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12:22 AM
She learns from the best.
stomp stomp stomp
2 hours later…
2:10 AM
corporatism in US when
It's decided. When I'm ready to diversify, I'm going to build corporate bodies and the fact that it exists within a capitalist system is merely accidental.
1 hour later…
3:22 AM
. o O ( $\int_0^{2\pi}r e^{-i\theta}r e^{i\theta}i\mathrm{d}\theta$ )
@TedShifrin I ate pie!
3:56 AM
Why not $\int_\gamma \dfrac {r^2}z\,dz$?
that's imaginary pie, surely?
4:14 AM
@TedShifrin to me, that would indicate that the integral is independent of $\gamma$
Blah: You know it's the circle of radius $r$.
1 hour later…
5:18 AM
@Jakobian Sage has some nice continued fraction features. doc.sagemath.org/html/en/reference/diophantine_approximation/… Here's a short demo. If you give it expressions involving l sqrt() they will be represented exactly. It also accepts expressions involving pi
"expressions involving sqrt()". Sorry, my finger slipped, and it was too late to edit by the time I noticed.
Of course, you can also feed it expressions involving functions like sin, atan, ln, etc, and they'll be stored symbolically (as are rational numbers / expressions), not converted to floating-point. So the continued fractions should be valid to arbitrary precision.
3 hours later…
8:22 AM
@TedShifrin: As advised, I watched your lecture on polar coordinates. I have a better understanding of the polar coordinates now that I did before. Thanks a lot.
I understand now that $S=\{(x,y)\in \mathbb R^2: x^2\le y\le 1, y\ge 0, -1\le x\le 1\}$ will be $\frac\pi 4\le \theta\le 3\frac\pi 4, \color{red}{\tan\theta \sec\theta \le} r\le \csc\theta$ in the polar coordinates plane. The red colored part is true by mapping $y=x^2$ using $r\sin \theta=r^2\cos^2\theta$ which is same as $r=\tan\theta \sec\theta$. Is my understanding correct? Thanks.
Why is $\tan\left(\frac{A-B}{2}\right)=\left(\frac{a-b}{a+b}\right)\cot \left(\frac C2 \right)$ called Napier's analogy?
9:16 AM
Hello everyone. Please guide me on two questions.
1. I have a polynomial.





Calculate the range of parameter $x$ such that polynomial $p$ has only real solutions for any value of $y$ in the specified range $-2<y<2$.
How are such tasks solved?

2. There is an arbitrary function of 4 variables $f(u,v,w,x)$ and:




From these ranges, an arbitrary combination of $uvw$ is taken, and this combination is then substituted into the function $f$, and t
$p$ looks like a fifth degree polynomial in two variables
Silly question : I want to calculate $\inf_{x\in X} F(x)$ where $F$ is some function and $X$ is say a topological space.
Can I say $\inf_{x\in X} F(x) = \inf_{x\in Y} F(x)$ where $Y$ is a subset of $X$ for which $F$ is uniformly bounded on $Y$ ?
9:43 AM
Do you have more assumptions on $Y$? The function is uniformly bounded on a singleton for example
10:03 AM
Alas, the region obtained in r theta plane here is not correct.
Ted, but now I have understood how to correctly convert the region $S$ here to the right region in polar coordinates plane with robjohn's help. :-)
@AlessandroCodenotti yes I want $Y$ to be any level set of $F$
The function is constant on its level sets, so taking the inf there does nothing
10:30 AM
sorry sublevel sets
11:12 AM
@Koro has your problem been solved yet?
I left right after my message so I didn't see yours yesterday
11:50 AM
Q: How do we know the position of fixed point in this Q?

S.M.TQ: A particle moves on a given straight line with a constant speed v. At a certain time it is at a point P on its straight line path. O is a fixed point. Show that (OP×v)is independent of the position P. My solution: I considered the axis to be X-axis I.e as 1D motion. Points P & O on it. O to be...

12:44 PM
@Astyx yes :). Thanks to robjohn and Ted. :)
12:57 PM
@Koro do you have enough rep to remove answers on main?
I guess not
1 hour later…
2:08 PM
@Astyx not sure, can one remove answers singlehandedly?
I think I can vote for closing a question.
2:20 PM
Hey, I'm from the physics side of things and a buddy of mine and I have been thinking about a math "proof" (I use the word very liberally) that I had thought of one day. It can't possibly be correct, and I' worried I'll be crucified for this, but we can't find any holes in it. Thought maybe you guys could point out where the holes in my swiss cheese are
it's a bijection from natural numbers to the reals on [0,1)
we tried applying cantor's diagonalization, but we failed
@Koro I don't think so
it goes like this, for all n in naturals, write n.0, then write the digits in reverse. 1 becomes 0.1, 2 becomes 0.2, 1056 becomes 0.6501, etc. This gets you any and every number in [0,1) and is inversible.
No, it gets every number with finite decimal expansion
1/3 is not in the image of your map
2:29 PM
is it not?
are infinitely long integers not in the set of integers?
well, what would its preimage be?
@Jim no, "infinitely long integers" does not make sense
@Astyx what is $6^{\infty}$? I don't know, it's undefined, but it's an integer whose last digit is 6 in base 10
I have not seen this notation before
I have chatjax working
I do not know what $6^\infty$ means
how is it defined?
2:32 PM
yes you do, you just know it's undefined
If it's undefined it's not an integer
so, if there are no integers of unending length, then there are only finite integers?
Yes, an integer is finite
There is a surjective map between sequences of integers to reals
so then, for any ennumeration of a subset of integers, could I not find an element in the natural numbers that's not in the ennumeration?
which is maybe what you're looking for
But the former does not have the cardinality of the integers
@Jim Yes, integers are countable
Well I'm not 100% sure I understand what you mean by this statement to be honest
2:37 PM
@Astyx then, necessarily there must be some integers of indfinite length, otherwise I'd always be able to find an element not in your ennumeration by continuously increasing the length of my chosen outlier
Oh I understood the opposite of your statement sorry
Let me rephrase
There is an enumeration of the integers
Namely, $(0,1,2,3,4,5,\dots)$
agreed. I mean, that's the definition
So what's your point?
but the allowable length of integers must be transfinite
2:40 PM
start with the number 3
put another 3 in front of it
continue until it is no longer an integer
Hello math folks
the length is transfinite
Does anyone know of a reference for the relation between the diffeomorphism group of a manifold and its jets?
ie between $\mathrm{Diff}(M)$ and $J^k(M, M)$
The resulting object is not an integer
This sounded promising but it does not have any references for it : math.stackexchange.com/questions/2088706/…
2:42 PM
@Astyx defend
Do you know the construction of the integers?
What is an integer, according to you?
an element of the reals having no decimal places
What is a real then?
all numbers that, when squared, are positive definite
What's the inverse of $\dots33333$?
2:44 PM
is that an ellipsis or some decimals?
so 33333?
seems easy to inverse that
@Jim I'm not sure why you want me to defend that the resulting object is not an integer when you've written it yourself
@Astyx when did I write that?
6 mins ago, by Jim
continue until it is no longer an integer
2:47 PM
yes, my point is that you'd continue forever because it's never not an integer
but at no point does it have an infinite number of digits
it remains finite but continues growing forever. It have transfinite length
So what's the inverse of the resulting "integer"?
a very small number
Which number?
2:51 PM
this is specious reasoning. My inability to tell you the number doesn't imply it isn't an integer any more than my inability to tell you Cantor's first name implies he doesn't have one
That's because your definition of integers as a subset of reals is circular and thus flawed
Formally, one defines integers first, then rational numbers, then reals
so my number would be real, it has no decimals, but not an integer?
How do you define real numbers?
I said it is all numbers whose square is positive definite. I define it using a necessary condition
What is a number?
2:56 PM
anything used for quantification
Anyway, if you want to know what is commonly called an integer, and understand why what you claim are integers are in fact not, I recommend reading something about Peano's axioms, the construction of the integers, the rationals and the reals
I see, I shall do this. I am interested to learn how there can be no largest integer and yet no integer of transfinite length
3:12 PM
I'm wondering the following question
Let $X_n$ for $n = 1, 2, ...$ be Baire spaces such that any finite product of them is Baire.
Does it follow that $\prod_{n=1}^\infty X_n$ is Baire?
Can someone help me with the Conway-chain arrow notation , rules are here ? If $$M=3\rightarrow 3\rightarrow 3\rightarrow 3$$ , how can I compare the numbers , for example , $$M\rightarrow 3\rightarrow 3\rightarrow 3$$ with $$3\rightarrow 3\rightarrow 3\rightarrow 4$$ ?
I don't think that's really possible, it's already challenging sometimes to determine whetever $a^b > c^d$ for natural numbers $a, b, c, d$.
hello everyone
I was looking at a game called Grepolis, in which there is an event Winter Grepolympia
in the event, each player has an athlete that participates in the contest
each athlete has 3 attributes (for example balance, control and speed) (all non-negative integers)
the game calculates the score of the partipation based on the attributes of the athlete (and a small bonus/penalty due to luck)
now I am wondering how the formula to compute the score would look like
from pure testing against the game, the community figured out that a 85%-5%-10% distribution for the first contest is the most ideal distribution of the attribute points
for example, 17 balance, 1 control and 2 speed would get the highest score out of any combination of 20 points
I don't think this classifies as math
3:28 PM
Games usually have a lot of math behind them.
given attribute points a, b and c, how could a score be calculated to produce such a favouring of a particular attribute?
I am not really looking for the exact formula the game uses, but I am just wondering how the formula would theoretically work
given that
- higher attributes should always result in higher score
- better balance (to the intended distribution) should always result in higher score
@robjohn Sounds to me like this question would fit better in some programming-related chat though
perhaps, perhaps not
besides, discussions here are not restricted to math only, though preference is given to math.
sure thing
I assume it is not a common formula then...
regardless of whether this is maths or programming, do you have an idea of how the formula could look like?
3:47 PM
If you couldn't come up with it yourself, then I guess I wouldn't be able to either
i. e. if it's not something simple then I have no clue
I am not that good with mafs though :D
one thing I can conclude is that it is probably not an aggregation of the results of a computation of each parameter
\frac{s^4}{s^2+(x-as)^2+(y-bs)^2+(z-cs)^2}\text{ where }s=x+y+z\text{ and }a+b+c=1
for example, it isnt something like
a*0.85 + b*0.05 + c*0.10
because if they are in isolation, you would just favour the one attribute with the highest result
Among equal total scores (identical $s$), the greatest score is when $x:y:z=a:b:c$
as the total score gets higher, the highest value gets higher (the square of the total score)
so, it might just be pythagoras?
3:55 PM
who knows, but this meets your criteria, I believe
sqrt((a*0.85)^2 + (b*0.05)^2 + (c*0.10)^2)
I think it does meet the criteria as well, any higher number in any of a/b/c would result in higher total
and the highest total would be produced by the best even distribution
the input numbers would just have to be "normalized" to allow this even distribution
@robjohn although, I have no clue what this is
@Wietlol you should install ChatJax. The link is in the room description
I've asked a question about topology up there btw. About Baire spaces
where xyz are the attributes
and abc are the intended distributions?
4:11 PM
@Wietlol yes
another would be $\frac{ax+by+cz}{\sqrt{x^2+y^2+z^2}}(x+y+z)$
yea, this is all too much maths for me :D
I can understand $\sqrt{ax^2+by^2+cz^2}$ but for your formulas, I just dont grasp the relation between the values
in any case, I have at least something that works, I can always replace it with a different formula later on
4:41 PM
Wow, it's been forever since you were in here, Wietlol
4:57 PM
I wouldnt classify a few years as "forever" but... hi :D
5:07 PM
Yes, long time no see!
What were we discussing last time?
looks like probability in relation to infinities
I just realized certain boolean operations hold for multiplicative distributivity. $k (x \land y) = kx \land ky$. $k (x \lor y) = kx \lor ky$.
I could prove this by restricting $k$ to powers of two then using the binary decomposition of any arbitrary integer $k$ and the distributive property to show that it holds.
Ah, my mistake. It only holds for powers of two.
And I suppose a certain subset of x and y for any k
6:15 PM
well... back to square 1
pythagoras does not actually work...
I guess I will try the 2 other formulas mentioned
$a=0.85$, $b=0.05$, and $c=0.10$
The max of $\sqrt{0.85^2+0.05^2+0.10^2}(x+y+z)$ is reached when $x:y:z=0.85:0.05:0.10$
@robjohn this is ((ax+by+cz) / sqrt(xx+yy+zz)) * (x+y+z) right?
6:26 PM
thanks, ill try it out
6:40 PM
@Slereah You might look at Guillemin and Golubitsky for basics on jet spaces. I no longer have it, so I cannot double-check.
@TedShifrin I'll give it a look, thanks
It's also in Hirsch's Differential Topology, @Slereah.
7:12 PM
Hi professor Ted, I'm able to work my way through exercise 11.28 in Apostol's vol 2. The exercises involve double integrals to be solved by converting to iterated integrals in r theta plane. I saw both the videos you suggested and that helped a lot. Thanks a lot professor :).
You're welcome :) I'm surprised you didn't learn all that stuff years ago in your regular multivariable course, but it takes practice.
In actual applications, knowing how to set up the circle $(x-a)^2+y^2=a^2$ in polar coordinates is quite important. (Similarly in spherical coordinates.)
That was amazing. Diameter subtending ninety degree on circumference.
Also a very nice vector proof with dot products :)
I also noted how $x^2+y^2=a^2, a\ne 0$ is a straight line in r theta but it's surprising how circle of radius a centred at (a,0) is not mapped to a straight line :).
Professor I may have told this earlier that I come from engineering background. We did have multivariable calculus but it was probably assumed that everyone knew polar coordinates. By knowing I mean that $(x,y)$ becomes $(r\cos\theta, r\sin\theta)$.
That was all about polars that I knew. :(
But now I have much better understanding of polars :).
Hmm, engineers and physicists use different coordinate systems all the time.
Anyhow, glad you're finally learning stuff and enjoying it.
7:22 PM
7:33 PM
@Derivative So your Faa di Bruno pointer did help somewhat, I reduced it to something relating to Bell polynomials so I can continue with that now. Thanks for your assistance
8:27 PM
how do I write a map in latex like X \to Y but label the points under the map x \to y. If anyone even know what I am trying to do?
$x\mapsto y$?
$f:\begin{cases}X\to Y\\x\mapsto y\end{cases}$
yes but except the big bracket
$f:\begin{align}X&\to Y\\x&\mapsto y\end{align}$
I've always used and seen used the one with the bracket
8:44 PM
@Astyx how do I extend that if there's more than one map? for example, $X \to Y \to Z$?
Oh u can still use & multiple times?
I don't think so, but you don't need to use & at all
$f:\begin{align}X\to Y\to Z\\x\mapsto y\mapsto z\end{align}$
just found that out lol
It's not usual to denote map compositions like this
okay I jsut found out a flaw with this writing on latex, it doesn't align everything if the bottom line is too big.
that's what the & is for
9:39 PM
Does anyone know where I can go to check a coding theory proof? Like we do here on Math.SE chat?
Nevermind, there's a tag on Math SE
Salut @Astyx
Howdy, coding @Under
Heeeyy :D
10:06 PM
Maths doesn't get enough humor smh
A reviewer of the first textbook I published complained that humor has no place in mathematics texts. Screw that.
That's funny.
Which makes it irony
I propose that in addition to surjective, injective, bijective, etc. we must also allow for objection and subjection between sets where objection between a set A and a set B is when A asserts dominance over B and subjection when B crawls over and submits itself to A.
Lol, he must have been boring to hang around.
ted it's a funny thing. i do think a little bit goes a long way. if taken to an extreme, it turns the already troubling job of learning math into the more vexatious task of having to tolerate the author.
although most authors are more tolerable than i am, so what do i know.
Studies show that high functioning groups often have a 'clown' member who may not be the greatest direct contributor but enables the group to interact in a more productive way. Appropriate humour is always good.
10:13 PM
Salut Ted
i have enjoyed some math humor in books. legal humor is never good. just never. no exceptions.
any idea why this has 2 votes to delete? math.stackexchange.com/a/4360637/377528
someone downvoted a question i asked. always curious why.
i did have a psq answer closed, but that was not unreasonable.
astyx: "this" meaning your answer? low rep users can't see votes to delete on other users' posts. beats me.
@leslie I agree, of course. Believe it or not, he was complaining about a "Whew!" at the end of a stupid algebraic calculation.
10:15 PM
there is a long running division of thought on the propriety and desired content of 'hint' answers.
I do have some puns, especially in a few terminology footnotes. I pun more spontaneously in lectures.
ted: well he's just no fun at all. "Whew" at the end of corresponding with him.
For example, the relationship between the irrationals and rationals is objective from the irrationals because it beats the rationals into submission as opposed to the rationals striving to be irrational ($\forall x\in \Bbb{Q}, \exists \lim_{x\to y} = z, z \in \Bbb{R\backslash\Bbb{Q}}$) which would make the relationiship subjective from rationals to irrationals.
@Astyx 2 votes to delete what?
I mean my answer is nothing spectacular
10:16 PM
I still laugh over the student in my differential geometry class who, after a not-fun Christoffel symbol computation, said, "I'm throwing in the tau."
but I don't think I'm breaking any standard in terms of answer
i have often found that high mathematical ability and appreciation of humour are rarely found in the same head. certainly not mine.
$\tau_\ell$ ?
@Astyx sans subscript :P
@Astyx what two votes are you referring to?
10:18 PM
i was once scolded at work for writing a memo that suggested a person's name was probably misspelled, i couldn't resolve it. we had documents both ways. so i put "Smith/Smyth" everywhere after identifying the problem. someone thought i was trying to be funny. i just didn't want to get a phone call some day asking why we spent money trying to track down a wrongly spelled name.
I would have voted to resolve directly first.
That (2) means 2 votes for deletion, right?
@Astyx that's odd, i don't see that
Anyway, I don't mean to make a fuss about it
10:19 PM
you often have stuff in my job where people get their name on a patent and then fall off the face of the earth. especially if they're in other countries.
probably i don't have enough rep
I was just wondering if I was going against community standards
maybe you did sumthing bad
@copper.hat need to keep grinding those points
astyx: some people don't like "hint" answers, i think that is the high level thing that is going on. it is a difference of opinion but not a community standard.
10:20 PM
only 850k to go
It's the community as usual imposing its subjective and ever changing standard of quality on your question I presume.
as i say when i am scolded, "hey man, i do what i do, you don't have to like it." then i slick back my hair with a comb and ride off on a motorcycle.
and into the sunset.
leslie: that was my guess as well. I was surprised because the two votes arrived quickly, which seemed to imply some kind of agreement that it deserved to be deleted
but you don't see the votes copper? that's really weird
Hi. I have a quick question. Is a class of groups closed under quotients iff it is closed under homomorphic images? I think it follows from the first isomorphism theorem . . .
10:23 PM
i'm special apparently
what does it mean for a class of groups to be closed?
something like this was in rose's book. this class of groups business.
@copper.hat It means they got covid
astyx: i sometimes get a random downvote with no explanation, i assume someone just doesn't like my style. which is fine, i probably don't like their style. we all get to click on buttons.
A class of groups is closed under quotients if, whenever $G$ is in the class and $N\unlhd G$, then $G/N$ is in the class.
ahhh, little oh
Thanks @Shaun
10:26 PM
i think i've done about one or two downvotes in 10 years. for something that was abusive. maybe i'm not pulling my weight by not downvoting other stuff.
downvoting without comment seems a little rude unless the problem is obvious.
you sometimes see fairly well-posed problems being downvoted i think mainly because they are what the downvoters regard as stupid questions. similarly a crummy question will get upvotes and answers if it is 'advanced' enough to be interesting.
downvoting costs points -- why would I waste precious rep to make this site a better place?
i have some points on the legal stack exchange
astyx: does it? i didn't know that. makes sense, i guess.
sometimes i downvote if i get annoyed at pure laziness.
@leslietownes I think it's one point per downvote, I might be wrong
10:29 PM
i don't really care about rep, but the -2 bugs the ship out of me
now that i have my se t-shirt
I, too, hate being told I'm wrong and not knowing why.
my threshold for downvoting is far beyond annoyance. you have to be worse than annoying.
and i mailed my se socks to my daughter in the uk
i am following the tony robbins replacement of crazy mad with slightly peeved
i think i'm free riding on the downvotes of others.
other people clean up the horizons so i don't have to look at ugly things.
i am always curious that laws are not examined for logical consistency
i am sure i am not the first to ask that
10:34 PM
yeah, there's no meaningful logical limit on the law.
or appellate review, for that matter.
they can all be as inconsistent as they want.
i am using zfc your honor
there was a good supreme court case in the 1980s. there was a cap on what an attorney could charge in some kind of proceeding involving, i think, veteran's benefits, of about $100. this meant essentially that nobody could get legal assistance for those claims. it was upheld on the rationale that congress might have wanted these proceedings to be non-adversarial. the $100 figure dated from the civil war era when it was a reasonable amount of money. the majority did not engage with this at all.
^ incentive to adopt the euro
they could've made the usual move which was 'if congress doesn't like this they can change it,' but they adopted this counterfactual narrative instead.
I don't know if you have chatjax on leslie, but it's messing with your message
10:38 PM
'if congress doesn't like this they could change it' is actually a very persuasive narrative in a world where congress actually does things. in a world where it doesn't, it is harder to stomach.
That comes from throwing around $ willy-nilly.
making it rain in the chat.
$yeah man. dollar sign, am i right.$
There's got to be an algebra joke somewhere about equations and words
cash rules everything around me
see copper?
Now I put the "hint" in the comments and someone else posts it as an answer
10:42 PM
"Now we can rewrite $o^2 h n$ as $ohno$ to find that..."
@Astyx happens a lot. the value is you are helping someone.
@leslietownes I have a copy of Rose's, "A Course on Group Theory" handy. Whereabouts in the book should I check? There's nothing obvious about it in the index.
@copper.hat yes I know, I'm just kidding
@Astyx, it sounds like you're joining me in the much-maligned MSE corner.
I have my MSE-fury moments
they usually don't last very long
10:45 PM
Mine are coming far more frequently.
what amazes is that some answers get a huge number of upvotes, certainly not in proportion to mathematical difficulty or novelty
This one is a humorous insult.
Oh, I learned that one in my first month here, @copper. Trivial answers get the most votes.
shaun: this is called 'calling my bluff.' my copy is gone, i think. i think it occurs later in the book, where he starts to do things relating to classification of groups. it's not near the beginning.
not my trivial answers :-)
if i can't easily read a question i don't even try anymore
@leslietownes is making my screen shift left every time i hit enter
nobody will believe this but i did not intend to mess up anybody's chatjax formatting.
10:50 PM
millions might not believe you, but i do
shift left
@leslietownes Thank you nonetheless :)
You are always villainous, @leslie.
$P(X=\phi) = 1/\pi$ and $P(X=-(1-1/\pi)\phi)=1-1/\pi$ @copper
i am clearly in a trite mood
Time to practice contrition.
10:55 PM
bless me father for i have sinned
never found it cathartic, but mother guilt used to make me go
then again, i was an angel child
The result does indeed hold (if the following is to be believed).
Q: Is there a way to syntactically characterize homomorphic images and products?

Maxime RamziBirkhoff's HSP theorem states that if a class of algebras (for a given type) is closed under products, subalgebras and homomorphic images ($\iff$ quotient), then it is actually defined by some equations. I was wondering what could be said about other combinations of these operators $H,S,P$, in ...

11:13 PM
Does there exist anything that might be described as a temporal density function that relates the distance over time from one point to another in the form $(x(t), y(t))$?
If there were such a thing, and we had an algebra surrounding it, we could compute inverses of functions by way of their arc length parametrization and temporal density function.
11:29 PM
@Ted: I made a commutative diagram. I may need to be disinfected.
Next you'll be computing cohomology, @robjohn.
Nooooo!!! Get me iodine!! Get me alcohol!!
isn't that what math is all about?
Iodine and alcohol... I'm not sure those mix
@AMDG not talking about Jim Beam here...
11:33 PM
I don't drink Jim Beam lol
I like my spiced rum
Tastes like cream soda but without like 40g of glucose/sucrose/whatever
ethanol any day over sugar water.
They seem to be lacking basic algebra
Indeed. But there's probably also language issues. I would not say "remove." I would say explicitly: Multiply the first equation by blah. Now subtract this new equation from the second equation.
Meanwhile, I'm fighting this battle. The picture seems to be completely out of whack. My computations, which I've checked, seem to put the bottom vertex of his "rectangle" below the $x$-axis.
It seems they are writing down the correct computation, but are not able to carry them through
Having all those yucky $\sqrt6$s doesn't help.
Maybe get them to do a simple one, like $x+y=2$, $2x+3y=5$.
11:49 PM
I did suggest them to try that
I have to go to sleep though
Unrelated: is there a name for the tensor algebra quotiented by commutative relations (like you would do to get the exterior algebra, except you impose commutativity instead of anticommutativity)

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