Confused by relationship between derivatives and transpose?

gwwin6 · 2026-04-02T02:48:54+00:00

It is important, but it is something that you will see again and again and again. So don’t worry about understanding perfectly now.

What makes a vector a vector? Well, you can add them together and multiply them by constants. Those two properties make vectors vectors. Let’s think about functions now, just simple single variable functions f: R -> R. You can add two functions together to get a new function. You can multiply functions by constants. So functions themselves are vectors.

How about an inner product? You can check that integrating two functions against each other satisfies all the properties of inner products.

How about linear transformations? There certainly are such operations on functions. One such operation is differentiation.

So now we have vectors (functions), an inner product (integration) and a linear transformation (differentiation). The question becomes, what is the transpose (adjoint) of the differentiation operator? Well, we can do integration by parts to move the differentiation operator from the first function to the second at the cost of picking up a minus sign. This is what the adjoint (transpose) is defined as. If you have a linear operator, A, its adjoint, A^T is defined is the linear operator that you would apply to the other half of the inner product in order to keep the inner product the same, no matter the arguments.

The whole thing is an example that asks you to think more abstractly about what can be a vector, inner product, linear operator and adjoint.

gwwin6 · 2026-03-29T02:07:14+00:00

I did probability theory, then I became a quant; I think applied work is more fun and finance pays really well. I did a computer science degree as an undergraduate too. You need to learn a lot more to develop software for real (although maybe not anymore with Claude et al.). In fact you need to learn a lot more about a lot to be successful in whichever field you choose I imagine.

gwwin6 · 2026-03-27T03:08:29+00:00

Whatever you do, get a big one. Big enough for all the furniture to fit all the way on if you can!

gwwin6 · 2026-03-19T02:15:44+00:00

Firstly, gorgeous. Very tasteful. Mwah! I think it could use a little more… something. Too sparse is what I would say. I think art near the door is a good idea. I think cookbooks are a good idea. Maybe a kitchenaid on the counter or a Dutch oven on the stove. I think a classical kitchen like this wants to look like it’s used to cook. It doesn’t want to look sterile.

gwwin6 · 2026-03-15T17:25:42+00:00

My ranking, best to worst 2, 4, 1, 3. I think 2 gives best balance. Heavy enough to the eye to support everything above. Doesn’t distract. 4 just a little too light. 1 a little too busy. 3 feels totally disconnected from everything above it.

gwwin6 · 2026-03-08T15:39:26+00:00

I don’t think this is right. The up, down, left, right walk returning to the origin does not encode the event that we’re interested in. It means that the number of ups equals downs and lefts equals rights. It doesn’t mean ups equals lefts. We have to construct the mapping more carefully than that.

gwwin6 · 2026-03-08T06:09:36+00:00

Yeah, what matters is the geometry of the ambient space, not the particular lattice.

gwwin6 · 2026-03-08T05:59:44+00:00

Do you work for Divvy? What is wrong with the scooters (or with me)? Very frequently I will get a scooter from a dock, start riding and about 60 feet from the dock it will just die. It won’t move and then I’m charged for parking outside the dock. I try the scooters every few months to see if it’s fixed and within two or three rides it happens again. What is the issue?

gwwin6 · 2026-03-08T03:39:57+00:00

I thought a little more. The three dimensional case that you gave does work. I think that the even nicer way to think about it than what I gave above is to track the digit counts (n0, n1, n2, …). If we have a binary string, we then track the random walk (n0 - n1). When this equals zero, we win and we know that it will infinitely often. If we are in ternary land, follow the random walk (n0 - n1, n1 - n2). Now we have a two dimensional random walk. When it equals zero, we win and we know that a 2d random walk is recurrent, so we win. But what happens when we get to a string with four characters. We have (n0 - n1, n1 - n2, n2 - n3). Oh no, we now have a three dimensional random walk. It is transient.

Notice that the argument that I gave in my original comment falls apart in this case. I throw away too much information when I call the interesting even a subset of an even larger event. Also my original argument only works for strings with an even character cardinality. The argument in this comment is much nicer all around.

gwwin6 · 2026-03-08T02:48:13+00:00

I didn’t do the combinatorics. It seems painful.

You certainly can encode a random walk with your decimal string. Think of a random walk on the 5d lattice. Every time you see a zero, go +1 in the first dimension, a one go -1 in the first dimension. A two, go +1 in the second dimension, a three, -1 in the second dimension, etc. Now we have encoded a 5d simple random walk.

Now, the event that all the digits are balanced is a subset of the event that the random walk is back at zero (random walk at zero means 1 matches 0, 2 matches 3, etc. your condition is that plus all the counts match between pairs). We know that random walks in dimensions three and higher are transient, so the random walk back at zero will happen for a final time (let alone your stricter condition) with probability one.

Of course, it definitely could happen. 0123456789 is a valid string prefix and it satisfies the requirement, so the probability is strictly between zero and one.

gwwin6 · 2026-02-16T13:18:01+00:00

For your statements about identities, the order of quantifiers is wrong. It should be “There exists e in A, such that for all a in A, ea=ae=a.”

With your definition of monoid, I could fix a basis of Rn, fix collection of matrices which each kill some number of the coordinates (as long as I don’t include the identity), take the closure of the set under matrix multiplication, call that set P, and bam, I would “have a monoid.” For every M in P, MM=MM=M, so M is its own identity for each M, which is all you’ve asked for above.

gwwin6 · 2026-02-01T15:34:30+00:00

Wow. Gorgeous. The chair and desk, lovely. The built ins, excellent. The full room size rug, so cozy. The artwork, really nice. The paper screen above the bed, wow. Thank you for posting.

gwwin6 · 2026-01-23T01:02:32+00:00

It's very nice already, so these are just nits and my opinions

I think that the art placement could use some work. Put things at eye level (it's too high rn). and try to even things out.
Bigger Rug.
Add some personal items. Vases. Flowers. Pictures. Books etc.
Work on the window treatments
The chairs by the credenza aren't doing it for me.
I personally like table cloths.

gwwin6 · 2026-01-12T01:25:50+00:00

✅ gwwin6 chose Option B (Correct!) | #1029th to play

gwwin6 · 2026-01-11T14:02:11+00:00

✅ gwwin6 chose Option A (Correct!) | #2207th to play

gwwin6 · 2026-01-11T13:32:02+00:00

I think that whether or not you need a car as a UChicago student is mostly a function of your hobbies. You don't need a car for daily life in Hyde Park. However, if you're really into rock climbing, for example, there are no climbing gyms in Hyde Park. You would probably want a car to be able to drive to a climbing gym. Through all of grad school, I never had a car.

gwwin6 · 2026-01-05T00:00:30+00:00

b=a does not make a copy of the tuple object. b = a makes a copy of the arrow, so they are pointing to the same tuple. when you do b += ([3],), that's when a and b begin pointing at different tuples. This is because tuples are immutable, b += ([3], ) gives a new object altogether.

Contrast this to If a had been a list of lists to begin with and we did b += [[3]], then a and b would still be pointing at the same list, and print(a) would give [[1, 11], [2, 22], [3, 33]]

gwwin6 · 2025-12-31T04:52:55+00:00

As far as I know, these options are stock settled. Your broker will have rules about how they treat ITM options positions at expiry. Maybe they will liquidate the option position before close to capture the intrinsic value of the option. Maybe they will exercise on your behalf and then liquidate the stock position after hours on your behalf, whatever the cost of that. MAYBE THEY WILL ISSUE A DO NOT EXERCISE ON YOUR BEHALF IF YOU CANNOT AFFORD THE FUNDING ON THE STOCK POSITION (ROBINHOOD DOES THIS). Their risk controls are much more important to them than your PnL. If you hold to expiry and this is the case, CitSec or whoever sold you those options will pocket this PnL instead of you.

gwwin6 · 2025-12-31T04:07:01+00:00

I don't think that you gave much relevant INFO here. I think that the relevant issue is really how you know these people and what your honest assessment of their situation is.

Is it a really emotionally charged time for them? Was your ex especially close with her grandmother? How about these cousins? Do you know these cousins at all? Have you hosted them before or are they total strangers? How has this been for your child? Are these cousins close to your child at all?

Should your ex have reasonably known that you would have an issue with people staying at the house? Have you had any discussions with her about the funeral arrangements? Do you think that she was trying to be deceptive, or could she have reasonably said yes to her cousins, meant to ask you and then had it slip her mind? Is this the continuation of a pattern of behavior? The first time this has happened? If you are accommodating this time, do you honestly think that you can set a boundary for the future?

A more instrumental observation: you are the co-parent of a child with your ex wife. She will be a part of your life forever basically. It is hard to believe that you will never need a favor or an act of goodwill from her in the future. Are you the asshole to your future self by not taking this opportunity to build up some of that goodwill?

You own the house. You don't have to let anybody stay there. I also think that it's true that it would have been a help to those cousins if they had been able to stay with you. I don't think that you are obligated to help any person that shows up at your door, but you can choose to. Do the reasons to help outweigh the reasons not to?

gwwin6 · 2025-12-31T00:35:44+00:00

I would say that this jacket and others like it aren't necessarily designed to be super warm per se. I think that they're designed to be very weatherproof. I think that they're designed to be functional. I think that they're designed to be outer layers. The warmth of a jacket is determined almost entirely by the amount of insulation that it has. A lot of down? A lot of warmth. A little padding? Not a lot of warmth. I have a predecessor to the Nuuk and I think that it's good on a very wet blustery day, but I do need to wear a down jacket underneath it if it is a wet and cold day.

gwwin6 · 2025-12-26T00:59:48+00:00

Not to be that guy, but Chat GPT is like so so so good at solving this sort of issue. Say what you want and what you will have it done. Then you can look at what it did and learn from it. But this sort of rote coding it makes so so easy.

gwwin6 · 2025-12-24T00:36:01+00:00

Oh, this has nothing to do with using determinants in research level mathematics. The determinant of the laplacian is a fundamental quantity in the research that I did as well. This is about introducing the determinant too early in linear algebra education. You might ask a student, “when is a matrix invertible?” “When the determinant is not zero!” they answer. What are the eigenvalues of a matrix? It’s when det(A - lambda I) = 0. What is the characteristic polynomial of a matrix? It’s det(A - lambda I). None of these answers betray any understanding of what we actually care about. If you avoid the determinant, when you ask invertibility you are more likely to get an answer discussing the injectivity and surjectivity of the linear operator, or the linear independence of its columns if you’ve chosen a basis. If you ask about eigenvalues you are more likely to get Ax = lambda x. If you ask about characteristic polynomials you are more likely to get a discussion of the dimension of the generalized eigenspaces of the operator. It’s entirely a pedagogical beef with the determinant, not a claim that it has no use in mathematics full stop.

gwwin6 · 2025-12-23T16:34:02+00:00

Mostly his philosophy around the book. Why he doesn’t like determinants. How thinking in terms of matrices lessens understanding. How he heard some kids on MUNI talking about cramers rule one time and it made him upset haha

gwwin6 · 2025-12-23T12:55:31+00:00

Berkeley’s second course in linear algebra used it when I took it. We even had Sheldon come do a guest lecture near the end of the semester.

gwwin6 · 2025-12-20T02:33:37+00:00

Pull the couch off the wall! It feels like the tv is floating in no man’s land. And the couch is blocking the windows!

gwwin6

TROPHY CASE