Only this you must obey
#help-39
naive zinc
edgy bane
Yes
low matrix
if there was only one minimal then it would be called least element?
edgy bane
Yes
naive zinc
So two incomparable elements, like your 2, 5. You can make their height <, =, > whatever you like
low matrix
So two incomparable elements, like your 2, 5. You can make their height <, =, > ...
cool
naive zinc
You want to find upper bound of what?
low matrix
low matrix
idk what upper bound is
edgy bane
idk what upper bound is
An upper bound is an element greater than all elements in a particular subset
naive zinc
An upper bound of a subset A of a poset B is {x in B: any y in A, y<=x}
low matrix
An upper bound is an element greater than all elements in a particular subset
can it have two elements
b and c both r higher than a
naive zinc
A subset can have many upper bound
edgy bane
Yes
low matrix
so b,c are upperbound of {a,b,c{?
naive zinc
Like your {2,5} has upper bound 10, 20
No
e is one, f is also one, and there are others
low matrix
e and f are not in the abc set tho
edgy bane
so b,c are upperbound of {a,b,c{?
No because b isn't greater than c (an element of {a, b, c} ) and c isn't greater than b
naive zinc
Four in total
edgy bane
e and f are not in the abc set tho
Upper bounds don't have to be in the set
Just in the main Poset itself
I'll give an example
low matrix
Upper bounds don't have to be in the set
so all the elements greater than a,b, and c?
naive zinc
Yeah
edgy bane
In this diagram the upper bounds on the 3-element subset {a, b, c} are
e, f, i, h
low matrix
naive zinc
Yeah those 4
low matrix
is it gonna be e,f,j, h
naive zinc
is it gonna be e,f,j, h
Yeah
edgy bane
{a, b, c} also has one lower bound which is a
naive zinc
If among them there is a smallest one, like this case , e, then e is called the least upper bound of {a,b,c}. least upper bound and greatest lower bound doesn’t always exist in a poset. Sometimes it’s an important property of certain posets. Like R, real numbers, it has least upper bound property (any non-empty subset of R has a least upper bound)
low matrix
If among them there is a smallest one, like this case , e, then e is called the ...
what if i want to find upper and lower bound of {j,h}?
naive zinc
Doesn’t exist
low matrix
there is no upperbound ig
naive zinc
Upper
naive zinc
Lower : every point except j,h,g
naive zinc
a,b,c,d,e,g,f?
Except g
low matrix
how?
low matrix
oh its an AND
naive zinc
…
naive zinc
You say, i check
naive zinc
Yeah
naive zinc
is the upper gonna be h
No, g,h
naive zinc
and lower is a?
a,b
naive zinc
Definition
naive zinc
Yeah
naive zinc
f,h,j
naive zinc
Same height doesn’t imply anything
low matrix
Same height doesn’t imply anything
it implies equal
low matrix
yuh
naive zinc
No polyline between two elements then those two are incomparable
naive zinc
it implies equal
So it doesn’t imply anything
5, 10, 15, 30 division for example, as long as you put 5 lowest, 30 highest. The height of 10 and 15 you can arrange them whatever you like, 10 higher or 15 higher or same height
pearl pondBOT
pearl pondBOT
Available help channel!
low matrix
a.) is a partial order right?
pearl pondBOT
a.) is a partial order right?
low matrix
cuz no shorter means taller than or equal to?
brb, (meanwhile react tick if u think im right or otherwise)
toxic lichen
"a is not shorter than b" means a's height >= b's height, yes
low matrix
sorry guys, Ann's message didntt load for me when i pinged the helpers
low matrix
"a is not shorter than b" means a's height >= b's height, yes
then it is partial order?
edgy bane
a.) is a partial order right?
Well this is a pretty nuanced case.
Technically it is not a partial order because it fails to satisfy anti-symmetry (two distinct people can be of the same height) but it is a total preorder
pearl pondBOT
@low matrix Has your question been resolved?
low matrix
Well this is a pretty nuanced case.
Technically it is not a partial order becaus...
if they are same height then they satisfy the relation
no shorter means either the height is greater than or equal to
as Ann said
edgy bane
if they are same height then they satisfy the relation
Partial orders must satisy a property called anti-symmetry which states that if $$ a \leq b ; \wedge ; b \leq a \implies a=b$$
jolly parrotBOT
eromorphism
low matrix
ppl with same height will be in the relation
ok
and due to that they dont satisfy anti symmetry
low matrix
Partial orders must satisy a property called anti-symmetry which states that if ...
people with same height are in the relation, lets say it is a and b such that a and b have same height
then does the relation have both of these ordered pairs: (a,b), (b,a)?
low matrix
.close
pearl pondBOT
pearl pondBOT
Available help channel!
maiden badger
I need help solving this exercise:
We are given to vectors and need to find the range of values for k where the angle they make is obtuse.
maiden badger
I've tried doing somethings but it's giving me k to powers bigger than 3 and I don't think that is supposed to happen
I'm starting to feel like this isn't leading nowhere
rugged niche
A/B < 0 and B > 0 => A < 0
So you only have to consider u * v < 0, no need to calculate the norms of these vectors.
maiden badger
Oh! That's the missing piece!
Oh yeah, I forgot obtuse vectors produce a negative product
Gee thanks!!
pearl pondBOT
@maiden badger Has your question been resolved?
maiden badger
.close
sharp smelt
I would like help solving this using the bisection method
pearl pondBOT
I would like help solving this using the bisection method
sharp smelt
upto 3 iterations
this is the method
the best I could do is conclude the root is in [0,0.125]
naive zinc
,w Table[Sqrt[x] - Cos[x], {x, 0, 1, 1/8}]
jolly parrotBOT
naive zinc
,w Table[(Sqrt[x] - Cos[x])/abs(Sqrt[x] - Cos[x]), {x, 0, 1, 1/8}]
shut elm
the best I could do is conclude the root is in [0,0.125]
Are you saying this isn't what you're supposed to get using the method to 3 iterations, or what 🤔
jolly parrotBOT
sharp smelt
well, it technically answers the question. But I don'tget how this gives me an approximate root
naive zinc
Why, wolfram can’t do absolute value?
sharp smelt
hmm? It is in the denominator
naive zinc
oh, f(x)/|f(x)| I expect it to show +/-1 turns out it doesn’t simplify
shut elm
well, it *technically* answers the question. But I don'tget how this gives me an...
It gets you that the x-value is approximately between those two, if you kept going you'd converge to so many decimal places
But I do wonder about your first step
naive zinc
Anyway, -1,-1,-1,-1,-1,-1,1,1,1,1
shut elm
Why did you go down to [0,1/2] and not up to [1/2,1]
naive zinc
Yeah first step wrong range
sharp smelt
But I do wonder about your first step
both f(0) and f(1/2) are less than 0
sharp smelt
oh
shut elm
It cannot lie on the interval below 0
sharp smelt
🤦
sharp smelt
right
shut elm
🤦
For context the actual 0 is at ||x=0.64171||
Try again and then compare to the actual 0 and you'll see you've approximated it
shut elm
Np!
sharp smelt
Anyway, -1,-1,-1,-1,-1,-1,1,1,1,1
what's this
naive zinc
sgn(f(x)): x =0,1/8,…,7/8,1
shut elm
what's this
Yeah the sign tells us the 0 lies between 5/8 and 6/8 (which we see it does)
naive zinc
Yeah
shut elm
Because the sign change happens between those two values
jolly parrotBOT
shut elm
May have been a bit easier
shut elm
And that's actually how you'll be determining your answer wai
The interval divided in half 3 times (so 0 to 1 in increments of (1-0)/2^3)
sharp smelt
so we essentially want a seqeunce of "solutions" that approach the actual soln from both sides
naive zinc
Yeah, nested intervals
sharp smelt
Yeah, nested intervals
oh yea, so if have enough intervals we can essentially use cantor's intersection theorm
naive zinc
Yeah
shut elm
Ooo smart
sharp smelt
shut elm
Ah now we reach the decimal places 💃
sharp smelt
After I solve this I'll probably write some python code to find the soln 😔
shut elm
(How many iterations do you need to split an interval of size 1 into pieces < 0.05)
shut elm
Right
So your answer will just be the closest of the two in the wolfram table where the sign changes!
sharp smelt
so I just have to repeat the method 5 times and indicate the interval in which the root lies
sharp smelt
and then chose the supremum or infimum of the intevral
sharp smelt
But doing that with a table is also much easier
true, but I'm prepraing for next sem so rn the aim is to learn as much as I can
shut elm
Do you still want me to generate the table (if not just for you to check your work) or you're good
I don't have to if you don't want me to
sharp smelt
lemme learn the table method sure
shut elm
,w Table[sgn(2+x+e^x), {x, 1, 2, 1/32}]
jolly parrotBOT
shut elm
Wait it can't be all 1's
Let me graph this that doesn't seem right
,w Table[sgn(2+x+e^x), {x, 1, 2, 1/2}]
jolly parrotBOT
shut elm
How can there be no 0's
Oh it's -e^x
Okay 😂
,w Table[sgn(2+x-e^x), {x, 1, 2, 1/32}]
jolly parrotBOT
shut elm
So I know the answer will be between 5/32 to 6/32
But the full process would of course be
sharp smelt
mhm
shut elm
,w Table[sgn(2+x-e^x), {x, 1, 2, 1/2}]
jolly parrotBOT
jolly parrotBOT
shut elm
You probably get the idea
pearl pondBOT
shut elm
👋
pearl pondBOT
Available help channel!
plush basin
hello
plush basin
can someone explain to me how for example, sin (x) is an odd function and cos (x) an even function
according to the definition of an even function where f(-x) = f(x)
and odd function that f(-x) = -f(x)
cinder flower
easy with power series
graceful leaf
i doubt that would be much easier for him to understand
graceful leaf
cos is symetrical over the y axis
cinder flower
i doubt that would be much easier for him to understand
just a little remark. not necessarily something that i think should be pursued
graceful leaf
just a little remark. not necessarily something that i think should be pursued
i can be ur bf......
cinder flower
i think i'll pass, mr pre-university math role
graceful leaf
noooooo
torpid matrix
can someone explain to me how for example, sin (x) is an odd function and cos (x...
i dont know if this is a good justification
but like
if you just sub in values into x
should give the same thing
plush basin
both functions have a domain of every real
so that is to say just for the cosine function that a value of a negative x value produces the same output as for a positive value
cinder flower
if you just look at the unit circle, can you see why cos(x) = cos(-x) when you look at some values of x?
frozen nova
Guys how do people back in the days know the value of sin and cos if there were no calculators back then
sharp vigil
,tikz[point/.style={fill,circle,inner sep=0pt, minimum size=4pt}, scale=2]
\draw[->] (-1.5,0) -- (1.5,0) node[right]{$x$};
\draw[->] (0,-1.5) -- (0,1.5) node[above]{$y$};
\draw (0,0) coordinate(O) circle[radius=1];
\coordinate (X) at (1,0);
\nodepoint, label=above right:{$\big(\cos(\theta),\sin(\theta)\big)$} at (30:1) {};
\node[point, label=below right:{$\big(\cos(-\theta),\sin(-\theta)\big)$}] (Q) at (-30:1) {};
\draw[dashed] (P) -- (O) -- (Q);
\draw pic[draw, angle eccentricity=2,"$\theta$"]{angle=X--O--P} pic[draw, angle eccentricity=2,"$-\theta$"]{angle=Q--O--X};
jolly parrotBOT
cloud ☁
sharp vigil
notice how the two points have the same x-coordinate but the y-coordinate is flipped
plush basin
is there a geometric proof
sharp vigil
that's what i gave isn't it?
plush basin
not really a sound proof
sharp vigil
the fact that the x and y coordinates must be the same absolute value comes from triangle congruence
pearl pondBOT
pearl pondBOT
Available help channel!
frank violet
pearl pondBOT
bitter herald
your question?
frank violet
Just sounds familiar to a graph theory problem but I have never learnt graph theory properly
frank violet
hint: ||try to translate this to graph theory||
bitter herald
Just sounds familiar to a graph theory problem but I have never learnt graph the...
think of the mathematicians as the vertices and the handshakes as edges
bitter herald
hmm
frank violet
think of the mathematicians as the vertices and the handshakes as edges
That's the only thing I know about graph theory 
, other theorems/ formula I have no idea
frank violet
i think you should try to count the paths of length 2?
What's the definition of length of a path 
bitter herald
What's the definition of length of a path <:sadcat:547124024585486359>
like
pick a random mathematician
lets call him Joe
you gotta count like how many "friends of friends" Joe has
so in how many ways can you go Joe -> Friend -> Friend's friend
frank violet
you gotta count like how many "friends of friends" Joe has
Okay, so how can I do that
bitter herald
well like
there are two ways
and you gotta use both
think of it this way
every vertex has d neighbours, correct?
bitter herald
number of edges = amount of neighbours = people you shook hands with = 3k+6 = degree
frank violet
Okay okay
bitter herald
gee a lot of jargon eh
anyways
so this is now regular ol' product rule from combinatrics
lets say you are at joe's vertex
how many paths can you go down on?
frank violet
3k+6?
bitter herald
yeah
ok now lets say you pick one
you go down to Joe's friend, Jill
how many paths can you go down on now?
frank violet
how many paths can you go down on now?
3k+6? does go down mean I can't go back to Joe?
If so 3k+5
bitter herald
<:hmmcat:831508801646755850> 3k+6? does go down mean I can't go back to Joe?
yes
to your second question
so how many paths in total, per the product rule?
frank violet
Uhh (3k+6)! ?
frank violet
bitter herald
its 3k+6 first step then 3k+5 second step
naive zinc
Not sure if you can solve it without graph theory. It’s a strongly regular graph (12k, 3k+6, c, c). This condition should let you express c using k. c being an integer will let you solve k. But I don’t know if you can get bypass graph theory
bitter herald
frank violet
Not sure if you can solve it without graph theory. It’s a strongly regular graph...
I hope we can cuz I checked our recorded lectures and non of them are about graph theory 
hmm yeah....
Hmm
What's wrong with (3k+6)!
Ohhh
(3k+5)! Yeah I forgot
bitter herald
no
frank violet
Okay lmao
bitter herald
like
lets do a side question real quick 
Kate has 52 playing cards
She gives one card to Grace and one card to BIll. In how many different ways can Kate do this?
bitter herald
no
frank violet
52*51
frank violet
idk where you are getting the factorial from
I thought you said keep going down and count how many
frank violet
Okay can you direct me to a video about graph theory and all I need about to know to solve this problem?
Sorry Imma go out for dinner with my family
It's gonna take a while
bitter herald
u dont need anything more tbh
teh rest is just general combinations
we can continue this after or something
eat well!
naive zinc
Oh I just wanted to type the only result you need regarding this question
For a strongly regular graph (v,d,s,t), (v-d-1)t=d(d-1-s)
(Proof:
fix a vertex x in G
A=A(x) be the set of other vertexes adjacent to x
B=B(x) be the set of other vertexes not adjacent to x
Count edges connecting one point from A and another point from B in two ways:
- any a in A, (ab) an edge where b in B. Since b is adjacent to a not to x, there exist d-1-s many such b, together |A|(d-1-s)=d(d-1-s)
- any b in B, (ab) an edge where a in A. Since a is adjacent to both b and x, t many such a. Together |B|t=(v-1-d)t
Thus d(d-1-s)=(v-1-d)t
Not to keep you I don’t have anything else to add
You got k already? I can check
pearl pondBOT
@frank violet Has your question been resolved?
pearl pondBOT
pearl pondBOT
Available help channel!
tawdry ember
pearl pondBOT
autumn fossil
autumn fossil
Can you send your diagram?
tawdry ember
i tried power of a point
tawdry ember
Can you send your diagram?
i did on a rock 😭
sorry
i was outside
a friend had sent
tawdry ember
yeah
autumn fossil
okay, cool, so we know that the angle subtended by the arc is 90°
tawdry ember
mhmm
autumn fossil
You also mentioned working with triangles, so there is one more triangle we can draw
tawdry ember
rightt
tawdry ember
i actually drew the line from the center to x
and tried to find a relation
bw those 2
but yeah
its wrong
tawdry ember
i didnt end up with anything nice lol
tawdry ember
it's not wrong, we will use it as well I think
oh
autumn fossil
Once you find the blue length, you can get the radius (red line)
autumn fossil
i actually drew the line from the center to x
and then you can use this
tawdry ember
Once you find the blue length, you can get the radius (red line)
yup
thank you
the 3rd triangle didnt strike my dumb ass
you can close it
royal galleon
For this I was thinking of setting u = sqrt(1-x^2). and then we also have x^2 = 1-u^2. then we have 1 - u / 1-u^2 and then simplifies to 1/1+u. but I don't know how I can say x to u for the limit itself
still hamlet
For this I was thinking of setting u = sqrt(1-x^2). and then we also have x^2 = ...
as x->0 where does u go
proper nova
For this I was thinking of setting u = sqrt(1-x^2). and then we also have x^2 = ...
Multiply by the conjugate
Should work
prisma kernel
autumn fossil
conjugate (almost) always works
royal galleon
as x->0 where does u go
1
autumn fossil
the substitution could work as well, but conjugate is the safest way
and it seems like the sub indeed does work
royal galleon
1/2
royal galleon
so by that you guys mean multiply by sqrt(1+x^2)/sqrt(1+x^2)
proper nova
but yeah a conjugate is safer
i definitely use the god dang useful hint they gave you
royal galleon
I don't see how their hint is helpful
proper nova
so by that you guys mean multiply by sqrt(1+x^2)/sqrt(1+x^2)
No, multiply by $1 + \sqrt{1 - x^2}$
jolly parrotBOT
1 divided by 0 equals Infinity
proper nova
And use the hint
pearl pondBOT
proper nova
ok I tried out the hint. thx guys
No problemo
proper nova
oh I see its of the same form
Get the grasp of algebra first, then these kinds of exercise is just a breeze
pearl pondBOT
Available help channel!
stoic seal
i never been so confused in my life. There are two spheres that partially overlap the distance from one center to the other is a vector called d . im pretty sure i need to integrate a bunch of circles along a normal vector centered d/2 in order to get a volume.
pearl pondBOT
i never been so confused in my life. There are two spheres that partially overla...
@stoic seal Has your question been resolved?
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.
sharp vigil
do you know what the electric field is inside a single uniformly charged sphere?
median leaf
oops wrong server mb
stoic seal
saying r<R then gaussian surface is inside the single sphere
you also get the volume for the density calculation
sharp vigil
yes
once you have it for a single sphere, you can use the principle of superposition
which says that the electric field due to two different charge distributions is the vector sum of what they would contribute individually
stoic seal
so would that just mean its two times Q_enc radially out but then for the inside its 0?
i thought the gaussian surface was supposed to be a mini version of that overlap. im just not getting how the two sphere sit at the center. like some of 1 surface goes in the overlap but some doesntr
sharp vigil
i don't think it would be possible to get a gaussian surface that works for both at the same time
instead focus on getting an expression for what the contribution to the electric field is for both spheres individually
stoic seal
i wanna just say -E-E=-2E from positive to negative because it would just be adding another field of opposite magnitude
im still very confused
sharp vigil
it's not that simple because both of them contribute vector fields of varying magnitude and direction
so you need to come up with a full vector formula for the electric field inside a uniformly charged sphere (your book will probably have it somewhere in the examples)
stoic seal
E=rhor/3epsilon_0 r^ for the postive and E=rhor/3epsilon_0 -r^
the d vector is r plus some other vector
sharp vigil
ok so at the moment you have two different r vectors, one pointing from the center of each sphere
it would be nice if you could relate them so that you only need one
stoic seal
well the d vector is the one im breaking apart because thats the one i was given. i have an r peice and if its just like a vin diagram D=(r-l) where l is the distance of the edge one sphere to the end of the overlap. or D is the to the top of the vin diagram to trhe other center
sharp vigil
ok so just to rewrite for clarity what you have so far, your two electric fields are
\begin{align*} \vec E_+ &= \frac{\rho r_+}{3\varepsilon_0} \hat r_+, \
\vec E_- &= - \frac{\rho r_-}{3\varepsilon_0} \hat r_- \end{align*}
where $\vec r_+$ is the vector pointing from the center of the positively charged sphere to any given point and $\vec r_-$ is the vector pointing from the center of the negatively charged sphere.
It would be useful to relate $\vec r_+$ and $\vec r_-$ in some way so that you can substitute in your relation and have both formulas in terms of only one of them
jolly parrotBOT
cloud ☁
sharp vigil
you can refer to the diagram if it helps
stoic seal
ok d=(r+ -r) ? and r+ is fine but the negative is written the r^- can be (d-r) so its 2rho/3epsilon_0 r^ so its just the whole magnitude with no r in the d direction?
like E = rho/3 epsilon d^ which is a constant!
sharp vigil
pretty much yes
you should be getting $\vec d$ rather than $\hat d$ in the formula, though
jolly parrotBOT
cloud ☁
stoic seal
oh yeahh d is a regular vector lol. that problem seems so much easier than it should be... just place a test charge and do the sum
that was very very helpful. i had like an axis along d where i was trying to integrate disks proportional to a cicular arc...
thank you!
sharp vigil
yeah it usually helps with these sorts of complicated charge distributions to break them down into more familiar parts and then add up
sharp vigil
.close
exotic gale
If four distinct positive integers sum to 20, how many such choices of positive integers, without regard to order, include at least one integer that is divisible by 5?
pearl pondBOT
If four distinct positive integers sum to 20, how many such choices of positive ...
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
2
naive zinc
Answer checking?
exotic gale
no i got stuck midway
modest tartan
just asking but is the a permutations and combinations question?
exotic gale
i was thinking about using complimentary counting but i think that just adds unnecessary work
modest tartan
i was thinking about using complimentary counting but i think that just adds unn...
you could make cases
but that would take a while
exotic gale
i know that the only multiples of 5 that you can use are 5, 10, or both
naive zinc
How about you separate it to smaller problems
Like distinct positive a,b,c not equaling 5 such that a+b+c=15
Two others are simpler
exotic gale
yeah i thought about doing that but idk if there's too many cases
naive zinc
Only three cases and
naive zinc
How about you separate it to smaller problems
Like distinct positive a,b,c not ...
Only this case you need to avoid some number, like 5 this case
Number of solution of distinct m positive a_j such that a_1+…+a_m=N has a general method: assume a_1<…<a_m, let b_i=a_i - (i-1), then it becomes positive integers b_1+…+b_m=N’, N’=N-(m-1)m/2. this you know, right? binom(N’-1, m-1).
naive zinc
You mean positive adding up to N’, how many solutions this you don’t know?
exotic gale
no? is that supposed to be common sense or something?
naive zinc
positive b_1,…,b_m , their sum being N’. It’s the same as
You place N’ many balls on one line, and use m-1 obstacles to separate them
exotic gale
oh stars and bars are you talking about that
naive zinc
N’ many balls, N’-1 many spots between them, place m-1 obstacles I meant
Yeah
So you do know it
exotic gale
yeah but how does that help in this case
naive zinc
?
naive zinc
Number of solution of distinct m positive a_j such that a_1+…+a_m=N has a genera...
I just told you how to calculate
m distinct positive integers adding up to N without order is binom(N’-1,m-1) many
So you obtain three numbers , m=3, N=15,10,5 respectively, minus number of solutions of distinct a+b+c=15 when a=5
pearl pondBOT
exotic gale
thanks for helping
pearl pondBOT
Available help channel!
rocky crow
If I have a flag, and it needs to be a ratio of 5:8. and I want the shorter side of a rectangle to be 20 cm how much would the larger side need to be?
rocky crow
I forgot how’d you do ratios
lilac quartz
5:8 means for every 5 units of shorter side there are 8 units for longer side.
so question, for 10 units of shorter side what should be the longer side?
@rocky crow
toxic lichen
so question, for 10 units of shorter side what should be the longer side?
op wants 20 cm not 10
lilac quartz
I mean, I was trying to give some intution
cyan aspen
op wants 20 cm not 10
Teacher going Step by step 😏
rocky crow
,calc 22/5
jolly parrotBOT
Result:
4.4
so what is the shorter side in 5:8?
rocky crow
,calc 4.4*22
jolly parrotBOT
Result:
96.8
96,8?
cyan aspen
,calc 22.5*8
jolly parrotBOT
Result:
180
,calc 180/5
jolly parrotBOT
Result:
36
,calc 13/22.5
jolly parrotBOT
Result:
0.57777777777778
i’m out bye
rocky crow
,calc 8/5
jolly parrotBOT
Result:
1.6
,calc 1.6*22
jolly parrotBOT
Result:
35.2
So when doing ratios u got to put the value u know on the bottom. It’s called dimensional analysis.
[know. Value] * [ratio of value u want] / [ratio of value u know]
rocky crow
35,2?
wraith stag
short side 22, and long/short must be 8:5
wraith stag
yeah 35.2
rocky crow
OMG I’m so smart
cyan aspen
,calc (22*8) / 5
jolly parrotBOT
Result:
35.2
35.2 in other parts of the world
rocky crow
35.2 in other parts of the world
I mean centimeters
wraith stag
just do whatever your country does
even numbers like "1 billion" don't have universal meaning
fiery willow
What’s it about
lilac quartz
If I have a flag, and it needs to be a ratio of 5:8. and I want the shorter side...
are you guys saying the longer side is 35.2?
fiery willow
Oh that’s easy I’ll assist you dot
rocky crow
are you guys saying the longer side is 35.2?
Yes
fiery willow
What’s the question are you working on
storm venture
Now do it with shorter side as 20 cm
pearl pondBOT
@rocky crow Has your question been resolved?
restive garnet
How to find inverses in modular arithmetic? Like if i have $5x \equiv 7 (\text{mod } 6)$, is there any systematic way to find the value of $x$?
jolly parrotBOT
Ishmam
wet osprey
Extended Euclidean algorithm?
restive garnet
Does it matter man
last summit
No its just funny
restive garnet
I just made something up with 567
wet osprey
Nah just different representatives
last summit
67
restive garnet
Extended Euclidean algorithm?
What's that?
wet osprey
The answer to your question
Is there a way to systematically find the inverse
Yes
Use the extended Euclidean algorithm
pearl pondBOT
pearl pondBOT
Available help channel!
spiral coyote
Let (E = ax_{1} + by_{1} + c), where (b \neq 0).
If a point (P(x_{1}, y_{1})) lies above the line (ax + by + c = 0), then
[
\frac{E}{b} > 0
]
If a point (P(x_{1}, y_{1})) lies below the line (ax + by + c = 0), then
[
\frac{E}{b} < 0
]
jolly parrotBOT
ch3rry
spiral coyote
Let $A(-1, 1)$ and $B(2, 3)$ be two points and P be a variable point above the line $AB$ such that the area of $\triangle PAB$ is $10$. If the locus of P is $ax + by = 15$, then $5a + 2b$ is:
jolly parrotBOT
ch3rry
spiral coyote
Where did i go wrong? Isnt $2p-3q=-25$ the locus of $P$?
jolly parrotBOT
ch3rry
naive zinc
Obtained the same, |2x-3y+5|=20. I saw no problem
spiral coyote
how do you quote replies?
uneven cypress
The stuff inside the modulus could be +- did you consider both? @spiral coyote
spiral coyote
The stuff inside the modulus could be +- did you consider both? <@1438816655399...
We have an inequality. So there's only one case
uneven cypress
Ooh ic mb
frozen bluff
i think you just need to normalize it to match the required form
uneven cypress
Then it seems correct
spiral coyote
$ax+by=15$ how do i rewrite it in this form then. %
jolly parrotBOT
ch3rry
uneven cypress
You can multiply an entire equation by a constant not zero without altering it
frozen bluff
multiply both sides with -3/5
spiral coyote
Oh right tht was a stupid ques
frank violet
Let \(E = ax_{1} + by_{1} + c\), where \(b \neq 0\).
If a point \(P(x_{1}, y_{1...
-# I'm back, do you still need help with this?
spiral coyote
**ch3rry**
I wanted to know where this comes frm btw
spiral coyote
-# I'm back, do you still need help with this?
Yesyesyes please
spiral coyote
multiply both sides with -3/5
-3/5 right im an idiot
frank violet
**ch3rry**
without the loss of generality we can assume that b>0
frozen bluff
-3/5 right im an idiot
yeah yeah idk why i did that 😭
frank violet
without the loss of generality we can assume that b>0
well we can multiply both side by a constant to get b>0 that's why
spiral coyote
without the loss of generality we can assume that b>0
Ok ig
frank violet
construct a line l2: ax+by+c+d=0
so l1 is parallel to the given line
Call the given line l:ax+by+c=0
P is the given point with the coordinate (x1,y1)
Let Q be a point lies on l that have the same x-value
Q= (x1,y2)
spiral coyote
Ehh there r three lines? $l,l_1,l_2$?
jolly parrotBOT
ch3rry
frank violet
Ehh there r three lines? $l,l_1,l_2$?
No I call the given line: l
From the set up: ax1+by1+c+d=0
and ax1+by2+c=0
subtract the first one by the second one
b(y1-y2)= -d
spiral coyote
Wait im not following
frank violet
okay
spiral coyote
Given line $l$ is $ax_1+by_1=-c$?
jolly parrotBOT
ch3rry
frank violet
Given line $l$ is $ax_1+by_1=-c$?
no the given line is ax+by=-c
spiral coyote
Ah kk so we hv three lines ryt?
frank violet
**ch3rry**
This shows a point with coordinate (x1,y1)
that lies on l a.k.a the given line
spiral coyote
Ahhhh okok
frank violet
b(y1-y2)= -d
so here uhh
if -d > 0 then P is above the given line
but ax1+by1+c+d=0 => ax1+by1+c=-d
so yeah
spiral coyote
I think a diagram might help 
frank violet
I think a diagram might help <:sully:651816820122189834>
okayy
spiral coyote
frank violet
frank violet
It looks kinda weird with high resolution
spiral coyote
Wait i still dont get it
glacial charm
Where did i go wrong? Isnt $2p-3q=-25$ the locus of $P$?
there are two loci
frank violet
Wait i still dont get it <:blobcry:971797574916907028>
Which part
spiral coyote
Which part
What significance does d hold
spiral coyote
there are two loci
The ques is pinned.
spiral coyote
$d$ is basically $E$?
jolly parrotBOT
ch3rry
spiral coyote
Yh its a constant
frank violet
Yh its a constant
yup
spiral coyote
Wht is the point is below x axis😭
pearl pondBOT
pearl pondBOT
Available help channel!
fluid cosmos
Confused as to how the statistics for this works. In the rightmost block, the statement "the reseacher sees 125 are true and 45 are not" makes me think that we can act as if this scenario of events was carried out in real life. But looking at the second block, we perform 5% of 900, which implies we know which 100 of the 1000 are actually true hypothesis. But if we knew that then obviously those 100 would be posted.
fluid cosmos
I just don't really understand what the point of this is because it doesn't make any sense
desert wadi
Which right most block?
fluid cosmos
before this example, he was talking about power analysis, statistical significance and effect sizes
but it was mostly pretty surface level since its a HCI course
desert wadi
whats the question
fluid cosmos
Confused as to how the statistics for this works. In the rightmost block, the st...
This example doesn't make any logical sense to me
im wondering if im misunderstanding something
desert wadi
so there are 1000 squares correct
fluid cosmos
yep
desert wadi
What do you not understand everything?
fluid cosmos
mm okay wait it might be easier if i explain how i understand things
desert wadi
yea
fluid cosmos
the premise starts off with 1000 squares, where 100 are defnitely correct hypotheses. It says the tests have a false positive rate of 5% and does 5% of 900 to get 45. This makes sense since for something to be a false positive, it has to defnitely be negative but we incorrectly detect it as positive. But this implies that the tests weren't done to the 100 that were actually true which would imply those 100 are known. Then they talk about power so only 80% can be confirmed to be true out of all the true hypothesis, further implying those 100 are known.
The last block then says "not knowing what is false and what is not, the researcher sees 125 hypothesis as true and 45 as not.". Its mainly this part that confuses me.
It makes it sound like it was carried out by a researcher, but the steps done implies they have knowledge over which are true hypothesis and which are not
which makes the whole process redundant because if you knew which were true and false, then there's no point in doing any of this
im trynna figure out if ive misunderstood the premise or if this is a bad example
desert wadi
This is the fact: The researcher see 125 hypotheses as true and 45 as not. Doesnt mean it is true though thats just his pov. That is why its a hypothesis. Im assuming, its based off some sort of probablity.
Am I right to assume
?
fluid cosmos
wdym by its based off some probability?
desert wadi
His hypothesis
Is based off a stat otherwise why would he make the hypothesis
Or some sort of observation
fluid cosmos
its based off the paper "why most published research findings are false"
i assume its based off ideas discussed from the paper
my interpretation of the premise makes it seem nonsensical but i struggle to see it any other way
fluid cosmos
the lecturer doesn't show anything else
he briefly mentions the paper
then talks about the image i sent above
desert wadi
Then I wouldnt worry to much about it unless there a problem about that
What exactly is the problem you just got confused
I mean without a proof
or formula
I would get confused too
fluid cosmos
its the sequential steps that were taken
i thought maybe i was misunderstanding something about the statistics
and how it can be applied
desert wadi
I dont think this is a widespread stat
cuz without a concrete formula
I think its just off of 1000 articles they read
Did I help or nah
fluid cosmos
ig helped in the sense that it doesn't really make sense to you too
so at least im not going crazy
desert wadi
Well It just seems like an observation
fluid cosmos
nah thee's no question about it
he brings it up as an example and leaves it at that
essentialy just read what's said on the image
desert wadi
is there more context
Cuz I dont get one part
It says the "new true"
What does that mean?
fluid cosmos
unfortunately there isn't
he really just reads what's said on the image
doesnt really explain it
desert wadi
Yea idk now im starting to trip over it lol
fluid cosmos
at least its not just me 😭
im just gonna move onto the next lecture
waste of time at this point
pearl pondBOT
@fluid cosmos Has your question been resolved?
pearl pondBOT
pearl pondBOT
Available help channel!
fading ledge
pearl pondBOT
pearl pondBOT
@fading ledge Has your question been resolved?
frozen bluff
have u tried anything?
fading ledge
Actually I didn't understand properly
frozen bluff
ah okay
first look at the equation, can you try rearranging it (move the negative terms to the other side)
fading ledge
5p+6r=9s+2q
frozen bluff
right now the coefficients on each side add up to 11 so we divide by 11
fading ledge
Sure
frozen bluff
you understood why right?
frozen bluff
okay so forget about formulas for a min
on the left you have 5 copies of point P and 6 copies of point R
so in total you have 11 copies right?
so now if i want one single point what should i do?
frozen bluff
yeah that's right
notice that coefficients add up to 1
when the coefficients of position vectors add to 1 the expression represents a point on the line joining those points not just a random vector
pearl pondBOT
@fading ledge Has your question been resolved?
sharp smelt
pearl pondBOT
sharp smelt
The likelihood function is $\prod_{i=1}^{n} \theta^2 x_i e^{-\theta \cdot x_i}$.
Which simplifies to to $ \theta^{2n} \left(\prod_{i=1}^{n}x_i\right) \cdot e^{- \theta \left( \sum_{i=1}^{n} x_i \right)}$
jolly parrotBOT
waimas
sharp smelt
now I have to maximise this wrt theta keeping x_i constant, right
waxen agate
yeah
sharp smelt
Okay, so differentaiting the likelihood function and equatiing it to zero, we get.
$2n \theta^{2n-1} e^{- \theta \sum_{i=1}^{n}x_i} + \theta^{2n} \cdot (-\sum_{i=1}^{n} x_i) e^{- \theta \sum_{i=1}^{n}x_i}=0$
jolly parrotBOT
waimas
sharp smelt
wait
waxen agate
consider using \[ ... \]
sharp smelt
I could also take logs first
sharp smelt
okay, so that gives $\frac{2n}{\theta} - \sum_{i=1}^{n} x_i=0$
jolly parrotBOT
waimas
waxen agate
correct
sharp smelt
so $\theta= \frac{2n}{\sum_{i=1}^{n} x_i}$
jolly parrotBOT
waimas
sharp smelt
Thanks a lot!
waxen agate
sharp smelt
.close
pearl pondBOT
pearl pondBOT
Available help channel!
proper grail
relative direction of the axes is important to describe any co ordinate systems for when describing so, to an object/point that is already in the 3 dimensional world and the situation isn't a mathematician "creating" the world, and both worlds follow properties without the loss of generalization.
toxic lichen
where is your question mate
waxen agate
In a real 3-D physical situation the point/object already exists out there. A coordinate system is just a convention you attach to the world so you can label positions and directions.
also
slow oak
@waxen agate lets not be derogatory
unborn abyss
relative direction of the axes is important to describe any co ordinate systems ...
hey so after yesterday, we have given you all of the help that we can regarding this concern, and as I mentioned you will need to talk to someone in real life about this instead.
.close
frank violet
Write a number on the board, then compute the sum of its digits and write the resultant number on the board, keep doing this until we get the final number less than or equal to 10. It seems to me that if the initial number is 8^{2n} then the final number is 10 and on the other hand if the initial number is 8^{2n+1} then the final number is 8 but I have no idea if that's true and if so how to prove it
frank violet
I guess it has something to do with taking mod 10^k
bitter herald
Hmm, so if you let the number's digits be $S(n)$ then you have [
n\equiv S(n) \q\op{mod} 9
]
jolly parrotBOT
frank violet
hmm
bitter herald
in your case you have [
8^{2n+1}\equiv (-1)^{2n+1} \q(\op{mod} 9)
]
jolly parrotBOT
bitter herald
so the odd case makes sense
same conclusion you can draw with the even case but with ^2n instead
frank violet
Yeah I can see that
bitter herald
Yeah I can see that
thats it really i guess
frank violet
I'm thinking about this
Okay I don't know why this's true
Sorry I'm a bit slow with modular , I just have learnt it for 2 weeks
bitter herald
Sorry I'm a bit slow with modular <:hmmcat:831508801646755850> , I just have lea...
well try to consider some examples
so like
ok try to see if what what sent makes sense if not i can try to give you an actual example
jolly parrotBOT
what
frank violet
ok try to see if what what sent makes sense if not i can try to give you an actu...
Honestly tho
It does lol
frank violet
**what**
This makes much more sense
How did you all come up with this bruh
anyway, Thank you all 
.close
low matrix
Isomorphism question
pearl pondBOT
Isomorphism question
sullen cedar
Look at the degrees of certain vertices.
low matrix
I got an idea, but idk if it is rigoroous enough,
In G1 we have the only vertex of degree three u4, and in G2 we have v_3, the only vertex with degree three.
v_3 has two vertices connected to it, each of those vertices has degree 4.
But u_4 has a vertex of degree 4 and another vertex of degree 1 connected to it. Hence they cannot be mapped on to on another
Hence not isomorphic
ivory basin
That is basically the argument yeah
low matrix
how long did it take you guys to see it btw
ivory basin
If two graphs are isomorphic, they should have the same number of vertices of each degeee
low matrix
i first tried writing the degree sequence
ivory basin
(hope you can see why)
autumn trellis
Shorter would be:
There are two adjacent degree 4 vertices in G1, but not G2.
unborn abyss
If two graphs are isomorphic, they should have the same number of vertices of ea...
(which these do)
ivory basin
(which these do)
Oh shit I miscounted lmfao
unborn abyss
graph isomorphism is NP-hard so it's not an easy thing
ivory basin
Yeah
low matrix
graph isomorphism is NP-hard so it's not an easy thing
how long did it take you to confirm non-isomorphism
unborn abyss
how long did it take you to confirm non-isomorphism
i mean i could see immediately that they weren't, but i hadn't identified an invariant before what mentioned the adjacent vertices thing
low matrix
i mean i could see immediately that they weren't, but i hadn't identified an inv...
what invariant?
sullen cedar
Shorter would be:
There are two adjacent degree 4 vertices in G1, but not G2.
This is the invariant I had in mind as well.
sullen cedar
It's a fancy word for property.
low matrix
I mean I studied 11 other invariants in our class but not this one
sullen cedar
Isomorphic graphs will preserve these kinds of properties.
ivory basin
I miscounted degrees 
ivory basin
I mean I studied 11 other invariants in our class but not this one
Damn which ones
ivory basin
Maybe this violates one of those as well
low matrix
i dont wanna bother checking 💀
sullen cedar
There aren't many circuits here, it's just how the degrees relate with each other.
low matrix
i dont think its a good idea to state an invariant which wasnt ttaught to us in the class
the prof might mind it
low matrix
I got an idea, but idk if it is rigoroous enough,
In G1 we have the only vertex ...
i'll just write this
raw logic
low matrix
i mean i could see immediately that they weren't, but i hadn't identified an inv...
how did you do that btw
sullen cedar
I'm sorry, this is the same style of argument as what you said.
low matrix
like what did you try
sullen cedar
It's just a lot shorther.
low matrix
identical displays?
unborn abyss
i guess i identified the fact that the degree-3 vertex was stuck and i couldn't move it around
low matrix
It's just a lot shorther.
can you state the invariant in general form
unborn abyss
like on one side there's a path that goes 1-4-4-3-1 and on the other that path doesn't exist
sullen cedar
There isn't any invariant here. I'm just literally using the fact that isomorphisms are relabellings of the graph.
low matrix
There isn't any invariant here. I'm just literally using the fact that isomorphi...
oh
cool
sullen cedar
In any case, a relabelling cannot change degrees of adjacent vertices.
vital estuary
degrees of adjacent vertices?
low matrix
This is the invariant I had in mind as well.
whats this then
sullen cedar
I'm just using the word that everyone else is using.
sullen cedar
In any case, a relabelling cannot change degrees of adjacent vertices.
This is as basic as it gets. Just the defintion of isomorphism
low matrix
I'm just using the word that everyone else is using.
so is it an invariant or not 😭
vital estuary
why do you want an invariant so badly?
sullen cedar
It is an invariant, but I don't see why that word label is bothering you so much.
sullen cedar
This isn't some incredibly advanced concept that you can't use if you didn't learn, the whole point of an isomorphisms is that the two graphs are the same upto relabelling, and relabelling cannot change the properties that vertices have in relation with each other (which have nothing to do with relabelling).
Your logic is fine as well, if you are so hellbent on not using the fact that in G1 there are two adjacent degree 4 vertices but not in G2.
pearl pondBOT
pearl pondBOT
Available help channel!
royal galleon
can someone explain to me what he is doing here. If we are trying to use epsilon delta to show that a limit of a quotient is the quotient of the limits but then we would need something like |f(x)/g(x) - A/B | < epsilon whenever 0<|x-p|< delta. but here we first look at f(x)/b and b/g(x) which makes snese because we want to use the product rulel which we had just proven. But then he says we want to show that b\g(x) goes to 1. I don't see how this does anything. Next he says to do this we need assume that h(x) = g(x)/B and that lim x-> p h(x) = 1. How he gets this I don't know and if we are assuming stuff why not just assume lim x->p b/g(x) = 1
little merlin
i suppose B is the limit of g as x approaches p
the purpose of doing this is just to simply calculations
if you want to you can still just show 1/g tends to 1/B as x approaches p
the only difference would be the presence of absolute value of B in the denominator
and the upper bound of 1/g will in terms of B
all of this will just make the proof look a little uglier imo
royal galleon
if you want to you can still just show 1/g tends to 1/B as x approaches p
why? we already assumed that lim x-> p f(x) = A and lim x-> p g(x) = B
little merlin
why? we already assumed that lim x-> p f(x) = A and lim x-> p g(x) = B
you want to show f/g tends to A/B using product rule
so
f/g=f*(1/g)
royal galleon
yes that the only step I understand of his proof
little merlin
g tends to B as x approaches p is true (it's the hypothesis) but that doesnt immediately prove 1/g tends to 1/B
royal galleon
how are we allowed to have B/g(x)
little merlin
f/g=(f/B)*(B/g)
B is just a real number
like i said
you dont have to proceed like this
you can just prove either B/g tends to 1 or 1/g tends to 1/B
royal galleon
ok so then we have B/g(x) approach g(x) = B so 1
also how do we know that 1/h(x) appraoches one there is nothing to prove
h(x) beocmes 1
royal galleon
h(x) = g(x)/B tho?
little merlin
and you know g tends to B as x approaches p
royal galleon
oh wait nvm
royal galleon
yes
little merlin
now you have to prove that 1/h tends to 1 as x approaches p
little merlin
(1/B)*g tends to 1
and you already know this
can you see where this is going
you can choose an epsilon that gives an upper bound on 1/h
in a delta neighborhood
royal galleon
ok so I see for 3.14 that we set up our epsilon condtiion and then our deltla condition. 3.15 is just rewriting. now for 3.16 where is |h(x)-1| <1/2 coming from?
little merlin
let epsilon equal to 1
you can do that because you know the limit h(x) as x tends to p exists
royal galleon
but we need it to exist for all epsilon if we set epsilon to 1 how do we know this will still be the case?
or you saying that since we have proven it for a general epsilon
little merlin
but we need it to exist for all epsilon if we set epsilon to 1 how do we know th...
you already know |h(x)-1|< epsilon for all epsilon
this follows from the product rule
royal galleon
wait can you explain again how we can sub in epsilon /2 again?
little merlin
what you have to show for arbitrary epsilon is |1/h(x)-1|< epsilon
royal galleon
what you have to show for arbitrary epsilon is |1/h(x)-1|< epsilon
yes
little merlin
wait can you explain again how we can sub in epsilon /2 again?
in the inequality it does not matter if you multiply epsilon with some positive constant since the inequality is true for all positive values of epsilon
the thing is you can actually just let epsilon be itself and choose epsilon= 1/2 in the next step
royal galleon
but wouldn't epsilon/2 < epsilon always and that doesn't imply that |something| < epsilon /2
little merlin
you have show |h(x)-1|< epsilon to be true for all epsilon>0
if you multiply epsilon by a positive constant it is still positive
royal galleon
so we don't actually care if it is epsilon in the sense that it is a fixed value
since we could get all the values of plain epsilon by increasing epsilon in epsilon /2
little merlin
yes exactly
ok so you understand what is going in 3.16, right?
let epsilon be equal to 1 then by definition of limit there will be some delta neighborhood of p for which |h(x)-1|<1/2
after that you can apply triangle inequality
btw @royal galleon the reason why epsilon/2 is a more natural choice over just epsilon becomes obvious when you combine everything from 3.16 with 3.15
royal galleon
ok so you understand what is going in 3.16, right?
no
he doesn't use the triangle inequality he some how gets h(x)>1/2
little merlin
reverse triangle inequality
then there are some basic theorems involving inequalities in absolute values
use c after applying reverse triangle inequality
royal galleon
so our reverse triangle inequality gets us | |h(x) | - |1| | <= |h(x) - 1|
but where does this get us?
little merlin
| |h(x) | - |1| | < 1/2
royal galleon
-1/2<|h(x)|-1<1/2
what is your a?
little merlin
no its not for products