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12:02 AM
@Obliv that's why you get real books.
You can dog ear and/or bookmark pages.
12:22 AM
@Jakobian Maybe I'm forgetting, but this exercise I don't think we did discuss a proof strategy? This exercise is about showing that if $x_n \to x$ then for $r$ a rational function we have $r(x_n) \to r(x)$
@Thorgott I guess what I mean is that for a sequence $(x_n)$ wherein we wouldn't otherwise be able to define the corresponding sequence $(1/x_n)$, we instead define the latter via a subsequence of $(x_n}$ (options include deleting all $x_n = 0$ or deleting everything before and including the last $x_n = 0$
the relevant part here is that this is not just any subsequence (which is why I find that perspective to be slightly misleading)
perhaps you like this formal perspective: define a sequence in a set $X$ to be a function $f\colon S\rightarrow X$, where $S\subseteq\mathbb{Z}$ is some subset s.t. there is an integer $N$ s.t. $n\in S$ for all $n\ge N$. we also denote this as $(fx)_{x\in S}$. say two sequences $(x_n)_{n\in S}$ and $(y_n)_{n\in T}$ have the same tail end if there is an integer $N$ s.t. for all $n\ge N$, we have $n\in S,T$ and $x_n=y_n$. this defines an equivalence relation on sequences.
if $X$ is a metric (or topological) space, any statement about the convergence of a sequence only ever depends on its tail
12:43 AM
The non-equivalence of Borel measurability and Lebesgue measurability always perplexes me.
Thanks, Cantor.
1 hour later…
1:47 AM
@DannyuNDos I believe this is a typo
2:05 AM
You mean this shouldn't be perplexing at all?
There are Lebesgue measurable sets that are not Borel.
2:22 AM
I've got a good joke for the room
The clock struck 4:20 and I said to it "No duh! I was already high at the latest primorial" : )
2:10 for those who don't do primes as much as an addict
*Ideally* I get high at $0, 2:10, 4:20, 8:40$ ... get it!!!!

Only for work though. I can do math without it
That was a joke. didn't really happen
@Shaun lol I tell the worse jokes man. I put math in there
math is everwhere
2:45 AM
And this, children, is why we don't drink and derive.
OMG!!! i like that one
When is a Grothendieck group trivial?
What's an equivalent property?
I guess for $LCM(x,y) = x\cdot y$ you can't really invert
So the Grothendieck group is trivial or some such
But with $\text{LCM}(x,y)/\text{GCD}(x,y)$ you get a boolean group on the set of square-free numbers, it's weird
4 hours later…
6:43 AM
I am trying to edit a document in word, but having a problem
say I need to write my name where an underlined space is provided to write it. But when I try to write the name, the underline is getting erased. I want to keep that underline.
7:04 AM
nvm, found on yt
7:16 AM
@DannyuNDos then I'm not exactly sure what you mean by equivalence. All I've seen from skimping at the article is that they cite article by Oxtoby which was talking about null sets vs meager sets
1 hour later…
8:25 AM
I just meant equality, yeah...
that's a really confusing way to say equality
but I suppose that's fair
3 hours later…
11:14 AM
@DanielDonnelly Grown :P
11:26 AM
I have a silly question maybe. Recently I was looking Folland's text again. In chapter 1, he very nicely derives the Lebesgue-Stieltjes measure on the real line. Then, in chapter 2, he discusses the integral, but he never really says anywhere that we fix some Lebesgue-Stieltjes measure. He does say we fix some measure at the beginning of section 2.2, and at one point on p. 56, he says that if the measure is Lebesgue measure, we get the Lebesgue integral.
So is it reasonable to assume Folland presents the Lebesgue-Stieltjes integral though he never drops this name?
Reading different texts on measure theory, I'm trying to synthesize the different construction of measures and how the integral follows from the different constructions. It's complicated I think...
For example, not every text uses the Caratheodory construction of measures, which will at some point include Caratheodory's extension theorem. I have seen some texts that don't include the theorem. Some texts also don't talk about premeasures at all.
what is the $\max\{a_i b_i\}_{i=1}^n$? If $a_i,b_i>0$? Can I write $\max\{a_i b_i\}_{i=1}^n=$\max\{a_i\}_{i=1}^n$\max\{b_i\}_{i=1}^n$?
One inequality is straight forward $\max\{a_i b_i\}_{i=1}^n\leq $\max\{a_i\}_{i=1}^n$\max\{b_i\}_{i=1}^n$
Cohn's book is an example that doesn't include any premeasures (and neither Caratheodory's extension theorem, I believe). I don't know how Cohn chooses to construct measures. I don't really know what the alternative route is. I know there is a corollary to the $\pi-\lambda$ theorem that gives one a uniqueness tool for measures, which is then frequently used. This is something that isn't in Folland's text for example.
For other inequality, we have $a_i b_i\leq \max\{a_i b_i\}_{i=1}^n, \forall i.$ Hence it is true for $\max\{a_i\}_{i=1}^n \max\{b_i\}_{i=1}^n\leq \max\{a_i b_i\}_{i=1}^n$
Am I correct?
11:44 AM
@Unknownx No, you can't write that because it is false.
I was taking a look at this exercise but I have some doubts
@Unknownx Take n=4. $a_i=i$, $b_1=4/5, b_2=3/4, b_3=2/3, b_4=1/2$
Let the following linear application be given
$$f:\Bbb R^3 \to \Bbb R^2$$defined as $f(x,y) = (3x+2y-z,x-z)$. Given the bases $B = \{e_1,e_2,e_3\}$ and $B'=\{(2,2),(-1,2)\}$, determine the matrix associated with the application with respect to the bases $M ^{B'}_B$.
max(a_i b_i)= 2. max(a_i)=4, max(b_i)= 4/5.
Since the starting set is $\Bbb R^3$, shouldn't it be $f(x,y,z)$?
11:46 AM
@Pizza yes
if it were R^2 -> R^3, an expression is missing in the linear application
@Pizza yes
i have another doubt
$M ^{B'}_B$.
shouldnt the associated matrix goes from $B$ to $B'$, because the base $B$ has dimension $3$ and the base $B'$ has dimension 2 and therefore is the arrival one, so I should get a 2x3 matrix?
what do you mean with $M_B^{B'}$?
the associated matrix go from B' which should be the starting base to B which is the arrival base?
i think so
12:01 PM
you should make clear your notation
shouldn't it be from B to B'?
represents how the coordinates of a vector change when moving from one base to another.
it has to be written somewhere in your notes what is $M_{B}^{B'}$
12:29 PM
@SineoftheTime $M^{B'}_B$ is the matrix representing the linear map $f: V \rightarrow W$ with respect to the bases $B$ and $B'$. The columns of this matrix are the destination basis vectors $B'$ expressed relative to the starting basis $B$, and each column corresponds to the image of a basis vector under linear application.
so now you should have solved your doubt
@Pizza Ugh... what terrible notation. :(
what notation do you use @Xander ?
@SineoftheTime Off the top of my head, I have no idea. I just really don't like that notation, as it doesn't make clear the domain and codomain.
my teacher used $[f]_{B}^{B'}$
which is similar to the one used by Pizza
12:38 PM
Maybe $L(A,B)$, for the set of linear maps from $A$ to $B$. Though that looks like the $L$ from $L^p$ spaces, so it may not be the best.
Or $M(A,B)$, for matrices which represent linear maps from $A$ to $B$.
Suppose a professor has to write recommendation for a student. The given format mentions both strengths and weaknesses of the student. Do you think a professor should only write the strengths or should write about both?
but how do you specify the basis?
@SineoftheTime Oh, I didn't see that this was part of the notation. Christ, that's terrible.
I wouldn't. ;)
But perhaps $M( (U,B_U), (V,B_V) )$.
For matrices which represent linear maps from $U$ (with basis $B_U$) to $V$ (with basis $B_V$).
I don't think that there is ever going to be a nice way of writing this down, but the subscript-and-superscript is awful. :/
(Though I am sure that people use it).
@SineoftheTime so if I invert B' and B would become
f^-1 : W -> V?
how do you know $f$ is invertible?
$B'$ is not a basis of $\Bbb R^3$, so it does not make sense
12:48 PM
@SoumikMukherjee I think neither but I'm not sure. I think the student should fill up the form himself by adding what they think their strengths and weaknesses are, and get it signed.
@SoumikMukherjee If the form asks for both, the recommender should address both.
@SineoftheTime I mean generally, not in this case
Is this thing I wrote wrong?
if it was $M^B_{B'}$
associated to which linear application?
@XanderHenderson Okay
$M^B_{B'}$ is the matrix representing the inverse of a linear map $f^{-1}: W \rightarrow V$ with respect to the bases $B$ and $B'$. Each column of this matrix represents the coordinate of the corresponding basis vector $B'$ expressed in the basis $B$.
12:55 PM
@Koro Does this happen in your institute as well? I know some professors who would ask the student to write it themselves and only sign it. I thought it was a local phenomena.
@XanderHenderson This is funny XD
@Pizza is $\dim W=\dim V$?
you mean in my case?
what is $f^{-1}$ and how do you know it exists?
@SineoftheTime The notation $f^{-1}$ refers to the inverse linear transformation, which exists only if $f$ is invertible. A linear transformation is invertible if and only if it is an isomorphism, meaning it is both injective and surjective.
you can have a linear bijective maps only between spaces with same dimension
so $f^{-1}$ in you example does not make sense
1:03 PM
so if B and B' are reversed, only the basic change process is reversed?
For those familiar with Hoffman&Kunze's book, I'm just at the 3rd chapter of Hoffman and Kunze, but I was curious as to where the proof for the following theorem comes in the book: " if Ax=0 and Bx=0 (for A and B, $m \times n$ matrices over field F) have the same solution space, they are row equivalent" I have flipped through the book cursorily, but I haven't found it. Is it something that's very obvious once the theory is developed that the author didn't bother taking the space to prove it ?
@Pizza what do you mean?
@Pizza use this
@SineoftheTime $M^{B'}_B$ if it was $M^B_{B'}$
only the basic change process is reversed?
it does NOT make sense since $B'$ is not a basis of $\Bbb R^3$
I mean, can the second notation come out in some exercises?
not here
perhaps I expressed myself badly, but I was saying generally, not in my exercise
1:16 PM
if $B'$ is a basis of the domain then yes
ah ok
However, even before I said generally not in my case
just take $g:\Bbb R^2\to \Bbb R^3$ linear and then you can make sense of $M_{B'}^B$
3 hours later…
3:58 PM
@psie he does provide the foundations for it then, but I wouldn't say thats presenting the Lebesgue-Stieltjes integral, but presenting the Lebesgue integral (but not as Folland meant it, I mean literally integration with respect to a measure). There are important theorems such as integration by parts for Lebesgue-Stieltjes integrals which are pretty much a must when trying to present this integral, so its not a presentation as how I look at it, rather an example of sorts
Yes, $\int f\mathrm{d}\mu$ is still called a Lebesgue integral, even if $\mu$ is not the Lebesgue measure
@psie I think Caratheodory's extension theorem and premeasures are definitely important topics, for example for purposes of geometric measure theory, but nonetheless for most people you probably only have to go through the construction of the Lebesgue measure once in your life and then never again, and choosing a path of how you go about constructing it, so to speak, is not that important in my opinion
ok, thanks πŸ‘ regarding premeasures, I have seen a couple texts not mention these at all (...or maybe they implicitly do). I think one can cleverly omit them in some way.
Different people achieve the same thing using different tools. I've seen, for example, multiple introductions of Stone-Cech compactification of a space
All are equivalent, using different tools, and all are valid
I mean okay its an example from topology but this was my first example
@psie I'd have to look at them to see if they really omit them, or just not name them
I'm experienced with the Caratheodory construction just like you are
ah ok :)
4:28 PM
macbook numbers app (like excel in windows) is so annoying.
apparently there is no way to import a wikipedia table into it.
4:44 PM
Let the following linear application be given
$$f:\Bbb R^3 \to \Bbb R^2$$defined as $f(x,y) = (3x+2y-z,x-z)$. Given the bases $B = \{e_1,e_2,e_3\}$ and $B'=\{(2,2),(-1,2)\}$, determine the matrix associated with the application with respect to the bases $M ^{B'}_B$
but in this case the starting basis is B, right?
I had a doubt on YouTube, someone always wrote the starting basis above and the destination below
$M ^B_{B'}$
however on the track it writes it differently...
5:35 PM
there is no standard notation for that. focus on whether they end up with the same matrix
every linear algebra textbook should have, in the margins near where they all introduce notation for stuff like that, "by the way, there is no standard notation for this" and maybe even give other examples for other versions of what is out there
pizza: the thing that has meaning (independent of notation) is, what basis are you putting with the domain, and what basis are you putting with the codomain. the problem needs to somehow specify that. unfortunately, that is mostly where the various notations differ
this ambiguity is less of an issue with linear maps from spaces of different dimension, because in the problem above, you know (for example) that B is the basis for the domain and B' is the basis for the codomain, for reasons of dimension alone
but in the (common) situation where the domain and codomain are the same, if someone asked what M^B_B' was, with B and B' bases of that space, my first question would be, in this notation, which these is being used as a basis for the domain
6:07 PM
Currently reading on multivariate random variables and I'm a bit perplexed about the definition of the Lebesgue integral of such functions, i.e. vector-valued functions. Let $\mathbf{e}_1,\mathbf{e}_2\in\mathbb R^2$ be the standard basis vectors and $X,Y$ be $\mathbb R$-valued functions. Let $P$ be a probability measure on $\Omega$. Does then $$\int_\Omega (X\mathbf{e}_1+Y\mathbf{e}_2)\, dP=\int_\Omega X\mathbf{e}_1\,dP+\int_\Omega Y\mathbf{e}_2\,dP?$$
The fact that the summation on the LHS contains vectors confuses me.
$X,Y$ are of course random variables and thus measurable.
After all, you prove linearity first for simple functions, then for nonnegative functions, then for real-valued functions. So I don't know if there is such a thing as proving it for vectors?
@psie yes because by definition, $\int_\Omega (Xe_1+Ye_1)dP = \left(\int_\Omega XdP\right) e_1 + \left(\int_\Omega YdP\right) e_2$
i.e. integration is taken over coordinates
ok, so I should probably view it more as a definition rather than having something to do with linearity
I suppose its a particular case of linearity, $\int_\Omega (Z_1+Z_2)dP = \int_\Omega Z_1dP + \int_\Omega Z_2dP$ for integrable vector valued random variables $Z_1, Z_2$
but this specific case follows directly from definition
the whole machinery of simple, non-negative etc. is not needed
for either
6:27 PM
ok, thanks
psie: a lot of vector valued stuff comes more or less 'for free' from the scalar case by working in terms of components. this may not be how your specific text has set it up, but it is a very efficient way of setting things up
the 'real' difficulties in definitional setup come with giving your functions complex domains (i.e. you may need more than calc 1 to integrate over a space that doesn't look like an interval), or if you want the integral to have a different 'type' from the thing being integrated (think e.g. integrating a vector field over a curve to get a number)
but if you can integrate a real valued function over Omega to get a number, you can integrate a vector valued function over Omega to get a vector
surprised that nobody has mentioned categories yet, so i'll do it
@leslietownes really, you just have to think categorically.
6:56 PM
Since we are discussing this
I'll mention that while integrating $Z:\Omega\to \mathbb{R}^n$ coordinatewise works well, if you replace $\mathbb{R}^n$ by a Banach space $X$, you want to use a different form of integration e.g. Bochner integrals, instead of integrating things "coordinatewise" whatever that means (weakly integrable)
The interesting part is generalizing this setting by changing the codomain, rather than the domain
7:12 PM
now say it in categories
maybe there's a third integral that's better than the options people ever actually think about because it has better categorical properties
I did see a categorical integral of some kind
I imagine this is what appeals to category theorists
7:34 PM
A: Expressing the Lebesgue integral using categories + the difficulty of describing estimates in category theory

Urs SchreiberA beautiful ∞-category theoretic formalization of integration of differential forms and of the Stokes theorem has recently been given in (Bunke-Nikolaus-Vâlkl 13). I had observed in (Schreiber 13) that if an ∞-category $\mathbf{H}$ is a cohesive ∞-topos then for group objects (deloopable objects...

I don't know if its this thing, but this is some nice looking nonsense. Abstract nonsense
7:45 PM
oh, that is top notch nonsense
8:28 PM
I want to review/write down some math from the "bottom up" but I'm wondering where to start. I guess I could start with set theoretic axioms? I want to then review relations, operations, then algebraic structures starting from simple stuff like partial magmas etc
If anyone has a book or something for this to recommend, I'd welcome it
oh god. is EE18 here?
not that rigorous
Just basic stuff to write down
I just wanna organize some things :P
well, literally nobody knows what "partial magmas" are, let alone uses them. they might be relevant to some random classification of structure in some abstract hierarchy, but that is a very niche thing of interest only to a few people. it isn't foundational for any math that people actually use.
Probably, but I want to memorize the dictionary so to speak
there are a lot of things that might naturally pop up if you just said, what happens if i filter math from most structure to least structure, that don't actually pop up outside of asking that exact question.
8:31 PM
do not repel me from cantor's paradise or whatever dave hilbert said ok
is there some subject matter that particularly interests you? everyone always wants to start "bottom up" but most of the time this is a horrifically bad idea. it's like if you decided to start by studying CS by heating up sand in your backyard to make silicon because "that's where it all starts." in some weird sense, sort of, i guess. but in any real sense, no.
Because my knowledge is swiss cheese already so I'm trying to de-swissify it
and I also just genuinely want to know what the hierarchy of structure is
no offense to any magma people, it should go without saying that this is just my opinion. but it might help to start with a book or topic that you would like to understand, and work backward from there. instead of "let's just begin as a blank slate and build Mathematics"
I also want to learn what categories are so I guess I am a lost cause now
yeah, i am just the wrong person to give advice on someone who wants to learn categories. in my (biased) view, the only reason to learn categories is because you want or need those tools to study or talk about something else.
8:36 PM
Fair enough. I will just scour wikipedia until I am content :P I know there isn't much learning going on when just memorizing definitions
but I'd have to learn it at some point anyway
at least I'm not into universal algebra
that just sounds bad
Which chat room has THE MOST all time users and all time messages? This one has the most of all the rooms on the first page, but which room has the most of ALL of them?
In mathematics, many types of algebraic structures are studied. Abstract algebra is primarily the study of specific algebraic structures and their properties. Algebraic structures may be viewed in different ways, however the common starting point of algebra texts is that an algebraic object incorporates one or more sets with one or more binary operations or unary operations satisfying a collection of axioms. Another branch of mathematics known as universal algebra studies algebraic structures in general. From the universal algebra viewpoint, most structures can be divided into varieties a...
This seems like a good starting point but doesn't get into the set theory stuff
side note, "unneeded" is such an ugly word, although in the abstract, i do like the consecutive pairs of double letters
@user402514 why is it relevant?
idle curiosity, i would imagine. with a high enough level of rep, there might be a way of directly posing that query to an SE database, instead of just browsing around looking at chats.
8:46 PM
set theory is so weird man
what's weird to me is how many alternative set theories there are en.wikipedia.org/wiki/List_of_alternative_set_theories
I wonder if the set of all alternative set theories is finite
I am so weak with set theory I kind of just want to gloss over this and never return to it .. yikes
well, most people who work in math never think about or use any of that stuff, if they even know it exists
You don't need to learn much of it anyway. Basic set operations, csb theorem, ordinals and cardinals and I think that's enough.
you get kind of a weird picture of math from wikipedia. wikipedia is structured the way it is because it is an encyclopedia. everything and the kitchen sink is in there
And AoC
there's always something you can click at the bottom of the page to go broader or narrower or both, and while that choice maybe also always exists at a metaphorical level in math, most people don't experience it that way
8:55 PM
I imagine we're just stretching/widening the goal posts by making set theory rigorous but at some point there is a boundary b/t philosophy and mathematics and that probably won't ever change
but yeah it gets really gritty
well if you did the same deep dive on the english language or something, you could easily convince yourself that it's impossible to ever read or write because there are always words you don't know, words you know whose histories you don't know, rules of grammar and usage that you know but don't know how to express, and rules of grammar and usage that you don't know
reading godel's incompleteness theorems, for example, makes 0 sense to me and makes me feel insane just trying to understand it
a word i use a lot in here (maybe overuse) is "focus"
Q: Verifying Math behind Full Bundle Adjustment for Multi-Camera Extrinsic Calibration

hunterlineageI'm working on an extrinsic calibration problem involving multiple monocular cameras. Each camera has known intrinsic parameters, and I have captured timestamp-aligned images of a calibration target from these cameras. The calibration target may sometimes go out of view. My goal is to compute the...

with the right level of focus you can meaningfully engage with something without knowing "the fundamentals" of some background setup, or the most general version of where you might push it
9:00 PM
I never thought these words can be used in a single sentence altogether
@SoumikMukherjee this "self-driving" car is going to hit somebody
@SoumikMukherjee top 10 moments caught on camera before disaster
Trees, people; trees. I see you cannot see the forest for the trees!
@leslietownes dude I've opened a can of worms in wikipedia. There is so much stuff I've never heard of D: does this ever happen to you
9:06 PM
Rabbit holes are everywhere on the internet.
aka blackholes of time management
Use your time wisely.
obliv: it definitely does happen, if i let it happen
I figured out why, it's because the stuff I was reading were topics in functional analysis/topology
Stuff I've never touched so no wonder
if I have a curve non parametrized by arclength, I can built a new curve with the same trace by composition parametrized by arclength. In general, is it possible to express explicitely the new curve?
math really be like different continents of a world and I've been exploring them with a sail boat :P
@SineoftheTime I don't think so
9:13 PM
in fact it feels stange
I was reading this
sine: that "explicitly" is kind of vague, but in the sense that i think you mean, in general the answer is no, the expression for the new parameter in terms of the old will not be explicit
I just write it as a composition I guess
a secondary question might be why you are writing it at all, much of the use of the arc length parametrization is conceptual enough that you wouldn't need a formula
or if you needed to compute some specific thing at some specific point, you'd probably just resort to (admittedly uglier) formulas that do not need an arc length parametrization
yes, I was just a curiosity.
There are the generic formulas for curvature, torsion, etc which do the job
10:07 PM
til you can even have a different choice in logic

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