3 ≡ 5 (mod 2) can be expressed as 3 = 5 + k2 so k = (3-5)/2 therefore k = -1
#help-38
floral solstice
lusty pike
Yeah I think we both understand what it's trying to explain now
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zenith summit
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zenith summit
I answered a and b
rugged latch
What do u have so far for part c
wraith hinge
I answered a and b
Solve a system of equations
zenith summit
I am not sure about which line equations to choose
rugged latch
What did u get for b
zenith summit
y=-1/3x+9
rugged latch
Mhm
zenith summit
isn't the value of x = 6
rugged latch
Well, that’s a point on only one of the lines
We’re trying to find the point where both of the lines meet
rugged latch
Hm
rugged latch
The “intersection point” is (1.5,8.5) yes
rugged latch
Midpoint is when we want to find the point in the middle of 2 points
zenith summit
yes
rugged latch
Intersection point is when we want to find where 2 lines meet
zenith summit
so from which point of the line am i trying to solve
rugged latch
Wdym
zenith summit
i am trying to focus with
rugged latch
?
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plush heron
https://media.discordapp.net/attachments/903486481929224232/1025873673011351673/unknown.png I'm confused how it got the red part. Can someone please explain? 🙂
plush heron
<@&286206848099549185>
lusty spindle
maybe try a proof by induction
lusty spindle
plush heron
thanks hello kitty, makes more sense now
steel osprey
sorry but what math is this
plush heron
idk, you tell me 😂
plush heron
Correct hah
I'm not a fan of it.. that's for sure 😅
but on a positive note, its pretty useful for stuff in reinforcement learning
Have you done markov chain before? @lusty spindle
lusty spindle
a little bit
plush heron
M/M queuing?
lusty spindle
yeah i heard about that
plush heron
like that 😛
lusty spindle
btw i think it's preferable to close this help channel now...?
lusty spindle
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plush heron
Thanks hello kitty for the help 😂
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green trench
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hybrid arrow
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hybrid arrow
<@&286206848099549185>
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mint peak
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mint peak
can someone help me with time complexity of this algorithm? outer loop is log_2(n), but i'm struggling with the inner loop
i need to express time complexity of entire program in terms of N
dark quest
First of all, you should have $$T = \sum_{i = 1}^{\log n} \text{inner}(i)$$
solid kilnBOT
jimmy1234
dark quest
Since $$j * j < i$$, you will have $$j < \sqrt{i}$$, hence it should be obvious that $$\text{inner}(i) = i^{\frac{1}{2}}$$.
solid kilnBOT
jimmy1234
dark quest
Do you know how to proceed from this?
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dark quest
Another Hint for you 🙂
mint peak
Another Hint for you 🙂
can you elaborate on that?
mint peak
ahh, dont get it
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@mint peak Has your question been resolved?
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@mint peak Has your question been resolved?
mint peak
what's time complexity of the inner loop based on i
I don't know how to find it out
I was told that I can't multiply time complexity when the loops are dependent
wraith hinge
Maybe as a warm-up replace i in the fourth line with n: j * j < n
dense breach
^
dense breach
I was told that I can't multiply time complexity when the loops are dependent
cause one loop feeds the index to the next
but if we pretend like the first loop is by itself how long does it take based on i
mint peak
but if we pretend like the first loop is by itself how long does it take based o...
sqrt(i)
dense breach
yeah
so every time it's called it takes sqrt(i) time
so the approach that you can't do
you can't just say it's called log(n) times so the overall time complexity is log(n)*sqrt(i)
notice that time complexity depends on i when it should only depend on n as well
so what you should do
look at the number of times it tends to be called and the values it's called with
wraith hinge
Maybe think about the value of i in the last step of the outer loop.
mint peak
yeah
I need to express it only it terms of n
This is my observation, but I don't know how to express it mathematically
dense breach
so this is for some n value between 256 and 511, right
mint peak
so this is for some n value between 256 and 511, right
512 I think
dense breach
257 to 512 then
dense breach
512 I think
but so
our runtimes are like
0,1,1,2,3,5,7,11,15
well we already know it's sqrt(i)
so if i assumes every power of 2 between 1 and n
dense breach
0,1,1,2,3,5,7,11,15
we need some way of approximating the sum of all these values for any n
just finding the order of growth
nothing precise
O(n) stuff
mint peak
hmm, but how
upper bound maybe can be sqrt(n) but don't know if there can be something better
dense breach
well we need to understand the sequence
does it make sense if i write it as around like sqrt(1), sqrt(2), sqrt(4), sqrt(8), ..., sqrt(2^n)
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dire charm
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wraith hinge
one question abt elimination
trim joltBOT
one question abt elimination
wraith hinge
When u do elimination, can u switch the equations
trim lichen
switch how exactly?
trim lichen
??
trim lichen
i was waiting for you to formulate your question...
wraith hinge
I THINK IMLIKE TOO HYPER
trim lichen
but ok
trim joltBOT
trim lichen
it sounds like the issue resolved itself
spare flint
hey um @trim lichen i need some help
trim lichen
- don't ping individual users like this
- #❓how-to-get-help
spare flint
i know i know
spare flint
1. don't ping individual users like this
2. <#488120190538743810>
but the channel doesnt go back to the available section
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wraith hinge
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wraith hinge
I have no fucking idea pls help
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novel kiln
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novel kiln
do we just write 22 in its factors
find the highest power of the greatest prime factor in 2006
ik that's how we do it
but what's the logic behind that
<@&286206848099549185>
novel kiln
is that just 2006 or is there more to it
that's 2006!
different way of writing factorial lmao
odd charm
How many ____ what does it say there
novel kiln
zeroes
odd charm
Ah
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wary thicket
How do I find help
proven chasm
I was trying to find all a and b integer solutions such that a^2+b^2=25. a^2+b^2 can be factored into (a+bi)(a-bi). If you write that into polar you get sqrt(a^2+b^2)e^iarg(b/a)+sqrt(a^2+b^2)e^iarg(-b/a). You can simplify this to get (a^2+b^2)e^i(arg(b/a)+arg(-b/a) and this equals 25.If you divide both sides by (a^2+b^2) then you get c/(a^2+b^2) which is one meaning e^i(arg(b/a)+arg(-b/a) is one and that means (arg(b/a)+arg(-b/a) this is equal to 0. if you solve for a and b you get a can be any number and b is 0. But there are other solutions like 3 for a and 4 for b.
proven chasm
my bad
trim lichen
and what's "arg(b/a)" exactly?
proven chasm
its
every number can be
written as
ze^itheta
z is sqrt(a^2+b^2)
theta is just
arg(b/a)
arg is ussually inverse tangent
but that sometimes does not work
so I used arg
lament jewel
isit just me or are these not rly int solns anymore when u move into C
proven chasm
what do you mean
lament jewel
isnt int solns just 0,5 and 3,4
trim lichen
arctan(b/a) + arctan(-b/a) = 0 for all values of a and b (where defined), not only when b=0.
proven chasm
isnt int solns just 0,5 and 3,4
negatives also work
lament jewel
ok yea ig u cld, mb
trim lichen
so what you arrived at is exactly nothing!
lament jewel
negatives also work
oh ye
proven chasm
arctan(b/a) + arctan(-b/a) = 0 for all values of a and b (where defined), not on...
I checked
on wolfram alpha
trim lichen
show exactly what you input into WA
trim lichen
would think WA does not understand your unorthodox notation
trim lichen
yeah but you're passing a real number to it as input
proven chasm
and then you can check the defenition
trim lichen
arg(a+bi) != arg(b/a)...
proven chasm
a and b are real numbers
oh
ok
so it does not work
like inverse tangent
one min
why does bi become
bigot
when i put it in wolfram alpha
I put it in as you said
and it did not give 0
trim lichen
it does, WA is just trying to avoid bullshit when a+bi is a real negative number...
trim lichen
jesus christ
proven chasm
say true when a and b are real here
trim lichen
no, b itself isn't complex and i never said b was complex
trim lichen
why is what not 0?
trim lichen
because, if it just so happens that a+bi is a real negative number, then arg(a+bi) and arg(a-bi) both equal pi, so the sum is 2pi and not 0...
trim lichen
.................
it can only happen if a < 0 and b = 0.
you yourself declared a and b as real
integer, even
proven chasm
ok
timber spoke
Is this what you are trying to do?
proven chasm
Is this what you are trying to do?
ya
but I get
e^i(arg(a+bi+arg(a-bi)=1
and so
arg(a+bi+arg(a-bi)
if that is 0
it works out
timber spoke
if the question is asking for all real solutions of a and b, it is safe to assume that a and b are real, since a and b are real, we can let arg be inverse tan.
then your line simply becomes, tan theta + tan (-theta) = tan theta - tan theta = 0
trim lichen
no the question is asking for INTEGER solutions of a and b
unless OP wants to amend that
proven chasm
ya it integer
timber spoke
yes, integer solutions are also real
proven chasm
solutions
what I want to do
is find all integer sol
and the add up how many I get
timber spoke
So can I confirm if a and b can be complex numbers with integer Re and Img coefficients? or does Integer solution here only refer to Real integer solutions
proven chasm
a and b are integers.
and when you square them both and then
add them
you get another integer
edgy cedar
For integers a, b a^2 + b^2 = 25 this should remind you of a certain triangle
proven chasm
For integers a, b a^2 + b^2 = 25 this should remind you of a certain triangle
I know
3,4,5
But I am trying to generlize
where if you get any numebr c
you can find how many integer a,b satisty
edgy cedar
Got it
proven chasm
a^2+b^2=c
anyway my doubt was cleared
I entered the wrong thing
on wolfram alpha
.close
lament jewel
hi so i was wondering
trim joltBOT
hi so i was wondering
lament jewel
for some contour C being D(0,R), R>0
how do we get cot(z) is upper bounded by 2
i mean i searched n thr r some nice elab proofs
but is there something nice i can just state
cuz i think its said i cant take it for granted?
or must i write a long elaborate proof for that
lament jewel
yea
maybe
ill rewatch
i feel it said
to NOT
but the proof looks
bit long
oki
amber python
no the
lament jewel
ty again
amber python
proof is in my notes
lament jewel
yea ik
amber python
i have the notes open
lament jewel
i have the notes open
yea i mean
amber python
a terrifyingly bad calculation
lament jewel
a terrifyingly bad calculation
TnT
amber python
no i
lament jewel
which is hard to get
lament jewel
u just
amber python
maybe not for the later part of the subj
lament jewel
at the summary bit
amber python
because i didnt write notes for it
lament jewel
D:
is ok
thrs
no more ass aft this
wait crap
ur
notes prob dont even cover the ass
this ass
idk
im not even at the ass yet
im just looking at the lects even
oki anw
tytyytyty
💕
.close
atomic herald
Work your way backward
trim lichen
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polar oriole
hi
polar oriole
i need help figuring out this q
- Jennifer buys a very large box of mixed-colour drawing pins. 25% are blue, 15% are red, and the
remainder are brown.
Find the probabilities that: (i) all 6 drawing pins selected are brown
what method do i use to do this
what type of probability is this
do i create 6 instances
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short kiln
is there any other faster way of solving this question?
trim joltBOT
is there any other faster way of solving this question?
warped kestrel
$$\frac{13.442}{4.7} = \dfrac{13442 \times \dfrac{1}{1000}}{47 \times \dfrac{1}{10}}=\dfrac{13442}{47}\times\frac{1}{100} = \frac{286}{100}=2.86$$
solid kilnBOT
秋水
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mighty kindle
Sin7=cosec83 ?
trim joltBOT
Sin7=cosec83 ?
surreal flume
csc x is the reciprocal of sinx, so csc 83 = 1/sin83
marsh crystal
Sin(90-Q) = cos Q
jip
mighty kindle
I mean if tan21=cot69, isnt sin21=cosec69?
marsh crystal
What's that?
oh sorry it means *yup
vital saddle
And once compare
vital saddle
oh sorry it means *yup
Np, Kaori
marsh crystal
Sin7=cos83
if you're working in degrees then yes
marsh crystal
So is sec81=cosex9?
yupp
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desert bane
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valid moss
what's your question
desert bane
I don't get this at all
according to the book the inverse of F(11) and regular f(11) should both be 8
but I get different answers
valid moss
you're supposed to find the x value for which f(x) is 11
not find the value of f at 11
desert bane
what does that mean
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coral cliff
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coral cliff
hi need help with this
silent canyon
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dense zealot
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dense zealot
dunno how to do this
wraith hinge
rational root theorem might help
i was thinking of that too
but isnt that for rational
numbers
ig interger can be represented
leading coefficient is 1
too
hmm so q or 1 and their negatives
no dont i have to sum p and q first
actually, maybe use vieta's
pq is the sum and p+q is the product
what then
idk
ig it tells you that p,q are also roots to a similar quadratic
reducing mod p and q gives that x must be a square mod pq, but idk if that helps
dense breach
give the roots names
dense zealot
so you sub in X1 and X2 in the x in the polynomial?
dense breach
no
dense zealot
wdym then
dense breach
(x-x1)(x-x2)
dense breach
now expand
dense zealot
(x-x1)(x-x2)
does this equal to the polynomial?
like both are factors of the
polynomial
dense zealot
ok wut after expansion
dense breach
compare it to the form you have
dense zealot
compare it to the polynomial??
dense breach
yes
dense breach
yeah
dense zealot
and -x2x-x1x is the mid term
dense breach
yeah
dense zealot
isnt this just vieta
dense breach
pq = x1+x2
dense breach
you know what
screw all this
a polynomial has integer solutions when it's discriminant is a perfect square
maybe that'll help lol
dense breach
oof
dense zealot
plus these questions are made by my prof himself
so i cant get online solutions
he lit has the copy right OZ math there lol
wraith hinge
Question 17?
dense breach
p+q = x1*x2
This condition is fun
wraith hinge
i just wanna give up
dense breach
Nah not 17 the old Q
wraith hinge
on all the remaning
dense breach
I feel like maybe it's typed wrong or something
or maybe I just can't see something
Anyways 17 ig
wraith hinge
any question
would be fine
😅
def is_square(integer):
root = math.sqrt(integer)
return integer == int(root + 0.5) ** 2
i=10
for p in range(-i,i):
for q in range(-i,i):
test=p**2*q**2+4*(p+q)
if test>=0:
if is_square(test):
if (p*q+test)%2==0:
print(p,q)```how do you take the discriminate for a cubic or is there such thing. Im looking at 18
wraith hinge
```import math
def is_square(integer):
root = math.sqrt(integer)
return ...
returns a lot of things
dense breach
No obvious pattern?
wraith hinge
not that I could see
just that p=1 works for anything
actually, maybe only p=-q away from 1 and 0
dense breach
hmm
dense breach
how do you take the discriminate for a cubic or is there such thing. Im looking ...
I know there's some generalized discriminant but idk what
Or if it'll help
My first observation is it's not true for every p,q,r
for example x^3-1 only has 1 real root
wraith hinge
ok it's fine, ill just ask him to explain all the questions
.close
wrontg
account
dense breach
well I can do 17 probably
wraith hinge
how
dense breach
I know there's some generalized discriminant but idk what
This is some graduate stuff you don't need it
Might not even help
The function is always negative for negative x
wraith hinge
moni i need you tonight for the reimann thing yesterday btw, maybe not tonight
i should wrap up my inequality notes tonight
wraith hinge
ok
dense breach
Has 3 real roots
dense zealot
ok
dense zealot
what
dense breach
great question professor
dense breach
Brb
dense zealot
dense breach
ok
trim joltBOT
dense breach
18 I meant I can do
trim joltBOT
✅
dense zealot
nice nice
dense zealot
the roots for 18 are 1, i, and -i?
dense breach
it's an arbitrary polynomial
dense zealot
quite so
dense breach
Based on the choices of signs
I assumed (though it would be much nicer if the professor specified)
p,q,r have to be positive
So I found 2 polynomials, 1 which has 1 real root and one which has 3
that satisfy that form
dense breach
is 2x^3-12x^2+22x-12
this has roots 1,2,3
dense breach
2x^3-2x^2+2x-2 has 1 real root ({1,i,-i})
This has roots 1,i,-i
dense zealot
how did u find them
dense breach
First one was just 2(x-1)(x-2)(x-3)
I figured that probably had the alternating form needed, and of course it has zeroes at 1,2,3
dense zealot
nice ill write that for my answer
dense breach
any equation with all positive roots will have the same alternating form
But not vice versa
dense breach
U need 2 answers really
You need to show it can have 3 or 1 real roots
Or maybe the answer is just "no"
because they're not always real
dense zealot
my class is starting, ill message u the solutions for the other ones
dense zealot
moni
dense breach
Yes
dense zealot
what is lim sup
dense zealot
dense zealot
most of the stuff he wrote under the topic limit of function
i dont quite get
ill prob have to relisten
to recording
dense breach
ok Lim sup isn't important here really
it's like
If you have a sequence that doesn't converge, but oscillates infinitely within a certain region
dense zealot
ok
dense breach
Lim sup is the upper bound of that region and Lim inf is the lower bound
So like
1,-1,1,-1,1,-1,...
dense zealot
upper bound is 1?
dense zealot
ok sick
dense breach
And Lim inf is -1
And unlike limits, which don't always exist
Lim sup and Lim inf always exist for a sequence if you allow them to be +-infinity in some cases
dense zealot
He’s roasting me so bad lmao
dense breach
So using them you actually get an exact condition for the radius of convergence
dense breach
He’s roasting me so bad lmao
lmao
dense breach
why am I teaching all the material you're missing because you're talking to me instead lol
this seems cyclic
dense zealot
Hopefully I'll be a good professor
You will!🥰
dense breach
:)
dense breach
Idk wasn't the problem originally -px
instead of -q
Maybe he's solving a different problem
dense breach
I want you to speak up and ask him if p^2 = p^2+0 is valid lol
if he asks about sum of squares
a²+b²
dense zealot
Oh he says it’s very hard to think of the solution he’s writing
dense breach
lmao
dense zealot
What is he doing moni
dense breach
Wild
dense zealot
As usual
dense zealot
What do I say
dense breach
P is an integer
dense zealot
Write out full sentence
dense breach
Is p²+0² also a valid solution
dense breach
scream
dense breach
ask him when you can
dense zealot
Is p²+0² also a valid solution
Afraid
dense breach
Trying to prove (M-N)/2 and (m+n)/2 are integers
dense breach
can't write q
dense zealot
no but it's still a valid solution to the first part
Yes
He said that
He said many other things
I didn’t comprehend
dense breach
You shouldn't need to use the details of later questions to deduce which correct answers to earlier questions are actually correct lol
But I'll give him benefit of the doubt here
lol
dense zealot
Bro what is his solution tho😭
Lol he said question 12
Is wrong too
Cogwheel did that one
dense breach
that's cause you can find exact values of a,b,c
dense breach
well yeah that's the start of what looks like a vlid proof
dense zealot
Rotational symmetry equation he calls
dense breach
that's cause you can find exact values of a,b,c
i misread this one
that's 13 nvm
yeah the solution to 12 was perfectly valid
dense zealot
dense zealot
yeah the solution to 12 was perfectly valid
He said it’s wrong ???
Like what I wrote
dense breach
maybe you didn't elaborate enough
dense zealot
Hmm he said it’s completely wrong 😭
dense zealot
Idk read his stuff, I also don’t know 😅
He’s not done yet
He’s literally stuck rn 🐓
It’s hilarious to watch him not solving the question 🤩
He says that my solution is using different cases which is logically wrong
X is a variable
You can’t assume f(x) is larger than x
Or smaller
dense breach
but the alternative is f(x)=x
dense zealot
for every x f(x)>x or f(x)<x
He says u can’t assume that. What do I say
dense breach
does f(x)=x then?
i'm not saying (for all x f(x)>x) or (for all x f(x)<x), we'll prove that later
i'm saying for all x (f(x)>x or f(x)<x)
we know f(x) =/= x, right
that's given
dense zealot
right
dense zealot
Right
dense breach
so
suppose there were 2 values a,b
such that f(a) > a, and f(b) < b
consider g(x) = f(x) - x
g(a) > 0, g(b) < 0
dense zealot
Ok I’m asking him
dense breach
g is continuous, since it's a difference of continuous functions
so by the IVT, there exists c between a and b such that g(c)=0
then f(c)=c, a contradiction
dense breach
i don't think so
dense zealot
I said exactly what you said. I think he got
dense breach
tell him the proof you saw involved the use of IVT maybe
dense zealot
What you mean
dense breach
ah ok
dense zealot
Ok I don’t get the details but I’ll figure that out later
dense zealot
Yay he says that you have to be explicit about the cases
And without being explicit, it’s logically wrong or smth
@dense breach
18 uses the discriminate of
Cubic
😅
dense zealot
Idk He’s using cardano’s method(?
dense zealot
He’s not done
dense breach
oml
you should bring up to him 2x^3-2x^2+2x-2 when he is
as that's 2(x-1)(x^2+1) and thus has 1 real root
unless he acknowledges the answer to the problem is "sometimes"
dense zealot
as that's 2(x-1)(x^2+1) and thus has 1 real root
I put that as my original answer
Think he’s aware
This is cardano
It’s basically discriminate for quadratic
But he turned cubic to quadratic already
dense breach
ah ok
dense zealot
If d<0, there are 3 real roots
If d>0, 1 real root
And 2
Conjugate imaginary roots
dense zealot
If d=0, at least 2 real roots are the same
dense breach
maybe not
vagrant marsh
What problem are yall doin
dense breach
What problem are yall doin
all of them ;-;
vagrant marsh
Where
dense zealot
All kinds of questions we know the method but can’t apply them well👍
vagrant marsh
Where
dense zealot
Lemme resend
dense breach
here
dense breach
feel free to look if ur curious but this is really more of a chat at this point lol
dense zealot
Lol yes
Indeed
Well it’s still my questions
Cause I can’t understand a shit
He’s writing
Mini is just helping
For intepretation
vagrant marsh
Vietas states that if we write the roots as r and s,
(x-r)(x-s)=x^2-(r+s)x+rs
Matching coefficients we get
r+s=pq
rs=p+q
Add together you get
rs+r+s=pq+p+q
We use sfft
(r+1)(s+1)=(p+1)(q+1)
vagrant marsh
There isnt if one of them =1
dense breach
ah you add and subract hmm
vagrant marsh
Add them
vagrant marsh
Ah i didnt think about subtracting 😅
vagrant marsh
Interesting, is this live
dense zealot
Yeppp
dense zealot
then subtract the 2nd from the first
I don’t think it works ???
vagrant marsh
Oh bruh listen to your prof
dense zealot
Can’t understand 😅
I always listen to the recording
@dense breach I’m super confused
dense breach
ok im here lol
dense breach
I don’t think it works ???
yeah i didn't know where to go either
dense zealot
Just different cases
And it’s solved somehow
Don’t need to consider p because when q=1, it already proves the p part
Somehow
Btw he’s speaking Chinese. So I’m trying my best to
Translate
dense breach
o dang
dense zealot
Number 17 now
😭
He subsided all x to 1/y
Substitute
And it’s the same
This is magical
So this means that if r is a root, 1/r is also a root
dense breach
wow
dense zealot
He’s introducing a concept called symmetric polynomials
dense zealot
i saw it in your work
Should I ask him 19-22
I didn’t even look at those
They seem quite unsolvable
dense breach
Should I ask him 19-22
i think that's negative 3
just let him talk lol don't ask him if it's just about every single question
i'mma not be very active here for a while
dense zealot
ClAss ends at 7
I can only fit in about 2
Questions
Moni give me two questions from 19-25
and you can leave
dense zealot
i'mma not be very active here for a while
A while. , like for a long time ?
dense breach
maybe a half hour to an hour
dense breach
lol
dense zealot
all right
Lmao he said 19 is super obvious and why I don’t know how to do it
😭😭
He doesn’t want to explain cause it’s too obvious
Wtv
I’m asking 25
dense breach
I tried to pick ones so you'd understand the explanation
lol
20 maybe them
then
dense zealot
Okok