Mathematics - 2021-11-24 (page 1 of 2)

12:09 AM

anakh: yes, me and my fellow teens all enjoy inhalants. it's the main thing we do when we aren't posting videos of ourselves doing it on TickTock

Leaky Nun

12:23 AM

@TedShifrin is the genitive suffix -го always pronounced like -во?

Ted Shifrin

Leaky. I appreciate your faith in me, but it’s been 50 years. I don’t remember such an anomaly.

copper.hat

languages with genitives should be abolished.

that would include my supposedly native language

Ted Shifrin

12:39 AM

Leaky, yes, you’re right. Also, other words.

copper.hat

that ambiguity between v and b again

it bugs me to see questions that can be answered (or at least guessed) by just plotting a formula.

is the notion of plotting not taught anymore?

too infradig maybe?

leslie townes

it is but for many it's too hard and something you don't do unless you're asked to do it.

if you mean that thing with absolute values, i'm not sure that it is always taught.

Leaky Nun

@TedShifrin what other words

copper.hat

a few kids i tutored over the years would spend forever trying to solve silly little problems. me: "just plot it". them: "we're supposed to work it out."

reminds me of the mathematician with constipation...

leslie townes

you see this at higher levels too. i don't want that semi-calculational way to do it, i want a shorter way to do it. a more elegant way. ideally one that takes me an hour to find but ends up being shorter in written length, instead of the way i could finish in 2 minutes at the cost of a few lines on paper.

copper.hat

12:46 AM

i suppose i used to be a bit like that.

leslie townes

which i understand if you're writing lecture notes or a book or presentation or something. if you want an example or exposition to be just so. but if someone gave you a dozen of these and your only job is to finish them . . . . . .

copper.hat

apart from anything else it is a reality check

like casting out the $8. \dot{9}s$ or something

Ted Shifrin

Not changing keyboards. The word for today springs to memory.

anakhro

@copper.hat The version of the joke I am familiar with is for an accountant.

copper.hat

work it out with logs?

accountants use pencils...

leslie townes

12:50 AM

what's a pencil

anakhro

@copper.hat Oh I thought you meant the work it out with a pencil joke

copper.hat

mom: "i found a contraceptive behind the radiator". daughter "what's a radiator"

@anakhro i did :-)

some of these old jokes just don't work anymore

leslie townes

"here's [coin of small denomination], call someone who cares"

(1) what are coins (2) what am i supposed to do with whatever coins are

copper.hat

farthing?

not to be confused with

i did have a farthing and a few halfpennies but i gave them away. what was i thinking

i have a severely corroded silver dollar

and always had a childish fascination with coins that had holes in them

anakhro

We don't have pennies anymore in Canada. :(

copper.hat

12:55 AM

really? how do you go to the bathroom?

doesn't translate. one expression for go to the bathroom was to spend a penny.

generational and location specific. my bad. too much starmucks

anakhro

Imagine having to pay money to go to the washroom

leslie townes

i know it from old movies.

anakhro

I always put a dollar charge on my washrooms in roller coaster tycoon.

copper.hat

public restrooms were often like that

i loved sim city

and derivatives.

frechet derivatives.

leslie townes

beat me to it.

copper.hat

12:58 AM

have your gateaux and eat it

anakhro

That was a leslie trap, if i ever saw one.

copper.hat

never understood the point of that expression

ffs, if you have a cake why would you not eat it. really

i have forgotten what the ffs stands for, too lazy to scroll back

right now i am too lazy to go to acme to get bread

anakhro

finite field something

copper.hat

:-)

i must remember that for the next time i utter ffs

anakhro

I think the point is that if you ate your cake, you'd not have it any longer.

copper.hat

1:00 AM

some infinity thing going on then?

anakhro

But I think it breaks down if you can eat part of your cake, too.

copper.hat

if i leave my cheese long enough it grows

i'm getting a "step away from the keyboard" alert

anakhro

I have one of those, too, but it is an "go do something more productive than chatting" alert.

copper.hat

is your name of Greek origin?

mine (joe) is apparently. sifis.

off to get a baguetttttttte

anakhro

@copper.hat verily

dc3rd

1:12 AM

@leslietownes the romanticism of formalism....."Yes I see that you showed me what intuitively is supposed to happen, but I didn't understand it but demand the rigourous proof of the claim"

Ted is the one that beat it out of me..

leslie townes

we'll make a symbol-pusher of you yet.

everything ted told you was lies. except when he recommended investing heavily in lesliecoin. that was sound advice.

dc3rd

symbol pushing is in the blood, I'm attempting to control it the way Bruce Banner controls the Hulk...

leslie townes

1:26 AM

i.e., not well?

Semiclassical

theguardian.com/science/2021/nov/23/…

money money money

leslie townes

on closer inspection, the document revealed a shopping list, and einstein's notes on how to explain how to row reduce a matrix, intended for correspondence with the 'math post-exchange,' a small group devoted mainly to doing the homework of all of switzerland

Semiclassical

@XanderHenderson Descarte can be a nice tool for eigenvalue problems, e.g., use it to detect a positive real root

of course, there's higher-powered tools for doing that

argument principle etc

i also did have a use for it in a physics problem recently. the question came down to (numerically) solving a cubic equation, and you could use Descarte to deduce that it indeed had a single positive real root.

Ted Shifrin

@dc3rd Probably a false claim!

@Semiclassic DescarteS

French: of the mapS.

Semiclassical

and here's Ted keeping me honest :P

Ted Shifrin

1:35 AM

It’s a hopeless task, admittedly.

Xander Henderson

@Semiclassical Yeah, that is kind of my point. By the time you actually want to find the roots of polynomials, you have better tools, e.g. Gershgorin seems more helpful than Descartes.

Semiclassical

for matrices yeah

though in truth I didn't see Greshgorin until grad school

leslie townes

and they let you in anyway, huh? hmm.

Xander Henderson

@Semiclassical Same here. Though it was in a class on numerical analysis where the primary text was in the AMS Undergraduate Texts series.

Ted Shifrin

Yeah, it’s not often taught. It’s an exercise in the complex matrices section in my lin alg book, but no one ever gets there.

Semiclassical

1:40 AM

for the problem context here, the equation is of the form $x y (2+3 y+6 x y^2)=z$, and for the answer to make sense it should be the case that you get a positive real root in $y$ whenever $x,z$ are positive reals

with Descartes it's immediate: $(6x^2)y^3+(3x)y^2+(2x)y-z$ has exactly one sign change if $x,z$ are positive

copper.hat

symbols just make clashing sounds

Ted Shifrin

This should come just from IVT

2

presumably, yeah

Glockenspiel @copper

love the sound

1:42 AM

Evidently, Semiclassic … real cubic with positive leading coeff

dc3rd

Of course the Greshgornian thm is in the Markov Chains chapter of my text that I skipped over to return to later..........I'll contribute to the discussion on it in a month

Ted Shifrin

Making Xander’s point for him …

Xander Henderson

Heh.

Semiclassical

if you modify the problem a bit you can get anything with $xy(a+by+cxy^2)=z$ for positive $a,b,c$

and Descartes still applies

Ted Shifrin

The depth of Descartes is when it involves Rolle on steroids

My remark still holds, Semi

Semiclassical

1:45 AM

yeah, it's trivial with IVT too. LHS is zero when y=0 and grows without bound as y increases, so there's definitely a positive real solution

and that argument has the advantage of only needing $c$ to be positive

Ted Shifrin

Ayup

Semiclassical

Descartes loses most of any luster it has once you realize how much zero coefficients messes with it

Ted Shifrin

Nah … why?

Semiclassical

tbh i'm vaguely remembering the issue. but for instance something like $x^2+\epsilon x-1=0$.

oh. that's a terrible example, isn't it

if $\epsilon \neq 0$ then there's always one sign change

guess my memory of high-school math is playing tricks on me :P

Ted Shifrin

Same with 0. 0 terms don’t count.

Semiclassical

1:52 AM

i guess not yeah. i could see things getting a bit strange if you had two consecutive zero coefficients, maybe, but now i'm reaching

looking at wikipedia though it seems you do just ignore zeros entirely

leslie townes

more like descartes' rule of wasted times, am i right

Semiclassical

eyyyyy

leslie townes

yeah, i really couldn't do better than that

Ted Shifrin

Cross the continental divide?

Semiclassical

i'll admit, while i know what the argument principle says (i.e., you can integrate $f'(z)/f(z)$ to count roots/poles of $f(z)$) i remember not being able to understand how to use it in practice

e.g. how to actually estimate anything with it

Ted Shifrin

2:01 AM

Geometrically, look at winding numbers.

leslie townes

well if it's gonna be an integer you can use a fairly crummy numerical integration algorithm to quickly figure out what it might be.

Semiclassical

sure, but using that to count roots in the first quadrant for instance

like, as a numerical method sure

leslie townes

i dunno how physicists would use it.

probably to desecrate all i regard as holy

Ted Shifrin

Take a quarter circle and see how its image winds around 0.

Semiclassical

but i know i've seen problems in the past to the effect of "look at this polynomial and use the argument principle to find out how many roots are in the first quadrant"

and i was never sure how one was supposed to actually do that, absent a plot of $f'(z)/f(z)$ as $z$ varies

Ted Shifrin

2:04 AM

You keep ignoring me, shrug.

I mean image under $f$, of course.

Semiclassical

not sure how i am. how does one 'see' the image under $f$ of some polynomial?

that's the issue for me

like, if i have access to plotting software, sure, but that seems implausible :P

Ted Shifrin

Estimates on positive reals and positive imaginaries and for large R.

Semiclassical

bleh

leslie townes

4

Q: Application of Rouché's Theorem. Show polynomial has exactly one root in each quadrant.

Show that the complex polynomial given by $z^4+2z^2-z+1$ has exactly one root in each quadrant. I know by the fundamental theorem of Algebra, that the polynomial has exactly four roots. Now, to show it has exactly one root in each quadrant, it suffices to show that there are two non-real r...

complex-analysis

one example in the answer.

Ted Shifrin

Plotting software is not far-fetched.

Good example.

AMDG

2:08 AM

@robjohn I forgot to distribute the negative. My bad. Still, it isn't what I'm looking for.

leslie townes

stuff like that sometimes showed up on berkeley prelim exams. not my year, but i remember seeing it when studying for it.

AMDG

@robjohn Close. I want to compute $\Gamma(z)$ using some function of $\sin(z)$.

Semiclassical

hmm, okay. i can see how that does reduce to seeing how the argument changes around the boundary

Ted Shifrin

We accept your apology.

Semiclassical

lol

how to actually compute those changes still slips past me (aside from on [0,R] ofc)

Ted Shifrin

2:12 AM

On the large circles, it’s the usual game that the leading term dominates.

Semiclassical

i can sorta buy that: in the example you end up arguing that $1+(2z^2-z+1)/z^4$ is not going to produce any additional winding as $z$ varies

which seems plausible insofar as for large $|z|$ that's not going to get too far from $1$

for the other contour you instead end up looking at $1-iR/(R^4-R^2+1)\to 1$ as $R\to\infty$

it doesn't fit nicely into my head, but i guess that's to be expected for something i've never had to actually use

leslie townes

it's certainly no lamecartes' misrule of signs.

i'm trying, i really am. it's not a lot to work with.

Semiclassical

you certainly are trying

leslie townes

i won't photoshop a jester hat onto a picture of descartes, but that's what "descartes, fool of signs" would be.

Semiclassical

2:34 AM

wouldn't that make him a fooler of signs

leslie townes

i don't even know anymore

copper.hat

3:04 AM

didn't des cartes write "i'm pink, therefore i'm spam"?

copper.hat

3:21 AM

end domination of the convergence theorem now!

Semiclassical

convergence uber alles

copper.hat

the new pay as you go app for reducing error

user178758

3:44 AM

o/

leslie townes

4:10 AM

o/

copper.hat

4:27 AM

o///o

... --- ... --- ... ---

Koro

6:05 AM

I think LADR doesn't cover bilinear forms.

leslie townes

this coincides with my memory of an earlier edition. just inner products.

Koro

where can I study bilinear forms?

Jordan form is there in Axler's.

copper.hat

what is ladr

Koro

The text Linear algebra done right by Axler

leslie townes

why do you want to study bilinear forms? is there a goal in mind?

Koro

6:17 AM

because in past I watched prof. Strang's lectures and there I learnt positive definite matrices etc. and I liked how conics equation was written in the form of matrix. Now, that I'm studying linear algebra, I thought about completely studying those.

leslie townes

i think there's a chapter on them in shilov

Koro

copper, there is also a book called linear algebra done wrong.

copper.hat

:-)

Koro

math.brown.edu/streil/papers/LADW/LADW.html

Leslie: noted.

Thanks.

copper.hat

In finite dimensions there's not a huge amount to deal with that you have not seen with linear operators.

questions like this math.stackexchange.com/questions/4314816/… make me realise that some large part of a maths education is being omitted nowadays.

leslie townes

6:22 AM

i don't think a lot of the stuff we see on there is symptomatic of 'maths education.' at least, i hope it isn't.

copper.hat

true.

on my desk are symptoms of my procrastination today, kantorovich & akilov's functional analysis, Ted's abstract albgebra, Clarkes nonsmooth analysis, boyd et al's convex optimization and rockafellar's convex analysis

leslie townes

that bad, huh.

copper.hat

and stark's intro to number theory

leslie townes

i cleaned several of the sinks and countertops in our house today. that was my procrastination.

copper.hat

yup. for reasons i cannot disclose i cannot charge hours at present, so things are getting antsy here

i am buying anchovies & jalapenos at the store without any need

this can't end well

i saw a yoga teacher/psychologist today.

Koro

6:31 AM

Robjohn's avataar looks like Sun.

leslie townes

hah, i hadn't noticed. someone's in the holiday mood.

maybe grandad needs a pilgrim hat

copper.hat

where is grandad

Koro

haha, it looks good. I noticed it sometime back.

copper.hat

i think it looks mean which is at minimum misleading

i don't mean average

leslie townes

my daughter accuses us of being mean when we deny her things that she wants

copper.hat

6:34 AM

she knows she's getting to you

working on our holiday card. i gave up being sensitive so it says merry christmas even though i am not a believer

i was going to include the picture of my daughter in the back of a police car during her recent ucla visit

leslie townes

today after dinner i asked what happened at school and she said "I want some vitamins, and I want some chocolate, and then I'll tell you what happened at school. OK?"

she does take one vitamin supplement her pediatrician recommended, the chocolate is just an add-on

copper.hat

just give up now

robjohn

@Koro It's a pumpkin pie... a mean pumpkin pie.

leslie townes

was it a staged photo or something screencapped from the tv news?

robjohn

@leslietownes That sounds scary.

copper.hat

6:38 AM

it needs some bézier curves for eyebrows

Koro

haha

Wolgwang

@robjohn I was going to write that...

Mouad

hilo :)

copper.hat

the irish language has one diacritical, but i could never remember where they went, so i used to just add them randomly.

time for a jalapeno & anchovy snack

robjohn

@Mouad highest in, lowest out? a city in Hawaii?

copper.hat

6:40 AM

it is a falling edge in logic

Mouad

@robjohn man I have no idea what's this, im newbie :(

robjohn

@Mouad asking about "hilo"

copper.hat

@Mouad you wrote "hilo", hilo is a city in Hawaii

Wolgwang

Is there a user here whose nickname is grandad?

Mouad

AAAAH! no, that's my accent! the way I say hello :)

copper.hat

6:41 AM

yes, @leslietownes

@Mouad a lot of odd word play goes on here

Mouad

I figured that out

copper.hat

sometimes even word play

robjohn

odd words, like seven, or thirteen.

copper.hat

i am still searching for the second even prime

Mouad

if im not gonna have to solve equations here, we're fine @copper.hat

leslie townes

6:43 AM

wolgwang, i don't think so

copper.hat

@Mouad you never know :-).

Koro

when we say open, mouth closes and when we say closed, mouth remains open. I don't remember where I read that.

leslie townes

everything's placid when you add the water to the acid

Mouad

@copper.hat

ready for it brother :D

copper.hat

@Koro that's better than kobayashi's " an open set is not like a door, it can be open and closed at the same time".

my mom accidentally put hydrogen peroxide in the holy water font once, it was rather amusing.

leslie townes

6:46 AM

it's like the mind, man.

copper.hat

my mind is so open that my brain falls out regularly

leslie townes

when you're being sworn in to testify in court, i think a good gag would be to pretend like the bible is sizzling hot

copper.hat

could i swear on my copy of rockafellar instead?

leslie townes

use multivariable mathematics

copper.hat

the epigraph, the whole epigraph and nothing but the epigraph

there was a funny episode when dick feynman almost got contempt for taking the truth line literally.

loved his books.

now i realise the real reason for my strange mood. trader joes has been out of cotswold cheese for 2 weeks and i am in withdrawal.

leslie townes

6:54 AM

mm, ours had some the other day. drive south for about 6 hours. can't miss it.

i didn't buy it, but i noticed it.

copper.hat

see you shortly

i'll take the lear

leslie townes

you could take a few pit stops along the way, i odn't think it'll be open until 8 anyway

copper.hat

i have certain skills

my brother was going to visit to do the sf chrissy field park run parkrun.us/crissyfield but they just cancelled it :-(. so he won't be coming out this year.

robjohn

I don't think I've ever noticed if ours has any of that cheese

leslie townes

sorry to hear it, copper.

my dad keeps on asking when i'll visit. i say, you can come down. i don't know what you may have read about traveling with a toddler, but it isn't nearly as fun as they make it sound.

robjohn

6:59 AM

@leslietownes short trips are good for getting them to go to sleep, but extended trips are pretty bad.

leslie townes

this would be a good 8 hours on the interstate. or maybe 3.5 with airport travel, but with airplanes we'd have to leave most of her stuff at home or deal with tons of luggage. i'll pass on that.

copper.hat

@robjohn i love cheese in general. cotswold is like crack. i feel like it is a bit of a cultural betrayal, but it is really good...

leslie townes

don't wanna deal with going potty on a plane, or in an airport.

robjohn

@leslietownes plus airplane travel can be hellish if they are not used to the pressure changes.

leslie townes

cheese is the one thing i sometimes impulse buy at trader joe's.

copper.hat

7:02 AM

my son used to get headaches on the descent

i will go to tjs just to buy cotswold

leslie townes

robjohn, yeah. that's a real unknown. she has flown across the country before (twice) but around age 1. did not complain, but also didn't have an attention span and maybe had no idea what was happening.

Koro

pressure changes, yeah. Even I get pain in my ears during airtravel.

robjohn

@copper.hat I use Smoked Seaside cheese from Whole Foods. It is a smoky cheddar imported from Ashley Chase Estate in England.

My wife thinks it is too smoky, but I love it

copper.hat

@robjohn something to add to my list when i go to whole paycheck

i like dubliner, but there is some bias there

Koro

but i have noted that if I'm chewing something then pain in ears is less during airtravel.

robjohn

7:04 AM

@Koro chewing and yawning help

copper.hat

airlines used to give out candy to alleviate the pressure change

they used to serve a full (not continental) breakfast on irish airlines

leslie townes

i just found out that a relative passed away... over a year ago. this is very on brand for my family.

Koro

Have you every tried chicken biryani in air? It tastes good :)

copper.hat

@leslietownes sorry to hear it.

robjohn

@copper.hat Yeah, airlines used to serve complete meals on a flight. Not any more.

leslie townes

7:07 AM

the best airplane meal i ever had was a biryani.

the dry environment affects the sense of smell and taste buds in a way that messes with a lot of food.

copper.hat

@robjohn i miss the days when flying was a luxury :-) they would give out toothpaste, eye covers, etc.

leslie townes

copper: my mom was like "didn't i mention that?" "no" "oh"

copper.hat

yeah, that can happen.

leslie townes

particularly if people are somewhat estranged from one another.

copper.hat

i had a little social faux pas as a result of one of those forget to inform moments. how's your mum? ohh, she passed away last year.

all books with analysis in the title are being repatriated to my basement library tonight.

user178758

7:11 AM

may i ask why?🤔

copper.hat

too distracting. i feel the need to learn something new and then get lost revising the things i previously knew.

robjohn

@leslietownes that reminds me that I need to reach out to an uncle that I haven't heard from in a long time.

leslie townes

google first. just a tip. :)

user178758

usually, of the "iceberg"

leslie townes

this is true. don't go too far. turn 'safesearch' ON.

user178758

7:22 AM

👍

copper.hat

in some countries a tip means dumpster

leslie townes

you can skip the tip, or tip the skip

copper.hat

indeed, you can even tip the tip, not sure about skipping the skip though...

@robjohn chat just beeped me about your message from 30+ mins ago!

robjohn

@copper.hat yes. There are privileges to having a diamond :-)

copper.hat

ahh, i see!

robjohn

7:36 AM

I added a picture link

copper.hat

will try that shortly

robjohn

sorry, I altered the image to remove some private info

so you were beeped again

copper.hat

np :-), i was wondering, i certainly did not notice the private info

leslie townes

why would you stamp your SSN on a block of cheese anyway, robjohn?

robjohn

@copper.hat It was in the EXIF data of my image, but actually, it was removed by imgur on upload, so I needn't have changed it.

user178758

7:48 AM

joke: therefore, you didn't need to cut the cheese in the breakroom?

robjohn

@leslietownes The image I uploaded had GPS info in it, and I didn't really want that in general distribution

However, imgur removed it on upload

leslie townes

this is why i wrap my phone in a thick layer of foil before i take a picture of anything

robjohn

@leslietownes transparent aluminum foil?

leslie townes

only the best for me

robjohn

regular aluminum foil would be bad for the lens

leslie townes

7:51 AM

no, whatever might hit the lens, that's what i'm most worried about

that and the EXIF data

robjohn

regular aluminum foil gets in the way of photons, for the most part.

copper.hat

do you mean aluminium?

vs tin foil

or copper hats

8:07 AM

A tin foil hat is a hat made from one or more sheets of aluminium foil (commonly called "tin foil"), or a piece of conventional headgear lined with foil, often worn in the belief or hope that it shields the brain from threats such as electromagnetic fields, mind control, and mind reading. The notion of wearing homemade headgear for such protection has become a popular stereotype and byword for paranoia, persecutory delusions, and belief in pseudoscience and conspiracy theories. "Tin foil" is a common misnomer for aluminium foil; packaging metal foil was formerly made out of tin before it was replaced...

robjohn

8:28 AM

@copper.hat depends on your country of origin ;-p

leslie townes

he's a spy! get him

robjohn

8:39 AM

@Semiclassical this answer uses trig and hyperbolic functions to get solutions for cubic equations. It shows when there are 3 and when there is 1 real solution.

flowian

10:47 AM

Hello

user178758

hi

Prithu biswas

12:14 PM

@robjohn Idk why , but your profile picture looks like a Digger.

A Mean Digger.

love_sodam

12:49 PM

What is the latex code of the above matrix?

especially the code for that vertical lines

monoidaltransform

1:11 PM

the following holds: $\mathbb{Q}(a\sqrt{b},a\sqrt{b} b )=\mathbb{Q}(a,b)$ right?

love_sodam

why not?

monoidaltransform

how do I prove it though?

love_sodam

Wait, I could be false

Yes if $b = \pi$ and $a =1$ then $\Bbb Q(\sqrt{\pi},\sqrt{\pi}\pi) =\Bbb Q(\sqrt{\pi}) \neq \Bbb Q(\pi)$ if I'm understanding correct.

monoidaltransform

what about if $a=\sqrt[4]{2}$ and $b=i$

?

Leaky Nun

1:58 PM

@love_sodam $\begin{pmatrix} \mid & \mid && \mid \\ v_1 & v_2 & \cdots & v_r \\ \mid & \mid && \mid \end{pmatrix}$

@monoidaltransform that's just a coincidence

as others have pointed out, there's no way to generate $\sqrt{b}$ using $a$ and $b$ in general

love_sodam

@LeakyNun That's pretty intuitive. Thanks.

Leaky Nun

I just typed what I saw lol

love_sodam

@monoidaltransform I don't believe $\sqrt{i}\in \Bbb Q(\sqrt[4]{2},i)$

Mad Spaces

2:16 PM

i do not understand why this seemingly trivial inequality holds:
if $x \geq y \forall x \Leftrightarrow inf(X) \geq y $
i guess we can write $ x - inf(X) \geq \epsilon , \epsilon > 0 $ and in the limit of $ \epsilon -> 0 $ the inequality $ y \geq x- \epsilon \leq inf(X)) $ would result in a contradiction but i mean we never actually reach zero we just get arbitrarily close so why does y still have to be smaller then the infimum

If we say that there is a sequnce of $x_n$ that approaches inf(X) then for the elements the inequality holds, but i really dont get how it holds once it reaches it. or when we write inf there this limit process is messing up my mind

oh wait i am stooooopid

because if it were $y \geq inf(X)$ and $\forall x, x \geq y$ then this means y is a lower bound and the infimum of x ist not the greatest lower bound

ahhhhhhhhhhhhh ... kk

Answering questions and answering them myself.. i think it a thing when you type it in latex and clean you kind of get a good look at what you are asking :dDD

Leaky Nun

Rubber duck debugging

In software engineering, rubber duck debugging is a method of debugging code by articulating a problem in spoken or written natural language. The name is a reference to a story in the book The Pragmatic Programmer in which a programmer would carry around a rubber duck and debug their code by forcing themselves to explain it, line-by-line, to the duck. Many other terms exist for this technique, often involving different (usually) inanimate objects, or pets such as a dog or a cat. Many programmers have had the experience of explaining a problem to someone else, possibly even to someone who knows...

Mad Spaces

oh well now i know the name of this phenomena thanks Leaky nun

AMDG

2:40 PM

I'm supposing I might implement the Riemann Zeta function in order to implement the Gamma function.

Daniel Adams

3:42 PM

Anyone around to help explain to me the definition of the path integral operator in this paper sciencedirect.com/science/article/abs/pii/026689209290018D

Allie

5:23 PM

hi guys

Koro

6:24 PM

V is FDVS and $P\in L(V)$ is such that $P^2=P$ and $||Pv||\le ||v||$ for every $v\in V$. Prove that there exists a subspace $U$ of $V$ such that $P=P_U$.

Proof: Let $e_1,...,e_k$ be an orthonormal basis of range P=:U, then the list can be extended to ONB of $V$. Let the basis of $V$ be $e_1,...,e_k,f_1,...,f_n$. So the inequality holds. Now, defining $P_U:=P$ works.

What is wrong in the proof? Thanks.

leslie townes

6:47 PM

when you say "so the inequality holds" what do you mean. why is P an orthogonal projection?

Koro

Hi Leslie.

copper.hat

what is $P_U$? the orthogonal projection?

presumably you have an inner product?

leslie townes

i agree that if the result is true then P will be the orthogonal projection onto its range (which is known to be a projection and satisfy the given inequality), but the exercise is to prove that, i.e., to show why the norm thing implies that P can't be some other projection onto its range

Koro

I thought: since $e_1,e_2,...,e_k$ is a basis of $U$, we can extend it to a basis of $V$ and then on that basis we can apply Gram Schmidt process to produce ONBs of V. Here we may note that span$(f_1,...,f_n)$ is orthogonal subspace of span $(e_1,...,e_k)$. So every $v\in V$ can be written as a unique sum $v=\langle v,e_1\rangle e_1 +...+\langle v,e_k\rangle e_k+\langle v,f_1\rangle f_1+...+\langle v,f_n\rangle f_n$ then $Pv=\langle v,e_1\rangle e_1 +...+\langle v,e_k\rangle e_k$

leslie townes

if you assume Pf_j = 0 you're assuming P is an orthogonal projection

all you know from the algebraic condition P^2 = P is that P sends f_j into the span of the e_j's

Koro

6:54 PM

yeah I think I messed up.

I'll think more on this.

leslie townes

gotta use that norm hypothesis somewhere :)

Koro

yeah, I thought that to be trivial :D

thanks. now I know what's wrong with my alleged proof.

yes copper, $P_U$ is orthogonal projection.

leslie townes

this actually points out one approach to the exercise. it's enough to show that if u in range P and v in ker P, $\langle u, v \rangle = 0$

Koro

Ah, that was previous exercise. I could solve that one.

copper, i thought that since orthogonal projection is being asked about in the problem so inner product must be assumed. But I think you are right, what if the norm doesn't come from inner product.

So this problem is more deep than I thought.

leslie townes

koro, if this is axler, the norm comes from an inner product.

Koro

6:59 PM

I see. :)

leslie townes

i really think you gotta say when these problems come from axler, and insert "inner product space" wherever he is silent on it.

Koro

noted. :)

leslie townes

"P_U" doesn't have unambiguous meaning without an inner product. that's another clue.

i'm not sure i've ever seen this result 'in use' (i.e. that something is shown to be an orthogonal projection this way) but it's a fun exercise.

it didn't seem right to me for a minute, but then i saw the light

copper.hat

7:31 PM

@Koro you are correct, but it is better to just be clear, no need to follow rudin :-)

leslie townes

i'd completely forgotten that axler (and indeed most/many intro linear algebra books, even proof oriented ones) didn't do general normed spaces. that's how far down the rabbit hole i've gone.

probably because outside of the orthogonal setting you tend to run immediately into stuff that is only interesting with a bit of analysis

user178758

are you guys gonna participate in winter bash?

leslie townes

where my wife and i drive around the neighborhood knocking over people's mailboxes with a baseball bat? yes. how did you know about that?

user178758

7:48 PM

🧢👑⛑️☃️

...lucky guess

leslie townes

tis the season!

user178758