irchans

I think the following sentence in the article is wrong "Applying Noether's theorem gives us three conserved quantities—one for each degree of freedom in our group of transformations—which turn out to be horizontal, vertical, and angular momentum.”

I think the correct statement is "Applying Noether's theorem gives us three conserved quantities—one for each degree of freedom in our group of transformations—which turn out to be translation, rotation, and time shifting.”

I think translation leads to conservation of momentum, rotation leads to conservation of angular momentum, and time shifting leads to conservation of energy (potential+kinetic). It's been a few decades since I saw the proof, so I might be wrong.

On the other hand, if the "shortest" proof of halting is more than 10^20 pages long, then we would have major problems "writing" it.

Google found this claim "According to the report, Tesla cars with Autopilot turned on are much safer than cars driven by people. In Q4 2022, there was only one crash for every 4.85 million miles driven."

Here is an article that discusses the safety of driverless cars. https://arstechnica.com/cars/2023/09/are-self-driving-cars-a...

Six months ago, I did not think that we had reached that level of reliability, so I paid my friend.

If AI significantly surpasses humanity in cognitive ability, then I think it will have a much bigger impact than the wheel. (I loved GEB and DH's other writings.)

LLMs have really improved a lot of the last two years and they have shown many unexpected capabilities. I am guessing that they will get some more good input (text mostly), a lot more compute, and algorithmic improvements, so that may very well be enough to become better than 99% of humans at tasks that involve only text. Tasks that require video or image processing may be a little bit more challenging. Having very smart AI's controlling robots may just be five years away. (I recently lost a bet about autonomous driving. Five years ago, I thought that autonomous cars would be better than human drivers by now.)

I'm frightened by what AI will become over the next 10 years.

(The last time I taught math was vector calc 4 years ago. The last time I taught the first semester intro to calc was 30 years ago OMG-LOL.)

https://www.sciencedirect.com/science/article/abs/pii/S03032...

See also

https://en.wikipedia.org/wiki/Multi-armed_bandit

Thanks for the idea.

I don't know if this is really similar, but maybe it's a bit related. One of my friends and I decided to write 100 words a day for 30 days. Two other friends ended up joining our little project. I kept it up for about 100 days, and it made writing much easier for me. Also, about 2 months ago, I decided to spend about 30 minutes a day writing. It is amazing how much easier writing became for me. (When I was young, I found writing to be very painful. The only C's I got in college were in English. Nowadays, I think of my self as a writer --- not a good writer, but a writer.)

    double numerator = 1.0;

    double denominator = 1.0;

    double pi = 3.14159265358979323846;

    int i;

    for (i = 1; i <= n; ++i) {
        numerator   *= pow((x - n * pi / 2.0), 2);
        denominator *= pow((n * pi / 2.0), 2);
    }

    return numerator / denominator;

(Edit for format)

Chernobyl was pretty stupid. Bad reactor design and rather bad operator decisions.

Part of the Fukushima problem was an insufficient sea wall. They did get unlucky with a 7.4 magnitude quake. It was stupid to put the back up generators in the basement where they were flooded. There is a summary of some of the human errors and corruption here

https://escholarship.org/uc/item/30x14946

Three mile island was largely operator error. There also appeared to be corruption during the clean up.

Private companies are always trying to cut costs. Government organizations are often subverted by the industry. America and Western Europe have a fairly decent nuclear safety record. I think it is just plain difficult to avoid corruption for any regulatory organization.

The potential for a serious nuclear incident involving 100,000+ deaths exists. Both Chernobyl and Three mile island could have been much worse. If things had been worse, there was a potential for hundreds of square miles to be contaminated for several decades.

It is very difficult to design a nuclear reactor the can survive operator error and government corruption. There is also the potential for terrorist attacks on reactors.

On the other hand, we do seem to have a very big CO2 problem and maybe we need to take the nuclear risk. As far as I can tell, we cannot rely on solar and wind for more than 70 or 80 percent of our energy needs. Maybe geothermal or wave power combined with solar and wind could be used to eliminate the need for fission nuclear power.

1) Practice makes you better. 2) You can make decisions that affect the rest of your life. 3) Programming (the idea that many procedures can be broken down into precisely defined steps which can be done by a computer.) 4) People are mostly kind. women are somewhat more likely to be kind than men.

5) Your conscious mind can make changes to your brain and your body. You can exercise. You can teach yourself to like or dislike things more. You can change your habits. 6) You can learn skills from a book or other written materials. 7) The idea and laws of probability (including Baysian) 8) The ability to estimate (Fermi Questions).

9) Information Theory and Kolmogorov Complexity (Algorithmic Complexity) 10) Polynomial and Exponential Growth (related to Black Swans) 11) You can use algebra to solve for one variable given a few equations 12) Calculus (Esp Fundamental Theorem of Calculus)

13) Object oriented Programming (Inheritance, Polymorphism) 14) Mathematical proof - the ideas that you can take a bunch of axioms and use logic to reach many amazing and useful conclusions. Furthermore, you can have a high degree of certainty that the result (theorem) is true especially if it is verified by another mathematician or a computer. 15) Game Theory (The Nash Equilibrium often exists and it (or they) can often be found.) 16) Energy and Momentum are both conserved. That fact arises from the symmetries of the universe (invariance under translation by space time, and rotation (see Emmy Noether).

17) Optimization ( many problems have an optimal solution and often calculus or math can be used to find it.) 18) You can guess many of the formulas of physics 19) Thermodynamics (Laws of Thermodynamics) 20) Recursion

21) Extending the idea of the pythagorean theorem (Application of Hilbert Spaces) 22) Declarative Programming

Reposted here

http://artent.net/2022/12/08/ideas-that-changed-my-life-morg...

Over the last 60 years, this has been the most commonly used physics text book in college. It's rather well written, and there are many very good homework problems. It covers the basics of mechanics and Electromagnetism and a few other topics also. The older editions have harder problems that are more instructive. By doing the homework problems, the reader learns physics, calculus, and the ability to manipulate and derive formulas.

This book is about the following problem,

"Given a point P and a set S in a vector space, find the point in S closest to P."

(The most common example of a vector space is the Cartesian Coordinate Plane which is also called R^2 because every point is of the form (x,y) where x and y are real numbers.)

This book is very simply written. All the details of the proofs are provided. Also, Professor Deutsch covers the history of many of the theorems in the book. Applications to Linear Regression, Interpolation, and Control Systems are given. The main prerequisite for reading the book is a course in mathematical analysis (Rudin or Royden) and a course in matrices or linear algebra. The book is based on the graduate level Approximation Theory course which Dr. Deutsch taught for several years.

I do sometimes. Sometimes I test the result, and if it is not correct, then I use a slower, more numerically stable algorithm. I'm not sure if I have ever written code that repeats the same code with quad precision, but of course that would often work. Sometimes you can compute a correction from the "residual".

>>One that uses a cost function iterative algorithm to solve the matrix (this is what everybody does), and a second algorithm using the brute force approach the author showed (known not to work).

We do often use Jacobi or Gauss-Seidel, or something similar (esp for computing estimates of A^(-1)b ). In those cases we never revert back to QR. We do sometimes use Jacobi or GS to improve a result obtained from an ill conditioned QR, but rarely.

Many years ago, I would have the code raise an exception if the result is bad so that I would get a call. When I got such a call, I would think about how to make the algorithm more stable.

Yes, we are too lazy to be 100% formal and many times we are too lazy to be mostly formal. This is mostly because we target our writing to other mathematicians who have no need to see every small step and including every step makes the proofs long. On the other hand, I do feel that generally speaking mathematicians should show more of their work and skip fewer steps.

I find your statement "People can't be bothered to be exact about things because being exact is hard and people avoid hard work." to be very true. Being precise is difficult.

When I left the Navy at age 25, I had just been overseas. I had applied to a Master's degree program at the same university where I got a BS. There was a mix-up and no one had ever looked at my application, but several people knew me and they had my grades from my BS, so when I showed up in person one week before classes, they accepted me (I was paying my own way) and I started. I ended up getting a Ph.D. instead.

In 1999 at age 34, I quit my job as CFO for a small unsuccessful hedge fund. I applied at Raytheon where I had worked briefly before (they were impressed with me) and I applied to work at a lab at the university where I got my degrees. I got offers from both places within a month, and I was working again.

In 2004 at age 39, I quit my research job at another small unsuccessful hedge fund (run by the same people). I once again applied at Raytheon and at the same university job and again got two offers. I was working again in two months.

In 2006 at age 41, I quit again. I spent a year taking fun classes in Machine Learning and Astrophysics. In 2007, the people at the small hedge fund convinced me to work again for them.

In 2009, I quit again because my son was having very serious health issues and I wanted to stay with him in the hospital for six months (he is OK now). In 2010, I applied at the university lab where I had worked before and I ended up working there for 7 years. After 7 years, I was laid off.

   About a year later, I ended up getting a job a huge prestigious hedge fund as a quantitative analyst.  I quit about six months later in 2017 (I just could not hack living in New York City).  The same night that I quit, my old boss from the small hedge fund called me just to say hi.  When he heard that I quit, he immediately offered me a position which gave me a small salary but also made me a partner in the firm.  I still work there now.