Mathematics - 2021-10-14 (page 1 of 2)

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12:32 AM

Hello!

For binary number multiplications similar to decimal number multiplications, if we have a number, then we can split it into two as follows $a=a_1*10^{n/2}a_2$, where $n$ is the number of decimal places in the original number, by doing this base 10 will assure that we shift places for adding $a_2$ to obtain $a$. Is it the same for binary numbers please? I mean we can represent binary number $a=a_12^{n/2}a_2$?

Shin’s conundrum is why I insist analysis is harder than algebra. The tiers of quantifiers!

I know this is not meant for me :/

avra, do you mean to add a + in there? this is a little goofy in that we might be mixing symbols for digit strings with symbols for numbers.

I mean if we try to multiply two integers recursively, one approach is to divide a number into two halves, so I guess $10^{n/2}$ is there for shifting

So, I see that same for binary numbers $2^{n/2}$ for shifting but it's not clear as in decimal numbers as in decimal $10^{n/2}$ is to do shifting clearly

do you mean like in the usual hand algorithm for decimal multiplication, there is shifting to the left that corresponds to multiplication by 10? yes, the same thing works in any base, with shifts corresponding to multiplication by b.

12:45 AM

123456 = 123 10^3 + 456

writing [x]_2 for the number represented by x in binary, [111010]_2 is indeed [111]_2 * 8 + [010]_2, for example.

For binary, 111010 = 111*2^3 + 010!

I am not sure if $2^3$ plays same rule as $10^3$ does

wow!!!!!

@leslietownes. I swear to gd that I mean the same example 111010 and we wrote it together at the same time

:0

That made my day :/

Mathematician think like me

yeah. it's basically the definition of the positional notation, and the distributive law. the base-b "digit" string [a_5 a_4 a_3 a_2 a_1 a_0]_b is shorthand for a_5 b^5 + a_4 b^4 + a_3 b^3 + a_2 b^2 + a_1 b + a_0, which by the usual distributive law is b^3 (a_5 b^2 + a_4 b + a_3) + a_2 b^2 + a_1 b + a_0, which by the notation convention is also b^3 [a_5 a_4 a_3]_b + [a_2 a_1 a_0]_b.

Akiva Weinberger

why did it take me so long to realize why subgroups of index n must contain $g^n$ for all $g$

@leslietownes. I see what you mean above by $111_2 * 8 + 010= 11100 + 010 = 11110 \ne 111010$?

Akiva Weinberger

12:51 AM

'cause if $G/H$ has index $n$ then $G/H$ has order $n$ so everything in it equals the identity when raised to the $n$th power

so $(gH)^n$ is the identity so $g^n\in H$

[111]_2 times 8 is [111000]_2. three zeros at the end there.

Akiva Weinberger

Basically the dual of "if $g$ is in a group of order n then $g^n=e$"

you want $H$ to be normal, though

@leslietownes. I forgot $8 = 1000$ in binary :(

@leslietownes. Do you forgot formulas sometimes?

you do want H to be normal to compute in G/H

avra, most of the time.

12:55 AM

$(12) \subset S_3$ doesn't contain $(23)=(23)^3$

@leslietownes. :/ Do you prefer to write a summary of theorems, lemmas, and so on for a topic or you just to look up a book?

@leslietownes. #1 student in our math department used to write all theorems, definitions, lemmas, ...for each course we took in papers and only revise those before exam :/

avra, i no longer work in math, but when i did, i looked a lot of stuff up. i did end up committing most definitions and some theorems to memory, which i distinguish from 'memorizing' them because it came via osmosis.

Akiva Weinberger

@LukasHeger oh that's true

Luckily in the original context everything is abelian

that kind of thing would be a waste of time for a lot of people. some people have studying things they do that are more to put themselves at ease than to learn the material.

rewriting a list of definitions/theorems/whatever seems like something in that category, but what do i know.

@leslietownes. Yeah she used to write all on separate sheets for all math courses we had

She only came with these sheets on the exam day without any further notes!

1:00 AM

when i taught i would sometimes supply that material myself on exams, precisely to prevent people from feeling that they needed to do it.

Wow!

@leslietownes. So you also did the same as she did and give them to students to save their time!

also to aid in citation. most theorems don't have names, but on the list there would be theorem 1, theorem 2, theorem 3, or whatever. so people could just say which theorem they were using.

But you did not allow these on exam day!

i would pass this out with the exam.

and make it available beforehand so people knew exactly what they'd have with them when they took the exam

Wow you are awesome teacher

So, you also did use to write them up on papers when you was studying math :/

Maybe now becuuse you use them all time, you already memorize them

1:04 AM

i sometimes asked difficult questions :) but i didn't want people spending too much time trying to commit things to memory, or filling a 'note sheet' with stuff that is just right out of the textbook.

can I ask mathematical physics lol

perhaps more useful in some subjects than in others. with something like calculus, kind of neutral. a general statement of the chain rule, or whatever, isn't going to help someone who isn't familiar with how to use it.

@leslietownes. You mean some courses are more practical than others, so this approach won't be good

banned people do bring up physics questions although the luck they have with getting assistance varies. it helps if math is involved. :)

But for others heavy proof courses, proably this is good approach?

1:06 AM

I have a hot air balloon I am curious why book writes as pressure in balloon is same as outside when temperature increase.

PV/nT=R

Equilibrium, banned?

P for pressure V for volume n for amount of particle T for temperature R for constant

well when the balloon is being filled and the balloon material is moving around, you do have a pressure differential that is moving the balloon.

if you've filled it and are floating above the vineyards, that's the kind of equilibrium i think ted is asking about.

balloon tourism was a huge thing in my hometown, i saw balloons in the sky most days of the year. this is a near and dear topic to me.

@leslie In my undergrad diff geo course, I provided formulas on exams that no one could reasonably remember, but expected them to know the important and easily remembered ones.

one time we had a visitor from japan and he wanted to go on a balloon tour, and weirdly enough his balloon had some issue and ended up somewhat crashing into a river. nobody was hurt. true story.

1:11 AM

Unlike the Cessna that crashed and burned in Santee, 20 miles from here, yesterday.

ted: yeah, i was about the same. i tended not to include definitions on my 'theorem sheet.' if you didn't know those, good luck.

In many courses, 1/6 of the final was stating definitions of salient terms.

“50 free points” …. Right.

And I warned them about pitfalls (like nonzero in the defn of eigenvector)

sorry I was busy in lecture

If the balloon is in equilibrium configuration, then pressure inside equals pressure outside. Otherwise not.

I am very sorry I will be back in 30 min

and read all the chat

1:32 AM

rewrites chat in invisible ink

2

Xander Henderson

$\color{white}{\text{Invisible ink?}}$

Cute :) I remember writing in lemon juice when I was very young.

Let’s not expect too much.

$\, \,$

That was too much.

Xander Henderson

1:47 AM

$%\text{Is this too much?}$

Past.

Xander Henderson

:D

From https://medium.com/@littlebrown/i-wore-the-juice-the-dunning-kruger-effect-f8ac3299eb1 Wheeler told police he rubbed lemon juice on his face to make it invisible to security cameras. Detectives concluded he was not delusional, not on drugs — just incredibly mistaken.

Wheeler knew that lemon juice is used as an invisible ink. Logically, then, lemon juice would make his face invisible to cameras.

Of course!

@leslie Quite the backups at LA and LB ports.

@PM2Ring Plus tremendous for the complexion.

@TedShifrin I suppose so, but it probably stings a bit if you get it in your eyes. ;)

@leslietownes Here's another avian-themed tune for you. Rocky, written & performed by 13 year old Emily Linge. Her Dad's English, her Mum's Norwegian, but they live in Dubai.

The song's a bit corny & clunky, but it is rather sweet, and I reckon it's a pretty good effort for a kid. She's quite a talented singer, but she has recently acquired braces on her teeth, which has made a slight impact.

Here Emily teams up with Sina the drummer & Chiara Kilchling (another multi-instrumentalist from Germany) on A-Ha's Hunting High And Low. I love the way their voices harmonize.

2:08 AM

ted i took some photos the last time we were at the beach. containers as far as the eye could see.

although if i'm being honest i was surprised the port wasn't already 24/7

Akiva Weinberger

2:34 AM

Wait why does the quotient rule look like a determinant

(u/v)' = −det(u, v // u', v')/v²

I'm almost certain I've seen this pointed out before but I can't remember what the answer was

@BannedUser Take a small portion dA of the balloon and then draw a free body diagram of the portion dA. Let the pressure inside the balloon be $P_i$ and outside be $P_o$. Since the balloon is filled up, net force on the portion is 0 that is, $P_o dA-P_idA=0$ whence $P_o=P_i$.

Akiva Weinberger

Oh you know what, I might actually see why

Only partly

2:50 AM

Sorry, I’ll break up the message

Hey guys, is there any simple of resolving this problem algebraically?

I was asked to approximate $(1+x)^{\frac{1}{3}}$ as a linear function about x = 0 and provide the bounds for x so that the error for the approximation is less than 0.01.

In other words, I just have to expand $(1+x)^{\frac{1}{3}}$ to two terms about x = 0 and figure out the bounds for x so that the remainder is less than 0.01.

The expansion is extremely simple and the remainder in Lagrange form can be written as:
$$-\frac{x^2}{9(1+\theta x)^{\frac{5}{3}}}$$
where $\theta \in [0, 1]$.

Since I don’t know what the exact error is, I thought it made sense to bound the x so that the maximum possible error is less than 0.01, as that makes it certain the error is less than 0.01.

The case for $x \geq 0$ is simple as for a given x the error term obtains its maximum value at $\theta = 0$ and I just have to solve:
$$\frac{x^2}{9} \lt 0.01 \\ \therefore x \lt 0.3 $$

The case for $x \lt 0$ is the one I’m having trouble with. For a given x, the error term in this case obtains its maximum value at $\theta = 1$ and the inequality I have to solve becomes:
$$\frac{x^2}{9(1+x)^{\frac{5}{3}}} \lt 0.01$$

The only way I could think of solving the equality above is to find the roots of:
$$\frac{x^2}{9(1+x)^{\frac{5}{3}}} - 0.01 = 0$$ using Newton’s method, as I don’t think it’s possible of obtaining an algebraic solution.

But at that point it’ll be just better to apply Newton’s method to:
$$\frac{x}{3} + 1 - (1+x)^{\frac{1}{3}} - 0.01 = 0$$
to find the exact bounds for x.

Am I over complicating this? I feel like I’m making a silly mistake here. I would appreciate any help?

3:04 AM

You could use the fact that ${1 \over (1+x)^{5 \over 3}} \le 1$.

@copper.hat that isn’t true for $x \lt 0$ right?

Yes, I goofed.

Don’t worry i thought the same when I first glanced at the question

But you could add an additional bound like $|x| \le {1 \over 2}$ and solve for the bound.

I was thinking positively...

Also, the best (as in $L_\infty$) linear approximation over a range and the Taylor's approximation are different things (in general). I presume you are looking at a Taylor's approximation?

Why is Taylor series expansion around x=0 called MacLaurin series?

3:18 AM

because of an immortal trademark lawsuit, MacLaurin v. Taylor.

honestly i don't know. both used series. some books just say 'taylor series' for everything.

i should read before writing.

something similar with rolle's theorem. no real special reason for it to have its own name, except maybe history. it could just be the MVT. i never knew there was something called rolle's theorem until graduate school.

it was one of the first theorems we were taught in my first year maths class in college.

i did not remember, i just happened across my old notes a few weeks ago.

2

Q: Why was Taylor series around zero named Maclaurin series?

I am eager to know why was Maclaurin series ,which is a special case of Taylor series, named after Taylor's finding? I myself guessed that Maclaurin found his series for functions around zero, and then Taylor expanded that series for the other points; but, after googling the history of Maclaurin...

tl;dr is that Maclaurin used the special case at zero a lot, and this was noted.

@copper.hat hmmm but doesn’t the question become what extra bound should I set?

3:25 AM

@leslietownes I convinced myself long time back that this kinda made sense because Rolle’s theorem came before Lagrange’s theorem.

then what do you mean by approximation by a linear function?

I personally wanted the best bound

The question doesn’t specify

depending on the notion of approximation, it may not be possible to derive the best bound 'by hand.'

@DavidChoi Newton's method on that last equation sounds good to me. FWIW, here's a plot

you need to define 'best bound'.

3:27 AM

depending on your notion of approximation, combining a point estimate near -1 and something derivative-based away from -1 (to avoid that junk with the derivative at -1) could produce an approximant that might not be best and might depend on the choice of parameters.

almost certainly the best linear approximation will not match the derivative at $x=0$. So you need to elaborate if you want to chase this tail.

You can do slightly better using a minimax polynomial, i.e., one that minimizes the maximum error in the interval.

Woah I never even thought abt that

tcheeheyhchecshev

Wait I thought u could define the derivative as the best linear approximation of the function at that point

3:29 AM

that is how it is pronounced in ireland after 5 pints.

Incidentally, if you use a quadratic approximation rather than a linear one, you get an order of magnitude less error.

you need to be clear what you are looking for.

which is what i meant when i wrote "what do you mean by approximation by a linear function"

Ok ok due to my limited knowledge I’m just gonna describe what I want

For a given interval $[-c,c]$, you could define the best linear approximation as a solution to $\min_a \max_{|x|\le c} |f(x)-ax|$.

I want to find the straight line that is very similar to the function $(1+x)^{\frac{1}{3}}$ about x = 0

3:33 AM

too many to choose from.

@AkivaWeinberger Not a complete answer, but any change of $(u,v)$ parallel to itself keeps $u/v$ constant), so the determinant removes that redundancy. Somewhat like differentiating a curve in projective space.

I am confusion

alternatively, you could define $e(c) = \max_{|x|\le c} |f(x)-f(0)-f'(0)x|$ and choose the largest $c$ such that $e(c) \le 0.01$.

@copper I’d like to hear you guys mangle Worcestershire :)

It is not too hard to see that $e(c)$ is attained at the boundary points.

wooster

3:35 AM

in what sort of context do we encounter n-dimensional log-determinants?

trying to figure out a proof and I've never seen one before

I don’t know what that means.

N-dim wronskians?

I'm not finding its definition hehe, I need to prove it's convex and I'm trying to find out what it is

they're the mathematical equivalent of a DEAD END sign. or a therapist inviting you to go back and evaluate the life choices that led you to that place.

@copper.hat I’m sorry but pls bear with me

it's like hitting rock bottom, except you hit n-dimensional log determinants.

3:39 AM

@copper.hat this definition I understand it’s saying:find the bounds for x so that the maximum distance between the tangent line at x = 0 and the given curve is less than some number right

Never heard of this … must be some econ thing.

@copper.hat this definition I don’t really understand: what is the min part about?

it's from a convex optimization book

the log determinant appears in convex programming.

Then copper must know.

3:41 AM

i'll try a new answer to the question. we encounter them in convex optimization books.

WTF is it?

it is concave on positive semi definite matrices.

$f(X) = \log \det X$.

Oh, oh, log det A

author hasn't given a definition :'(

sounds like a great book

3:43 AM

I bet it appeared somewhere.

@DavidChoi Eyeballing that graph I posted earlier, we get an error of 0.01 at the endpoints of the interval [-0.29, 0.31]. The minimax linear poly on that interval is ~ y = poly 0.336786735*x + 0.994903449, which has max error ~.0.005123. Here's its difference graph:

user image

there are a number of operator determinants defined in terms of formally inverting classical formulas involving det where all the ingredients except det still have meaning. sometimes they have det-like properties. often not, but you can still define them.

@shintuku it is concave. the function is useful as it is a proxy for volume.

noted, but what exactly is it? i've searched two convex optimization books and google and nothing shows up

@DavidChoi the graph from @PM2Ring is essentially what $e(c)$ above gives.

3:46 AM

We both said it above.

@shintuku The definition is $f(A) = \log \det A$.

it's just the logarithm of the determinant? i thought it was odd because I couldn't find a base for the log

Use properties of det to relate to the characteristic polynomial….

@shintuku yes. were you expecting the log determinant to be something different?

well, i've usually seen logarithms state bases before, or otherwise be written $ln$

is it implied it is base $e$?

3:51 AM

it doesn't really matter, but yes.

@copper.hat yep I can see that; if it isn’t too much trouble could you explain the first definition you provided for linear approximation? I don’t understand the what $\min_a$ means together with the max part.

thanks!

Here’s the defintion so u don’t have to scroll up:

For a given interval $[-c,c]$, you could define the best linear approximation as a solution to $\min_a \max_{|x|\le c} |f(x)-ax|$.

@DavidChoi if you pick some slope $a$ and an interval $[-c,c]$ this defines a $\max$ error. So now pick $a$ such that the $\max$ error is minimised.

This gives a function of $c$, and you pick the largest $c$ such that the error is $\le 0.01$.

From [Wikipedia](https://en.wikipedia.org/wiki/Minimax_approximation_algorithm) Polynomial expansions such as the Taylor series expansion are often convenient for theoretical work but less useful for practical applications. Truncated Chebyshev series, however, closely approximate the minimax polynomial.

One popular minimax approximation algorithm is the [Remez algorithm](https://en.wikipedia.org/wiki/Remez_algorithm)

3:53 AM

chevychasetycheb again

Oh I seeee; I’m sorry abt that I wasn’t familiar with the notation

Akiva Weinberger

@TedShifrin Realized that that determinant is the Wronskian, and also the angular momentum (assuming unit mass).

i caught my wronskian in my zipper once. hurt like hell.

Akiva Weinberger

So I suppose that means the rate of change of the slope (u/v) is the same as the angular momentum over the square of one of the coordinates.

i was linearly dependent for ages afterwards.

3:55 AM

@copper.hat Yep. The Chebyshev is a good 1st approximation to the minimax poly, and is often used to "kickstart" the Remez algorithm.

Akiva Weinberger

@copper.hat Oof. Be careful with your Schläfli

More properly, the derivative of y/x is the angular momentum over x^2.

@PM2Ring I broadly refer to such minimax problems as Chebyshev problems. I am in an imprecise mood.

Akiva Weinberger

(time derivative)

Hm. Also $(y/x)'/(x/y)'=-y^2/x^2$?

Wait that's just the chain rule lol

amazon.com/Introduction-Minimax-Dover-Books-Mathematics-ebook/… is a nice (if slightly dated) intro.

Akiva Weinberger

In any case, I was thinking about curvature

4:00 AM

i think about curvature a lot.

I don’t see the chain rule.

i see division but no chains

Akiva Weinberger

It's $(1/f)'=-f'/f^2$ rearranged, with $f=x/y$

@copper.hat That's fine. The Remez algorithm isn't hard to code, but it's a bit sensitive, and you generally need to run it at a much higher precision than the final desired precision of the coefficients. And in actual practical use, the minimax poly may actually be worse than the Chebyshev poly, if many of your x values happen to be close to where the minimax poly's maximum errors are.

Akiva Weinberger

$(1/f)'/(f)'=-1/f^2$

4:03 AM

Looks like reciprocal rule, which is what we’re doing, of course.

Akiva Weinberger

It's the chain rule with $g(x)=1/x$

sure, but reciprocal is the key

@PM2Ring there are a whole class of exchange algorithms that are broadly referred to as remez type. i'm still in an imprecise mood.

Akiva Weinberger

Right

Curvature boils down to unit tangent, of course.

Akiva Weinberger

4:06 AM

Derivative of.

Sure.

Akiva Weinberger

LaTeX test $\angle$

OK so if $\angle(u,v)$ is the angle between those vectors,

curvature is roughly $\dfrac{\angle(r(t),r(t+\epsilon))}{\|r(t)-r(t+\epsilon)\|}$ as $\epsilon\to0$

which isn't news

@DavidChoi Here's another link, with nice graphs. :) en.wikipedia.org/wiki/Approximation_theory#Optimal_polynomials,

Akiva Weinberger

but swapping out $\angle$ for $u\times v/\|u\|\|v\|$, which you can do for small angles, gives a neat way to reconstruct the usual formula

and you can use that directly for, say, the vertex of a parabola in your head

I wrote that wrong

$\dfrac{\angle(r'(t),r'(t+\epsilon))}{\|r(t)-r(t+\epsilon)\|}$ as $\epsilon\to0$

at Ted: can i have another hint for using the properties of characteristics polynomials and determinants?

Akiva Weinberger

4:14 AM

So for the parabola, $(t,t^2)$ at $(0,0)$, that's $\angle((1,0),(1,2\epsilon))$ over $\|(0,0)-(\epsilon,0)\|$ (roughly)

@PM2Ring thanks for the link! Honestly the more I study math, the more I just go damn… there’s so much I don’t know and never will know

Akiva Weinberger

which is $2\epsilon/\epsilon=2$

@DavidChoi It can be a bit daunting, but it also means you'll never run out of interesting things to learn. ;)

Akiva Weinberger

By the way, we can define a group to be a function from the set $F(S)$ to $S$ satisfying certain properties

where $S$ is a set and $F(S)$ is (the underlying set of) the free group on $F$

($F(F(S))$ is strange to think about. It's not the same set as $F(S)$! But there's a natural injection map from $F(S)$ to $F(F(S))$, and a natural surjection map from $F(F(S))$ to $F(S)$.)

ffs

Akiva Weinberger

4:28 AM

The ingredients for writing those properties are that map $F(F(S))\to F(S)$, the injection $S\to F(S)$, and the concept of $Ff$ (if $f:S\to T$ is an arbitrary function between the sets $S$ and $T$, there's a natural map $Ff:F(S)\to F(T)$ between the sets $F(S)$ and $F(T)$)

Assembling these into the properties that make the definition work is left to the reader

4:51 AM

For some reason I was trying to prove that ∀x ∈ ℝ[≠0] (x / 0 = 0).

Any data scientist here? What is the name of the 25 questions we should ask when someone presents data/research

it's just four questions. (1) who are you, (2) how did you get into my house, (3) why are you presenting data and research in my house, (4) no really, enough with this data and research, who are you

5:07 AM

that is a peculiar question to ask in a maths chat room?

to be more serious, i'm unaware of any standard list of questions, or any name for such

25 seems like too many

5:26 AM

Data scientists like to have lots of data. Why worry about quality if you have plenty of quantity? ;)

(1) ask them how they obtained their data - which sources they got the data from

From en.wikipedia.org/wiki/GPT-2 GPT-2 translates text, answers questions, summarizes passages, and generates text output on a level that, while sometimes indistinguishable from that of humans, can become repetitive or nonsensical when generating long passages. It is a general-purpose learner; it was not specifically trained to do any of these tasks, and its ability to perform them is an extension of its general ability to accurately synthesize the next item in an arbitrary sequence.

The GPT-2 language model has 1.5 billion parameters. Its successor GPT-3 has 175 billion.

5:47 AM

@leslietownes actually I remember seeing in political SE/Cynical SE or whatever it is called. the 25 questions were named after a dude but I don't remember his name, wait lemme search that SE then

10

A: What is it that "97% of climate scientists" actually believe about global warming?

At SkepticalScience.com, we see this infographic: With regard to the National Council on Public Polls guidelines outlining the 20 questions that journalists ought to ask when reporting on polls, I will address all 20 questions here with regard to: Examining the Scientific Consensus on Climate C...

Yess i found it, my stupid brain thought it was something else

politics.stackexchange.com/questions/3091/… oh and i found this question too that some may find interesting

1 hour later…

7:10 AM

2

Q: Proving that dim $\cap_{i=1}^m N(\phi_i)=\dim V-n$, where $V$ is a finite dimensional vector space and $\phi_i$'s are L.I. in $V'$.

Let $V$ be a finite dimensional vector space over field $F$. Let $\phi_1, \phi_2,\ldots,\phi_m$ be linearly independent (L.I.) in $V'$, the space of all linear functionals on $V$. Let's denote nullspace of linear map $A$ by $N(A)$ and dimension of space $S$ by $D(S)$. Then it is to be proven th...

linear-algebra linear-transformations

In the answer, why is $\Phi$ surjective?

If I can take infinitely many classes, is it possible that my GPA converges to $\pi$?

koro, it suffices to show that if f is a linear functional on F^m and f(Phi(V)) = 0, then f = 0...

I like love_sodam's question on GPA :)

@leslietownes but that seems counter-intuitive to me. We may have $F^m \setminus \Phi(V)$ non trivial.

yes, but then there will be a nonzero functional that is zero on Phi(V)

general proposition, if W is a finite dimensional vector space, U is a proper subspace of W, then there is a nonzero linear functional on W whose kernel contains U (in fact, whose kernel is U)

7:28 AM

Suppose for some $f$ under these premises, we have $f(x)\ne 0$ for some $x\in F^m$, where $x$ is not in $\Phi(V)$. There doesn't seem to be any contradiction. :(

what are 'these premises'

@leslietownes I agree. Proof: Let basis of $U$ be $u_1,u_2,...,u_k$,we extend it to a basis $u_1,...,u_k,v_1,...v_n$ of V we can define $f:V\to W$ as $f(u_i)=0$ and $f(v_1)=1$ and $f(v_i)=0$.

@leslietownes this "f is a linear functional on F^m and f(Phi(V)) = 0, then f = 0..."

i'm also assuming your hypotheses (e.g. the definition of Phi and that the phi_j are linearly independent)

i.e. Phi is not an arbitrary linear map from V to F^m, but the one of your problem statement

if f is a linear functional on F^m that vanishes on Phi(V), then for all v in V, 0 = f(Phi(v)) = f(sum phi_j(v) e_j ) = sum f(e_j) phi_j(v), meaning that sum_j f(e_j) phi_j = 0 in V'. from linear independence of the phi_j, deduce f(e_j) = 0 for all j, and therefore f = 0 in (F^m)'. here the e_j's are the standard basis for F^m

conclusion from above, Phi(V) is not a proper subspace of F^m

@Koro one thing that you didn't take is the balloon also apy force

so I think pressure won't be same

the problem is density changes too and force inside balloon changes too

7:47 AM

someday i will become a regular user of chatjax, but with the time i save by not writing dollar signs and backslashes, i get more time to hang out with my cat

also air can escape from balloon

@leslietownes it also extends life of your shift key.

yes. it has many benefits.

it also lets you fit more in a math.SE comment, i think. pretty sure that markup counts against total characters

while others are using mathjax, i am taking over the world

for differentiable where needed single variable functions, is there any case where $\lim \limits_{h \to 0} \frac{g(f(a+h))-g(f(a))}{f(a+h) - f(a)}$ does not exist?

I’m still trying to understand your argument above. I’ll get back to that soon. @Leslie

@leslietownes ahh, I got it. I somehow didn’t use the fact that $(x,y)=xe_1+ye_2$

Thank you so much @Leslie :)

8:18 AM

I wonder what it means by equilibrium I forgot a lot of differentir

equation

I thought my question is related to Cesaro mean but seems not.

@Koro I was talking with my friend and I got this question.

8:56 AM

Since $3<\pi<4$ and each grades are rational numbers, maybe we can construct any rational number between 0 and 4.

1 hour later…

10:05 AM

@Koro Now I understand that suppose balloon got pressure equal to air pressure and when volume of balloon is maximum then this implies that balloon cover is weightless

thanks now I understand

1 hour later…

11:26 AM

Is it normal that I could formalize a basic proof easily but can't understand a bit about the intuition behind that proof ? Like for example the existence of root(2) in ℝ?

That proof just feels extremely adhoc for me to learn from.

Relatively easy to formalize, But very hard for me to intuitively understand.

11:58 AM

No I was wrong because pressure decrease as balloon rise up

when hot air balloon reach it's maximum height the pressure is same as atmosphere

12:40 PM

I was wrong again

should take account that air pressure changes toi

so the pressure stays same

12:53 PM

@TedShifrin I think I finally understand the answer lol

feels so stupid to think this for hours and now I finally realize I didn't take atmospheric pressure change as height changes

1:44 PM

@leslietownes who was immortal: MacLaurin or Taylor?

1 hour later…

2:48 PM

Does the fact that a group $G$ of order $385 = 5\cdot 7\cdot 11$ has a central $7$-Sylow subgroup and a normal $11$-Sylow subgroup imply there are exactly two such subgroups $G$ up to isomorphism?

2:59 PM

I have been banned

didn't realize that you couldn't make two non-interacting accounts on physics stack exchange like you can on math stack exchange

2

A: Proving that dim $\cap_{i=1}^m N(\phi_i)=\dim V-n$, where $V$ is a finite dimensional vector space and $\phi_i$'s are L.I. in $V'$.

One simple approach to the problem is to consider the map $\Phi:V \to F^m$ given by $$ \Phi(v) = (\phi_1(v),\dots,\phi_m(v)). $$ Verify that $\bigcap_{i=1}^m N(\phi_i) = N(\Phi)$. Because the $\phi_i$ are linearly independent, we can conclude that $\Phi$ is surjective. From there, we conclude tha...

with reference to the comment below the answer to the question linked above, can anyone please help me understand why "$\Phi$ fails to be surjective if and only if there is a non-zero column-vector 𝑐 such that for all $v\in V, c^T \Phi(v)=0$."?

3:19 PM

hmm

3:45 PM

@Koro the range of $\Phi$ is a subspace.

4:00 PM

koro, same answer as last night :) it's the dual characterization of proper subspaces, using the dual of F^m

if Phi(V) is proper, there's a nonzero linear functional f that vanishes on Phi(V). then you can take "c" to be (f(e_1), f(e_2), ..., f(e_m))^T (where e_1, ..., e_m is the standard basis)

Might as well just go back to thinking about matrices and the orthogonality of the rowspace and nullspace, orthogonality of the column space and nullspace of the transpose.

Happy Thursday to the munchkin, @leslie.

i was trying not to use matrices last night because thorgott made me ashamed of using them.

The hell with Thor.

or unhappy thursday. lots of protest against going to day care this morning. "i want to draw!" (you can draw later.) "the cat is roaring at you!" (she's pointing to yesterdays drawing of a cat.) "i don't want to go to school! roar!"

If so, might as well use my dual basis argument (which does call for double-duality).

Ah, so things are getting back to normal.

4:08 PM

same with last night's bath. "i don't want to take a bath! i don't want to take a bath! i already took a bath! aaaghghhh!" then the minute she's in the bath she demands tons of bath toys. then she doesn't want to get out of the bath because her rubber duck needs a bath.

In a few short years, you'll be worrying about her romantic entanglements.

hi, demonic @Alessandro

Alessandro Codenotti

Hi @Ted

I started following a course on Lie groups today

That's cool :)

Alessandro Codenotti

Ok maybe selling it as a course on Lie groups is a bit of a Lie. The aim is proving Gleason-Yamabe

Watch out lest we wash your eyes out with lye. ... What is Gleason-Yamabe?

Alessandro Codenotti

4:20 PM

There are a few ways to state it. The one we're going with is that if $G$ is a locally compact group and $U$ is an open nbhd of the identity then there is $G'\leq G$ open and $K\leq G$ compact normal with $K\subseteq U$ and $G'/K$ Lie

Another way to say this is that if a projective limit of Lie groups satisfies the no small subgroups condition then it is Lie

Why is it interesting to have a course for which that is the goal?

Alessandro Codenotti

That's the solution to Hilbert's 5th problem

This is not the sort of stuff I've ever thought about.

Oh.

Alessandro Codenotti

There will also be a part on approximate subgroups if time allows it

The statement of Hilbert's 5th problem on wiki looks quite different to me.

Alessandro Codenotti

4:24 PM

Yes there are various ways to interpret the problem that give different answers. Gleason-Yamabe is very general and most likely not the statement Hilbert would have liked

there's a line on wiki saying that at least some people characterize hilbert's 5th differently, but also suggests that this might be a minority view.

Oh, I see ... your "another way to say this is ..." is the global point.

Hilbert's point was, I guess, to what extent does smoothness destroy anomalies?

4:40 PM

@leslietownes Leslie, I’d understood your answer :). After seeing Mr. Ben’s comment in response to my comment (post which I asked my doubt here in chat), I was thinking if Mr. Ben had something else in mind. But as per discussion with Mr. Ben in the comment section, he had the same theorem in mind. :)

Hi all.

Can I ask a question about studying baby ℝ analysis?

as long as it isn't toddler real analysis.

Is ∃x ∈ ℝ (x.x = 2) toddler?

Hi, a @Balarka.

4:46 PM

no, i don't think it is.

But using MathJax here is required.

How's your day, @Ted?

5 hours ago, by Prithu biswas

Is it normal that I could formalize a basic proof easily but can't understand a bit about the intuition behind that proof ? Like for example the existence of root(2) in ℝ?

if you want to prove that, it unfortunately touches upon what your definition of $\mathbb{R}$ is, which is likely tied to what your definition of $\cdot$ is.

Mostly dealing with a cat who's recently been spayed, @Balarka. Nothing fascinating, but about to head off to a massage to deal with my horrid back and neck.

@leslie Are you about to embark on the road of Dedekind cuts?

4:48 PM

@leslietownes I am using this axiom system.

Gotcha, @Ted. Enjoy the massage!

i don't see the details of contextual choices about definitions of $\mathbb{R}$ as being 'intuitive.' but if you do formulate and choose definitions of those things, i think techniques for proving that result naturally suggest themselves.

And no I haven't constructed ℝ yet. That is left for later. :P

e.g. if it's cauchy sequences of rationals, you construct a sequence of rationals. if it's completeness in some order property, you identify a set with the candidate for $\sqrt{2}$ as its inf or sup.

So you are using the least upper bound property to characterize $\Bbb R$.

4:49 PM

Here is the proof I am trying to formalize.

and then proving that those things give you what you want is a chase through the definition of multiplication.

@TedShifrin Yes the completeness axiom.

The problem is , I can formalize it , but cant internalize it.

i think this is the approach taken in rudin, if i'm not mistaken

i may be in the minority on this, but i don't think the details of constructions of $\mathbb{R}$ are very instructive or worth internalizing. the properties of these constructions are worth internalizing.

if you write the ideas of that proof in human language it makes a kind of sense. by completeness and some complicated reasoning with the axioms, there's a real number that can't square to anything less than 2, or anything greater than 2.

Basically proving the intermediate value theorem ad hoc in this case.

@leslietownes So the completeness axiom "fills in" the gaps in the rationals in some sense?

4:57 PM

yeah, in some sense.

The only issue is where everything lives, but intuitively ...

0

Q: Constructing $\mathbb R$ from $\mathbb Q$ using Dedekind Sections/cuts.

I'll try to present my understanding of construction of $\mathbb R$ and request your help in doubts that I have in the same. So we consider a straight line $\Gamma$ and mark origin (reference point) on it as $O$, we'll measure all distances (length/taxi-cab distances) from $0$. We mark integers ...

This may interest you @Prithubiswas

We do NOT want to do Dedekind cuts. Go away.

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