Mathematics - 2023-08-08 (page 2 of 2)

Axoren

17:00

You have failed to reject the Troll Hypothesis.

Xander Henderson

@Jakobian Hades very often tells me that I am not funny. I feel the same way about his "jokes". :/

冥王 Hades

Well I never said I’m any funnier

Jakobian

this one was alright, I didn't laugh or smirk but I appreciated it

Axoren

Ugh... why is modeling physical things hard?

I don't want to post on physics.SE, they always tell me my questions are engineering and then the engineering SE tells me my questions are theoretical.

Jakobian

@Axoren because you need to think about what you're doing

Axoren

17:11

@Jakobian That should be illegal.

冥王 Hades

This is why I prefer math classes more than physics or engineering

Jakobian

I think that it boils down to experience

if I were advanced in physics I'd have already plethora ways of making a physical model

so I wouldn't need to think much about it

it's similar to how a mathematician can do research, you don't need to come up with new things, but provide new results

model of physical phenomena I mean (not sure if that's the same thing as physical model)

so "modeling" in mathematics is not as prevalent and not as needed of an ability

Axoren

I can easily do a 3-body model with treating the three bodies separate from the vector space they are located in, but I'm trying to model them as values in the vector field directly. Mass warps space time, so I was trying to model the objects themselves as just being the curvature they induce on the space. But there's just so many reasons to treat them independently as points with properties at those points.

Jakobian

but arguably it's not as needed of an ability in physics either, since they probably use models that they already know (just that learning those models is the challenging part of physics)

Axoren

Things break down so quickly.

Ted Shifrin

17:18

What you said doesn’t make much sense to me.

Axoren

Instead of "There is a particle at point $p$ with mass $m$", I'd like to just say "This space is curved at point $p$ in a way that is consistent with a particle at $p$ with mass $m$."

And never referencing the particle directly

Ted Shifrin

There’s a singularity in the field/curvature … so how do you quantify that carefully?

Axoren

I misspoke, then. It's not a singularity. That would only be true if the particle was actually point-charge in size. I'm considering a particle to be real-life particle sized for the particle that it is.

Like a proton, for example

冥王 Hades

17:35

I learned that my body simply can’t handle alcohol

Jakobian

@冥王Hades ncbi.nlm.nih.gov/pmc/articles/PMC6875758

this might be why

not sure how good is this study though

Soumik Mukherjee

Just wondering who are the representation theory experts here?

Ted Shifrin

None.

Soumik Mukherjee

:(

Axoren

I remember trying to find someone who could explain representation theory to me back in college because I saw potential in it for sorting theory. I can only hope you find more resources than I did back then.

Ted Shifrin

17:50

@Axoren So you’re going to have approximations to the Dirac delta based on the radius of the particle,

I recommend Fulton and Harris’s book.

2

Axoren

@TedShifrin That is one way to put it. As the particles get bigger, it would be a lot more awkward to address them as Dirac approximations, wouldn't it?

Ted Shifrin

Why? I’m assuming spherical symmetry for simplicity.

The issue is what the model you’re using is for the force,

Axoren

I'm anticipating that force will naturally arise from the curvature itself, rather than needing to be simulated separately.

Operating in units of G, so I can keep things simple.

Or rather, sqrt(G)

Also, "keep things simple" what a bold-faced lie I just told.

This is the least simple thing I've done this year

Ted Shifrin

Do you even know what curvature means?

Jakobian

@SoumikMukherjee you can try asking Thorgott. I think he's an algebra guy

Ted Shifrin

17:58

Yeah, we used to have more algebra representation, but they aren’t around so much.

Axoren

@TedShifrin I mean G as the gravitational constant, not as any other existing symbol for a curvature related variable. It's the tensor that indicates the paths geodesics take along a surface of a manifold, in the context I'm using it in.

Ted Shifrin

I’m a differential grometer, remember? You need a metric tensor or another affine connection to even talk about curvature.

I think you’re throwing words around.

Axoren

I'm misusing curvature and the metric tensor which grants it interchangibly here.

Ted Shifrin

Well, don’t.

Axoren

As for the affine connection, I didn't even stop to think that there might not be a natural one.

Yeah, I'll correct that. I can see it being confusing.

Ted Shifrin

18:08

If you have a metric, there is a canonical connection, called the Levi-Civita connection, but curvature exists without metrics if you define a notion of parallel transport.

PM 2Ring

You can talk about GR stuff in The h Bar, the main Physics.SE chatroom. One of the room regulars is currently writing a book on GR.

Axoren

That name is absolutely cool

PM 2Ring

:)

Axoren

I might check it out after work

My break is almost over

Jakobian

and what do we get? "Mathematics"

Axoren

18:13

If we let the Brits pick the name, it would have been "Maths", even less letters, and none of the cool ones.

(ice)

PM 2Ring

Another option was Schrödinger's Chat physics.meta.stackexchange.com/q/1028

Ted Shifrin

Good suggestions, PM2.

Axoren

Talking there would have been helpful and not at the same time.

Ted Shifrin

Huh?

Axoren

Superimposed state of $\text{helpful}$ and $\neg \text{helpful}$ at the same time.

With regards to Schrodinger's Chat.

You wouldn't know until you got there.

user 726941

18:19

Should we try to start a vote on a room name change suggestion contest?

HERE

Axoren

I think if you changed it now, people might not realize, Wouldn't it change the link?

user 726941

Nope.

PM 2Ring

I don't know much GR, but I can do photon trajectories in the Schwarzschild metric. We recently had a question about Schwarzschild trajectories for a massive test particle, physics.stackexchange.com/q/772180/123208

Axoren

Hmmm, looks like you can name the chatroom whatever you want in the link and it still works: chat.stackexchange.com/rooms/36/spookyscaryskeletons

user 726941

It goes by number.

Axoren

18:21

(Also, I am not voting for spookyscaryskeletons)

geocalc33

I kinda like the name how it is: Mathematics

Axoren

Esoteric Numerology.

It's hard to beat Mathematics.

user 726941

This is definitely a meta topic.

Axoren

Like, anything else seems like false advertisement.

user 726941

Physical Mathematics

Axoren

18:23

There's plenty of Pure Mathers here, though

user 726941

True dat.

Jakobian

Math Wizardry

Axoren

Fundamental Theorem of Socializing.

PM 2Ring

Pointless Topology

Jakobian

I like themes that combine mathematics and sorcery

Axoren

18:25

Mathic Missile

... Sounds like I'm making fun of someone with some missing teeth... Let's not use that one.

Jakobian

@PM2Ring yeah. I know. lol

Axoren

Homework Support Team

user 726941

^

Jakobian

the fact that so little people seem to know about math.se chats is a blessing in itself

I'd hate it if there were teenagers running around begging for help

Axoren

... there used to be quite a few

Did that change?

Are they asking for help on TikTok now?

user 726941

18:29

They could create their own room(s)

Jakobian

I don't really see any. If people ask about something it's more like early university

Axoren

@user726941 If they create their own rooms, who would they ask?

user 726941

Anyone who wants to create it with you.

Jakobian

I mean, it's different wanting to get help and understand something, and being pressed into understanding something but being unwilling to understand it

Axoren

"That sounds like effort. I'll just pay for Ch*gg."

Or whatever it's called now

user 726941

18:32

@Axoren "Start a new room with this user" is an option in the drop down menu

Jakobian

@Axoren ChatGPT

Axoren

@user726941 That sounds like a hostage situation.

user 726941

LoL

Axoren

"I have 20 Homework Problems tonight. I demand your time as tribute."

"Whatever you were doing? Less important than this 2% of my GPA."

I have had to fail students for cheating on homeworks that were individually worth 0.5% of their final grade.

and I scaled

user 726941

Ouch

Axoren

18:35

I even gave the whole class a second chance. "I've caught over 50% of you cheating. I will not fail anyone for past cheating attempts, but if I catch any of you cheating again, instant F." Ended up failing everyone who cheated the first time.

user 726941

The scourge of the internet.

PM 2Ring

@Axoren What was the class?

Axoren

Introductory-level Programming in C, as a Sophomore Course. For programming, it's easier. The cheaters obviously trip source-code similarity metrics by such a wide margin, you can't ignore it. Especially when they all copy-paste the same mistakes and comments from the original coder.

I can't imagine how hard it is to detect with Math

user 726941

Things are too easy these days to find.

Axoren

The problem with that was that the professors teaching the first year programming courses didn't weed out cheaters earlier on in the degree path, so the pond scum rose to the top.

冥王 Hades

18:39

@Jakobian seems accurate, most of my Japanese peers don’t drink a lot. On the other hand, other international students can effortlessly chug through several glasses without problems

Jakobian

yeah, I remember being surprised at how good tolerance for alcohol I have
it's probably average though

I don't drink at all, alcohol is literally poison

Axoren

The professor who taught the course after me was overly optimistic and planned to get students working on Operating System code in the second half of the course. We bonded a bit over how woefully unprepared students were after a year of paying college tuition for what amounted to nothing.

@Jakobian It's also just gross.

Mixed drinks taste like the stuff you mixed with them, so why not just drink the mix without the alcohol?

PM 2Ring

It scares me to think how many people have passed programming courses who don't really know how to code. Sure, you can get by with copy & paste for a lot of stuff, but if you don't actually understand what you're doing, it can lead to chaos.

Jakobian

@Axoren I don't do mixed drinks so I don't know, but I do like drinking whiskey. Well, I don't drink whiskey because it's unhealthy but I used to drink it for a bit

user 726941

Chaos reigns supreme on overflow...

Jakobian

18:44

so I wouldn't say it's gross

Axoren

One of the critical things I always saw, and it's informed how I work with subordinates in work and in other team-based activities, is that not everyone has sufficient Critical Reasoning skills for the tasks they're interfacing with. Some people just don't get things, and they shutdown instead of exploring outside of their knowledge base independently.

冥王 Hades

They’re usually consumed in Karaoke events here. I’ve been to them, and I’m basically a background character. I can neither sing nor drink alcohol

Axoren

So, I always give people an A to B to C problem, and give them how to get from A to B, and then how to get from B to C. Then, I ask them to get from A to C. A majority of the students I had in college? Couldn't solve for A to C.

It was a surreal observation the first time I saw it live.

user 726941

Sounds like old school algebra textbooks

Axoren

The intermediate step B is just invisible to them.

PM 2Ring

18:47

I basically gave up on SO. I like helping people who genuinely want to learn how to code, and who have adequate reasoning skills. I have no interest in helping people who just want to learn how to pass their exam.

Axoren

Yeah, so what if you graduate with a coding degree? People are hiring you to write code that doesn't exist. There's no where to copy from. If the code already existed, they wouldn't hire you, they would buy the code. You need some sort of ability to act independently and critically.

Jakobian

ability or aptitude?

user 726941

One builds the other.

PM 2Ring

When I first started writing answers on SO, sure we got a lot of low quality questions from homework "help vampires", but there were lots of good quality questions too. But the sheer quantity of questions kept growing, and the quality reduced...

冥王 Hades

@Axoren This should be extremely obvious to anyone who has even a passing understanding of vectors. How did majority of college students fail this

Axoren

18:51

It's hard to say. I personally believe critical thinking can be taught, but it requires an internal source of drive to be activated. If you have the ability, but don't use it, you may as well not have it.

@冥王Hades I was tutoring a girl who went through an entire linear algebra class without understanding vector addition.

When I discovered THAT was the piece she was missing, I went to a whiteboard, drew a diagram of adding two vectors, and then we breezed through the textbook re-calibrating everything she didn't fully learn the first time.

PM 2Ring

Jun 21, 2021 at 13:42, by PM 2Ring

So instead of people learning how to think like a mathematician, they learn, parrot-fashion, how to pass mathematics exams. Physicist Richard Feynman encountered this problem (in connection to physics), when he spent a year teaching in Brazil, not long after World War 2.

v.cx/2010/04/feynman-brazil-education

Jakobian

I think it's because people are too much of an idealists

They want to be this, that, but the knowledge won't magically appear

user 726941

This kind of discussion gets deep into pedagogy.

Jakobian

or just being greedy bastards...

Axoren

That's something a lot of people have trouble parsing too. "How old do you have to be to be a scientist?"

"What do you want to be when you grow up?"

Your birthday isn't a Level Up screen. You don't just unlock new skills.

But people think like this and psychologically hold themselves back from pursuing knowledge because they think it will just come with age, or time.

Jakobian

18:57

well, parents do tell children things like that, could we say it's fault of parents oversimplifying how world works

冥王 Hades

My father started teaching me some basic algebra at about 3rd grade level as far as I remember

user 726941

Good question.

Axoren

My nephew wanted to do something like Greta Thunburg when he was older, so I told him, "Have rich parents and go yell at politicians and you can do it right now."

Jakobian

I don't really blame people for believing that, it's only easier to think more independently as you grow up, and it's hard to change opinions from the one you grew up with

but here, I do believe that age helps

Semiclassical

@PM2Ring i've gotten more sympathetic to students like that over time, because there's a sense in which they understand their actual situation pretty well: not education as a means of becoming critical thinkers, but as a means for economic advancement/security

Axoren

19:00

Age helps stochastically. You're more likely to have experienced the a-ha! moments that take you to the next level of understanding and your brain chemistry has likely developed far enough for you to retain those ideas.

冥王 Hades

That said I find it hard to believe that you can get a graduate with a computer science degree with at least a decent GPA without having some critical thinking ability

Axoren

At some point, as a young adult or even younger, you can't be expected to learn much more passively.

Jakobian

@Axoren I don't think any of my realization moments sounded this dramatic

Axoren

Well, now most schools don't offer Bachelor's of Arts degrees in Computer Science.

Semiclassical

i find it harder to be idealistic about education when it really is so tied up with very material concerns

Axoren

19:02

@Jakobian You've never had a lowercase "a-ha"? They're pretty great.

Just exciting enough to make you smile.

Semiclassical

both in terms of what the students hope to get out of it, and what the universities extract in turn

Jakobian

No, if I get something it's more because of my gradual increase in knowledge and understanding, not because of some sudden shock I had

user 726941

^

Jakobian

in fact, I think I'd be worried if something came to me too suddenly, because the more probability it's completely wrong

冥王 Hades

It’s pretty common to have that “a-ha” moment during math competitions

PM 2Ring

19:04

@Semiclassical Fair enough... as long as they aren't in a position to cause damage due to their lack of genuine understanding. Or at least, they're in an organisation that can detect & correct their mistakes.

user 726941

Yeah, the "doh" moment

Semiclassical

i'm not saying it's a good state of affairs, to be clear

冥王 Hades

in fact that’s usually how I came up with some of my best solutions

Axoren

I've even had moments are potent as a full-on Eureka!

Semiclassical

but that it's inevitable so long as education is something you really can't pursue outside of an economic context

Axoren

19:05

I once did a Shoryuken in my lab out of excitement and I have not been fit enough to repeat the maneuver.

Jakobian

why

Axoren

Well, because how I moved generated enough upward momentum for me to lift off of the ground in a specific pose.

I had resolved a numerical issue with one of my machine learning models and found a way to shorten the runtime by a factor of 100x or something, iirc.

All I clearly remember now of the event was the lift-off

user 726941

coolio

冥王 Hades

Hyperfocusing helps a lot here. Love my Asperger’s syndrome

Jakobian

I hope that one day, we can establish Hades' full medical record.

user 726941

19:09

LoL

Axoren

Family history:
Eaten by father.

冥王 Hades

@Jakobian That’s gonna be a big list. I have many medical problems/disorders

Axoren

@PM2Ring Unfortunately, lack of skill serves as damage even by itself. A company is paying for a skill that they are not delivering. No only is the company losing money by paying the person, but they also still lack the service they now need to pool separate resources to.

mick

@algbr periodicity is conditional

Axoren

I had a coworker who was so oblivious to his role, he was asked to write down a list of deliverables he was expected to provide to his team. He wrote "Show up 30 minutes early every day, spend 1 hour studying for Java course" and that's it. After 3 months of being tasked with critical asks, his idea of what he was responsible for was "show up and self-study"

He had to be let go because his lack of performance required his other team members to do his job on top of their own.

He is, by far, an extreme case. Don't expect to find people like him out in the world.

冥王 Hades

19:17

That makes me wonder how someone like that can even graduate with a relevant degree and a decent GPA to land a job

Axoren

They had job experience (unverified on paper) and had a professional reference.

user 726941

Connections.

Axoren

The team's budget couldn't afford a dev, so they were simplying trying to fill an assistant/operator role and were willing to train someone up.

They were expecting to get someone who could learn from his interview and resume.

user 726941

Had he been a talented independent learner he would've been elected CEO.

冥王 Hades

suddenly Google, Microsoft, Apple, Amazon and more asking brain teasers and puzzles during interviews makes a lot of sense

Even the mere fact that someone could train themselves enough to reasonably solve these puzzles implies that they had some critical thinking abilities as well as the drive to learn this

Axoren

19:23

Or they found the answer in the LeetCode forums, so the company needs to ask even more questions

user 726941

Competition weeds out the fakes, eventually.

Darwin once said.

Axoren

The problem is that once metrics are a target, they stop being useful metrics.

冥王 Hades

@Axoren I doubt they’ll ask exactly the same questions found on leetcode. In fact I’m pretty sure they go out of their way to make sure such questions are blacklisted

Axoren

@冥王Hades What happens, is their questions get leaked by interviewees

Are there more doors or wheels on Earth?

How many ping pong balls fit in a standard school bus?

These questions are famous, so using them in an interview is no longer worthwhile

冥王 Hades

Not sure about th doors or wheels part but dividing the volume of the bus by the volume of the ping pong balls gives a reasonable estimation

Axoren

19:27

@冥王Hades Can you estimate it closer? Don't the chairs/benches deduct from that volume?

If I wanted just an estimate, I could have said "Seven". The goal of those questions is to see how the interviewer thinks about modeling the problem. However, if someone shares a model of the problem online, the interviewer is going to be able to parrot someone else's answer.

sunny

I want to study the differentiability of $$f(x)=\sum\limits_{n=0}^\infty\frac{e^{-nx}}{n^2+1}, \quad x>0.$$ Right off the bat, I'm unsure if we either have $\left|\frac{e^{-nx}}{n^2+1}\right|\leq e^{-nx}$ (a geometric series) or $\left|\frac{e^{-nx}}{n^2+1}\right|\leq \frac{1}{n^2+1}$ (a p-series).

Probably both, or? I guess we prefer the latter, since it is independent of $x$. But for studying the differentiability, I could use the former inequality too, since we only require pointwise convergence of $f$ (while uniform convergence of $f'$).

PM 2Ring

Goodhart's law

Goodhart's law is an adage often stated as, "When a measure becomes a target, it ceases to be a good measure". It is named after British economist Charles Goodhart, who is credited with expressing the core idea of the adage in a 1975 article on monetary policy in the United Kingdom:Any observed statistical regularity will tend to collapse once pressure is placed upon it for control purposes. It was used to criticize the British Thatcher government for trying to conduct monetary policy on the basis of targets for broad and narrow money, but the law reflects a much more general phenomenon. ��2�...

2

geocalc33

Interviewer: Give me an interval of 100 consecutive natural numbers where there are no primes in it and s.t. the first number of the interval is minimized.

Axoren

So, I give interviewees very difficult problems I don't expect them to solve. Something for which the answer is too involved to answer in 30 minutes, but could be done in a full workday. And I ask them to go as far as they can, and watch their initial modeling of the problem. This lets me see: 1) if they freeze up or do well under pressure, 2) if they can actually use the skills they claim to have, and 3) if they ask questions or struggle alone and fruitlessly.

I've given a positive review to at least one interviewee who has not solved my problems in the allotted time, and I now work with one. I'd be incredibly suspicious if my problems were solved in time, but that opens up a conversation about their experiences that lead to it being an easy problem for them.

I've found it much more successful than leetcode.

PM 2Ring

So when passing the exam (or doing well in a competition) becomes the target, some people will memorise tricks that let them achieve the target, without necessarily having a deep understanding of the material. So the exam ends up just being a measure of what tricks you've learned, not your true understanding. Sure, you need some understanding to figure out which particular trick is applicable. But sometimes you need actual understanding & critical thinking.

Jakobian

19:36

@sunny yes, both

sunny

ok, cool

Jakobian

this should be infinitely differentiable

Axoren

@PM2Ring Like how people learn the polynomial derivative pattern and give up on the definition of it.

Jakobian

because no matter how many derivatives you take, it'll still converge uniformly on $[d, \infty)$ for $d > 0$

the series for $f$ converges uniformly as well, I'm guessing the series for $f'$ doesn't

PM 2Ring

@Axoren Yes. Of course, when I take the derivative of a poly I just do it mechanically. But I know I can do it from first principles if I want to.

冥王 Hades

19:39

@Axoren Sure, if we can compute the volume of one of those benches and deduct that. Also since the balls are perfectly spherical there is bound to be “wasted” space as well so the actual estimate will be quite a bit lower

PM 2Ring

"Give me an interval of 100 consecutive natural numbers where there are no primes in it" Easy. "and s.t. the first number of the interval is minimized.". Not so easy. But I can write Python code to do it in a couple of minutes. :)

Axoren

@冥王Hades Here's some examples of refinements of the Bus Model for the problem: Are there people inside? How big is the steering wheel? Which brand of Bus is it? What if the windows are open? If there are people, what if they have backpacks? What brand are the balls? Are they all the same size? Can you give me an actual answer in cm^3?

The interviewer's goal is to see how much of this you extrapolate on your own in an effort to get as accurate as possible within the hypothetical.

They don't have a number in mind, they want to see the depth of your thought process.

... Also you get brownie points for mentioning the Packing Coefficient of Spheres...

冥王 Hades

@Axoren I assumed that the bus is completely empty, no people or luggage so I also take into account the fact that I can fill up the luggage compartment as well, as for brands, I’m not really familiar with that at all.

Axoren

If you assume the bus is a point-particle at rest without air resistance, the problem is even easier.

冥王 Hades

I don’t know how significant of a difference there is in size between different brands of buses. I assume not much

Axoren

19:44

@冥王Hades Is it a double-decker?

冥王 Hades

@Axoren Also if they have backpacks, why can’t I fill those backpacks with balls too? Do they have books in them?

can I fill their lunchboxes too?

Axoren

@冥王Hades Good! That would be received positively during such an interview.

冥王 Hades

If they’re returning home then filling their lunchboxes with balls should be fine since they’ll be empty

Axoren

Because it shows you're properly modeling the very nebulous vast state space of the problem.

冥王 Hades

@Axoren Also what about the hood? Surely it has some free space too, why not fill that up too?

Axoren

19:47

Heat the bus to make more space with Thermal Expansion. Flatten the balls so that you can pack more per unit volume. Etc.

If you melt the balls, you can turn the problem from "how many balls" to "how much ball (material)" you can fit in a bus.

冥王 Hades

Why not remove the benches or chairs too? Also some buses seem to have storage compartments for repair kits and other stuff, why not throw all that out and fill it with ping pong balls?

In fact why not remove the engine too and fill that space with more balls?

Nobody said the bus has to go anywhere, we don’t need an engine @Axoren

I saw a similar problem about how much you’d charge to clean every window in Seattle. Personally the first thing I’d do is increase price per window the higher up I have to go. Say $10 more for every 50 feet

sunny

@Jakobian hmm, consider $$f'(x)=\sum_{k=1}^\infty -\frac{ke^{-kx}}{1+k^2}.$$ Can we bound these terms to get a convergent series and use the Weierstrass M-test? I guess we need to consider an interval of the form $[d,D]$, then...$$\left|-\frac{ke^{-kx}}{1+k^2}\right|\leq \frac{De^{-kd}}{1+k^2}\leq \frac{D\cdot 1}{1+k^2}.$$

Axoren

@冥王Hades The first thing I would do is buy every window cleaning company in Seattle and hold all the dirty windows hostage with my monopoly.

If you control the supply, you set the price.

冥王 Hades

@Axoren yeah well what if you don’t have the money for that

Axoren

Charge more for cleaning windows. :P

冥王 Hades

19:56

exactly, that’s the problem

Jakobian

@sunny why is $k$ getting bounded by $D$

sunny

if we only consider $[d,\infty)$, then the $k$ in the numerator could get arbitrarily large

Jakobian

what does it matter if you have an exponential

it'll die to $0$ way faster

either way you can't bound by this $D$ as you did

the exponential guarantees all sorts of convergence

and you don't want to, under any circumstance, bound it by $1$

sunny

well, for $f$ to be differentiable, we need uniform convergence of $f'$. How would you have bound the terms?

Jakobian

you need to be more clear about what you're trying to say

well either way what I was getting at is the series for $f'$ doesn't converge uniformly on $(0, \infty)$

that's why I mentioned $[d, \infty)$

differentiability is a local property so this is enough

sunny

20:13

@Jakobian are you're claiming $f'$ is uniformly convergent on $[d,\infty)$?

Jakobian

yes!

sunny

I'd be grateful if you could show that :) I was claiming $f'$ is only uniformly convergent on $[d,D]$, where $D>d>0$.

Jakobian

I already told you why your argument is bad

$k$ gets bounded by $D$

the same $k$ that should grow arbitrarily large

gets bounded by a constant

I'd rather you look at your own argument first before we proceed

Akiva Weinberger

@SoumikMukherjee I don't know if you're still thinking about it, but: show that $\cos(nx)$ can be written as a polynomial in $\cos(x)$. As a consequence, show that if $f(x)=a_nx^n+\dotsb+a_0$ is a polynomial, then $f(\cos x)$ can be written as $b_n\cos(nx)+b_{n-1}\cos((n-1)x)+\dotsb+b_1\cos(x)+b_0$ for some new coefficients $b_n$, and vice versa.

sunny

@Jakobian I don't see though how you'd use the Weierstrass $M$-test on $[d,\infty)$. Then you'd need to go through some sort of bounding of terms as well, i.e. $$\left|-\frac{ke^{-kx}}{1+k^2}\right|\leq\ldots$$

Jakobian

20:21

yeah, and of course this is bounded by $\frac{k}{1+k^2}e^{-kd}$

If $h(x)$ is a rational function, then $\sum_{k=1}^\infty h(k) e^{-kd}$ converges

To see this you can e.g. observe that $h(k)e^{-kd} = (h(k)e^{-kd/2})e^{-kd/2}$

while $h(k)e^{-kd/2}\to 0$

thus, for example, for large enough $k$ we have $h(k)e^{-kd}\leq e^{-kd/2}$

put absolute values where its needed

either way the series converges, and does so absolutely

sunny

ok, thanks, that clarified it. I guess my argument isn't wrong then, it's just "bad" :)

Jakobian

those are synonyms for me

either way I used the word bad not to make you feel lesser, but because I thought it was misdirected too, together with being wrong

hope you're not mad at me for that

sunny

I'm not :) but I still don't see though why we want $k$ to grow arbitrarily large. Anyway, you don't have to explain if you don't feel for it.

PM 2Ring

@AkivaWeinberger Ah, right. :) The (0, 4) solutions look like this, and its cyclic permutations of the columns:

Jakobian

@sunny I'm not sure I understand your question. It's not that we want $k$ to grow arbitrarily large, it just does grow. After all we're interested at what's happening as $k$ gets larger and larger

$k$ is the variable we are summing over

$k = 1, 2, 3, ...$ to infinity

sunny

20:32

I finally see my mistake!

I didn't realize I substituted $k$ for $D$, that's definitely wrong haha

Jakobian

ah, alright

you can try proving that the series for $f'$ nor the consequent series for $f^{(n)}$ don't converge uniformly on whole of $(0, \infty)$

although $f$ does converge uniformly on $(0, \infty)$

sunny

I could try indeed, thanks for all the help so far though

Couldn't we bound on $[d,\infty)$ like this as well: $$\left|-\frac{ke^{-kx}}{1+k^2}\right|\leq\frac{ke^{-kx}}{1+k^2}\leq\frac{e^{-kx}}{k}\leq e^{-kx}\leq e^{-kd}.$$ Then we would have geometric series instead of $\sum_{k=1}^\infty h(k) e^{-kd}$, which i think is less known.

Jakobian

20:50

wdym by less known

It's something you could come up with in a matter of seconds yourself

sunny

I meant a geometric series is more known than $\sum_{k=1}^\infty h(k) e^{-kd}$, where $h(k)$ is a ratioanl function

Jakobian

well sure, I guess I was thinking more about how to approach it for higher derivatives as well

sunny

ok, I see

Jakobian

still, the argument is pretty much the same

Sonozaki

I proved that $x \mapsto \lfloor x \rfloor$ is discontinuous for each $x \in\mathbb{Z}$. Can I deduce that $\{x\}=x-\lfloor x \rfloor$ is also discontinuous for each $x \in\mathbb{Z}$ by saying that if by the sake of contradiction it is continuous, then $-(\{x\}-x)$ would be continuous for each $x \in\mathbb{Z}$ as well. But this is false, hence ${x}$ must be discontinuous for each $x \in\mathbb{Z}$.

Jakobian

21:03

yes

Sonozaki

Thanks!!

PM 2Ring

21:37

@geocalc33 oeis.org/A002386 A couple of weeks ago I was experimenting with various prime generators in Python. This one is quite compact, and it's ok for small primes, but it gets a bit sluggish for primes > 1000000. I have much faster generators that easily handle larger numbers.

sagecell.sagemath.org/…

PM 2Ring

22:06

Chris Caldwell has extensive info on this topic: The Gaps Between Primes. OEIS has some old dead links to Chris's material.

Shaun

22:57

Hi :) Please may I have some feedback on the following?

-1

Q: Is there a maximum length of the chief series of $SL(n,q)$?

Motivation: I'm studying certain properties of conjugation in $SL(n,q)$. There's a nice number, a bit like a covering number, that one can associate with an arbitrary group. In writing a programme in GAP to compute this number, a shortcut comes to mind - I'm not going to share what exactly - that...

gr.group-theory

Dannyu NDos

23:38

It's quite annoying that Munkres insists "imbedding" as the spelling, while most dictionaries only have "embedding".

Jakobian

@DannyuNDos both are in use

embedding is a little more common

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