Mathematics - 2021-12-21 (page 1 of 3)

Wolgwang

3:05 AM

Was some answer deleted here?

copper.hat

@Wolgwang there are two deleted answers

Wolgwang

3:32 AM

Ah! thanks :)

love_sodam

Is there any problem using L'Hopital's rule to prove something rather than to calculate limits (or simplifying limits)? Only example I know is math.stackexchange.com/q/407654/668308 I can't find anything else.

leslie townes

3:57 AM

that... is a use of l'hopital's rule to calculate limits? i'm not sure i understand the question.

anywhere there are limits, there may be applications of l'hopital's rule. it's not just for explicit calculations where there's a formula for the thing you're taking the limit of.

koro was deep into this a while ago, he may have more information.

Koro

@love_sodam: One more use could be: if $f$ is convex on $\mathbb R$ and is differentiable everywhere then $\lim_{x\to \pm \infty}f'(x)= 0\implies f$ is constant on $\mathbb R$.

I think that lot more such examples can be created if you do want to force-enter L'Hospital's rule in some theorems.

leslie townes

you can use it to compute derivatives of C^1 functions! lim x->a (f(x) - f(a))/(x - a) = lim x->a f'(x)/1 = f'(a)

👌👍👍👍😉😉

Koro

😄

love_sodam

4:14 AM

@Koro How L'Hopital rule applied here?

Koro

@love_sodam In convex function example?

love_sodam

@Koro Yes the problem you wrote

Wolgwang

@leslietownes Oh emojis 🤩🤩🤩

Koro

@love_sodam because then we'll have $\lim_{x\to \pm\infty} \frac {f(x)}x=0$

Prithu biswas

Is the epsilon choices for proving the limit laws adhoc or intuitive?

copper.hat

4:20 AM

are there more answers we can select from?

leslie townes

prithu: given epsilon, the choice of delta is never unique. this is an artifact of the definition itself. in some contexts it is possible to find the best delta for an epsilon (which arguably is not 'ad hoc') but mostly it is not

copper.hat

i'm in a vile mood today

Koro

cheer up, copper!

leslie townes

so often there is some flexibility in how and why delta is chosen. copper is suggesting that there may be a wide gulf between having nonuniqueness in how you choose something, and something being completely ad hoc

copper.hat

Thanks @Koro!

leslie townes

4:22 AM

in fact a lot of epsilon-delta and epsilon-N proofs follow inequalities whose forms are so intuitive that books will refer to them by name, e.g. "this is a delta = epsilon/3 proof"

which makes no formal sense whatsoever but is suggestive of a surrounding context that makes a particular argument a logical choice

even if it is not the only choice

the need to frequently make choices that turn out to be semi-arbitrary is one of the main things i see that separates introductory analysis from introductory algebra

copper.hat

the choice may be made for later appearance (so the result is $< \epsilon$ rather than $2(L+1)\epsilon$ or some other junk like that).

leslie townes

and some level in the galaxy brain hierarchy is definitely realizing that it's fine if you can make it < 2 (L+1) e

Koro

(Rudin does that in many proofs in Riemann integration chapter)

PM 2Ring

That's part of the beauty. You often don't care what's the best delta, you just want a delta that works.

leslie townes

in my own research area, people were fine publishing results where they chose parameters to get < [an enormous number of bounded things]*[a function of epsilon known to go to zero by some other argument]

love_sodam

4:26 AM

@Koro Hmm essentially the problem that convex $f$ with $\lim_{x\to\pm\infty}\frac{f(x)}{x} = 0$ implies $f$ is constant.

leslie townes

the only reason you don't see this done in beginning books is that formalizing what you can get away with in epsilon-delta proofs is very close to formalizing what you mean by limits in the first place

copper.hat

personally, i often find it instructive to see the 'unclean' expression as it sometimes highlights the 'sensitivity' to certain components of an expression.

probably more engineering than mathematical interest.

love_sodam

Any basic analysis book (except PMA) you want to suggest @Koro ?

PM 2Ring

People did calculus for a couple of centuries before Weierstrass & friends invented epsilon-delta and introduced the modern formal understanding of limits. Before that time, the foundations of calculus were a bit shaky, and philosophers were understandably critical of the justifications that mathematicians gave for the whole game. ;)

Koro

@love_sodam I'd solved this problem yesterday night and then got stuck at a Legendre transform question, which also had some "convex" function related properties under certain conditions. I have postponed the exercise for some other day.

For analysis book, you could refer to Zorich's volume 1. I don't know what you mean by basic. @love_sodam

love_sodam

4:36 AM

@Koro I mean elementary (or introductory) analysis textbook like PMA.

I forgot analysis a lot

shintuku

I like Elementary Real Analysis by Thomson and Bruckner

it's free on the author's website

Koro

I think you could refer to Zorich's :). It starts with "logics, set theory" and starts sequences etc.

leslie townes

copper: you see that in a lot of analysis too. also, explicit attention to whether something like "C" is an absolute constant or implicitly depends on the dimension of something or some other intrinsic property.

Koro

So this starting section -logics, set theory will help you write definitions in symbols and also help you understand countable/uncountable etc.

:)

This book also defines Big O and small o and uses them in solving limits etc. :)

shintuku

Koro did you do all zorich o.O

Koro

4:40 AM

not all of it yet. :(

shintuku

it would take a while lol

Koro

And I don't think that the book has a solution manual :)

shintuku

i am pro-solutions

i don't know why some authors insist that it is better not to have them

Koro

@shin: Theorem: Every bounded sequence in R has a convergent sequence.
Zorich's style proof: if $(x_n)$ is a bounded sequence in R, then the sequence $\inf_{m\ge n}x_m$ is convergent. QED.

<3

love_sodam

@Koro Looks great. Thanks

leslie townes

4:49 AM

koro: that's certainly a limit of a subsequence of (x_n), but the inf formula does not in general define a subsequence of x_n

maybe i should take this up with zorich and not you

shintuku

isn't the infimum here a single number, or did i misunderstand the notation?

Koro

$\inf_{m\ge n} x_m:=\inf \{x_m: m\ge n\}$

leslie townes

koro: the constant sequence 0 is not a subsequence of x_n = 1/n

0 is the limit of every subsequence of x_n, but it's not a subsequence of x_n

shintuku

hm it seems to me Koro that this is a single number that might not be in the sequence

PM 2Ring

@shintuku Some books compromise, eg they only give the solutions to odd-numbered exercises.

Koro

4:52 AM

Ohh no, I made some mistake. I'll look up again and state the correct version.

leslie townes

if solutions helped textbook sales, they would all include solutions. from this we can infer that some people who choose what textbooks to assign don't want there to be solutions

shintuku

at PM: yeah that makes sense

when self-studying i have nothing else to check how i'm doing

but i see how it makes sense for a class with assignments

PM 2Ring

Well, if you're doing integrals you can always differentiate to see if you get back to the original equation.

shintuku

for those i double check with sagemath hehe

PM 2Ring

It's nice if you can make some kind of graph to verify that your solution is at least sane. Sage (etc) is good for that. And the process of creating the best graph may give you further insight into the function(s).

leslie townes

5:01 AM

koro: you do see a general split between purely order-theoretic treatments of analysis (which if they construct R from something else often choose dedekind cuts) and sequence-focused treatments (which prefer cauchy sequences). i.e. whether "lim sup" is defined in the first instance as an abstract symbol for the sup of a set of limit points (rudin style), or the actual limit of a sequence of suprema of a decreasing sequence of subsets.

my rough impression is that the sequence-focused way is easier for a lot of people to grasp, but that the difference between the two choices is less significant than the quality of the instructor who is implementing that choice.

Koro

I misunderstood earlier and thought that $\inf _{m\ge n} x_m$ is a subsequence of $(x_n)$, which is wrong as I realized today and hence I interpreted the proof in the book in my (wrong) way.
Theorem: If $(x_n)$ is bounded in $\mathbb R$ then it has a convergent subsequence.
Proof: The proof in the book is along these lines: Since $(x_n)$ is bounded the sequence $\inf_{m\ge n} x_m$ (and it's not in general )a subsequence of $x_n$) is increasing and bounded and hence convergent to L,say. Now by proposition (proven earlier in the book), there is subsequence of $(x_n)$ converging to L. Done.

@shin

shintuku

oh right, and you can use the sequence of supremums for the rest of the cases

Koro

sorry for the confusion caused earlier. But you don't need sups here. Sequence is bounded so either inf or sup should suffice. Also, in the book B-W theorem comes out as a corollary :)

leslie townes

5:19 AM

pathetic the kind of stuff you could get your named attached to if only you were born at the right time

shintuku

@Koro doesn't a decreasing bounded sequence have a sequence of infimums outside the sequence?

leslie townes

it can, yes. depends on what you mean by 'decreasing.'

there are a number of classic arguments for the subsequence selection.

copper.hat

are you being strict?

leslie townes

i hate doing free advertising but i like the approach in koerner's companion to analysis.

shintuku

either strictly decreasing, or a wavy sequence whose tips are decreasing. both cases, if bounded, have an infimum outside the sequence

leslie townes

5:23 AM

he tends to give more than one standard textbook approach, maybe one worked out and others as exercises with multiple sub-parts.

thank god you didn't use my least favorite word in the context of limits, which is 'oscillating'.

Koro

shin, the highlight is that (see, I was also confused at it earlier) we want "limit L of that sequence of infimums" and argue that L can be approached by a "subsequence of $x_n$".

shintuku

right, but if i understand you correctly, note that if we take a strictly decreasing bounded sequence, the sequence of infimums does converge to the same limit but is not a subsequence

Koro

"that sequence of infimums" does not have to be a subsequence of $x_n$.

Ishwaran

Can anyone help me in my math problem ?

Ted Shifrin

Ask the question.

Ishwaran

5:35 AM

Find the length of the line of sight of the horizon, where you are standing on top of a mountain with slope of 10 degrees and with height 9.1 kilometer taking the radius of earth is 6300 kilometer

PM 2Ring

BTW, that mountain is higher than Mt Everest. And Earth's radius is closer to 6371 km.

But anyway... Have you tried drawing a diagram?

Ishwaran

@PM2Ring yes the mountain is taller than the mount everest :p But we are approximating the earth radius for ease of calculation

@PM2Ring yes

Koro

@shintuku Lemme state the proof to make it clear: Suppose $(x_n)$ is bounded. So $\inf_{m\ge n} x_m=:y_n$ converges to some $L$. For every $n$ there is a ${k_n}$ such that $y_n \le x_{k_n}\le y_n+1/n$ (by definition of inf). We choose $k_n$ such that $k_n\lt k_{n+1}$. Now by squeeze theorem, $x_{k_n}\to L$.

shintuku

cool yeah that makes sense

PM 2Ring

@Ishwaran Great. So can you work out the line of sight distance if you ignore the slope of the mountain?

Ishwaran

5:49 AM

@PM2Ring definitely i can workout if the slope is neglected, But the question is will the slope impact the sight since if the field of view of the eye is also 10 degress

typo FOV of eye is 10 degrees only

PenAndPaperMathematics

0

Q: Mathematics in $\textbf{Prop}$, the ring of propositions? Is variable substitution a ring homomorphism?

Consider the ring $R = \textbf{Prop}$ of proposition in which $+$ is XOR of two propositions and $\cdot$ is AND of them. Since $\textbf{Prop}$ is a countable set, we have that $R$ forms a countable ring. The basic inference rule $f:A \to B \implies \forall a \in A, fa : B$, can be encoded as the...

Today's my birthday

shintuku

hbd

PenAndPaperMathematics

Thx, upvotes are a great gift :D

shintuku

that's illegal and i'm a law abiding citizen

PenAndPaperMathematics

@shintuku this is about the problem I posed the other day

Remember the ring of propositions / atomisation you linked to?

I can't figure out what to modulo by and where that would be useful, but if you have a principal ideal in this ring, then $x \in \mathfrak{a} \iff x = ay$ for some proposition $y$ meaning $x \iff ay$ i.e. they're logically equiv.

Where multiplication is anding

So it's like saying $a$ could be useful as part of an equivalence condition for $x$

PM 2Ring

6:01 AM

@Ishwaran Eyes generally have a wider FOV than 10°. ;) Does the question really say that? You didn't mention that before. You just said the mountain's slope is 10°.

Wolgwang

@PenAndPaperMathematics Happy Birthday 🎂

PenAndPaperMathematics

Thanks, mon!

PM 2Ring

Hippy bath day!

PenAndPaperMathematics

True dat

I did shower today

Had a job interview for tutor

robjohn

@PM2Ring extreme tunnel vision?

PenAndPaperMathematics

6:03 AM

I could be making $18 / hour soon, I think they're going to train me

with curriculum soon

Ishwaran

@PM2Ring The question actually didn't include that, but will it affect the strategy ?

@PenAndPaperMathematics Happy birthday !

PenAndPaperMathematics

Thanks, all! You are too kind :)

copper.hat

find it a little irritating when someone answers a question some time after yours with essentially the same content.

PenAndPaperMathematics

I borrowed money from parents for dog vet bill, so I think that's a good present

and my dad gave me his old work desktop computer, so I'll be coding / mathing with dual monitors from now on

Pecan pi was my cake

@copper.hat could be the rewrite elucidated some parts of the other answer, but they should have stated that as the purpose of it

leslie townes

it's the same answer as far as i'm concerned.

PM 2Ring

6:07 AM

@Ishwaran Don't make up stuff & add it to the question! You need to answer the actual question they're asking you. That's why it's important to read questions carefully so you don't accidentally misinterpret things. Forget FOV. What's the base & hypotenuse of a right triangle with a base angle of 10° and a height of 9.1 km?

leslie townes

there ought to be a way of closing answers as duplicates. :D

copper.hat

i'm just in a foul mood anyways :-)

Ishwaran

@PM2Ring yes

leslie townes

the charitable interpretation is they began an answer and just took longer to finish and ignored any prompts that the question had been answered. could totally happen, on this time scale.

copper.hat

it is possible, i have done that in the past. ok, i'll redirect my foulness :-)

PM 2Ring

6:16 AM

Here's a nice Irish Christmas song for you. Chris Thile & Aoife O'Donovan doing The Pogues Fairytale of New York. They dropped the non-PC verse though, for the benefit of the Prarie Home Companion audience.

copper.hat

love that tune, makes me sad. last time i saw the pogues was in sf, and if i recall correctly, joey strummer took shane's place who was out of sorts.

leslie townes

tired and indisposed from the long flight no doubt

PM 2Ring

No doubt.

copper.hat

indisposed certainly...

PM 2Ring

That song's a work of art, combining joy and sadness. I was never a huge Pogues fan, but I love Kirsty MacColl, and Shane & Kirsty's duet is priceless.

FWIW, The Pogues song Dirty Old Town was written by Ewan MacColl, Kirsty's Dad, whose most famous song is First Time Ever I Saw Your Face, which is definitely not in the same universe as The Pogues. :)

copper.hat

6:30 AM

pogues were on the scene long before the whole celtic tiger and all that rah rah, they were sort of an emblem for many irish in the usa (not necessarily legal) when things were a little rougher.

leslie townes

kirsty maccoll is sadly the subject of one of the few youtube comments that made me laugh out loud. on one of her hits, someone commented: "Wow great song i cant believe she was killed by a boat"

that's my sad sense of humor in a nutshell

if it can be put in an artless manner, youtube commenters will find a way

PM 2Ring

Yeah, she drowned while saving her kids in a water skiing accident. :(

Wolgwang

@leslietownes Do you flag questions? (Asking for spotting scope)

copper.hat

roberta flack (one of my faves) sang one of his songs, First Time Ever I Saw Your Face

leslie townes

wolg: i may have done once or twice, in ancient memory, but not recently enough for it to trigger any hats.

copper.hat

6:33 AM

i have flagged many things,

Wolgwang

@leslietownes Then you must have CVed some questions?

leslie townes

CVed meaning what?

copper.hat

what the cv?

shintuku

i think it is shorthand for covfefe

copper.hat

:-)

Wolgwang

6:34 AM

@leslietownes Close voted (☉｡☉)!

copper.hat

this one needs urgent attention math.stackexchange.com/questions/4338869/…

leslie townes

from the name of the thing alone, i wonder if it might be triggered on being the first to identify a post that is later closed or deleted.

yes, i do vote to close. probably an average of 5 times a day.

PM 2Ring

I mostly flag Not An Answer, especially on Astronomy, where we get a lot of 1 rep newbies posting comments or related questions as answers.

leslie townes

copper despite his penchant for calling me dr death or whatever on account of a number of close votes is also a voter to close.

PenAndPaperMathematics

\o/ I got upvoted. Celebration time

copper.hat

6:37 AM

i do close from time to time, but only if there is pure lack of effort :-)

PenAndPaperMathematics

If you get upvoted, it means 10 newbies got the downvote :O

copper.hat

or if the number $-{ 1 \over 12}$ appears.

PM 2Ring

Flagging spam is fun, but you don't see much of that on the busier sites. It usually gets killed pretty quickly by regular users & by Smokey.

leslie townes

i vote to close for vagueness or lack of effort. if someone posts a pure PSQ that is perfectly phrased (e.g. no ambiguities suggesting ignorance or a lack of effort) i usually frown but do not always vote to close.

copper.hat

i vacillate between hardnosed and softie

i am a sucker, as i have repeatedly discovered...

leslie townes

6:40 AM

if there's any hint of something coming from a textbook or a multi-part exercise where they're treating parts (b), (c), (d) as their own 'efforts' i similarly frown but remain silent on the voting front

but christ, that's obvious when people do it

PenAndPaperMathematics

What's worse is when you get upvoted thrice, and starred 5 times and no one comments any longer or posts a correct answer. It's as if the post was too perfectly made.

leslie townes

hey guys i was wondering [likely paste from textbook]. all i could think was [textbook hint], and maybe this is relevant to [part (b)]? or like maybe if we knew it we could [part (c)]? just wondering!!!! lol please answer, if this were due it would probably be due in the next 24 hours

pen if you want me to balance out the universe by downvoting these upvotes, just let me know. :)

Wolgwang

"This is our final review question for freshmen" What does review question mean?

leslie townes

beats me

shintuku

a question about previously seen material, usually

PenAndPaperMathematics

6:44 AM

I'm referring to my Twin Prime counting formula, 3 upvotes, 5 stars, no answers or any more comments. I hope people like it though. I figure, the twin primes has already been solved, but the person that solved it first will not get the money. You have to be in accademia to receive the cash. So why persist. I did come up with the proof of that formula though on my own, so got to practice a few ideas. I learned some stuff..

Not saying I solved, but just not going to waste time on money that doesn't exist

for me

It is indeed a surprisingly succinct formula

PM 2Ring

@PenAndPaperMathematics The answer to that isn't something that you could write in an hour or two. People have dedicated substantial chunks of their entire career to twin primes. ;)

PenAndPaperMathematics

Not saying it's impossible to win a Millinium prize as a non-accademician, just next to impossible to get published in proper journal or even recognized

That took more than an hour or two to come up with

copper.hat

ok, since we're griping. it bugs me when what constitutes an 'effort' for the psq fast closers is nothing other than some obvious statement that has no bearing on a solution.

PenAndPaperMathematics

Try 5 years lol

It was a culmination of thought

copper.hat

what is the fascination with twin primes. i can see fascinations with such things in other arenas, but not primes...

PenAndPaperMathematics

6:49 AM

It's the million dollars, the late night parties and you're a celeb then

copper.hat

like all the other maths celebs :-)

PM 2Ring

@PenAndPaperMathematics Exactly. So why do you think some random person on MSE is going to be able to post an adequate answer, unless they've also done at least the same amount of work on stuff related to counting primes?

PenAndPaperMathematics

It was not to prove TP, it was to answer a subproblem

copper.hat

so its a mount everest sort of thing

Wolgwang

Can someone temporary (for 5 minutes) upvote this comment? (Testing some trigger)

copper.hat

6:53 AM

done

PenAndPaperMathematics

I upvoted your two comments

copper.hat

wait, are you expecting me to downvote in 5 mins? i'm not sure my little glass of kona will last that long

Wolgwang

Ok thanks. Fingers crossed

PM 2Ring

@PenAndPaperMathematics Yes, I know it's just a sub-problem. It's still a pretty deep sub-problem. BTW, if you had claimed to have solved Twin Primes, your question would be closed as off-topic.

PenAndPaperMathematics

🎂 mah birthday today

There should be a site like MSE for formal proofs so that people can post new results on the site, and the community can see that for certain it is true, without the lengthy peer review process

shintuku

6:57 AM

there's academia.edu

PenAndPaperMathematics

I know that Lean community is moving over to SE from Zulip

but it's still not a "proof site"

we need a dang proof site, people!

shintuku

there's also proofwiki

PenAndPaperMathematics

not good enough / too many ads / not complete enough / no built-in math machinery

though I do refer to it some times

PM 2Ring

@PenAndPaperMathematics Do you mean a Stack Exchange site?

shintuku

it does suck navigation-wise

PenAndPaperMathematics

7:00 AM

It could be that format, but the math machinery part takes priority in the design, which might dictate the UI format

PM 2Ring

It wouldn't be possible on Stack Exchange, even without the machinery. It just doesn't fit the Stack model.

shintuku

it would be nice to revamp proof-wiki and add voting/discussion

make it more easily navigable

PenAndPaperMathematics

Yes, there should be voting categories like "most elegant" etc

shintuku

proofwiki.org/wiki/Category:Proofs

look at that page. it sucks

how do i even find what i want

or browse for fun

PenAndPaperMathematics

it has two big ads for me

Keanu Reeves is on there

:D

Wolgwang

7:04 AM

@Wolgwang You can revert the votes. It didn't work 😞

PM 2Ring

The votes are probably locked by now.

Wolgwang

@PM2Ring Comments votes are locked?

PM 2Ring

Most certainly. You only have a window of a few minutes to change them. And that window closes if you navigate away from the page, IIRC.

copper.hat

and then the whole world ends in a little bang...

2

much to the disappointment of all those singularity fans

and bill & elon sit around shaking their heads

why did we waste so much time putting those chips in people they ask themselves

ok, wine if finished, time to go to bed. good night folks!

PM 2Ring

Night, copper.

Koro

8:09 AM

@PenAndPaperMathematics Happy Birthday 🎂

Odestheory12

The set $\{1/n : n \in \Bbb N\}$ is finite right?

shintuku

no

Odestheory12

Its a bit anti-intuitive

shintuku

it is infinite

Odestheory12

How so?

Its majorized and minorized hence its bounded and hence finite

shintuku

8:21 AM

a set is (countably) infinite if there is a bijection between the natural numbers and that set

here's a bijection: $n \mapsto 1/n$

Odestheory12

Uhm.

shintuku

also, boundedness does not imply finiteness

Odestheory12

I've seen that if a subset of Z is bounded then its finite.

shintuku

notice that the set you mentioned is not a subset of Z

1/2 is not in Z

Odestheory12

Ah, right, my bad.

Wow, :D

How did I miss that haha.

Thanks

shintuku

8:24 AM

np

Odestheory12

@shintuku Uhm is it enough to be injective?

Well if its $f:A\to \Bbb N$ I guess

shintuku

no, since the function from the set $\{1,2,3\}$ to three random numbers in the natural numbers is injective

but it is finite

Odestheory12

Ah, you wrote infinite :P

shintuku

yes, in order to get countable infinity you need a bijection

Odestheory12

Yeah, i was thinking only in countable sets

shintuku

8:35 AM

if what you want is to prove finiteness, and you found an injection, this is not enough. you have to prove there is no bijection

Odestheory12

Yeah, because it could be either finite or countable infinite

shintuku

yep

PM 2Ring

8:54 AM

@Odestheory12 Yes, your set is bounded, but it has no lowest element. That's a sure sign that it's infinite.

Odestheory12

@PM2Ring Yeah, for some reason I was thinking the set was in Z and hence with a min.

404 mode.

PenAndPaperMathematics

10:54 AM

What do you think of this design for QuiverDatabase?

:en.wikipedia.org/wiki/Type_theory#Difference_from_set_theory

nvm that

That's the design

The user enters in $C\in \text{Ob}(\textbf{Cat})$ and clicks on it, and it auto-gos to the definion, expanding in another tab

Mad Spaces

11:26 AM

Hello. In this proof, where is the continiousness of the function used? the only time i see that is where i highlighted it, but why is it important there? what would happen if it would not have been continious?

robjohn

11:38 AM

Let $K=[0,1]$ and $f(x)=\left\{\begin{array}{}x&\text{if }x\le\frac12\\x+1&\text{if }x\gt\frac12\end{array}\right.$

what is $f(K)$?

Mad Spaces

$[0,1/2] \cup (3/2,2]$ "not closed" thus not compact.

robjohn

so obviously the continuity is needed.

Mad Spaces

But i mean, in the proof, where is it called?

robjohn

right where it is highlighted.

Mad Spaces

Yea, but i do not understand that part. Howcome we need to call the continiousness at that part. What happens if i omitted it?

robjohn

11:43 AM

$\lim\limits_{n\to\infty}f(x_n)=f(x)$

Mad Spaces

Oh that.

robjohn

that's sort of the main point of continuity

Mad Spaces

In the main theorem, it is required from the function to be monotonious, not continious. Is Monotony on a compact set means also continiousness?

robjohn

my function was monotone.

Mad Spaces

Oh

Excuse me.. i had really misundersanding when i translated the terms from German. i am looking at two different theorems..

Thanks for the help John!

robjohn

11:48 AM

np

Mad Spaces

If a function is called monoton, it means its either decreasing or increasing. But not in spesific interval one and in another the other, right?

I had this strange idea, that it could be that... stupid

robjohn

@MadSpaces yes

under the second, $\sin(x)$ would be monotone.

Mad Spaces

12:04 PM

General question. I seem to be forgetting simple stuff i learned and viewed at some point in mylife trivial. Is this normal? Or does this mean i am learning wrong? Do you experiance this?

robjohn

12:45 PM

That sounds like a question for a psychologist. Personally, I don't know if there is a "wrong" way to learn things.

Prithu biswas

@MadSpaces example?

Mad Spaces

I do not know what example to present. I just feel constantly that i am not "sharp" in some subjects, or topics, that at the time, i used to be extremly well at.

I wish we could remodify the brain to make it part maschine, so we can improve memory and learning :D

Prithu biswas

@MadSpaces Personally , when I finish studying a topic and then do something else for a couple of months , I find out I have forgotten a lot of things about that subject.Then when I revise it again , I discover some new insights that I didn't found before. I guess I can't "master" with just one revision. It takes a lot of revisions and a couple of years to "master" a topic.

Mad Spaces

1:34 PM

Well. Yea. Still. one forgets. nevertheless

Is there a reason that the spesific case of g(x)= x for the mean value theorem a special theorem? " look at general case " ..
Maybe historical reasons? i wouldnt call it a theorem, more like.. a consequence. Or a corollary

shintuku

i use anki for keeping what i've learned previously

Koro

what's that @shin?

shintuku

apps.ankiweb.net

it's a spaced repetition program

i put in exercises and proofs that were difficult or i was unable to solve

Mad Spaces

thanks for sharing. i will check that out. i didnt know about it.

Koro

Ooo we can use latex also in it

shintuku

1:45 PM

also useful for memorizing formulas or facts

yeah mine is full of latex

Koro

shin: iOS version seems to be a paid app and android version is free?

shintuku

yep. i use pc version

e.g. i just recently did an analysis problem i put in last year. so it's still fresh throughout the year

you don't put allll of the problems you solve, only those that are particularly difficult or instructive

Xander Henderson

@shintuku I prefer nitrile.

shintuku

ah, yes, sniffing glue

Xander Henderson

2:04 PM

@shintuku Nonono... a lot of people are allergic to latex, so it is better to use nitrile (eg those sexy blue gloves).

shintuku

i guess whatever works, as long as those gloves are airtight when tied. they have to keep the glue's gas from leaking out

Semiclassical

2:31 PM

How to get the grader to roll their eyes: complain about getting 9/10 instead of 10/10 in a grad level course for not showing your work

Xander Henderson

@Semiclassical "Oh! You're right! It looks like I made a mistake here. 6/10"

"But!---"

Semiclassical

How to annoy a grader further: opine about how you’d have split the two HW problems as 1+9 instead of 4+6

Xander Henderson

"5/10"

"But my answer is correct!"

"4/10. Wanna go for less?"

Semiclassical

“But I can explain why it’s right!” Then you should have done that

Xander Henderson

@Semiclassical I had a student spend the entire semester this fall telling me how he, a community college calculus student, would teach the class completely differently.

@Semiclassical "3/10"

Semiclassical

2:35 PM

lol

Xander Henderson

Also, don't graduate students know that their grades don't matter?

Semiclassical

Exactly!

And they’re already at 95%

So it’s not as though this has any possibility of changing the grade

There is an argument to be made for a 3+7 split, maybe 2.5+7.5

But going to 1+9 in my mind shows contempt for the prof in setting the HW

If it was as trivial as that, then why would the prof even bother to split it up at all?

Xander Henderson

Indeed.

Though I am not sure what you mean by a 3+7 split.

Semiclassical

There were two problems on the HW, and I’ve been assigning 10 pts per HW

Xander Henderson

Ah.

Makes sense.

Semiclassical

2:43 PM

Also, come to think of it I could have complained about a different step he left implicit

The computation amounts to $\sum_{n=0}^\infty (-1)^n \exp(-x(2n+1)}=1/(2\cosh x)$

Which is simple—it’s a geometric series

But to just state that that’s what the sum equals is pretty arrogant imo

Xander Henderson

Agreed.

dtn

Hello everyone. I need help from experts in matrix analysis. The problem is the following: there is a symmetric matrix and I need to determine if it is positive definite or not. There are many ways, and I decided to go with the next one. I wrote the code in Mathematica below and found a contradiction.

Among the eigenvalues of the matrix there are 3 positive and 1 negative. Those the matrix is not positive definite. At the same time, the matrix satisfies the specified criteria, i.e. is positive definite. Please clarify this case for me.

H = {{4.`, 0.5`, 10.25`, 16.625`}, {0.5`, 10.25`, 16.625`,
57.0625`}, {10.25`, 16.625`, 57.0625`, 141.90625`}, {16.625`,
57.0625`, 141.90625`, 395.890625`}};
eigenvalues = N[Eigenvalues[H]];
Table[Abs[H[[i, j]]] <=
Sqrt[H[[i, i]] H[[j, j]]] <= (H[[i, i]] + H[[j, j]])/2, {i, 1,
n}, {j, 1, n}] // Simplify // MatrixForm;

Please see for yourself.

leslie townes

dtn, i can't read mathematica very easily, but i note that while the circled trace inequalities are a consequence of positive definiteness, they are not the definition of positive definiteness

seems totally possible for the inequalities to hold for a matrix that is not positive definite, if that's what your code suggests is happening

Koro

Every convergent sequence in R has either a maximum or a minimum term in it.

(fact not so common)

dtn

@leslietownes I also assumed this and, therefore, these criteria cannot be criteria for positive certainty. I was attracted to these formulas by their simplicity and work with matrix components without the need to connect traces, determinants, etc.

leslie townes

3:01 PM

yeah, they're cool. with more or different hypotheses you can come up with similar stuff that can be used to get info about eigenvalues from matrix entries.

schur's inequalities and the gershgorin circle theorem come to mind

dtn

0

Q: Extension of the Gershgorin circle theorem for symmetric matrices and localization of positive eigenvalues

In mathematics, the Gershgorin circle theorem can be used to localize eigenvalues of a matrix (including symmetric). Let $A$ be a real symmetry $n × n$ matrix, with entries $a_{ij}$. For $i∈{1,…,n}$ let $R_i$ be the sum of the absolute values of the non-diagonal entries in the $i$-th row: $R_{i}...

matrices inequality eigenvalues-eigenvectors matrix-calculus gershgorin-sets

@leslietownes I worked a little with the Gershgorin theorem, it also has its own peculiarities, I wrote about it here. Those. there are cases when one of the eigenvalues of the matrix lies in a small piece of a circle that "crawls" a little into the negative area

napstablook

why cant I solve this qn? Basically I can prove that for a>3 this series aint gonna converge

but I cant prove that it converges for sure inside |a|<=3

My attempt: $x_{n+1} - x_{n} =\frac({1}{4})(x_{n}^2 - 4x_{n} + 3) $

Wolgwang

1

Q: Examining the convergence with parameter $a$

For $a \in R$, let $x_1=a$ and $x_{n+1}=\frac{1}{4}(x_{n}^2+3)$ for all $n≥2$. Examine the convergence of the sequence ${x_n}$ for different values of $a$. Also, find $\lim x_n$, whenever it exists. I am having problems on how to take $a$ as a parameter. I am unable to think which all values of...

calculus real-analysis sequences-and-series convergence-divergence

napstablook

thx

Semiclassical

4:49 PM

@XanderHenderson oh g********. i'm finally looking through my students last lab reports. I gave students two options for what to write their report on. This student? they wrote a decent-looking report...on a lab that wasn't assigned

Xander Henderson

@Semiclassical ZERO

And report it to the relevant body.

robjohn

@XanderHenderson In a bad mood this morning?

Xander Henderson

Nope. Just salty.

robjohn

Ah

Semiclassical

That’s what I’m inclined to do

The issue is that they missed enough other lab stuff that this would result in a failing lab grade