yeah
#help-23
buoyant shadow
flat frigateBOT
Zex
buoyant shadow
you find the two ns and subtract
lean otter
ohk but does finding lower bound and upper bound follows approximation always?
or is there a direct method?
rigid hazel
was thinking about what 2^50 means and if you look into n! all numbers with factor 2 add 1 to the exponent
ebon mesa
i know a method but it’ll take a long time
rigid hazel
was thinking about what 2^50 means and if you look into n! all numbers with fact...
4 adds even 1 more and 8 adds 2 more
and you find for what value n you have 50 factors of 2
buoyant shadow
you have to guess
ebon mesa
check how many factors there is in an n!
rigid hazel
i dont do this stuff though but i would approach it this way
lean otter
i know a method but it’ll take a long time
it's okay take your time but let me know
ebon mesa
my first try would be on 30
lean otter
was thinking about what 2^50 means and if you look into n! all numbers with fact...
that also requires guesssing ...
rigid hazel
that also requires guesssing ...
not as much
just counting how many and its not many
ebon mesa
like 30!=302928… skip the odd numbers
30 has 1 factor
28 has 2 factor
26 has 1 factor
…
sum them all up to check if it has 50 factors or not
rigid hazel
there should be an efficient way to count how many 2s there are
50 for example has 25 2s, considering the multiples of 4 go up until 48, so 12 more 2s, considering multiples of 8 up until 48 again you have 6 more 2s
25 + 12 + 6 not quite 50
so n is bigger than 50
lean otter
that is again the same method i believe, repititive division
rigid hazel
yea
rigid hazel
50 for example has 25 2s, considering the multiples of 4 go up until 48, so 12 m...
wait 16 and 32 too consider
lean otter
yes
lean otter
Okay I got it! This question follows identifying lower and upper bound and then simply subtracting it
ebon mesa
find the lowest n where n! is divisible by 2^50
find the highest n where n! isn’t divisible by 3^40
those are the bounds
it’ll take some guessing no matter what
lean otter
okay!
Thank you soooo much everyone! @rigid hazel @buoyant shadow @queen ingot @ebon mesa !!!!!
.close
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ashen ruin
The drawing below represents a model of the stained glass windows in a church that need to be restored. A special glass paint will be used, which costs R$80.00 per can and each can covers approximately 100 cm2 of area. The dark area in the figure is the part of the stained glass window that needs to be painted with the special paint. Considering R = 40 cm and r = 10 cm respectively as the radii of the largest and smallest circles, and that the inscribed triangle is equilateral and the inscribed hexagon is regular, the approximate amount that will be spent on the restoration of each stained glass window is:
pi = 3
sqrt(3) = 1,7
safe radishBOT
The drawing below represents a model of the stained glass windows in a church th...
@ashen ruin Has your question been resolved?
ancient socket
The drawing below represents a model of the stained glass windows in a church th...
find the triangle's side length
that will be the opportunity to use the given sqrt(3) value
ancient socket
find the triangle's side length
to therefore get its area
ancient socket
that will be the opportunity to use the given sqrt(3) value
u can use a certain special right triangle property
lavish storm
Help
safe radishBOT
@lavish storm Has your question been resolved?
plucky elk
What does converse mean
desert juniper
i'd expect "convexe" to mean "convex"
desert juniper
!show
safe radishBOT
Show your work, and if possible, explain where you are stuck.
@lavish storm Has your question been resolved?
lavish storm
i'd expect "convexe" to mean "convex"
convexe in french
lavish storm
Show your work, and if possible, explain where you are stuck.
No clue where to start
lavish storm
What does converse mean
💀💀💀💀
prime tundra
safe radishBOT
prime tundra
How did this get simplified to secx
junior raven
factor a sec x out the numerator
prime tundra
Oh tyy
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@prime tundra Has your question been resolved?
dusky raptor
Hi everyone, I'm struggling to prove this inequality about markov chains. I think that it's quite basic but I cannot conclude for the life of me, is there something I'm missing? Many thanks in advance, here is my work.
safe radishBOT
Hi everyone, I'm struggling to prove this inequality about markov chains. I thin...
@dusky raptor Has your question been resolved?
safe radishBOT
@dusky raptor Has your question been resolved?
dusky raptor
any ideas ?
safe radishBOT
@dusky raptor Has your question been resolved?
plucky elk
Can also try in #advanced-probability
safe radishBOT
@dusky raptor Has your question been resolved?
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@dusky raptor Has your question been resolved?
flat frigateBOT
ƒ( wai ina teacup)= I don't know
split holly
this is just squeeze theorem
just consider \epsilon from both sides
$|x_n - L| \leq \epsilon$
flat frigateBOT
BrandenXia
split holly
we have $L-\epsilon \leq x_n \leq L + \epsilon$
flat frigateBOT
BrandenXia
split holly
and $L-\epsilon \leq z_n \leq L + \epsilon$
flat frigateBOT
BrandenXia
split holly
so we have $L-\epsilon \leq x_n \leq y_n \leq z_n \leq L + \epsilon$
flat frigateBOT
BrandenXia
split holly
which means $|y_n - L| \leq \epsilon$
flat frigateBOT
BrandenXia
split holly
so the limit of y_n is L
desert pasture
Got it
desert pasture
so the limit of y_n is L
Thanks
safe radishBOT
safe radishBOT
Available help channel!
wind dagger
hi
wind dagger
5c-7X=
light shoal
explain yourself
wind dagger
HI
burnt notch
Are you a troll?
safe radishBOT
@wind dagger Has your question been resolved?
random knot
this is decision math, for question c my answer was originally 380 but when i looked at the actual answer it was 190 (which is what you get if you divide 380 by 2), why do we have to divide by 2?
safe radishBOT
this is decision math, for question c my answer was originally 380 but when i lo...
random knot
for context Kn is a graph where n is the number of nodes and each node had to be connected to the other
lean otter
curved connections
random knot
what do u mean by arcs??
arcs is another word for edges
random knot
what
lean otter
u are coounting the one egde as two
random knot
how
lean otter
thats why the problem is arising
random knot
K4 for example
each node in K4 has a valency of 3
which means if you wanna find the total arcs of Kn its (n - 1) x n
lean otter
K4 for example
each node in K4 has a valency of 3
which means if you wanna find ...
yes but that will give you answer to the shared egdes
lean otter
the correct solution is (n - 1) x n/2
random knot
i get it now
lean otter
exactly
random knot
it all makes sense now
lean otter
lol hope that helps
random knot
lol hope that helps
it did thank you bro
lean otter
geez np : )
random knot
!close
random knot
i really like helping out people with math lmao
you're smart for figuring that shit out
do you take decision math aswell
lean otter
do you take decision math aswell
i dont actually
but
i solve International math olumpiad problems regularly
so my intuition is strong in that regard
red shore
exotic crag
easy pizy
red shore
can u pls show it
exotic crag
try with Venn diagram
-(-p^q) = (pV-q)
and let the other be as it is
draw the venn diagram for (p V -q) ^ (p V q)
you will see it is simply p
exotic crag
is that boolean alzebra?
red shore
draw the venn diagram for (p V -q) ^ (p V q)
ik it can be done but
red shore
is that boolean alzebra?
yes
exotic crag
computer?
red shore
yes
red shore
>< lol y in maths server
cause computer is pure maths..ff
red shore
i studied it last yr
in computer?
exotic crag
draw the venn diagram for (p V -q) ^ (p V q)
see the 12th law
exotic crag
draw the venn diagram for (p V -q) ^ (p V q)
it can be written as .... p V (-q ^ q)
red shore
lemme check qick
exotic crag
now see the last law .... (-q ^ q) ... can be written as False
now we have .... p V false
see the second law
exotic crag
in computer?
yea...
exotic crag
any issue?
?
exotic crag
yus
safe radishBOT
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Available help channel!
gusty pendant
i have a set of N points and a canvas height and width as an example case horizontal_lines = [[(0, 1311), (1920, 1311)]]
vertical_lines = [[(920, 0), (920, 1311)]] image_width=1920, image_height=2490 which forms a T and 3 rectangles. I am looking for the maths that get me from my inputs to the top_left and bottom_right coordinates of each of the 3 (or more or less) rectangles specefically i am guessing that there is some formula or wikipedia page with the answer?
safe radishBOT
i have a set of N points and a canvas height and width as an example case horizo...
@gusty pendant Has your question been resolved?
gusty pendant
<@&286206848099549185>
topaz wagon
safe radishBOT
The purpose of this server is to help you learn, not to hand out answers. Do not ask someone to give you the answer directly.
gusty pendant
uh thats your question @topaz wagon since you asked it in your own channel could you delete it from here as i already had a question here
topaz wagon
ok chill
safe radishBOT
@gusty pendant Has your question been resolved?
gusty pendant
<@&286206848099549185>
safe radishBOT
@gusty pendant Has your question been resolved?
gusty pendant
<@&286206848099549185> i have a set of N points and a canvas height and width as an example case horizontal_lines = [[(0, 1311), (1920, 1311)]]
vertical_lines = [[(920, 0), (920, 1311)]] image_width=1920, image_height=2490 which forms a T and 3 rectangles. I am looking for the maths that get me from my inputs to the top_left and bottom_right coordinates of each of the 3 (or more or less) rectangles specefically i am guessing that there is some formula or wikipedia page with the answer?
gusty pendant
<@&286206848099549185>
safe radishBOT
@gusty pendant Has your question been resolved?
gusty pendant
hrm
gusty pendant
<@&286206848099549185>
main mural
<@&286206848099549185> i have a set of N points and a canvas height and width as...
i'm sorry but i do not understand this question at all. maybe a visual example would be helpful?
main mural
no guarantee that i'll be able to help tho, but i'll try
gusty pendant
thanks
heres the image but if its still not clear i can try to explain better...
gusty pendant
for some reason plot drew the thing upside down but yeah
been strugling with this for 2 days
main mural
you want to get the coordinates of these points right
gusty pendant
no i only care about the top left and bottom right for each formed rectangle
gusty pendant
some what yes
main mural
can the lines intersect and pass through each other
gusty pendant
i am trying to create a data set of sewing patterns (since none exist) and the only viable source is a german book from 1957 updated with updates until 2024 i want to extract those patterns but they for some reason made like 3 patterns in a block instead of keeping them seperate
so i wrote code which identifys the lines that divide them
but now i need the rectangles so i can actually cut them out
yes there could be 4 rectangles
or there could be just a single line and 2 rectangles
my end goal is to take the patterns and put them through an ai system which will turn them in to mesh objects
main mural
ah okay
gusty pendant
dressing the metaverse basically lol
i didnt do so well with maths as a kid but i did catch up a bit learned graph mathematics etc
i am assuming there should be something like Pythagoras a squared x b squared = c that just gives the answer
gusty pendant
well at least i know it is tricky now and i wasnt just being dumb
i tried to solve it with shapely r omlette wolfram octave manifold cgal all with out success most often 3 rectangles are counted as 4
main mural
well in that case
first get the top left and bottom right corners of the picture (i assume you have that sorted out)
in case you have 3 or 4 pictures
you take each line that starts at 0 and goes to some coordinate that isn't on the border of the picture, let it be called c
that c must necessaly be in the middle of the picture and thus must be one of the corners
you match those coordinates up with the corners of the picture and figuring out whether it's a bottom right or a top left corner shouldn't be too hard
gusty pendant
i had that idea worked somewhat with 2 lines
tried with 8 lines and it totally failed
main mural
oh so it can be 8?
gusty pendant
but i might of missed something there
main mural
i thought it was max 4
gusty pendant
yeah N horizontal and M vertical
main mural
but i might of missed something there
no it's far from a universal solution
main mural
yeah N horizontal and M vertical
aw crap
gusty pendant
i think 4 could be a sensible max i mean thats better than 0
main mural
tried with 8 lines and it totally failed
can you show an example of that
gusty pendant
but even then i tried that method with 3 lines and i could solve all but bottom_right which i knew was 1 of 2 values
yes moment
the ? was where i got stuck because we dont know if the line goes up in the top half or down so we cant know directly the bottom left with greater than a 50/50 probability
i thought there might be a way to solve that so then i tried increasing from 3 lines to 8 and thats when that idea went to a dead end for me but maybe i am missing something
sorry for my terrible hand writing hope you can read that ok
basically it says first rectangle will always be the canvas top left so we have it 0,image_height
last rectangle will always have a bottom right of image_width, 0
main mural
basically it says first rectangle will always be the canvas top left so we have ...
bottom_right will be maximum y coordinate of all horizontal lines
gusty pendant
yeah
one observation i made was this
if you list the start and end coords of all lines and the 4 points of the canvas
the answer will always be a subset of that
main mural
the answer will always be a subset of that
this might be it
gusty pendant
ive heard of set theory if that helps i know its a thing
main mural
if you make a list of all points like that and do a vertical and a horizontal "scan" of the picture i think you'll be able to get all the data you need
gusty pendant
i read something about this approach
main mural
oh
main mural
if you make a list of all points like that and do a vertical and a horizontal "s...
(i think)
gusty pendant
trying to think what it was called
yes sweep line algorithm or plane sweep algorithm
is this what you are talking about using ?
main mural
i think so, i literally just came up with this 💀
gusty pendant
the beginning point of horizontal lines and the end point of vertical lines will always be a top left and the top left of the canvase will also that fast fills some of the data
also the bottom right corner will always be the last rectangles bottom right
given that we know the top left of all rectangles we also know how many rectangles are formed and thus we know how many bottom right values were searching for from the subset of the set (the unused coordinates)
maybe theres some elaborate way to take that first horizontal slice and consider IF the vertical line is in the top or bottom thus revealing which coord is the bottom right... the other one goes in the remaining slot
could that work?
that should be first full slice rather than limited to horizontal
Does that make sense?
main mural
Does that make sense?
it might work but i am a little too tired to comprehend this now, i'm going to bed.. i think you should close this channel now and ask again tomorrow because now at least you can rephrase the question differently
gusty pendant
ok one final idea
if we guessed the rectangles could we do a reverse algo to get the lines
safe radishBOT
@gusty pendant Has your question been resolved?
safe radishBOT
@gusty pendant Has your question been resolved?
gusty pendant
heres some relevent code im working on https://colab.research.google.com/drive/14M-cxGMUUODjInf779aI1AL-aLkiQ2_G?usp=sharing
gusty pendant
R = \left{ \left( (x_1, y_1), (x_2, y_2) \right) \mid x_1 \leq x \leq x_2, , y_1 \leq y \leq y_2, , x_1 < x_2, , y_1 < y_2 \right}
does this solve it ?
@main mural
<@&286206848099549185>
i think if im right i need to do a mathmatical proof or something like that
humble egret
guys anyone here to help ??
covert yoke
guys anyone here to help ??
!noadvert
safe radishBOT
guys anyone here to help ??
Please do not advertise your help channel or thread in other parts of the server. There are many people who need help, so advertising can quickly turn into spam.
gusty pendant
@humble egret do you not see the channel clearly says help scatterp how do you expect to do mathmatics if you cant add that up lol
given an input of coordinates for lines is R the top left and bottom right cordinates of all formed rectangles using this: R = \left{ \left( (x_1, y_1), (x_2, y_2) \right) \mid x_1 \leq x \leq x_2, , y_1 \leq y \leq y_2, , x_1 < x_2, , y_1 < y_2 \right}
<@&286206848099549185>
wind laurel
$$R = \left{ \left( (x_1, y_1), (x_2, y_2) \right) \mid x_1 \leq x \leq x_2, , y_1 \leq y \leq y_2, , x_1 < x_2, , y_1 < y_2 \right}$$
flat frigateBOT
woomy
Compile Error! Click the 
reaction for more information.
(You may edit your message to recompile.)
wind laurel
By definition, the left top corner of a rectangle LT = (x1,y1) has x coordinate smaller than any other x coordinate in the rectangle, and y coordinate bigger than any other y coordinate in the rectangle
wind laurel
Consider the following rectangle, where in red we have LT (Left Top), and in blue RB (Right Bottom).\
Clearly LT is as far left and high up you can go in the rectangle, meanwhile RB is as far right and bottom you can go, in the language of mathematics we would right:\
$LT = (x_1, y_1),\quad x_1 \leq x\quad y_1 \geq y\quad \forall x,y \in$ Rectangle\
And:\
$RB = (x_2,y_2),\quad x_2 \geq x\quad y_2 \leq y\quad \forall x,y \in$ Rectangle\
Combined we do in fact achieve the formula you are looking for, but this only works if you can know whether or not a certain point is in one rectangle or another:
$$R = ((x_1, y_1), (x_2, y_2)), x_1 \leq x \leq x_2, y_2 \leq y \leq y_1\quad \forall x,y \in \text{Rectangle}$$
Notice how we dont need to specify that $x_1 < x_2$ and $y_1 < y_2$ As this is already clear from the definition of $R$.
Worst case scenario your rectangle is 1 pixel wide/tall in that case $x_1 = x_2$ or $y_1 = y_2$.
Sorry! Made a mistake, corrected it now!
gusty pendant
ok
flat frigateBOT
woomy
wind laurel
But keep in mind that all of this works only if you know when a specific coordinate lies withing one rectangle or another
gusty pendant
ok trying to umm keep up...
wind laurel
If i were to program this, i would first chose which coordinates i want to have as corners, and then draw the lines
gusty pendant
so its not nessasary to identify the rectangle in all corners initially ?
finite dune
Can I ask my doubt here.... Not sure where to ask
wind laurel
Can I ask my doubt here.... Not sure where to ask
!help
safe radishBOT
Can I ask my doubt here.... Not sure where to ask
To ask for mathematics help on this server, please open your own help channel or help thread. See #❓how-to-get-help for instructions.
wind laurel
so its not nessasary to identify the rectangle in all corners initially ?
What do you mean by identify? In general a rectangle is defined by its 2 opposite corners, as its enough to then create the rest of the rectangle with only this information
gusty pendant
yes as you said define ..
wind laurel
Okay, so yes we only need 2 corners, the opposite ones, to identify a rectangle
gusty pendant
oh i mean
given a canvas with N rectangles
the tl and br should be identified first for the corners of the canvas
wind laurel
Yes, although thats just (0,height) and (width, 0)
gusty pendant
hmm one sec sorry i suck at explaining stuff... will draw on paint moment
wind laurel
Okay! No problem :)
gusty pendant
i am saying that if i am correct its best to define by oposing corners the red areas first
then the remaining points from all points form the white areas where the area >1
wind laurel
oh yes, those are way easier as they have a corner in common as the canvas
what do you mean by >1?
gusty pendant
so my original formula R = \left{ \left( (x_1, y_1), (x_2, y_2) \right) \mid x_1 \leq x \leq x_2, , y_1 \leq y \leq y_2, , x_1 < x_2, , y_1 < y_2 \right}
gusty pendant
so my original formula R = \left{ \left( (x_1, y_1), (x_2, y_2) \right) \mid x_1...
is this right ?
gusty pendant
what do you mean by >1?
if i take the points marked blue ill get a rectangle of zero size thus the points go together in a diffrence descernable configuration
wind laurel
oh now I get it, would this be the formula?
$$R = \left { ((x_1, y_1), (x_2, y_2))\mid x_1 \leq x \leq x_2, y_2 \leq y \leq y_1, x_1 < x_2, y_1 < y_2 \right}$$
flat frigateBOT
woomy
gusty pendant
wind laurel
So R is the set containing the corners, i couldnt see the brackets before
wind laurel
Yes, the corners being on the same line define a rectangle with 0 area, i.e. a line
gusty pendant
yes so in that way i theorize that the definition of the white rectangles is discernable from the line coords
and thats more or less how i came up with the formula
but my maths is not good enough to know if its done correctly
gusty pendant
wow
gusty pendant
yeah umm i dont even know where those buttons are on a calculator
so not sure how i might do that
but isnt the next step a mathmatical proof ?
wind laurel
It really depends what you are trying to do
from the question you posed it seems like you are trying to program something
and for that I'm afraid I cannot help
gusty pendant
yes sorry let me give the background
i have some book from 1957 which has sewing patterns trying to cut them based on identified cut points and stick in a database which ill then use to create 3d clothing for the metaverse
its 13000 patterns
wind laurel
The cut points being the corners or lines?
gusty pendant
lines
wind laurel
and from the lines you want to find the corners
gusty pendant
yeah to make the cuts
hmm so does the formula work for any amount of lines horizontal or vertical including edge cases like 0 horizontal or 0 horizontal and vertical lines ?
wind laurel
It seems like it, but I'm not sure i understand how the data is presented to you, could you give an example for a pattern in the book?
gusty pendant
sure
moment
Width: 1921, Height: 2491
[(1920, 1311), (0, 1311)]
[((920, 1307), (920, 2490))]
wind laurel
huh
gusty pendant
Rectangles formed by the lines:
Rectangle: ((0, 1307), (920, 2490))
Rectangle: ((920, 1307), (1920, 2490))
Rectangle: ((920, 1307), (1920, 1311))
wind laurel
<@&268886789983436800>
wind laurel
Rectangles formed by the lines:
Rectangle: ((0, 1307), (920, 2490))
Rectangle: (...
Okay so, you have pdf scans of pages in the book, and you are trying to seperate a page into other pages, based on the lines
gusty pendant
well destinct patterns yes
wind laurel
the goal is too find the corner of these rectangles based on the coords of the lines, so you can create the other pages
gusty pendant
yeah
wind laurel
is every page formatted this way? Or can one page be cut into more than 3 rectangles
gusty pendant
there quite random
pages are from 1957 - 2024
the only constant is lines as seperators and those circles with numbers
which are an ID tag
i,e BL rectangle is pattern 154
to make my life worse theres images like the above one which have broken lines with data from the left image placed in the right image
but im ignoring that part for now
wind laurel
how about
Another idea is to take the rectangle
and look first at lines starting and ending at the end of a page
So here it would be in blue
Now, cut the page based on these lines
basically we would create from the canvas, 4 rectangles in this case
Now, the new lines (in red) start and end on the borders of the new canvas
all you have to do now, is cut the page again following the same algorithm
Does this make sense?
gusty pendant
sorry well initially zero but im starting to see it now
wind laurel
if you have any questions feel free to ask
gusty pendant
well it sort of comes back to same issue i cant cut a line
i can only cut a rectangle
so i would need to calculate potentially existing rectangle in the 4 corners
wind laurel
Why cant you cut a line through the canvas? What software are you using to do this
gusty pendant
moment let me check
omg your right
i can cut by lines
so i thought it was only possible by rectangles
gusty pendant
3 days of life ill never get back
wind laurel
lol happens to the best of us
gusty pendant
i am kinda happy my maths checked out
wind laurel
Yea atleast you learnt something :)
gusty pendant
well thanks for all the help hope you have a great new year
wind laurel
Goodluck creating clothes for the metaverse!
gusty pendant
.close
safe radishBOT
kind tinsel
@vernal girder can you stop react spamming on every message please
safe radishBOT
Available help channel!
opaque fern
safe radishBOT
opaque fern
Could someone tell me how they got that density function
seems like it just popped out of nowhere
opaque fern
Geez, I haven't seen the Mellin transform in like forever
Yeah nevermind, that is some very crazy math right there
thanks, though!
.close
coral dagger
Is this correct?
safe radishBOT
Is this correct?
coral dagger
Start = 0, end = pi/2
What I'm confused about is where to draw the 2nd cosecant curve that is in the bottom. Would it just be the midpoint of the period?
merry zenith
looks like you drew the csc curve just fine
the vertical asymptote would be in the middle of the period (sin = 0 in the middle of its period)
safe radishBOT
safe radishBOT
Available help channel!
waxen gale
Anyone help me with the quesrion
safe radishBOT
Anyone help me with the quesrion
devout shale
what da quesrion
lean otter
whats the question though?
devout shale
whats the question though?
quesrion*
safe radishBOT
@waxen gale Has your question been resolved?
fervent hatch
When making the jump from Fourier series to complex Fourier series and then to Fourier transforms (I assume this is the order), where does the importance of complex variables come in? I mean like why do we need to start using cosx + isinx? What was wrong with just real sinusoids like in the regular Fourier series? And why can’t we use that same real approach in the Fourier transform? I get that eurlers formula decomposes into a real and imaginary cos and sin, but isinx isn’t the same as sinx, some new aspect is being added when we jump to the complex form.
safe radishBOT
When making the jump from Fourier series to complex Fourier series and then to F...
fervent hatch
I guess an equivalent question would follow as to why I can’t just say $$\int^{\infty}_{-\infty}(x(t)(cos(x) + sin(x))dx$$
Why can’t I say this is the Fourier transform
flat frigateBOT
Nathan
fervent hatch
Rather than $$\int^{\infty}_{-\infty}(x(t)(cos(x) + isin(x))dx$$
flat frigateBOT
Nathan
fervent hatch
What’s so special about the complex variable that changes the meaning of the Fourier transform and makes it different from the regular real Fourier series
stoic dune
The complex numbers are generally "richer" than the real numbers, and you gain a lot of power when using them
That being said, there's nothing special about i here.
fervent hatch
That being said, there's nothing special about i here.
Then why not use real stuff
Why is the transform the way it is
peak estuary
because it turns out to be useful
stoic dune
This is kind of "why do we call a dog a dog"
The transform is useful, and there's no real equivalent
peak estuary
its not like someone just came up with it and decided "lets teach this to all students for no reason"
stoic dune
Might be worth seeing the Laplace transform, which is another integral transform, but is entirely real
Also has useful properties, but is used in other places
peak estuary
if you have the usual real fourier series with its coefficients a_n,b_n, then you always have pairs of numbers
peak estuary
**Nathan**
if you decided to do this, then you would have sums a_n+b_n (modulo details) from which you cannot recover a_n and b_n
however with the i in there you can recover them
fervent hatch
hmm
peak estuary
thats not like "the reason" why its useful but it maybe gives an idea
of why C can be nicer than R
fervent hatch
Might be worth seeing the Laplace transform, which is another integral transform...
seen it
just begs the question
I still dont really get it
All im hearing is it is because it is
fervent hatch
if you have the usual real fourier series with its coefficients a_n,b_n, then yo...
I dont know what youre getting at what do you mean
fervent hatch
Are you uncertain about something it does?
Its derivation seems random
why use complex exponents
why not stick to real sinusoids
stoic dune
Maybe the derivation is pretty random, bad book?
fervent hatch
i dont have a book
youtube has taught me everything unfortunately
probably why this isnt making sense
peak estuary
you cant forget that lots of math things were discovered over a lot of years
the derivations that you see today are not how the things were discovered
fervent hatch
sure
stoic dune
Complex exponents are not sinusoids. Sinusoids hold less information than complex exponents. That gain of information isn't to be ignored
fervent hatch
sure, complex exponents hold exponents and sinusoids, i get that, laplace is generalized fourier blah blah, I get it, Im confused why we even go there at all, why not just say e^x(cos(x) + sin(x))
fervent hatch
and what makes complex so useful
stoic dune
Well, you could! And there may be results for doing so.
fervent hatch
Well, you could! And there may be results for doing so.
how would one?
stoic dune
e^x(cos(x) + sin(x)) is not the same as e^(x)(cos(x) + isin(x))
peak estuary
everything
fervent hatch
okay but what
stoic dune
We can use it! But it wouldn't be the same transform.
fervent hatch
We can use it! But it wouldn't be the same transform.
i dont get it
stoic dune
Like, are you uncertain why e^x(cos(x) + sin(x)) is not the same function as e^(x)(cos(x) + isin(x))?
fervent hatch
thats part of it
theyre obviously different
I still dont really get what isin(x) is supposed to mean
other than the "spiral" in the real imaginary x plane
peak estuary
you do know that cosx+isinx runs around the unit circle as x increases, yes?
peak estuary
in that way it perfectly represents rotations and all that stuff
peak estuary
meanwhile cosx+sinx just kinda doesnt do it that quickly
fervent hatch
wdym
peak estuary
like sure they are still somehow linked to a circle and stuff
but its just way less nice
fervent hatch
well i mean we completely change the definition of a point in the complex plane its not really a fair comparison
you could have to compare (cos(x), sin(x)) to cos(x) + isin(x) because of the way stuff gets plotted
peak estuary
but now you have to introduce tuples
fervent hatch
?
peak estuary
and whats easier to deal with, a single number or two numbers at the same time
fervent hatch
a single number
peak estuary
aka cosx+isinx instead of (cosx, sinx)
stoic dune
Mind you, that is exactly the same, and that would be another Fourier transform
stoic dune
But your (cos(x), sin(x)) is just another way of writing cos(x) + isin(x)
stoic dune
That is complex
peak estuary
working in R^2 is just way more annoying than working in C
you can absolutely do it
but working in C is so much easier
fervent hatch
So youre telling me the only difference is that one is neater
stoic dune
In terms of mathematical machinery, there is no difference
peak estuary
the fact that you have a multiplication in C is insane
one that feels "natural"
instead of weirdly writing it in R^2
fervent hatch
what does the real fourier transform look like then
peak estuary
well integrate f(x) cosx or f(x) sinx
stoic dune
The transform you are suggesting is exactly the already existent complex fourier transform, just you aren't writing i
peak estuary
wdym
fervent hatch
I dont really know what this means
stoic dune
The i is there, you just aren't writing it haha
fervent hatch
well integrate f(x) cosx or f(x) sinx
okay but
peak estuary
one is in terms of cosx and sinx
peak estuary
the other in terms of cosx+isinx (or more easily, e^ix)
fervent hatch
why $$\int{f(x)isin(x)}dx$$ instead of $$\int{f(x)sin(x)}dx$$
flat frigateBOT
Nathan
fervent hatch
since you can break it apart to that point
thats all it comes down to
thats literally the only difference
peak estuary
from the complex number a_n+ib_n you can recover a_n and b_n
fervent hatch
how?
peak estuary
real part and imaginary part
stoic dune
The integral doesn't eat the i, to be very clear. The i is still there, but outside of the integral
fervent hatch
from the complex number a_n+ib_n you can recover a_n and b_n
and the same can be done in regular fourier series no?
recovering an bn that is
peak estuary
not if you are just adding the numbers
if you add 3+4=7, how can you tell that you didnt add 2+5
fervent hatch
The integral doesn't eat the i, to be very clear. The i is still there, but outs...
okay, sure, what does it mean?
what does it change by being there
peak estuary
meanwhile if you add 3+4i you can tell
fervent hatch
I see it and it just kind of confuses me because it has no literal meaning
stoic dune
We're back to "cos(x) + isin(x) is not cos(x) + sin(x)"
peak estuary
if you compute the integral of f(x)(cosx+sinx), you find out what a_n+b_n equals
fervent hatch
you can derive a and b easily with trig identities and go back and forth between forms
peak estuary
so for example you find out it equals 7
peak estuary
but what you actually wanted was to find out a_n and b_n individually
which you cant from this information
fervent hatch
yeah
peak estuary
meanwhile, if you compute the integral f(x)(cosx+isinx) you get a_n+ib_n
so if you find out that this integral equals 3+4i, now you found both a_n and b_n
with just a single integral
stoic dune
I see it and it just kind of confuses me because it has no literal meaning
This is a confusing thing to say. The i can't just be removed. It becomes a different expression.
fervent hatch
so if you find out that this integral equals 3+4i, now you found both a_n and b_...
okay why not just remove the i and not add them
stoic dune
Of course the i has meaning, it's an algebraic element
peak estuary
the i models that you are doing tuples. the cos and sin get treated differently
fervent hatch
This is a confusing thing to say. The i can't just be removed. It becomes a diff...
Yes but one that doesnt seem to make any difference?
peak estuary
just like in a tuple (cosx,sinx) you have a first and second element
if you are just adding cosx+sinx you are losing that information
stoic dune
3i + 4 is not the same as 3 + 4. The i matters
fervent hatch
if all you care about are the coefficients does it really
3+4 tells me the same information
fervent hatch
if you are just adding cosx+sinx you are losing that information
what information??
what am i losing
stoic dune
The fact that you can see two coefficients is proof the i matters haha
fervent hatch
?
stoic dune
There's no two coefficents in the real version
fervent hatch
sure but each integral will have its own value
stoic dune
That gets mixed into one coefficient
fervent hatch
before adding them those real values are the coefficients
peak estuary
3+4 tells me the same information
7 does not tell you the same information has two numbers (3,4)
fervent hatch
just dont add
fervent hatch
7 does not tell you the same information has two numbers (3,4)
why add it
dont add it
split the integral
stoic dune
You got me for a while haha. No more bait for me
fervent hatch
you get the same numbers
fervent hatch
You got me for a while haha. No more bait for me
what?
fervent hatch
you wanted to add
Yes but just split the intgral
dont add them, the first term and the second term are your coefficients
peak estuary
but the beauty with complex numbers is that you dont have to
you can just integrate it as one
as f(x)e^ix
one thing
fervent hatch
no I cant? to integrate that I have to break it apart and evaluate it
peak estuary
nah, depends hugely on what you are actually integrating
fervent hatch
idk let f(x) be x^2
how is that any easier than me doing integration by parts for two terms
peak estuary
integrating x^2 e^ix is very easy
fervent hatch
how?
peak estuary
well ibp like you said
fervent hatch
the entire point of this is just it makes it quicker and easier
peak estuary
once for cos, once for sin
peak estuary
the entire point of this is just it makes it quicker and easier
but like miles easier
fervent hatch
cleaner and faster
peak estuary
so much easier
fervent hatch
yes
okay
so for the complex fourier series
they mean the exact same thing
ones just obviously nicer looking
peak estuary
yes
fervent hatch
eye opening stuff
so if i really wanted to
i could do the same thing for laplace
do only real stuff
peak estuary
you can always try and only work in the real world
it just turns out to be so much uglier
fervent hatch
ok
so then
what about the linkage between the complex fourier series and the fourier transform
peak estuary
for the fourier series you always have e^inx
(or some other scaled version)
where n is an integer
now you just take all the values
fervent hatch
is that it?
fervent hatch
what about how one exists to approximate something while the other transforms to a frequency domain
peak estuary
in the series, the n means frequencies
fervent hatch
i feel like one does a different thing
peak estuary
so its itself now a variable in the "frequency domain"
peak estuary
yes kinda
fervent hatch
a discrete one
peak estuary
but you didnt really think about it that way
you thought about it more like an index or something
fervent hatch
i see
then whats the difference between the discrete fourier transform and the fourier series??
peak estuary
so and depending on the value of n the integral f(x)e^inx has some different value
so you have a function which takes in the input n and outputs the value of that integral
and that function you call the fourier transform of f
fervent hatch
and that function you call the fourier transform of f
and this output
what is its input?
frequency?
safe radishBOT
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keen moon
safe radishBOT
keen moon
I got until the very end but at the end I need to find
which is basically infinity - infinity 
did I do something wrong
how am I gonna turn that value into infinity/infinity
ohh that makes a lot of sense
thank you
.close
safe radishBOT
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lilac sun
is this correct?
safe radishBOT
is this correct?
versed wave
yes it is indeed correct
lilac sun
yes it is indeed correct
why? there are intersections tho
wouldn't it be parallel if its inconsistent
versed wave
but there are no intersections of 3 lines altogether
the solutions of the three-equation system is the set of all points that all 3 lines intersect at
versed wave
is this correct?
the intersections you are talking about are merely the solutions of the subsystem of equations with only the relevant line equations
safe radishBOT
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zealous gate
how would the initial tension be stronger than mg?
safe radishBOT
how would the initial tension be stronger than mg?
zealous gate
the answer is B
but isnt the block of mass m held in place?
so the vertical components will be equal?
empty gyro
so the vertical components will be equal?
Correct. As will the horizontal components
vagrant ice
Yes T1 has mg and some rightward force, so it must be more than mg alone
desert juniper
i'm not seeing how T2 is below mg tho
zealous gate
Okay
how woudl T2 be below mg tho?
this is it after the horizontal rope immediately cut
oh wait
wait why would T2 balance with mgcos(theta)
vagrant ice
Ah they'd be equal and opposite by Newton's 3rd law
That's a very clever force diagram
zealous gate
yeah
vagrant ice
wait why would T2 balance with mgcos(theta)
You mean mg/cos theta or something
zealous gate
no
mg would be the hypoteneuse if im not mistaken
so it would be cos(theta) = ?/mg
mgcos(theta) = ?
vagrant ice
Ah they'd be equal and opposite by Newton's 3rd law
I know this logic is iffy because there's technically no rope in the opposite direction
zealous gate
yeah
vagrant ice
Immediately after
vagrant ice
Immediately after
Yeah it has to do with this
safe radishBOT
vagrant ice
In my mind it's just easier to think that the particle would go up and to the right cause of inertia, as that's what happens when the left and downward force of the horizontal string is removrd
But since the original downwards force is mg, and you're reducing the vertical and the horizontal components of the tension
Yeah must be B
@zealous gate
zealous gate
oh okay
safe radishBOT
Available help channel!
stable halo
Hello, new here. I've been going through Mendelson's Intro. to Math. Logic textbook and I am on problem 2.55c. "A wf φ is k-valid iff it is true for all interpretations that have a domain of k elements." and I'm supposed to prove that (k+1)-validity implies k-validity. I initially attempted trying to disprove it but I think I might be misunderstanding what is being said. My initial thought is - say φ says there are k+1 elements in the domain. Then φ is true for all interpretations in a domain of k+1 elements but not in a domain of k elements.
safe radishBOT
Hello, new here. I've been going through Mendelson's Intro. to Math. Logic textb...
stable halo
The full statement from the textbook is: "Let a wf φ be called k-valid if it is true for all interpretations that have a domain of k elements. Call φ precisely k-valid if it is k-valid but not (k+1)-valid. Show that (k+1)-validity implies k-validity and give an example of a wf that is precisely k-valid. (See Hilbert and Bernays (1934 § 4–5) and Wajsberg (1933).)"
I just realized there is no equality in the logic (first order logic without equality) so perhaps no wf can express this..?
Wait sorry, I just realized this channel is for preundergrads.
.close
safe radishBOT
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safe radishBOT
lean otter
These channels are usually for more specific questions, so if you got any specific questions you need help with, we gotchu
Also, the first msg you type when you make this channel is pinned, so usually most people put their questions there
Just a lil fyi
wet frost
you can type .close if you do not need help anymore
safe radishBOT
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Available help channel!
hard orchid
Need help with this question pls
safe radishBOT
Need help with this question pls
hard orchid
Ive tried getting y = ln(x+1/2) then subbing into the bottom equation but im just confused i cant lie
brave wolf
I'd probably substitute the second equation into the first one instead
rugged arch
Ive tried getting y = ln(x+1/2) then subbing into the bottom equation but im jus...
just solve for 2y from the second equation right
instead of y
hard orchid
Ok thanks ill try
brave wolf
Ive tried getting y = ln(x+1/2) then subbing into the bottom equation but im jus...
(This would work too btw, you would just have to apply exp on both sides and it would have few extra steps)
brave wolf
I feel like you did way too many substitutions there
hard orchid
confused on this step
brave wolf
and in the end, the substitutions "cancelled" each otehr
hard orchid
why can you put e2y over e like that
brave wolf
why can you put e2y over e like that
that's an exponent rule
specifically the quotient rule
a^(m-n) = a^m / a^n
e^(2y - 1) = e^2y / e^1
safe radishBOT
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mild jolt
When your doing this do, which domain do you use, do you use the domain of the inverse function (x-1)^2 + 1 or the domain of the function showed?
safe radishBOT
When your doing this do, which domain do you use, do you use the domain of the i...
lean otter
the domain of the first one
safe radishBOT
@mild jolt Has your question been resolved?
mild jolt
the domain of the first one
why
lean otter
bcz the inverse exists only on the domain of the first function
otherwise it is not defined
ofc the function has to be bijective
zealous glen
The tangent to the curve x=2sint y=2+cos2t at the point (2,1) meets the x axis at P and y axis at Q.Find the distance PQ
safe radishBOT
The tangent to the curve x=2sint y=2+cos2t at the point (2,1) meets the x axis a...
sacred sentinel
show your work and where you are stuck
zealous glen
I found dy/dx=-sint .I don't know how to continue from there
sacred sentinel
let's say we have y=x², how would you find the tangent to the function there at x=2?
zealous glen
I would differentiate the curve to get 2x and substitute for x to get 4 as the gradient
So I have x as 2 and y as 4 and gradient as 4 so I can substitute in y=mx+c to get y=4x-4
sacred sentinel
yes
very good
so, what we need is:
- the slope at (2,1)
then, we find the equation of the line with the point we have
i just realised, your derivative isn't quite right
let's do it together
zealous glen
Ok
sacred sentinel
$\dv{y}{x}=\frac{-2\sin(2t)}{2\cos(t)}$
flat frigateBOT
pixel
sacred sentinel
right?
the 2 in the numerator appears because of the chain rule
,,=\frac{-\sin(2t)}{\cos(t)}
flat frigateBOT
pixel
zealous glen
Yes
sacred sentinel
,,=\frac{-2\sin(t)\cos(t)}{\cos(t)}
zealous glen
Then I tried to simplify it by expanding sin2t
flat frigateBOT
pixel
flat frigateBOT
pixel
zealous glen
Oh yes
sacred sentinel
we have the point (2,1), but to find the slope we need a value of t, how do you think we could find what that value of t is?
zealous glen
Substitute for x in x=2sin t but I'm not sure if we would get the same value if we used y=2+cos2t
flat frigateBOT
pixel
sacred sentinel
oh wait
i mixed it up
its (2,1)
so we should use 2 for x
,,2\sin(t)=2\\sin(t)=1\t=\frac{\pi}{2}+2\pi n
flat frigateBOT
pixel
zealous glen
Am getting t as 90⁰ which is pi/2 I don't get the +2 π n part
sacred sentinel
the 2pi part is there because sine is periodic, but it doesn't matter in this case
,,1=2+\cos(2t)\-1=\cos(2t)\2t=\pi\t=\frac{\pi}{2}
flat frigateBOT
pixel
sacred sentinel
so we see that it doesn't matter whether we use x or y
queen ingot
you get twice as many solutions for y
zealous glen
I tried y and I'm getting multiple values of t
queen ingot
but you can only take the solutions that satisfy both equations
zealous glen
but you can only take the solutions that satisfy both equations
Okay
sacred sentinel
we could have also used $2\frac{1}{2}\pi$ but $\frac{\pi}{2}$ is easier
flat frigateBOT
pixel
sacred sentinel
now that we have a value of t, can you complete the problem?
zealous glen
Let me work it out, help me to verify it
sacred sentinel
I'm getting (3 sqrt 2) is that right
for the slope you mean?
sacred sentinel
How did u get the 2.5 π
use n=1 for pi/2 + 2npi but doesn't really matter right now
zealous glen
for the slope you mean?
No as distance PQ
zealous glen
Thank you😭
sacred sentinel
np also thanks to the other guy
zealous glen
np also thanks to the other guy
yeah
There's another one I haven't been able to do too
Can I ask it
zealous glen
well according to the rules not but its fine for now
thx
So I was able to find dy/dx but the second derivative is quite challenging
sacred sentinel
yeah
we have a fraction as first derivative
so what differentiation rule do you think we need to use?
zealous glen
Quotient rule
sacred sentinel
very good
can you show what you did, then i will help you find the mistake or help you on the step you're stuck on
zealous glen
Ok
I don't know how to go about simplifying that to get the desired second derivative
sacred sentinel
uhh i'm not really familiar with your way of computing it i just realised i don't actually know how to calculate second derivatives (i didn't learn this in school yet)
so id recommend asking it in another channel (thats exactly why the rule for making a new channel exists)
@zealous glen
boreal totem
$\int \frac{1}{1+e^x},dx$
safe radishBOT
$\int \frac{1}{1+e^x}\,dx$
boreal totem
can i solve this by subbing $x=\ln u$?
flat frigateBOT
109105116
fathom adder
dx = 1/u du ?
boreal totem
yea
boreal totem
i got this $\int \frac{1}{u(1+u)} dx$
flat frigateBOT
109105116
fathom adder
du
boreal totem
du*
yea
and then plugging back e^x
to get
$\ln \left\lvert\frac{e^x}{e^x+1}\right\rvert+C$
flat frigateBOT
109105116
fathom adder
Ah yeah cuz du = e^x dx
boreal totem
is that right? how do i differentiate it to check
fathom adder
You would need 1/2
boreal totem
i forgot to add the extra e^x term
$\ln \left\lvert\frac{e^x}{e^x(e^x+1)}\right\rvert+C$
flat frigateBOT
109105116
fathom adder
No
fathom adder
You have
boreal totem
**109105116**
from this
fathom adder
1/u+u² du
boreal totem
i used partial fraction
fathom adder
No need
fathom adder
du = e^x dx
So you have that
e^x/(e^x+e^2x) dx
But
Yeah never mind
What did you get with partial ?
boreal totem
$\int \frac{1}{u(1+u)} du = \int \frac{1}{u} , du - \int \frac{1}{u+1} , du$
with an integral
du*
fathom adder
Indeed
flat frigateBOT
109105116
boreal totem
which is $ \ln |u| - \ln |u+1| $ right?
oh that's right i just differentiated it to check
thanks @fathom adder
.close
safe radishBOT
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fervent hatch
Can I get some help on 9
safe radishBOT
Can I get some help on 9
safe radishBOT
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
Dunno how to set up the integral
I’m assuming the bounds are 0 to pi/2 when looking at it graphically but it evaluates to zero
Then I tried 0 to pi/4 and multiplying it by two and just got the wrong answer
queen ingot
With the second method you'll only need one of the two equations
Can you show your work?
fervent hatch
With the second method you'll only need one of the two equations
Probably put them in the wrong order cause of the negative but it doesn’t matter
fervent hatch
Wdym
queen ingot
Oh you're doing 0 to pi/2 here
fervent hatch
Yea
queen ingot
I think you need to break it up
fervent hatch
How come
queen ingot
Draw a radius
fervent hatch
?
queen ingot
A ray from the origin
fervent hatch
Okay
fervent hatch
I don’t see what you mean
queen ingot
OK get rid of one of the circles
fervent hatch
Yeah