Mathematics - 2015-02-12 (page 1 of 3)

Committing to a challenge

00:00

@Chris'ssis Congratulations, that's a straw man

3

Chris's sis

lol, I star you too for the speed (although I wouldn't say that at all)

:-)

Julian Rachman

^^

Committing to a challenge

@Chris'ssis So you were an undergraduate student in your third last year of highschool?

Chris's sis

@Committingtoachallenge do you proceed in the same way in the real life? Do you stalk people as you read my 12000 messages?

Committing to a challenge

@Chris'ssis In debating environments at uni, yes. "Do you stalk people as you read my 12000 mesaages?"(Editted in), no.

Chris's sis

00:03

@Committingtoachallenge I mean it doesn't sound nice to do that. I suppose you're not a happy person too often.

Committing to a challenge

@Chris'ssis Why don't you answer my question? It has a (yes/no) because it is true or false. Why don't you see that?

Chris's sis

@Committingtoachallenge This obession (or how to call it?) it's not good, seriously speaking.

I hope you can improve that in the future. :-)

Committing to a challenge

2 mins ago, by Committing to a challenge

@Chris'ssis So you were an undergraduate student in your third last year of highschool?

Chris's sis

Oh, now I see ...

@Committingtoachallenge I let you unreveal the mysteries alone. I think this is the best way. ;)

Committing to a challenge

@Chris'ssis I have revealed your false history already, and I am enjoying you avoiding answering to it

Bitrex

00:06

who cares

Julian Rachman

^^

Chris's sis

@Committingtoachallenge lol, you reveal nothing. People do not come here for the comments that are not related to mathematics.

Julian Rachman

^^

Committing to a challenge

@Chris'ssis Then why build such a massive history and run multiple accounts?

Chris's sis

@Committingtoachallenge My history doesn't matter here you know. Who cares my comments history here excepting you? As I told you, you do not seem interested in mathematics.

Julian Rachman

00:08

It is not like this is going to make or break your mathematics career

Committing to a challenge

@Chris'ssis Jasper Loy?

@Chris'ssis Venus(you claim not to be you)

@Chris'ssis Aren't you solely interested in series, limits and integrals? I am interested in many more areas than you?

Chris's sis

@Committingtoachallenge Well, I do it for you. I mean it's OK to ask some questions, but not to be obsessed. Look, if I asked someone a personal question and that person wouldn't tell me the truth here, that would be OK. It's about anonymity!

Committing to a challenge

@Chris'ssis Wouldn't tell me the truth, meaning to not respond, or to lie?

Chris's sis

@Committingtoachallenge well, it depends on the situation. Some would say nothing, some would say another thing. It depends on the person.

Committing to a challenge

@Chris'ssis Anyway I have to go, study group for next semester meeting up for first time

@Chris'ssis Was an eye opening talk ;) Thank you

Chris's sis

00:12

@Committingtoachallenge lol, OK :-)

Julian Rachman

Cool. Let's talk math!

Chris's sis

@Committingtoachallenge Remember, we're people not computers. ;)

Bitrex

@Chris'ssis can you link to the question regarding the integral you posted earlier?

Committing to a challenge

@Chris'ssis ;)

Chris's sis

@Bitrex That one is an integral that is going to appear in my book.

Bitrex

00:14

Ah

Can you post it again? I'd like to F around with it

Or if it's a secret....:(

Chris's sis

OK, but don't post it on main

Bitrex

Roger that.

Thnx

Chris's sis

@Bitrex Welcome. The answer is really unexpected! :-)

Pyraminx

Hello!!!!

Julian Rachman

Can anyone help with toplogy?

Bitrex

00:16

oh it's a little different than I thought I thought it was sin(x_1)sin(x_1 + x_2)...etc

ok

anon

@JulianRachman dunno, can we?

Julian Rachman

@anon I believe you can...

basis is still confusing me but I am getting closer to understanding

Chris's sis

@Bitrex Yeap. Do you have in mind an approaching way?

Bitrex

@Chris'ssis I'm not that quick! :)

Chris's sis

@Bitrex hehe, no hurry with that.

Bitrex

00:21

I'm just an electronics person...

not a SUPER MATH PERSON

anon

@JulianRachman so do you have a question?

Display Name

Is there a query I can run for MSE for number of questions with 0 votes and 0 answers?

Bitrex

@DisplayName you could try math.stackexchange.com/help/badges/40/tumbleweed

Chris's sis

@Bitrex It's not an easy question, one needs some experience for dealing with it. So, no worry with that.

Display Name

@Bitrex I meant using this data.stackexchange.com/math/queries

Julian Rachman

00:26

@anon not in particular. I just want an in-person-for-"first"-grader explanation of a basis which leads me to the definition and a very concrete example.

anon

were you the one I was talking with about top bases the other day?

Julian Rachman

i believe so.

anon

a collection of subsets is a basis if every open set is expressible as a union of said subsets

which is equivalent to the conditions (1) and (2) given in whatever reference you gave

Julian Rachman

Ya. So say I had $X=\{a,b,c\}$ and had a topology $\tau=\{\emptyset,\{a\},\{b,c\},\{c\},X\}$ on $X$. Then what would be the basis of this topology?

I thought it would be the set of all singleton subsets of $X$ but that is only for discrete topologies.

@anon

anon

@JulianRachman that's not a topology

Julian Rachman

00:34

@anon Oops. I meant $\tau=\{\emptyset,\{a,b\},\{b\},\{b,c\},X\}$

anon

why are you putting the word the in front of the word basis? "the basis"?

Julian Rachman

Sorry. Is "a" the right word to replace it?

anon

yes. a space can have many bases.

Julian Rachman

sorry. my terminology issues. :(

anon

if you thought spaces had unique bases, that would be a conceptual issue not a terminological one

anyway the bases for that space are pretty trivial

Julian Rachman

00:38

no. Like I typed it without taking the difference between "a" and "the" into account

anon

there's {{a,b},{b},{b,c}}, and the only other three are obtained by adjoining either {} or X or both

Julian Rachman

?

anon

do you have a question?

Julian Rachman

how does that apply to finding a basis for that specific topology I had defined?

Bitrex

@DisplayName I've asked one of my SQL dude friends if he can whip something up

anon

00:41

@JulianRachman You first asked "what would be [a] basis for this topology" [before correcting your definition of the topology]. I answered that by giving you every possible base.

I take it your follow-up question is to ask how I obtained it? Use your words!

Julian Rachman

Sorry.......... Yes, that is my follow up question.

anon

hold on need to reboot my browser. start thinking about it. use logical reasoning.

Julian Rachman

Ok.

anon

k back

Julian Rachman

k

Mike Miller

00:44

♫ welcome back ♫

anon

since {b} is an open set, it must be a union of sets in whatever base you want. the only way that's possible is if {b} is in the base.

can you use logical reasoning to deduce {a,b} must be in any base?

@MikeMiller I just learned the zeta-regularized cardinality of a compact p-adic Lie group is zero

Mike Miller

explain

anon

$\sum(\dim V)^2=|G|$ for finite groups, so just zeta-regularize the sum for G in which there are only polynomially many irreps of dim below a given bound

(I wasn't looking for p-adic Lie groups G at all but that's the kind of group the literature seems to be focused on)

Mike Miller

can you be explicit about what you mean there

zeta regularization is not, like, a thing i've ever thought abuot

anon

oh, you analytically continue $\sum (\dim V)^{-s}$ as a function of complex $s$ in order to evaluate at $s=-2$

Mike Miller

00:49

ah, i see

cannot say I see why one would care that this ends up evaluating to zero other than idle curiosity

anon

it was just idle curiosity

Mike Miller

great!

i love idle curiosity

anon

although something like $\sum (\dim V)\chi_V(g)$ seems to be a "Dirac delta function" on $G$, and I think works like a Green function for the heat kernel. in particular for finite $G$ if we call if $\delta(g)$ then $\int_G \delta(g)f(g)dg=1$ (normalized counting measure). and a formal reason to believe it should function similarly for compact in general came up when I was trying to prove matrix coefficients densely span L^2(G).

Julian Rachman

@anon I can say that it must be a union of sets within any base of the topology. But other than that, I do not know.

Mike Miller

neat.

not sure how that only snuck in.

anon

00:52

@JulianRachman can any of those sets that {a,b} is a union of contain c?

Chris's sis

Done with my proof. Out for some sleep.

Julian Rachman

NO

@anon

anon

(err that should be $\int_G\delta(g)f(g)dg=f(e)$ in my comment above)

@JulianRachman can any of those sets that {a,b} is a union of contain a?

Julian Rachman

Yes

Mike Miller

i guessed, @anon

anon

00:54

@JulianRachman can {a} be one of those sets?

Julian Rachman

Of the union?

anon

yes

Julian Rachman

then yes?

anon

is {a} an open set?

Julian Rachman

yes

anon

00:55

waits for Julian to go back and check

Julian Rachman

no?

anon

21 mins ago, by Julian Rachman

@anon Oops. I meant $\tau=\{\emptyset,\{a,b\},\{b\},\{b,c\},X\}$

Julian Rachman

Because it is not an open set of the topology

yep

I figured.

Bangs head on wall

anon

so can {a} be one of the sets we are unioning to get {a,b}?

Julian Rachman

no

anon

00:57

correct

is it possible for none of the sets we are unioning (to get {a,b}) to contain a?

Julian Rachman

no

anon

so one of the sets we are unioning must contain a, and also must contain something other than a, but can't contain c

therefore _____ is one of the basis sets

Julian Rachman

correct. to get {a,b}

anon

one of the sets we are unioning must have been {a,b} itself

symmetrical reasoning applies for {b,c}

thus a base necessarily must contain {a,b},{b}, and {b,c}

next you can prove that is sufficient: {{a,b},{b},{b,c}} is a base

Julian Rachman

Oh. So if we were to formally write what THE bases of the topology are, how would we formally write them in terms of basis elements $B$ and a collection $\mathcal{B}$

?

anon

01:02

24 mins ago, by anon

there's {{a,b},{b},{b,c}}, and the only other three are obtained by adjoining either {} or X or both

Julian Rachman

So {{a,b},{b},{b,c}},

{{a,b},{b},{b,c},{}},

{{a,b},{b},{b,c},X},

{{a,b},{b},{b,c},{},X}

anon

are the four bases yes

Julian Rachman

So then how would we show that these basis generate the topology?

anon

let's pick {{a,b},{b},{b,c}}. show that it's a base

Display Name

@Bitrex Very cool. Thanks

Julian Rachman

01:10

@anon of what topology?

anon

@JulianRachman the topology you put on X........................................

Julian Rachman

If you take the union of the given base, you get {a,b,c} which is equivalent to the set X.

Just an "educated" guess....

anon

Is that supposed to be a full proof or have you only just started writing out a proof?

Julian Rachman

I dont know how to prove it

and I didnt know I had to prove it

anon

you asked how we would show these are in fact bases, that these collections of subsets generate the topology

I am making you do it for {{a,b},{b},{b,c}}

Julian Rachman

01:15

Yes. ANd I dont know... :(

anon

There are certain definitions in mathematics that tell you how to prove if something is a blah or not. In this case, a collection of open subsets is called a base if every open set in the topology can be written as a union of subsets in this collection. That tells you what to do!

Julian Rachman

So if we took a union of subsets of each open set of the topology I had defined, I can see that the result is the base I had originally wanted to prove?

anon

huh?

is {} a union of elements of {{a,b},{b},{b,c}}?
is {a,b} a union of elements of {{a,b},{b},{b,c}}?
is {b} a union of elements of {{a,b},{b},{b,c}}?
is {b,c} a union of elements of {{a,b},{b},{b,c}}?
is {a,b,c} a union of elements of {{a,b},{b},{b,c}}?

Julian Rachman

Wait. If I took the intersection...

I took the intersection of all possible bases and got the topology I defined

(I still feel like I am doing it wrong...)

@anon

anon

why are you intersecting different putative bases?

Julian Rachman

01:33

Idk......................................

anon

I told you to prove {{a,b},{b},{b,c}} is a base. go and do it.

see my 5 "is ..." questions?

do you understand why answering those 5 questions is how you determine whether or not {{a,b},{b},{b,c}} is a base?

Julian Rachman

Because those are the elements of the topology.

anon

the topology also includes {} and {a,b,c}. I think you are getting confused because the base and the topology itself have nearly the same sets. maybe you should pick a bigger topological space so you don't make yourself dizzy.

or by "those" were you referring to the sets at the beginning of my "is..." questions?

Julian Rachman

Yes. My head is a little dizzy but I can bear it. (For math!)

anon

in any case your study might be more fruitful if you picked more meaningful topological spaces

for instance, you're aware every open set in R^n can be written as a union of balls, or a union of boxes?

Julian Rachman

01:38

Yesw

anon

I don't see why you would need any more examples than those. Those perfectly illustrate the point of bases.

Bitrex

@DisplayName I have a query script for you, just tidying it up a little bit

Julian Rachman

@anon I just wanted to develop an example for future reference.

Display Name

@Bitrex That was fast!

Bitrex

@DisplayName I think the script may also return questions that are in the negative and have no answers and aren't closed, getting rid of those may take a little more work but there don't seem to be too many of them

Display Name

01:44

@Bitrex Yeah that's fine. negative with no answer questions are autodeleted

Bitrex

@DisplayName please try this: data.stackexchange.com/math/query/272758/…

2

Display Name

Looks good! 16216 questions with these qualities

Bitrex

@DisplayName ya lol

Display Name

Crap, ones at bottom of list have answers

Bitrex

:(

Display Name

01:47

Oh wait nvm

It works

The ones at the bottom are ones that were just put up and the system doesn't know yet that they are answered

Bitrex

Cool!

Enjoy!

Display Name

@Bitrex Thank you!

Bitrex

:D

@DisplayName (made possible by Xenos in #cobol on synirc.net)

Display Name

Good to know :)

Modded Bear

does anyone know how to solve this with a computer?

0

Q: How many such families exist? (Placing pigs into pens)

Consider the set $[n]$ and a positive integer $k$. Now consider the set $F$ of families of subsets of $[n]$ exist such that: There are exactly $k$ elements in $F$ Subsets can appear more than once Each subset has an odd number of elements Two intersecting subsets $A$ and $B$ satisfy $A\subsete...

combinatorics

anon

02:32

lol'd

6

@ModdedBear having some trouble understanding that

Modded Bear

great, someone actually tried.

what part is unclear?

anon

let's start with bullet point 2, subsets of what appearing in what?

Modded Bear

subsets of $[n]$

$[n]$ is the set $\{1,2,3,4\dot n\}$

anon

yes

so every element of F is a multiset of subsets of [n]?

Modded Bear

no, $F$ is a multi-family of subsets of $[n]$

anon

02:36

multi-family is not a word

or are you yanking my chain?

Modded Bear

ok, $F$ is a family which contains subsets of $[n]$, but each subset can appear more than once

so for example $F$ could be $\{1,\},\{1\},\{2\},\{3\}$ when $n=3$ and $k=4$.

anon

you said "the set F of families of subsets of [n]" which implies every element of F is a "family of subsets of [n]," although the word "family" should be amended to "multiset" because multisets indicate the possibility of multiple membership whereas families do not

Modded Bear

Oh dang, then what you said at the beginning is correct

yeah, $F$ is the set of all such multisets of subsets of $[n]$

JMoravitz

@Modded you seem to be going back and forth describing two different things. Lets take a small example, let $n=2$. Is $F=\{~\{\{1,2\},\{1,2\}\},~\{\{1,2\},\{1,2\}\}~\}$ allowed?

(ignoring the condition that there must be an odd number of pigs)

anon

this is insane, maybe you should describe the problem in terms of pigs and pens first

then I'll try to figure out how I'd express the problem without reference to them

Modded Bear

02:41

I think it is ok now anon

srry for confusion

anon

ah good I was going to ask about "F" in your first bullet point later

but you've fixed it so I can read on

you should fix it in your second bullet point too

JMoravitz

"for example F can contain..."

Modded Bear

@JMoravitz If you give $n$ and $k$ $F$ should be automatically defined.

the elements of $F$ are going to be the multisets $M$ satisfying the properties in the bullets

JMoravitz

Yea, I wasn't looking at the question at the time, just trying to help sort through if it was a multiset of multisets of subsets of [n] or if it was a set of multisets of subsets of [n]

Juan

03:08

Hello everyone

Is anyone online?

I need some help with probability

0

Q: Expected Value - Probability - COVARIANCE

I need some help about calculating a Covariance having the density function (I hope this names are OK, I'm a Spanish speaker). The density function is: $$f_{X,Y}(x,y)=\frac{1}{x}.exp(-\frac{4x^2+y}{2x}).I{x>0, y>0}$$ I need to calculate $E[X.Y]$ and I don't know how. I know how to calculate $E...

probability probability-theory probability-distributions

Quaxton Hale

03:21

What are some good resources on $L_1$ approximation?

Semiclassical

ugh. trying to do reason correctly re: contour integrals with branch cuts is tricky when your brain refuses to remember obvious things

Axoren

03:53

Is there a notation for all finite conjugations of truth expressions in a set?

Shahab

04:17

hi

Do you think that in doing mathematics definitions are more important then theorems?.

I feel that definitions are the chief reason why these theorems exist in the first place and the inherent beauty of the subject lies in capturing hidden mathematical ideas through carefully crafted definitions. In a more ethereal sense the definitions already exist whereas the theorems come into existence when one is convinced of the justification provided by the proof. Hence definitions preempt theorems and should occupy the center stage of mathematics. (When I talk of definitions I also mean the motivat…

Semiclassical

that would seem to imply that (for example) while the concept of integers and primes would exist from the moment they were defined, the infinitude of the primes would not come into existence until someone were to prove it.

so to separate the theorems of an axiomatic system from the axioms themselves seems a bit strange to me

(were i better versed in the subject, now would probably be the right moment to raise the specter of Godel's incompleteness theorem. but i'll leave that to someone with better expertise)

Axoren

Definitions are like theorems that have no predecessors.

Semiclassical

what about a scenario wherein there are multiple but equivalent ways to state the axioms of a system? say, Euclidean geometry with the parallel postulate replaced by something which is expressed differently but is logically equivalent (like playfair's axiom, if i remember right)

then one would seem to be in the situation of saying that each possible 'grounding' of Euclidean geometry, in that fashion, would possess a different set of basic axioms. but any statement that's true in one such formulation would be true in any other

not sure that's insensible, just a bit strange to my brain

Mike Miller

04:36

if you fiddle with the things you just said just a bit, you've stumbled on the idea of model theory and elementarily equivalent models, I think

Semiclassical

heh, neat.

Axoren

@MikeMiller Do you know if notation questions are appropriate on Math.SE?

Semiclassical

i've read stuff on godel in the past, so i'm probably a bit conditioned to be able to talk in that spirit (though with no pretenses at rigor!)

Mike Miller

I think so, @Axoren.

Semiclassical

i'd put it under the terminology tag

Axoren

04:38

@MikeMiller @Semiclassical Thanks

Mike Miller

@Semiclassical When you say read stuff, do you mean actual mathematical writing, or more pop-sci writing?

Semiclassical

closer to the latter than i'd like to admit

Mike Miller

The latter, sadly, does a terrible terrible job of explaining things, and it's easy for it to lead one astray.

Semiclassical

mostly Rachel Goldstein's biography, though, which i thought did a good enough job

Mike Miller

(in the same way that I want to read some physics... eventually) you should try to read a mathematical logic book sometime if you enjoyed thinking about Godel's

usually at the introductory level it's all quite simple, simple enough that I could grasp it as a sophomore even though I was dumb as bricks then

Semiclassical

04:40

heh, maybe. i'm decidedly a dilettante at such things

i'm trying to remember, though, what a relatively concrete example of model theory would be. does ZF set theory + axiom of choice versus ZF+ negation of choice count?

Mike Miller

no, no, a model is like, a concrete realization of an axiom system

Semiclassical

ohh

Mike Miller

and model theory, then, is the study of models

Semiclassical

right. hmm

infinitesimal

05:14

For you gamers :-)

Axoren

05:35

The problem with Minecraft being the "go to" for creativity is that everyone always makes 100% symmetrical dirt dildos half the time

There are probably far more creative games that are twice as fun, but less popular.

infinitesimal

06:20

For you thinkers :D

Committing to a challenge

06:50

@Axoren 100% symmetrical dirt dildos? Is this a thing you have seen built?

First post on the the Mathematics chat with the total chat message count over 20,000,000(He got 20,000,068) goes tooooooooo:

yesterday, by Julian Rachman

Hey. I had a quick question: If you had a topology $\tau$ on set $X=(0,1)$, does that mean that we can form a basis with all singleton sets of $(0,1)$? Moreover, can you do that with all other topologies and take the collection of singleton sets as a basis?

Mike Miller

hell, and it's even a math question!

Committing to a challenge

@MikeMiller Happy days :)

For the record, the one who actually got it...:

in The 2nd Monitor, yesterday, by rolfl

Chris's sis

Greetings

Committing to a challenge

@Chris'ssis How are you :)?

Chris's sis

@Committingtoachallenge Great! Ready for some work.

Committing to a challenge

07:00

@Chris'ssis That is wonderful to hear!

Chris's sis

Not before taking some milk and cocoa. (brb)

Committing to a challenge

@Chris'ssis I need coffee myself, be back

Chris's sis

@Committingtoachallenge OK

Mike Miller

07:16

Morning, @Huy.

Huy

It's high time you went to bed, @MikeMiller.

Mike Miller

No, it's only 11.

Huy

Ok.

I have a cold.

I hate it.

On the plus side, the power saving light bulbs I got for free from IKEA are really good, so far.

@MikeMiller: How are you progressing with your work?

Chris's sis

07:49

hmmm, I've just developed a new kind of integral ...

infinitesimal

:-)

Chris's sis

@infinitesimal Does it sound better now?

Axoren

@Committingtoachallenge Yes. All the time.

Sometimes, they put in the effort to use Pink Wool instead. Creative Multiplayer is the worse thing possible.

I completely forgot to submit a peer review I've been writing for a month and a half, only to remember that I reformatted the laptop I was writing it on.

It's due today.

infinitesimal

@Chris'ssis Greetings.

Chris's sis

@infinitesimal Greetings :D

infinitesimal

07:55

How are you @Chris'ssis?

Chris's sis

@infinitesimal Great these days, very creative! I feel I'm going to publish a very good book. :-) You?

infinitesimal

@Chris'ssis fine thanks.

Chris's sis

That's fine!

Committing to a challenge

08:12

@Axoren wtf Holy crap man that doesn't sound good at all

Axoren

@Committingtoachallenge It all started when I tried to install xubuntu-desktop. Then, when I tried to remove it.

It never left and everything broke.

Logging in was no longer an option, because logging in would just being me to xubuntu's login screen

Which was no longer installed

Committing to a challenge

08:39

@Axoren So what are you going to do about the due date?

Axoren

@Committingtoachallenge Writing it from memory right now.

To the best of my ability.

Committing to a challenge

@Axoren That's crazy!

Axoren

It's due today, which according to my watch, it is only 3:40am right now

I'm almost done.

Committing to a challenge

@Axoren Are you the guy who made the coffee tex tool I was testing?

Axoren

Nope. Why would I be that guy?

infinitesimal

08:41

You must have a good memory.

Committing to a challenge

Oh sorry that was Kasper

Axoren

@infinitesimal Only for things I've written or things I will write.

I once memorized the constants file in my compiler's course, years ago.

Gone now, but I knew the actual integer values of every constant.

It's not hard, it's like photographic memory, but much easier.

Because it's just text.

infinitesimal

:O

Committing to a challenge

@Axoren What was the first thing we talked about :P

Axoren

That was a while ago. I think it was about "iwriteonbanana's" name.

I only remember things that are locally important temporally, I guess I'd say.

Committing to a challenge

08:46

@Axoren Fair enough

Axoren

local temporally?

I don't even know how I'd combine those words.

To mean what I mean

Committing to a challenge

@Axoren What time do you normally go to sleep?

Axoren

Lol

Let's just say that distribution has no defined expected value.

infinitesimal

Temporally important.

Committing to a challenge

Your talking on chat graph seems suggestive :P

Axoren

08:48

@infinitesimal But only local time.

Like recent within some neighborhood.

Or timehood

A Timeberhood.

Committing to a challenge

@Axoren You seem to sleep about now for the next 7 hours

chat.stackexchange.com/users/134723/axoren

Axoren

That's just when I get off chat.

And hop on my laptop and write stories in bed.

Committing to a challenge

@Axoren But you talk at every other point in the day

Axoren

That's just when I have things to say.

Or when people are here.

Committing to a challenge

@Axoren In the most part our times on are opposite

Axoren

08:50

If I had to describe a subset of when I fall asleep, it's usually at ~9:00pm

That's when I'm not working on anything during the week.

Otherwise, it's sporadic and all over the place.

Committing to a challenge

@Chris'ssis Out of curiosity, how is your integration method new?

Chris's sis

@Committingtoachallenge I said a new kind of integral, not a new integration method. There is a difference between them.

Committing to a challenge

@Chris'ssis What does that mean still?

Transcript for

$Mathematics$ Mathematics