meager delta
why
#help-27
gritty terrace
i'm not here to hand out answers but i can give hints
gritty terrace
i'd recommend rewriting the integral as $\int x^2 \sqrt{x} \sqrt{4x^2 + 1} dx$
woven radishBOT
Pseudo (Cat theory #1 Fan)
gritty terrace
i.e. $\int x^{\frac 52} \sqrt{4x^2 + 1} dx$
woven radishBOT
Pseudo (Cat theory #1 Fan)
gritty terrace
that's my hint
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mortal vault
I dont know where to begin with this question.
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I dont know where to begin with this question.
hollow jolt
Do you know trig identies?
hollow jolt
Cos(A+B) and sin(A+B)
Oml seriously
Do all these accounts fall for that and get hacked
supple knot
Cos(A+B) and sin(A+B)
Show what you get after you use the identity
mortal vault
I got -1n for both
sand dove
I got -1n for both
That's good
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dusky shore
Hi! I'm working on the proof of Leibniz's rule by induction: $(fg)^{(n)} = \sum_{k=0}^n \binom{n}{k} f^{(n-k)} g^{(k)}$.I have already done the differentiation and split the sum into two parts. I also re-derived Pascal's Identity at the bottom of my sheet to make sure my algebra was correct.Where I'm stuck:I'm struggling with the final "merge" of the two sums. Specifically, I'm having trouble showing the clean transition from:$$\sum_{k=1}^{n+1} \binom{n}{k-1} f^{(n-k+1)} g^{(k)} + \sum_{k=0}^{n} \binom{n}{k} f^{(n-k+1)} g^{(k)}$$to the final form:$$\sum_{k=0}^{n+1} \binom{n+1}{k} f^{(n+1-k)} g^{(k)}$$I’m a bit confused about how to properly handle the boundary terms ($k=0$ and $k=n+1$) while applying Pascal's rule to the middle terms. Any guidance on how to write this step rigorously would be appreciated!
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Hi! I'm working on the proof of Leibniz's rule by induction: $(fg)^{(n)} = \sum_...
woven radishBOT
ϕ(x)=ϕ0+h(x)
dusky shore
<@&286206848099549185>
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deep heath
C h a t g p t
dusky shore
C h a t g p t
Gnagnagna (it just means that you have no idea)
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deep heath
Gnagnagna (it just means that you have no idea)
It works tho 🥲
So idk but what is middle term
Is it the casework or the f and g
deep heath
Gnagnagna (it just means that you have no idea)
French?
deep heath
**ϕ(x)=ϕ0+h(x)**
E x p a n d and s i m p l i f y yah?
worldly tapir
really small question wondering if this is 2 cycles or not
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really small question wondering if this is 2 cycles or not
@worldly tapir Has your question been resolved?
vital edge
Yes
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raven kayak
Hello, could someone check if this proof looks good please?
\begin{Definition}[Natural numbers]
The set of the \emph{natural numbers} is the set $\bN = \{1,2,3,4,5,\dots\}$.
\end{Definition}
\begin{Theorem}
Suppose $n \in \bN$. Then, the sum of the first $n$ odd equals $n^2$.
That is, $1 + 3 + 5 + \dots + 2k - 1 = n^2$.
\end{Theorem}
\begin{proof}
Let $n \in \bN$. We proceed by induction.\\
\underline{Base Case.} The base case is when $n = 1$, and $1 = 1^2$ as desired.\\
\underline{Inductive Hypothesis.} Let $k \in \bN$, and asume $1 + 3 + 5 + \dots + 2k - 1 = k^2$.\\
\underline{Induction step.} We aim to prove that the result holds for $k + 1$.
That is, we wish to show that $$1 + 3 + 5 + \dots + 2(k + 1) - 1 = (k + 1)^2.$$
Written slightly differently, $$1 + 3 + 5 + \dots + (2k - 1) + 2(k + 1) - 1 = (k + 1)^2.$$
To do this, we begin with the expression on the left, we apply the inductive hypothesis to the sum
of the first $k$ numbers, and from there simplify:
\begin{align*}
1 + 3 + 5 + \dots + 2k - 1 + 2(k + 1) - 1 &= k^2 + 2(k + 1) - 1 \\
&= k^2 + 2k + 2 - 1 \\
&= k^2 + 2k + 1\\
&= (k + 1)^2\\
\end{align*}
Therefore, by induction, $1 + 3 + 5 + ... + 2n - 1 = n^2$ for all $n \in \bN$.
\end{proof}
Mor Bras
drifting sierra
Apart from some wording, that looks fine
raven kayak
Which wording?
drifting sierra
the sum of the first n odd
asume
The "written slightly differently" part is useless, you may as well just write it that way first
wicked rover
another typo in the theorem's statement, you used n and k in the same equation
drifting sierra
Oh yeah good catch
I probably would have put parentheses around the last term for k+1 too, so (2(k+1)-1)
(to show it's one single term and not something followed by - 1)
raven kayak
\begin{Theorem}
Suppose $n \in \bN$. Then, the sum of the first $n$ odd natural numbers equals $n^2$.
That is, $1 + 3 + 5 + \dots + 2n - 1 = n^2$.
\end{Theorem}
\begin{proof}
Let $n \in \bN$. We proceed by induction.\\
\underline{Base Case.} The base case is when $n = 1$, and $1 = 1^2$ as desired.\\
\underline{Inductive Hypothesis.} Let $k \in \bN$, and assume $1 + 3 + 5 + \dots + 2k - 1 = k^2$.\\
\underline{Induction step.} We aim to prove that the result holds for $k + 1$.
That is, we wish to show that $$1 + 3 + 5 + \dots + 2(k + 1) - 1 = (k + 1)^2.$$
Written slightly differently, $$1 + 3 + 5 + \dots + (2k - 1) + (2(k + 1) - 1) = (k + 1)^2.$$
To do this, we begin with the expression on the left, we apply the inductive hypothesis to the sum
of the first $k$ numbers, and from there simplify:
\begin{align*}
1 + 3 + 5 + \dots + 2k - 1 + (2(k + 1) - 1) &= k^2 + 2(k + 1) - 1 \\
&= k^2 + 2k + 2 - 1 \\
&= k^2 + 2k + 1\\
&= (k + 1)^2\\
\end{align*}
Therefore, by induction, $1 + 3 + 5 + ... + 2n - 1 = n^2$ for all $n \in \bN$.
\end{proof}
Mor Bras
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raven kayak
Hello, could someone check if this proof looks good please?
\begin{Definition}[Natural numbers]
The set of the \emph{natural numbers} is the set $\bN = \{1,2,3,4,5,\dots\}$.
\end{Definition}
\begin{Lemma}
For any $n \in \bN$, $1 + 2 + 3 + \dots + n = \frac{n(n + 1)}{2}$.
\end{Lemma}
\begin{Theorem}
Suppose $n \in \bN$, then $n^2 - n$ is even.
\end{Theorem}
\begin{proof}
Let $n \in \bN$.
By lemma 2, we have that $1 + 2 + 3 + \dots + (n - 1) + n = \frac{n(n + 1)}{2}$.
Then,
\begin{align*}
1 + 2 + 3 + \dots + (n - 1) + n &= \frac{n(n + 1)}{2} \\
1 + 2 + 3 + \dots + (n - 1) &= \frac{n(n + 1)}{2} - n \\
&= \frac{n^2 + n - 2n}{2}\\
&= \frac{n^2 - n}{2}\\
2(1 + 2 + 3 + \dots + (n - 1)) &= n^2 - n\\
\end{align*}
Since $1 + 2 + 3 + \dots + (n - 1)$ is an integer, so is $2(1 + 2 + 3 + \dots + (n - 1))$.
Therefore, by the definition of an even integer, $n^2 - n$ is even.
\end{proof}
Mor Bras
jade pecan
looks good to me
vital edge
The second last line could be changed a bit
Since 1+2+.. is an integer, 2(1+2+..) is an even integer
raven kayak
Since 1+2+.. is an integer, 2(1+2+..) is an even integer
For sure, by the definition of even integers
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void fox
were you told explicitly to use the lemma?
because the usual proof should just be observe that one of the factors in n(n-1) has to be even
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raven kayak
were you told explicitly to use the lemma?
Yes
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barren dune
How do i do part i
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How do i do part i
pseudo basin
How do i do part i
first find the probability that a randomly selected biscuit is chocolate.
you apparently tried to figure out how many biscuits of each flavor one box contains on average? that's no good here.
then write down the n and p for the distribution of the number of chocolate biscuits in the box
@barren dune are you still here
barren dune
<@1055452561211146312> are you still here
Yes
barren dune
first find the **probability** that a randomly selected biscuit is chocolate.
...
That’s what i tried to find out…is it 1/4?
barren dune
So cream is 3/4
pseudo basin
So cream is 3/4
so the probability of a cream biscuit is 3/4 yes.
barren dune
and in fact that's going to be more relevant than your 5/2
I see, but the 5/2 was necessary to get the probability?
pseudo basin
no, the probabilities can be obtained directly from the given 3:1 production ratio
pseudo basin
do you understand how to continue from here?
pseudo basin
no
if the count of chocolate biscuits is 5 it already implies there will be 5 cream biscuits
so you should not do the same thing twice
pick ONE flavor and stick with it; find the probability that the box of 10 will contain exactly 5 of that flavor.
barren dune
pick ONE flavor and stick with it; find the probability that the box of 10 will ...
I get that, but if there’s five of one flavor, there’ll definitely be 5 of the other flavor…so why is prob of 5 chocolate and prob of 5 cream wrong?
pseudo basin
because "there are 5 chocolate" and "there are 5 cream" aren't independent (which would justify multiplying their probabilities), they're the same event
"and = multiply" is a bit of a cognitohazard
barren dune
Aha, so the prob of x AND y happening = prob(x) * prob(y) can only happen if they’re two different events
pseudo basin
Aha, so the prob of x AND y happening = prob(x) * prob(y) can only happen if the...
no
they need to specifically be independent
merely being different isn't enough
barren dune
"and = multiply" is a bit of a cognitohazard
How do I decide if i need to multiply or add?
pseudo basin
here you don't need to do either. you need to work out one and only one binomial probability.
barren dune
they need to specifically be ***independent***
So i should get 5 choc in a box of 10 and 5 cream in another box of 10. Having 5 and 5 in the same box wont do?
pseudo basin
So i should get 5 choc in a box of 10 and 5 cream in another box of 10. Having 5...
no!!!!
that is not what i said to do at all!
barren dune
pseudo basin
pick ONE flavor and stick with it; find the probability that the box of 10 will contain exactly 5 of that flavor.
i said to pick ONE flavor
please read what i am saying carefully and don't omit key parts of it
pseudo basin
i don't want you insisting upon doing twice the work
pseudo basin
I meant this is what it would look like if i did it in real life
you misunderstand.
let X be the number of chocolate biscuits in the box. then X ~ Bin(10, 1/4), and the event X=5 already means a 5-5 split. by itself. THERE IS NO SECOND EVENT.
pseudo basin
So my AND rule is to blame here
yes, rather i would call it a misconception and misunderstanding of the concept of independence
barren dune
How do I decide if i need to multiply or add?
Is this question valid?
If so could you please answer it
pseudo basin
if you are working with two events and:
- they are disjoint, and you want the probability of their union, then you add their probabilities
- they are independent, and you want the probability of their intersection, then you multiply their probabilities
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deep heath
Yes i'm a fucking french dude
CA VA BIEN or smth
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CA VA BIEN or smth
wicked turtle
sacre bleu
pseudo basin
CA VA BIEN or smth
do you have a math question to ask?
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restive river
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restive river
i need help with this proof i have solved something similar where i had to prove that inn G is normal subgroup of Aut(G)
but i have no idea how to approach this
vital edge
Well what's an inner automorphism
gray tartan
can someone teach me something that sounds cool to flex
devout snowBOT
can someone teach me something that sounds cool to flex
Someone else is already using this help channel. If you need help with a question, please open your own help channel/thread (see #❓how-to-get-help for instructions).
gray tartan
Someone else is already using this help channel. If you need help with a questio...
im sorry
vital edge
ig(x) _gxg^-1
Actually lemme go brush up on my group theory cuz I'm not sure if that's correct
Oh okay it's conjugation by a specific group element
gray tartan
Actually lemme go brush up on my group theory cuz I'm not sure if that's correct
how are you so intelligent
vital edge
Okay I am confused at what exactly the question is asking you to do here
twilit field
quick question: wouldn't I(G) be a subgroup of Aut(G) instead of G itself?
I'm sorry if I'm imposing on this channel - please let me know if I am.
vital edge
quick question: wouldn't I(G) be a subgroup of **Aut(G)** instead of G itself?
I...
That's what I was thinking too
restive river
Okay I am confused at what exactly the question is asking you to do here
How do i do the proof
restive river
quick question: wouldn't I(G) be a subgroup of **Aut(G)** instead of G itself?
I...
Yes i thought thay
That too
vital edge
Inn(G) is a normal subgroup of Aut(G)
restive river
Yup i know this
restive river
Yup you think the question is wrong?
vital edge
It very much isn't a subgroup of G
vital edge
quick question: wouldn't I(G) be a subgroup of **Aut(G)** instead of G itself?
I...
Also don't worry about imposing if you're adding to the conversation lol
restive river
Yeah cause the automorphism is a fucntion
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void fox
What do they mean by saying it's a su group of set G
it is not uncommon to abuse notation and drop the "an isomorphic copy of" in saying that G contains Inn(G)
although I agree it's probably a typo here
vital edge
Yeah if it was isomorphic to a subgroup I'd understand that
But it's isomorphic to a quotient
(also please mention me when you reply so I can see)
vital edge
Nwnw
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restive river
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restive river
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severe sparrow
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ionic harness
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ionic harness
damnit
One second the image did not send
wth is a .data file anyway
Does this look good?
twilit field
you did not show Proposition 1.6, by the way. just a heads-up.
ionic harness
I didn't think I needed to since the property is obvious, but for context it states that if n is an integer, there is no integer on (n, n+1)
sand dove
I didn't think I needed to since the property is obvious, but for context it sta...
Well the statement might seem trivial but the proof isn't
Try the proof with n = 0, the other values of n will follow suit
ionic harness
Try the proof with n = 0, the other values of n will follow suit
Thats what the book did as well
sand dove
(You have to use well ordering principle)
ionic harness
But what about this 1.7
ionic harness
Are you allowed to use properties on R?
I thought I cited all the properties I used
ionic harness
If you give me a minute I can screenshot exactly what we have so far
Field Axioms and their basic implications (0 is unique, ax = 1 and x + a = 0 have a unique solution, 0a = a0 = 0, etc)
And the relevant proposition 1.6
ionic harness
(Hopefully) the only thing(s) I didn't cite in the proof are the field axioms
Any other assumption should be mentioned
(supposing the proof is valid in the first place)
This is in order of the book as well so Z was introduced after R
Oh I just noticed the definition depends on the naturals
Which they defined as 1 and n+1 for all n
Also after the reals
polar chasm
More importantly, how do u apply P1.6 if ur lub isnt a natural?
There is an integer between e.g. 2.5 and 3.5
Ur proof is very sus even if we allow lub
ionic harness
Good point i just assumed s is an integer
Without stating why it should be
I think I can generalize out of that
I will come back once that's done
Thank you
.close
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oblique jetty
I’m sorta confused on this (I’ve been getting mixed answers)
If I have an autonomous differential equation (I think it’s called) whose graph looks like this, are the limits here correct?
fickle magnet
young spade
I’m sorta confused on this (I’ve been getting mixed answers)
If I have an auton...
yeah, they are correct
oblique jetty
I swear at some point I heard the the limit going to infinity for the top one would somehow magically go back to 2
But yea if there’s no problem then I can retain my sanity 😂
Thanks
young spade
np
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visual stone
-1 = log(c) 2
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-1 = log(c) 2
glass canopy
Is this in complex analysis?
Sorry, fried brain at the moment.
glass canopy
is it c^-1 = 2?
What are your steps to get here?
cold haven
Is this
- -1 = log_c(2)
- -1 = 2 log(c)
If it’s the second one, what is the base of log?
visual stone
first
cold haven
is it c^-1 = 2?
Not quite.
visual stone
what is it then
cold haven
what is it then
I can’t solve it for you. What steps did you take?
visual stone
it started with -1 = log(c) 0.4(6-1)
-1 = log(c) 2
then i have to remove log
c^1 = 2
solid osprey
Not quite.
ok sorry am i missing something, is that not correct?
cold haven
ok sorry am i missing something, is that not correct?
The question was log_c(2) = -1 right? 
solid osprey
The question was log_c(2) = -1 right? <:nervoussweat:789206248419491900>
yeah and you raise the lhs and the rhs by c no?
urban harbor
is it c^-1 = 2?
yes
visual stone
cold haven
i did not just do that
visual stone
i did not just do that
what happend
cold haven
what happend
nothing you’re right. Excuse me while i step back. You were correct.
urban harbor
$c^{-1}=2\implies c=\frac{1}{2}$
cold haven
I got c = ¹⁄₂ and completely forgot that you could rearrange.
woven radishBOT
Soosh
cold haven
So sorry. That’s on me.
visual stone
I got c = ¹⁄₂ and completely forgot that you could rearrange.
well its the right answer
according to this bot
urban harbor
but c^-1 = 2 i dont know'
are you familiar with negative exponents? you really should be if you are learning logs
visual stone
are you familiar with negative exponents? you really should be if you are learni...
well no tghis is my first time dealing with that
maybe i did some negative exponent before but i forgot
urban harbor
you can think of negative exponent as reciprocal
$\left(\frac{a}{b}\right)^{-1}=\frac{b}{a}$
visual stone
ohh
woven radishBOT
Soosh
urban harbor
so $c^{-1}=\left(\frac{c}{1}\right)^{-1}=\frac{1}{c}$
woven radishBOT
Soosh
urban harbor
that's it
urban harbor
what if its ^-2
c^(-2) = 1/c^2
you can transfer factors from numerator <--> denominator by changing their signs
urban harbor
c^2 = 1/2
c = sqrt(1/2)
you really need to like properly review exponents
logs wont really make sense if you dont remember basic exponent rules
or youre getting taught this in a weird order idk
but learning logs when you arent familiar with negative exponents seems egregious
visual stone
but learning logs when you arent familiar with negative exponents seems egregiou...
yeah
visual stone
c^2 = 1/2
c = sqrt(1/2)
where did 1/2 come from
urban harbor
thats just basic arithmetic
visual stone
logs wont really make sense if you dont remember basic exponent rules
no i knew that if u have a negative exponent on a fraction u can switch the numerator from denominator
i just forgot that u can use it
i didnt think about it
visual stone
thats just basic arithmetic
but it starts with 1/c^2 = 2
then the 1 goes the other side? and becomes 1/2?
woven radishBOT
Soosh
urban harbor
and i ask you to solve for x, how do you do it?
visual stone
2x = 1
urban harbor
and?
visual stone
x = 1/2
urban harbor
ok so, its the exact same situation
instead of x we have c^2 and we are just isolating it so we can take the square root
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soft nova
New to this concept and missed class, for C do you just add the pieces like I tried? Or is there something I’m missing
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lucid pumice
Mostly right. What you missed in class is that area under the x axis counts as negative for integrals.
soft nova
so I'd subtract that amount instead of adding
soft nova
thanks a ton, preciate it
lucid pumice
Other rule before you go: If you go backwards, the sign flips. Like, if you integrate from 7 to 0 instead you'll get the negative of the original answer when you went from 0 to 7.
soft nova
so if I want to flip I can do that and just throw a negative sign outside?
lucid pumice
Yep. You got it.
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graceful cosmos
<@&268886789983436800> scammer
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graceful cosmos
You're the best
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ionic jasper
Guys, what is (27+33):10×2
devout snowBOT
Guys, what is (27+33):10×2
graceful cosmos
You need more parenthesis to fully express this. Can you add the ones you are missing?
ionic jasper
Wym there are parenthesis missing
graceful cosmos
These guys ( )
ionic harness
ionic jasper
Ye but why are they missing
cold haven
Wym there are parenthesis missing
Your question can be interpreted multiple different ways.
Is it (27+33)/(10) × 2? Or (27+33):(10×2)?
ionic jasper
I get what yall are saying, so what do i do if theres not more info given in the task
Just do the interpretation i like more?
graceful cosmos
Is this off the internet? Like some Reddit post or something? This is a common internet ragebait question.
Or is this legit from a class?
violet wind
I'm assuming : is just an alternate divided by sign
violet wind
In which case order of operations is unambiguous
ionic jasper
I saw it on an tiktok and the comments gor crazy
graceful cosmos
Man, I am so good. I can just call this
ionic jasper
Ye but i rhink the task was on an class test
graceful cosmos
Basically, just because it has numbers and operations doesn't mean it's well formed. It's meant to have no single solution
graceful cosmos
Different schools in different countries have different BEDMAS/PEMDAS rules. You'll get a different answer depending on where you grew up
ionic jasper
Okay
violet wind
I would expect operators of the same precedence level (such as multiplication and division) evaluate left to right
So it would be ((27+33):10)×2
ionic jasper
Thats what i thought to
violet wind
But ig it depends on whatever your teacher wants
graceful cosmos
We use the overbar / to be more clear
worthy raft
If I was really unsure on a test I'd do it both ways, with a comment explaining why I did that
violet wind
÷
graceful cosmos
$\frac{27+33}{10} * 2$
woven radishBOT
Kaynex
worthy raft
lmao
graceful cosmos
That one can't be misinterpreted
violet wind
$\times(\div(+(27,33),10),2)$
woven radishBOT
Dreyuk
worthy raft
Prefix notation my beloved
violet wind
parentheses optional I think
worthy raft
Hell you don't even need brackets there so long as everything is well defined
Yeyeye
graceful cosmos
27, 33, +, 10, ÷, 2, ×
Completely readable
violet wind
now to make a parse tree
worthy raft
Hell of a lot easier to write a computer parser for that than this nonsensical infix notation
Clearly this is how we should be teaching math /j
graceful cosmos
No downsides at all
ionic jasper
Can i close now?
worthy raft
If you're done then go ahead
ionic jasper
.close
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visual stone
how can i find L(t) if the equation is
devout snowBOT
how can i find L(t) if the equation is
visual stone
urban harbor
seems like you already have an equation for L(t), what exactly do you mean by finding it
visual stone
like when i do L(t) = log(7.5)
-50+50
@urban harbor
and i need to L(t)
so its log(7.5) + 5
?
because the -50+50 is 0
fierce viper
!xy
devout snowBOT
Please show the original problem, exactly as it was stated to you, with the entire original context. A picture or screenshot is best. If the original problem is not in English, then post it anyway! The additional context might still be helpful. Do your best to provide a translation.
visual stone
Please show the original problem, **exactly as it was stated to you**, with the ...
the question asks
"Does this rule allow to determine the length of the metal rod at -50 degree celcius?
Explain your answer
L(t) = metal rod in cm
urban harbor
"screenshot is best"
visual stone
t = temperature in celcius
visual stone
"screenshot is best"
gg
urban harbor
can you show what the equation given to you actually is, without you substituting any value for t etc., like what does the problem actually say?
crystal dawn
also purportedly this is a function in t, yet if I didn't misread the function statement there is no t
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ionic harness
More importantly, how do u apply P1.6 if ur lub isnt a natural?
Okay, I have revised the proof:
devout snowBOT
Okay, I have revised the proof:
ionic harness
Above the replied message one can find all of the resources I'm allowed to use (and should have cited if used)
Minus the Distribution of the Integers which says that for any real c, there is a unique integer on [c, c+1)
wicked turtle
is this real handwriting? if so you could sell it as a work of art
uncut crow
beautiful handwriting disciplud
ionic harness
beautiful handwriting disciplud
Thank you 🙏
ionic harness
is this real handwriting? if so you could sell it as a work of art
Haha yes this is how I write all my notes
I mean write in general but if you saw a sample you'd get what I mean
Which I would post but I don't wanna distract from the question yet
zenith spoke
i assume you meant to write n\notin S here, otherwise the proof looks good to me
young spade
The proof looks good, also, i suppose you dont have access to the well-ordering principle, right?
ionic harness
i assume you meant to write n\notin S here, otherwise the proof looks good to me
Yes 😅
Not in this book sadly
Otherwise I would have said LOOK we can choose a maximal element and compare it to the LUB
zenith spoke
btw what are you studying? real analysis?
ionic harness
Or I guess LOOK there exists a maximal element since Z is totally ordered it's a maximum done
ionic harness
btw what are you studying? real analysis?
Yeah
zenith spoke
nice
ionic harness
How did you know from this 👀
zenith spoke
also dont worry about reply pinging me
young spade
Yeah, you can always argue that theres some reflection + shift you can pull off to make the set into a non-negative set in Z that keeps the order**
zenith spoke
How did you know from this 👀
its typical to see the axiom of completeness etc.. in real analysis
ionic harness
Yeah, you can always argue that theres some reflection + shift you can pull off ...
Yeah, this makes sense
zenith spoke
its typical to see the axiom of completeness etc.. in real analysis
its usually discussed there
ionic harness
is this real handwriting? if so you could sell it as a work of art
The rest of the page
For bungo
zenith spoke
which book are you using?
ionic harness
Advanced calc Fitzpatrick
zenith spoke
next it should be distribution of primes 
zenith spoke
Advanced calc Fitzpatrick
oh i have never heard about this
winter torrent
the fact that you can write that well on a screen is insane
zenith spoke
why didnt you go with a classical one like abbott or rudin
ionic harness
Sorry the internet is out it'll take a minute to get my message
zenith spoke
the fact that you can write that well on a screen is insane
wicked turtle
The rest of the page
beautiful!
zenith spoke
the fact that you can write that well ~~on a screen~~ is insane
if you see my handwriting you will know what i mean
ionic harness
oh i have never heard about this
It's probably a knockoff or something 😆
ionic harness
the fact that you can write that well on a screen is insane
That's what all my friends say
They buy paper screen protectors and fancy tips for their pencils
I never understood it
I just write 😆
crystal dawn
dear god
ionic harness
why didnt you go with a classical one like abbott or ~~rudin~~
Ah because that's what the class called for
crystal dawn
zenith spoke
thats real btw, when i started my thread he joined but he left after a few mins (exactly when i declared that i will be using rudin)
zenith spoke
Ah because that's what the class called for
ohhh i see
ionic harness
next it should be distribution of primes <:sotrue:826971355542192148>
It would be backwards 
instead of for all intervals [a, a+1) there exists an integer it would be there exists an interval [a, a + n^2] on which there are no primes
Okay looks like we've wrapped up now 💃 thanks everyone!
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shut anchor
Hello
shut anchor
Im just a bit confused on how I know that the matrix is solved, and how to interpret it
Im working from linear algebra 5th edition by Johnson Riess and Arnold
keen hawk
Im just a bit confused on how I know that the matrix is solved, and how to inter...
ah, there's 3 variables but only 2 linear equations
so that means you (can) have infinitely many solutions
shut anchor
Always?
keen hawk
Always?
not always
some of the equations can be redundant
if you have x + y = 3 and y + z = 2, then x + 2y + z = 5 is not a new equation
the only way to know for sure if you have infinitely many solutions, exactly one solution, or no solutions is to find the RREF
shut anchor
Which stands for
keen hawk
the reduced row-echelon form
shut anchor
So the bottom matrix has.. one solution?
keen hawk
no
okay I'll explain this next bit
that means you have the equations y + 2z = 4, x + z = -3
we can write the other variables in terms of z
so that's y = 4 - 2z, and x = -z - 3
and then z can be any real number
whatever number you choose for z, you can generate a new (x, y, z) solution to the equation
lyric hornet
so that means you (can) have infinitely many solutions
-# For more info, see here.
I mean to the OP
blitzchamp is right, my bad if that was confusing lmao
shut anchor
I'm sorry for not responding for a bit I 'm just trying to reread to understand
Okay, sure, but this has to do with rewriting the equation, how does it help me know if a particular problem has a certain amount of solutions
Is what you mean, if I combined the two equations, knowing the sum of the columns is 1, i know what variables are dependent and independent and therefore determine the amount of solutions?
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remote copper
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pseudo basin
!status
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What step are you on?
1. I don't know where to begin.
2. I have begun but got stuck midway.
3. I got an answer but I was told that it's wrong.
4. I got an answer and would like my work checked.
5. I have a question about someone else's work/solution.
6. I have completed the problem and don't need help anymore. Thank you.
7. None of the above
is there any formula like det=0
i remember a det which works for three points i guess
pseudo basin
calculate differences between these 4 numbers and show that those differences are real multiples of each other
remote copper
pseudo basin
or if you really want to use this then apply it to z_1, z_2, z_3 then apply it to z_2, z_3, z_4
or any other two trios that collectively cover all your pts
imo kinda overkill though
remote copper
calculate differences between these 4 numbers and show that those differences ar...
i did not get it properly because
there can be many cases
z2-z1
z3-z2
z4-z3
z3-z1
pseudo basin
calculate enough differences to involve each number at least once
pseudo basin
or you can spend more time listing all 12 differences if you really want
remote copper
yeah 4c2
pseudo basin
calculate enough differences to involve each number at least once
actually not exactly true as written
calculating z2-z1 and z4-z3 only would not be enough
remote copper
what?
pseudo basin
the actual way to tell if you've done enough differences is to make a graph (in the graph theory sense) with a vertex for each z_k and connect each pair for which you've calculated a difference with an edge. then what you want is for the resulting graph to be connected.
and if that made no sense to you then here is an easier and foolproof method: calculate the differences of all your points with z1
pseudo basin
yes that is what i just said
remote copper
and check the distance?
pseudo basin
not distance.
remote copper
let me calculate it
pseudo basin
i said difference like just subtract them only, no modulus
and then if you exercise your memory for sth i said 6 minutes ago:
and show that those differences are real multiples of each other
pseudo basin
i said difference like just _subtract_ them only, no modulus
i am amazed that you hallucinated a mention of modulus where there was never one
remote copper
i got
(-2-6i),(1+3i),(3+9i)
so i can take common -2,3 and it seems one point is multiple of -2
and other one is 3
so they are lying over a line
pseudo basin
i got
(-2-6i),(1+3i),(3+9i)
why are you apparently multiplying them together
pseudo basin
why not separate them with commas
(-2-6i)(1+3i) reads like multiplication
there we go
harsh halo
now clear?
yeah, so if you have parentheses together, for example
(a^2+b^2)(-a^2+-b^2), it will be read as
(a^2+b^2) • (-a^2+-b^2)
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green crypt
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green crypt
why do we give arbitrary values to x3, x5 and x6 and not x1 x2 and x4
alpine python
maybe because there are three equations? so if you give arbitrary values to three of the variables it could still be solvable
3 (linear) equations in 3 variables usually has a solution
green crypt
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viral kernel
[
\mathbb{E}[X]
= \sum_{x\in\mathcal{X}} x,\mathbb{P}(X=x)
]
[
\mathbb{P}(X=x)
= \sum_{m=1}^M \mathbb{P}(X=x \mid E_m),\mathbb{P}(E_m)
]
[
\mathbb{E}[X]
= \sum_{x\in\mathcal{X}} x \left( \sum_{m=1}^M \mathbb{P}(X=x \mid E_m),\mathbb{P}(E_m) \right)
]
[
\mathbb{E}[X]
= \sum_{m=1}^M \mathbb{P}(E_m)\sum_{x\in\mathcal{X}} x,\mathbb{P}(X=x \mid E_m)
]
[
\mathbb{E}[X\mid E_m]
= \sum_{x\in\mathcal{X}} x,\mathbb{P}(X=x \mid E_m)
]
[
\mathbb{E}[X]
= \sum_{m=1}^M \mathbb{P}(E_m),\mathbb{E}[X\mid E_m)
]
woven radishBOT
Goofy Joe
viral kernel
Why do they exchange sums?
step 4
It's not so much why, but why it can be done.
X is finished, M too
pseudo basin
"finished"?
did you mean finite?
"finished" is like your state after a fatality in mortal kombat
anyway what you're asking about basically is the legality of $$\sum_{x \in X} \sum_{y \in Y} f(x,y) = \sum_{y \in Y} \sum_{x \in X} f(x,y)$$ right
woven radishBOT
Ann
viral kernel
yes
pseudo basin
it's a finite number of terms in a rectangular array all added together
you add them up either column-wise or row-wise, it's going to be the same sum
viral kernel
lemme see
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willow helm
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coral forum
Find $\sin v$ when $v$ is an angle in the 3rd quadrant and $\tan v = 2.$
winter patrol
what have you tried?
woven radishBOT
dream
coral forum
what have you tried?
$\tan v = \frac{\sin v}{\cos v}$
woven radishBOT
dream
coral forum
$2 = \frac{\sin v}{\cos v}$
woven radishBOT
dream
coral forum
$2\cos v=\sin v$
woven radishBOT
dream
coral forum
Thats what i have done till now
winter patrol
ugh, don't need to do that
coral forum
Oh
winter patrol
two main approaches, to start
one would be to draw a reference triangle where tan(t) = 2
or alternatively draw the whole trig circle with tan(v) = 2
up to personal preference which one to use
woven radishBOT
dream
zealous comet
two main approaches, to start
one would be to draw a reference triangle where ta...
yea that's good
oh they want you to find sin v
mb mb
coral forum
$2^2 * \cos^2 v= \sin^2 v$
woven radishBOT
dream
winter patrol
i suppose you can do the algebra that way too
coral forum
$4 * (1-\sin^2 v)= \sin^2 v$
woven radishBOT
dream
coral forum
$4 = 5 \sin^2 v$
woven radishBOT
dream
coral forum
$\sin^2 v=\frac{4}{5}$
woven radishBOT
dream
coral forum
Now i can take the sqrt?
winter patrol
yes
coral forum
So the answer is just +-sqrt(4/5)?
winter patrol
not quite
winter patrol
yes
winter patrol
yes
coral forum
Well thank youu
winter patrol
i didn't really do anything.
minor shell
I chose personnaly $x=-1$ and $y=-2$ because the formula is $\frac{x}{y}$, and we know that both $x$ and $y$ are negative (third quadrant).
woven radishBOT
Nallanos
minor shell
I that correct ?
$$
r=\sqrt{(-1)^2+(-2)^2}=\sqrt{1+4}=\sqrt{5}
$$
$$
\sin v=\frac{y}{r}=\frac{-2}{\sqrt{5}}=-\frac{2\sqrt{5}}{5}
$$
woven radishBOT
Nallanos
coral forum
Cause then tan v would be 2
coral forum
I mismatched them when writing
No worries
winter patrol
And y = -1?
x=-1 and y=-2 was valid,
issue was that the ratio is y/x, not x/y
minor shell
yes
winter patrol
Hmm but i got - sqrt(4/5)
yes,
you should simplify your value and that'll lead to
-2sqrt(5)/5
coral forum
Cause tan v = sin v/cos v
coral forum
yes,
you should simplify your value and that'll lead to
-2sqrt(5)/5
Ohh got it!
quasi vessel
yo
quasi vessel
so this is a question
and the asnwer key says that medium b is more dense than a
am i trippin?
dusty fox
What topic is this
quasi vessel
chapter light
class 10
CBSE
this is the question
answer key says this
I CANT SEE THE BENDDD TOWARDS NORMAL
CAN U?
untold ravine
this answer looks like it was written by ai
quasi vessel
this answer looks like it was written by ai
OH
grok is saying the same shi
like is the bend of 1 pixel
🙏🏻
turbid bane
So the question answer is that the medium B is more dense even though the light doesn't bend (at least visibility) when it enters it. And you're asking why?
quasi vessel
So the question answer is that the medium B is more dense even though the light ...
yeah the answer says it is more dense, hence it shall bend towards normal but it doesnt ( atleast in the fig)
wrong figure maybe?
turbid bane
Sonethings wrong
fair fable
what
quasi vessel
PHYSICS
fair fable
did i do something wrong
quasi vessel
nah
turbid mauve
dusty fox
!occupied
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formal sand
<@&286206848099549185>
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<@&286206848099549185>
formal sand
<@&286206848099549185>
supple knot
!15m
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mystic scarab
<@&286206848099549185>
CAN YOU READ THIS? #help-27 message
If not, you must be surely trolling
<@&268886789983436800>
wicked rover
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fierce viper
test
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vital edge
.close
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pseudo basin
in #1, why didn't you already conclude the solution set is empty by the time you got to "2 > 6"?
are you not allowed to do that somehow by your teacher?
(not that it was wrong to subtract 2 but still)
devout snowBOT
unique thorn
native gyro
All of then are correct
unique thorn
Okay thank you guys 🙏
native gyro
All of them are correct, some of them have more step that needed
zealous comet
yes
pseudo basin
!msgdel
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unique thorn
All of them are correct, some of them have more step that needed
gotcha
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knotty igloo
anyone can help me solve this problem when only Menelaus’ Theorem,Similar triangles,Ceva’s Theorem ,Pythagorean Theorem or Some Grade 8 advanced-level knowledge in English. bc Im a 14 year old student PLs
devout snowBOT
anyone can help me solve this problem when only Menelaus’ Theorem,Similar triang...
devout snowBOT
@knotty igloo Has your question been resolved?
knotty igloo
uh, so you only need to prove that orthocenter exists?
yess pls help
charred eagle
how do i do thi
devout snowBOT
how do i do thi
pseudo basin
there are two ways but ultimately you need to know how the sequence of powers of i looks like.
charred eagle
there are two ways but ultimately you need to know how the sequence of powers of...
is that what u mean
pseudo basin
your description isn't precise enough imo
can you make a table of n and i^n with n counting up from 0 to at least 7
on your paper or wherever you are writing these
\begin{tabular}{c|c}
$n$ & $i^n$ \\
\hline
0&\\
1&\\
2&\\
3&\\
4&\\
5&\\
6&\\
7&\\
\end{tabular}
Ann
charred eagle
yes
pseudo basin
can you also tell me the sum of any 4 consecutive terms here?
charred eagle
0
i see
so u end up with 0 because 100 terms is divisible by 4
wait no u end up wit h1
wjy does the answers say this then
pseudo basin
there are 101 terms not 100
charred eagle
or is this the other way
pseudo basin
that was the other way
pseudo basin
they're treating it as a geometric series
charred eagle
if i recognise ur pattern
pseudo basin
which it is
charred eagle
wjy does the answers say this then
do u think i shuodl emmorise this formula?
pseudo basin
what, the geometric sequence sum formula?
charred eagle
yea
pseudo basin
i mean, it'll be good to practice it enough to commit it to memory
but i will recommend against memorizing it blindly
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jade reef
I need some help
devout snowBOT
I need some help
jade reef
Can someone complete the square, bc its not working when I tried
devout snowBOT
Show your work, and if possible, explain where you are stuck.
pseudo basin
yeah bc you added and subtracted the wrong thing
jade reef
Wdym
pseudo basin
first factor out the 3, then do the (b/2)^2 thing
that way a perfect square will form
jade reef
Then does the -9/4 go outside the bracket?
round raptor
correct
round raptor
Nice pfp BTW
thanks
round raptor
This was very helpful
thank to Ann
jade reef
thank to Ann
Yep
round raptor
lmao i missed
jade reef
oh your pfp is of him
Yes
jade reef
missing the ^2 at the very end
Oof, thanks I didn't see that
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twilit field
<@&268886789983436800>
robust ermine
hippity hoppoty
devout snowBOT
hippity hoppoty
robust ermine
this channel is now my property
#help-6 message
this is where the initial conversation started
like im trying to extend the definition of sin
fathom verge
Don't delete modpings please
swift urchin
https://discord.com/channels/268882317391429632/486841085210132490/1464950084947...
there is the .reopen
swift urchin
Don't delete modpings please
sorry my bad
robust ermine
i cant msg in that channel
swift urchin
i cant msg in that channel
im sorry excuse me
robust ermine
all goot
fathom verge
You have claimed this channel now, so shoot with your question 
fathom verge
220°?
robust ermine
yes sir
fathom verge
Why wouldn't it make sense?
fathom verge
What definition of the sine do you have yeah?
fathom verge
Oh I see
restive river
Is 1C3 correct or not
robust ermine
where opposite is the side opposite to x
robust ermine
Is 1C3 correct or not
i do not know friend
fathom verge
robust ermine
sin x = opposite/hypotenuse
uhm so if we use the unit circle
robust ermine
,tikz\draw(0,0)circle(1);\draw(220:1)node{$\bullet$};
\draw({cos(220)},0)node[be...
holy fuck
ur a genie
HOW DID U DO THAT
woven radishBOT
Matplotlib
fathom verge
This is the definition you wanna use
robust ermine
im not sure what im supposed to be looking at
restive river
This is the definition you wanna use
Is 1C3 correct or wrong
fathom verge
Wdym by 1C3?
restive river
Wdym by 1C3?
C means combinations
twilit field
why are there combinatorics in a trigonometry question?
fathom verge
Indeed I don't see a relation between both
twilit field
perhaps; I apologize for randomly entering this channel as well.
robust ermine
nah bro all good
restive river
perhaps; I apologize for randomly entering this channel as well.
Hi Nicole
glossy dew
**Matplotlib**
uh is it how its supposed to look like
robust ermine
**Matplotlib**
ive seen this before
fathom verge
Okay I can make it look better
robust ermine
is cos and sin like x and y coordinates?
fathom verge
OP, have you seen this before?
fathom verge
Okay, so the idea is that you can associate a point on the circle to any real number
You stick a real line tangent to the right of the unit circle (circle of radius one and whose center has (0,0) as coordinates)
And then you "loop it around" that circle
This means that the image of the real, say, pi/2, is obtained by strating at the point to the right and moving for a length that equals pi/2
robust ermine
wwwwwwaaaaat
yueah i have not seen this before
le tme process this image and wat u just said
fathom verge
What grade are you in?
robust ermine
9
fathom verge
Okay so you'll see it in due time then!
robust ermine
do i just shelve this for now?
fathom verge
I don't wanna bash a lot of things to your head right now 
robust ermine
wait till im older to understand?
fathom verge
I mean you're allowed to try and be curious about it now too
robust ermine
mmm
ive been banging at this for 4 days
i aint the brightest crayon in the box
i now just realised that
sin 220 doesnt make sense to me
becaduse u cant have 220 in a right angled triangle lOL
fathom verge
This video makes the explanation I was trying to give to you, along with the connection to SOHCAHTOA and right triangles
fathom verge
It'll probably be better than any lengthy text explanation
robust ermine
ia havent seen u in so long
robust ermine
It'll probably be better than any lengthy text explanation <:catGiggle:974296908...
for real
ive been
hacking at chatgpt
for 4 days
fathom verge
robust ermine
LOL
twilit field
this is why, once you are comfortable with the basic definitions of the trigonometric functions, you should divorce the trigonometric functions from the existence of a right triangle and treat the trigonometric functions sort of as properties of the angles themselves.
this can be shown by the unit circle, as Matplotlib has shown.
robust ermine
this is why, once you are comfortable with the basic definitions of the trigonom...
sohcahtoa has been drilled into me so hard
its hard
fathom verge
Yeah, well there is a connection to SOHCAHTOA!
robust ermine
i shall cook with this video youhave given me
fathom verge
It's just that it goes the other way around as what you're expecting
From the circle definition, you can recover SOHCAHTOA
twilit field
let me try digging up a diagram from my friend.
robust ermine
i am too sleepy now to learn anytthing
twilit field
here.
robust ermine
that is an absolute masterpiece
LOL
just kidding
thank u
ill close this channel for now and save the messages to go over tomorrow
fathom verge
Whenever you're feeling like it, you may try to derive SOHCAHTOA on triangles whose hypothenuse isn't just 1 from Thales
Glhf!
robust ermine
Whenever you're feeling like it, you may try to derive SOHCAHTOA on triangles wh...
this thales name sounds familiar
okay thank u again peeps
.close
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robust ermine
goats as usual
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sand quarry
\lm
I am modeling a switched dynamic system where a state discontinuity coincides exactly with an impulsive input, basically requiring the multiplication of a Heaviside step $\Theta(t)$ and a Dirac delta $\delta(t)$
\medskip
Since the associative multiplication of distributions is impossible in the standard Schwartz space $\mathcal{D}'$, I am attempting to embed this into a Colombeau algebra of generalised functions, $\mathcal{G}(\vb R)$.
\medskip
\textbf{My question is}: Does the 'association' (the weak limit) of the product $\Theta \2 \delta$ in $\mathcal G$ yield a unique canonical distribution, or is the result strictly dependent on the specific class of mollifiers used to regularise the singularity?
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\lm
I am modeling a switched dynamic system where a state discontinuity coincide...
woven radishBOT
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vagrant skiff
no
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supple knot
Probably ask in #advanced-analysis
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idle dune
problem 1.4
