particularly integration part
#help-23
rich elm
rich elm
dats good
split fulcrum
the advanced maths 2u integrals are soo boring
even the 4u integrals are pretty easy in itself
rigid hazel
even the 4u integrals are pretty easy in itself
what are those
rich elm
i would say most 4u integrals are easy unless they are towards last question
desert pasture
the advanced maths 2u integrals are soo boring
Do jee integrals
rich elm
recursive series might get some ppl but it depends
4u integrals used to be hardb ut they toned down the diffculty quite a bitn ow
calculus isn't really the 'flavour' of current 4u examiner
split fulcrum
i would say most 4u integrals are easy unless they are towards last question
yeah true
the only hard 4u ones are the recurrence relation ones
even in 3u where they tell you what substitution to make, i feel like that takes away all the fun
rich elm
do u know all of ur definite integral stuff
split fulcrum
do u know all of ur definite integral stuff
i can solve definites yeah
like wdym exactly
rich elm
all the geometric tricks
shifting graphs
flipping it
think its called kings rule or something
desert pasture
think its called kings rule or something
Yes
split fulcrum
think its called kings rule or something
ya
i can do a bit but my trig is clapped so im not fluent in those trig tricks
split fulcrum
do u know all of ur definite integral stuff
have u done improper integrals yet
desert pasture
@split fulcrum , try this integral
rich elm
have u gone through partial fraction btw?
or not up to there yet
im just curious what textbook u using
desert pasture
Please don't say Stewart
split fulcrum
have u gone through partial fraction btw?
yes
ill probably finish trig sub by later today
and then move onto improper
rich elm
okok
split fulcrum
im just curious what textbook u using
daniel J velleman a rigorous first course in calculus
rich elm
hmm ok
also this might be slightly controversial
dont care too much about 'rigour' as long as u understand the concepts
it doesn't really mean much in the learning process for u at the moment (at least not worth the time) and i think its better taught with actual professors
desert pasture
daniel J velleman a rigorous first course in calculus
Nice
rich elm
somethings are just really difficult to self-teach
split fulcrum
dont care too much about 'rigour' as long as u understand the concepts
oh im skipping the stupid epsilon delta crap
im not going near it
rigid hazel
<@722683408190406667> , try this integral
what does f:(0,1) mean
split fulcrum
🤣
desert pasture
dont care too much about 'rigour' as long as u understand the concepts
I disagree, getting used to rigor earlier , makes stuff easier later
desert pasture
im not going near it
You're missing out on a lot
split fulcrum
ill learn it properly in uni
rigid hazel
oh im skipping the stupid epsilon delta crap
its actually pretty easy
desert pasture
what does f:(0,1) mean
The domain is (0,1)
rigid hazel
The domain is (0,1)
oh alrr
rigid hazel
oh im skipping the stupid epsilon delta crap
theres a short bprp vid over btw if u want
rich elm
I disagree, getting used to rigor earlier , makes stuff easier later
each to their own i suppose. It is a hump u have to get over eventually but i think rigour is easier to grasp once u have sufficient other mathematical knowledge built up
i just dont think it makes sense to focus rigour in highschool math
desert pasture
each to their own i suppose. It is a hump u have to get over eventually but i th...
Yeah, so water beam should do book of proof first
split fulcrum
the only thing i can prove is sqrt2 is irrational 
desert pasture
the only thing i can prove is sqrt2 is irrational <:opencry:586078614865444865>
Prove that sqrt(2) exists 🫵
split fulcrum
Prove that sqrt(2) exists 🫵
every number exists - QED
rich elm
question left as an exercise for the marker
split fulcrum
rich elm
but on a more serious note
desert pasture
every number exists - QED
Close , the completeness axiom along with supremums helps
split fulcrum
i havent touched polar coords and parametrics like ever before but thats covered in my book so i wonder if ill have a hard time with that
rich elm
uhhh
rigid hazel
i havent touched polar coords and parametrics like ever before but thats covered...
its not hard dw
rich elm
should be fine i think
make sure u learn ur complex numbers tho
apparently a lot of ppl avoid it until the end
of their bridging courses or whatever
rigid hazel
polar coords are pretty easy to visualize and grasp
split fulcrum
of their bridging courses or whatever
na i dont need a bridging course mainly cuz ive got most things covered
plus my friend took one of those for math cuz he didnt do ext
rich elm
tbh order of topics i would recommend u to learn is
calc (whatever u are doign rn)
then basic proofs + logic (learn logic, quite important)
then vector
then complex numbers
split fulcrum
he said it was pre useless
rich elm
uni maths eventually goes through them anyways
split fulcrum
ye
split fulcrum
tbh order of topics i would recommend u to learn is
calc (whatever u are doign r...
i dont think proofs and logic is a part of the math course ill take
i believe its jsut calculus + linear algebra
so ill probably go thru vectors and complex numbers next?
rich elm
i believe its jsut calculus + linear algebra
proofs and logic will be part of maths just by nature of what the maths studied at uni entails
theres a lot of proof exercises
rich elm
so ill probably go thru vectors and complex numbers next?
but this definitely
go through 2d 3d vectors + complex numbers
desert pasture
Linear algebra has a ton of proofs
safe radishBOT
@split fulcrum Has your question been resolved?
queen mango
safe radishBOT
queen mango
Trying to solve the latest question
Don't bother with the french
Everything above the latest question is given
And the latest question is asking to prove that their is a P that verifies the condition
Not sure how to start even
inner sable
221 c'est égal à 13 fois 17, donc d'après le lemme des restes chinois, $10^p$ congru à 18 mod 221 est équivalent à dire que $10^p$ est congru à 18 mod 13 et $10^p$ est congru à 18 mod 17
flat frigateBOT
TimourX
inner sable
tu décompose ton equation en un système d'équations plus simples
queen mango
221 c'est égal à 13 fois 17, donc d'après le lemme des restes chinois, $10^p$ co...
Je ne comprende pas , lemme des restes chinois?
inner sable
ah mince, tu connais pas ce théorème ?
inner sable
ah oui j'avais pas lu ce qu'il y avait au dessus
queen mango
ah mince, tu connais pas ce théorème ?
Non je ne le connais pas , mais je peut l'explorer
raven heart
Donc tu veux trouver p tel que 10^p = 5 mod 13 et 10^p = 1 mod 17
T'as fait le 3b ou pas ?
queen mango
Donc tu veux trouver p tel que 10^p = 5 mod 13 et 10^p = 1 mod 17
Pourquoi?
raven heart
Trying to solve the latest question
Faut être plus clair dans ce cas
queen mango
Je veux faire la dernier question
raven heart
Ah non , je fais 3.a
Et qu'est-ce qui bloque avec la 3a donc ?
?
raven heart
Ok mais t’en es à quelle question là, je pige rien à ce que tu recherches
queen mango
I want to solve 3,c
raven heart
Ok oui
raven heart
You said I need to solve b ?
Je me dis que la réponse à la b doit aider pas mal oui
On est pas en examen là, t’as tout le temps du monde pour bosser ta question, je vois pas l’intérêt de skip une question dans ce contexte
queen mango
Je me dis que la réponse à la b doit aider pas mal oui
avoir du sens
queen mango
Donc tu veux trouver p tel que 10^p = 5 mod 13 et 10^p = 1 mod 17
Donc je peut faire la multiplication ICI?
et je doit utilise la division Euclidien pour que je trouve "p"
raven heart
Donc je peut faire la multiplication ICI?
Oui tout l’intérêt du theoreme des restes, c’est que tu peux un calcul avec un gros modulo (228) par deux calculs sur des modulos plus simples (13 et 17)
raven heart
et je doit utilise la division Euclidien pour que je trouve "p"
Oui pour arriver à passer des équations du type 10^p = ??? vers p = ???
safe radishBOT
@queen mango Has your question been resolved?
safe radishBOT
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vapid cypress
limit x tends to 1+ of (arctan(x)ln(ln(x)))/tan(pi*x/2) =??
safe radishBOT
limit x tends to 1+ of (arctan(x)ln(ln(x)))/tan(pi*x/2) =??
vapid cypress
solve without lhopital
desert pasture
$\lim_{x \to 1^{+}} \frac{\arctan(x) ln(ln(x))}{ \tan{\frac{ \pi x}{2}}}$
flat frigateBOT
ƒ( wai ina teacup)= I don't know
desert pasture
this?
safe radishBOT
@vapid cypress Has your question been resolved?
ocean crypt
ebon mesa
ln(lnx)=ln(x-1)?
desert pasture
I don't think there's any way without LH
spiral saddle
ln(lnx)=ln(x-1)?
its a taylor expansion of ln(x) around x = 1 i think, the first two terms
since op said no lhopital allowed...
split moss
Compute the limit in the sense of distributions of the sequence ( (T_n) ), where ( T_N = e^{n^2} \delta_n ), and ( \delta_n ) is the Dirac delta concentrated at ( n ).
safe radishBOT
Compute the limit in the sense of distributions of the sequence \( (T_n) \), whe...
flat frigateBOT
jandro0103
split moss
Hint ?
safe radishBOT
@split moss Has your question been resolved?
safe radishBOT
@split moss Has your question been resolved?
split moss
<@&286206848099549185>
obsidian oracle
Hint ?
here's a hint: while the concentration point of the dirac distributions grows bigger, the functions they're evaluated on keep the same compact support...
flat frigateBOT
jandro0103
split moss
<@&286206848099549185>
plucky elk
**jandro0103**
The same argument proves that delta is also the zero distribution
split moss
The same argument proves that delta is also the zero distribution
But then T_n -> 0 right?
split moss
<@&286206848099549185>
split moss
<@&286206848099549185>
safe radishBOT
@split moss Has your question been resolved?
split moss
<@&286206848099549185>
split moss
<@&286206848099549185>
icy birch
How can i help you?
split moss
How can i help you?
Read above
split moss
<@&286206848099549185>
safe radishBOT
@split moss Has your question been resolved?
fallen heart
That's a lot of helper pings
safe radishBOT
safe radishBOT
Available help channel!
safe radishBOT
plucky elk
Just plug N to the right side of equation 4
near tendon
equation 4?
plucky elk
equation 4?
In your image
safe radishBOT
@near tendon Has your question been resolved?
near tendon
In your image
im ngl i still dont get it
how would it give the same equation they ask for
plucky elk
how would it give the same equation they ask for
Do the algebra and find out
And find dN/dt
safe radishBOT
@near tendon Has your question been resolved?
near tendon
How does subing n change it
plucky elk
How does subing n change it
Change what
near tendon
Like how does my equation go from this with n instead of t
near tendon
To the one they ask for
plucky elk
Just plug N to the right side of equation 4
Do this
N=(stuff)
Plug in stuff for N in the right side
safe radishBOT
@near tendon Has your question been resolved?
safe radishBOT
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Available help channel!
deep rapids
someone help with C-8
safe radishBOT
someone help with C-8
deep rapids
im not sure if im allowed to ask these questions here
sacred heron
Phew!
buoyant shadow
i would think it's (A) v (but maybe v²?)
like it doesn't matter what happens in the frame of the plate
it's relative to me
but that's "exerted by the jet"
very confusing
0 is not an option
in the direction of the jet?
deep rapids
i think i got the answer
buoyant shadow
lmao
deep rapids
jet water = roh * a * v^2
and we have to use relative velocity
so it was just a formula based question
buoyant shadow
this is very interesting
deep rapids
bruh
deep rapids
this is very interesting
nah not really
buoyant shadow
so it's d?
deep rapids
its just a formula based question
deep rapids
so it's d?
yea
u can also conserve energy
Energy will be 1/2m(v-V)^2
so that gives the answer
buoyant shadow
ok
deep rapids
wait
i forgot abt the pressure energy
but presusre energy is constant
so doesn't matter
.close
safe radishBOT
buoyant shadow
ok it's too hard, i've no clue
last geode
someone help with C-8
Why not ask in PhoDs?
safe radishBOT
Available help channel!
deep rapids
Why not ask in PhoDs?
whats this
safe radishBOT
whats this
desert pasture
someone help with C-8
Is it c
deep rapids
Change in momentum
momentum of?
deep rapids
Nah i got the answer
it was formula based
but u can also get it through energy conservation
idk about momentum conservation tho
ig the momentum isn't conserved
safe radishBOT
@deep rapids Has your question been resolved?
lean otter
Good day and Merry Christmas 🎄
I’m currently writing a mathematical research paper but I don’t know what format or what kind of structure that I need to follow. I already have the proofs of my work but I don’t have the paper. What format do I need to follow that is commonly used by modern mathematicians? Thank you in advance!
lean otter
I know how to use LaTeX with Overleaf but I don’t know the format.
Uhhh I’m confused, what does that mean? 🤣
peak estuary
you can look on the website arxiv and see some papers
and try to copy that
abstract, introduction, then the general stuff of the paper, maybe conclusion, references
lean otter
Oh cool! Tysm!
What kind of paper do they use? A4?
Nvm they have a detailed guidelines. Anyways, thank you!
.close
safe radishBOT
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Available help channel!
sour igloo
Am i getting trolled
i can't see the ibp
safe radishBOT
Am i getting trolled
i can't see the ibp
sour igloo
ibp gives $$\int{x^2\sin{(\ln x)} \mathrm dx = \frac{x^3}{3}\sin{(\ln x)} - \int{\frac{x^3 \cos{(\ln x)}}{3x}} \mathrm dx$$
f
topaz tree
x=e^u, dx = e^udu
this gets the answer
versed wave
Am i getting trolled
i can't see the ibp
u = x^2, dv = sin(lnx)
flat frigateBOT
Pro_Hecker
Frosty
ibp gives $$\int{x^2\sin{(\ln x)} \mathrm dx = \frac{x^3}{3}\sin{(\ln x)} - \int{\frac{x^3 \cos{(\ln x)}}{3x}} \mathrm dx$$
```Compilation error:```! Missing } inserted.
<inserted text>
}
l.49 ...{\frac{x^3 \cos{(\ln x)}}{3x}} \mathrm dx$
$
I've inserted something that you may have forgotten.
(See the <inserted text> above.)
With luck, this will get me unwedged. But if you
really didn't forget anything, try typing `2' now; then
my insertion and my current dilemma will both disappear.```
versed wave
you need to do IBP 3 times
versed wave
u = x^2, dv = sin(lnx)
the first time is like i said here
you should have an integral of cos(lnx)
here, take u = 1 and dv = cos(lnx)
perform ibp 2 times and see
sour igloo
so ibp first time then u sub
sour igloo
the first time is like i said here
im not sure im seeing how to directly integrate sin(lnx)
is u = x^3 and dv = sin(lnx)/x acceptable
?
topaz tree
yes that is acceptable
sour igloo
and then ibp $-x^3\cos(\ln x)-\int-3x^2\cos(\ln x)$ again should give the answer right
flat frigateBOT
Frosty
faint seal
You have to use ibp again
sour igloo
yeah
safe radishBOT
@sour igloo Has your question been resolved?
safe radishBOT
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spiral saddle
safe radishBOT
spiral saddle
safe radishBOT
@spiral saddle Has your question been resolved?
spiral saddle
,w {{0,-2,1},{1,0,-1},{0,1,-1}}^(-1) * {{1,3,-1},{-2,1,0},{0,1,2}} * {{1},{1},{1}}
like
this is nonsense
which change of basis matrix do I need to multiply to get the right answer
can someone help me please
<@&286206848099549185>
spiral saddle
it means
f(b1) = (1,-2,0)_(B')
f(b2) = (3,1,1)_(B')
f(b3) = (-1,0,2)_(B')
where B = {b1,b2,b3}={v1,v2,v3}
spiral saddle
What is meant by M_BB'(f)?
it means f is applied to basis vectors in B and the image of f under each basis vector of B is expressed in coordinates B'
by image I mean the result of the linear transformation
snow robin
$f(\vec{v_1}) = \begin{pmatrix} 1 \ -2 \ 0 \end{pmatrix}_{\text{B'}}$
spiral saddle
yeah and b1 = v1
flat frigateBOT
Katharine
snow robin
ok
spiral saddle
yeah 
snow robin
so if you apply the basis transformation from B' to B
you get f(v_1) in basis B
right?
which you can then use
spiral saddle
you mean M_(B'B)(f)?
spiral saddle
how is P defined
flat frigateBOT
Katharine
snow robin
$[v]{B} = P{B' \rightarrow B} [v]_{B'}$
flat frigateBOT
Katharine
snow robin
where in our case
spiral saddle
**Katharine**
I agree with this
snow robin
$[v]{B'} = \begin{pmatrix} 1 \ -2 \ 0 \end{pmatrix}{B'}$
flat frigateBOT
Katharine
spiral saddle
yes we need to put P in terms of what we have
snow robin
well you have both bases
spiral saddle
still dunno what it is
snow robin
so you can construct P
spiral saddle
like it could mean a projective linear transformation or it could mean a matrix change of basis with respect to the identity linear transformation or it could mean a matrix change of basis with respect to f
like there is a projector operator called P aswell
spiral saddle
wrt f or Id
snow robin
you have the coordinate vectors in the bases
snow robin
**Katharine**
Defined by this
If you have $\begin{pmatrix} 1 \ 0 \ 0 \end{pmatrix}_{B'}$
flat frigateBOT
Katharine
snow robin
then using the transformation matrix you know it is
$\begin{pmatrix} 1 \ 1 \ 0 \end{pmatrix}_{B}$
flat frigateBOT
Katharine
snow robin
if i am not mistaken
spiral saddle
P i need to know if its a change of basis matrix with respect to identity linear transformation or f
snow robin
it's not wrt f
spiral saddle
if you prolly mean it as a regular change of basis matrix with respect to anything then its with respect to identity operator
spiral saddle
how do I find this matrix P that changes coordinates from B' to B
snow robin
$\begin{pmatrix} 1 \ 1 \ 0 \end{pmatrix}{B} = \begin{pmatrix} . & . & . \ . & . & . \ . & . & . \end{pmatrix} \begin{pmatrix} 1 \ 0 \ 0 \end{pmatrix}{B'}$
flat frigateBOT
Katharine
spiral saddle
B'={v1+v2,-v1+v3,v3}
B={v1,v2,v3}
snow robin
if basis B was made using v_1, v_2, and v_3 and basis B' was made using w_1, w_2, and w_3 we would not be able to do this
snow robin
You might be able to use this $f(\vec{v_1}) = \begin{pmatrix} 1 \ -2 \ 0 \end{pmatrix}{\text{B'}} \ f(\vec{v_2}) = \begin{pmatrix} 3 \ 1 \ 1 \end{pmatrix}{\text{B'}} \ f(\vec{v_3}) = \begin{pmatrix} -1 \ 0 \ 2 \end{pmatrix}_{\text{B'}}$
in that you know what a coordinate vector means in terms of v_i
a coordinate vector in B'
if that makes sense
but i'm not sure how to do the composition without the change of basis in between
as i'm guessing M_BB'(g) is defined similarly
$f(\vec{v_1}) = \begin{pmatrix} 1 \ -2 \ 0 \end{pmatrix}{\text{B'}} \ f(\vec{v_2}) = \begin{pmatrix} 3 \ 1 \ 1 \end{pmatrix}{\text{B'}} \ f(\vec{v_3}) = \begin{pmatrix} -1 \ 0 \ 2 \end{pmatrix}{\text{B'}}$ and $\$
$g(\vec{v_1}) = \begin{pmatrix} 0 \ 1 \ 0 \end{pmatrix}{\text{B'}} \ g(\vec{v_2}) = \begin{pmatrix} -2 \ 0 \ 1 \end{pmatrix}{\text{B'}} \ g(\vec{v_3}) = \begin{pmatrix} 1 \ -1 \ -1 \end{pmatrix}{\text{B'}}$
flat frigateBOT
Katharine
snow robin
so in order to have $g(f(\vec{v_1})) = \begin{pmatrix} ? \ ? \ ? \end{pmatrix}_{\text{B'}}$
flat frigateBOT
Katharine
snow robin
you need something inbetween
so let's continue by writing out P
$\begin{pmatrix} 1 \ 1 \ 0 \end{pmatrix}{B} = \begin{pmatrix} . & . & . \ . & . & . \ . & . & . \end{pmatrix} \begin{pmatrix} 1 \ 0 \ 0 \end{pmatrix}{B'} \ \begin{pmatrix} -1 \ 0 \ 1 \end{pmatrix}{B} = \begin{pmatrix} . & . & . \ . & . & . \ . & . & . \end{pmatrix} \begin{pmatrix} 0 \ 1 \ 0 \end{pmatrix}{B'} \ \begin{pmatrix} 0 \ 0 \ 1 \end{pmatrix}{B} = \begin{pmatrix} . & . & . \ . & . & . \ . & . & . \end{pmatrix} \begin{pmatrix} 0 \ 0 \ 1 \end{pmatrix}{B'}$
flat frigateBOT
Katharine
snow robin
matrix multiplication
spiral saddle
yeah
snow robin
When we apply that matrix to the output of f
of M(f)
we can then apply M(g) to it
and again apply it
and the first component should then be the multiple of v_1
so it will be
$P_{B' \rightarrow B} M_{BB'}(g) P_{B' \rightarrow B} M_{BB'}(f) (\vec{v_1} + \vec{v_2} + \vec{v_3})$
flat frigateBOT
Katharine
snow robin
And as a result we should get a coordinate vector in the B basis
with the entries being the multiples of v_1, v_2, and v_3
spiral saddle
,, P_{B' \to B} = \begin{bmatrix} 1 & -1 & 0 \ 1 & 0 & 0 \ 0 & 1 & 1 \end{bmatrix}
flat frigateBOT
938c2cc0dcc05f2b68c4287040cfcf71
spiral saddle
$P_{B' \rightarrow B} M_{BB'}(g) P_{B' \rightarrow B} M_{BB'}(f) (\vec{v_1} + \v...
we need to multiply all this matrices together
snow robin
yes
spiral saddle
also, v1+v2+v3 is (1,1,1)_B
snow robin
yes
apply it to that vector
and you will get the answer
i think
i'm never entirely sure about basis transformations
but if i am correct then this will give you the answer
spiral saddle
,w {{1,-1,0},{1,0,0},{0,1,1}} * {{0,-2,1},{1,0,-1},{0,1,-1}} * {{1,-1,0},{1,0,0},{0,1,1}} * {{1,3,-1},{-2,1,0},{0,1,2}} * {{1},{1},{1}}
flat frigateBOT
snow robin
I was hoping it was correct
lyric monolith
just a quick question when calculating limits, if i come across 0/0+ is that indeterminate or +infinity??
safe radishBOT
just a quick question when calculating limits, if i come across 0/0+ is that ind...
lyric monolith
like does 0 count as a real number in the whole anything divided by 0+ or 0- is + or - infinity
this is the limit im calculating and i do know that it is infinity
x+1 in the first line**
thin bridge
if you have something that's outright 0,
e.g 0/x
the limit will be 0 when taking the limit
lyric monolith
yeah but in this case its 0/0+ so would that be any diff
the limit is infinity so i just am not sure if this is the right way to get to it
lyric monolith
hmm so i would be allowed to just write +infinity right after that
thin bridge
no
if both the numerator and denominator approach 0, you'd have an indeterminate form and more work is required
lyric monolith
yeah but this is limit from the right and the denominator approaches 0 from the right so how would that work
usually if the denominator approaches 0 from the right or left it is +/- infinity but in this case numerator is also 0
lucid stream
usually if the denominator approaches 0 from the right or left it is +/- infinit...
indeterminate. any 0/0 is indeterminate
thin bridge
as mentioned above, it depends. the specifics are important (there's a distinction between approaches 0 and is 0)
lyric monolith
yeah thank you!! i kind of messed up some sign stuff and couldn't figure out how to factor it but turns out its kinda easy
.close
safe radishBOT
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slim plinth
how do i integrate tan-1(x)/x from 1/2 to 2? tried by parts but dosent seem to work
safe radishBOT
how do i integrate tan-1(x)/x from 1/2 to 2? tried by parts but dosent seem to w...
median vigil
i don't think that has an elementary antiderivative
slim plinth
but it has bounds right?
i was thinking of using kings, but that dosent seem to help
plucky elk
Try t = 1/x substitution
pseudo scroll
safe radishBOT
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lethal walrus
are dx and integrals of trigonometric functions a rotation anticlockwise and clockwise respectively? like in this graph?
safe radishBOT
are dx and integrals of trigonometric functions a rotation anticlockwise and clo...
empty gyro
are dx and integrals of trigonometric functions a rotation anticlockwise and clo...
You could say that. Sure
safe radishBOT
covert yoke
There is a connection here between this and Euler's formula
Which is pretty interesting and enlightening if you aren't aware
@lethal walrus ^
lethal walrus
i am not aware no, whats the connection?
covert yoke
So e^(iθ) = cos(θ) + i sin(θ)
So Re(e^(iθ)) = cos(θ)
If you apply d/dx to this it's like multiplying by i
So you get i cos(θ) + i^2 sin(θ) whose real part is -sin(θ)
We can relate multiplication by i with a 90° rotation in the complex plane
Hence your circle presentation is more on the nose than you imagined.
rigid hazel
ay thats cool
covert yoke
If you exchange cos and sin and just place it on a complex plane and rotate this around voila
lethal walrus
Hence your circle presentation is more on the nose than you imagined.
why dont they explain it like this in the first place lol, its pretty interesting thanks
i still have to study complex numbers better but i more or less understand
covert yoke
Well, it's one of those things where simply presenting it up front might be a little difficult to explain.
safe radishBOT
Available help channel!
exotic charm
when ur sketching fields eg from let's say R to R^3
safe radishBOT
when ur sketching fields eg from let's say R to R^3
exotic charm
how can u differentiate when it's a vector field or when u get a surface?
cz in my eyes both take an input and output a vector but vector fields when sketched will show only vectors whereas surfaces will be like actual 3d structures
so im a bit confused where to make the distinction
safe radishBOT
@exotic charm Has your question been resolved?
median vigil
a vector field would be something like a function R^3 to R^3 (you take each point in 3D space as an input and show an arrow as output)
a function from R to R^3 would usually be visualized as a curve in R^3 (which shows all outputs of the function for a given set of inputs)
and a surface would be a function R^2 to R^3 (again only showing the outputs)
exotic charm
a vector field would be something like a function R^3 to R^3 (you take each poin...
but
in this qu it's not from R^3 ?
ik it says it's a vector field but i wanna know like how u would determine it without it saying it
median vigil
here they are interpreting it as a function from (R,φ,z) but which only changes based on the first coordinate
alternatively you can view it as a function which depends on (x,y,z) but rewritten into a different coordinate system
safe radishBOT
@exotic charm Has your question been resolved?
exotic charm
here they are interpreting it as a function from (R,φ,z) but which only changes ...
but surfaces are normally given as
a subset of R^2 to R^3 for instance
or is it only a vector field when it's R^3 to R^3
median vigil
you can interpret the same formula for different things potentially. so in this case we have a vector field because we are interpreting it as a function R^3 to R^3 even though the formula it uses only depends on one coordinate
exotic charm
that makes sense
fathom jewel
hi gang
fathom jewel
can someone check my proof please 🙏🏻
flat frigateBOT
𝔸dωn𝓲²s
arctic raven
can someone check my proof please 🙏🏻
you need an actual example of such a group
fathom jewel
I gave one I thought
arctic raven
the "for example" doesn't read well at all; just present the example
however, it's not immediately clear how + is defined in this set G (you're calling it D too?)
fathom jewel
the first part isn't even necessary then though
Why, if G is cyclic with order 163 then there exists such group, right?
arctic raven
Why, if G is cyclic with order 163 then there exists such group, right?
this doesn't make sense syntactically
frank horizon
why not just take Z_163
fathom jewel
however, it's not immediately clear how + is defined in this set G (you're calli...
D is a subset of integers
addition works like in Z
arctic raven
why give it two names though?
arctic raven
addition works like in Z
then it doesn't work. what's 81+1?
fathom jewel
fathom jewel
why not just take Z_163
this is actually the best?
stoic dune
The proof is just a single example of said group.
There is, of course, a cyclic group of any order
arctic raven
cyclic groups of fixed order are unique up to iso, so you could say Z_163 is the only one
stoic dune
I figure the book you're using should state that so I'm not sure how much more you could add
sonic dragon
**𝔸dωn𝓲²s**
This math looks intimidating
stoic dune
Group theory is fun af
fathom jewel
This math looks intimidating
My favorite isomorphism joined 🥰
sonic dragon
I just wanted to see what real mathematician would do for math
That is complicated and igtg
What is group theory
arctic raven
well 163 being prime doesn't really affect the proof, but it does imply that there's only one group of order 163 up to iso
stoic dune
What is group theory
A group is a "symmetry" of an object.
For example, there's a bunch of ways you can take a square, pick it up, flip it around, and put it back down the way it was. There's a group that represents this.
Groups get a neat algebraic structure, studying them can be fun.
fathom jewel
well 163 being prime doesn't really affect the proof, but it does imply that the...
ahh
arctic raven
so if all groups of prime order are cyclic. and if any two cyclic groups of the same order are isomorphic, then there's only one group of each prime order
sonic dragon
A group is a "symmetry" of an object.
For example, there's a bunch of ways you...
That sounds super interesting if I were a math student I definitely would take it
fathom jewel
so if all groups of prime order are cyclic. and if any two cyclic groups of the ...
can you give an example for Z_2
fathom jewel
yes sorry
arctic raven
why so small though?
fathom jewel
😭 I just wanna understand how to construct okkk
arctic raven
you don't have to prove it
there's only two bijections between the two anyway
where does 0 go, where does 1 go
fathom jewel
0 goes to 1
1 to 2?
fathom jewel
how?
arctic raven
you can prove it
let G, H be groups with p : G -> H a group hom. show that p(e_G) = e_H
e_G id in G, e_H id in H
in fact, homomorphisms are meant to do this anyway; they're meant to preserve all the structure (multiplication, identities, and inverses)
it just so happens that every function between groups that preserves the multiplication bit implies preservation of id's and inverses
fathom jewel
Define $\Phi : \mathbb{Z}_2 \to \mathbb{Z}_3^$ with $a+b \mapsto ab, : a,b \in \mathbb{Z}_2 \$
Then [ \Phi(a+b) = ab = (a\cdot 1)(b\cdot 1) = \Phi(a+0)\Phi(b+0) = \Phi(a)\Phi(b) \quad \forall a,b \in \mathbb{Z}2. ]
But this already assumes that $0 \mapts to 1$ or rather $id{\mathbb{Z}2} \mapsto id{\mathbb{Z}_3^}.$
arctic raven
monoids are groups without inverses, and a monoid homomorphism actually needs to be defined around mapping ids to ids
arctic raven
Define $\Phi : \mathbb{Z}_2 \to \mathbb{Z}_3^*$ with $a+b \mapsto ab, \: a,b \in...
sorry, this isn't well-defined
0 isn't in Z_3*
stoic dune
I like using Cayley tables for the small groups
You have to play them like a Sudoku. Try making a 2×2 sudoku, you don't get any choices on how you fill it in. There's only one group with two elements
Oh but you're asking about all prime groups, mb
fathom jewel
Define $\Phi : \mathbb{Z}_2 \to \mathbb{Z}_3^$ with $a+b \mapsto ab, : a,b \in \mathbb{Z}_2 \$
Then [ \Phi(a+b) = ab = (a\cdot 1)(b\cdot 1) = \Phi(a+0)\Phi(b+0) = \Phi(a)\Phi(b) \quad \forall a,b \in \mathbb{Z}2. ]
But this already assumes that $0 \mapsto 1$ or rather $\iota\delta{\mathbb{Z}2} \mapsto \iota\delta{\mathbb{Z}_3^}.$
arctic raven
Define $\Phi : \mathbb{Z}_2 \to \mathbb{Z}_3^*$ with $a+b \mapsto ab, \: a,b \in...
same thing, phi(0+0)=0 is not in Z_3*
fathom jewel
The definition is flawed argh
arctic raven
but also, in defining phi around sending sums to sums, you need to verify that it produces a well-defined function
queen ingot
i'm a little confused why you're defining phi by a+b maps to ab
normally you would define it by phi(a) = something
arctic raven
consider a mapping Z_2 -> Z_2 defined the same way. then 0+0 = 1+1 = 0, but 0*0 = 0 != 1 = 1*1
fathom jewel
In my script there is this def.
(G, theta_1) and (H, theta_2)
Phi(a theta_1 b) = Phi(a) theta_2 Phi(b)
stoic dune
You're trying to come up with a clever way to make this mapping happen, but all you need to do is say where each element goes
fathom jewel
for all a,b in G
fathom jewel
You're trying to come up with a clever way to make this mapping happen, but all ...
yea i know
i jsut wanted to generalize it
arctic raven
you mean, finding all the isomorphisms between two groups?
fathom jewel
Like Z_m isomorph to Z_(m+1)*
arctic raven
hm
fathom jewel
actually thats false
arctic raven
counterexample?
queen ingot
we saw an example of a multiplicative group earlier that had the klein 4 group as a subgroup
so that group isn't isomorphic to any Z_m
because Z_m is always cyclic
fathom jewel
I just wanted to show (Z_2, +) ~= (Z_3*, *) is indeed an isomorphism using by showing first there is an homomorphism
queen ingot
so to define phi you only need to define phi(0) and phi(1)
arctic raven
I just wanted to show (Z_2, +) ~= (Z_3*, *) is indeed an isomorphism using by sh...
in such small cases, coloring the cells of the two operation tables suffices
fathom jewel
so to define phi you only need to define phi(0) and phi(1)
so phi(a) = a+1?
fathom jewel
ok
queen ingot
phi(a+b) = phi(a)*phi(b)
arctic raven
it's a little wacky because you kinda need to view each of the elements as entries of Z (1 who?)
but it seems alright, just funny
queen ingot
it gets confusing with all the operations flying around
to be more precise you can just say
phi(0)=1
phi(1)=2
arctic raven
yeah because addition in Z2 compared to Z3 is very different
the "a+1" is very specific about treating a as simultaneously an entry in Z2 as well as Z3
queen ingot
i mean you're really taking addition from neither {Z2, +} nor from {Z3, *}
you're taking it from Z i guess
arctic raven
right
queen ingot
this is one of the confusions i have about group theory
fathom jewel
I am actually having trouble lmao
queen ingot
have to think carefully about the operations
fathom jewel
have to think carefully about the operations
I dont think it works 
Define $\Phi : \mathbb{Z}_2 \to \mathbb{Z}_3^*$ with $u \mapsto u+1\$
Then [ \Phi(a+b) = (a+b)+1 = ab+a+b+1-ab = ab+a+b+1 = (a+1)(b+1) = \Phi(a)\Phi(b) \quad \forall a,b \in \mathbb{Z}_2. ]
flat frigateBOT
𝔸dωn𝓲²s
fathom jewel
It's wrong btw
I can't believe MATH
such simple sets so hard to find a proper function
arctic raven
there's nothing wrong with saying phi maps 0 to 1 and 1 to 2
then checking all four combinations
fathom jewel
but when I had (Z_2 x R_(>0)) ~= R\{0} it was so simple to say (z,r) |-> zr
arctic raven
or three if you trust they're both abelian
you can also define your function inductively
arctic raven
yeah actually I just noticed you killed a poor innocent -ab
fathom jewel
haha i was working from both sides
but just doesnt work, maybe you cant find one and you have to go all combinations
safe radishBOT
@fathom jewel Has your question been resolved?
fathom jewel
I AM SEXY
fathom jewel
Define $\Phi : \mathbb{Z}_2 \to \mathbb{Z}_3^*$ with $u \mapsto 2^u. \$
Then [ \Phi(a+b) =2^{a+b} = 2^a \cdot 2^b = \Phi(a)\Phi(b) \quad \forall a,b \in \mathbb{Z}_2. ]
flat frigateBOT
𝔸dωn𝓲²s
devout shale
show that Phi is well-defined
devout shale
so it is a finite surjection and hence..? Also, show that Phi is well-defined
arctic raven
this wouldn't produce a valid mapping in any other context
Z_3 -> Z_4* for instance
devout shale
Sir all that is left is to do what I said (and fill in the ...)
arctic raven
I really don't think that there's a lot of insight you could get from describing your function this way
fathom jewel
why not
arctic raven
it's too small
the groups are too small rather
you could just draw their operation tables
and color the cells that are friends via a mapping
squint so you can't see the symbols, and the colors match
fathom jewel
ye i just wanted to find a function and show that way there exists a homomorphism between Z_2 and Z_3*
devout shale
if you could first prove that any group of prime order is cyclic, then it is easy to prove that any two groups of the same prime order are isomorphic by sending the generator of one group to the generator of the other
fathom jewel
even if it's ridiculous
fathom jewel
if you could first prove that any group of prime order is cyclic, then it is eas...
hmm okay
arctic raven
squint so you can't see the symbols, and the colors match
I did this in a "lesson" once to show that two instances of K4 (rectangle symmetries & 2-switch light panel) were isomorphic
it quickly and pictorially shows operation preservation for every combination of elements mashed together, plus bijectivity
queen ingot
you can handle it with cases
fathom jewel
if you could first prove that any group of prime order is cyclic, then it is eas...
Let $\abs{G} = p \in \mathbb{P}$ and $a \in G$ such that $a \neq e.\$
Then $[a] \leq G.$ The Theorem of Lagrange implies that $\abs{a} \mid \abs{G}.\$
Since $\abs{G}$ is prime it implies $\abs{a} = 1$ or $\abs{a} = \abs{G}$ but $a\neq e.\$
Thus $\abs{a} = \abs{G} = p$ implies $[a] = G.$
flat frigateBOT
𝔸dωn𝓲²s
queen ingot
$\phi(0+0)=\phi(0)=1=1\cdot 1=\phi(0)\phi(0)$\
$\phi(0+1)=\phi(1)=2=1\cdot 2 = \phi(0)\phi(1)$\
$\phi(1+1)=\phi(0)=1=2\cdot 2 = \phi(1)\phi(1)$
flat frigateBOT
Axe
fathom jewel
yea but i didnt want cases lol i was interested into finding a generic function and it worked out even if it's for some ludicrous lol
fathom jewel
$\phi(0+0)=\phi(0)=1=1\cdot 1=\phi(0)\phi(0)$\\
$\phi(0+1)=\phi(1)=2=1\cdot 2 = ...
Or what do you think
fathom jewel
**𝔸dωn𝓲²s**
About this
queen ingot
idk, have you really thought about what it means when a=b=1?
what operation does the dot represent?
fathom jewel
Multiplication
queen ingot
in Z or in Z*_3 ?
fathom jewel
Z*_3
queen ingot
so implicit here is that 2^(1+1) = 2^0 = 1 = 2*2
or something like that
for the case a=b=1
lean otter
Hi
queen ingot
yeah that's how i got from 2^(1+1) to 2^(0)
queen ingot
do we have power laws for addition mod 2 and multiplication in Z*_3 ?
if so then i think your proof works
queen ingot
...sleep?
fathom jewel
do we have power laws for addition mod 2 and multiplication in Z*_3 ?
This is keeping me awake
queen ingot
fathom jewel
<:KEK:586240877358350341>
Let a,b in Z_2 then 2^a2^b = 2^(a+b).2^0 * 2^0 = 1 * 1 = 2^(0+0) = 2^0 = 1
2^0 * 2^1 = 1 * 2 = 2^(0+1) = 2^1 = 2
2^1* 2^1 = 2* 2 = 4 = 1 mod 3
= 2^(1+1) = 4 = 1 mod 3
Seems to work
fathom jewel
Right
devout shale
is this ruckus about whether or not the power law holds in Z3*?
queen ingot
and like, is it a fluke that it works for these operations, or will this line of thinking help you with bigger additive and multiplicative groups
i guess
devout shale
This homomorphism will not be helpful for bigger groups
yet the power law does hold
fathom jewel
It works for this purpose
devout shale
indeed
fathom jewel
Austinnnn
queen ingot
i guess i'm just trying to understand if this makes sense
cause i'd rather just treat it with cases
fathom jewel
Homomorphism
devout shale
?
queen ingot
it's an argument that phi is a homomorphism
fathom jewel
Yes
frozen shoal
i guess i'm just trying to understand if this makes sense
which part are you having trouble with?
fathom jewel
😭
fathom jewel
This is really important to me ok
Everybody saying this is stupid
queen ingot
so implicit here is that 2^(1+1) = 2^0 = 1 = 2*2
how is this step justified
fathom jewel
Let a,b in Z_2 then 2^a2^b = 2^(a+b).2^0 * 2^0 = 1 * 1 = 2^(0+0) = 2^0 = 1
2^0 *...
.
queen ingot
2^(a+b) = 2^a * 2^b
frozen shoal
I think a proof of the power law would go like the quotient map f: Z-> Z/nZ is a ring homomorphism, so since power law equations hold on the left they hold on the right
fathom jewel
2^0 * 2^0 = 1 * 1 = 2^(0+0) = 2^0 = 1
2^0 * 2^1 = 1 * 2 = 2^(0+1) = 2^1 = 2
2^1* 2^1 = 2* 2 = 4 = 1 mod 3
= 2^(1+1) = 2^0 mod 2 = 1
I am done
I think I will ask tomorrow Denascite
Still thanks for the help ❤️
.solved
safe radishBOT
safe radishBOT
Available help channel!
red lotus
hi
red lotus
Statistics. so i was calculating for XLB or the lower class boundary
i'm finding 6th decile.
i
53-56
57-60
f
5
4
cf
5
9
(all respectively)
with n = 50.
I was diving 50/10 then it resulted 5. so the question is what would be my cfb?
wait no, what would be my xlb and by cfb then?
is my XLB 56.5 or 52.5?
if my XLB is 52.5, then I have cfb = 0
if my XLB is 56.5, then my cfb would be 5.
so which is which?
lean otter
red lotus
this is not an assignment or test. i'm creating my own reviewer all about the lessons of statistics
can someone help me please
safe radishBOT
@red lotus Has your question been resolved?
mighty mango
It (A,B) is a point on y= 2^x then which of the following points is on y= 2^(x+2)
safe radishBOT
It (A,B) is a point on y= 2^x then which of the following points is on y= 2^(x+2...
earnest wagon
Seems soo
mighty mango
Its 8B , A
earnest wagon
wait
mighty mango
Somehow
mighty mango
Yeah? I think
earnest wagon
so for x = A, y = 4B
mighty mango
He didnt mention anything about that anyways
fallen heart
What quadratic
mighty mango
Wheres the quadratic
earnest wagon
exponential function always has 1 y for 1 x
so 4B is supposed to be the only y for x = A
earnest wagon
So what- no way my book is just straighr up wrong
So your book answer is indeed wrong, I think
mighty mango
so 4B is supposed to be the only y for x = A
What if the function is y=2^x+3
mighty mango
No i mean- i might be seeing the question wrong 😭
earnest wagon
Lol
earnest wagon
I see
mighty mango
So like wth
earnest wagon
So like wth
I asked chatgpt
what 2 and 3 looks like in arabic
ChatGPT said:
ChatGPT
In Arabic, the numbers 2 and 3 are written as:
2: ٢
3: ٣
These are the Arabic numerals used in the Arabic-speaking world, which are different from the standard Western digits.
Not same at all, focus on letter
mighty mango
what 2 and 3 looks like in arabic
ChatGPT said:
ChatGPT
In Arabic, the numbers 2...
Ik my language xd anyways so if y=2^x+3 wouldnt tbat worj
Work
Like it would be y=8b
earnest wagon
Ik my language xd anyways so if y=2^x+3 wouldnt tbat worj
- 3 on exponent or bottom?
mighty mango
+ 3 on exponent or bottom?
Exponent
mighty mango
It says (8B , A) tho
mighty mango
definitely 3 then
Well it is not a three and the question is wrong
earnest wagon
there r 2 possibility: either the answer is wrong or the question has misprinting
earnest wagon
Well it is not a three and the question is wrong
es
earnest wagon
Np
mighty mango
.close
runic tinsel
does this proof for proving limit law seem ok
safe radishBOT
does this proof for proving limit law seem ok
runic tinsel
im tryna prove that lim (fx + gx) = lim fx + lm gx
there is a way to do this using limit definition, but i tried to prove using continuity
lemme know if this seems right
split holly
emm
lemme find my analysis textbook
This is the prove given in Analysis I from Terence Tao
Hope this helps
To prove the limit laws for sequence, just consider two Cauchy Sequence, and take epsilon/2 for each
safe radishBOT
@runic tinsel Has your question been resolved?
runic tinsel
To prove the limit laws for sequence, just consider two Cauchy Sequence, and tak...
thank you
left gyro
.close
dreamy snow
safe radishBOT
ebon mesa
!status
safe radishBOT
What step are you on?
1. I don't know where to begin.
2. I have begun but got stuck midway.
3. I got an answer but I was told that it's wrong.
4. I got an answer and would like my work checked.
5. I have a question about someone else's work/solution.
6. I have completed the problem and don't need help anymore. Thank you.
7. None of the above
2
ebon mesa
can you share work
dreamy snow
I tried simplifying it but Im not getting close
yes it's not in English so might take a second to translate it
rocky island
hi guys
noble kiln
Btw welcome to mathcord @dreamy snow and @rocky island
rocky island
can anyone send me properties of logarithms like very basic
dreamy snow
thank you
safe radishBOT
can anyone send me properties of logarithms like very basic
To ask for mathematics help on this server, please open your own help channel or help thread. See #❓how-to-get-help for instructions.
safe radishBOT
To ask for mathematics help on this server, please open your own help channel or help thread. See #❓how-to-get-help for instructions.
noble kiln
Rip
safe radishBOT
To ask for mathematics help on this server, please open your own help channel or help thread. See #❓how-to-get-help for instructions.
versed wave
rewrite tanx as sinx/cosx
dreamy snow
rewrite tanx as sinx/cosx
did that
rocky island
wait ill get back
dreamy snow
Did multiplying both sides by cos x help
oh wait lemme try that
ebon mesa
Did multiplying both sides by cos x help
i dont see how that helps
versed wave
rewrite tanx as sinx/cosx
just do this first
i assume you should have something like
versed wave
$\int\frac{\sin x}{\cos x-\cos^2x}\dd x$
flat frigateBOT
versed wave
for which you can simply do u = cosx
dreamy snow
for which you can simply do u = cosx
I don't understand that honestly
versed wave
do you know what is u-substitution?
dreamy snow
but dividing by cos x worked
dreamy snow
do you know what is u-substitution?
I don't study in English so I'm not sure
versed wave
what language are you native in
dreamy snow
Arabic
ebon mesa
u sub is simple
just let u=cosx and rewrite your integral
also if you differentiate both sides you get du=-sinxdx
isolate dx so that dx=du/-sinx
noble kiln
O ya i didn't see that
ebon mesa
then plug that in to the integral as well
versed wave
isolate dx so that dx=du/-sinx
why would you do that
ebon mesa
until you have everything in terms of u in the integral
ebon mesa
to rewrite dx?
dreamy snow
what does 1 over cos mean in English again
gotta start learning maths in English Arabic isn't helping
versed wave
to rewrite dx?
that's just a needless step, you can immediately spot sinxdx = -du
ebon mesa
yeah the same thing
versed wave
same thing, but still a needless step 
versed wave
what does 1 over cos mean in English again
sec?
ebon mesa
sometimes its not as easy to see as in this integral
thats why i always do it as a seperate step
ebon mesa
separate
dreamy snow
yeah
ebon mesa
its abs
noble kiln
i think it should be sec-1
Is absolute value
dreamy snow
Arabic is written from right to left so I mixed this up
dapper star
doesn't matter, its inside absolute value here
junior raven
not really
dreamy snow
cuz that took too long to figure out
junior raven
unless you can see the result
ebon mesa
thats the beauty of integrals
junior raven
you just throw random stuff at it and pray it works
ebon mesa
no systematic way to solve them
dreamy snow
thats the beauty of integrals
beauty indeed
ebon mesa
kinda like geometry
dreamy snow
I just finished the derivative applications
are they a lot easier than integrals or am I still not used to this
junior raven
differentiation is definitely easier
noble kiln
are they a lot easier than integrals or am I still not used to this
Differentiation is like expanding in algebra but integration is like factorising
One is tedious but doable and the other is impossible if you can't see it
dreamy snow
yes that's it, there's always something small that I just gotta figure out
noble kiln
Fr
dreamy snow
I mean its fun but getting stuck like that on an exam isn't gonna feel good 😭
I think I shouldnt be chatting in a help channel so thank you guys for helping me
really welcoming too I appreciate that
noble kiln
I think I shouldnt be chatting in a help channel so thank you guys for helping m...
Is ok im bad at calc 😭
dreamy snow
Is ok im bad at calc 😭
I don't think many people can say they're good at it😭
noble kiln
true
chrome owl
,rotate
ebon mesa
can you see why AF=AD, DB=BE and so on
gentle edge
can you see why AF=AD, DB=BE and so on
Yes
ebon mesa
mark them x, y and z
gentle edge
mark them x, y and z
I did
ebon mesa
now what is 2x+2y+2z
gentle edge
perimeter*
Ye
ebon mesa
now what is just x+y+z
gentle edge
now what is just x+y+z
15
ebon mesa
what is x+y?
gentle edge
15-z
ebon mesa
yeah
ebon mesa
np
gentle edge
.close
safe radishBOT
safe radishBOT
Available help channel!
lean otter
find the number of possible values for the natural number n such that $2^{50}$ is a factor of $n!$ but $n!$ is not a multiple of $3^{40}$ ?
safe radishBOT
find the number of possible values for the natural number n such that $2^{50}$ i...
buoyant shadow
darn i'm hearthstoning
wait 12 minutes
i mean playing the game
not commuting home
queen ingot
i think the first condition gives you a lower bound for n, and the second condition gives an upper bound