Shulou(Shulou.com)11/24 Report--

Following the "endorsement" for GPT-4, Copilot was also crazy Amway by Tao Zhe Xuan.

He said bluntly that when programming, Copilot could directly predict what he would do next.

With Copilot, research becomes more convenient, and Tao Zhe-Xuan also uses it to help him complete the latest research results.

Tao Zhe Xuan said that in this paper, there is only one page about this part.

But to prove that he had completed this page, he wrote more than 200 lines of code, using the newly learned programming language Lean4.

On Tao Zhe Xuan's public code GitHub page, Copilot has increased the speed of writing code by more than half.

Tao Zhe Xuan said that the reason why he chose Lean4 is its "rewriting strategy", that is, a targeted local replacement of a long expression.

For example, if we define a complex function f (x), when we want to enter the expression of f (114514), we can simply "rewrite" x to 114514 in code.

Tao Zhe Xuan said that this feature is simply not too convenient compared to LaTeX, which requires repeatedly entering formulas.

So what new achievements has Tao Zhe Xuan's "one-page proof" brought to us?

A page to prove a new inequality this paper deals with the McLaughlin inequality.

McLaughlin inequality is a classical inequality in mathematics, which is derived from the law that the arithmetic average of non-negative real numbers is greater than or equal to the geometric mean.

Let Y1... Yn is a nonnegative real number. N, define the mean Sk as (the number of terms whose denominator is the numerator):

It appears as the normalization coefficient of a polynomial of degree n with roots.

(remember this formula, we call it formula 1)

Then McLaughlin's inequality can be expressed as:

Where the equal sign holds if and only if all yi are equal.

In calculus, there is also a classic Newton inequality:

For any 1 ≤ k0 and 1 ≤ℓ≤ n, there must be formula 2 or 3.

This is what Tao Zhe Xuan is trying to prove on this page, and the specific proof process is as follows:

You might as well construct a polynomial P (z) about the complex variable z:

From the previous formula 1 and trigonometric inequality:

So all you need to do is establish a lower bound:

By taking the absolute value of P (z) and then taking the logarithm, we can get:

Because t ↦ log (et+a) is convex for any real number and a > 0, an inequality can be obtained:

When aaudr2dint twee2log yj, you can get the following:

The above is the proof process given by Tao Zhe Xuan, but when the normalized | Sn | = 1, the following formula holds:

Next step: build a refined version in addition to the "one-page proof" mentioned this time, Tao Zhe-Xuan proposed another new theorem in this paper, that is, for any 1 ≤ k ≤ ℓ≤ n.

In his blog post, Tao Zhe Xuan revealed that his next step is to come up with a refined version of this inequality.

Tao Zhe-Xuan said that the process of proof is "like practice" and can be done with calculus.

However, he also mentioned that there would be a small difficulty because the progressive symbol was used in this part of the argument.

Let's wait and see what the new conclusion is.

One More Thing Tao Zhe Xuan is a big fan of AI tools, and Copilot, GPT-4, and other assistive tools have been recommended by him.

This time, he also put forward new expectations for the development of the large model, hoping that one day the model can directly generate variants of inequalities.

Paper address:

Https://arxiv.org/abs/2310.05328

Reference link:

Https://mathstodon.xyz/@tao/111271244206606941

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