im not sure how to start
#help-49
azure bolt
azure bolt
its gotta be c + something
azure bolt
Can you identify what's given and what the question wants you to find?
their speeds are given
and it wants me to find the distance that johnathon walks once he overtakes anna
past jay
Alright
azure bolt
i konw that
know that
d = c + ...
because he has to surpass the distance of their gap before he reaches her
past jay
Well what's the definition of speed?
azure bolt
Well what's the definition of speed?
s = distance / time
past jay
That's Anna's distance
past jay
Yeah
past jay
for johnathons distance
Alright Jonathan's distance = pt = ???
azure bolt
yes
past jay
Based on the fact that he has to walk the distance c and overtake anna
azure bolt
so then the total distance will be d = c + pt
past jay
But pt is Jonathan's distance
azure bolt
so its qt?
past jay
Yeah
past jay
Since Jonathan walks faster
past jay
pt = c + qt
Exactly
sick jay
Hello, claim your own channel
How
azure bolt
this is trippy
sick jay
Oh ok
past jay
!occupied
midnight plankBOT
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).
sick jay
!claim
pearl hull
How
Go type a message in https://discord.com/channels/268882317391429632/1018700692027883620
past jay
!claim
Check #❓how-to-get-help and go to an empty help channel and type your question there
There are plenty so don't worry
midnight plankBOT
past jay
ok thanks vulcan
Keep in mind
The time they took is the same for Anna and Jonathan
That's why it's pt = qt + c
azure bolt
thanks
midnight plankBOT
Available help channel!
primal owl
Can someone help me with this question and tell me where I messed up? ;( ig I do understand that the derivative of -cot2x should be cosec²2x/2 instead (right?) but idk how that'd help :( mb if I made a silly mistake im kinda new to integrating/differentiating with trig functions
midnight plankBOT
Can someone help me with this question and tell me where I messed up? ;( ig I do...
shell wigeon
The derivative of -cot(2x) is not 1/2 cosec(2x)^2 if that's what you wrote
primal owl
The derivative of -cot(2x) is not 1/2 cosec(2x)^2 if that's what you wrote
nahh thats not what i did then
i asked my classmate she said thats my mistake but i was like 'that doesnt help either'
i used cosec²2x according to what i proved in part a ig
but its probably wrong
shell wigeon
Well it does help, kinda
primal owl
Well it does help, kinda
YEAH i mean obv the correct derivative would help but she said thats the derivative which is not the case according to what u said :p
shell wigeon
I see two ways of doing this: change of variable in the integral, or figuring out the antiderivative of cosec²(2x)
primal owl
I think I'm supposed to do the latter
shell wigeon
So it's either the chain rule or the "inverse" of the chain rule
primal owl
How does the first one work tho
primal owl
So it's either the chain rule or the "inverse" of the chain rule
If ur leading me toward u sub idk how to do that yet 😔
Is there a way to solve it without using it
shell wigeon
Yes, the second method
primal owl
Yes, the second method
then could u tell me where i messed up? ;(
shell wigeon
Well, here of course
primal owl
Well, here of course
yeah ikk but idk how to correct it
im kinda ignoring the chain rule here ik
kinda dumb
shell wigeon
Can you state the chain rule?
primal owl
uhh if u have a composite function f(g(x)) then its derivative would be f'(g(x))*g'(x) right
😭
shell wigeon
Yes
shell wigeon
Exactly
primal owl
and then u divide two on both sides
BROO 😭
i knew it was smth small
ok i have my answer now
u dont know how happy this makes me tho
this was a fun question
.close
vernal field
My Q is it possible to divide and multiply by 2pi*root(1+n^2)
midnight plankBOT
My Q is it possible to divide and multiply by 2pi*root(1+n^2)
dawn dagger
My Q is it possible to divide and multiply by 2pi*root(1+n^2)
yes
vernal field
yes
but n must be tending to 0 right
here even 2pi(root (1+n^2)) must be as the new "n" too.. if u get what im saying
shell wigeon
It's pretty obvious this is indeterminate
dawn dagger
It's pretty obvious this is indeterminate
This limit isn't over the reals
tender dew
It's pretty obvious this is indeterminate
It's not
shell wigeon
Oh, I should learn to read
vernal field
This limit isn't over the reals
can u also pls
vernal field
I don't know the answer
no but i cant divide it by 2pi*root(1+n^2) right?
tender dew
but n must be tending to 0 right
about this yes, dividing by 2pi sqrt(1 + n^2) won't be useful as you can't use sin(x) / x limit
vernal field
about this yes, dividing by `2pi sqrt(1 + n^2)` won't be useful as you can't use...
im just asking not in terms of q
but
lim 2pi(1+n^2)---> 0 must be there right
tender dew
well that doesn't go to 0
vernal field
for sin(2pi(1+n^2))/2pi(1+n^2)
tender dew
right
dawn dagger
I only remember a trick of adding and subtracting n inside sine
vernal field
I only remember a trick of adding and subtracting n inside sine
he subracts -2pin here.
brisk iris
This does not have a limit …
vernal field
This does not have a limit …
...?
brisk iris
Look at it. Whatever is inside the sine goes to infinity
vernal field
Look at it. Whatever is inside the sine goes to infinity
this has
a limit
ans is pi.
brisk iris
??
dawn dagger
It has a limit, it's sequence not a real function
,, \sin(2\pi n+2\pi(\sqrt{1+n^2}-n))=\sin(2\pi(\sqrt{1+n^2}-n))=\sin(\f{2\pi}{\sqrt{1+n^2}+n})
grand pondBOT
vernal field
again 2npi/root(1+n^2) + n must be used as limit right
vernal field
u r smart
vernal field
but he divides by the 2pi/root(1+n^2) + n
my doubt is how can u do that as lim 2pi/root(1+n^2) + n --->0 must be there but its n--> inf
@dawn dagger @tender dew
dawn dagger
but he divides by the 2pi/root(1+n^2) + n
my doubt is how can u do that as lim 2...
Since this goes to 0 we have that sin(x) ~ x
vernal field
Since this goes to 0 we have that sin(x) ~ x
works but
he divdes here..
and says 1.
which cannot be done right?
tender dew
Look at the argument inside the sine in the numerator, you have $$\frac{2\pi}{\sqrt{1+n^2} + n}.$$ This obviously goes to zero, agreed?
grand pondBOT
Peter
dawn dagger
Is your issue that he is dividing by 0 or what?
tender dew
||what a cute problem btw||
vernal field
**Peter**
i get it its approx x
that sin theta approx theta for very small theta
but
according to soln i sent
he divides and multiplies by the theta and says its one.
Oh wait..
you say Theta/theta = 1.. so..
ok damn @tender dew @dawn dagger smart as heck
dawn dagger
Me when algebra
vernal field
Me when algebra
one doubt tho
here lim n---> inf sintheta/theta (theta smth in terms of n ) it ISNT one right?
dawn dagger
It is
tender dew
in the image you sent, I believe this n (underlined red) shouldn't be there
dawn dagger
in the image you sent, I believe this `n` (underlined red) shouldn't be there
Yeah you are right
Else it would go to 0 actually
instead of pi
@vernal field
tender dew
it shd be theta-->0 right
yes, and theta DOES go to zero
where
$\theta = \frac{2\pi}{\sqrt{1+n^2} + n}$
grand pondBOT
Peter
vernal field
im telling not for small theta
dawn dagger
atp just do a sub xd
tender dew
$\lim_{\theta \to 0} \frac{\sin \theta}{\theta} = 1$
vernal field
i got thatcleared
grand pondBOT
Peter
vernal field
**Peter**
Yes. This is right.
Now say theta = x^2 +1
x isnt small.
now:
Lim x--> infinity (sin theta)/theta
it isnt 1 right
tender dew
correct, but why do you keep insisting theta doesn't go to zero, when in our cases it does?
vernal field
correct, but why do you keep insisting `theta` doesn't go to zero, when in our c...
i understood prev case, i provided a new case to get clarity.
dawn dagger
Yes. This is right.
Now say theta = x^2 +1
x isnt small.
now:
Lim x--> infinit...
if theta=x^2+1 then x goes to x²->-1 not infinity
tender dew
aha, yes then you are correct
tender dew
it goes to *i*
i think they meant to ask what happens to sin(x^2 + 1)/(x^2 + 1) when x -> infinity,
dawn dagger
oh
tender dew
anyways, if the original problem is resolved, we can close the channel
midnight plankBOT
@vernal field Has your question been resolved?
vernal field
i think they meant to ask what happens to `sin(x^2 + 1)/(x^2 + 1)` when `x -> in...
it doesnt go 1 right
dawn dagger
it doesnt go 1 right
no
midnight plankBOT
dawn dagger
,, \abs{\f{\sin(x^2+1)}{x^2+1}} \le \f{1}{x^2+1} \ra 0
grand pondBOT
midnight plankBOT
Available help channel!
inner marsh
ok in doppler effect:
example ambulance moves past a cyclist then th cyclist is moveing after the ambulace the formula is gonna be
midnight plankBOT
ok in doppler effect:
example ambulance moves past a cyclist then th cyclist is ...
inner marsh
its not gonna matter the ambulance is faster then the cyclist right
since the cyclist is still moving toward the ambulace
?
midnight plankBOT
@inner marsh Has your question been resolved?
inner marsh
<@&286206848099549185>
stark tartan
,,y''(x)+9y(x)=0, y(0)=0, y(\pi)=1
midnight plankBOT
,,y''(x)+9y(x)=0, y(0)=0, y(\pi)=1
grand pondBOT
MichaelRafto
stark tartan
i have problem, how do i find the constants on this one?
i have this:
,,y(x)=c_1cos3x+c_2sin3x
grand pondBOT
MichaelRafto
grand pondBOT
MichaelRafto
stark tartan
,,1=-c_1
grand pondBOT
MichaelRafto
dawn dagger
The conditions can't hold at the same time I think
stark tartan
these are correct though, right? this is what i get
shell wigeon
Yeah looks like a trick question; original?
stark tartan
for roots $\pm3i$
grand pondBOT
MichaelRafto
dawn dagger
yes
stark tartan
Yeah looks like a trick question; original?
it is from an exercise paper
lyric charm
!original
midnight plankBOT
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.
lyric charm
are you absolutely sure that it said y(0) = 0 and y(pi) = 1?
stark tartan
aight aight
lyric charm
any sol to this is guaranteed pi/3 periodic and so guaranteed not to satisfy these conditions both at once
lyric charm
can you translate the greek
i can only understand problemata without translation 
and ig the first word says "Characterize"?
stark tartan
ahhhhh yes my bad
lyric charm
wait does this literally only ask you to classify these problems into initial value problems vs. boundary value problems
stark tartan
"Classify the problems as problems with initial values and problems with boundary values"
yes my bad
lyric charm
yeah youre not even asked to solve anything 
midnight plankBOT
midnight plankBOT
Available help channel!
thorny delta
is this a horizontal compression of √x ?
midnight plankBOT
is this a horizontal compression of √x ?
thorny delta
same qu stion with this
steel crest
is this a horizontal compression of √x ?
i think we use the word "compression" only if the absolute value of the number is < 1
or perhaps <= 1
here sqrt(3) > 1
thorny delta
i think we use the word "compression" only if the absolute value of the number i...
i didnt understand 😢
steel crest
"compression" as far as I know, implies you're multiplying by a number c such that -1 <= c <= 1
and if c < -1 or c > 1 I would not call it a compression
midnight plankBOT
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Available help channel!
grave kernel
given integers a,b,c satisfying a² + b² = c² prove that abc is even
a)consider various parity cases
b) argue by contradiction
in the first part
obviously if even one of a,b,c is even, the product is even
but then i need to prove that all 3 arent odd
so considering them like (2n+1) and then proving stuff
is that what is meant by considering different parity cases
midnight plankBOT
given integers a,b,c satisfying a² + b² = c² prove that abc is even
a)consider ...
spiral rock
abc is even onl if one of a,b,c is even
eternal pawn
look mod 4
spiral rock
so what you want to show is that its impossible for a,b,c to be odd and a^2 + b^2 = c^2
grave kernel
look mod 4
mhm i get that
but i just wanted to know what they mean by looking at different parity cases
dreamy lichen
What's wrong on looking mod 4?
spiral rock
unnecessary
dreamy lichen
unnecessary
simple enough for a one liner
unless you have a proof in one word or sth like that, it cant get much better
spiral rock
the entire proof can be in mod 2 also
eternal pawn
oh yh lol
dreamy lichen
mhm i get that
but i just wanted to know what they mean by looking at different ...
isnt that kind of what u did?
you considered even-odd-odd, odd-odd-odd...
grave kernel
so when im doing it by contradiction tho, its basically the same thing what im doing right
not much of a difference in the main step
dreamy lichen
I'd say so
eternal pawn
same
dreamy lichen
if you want to avoid unnecessary algebra with that 2n+1 thingy, consider modulo instead
grave kernel
mhm yes got you
dreamy lichen
though the algebra will work too
trim current
Brushing up on integration. What is the approach to integrating the first term? I know that the other two terms come out to be 4pi and is said to be the volume element, however my book does not discuss the derivation.
midnight plankBOT
Brushing up on integration. What is the approach to integrating the first term? ...
civic lynx
i.e. just the integral
$$c_i c_j \int \left(-6\alpha_j + 4\alpha_j^2 r^2 \right) e^{-(\alpha_i + \alpha_j)r^2} ; \dd \vec{r},$$
where $c_i, c_j, \alpha_i,$ and $\alpha_j$ are constants?
grand pondBOT
haseeb
trim current
Yes
civic lynx
dr in exponent
hmm ok
wait, you said the other terms come out to 4pi. are there bounds on the integral?
trim current
i guess im getting confused as to how $\dd \vec{r}$ is supposed to be applied?
grand pondBOT
Hamp
civic lynx
well, if it is an integral $\dd\vec{r}$, then it's probably a path integral/line integral over some path $C$
grand pondBOT
haseeb
civic lynx
does the original q have something like "over the unit circle" or "over the triangle connecting the points..."
trim current
I dont know if its just the two terms that come out to 4pi but the system is sphereically symmetric. so in the end it comes out be $4 \pi * \int{g(x)} \dd r$
grand pondBOT
Hamp
civic lynx
where g(x) is our integrand?
trim current
The original problem stems from generating this matrix T where $T_{ij}=\int{\phi_i \nabla^2 \phi_j} \dd \vec{r}$
grand pondBOT
Hamp
trim current
where g(x) is our integrand?
g(x)
grand pondBOT
Hamp
civic lynx
ah ok, but they drop the $\dd\vec{r}$ and replace it with just $\dd r$
grand pondBOT
haseeb
trim current
I had analytical solutions, but I dont know if they were given correctly. T_ij should be a symmetric matrix and my python was giving me an unsymmetric matrix. so i thought it best to derive the analytical
civic lynx
i think because you have that symmetrical $1s^2$ orbital, the values of $\theta$ and $\phi$ "dont matter" in the spherical coords, so you can just reduce this problem to single-variable
grand pondBOT
haseeb
civic lynx
dont quote me on the physics of it, though
trim current
then integrateing wrt r. any tricks needed for the first and 3rd terms?
or is it more or less just straightforward?
$T_{ij} = c_i c_j \int{4 \alpha_j^2 r^2 e^{-(\alpha_1+\alpha_j)r^2} - 6\alpha_j e^{-(\alpha_1+\alpha_j)r^2} + 2/r^2 e^{-\alpha_i r^2} (\frac{\cos{\theta}}{\sin{\theta}^3})} \dd r$
grand pondBOT
Hamp
civic lynx
looks like IBP + u sub could work for the first two terms (i combined them by factoring the exponential)
you do end up with erf(x) which is 
, but from 0 to infinity you should get a nice closed form
no idea for the last term ill be honest, this just seems like a tough integral, probably why they derived it
trim current
you do end up with erf(x) which is <:sully:651816820122189834>, but from 0 to in...
They all come out to be beautiful curves, but getting there seems to be the trick
civic lynx
I had analytical solutions, but I dont know if they were given correctly. T_ij s...
if you just wanna check it's symmetric, see if you can show that T_ij = T_ji
truly very nice curves 
yeah i checked it via calculator and there doesnt seem to be any amazing trick
civic lynx
if you just wanna check it's symmetric, see if you can show that T_ij = T_ji
assuming real entries*
trim current
gotcha. i appriciate it. Ill get those solved out and close this. with an analytical solution they should be. with the given equation they are all over the place which is why im on the adventure. I had originally solved numerically but that has FP errors.
As you can see thats very unsymmetric haha
civic lynx
truly! yeah i think algebraically checking symmetry of your entries should be enough
but i do wonder if the entries are real or complex with just the real part 🤔
because the latter would not be symmetric without the imaginary part
trim current
You can close this whenever. I appriciate the help. In theory we allow for psi/phi to be imaginary. technically phi_i is the complex conjugate. but we live in the real world so all functions are real
That is how it has been explained to me
civic lynx
ah ok well that makes life easier
generally the helpee can do .close when they're done, so we dont accidentally cut you off :)
fallow scarab
You can close this whenever. I appriciate the help. In theory we allow for psi/p...
you can do .close
midnight plankBOT
midnight plankBOT
Available help channel!
oblique smelt
midnight plankBOT
oblique smelt
I need to factoirse and simplify this into -1/B
surreal moon
!status
midnight plankBOT
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
!show
midnight plankBOT
Show your work, and if possible, explain where you are stuck.
oblique smelt
im at -17/(17x)^1/2(17(x+h)^1/2((17x)^1/2+(17(x+h)^1/2)
i dont know how to simplify the denominator
midnight plankBOT
@oblique smelt Has your question been resolved?
zenith snow
im at -17/(17x)^1/2(17(x+h)^1/2((17x)^1/2+(17(x+h)^1/2)
Can u write it down on a paper
At first, add the fractions on numerator, and get a common denominnator and then multiplying both numerator and denominator(of inimtial numerator) with the conjugate of numerator and then u will notice that "h" cancels out. And u get soonething like -1/B ,like u wanted
zenith snow
The common denominator should be sqrt(17x(x+h))
zenith snow
$\frac{\sqrt(x) -\sqrt(x+h)}{
\sqrt(17x(x+h))h}$
grand pondBOT
Victimizer
zenith snow
This is what u get b4 multiply with conjugate
The denominator should be this
Sqrt 17 is present jn both fractions, so it's common u dont meed to take it twicw
oblique smelt
wait let me think
this through
why is it 17 and not 17x square
wait nvm i see
it
thx a lot
.close
midnight plankBOT
midnight plankBOT
Available help channel!
opal hollow
Can someone help me with this problem? Had to redraw cause i drew all over it
fallow scarab
Can someone help me with this problem? Had to redraw cause i drew all over it
the height is f(2)
width = 5-2. so area is height * width
shell wigeon
What would be 6?
opal hollow
area
shell wigeon
No it's not
fallow scarab
,calc 2^3 - 2 + 7
grand pondBOT
Result:
13
your height looks incorrect
opal hollow
ok im confused
shell wigeon
The height is f(2), not 2
fallow scarab
f(2) = 2 plugged into f(x)
opal hollow
oh but why
shell wigeon
Because every point on the curve of a function is (x, f(x))
fallow scarab
oh but why
opal hollow
Because every point on the curve of a function is (x, f(x))
also still dont understands
like i get it but i dont
grand pondBOT
Result:
39
i see
fallow scarab
Result:
```
13
```
yes 13 is the correct height
opal hollow
area is 39 then
shell wigeon
also still dont understands
To draw the curve of a function f, you take all possible x-coordinates and place a point at the corresponding y-coordinate. If the x-coordinate is x, then the y-coordinate is f(x). In your picture, there is a specific point on the curve at x=2, so the y-coordinate is f(2) = 13; that's the height of the rectangle
midnight plankBOT
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Available help channel!
uneven inlet
I need some help with
- Math 9
- Foundations: Pre Calc 10
- WAM 10
These are the classes I need to graduate and I'd like to pass to get a degree in the future and to set me up for success, can anyone help??
midnight plankBOT
I need some help with
- Math 9
- Foundations: Pre Calc 10
- WAM 10
These are th...
scenic wyvern
open questions are generally best asked in #study-discussion
uneven inlet
I tried
scenic wyvern
you don't seem to have a single message in that channel
scenic wyvern
that is probably not the best place to ask, considering most people treat that as a general chat
midnight plankBOT
midnight plankBOT
Available help channel!
proud abyss
The overall run time complexity should be O(log (m+n)).``` how does one approach this while keeping the expected time complexity in mind?
bold peak
Like median of the combined array after merging?
bold peak
I believe that's a slight overapproximation
What you end up actually getting is O(log(min(m,n)))
Is that a hint
proud abyss
there is no hint given on this one
bold peak
What you end up actually getting is O(log(min(m,n)))
This is a small hint for what to do
bold peak
Well what is an algorithm in which log shows up in the time complexity
proud abyss
What you end up actually getting is O(log(min(m,n)))
how is this the same as O(log (m+n))?
bold peak
It isn't, it's less
proud abyss
i mean there are many
proud abyss
it visits n elements but twice? not 2n; i do not know how to formulate the answer
bold peak
Okay no
proud abyss
but like, when we use nested loops (both from 0 to < n)
proud abyss
The time complexity of an algorithm is a function that describes how the running time grows with respect to the size of the input n
proud abyss
worst case?
bold peak
Ehhh not exactly, it's not a CS exclusive term
If a function is O(n²) then it means that
There exists a k such that the function is always smaller than kn²
proud abyss
this is new information
proud abyss
worst case?
we were always taught this
bold peak
I'm surprised considering you're taking a DSA course
proud abyss
not in uni
bold peak
Not even a passing mention of what big O actually is, is weird
Well anyway now you know
So do you see how O(n) is still O(n²)
proud abyss
yeah
bold peak
And in particular, how O(log(min(m,n))) is still O(log(m+n))
proud abyss
okay, sure, but how did you reach that conclusion?
bold peak
How did I reach the conclusion that it would be the time complexity I said?
I've done this problem in a course last year lol
proud abyss
yeah
bold peak
So I remember it from then
proud abyss
fair enough
bold peak
Anyway back to where we were
What's the first algorithm you think of when you see logarithmic time
Another hint is that you have sorted arrays
proud abyss
There exists a k such that the function is always smaller than kn²
also, does this still not mean it is the worst case?
binary search
bold peak
also, does this still not mean it *is* the worst case?
You can do worst case analysis and give your answer in different ways
Big O is just notation
bold peak
binary search
Good
proud abyss
right
bold peak
Now is there any way you can think of to use binary search here?
We don't wanna merge and sort, that's too much time
bold peak
Go ahead
proud abyss
but it is probably incorrect
give me a minute to think of how i am going to write it down
bold peak
Sure take your time
proud abyss
[1, 2, 3, 4], [15, 26, 37, 48, 59]
so array 1 has 4 elements and array 2 has 5 elements
what we could derive from here is that the median would lie in the bigger array, but, this would not work when both of them contain even number of elements
but what we could conclude is that
bold peak
what we could derive from here is that the median would lie in the bigger array,...
That need not be true in general
proud abyss
oh, right
bold peak
Cuz it never said the arrays are sorted wrt each other
The arrays are only sorted within themselves
proud abyss
is the first step i should be thinking of related to m and n?
bold peak
What I said earlier about time complexity should be a hint
bold peak
*is* the first step i should be thinking of related to m and n?
Also yes, sorry forgot to answer that
proud abyss
What you end up actually getting is O(log(min(m,n)))
this reads to me as, "think of binary search on the smaller array", but i do not get why
binary search only because you said yes
bold peak
this reads to me as, "think of binary search on the smaller array", but i do not...
You are correct
proud abyss
right
bold peak
It's in some sense a simultaneous binary search
proud abyss
eh
bold peak
God I can't fucking read my own notes, my handwriting was so bad
proud abyss
notes on? this question?
proud abyss
i see
proud abyss
well, back
midnight plankBOT
@proud abyss Has your question been resolved?
lusty python
```Problem Statement: Given two sorted arrays nums1 and nums2 of size m and n re...
If i remember correctly
The median is the average of middle 2 elements if the size is even
Right?
And the value is the middle element if the size is odd?
Depends on what the question asked, maximum time complexity should be O(m+n)?
lusty python
```Problem Statement: Given two sorted arrays nums1 and nums2 of size m and n re...
The crucial thing here is to find the value of the middle indices
Probably not
proud abyss
right
proud abyss
Depends on what the question asked, maximum time complexity should be O(m+n)?
O(log (m+n))
lusty python
Are you required to find the median of the merged arrays?
lusty python
Probably not
Oh wait im wrong
proud abyss
would not that result in greater time complexity
lusty python
Do you have to merge the arrays and then find the median of that?
But is that the result you are looking for?
proud abyss
sure
lusty python
Place the two arrays on top of each other
I based off this to think
lusty python
sure
So still
You still wants 2 seperate arrays
And 2 variables i and j
Wait i think that's not O(log(m+n))
proud abyss
i do not want you to tell me what to do directly
that does not help in learning at all
lusty python
i do not want you to tell me what to do directly
Yeah
lusty python
that does not help in learning at all
I see
But for log complexities, binary search would be the way to go
midnight plankBOT
lusty python
i think i will take a break and close this for now
Don't tell me this is from leetcode lol
proud abyss
it is
proud abyss
and how is that relevant here?
midnight plankBOT
Available help channel!
night gyro
where did the OH go this drawing isnt making sense to me
midnight plankBOT
where did the OH go this drawing isnt making sense to me
sudden yacht
Do you know what polymerization is?
night gyro
is joining small monemers to from polymers
sudden yacht
Do you know that the condensation of two molecules results in a loss of a molecule of water?
night gyro
yes
sudden yacht
Then that's the answer to your question 😅
sudden yacht
sudden yacht
where did the OH go this drawing isnt making sense to me
These two "pieces" form the H2O molecule that vanishes
night gyro
.close
midnight plankBOT
fathom onyx
Freesciencelessons my man 🦾
midnight plankBOT
Available help channel!
twilit field
If $G$ is a finite element, and $a \in G$ . Then $o(a) \mid o(G)$
\
Consider $H=\langle a \rangle$. Then as $o(H)|o(G)$. and $m=o(H)$, we have $o(a) \mid o(G)$
midnight plankBOT
If $G$ is a finite element, and $a \in G$ . Then $o(a) \mid o(G)$
\\
Consider $H...
grand pondBOT
wai
candid gull
!original
midnight plankBOT
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.
lyric charm
If $G$ is a finite element, and $a \in G$ . Then $o(a) \mid o(G)$
\\
Consider $H...
G is a finite what?
spare ferry
If $G$ is a finite element, and $a \in G$ . Then $o(a) \mid o(G)$
\\
Consider $H...
Yes
manic bison
I have no clue how to solve this
midnight plankBOT
I have no clue how to solve this
manic bison
I tried taking the sum 4-100 / 97 but that's obviously wrong
lyric charm
first you need to find the number of elements in S
then for each possible value of d (which seems to be between 4 and 98 inclusive) you need to find the number of quintuples that contain that value as their d
manic bison
then for each possible value of d (which seems to be between 4 and 98 inclusive)...
I kind of get it
Like d will have 4 for only one quintuple
And 5 for 2
But I can't seem to grasp how to calculate that for higher values?
lyric charm
first you need to find the number of elements in S
did you do this and if so how
manic bison
Is it 100C5?
midnight plankBOT
@manic bison Has your question been resolved?
vague seal
With the a, b, c, d, and e being limited by 20 the number of elements in the set is ~10000
midnight plankBOT
With the a, b, c, d, and e being limited by 20 the number of elements in the set...
vague seal
i couldn't do 50 with python unfortunately @manic bison
manic bison
With the a, b, c, d, and e being limited by 20 the number of elements in the set...
Yeah it has a lot
manic bison
i couldn't do 50 with python unfortunately <@703813252013883444>
It's alr
vague seal
40 is 4688296 elements
i don't know how to help get the actual number using maths
i'm terrible at combinatorics
manic bison
i don't know how to help get the actual number using maths
It's 100C5 most probably
I'll just ask my teacher later
shell wigeon
a can't be 0 and e can't be 100, so the elements to choose from are only 1 to 99
It's not quite 100C5
vague seal
100C5 is 75m
shell wigeon
Do you still want help or is that it
shell wigeon
Ok, how many tuples have d=4?
manic bison
1
shell wigeon
No
vague seal
d = 4 is only once
shell wigeon
Are (1,2,3,4,5) and (1,2,3,4,6) the same tuple?
manic bison
ohhhhhhh
manic bison
Are (1,2,3,4,5) and (1,2,3,4,6) the same tuple?
so it'll have 96?
vague seal
last element of the element
manic bison
95 mb
manic bison
94?
shell wigeon
No
shell wigeon
If we ignore e and only consider 4-tuples (a,b,c,d), how many have d=5?
manic bison
94 times 4C3?
manic bison
93 times 5C3
shell wigeon
Yeah, can you make a weighted sum?
shell wigeon
Yes, that's the number of tuples weighted by d
shell wigeon
Yes
manic bison
Tysm
shell wigeon
So what would the sum look like
manic bison
So what would the sum look like
Summation of something
shell wigeon
i don't get 99C5 when i do it using python
Can you show the code?
vague seal
i got 586537416
size = 100
proto_set = [(a, b, c, d, e) for a in range(1, size - 4) for b in range(2, size - 3) for c in range(3, size - 2) for d in range(4, size - 1) for e in range(5, size) if a < b and b < c and c < d and d < e]
print(proto_set.__sizeof__())
__sizeof__() is the size in bytes, just do len(proto_set)
Your answer is about 8 times 99C5 probably because integers are 8 bytes
bold peak
What are we doing here, the pin doesn't explain anything
shell wigeon
I have no clue how to solve this
vague seal
nvm i'm stupet
bold peak
There has to be some sort of trick to do this no
shell wigeon
nvm i'm stupet
Is ok
bold peak
Lemme put some thought into this
bold peak
And read through the channel
bold peak
i was just gonna sum up all d's and divide by number of elements
Run the code for a day?
vague seal
it's not that slow
shell wigeon
There's only 71 523 144 tuples, modern computers can do that
vague seal
i got 99C5 now
bold peak
What I'm trying to think is, there's only 1 time 99 can occur there
vague seal
it's correct
shell wigeon
@manic bison did you get the solution?
manic bison
<@703813252013883444> did you get the solution?
Yeah I got the summation
I'm too tired to finish the solution rn but I'm sure I can solve it now
Ty
shell wigeon
Okay
bold peak
Ah cool
vague seal
it's crunching now
bold peak
Have fun
shell wigeon
I'll just write the answer down I guess, it's not really a spoiler now
vague seal
and due to lack of physical RAM i had to change my approach
bold peak
Random guess, is it like 66.6 or smth
vague seal
not that far off
shell wigeon
66, yes
vague seal
well
shell wigeon
$\frac{1}{\binom{99}{5}} \sum_{d=4}^{98} \binom{d-1}{3} \cdot (99-d) \cdot d$
grand pondBOT
Nel
bold peak
...
shell wigeon
,w 1/binomial(99,5) * sum_{d=4}^{98} binomial(d-1,3) * (99-d) * d
grand pondBOT
shell wigeon
Unless I made a mistake somewhere
vague seal
i summed over all d
added them all together
and then i divided by number of elements of S
that is correct right?
shell wigeon
Yeah
shell wigeon
,w 1/binomial(9,5) * sum_{d=4}^{8} binomial(d-1,3) * (9-d) * d
grand pondBOT
bold peak
Is it just 2n/3 in general
vague seal
i get 8.33333
bold peak
,w 1/binomial(n,5) * sum_{d=4}^{n-1} binomial(d-1,3) * (n-d) * d
grand pondBOT
shell wigeon
Let's try 7 manually:
(1,2,3,4,5)
(1,2,3,4,6)
(1,2,3,5,6)
(1,2,4,5,6)
(1,3,4,5,6)
(2,3,4,5,6)
6 total, sum of d is 4*2+5*4
,calc (42+54)/6
grand pondBOT
Result:
4.6666666666667
,calc 2(6+1)/3
grand pondBOT
Result:
4.6666666666667
Yeah I think I'm correct
vague seal
hurdur
shell wigeon
Oh lmao
shell wigeon
$\frac{5!(99-5)!}{99!} \sum_{d=4}^{98} \frac{(d-1)!}{3!(d-1-3)!} \cdot (99-d) \cdot d$
grand pondBOT
Nel
shell wigeon
$\frac{5! \cdot 94!}{99!} \sum_{d=0}^{94} \frac{(d+3)!}{3! \cdot d!} \cdot (99-d-4)(d+4)$
grand pondBOT
Nel
shell wigeon
Something like that?
vague seal
it looks like a sum identity
shell wigeon
$\frac{5! \cdot 94!}{99!} \cdot \frac16 \sum_{d=0}^{94} -d^5 + 105 d^4 - 605 d^3 + 1095 d^2 - 594 d$
grand pondBOT
Nel
vague seal
me neither
shell wigeon
Like each sum has a formula now, but damn the numbers are not fun
vague seal
i'd rather just use the computer
lol
but then again we can just use the computer to compute the binomials
ok it's done
66,6666666
shell wigeon
Oh you were running the code all along
vague seal
ye to double check i did it correctly
shell wigeon
Alright well there isn't much more to it
twilit field
midnight plankBOT
twilit field
So $\sigma X$ simply becomes $\sigma F$
grand pondBOT
wai
spiral rock
no,
if what you say is true then P(Y <= t) is somehow larger then 1 if sigma is large enough
twilit field
why not, probability at each point is just multiplied by \sigma , right
spiral rock
what if sigma is very large? then the right side is larger then 1
grand pondBOT
wai
twilit field
what if sigma is very large? then the right side is larger then 1
\sigma cannot be any value now, can it
tender dew
it can
spiral rock
it can be
tender dew
no it is not
spiral rock
still incorrect
twilit field
okay, then I've understood PMFs and random variables incorrectly
nova yoke
geometrically, what do you expect should happen to the CDF if you increase sigma?
twilit field
geometrically, what do you expect should happen to the CDF if you increase sigma...
It should widen?
spiral rock
$Y \leq t \iff \sigma X \leq t - \mu \iff X \leq \frac{t - \mu}{\sigma}$
grand pondBOT
ExpertSqueeSQUEE
nova yoke
yea
nova yoke
whereas you're multiplying F by sigma, which would stretch it vertically
twilit field
and grow shorter
quiet verge
**ExpertSqueeSQUEE**
ooooo
nova yoke
it can't grow shorter, it still has to span (0,1) on the y axis
spiral rock
but $P(X=i)=p(i), right
what is p(i)?
twilit field
probability of when x=i
spiral rock
ok so what you said is trivially true
twilit field
what is p(i)?
so P(X) = P(t-µ/σ)
proud abyss
(re-post; reason: did not get an answer before) can someone please check where have i gone wrong? this is apparently incorrect but i cannot figure out why
midnight plankBOT
(re-post; reason: did not get an answer before) can someone please check where h...
bold peak
Are you trying to take negation of B6?
Cuz I don't think the negation of P implies Q is not P implies Q
¬(P => Q) = P ∧ ¬Q
proud abyss
which line?
bold peak
The first
Ofc I'm assuming that you actually wanna write out the negation of B6
Although my sleep meds have started kicking in, so please excuse me if I'm being dumb
(and also if I disappear suddenly, please assumed I crashed out)
bold peak
which line?
B6 is of the form P => Q
bold peak
¬(P => Q) = P ∧ ¬Q
So ¬B6 will be of the RHS form
proud abyss
B6 is of the form P => Q
no, it is inside one whole pair of brackets
my formatting is trash
proud abyss
is okay
proud abyss
good night
i wonder if this whole thing can be written neatly with latex
please ping if replying
midnight plankBOT
@proud abyss Has your question been resolved?
proud abyss
.close
midnight plankBOT
midnight plankBOT
Available help channel!
tidal turret
midnight plankBOT
tidal turret
rapid violet
calculus early transcendental functions 5th robert pdf
tidal turret
calculus early transcendental functions 5th robert pdf
say what?
we dont need polar thingy
we can just, use sandwich
dawn dagger
i will sandwich you
tidal turret
and use that
e^(u+2). u^2 + v^2 >= u^2 + v^2
visual panther
Btw you are missing a factor of 9 in the numerator.
grand pondBOT
tidal turret
tidal turret
The numerator can never become 0 in fact
WDYM?
dawn dagger
btw what do you do if e^z < 1
tidal turret
btw what do you do if e^z < 1
i thought e^x > 0
dawn dagger
ok, what's your point?
dawn dagger
if e^z < 1 then you would scale down y^2
tidal turret
ok, what's your point?
oh, is no guarantee that e^x > 1
tidal turret
if e^z < 1 then you would scale *down* y^2
yes im stupid
dawn dagger
but i think you can still use squeeze theorem locally meaning you can consider a small neighborhood around the origin, so that the inequality holds
If we consider u^2+y^2 =< 1 then e^(u+2) > 1 because the worst u can be is -1
dawn dagger
So if you use what I just said, you can then apply polar coordinates and show it goes to 0 easily
dawn dagger
I mean its strictly increasing
you just said this
so the min(e^(u+2)) on the disk would be e^(-1+2)=e^1=e>0
dawn dagger
the range of f(x) = e^x, do you know it?
how the tables turned
tidal turret
so the min(e^(u+2)) on the disk would be e^(-1+2)=e^1=e>0
what?
how is u = -1 the minimum of e^(u+2)
dawn dagger
you just said e^x is strictly increasing
so is e^(u+2)
and on the disk with radius 1 around the origin, you have u in [-1,1]
so logically, the smallest value (which is u=-1) gives you the minimum e which is positive
so it can only become more positive
,w Plot[Exp[u+2], {u, -1, 1}]
dawn dagger
why you use =<
because <= is implication duh
tidal turret
but i think you can still use squeeze theorem *locally* meaning you can consider...
anyways, can you help me accomplish this?
are you planning on using polar coordinates either way?
grand pondBOT
dawn dagger
are you planning on using polar coordinates either way?
no (well after the squeeze)
dawn dagger
anyways, can you help me accomplish this?
duh i just told you
then you kinda lost it
tidal turret
but i think you can still use squeeze theorem *locally* meaning you can consider...
we lose generality and the limit can approach anything at that point
dawn dagger
wdym generality
tidal turret
idk if I am explaining myself correctly
dawn dagger
nobody cares what happens with the limit if you are in x=100 and y=-1000
continuity is a local property
funny stuff can happen, as long as we find a safe neighborhood around the origin, we are good
tidal turret
the problem is dat
dawn dagger
well i suggested a disk, and it seems to work
tidal turret
how do you know x^2 + y^2 <= 1
dawn dagger
you just have to continue and finish the proof, in order to see
dawn dagger
how do you know x^2 + y^2 <= 1
because you wanted e^(u+2) >= 1 not e^(u+2) < 1
Only then your upper bound works!
so you think when u+2>=0 which is the case for u>=-2
so even a disk with radius 2 would work
tidal turret
this aspect of proof writing that a limit approaches 0 I dont get it, you are considering specific cases when the disk is radius between 1 and 2
dawn dagger
what?
tidal turret
what am I supposed to write?
dawn dagger
maybe you ought to understand before you write? ig
tidal turret
you are saying I can still use my upper bound, but only when x^2 + y^2 <= 1
otherwise the inequality falls apart
dawn dagger
crazy stuff can happen but we don't care, we can go as close to 0 (or in better terms find a safe neighborhood)
tidal turret
how did you found that x^2 + y^2 <= 1 makes the inequality e^x . y^2 + x^2 >= y^2 + x^2 work?
lets begin from there
dawn dagger
dude
dawn dagger
because you wanted e^(u+2) >= 1 not e^(u+2) < 1
and here again
dawn dagger
yea
tidal turret
but, I got lost after dat
dawn dagger
then you pick a nice neighborhood where u>=-2 for example a disk with radius 1 around the origin satisfies that
u^2+w^2 =< 1
dawn dagger
the upper bound is independent of w so a disk is just a simple pick (for a region of interest)
tidal turret
by disk, wdym exactly? like sort of a circle but instead of = r you got that =< r
dawn dagger
yea the interior is included
💿
literally a disk
when it comes to limits we only care what happens close enough to the limit
tidal turret
how do you know a disk with radius 1 works
dawn dagger
the upper bound is independent of w so a disk is just a simple pick (for a regio...
..
disks and balls in general are just "nice"
you could also use a square or cubes or whatever (not an expert in extensions)
dawn dagger
the upper bound only depends on e^(u+2)
the limit depends on w that yes
but for bounding, it all comes down to the factor of e^(u+2)
tidal turret
but for bounding, it all comes down to the factor of e^(u+2)
yes
dawn dagger
it all starts with e^(u+2) * w^2 >= w^2
tidal turret
yes
dawn dagger
it all starts with e^(u+2) * w^2 >= w^2
that must be necessarily true, so we make sure e^(u+2)>=1
tidal turret
are there any disks that make the inequality false
dawn dagger
well you derived u >= -2
pick any disk with greater radius and you got points for which the inequality fails
tidal turret
but dude, if u -> 0, then e^(u+2) is never between 0 and 1
dawn dagger
dude that's the point
but you didnt justify that part
if, you assumed it to be trivially true
"by observation" or what you wrote
tidal turret
pick any disk with greater radius and you got points for which the inequality fa...
radius of a disk cant be negative, correct?
tidal turret
then just pick disk with radius 1
dawn dagger
ok sure
tidal turret
then, the upper bound works
dawn dagger
yes
tidal turret
what I dont understand is what do we do with u = -2
that's the minimum that e^(u+2) can go without the inequality getting broken
dawn dagger
depends on what you choose
choosing a disk with radius 1, then yes
doesn't ever happen
tidal turret
let u < -2 , now pick a disk with radius 1, then the upper bound fails
dawn dagger
?
dawn dagger
let u < -2 , now pick a disk with radius 1, then the upper bound fails
if u<-2 and you pick a disk with radius 1, then u^2+w^2=<u^2+w^2*e^(u+2) is vacuously true (it's meaningless)
you can never have u<-2 and u^2+w^2=<1 at the same time