00:00 - 17:0017:00 - 21:00

12:19 AM
@Asinomás AOPS is books/classes for kids from a young age to high school). It has a competition bias because the guys who started it were in love with competitions — all levels, not really olympiad.

1:17 AM
@Aplateofmomos once in a while i check in on math.codidact.com . At the moment it looks like a Math.SE clone with some people banned from Math.SE. Perhaps you can suggest points 2 and 3 to them so that they can differentiate from math.se. (points 4 and 5 sound like we need to wait for the tech to appear or make it yourself)

People banned! I’m shocked! Are they wearing MAGA hats?

Truth Social has an active puzzling community

The puzzle is: Does the sum of all their IQs exceed 1000?

if it did, i'm sure they'd be the first to tell you
[x] coyote visit checks box for today

The coyote subscribes to the greatest integer function?

1:27 AM
yes

1:37 AM
Who carried whom up the stairs today?

she's getting way too old for that

The cat or the punk?

both of 'em
when my wife works from home, livvy spends her days stretched out like a log behind her laptop. and sometimes visits zoom calls.

Screech likes to sleep on my desk when I’m on the computer and interfere in whatever possible.

1:53 AM
livvy used to perch on my desktop tower and loom over my computer, but she doesn't do that much anymore. she does suddenly leap into my lap, or vault up on the ledge behind me, when i'm on zoom.

Clever amunals.

her favorite is to vault up to the ledge, make her entrance, then climb onto my shoulders and then down into my lap, while people praise her on zoom. they shouldn't do that, but they do.

If I bend over at the sink, I soon find Screech perched on the upper part of my back.

Hello folks. Sweltering here in the Bay Area 20C.

2:16 AM
oof.

i'm sensitive to the heat :-)

i remember that about living there. 'normal' temps felt oppressive.
i used no heat or ac from 1998 to 2005. my wife made us use the apartment's heater when we moved in together.

yep, we're getting into the Bay Area 'hot' season.
similar spousal situation here.

we're in the sweet spot right now down here. no heat or AC for about two months now.

excepting holidays, i have never lived in a place with ac

2:26 AM
my first AC was in iowa. and the idea of people living without it in iowa is just baffling.

in ireland now, a lot of newer houses have ac, mainly because energy requirements reduce natural airflow.

same with the pilgrims settling near boston. imagine arriving there and thinking "yeah, let's build a city right here. should be fine in the summer. at least we aren't being religiously persecuted by the humidity."

i appreciate the idea of saving energy, but somehow i like the natural air more

i never use AC in my car, i roll the windows down. nobody does this anymore.

:-). i did appreciated ac in the car with small kids

2:28 AM
oh, good point. i use it when my daughter yells at me about having the windows open.
i never use it by myself, i should say.

i rented a convertible mustang when in one of the Hawaiian islands once. big mistake.
many mistakes. why go to an island where everything costs twice as much, the entire bay area vacations there and local culture has all but been eliminated?

that's funny. i get why people would want to vacation in hawaii. i don't get why someone who already lives in california would want to vacation in hawaii.

but the sunlight was too much so the roof came up but then the kids could not see out properly and they decide to go in the cousin's minivan instead

at least you had 15 minutes of magnum PI experience. or whatever you were hoping for with the convertible.

i would prefer mexico, cheaper, more fun, real people & the food...
i loved magnum pi. even the latest incarnation.

2:32 AM
i've never been to hawaii.

it helped that the caretaker had the same last name

did you watch the reboot of hawaii five-o?

yes, i am a fan of whatshername park
grace park. very cute

grace park is indeed a talented actress.

oh? she is?
battlestar galactica
i used to have a picture of jack lord beside my bed growing up

2:34 AM
you can probably pick up his original art for cheap. he was a painter when he wasn't hawaii five-o-ing.
not a good one, in my opinion, which is why you can probably buy it cheap.

i thought hawaii 5-0 was a telephone number, our phone number at one point was piltown 19
everything about hawai-5-0 was wonderful to me back then
like star trek
my mom saved the jack lord picture. i have it somewhere.

ha, this is funny. one of the theories of the show's origins is that the producer wanted to set a show in san pedro and was talked out of it.
that would have been a different show.
cheaper to film, though.

at the risk of being a non-pc idiot, both shows were completely exotic to me.

they definitely played that up on five-o.
the footage that accompanies the theme song is like, every hawaiian tourist stereotype at once, plus jack lord.

well, coming from homogen-o-town, it was basically porn
well, to be fair, we had an influx of french & spanish girls during the summer. but they were even less accessible than the local girls.
basically anything with a tan was exotic.
i'm going for a psq. its been a day since a downvote

3:04 AM
when i was living in oakland i once had a cousin visit from nashville. he said "i like how there's girls with all kinds of hair colors here." i thought he meant that he'd seen someone with purple or neon green hair or something.
he meant "not bleached blonde"
i said, yes, if that is your metric of diversity, the bay area does supply that.

at this point i'm happy if people don't back away slowly when i start talking to them :-)
i was a bit dismayed at how the kids at the local hs seemed to clump by presumed ethnicity.

where i lived in oakland i think you would have struggled to find a bleached blonde. you do see them around here. in OC it is something of a standard look.
that's interesting. at my hs it sorted mostly by income level, which had a component of ethnicity.

3:30 AM
i suspect the same in albany.

which is kind of weird, because i don't remember being acutely aware of family wealth in high school. it wasn't that stratified by income, either. but that's how people sorted.
my best friend from hs was a rich kid and she only hung out with the other rich kids on the weekends and we both thought this was normal.

i suppose normal when one is younger encompasses much more
although, to be fair, my standards for normal are fairly low (no open carry, washes hands after bathroom).

when i went to law school there was a group of kids like this. people who knew each other maybe as far back as high school, only hanging out with each other and other very wealthy people.
it was weird to see something i sort of recognized from high school, but 'for real.' like, the orthodontist's son at my school didn't have money like these kids did.
just a concealed carry man. i see.

my standards have dropped :-)
i added a comment to math.stackexchange.com/q/4457354/27978 of the form "This is next up for the Supreme Court to consider." but thought it might be a bit inflammatory. i have become very pc.

hah. i personally find that funny but a lot of people are understandably on short fuses these days about that stuff. it might be less funny 5 years from now.

3:40 AM
no irish need apply soon

they might be mulling over the basic right to exist for various groups of people pretty shortly.

a bit scary how the court is leaning. i thought that would be a bastion of principle.
even among the right leaning folks.

some of the stuff lately is extreme even by right wing standards. i am no fan of justice scalia but he was far more cautious about letting judges upend things than the present batch seems to be.

trump is an idiot, but he certainly knew how to leave his mark

the recent decision by a lower court that attempts to deprive the SEC of any enforcement power and the administrative state of any reason to exist. scalia might have liked that result, but he hated the idea that judges knew better than congress what it had done. he wrote articles about it when he was an academic.
which isn't to say he held to those views once he was on the court.

3:47 AM
more anarchy than libertarian

justice thomas is pretty sharp on cases with no obvious political valence (e.g. IP cases we cite him all the time) but has a lot of blind spots elsewhere.
gorsuch and roberts are sometimes not insane, although i think they would be fine with a world that is insane.
i'm in the middle of a series of books about life in germany during the 1930s and what happened after you know who came to power. the whole legal system continued. it hollowed out, and became less relevant, but it was there the whole time.

it certainly has that air about it
thankfully my kids have dual citizenship.
need to talk to them about offshoring.

i don't mean to be histrionic again, but, it's not the first time a world power has had an extremist minority party that acted like opposing viewpoints don't exist, over and over again, over parliamentary and procedural objections from people who should have known better and seen the train that was coming to run them over.
i wrote in a cat for the sheriff of LA county. that's where i am these days.

i will mostly be leaving the choices blank.
i am loath to vote for noise some, but need to check the alternatives

LA local government is mostly dysfunctional. lots of dynasties and people with debts to pay to unknown forces. tons of little fiefdoms.
my strategy for a lot of it was to use the extremist mailers i got as a guide to vote

3:52 AM
ever see the movie chinatown?
jack nicholson

yes! it's a documentary.

in many (scary) ways

james hong, who played the butler in chinatown, is in "everything everywhere all at once" which just came out. he is in his 90s now. i haven't seen it but hear great things.

i just watch netflix at the moment before dropping off.

he was also in a very funny key and peele sketch.
some character actors become immortal.

4:00 AM
the drivel i'm watching atm is lincoln lawyer. pure kitsch rubbish

i think training day was the last best LA-based crime movie

i'll add it to my list :-)

oh, you're in for a treat

Vapid claims to respect precedent.

4:06 AM
ted: well. there's precedent (smiles, nods head) and there's precedent (frowns, shakes head 'no')
i hope that clears it up.

Perspicaciously.

then there's past precedent
and that is uuugh

You mean as opposed to future precedent?

i'm afraid the past precedent and future precedent might end up being the same thing

4:22 AM
Given a square matrix $A$ of order n by n such that $A^t A=I_n$, if $[1]_{n\times 1}$ is an eigenvector of A, what can be said about the corresponding eigenvalue?
Noting that A is isometry, it is clear that all eigenvalues of A must be of mod 1.
Can anything more specific be said about the eigenvalue?

$Q^T A^T A Q = I$ as well, so knowing an eigenvector doesn't tell you anything about the corresponding eigenvalue.

as copper said, nothing in general. you sometimes see [1]_{nx1} singled out when a matrix has nonnegative entries representing transition probabilities and such.

Sum of all entries in each row is the same.
@leslietownes hmm

hahaha
i foretold the future
good to know but not part of the toolkit of many people.
en.wikipedia.org/wiki/Stochastic_matrix more fodder for the wikipedia wormhole

i'll markov the list of things to do
sorry

4:36 AM

i'm blaming tiredness

@leslietownes I must, with regrets, concur.

seems to be a recurrent state

we live in a fallen world.

4:42 AM
@leslietownes nope :(

it's a good theorem.
i was refereeing a paper and someone announced the perron frobenius theorem as a corollary to something they had done. i said, uh, no, some guys figured that out already.
it's hazardous to publish outside of pure math, you get weird requests from people who genuinely don't know any better.
but, one less howler in the applied econ literature is a public service.

An alternative proof might be interesting.

agreed, but it wasn't even a proof. it was more of a notion.
i do like seeing proofs of the theorem.

leslie: I read Gershgorin theorem though. It sometimes helps finding largest eigenvalue.

it does, although sadly i think the awareness of that theorem has led to people cooking up problems to which it is applicable
i like the theorem though.anything relating matrix entries to other things is cool.

4:59 AM
Suppose A is an m ×n matrix, V an m ×1 matrix, with both A and V having rational
entries. If the equation AX = V has a solution in R
n, then the equation has a solution with rational entries.

I used Gershgorin once for something, but that was eons ago and I forget.

(asked in an entrance exam in 2010).

entrance to what?

leslie: I discussed a similar question the other day.
@TedShifrin to a Masters in Mathematics program in India.

hmm … not difficult at all.

5:01 AM
yeah, I realize that too :)

People who are all about theory and can’t calculate things may struggle.

If I were to write that exam now, I would have skipped complex analysis and Galois related questions.
:(

They’re both easy. Just write down a cubic with a pair of conjugate complex roots.
Integer coeffs.
the complex question is trivial once you remove its disguise.

similar in spirit to the berkeley prelim, except shotgun style with more questions and maybe fewer expectations as to the form of the answer.
i'm in a dark mood. has anything been funny recently? ted, have you seen any good shows or movies?

Just lots of things about Julia Child, one of my idols.

5:13 AM
oh yeah, there's that new thing out
hrm.
she did a lot for the representation of red headed tall people. i like her.

Was she a redhead? Never noticed that.
great teacher, too

maybe it was artificial. anyway, she is a positive role model for my daughter.

5:32 AM
Her cookbooks are superb.
math profs should teach so well in their books …

i like Julia Child. Her cookbooks were mostly beyond my capabilities.
She had an interesting history, her husband was in the diplomatic service.

Hmm, looks like they’ve removed the differential topology tag?

i don't think i've ever used a julia child book. i've worked out of james beard, marcella hazan, and deborah madison.

so was she … that’s how they met

I liked a few of Molly Katzen's.

5:36 AM
marcella hazan has you use butter like there's no tomorrow. i have to adjust all of her recipes for that. i want to live to see next week.

spice, salt, fat make anything taste good.

yes, we call it cheating.

Oh, my app is malfunctioning with all tags.

i suspect cheating is what most folks did to make palatable

that's the sign of a bad taqueria, they load up anything you order with cheese.
or (in the spice column) they have a weirdly extensive salsa bar. red flag.
they're cheating.

5:38 AM
i can live with it, albeit my LDL is a wee bit high
lack of exercise due to hip

@robjohn Any idea what’s going on with the iOS app?

roux, stock & cast iron pans were always a bit of a mystery to me.
the pans because of the beating i got when i washed it

i've never worked out non-vegetarian stock, but vegetarian stock is mandatory and a game changer.

yep, but simple vegetable stock is quick and painless.

OK, so is a can of campbells soup. boo to that.
i need some lemongrass in my stock

5:42 AM
it is like pesto sauce, its nice to make but a fuss to make and the tubs from colusa market are much the same

you do end up throwing out a lot of vegetable matter if you are serious about your stock.
it feels wasteful, even though it isn't.

Easy to make with food processor .

dang, i've done my teeth and this discussion is giving me the munchies

i just had a midnite helping of triscuits with hummus.

Midnight?

5:44 AM
close enough.

i had a tin of sardines in olive oil about an hour ago to stave off the hunger

any time after the toddler goes to bed feels like midnight

enjoy the time. it will pass.

Especially if you transport yourself eastward.

i still carry her down the stairs when she asks me to in the morning, because one day, she won't.
on the plus side, her conversation may be more interesting at that time.

5:47 AM
She is good at sass already. Just imagine!

i have a hankering for pickled herring. unfortunately the store bought stuff is not up to snuff.

oh, i can't wait.

I love pickled herring. But I can’t find it many places.

A Swiss friend used to make it for me, she was a good drinking companion as well. Unfortunately passed.

Aw.

5:54 AM
She was the grandmother of one of my son's friends. Combined exquisite manners with the ability to let loose a stream of invective that would make Brendan Behan proud.

sounds wonderful.
i love a person without a filter.

me too. seems to be a characteristic of many of my friends.

my mom was still working as a nurse when my now-wife met her. we were staying at her place, and my mom came home after a night shift at the ICU and said some very colorful things about the dead and dying. you couldn't print any of it.
i think my wife understood me better after that.
just the worst, worst, worst jokes about death.

i think i have become more sensitive with age, unfortunately

unrelated to above, ICU charges are exhorbitant.

5:58 AM
koro: yes.

try america

my wife and i were once billed over $1000 for an ambulance ride from one part of a parking lot to a different part of the same parking lot. this was after insurance. that is robbery did they tell you up front how much it would cost? surely that is a precondition of charging? no, and no. i hate medical stuff here. 6:00 AM the guy driving the ambulance doesn't know how much it will cost. nobody does. you find that out later. it doesn't make any sense!! 1000$

we paid it, too!
maybe i'll find out some theory to sue them on later.

i made an appointment to see the ortho chap today. the admin asks "what body part?", I answer "hip and knees". "we have a policy of one body part per visit, which is it?"
i hate when an admin is making medical decisions for me
when my daughter was born the bill was apparently $10k. i don't like how sometimes they force you into particular things while letting you take the bill. when we had our expensive 500 foot trip, they wouldn't let my wife walk. literally would not let her out of a stretcher. but the made such a shambles of it that they knew i would come hunting so they dropped it. 6:02 AM and then it's a charge that we incurred because we wanted ambulance services. stupid stuff. that seems wrong the great medical oligarchy our daughter being born was pretty cheap actually. maybe$1200.
$200 more than a drive across a parking lot. good value. i had to get rabies shots once. they would not give them to me because the insurance would not cover it. i gave them my cc. a month later i got the$1k back.

nice!

it had the same feeling when a cory hall admin said that i needed to take the toefl.

6:05 AM
hahaha.
you stupid potato man. take the test.

i'm surprised i didn't explode. i think i amped up my vocab for a few sentences
there was an irish admin working there, she just looked and shook her head.
i suspect she might have been my saviour

they're nonverbal people, the irish.

we have difficulty with words

with a lot of medical stuff i just give into it. my mom is on VA/medicare, and they have essentially no dental coverage and i just give in to these outrageous bills for normal dental stuff.

ugghm, pint, ugghgh

6:07 AM
FECK groaning ARSE
like father so-and-so

:-)
my vocab is shrinking
partly age. partly lack of practice
my best vocab comes out when i am criticising.

partly drinking toilet duck, like father jack

ever watch father ted?

yes. a million times.

i really think creativity comes out of some form of pain

6:09 AM
my mom has the whole set on dvd. her favorite episode is 'kicking bishop brennan up the arse'

the sad part is how close to reality some of it is

bishop brennan having a secret family somewhere
haha

brilliant
the funny part was that irish folk were not bothered by the fact that he had a child. but by the fact that he would not meet the child.

hahah of course

casey
the whole orphanage thing is a big deal atm. some time i despair of my people.

6:13 AM
the funds were merely resting in my account

this is where i favour open carry

oh that stuff is awful. it's disgusting that people are still alive who did that.

it feels like ancient history in a way and some people are just hanging out! like it was yesterday.
i don't get it.

it is so far from my understanding that it seems unreal

6:15 AM
i could live to be 500 and never understand how someone would enable that.

that is part of the problem and the reason for my despair. it is implicitly supported by many people who do not have the cojones to do the right thing.

this is why i tell my daughter at every opportunity that she was born into an empty and senseless cosmos and anyone who tells her otherwise is a suspicious charlatan.

:-). i sort of did the same for my daughter.
poor thing.
As Senator Moynihan said, To be Irish is to know, in the end, the world will break your heart
And on that happy note I will say Goodnight :-)

cheers.

Alien worlds is a good show :)

2 hours later…
7:58 AM
@CalvinKhor I found some error in my solution.
19 hours ago, by barista
$n(x,y,z) = {1\over 3}(x,y,z)$ and $\nabla u = \left({\partial u\over\partial x},{\partial u\over\partial y},{\partial u\over\partial z}\right)$. Since ${\partial u\over\partial x} = {\partial u\over\partial \rho}{\partial\rho\over\partial x} = u_\rho {x\over \rho}$, $\nabla u = \left(u_\rho{x\over\rho},u_\rho{y\over\rho},u_\rho{z\over\rho}\right)$. Hence, $\nabla u\cdot n = 0\iff {\rho u_\rho} =0\iff \left(-{c_1\over\rho^2}+{1\over 3}\rho\right){\rho} =0$. Since $\rho = 3$, $c_1 = 9$.
During the computation of ${\partial u\over\partial x}$, I only considered $\rho$ term but there are two extra $\phi$ and $\theta$ terms.

what do you mean?
you mean $u = (u_\rho,u_\phi,u_\theta)$?
@barista

Oh but $u$ only depends on $\rho$ so no problem. Sorry

oh. :)
yeah, uniqueness of solutions means you can assume it depends only on $\rho$
@Aplateofmomos your pic is literally a plate. Is it a plate of momos? what is momos?

8:21 AM
:( didnt even tell me to just google it
well...found it. not salty
Momo म:म: - also known as momo-cha ममचा are bite-size dumplings made with a spoonful of stuffing wrapped in dough. Momo are usually steamed, though they are sometimes fried or steam-fried. Meat or vegetables fillings becomes succulent as it produces an intensively flavored broth sealed inside the wrappers, originated in Tibet and made popular by Newar community of Kathmandu Valley. Variants of the dish developed later in Nepal after it became popular among Asians. == Names == Momo is the colloquial form of the Tibetan word "mog mog". The different names for the dumpling include Assamese: মম...

dayum
I'm back
It's like in India, you ask "one plate of momos"
and they give you an aluminium plate usually with 5-6 pieces
With momos, as with mathematics itself, there are many levels of sophistication that you can eat the same idea at
My picture looks more like a sophisticated plate
at a five star restaurant or smthn

hola
Momo so good to eat
It's called "Box of Chinese chives"(word by word translated) in my country.

8:38 AM
hmm
chineese?
o_o
I thought it orginated from tibet
@CalvinKhor regarding that coddict forum, I had went there and the design just killed my inetrest
it's like a worse version of MSE

8:51 AM
@Aplateofmomos yes, 100% agree. but IMO there is potentially one saving grace: see their physics site which has "questions", "meta", and another section, "problems". Or photography with a FAQ section. I.e. it is already technically possible to implement your points 2 and 3 there

9:01 AM
I haven't heard about that site before. Would that be a suitable addition to the post "Useful mathematical fora" on the main site?
I mean the Mathematics section (community).
And maybe, if somebody has positive experiences with that site, it could be added to this post on MathOverflow Meta: My question was closed on MO because it is not research level. Where should I ask instead?

@MartinSleziak you're the last person I expected to introduce a new site to :) I didn't suggest it because I was hoping for it to improve out of 'infancy' first. But maybe that's a self-fullfilling prophecy. One of the mods there is a MO user (Peter Taylor)

Why the last person? Because I'm so inept in math... so it doesn't make sense for me to look at math-related sites? :-)

no, because you're omnipresent and all-knowing :)

@linear_combinatori_probabi hello.
I hope you are doing well =)

9:08 AM
I have to admit that I used to visit several of the fora listed in "Useful mathematical fora" (And sometimes contributed there.) It's probably not a good thing that for a very long time I only visit Mathematics and nothing else.

I'm still essentially only on math.SE, but occasionally peeking at the others

If there was like a hierachy of needs
MSE has all the basic needs done perfectly, the middle to higher one is where there is lack
Codaddict maybe advanced in the area you said, but like the most basic thing before one does that is look at the site. If it looks like it is, then people will be turned off from using the site in totality

9:30 AM
@Aplateofmomos I call it...Momos' Hierachy of Needs

I like samosas.

9:42 AM
@Prithubiswasleftmse i suddenly want to eat some bhajis

10:20 AM
@CalvinKhor Can you check whether this question is fine?
0

Here are some theorems about Continuous Functions which were taken from Spivak Calculus: $(a)$ There is no continuous function $f$ on $\mathrm{R}$ which takes on every value exactly twice. $(b)$ There is no continuous function $f$ on $\mathrm{R}$ which takes on each value either $0$ times or $... I mean so far my intuition is telling be that it is fine, but it has been 7-8 months since I have asked anything, so I might be quite rusty. I'm currently trying to solve the following boundary value problem: $$\begin{cases} \Delta u = 0 & \Omega = \{(x,y)\in\Bbb R^2: 0<x^2+y^2<1, 0<{y\over x}<1,x>0\}\\ u(x,y) = 0 & \{(x,y)\in\overline{\Omega}:{y\over x} = 0\ \text{or}\ 1\}\\ \nabla u\cdot n = h(x,y) & \{(x,y)\in\overline{\Omega}:x^2+y^2=1\}\\ \end{cases}$$ where$h$is a smooth function on$\Bbb R^2$and$n$is an outward unit normal vector on$\{(x,y)\in\overline{\Omega}:x^2+y^2=1\}$.\\ We take a coordinate transformation into polar coordinate. Let$u(r,\theta) = P(r)Q(\theta)$. Then from$\Delta u =0$and solving eigenvalue p I don't know what to do next and I wonder if I did it correctly so far. 1 hour later… 11:38 AM @PrithuBiswas sorry, was away. question looks ok, but maybe try to put the question closer to the top of the page? (I would also prefer$\mathbb R$instead of$R$) 12:06 PM There's a silly mistake on the last part.$u(r,\theta) = \sum_{n=1}^\infty B_{4n}r^{4n} \sin 4n\theta$and$u_r(1,\theta) = \sum_{n=1}^\infty B_{4n}(4n\sin 4n\theta) = h(\theta)$. I wonder if$B_{4n} = {1\over 4n\pi}\int_0^{\pi/4} h(\theta)\cos 4n\theta\ d\theta$. No ignore the question it's completely wrong lol 12:39 PM @TedShifrin I don't use the iOS app. I assume it is not working? 1 hour later… 1:58 PM @barista it looked approximately right but i'm a little too busy to check details 2:19 PM if$F_1, F_2$are fields over$K$such that$F_1$and$F_2$are isomorphic as fields then$|F_1:K|=|F_2:K|$right? 1 hour later… 3:42 PM @TedShifrin: I found this. 4:06 PM @robjohn Wow. I had no idea. Well, the thing that malfunctioned last night is working fine now. But good to know. Ugh. It looks as if notifications are still going, but that is about it. 4:39 PM @leslietownes I see. 4:56 PM$U,V$are finite dimensional vector spaces.$W$is a given vector space. Let$T\in L(U,V), S\in L(V,W)$be linear transformations then it is to be shown that$\rm dimrange \,ST\le\min (dimrange\, T, dimrange \,S).$**Proof:** For any$\rm x\in range \,ST, \exists y\in U$such that$\rm STy=x\implies x=S(Ty)\in range \,S$. It follows that$\rm range\, ST\subset range\, S$, whence $$\rm dimrange \, ST\leq dimrange\ S.\tag 1$$ The following shows that$\rm dimrange \, ST\leq dimrange\, T$Let$\rm v\in range\, T $. There exists a$u\in U$such that$Tu=v$. It follows that$\rm STu= Sv\in r

00:00 - 17:0017:00 - 21:00