choose k=1 then that would imply all a are in the kernel
#help-33
regal coral
red nimbus
yea
willow owl
red nimbus
lol I mean sqrt(o²) but actually I understand now
elfin berryBOT
𝔸dωn𝓲²s
upbeat sable
what exactly are you confused about?
red nimbus
Actually I think I am done, I will finish it later.
I will consider the cases of phi being bijective or not and G being of finite or infinite order
thanks for the help!
.solved
manic stirrup
i get the result 2e^-2(1+2sin^2 t) :/
marsh citrusBOT
i get the result 2e^-2(1+2sin^2 t) :/
manic stirrup
no we are just simplifying this.
yea
(2cos^t + 4sin^2 t)e^-2 = 2e^-2(1+sin^2t)
marsh citrusBOT
@manic stirrup Has your question been resolved?
hidden dawn
marsh citrusBOT
hidden dawn
limber hearth
Soit une fonction phi : [0,pi/2] -> R et de classe C1, et avec une ipp tu montres que l'intégrale de 0 à pi/2 de phi(t) sin((2n+1)t) dt converge vers 0
Pour la 2
Tu laisses phi(t) = u
Dans la formule
limber hearth
Donc
On a le produit d'un truc qui converge vers 0 et d'une suite bornée
Ca déduit la convergence vers 0
Ensuite
Après
Tu peux montrer que la fonction 1/t - 1/sin(t) vérifie les conditions de phi
Donc que c'est une classe C1 sur 0,pi/2
(Avec des équivalents et des continuité de dérivées, prolongement...)
Et après
Enfin
hidden dawn
Tu peux montrer que la fonction 1/t - 1/sin(t) vérifie les conditions de phi
jai pas trop compris ça
elfin berryBOT
limber hearth
Autrement dit
hidden dawn
ahh
limber hearth
C'est la suite qu'on a dit qui converge vers 0
limber hearth
C'est pas vraiment majorée ici
hidden dawn
comment?
limber hearth
C'est surtout que ça converge par produit de suite bornée par une suite convergente
En particulier vers 0
Et comme c'est les même
Bah Un - Jn -> 0
Donc
Jn -> Un
hidden dawn
ah
hidden dawn
C'est surtout que ça converge par produit de suite bornée par une suite converge...
test de la comparaison directe
j'ai compris ta méthode
limber hearth
Je ne savais pas qu'il portait un nom
limber hearth
Good to know
hidden dawn
of course, my friend
limber hearth
Et pour le dernier tu pose u = (2n+1)t
Fais attention aux bornes
Et tu auras ton résultat
hidden dawn
bah j'ai cru que le dernier était le plus dur
limber hearth
Non pour le coup c'est plus le deuxième
Peut être qu'il y a une méthode plus courte
Mais la j'avais que celle là en tête
limber hearth
Tout va bien pour la 2 ?
hidden dawn
oui, mrci bcp
limber hearth
Parfait
hidden dawn
juste qlq chose avant que tu quittes
limber hearth
Oui ?
hidden dawn
ça c'est un exercixe qui n'est pas de mon niveau
limber hearth
Celui ci ?
hidden dawn
ma sœur m'a proposé ça
ppur m'aider à résoudre l'integrale de dirichlet
(c'est la dernière integrale, I)
hidden dawn
mais comment je peux continuer
je veux montrer que I = pi/2
tout ça comment va m'aider?
spark otter
Un
hidden dawn
mais il y a pas de but pour 1)?
spark otter
non l inverse
limber hearth
Ah
spark otter
C'est In qui existe pas
spark otter
XDD
hidden dawn
damn
elfin berryBOT
hidden dawn
bravoo
limber hearth
Ah la on est bien la
limber hearth
Bah merci à rafilou aussi pour le coup
marsh citrusBOT
limber hearth
Strangequark has something to say i think
velvet cedar
Nah I've just been translating the conversation
limber hearth
The goat
marsh citrusBOT
Available help channel!
honest perch
gyato
honest perch
.close
marsh citrusBOT
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weak wind
can someone help me on this question:
marsh citrusBOT
can someone help me on this question:
valid cape
!15min
marsh citrusBOT
Please only use the <@&286206848099549185> ping once if your question has not been answered for 15 minutes. Please do not ping or DM individual users about your question.
lethal bridge
I understand up to the box highlighted in red where they convert things back in terms of t; I don’t get how they found that
marsh citrusBOT
I understand up to the box highlighted in red where they convert things back in ...
@lethal bridge Has your question been resolved?
lethal bridge
can anyone explain how they got this, i understand everything up to the bit highlighted in the red box
marsh citrusBOT
can anyone explain how they got this, i understand everything up to the bit high...
main idol
They substituted back in for t using t =√3 tan(theta) + 1
Solve for tan(theta) then use Pythagorean identities to solve for sec(theta)
marsh citrusBOT
@lethal bridge Has your question been resolved?
marsh citrusBOT
marsh citrusBOT
Available help channel!
fathom sun
prove that there is no integer n such that $3^n \equiv 99 \pmod{100}$
elfin berryBOT
Dork9399
sweet pawn
marsh citrusBOT
@fathom sun Has your question been resolved?
hollow glen
what is 99 mod 20?
fathom sun
19?
marsh citrusBOT
marsh citrusBOT
Available help channel!
void elm
okay so, I need to find a counterexample to the conclusion of this lemma for when A is not closed, but I'm kinda doubting my negation of this statement
marsh citrusBOT
okay so, I need to find a counterexample to the conclusion of this lemma for whe...
spark otter
okay so, I need to find a counterexample to the conclusion of this lemma for whe...
hint: ||think about 1/x||
void elm
am I proving that
for any open U containing A, there doesn't exist a smooth function st both of the two properties hold; or am I proving that there exists an open U containing A st there isn't a smooth function st both of the properties hold
void elm
hint: ||think about 1/x||
I just need the negation rn so I know what I'm proving 
idk why these statements always trip me up
spark otter
you don't have to prove the negation
you have to prove the original statement becomes false in general
if A is not supposed closed
void elm
you have to prove the original statement becomes false in general
but which statement is the one that becomes false?
spark otter
"false in general" = not(true for every instance of A, f)
void elm
that's my confusion rn
void elm
"false in general" = not(true for every instance of A, f)
what about U?
is U arbitrary, or can I choose one?
that's primarily my confusion I think
spark otter
well that could also be part of your counterexample
void elm
right, okay
spark otter
if you show that there is a U such that this doesn't apply
then the statement is false in general too
void elm
hmm, can I share smth?
spark otter
sure
void elm
this surely doesn't work, right?
I chose A, f, and U here
ah, do I need to prove that every smooth extension doesn't have at least one of the desired properties?
because I only proved it for one, I think
namely the identity
void elm
this surely doesn't work, right?
also wait, the support of f~ is R, not R \ 0
@spark otter hm, does this work better? 
hm, I guess this works 
it makes sense to me
thanks for the hint @spark otter 
.solved
marsh citrusBOT
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obtuse plume
I'm curious what rules we can apply here to find out the answer?
ln(e^3)
According to my book there is a rule we can use so-called e^ln(x)=x, but I don't see the relation between those 2
limber hearth
What do you mean by the relation between those two ?
worn sluice
$f^{-1}(f(x)) = x$ for appropriate domain and range
elfin berryBOT
Pro_Hecker
limber hearth
Actually you also have that ln(e^x) = x
obtuse plume
Well how would one solve this in a logical order?
safe badger
ln(e^3) = ln(eee)
obtuse plume
Because also in the book it says that
10^x = a <=> x = lg(a), can we just simply distribute 10^x as a?
So we would get x = lg(10^x), make 10 a constant instead so it becomes lg(a^x) = x?
limber hearth
Ln is a log base e
onyx reef
Because also in the book it says that
10^x = a <=> x = lg(a), can we just simp...
Here it is log_10(10^x) = x, the base of the log needs to be the base inside
nimble prairie
Are you familiar with the usual rules for powers?
obtuse plume
Are you familiar with the usual rules for powers?
yes
nimble prairie
yes
Well, log rules are just power rules in disguise, try and translate the problem into a power rule problem
obtuse plume
In e^ln(x) = x
The question would be, what number does x have to be in e^ln(x) to get x
but in ln(e^3), it's a different format
nimble prairie
Actually you also have that ln(e^x) = x
I mean also this is just this btw, it’s almost by definition
obtuse plume
But how can you explain that rule?
If you'd start with ln(e^3), what would your next step be, if you weren't familiar with that rule
nimble prairie
If you’re aware that log and exp are inverses of each other this becomes easier, but alas.
Call ln(e^3) some number y, that is
ln(e^3) = y,
now using the rule you only know somehow; namely that e^ln(x) = x, then we get that
e^3 = e^y, what number y satisfies this?
obtuse plume
If you’re aware that log and exp are inverses of each other this becomes easier,...
I understand that part, but what are the steps you took between going from ln(e^3) = y to e^3 = e^y. You used e^ln(x) = x, but in what step?
obtuse plume
Hm I think I start to understand as
lg_a(a) = 1, meaning that
e^ln(e^3) = e^y, would become
e^3*ln(e) = e^y (multiply with 1)
e^3 = e^y
nimble prairie
I’m only using the rule you gave us, that e^ln(x) = x
sharp harness
$e^y = \textcolor{red}{e}^{\textcolor{red}{\ln}\textcolor{red}{(}e^3\textcolor{red}{)}} = e^3$
elfin berryBOT
bee [it/its]
obtuse plume
I’m only using the rule you gave us, that e^ln(x) = x
Ah I think I understand now
I wasn't really aware that we could just lift the whole equation up with e^, so that you'd get the right format of the rule I gave
nimble prairie
I wasn't really aware that we could just lift the whole equation up with e^, so ...
It might be easier if you think of it as just applying a rule to both sides, namely if e^ corresponds to exp() of something, would that seem more comfortable?
obtuse plume
I'm not quite sure what exp() does, as this book first covers log and ln
I'd assume it's just the opposite of log
nimble prairie
Oh it’s still e^ but with a different notation, I’m simply wondering if you would think it’s more comfortable that you’re allowed to apply a rule to both sides of an equation, that is going from x = y to f(x) = f(y), where f() is some rule, like e^
obtuse plume
Ah right, just as we can apply it with logs you mean? that a = b <=> lg(a) = lg(b)
nimble prairie
Right, you’re just applying something to both sides
The same thing that is of course
obtuse plume
So what you did, is that allowed with all sorts of numbers?
that if a = b <=> 10^a = 10^b for example
nimble prairie
As long as the operation/rule you’re applying is defined for the numbers
Going from -1 = -1 to log(-1) = log(-1) might be weird
obtuse plume
That makes sense, I think I've just more struggled with it as so far we haven't come across a question that makes you take the base equation as the exponent
nimble prairie
Keep at it, you’ll be more comfortable at it with experience
obtuse plume
Keep at it, you’ll be more comfortable at it with experience
Thanks for the help anyways! I really appreciate it, makes a lot of the questions a lot more easy to solve 😄
.close
marsh citrusBOT
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Available help channel!
red nimbus
. @hard berry
hard berry
Yess
red nimbus
,, x^2y''-xy'-3y=x^2\log x
elfin berryBOT
𝔸dωn𝓲²s
red nimbus
This was your ques
hard berry
This was your ques
Yess can you help with it
red nimbus
no sorri
hard berry
Ok
red nimbus
Ok
It's a Cauchy-Euler equation if you look at the x^n terms, so maybe try the substitution x = e^u
odd crest
This is an equation of the form $P(x) y'' + Q(x) y' + R(x) y = S(x)$
elfin berryBOT
Katharine
odd crest
smart people have solutions for these kinds
:D
have found solutions
idk if you have to use a specific method
marsh citrusBOT
@hard berry Has your question been resolved?
red nimbus
If you consider $x = e^u$ you can turn the differential equation into one in terms of $u.\$
So $y(x) = y(e^u) =: z(u)$, now you need to calculate $y'$ and $y''$ wrt. $u$ using the chain rule and you will turn it into a linear differential equation with constant coefficients.
elfin berryBOT
𝔸dωn𝓲²s
patent fjord
Help
red nimbus
(x,y) is in R² not R³ btw
patent fjord
(x,y) is in R² not R³ btw
I am so clueless with these proofs 😭😭
red nimbus
Your proof is sexy imo
patent fjord
Like rigorous you mean
patent fjord
you wrote down x²+y² instead of x²+y^4
Yes you’re correct 😭😭😭
red nimbus
But you should still be able to adjust these minor things
patent fjord
I am on it
red nimbus
Like the overall approach is good I think
marsh citrusBOT
marsh citrusBOT
Available help channel!
patent fjord
Help
patent fjord
I have a simple continuity proof which I have just crafted. Please help me to check if it’s rigorous
void elm
mm, the proof works, but it's a formatted in a pretty clunky way
patent fjord
mm, the proof works, but it's a formatted in a pretty clunky way <:kongouderp:48...
How to improve it because I am not studying mathematics as major, so formatting is hard for me 😭😭
void elm
what definition? what is "it'?
red nimbus
I have a simple continuity proof which I have just crafted. Please help me to ch...
From where is this definition?
void elm
but you don't know that 1/x^2 is uniformly continuous yet
really, what you're doing in the first part is scratch work in order to find delta
void elm
if, yes
but you can instead just do the scratch work on the side
and in the final proof, omit it
like
patent fjord
Like the later part?
void elm
Let $\varepsilon > 0$. Choose $\delta = \frac{\lambda^3 \varepsilon}{2}$. Then for any $x, y \in [\lambda, \infty)$, if $|x - y| < \delta$, then (your computation goes here), so $\left|\frac{1}{x^2} - \frac{1}{y^2}\right| < \varepsilon$, which means that $\frac{1}{x^2}$ is uniformly continuous on $[\lambda, \infty)$.
elfin berryBOT
higher!
void elm
this is the kind of structure that the proof should ideally take
patent fjord
Let $\varepsilon > 0$. Choose $\delta = \frac{\lambda^3 \varepsilon}{2}$. Then f...
Can I write the conclusion and then use it to implies the uniform continuity if it is required to write all steps
red nimbus
Can you explain this step
patent fjord
I mean I am probably going to take real analysis next year
void elm
Can I write the conclusion and then use it to implies the uniform continuity if ...
I can't parse this sentence
patent fjord
Can you explain this step
This is easy, bcs I factor out (x-y) and then I separate x/x^2y^2 +y/y^2x^2
void elm
This is easy, bcs I factor out (x-y) and then I separate x/x^2y^2 +y/y^2x^2
adonis is probably asking about the <=
red nimbus
ikr but then you do some lambda wamba trick
void elm
And since lambda is smallest in the interval
so what?
patent fjord
So the expression is bounded
void elm
why does this mean that 1/x^2y < 1/lambda^3
patent fjord
Because it is bounded from above
patent fjord
Like it can greater than 2/lambda for that part right
I mean I ain’t studying math 😭😭
void elm
you can say that since x and y are in [lambda, infty), x, y >= lambda
so 1/x and 1/y <= 1/lambda
patent fjord
Noted
patent fjord
Yes next time I will refine my phrasing
I am definitely going to fail real analysis if I were to choose it next year
I feel like it’s just so difficult
And challenging
void elm
I am definitely going to fail real analysis if I were to choose it next year
I doubt it
you're much more capable than you think
red nimbus
I am definitely going to fail real analysis if I were to choose it next year
lmao even me wouldn't come up with that what higherino explained
patent fjord
Hopefully I will jump into another one
void elm
it takes some getting used to, but you will get used to it
void elm
lmao even me wouldn't come up with that what higherino explained
higherino?! 
that's such a silly nickname :p
patent fjord
Thanks for that kind words 🥰🥰🥰
void elm
Thanks for that kind words 🥰🥰🥰
good luck with your real analysis studies c:
red nimbus
I mean I ain’t studying math 😭😭
that's not the last time i will hear from you
you just happen to be an economist and math prodigy 
i get it
😂
patent fjord
that's not the last time i will hear from you
Which is true and one of the biggest disadvantage of mine 😭😭
The nuanced formalism is too hard for those not studying it
void elm
The nuanced formalism is too hard for those not studying it
hell, I didn't understand the formalism until after my RA course
red nimbus
The nuanced formalism is too hard for those not studying it
you seem to have it mastered already
void elm
and yes, you're doing much better than I was at the time
it's just a few details you need to polish up
patent fjord
That’s so much flattering thanks I will definitely try harder and next year around this time I’ll be confident enough to take real analysis 🥰
patent fjord
There must be a reason for 50% dropout rate though
red nimbus
There must be a reason for 50% dropout rate though
you're just built different trust me
patent fjord
Better prepare in advance
red nimbus
you are def. not ordinary
void elm
every RA course in existence has a 50% drop out rate
that's mostly because it's
a) often students' first introduction to proofs
b) often used as a weedout course by unis
patent fjord
every RA course in existence has a 50% drop out rate
Btw I have another work
void elm
what other work? 
patent fjord
patent fjord
what other work? <:catthimc:732011802459635713>
Can you check if this indeed proves Dini theorem
red nimbus
Can you check if this indeed proves Dini theorem
Can you teach me how to make money, as the economist that you are 😄
patent fjord
Buy index fund and never withdraw
void elm
Can you check if this indeed proves Dini theorem
hm, I could probably make an attempt, but I'm also not very confident in my ability to check analysis proofs in general
this subject is my worst haha
patent fjord
I definitely must try harder than…
void elm
I think you are better at analysis than I am 
red nimbus
real analysis
patent fjord
I must say baby rudin is too hard since I am studying it without lecturing with it maybe there’s easier book?
void elm
What does a real analysis course do?
uhh, lots of different things tend to be called "real analysis", unfortunately
the most standard kind of course I see covers things like
continuity, differentiation, integration, sequences/series, convergence of sequences of functions, some basic topology, etc
patent fjord
What does a real analysis course do?
Basically a part of math55
odd crest
no idea what math55 is
void elm
some courses are called "real analysis", but are more like measure theory
red nimbus
I definitely must try harder than…
void elm
no idea what math55 is
that would be one course in Harvard's hardest math sequence for freshmen
patent fjord
Just a lot of school uses that math for a combination of math subjects
void elm
I don't know too much about it though
void elm
some courses are called "real analysis", but are more like measure theory
fwiw, my uni has like
patent fjord
Real complex analysis and abstract algebra and LA
void elm
4 courses that are called real analysis or analysis or whatever, in one form or another
patent fjord
that would be one course in Harvard's hardest math sequence for freshmen
Not only Harvard I think
void elm
but they cover pretty different material
patent fjord
Is there an easier book than rudin s principe of mathematical analysis
patent fjord
This book is so hard for self study
red nimbus
This book is so hard for self study
be harder
void elm
literally anybody
baby Rudin is really hard to self-study from
the man writes like the reader already knows analysis
patent fjord
Because I feel like it wasn’t that hard at beginning but gradually it becomes excruciating
patent fjord
Like I was using calculus from Micheal spivat for calculus 2 so maybe it helped me a bit for self studying?
I will definitely buy a easier book
void elm
ah, if you've read Spivak, then you already know most of basic analysis 
odd crest
in my university we have infiniaalrekenen 1 and 2
which is everything calculus
and analyse
which is i think a bunch of real analysis stuff
void elm
in my university we have infiniaalrekenen 1 and 2
Dutch?
odd crest
yis
void elm
icic 
odd crest
i have done only 2 mathematics courses
proofs in mathematics and introduction groups and rings
:D
i'm a physics students
s
void elm
mm, but you'd learn quite a bit of math via physics anyhow
odd crest
ye
patent fjord
i'm a physics students
I must say that the way you choose N is super professional
void elm
I'm partially a physics student
patent fjord
I am an art student
void elm
I'm partially a physics student
mechanics nearly blew up me the last semester
but I managed
odd crest
and i really wanna study fields and galois theory
void elm
algebraist? 
I know literally zero algebra still
I should really get to studying it but... idk I just can't be bothered to
whenever I see adonis and you talk algebra, idk what's going on at all
red nimbus
Algebra is a rough love
void elm
it's kind of a shame that manifolds don't seem to be standard in math curricula
it takes time for somebody to help me usually
patent fjord
I also know 0 algebra
void elm
which is okay, but it's also kinda 
red nimbus
I also know 0 algebra
And yet you use it all the time
void elm
it takes time for somebody to help me usually
I think I pretty much wait for a select group of 6 or so people to eventually see my help channels
patent fjord
But I don’t know things like sets containing binary operations and that kind of algebra
void elm
my motivation for manifolds is GR
patent fjord
i really wanna learn about differentiable manifolds
Ricci flow?
void elm
nahh, geometry/topology is easier than analysis
red nimbus
That sounds just too hard
dw life gets pretty abelian with time
void elm
everything is easier than analysis
void elm
nahh, geometry/topology is easier than analysis
don't emote me cat bit 
odd crest
what book do you use to study manifolds
patent fjord
dw life gets pretty abelian with time
What if it’s not commutative anymore?
void elm
what book do you use to study manifolds
Lee's Introduction to Smooth Manifolds
though it's overkill if you just want enough to do GR
I'm probably going to read Wald's book eventually
for GR, that is
red nimbus
My admiration for y'all is very great
I am the least capable person suitable for university
patent fjord
My admiration for you all is very great
You guys broden my horizon and helped me so much
void elm
nah, I scrolled through Lee yesterday and nearly got a heart attack looking at how much material there is
I don't think I'm gonna be able to get through this book in less than 2 years
hyzae the doctor returns c:
odd crest
i'm currently very stuck in a short book and it's doing my head in
void elm
which book?
void elm
ah, I don't know very much about GA
patent fjord
Linear algebra I will recommend the all time classic linear algebra done right
That’s best book
odd crest
i can do it if i take something from the bit after the exercise i am stuck at
if that makes sense
patent fjord
Even Econ students who take la will use that book
patent fjord
<:angerywoog:545476186235731978>
Isn’t that the best book?
void elm
I'm just playing around haha
red nimbus
Linear algebra I will recommend the all time classic linear algebra done right
Even though I am done with my LA courses, is it really worth that book?
patent fjord
Even though I am done with my LA courses, is it really worth that book?
I feel most U.S. universities use it
void elm
the man acts like determinants killed his family
patent fjord
Many students of engineering or economics and other math demanding fields will use it
All time classic
No doubt
LA done right
red nimbus
the man acts like determinants killed his family
what 😂
void elm
I'm a FIS shill
void elm
what 😂
so the thing is like, Axler wrote his book with the intention of excluding dets from the material until the very end
but for literally no good reason
odd crest
hold on the best algebra book is abstract algebra
patent fjord
You can use invariant to solve most of concepts in LA though
odd crest
by dummit and foote
void elm
he says that it's how linear algenra "should be done", hence the name
but like
did the dets kidnap your family or what
patent fjord
Leveraging Cayley Hamilton theorem
void elm
I just don't understand why
patent fjord
You can even craft formula for characteristics polynomial for n x n matrix easily
void elm
by dummit and foote
there are two principal things I've heard about D&F
a) it's a really good book to read before bed, because it'll make you fall asleep within minutes
b) the only way to make it less dry is to dunk it in water
patent fjord
Linear algebra is a small part of algebra and the only thing I studied as Econ students at mathematics department
red nimbus
true (waiting for Quadratic Algebra)
void elm
there are two principal things I've heard about D&F
a) it's a really good book...
I cannot verify these claims, because I've not read it, but many on this server and elsewhere keep iterating this
or variants of these
patent fjord
Yes and cal 1 cal 2 cal 3
red nimbus
Yes and cal 1 cal 2 cal 3
Anyway I think it's time for an integral 😄
void elm
maybe I'll find a reason to open it up
void elm
Like where are you from you guys take linear algebra after analysis
I did LA simultaneously
we did most of FIS
void elm
other algebra books do? 
odd crest
idk if other algebra books do
void elm
ic
patent fjord
This is also a nice book I think
void elm
Aluffi is recommended a lot
again though, I can't speak to it's quality 
algebra just seems kinda boring to me, so I can't bring myself to go and study it until I'm forced to 
@red nimbus have you been enjoying algebra?
patent fjord
You should read my Econ books
void elm
econ is analysis so nty 
patent fjord
They are what the definition of boring is
red nimbus
<@587715738484211754> have you been enjoying algebra?
yes
red nimbus
none, it's a script that the prof created
patent fjord
thankfully I don’t have to study harder math
Analysis is considered the last part for Econ student 🥰
red nimbus
thankfully I don’t have to study harder math
If this was hard for you, just wait 
void elm
not probability?
don't econ students need prob/stats?
and mathematically, they are harder than basic real analysis
patent fjord
There’s a tradition that Econ students take real analysis to show their capability
An U.S. thing I guess
red nimbus
There’s a tradition that Econ students take real analysis to show their capabili...
you are very capable 🥰
patent fjord
don't econ students need prob/stats?
They are harder?
void elm
They are harder?
if you do them mathematically rigourously, I'd say so
patent fjord
You mean Lebesgue? I doubt I don’t think mathematical finance
void elm
usually probability requires measure theory
void elm
You mean Lebesgue? I doubt I don’t think mathematical finance
yes, Lebesgue would be a part of measure theory
patent fjord
It’s a path though, but usually it’s more of economics theories…
red nimbus
It’s a path though, but usually it’s more of economics theories…
What's the part you really like about math
patent fjord
If I only focus on economics side then probability (classic) statistics, and real analysis will be super sufficient for my math
patent fjord
What's the part you really like about math
In comparison with economics I like all parts of math
Economics is pretty empty worded thing I guess math is fun… but economics does give me more monetarily rich future
void elm
when you become rich, don't forget about us :c
patent fjord
To some extent I feel like it’s really just to take advantage of people with basic math but it doesn’t matter as long as it gives me money
red nimbus
when you rich remember to give me a penny when i live on the street and u pass by
patent fjord
If only it happened sure! I doubt I will be rich tho I have the dream
void elm
I cannot verify these claims, because I've not read it, but many on this server ...
@odd crest case in point
#book-recommendations message
I literally just saw that too
patent fjord
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tropic vessel
Determine if the following statement is true: $x_n \rightarrow x$ if and only if for every $\epsilon > 0$ it is $|x_n-x| < \epsilon$ for an infinite amount of terms of $x_n.$
marsh citrusBOT
Determine if the following statement is true: $x_n \rightarrow x$ if and only if...
elfin berryBOT
-KlariNoX-
tropic vessel
So im facing this problem, and I tried this approach: I defined a sequence x_n, with x_n = 1/n for odd n, and x_n = 1 for even terms. Then for infinite terms of that sequence it is true that |x_n-1|<ε, for any ε. However, I'm pretty sure I can show that this sequence doesn't converge to anything. Would that be a correct approach?
mellow crag
So im facing this problem, and I tried this approach: I defined a sequence x_n, ...
or even easier, x_n = 0 for odd n, x_n = 1 for even n
tropic vessel
or even easier, x_n = 0 for odd n, x_n = 1 for even n
Hm, right! that will definitely make my arguement easier, thanks!
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junior blade
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junior blade
have I done anything wrong here ?
marsh citrusBOT
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inland thicket
Only whole numbers as input and only whole numbers as output is there more solutions? $f: \bN\mapsto\bN$ where $f=\sqrt[4]{\frac{x^2+1}{2}}$
marsh citrusBOT
Only whole numbers as input and only whole numbers as output is there more solut...
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SilentALume
normal moss
x=13, y=239
Lol
I do not know about in general. It is not hard to show x cannot be even if f(x) needs to be an integer.
But I didn't see an obvious stronger condition.
Besides just saying (x^2+1)/2 is a fourth power of an integer.
marsh citrusBOT
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red nimbus
I have a question, if a group has subgroups, how do I know based on the orders of these subgroups whether they are cyclic?
marsh citrusBOT
I have a question, if a group has subgroups, how do I know based on the orders o...
red nimbus
For example: (Z*_33, multiplication)
I read that cyclic are such subgroups that have a (prime) order
vital oracle
Don’t know much about this, but presumably they mean that a prime ordered group can only be cyclic
red nimbus
So if the order of a subgroup is not a prime number it can be cyclic, but if it is a prime number then it must be cyclic?
Ok thanks!
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crisp sigil
can someone help me w this question
marsh citrusBOT
can someone help me w this question
@crisp sigil Has your question been resolved?
main idol
can someone help me w this question
marsh citrusBOT
crisp sigil
https://math.libretexts.org/Bookshelves/Differential_Equations/A_First_Course_in...
I was looking at a resource online but I was still confused
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quiet plinth
howe
weary karma
toxic zinc
\begin{align*}
b \lim_{x \to \infty} \frac{ \frac{1}{1+a/x} \cdot \frac{-a}{x^2}}{-1/x^2} &= b \lim_{x \to \infty} (-x^2) \frac{-1}{x^2} \frac{-a}{1+a/x} \
&= ab \lim_{x \to \infty} \frac{1}{1+a/x}
\end{align*}
elfin berryBOT
kipf
toxic zinc
the -1/x^2 on the denominator becomes a (-x^2) on the top
which cancels with the 1/x^2 already on the top
bring the a out the limit
you get the right thing
quiet plinth
how abt this one
spark mason
probably the same story
quiet plinth
yh
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tiny patrol
How do I find whether the series converges or diverges?
marsh citrusBOT
How do I find whether the series converges or diverges?
digital hound
But why am I given the first value of the sequence
because its a recursive formula, you need to know the base case
a2 = (3n-1) / (2n+5) * a1
tiny patrol
Ah. Although I will not use it?
digital hound
a1 = (3n-1) / (2n+5) * a0, but there is no a0
tiny patrol
Hm
digital hound
so we need to define what is a1 (the base case)
digital hound
Ah. Although I will not use it?
yes
tiny patrol
Ohhh alright got ya
Ok so I will divide by an correct?
How do I know that an won't be p
0*
digital hound
Ok so I will divide by an correct?
yea you can use the ratio test
tiny patrol
How do I know that an won't be 0 for some value of n
potent reef
Wdym 'increasing' or 'decreasing'?
lunar heath
How do I know that an won't be 0 for some value of n
you are given a1 is positive
not
tiny patrol
?
lunar heath
anyways you know a1 is positive and an+1 is always positive
tiny patrol
Wdym increasing or decreasing?
The sequence
If the terms are increasing or decreasing
lunar heath
so an wont be 0
lunar heath
so an wont be 0
for any real n value
lunar heath
How though
look at the formula of an+1
3n-1 when n is 1 is positive and bottom is positive
when n goes higher and higher it is still going to be positive
potent reef
3n - 1 > 2n + 5 for 'n' > 6.
So from n = 7, the terms will start increasing since the ratio will be greater than 1. I think so.
lunar heath
and the previous value is still positive
because you are given that a1 is 1/3 which is positive
tiny patrol
Ohhh, I just have to plug in values for n make some observations say that an is increasing so it won't be 0
?
lunar heath
yeah you can also mathematically prove it probably
denominator always positive for n>1 and also the numerator always positive for n>1
since a1 is positive, a positive number times a positive number will always give a positive number for n > 1
tiny patrol
Alright. Thanks for your help!!
lunar heath
np
tiny patrol
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dry zodiac
i am doing an exercise where i have to factorize and the equation is y(x^2-y)-(x-y)^2 the common parenthesis would be (x-y) if the x in b(x^2-y) wasnt squared can i remove one x outside the parenthesis making it yx(x-y)-(x-y)^2
marsh citrusBOT
i am doing an exercise where i have to factorize and the equation is y(x^2-y)-(x...
dry zodiac
if not how could i factorize it
desert dirge
no, you indeed cannot do that
dry zodiac
i figured
desert dirge
im also not sure there is all that much you can do beyond expanding and collecting like terms
dry zodiac
im also not sure there is all that much you can do beyond expanding and collecti...
yeah thats what i have done so far but i think that,that may be the wrong way-not intended solutions since the exercise is about factorisation
desert dirge
do you happen to have the solutions
dry zodiac
no we are only given the question
thats actually not the question i just simplified it to see if what i wanted to do was possible
desert dirge
oh, what was the original
dry zodiac
desert dirge
,rccw
elfin berryBOT
desert dirge
(x-1)[ (3x-6)(x+1) - (5y-10)(x-1) ]
think thats honestly it, not much to do with the interior of []
dry zodiac
oh by doing difference of 2 squares on (x^2-1)?
desert dirge
yeah
dry zodiac
hadnt thought of that
yeah there is probably not much in the parenthesis
unless you make it way more complicated
thanks!
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dry zodiac
can you use difference of 2 squares on x^3-1
you could use difference of cubes but i am curious if you could use difference of 2 squares as you could on x^4-1 by making it (x^2)^2-1=
(x^2-1)(x^2+1)
marsh citrusBOT
can you use difference of 2 squares on x^3-1
you could use difference of cubes ...
pliant siren
well you could make it into $(x^{\frac{3}{2}})^2-1$
elfin berryBOT
DerTheo
dry zodiac
intresting but how is 3/2^2 equal to 3
odd orchid
3/2 * 2 is 3
pliant siren
well you know that if we „stack“ powers they multiply
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buoyant pond
What have you tried
buoyant pond
What is the center of the new circle that you will draw
median stag
What is the center of the new circle that you will draw
the questions language is not clear
What is the diameter?
buoyant pond
The given equation will form a chord to the given circle
median stag
can u solve by the normal form
buoyant pond
now the chord is taken as the diameter and a new circle is formed
buoyant pond
I’ll give you a hint: the foot of perpendicular from 0,0 on the line is the center of the new circle
median stag
I’ll give you a hint: the foot of perpendicular from 0,0 on the line is the cent...
idk
median stag
ok
buoyant pond
The white circle is x^2+y^2 =a^2
buoyant pond
The line perpendicular? I’m not sure
median stag
The line perpendicular? I’m not sure
they got the coordinates of centre by it
Its (pcosalpha,psinalpha) but how?
magic condor
I HAVE MY MATHS HIGH SCHOOL EXAMINATION WITH 14 CHAPTERS IN 7 DAYS, WITH NOT GOOD PREP WHAT SHOULD I DO??
pliant siren
I HAVE MY MATHS HIGH SCHOOL EXAMINATION WITH 14 CHAPTERS IN 7 DAYS, WITH NOT GOO...
2 chapters per day
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median stag
I HAVE MY MATHS HIGH SCHOOL EXAMINATION WITH 14 CHAPTERS IN 7 DAYS, WITH NOT GOO...
you're cooked
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covert scarab
$\int_{-40}^{\infty}\frac{49cosc}{-20c}dc$
marsh citrusBOT
$\int_{-40}^{\infty}\frac{49cosc}{-20c}dc$
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Nyxzore
elfin berryBOT
covert scarab
What am I doing wrong?
deep anvil
What am I doing wrong?
Hi Nyx, it appears to be the case that you used the product rule for derivatives (which does not apply to integrals)
So the mistake happens early on when you computed the antiderivative.
covert scarab
I did integration by parts?
deep anvil
Oh I read top to bottom of the first page. Let me check again where the mistake is then.
deep anvil
But also at the same time, you should see this:
But I'll still try to find where the mistake is. Because there has to be.
deep anvil
It'd be helpful to know where I went wrong
Okay I spotted a potential mistake, (can't be sure since it gets cut off)
Here the -2c^-1 should be -2c^-3 if the rest is what I believe it is.
marsh citrusBOT
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covert scarab
I'll work on it and dm you how far I get
marsh citrusBOT
deep anvil
I'll work on it and dm you how far I get
Sure but remember you cannot actually compute that antiderivative!
covert scarab
Why?
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weak wind
need help
weak wind
I was thinking drawing a diagram might help
marsh citrusBOT
@weak wind Has your question been resolved?
charred mirage
Show some work bro else give up
unborn condor
i expect you to know that distance = speed * time
you have two trips, one SD->H, and one H->SD
ground speed is gonna be airspeed (i'll call this one S) + wind
so speed going SD->H is S+50, while going H->SD is S-50
is this enough?
weak wind
yes, thank you
i was just confused on the extra 50 mph from the wind
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vivid beacon
pls help
vivid beacon
i have no clue wht to do
odd orchid
substitute x and y in
vivid beacon
wht does suitable degree of accuracy mean
vivid beacon
substitute x and y in
?
odd orchid
you know what x and y are
vivid beacon
2,23x10 power 6
odd orchid
thats the answer then
vivid beacon
thats the answer then
but what does the term to suitable degree of accuracy mean
vivid beacon
wht do i write it as then
odd orchid
yours was fine
vivid beacon
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untold wyvern
Im having trouble pinning down exactly what the last line is saying
what do they mean by dual group?
oh wait nvm thats just defintiio
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untold wyvern
Having trouble understanding why it would be g^{-1}
marsh citrusBOT
Having trouble understanding why it would be g^{-1}
untold wyvern
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lunar cedar
I don't get it. How would I obtain w?
marsh citrusBOT
I don't get it. How would I obtain w?
lunar cedar
oh ok I think I got it, but I don't understand how it's simplified to that last step
why isn't it 1/(||v|| * cos(theta))
why 1/(||u||)
glass silo
Isn't that a v they wrote there (but what terrible handwriting 
)
Their point being that v/norm{v} is a unit vector, then you scale it afterwards to fit what you want 
glass silo
cos(theta) is still there, as is norm{u}, they just highlighted that v/norm{v} is a unit vector
lunar cedar
I'm thinking he forgot to write cos(theta) didn't he
can't you cancel out the cos(theta) as well or am I tripping?
glass silo
Either that bottom is them writing cos(0) (= 1) in that denominator, rather than cos(theta), or they wrote a cos(theta) in error
After all, v^t * v is effectively norm{v}^2
lunar cedar
ah you're right it is a 0
this entire thing is v? I thought it was to the right of w
glass silo
The whole thing is v, yep, from that left to the end 
lunar cedar
damn wtf I still need to find the angle
gah alright. it's the u \dot v = norm{u} norm{v} cos(theta) right?
glass silo
Yep, that way 
lunar cedar
how do you remember this stuff man. After I take this class I'll probably forget it
glass silo
I have no idea, probably helps that I found it somewhat interesting at least, but other than that, don't know if I have any useful advice 
There's a whole bunch of shit that I forgot the moment I didn't have to use it, after all 
lunar cedar
I don't get it. How would I obtain w?
idk if you solved it but my answer is 1/6
@glass silo could you verify it if you can?
I seem unsure about this answer
oh I forgot to multiply with v
glass silo
The coefficient you're gonna multiply v by is 1/6, at least I agree with that one
lunar cedar
alright I got Dimension Error. hbu
glass silo
Dimension error? 
glass silo
Sounds about right, if I can even compute anything mentally 
glass silo
Well, some of it at least 
working out the "coefficient" was relatively easy to do, mind you, I can barely remember my own name as it is
lunar cedar
sounds great. forgot how to do the thank you think in the server
.thanks
?thanks
!thanks
lunar cedar
great customer service 5/5
glass silo
Awwwww 
why thank you 
lunar cedar
can I friend you in case I have any other questions this semester?
glass silo
You can though sometimes I'm terrible at replying when it gets busier for me
covert basin
Can someone please help with 11b?
marsh citrusBOT
Can someone please help with 11b?
covert basin
The answer is here
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covert basin
no
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spark siren
what do you have for (a)?
marsh citrusBOT
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severe shore
minimizing |z| corresponds to finding the point on the line that is closest to the origin
How can we find the closest point of a line to the origin
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modest jolt
I am posting the full question and answer here but I am only asking about part iii c. How do they conclude that (something) makes an angle of - pi/12 ? Also can I not simplify the transformation to just reflection in y = x as the + pi/3 and - pi/3 will cancel?
marsh citrusBOT
I am posting the full question and answer here but I am only asking about part i...
fervent rampart
it would be invariant if after the initial rotation it landed on the line y = x (angle pi/4), making the reflection not do anything and subsequently rotating back to its starting position
so if it lands on pi/4 then its starting position must have been pi/4 - pi/3 = -pi/12
and the rotations don't cancel if there is another transformation between them. if we consider the point on the positive x-axis (angle 0) then in the first place it:
- rotates to pi/3
- reflects across pi/4, giving angle pi/6
- rotates to -pi/6
whereas with just the reflection it would end up at pi/2
modest jolt
Oh, I see so the line of invariant points is a line that makes an angle of -pi/12 with the positive x axis right?
modest jolt
and the rotations don't cancel if there is another transformation between them. ...
Ah that makes sense.
Alright, thank you.
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gritty cypress
At least 2 of the last 3 questions out of 8 must be solved. and a total of 5 questions will be answered. How many different ways can the exam be answered?
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At least 2 of the last 3 questions out of 8 must be solved. and a total of 5 que...
gritty cypress
guys its just a basic question. and books uses a combination with [5,3].[3,2] but it dont makes sense? there is 6 questions left if 2 of them was answered so arent we need to use 6 3 combination?
proud basin
well its 2 out of the 3
gritty cypress
yep
proud basin
so from the 3 last, its already decided you choose 2 from 3
gritty cypress
oooooooooooooooooooohhhhhh
yeah maybe studying late at night was not the best idea huh
thank you though. that little sentence you made just opened my eyes
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novel juniper
Let $(x_n) \to x$ and let $p(x)$ be a polynomial, show that $p(x_n) \to p(x)$.
\
Let $p(x)= \sum_{i=0}^{m} a_i x^i$
\
Then using the algebra of the limits of sequences, we have $x_n^i \to x^i$ (product of convergent sequences) and $ax_n^i \to ax^n$. We thus find $p(x_n) = \sum_{i=0}^{m} a_i x_n^i \to \sum_{i=0}^{m}a_i x^i$( sum of convergent sequences )
marsh citrusBOT
Let $(x_n) \to x$ and let $p(x)$ be a polynomial, show that $p(x_n) \to p(x)$.
\...
novel juniper
does this suffice
tired oxide
as long as you have that products of convergent sequences converge to the product of the limits, and also that sums of convergent sequences converge to the sum of the limits, then yeah that's fine
but like cite that if that's what you're using
novel juniper
I do , yea
tired oxide
then you're all set
maybe if you wanted to make it more formal then you'd go by induction
elfin berryBOT
ƒ( wai ina teacup)= I don't know
novel juniper
maybe if you wanted to make it more formal then you'd go by induction
hmm, okay
novel juniper
but that would ultimately mean proving what I've already proven, no?
nvm, wrong question
tired oxide
i mean you have to show that x_i^k converges to x^k where x is the limit of the x_i
but like
that's induction
pastel horizon
Nice
tired oxide
if you only have the product
pastel horizon
Dang this hard
tired oxide
Dang this hard
we do math pretty much of all difficulty levels. see #❓how-to-get-help for math help.
pastel horizon
Nice
pastel horizon
we do math pretty much of all difficulty levels. see <#488120190538743810> for m...
What math is this? Is this difficulty level on polygons?
pastel horizon
**ƒ( wai ina teacup)= I don't know**
.
novel juniper
this is introductory real analysis
pastel horizon
this is introductory real analysis
Thanks ill learn this
tired oxide
if you only have the product
@novel juniper does this part make sense?
novel juniper
<@796939110815629322> does this part make sense?
You mean I'll need induction if I only have the product rule, right