Mathematics - 2021-12-08 (page 1 of 3)

12:00 AM

@TedShifrin Thank you for notifying me. I was confused by Xanders posts in CURED. I did not vote to close the "good post", I just deleted the comment you condemn me for, automated when I voted to close as a dupe, when I had the two posts mixed up. I upvoted the "good post", and downvoted the "bad post". What next will you excoriate me for, @Ted???? Adios!

I did not intend to disturb your kingdom. But you were willing to sit on your resentment about the one post, vs. the other post, and two and more of us tried to do something about it. The way the posts were presented to me confused me. And I corrected the mistake you scathed me for. Let me now, having learned I am not welcome in your Kingdom depart.

Axoren

@AkivaWeinberger Okay, so I think my question is more about this. We, in this context, know that $\alpha$ is $\sup a_n$. However, let's say we met $\alpha$ on the street and it was dressed like $\beta$ that is the output of a program $P$.

Akiva Weinberger

We interrogate $\beta$'s $n$th digit, using $P$

That gives us an algorithm to decide if $n\in\{c_i:i\in\Bbb N\}$

Axoren

Okay, so $c_n$ is an uncomputable sequence?

Akiva Weinberger

but the hypothesis was that, though $\{c_i:i\in\Bbb N\}$ was computably enumerable, it was not computable. (Two explicit examples are Gödel numberings of halting Turing machines and Gödel numbers of provable statements in arithmetic.)

Yeah

$c_n$ is a computable sequence, but if we rearrange it to be increasing, it is not computable anymore, essentially.

Think about how you would make a computable sequence that outputs a list of halting Turing machines

You would run machine 1 for one timestep, then machines 1 and 2 for one timestep, then machines 1, 2, and 3 for one timestep, etc

and whenever a machine halts, you write its ID down on a list

This list will not be given in order, because some short programs take a long time to halt and some long programs halt very quickly.

Axoren

12:16 AM

It's been quite a while since I've done any computability theory.

This just shows I'm getting lame.

My CS legs aren't what they used to be.

Akiva Weinberger

If you want a better description then what I've given here, I recommend the book Reverse Mathematics: Proofs from the Inside Out by John Stillwell

He is a much much better teacher than I am

Axoren

I'll add it to the list.

I recently cleaned out my book wishlist so it's not that long anymore.

Ulysse Corbeil

1:13 AM

Anyone would be willing to help me fix my kernel panic problem.. i'm desperate

Saad

← 1 message moved from CURED

Elchanan Solomon

@UlysseCorbeil what's the question?

UnderMathUate

1:54 AM

Let $B$ and $A$ be nonempty sets such that $B \subset A$. Prove that $|A - B| \leq |A|$.

So, I know that I need to show an injection between the sets in order to prove this. My thinking is that $A - B = A$. So we can define a function $f: A - B \rightarrow A$ such that $f(a) = a$. And from here, an injection is simple to show.

leslie townes

that rule will work even without the assumption (not needed, and not implied by the given information) that A - B = A

UnderMathUate

Oh, hmm

leslie townes

e.g. B = {1,2} is a subset of A = {1,2,3,4} and A - B = {3,4} is not A

but the rule f(x) = x does define an injection from A - B to A

UnderMathUate

Frick

That always trips me up

Alright, that's good at least.

leslie townes

note that this is equivalent to showing that $|S| \leq |A|$ for any subset $S$ of $A$. the fact that $S$ has the form $A - B$ is kind of beside the point (although any subset $S$ of $A$ does have this form, for some $B$)

UnderMathUate

1:58 AM

Oooh, that point becomes much more clear when you present it that way.

Joseph Sible-Reinstate Monica

3:56 AM

Is there a term for functions that feel cheaty, in that they're defined to be the solution to something we don't otherwise have a solution for, and also don't have another way to evaluate? E.g., the error function and the lambert W function

copper.hat

injections hurt sometimes.

you mean that they are implicitly defined?

leslie townes

i can't think of a term. it isn't really a property of a function so much as a property of a way of defining a function

e.g. ln(x) can be defined in a cheaty way as the solution to e^y = x, or as int 1...x 1/t dt which maybe doesn't feel as cheaty

i think copper's notion of implicit definition captures a lot of what you might not like about W and erf

at least as defined in the usual ways

Joseph Sible-Reinstate Monica

I don't think it's implicitly defined though. Doesn't that mean when f(x) doesn't appear on one side of the definition by itself? But for erf, it does, it's just that the other side is an integral we can't otherwise solve

leslie townes

some people seem more OK with some inverse definitions than others, e.g. i think many people who don't think arcsin or ln defined as inverses are cheaty, but would agree that W is.

being 'implicitly defined' isn't really a property of a function. it's a way of defining a function

it's like 'piecewise function' - that's not really a property of a function either, more of a property of a method of definition

you might be asking about which functions can or cannot be expressed in terms of 'eelmentary functions' which are a somewhat arbitrary but well known collection of functions. liouville proved some theorems on this

Liouville's theorem (differential algebra)

In mathematics, Liouville's theorem, originally formulated by Joseph Liouville in 1833 to 1841, places an important restriction on antiderivatives that can be expressed as elementary functions. The antiderivatives of certain elementary functions cannot themselves be expressed as elementary functions. A standard example of such a function is e − x 2 , {\displaystyle e^{-x^{2}},} whose antiderivative is (with...

so that's one way of making the notion rigorous, it might not quite match up with what feels 'cheaty'

Joseph Sible-Reinstate Monica

Ah, that gets close to what I'm going for. I think "functions that can't be expressed in terms of elementary functions" would be a necessary but not sufficient condition for what I'm calling "cheaty"

leslie townes

4:05 AM

arxiv.org/abs/math/9805045 is a kind of fun article that speaks about numbers instead of functions

W is not an 'elementary function,' nor is erf

Random Variable

The redesigned Activity page has major MathJax issues.

leslie townes

people have been mentioning this

Random Variable

It was clearly designed without MathJax in mind.

UnderMathUate

4:32 AM

Are there any theorems that state that an uncountable set unioned with a countable set is uncountable?

leslie townes

yes

UnderMathUate

omg ;-;

Does it have a specific name? I'm not finding anything about it in my textbook.

leslie townes

you proved something like it earlier. if S is a subset of A then $|S| \leq |A|$

i don't think it has a name

[uncountable set] is a subset of [uncountable set] union [countable set]

UnderMathUate

OH

I keep telling myself I'm never going to complete a problem set the night before it's due ever again. This takes all of the fun out of math.

leslie townes

that's pretty common. everybody does it.

i don't know if it's that bad. if the instructor is not assigning insane stuff on a weird timetable, it ought to be workable.

i don't recommend a similar approach for studying and exams.

once i graded homework for a prof who assigned homework due every monday, wednesday, and friday. they were short assignments, so it wasn't an unreasonable as an amount of work to assign in a week. but the sheer amount of paper that everyone had to deal with was unreasonable. and you have to allow people a little more flexibility in their scheduling than that.

UnderMathUate

4:43 AM

I try not to let this habit bleed over into exams, but sometimes it does.

And that many assignments in a week for one class is crazy.

leslie townes

the pedagogical purpose, i think, was to keep people engaged with the class on a near daily basis. but it also discouraged deeper thinking and led to a lot of hastily done work.

it also sort of increased the feeling among students that they were constantly being evaluated, which i didn't think was helpful. people need space to practice. the same amount of work due once a week would have been just fine.

UnderMathUate

I agree.

I found it. It's labeled as "Theorem 11.10" in the text.

"If $A \subseteq B$ and $A$ is uncountable, then $B$ is uncountable."

leslie townes

it's a practice that might only make sense in a setting where all students have the same life. i think frequent homework in high school can sometimes work on this basis. with college students people can have much more widely varying courseloads, jobs, etc. it's not reasonable to expect work every MWF.

UnderMathUate

Oh definitely. Especially these days where pretty much everyone works.

leslie townes

i'm against takehome exams in college for this reason. they really exacerbate any differences in the amount of time that people have available outside of class.

UnderMathUate

4:48 AM

I think there are a lot of facets of higher education that could stand to be adjusted for the times.

Wow, I was just telling someone I wish I had take-home finals today. I hadn't even considered this.

leslie townes

i mean, if i have the time, i definitely want a take-home final, as a student.

but in a group of 25 people, some of them won't. so it's harder as a prof.

UnderMathUate

I would prefer just because of the more relaxed environment -- assuming you start within a reasonable time frame.

leslie townes

if everyone has the time, they eliminate the time element. which i think is helpful.

but it's tough if it's like "you have a week to do this!" and some of your students have three finals in three other classes during that week, and some of your students don't.

if ted were here he would say that if you study properly along the way, it should not make that much of a difference, which is true.

copper.hat

how boring

leslie townes

yes. nobody should be studying, they should be at the bar with copper.hat, mortgaging their futures.

UnderMathUate

4:56 AM

I tend to get stuck on the whole "studying properly" part

I never had to study before college, so the adjustment has been really rough

leslie townes

well, you recognize the need for adjustment. that is good fortune. for some, that comes too late, or never.

UnderMathUate

Yeah, I guess that's true.

copper.hat

the pub...

leslie townes

math is deceptive because if you 'get it' early on, you definitely can cultivate habits that aren't sustainable, and it works for a time

the low page numbers have a lot to do with it. i don't think anyone would try to cram 1500 pages of reading into a few nights' reading, but if an exam is just on 25 pages of a math textbook, it seems like less of a bad idea to wait. it's only 25 pages!

UnderMathUate

That's so true!

leslie townes

5:01 AM

and maybe actually that's not true, people actually did try to cram hundreds of pages into last minute reading in law school.

studying just sucks and people don't like to do it.

UnderMathUate

Even today for my calc quiz. I waited until 30 min before to study one of the sections because my notes were only 3 pages. But it was very dense and I ended up needing to ask my professor for a clarification before the quiz.

leslie townes

it's only three pages!

:D

UnderMathUate

Only three ;-;

Law school kind of sounds like a nightmare.

leslie townes

there are only like, what, not even 10 axioms for an order-complete ordered field? i've got this. real analysis, piece of cake.

i tend not to recommend it to others.

UnderMathUate

Lol

copper.hat

5:20 AM

@Fargle i think it is fairly easy to compute a solution to your problem, but it is not an explicit formua.

subdifferential calculus has got to be good for something.

leslie townes

it's good for domdifferential calculus. it has to be.

copper.hat

it involves tying up parts of one's differentials, preferably while wearing black thigh highs.

did the round trip to santa cruz today!

leslie townes

quite a bike ride.

or does that mean you made contact with copper jr

copper.hat

by car, unlike my 2 yrs younger brother, i am not up for such cycles :-). yes, he's home. for some reason he does not like the little badges i collected yesterday at the trade sho.

however, gordos went down well.

leslie townes

i wouldn't mind a trip to gordos.

copper.hat

5:31 AM

on another note, i overheard my wife talking to my daughter who had he ears pierced recently

leslie townes

i was in berkeley the week before the pandemic really took off and got a burrito at the cactus location in rockridge.

copper.hat

the line i hear was "the bigger the hoops the bigger the ho"

leslie townes

goodness.

copper.hat

that cactus is better than the solano one

the solano one serves burritos for dieting moms

leslie townes

i remember housesitting in north berkeley and going to the solano one and it was like a different restaurant.

copper.hat

5:33 AM

just heads up when your daughter gets here ears pierced...

leslie townes

most of my female friends either have no piercings whatsoever, or quite a lot of them. there's no middle ground. i don't know what this means.

copper.hat

i was against ear piercing

labia is out of sight (mostly)

im joking.

leslie townes

we expect this. it's to be expected.

copper.hat

i'm fine with visible labia

leslie townes

my wife is very anti-piercing. i'll let her handle those conversations with my daughter.

copper.hat

5:36 AM

there used to be a clothing optional section behind the bleachers at the RSF

leslie townes

my friends would be bad influences.

copper.hat

i had a friend who got a long lasting infection as a result of a shady piercing

why is your wife opposed?

leslie townes

i honestly have no idea. but she's opposed.

copper.hat

i just don't like permanent body modifications

except for the chip injections

leslie townes

my wifi was starting to suck, so i did have to get the booster.

but yeah, other than that.

copper.hat

5:38 AM

whew, i read wife

leslie townes

of course you did.

copper.hat

old age, sight and all that

strangely, my son was interested in a cadence branded laptop bag

you just never know...

personally i think the "handshake approved" badge is pretty darn cool

leslie townes

huh. i don't know cadence except from ip litigation.

are they a real company? in my job you lose track of which companies are real and which ones just sue people a lot.

copper.hat

my life is cadence.

i don't work there, however, their versions of my products are still generating mullions yearly...

and they bought my company

there were two people there that inspired me, both long gone.

i have a marketing mug that was inspired by me. the words on the side of the mug are "size matters"

and i have my tempus fugit balls.

hmm, he wasn't so impressed with the siemen's hand sanitiser

it was tim cook that nixed our request to have the apple logo on our tempus fugit site (they were using our product). i'll never forgive him

we were allowed to say that apple was a customer, but not the nice logo.

nvidia let us put the logo up. i like nvidia.

and sun.

its wine time i think

leslie townes

i have some sun swag from something in the 90s. i forget why

and a silicongraphics frisbee

copper.hat

5:58 AM

the trade show (DAC) used to be a great supply of kiddie soft toys/animals.

flashlights, etc

robjohn

@copper.hat when covid first hit, I remember gifs suggesting rolling back to UDP to reduce handshakes.

copper.hat

:-)

Avra

Hi

robjohn

@Avra are you really?

copper.hat

@robjohn excuse the repeat post, badges from a trade show yesterday: imgur.com/Jt2PXK3

UnderMathUate

6:02 AM

Prove or disprove that $A = \left\{\frac{m}{2^n}: m, n \in \mathbb N\right\}$ is countable.

I'm pretty sure this is true if you think of this as the product of two countable sets.

Avra

@robjohn. Yes Professor. It's me :|

Passed exam with your efforts and patience

copper.hat

@UnderMathUate surely you can find a surjective function from $\mathbb{N}^2$ to $A$?

robjohn

@copper.hat It's been a long time since I've been to an electronics trade show.

copper.hat

@robjohn the show yesterday (and tomorrow if I can generate the energy) was the deadest show I have ever been to

they used to be the highlight of the working year

UnderMathUate

Yeah, that was my first thought. I was just trying to find a possibly shorter work around.

copper.hat

6:04 AM

what could possibly be shorter?

UnderMathUate

I honestly don't know.

leslie townes

under: it's a subset of Q. do you know Q is countable?

it's hard to rank levels of shortness on this time scale

UnderMathUate

Ah, yeah. We proved that in class.

And on that note, I'm going to bed. I'll write it up in the morning.

copper.hat

so you are going to spent longer trying to find a marginally shorter proof?

leslie townes

copper's point of course is that if S is any set and f is any function from N^2 to S, {f(m,n): (m,n) in N^2} is countable

copper.hat

6:07 AM

I liked Q in TNG

speaking of countable

UnderMathUate

I see it now

copper.hat

i can die peacefully npw

leslie townes

nqw

copper.hat

oops pwned myself

Avra

@robjohn. What are you doing for Christmas? Hopefully not math?

robjohn

6:08 AM

@UnderMathUate $(2m+1)2^n$ uniquely determines an element of $A$

UnderMathUate

I thought everyone did math for Christmas

copper.hat

i will be summing a turkey

robjohn

@Avra math is a Christmas present, why would I not want to get another present?

copper.hat

sorry, consuming

Avra

ahhh

UnderMathUate

6:09 AM

@robjohn You're referring to what I said about the product of two countable sets? Sorry if I'm being slow.

Avra

you are addiction guys

cool

copper.hat

maths is a dangerous siren

nothing but pain

its a universal property

UnderMathUate

nothing but pain = infinite gain

copper.hat

infinite gain is a recipe for instability, need to keep eastern Europeans out of the starboard side of the aircraft.

Avra

@robjohn. Can you use your profile photo :|?

UnderMathUate

6:15 AM

@copper.hat I'm so tired, I literally cannot parse this sentence.

copper.hat

in linear control, one needs to ensure that there are no poles in the right half plane.

old, really bad joke.

leslie townes

that's old even to me.

copper.hat

but i qualify on all counts.

UnderMathUate

:/

Avra

haha

copper.hat

6:16 AM

da da da dumf

Avra

you should have your own big bang show

UnderMathUate

Alright, good night guys. Thanks for the help.

copper.hat

i'm too old for stage shows.

Avra

gn

copper.hat

gn

leslie townes

6:17 AM

gn

copper.hat

been a while since i was in amsterdam...

but it used to be walking down the street you would get dragged into stage shows

even if you were with a girl...

Avra

hahaha

Amesterdam is a unique city

copper.hat

i loved the Netherlands

i have family there now

Avra

Netherlanders are very similar to Germans

copper.hat

paving the driveways

Avra

6:22 AM

:|

copper.hat

except that Germans can't pronounce Scheveningen properly

the paving comment was a rude remark about being Irish

Avra

haha

Irish people are similar to what? I would never say French.

copper.hat

it was very non pc. it was really aimed at Lesie

leslie townes

i love it

copper.hat

i don't feel comfortable explaining it. discussing labia piercings is fine, but travelers, no....

Avra

6:24 AM

lesile you are Irish?

copper.hat

in spirit

my most irish friend is actually african.

Avra

leslie is definitely from United Kindgom

copper.hat

untied

i have 3 close family members living there.

leslie townes

i think the irish are going through a major image change. i mean, the old image of leprechauns, shamrocks, guinness. horses running through council estates. toothless simpletons. people with eyebrows on their cheeks. badly tarmacked drives, in this country. men in platform shoes being arrested for bombings. lots of rocks, and beamish. i think people are saying "yes, there’s more to ireland than this."

copper.hat

and a brother's brother-in-law whom i say next to in high school

because nowadays, everybody knows where everybody lives

robjohn

6:27 AM

@copper.hat $\sum\unicode{x1F983}$

@Avra for what?

copper.hat

@robjohn i am trying to interest my son in the idea of cooking a turkey for Christmas dinner

robjohn

@UnderMathUate yes

Avra

@robjohn. Can I use your photo as my profile photo :|

copper.hat

I think he wants Indo food, but that is so passe.

it was the drink driving laws that ruined Ireland.

leslie townes

sorry, i was reciting something from alan partridge.

robjohn

6:29 AM

@copper.hat My wife cooked our turkey breast side down, and the white meat was much moister.

copper.hat

@robjohn i hate to admit it, having tried many techniques, but cooking on one of those oven bags seems to be the trick.

robjohn

@leslietownes don't get rid of the Guinness!

@copper.hat it was in an oven bag as well

copper.hat

My lifetime integrated consumption of Guinness is less than a pint.

@robjohn I want to stuff the turkey, but my wife is afraid of some microbes.

robjohn

@copper.hat cook it to the required temp (165°F, I believe) and it should be okay

copper.hat

just realised i forgot to pick up my frozen tamales on my run to tjs an hour ago. blast.

@robjohn i agree. but i have no say when it comes to such matters.

her family regularly cooks meat to the point of dessication.

robjohn

6:32 AM

@copper.hat You need to have the stuffing from inside the turkey to mix with the stuff made outside the turkey to add the flavor.

copper.hat

signs of pink in a steak are clear warning signs.

robjohn

@copper.hat ACK! I like my steak to moo!

copper.hat

@robjohn i think the turkey fat adds flavour

My mom used to overcook stuff as well. I think it is cultural.

I loved my Mom's cooking, but I like it a lot less cooked too.

robjohn

It might be a generational thing. My mom and her friends liked shoe leather for dinner

copper.hat

I only discovered spinach when I was 40+

before that i thought it was dark green slop.

god forbid, raw spinach

robjohn

6:35 AM

I like raw spinach okay, but cooked spinach makes me gag, actually.

copper.hat

me too

robjohn

My mom stopped giving it to me when I threw it up

copper.hat

i 'discovered' cabbage the same way

it does not have to be a sodden mass.

robjohn

Now, cooked cabbage is fine with me

Avra

cooked spinach is amazing with lemon and chickpeas

copper.hat

6:36 AM

i like it cooked, but not to mush

robjohn

sure, no mush

leslie townes

i only like cooked spinach in palak paneer

i love garbanzos

raw is better

they are so chic

6:37 AM

too chic for me

copper.hat

just hummusing along...

Avra

Have you ever went to gym guys or you prefer running, jogging?

robjohn

three bean salad is fine, but they keep giving me more than three of the beans.

copper.hat

@Avra i'm 61. i have done all the above.

:-)

Avra

That's great. Wish you live till 1000

copper.hat

6:39 AM

i prefer to exercise outside.

Avra

They now are studying methods to increase life up to 200 years :|

copper.hat

i appreciate the wishes, but it this is what it is like at 61 i don't want 100

Avra

even young guys once they stop walking, they get heavy. That's normal :|

copper.hat

heavy is not the issue. wear & tear is :-)

Avra

nothing stops math

robjohn

6:43 AM

@UnderMathUate did you understand the idea?

$\frac{m}{2^n}\to(2m+1)2^n$ is an injection from $A\to\mathbb{N}$

You don't even need to worry about reducing fractions.

BannedUser

How does it feel when God always roll unlucky dice on you?

copper.hat

Cormac McCarthy had something to say about that

Slereah

7:23 AM

Is there a name for the space of matrices that are the sum of skew symmetric matrices and symmetric matrices of signature $(p,q)$?

Mad Spaces

7:42 AM

@Astyx I think i might be starting to understand this better. this Ismorphism you presented. Is it now between R^2 and C as fields or as vector spaces? Since we can identify them "depending on the operations acting" as such.

This function you have defined, we used " i checked my old linear algebra papers" to define R^2 as a FIELD with the vector addition and vector multiplication, then with identifying (0,1) to be the element i , we recieve the complex field.

However if we speak about them as vector spaces, then we are not using this multiplication anymore right? because we have to identify then scalar multiplication and vector addition for them.. or am i wrong?

Random Variable

8:03 AM

@robjohn I remember there were MathJax issues the last time they redesigned the Activity page, but it wasn't this bad.

robjohn

@RandomVariable The layout is just horrible

Alex

Hi

robjohn

I was just saying elsewhere that even the mobile interface has too much white space. All I can see is my reputation and badge totals. everything else requires scrolling down

@Alex hey there

@RandomVariable It is like in both interfaces, someone has decided what I want to see without scrolling, but gotten it wrong.

In the full interface (or whatever it is called currently), things also have too much space around them or frames. It eats up the available screen real estate.

Alex

I'm new to this section, it's a nice place to talk about math. I'm on my cell phone, but I can't use LaTeX :-(

robjohn

Things may be "neater", but you have to scroll a lot to see anything

Alex

8:10 AM

in this room

Koro

I think it’s bit complicated to render latex in android.

robjohn

@Alex is it that you cannot install ChatJax, or that you don't know LaTeX?

@Alex I know in the mobile interface, one does not see the sidebar which has the installer link for ChatJax.

Alex

@robjohn Thanks, I know LaTeX but here I cannot use it. For example I don't see the LaTeX on this line $(0,1)$.

robjohn

@Alex Have you installed ChatJax?

Alex

@robjohn No :-( Just right now I'm reading about this

robjohn

8:14 AM

See if that helps.

@RandomVariable Each entry has a frame around it in each panel: each answer, question, reputation, badge total, bookmark, etc has a frame around it, which adds white space as well as the frame.

This is a misuse of screen real estate, IMO

If I were to use my 4K TV as a monitor, maybe the layout might look neater, but to get enough stuff on my screen to be useful, the resolution would require me to stand a few feet from the screen to read anything.

also, the sidebar with "Summary", "Answers", "Questions", etc. links takes up a small vertical space, but it takes up horizontal space along the whole page Rather than have those set out like that, maybe a pop-up menu would be better or just make the titles to the same sections in the main part of the page should be live links.

In case anyone is wondering, I think the new layout for the Activity and Profile pages is horrid

Random Variable

@robjohn Even if they fix the MathJax issues, it's still going to look awful.

robjohn

There is a huge white space along the left side for a small amount of data at the top of the left side.

@RandomVariable You mean the multiple lines for a single line of MathJax?

That is bad, too, but even when there is no MathJax, the layout is atrocious

Quite wasteful of screen real estate

Random Variable

8:29 AM

@robjohn Yes. I've seen titles spread out over like 10 lines.

robjohn

@RandomVariable more than that for some that I have on my screen

As I say, it is horrible

Alex

I couldn't install LaTeX since Chrome on my phone, I'm sad about that. By the other hand, in the last part in this solution math.stackexchange.com/a/4326466/941425 is it correct? I think should be the denominator above in this fraction.

Random Variable

@robjohn Empty space seems to be the new design trend.

Balarka Sen

they butchered the activity page

robjohn

@Alex That should not prevent installing chatjax

@BalarkaSen they butchered that and the profile page

Balarka Sen

8:45 AM

at least the top questions/answers section doesnt have massive fonts with broken TeX

but you have to scroll so far to see those

Alex

@robjohn I think I did not understand the instructions, in the link.

Here it's says that: Using with Chrome on Android

Evidently, Chrome on Android does not remember the current page when
opening a bookmark from the standard bookmark menu. The bookmark can
still be used; however, the procedure is a bit more involved than simply
opening from the bookmark menu.

Go to the Search bar and start typing the name you've given to the bookmark
(e.g. "start ChatJax"). The bookmarklet should appear in the list of items
found. Open that item and ChatJax should start running on the current page.

Random Variable

When you refresh the page, the MathJax issue goes away for like a split second.

Alex

Where can I find "search bar" here?

Wolgwang

@Alex Related : chat.stackexchange.com/rooms/9369/conversation/…

Have they increased the size of LaTeX? O_o

Alex

@Wolgwang thank you, the video in your link it helped me a lot!

Now, I suppose that $\frac{1}{2^{\frac{n-1}{n}}+\cdots+1}\cdot \frac{2^{\frac{1}{n}-1}}{2^{\frac{1}{n}-1}}=2^{\frac{1}{n}-1}$.

About this answer here math.stackexchange.com/a/4326466/941425

robjohn

9:24 AM

@Alex do you not have a field to search for things (files, etc.)? (often it says search or has an icon of a magnifying glass, but not always)

you should be able to start typing the name you gave to the bookmark there, and it should show up.

@RandomVariable Yeah, like in interior design. Lots of empty space is what is desired these days. On screen, it is a waste.

@Alex Ah, so I assume you have ChatJax (or some other MathJax enabling script) installed and working?

ACK!! I just noticed that the dates have been removed from the posts on the activity page! This, probably to make room for the borders and frames.

SultanDeGranada

9:41 AM

Is any of you aware of how to find a math pen pal?

3

robjohn

Ugh! I just looked at the Answers page. The same horrible layout and space waste has been added there, too.

Alex

@robjohn Yes, now I can use ChatJax here. Thank you so much :-)

robjohn

Great!

Alex

Is my observation about the answer in the link correct? I think the sequence of functions is UC 🤔

robjohn

10:07 AM

@Alex see my comment to that question.

Wolgwang

10:21 AM

$AXB$ is a semicircle and $AYB$ is some curve. Is the length of $AXB$ always greater than all curves like $AYB$ ?

robjohn

@Wolgwang No. The inner curve can wiggle and have a much greater length. With constraints on the tangents to the inner curve, it might be possible.

Leaky Nun

maybe if you require it to be concave then it's ok

Wolgwang

@robjohn If AYB is an arc of another semicircle/circle(with different diameter) then...?

robjohn

@Wolgwang That would be enough

Wolgwang

Ohk Thanks :-)

Wolgwang

10:39 AM

Is there any proof ?

Astyx

11:09 AM

@MadSpaces what do you mean by vector multiplication? $\mathbb R^2$ isn't a field per se (although it inherits a field structure from the isomorphism with $\mathbb C$)

robjohn

@Wolgwang yes. note that the triangle in the wedge has $2r\sin(\theta/2)=AB$

For $AXB$, $\theta=\pi$, for $AYB$, $\theta\lt\pi$. The length of the arc is $r\theta=2r\sin(\theta/2)\frac{\theta/2}{\sin(\theta/2)}=AB\frac{\theta/2}{\sin(\theta/2)}$.

robjohn

11:25 AM

Note that $\frac{\theta/2}{\sin(\theta/2)}$ decreases from $\frac\pi2$ to $1$ as $\theta$ decreases from $\pi$ to $0$

Leaky Nun

11:50 AM

@Wolgwang basically the arc length formula $\int_a^b |f'(x)|^2 \ \mathrm dx$

so if AYB is always flatter than AXB, then the f'(x) would be smaller, so the whole integral would be smaller

Mad Spaces

12:08 PM

@Astyx we can define a multiplication on R^2 like that of a complex vetor with another, then it is a field.

Mad Spaces

12:41 PM

Algebra question. Suppose we have a monoid.
Can an element exists , that behaves with all other elements just like the nutural element of that monoid but when it is paired with the nutural element it just gives itself back and not the natural element?
$ c \circ a = a = a \circ c $ but $ c \circ 1 = c = 1 \circ c \neq 1 $

I know that if it behaved like the nutural element for all elements then it is because the nutural element is clearly defined, the same as the nutural element, but what about that little twist i added? can we have such a meme?

Leaky Nun

12:53 PM

@MadSpaces consider {0,1} with multiplication

UnderMathUate

@robjohn Ah, yeah. I think I see it now.

Astyx

@MadSpaces that's what I meant by the phrase between parentheses

Mad Spaces

So your isomorphism you defined, is defined on R^2 and C as fields..and the perspective multiplication on those fields.. not as vector spaces.. which have different multiplication.

Thus C and R^2 are isomorph to each other as fields as well

Btw i was able to answer my algebra question so ignore it :D

The answer "apprently" is "Yes"

Astyx

The isomorphism I gave is a R-linear isomorphism, so the structure if vector space

Mad Spaces

Oh Okay.

Transcript for

$Mathematics$ Mathematics