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There are many objects and structures in mathematics, and we want to do something about them. For example, given a number, we will double it, square it, or calculate the reciprocal according to the context; given an appropriate function, we may want to differentiate it; given a geometric figure, we may want to make a transformation, and so on.

If we define a mathematical program, then inventing the skill to execute the program is an obvious mathematical plan. This leads to so-called direct questions about the program. However, there is a more profound class of so-called inverse problems, which take the following form. Assuming that the program is given and the answer to the execution program is given, is it possible to figure out what mathematical object the program is acting on? As an example, it would be very clear if I told you that there was a number and squared it, and the result was 9. Can you tell me what this number is? Quite simply, the answer is 3 or-3.

If you want to discuss this problem more formally, you will say that you were studying the equation x ^ 2 = 9 and found that it has two solutions. Such examples raise three general questions:

Does an equation have a solution?

If so, does it happen to have a solution?

What kind of set are these solutions in?

The first two problems are called the existence and uniqueness of the solution. The third problem does not make much sense in the case of the equation x ^ 2 = 9, but it may be very important in more complex cases, such as for partial differential equations.

In more abstract language, if f is a function, there is such a proposition in the form f (x) = y. The direct problem is to find y when x is given, and the inverse problem is to find x when y is given. This inverse problem is called solving the equation f (x) = y. The problem of solving this form of equation is closely related to the reversibility of function f. Because x and y may be much more than ordinary objects, the concept of solving the equation itself is very general, so it is one of the central problems of mathematics.

Linear equations the equations that pupils first encounter are typically equations like 2x+3=17. To solve such a simple equation, we regard x as an unknown number, and unknowns have to obey the usual laws of arithmetic. Using these rules, you can transform this equation into a much simpler equation, ∶, subtract 3 from both sides of the equation, get 2x=14, and divide the two sides of the new equation by 2, and you get xS7. We actually proved that if ∶ has a number x that makes 2x+3=17, then the number must be 7. What we haven't proved yet is that ∶ does have such a number x. So, strictly speaking, there should be the next step, which is to verify 2 × 7 + 3x17. Here, it is obviously true, but for more complex equations, the corresponding conclusion is not always true, so this last step is important.

The equation 2x+3=17 is called a linear equation. This is because the function acting on x (multiplied by 2 and then plus 3) is a linear function. As you have just seen, it is easy to solve a linear equation with only one unknown, but it is more complicated if you want to solve an equation with more than one unknown. Consider a typical example of an equation with two unknowns, namely the equation 3x+2y=14. There are many solutions to this equation. After you choose a y, you can make

So there is a pair of (xmeme g) to satisfy this equation. To make the problem a little more difficult, add another equation, such as 5x+3y=22, and then try to solve both equations at the same time. The result at this time is that there is only one solution (a set) of x _ 2 and y _ 4. In general, two linear equations with two unknowns happen to have a set of solutions. If you look at this from a geometric point of view, it is easy to understand. The equation in the form of ax+by=c is the equation of a straight line on the xy plane. Two lines normally intersect at one point, except that they are the same, when they intersect at an infinite number of points, or they are parallel, and they do not intersect at all.

If there are several equations that contain several unknowns, it would be easier to think of them as an equation containing an unknown. This sounds impossible, but it is entirely possible to allow the unknown to be a more complex object. For example, the equations 3x+2y=14 and 5x+3y=22 can be written as a single equation with matrices and vectors.

If you use A to represent the matrix above, x for unknown vectors, and b for known vectors, then the equation becomes Ax=b, and it looks much simpler, even though in fact it just hides complexity in symbols.

However, the process is not just "sweeping the garbage under the carpet to hide", but there are more things. On the one hand, the simple symbol masks many specific details of the problem, but on the other hand, it also reveals something that is invisible: there is a linear mapping from R ^ 2 to R ^ 2. What I want to know is which vector x is mapped to vector b (if there is such a vector). If you encounter a specific set of simultaneous equations, it doesn't make much difference, and the calculations we need to do are still the same. But if you want to make general reasoning, then the matrix equation with a single unknown vector is much easier to consider than the simultaneous system of equations with several unknowns. This phenomenon will appear in the whole mathematics, and it is the main method to study high-dimensional space.

We have just discussed the generalization of linear equations from one unknown to several unknowns. Another direction of generalizing them is to regard linear equations as polynomials of degree 1 and consider functions of higher degree. For example, in middle school, we learned how to solve problems such as

Such a quadratic equation. More general polynomial equations are shaped like

To solve such an equation is to find the value of x to satisfy the equation. This seems like an obvious thing, but when it comes to something as simple as

When it comes to such an equation, it's not so obvious. The solution of this equation is

So, what is the root sign 2? It is defined as a positive number equal to 2 after being squared. But if x equals positive or negative and the square is 2, it seems that the equation has not been "solved" yet. Even if we say Xerox 1.4142135... It is not entirely satisfactory, because it only writes the first paragraph of an endless formula, and there is no discernible pattern in it.

Two introductions can be obtained from this example:

One is that for an equation, it is often the existence and properties of the solution that matters, not whether the solution can be found or not.

Although when we say

It doesn't teach us anything, but this argument does include the fact that ∶ 2 has a square root. This point is usually put forward as a corollary of the so-called intermediate value theorem. This theorem points out that if f is a continuous real-valued function, and f (a) and f (b) are on the side of zero, then somewhere between a _ line b, there must be a real number c such that f (c) = 0. This result can be applied to the function f (x) = x ^ 2-2 because f (1) =-1 and f (2) = 2. So there must be an x between 1 and 2 so that x ^ 2-2 is zero. For many purposes, it is sufficient to know the existence of this x, plus to know that the property of defining this x makes it positive and squared 2.

With a similar argument, we know that all positive real numbers have positive square roots. But when we try to solve a more complex quadratic equation, the situation is different. There are two ways to choose at this time. For example, consider the equation.

We will notice that when x is 4, its value is-1, and when x is 5, its value is 2, so we know from the intermediate value theorem that this equation has a solution between 4 and 5. But if you use the matching method

You will get two solutions, which is more information than using the intermediate value theorem. We have proved the existence of the root sign 2 and know that its value is between 1 and 2. Now we not only know that the equation x ^ 2-6x+7=0 has a solution between 4 and 5, but also know that this solution is closely related to the solution of the equation x ^ 2 = 2. It can even be said that this solution is constructed from the solution of the equation x ^ 2 = 2.

This proves that there is a second important aspect of solving the equation, that is, in many cases, the explicit solvability of the solution is a relative concept. As long as a solution of the equation x ^ 2 = 2 is given, when solving the more complex equation x ^ 2-6x+7=0, it is no longer necessary to get any new input from the intermediate value theorem, only a little algebra is needed.

But the root sign 2 in this expression is not defined by an explicit formula, but as a real number. This real number has some properties, and we can prove its existence.

Solving higher order polynomial equations is much more difficult than solving quadratic equations, and many attractive problems arise. In particular, there are complex formulas for solving cubic or quartic equations, but solving quintic and higher equations has been a famous unsolved problem for hundreds of years, until the 19th century. Abel and Galois proved that the formula of the explicit solution could not be found.

The polynomial equation of multivariable has such an equation.

We can see that it has many solutions ∶ if x and y are fixed, we get a cubic polynomial equation of z, and all cubic polynomial equations have (at least one) real solution, so for every fixed x and y, there is a certain z such that the triple (x ~) becomes the solution of this equation.

Because the formulas for the solutions of cubic equations are very complex, it makes no sense to accurately describe the sets of all these triples. However, if we regard this set of solutions as a geometric object, that is, a two-dimensional surface in space, and consider some qualitative questions about it, we can learn a lot from it. For example, we may want to know its general nature, and we can explain these problems clearly in the language of topology.

Of course, it can be further extended to consider the simultaneous solution of several polynomial equations. Understanding the solution set of these equations belongs to the field of algebraic geometry.

Diophantine equation whether a particular equation has a solution depends on where it is allowed to be solved. If only x is allowed to be a real number, then the equation x ^ x + 3 ^ 0 has no solution, but in the complex number, it has two solutions. The equation x ^ 2 + y ^ 2 = 11 has infinitely many solutions, but if both x and y are integers, the equation has no solution.

The above example is a typical Diophantine equation, and the sight of this noun means to require its integer solution. The most famous Diophantine equation is Fermat equation.

Thanks to Wiles' work, it is now known that when n is greater than 2, it does not have a positive integer solution. In contrast, the equation x ^ 2 + y ^ 2 = z ^ 2 has an infinite number of integer solutions. A large part of modern algebraic number theory is discussing Diophantine equation directly or indirectly. As for real or complex equations, it is fruitful to discuss the structure of the set of solutions of Diophantine equations. This kind of research belongs to the field of arithmetic geometry.

A noteworthy feature of Diophantine equations is that they are extremely difficult. So it's natural to wonder whether there might be a systematic approach to them, the 10th question in Hilbert's famous list of questions in 1900. But it was not until 1970 that Yuri Matiyasevich pointed out that the answer to this question was no.

An important step in solving this problem was made by Church and Turing in 1936. Only by formalizing the concept of the algorithm (in two different ways), thus clarifying the concept of "systematic processing", did we take this step. Before the computer age, this was not easy, but now we can restate the solution to Hilbert's 10th problem as follows: ∶

Want to find a computer program so that after entering any Diophantine equation, if the equation has a solution, it must output "YES", when there is no solution, it will output "NO", and never make mistakes, which is impossible.

What does this tell us about the Diophantine equation? We can no longer dream of a final theory that includes all such equations; instead, we are forced to focus on special categories of these equations and develop different solutions to them. If it were not for the noteworthy connection between the Diophantine equation and the very general equations of other parts of mathematics, it would seem to make the Diophantine equation uninteresting after solving the first few equations.

Such as equations.

It looks very special, but in fact, the elliptic curve it defines is the central problem of modern number theory (including the proof of Fermat's great theorem). Of course, Fermat's Great Theorem itself is also a Diophantine equation, but its research has led to the great development of other parts of number theory. The correct conclusion that should be drawn may be that ∶ solves a special Diophantine equation, and if the result is not just adding another one to the list of equations that have been solved, then it is attractive and worth studying.

Differential equation

So far, the equations we consider are based on numbers or a point in n-dimensional space as unknown things. To generate such equations, we make different combinations of the basic operations of arithmetic and then apply them to the unknown.

Here are two famous differential equations to compare with the equations discussed in the past, ∶.

The first is the "ordinary" differential equation, which is a simple harmonic equation of motion, which has a general solution.

The second is the partial differential equation, which is the heat equation.

There are many reasons why solving differential equations is a leap in ingenuity.

One reason is that what is now unknown is a function, which is much more complex than numbers or points in n-dimensional space.

The second reason is that the operational differentials and integrals applied to functions are far less "basic" than addition and multiplication.

The third reason is that differential equations, even very natural and important equations, can be solved in a "closed form", that is, an unknown function f can be expressed by a formula, only as an exception rather than as a rule.

Now go back to the first equation.

This means that the differential equation can be regarded as a matrix equation extended to infinitely many dimensions. The heat equation has the same property ∶ if ψ (T) is defined as

Then ψ is another linear mapping. These differential equations are called linear, and their obvious connection with linear algebra makes them much easier to solve. A useful tool in this respect is Fourier transform.

What about the more typical equations, that is, those that cannot be solved in a closed form? At that time, the focus once again shifted to whether there was a solution. If so, what are their properties? Like the polynomial equation, it depends on what is considered a permissible solution. Sometimes, it is as if we are studying the equation x ^ 2 = 2 again. It is not difficult for ∶ to prove the existence of the solution. We just need to give it a name. Equation

Is a simple example. In a sense, it cannot be solved, and it can be proved that you cannot find a "basic" function constructed from polynomials, exponential functions, trigonometric functions, etc., and you will get e ^ (- x ²) after differentiation. However, in another sense, the equation is easy to solve, you only need to integrate the function e ^ (- x ²), and the resulting function is the normal distribution function. This function is of fundamental importance in probability theory, so it is given a name (notation):

In the vast majority of cases, it is hopeless to write a formula for the solution. A famous example is the three-body problem ∶ gives three moving objects (particles) in space, and suppose that they attract each other by gravity, asking how they will continue to move? The differential equation describing this situation can be written by Newton's law. For two moving objects, Newton solved the corresponding equations, which explained why the planets move in an elliptical orbit around the sun, but for three or more objects, these differential equations proved to be very difficult to solve. It is now known that there is a very profound reason for this difficult situation ∶ when these differential equations will lead to chaotic behavior. However, this opens the way to study very interesting questions such as chaos and stability.

Sometimes, there is a way to prove the existence of solutions, even if they cannot be easily determined. At this time, we can not ask for an exact formula, but only want to get a general description. For example, if the equation is time-dependent (for example, both the heat equation and the wave equation), people will ask, does the solution decay, blow up, or remain roughly the same with time? These more qualitative problems are called asymptotic behavior problems, and there are some techniques to answer some of these questions, although there is no explicit formula to give the solution.

As in the case of Diophantine equation, partial differential equations, including nonlinear partial differential equations, have some special and important classes, and the solutions can be written accurately. This gives a very different research style ∶ people once again focus on the properties of the solution, but this time it is a more algebraic property in nature, that is to say, the formula of the solution will play a more important role.

This article comes from the official account of Wechat: Lao Hu Shuo Science (ID:LaohuSci). Author: I am Lao Hu.

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