Is there a name for a sequence of functions connected by successive inverse integral transforms? For example a famous "Mellin pair" is $\Gamma(z)$ and $e^{-z}$ but if you again take the inverse Mellin transform of $e^{-z}$ you get a Dirac delta function, so you get a sequence of 3 functions connected by inverse mellin transforms: $\Gamma(z) \to e^{-z} \to \delta(z)$

What are some other important "Mellin sequences?" What is the longest mellin sequence?

Good morning! It's said that when determining the number of significant figures in addition and subtraction of numbers, the answer is rounded to the least number of decimal places in the original numbers added.

But how to determine the significant figures in something like 380000+4.25?

there is a lot of ambiguity around integers with lots of zeros before the decimal point. i've seen at least one book where they would write "38000." to mean 38000 to five significant figures. i forget how they would denote 38000 to three or four figures.

i've missed one 0 from your example but you get the point.

as far as i can tell, these are more or less arbitrary rules invented for testing students, not fully investigated by the people who invented them, and this does not come up in 'real life'. my instinct would be to interpret 380000 as having two significant digits and the sum would just again be 380000. maybe someone who has taught this material for a living can weigh in with a second opinion.

Thanks a lot @leslietownes then we are looking at the number of significant figures isn't it? Not the number of decimal places?

Because the rules says we have to look at the number of decimal places when adding/subtracting and have the answer in the least number of decimal places... This is what caused me a problem when doing the above sum

yeah, as i indicate above i am not sure. there's no standard way of specifying significant figures for integers with lots of 0s in them, so i'd default to 2 for "380000" but that might not be orthodoxy.

i do remember spending way, way more time on this in high school than i ever spent measuring anything.

Hmmm okay :)

But why we look for significant figures instead of decimal places in this type only is bit unclear to me...

well, you're the one who brought up significant figures :) "we" don't look for them. i'd ask whoever asked you if they had something in mind, or were just testing an edge case of a weird set of rules.

i've seen engineering situations that involved measurements over several scales of magnitude (roughly, dozens of inches and also microns) but the way people calculated with numbers was largely determined by industry needs, and not an invented set of abstract rules about what the measurements looked like.

i may be going too hard on "significant figures" here. i don't hear anyone chiming in to defend them. :D

Okay

Thank you very much @leslietownes :)

Have a nice day!

you too.

This is a science/engineering measurement issue, not a mathematical one. If we can measure 4.25 with precision and expect to add it to 380000, then either that should be 380000.00 or, if not, the 4.25 should be 4 or 0. It makes no sense to add something with error $\pm 100$, say, to 4.25.

@leslie I would assume exponential notation distinguishes between $3.8\times 10^3$ and $3800$ in terms of expected accuracy, if one’s playing that game.

yes. although i remember being asked about stuff like 380000 in high school, just to test if i had learned whatever rules. kind of like, how many fingers am i holding up, and the thumb is or is not a 'finger' depending on how you answer. that's how it felt to me.

the "real life" docs that i see often include explicit \pm error bounds also. i do not remember ever having to use these rules to figure out what i was looking at in 'real life,' which suggests to me that maybe they shouldn't be teaching them.

They were taught for the chem and physics support.

even if it can be made coherent when treated carefully, nobody seems to be doing that, or relying on people to remember it. it's like relying on someone to remember whether 0 is a natural number or not. you're gonna hafta tell them.

i could see teaching it in a class where people actually measure things in a way that is sensitive to whatever they are measuring. it seems like something that should not be pushed into the math department (or as it was in my high school, math class)

i'm just ranting. bad mood today.

The country makes me in a semi-permanent bad mood.

Did Munchkin torture Olivia again?

upon getting home, she picked her up and marched her around the house.

so, yes, i guess.

Olivia needs Screech’s sharp claws.

Oh, great. Lily Tomlin on tonight’s Carol Burnett rerun.

Seems another person deleted his/her post (after my comment). I was looking to check up on a response and can't find the posting any more. It was a not a bad question, although there may be multiple posts on this sort of thing. If $g\colon\partial\Bbb D\to\Bbb C$ is continuous, is the function given by integrating against the Cauchy kernel continuous up to the boundary? My pseudo-hint was: What happens if you start with $g(z)=\bar z = 1/z$?

@leslietownes Is there a set with an odd maximum which is not constructable?

@D.C.theIII Matrix of TS = matrix of T $\times$ matrix of S

where T,S are in L(V) where V is n dimensional vector space.

@Koro right.....

$T$ is normal means by definition that $TT^*=T^*T$.

Now, I take matrices both sides w.r.t. basis $\beta$

sounds good to me. and then use the fact koro mentioned and the theorem.

I think that you want to show that matrix of $T^*$ w.r.t. ONB $\beta$ is complex transpose of T.

that's his theorem 6.10, i think.

I'm glad that it sounds good to leslie :-)

it's a little tricky in that * means different things on both sides of the equation in that theorem, but we discussed that the other day.

Maybe I'm over complicating it...but I'm thinking that I need to show the entry of both products of matrices are equal

no, you don't need elementwise computation. you might be repeating some of the proof of 6.10 if you do that.

[T] [T]* = [T] [T*] = [TT*] = [T* T] = [T* ] [T] = [T]* [T].

6.10 on the outer equalities, koro's fact on the next outer equalities, normality of T in the middle.

hmmm...I definitely over complicated it...

6.10 is not trivial. we were thinking about this the other night.

but, if you mentally delete everything about the proof of 6.10 and just use the result, that's enough for this exercise.

the result of 6.10 isn't trivial and I think I'm going to go back and look at the proof because of our discussion of the meaning of * on both sides of it. The proof is some straightforward symbol pushing that you might appreciate:

a whole lot of operator theory involves finding non-elementwise ways of looking at stuff. it's not a natural way to look at it. a lot of fields that use matrices but do not consult with applied mathematicians do everything elementwise, and confuse themselves, and fill pages of journal articles with long formulas.

but yes, I was trying to use the ideas from this proof to show the above result

i do like the straightforward symbol pushing. it puts me at ease.

Something is seriously wrong with my computer, a lot of user icons are broken, pictures in messages don't show up =(

I did too, but I've been taken by the dark side....now I want meaning

well, the meaning is there too. it's just what matrix entries are.

no more, no less.

@leslietownes isn't part of the hustle to make yourself look "smart" by pumping out a long complex formula so the regular folk will be intimidated?

The key thing to note with the theorem is that if the basis is orthonormal, then the inner product satisfies $\langle \sum_i x_i b_i , \sum_j y_j b_j \rangle = \sum_k \overline{x_k} y_k$ (unless you are an antilinear pervert).

in applied fields? no, i don't think so. i think more often they write the long formulas because they don't trust anything that isn't a long formula. so if they have some way of grasping it, it becomes THE way to grasp it. it would be suspicious if it were a short formula.

^^ this has encapsulated my way of thinking....I'm not sure how to reconcile it....

self imposed masochism

maybe some of them optimize for formula length to intimidate, but most folks i think just don't feel right if a 'big' result doesn't have a 'big' list of equations behind it.

leslie: any possible reason why normal operators are called normal operators?

while saying normal -'some dot product' is zero comes to mind.

koro: i don't know, except 'normal' is one of those words that is basically free to use. name anything and call it 'normal.' it's the most used word of all time.

its normal to use normal as many places as possible.

seems like the normal thing to do in normal circumstances...

I see so this is normally due to historical reasons.

What's good for the gander is good for the Gauss

@copper.hat alan sokal quote?

@PrithuBiswas Nothing so erudite I am afraid, it represents the depths I need to reach to continue the normal punning.

But I liked the Sokal hoax.

It's wine O'Clock.

i just accepted a full time position this morning. it feels like i have entered prison again.

@copper.hat Me too =) I recently watched a video which had the sokal hoax and the bogdanoff affair.

@copper.hat hope those joints of yours are well and oiled up to run on the hamster wheel again

@D.C.theIII I am afraid my joints are in need of replacement

The Bogdanov bros. are the academic Kardashians

If, indeed, they were actually real.

@copper.hat XD

for those who remember my little saga, my daughter was/is able to sit her exams, so while not optimal, it is better than not being able to take them

so, fingers crossed that her results represent her appropriately

did her being sick prevent her form studying?

well, the way it works is that all her years of study are examined at the end, and a few days before her exams she tested positive for covid

normally (again) she has to dress formally to take the exams (subfusc)

this is the end of her undergraduate and a lot is riding on it

job, etc

given the number of times she was exposed to covid positive over the years and walked away unscathed, it seemed awfully unlucky for her to get it when she did

@D.C.theIII yes,she was basically asleep for 2 of the few days before the first exam.

not sure what the purpose of having to dress formally for exams achieves. I could imagine being in the UK. I went in Decemeber at it was a mad house and cavalier with regards to Covid

@copper.hat but them being so cumulative she was most likely already prepared.

There was no dress code at my college for exams.

@D.C.theIII she is in the uk. she is smart, hard working & diligent, so hopefully it will work out

she certainly did not inherit at least 2 of those traits from me

Just dress in a way that you don't look too different from ID card photo.

oh yes I'm aware. you mentioned it and subfusc as well implied it to me

surprisingly as an irish citizen (she got that from me :-)) she can use the nhs

and vote in some of their elections

The Bogdanov brothers have passed away

Together.

who are even bogdanov

how did the pass away together seems so death dates are 1 yr apart

COVID

Given an n dimensional vector space V and its two bases $\beta=\{a_1,...,a_n\}$ and $\beta'=\{a_1',a_2',...,a_n'\}$, suppose that $v\in V$, then there exist a matrix P such that $[v]_\beta=P[v]_{\beta'}$

$[v]_\beta$ denotes coordinate column matrix of v w.r.t. basis $\beta$.

From this information, I want to show that P is invertible.

Noting that $[v]_\beta=0\iff v=0$, the following holds: $P[v]_{\beta'}=0\iff v=0\iff [v]_{\beta'}=0$

So I think that it can't be deduced from here that P is invertible.

For, the above shows $Pw=0 \iff w=0$ for only a certain type of w's and not for all w.

nope, it proves for all types of w. That is, if $Pw=0$ for any w in $F^{n\times 1}$ then w=0

So P is indeed invertible.

@Koro I think that $P$ should be introduced before $v$

Can't really follow your reasoning

@CalvinKhor I should have added more context. P is the base changing matrix. Its construction is as follows: There exist scalars $P_{ij}$ such that $a_j'=\sum_{i=1}^n P_{ij} a_i'$.

Hi

Hi

I bought some noodles for dinner

with extra noodle

and realized that I cannot eat all of it

@Koro yes, but if you say, for any v, there exists P…. Then this matrix could be a different matrix and not invertible

Now, the matrix P used earlier is basically $[P_1, P_2,..., P_n]$, where $P_j$'s are column matrices with entry $P_{ij}$ in ith row.

@CalvinKhor yes.

With complete context in place now, is my understanding correct?

(in concluding that P is invertible.)

Since $\beta$ is a basis, you can write $\beta'=Q\beta$.

What is the relation of $P$ to $Q$?

@robjohn the symbols are not clear.

$\beta$ is a set (ordered).

I am using the fact that: For a square matrix A, $Ax=0\iff x=0$ is equivalent to A being invertible.

I am loosing my mind over a simple calculus post.

So I resorted to my final weapon

If this doesn't work I will forget about the problem and move on with my life to save my sanity.

@Koro sorry for late reply, I think the main ideas are all there. I'd write it like this: you want to show that $P h = 0$ implies $h=0$ by the result you already know that this is equivalent to invertibility. So suppose $Ph=0$. Then $h=[v]_{\beta'}$ for $v = \sum_i h_i a'_i$. The given property of $P$ implies that $[v]_\beta = 0$, which implies $v=0$, and hence $h=[v]_{\beta'}=0$, as needed.

@CalvinKhor yes, I’d understood that earlier when I said ‘nope,…”. h is a vector in F^n where F is the field over which V is a vector space :-).

Thanks a lot for the review :-).

Torus is a correct generalization of circle S^1. If we define $S^1$ as $\Bbb R/\Bbb Z$, then $\Bbb R^n/\Bbb Z^n=T^n$ looks correct generalization of $S^1$. Also, $S^1$ is simply connected but $S^n$ is not simply connected for $n\geq 2$ whereas $T^n$ is also not simply connected. Even more, $S^1, T^n$ is a group but $S^n$ is not a group for $n\geq 2$.

Is there any analogue of torus version of higher homotopy group?

Maybe maximal torus in compact lie group theory?

So formal linear sums of subvarieties correspond tou subvarieties again and if yes then for example what would be a[c]+b[k] the sum of 2 points in a variety with some coefficintes . This sum is in the Group of cycles.

@Koro If $\beta$ and $\beta'$ are both bases, then you should be able to write each vector of $\beta$ as a linear combination of of the vectors of the other. That is, $\beta'=\beta P$ and $\beta=\beta'Q$, for some matrices $P$ and $Q$.

Since $v=\beta\,[v]_\beta=\beta'\,[v]_{\beta'}=\beta P[v]_{\beta'}$, we have $\beta\left([v]_\beta-P[v]_{\beta'}\right)=0$. Since the vectors of $\beta$ are linearly independent, we have $[v]_\beta=P[v]_{\beta'}$. Similarly, $[v]_{\beta'}=Q[v]_\beta$.

From either equation, we get $PQ=QP=I$

@Koro That means that the columns of $A$ are linearly independent.

@onepotatotwopotato I think you swapped $S^1$ and $S^n$ in saying which is simply connected and which is not

Also why tori instead of solenoids? Solenoids are also groups with trivial fundamental group and in some sense they are closer to $S^1$ than a torus is

I think we discussed this briefly earlier, but I ask again. What tips or suggestions do you have for organizing / building your mathematical "filing cabinet"? I feel I'm in this place that I got theorems scattered all over my mind and though they do have some semblance of structure, it still feels disorganized. For instance if I wanted to recall a certain theorem or idea to be applied to a question the theorems to use are papers scattered all over my mental desk...

Let $X$ and $Y$ be matrices. If $e^Xe^Y=e^Ye^X$, must $XY=YX$?

I know the converse is true

Ah - no

Various matrices satisfy $e^X=I$, $~X$ not diagonal (and therefore something doesn't commute with it)

i think you could get something like that if you assumed the semigroups generated by X and Y commuted. at least for matrices X and Y, with unbounded operators you might have trouble with the domains

@geocalc33

@AbstractSpaceCrack

that's right son

🌞😎

Hey, @geocalc33 want to see the C++ app I'm working on?

A screenshot maybe

hi @Alessandro I have set theory question

@AbstractSpaceCrack sure

do you know how strong the statement "two bases (assuming they exist) of a vector space always are in bijection" is over ZF?

The running app is just the window lower right

lukas there was a twitter thread on this a few months ago. somewhere. or at least related to it.

@leslietownes do you remember the conclusion?

the code you see with "ticketNumber". Because the rendering takes a second or two and the user is pretty fast, each item I place has to "take a ticket number" for the current source text about to be rendered, then later receives a signal from the renderer when the image is ready

I have a feeling that the ultrafilter lemma is strong enough to prove this, so you don't need full choice

I call it the "help desk" algorithm

lukas: no. googling my history i found that BPIT is enough, which i think is short of full choice. mathoverflow.net/questions/93242/…

I don't know why KaTeX runs faster in usual browser than when embedded into a QWebEngineView using Qt framework

but that's not the thread. twitter is so hard to search, ugh.

@leslie thanks. BPIT is in fact equivalent to the ultrafilter lemma, so my guess was correct

@geocalc33 it will be a visual, assistive diagram chaser, the first really in the world that does that, purely a desktop app because I'm not great at mobile yet

@AbstractSpaceCrack ahctog s'taht looc

So you may have picked up, I'm using KaTeX instead of MathJax for rendering (MSE uses mathjax). I used to think KaTeX had some sort of speedup over MathJax

Anyway, these are browser / Javascript techs so that editor you see to the right with the rendered image, is actually a mini embedded Qt web browser displaying a simple page that imports KaTeX css & javascript, and calls KaTeX to render the page's contents. I then simply screenshot the widget after zooming and that will determine the best resolution of an object or arrow label that you can display on the grid to the left

I hacked it together from a markdown editor example in the Qt Creator examples collection

So my old prototype code was done in Python, but after some scene complexity builds, it's no longer okay for the user to edit with. This C++ approach typically runs 100-1000x faster depending on the situation

So for instance in C++ I was able to crop the screenshot image by doing a brute force search through all pixels searching for non-white pixels. You have to make a call to C++ side somehow (there are ways). Anyway, the whole app generally needed a speedup, and a rewrite for technical design reasons.

so basically you're programming?

*ie. you have to call C++ side from Python if you want to speed up just computationally intensive chunks of code

I am programming bro

Qt Creator is a programmer's IDE with lots of tools for working with their forms, QML, etc tools and also editing the C++ code itself.. The debugger is decent enough. Nothing is as good as Visual Studio, but I'm working with Qt since I worked with PyQt5 so mostly know it

nice

I can show you how I put a feature in, if you want over Zoom. It's not easy but also not as hard as you think

Takes time

Firstly, you think, oh I could easily do this. Then it's googling how you're going to do that and finding best boilerplate outthere that already works

Then this feature is going to end up interacting in buggy ways with all your other related features

So you have to debug each place, and in the appropriate place in code per your design skills

Then it's not going to work the way you want it, even though it's bug-free. So you have to rewrite it some, and repeat the above. Making use of breakpoints when debugging and checking values in the IDE.

@LukasHeger mathoverflow.net/questions/93242/… some discussion here

Follows from ultrafilter lemma/BPI, seems unclear what the sttength is

maybe worth noting that if you assume BPI, then you also prove this for free modules over all commutative rings

since BPI implies the existence of a prime ideal and you can tensor with the fraction field of the quotient ring

Nice, you said prime while you smoke ultrafilters

Roll up, smoke adjoint

🚭

@geocalc33 our AC is broken and my PC doesn't like > 80*F

For my app icon I used the unicode milkyway galaxy emoji

🌌

I don't think that's copyright violation

lol

@AbstractSpaceCrack yikes. Where are you? It is 91°F outside right now, I haven't turned on the AC yet, and it is only just now starting to get uncomfortably warm inside. I'll probably turn the AC on in an hour or two (around 3 pm).

Oh, in Sedona, so it's only 89*F here outside but the cave gives a little bit of drop so it's about 86*F maximally today, but of course it's gonna get hotter.

Oh, right. We are neighbors.

Yes, my algebro

Do you like diagram chase proofs @XanderHenderson?

@AbstractSpaceCrack No.

I'm an analyst precisely so that I don't have to chase diagrams.

I've heard that homological algebra can be applied to analysis

That sux, and my tool will be aware of that. It will teach you a more pleasant way to do it. Well even analysis has been categoretized in some papers

@LukasHeger I'm sure it can, but that isn't the kind of math I want to do.

Peter Scholze has something called Liquid Modules

I love CD's!

Peter Scholze totting up dem numbas

He's smaht

@geocalc33 I plan to develop this in 1-2 years before ready for sharing and I want to sell licenses to Universities. Maybe $20/(seat * year) for a student, which will include feature & bug updates. If it can help you learn how to navigate categorically, then it will be a valuable tool to have as a researcher or student. More so the student at first, because as you start out your code won't even touch the more advanced proofs just because it needs more things to be coded to do that.

It will probably understand /some/ theorem encodings but not everything in math. For exmaple, I won't be dealing with polynomial expressed elements at first

Probably just module elements.

Also it's not a full-fledged proof engine. Meaning somethings are derived on paper, and then hard-coded without formal proof. The goal is to generate human-readable and elegant-looking visual + textual proofs.

That's one of the things it can do. For example given a CD, I can let you delete any arrow you want and the thing still commutes. I don't require that the system prove that to you

But the proof step will be listed in a Proof Step dock widget, you can go back and forth. It will even jump between relevant tabs for you as you step through. All "proof steps" such as move the object visually (-1,0) in the grid will be hidden in the proof, because they have no mathematical importance.

So the proof ties in directly to the Undo/Redo stack (works just like MsPaint or Notepad undo / redo - ie. the command pattern).

it's 72 degrees here

66 and breezy.

Yeah, here I can fry an egg on my PC case

76° here

is that the same as 436°? would we care?

relying on those things to be the same appears to be our climate change plan

The nation is taking a total 360° change on climate controls.

very tempted to found the "S1 Institute" and sell the opinion that the climate can rise by 360 degrees without any effect on mankind

You mean $2\pi$ radians, of course.

you can also smoke $2\pi - \epsilon$ cigarettes and get healthier

I have never smoked nothing.

Nor have I

we do light incense in the house, which might be like second hand something.

@TedShifrin but imagine: smoking -1 cigarettes would make you healthier...

I’m trying to understand the physics/biochemistry of unsmoking.

just apply a time reflection to the physics/biochemistry of smoking

I am having issues.

Uneating doesn’t seem to work like that.

pamphlets that explain everything are available from the S1 Institute for a nominal fee

and, of course, shipping and handling

Ask Munchkin to deliver mine …

@TedShifrin Oh, you want to understand? Don't worry about that ;-)

A probability question I liked was removed by its OP because there were questions about no effort (including my query where his/her analysis had gotten). But I love it because it shows that no matter what question you ask about the card I've picked at random — is it the queen of spades? or is it a black card? or is it a 5 of some suit? — you have the same probability of guessing the card after hearing the answer. Always double the probability with no information. I love it.

as long as you have a link to it, you can see it. You could ask a self-answered question about it, too.

the S1 Institute sells a pamphlet about this called Double Your Probability, Guaranteed! The Easy Way

@robjohn I can still see it, but no one can write on a deleted question, right?

right

10 Secrets About Probability Doubling The Government Doesn't Want You To Know (#4 Will Blow Your Mind)

Leslie is off to make lesliecoin profit from every tidbit I know.

US Income Tax Is A Myth, published by Leslie Pioneer Publications on the offshore oil rig of LeslieLand

@Ted I got another downvote today. Although no comment was made explaining what was wrong, I expanded the answer a bit to possibly make things better. Not that I expect them to remove the downvote since I have no idea why it was downvoted.

i did that because it made me feel big.

@leslietownes I hope it helps. I'd hate to think it didn't.

I still get the occasional one, @robjohn. You might find this an interesting question/post. One of our usual geometers has answered it.

Someone I've never encountered has summoned me to a private chat. Why?

catfishing scam?

On MSE?

maybe the NSA is recruiting in really weird ways now, i don't know

Hmm, I'm trying to back up my essential files to a zip drive and it won't let me copy the disk image of Mathematica (about 5 GB). "Too large." Something must be fishy.

well, "zip drive." goodness. they may not have anticipated the idea of 5 gb file transfers.

It's a 128GB zip.

I'm starting to get signs that there may be corruption in my hard drive — probably a Leslie spy.

maybe an old version of windows goofing up on large file transfers.

Windows? Whom do you think you're talking to?

our office has a great thing, it will let clients create folders in a shared file space that have filenames that are too long to make them accessible. or to make them rename-able.

it's some stupid thing in windows 10.

Oh yeah, you complained about that to me before.

i forgot you were a mac guy. i had trouble moving some of my stuff from windowses xp and 7 to 10 and it did choke on very large files. but i was able to split things up and move them over.

@TedShifrin It was me: I was just reading your "Multivariable Mathematics" book and I think in Exercise 8.2.11 (b) the answer should be $18u^2 dv$ instead of $18dv$ as written in the back of the book. Hope this helps.

LOL ... You don't need to summon me to a private chat, @Lorenzo. You're welcome to ask in here. There is no $u$ in that problem.

I think you are looking at (c).

@TedShifrin Sorry, this is my first time using the chat and I wasn't sure if I could ask here. Ah, I see I had copied the wrong exercise. Sorry to have bothered you for nothing then.

There is also an errata sheet on my webpage. A lot was corrected in the second printing, but I don't know which version you're looking at.

@TedShifrin Yes I had already looked at it and since I couldn't see what I thought was a typo it occurred to me I could notify you here. I am pretty sure I have the second printing.

Well, happy learning!

@TedShifrin thanks, I am finding your book and lectures on YouTube very useful. Bye!

Glad you're enjoying it!