Mathematics

12:00 AM

A cool, interesting, and practical word exercise would be as follows: "Find a game or activity which you enjoy, having a certain economy within it. Create an optimal algorithm for maximizing profit in that economy."

Virtually anything has something about it which has an application to mathematics and optimization with some sense of an "economy".

Obliv

We aren't computer programs, so sometimes mixing in a relevant word problem to a lesson in mathematics can be helpful. Everyone learns in different ways.

leslie townes

obliv: 'gist' is a matter of opinion, if you have a graphing tool of any kind it should seem plausible that the c_1 cos + c_2 sin function could potentially take that form. formally, should you suspect that such a function does have that form, trig identities give formulas that you can use to deduce those expressions involving A and phi.

note that without any surrounding context, i dunno if anybody would automatically prefer one representation over the other. it's not a situation where there's a natural direction of complexity in which one is clearly a "simplification" of the other.

just two different ways of representing the same thing.

Obliv

If asked for the amplitude, I don't necessarily have to find the phase angle and put it in that alternative form then?

AMDG

@Obliv Right... well math is math, not a computer program. There's something in math in any subfield thereof that is capable of appealing to anyone, but seeing as how most people these days shun intellectual pursuits in favor of satisfying concupiscence, that won't help without some encouragement. People don't know what they're missing out on. Instead of binging Netflix, they could become addicted to optimization and finding algorithms in discrete mathematics (like myself).

Obliv

I think the reason is because for $y'' + k^2y = 0$ the solution to these kinds of d.e. are naturally the first form

leslie townes

12:06 AM

obliv: if only asked for A (or |A|), you only need to find A, yeah. the answer will just be some number.

what if i'm addicted to optimizing netflix's recommendation algorithm by providing it with my usage data.

AMDG

Before I answer that, one more thing...

leslie townes

i'm kidding of course, purely rhetorical question.

Obliv

If the world was full of mathematicians, we'd be dead before the end of the month :) @AMDG

D.C. the III

Alright @TedShifrin I know you will not give me the answer so let me lay out my "stream of consciousness" and maybe this will show you where I'm stuck and what I can do to correct it:

"We have shown that all the good rectangles are contained in rectangles of $P'$ and as such the good $U-L$ is at most $U(f,P’)-L(f,P’) < \epsilon/2$. The bad rectangles cannot be described by using the partition $P'$. So what do I know?...I know that the total area of the dividing hyperplanes (which are the dark lines in the figure) is $A$...I want to cover these hyperplanes by rectangles of diameter $< \delt…

AMDG

It is not in fact true that everyone learns in his own way. Every man has the same process of reasoning and intuition. Intellectual capacity only changes one's capacity to reason and intuit by some quality or quantity of sorts like speed, awareness, etc.

Obliv

12:09 AM

@leslietownes I do the same for pandora radio, I'm addicted to music

AMDG

@leslietownes Meh. If it can be modeled mathematically (it can)...

Obliv

@AMDG Why is it so common for people to make assumptions like this? I wonder..

AMDG

@Obliv I question how you came to this conclusion.

@Obliv Excuse me, but what are you referring to as the assumption here, and what do you imagine to be so different between every single person in terms of how he obtains knowledge and understanding?

Obliv

Maybe not so common, but it sounds like you're basically equating humans to programs.

AMDG

Not at all. I'm just saying man learns by way of reason and intuition. Every man.

Obliv

12:14 AM

Okay, so by your logic there is an optimized learning process for all humans? *and word problems are definitely not in this process

I'd need a proof

AMDG

Yes, and that consists of presenting an essence followed by answering every question about what constitutes the essence until no more questions remain (which guarantees the student has attained to comprehension of the essence).

For memorization, the practice (application rather) of that knowledge is best.

Whether or not word problems belong depends on whether or not this kind of problem is appropriate in the first place.

I mean if you have to labor to formulate the word problem, then it probably shouldn't be a word problem.

Just have an end in mind and choose the appropriate means.

If I want to test for division, I don't contrive a scenario in which I have to divide. I just tell him to divide.

Obliv

@leslietownes I'm confused by this wording. A damping "force" shouldn't that be a 2nd derivative

what is a real life example of a damping constant? something like medium resistance?

but even that seems like it's a force

$\beta\frac{dx}{dt}$ doesn't look like a force

leslie townes

parameters like beta in models like that can carry units along with them, and they can even be physically meaningful. i'm not a physics person but think of a spring immersed in air, water, or a thick goo like cornstarch.

you jump on that spring in the cornstarch, it might not compress at all because fast movement against the cornstarch goo causes the goo to resist you more than slow movement would.

Obliv

Oh so it's dependent on velocity

leslie townes

or just think about dragging your hand in a swimming pool very slowly vs. trying to deal a knockout punch underwater.

thats how i see it, yeah.

Obliv

12:28 AM

Interesting, that makes sense thanks!

leslie townes

so the constant in this model would represent something like the thickness of the goo, and have units to match.

Obliv

Perhaps given in density

whatever the units for that is, this book doesn't use metric

instead I'm working with slugs

Ted Shifrin

@D.C.theIII No, all that matters is fhe $A$. In the picture, $A$ is the total length of the dividing lines, not counting edges. What is the greatest area the bad rectangles can have, and what is their worst-case scenario contribution to $U-L$?

Ted Shifrin

12:51 AM

@Obliv Think of friction.

Obliv

right, even air resistance would be proportional to velocity

in general if $a^2 + 2b + c^2$ where $b^2 = c^2$ it's a repeated root?

oh duh

$(a+c)^2$

for repeated root situation of such a spring we'd have $x(t) = e^{-\omega t}(c_1 + c_2 t)$ or $x(t) = e^{-\lambda t}({c_1 + c_2 t})$ where $\lambda = \frac{\beta}{2m}$

since $\omega^2 = \lambda^2$

I have spotted a rare typo in my textbook

leslie townes

1:06 AM

Critical Damping would be a good name for a podcast

Obliv

should be $x(t) = e^{-\lambda t}(c_1\cos(\sqrt{\lambda^2 - \omega^2} t + c_2\sin(\sqrt{\lambda^2 - \omega^2}t)$ :P

@LeslieTownes Welcome to Critical Damping where our guest today is the maximally difficult Leslie Townes.

for complex roots we have $e^{\alpha x}(c_1\cos(\beta x) + c_2\sin(\beta x))$ I forgot what do complex repeated roots look like again

I'm tempted to just throw an x randomly in there

That sounds right.

D.C. the III

if $A$ is the total length of the dividing lines is it correct to say that $A/n$ would be the length of a side for any bad rectangle? where $n$ is the number of bad rectangles. And if this is the line of reasoning, then $A^2/n^2$ would be the area of a bad rectangle and the worst case scenario would be if all the rectangles were bad rectangles.

Obliv

When in doubt multiply by x, $e^{\alpha x}(c_1\cos(\beta x) + c_2\sin(\beta x)) + xe^{\alpha x}(c_3\cos(\beta x) + c_4\sin(\beta x)) $

Do you get an honorary PhD if you finish all of the exercises in Ted's books? @D.C.theIII

D.C. the III

@Obliv That would be the dream........

Obliv

Ted just ships you a certificate that says good job

Ted Shifrin

1:16 AM

@leslietownes subtitle: Sodden diapers

Obliv

I don't know what you're doing other than telling apart "good" rectangles from "bad ones but god speed D.C.

D.C. the III

@Obliv I'll send a couple of stacks with it to bribe him for a letter of recommendation. Or a lifetime subsription to his favorite restaurant

Ted Shifrin

@D.C.theIII No, sizes are irregular. Just look at total width times height.

Obliv

It's been a few days of this eccentric Mr. Miyagi style teaching and I'm here for it.

D.C. the III

irregular sizes was my concern

Ted Shifrin

1:19 AM

What is the biggest possible width if diameter is at most $\delta$?

D.C. the III

If $A$ is total length then $A^2$ would be total area

Ted Shifrin

Grrr. No.

Look at the picture!

Obliv, this is how I taught for 40+ years. Shrug.

D.C. the III

It is painful but it is enlightening. I have made some discoveries in my understanding through this process, though I will detail them after I finsish this question.

Obliv

I'd be both terrified and grateful for a teacher like you :)

(especially when you growl at students lol)

Ted Shifrin

You do tend to fall in ravines and ignore most of what I say, repeatedly, to clamber out. Closed minds …

Obliv

1:24 AM

I like when Ted tells me to stop being lazy and figure it out myself. It's actually great advice for learning to critically think

I'm getting back into the groove

Ted Shifrin

Chat is not ideal, but I think people need to scroll back and reread, taking notes.

@Obliv My former students will tell you they got Grrr and Grrr$^2$ on their graded homeworks, along with other pithy words :)

Obliv

literally 2 inches above this they gave such an example that wasn't the sum of a non periodic function and a periodic one

Why must this book be so mysterious

oh nvm the first term is quasi periodic

damped periodic motion isn't periodic I guess

leslie townes

yeah. you can identify a parameter that plays a role somewhat like a period, but it's not a period in the formal sense.

Obliv

1:40 AM

Seems weird

Shouldn't that damping term still matter no matter how much time passes

if it's like the goo for example

I get that the forcing function is responsible for the periodic term

oh it's underdamped

the $\lambda > 0$ threw me off, it should also mention $\lambda^2 - \omega^2 < 0$ for it to be underdamped

An underdamped system could be like if the goo was being vacuumed overtime :P

Or getting less thick

leslie townes

goo not thick enough

Obliv

exactly

leslie townes

or gooey enough, i'm not sure "thick" suggests the right quantity

you could do a whole book around goo

Obliv

"Johnny get some thickness in this goo STAT" - some engineer, probably

Obliv

1:56 AM

Cool they use the le hospital rule to tune in the frequency of the driving force

they spell hospital funny

(i'm kidding don't kill me)

leslie townes

english books do vary in how they do that.

haha, wikipedia says it's sometimes called "bernoulli's rule." sure it is. i bet some descendant of a bernoulli edited that into the page.

Obliv

@copper.hat I just think of this message whenever I see L'hopital

leslie townes

you accept a sponsorship deal, you put the sponsor's name on the stadium. end of story.

Obliv

@leslietownes LOL

It's like encoded in their genes to try to take credit from each other if that were the case xD

My prof briefly mentioned the bernoullis i'm glad he did because now I get the joke

D.C. the III

😭😭😭😭

Obliv

2:08 AM

:(

Does the word rectangle frighten you D.C.

D.C. the III

my blindness does

Obliv

Going to have to go through some therapy to undo the damage that this problem has done on you lol

D.C. the III

except the follwoing problem also has me stumped and it has similarities in its thinking...

Ted Shifrin

2:41 AM

@Obliv damping says it cannot be!

Franklin

5:08 AM

Can anyone please help me with this ?

I think the option B is correct.

I found this answer with this reasoning:

First of all, in option A it's given $x\geq 1$ thus |x|+|y|\geq 1+|y|. But in the question it's given for any point in $K$, $|x|+|y|\leq 1$. Thus, option A is incorrect.

In option C, by a similar reasoning, it's incorrect

Option D is false by the same reasoning

Thus, we are left with option B

But I think, my reasoning is a little off beam. That's because, $A$ is not any point. It's assumed to be a point in $K$. Can anyone please help me with this question ?

leslie townes

have you drawn a picture?

draw a picture. convince yourself from the picture that the set (A) contains points whose closest point in K is something other than (1,0), and that there are points in the plane whose closest point in K is (1,0) that are not on the x-axis.

i'm guessing that you probably wouldn't be fiddling with inequalities if you had a picture, and that if you have a picture, the inequalities you want will suggest themselves.

PM 2Ring

5:27 AM

@Obliv Someone should make a movie. They could call it "Weekend at Bernoulli's".

leslie townes

Murder on the Bernoulli Express, where people on the train keep killing a Bernoulli, but they're all mistakenly killing different Bernoullis from the one that they want

"did you get bernoulli?" "yes, i got him." "you sure?" "yes" another bernoulli walks into the dining car

Franklin

@leslietownes I had an alternative argument. It goes like this: $(0,1)$ is a point in K and the distance between the point $F_A=(1,0)$ and $(0,1)$ is as least as possible i.e $0$. But $P=(0,1)$ is not in the set of points, specified by A,B and C. This is because, in A $x\geq 1$ while in P, the x-coordinate is 0. In C and D, y =0, but in P y is 1.

Thus, option B is left and it is the answer

PM 2Ring

With Leibniz doing the Hercule Poirot role.

Franklin

@leslietownes I think this argument is valid. What do you think?

Diagram makes it complicated in my case, maybe

@PM2Ring I thought up until now, Bernoulli was a single person, isn't it?

leslie townes

i don't understand your argument as written but you have the right answer so maybe it doesn't really matter? P = (0,1) is not very helpful for this purpose because it is not a point A for which F_A = (1,0), so i don't see the relevance of any of that discussion is

i don't care enough about the problem to micro-edit and tweak an argument that i don't understand, when it seems to have led you to the right answer

maybe someone else will have a different view

Franklin

5:38 AM

@leslietownes yeah. But how would you approach this problem

?

PM 2Ring

Bernoulli family

The Bernoulli family (German pronunciation: [bɛʁˈnʊli]) of Basel was a patrician family, notable for having produced eight mathematically gifted academics who, among them, contributed substantially to the development of mathematics and physics during the early modern period. == History == Originally from Antwerp, a branch of the family relocated to Basel in 1620. While their origin in Antwerp is certain, proposed earlier connections with the Dutch family Bornouilla (Bernoullie), or with the Castilian family de Bernuy (Bernoille, Bernouille), are uncertain.The first known member of the family was...

leslie townes

franklin: i would draw a picture

jacob, johan, and daniel are the 'main' bernoullis

no disrespect to the others

Franklin

@leslietownes Now, that you mention, there's a huge flaw in my argument. I admit

leslie townes

imagine having bernoulli as a last name and being not that good at school

in switzerland in say 1850 or so

hey bernoulli!!!!!! what's the answer!! tell us, bernoulli

PM 2Ring

The family would not be pleased...

Franklin

5:41 AM

@leslietownes I drew the same, with the usual coordinate system and plotted F_A. But no idea, how to proceed after that 🤔

@PM2Ring or maybe, his life would be unimaginably depressed ...

PM 2Ring

Gauss dealt with that problem by banning his kids from becoming mathematicians.

Franklin

@PM2Ring Why ?

Is it because, he did not want his kids to be compared with him

,whatsoever ?

I think that's the only possible explanation for the legendary Gauss.

PM 2Ring

He didn't want the name of Gauss to be associated with mathematics that wasn't at his standard.

Franklin

@PM2Ring A strict father ? 😂😂😂 Imagine the life of his kids. They might have hated the Gauss, we admire so....

PM 2Ring

6:06 AM

Here's a family that enjoyed doing mathematical / arithmetic work together. sciencedirect.com/science/article/pii/S0315086002000186

> Edward Sang (1805–1890), aided only by his daughters Flora and Jane, compiled vast logarithmic and other mathematical tables. These exceed in accuracy and extent the tables of the French Bureau du Cadastre, produced by Gaspard de Prony and a multitude of assistants during 1794–1801.

Franklin

@PM2Ring happy family !!!

PM 2Ring

It's pretty boring & tedious work. The calculations aren't difficult, but you need to have a good workflow that catches and eliminates errors. You need to have a lot of positive attitude and a strong belief that your work will benefit future mathematicians for ages to come.

Franklin

6:22 AM

@PM2Ring Today, in these days of quantum computers and supercomputers, if I am not mistaken, common calculations have become a joke...

But in those days, it was a great leap to explore something

Nevertheless, many useful calculations have contributed much in it's own way...

Your citation was just one of the many examples

Arthur

6:35 AM

@Franklin another good question

From where do you get these ?

one potato two potato

0

Q: Definition of meromorphic $q$-differential

In Farkas Kra riemann surfaces textbook, it defines a meromorphic $q$-differential as follows: Let $q$ be an integer. By a (meromorphic) $q$-differential $\omega$ on $M$ we mean an assignment of a meromorphic function $f$ to each local coordinate $z$ on $M$ so that $$f(z)dz^q$$ is invariantly de...

Arthur

@Franklin Did you get the answer u were wanting to know the other day ?

one potato two potato

Is it ok to understand $dz^q$ in $f(z)dz^q$ as a formal symbol? I remember that in Bott&Tu, differential forms are understood as a symbol at first.

Franklin

@Arthur Yeah. But do you know how to solve this? A weird problem, in my opinion.

Any ideas?

Arthur

@Franklin Well, as @leslietownes suggested, you might wanna draw a diagram

First of all, any point to the right of (1,0) can never be in K

By the phrase "right of ...", I mean, right of (1,0) in x-axis

So we may consider left of (1,0) in the left of x-axis

That's for starters

PM 2Ring

6:40 AM

@Franklin Certainly! Until the middle of the 20th century, most calculations were done manually, although we had simple calculating machines that could do addition & subtraction, and occasionally multiplication (but rarely division). The Industrial Revolution, the great engineering works of the 19th & early 20th century wouldn't have been possible without people able & willing to do manual calculations.

Arthur

@Franklin Thus, you can eliminate option A and C, for sure

PM 2Ring

And let's not forget the vital importance of good navigation tables. The British Empire achieved much of its power because its navigators had the best tables & charts in the world.

Arthur

@PM2Ring Ahh the Brits were bad guys at that time, colonial guys. They colonised India for a great time, and opressesed, looted , killed thousands, drove Indians to famines, crushed manpower. Divided India in my opinion.

But today's Britain is so different from that Britain. We can't compare them. Now, India and Britain have such a cordial relationship!

And not to forget, such a close cooperation

PM 2Ring

Sure, horrible things happened all over the world during the era of European colonialism. But eventually, the colonial powers (mostly) realised that they didn't have a God-given right to rule the world and treat other people like crap. Sometimes, bad things have to happen before people understand how & why they are bad.

Arthur

@PM2Ring you have spoken true words in a right way! If you are conversent with the history ofI Indian nationalism, you will find that they heavily took advantage to divide Indians on the basis of Aryan and Dravidian race. They asserted northern part to be Aryan populated and the southern part to be Dravidian populated.

(If I remember correctly, about these division on basis of race). An interesting thing was that, some Indian leaders took help from the Axis Powers at that time to gain freedom. The then world, was so full of political tensions and hardships.

PM 2Ring

6:56 AM

@Arthur Yes, I know a bit about Indian history. This chat room is probably not the best place to get into an in-depth discussion on this complex topic...

Arthur

@PM2Ring Yeah....I was chatting with you to kill my boredom😂😂😂

PM 2Ring

But consider: when the Europeans came to India it wasn't exactly a peaceful paradise. It was still in the process of being conquered by the Mughals en.wikipedia.org/wiki/Mughal_Empire

Arthur

7:14 AM

@PM2Ring True!

Arthur

7:27 AM

@Franklin Now, back to your question, you see, you can find a point closest to (1,0) from the left of (1,0) and as well as from above as well (i.e the points not on the x-axis). That eliminates D

The corect option is thus B

I would like @leslietownes to validate this as it seems he had looked at your problem and maybe, this is what he meant :)

First of all, any point to the right of (1,0) can never be in K

By the phrase "right of ...", I mean, right of (1,0) in x-axis

So we may consider left of (1,0) in the left of x-axis

That's for starters

Thus, you can eliminate option A and C, for sure

leslie townes

2 hours ago, by leslie townes

draw a picture. convince yourself from the picture that the set (A) contains points whose closest point in K is something other than (1,0), and that there are points in the plane whose closest point in K is (1,0) that are not on the x-axis.

Arthur

you can find a point closest to (1,0) from the left of (1,0) and as well as from above as well (i.e the points not on the x-axis). That eliminates D

The corect option is thus B

leslie townes

pretty much what i had in mind here :)

just looking at (1,1) and (2,1) for example is enough for the process of elimination. i think drawing a picture also makes clear why (B) would be correct even if you were not relying on the process of elimination

e.g. if it were not a multiple choice question, the picture would get you there eventually

Arthur

@leslietownes Hmm..I just restated your argument conpletely in words and posted them together, I mean all my comments made under this question together again to avoid confusion( Thus the intuitive solution you may want is : @Franklin Start reading from the sentence " First of all..." in my comment tagging leslie townes ) I assume my so-called solution are in agreement with @leslietownes as I take his comments to be an affirmation of the validation, I desired so much !

Franklin

@Arthur Thank you so much !

Franklin

7:51 AM

@Franklin I think @leslietownes also meant this approach , right ? 🤔🤔🤔

Ohh.. he has already, commented I hadn't looked it! Yes , it seems @leslietownes also meant it...that chat didn't load up here. Strange!

Ohh.. he has already, commented I hadn't looked it! Yes , it seems @leslietownes also meant it...that chat didn't load up here previously. Strange!

Transcript for

$Mathematics$ Mathematics