Stein and Rami Shakarchi

《实变函数论》·周明强·北京大学出版社

《实变函数论》第五版·那汤松·高等教育出版社

《张量分析》第2版·黄克智·清华大学出版社

《近世代数》第三版·杨子胥·高等教育出版社

《解析几何》第四版·吕林根·高等教育出版社

《高等数学》·同济大学数学系·高等教育出版社

• 数学史+科普类读物+工具书+数学思维等

Science and Method, by Henri Poincare

《数学史通论》第二版·【美】Victor J· Katz, 李文林译·高等教育出版社

《数学史概论》·【美】伊夫斯, 李文林译·哈尔滨工业大学出版社

《古今数学思想》·【美】·克莱因·上海科学技术出版社

《数学天书中的证明》·艾格纳 (Martin Aigner) , 齐格勒 (Gunter M.Ziegler), 冯荣权& 宋春伟& 宗传明&李璐译·高等教育出版社

《费马大定理:一个困惑了世间智者358年的谜》·【英】西蒙•辛格·广西师范大学出版社

《算术探索》·【德】高斯·哈尔滨工业大学出版社

《数学指南:实用数学手册》·【德】埃伯哈德·蔡德勒·科学出版社

《数学的语言：化无形为可见》·【美】齐斯·德福林·广西师范大学出版社

《天才引导的历程:数学中的伟大定理》·【美】William Dunham·机械工业出版社

《完美的证明：一位天才和世纪数学的突破》·玛莎·葛森·北京理工大学出版社

《什么是数学:对思想和方法的基本研究》第三版·【美】R·柯朗H·罗宾, I·斯图尔特, 左平&张饴慈译·复旦大学出版社

《数学恩仇录：数学家的十大论战》·【美】哈尔·赫尔曼·复旦大学出版社

《数学大师:从芝诺到庞加莱》·埃里克•坦普尔•贝尔·上海科技教育出版社

《数学世纪:过去100年间30个重大问题》·【意】皮耶尔乔治•奥迪弗雷迪·上海科学技术出版社

——小海cosea

——首稿于2016年10月17日

## 书籍购买

create is my all.魂荡魄，勇者雄心——小海cosea

## 我与她的恋情

天赋和伟大之间，隔着8446个昼夜