In analytic number theory, an arithmetic function is simply a function from the natural numbers to the real or complex numbers. (One occasionally also considers arithmetic functions taking values in more general rings than or , as in this previous blog post, but we will restrict attention here to…]]>

In analytic number theory, an arithmetic function is simply a function $latex {f: {bf N} rightarrow {bf C}}&fg=000000$ from the natural numbers $latex {{bf N} = {1,2,3,dots}}&fg=000000$ to the real or complex numbers. (One occasionally also considers arithmetic functions taking values in more general rings than $latex {{bf R}}&fg=000000$ or $latex {{bf C}}&fg=000000$, as in this previous blog post, but we will restrict attention here to the classical situation of real or complex arithmetic functions.) Experience has shown that a particularly tractable and relevant class of arithmetic functions for analytic number theory are the multiplicative functions, which are arithmetic functions $latex {f: {bf N} rightarrow {bf C}}&fg=000000$ with the additional property that

$latex displaystyle f(nm) = f(n) f(m) (1)&fg=000000$

whenever $latex {n,m in{bf N}}&fg=000000$ are coprime. (One also considers arithmetic functions, such as the logarithm function $latex {L(n) := log n}&fg=000000$ or the von Mangoldt function, that…

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I began thinking this question a long time ago, when I made up my mind to be a mathematician. To answer this question, we need to know the definition of math. So what is mathematics. Most people begin to learn math in kindergartens, and this process never pause even till university. I didn’t know what is math until I finished my undergraduate study. From the undergraduate education, we can generally define math as a subject that deal with quantity, structure, space and change.

Each of them comes from the process when human beings began to learn the world they live in. Math comes from the concrete objects, but not limited to special objects. Mathematicians always try to include more objects in one mathematical concept by subtracting more general properties from concrete things. Than to study mathematical concepts, we can obtain more general properties for many concrete things. From this point of view, we can see that one kind of good math is to construct new theory or concept in which we can include several old ones. This kind of taste is similar with that of physics. It is ambitious. Hundreds of physicists, including many famous ones like Einstein, have spent more than one hundred years to look for a unified theory to explain our world, but in vain. There seems always a balance in it. Paying attention to generality may sacrify special characters. Taking PDE as an example. After Hormander (who studied linear PDE in a systematic way), no one can find a general theory for nonlinear PDE. Because different equations have different properties, and the method for one equation usually does not work for another. This lead to the fact that nowadays, most of mathematicians in this field usually focus on one type of equations in the whole life time. It seems now in the field of PDE, the top work is to introduce new tools to solve problems.

One view is:

1st class is introducing new concept and building new theory.

2nd class is expressing new ideas and making new tools.

If judging it in this view, PDE is far from the good math.

Another kind of good math is the subject which can connect one branch with the other. One example is Gauss-Bonnet theorem in differential geometry. This is good because it connects geometry and topology. It provides a bridge to understand two different branches as a whole. Another field share the same character may be the geomety analysis, which is using analysis tools (mainly PDE) to study geometry.

To be continued……

]]>After these days learning, I have a strong feeling that PDE is a mass (I do not want to be rude here, so I did not use the word ‘shit’)! I say so because I found dynamical system is such a beautiful subject. It can describe the evolution system in such a detailed way, which can never be reached for PDE. In dynamical systems, the properties we care includes repeatability and stability. The phenomenon that just happen once and will not be repeated is not the centre of research. Judging from the history of DS, we can understand this easily. The dynamic systems used to be developed for understanding the motion of planet. Most planets have periodic orbits. They will ‘visit’ earth in some definite frequency. That’s why people care about the phenomenon that can be repeated.

After people realized the earth is moving around the sun, a concern is spread among some educated persons, that is, will the earth falling to the sun? The stability of dynamic system comes from this question. In the theory of hyperbolic dynamical systems, the manifold can be divided into two or three parts: stable part, unstable part, (sometimes we may have) central part. This properties can be preserved under a map, called Anosov map. If this map is a diffeomorphism, we can get structural stability, which is helpful to classify dynamic systems. More detailed analysis may reveal that many dynamical phenomena are only related with topological properties, not with differential properties. So much energy has been put into study of topological dynamical systems. Mathematicians have constructed a great mansion to describe the phenomenon in a very detailed way.

However, when we get back to PDE, we may find the situation is really dispointing. In PDE, at this moment, people is still struggling on the well-posedness. This is good, because mathematicians are usually very careful for the objects that may not exist. Except that, what we know for a system is rather limited. We may get some assymptotic properties, and also local wellposedness, but, what is the middle part? We may see it clearly through following example. One of the million dollar problem is global wellposedness of Navier-Stokes equation, which is used to describe the motion of fluid. Currently, we have some results for local wellposedness, and also some assymptotic estimate. But we know almost nothing about the situation of middle part (This may be improper, because some dynamical results have been obtained, such as bifercation, etc, but still very limited). If we think in this way, then we may find the turbulence is just something happen in the middle time period. But we cannot describe it by NS equation (Here I do not consider whether this model is good or not). So PDE theory is far from being perfect, especially for nonlinear PDE. It has been called ‘dirty math’, because we have no general mathods to deal with nonlinear PDEs.

So much about the frustrating side. Above discussion also bring us good news. There are still many open problems in PDE, and we need to introduce new tools, new ideas into PDE research to help us understand the solution better!

]]>I will devote this article to Cartan’s package.

Assume is a vector field on . The flow of is denoted as . Suppose is diffeomorphism, and its inverse is with push-forward .

**Definition1**: For vector field with flow , the Lie derivative of a tensor is defined as .

**Definition2**: For a vector , define interior product of tensor with vector as , satisfying

.

Cartan’s package is

.

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1) Assume

**Proof:**

2)

**Proof by induction:**

.

Suppose it holds for , then we can prove it also holds for

3)

**Proof:**

4)

**Proof:**

5)

**Proof: **

due to skey-symmetry.

6)

**Proof:**

7)

**Proof by induction:**

Supppose it holds for , prove it for

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When we look back, we may find it has many terms on RHS because the manifold is not flat, i.e. the frame is changing with manifold. So when we take derivative, we not only need to differentiate the component of the tensor fields, which is a function of manifold, but also need to differentiate the frame, which also rely on the space point. In the end, we get several terms when we calculate the derivative.

]]>;

while in the case of , we calculate Fourier coefficients .

These two cases are much similar, especially for the algebraic properties. However, the difference between and and other difference between and will show they are two different things.

In this set of posts, I will discuss something about Fourier series. The main topic may be focused on:

1) Decay of Fourier coefficients and smoothness of function .

2) Pointwise convergence.

3) Bochner-Riesz summation.

4) Lacunary series.

Fourier transform is mainly used to measure oscillation. We can look functions as signals. The higher frequency, the more oscillation. By calculating , we ‘take out’ the wave with frequency k. The larger , the larger the amplitude, and the heavier of this frequency weighs in the function .

By Riemann-Lebesgue lemma, we know for function , approaches to 0 as , and this rate of decay is arbitarily slow.

For smooth functions, we have the decay estimate

.

Conversely, if the Fourier coefficients decay as

for all , then has partial derivative of orer , where is the largest integer strictly less than s.

For bonded vatiational function, .

]]>Following last post of this series, we already have tangent and cotangent vector space at point , denoted as and . Collecting the space for all point on the manifold, we get bundles. We have known how to define tensor for finite dimensional vector space. Now we are going to define tensor on tangent or cotangent spaces, which are infinite dimensional vector spaces. Define type tensor space of manifold M at p as follows:

.

After collecting all tensor space for all point of $M$, we get tensor bundle . The natual projection from to is the bundle projection, while is the fibre of the bundle at point .

Assume is a smooth mapping, if , then we say is a smooth section of tensor bundle , or a type smooth tensor field. The definition is easy to extend to exterior differential form by anti-symmetrizing.

We can define exterior vector bundles and exterior form bundles as

.

So exterior differential form of degree r on M is a smooth map as a section of exterior form bundles. Another view is to look it as a smooth map , which is multi-linear and alternating.

]]>After we defined the manifold, we can define smooth functions on each subset of the manifold, however, all the smooth functions defined near a point p, under the operations + and multiplication with real numbers, cannot form a linear space, because the nul element is not defined uniquely. So we take the equivalent class, which is called germ of manifold at point p. All the germs form a vector space, denoted as .

By defining parametric curve , we can define a linear functional on , . Using this notation, we can define a subspace of by

,

where is the set of all parametric curve through p.

The quotient space is the cotangent space of manifold at p, denoted as . The element in this set is called cotangent vector, denoted as .

Define equivalent relation in as

then each equivalent class is a linear functional on cotangent space . All the equivalent classes comprise tangent space .

An element in tangent space can be viewed as a linear functional on by denoting , which we called directional derivative. Here we can also get another definition for tangent space , which is the set of all linear operators on satisfying .

It is useful to express tangent vector and cotangent vector by linear combinition of natural basis.

where

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Welcome and enjoy!

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