Category: Blarghing

Here’s a cute little problem: let be a polynomial of degree with real coefficients such that it has non-negative values. Show that for all real .

Notice that goes to positive infinity in both limits, as is an even order polynomial. Furthermore, it is also non-negative at 0. Now, the minima/maxima of exists where , but evaluating at such points reveals that the extremal values are all positive.

A6.  X is a subset of the rationals which is closed under addition and multiplication. 0 ∉ X. For any rational x ≠ 0, just one of x, -x ∈ X. Show that X is the set of all positive rationals.

I’m not sure if the Putnam was easier back in the day like the AMC/AIME, but this A6 was ridiculously simple. The problem popped up in Larson’s book 1.9.5 with some hints, which I’ll outline the solution with.

First, note that 1 is in our set. This can be seen several ways, with the official solution being much more elegant (only one of -1 or 1 can be in the set, and we also need to maintain multiplicative closedness). With this, we can prove that all positive integers are in the set.

Finally, assume that there exists a negative rational number in the set. Then simply add it to itself however many times is in the denominator, and it’ll be a negative integer; contradiction.

Going on vacation really puts into perspective a lot of things, though I want to focus on how far my mathematics and problem solving skills have developed since high school. I really think I was most creative at the younger age when doing all those math contests, but I’ve matured now.

It seems at Brighton’s age and level of a 8th grader, I can provide a level of guidance that’s similar of how I feel now with respect to my adviser. It’s really a nice feeling to think that I am indeed getting smarter.

It’s been awhile since I last posted, just wanted to document a little of what happened.

1. Passed my prelim exam. Overall, it went pretty well in my opinion. It seems the minor actually went much worse than my major. I’m glad that’s over.
2. Went the the Gene Golub summer school immediately after. It was an extremely fun program where the emphasis seemed to be on making connections rather than actually learning the mathematics. I believe I spent more time playing volleyball than actually doing math.
3. Traveled alone in Germany: first to Heidelberg, then to Boppard and finally back to Berlin. It was extremely relaxing, and I met some incredible people whom I hope to hopefully keep in touch in the future.
4. Back to work now…

I’ve gotta really buckle down in my prelim studying and those sorts. Literally only got a month left before the big day.

So in lieu of other stuff, there’s an ongoing PDF about semigroups which might help some poor bloke down the line.

Link

What a wonderful game. I’m glad I moved from the Witcher to this little gem; it really does show that you don’t need super graphics with a deep storyline to have a compelling gaming experience. Rather, Stardew Valley is a retro-style, “farming” simulator, with just a little story component.

I think the main draw of it is how well it’s implemented. The flow of each day feels so natural, with just the right amount of routine and just the right amount of surprises. The first year feels like you are moving into a brand new home, trying to find friends and start a new life. I’m finally feeling settled into my little farming house.

The main drawback is how time consuming the game is haha.

I’m playing through the incredibly addictive game of Stardew Valley, while also reading through Steinbeck’s East of Eden. It seems a common theme between the two is the setting of valleys; in the case of Eden, it’s the Salinas valley in California.

If Hidden Valley Ranch taught me anything, it’s that valleys hide magical places. In Stardew, it’s a dwarves and magic. In Eden? I’m not sure as I’m only 5

Anyhow, spring break is over.

(This is the last post about the Witcher 3.)

I finally finished the entire pack of Wild Hunt (WH), Heart of Stone (HT) and Blood and Wine (BW) two days ago. The world is absolutely incredible, with so many intricacies tied together. I never managed to complete RPGs before: the old Star Wars were too clunky, Torchlight just felt grindy (and too easy after a few spells), Dragon Age never really took hold of me. So I was surprised how much I enjoyed jumping into such a large universe, and loving every moment of it. By around 10 hours in, I was reading a bunch of the in-game texts and notes to figure out what’s going on.

Still, there are some issues. The combat mechanics felt broken in a sense. I hated how there’s instant kill mechanics for some of the bosses, and the monsters (and people) just felt like paper even on the hardest difficulty. The storyline for WH is so long that I forget the end goals a lot, while there’s pacing issues with WH and BW. After all, if there’s murders with a time limit or a global crisis which only you can solve going on, one shouldn’t be really chilling out and drinking.

But besides these, the game never felt grindy. The income balance felt fine for most of the game, and doing the side quests is actually a pleasure. There’s a few simple quests in the beginning, but by the end, it seems that all the quests had some twists going on. There wasn’t simply fetch or kill quests… it was kill, but note that this monster kills for a (almost) valid reason quest.

I’m really tempted to do a play through over the summer with a different build and ending.

 

I know. I talked about B&W just 3 days ago. But it’s spectacularly engrossing. The story so far is captivating, with characters that have just enough mystic to them. Of course, there seem to be a few background stories I missed from not playing the previous games, but the writers were smart enough to either make them non-consequential or (better yet) imply the past.

Spring break is coming up. I hope to finish this obsession and focus on wine math during.

Oh my goodness.

The Witcher 3 is blowing my mind again. The Blood and Wine expansion is gorgeous. AND I can take super nice pictures using Nvidia Ansel. It’s amazing.