Ok I just managed to make the matrix way more sparse
#help-26
dawn shadow
for (Volunteer v : volunteers) {
GRBLinExpr assignmentConstraint = new GRBLinExpr();
for (Task t : v.allowedTasks) {
assignmentConstraint.addTerm(1.0, x_vt[v.number][t.number]);
}
model.addConstr(assignmentConstraint, GRB.LESS_EQUAL, 1.0, "AssignmentConstraint_" + v.number);
}
Instead of going over 750 tasks, I go over v.allowedTasks.size()
strange whale
yeah that should be much nicer
now the suitability constraints should still be a bit of pain to add
dawn shadow
Im going to test out addTerms now
dawn shadow
yeah that should be much nicer
I did this for all 7 constraints
strange whale
ah so you can also use the sparse task matrix for the suitability constraints ?
dawn shadow
yes
I went from 45sec -> 35sec for a smaller instance with 5k volunteers
So it definitly helped
strange whale
like the naive way I see of modeling that constraint is $$\sum_{t\in \text{NotSuitableFor(v)}} x_{vt} = 0$$
thorny flameBOT
aPlatypus
strange whale
so you'd still have a whole bunch of terms in there
dawn shadow
Oh but the problem is way more complex then that
dawn shadow
You dont have to be suitable, a certain % of the volunteers that execute the task have to be suitable.
strange whale
it's not like you told me much lol
ah ok so you can have multiple volunteers on a single task
that's what I didn't get
dawn shadow
yes
Lets say the task has a demand of 10, then it can have anything below or equal than 10 volunteers assigned to it.
0 is also fine
Im just wondering how I can apply addTerms here without breaking the constraint:
for (Volunteer v : volunteers) {
GRBLinExpr assignmentConstraint = new GRBLinExpr();
for (Task t : v.allowedTasks) {
assignmentConstraint.addTerm(1.0, x_vt[v.number][t.number]);
}
model.addConstr(assignmentConstraint, GRB.LESS_EQUAL, 1.0, "AssignmentConstraint_" + v.number);
}
yeah with a sparse matrix it's prolly hard to use addTerms
you'd have to make the list of variables which should be together yourself
and idk if that would be much quicker
dawn shadow
Yep I am not sure how to accomplish that yet
strange whale
but hey we're here to try stufff
dawn shadow
Do you use other solvers often?
strange whale
I don't use them much
the most complex thing I did on my own was screwing around with Z3 to solve some CTF tasks
but I've had a few classes on optimization so I know the terminology and all, haven't used solvers a lot for actual problems tho
I mean your solver should take care of all that
they're outsourcing the assignment to dutch(?) CS labs damn
dawn shadow
Im Belgian
dawn shadow
The data is fully anonymized but the locations etc are all real
strange whale
there's certainly a french team working on it I hope
but hey whatever
as long as we have good assignments in reasonable time
dawn shadow
as long as we have good assignments in reasonable time
Well there are basically no rules on how we have to solve it, most students made a constructive heuristic to make their initial feasible solution
But I just did it with gurobi
I just double checked and that sparsity literally cut out 50% of the matrix
strange whale
I mean you could feed an initial solution to gurobi I suppose
strange whale
I just double checked and that sparsity literally cut out 50% of the matrix
I thought it would be much more tbh
but hey 50% is 50%
dawn shadow
I mean you could feed an initial solution to gurobi I suppose
The problem is that we get a certain amount of time to create a solution.
Lets say you get 20 minutes, the student who gets the best objective (being feasible ofc), wins
strange whale
ah yeah so it's not like "just give us a solution at this date, got all the time you want"
dawn shadow
ah yeah so it's not like "just give us a solution at this date, got all the time...
Nope they will let everyone's program run for X amount of minutes on their server
strange whale
well ofc don't spend 3hours of computer-time making a good first heuristic
dawn shadow
I can make a feasible solution for the big instance within 2 minutes, but I am not sure how I want to optimize it
strange whale
also, do you really need to define variables for impossible (volunteer, task) assignments ?
that's would cut a damn lot the number of variables in your model
maybe you already did that idk
dawn shadow
Yep currently I still do it like this:
GRBVar[][] x_vt = new GRBVar[volunteers.size()][tasks.size()];
The volunteers.size() is fixed, but instead of tasks.size() I actually need the volunteer it's amount of allowed tasks
Not sure how to do that easily tho
strange whale
yeah the task size would depend on the volunteer
you can't do the whole array at once
strange whale
I certainly hope you can
that would be weird that you can't add variables as you want to
dawn shadow
Its because only a % of a task needs to meet the criteria
strange whale
ah yeah right
dawn shadow
volunteers only get cut out because of locations / availability in that 50%
strange whale
yeah the task size would depend on the volunteer
and if you do that you could just use addTerms, there would be no indirection left like you had to do here assignmentConstraint.addTerm(1.0, x_vt[v.number][t.number]); with the v.number and t.number
dawn shadow
Ill try it out.
And let you know
I make the full square matrix, but it only optimizes the variables I add in the for loop
topaz sinewBOT
@dawn shadow Has your question been resolved?
strange whale
yeah you have a huge ass amount of variables which don't even exist in your array now, so it's a bit weird @dawn shadow
you do you
shy obsidian
I have no idea how to find theta with number 7
noble heart
Parallel to the axis of the cylinder, the height of which is greater than the diameter of the base, a section parallel to the axis of the cylinder is made. The area of the section is 6 cm^2, and its perimeter is 10 cm. Find the height of the cylinder and the radius of its base, if the section cuts arcs of 120 degrees from the bases of the cylinder.
topaz sinewBOT
Parallel to the axis of the cylinder, the height of which is greater than the di...
topaz sinewBOT
@noble heart Has your question been resolved?
noble heart
I don’t even roughly understand how to solve this problem
noble heart
<@&286206848099549185>
winged sapphire
Do you have a picture of the cylinder?
noble heart
Do you have a picture of the cylinder?
but I could be wrong because I don’t really understand how to draw it
winged sapphire
Well. Uh. I’m sorry I don’t exactly know how to help with this.
noble heart
🥲
topaz sinewBOT
@noble heart Has your question been resolved?
noble heart
.close
manic swan
Last exercise, what is exactly mean?
topaz sinewBOT
Last exercise, what is exactly mean?
manic swan
I mean, idk what is this2 numbers below
What i know is how to do multiple btw 2 IEEE754 but it's bot write like that (4280000) ..
.close
topaz sinewBOT
topaz sinewBOT
Available help channel!
solemn jolt
I understand what groups are and I also understand abelian groups. but how do I go about solving this question?
topaz sinewBOT
I understand what groups are and I also understand abelian groups. but how do I ...
vale furnace
consider (ab)^2
solemn jolt
aye bloodborne guy whats up
vale furnace
solemn jolt
xD
vale furnace
finshing elden ring
solemn jolt
and we are trying to use the fact that G is a group and G fulfils the four criteria for being a group to prove that it is also commutative, right?
vale furnace
correct
vale furnace
consider any two arbitary elements in G, say a and b
vale furnace
what platform?
ps5
solemn jolt
I just learned this stuff today so i'm not that familiar with it
solemn jolt
consider any two arbitary elements in G, say a and b
ok got it
vale furnace
by the axiom of groups it is close under the operation so we have a∘b is also in G
solemn jolt
ps5 <a:sky_evilelmoOwO:808717482394124308>
oh bro send me ur psn I finished the game on pc but im still early game on ps
solemn jolt
by the axiom of groups it is close under the operation so we have a∘b is also in...
true
vale furnace
oh bro send me ur psn I finished the game on pc but im still early game on ps
im pretty far in the game tho like 100+ hours in
solemn jolt
im pretty far in the game tho like 100+ hours in
damn then i gotta catch up.
yes
cuz two like elements in an operation give the neutral element
right?
vale furnace
yes it gives the identity
my bad that should be e not 1
so from (a∘b) ∘ (a∘b) = e we can do ∘ b from the left
so (a∘b) ∘ (a∘b) ∘b = e ∘b
solemn jolt
wait we can just treat this random operation as if it were multiplication?
vale furnace
when did i do that?
vale furnace
yeah its just applying the operation on both sides from the right side
solemn jolt
ah ok
vale furnace
can you simplify this (a∘b) ∘ (a∘b) ∘b = e ∘b
solemn jolt
since it's associative, we can just write it as a∘a∘b∘b∘b=e∘b
then itwould be e∘e∘b=e∘b
vale furnace
wait thats not right we can't just swap a and b since we dont know yet if this group is commutative
solemn jolt
oh yeah
vale furnace
more clearly we can say (a∘b) ∘ a∘(b ∘b) = e ∘b due to associativity
vale furnace
(a∘b) ∘ a∘(b ∘b) = e ∘ b then means ?
vale furnace
now do you see how to get to what we want which is a∘b=b∘a
vale furnace
whats the inverse of a?
vale furnace
yes
solemn jolt
cuz a∘a = e
vale furnace
so just do ∘a on the right side
vale furnace
yeah then it just comes out
vale furnace
lol its like a common question for starting off group theory
solemn jolt
still tho
vale furnace
later on you'll prove G is abelian iff there exists an automorphism f mapping a to a^-1 and you an use that to prove this pretty easily
but thats in the future
solemn jolt
automorphism? idek what that means
vale furnace
well youll learn about the phisms later then 
solemn jolt
phisms? sounds like something out of a soulsborne game
radiant tapir
automorphism? idek what that means
It's not as scary as it sounds 🙂
solemn jolt
and we all know what things that sound like stuff out of soulsborne games do to people
solemn jolt
It's not as scary as it sounds 🙂
thanks for the reassurance
vale furnace
vapid lichen
well youll learn about the phisms later then <:kekw:1037875930166743080>
phism?? 
solemn jolt
<:bleakkekw:913344117981999125>
ok thanks for the help bye 🙏
vale furnace
vapid lichen
@solemn jolt @vale furnace btw its morphism
vale furnace
wdym did i mispell it
vapid lichen
morphism not phism
vale furnace
ohhh 
vapid lichen
THEN you add on prefixes iso endo auto
jaunty dawn
hey, how do i proof that lim sup = 1+e, lim inf = 1-e?
topaz sinewBOT
hey, how do i **proof** that lim sup = 1+e, lim inf = 1-e?
jaunty dawn
I already showed that a_k converges to 1+e if k is even and to 1-e if a is odd. But how do i proof generally that a function doesnt have a greater accumulation point?
topaz sinewBOT
@jaunty dawn Has your question been resolved?
opal vault
you can show that a_(2n) is increasing and a_(2n+1) is decreasing
one is easier than the other
jaunty dawn
I tried it for a_(2n) but kept failing. But if this is the way, i do atleast know what to do. thanks 🙂
topaz sinewBOT
@jaunty dawn Has your question been resolved?
topaz sinewBOT
versed lily
anyone want to explain Accurately solve using routine/non-routine operations the approximate percentage error in the time-period calculation if your measurement of length is 3% high and G is measured 2% too small.
topaz sinewBOT
Available help channel!
delicate pike
Hello
delicate pike
Hello i need equation for this shape
here is more information on this
https://en.wikipedia.org/wiki/Harmonic_function
vale furnace
its laplace's equation
delicate pike
its laplace's equation
How would i find out its X Y Z equations
vale furnace
looks like an annulus with outer radius 4 and inner radius like 2
delicate pike
what would i have to type in that graph above to get this result ?
@vale furnace hope u dont mind the tag
topaz sinewBOT
@delicate pike Has your question been resolved?
delicate pike
anyone ?
topaz sinewBOT
@delicate pike Has your question been resolved?
quick blaze
I’m really confused in how to approach question 21.2 parts a,b,c I’ve done work for all I just don’t know the correct way to do it. All the online resources point towards row reduction but Stanford doesn’t teach us that
polar torrent
hey @quick blaze you still around?
For (a) and matrix A - we are working with 2-vectors, so the nullsapce of A must be a subset of $\mathbb{R}^{2}$.
thorny flameBOT
pencenter
polar torrent
For part (b), we know that the solution to $A\mathbf{x}=\mathbf{0}$ will form the nullspace. Multiplying that out gives the system of equations you need.
thorny flameBOT
pencenter
polar torrent
For part (c), solving this system I got $Null(A) = \left{ (2,1)t: t\in \mathbb{R}\right}$
thorny flameBOT
pencenter
topaz sinewBOT
@delicate pike Has your question been resolved?
neon iron
find the angle between the line and the plane (not intersecting nor parralel)
neon iron
Line: $(3,0,0) + t(1,1,2)$
thorny flameBOT
Derivative
neon iron
Plane: $2x - y + 3z = 5$
thorny flameBOT
Derivative
topaz sinewBOT
topaz sinewBOT
Available help channel!
glossy osprey
topaz sinewBOT
glossy osprey
I use a ti84 calculator to solve this but I can’t seem to get the right pval answer
I have the answer but they don’t match to what I get
waxen flame
it might depend on the specific test you use
topaz sinewBOT
topaz sinewBOT
Available help channel!
feral crypt
I need help
topaz sinewBOT
I need help
feral crypt
Ok so basically
we are doing 3D trig
anad like how does this make sense
when you are doing cos45 = (d/27.4) you'd turn it into 27.4cos45=d
where did the 1/sqrt2
come from
also this is an answer sheet
nvm
I just realised that I'ma dumbass
google didn't put it into degrees for me
fiery edge
Need some help with quaternions for a personal project
topaz sinewBOT
Need some help with quaternions for a personal project
fiery edge
I have a point K that gets rotated to a point S, and I need to find the quaternion Q that does that rotation
I know that (Q * K) * Q' = S, and I want to find Q knowing K and S
I'm doing this for fun, so I'm largely interested in the theory behind how to handle such a problem.
<@&286206848099549185>
pastel salmon
Well, this goal can also be achieved in the following way. Let's place our three-dimensional space inside H - the non-commutative field of quaternions on the last three axes, i.e. as a set of "purely imaginary" quaternions:
$x_{2}\overrightarrow{i}+x_{3}\overrightarrow{j}+x_{4}\overrightarrow{k}$
thorny flameBOT
Joanna Angel
fiery edge
I was thinking there was just some rule about quaternion multiplication that I'm missing, namely how to rearrange the algebra from the left
pastel salmon
and let vector w wil be :
$w_{2}\overrightarrow{i}+w_{3}\overrightarrow{j}+w_{4}\overrightarrow{k}\text{ whose length is 1, i.e. the unit vector}$
thorny flameBOT
Joanna Angel
pastel salmon
indicating the axis of rotation that we want to implement and let
$\theta\text{ will be the angle at which we want to rotate the space}$
thorny flameBOT
Joanna Angel
fiery edge
I'm sorry, I don't see what vectors have to do with this. I want to be thinking about quaternions as quaternions, points in 4D space. In fact, you can ignore what I said about rotation. I want to find a way to rearrange Q * K * Q' = S, to get Q in terms of K and S. Basically, I don't know how quaternion algebra works.
I'm more curious about how to do algebra with non-commutative operations
pastel salmon
i thought you wanted to know the formual of quaternion
fiery edge
No, I know how to do that.
pastel salmon
which helps you to raotate
so let me change th approach then
first you used geometrical terms
so if we tlak ab onyl quaterniosn lets forget
ab geometry
fiery edge
Yeah, I just wanted to provide some context for what this is ultimately going to be used for, but I suppose it isn't relevant
(Q * K) * Q' = S.
Since multiplying by the inverse of a quaternion is the same as dividing by that quaternion, then that should mean that
(Q * K) / Q = S.
Which should also mean that
Q * K = S * Q
Am I correct?
(And apologies for not using LaTeX, I'm not extremely familiar with it)
pastel salmon
it's ok, yes i may agree on it
but let me show you oen thing:
i want to show certain transformation from H toH where quaternion is
$q=cos\frac{\theta}{2}+sin\frac{\theta}{2}\left( w_{2}\overrightarrow{i}+w_{3}\overrightarrow{j}+w_{4}\right)$
thorny flameBOT
Joanna Angel
pastel salmon
and
$q=cos\frac{\theta}{2}+sin\frac{\theta}{2}\left( w_{2}\overrightarrow{i}+w_{3}\overrightarrow{j}+w_{4}\right)$
thorny flameBOT
Joanna Angel
pastel salmon
sorry
fiery edge
I'm aware of how this construction works, but this is for converting a point in 3d space to axis-angle representation in quaternions. I watched all the 3 blue 1 brown videos
pastel salmon
$x->q\cdot x\cdot q^{-1}$
thorny flameBOT
Joanna Angel
fiery edge
I'm looking for a formula of Q, a quaternion, in terms of K and S, which are also quaternions
fiery edge
No, I don't see how this is relevant, sorry
q isn't a point that I'm converting from, it's a quaternion that represents a rotation from K to S
pastel salmon
maybe also , the thign that disturbs me lil bit is my copign with virtual reality wher ei also sue roations as quaternions haha
forgiv eme
fiery edge
That's fair
Still need a formula for Q in terms of K and S
Is there a way to divide from the left?
Q * K * Q' = S
or rather the middle, i guess
pastel salmon
if you get a formula for k, that makes you happy ? just want to make sure )
we wud nd to check
fiery edge
for Q, in terms of K and S
You can also write that (Q * K) / Q = S, if that's helpful
fiery edge
Q * K = S * Q
fiery edge
S'?
pastel salmon
Q' or -Q' sin
fiery edge
So
Q' * Q * K = Q' * S * Q
?
fiery edge
Then the two Qs on the left cancel out and you get K = Q' * S * Q
pastel salmon
yes
fiery edge
I think they do cancel out
fiery edge
Since multiplication by Q' is equivalent to dividing by Q, that means Q' * Q = Q' / Q'
pastel salmon
yes
fiery edge
and Q' / Q' should be the identity quaternion, so we can eliminate that.
But now we just have K = Q' * S * Q
pastel salmon
you know , in yoru palce i wud try to prove it
inverse fomrula fo rQ is known
i can write it if needs
but next
yoru offe rof the new formual shud be proved
fiery edge
We can prove that by substituting K back in
pastel salmon
well yes
fiery edge
K = Q' * S * Q into
Q * K * Q' = S yields
Q * Q' * S * Q * Q' = S which should be
(Q * Q') * S * (Q * Q') = S
1 * S * 1 = S
S = S
pastel salmon
$q=a+bi+cj+dk\text{, then: }q^{-1}=a-bi-cj-dk$
fiery edge
Checks out
thorny flameBOT
Joanna Angel
pastel salmon
so in such way i wud try to prove it or check it
because oyu can t make proof
if you use proved assumtpion )
or how to say
fiery edge
Hm.
pastel salmon
hence were my concerns
fiery edge
No, i get what's happening here
fiery edge
If we plug in 1 for all the coefficients we get that Q * Q' = 4, and also that Q' * Q = 4.
Which might seem like a problem, but we have to consider that the magnitude of both Q and Q' is 2.
If you do this with unit quaternions then I think everything cancels out, which is perfectly fine since I'm only working with unit quaternions.
And yes, then it checks out
👍
fiery edge
can you say that again? there's a few typos in that message
can't understand what temotign means lol
pastel salmon
tempting
pastel salmon
tha is not my area in maths but i may evnetually do something in it
you may also treat quatrnions
as a pair of complex numbers
it can simply soemtimes
soem calcualtions
pastel salmon
yes tryign to write the multiplication in it
$\left( a,b \right)\cdot \left( c,d \right)=\left( ac-b\overline{d},ad+b\overline{c} \right)$
thorny flameBOT
Joanna Angel
fiery edge
I feel like we're a little off track here
Is there no way to just... pull K out from the middle?
Like, we have one equation and only one unknown here, this has got to be solvable
Yknow what might help?
If there's a way to relate Q x P and P x Q
Some way to flip the multiplication around
that would pretty much solve the problem
pastel salmon
yes 🙂 hence i prepare "weapons" to hit on it , such a brain-storming
pastel salmon
we need to create a key to enter the door
fiery edge
I don't think there's going to be a simple, direct construction for it, but i may be wrong
it's going to flip the coefficients of part of the k and i components, but not all of them
that's interesting
I don't know how to explain that part very well, it's just intuition
pastel salmon
yes , non-commutativity is not a nice thing
fiery edge
agonizing, really
pastel salmon
since we get to used to live in real world which is commutaitve moslty )
oru habits etc
fiery edge
not really! if you're putting on your socks and then your shoes that's not the same as putting on shoes and then socks
pastel salmon
but ppl often wear left sock on right, they cant see the difference, , haha it is metaphore
fiery edge
can't really explain how my mind got to it but this matrix out of it and you might be getting to it somehow
1 1 1 1 1 1 -1 1 1 -1 1 -1 1 1 -1 1
pastel salmon
yes there is matrix interpreation of quaternions
fiery edge
this relates the outer products of them, i believe
pastel salmon
ok
macierz przejścia, in my native lang, let me thikn how to say it in eng
transition matrix from base to another base
fiery edge
wait.... wait wait wait
i like where this is going but taking a step back for a second
presumably there's some operation O that we can perform on (P x Q) to get (Q x P)
we could do this on both sides, O(Q x K) = O(S x Q)
which would mean we get K x Q = Q x S
Is that true?
I know i'm getting ahead of myself
but still
fiery edge
my bad, yes
Wait, no, this is wrong
because it means K = Q x S x Q'
which is wrong, because it's the other way around
alright, my bad
pastel salmon
🙂 ok
fiery edge
also I apologize but it's nearly midnight here, I will have to stop soon
fiery edge
I still have some time if you want
but we should go faster
if possible
alright, I guess we're done for the night
.close
topaz sinewBOT
topaz sinewBOT
Available help channel!
neon iron
I'm having difficulty in trying to find a way to approach solving this recursive function. I thought of approaching it as a difference equation, but the form does not really fit. or does it? I don't really know.
topaz sinewBOT
I'm having difficulty in trying to find a way to approach solving this recursive...
neon iron
by the way, the values for time and Pgas are controlled by a data set I've made.
topaz sinewBOT
@neon iron Has your question been resolved?
topaz sinewBOT
topaz sinewBOT
Available help channel!
slender pulsar
A={1,2,3.......99,100}
B={{x∈A: a|x } : a∈N}
how do i know what objects are in group B and how many singletons does it have?
topaz sinewBOT
A={1,2,3.......99,100}
B={{x∈A: a|x } : a∈N}
how do i know what objects are in g...
vale furnace
Think about when a<=50 and a>50
slender pulsar
so its an empty group whenever a>50?
slender pulsar
oh
vale furnace
It would be empty if a>100 because then a doesn’t divide anything in A
But if you want to find singletons then think about what happens when a>50
slender pulsar
oh so whenever a>50 its a singletone of the number and when its smaller its all the numbers that devide it?
vale furnace
When a>50, a will only divide one number in the set A.
For example, if a = 51 then a|51 and the next number a will divide is 102 which in not in A.
so when 100>=a>=51 then we have singletons {a} in B
vale furnace
slender pulsar
.close
obsidian yew
how would I do this?
topaz sinewBOT
how would I do this?
sweet shard
it says "determine an equation to model the relationship"
Start by plotting
obsidian yew
bro idk how
sweet shard
bro idk how
topaz sinewBOT
@obsidian yew Has your question been resolved?
drifting bough
@frosty belfry in your solution, why is 3k^2 and not 3k^3?
frosty belfry
must be a typo
sage dust
sage dust
!status
topaz sinewBOT
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
pastel salmon
- 6 is wrong in denominator
and
you forgot ab parenthesis in nominator
plz correct it next when you write it down , then consider the factorisation of the trinnomial
sage dust
+ 6 is wrong in denominator
why?
pastel salmon
because you shud write x + 2 - (3x + 4)
lime vine
Numbers from 1 to 25 are placed arbitrarily on a circle. Prove that there exist 3 numbers next to each other whose sum is >= 41, and there exist 3 numbers next to each other whose sum is <= 37.
topaz sinewBOT
Numbers from 1 to 25 are placed arbitrarily on a circle. Prove that there exist ...
@lime vine Has your question been resolved?
lucid tiger
i dont understand the second last step on how to solve this
topaz sinewBOT
i dont understand the second last step on how to solve this
lucid tiger
Specifically this step
sweet shard
Specifically this step
do you know $cis = \cos + i\sin$
thorny flameBOT
riemann
sweet shard
so plug in pi/6 to the right side
lucid tiger
so plug in pi/6 to the right side
so pi/6 + i pi/6?
sweet shard
so pi/6 + i pi/6?
no
thorny flameBOT
riemann
topaz sinewBOT
topaz sinewBOT
Available help channel!
upbeat flume
hi
turbid fractal
type your queston
ancient dagger
Are you familiar with how exponents/squares are computed?
ancient dagger
You’re all good
the little 2 up there just tells you how many times you should multiply n by itself
So n^2 = n x n
upbeat flume
whats the value for n
ancient dagger
So they’re asking you to compute the first 5 numbers in the sequence
sullen sentinel
solve for n^2 + n using the values 1-5
ancient dagger
So real
sullen sentinel
did I do something wrong
ancient dagger
No I use the exclamation as an emphasis
upbeat flume
WHAT
sullen sentinel
oh
ancient dagger
You got it Kavish
upbeat flume
whats the value for N
sullen sentinel
lets look at (a)
the formulas that are listed are what you will use to solve for the nth term of the corresponding sequence
n^2 + n
if u want the first term of the sequence
u sub in 1 for n
and calculate
sullen sentinel
be careful it isn't nx2 it is n^2
sullen sentinel
n^2 means nn not n2
sullen sentinel
and then +1 right
upbeat flume
=1
sullen sentinel
basically u have n^2 + n, and u sub 1 for n so u get (1)^2 + (1)
upbeat flume
so it starts with 1
so that first and second (1) will change to 2
and then 3
then 4
and 5
sullen sentinel
wait where are you getting 3
upbeat flume
uh
sullen sentinel
ok let's regroup what we know
sullen sentinel
(1)^2 + (1)
upbeat flume
=3
sullen sentinel
solve this
upbeat flume
2
upbeat flume
yes
sullen sentinel
then for the second term we have to sub in 2 for n
so now n^2 + n becomes (2)^2 + (2)
so what is the second term
upbeat flume
6/
sullen sentinel
yes
sullen sentinel
so continue this process till u get the first 5 terms
upbeat flume
okk
sullen sentinel
lets do the 3rd term
and u do the rest
what do u think the 3rd term is
for (a)
sullen sentinel
for all of these equations u need to calculate the term values by subbing all the n values of 1-5
upbeat flume
im smartt
sullen sentinel
correct
sullen sentinel
so the answer for question (a) is 2, 6, 12, 20, 30
same process for the rest but the equation changes
upbeat flume
thank you sir
sullen sentinel
u got it
sullen sentinel
np
hasty vine
guys I need help with 3) answers for 1 and 2 are in pink
topaz sinewBOT
guys I need help with 3) answers for 1 and 2 are in pink
hasty vine
here's how I did 3) so far, not sure if it's good
topaz sinewBOT
@hasty vine Has your question been resolved?
hasty vine
<@&286206848099549185>
lone pier
Were you after a graphical answer?
Do you have "dot product" is your workbook?
hasty vine
I do have it
yea I was after a graphical answer
like I'm not sure what direction the arrows should be pointing
so I haven't added them yet to the graph
topaz sinewBOT
@hasty vine Has your question been resolved?
hasty vine
<@&286206848099549185>
golden ferry
topaz sinewBOT
golden ferry
hello, i dont know how to attack this question. could i be shown how to do this, or be provided with a link to a video that could explain this?
craggy haven
i see "any tangent line to the curve"
take an arbitrary point on that curve and calculate the eqn for the line tangent at that point
golden ferry
i see "any tangent line to the curve"
take an arbitrary point on that curve and ...
okay
golden ferry
i see "any tangent line to the curve"
take an arbitrary point on that curve and ...
so i put that sqrt(x) + sqrt(y) = sqrt(c) formula in desmos and it says i have to choose a value for c
wooden osprey
c is just a constant
craggy haven
well yeah, or you can add a slider
wooden osprey
it can be any positive integer
golden ferry
so should i say in my answer "for any c>0" or somehting?
but i guess the problem is idk how to choose a point
bruh
because the point depends on what c is
or is c the midpoint of the graph?? idk
@wooden osprey @craggy haven ^how would i choose an arbitrary point? should i just do like [1,2]
or is there a way to specifically find a point
craggy haven
you would just say "for point (x, y)" or you could give it some other names
but! you can pick a value for c, and then try a point on that curve, if it helps you see that the theorem is true, and maybe gives you some inspiration
I'd recommend picking a square number for c, like 9 or 25
golden ferry
tyvm🙏🏾🙏🏾
topaz sinewBOT
@golden ferry Has your question been resolved?
golden ferry
@craggy haven @wooden osprey so i tried at it here, i tried to find the tangent line by finding the slope first then plugging it into that y-y1 formula
is this looking any good? and how can i prove that the sum of the intercepts is c?
but the intercepts don’t exist on the graph😭unless i’m getting everything wrong
golden ferry
WHOOPSbruh
golden ferry
derivative of sqrt(c) isn't 1/2 c^-1/2
is this better?
cyan mesa
no
cyan mesa
second line
$0=\frac{1}{2\sqrt{x}}+\frac{y'}{2\sqrt{y}}\to\frac{y'}{2\sqrt{y}}=-\frac{1}{2\sqrt{x}}$
thorny flameBOT
Combustion
cyan mesa
$\frac{y'}{2\sqrt{y}}=-\frac{1}{2\sqrt{x}}\to y'=-\frac{2\sqrt{y}}{2\sqrt{x}}$
should be like this
thorny flameBOT
Combustion
golden ferry
ok hold on im trying to wrap around my head the 2 sqrt(x) on the bottom, ik its same as saying something i have written down but im tryna figure out exactly what
betttttttt
golden ferry
should be like this
is 3rd time the charm for getting tangent line eqn ?
golden ferry
so it should be -2
🙏🏾🙏🏾
cyan mesa
cyan mesa
you can prove it without plugging in numbers btw lol
golden ferry
bruh actually😭how so ??
golden ferry
🙏🏾🙏🏾
cyan mesa
the tangent should look like this $y=y'\left(x-x_{0}\right)+y_{0} \\$
subbing in y' gets us this $y=-\frac{\sqrt{y_{0}}}{\sqrt{x_{0}}}\left(x-x_{0}\right)+y_{0} \\$
first let's find the x intercept aka $ y = 0 \\$
$0=-\frac{\sqrt{y_{0}}}{\sqrt{x_{0}}}\left(x-x_{0}\right)+y_{0}\to\frac{\sqrt{y_{0}}}{\sqrt{x_{0}}}\left(x-x_{0}\right)=y_{0} \\$
$\frac{1}{\sqrt{x_{0}}}\left(x-x_{0}\right)=\sqrt{y_{0}} \\$
$x-x_{0}=\sqrt{x_{0}}\ \sqrt{y_{0}}\to\ x=\sqrt{x_{0}}\sqrt{y_{0}}+x_{0}$
thorny flameBOT
Combustion
cyan mesa
now that we found the x intercept, we should find the y intercept at $ x =0 \\$
$y=-\frac{\sqrt{y_{0}}}{\sqrt{x_{0}}}\left(0-x_{0}\right)+y_{0} \\$
$y=\sqrt{y_{0}}\left(\frac{x_{0}}{\sqrt{x_{0}}}\right)+y_{0} \\$
$y=\sqrt{y_{0}}\sqrt{x_{0}}+y_{0}\\$
that's the y intercept, now we find the sum of the x and y intercept $\\$
$y+x=\sqrt{x_{0}}\sqrt{y_{0}}+x_{0}+\sqrt{y_{0}}\sqrt{x_{0}}+y_{0}\\$
$y+x=x_{0}+2\sqrt{x_{0}}\sqrt{y_{0}}+y_{0}\\$
$y+x=\left(\sqrt{x_{0}}+\sqrt{y_{0}}\right)^{2}=c$
thorny flameBOT
Combustion
golden ferry
the tangent should look like this $y=y'\left(x-x_{0}\right)+y_{0} \\\\$
subbing ...
that first line says the tangent line should look like that, is that according to a specific rule? if so which one?
cyan mesa
🙏🏾🙏🏾
It’s the tangent line equation
You even used it in your image
rather than y - y_0 = y’(x-x_0) I just moved the y_0 to the right
golden ferry
rather than y - y_0 = y’(x-x_0) I just moved the y_0 to the right
bet
topaz sinewBOT
@golden ferry Has your question been resolved?
topaz sinewBOT
topaz sinewBOT
Available help channel!
manic skiff
Is there any method to find the set of integer that its sum is 17 and its squares sum is 87.
topaz sinewBOT
Is there any method to find the set of integer that its sum is 17 and its square...
manic skiff
We can write like this: find set of integer (x_1, x_2,..., x_n) so that Σx_n=17 and Σ(x_n)^2=87.
thorny flameBOT
Disorganized
hazy pumice
notice that the left formula is a factor of the right formula
if we can make this substitution and solve for n, and n is in the Naturals, then there is such a number n for which this is true.
manic skiff
It works for n consecutive number.
hazy pumice
It works for n consecutive number.
I don't understand your statement.
$\frac{n(n+1)(2n+1)}{6} = 87 \longrightarrow 17\cdot\frac{(2n+1)}{3} = 87$
thorny flameBOT
Disorganized
hazy pumice
not looking good
hazy pumice
Is there any method to find the set of integer that its sum is 17 and its square...
is the original problem talking about
the sum of consecutive integers being equal to 17
and the sum of consecutive squared integers being equal to 87?
manic skiff
For example, {1,5,5,6} is one of the solution
hazy pumice
because that is what I was setting up but the result won't be in the naturals
hazy pumice
For example, {1,5,5,6} is one of the solution
are those values of n or numbers in the set?
oh I see
sorry about that
I misread the problem
well that's a lot harder
manic skiff
For 1+5+5+6=17 and 1^2+5^2+5^2+6^2=87
manic skiff
well that's a lot harder
No. I try to solve it w/o program.
hazy pumice
For example, {1,5,5,6} is one of the solution
apparently repetition is allowed
manic skiff
Maybe algebraicly, or combinatorically.
hazy pumice
what class are you taking
where did this problem come from
that will help me know what tools you are expected to use
manic skiff
what course, what level
One of my lecturer give it to us, as a challenge. He dont specify the subject.
hazy pumice
One of my lecturer give it to us, as a challenge. He dont specify the subject.
what is the name of the class
is it Combinatorics?
Number Theory?
I can only think of an algorithm
and it would go something like this:
Begin with the trivial case:
{1,1,...1,1} (seventeen ones)
this will satisfy the sum and sum of squares. pfft! no it won't. What am I talking about? they're not both equal to 17
(actually it gets a lot trickier immediately, doesn't it?)
yuck
...
$\sqrt{87} \approx 9.3$
thorny flameBOT
Disorganized
hazy pumice
so no number in the set can be greater than 9, that's for sure.
"Find all sets of Natural numbers such that the sum of these numbers is M and the sum of the squares of these numbers is N"
I don't know, but maybe for whatever algorithm does work, you start with the highest n you test being no greater than sqrt(N).
Decrement n for every failure and add numbers to the set starting with a 1.
Proceeding like this should scoop up all valid sets, in smallest-to-largest length order.
Do you have another specific example that is not yet solved?
manic skiff
is it Combinatorics?
Yes, it is.
manic skiff
The previous problem given is "show that: if we drawn m lines which x_1 line paralel in a direction, x_2 line paralel in another direction,..., until x_n parallel in another direction, and x_1+x_2+... +x_n=m then there will be (m^2-x_1^2-x_2^2-...-x_n^2)/2 intersection points"
Then he ask (how to draw 17 lines that intersects at 101 points)
And that's where my question comes.
hazy pumice
The previous problem given is "show that: if we drawn m lines which x_1 line par...
I am confused again,
are you saying that there are two slopes for all these lines?
that is, you have one set of lines all pointing one way,
and the other set are all pointing another way?
manic skiff
You're right. If we will do "trial and error", it is better to use program.
hazy pumice
that's an interesting formula
manic skiff
This is one of the example. And The formula could be proven by using binomial coefficient C(n,r)
manic skiff
Then he ask (how to draw 17 lines that intersects at 101 points)
So, the final problem is, how to solve this.
manic skiff
Is there any method to find the set of integer that its sum is 17 and its square...
And by the given formula, it is equivalent to solbe this.
manic skiff
I don't get what do you mean by "higher magnitude slope"
hazy pumice
yeah this language barrier is rough
hazy pumice
are the slopes determined by the number of sets of lines with the same slopes?
hazy pumice
I need help gmm
you are out of pocket right now, man
limber mirage
you are out of pocket right now, man
Why
hazy pumice
!occupied
topaz sinewBOT
Someone else is already using this help channel. If you need help with a question, please open your own help channel/thread (see #❓how-to-get-help for instructions).
manic skiff
are the slopes determined by the number of sets of lines with the same slopes?
The slopes are arbitrary. But if there 4 sets of parallel line, there will be 4 distinct slopem
It goes like this:
If there n line, no concurrent lines, and no parallel line, there will be C(n, 2) intersection points.
If there n line, no concurrent lines, and some parallel line, there will be C(n, 2)-C(x_1,2)-C(x_2,2)... intersection points.
manic skiff
If there n line, no concurrent lines, and some parallel line, there will be C(n,...
I hope you can sense the way of the proof
hazy pumice
If there n line, no concurrent lines, and some parallel line, there will be C(n,...
it's the x notation that is throwing me off
topaz sinewBOT
@manic skiff Has your question been resolved?
hazy pumice
sorry @manic skiff I can't figure out this one soon
manic skiff
Thank you for your consideration, and your advice for writing a program.
pallid dune
hi
pallid dune
@lost laurel
sorry our conversation ended
I had a last question though I hope u dont mind
I was asked to find the max and minimum corresponding x values for these
lost laurel
is that the full equation ?
is the s significant or is it just a word
thats starting
topaz sinewBOT
@pallid dune Has your question been resolved?
hexed remnant
hello
hexed remnant
what is the dimension of the set of vectors
$(1,0,0,...,0,-1), (0,1,0,...,0,-1), (0,0,1,...,0,-1), ...,(0,0,0,...,1,-1)$
thorny flameBOT
george clooney real account
hexed remnant
is it just n-1 (where n is the length of each vector)?
odd pagoda
the dimension of the span of those vectors is n-1, yes
hexed remnant
okay
then i must be doing something wrong in a different part of the question
the question is $T:\mathcal{M}{n\times n}(\mathbb{R})\rightarrow\mathbb{R}, T(X)=tr(X)=\sum\limits{i=1}^n X_{ii}$
thorny flameBOT
george clooney real account
hexed remnant
i am meant to find rank and nullity of this map
if the dimension of the span of those vectors (aka the nullity) is equal to n-1
and the dimension of the image (which is dim(R), which is 1)
odd pagoda
those arent vectors in M_nxn(R)
hexed remnant
yeah thats the kernel
odd pagoda
no
hexed remnant
explain
hexed remnant
yeah the vectors in M_nxn(R) is a linear combination of those
that's the span of the kernel
odd pagoda
so the elements of the kernel also are matrices
hexed remnant
ohh ok
wait so how do i find the kernel
it's just "matrices whose trace is zero"
odd pagoda
(by rank nullity theorem its just easier to find the rank and nullity from that btw)
hexed remnant
yeah but i need to describe the kernel
lethal prism
💀 💀 💀 💀
odd pagoda
<@&268886789983436800>
lethal prism
Why is this thing so damn hilarious
hexed remnant
okay so i think my starting point would be to point out that n^2-n slots of the matrix are free
because if it's not on the diagonal it can be anything
odd pagoda
yes
hexed remnant
OH
once i know n-1 members of the diagonal
(which are free)
i know the last one must be the negative of their sum
odd pagoda
yes
odd pagoda
the diagonals are the vectors you wrote down earlier
hexed remnant
so those diagonals plus every other place
gotcha
thank u so much
linear algebra is a pain
analysis for life
.close
crisp raptor
what kind of trig identity is this
topaz sinewBOT
what kind of trig identity is this
crisp raptor
,w 1/sin(pi/5) = -(sin(-4pi/5)-sin(-6pi/5))/(1-cos(2pi/5))
thorny flameBOT
hushed kettle
who can help pls, im bad at match
from point C to plane B, two oblique lines are drawn, one of which is 2 cm longer than the other. The projections of the obliques are 9 cm and 15 cm. Find the distance from point C to plane B.
livid tusk
You should consider the ratio between them
Consider drawing a line perpendicular to plane B to see that they are similar triangles.
then you denote the projection of the smaller line as x, and the other one as x+2
then you write 9/15 = (x)/(x+2) = 3/5
thus 5x = 3x + 6 thus x = 3
hushed kettle
You should consider the ratio between them
Consider drawing a line perpendicular...
wow, thanks, can you draw a figure with changes pls 👉👈
livid tusk
no, on laptop
topaz sinewBOT
@hushed kettle Has your question been resolved?
hushed kettle
no, on laptop
use Paint pls
topaz sinewBOT
@hushed kettle Has your question been resolved?
neon iron
topaz sinewBOT
silk pewter
cannot be answered without a and b
worthy lake
A 26
it can be answered
they do cancel out
try drawing it out
you could then split the quadrilateral into 4 triangles
icy sky
they also dont specify what the expression is equal to
worthy lake
I don't have pen and paper so i can't really explain the entire process step by step
icy sky
So no, it cannot be determined
worthy lake
i assumed it to be 1?
icy sky
So youre telling me that if you scale a and b by a factor of 3, which implies the scaling of the ellipse by a factor of 3, the area of the square tangent does not change?
worthy lake
then it does change
since it is not given i just assumed to be 1 which does yield a valid answer
icy sky
No it does not
icy sky
x²/2² + y²/2² = 1 has a square tangents on all sides with area 4
x²/6² + y²/6² = 1 has a square tangents on all sides with area 36
This question is poorly written unless theres other prior information
thick arrow
for the tangents to form a square, the ellipse needs to be a circle. no?
if so, then a = b
so, the area really just comes down to the value of a and the unknown(?) constant on the hidden RHS
which varies
icy sky
Does not necessarily have to be a circle, the tangent square is tilted 45 degrees
If the axes of the ellipse were aligned with the xy axes
thick arrow
Does not necessarily have to be a circle, the tangent square is tilted 45 degree...
really? ellipse tangents can for a square?
i thought it was only rectangles
?
neon iron
for the tangents to form a **square**, the ellipse needs to be a circle. no?
if ...
in ellipse it is possible to
tilted one
neon iron
Excuse the poor drawing, but it would look something like this
exactly
cursive thorn
if you rotate the "rectangle" around the ellipse, some sides keep shrinking for a bit, and other get bigger, at some point they will be equal
cursive thorn
Oh hey jelle
hi
thick arrow
tilted one
oooh
icy sky
Still, the square could vary in area in so many ways
Are you sure there is no other information required to solve the problem?
thick arrow
ye, it still does vary with a and b values
icy sky
And also is not set equal to something
thick arrow
^
worthy lake
neon iron
Are you sure there is no other information required to solve the problem?
nah i dont see anything else
worthy lake
heres a graph to help you understand better
icy sky
Are you sure there is no other information required to solve the problem?
If there exists no other information, can you ask the person who made the problem or answer key?
neon iron
If there exists no other information, can you ask the person who made the proble...
yeah i can, but not rn
icy sky
Please do ask what they meant because the problem is unsolvable in its current state
neon iron
umm
icy sky
I gotta get to bed, good luck!
neon iron
.close
forest bloom
Hii, could anyone help me regarding this question. Why would the answer be B? Just the fact that (1.0) is literally on the circle already shows that there's a case where quantity A = quantity B
topaz sinewBOT
Hii, could anyone help me regarding this question. Why would the answer be B? Ju...
wary tulip
forest bloom
wary tulip
so one thing to note for this is that if (x,y) is on the circle then x^2 + y^2 = 1
forest bloom
right and i can't have any combinations where x = 0 or y = 0 right?
wary tulip
and |x| and |y| are both less than 1 (if either were 1, the other would be 0)
forest bloom
and |x| and |y| are both less than 1 (if either were 1, the other would be 0)
but how would i know the addition leads to a quantity less than 1?
addition of 2 quantities less than 1
wary tulip
so one thing to note for this is that if (x,y) is on the circle then x^2 + y^2 =...
well you know this
wary tulip
and |x| and |y| are both less than 1 (if either were 1, the other would be 0)
and this
forest bloom
oh then use triangle inequality?
wary tulip
ya
vale furnace
welcome
wary tulip
left flare
Can u please explain me this question?
wary tulip
Can u please explain me this question?
don’t ask questions in other people’s channels
vale furnace
i'll let it slide this one time but don't do this again eddie
wary tulip
pure you should speak to me like that more
vale furnace
forest bloom
wary tulip