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

Avra

00:32

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 $a=a_12^{n/2}a_2$?

Ted Shifrin

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

Avra

I know this is not meant for me :/

leslie townes

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.

Avra

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

leslie townes

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.

Avra

00:45

123456 = 123 10^3 + 456

leslie townes

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

Avra

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

leslie townes

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 $g$

Avra

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

Akiva Weinberger

00:51

'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 $g^n\in H$

leslie townes

[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 $g^n=e$"

Lukas Heger

you want $H$ to be normal, though

Avra

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

@leslietownes. Do you forgot formulas sometimes?

leslie townes

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

avra, most of the time.

Lukas Heger

00:55

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

Avra

@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 :/

leslie townes

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

leslie townes

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.

Avra

@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!

leslie townes

01:00

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

Avra

Wow!

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

leslie townes

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.

Avra

But you did not allow these on exam day!

leslie townes

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

Avra

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

leslie townes

01:04

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.

BannedUser

can I ask mathematical physics lol

leslie townes

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.

Avra

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

leslie townes

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

Avra

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

BannedUser

01:06

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

Ted Shifrin

Equilibrium, banned?

BannedUser

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

leslie townes

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.

Ted Shifrin

@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.

leslie townes

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.

Ted Shifrin

01:11

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

leslie townes

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.

Ted Shifrin

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)

BannedUser

sorry I was busy in lecture

Ted Shifrin

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

BannedUser

I am very sorry I will be back in 30 min

and read all the chat

Ted Shifrin

01:32

rewrites chat in invisible ink

2

Xander Henderson

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

Ted Shifrin

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

Let’s not expect too much.

leslie townes

$\, \,$

Ted Shifrin

That was too much.

Xander Henderson

01:47

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

Ted Shifrin

Past.

Xander Henderson

:D

PM 2Ring

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.

Ted Shifrin

Of course!

@leslie Quite the backups at LA and LB ports.

@PM2Ring Plus tremendous for the complexion.

PM 2Ring

@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.

leslie townes

02:08

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

02:34

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

Koro

@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

David Choi

02:50

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?

copper.hat

03:04

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

David Choi

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

copper.hat

Yes, I goofed.

David Choi

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

copper.hat

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?

Koro

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

leslie townes

03:18

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.

copper.hat

i should read before writing.

leslie townes

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.

copper.hat

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.

anakhro

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...

math-history

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

David Choi

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

Koro

03:25

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

copper.hat

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

David Choi

I personally wanted the best bound

The question doesn’t specify

leslie townes

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

PM 2Ring

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

copper.hat

you need to define 'best bound'.

leslie townes

03:27

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.

copper.hat

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.

PM 2Ring

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

David Choi

Woah I never even thought abt that

copper.hat

tcheeheyhchecshev

David Choi

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

copper.hat

03:29

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

PM 2Ring

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

copper.hat

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"

David Choi

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

copper.hat

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

David Choi

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

leslie townes

03:33

too many to choose from.

Ted Shifrin

@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.

David Choi

I am confusion

copper.hat

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

Ted Shifrin

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

copper.hat

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

wooster

shintuku

03:35

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

Ted Shifrin

I don’t know what that means.

N-dim wronskians?

shintuku

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

leslie townes

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.

David Choi

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

leslie townes

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

David Choi

03:39

@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

Ted Shifrin

Never heard of this … must be some econ thing.

David Choi

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

shintuku

it's from a convex optimization book

copper.hat

the log determinant appears in convex programming.

Ted Shifrin

Then copper must know.

leslie townes

03:41

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

Ted Shifrin

WTF is it?

copper.hat

it is concave on positive semi definite matrices.

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

Ted Shifrin

Oh, oh, log det A

shintuku

author hasn't given a definition :'(

leslie townes

sounds like a great book

Ted Shifrin

03:43

I bet it appeared somewhere.

PM 2Ring

@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:

leslie townes

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.

copper.hat

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

shintuku

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

copper.hat

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

Ted Shifrin

03:46

We both said it above.

copper.hat

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

shintuku

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

Ted Shifrin

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

copper.hat

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

shintuku

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

is it implied it is base $e$?

copper.hat

03:51

it doesn't really matter, but yes.

David Choi

@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.

shintuku

thanks!

David Choi

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 $\min_a \max_{|x|\le c} |f(x)-ax|$.

copper.hat

@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 $\le 0.01$.

PM 2Ring

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)

copper.hat

03:53

chevychasetycheb again

David Choi

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).

copper.hat

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.

copper.hat

i was linearly dependent for ages afterwards.

PM 2Ring

03:55

@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.

copper.hat

@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

copper.hat

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

copper.hat

04:00

i think about curvature a lot.

Ted Shifrin

I don’t see the chain rule.

copper.hat

i see division but no chains

Akiva Weinberger

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

PM 2Ring

@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

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

Ted Shifrin

04:03

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

Akiva Weinberger

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

Ted Shifrin

sure, but reciprocal is the key

copper.hat

@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

Ted Shifrin

Curvature boils down to unit tangent, of course.

Akiva Weinberger

04:06

Derivative of.

Ted Shifrin

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

PM 2Ring

@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 $\epsilon\to0$

shintuku

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

Akiva Weinberger

04:14

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

David Choi

@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 $2\epsilon/\epsilon=2$

PM 2Ring

@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 $F(S)$.)

leslie townes

ffs

Akiva Weinberger

04:28

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

Prithu biswas

04:51

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

Adil Mohammed

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

leslie townes

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

copper.hat

05:07

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

leslie townes

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

25 seems like too many

PM 2Ring

05:26

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

geocalc33

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

PM 2Ring

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.

Adil Mohammed

05:47

@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

Koro

07:10

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?

love_sodam

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

leslie townes

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

Koro

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.

leslie townes

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)

Koro

07:28

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. :(

leslie townes

what are 'these premises'

Koro

@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..."

leslie townes

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

BannedUser

@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

leslie townes

07:47

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

BannedUser

also air can escape from balloon

Koro

@leslietownes it also extends life of your shift key.

leslie townes

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

shintuku

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?

Koro

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 :)

BannedUser

08:18

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

equation

love_sodam

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

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

love_sodam

08:56

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

BannedUser