Category: Math/Programming

This post won’t have too much math, but mainly musings.

After a four month hiatus from research, it seems my dive back into the world of unknown math is being squandered by my ineptitude. My documentation of the progress made, and the comments in my code seems to be greatly lacking. I’ve spent a good chunk of time trying to recall progress.

I think one of the cooler things Bernstein polynomials can prove is that polynomials are dense (uniformly too IRCC) in continuous functions on a bounded domain. Durrett has a proof in his book using a probabilistic view, which seems much cleaner than an algebraic/analysis proof such as here.

There’s a problem I solved awhile back which I was quite proud of:

Determine, with proof, the largest number that is the product of positive integers whose sum is 1976. (IMO 1976)

Note that this was at a high school olympiad level. Two days ago, my little brother brought me a state Mathcounts preparation packet. One of the problems was

Write 33 as the sum of two or more distinct prime numbers so that the product of these prime numbers is largest. What is the largest possible product?

While a proof is not needed, the move from an high school international level to a middle school state level in 40 years is pretty is pretty impressive.

Let W XY Z be a square. Three parallel lines d, d’ and d” pass respectively through X, Y and Z. The distance between d and d’ is 5 and the distance between d and d” is 7. What is the area of the square ?

This problem was unusually tough after the geometry was done. The three lines gives a system of equations using Pythagorean:

2s^2 = x^2 + 44, s^2 = (x+y)^2 + 4, s^2 = y^2 + 25

But solving this system doesn’t seem to be trivial.

Started subscription to CM magazine. I will occasionally post solutions here

Show that if a 5×5 matrix is filled with zeros and ones, there must always be a 2×2 submatrix (that is, the intersection of the union of two rows with the union of two columns) consisting entirely of zeros or entirely of ones.

Solution: Any column of the matrix will have a number with 3 or more instances of that number in that column. WLOG, assume that number is 1, and choose 3 rows. We will now consider this sub 3×5 matrix.

If the remaining 4 columns contain a column with two 1s, then we are done. Else, note that there are only 4 remaining columns with one 1 or zero 1, with one of those columns being all 0s. Hence, we will have a 2×2 submatrix of 0s.

Convocation was today.

The speaker talked about how we, as Brown students, must be perceptive in this world. We must continue on the path of improvement, for both society and self. She spoke well. But the thing is, I couldn’t care less for this.

It’s not that I don’t have sympathy for the plight of African-Americans. There is just no way I can influence that sort of things. My personality is counter as to what such activists expect from one another. In another context, my opportunity cost of doing such social work is great.

I used to find these problems interesting. I used to be able to speak intelligent with my friends on these topics. But as time turned, the less I read about #BlackLivesMatter, inequality or gender issues. Now, I have but a cursory overview of these matters.

Specialization is a good thing. That’s what econ says.

I went to Boston again on Friday to meet up with Ashley. As usual, it’s a relaxing time. There is a sense that time stops when I am there; no overwhelming pressure to succeed, even when she is so successful with her endeavors.

Somehow our friendship has continued all these years. I am glad of that.

1. Oddly enough, Brighten has a tendency to get up in the middle of the night. Last week, at 12am, he woke up and came outside. I intercepted him, and asked what he wants to do. He proceeded to tell me he wants to brush his teeth.

   He then walks into the master bedroom’s bathroom, waking up mom and dad. He brushed his teeth.

   Now the weird part: he remembers none of this apparently.

2. Inspired by a podcast, I think I’ll start taking notes on the interaction with people that I don’t meet daily (or even daily, I’ll note important things).

Been stuck in my head lately.

After struggling with power cleans for a month, I’ve decided to not do them anymore. Trying to catch them is just too hard on my wrist, and I can’t seem to be more explosive doing it. If only I had a coach for a few sessions on this stuff to correct my errors.

On a similar note, Brighten is in the finals of the tennis tournament. He is even playing in the higher division! Quite proud of him for this.

I read these two books around one and a half weeks ago, and it made me realize how much I’ve been missing out. Not reading during the semesters stopped me from exploring stories which television or movies can ever tell. The details which books can delve into about the characters is simply amazing. What he wants to do, how he interacts, and how he feels. These are things which are explored in a deeper context in prose.

For A Spool of Blue Thread, it really has no plot. The whole book is a flowing family saga about the Whitshank family and their house, about their lives and their fears. Somehow, it entranced me to keep on reading. It made me realize again how intricate the stories must be for me to be here; how my great-grandparents met and fell in love, how they dealt with tragedy and happiness.

It also made me wish I have a better memory of stories. I don’t recall much at all of my early life, as trading stories have never been my thing. Maybe I’ll write them down from now on.

As for the Invention of Wings, I finished the book in the span of two days. That should be enough praise for any book.