yeah it's slowly all connecting for me
#help-27
daring bobcat
i'm learning :p
i might need some time to look through the wiki though
for now, last question - do you mean differential calculus?
lusty sapphire
Blake nailed the answer btw Ginger
You need the discriminant(s) of the quartic
That will tell you the nature of the roots
whether they are all real or not
But it's ginormous
daring bobcat
here and above?
lusty sapphire
The real question should be, what is your purpose for doing all of this?
daring bobcat
i'm definitely going to reread all of blake's answers here
lusty sapphire
Do you know how to determine when a quadratic has real roots?
daring bobcat
The real question should be, what is your purpose for doing all of this?
neutrinos in supernovae are only stable when they are described by a set of eigenvectors all with purely imaginary eigenvalues
daring bobcat
Do you know how to determine when a quadratic has real roots?
i guess i would use the discriminant
lusty sapphire
neutrinos in supernovae are only stable when they are described by a set of eige...
This might be simpler if you considered any symmetries in the matrix first
daring bobcat
i hadn't realized before how that relates to factoring a polynomial bad phrasing
here's the matrix
[-ik, Az, -Ay, 0, Bz, -By],
[-Az, -ik, Ax, -Bz, 0, Bx],
[ Ay, -Ax, -ik, By, -Bx, 0],
[ 0, Cz, -Cy, ik, Dz, -Dy],
[-Cz, 0, Cx, -Dz, ik, Dx],
[ Cy, -Cx, 0, Dy, -Dx, ik]
(A, B, C, and D are 3D vectors with x, y, and z components)
(this can also be written as a coupled equation for two 3D vectors that involves cross products)
(it's a 6x6 matrix, but i've got the characteristic equation factored into a quartic polynomial multiplied by a quadratic;
at least when A = B and C = D, we can just assume that)
lusty sapphire
there could be something there, there could not
With the x, y, z, it likely completely arbitrary
daring bobcat
With the x, y, z, it likely completely arbitrary
gotcha, yeah. there is a relevant case where all the x and y components are zero, but i think we've solved that one already. in general they can all be there
(A = B and C = D is also important)
regardless, i'll definitely think more about the quartic discriminant
arctic field
what is k
daring bobcat
what is k
an arbitrary (positive) real number
(everything is real in the matrix actually, except the explicit use of i)
arctic field
well at least you can conclude for B = C that all the eigenvalues are purely imaginary
daring bobcat
ok, yes, that itself is a nice result actually
how did you conclude that actually?
arctic field
spectral theorem
arctic field
anti-symmetric refers to A* = -A in this case
you have to take the conjugate transpose
arctic field
i'm talking about the whole matrix
daring bobcat
ah. then, right, the diagonal entries don't work (since they're nonzero), that's what i meant
i also mean, if B = C
arctic field
-(ik)* = ik though
daring bobcat
oh
oh!
sorry, i'm just not used to dealing with fully complex matrices i guess :p
arctic field
when B and C are not equal, there's nothing that jumps out immediately
you could try decomposing and analysing the effect of the symmetric part
daring bobcat
that is also an interesting idea
arctic field
the hope is that it's very singular but who knows
daring bobcat
why would that be a good thing?
arctic field
well the eigenvalues of a singular matrix are going to be mostly 0
but at the same time you lose control over this when you add things so it doesn't help too much
but you'd hope that it doesn't perturb the eigenvalues of the anti-symmetric part that much
daring bobcat
ah right, the symmetric part being singular (i mixed them up)
so the anti-symmetric part dominates
this is definitely helpful
i'm not sure if this will be the final method of calculation used but this is definitely very interesting
i think it will help me/us (other people working on this too) gain insight into what is going on
i'm just going to continue thinking things over, but in the meantime, thanks
arctic field
i would think this is quite a difficult thing to characterise in general
using the quartic discriminant on the char poly might be the most direct brute force method
daring bobcat
i would think this is quite a difficult thing to characterise in general
(definitely not trying to obligate you to keep working on this but) just want to write down 2 things that could help:
- we can just stick with A = B and C = D (corresponding to massless neutrinos, which they pretty much are)
- in this case, two of the eigenvalues are just +/- ik, and the characteristic eq. can be factored into a quartic multiplied by (lambda^2 + k^2)
which i guess i sort of already said... but i'm not sure if these facts can be leveraged while also thinking about the antisymmetric and symmetric contributions (to the 6x6 matrix)
daring bobcat
using the quartic discriminant on the char poly might be the most direct brute f...
either way, i'm very glad i understand this method (using the quartic discriminant) a lot more now
arctic field
which i guess i sort of already said... but i'm not sure if these facts can be l...
nothing jumps out
daring bobcat
👍 sounds good. i'll keep staring at it anyway
arctic field
👍
daring bobcat
(i really do appreciate all the time and help btw 
)
arctic field
👍
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lethal pollen
Yea that's the premise
fair juniper
my bad
dense jay
sure ill have a whirl
as long as theres no pesky statistics
any, i took an oath that i wont do statistics outside of physics work
oop
old carbon
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supple knot
What is orthogonalize
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solid osprey
let $p$ be an odd prime and $a,b$ be coprime positive integers, prove that $$\gcd(a+b,\frac{a^p+b^p}{a+b})=1 \text{ or } p$$
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let $p$ be an odd prime and $a,b$ be coprime positive integers, prove that $$\gc...
woven radishBOT
Skissue ping4response
solid osprey
starus 1
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solid osprey
when its odd right?
lament kraken
@solid osprey
valid silo
k|a^p+ba^p-1
lament kraken
Hey
solid osprey
what
solid osprey
how do i simplify
solid osprey
let us say the gcd is k
a and b is coprime so k=1
lament kraken
Divide the a^p + b^p by a+b
valid silo
a and b is coprime so k=1
no?
solid osprey
what?
lament kraken
Bye
solid osprey
oo
solid osprey
k|a^p+ba^p-1
how do you get this?
valid silo
if k|x k|xy
solid osprey
Divide the a^p + b^p by a+b
ok i tried dividing and using euclids division algorithm and got gcd(a+b,pb^(p-1))
this feels close
now how do you get rid of the b^(p-1)
devout snowBOT
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hollow ice
ok i tried dividing and using euclids division algorithm and got gcd(a+b,pb^(p-1...
If gcd(a,b)=1, then gcd(ab,c)=gcd(a,c) * gcd(b,c)
You can use it here
solid osprey
hm
well either gcd(p,b)=1 or p
if its 1, then gcd(a+b,pb^(p-1))=gcd(a+b,b^(p-1))*gcd(a+b,p)
gcd(a+b,p) can be 1 or p, not sure about gcd(a+b,b^(p-1))
hollow ice
you know gcd(a,b) = 1
solid osprey
1=gcd(a,b)=>gcd(a+b,b)=1?
hollow ice
Yes
solid osprey
oh
so if gcd(p,b)=1 then gcd(a+b,pb^(p-1))=1 or p
if gcd(p,b)=p then gcd(a,p)=1
of we use the previous we only need to worry about gcd(a+b,p)
gcd(a+b,p)=gcd(a+pk,p)=gcd(a,p)=1
ok isnt this solved
solid osprey
wat
hollow ice
If gcd(a,b)=1, then gcd(ab,c)=gcd(a,c) * gcd(b,c)
You can use it here
I mean the idea is right, but you cant use this result
hollow ice
if the gcd is p, you can just sub in b=kp
solid osprey
shush i did not write anything
solid osprey
if the gcd is p, you can just sub in b=kp
truw
gcd(a+b,pb^(p-1))=gcd(a+kp,k^(p-1)×p^p)
gcd(a+kp,p^p)=1 right?
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solid osprey
i just realized gcd(a,k)=1
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round talon
what did i do wrong
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what did i do wrong
urban spindle
x+2x+3x=/=5x sadly
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spring oasis
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938c2cc0dcc05f2b68c4287040cfcf71
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sonic smelt
**938c2cc0dcc05f2b68c4287040cfcf71**
!status
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**938c2cc0dcc05f2b68c4287040cfcf71**
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
the matrix M_(B)(f) is saying that
- f(0,1,-1) = (1,a,1)_(B)
- f(1,1,1) = (-1,3,0)_(B)
- f(0,0,1) = (b,1,2)_(B)
spring oasis
```
What step are you on?
1. I don't know where to begin.
2. I have begun but go...
1
sonic smelt
Well, first you need to ensure f(2, 3, 1) equals (4, 3, 6)
Can you find the coordinates of (2, 3, 1) and (4, 3, 6) in the given basis?
spring oasis
you want me to find (2,3,1)_B and (4,3,6)_B?
sonic smelt
Yeah
spring oasis
- (2,3,1)_B ==> (2,3,1) = a(0,1,-1) + b(1,1,1) + c(0,0,1)
- (4,3,6)_B ==> (4,3,6) = a(0,1,-1) + b(1,1,1) + c(0,0,1)
right?
sonic smelt
No, those the vectors that have coordinates (2, 3, 1) and (4, 3, 6) in the given basis
spring oasis
my bad
sonic smelt
(2, 3, 1)_B is the vector (x, y, z) s.t. x(0, 1, -1) + y(1, 1, 1) + z(0, 0, 1) = (2, 3, 1)
spring oasis
yeah, I wrote that first, but idk why I rewrote to the other equation
I am bad with coordinates calculations, I am very bad
sonic smelt
Here calculating coordinates is simpler than usual since there is only one basis vector with a nonzero first entry
sonic smelt
(2, 3, 1)_B is the vector (x, y, z) s.t. x(0, 1, -1) + y(1, 1, 1) + z(0, 0, 1) =...
So you immediately see that y should be 2
Actually I probably should have given those variables different names
Deduce the value of x by looking at the second entries and of z by the third entries
spring oasis
(2,3,1) = (b,a+b,-a+b+c)
sonic smelt
Right
spring oasis
b = 2
3 = a+ 2
a = 1
1 = -a + b + c
1 = -1 + 2 + c
c = 0
- (a,b,c) = (1,2,0)
sonic smelt
Right, so (2, 3, 1)_B = (1, 2, 0)
spring oasis
perfect
sonic smelt
What about (4, 3, 6)_B
spring oasis
(4,3,6) = a(0,1,-1) + b(1,1,1) + c(0,0,1)
i) 4 = b
ii) 3 = a + 4 ==> 3 - 4 = a ==> a = -1
iii) 6 = -a + b + c ==> 6 = 1 + 4 + c
c = 6 - 5 ==> c = 1
- (a,b,c) = (-1,4,1)
sonic smelt
So you have (2, 3, 1)_B = (1, 2, 0) and (4, 3, 6)_B = (-1, 4, 1), this tells you that multiplying the given matrix of f by (1, 2, 0) should yield (-1, 4, 1)
In order for the f(2, 3, 1) = (4, 3, 6) to hold
I.e.,
[ \begin{bmatrix} 1 & -1 & b \ a & \phantom{-}3 & 1 \ 1 & \phantom{-}0 & 3 \end{bmatrix}\begin{bmatrix} 1 \ 2 \ 0 \end{bmatrix} = \begin{bmatrix} -1 \ \phantom{-}4 \ \phantom{-}1 \end{bmatrix} ]
woven radishBOT
A Loner Bean
sonic smelt
You can rewrite this as a linear system regarding a, b
Actually you will find the value of a using the latter, the value from b will come from ensuring that f is not injective
spring oasis
,, \begin{cases} 1 -2 &= -1 \ a + 6 &= 4 \ 1 &= 1 \end{cases}
woven radishBOT
938c2cc0dcc05f2b68c4287040cfcf71
spring oasis
a = 4 - 6 ==> a = -2
sonic smelt
Great
sonic smelt
Actually you will find the value of a using the latter, the value from b will co...
Now you can go about this in many ways, I personally would set the determinant to zero
spring oasis
if we set the determinant to zero
the image is R3 and the kernel is dim 0?
what does it mean that is not injective?
in terms of the image and the kernel?
sonic smelt
You are describing the case when determinant is nonzero
sonic smelt
in terms of the image and the kernel?
A linear map is injective iff its kernel is trivial
spring oasis
the determinant being zero means we have a linear dependency
sonic smelt
Not injective linear map <=> nontrivial kernel
spring oasis
by the matrix which matrix you mean M_(B)(f)
I take the determinant of this matrix? the matrix representation of the linear transformation that also changes coordinates
we can start by using last row, since it has one zero
sonic smelt
by the matrix which matrix you mean M_(B)(f)
Yeah
spring oasis
$-1 \cdot \begin{vmatrix}-1 & b \ 3 & 1 \end{vmatrix} + 2 \cdot \begin{vmatrix} 1 & -1 \ 3 & 3 \end{vmatrix} = 0$
its b not k, I misread
a = -2
no wayy
sonic smelt
\begin{vmatrix} is missing
spring oasis
the determinant being zero means we have a linear dependency
.
woven radishBOT
938c2cc0dcc05f2b68c4287040cfcf71
sonic smelt
1, 3 instead of -1, 2
spring oasis
$-1 \cdot \begin{vmatrix}-1 & b \ 3 & 1 \end{vmatrix} + 2 \cdot \begin{vmatrix} 1 & -1 \ 3 & 3 \end{vmatrix} = 0$
woven radishBOT
938c2cc0dcc05f2b68c4287040cfcf71
sonic smelt
a = -2 yeah
spring oasis
**938c2cc0dcc05f2b68c4287040cfcf71**
so it should be like this
ahh you mean the coefficients outside
not inside the vmatrix
spring oasis
I was using row 3
1 times vmatrix -1 b \ 3 1 vmatrix + 2 times vmatrix 1 -1 \ a 3 vmatrix
right?
,, 1 \cdot \begin{vmatrix}-1 & b \ 3 & 1 \end{vmatrix} + 2 \cdot \begin{vmatrix} 1 & -1 \ -2 & 3 \end{vmatrix} = 0
woven radishBOT
938c2cc0dcc05f2b68c4287040cfcf71
spring oasis
spring oasis
spring oasis
Along which row/column are you expanding?
which row or column do you want me to expand? How do you guys take the determinant of a 3x3 in america?
sonic smelt
I am not from America, so I wouldn't know
spring oasis
oh ok
sonic smelt
I would expand along the third row too though
sonic smelt
Mostly cofactor expansion
spring oasis
ok
,, 1 \cdot \begin{vmatrix}-1 & b \ 3 & 1 \end{vmatrix} + 2 \cdot \begin{vmatrix} 1 & -1 \ -2 & 3 \end{vmatrix} = 0
woven radishBOT
938c2cc0dcc05f2b68c4287040cfcf71
spring oasis
so this is correct or am I tripping?
like we want determinant zero so we have a nontrivial kernel
sonic smelt
So far so good
spring oasis
ok
(-1 x 1 - 3b) + 2(3 -2) = 0
(1-b) + 2(1) = 0
1 -b = -2
1 + 2 = b
b = 3
ok, so that was all
I think
sonic smelt
How come -1 - 3b became 1 - b?
sonic smelt
Right, that's it
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glacial ether
what is nicht führende Hauptminoren - non leading principal minor?
rotund umbra
Let X be an NxM (possibly non-square) matrix.
- A "minor of the matrix
X" is the determinant of a submatrix ofXformed by deleting some rows and columns. For this determinant to exist, the remaining columns must equal the remaining rows. - A "principal minor of
X" is where 1. the matrixXis square, and 2. the indices of deleted rows equals the indices of deleted columns. - A "leading principal minor of
X" is where the submatrix consists of the firstkrows andkcolumns ofX. - A "non-leading principal minor of
X" is any principal minor ofXthat isn't principal
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glacial ether
how do i use it to find the definite of the matrix
rotund umbra
glacial ether
how do i use it to find the definite of the matrix
rotund umbra
how do i use it to find the definite of the matrix
devout snowBOT
@glacial ether Has your question been resolved?
glacial ether
i mean how do i know if the matrix is positive definite or negative definite
rotund umbra
i mean how do i know if the matrix is positive definite or negative definite
I don't think there's a short cut. You can calculate the determinant and see if it's negative, positive, or zero.
Well, there is a short cut to see if a determinant is zero, if it's covector components are dependent.
For example, the determinant of {{1,2}, {2,4}} is zero because the system of equations it describes (x + 2y and 2x + 4y) are the same.
That is, if you can get one row from a sequence of transformations of the others, the determinant is zero.
Another example;
1 2 3
2 4 5
1 2 1
the determinant of this matrix is zero because row3 = 2 row2 - 3 row1
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mystic spire
can anyone help me in my channel
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gilded herald
How would you go about finding the coordinates of the point of a line that is the nearest to another point in a 3d space?
In my problem, i have access to the line's equation, the other point's coordinates and the distance between the line and the point.
devout snowBOT
How would you go about finding the coordinates of the point of a line that is th...
gilded herald
The distance is sqrt(236)/sqrt(14)
supple knot
How would you go about finding the coordinates of the point of a line that is th...
Just show the whole problem
gilded herald
The line's equation is
[2 4 -3] + k[2 1 -3]
The point Q's coordinates are
(2, 1, 4)
The question is : which point in the line is the nearest to the point Q ?
The question is related to the last question where you find that the distance between the line and the point is sqrt(236)/sqrt(14) units
Thats the whole problem, I cant really send an image since it's not in english and there are like 12 questions that separate the info about the line and the point and the question
supple knot
The question is related to the last question where you find that the distance be...
Find the distance from point to line as a function of k and then minimize distance and solve for k
It'll help to minimize distance^2 since it gives the same answer
timber ore
You can do the minimisation route, but it's probably easier to do it purely as a vector job.
Pick any point on the line (e.g. the one you're given), find the vector from there to Q, and dot that with the (normalised!) direction vector of the line. That gives you the distance from the point you chose on the line, along to the point that's closest to Q. Then you just go along the line by that distance and you're there
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west depot
Can someone help me with this
devout snowBOT
Can someone help me with this
west depot
I have to simplify the x^2 + y^2 - 2xy -1 but there's a step I dont understand
This one precisely
olive snow
Its using identity
west depot
Its using identity
how do you go from (x-y)^2 -1 too [(x-y)-1)*[(x-y)+1]
west depot
why would you do that tho
olive snow
Cuz the question ask for ?
west depot
I feel like its doing the opposite of simplifying
olive snow
I have to simplify the x^2 + y^2 - 2xy -1 but there's a step I dont understand
You can't simolify anything to this
west depot
so like it's a formula I have to follow?
west depot
why you divide specifically (x-y) and -1
dont you have to consider the ^2 in the (x-y)?
olive snow
Divide ?
slender peak
è un doppio prodotto
olive snow
Well you can distribute (a-b)(a+b) to see what it gives
olive snow
sorry Im translating in my native language
Yeah no prob
Just making sure
To not gives useless thing
tall cypress
I have to simplify the x^2 + y^2 - 2xy -1 but there's a step I dont understand
There is a formula
(x - y)^2 = x^2 - 2xy + y^2
olive snow
But since an italian arrived
west depot
I dont know how I can struggle in an easy topic but can easily do way harder topics
west depot
There is a formula
(x - y)^2 = x^2 - 2xy + y^2
yes, that's the one that originates from
slender peak
se fai i calcoli, quindi fai la potenza, alla fine torna tutto
west depot
se fai i calcoli, quindi fai la potenza, alla fine torna tutto
mi dispiace ma per me è come se non seguisse alcun filo logico
aspe
tall cypress
yes, that's the one that originates from
Yes then you put it back there
west depot
(x - y)^2 - 1 is the answer
not really it says it's (x−y−1)(x−y+1)
west depot
I don't get how you go from what slender said to the result
olive snow
This one precisely
Also idk but if this is gpt, avoid it cuz its sometimes really wrong
tall cypress
I don't get how you go from what slender said to the result
Difference of squares
a^2 - b^2 = (a - b)(a + b)
west depot
shouldn't it be sonething like (x-y)(x-y) -1?
tall cypress
(x - y)^2 - 1 is the answer
So (x - y)^2 - 1^2
west depot
why do we put the 1 both times
tall cypress
((x - y) - 1)((x - y) + 1)
tall cypress
a^2 - b^2 = (a - b)(a + b)
Think of (x - y) ^2 as a^2
west depot
Im trying to think
okay I understand that [(x−y)−1][(x−y)+1] is [a-b][a-b]
so like we divide (x-y) and -1 because they don't share anything in common?
like (x-y) = a and -1=b?
@tall cypress
slender peak
coma scomporresti x^2-9?
tall cypress
okay I understand that [(x−y)−1][(x−y)+1] is [a-b][a-b]
No its (a - b)(a + b)
slender peak
che è esattamente la stessa cosa
west depot
coma scomporresti x^2-9?
fammi pensare
slender peak
giusto!!!
slender peak
ora è lo stesso principio, devi rifare la stessa cosa
west depot
[(x-y)+1][(x-y)-1]
ci credi che io riesco a fare la trigonomia senza difficoltà e poi mi blocco per argomenti del genere?
grazie mille comunque
I think I've understood now, I'll just try another similar exercise and I'll close if I dont need your help
slender peak
quindi x^2+y^2-2xy diventa, visto che bisogna scomporre il polinomio in più di grado più basso, diventa ((x-y)-1)(x-y)+1)
west depot
si grazie
slender peak
ci credi che io riesco a fare la trigonomia senza difficoltà e poi mi blocco per...
hahahahhaha
west depot
provo a fare un altro esercizio per conto mio
ok ho capito
thanks everyone I seem to understand now I did another exercise by myself
/close
.close
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clever sphinx
hellooo
clever sphinx
can someone help me with 12
tired tendon
What does it want me to do?
devout snowBOT
What does it want me to do?
supple knot
Use graphs of cos and sin
tired tendon
Do I just graph it?
queen vine
Do I just graph it?
yah
tired tendon
Okay, thanks
devout snowBOT
@tired tendon Has your question been resolved?
tired tendon
Can anyone compare answers with me rq
?
Graphing calculator is being uselesss
HELLO!?
ocean viper
I’m here
misty crest
@ocean viper
tired tendon
okay, thanks
ocean viper
Gimme one sec lemme get a piece of paper
tired tendon
Gimme one sec lemme get a piece of paper
Alright, thanks sir
ocean viper
What did u get for the first one
tired tendon
y = sin[36(x-4)] + 2.5
tired tendon
Max = 3.5
Min = 1.5
C = 2.5
A = 1
oh
k = 180?
I'm not sure
K is 360/period
from what I know
ocean viper
I looked at the scale wrong for the y-axis 😭😭😭
tired tendon
Can anyone compare answers with me rq
What about B?
ocean viper
Lemme solve that gimme 1 min
tired tendon
Alright, thank you
ocean viper
What did u get
tired tendon
y = 2sin(x-1) + 4
ocean viper
?
ocean viper
Oh ur right my bad
tired tendon
Alrighty
tired tendon
lol
ocean viper
At this point I should be getting help lol
tired tendon
tired tendon
At this point I should be getting help lol
You probably just haven't done them in a while
that's all
ocean viper
What have u done so far
ocean viper
Dang
tired tendon
I know to find period, I need to do 1.75 - 1, to find the diff
for the value betwen them
ocean viper
So it takes 0.75 seconds to go from min to max
tired tendon
yeah
ocean viper
But that’s half the period because we want a full cycle
tired tendon
then x2?
tired tendon
Yep
ocean viper
Which basically represents the amount of time for a full cycle to occur on her rocking chair
ocean viper
?
tired tendon
C = 1.375
ocean viper
Yep
tired tendon
For c, it's asking for the domain when period is 40
40 x 1.75
I think
wait
on
no
ocean viper
Uhh
It’s going through 40 cycles
And each cycle or period is 1.5 seconds
Which we identified before
ocean viper
Ye
ocean viper
And since range is in terms of y we need to find restrictions on distance not time
ocean viper
Yes
ocean viper
Yep
ocean viper
Should be right
devout snowBOT
@tired tendon Has your question been resolved?
placid trout
devout snowBOT
placid trout
for part c why didnt they check t = 1 during the candidates test
nvm
they check the bounds and then the critical point in part b
.close
devout snowBOT
lament schooner
if you were chovy this wouldve been easy
devout snowBOT
Available help channel!
normal merlin
devout snowBOT
normal merlin
you can use the ratio test and the nth term test to solve this right?
becasue if you use the nth term test you would get infinity right?
and if you use the ratio test you get e^5 which is greater than 1 which means it diverges
just trying to understand if I understand what im learning
if that makes sense
acoustic leaf
yes
rose wadi
i need help in a question
devout snowBOT
i need help in a question
rose wadi
should i just send it here?
simple seal
yes
rose wadi
Find all natural numbers n such that 2^n-n is a perfect square
i've only managed to solve for even n
simple seal
show us your working so far
rose wadi
wait
if n is even, n=2k, then 2k=(2^k+x)(2^k-x)
there exists a,b such that 2^k+x=a, 2^k-x=b, daí 2^k=(a+b)/2 e x=(a-b)/2\implies (a^2+b^2)/2=(a^2+b^2)/4+ab/2\implies a=b, which is a contradiction
simple seal
so there are no even n
devout snowBOT
@rose wadi Has your question been resolved?
devout snowBOT
@rose wadi Has your question been resolved?
devout snowBOT
@rose wadi Has your question been resolved?
devout snowBOT
@rose wadi Has your question been resolved?
modest pollen
Does anyone know the exact method archimedes used to approximate pi with his method of exhaustion? Writing about it in my dissertation and there is so much info on the internet that conflicts eachother, also quite hard to find a decent paper with it
devout snowBOT
Does anyone know the exact method archimedes used to approximate pi with his met...
simple seal
iirc he calculated it using polygons with many many sides
i believe he started with a hexagon
and started doubling the sides
acoustic leaf
here is a site which attempts to present the original method as written: https://web.archive.org/web/20191118171203/http://itech.fgcu.edu/faculty/clindsey/mhf4404/Archimedes/Archimedes.html
devout snowBOT
@modest pollen Has your question been resolved?
restive river
help
restive river
heeeeeeeeeeellllllllllllllllpppppppppppppppppppp
restive river
what is E(W), thats like random discrete variable
but i dont even know what that fully means
final drift
do you know what the cumalitive distribution function means
iron sun
That’s a number
final drift
what is E(W), thats like random discrete variable
E(W) refers to the expected value of W
restive river
do you know what the cumalitive distribution function means
nope
shi i think thats the one lesson i missed at skl😭
it was the last lesson
and i left skl early
final drift
nope
okay the cumalative distribution function "$F(w)$ for $w$" is the probability that $W$ is less than or equal to $w$
woven radishBOT
Arnavutköy
final drift
so basically, if we plug $w=7$ into that function, we will get the chance that $W$ is less than $7$
woven radishBOT
Arnavutköy
final drift
if we plug in $w=6$ into that function, we will get the chance that $W$ is less than or equal to $6$
restive river
that all outcomes
woven radishBOT
Arnavutköy
final drift
that it is less than or equal to
restive river
are less than that number
so E(6)
gives
us the probability
of it
being
5,4,3,2,1
restive river
but here, we know that it only can be 6,7, or 8
yh
restive river
also im describing F(W) when w=6, not E(6)
ok
F(W)
is everything
equal to and less than
final drift
so we have that when w=6, F(W) is the chance that w=6
restive river
what do you mean by E(6)?
in genreal
lik
expected
discrete value
like in a diff scenario
final drift
do you mean P(W=6)? "The chance that W=6?"
restive river
nah its fine
final drift
so plug in $w=6$ into the equation
woven radishBOT
Arnavutköy
final drift
plug in $w=7$ and $w=8$ also
restive river
yh
woven radishBOT
Arnavutköy
restive river
45/77
restive river
yes, so what is the chance that P=6?
45/77
final drift
okay, so what is the chance that the variable is equal to 7?
restive river
so chance of =7 is 15/77
restive river
and chance of it being =8 is 17/77
final drift
yep
restive river
ok
final drift
do you know how to calculate the expected value now?
restive river
no
modest pollen
here is a site which attempts to present the original method as written: https:/...
much appreciated
final drift
okay, so for the expected value, we basically want the "average" value the variable takes
final drift
so for example, if a variable was half of the time 0, and half of the time 1, we would say that the expected value is half
restive river
ok
final drift
so howd we do that
so for each number it can be
we take the probability that it is that number
and multiply it by the number itself
final drift
yes
final drift
7x15/77 or 60/77
7*15/77, since 15/77 is the chance that it is 7
restive river
k
final drift
we have already multiplied by the probabilities
supposing it had an equal chance of being 6,7, or 8
we would just take (6+7+8)/3
another way to think of this is (1/3) times 6+(1/3) times 7+(1/3)*8
as in we multiply every number by the probability
now we have done the same thing
except the probabilities are no longer all 1/3
restive river
yh
final drift
they are different for every number, but the three probabilities still all sum up to one
restive river
yh
bro
what is this
bro
im bugging
how do we even find this
is it that one formula
likeeeee
Z WORLD
final drift
so do you understand when a kiwifruit is large
restive river
Z- x-standard deviation over mean?
restive river
so do you understand when a kiwifruit is large
yh
final drift
so its when it is greater than equal to $70$ right?
woven radishBOT
Arnavutköy
final drift
and since mean is 80 and std.dev. is 8, do you know what z-score that will correspond to?
restive river
yh
restive river
and since mean is 80 and std.dev. is 8, do you know what z-score that will corre...
no
OH SHIIIIIIIIIIII
NOW
WE USE
THE GRAPH THING
IS IT NCD???
i got
0.89
restive river
abt b
final drift
0.89 seems to be about right guestimating without a calculator
final drift
upper limit
you don't need to now the upper limit?
fast oriole
its 37
woven radishBOT
Arnavutköy
final drift
do you know what the z-score is
restive river
idk what it even means
its like some code
but
i know the formula
icl might need to grind stats
im shambolic
final drift
so the z-score reperesents how many standard deviations you are away from the mean
so for 70 for example
the mean was 80
so we were -10 from the mean
and we are going to divide this by 8, our standard deviation
thus we are -1.25 standard deviations away from the mean
so we call this number the z-score, which was -1.25
and then, the thing is for the normal distribution, we can always find the probability based on the z-score
restive river
so the z-score reperesents how many standard deviations you are away from the me...
oh yh
final drift
it is a complex function which is quite hard to understand
but for our purpose we just use a calculator or google to find out what z-score corresponds to which probabiliy
restive river
but how did u do b
final drift
specifically we saw a z-score applies to roughly 10.5 percentile
final drift
but how did u do b
oh wait wait
so first find out what percentage of kiwis are 80 or greater
find the z-score corresponding to 80
final drift
the area
what do you mean the area?
we have the mean, the standard deviation, and the value
that is all we need to find the z-score
specifically, the z-score is the value minus the mean, all over the standard deviation
final drift
so what probability does that correspond to?
final drift
yeah
so 50% of the time, the kiwi is jumbo or mega
if 35% of the time the kiwi is jumbo, that means that 15% of the time is mega
or alternatively, the kiwi is mega if the kiwi is in the 85th percentile or above
so now, google says the 85th percentile corresponds to a z-score of 1.036
so using this, can you calculate the cutoff between jumbo and mega?
devout snowBOT
@restive river Has your question been resolved?
devout snowBOT
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Available help channel!
placid trout
placid trout
why did they use implicit differentiation here
simple seal
because you cant directly differentiate x^1/x
they used a more specific technique called logarithmic differentiation
bringing the exponent down with logs before differentiating
placid trout
bringing the exponent down with logs before differentiating
this is the same for x^x?
lament kraken
this is the same for x^x?
Yes
devout snowBOT
devout snowBOT
Available help channel!
jolly ravine
for this equation, I've already solved it by cross multiplying both sides and using the Pythagorean identity for sine and cosine (sin^2(x) = 1 - cos^2(x) then expand it)
not sure what other algebraic methods or identities I could use to verify the equation.
is multiplying the numerator and denominator by the conjugate of 1 + cos(x) which is 1 - cos(x) a valid way to verify the equation?
devout snowBOT
for this equation, I've already solved it by cross multiplying both sides and us...
jolly ravine
here's the supposed verification for what im asking:
i wasnt taught it so im just unsure of it's a correct way of verifying a trig identity
simple seal
yes it is valid
devout snowBOT
devout snowBOT
Available help channel!
void knot
Why can this indicate that the vector set a1,....an is linear indep?
simple forge
its definition
devout snowBOT
@void knot Has your question been resolved?
static ember
it's a definition, but intuitively, a system of vectors is linearly independent if none of the vectors are "superfluous". That means that in case it is linearly dependent, then there is some vector you can throw out of the system and they will still span exactly the same vector space. That is indicated by the fact that when a system is linearly dependent, it is possible to construct the same vectors with multiple different linear combinations. Like if you have the system (1, 0), (1, 1), (1, 3), then you can construct (2, 1) as (1, 0) + (1, 1) or as 3(1, 0) - 2(1, 1) + (1, 3), which are already two different ways. This indicates that at least one vector of our system is superfluous and when throwing it out, all linear combinations become more unique. If there are no more superfluous vectors and our system is linearly independent, then every linear combination is truly unique. In order for that to be true, it just suffices to show that the zero vector has a unique linear combination, which is exactly that definition
if it's not linearly independent and there was some non trivial linear combination of the zero vector $k_1 \alpha_1 + ... + k_m \alpha_m = 0$, then let's just assume $k_1 \neq 0$ and we can write $\alpha_1 = \frac{-k_2}{k_1} \alpha_2 + ... + \frac{-k_m}{k_1} \alpha_m$
woven radishBOT
rbit
static ember
so we can represent some vector of our system by a linear combination of the others, making it superfluous
plush grail
devout snowBOT
plush grail
so im having a problem with part b
i want someone to see my working out
if they dont mind
because it feels like i hit a wall and cant continue
this is what i got up to
and idk what to do
if there wasnt an x after the equal sign it wouldve been way easier
like a zero instead
idk
help please
devout snowBOT
@plush grail Has your question been resolved?
restive river
help please
still need help ?
plush grail
yeah
restive river
in the last line, the sum should start from 2, not 1
there is probably a mistake in the lines before
@plush grail actually, you are also right, it is ok if you write it from 1
but why are you stuck now ?
plush grail
im used to that after the equal is zero
not x
now i cant say what's inside the braket to be zero
restive river
if 2 polynomials are equal, it means that they have the same coefficient before x , x^2 , x^3 ... etc. but on the right, you have only x
restive river
on the right, it is a polynomial with a constant=0 and the coefficient before x^2, x^(anything) are 0, all except fox x^1
plush grail
so im comparing coeff ?
restive river
yea
plush grail
so everything inside the x power n braket equals zero
restive river
except when x=1
plush grail
mm
restive river
so guess what must be the coefficient before x^1 ?
plush grail
x power zero ?
wym
im confused
i should take the braket and equal it to zero
get cn+2
restive river
i should take the braket and equal it to zero
yes but inside the bracket for each power of n is equal to the coefficient of the polynomial: x
plush grail
i dont grasp what your trying to get at
restive river
the polynomial x is: 0x^0 + 1 * x^1 + 0x^2 + 0* ...
plush grail
i dont get it
will this differ in the way of solving this
or can i procceed with my thing
restive river
or can i procceed with my thing
you are almost right, im explaining the reason you are a little wrong
plush grail
mm okay
should i get the value of the series when n is 1 and sub it on the other side of the equation ?
dont think that would lead to anything
plush grail
uhm
no
like
we're comparing coefficients
so like
compare anything that's multiplied by x on the other side
restive river
compare anything that's multiplied by x on the other side
exactly, so why don't u agree withmy last message ?
plush grail
cause you mentioned other powers of x
x
it should only include the first power of x
restive river
so it is like if you have a + bx +cx^2 = x and u must find a,b and c