oh because I didn't account for repetitions
#help-49
crystal plover
mint ravine
i think you divide by 3?
crystal plover
by 3! no?
mint ravine
what? i said 3 not 3!
crystal plover
I am saying it should be divided by 3!
mint ravine
how did you get 3! tho
crystal plover
you make 3 picks ABC, but I didn't account that the order doesn't matter and there's 3! ways to permute ABC
midnight plankBOT
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Available help channel!
inland patio
Does anyone have any idea how I can approach this exercise? The $T$-annihilator of $v_1$ is the polynomial $q(t)$ of least degree such that $q(T)(v_1) = 0$.\
Here's a useful fact. Let $K_{\phi_1}$ be the space of all vectors $x\in V$ such that $(\phi_1(T))^p(x)=0$ for some positive integer $p$. Then the $T$-annihilator of a nonzero vector in $K_{\phi_1}$ is $(\phi_1(t))^q$ for some positive integer $q$, and the $T$-annihilator divides the minimal and characteristic polynomial. Yet I don't see how I can prove the existence of $v_1$ such that $\phi_1(t)$ is the $T$-annihilator, given the characteristic polynomial in the exercise.
midnight plankBOT
Does anyone have any idea how I can approach this exercise? The $T$-annihilator ...
grand pondBOT
psie
midnight plankBOT
@inland patio Has your question been resolved?
wheat nymph
Does anyone have any idea how I can approach this exercise? The $T$-annihilator ...
What's ur definition of a T-annihilator?
wheat nymph
O sry I just saw
I'm pretty sure I can generalize ur fact tho
It doesn't have to just be powers of monic irreducibles
inland patio
It doesn't have to just be powers of monic irreducibles
what does it refer to now? 🙂
wheat nymph
In general the T-annihilator should just divide the min poly, and hence the char poly
inland patio
In general the T-annihilator should just divide the min poly, and hence the char...
right 👍 it doesn't need to be irreducible
oak nymph
What is $\beta_v$ ?
grand pondBOT
Herzog
inland patio
What is $\beta_v$ ?
Yes, sorry. Though I haven't reached that part yet, $\beta_v$ is the $T$-cyclic basis generated by $v$. So ${v, T(v), T^2(v),\ldots,T^{n-1}(v)}$.
grand pondBOT
psie
inland patio
It doesn't have to just be powers of monic irreducibles
your comment is correct, however, for the space $K_\phi$ that I defined above, the $T$-annihilator is indeed of the form $(\phi(t))^p$ for some positive integer $p$.
grand pondBOT
psie
wheat nymph
yeah
i think u first should show that $f$ is also the min poly of $T$, and then prove that the $\ker\phi_i$ are nontrivial
grand pondBOT
Bob the Builder
inland patio
i think u first should show that $f$ is also the min poly of $T$, and then prove...
Ok, makes sense. So I know that the minimal polynomial and the characteristic polynomial share the same monic, irreducible factors. So except for $(-1)^n$ perhaps, the minimal polynomial is $\phi_1(t)\phi_2(t)$. However, I don't think I understand your second hint. Why are we interested in the null spaces of $\phi_i(T)$?
grand pondBOT
psie
wheat nymph
u can deduce from ur useful fact that if $v \in \ker \phi(T)$, then $v$ has $T$-annihilator $\phi(t)$
grand pondBOT
Bob the Builder
wheat nymph
ofc $v \neq 0$ as well
grand pondBOT
Bob the Builder
inland patio
u can deduce from ur useful fact that if $v \in \ker \phi(T)$, then $v$ has $T$-...
Hmm, from my useful fact we have that the $T$-annihilator is $(\phi(t))^p$ for some positive integer $p$. The thing is, I don't see how to exclude the case $p>1$. You see, if $p>1$, then it can still divide the minimal polynomial, the quotient will simply be $0$ though.
grand pondBOT
psie
wheat nymph
oic what u mean
maybe ill edit this fact to be that if $f(T)v=0$, then the $T$-annihilator of $v$ divides $f$ (alternatively, u could have defined the $T$-annihilator to be the monic poly of largest degree with this property)
grand pondBOT
Bob the Builder
wheat nymph
so if $v \in \ker \phi_i(T)\setminus{0}$, then as $\phi_i$ is irreducible, u can use this property
grand pondBOT
Bob the Builder
inland patio
so if $v \in \ker \phi_i(T)\setminus\{0\}$, then as $\phi_i$ is irreducible, u c...
ok, makes sense. So $v\in\ker\phi_i(T)\setminus{0}$ forces $\phi_i(t)$ to be the $T$-annihilator since it is irreducible, i.e. there is no polynomial $q(t)$ of smaller degree such that $q(T)(v)=0$ (for if there would be, then it would divide $\phi_i(t)$, impossible). I feel like I'm close now. I only need to show $\ker\phi_i(T)$ is nontrivial.
grand pondBOT
psie
inland patio
so if $v \in \ker \phi_i(T)\setminus\{0\}$, then as $\phi_i$ is irreducible, u c...
Do you think I could argue by contradiction in establishing that $\ker\phi_i(T)$ is nontrivial? Suppose it is trivial, i.e. ${0}$. Then $\phi_i(T)$ are both one-to-one, and so is $\phi_1(T)\phi_2(T)$. But then this possibly can't be minimal polynomial, contradiction.
grand pondBOT
psie
noble forge
Hmm, from my useful fact we have that the $T$-annihilator is $(\phi(t))^p$ for s...
The characteristic polynomial of a linear operator T on an n -dimensional vector space V is always a monic polynomial of degree n. If the quotient of the minimal polynomial is zero, this implies that the characteristic polynomial is degree 0. The problem asks for vectors ( v_1, v_2 \in V ) with specific T-annihilators. If ( V = {0} ), the only vector is 0, whose T-annihilator is the zero polynomial, which is not irreducible or monic
grand pondBOT
a guy with no beard
inland patio
The characteristic polynomial of a linear operator T on an n -dimensional vector...
ok, when you say the quotient of the minimal polynomial, do you mean c(t)/m(t) = q(t), where c(t) is the characteristic polynomial and m(t) the minimal polynomial?
noble forge
Yes, if c(t) is zero, then p can be > 1. But it makes no sense in the context of this problem
midnight plankBOT
Available help channel!
graceful ferry
$\lim_{(x,y) \to (0,0)} \frac{1-cos(\pi xy)+sin(\pi (x^2+y^2))}{x^2+y^2}$
midnight plankBOT
$\lim_{(x,y) \to (0,0)} \frac{1-cos(\pi xy)+sin(\pi (x^2+y^2))}{x^2+y^2}$
grand pondBOT
prograce
graceful ferry
$\lim_{(x,y) \to (0,0)} \frac{sin(\pi (x^2+y^2)}{x^2+y^2} +\frac{1-cos(\pi xy)}{x^2+y^2}$
grand pondBOT
prograce
graceful ferry
Does that help?
latent wadi
I think for sin(pi * (x^2 + y^2))/(x^2 + y^2) you could substitute u = x^2+y^2
graceful ferry
Yes I thought so too
latent wadi
meager wraith
the first seems to be pi and second term 0
graceful ferry
**prograce**
the 1 bothers me for the second expression
meager wraith
gotta check tho
graceful ferry
the first seems to be pi and second term 0
How did you do it for second term
latent wadi
wait for the second one you can fix x = 0 then it is equal to lim y -> 0 (1-cos(0))/y^2? I dont really know but maybe?
meager wraith
1-cos(2A)=2sin^2(A). changing em to polar coordinates help
U get the $\lim_{(t) \to (0)} \frac{1-cos(t)}{t}$
grand pondBOT
Kokkeeng
graceful ferry
wait for the second one you can fix x = 0 then it is equal to lim y -> 0 (1-cos...
Hm this doesn't give much, when you substitute values usually ur looking to show theres no limit
latent wadi
there was an "i dont really know" there
graceful ferry
U get the $\lim_{(t) \to (0)} \frac{1-cos(t)}{t}$
What about the pi ?
meager wraith
subsitute everything with t
graceful ferry
Can't do that because denominator is different
I mean the term is "homogenous" but yea like u said i could change to polar coordinates but idk what to do with the 1 in the numerator
graceful ferry
1-cos(2A)=2sin^2(A). changing em to polar coordinates help
Maybe i could use this
tired osprey
let x = rcos(theta) and y = rsin(theta)
osccilating nature will make this much easier
meager wraith
$t=\pi r^2 \cos\theta \sin\theta \newline \lim_{r \to 0} \frac{1 - \cos(\pi r^2 \cos\theta \sin\theta)}{r^2} {=} \pi \cos\theta \sin\theta \cdot \lim_{t\to 0} \frac{1-\cos t}{t} = 0$
grand pondBOT
Kokkeeng
grand pondBOT
graceful ferry
Is this right?
tired osprey
I think u could just say 1 - cos(pi r^2 sin(theta) cos(theta)) approaches 0, as sin(theta)cos(theta) doesn't approach anything because of it's osccilating nature, thus the final limit will be lim of r -> 0 of sin(r^2 pi )/ r^2, which wil approach pi.
so pi will be the final answer
graceful ferry
1-cos(2A)=2sin^2(A). changing em to polar coordinates help
I did the same thing basically but after substituting the 1-cos(theta) with the identity... is it still fine ? I think so no?
I proved it approaches 0 (not osicciltates) in the red colored writing using sandwich thm and changing to polar coordinates for the term on the right side
tired osprey
Yeah it's fine I think
midnight plankBOT
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arctic herald
i have abs no idea what to. do
midnight plankBOT
i have abs no idea what to. do
latent wadi
It's basically base 3
What is the biggest number (a, b, c, d, e, f, g, h) that is less than 2010?
snow ledge
They couldn’t write this problem in a more confusing way lol
latent wadi
Exactly right
latent wadi
What is the biggest number (a, b, c, d, e, f, g, h) that is less than 2010?
Wait forget this
First step translate 2010 into base 3
snow ledge
Notice that you have to use 2187 because if you don’t you can never get to 2010 (even if you add all the others together you are less than 2010)
arctic herald
yes
snow ledge
Oh if they know base 3 that makes way more sense okay
arctic herald
Notice that you have to use 2187 because if you don’t you can never get to 2010 ...
u would need trail and error that way
latent wadi
Just translate it into base 3 first
arctic herald
u just divide it as much as possible
latent wadi
We will guide u after u send the translated number
arctic herald
and like reverse remainder
latent wadi
Ye
snow ledge
u would need trail and error that way
Yeah base 3 is more systematic and reliable
arctic herald
could u explain the concept, i dont have anything on me that can help with translation
latent wadi
u just divide it as much as possible
This?
latent wadi
Huh
arctic herald
0112022
arctic herald
wait but how do u know to just use base 3 when u see this question
arctic herald
is it reversed
latent wadi
U mean it's 2202110?
Correct
latent wadi
so in theory if was power of 4s i would use base 4?
Yes
Ok, it says we can use the numbers at most one time right?
latent wadi
U mean it's 2202110?
Ex : the 2 at the front corresponds to 2 x 729
latent wadi
Expressing 2010 as a sum of a b c d e f g and h
arctic herald
yes but like how are you planning on doing it (not the solution)
latent wadi
Ok, is 2 x 729 = 2187 - 729?
arctic herald
yes
latent wadi
Note that 2187 = 3 x 729
latent wadi
yes
Nice
Now, let's expand 2202110 as : 2 x 729 + 2 x 243 + 0 x 81 + 2 x 27 + 1 x 9 + 1 x 3 + 0 x 1
Makes sense?
arctic herald
wow
latent wadi
Do you know what to do from here?
latent wadi
Ok
Just remember : 2x = 3x - x, example : 2 x 729 = 2187 - 729
Then rewrite everything which is in the form 2x
arctic herald
i c
latent wadi
Then simplify (cancel out things which are equal)
arctic herald
i see
midnight plankBOT
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Available help channel!
graceful ferry
given $g(x) =
\begin{cases}
e^{-1/x^2}, & x \neq 0 \
0, & x = 0
\end{cases}$ \
I need to prove for $f(x,y)=(x^2+y^2)(g(x)g(y)$ that $lim_{(x,y) \to (0,0)}f(x,y)=0$ and that $\lim_{y \to 0}\lim_{x \to 0}f(x,y) \lim_{x \to 0}\lim_{y \to 0}f(x,y)$ don't exist
midnight plankBOT
given $g(x) =
\begin{cases}
e^{-1/x^2}, & x \neq 0 \\
0, & x = 0
\end{cases}$ \...
grand pondBOT
prograce
graceful ferry
For $lim_{(x,y) \to (0,0)}f(x,y)=0$ this is what I did: \
Let $\epsilon > 0$ Choose $\delta = ?$ such that for every $\sqrt{x^2+y^2} < \delta$ : $|(x^2+y^2)e^{\frac{-1}{x^2}}e^{\frac{-1}{y^2}}| < \epsilon$
grand pondBOT
prograce
graceful ferry
How do I manipulate the expression so I can find the appropriate delta?
visual panther
Maybe use exp(-1/x^2) < exp(-1/(x^2+y^2)) = exp(-1/delta^2), same for exp(-1/y^2) as a starter.
midnight plankBOT
@graceful ferry Has your question been resolved?
grand pondBOT
visual panther
Looks fine. I would suggest limiting delta to 1 and you can solve exp(-2/delta^2) = epsilon for delta.
graceful ferry
How do I limit delta to 1 ?
visual panther
Your result will be something like d < min (1 ; "expression with Epsilon"). Just state, that delta has to be <= 1.
visual panther
In this case we care about very small values for Epsilon and Delta.
So it might be helpful, to limit Epsilon too. That way, you can easily apply the ln() on the equation.
graceful ferry
The other expression is sqrt(2/ln(epsilon)) so i could say epsilon <1 so ln is never undefined
But question I have is this allowed to limit epsilon for formal proof ? I just sau that since epsilon is a very small value then let epsilon <1 ?
visual panther
I do not see a problem, because we prove the limit for arbitrary small values of Epsilon.
graceful ferry
Okay!
visual panther
On the other hand, you could always make a seperate case fpr Epsilon >= 1, but I do not think that it is necessary.
graceful ferry
I will keep that in mind for other questions ty
Can I move on to the other parts pf the question, which I'm also struggling with ?
Where the red question mark is, is that step legal ? Because then the limit =0 and it does exist..
,rccw
grand pondBOT
graceful ferry
For e^-1/y^2 -> 0 so multiply everything by 0 ?
graceful ferry
It's ok
visual panther
I do not see, why these limits should not exist. I clearly miss something here.
graceful ferry
Maybe I did a mistake
Oh no I made a mistake i was supposed to find g(x) such that limit to 0 doesnt exist oops . I will retry from beginning I will coke back later
Is cos(1/x)+2 ok?
Is it bounded, limx->0 doesn't exist, g(x)!=0 for all x!=0
visual panther
DO I understand this correctly, this g(x) will be put into f(x,y) = (x^2+y^2)g(x)g(y) ?
graceful ferry
Yes
visual panther
This should work, the limit for (x,y)->(0,0) is 0 and the single inner limit will not exist, I think.
graceful ferry
Great
Sorry I have to go through the proof again I thought maybe I can use sandwich thm? Since the g(x) is bounded? I could try it with f(x,y) left side by 0 since everything is positive and right side idk (?)
Or maybe proof by definition again if I could manipulate the expression
visual panther
Proof by definition is easy, because you have |cos(1/x)+2| <= 3
Should be like 9|delta^2|< epsilon
graceful ferry
Okay
visual panther
But I have to go now. Hope everything works now.
hybrid crow
So
hybrid crow
I need to verify that the sum of the subspaces is itself a subspace
The criteria I'm having trouble verifying is closure under scalar multiplication
The only way I know how to prove it would be using distributive property, except I don't think I can use distributive property without first proving scalar multiplication.
nova yoke
"v is in V_i for some i" - why?
all you know is it can be written as v = v1 + ... + vn, where v1 is in V1, ... vn is in Vn
hybrid crow
Agreed Bungo
quiet hinge
that is not what the definition says, but ok (i think what i said is simpler)
hybrid crow
Using that we must prove that fv is in the sum
Where f is from the field
So we have f(v1 + v2 + ... + vn)
Now what...
nova yoke
this is also not the definition, the question asks us to prove that using the actual definition
hybrid crow
See if we could distribute here it would be real easy
quiet hinge
this is also not the definition, the question asks us to prove that using the ac...
well i took the blue box to be the definition
nova yoke
why do you think you can't distribute?
hybrid crow
Cuz we didnt prove scalar multiplication
nova yoke
well i took the blue box to be the definition
it has a proof right below it 🤣
hybrid crow
How can we distribute without proving scalar multiplication is closed
nova yoke
Cuz we didnt prove scalar multiplication
but any element of V1 + ... + Vn is an element of V
and you can do scalar distribution in V
nova yoke
then just show that the result (after distribution) is in fact in V1 + ... + Vn
nova yoke
Who said we can do it in V1 + ...
do it in V
hybrid crow
Yeah but then all we r proving is
"If we can use distributive property, then scalar multiplication is closed"
quiet hinge
it has a proof right below it 🤣
im not sure how that implies it is a proof of the blue box, but sure, i wont drag this further. i still think defining it as the smallest containing subspace is superior. maybe you are familiar with this book
nova yoke
well it's a subset of a vector space, anything that you can do in that vector space you can do in the subset
then just show that the "outcome" is in the subset
nova yoke
that's how you show things are closed
hybrid crow
U cant just do what u do in a vector space in a subset of the vector space and assume it works
nova yoke
im not sure how that implies it is a proof of the blue box, but sure, i wont dra...
yep, very familiar, it's axler's LADR
hybrid crow
U can only do it in a subspace
nova yoke
U cant just do what u do in a vector space in a subset of the vector space and a...
why not?
hybrid crow
We need to prove out subset is a subspace
nova yoke
yep
hybrid crow
Bungo
hybrid crow
but you can do scalar multiplication on *any* element of V, right?
Agreed
And it'll be in V
nova yoke
yep
then just look at your final answer after doing the distribution in V
and explain why in fact it's in the subspace
nova yoke
because V is a vector space
hybrid crow
But
nova yoke
and you are working with elements of that vector space
hybrid crow
Hold on
Let me think
So f(v1 + v2 + ... + vn)
Is in our Sum vector space
I mean
We need to prove it is
How can we apply the distributive propery
nova yoke
right we need to prove that
nova yoke
maybe the confusion is the following
look at the definition of subspace
look at what i highlighted in particular
hybrid crow
Yes
And idk if it has the same scalar multiplication as on V
Without first assuming it has the same distributive property as V
There is no reason to believe that f(v1 + ... + vn) = fv1 + ... fvn
nova yoke
i think you have things backward a bit
the idea is that you first consider U as a subset of V, with the same operations as in V, since after all every element of U is also in V
hybrid crow
Since we dont know that distributive property holds on our subset
nova yoke
then you show that if you add two elements of U (using the V addition) then the result is in U
nova yoke
similarly if you multiply an element of U by a scalar (using the V scalar multiplication) then the result is in U
hybrid crow
I understand now
nova yoke
cool
(oh, and for completeness, you also have to show that U contains the zero vector from V)
hybrid crow
Yeah i alr know the rest
Each individual subspace has 0
So we just add 0 + 0 +...
And get 0 in U
nova yoke
yep you got it
hybrid crow
Except
Now im confusing myself again
Wait nvm
I unconfused myself
Ok Bungo
How about this @nova yoke
nova yoke
well any set contains itself (as a subset)
crystal plover
Every space contains itself, just like every set, in the meaning $A \subseteq A$ for all $A$
grand pondBOT
Luigi
hybrid crow
crystal plover
you have to read the highlighted part well
nova yoke
they mean "as a subset"
hybrid crow
How do u know that
crystal plover
You can read it in the text
crystal plover
here is meant as a subset
nova yoke
yea he's being a tiny bit sloppy
with experience you know what they mean unless you're reading a set theory book where you better be more careful
nova yoke
in axler's proof?
nova yoke
let me quickly reread
nova yoke
nope, all except this one mean subset
hybrid crow
Are all contained in the sum
nova yoke
but the highlighted one means "contain as elements"
nova yoke
yeah
fortunately iirc in this book, there aren't many situations where he talks about sets as elements of other sets
(maybe just in the quotient space section)
so if he says one set contains another, he's gonna mean "as a subset"
hybrid crow
Okay
Do u have any tips
On dealing with book
Its been good so far but
Idk anything to look out for
nova yoke
just come here and ask questions any time you get confused
hybrid crow
Ok thanks
nova yoke
it's not an easy book for a first exposure to LA, and it will get harder
but it's a good book
oh and if you don't already know, there is #linear-algebra in case you want to ask any conceptual questions that aren't about a specific exercise or whatever
midnight plankBOT
@hybrid crow Has your question been resolved?
wary trail
We say R/Z is homeomorphic to S1
is there an equivalent for S2?
midnight plankBOT
We say R/Z is homeomorphic to S1
is there an equivalent for S2?
wary trail
or at least a neater one, I'd probably have to say R2/U
where U is the union of all lines with one integer coordinate in R2 right?
midnight plankBOT
@wary trail Has your question been resolved?
quiet hinge
where U is the union of all lines with one integer coordinate in R2 right?
are you looking for some quotient by a subgroup of R^2 to give S^2? i dont think that works because S^2 isnt a group so there probably wont be any nice maps. your U doesnt seem like a subgp. of R^2 either
last slate
pls help
midnight plankBOT
@wary trail Has your question been resolved?
midnight plankBOT
pls help
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).
wary trail
are you looking for some quotient by a subgroup of R^2 to give S^2? i dont think...
U is a subgroup under addition, but I'm speaking topologically here
quiet hinge
U is a subgroup under addition, but I'm speaking topologically here
can you exactly write your group U? i dont see how its a group as written
either way i dont know of "nice way" to write S2
wary trail
can you exactly write your group U? i dont see how its a group as written
Actually no it is not lol
(½,0) and (0,½) is not in U
quiet hinge
right so the "quotient by U" is also slightly problematic
midnight plankBOT
@wary trail Has your question been resolved?
midnight plankBOT
midnight plankBOT
Available help channel!
midnight plankBOT
livid python
<@&268886789983436800>
midnight plankBOT
hard shard
what was this
normal valley
wow derivada is fast
livid python
spammy scam
nimble copper
i need joia
velvet compass
🤨
dusty portal
The 4 stages of suspicion
velvet compass
And no answer, I guess we'll never know it
midnight plankBOT
Available help channel!
twilit field
Prove that the subsequence of a monotone bounded sequence is monotone and bounded.
\
I'm hoping to use the definition of a sub-sequence here.
\
Definition: Let $(a_n)$ be an seqeunce . Let $b_n: \N \to \N$ be a sequence such that $b_n$ is strictly increasing. We then define a sub- sequence $c_n= a_{b_n}$
\
\textbf{ Issue here : I can't use the fact that the composition of two increasing functions is increasing}
midnight plankBOT
Prove that the subsequence of a monotone bounded sequence is monotone and bounde...
grand pondBOT
wai
twilit field
Any way around that
I can't use that fact as I'm still doing basic RA
or can I prove that without calculus?
feral sedge
Calculus is irrelevant here; you are dealing with sequences, not functions
prime hornet
Prove that the subsequence of a monotone bounded sequence is monotone and bounde...
every subsequence of a monotone sequence is monotone?
feral sedge
or can I prove that without calculus?
ask yourself how do you prove a sequence is monotone? how do you prove a sequence is bounded?
chase through the definitions and you will have an easier time writing the proof
prime hornet
I will second that
carmine sigil
I... Don't think this is true
Whoa more colors
Ok so if you have a monotonic sequence, it's not true that a subsequence is monotonic and bounded
twilit field
Prove that
theany subsequence of a monotone bounded sequence is monotone and bounded.
carmine sigil
Oh, you meant that if it's monotone and bounded ok
twilit field
ask yourself how do you prove a sequence is monotone? how do you prove a sequenc...
I see, thanks
Let me try chasing definitions then
carmine sigil
I would reach for a proof by contradiction here, but that's a weakness of mine.
There's probably a way to prove it without that
twilit field
I would reach for a proof by contradiction here, but that's a weakness of mine.
one more "advanced" way I can see is the composition of two increasing functions is increasing so any subseqeunce is increasing
carmine sigil
That's not necessarily true
Well, hmmm
Sorry yes it is true
I was imagining something that was not a composition
But regardless I don't see a way to move from composition to subsequences
twilit field
Here though $c_n≥ c_m\forall n≥m$ as $c_{b_r}≥c_{b_q} \forall b_r≥ b_q$ where $c_n= c_{b_r}, c_m= c_{b_q}$
grand pondBOT
wai
twilit field
wait
I didn't mean that
oops
I've got this
give me 2 minutes
Let the sequence be $(a_n)$, we then define a strictly increasing sequence $b_m$. It follows that $a_{b_l}≥a_{b_r} \forall l≥r$ as $a_n$ is an increasing seqeunce
grand pondBOT
wai
prime hornet
There's probably a way to prove it without that
not super formal, but I think the idea is that ||each member of the subsequence a_n_k is a member of the bounded sequence a_n, so the subsequence is bounded, and because the original sequence is monotone, you'd have that a_n_k <= a_n_l for k < l once you treat them as members of the original sequence, so monotoneness follows too?||
wai ought to not read this.
twilit field
Let the sequence be $(a_n)$, we then define a strictly increasing sequence $b_m...
Is this is the key idea for incrasing
carmine sigil
not super formal, but I think the idea is that ||each member of the subsequence ...
That makes sense
twilit field
As for bounded the terms of the subsequence(c_n) are still terms of the same sequence(a_n) as the indexing seqeunce is a strictly increasing natural sequence (b_n) ,$c_n = a_{b_n}$
grand pondBOT
wai
Compile Error! Click the 
reaction for more information.
(You may edit your message to recompile.)
midnight plankBOT
twilit field
thanks everyone
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chilly cobalt
Hihi!! so i asked this yesterday but nobody helped me
translation: 'Let triangle ABC have I, O, I_A as its incircle, circumcenter, and excircle opposite A respectively. Let D be a point on BC such that DA = DI_A. Let the radius of its incenter be 8, and the radius of the circumcircle be 18. Suppose that IO// BC. Then whats the length of AD?
midnight plankBOT
Hihi!! so i asked this yesterday but nobody helped me
translation: 'Let triangle...
chilly cobalt
my figure
i only have that A,O,D collinear cause of similarity😓
can anyone help??
midnight plankBOT
@chilly cobalt Has your question been resolved?
viral dagger
IO//BC <=> tan(B)tan(C)=3 from googling :p
chrome vessel
i only have that A,O,D collinear cause of similarity😓
and can you prove that?
chilly cobalt
and can you prove that?
yea
chilly cobalt
IO//BC <=> tan(B)tan(C)=3 from googling :p
what did you even google
chrome vessel
this (since theyre parallel)
why do you have that the two red angles are equal?
the angle marked with single red
chrome vessel
euler line parallel to bc
incenter is not necessarily on the euler line
chilly cobalt
i have never heard of that
chrome vessel
incenter is not necessarily on the euler line
and the implication your drew earlier is false
viral dagger
viral dagger
incenter is not necessarily on the euler line
oh
chilly cobalt
im never getting better at geo😓
chrome vessel
anyways back to the question
chrome vessel
im never getting better at geo😓
what's your proof that AOD are colinear?
chilly cobalt
bleh dont have it💔
chrome vessel
bleh dont have it💔
ok ok. In this case let us make a conjecture: it seems like DI_A is perpenticular to BC, so how about we do this:
Let D' be the perpendicular from I_A to BC, and show that D=D'
untold sentinel
midnight plankBOT
untold sentinel
I'm unsure how to progress on d)
midnight plankBOT
@untold sentinel Has your question been resolved?
untold sentinel
<@&286206848099549185> Thank you
lyric charm
i'd try finding the length of AD (or DC -- these two are the same length)
that, or
actually
BD is parallel to BA+BC
you can use this + the length of AD to figure out the vector BD and from that you can get OD
visual panther
The picture for c) is wrong. |BA| != |BC|
visual panther
Took me a while too. I had the feeling some info was missing. 😉
untold sentinel
The picture for c) is wrong. |BA| != |BC|
Yeah I drew it wrong
ABC is obtuse also
visual panther
I would use a projection of e.g. AB onto the diagonal AC to get the intersection point of the diagonals.
visual panther
I would not call it O, because this is usually reserverd for the origin.
But it looks right.
untold sentinel
I would not call it O, because this is usually reserverd for the origin.
Ah, what would you notate it as then
visual panther
M, because it is the kind of the middle of the kite.
YOu can use any letter, but O is kind of special.
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mortal heart
honestly this is kinda simple, but im dumb
midnight plankBOT
honestly this is kinda simple, but im dumb
mortal heart
i equated sum of inf terms formula with r1 and r2
but idk what to do exactly after that
desert siren
i equated sum of inf terms formula with r1 and r2
Show your work
mortal heart
all i did was equate
desert siren
Ah ok
desert siren
Yeah
mortal heart
desert siren
Yeah keep going
mortal heart
ig ill cross multiply now
desert siren
Yeah
desert siren
$r_{2}^2 - r_{1}^2 = (r_{2}-r_{1})(r_{2}+r_{1})$
grand pondBOT
Executor (ask on server b4 DM)
desert siren
Yuh
last slate
Yup thats it
mortal heart
can you tell me what this solution meant
the reason i came here was i didnt understand that ^
desert siren
This forms a quadratic in r
Which has the roots r1 and r2
Since if you put r1 and r2 into the thing you go back to the original two eqns
desert siren
Yeah?
mortal heart
um its a hw one but its from jee adv
desert siren
Np
last slate
can you tell me what this solution meant
Ill explain it as if u consider( a/1-x)=x as an equation, r1 amd r2 solve the equations hence are roots
fallen sparrow
Ill explain it as if u consider( a/1-x)=x as an equation, r1 amd r2 solve the eq...
Most appropriate statement
mortal heart
Ill explain it as if u consider( a/1-x)=x as an equation, r1 amd r2 solve the eq...
ah okay
ty
desert siren
Most appropriate statement
last slate
Ms executor is right as well
last slate
🙏
desert siren
its from the same ch, this is a passage
!status
midnight plankBOT
its from the same ch, this is a passage
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
mortal heart
and this is the sub q
fallen sparrow
its from the same ch, this is a passage
This is a pyq right
last slate
Bruh
desert siren
Yeah I've done this I remember lmao
mortal heart
i found the coordinates of Dn and value of r
fallen sparrow
Yeah I've done this I remember lmao
all jeetards in one channel
desert siren
Rudy, you can take over if you wish- I have some work for a bit
fallen sparrow
Rudy, you can take over if you wish- I have some work for a bit
Same here
mortal heart
ye
fallen sparrow
But I'll stay for this one
fallen sparrow
just begun 11th
Ayo you shouldn't be solving pyqs rn
mortal heart
Ayo you shouldn't be solving pyqs rn
class hw
fallen sparrow
Do them after your syllabus is complete
mortal heart
since the ch is over we got hw
theres a section of qs from jee adv
lol
i believe that ones from 2021
fallen sparrow
i mean they're solvable ig....
That's not the point
last slate
Lol
fallen sparrow
When you do it under time pressure, questions that are generally solvable seem very tough
mortal heart
@fallen sparrow i think i solved this one
fallen sparrow
wait hold on
Yeah?
mortal heart
and this is the sub q
so if i take the critical case here
fallen sparrow
<@1356904580633596075> i think i solved this one
Alright
mortal heart
like the smaller circle touching the circumference
of the bigger one
1s
lemme show image of what i mean
fallen sparrow
Yeah
mortal heart
like this
center of inner circle P
center of bigger circle O
and the point on which the diameter of inner circle touches be Q
just assuming
desert siren
Ayo you shouldn't be solving pyqs rn
Fr.
mortal heart
so i can equate PQ + OP = OQ right?
mortal heart
i can prob find OP using distance formula
mortal heart
so i can equate PQ + OP = OQ right?
is that method correct?
mortal heart
Especially adv
bro its hw i cant help it
desert siren
so i can equate PQ + OP = OQ right?
Yeah
desert siren
U appeared for this year adv?
Yes
desert siren
Not so good, I got my college through a diff exam
mortal heart
bro if i suck so hard rn what will happen during my jee 😔
last slate
bro if i suck so hard rn what will happen during my jee 😔
Bro ur all good
desert siren
bro if i suck so hard rn what will happen during my jee 😔
You're just starting out dw
fallen sparrow
Not so good, I got my college through a diff exam
??? Might be sensitive but did you get into iit?
last slate
Just be persistent
fallen sparrow
bro if i suck so hard rn what will happen during my jee 😔
Bruh you got like 2 years ahead of you
mortal heart
yea ik but these qs kinda hard tbh
desert siren
??? Might be sensitive but did you get into iit?
I said adv not so good why would I get iit lmao
mortal heart
im needing like 10 mins or 15 to solve these
fallen sparrow
Right now you should be experimenting with tough books like pathfinder, loney sbt etc
desert siren
im needing like 10 mins or 15 to solve these
That's great not even going to lie
fallen sparrow
Olympiads
last slate
yea ik but these qs kinda hard tbh
Ur gloatng now 😑
fallen sparrow
I said adv not so good why would I get iit lmao
Oh shi mb
desert siren
That's great not even going to lie
Even in actual advanced tough problems take 7 minutes
last slate
Even in actual advanced tough problems take 7 minutes
True
last slate
Right now you should be experimenting with tough books like pathfinder, loney sb...
Omg pathfinder
fallen sparrow
Omg pathfinder
Yes pf
last slate
Brings me horrors fresh ones
fallen sparrow
Brings me horrors fresh ones
Lol
desert siren
Do room.examgoal for pyqs
mortal heart
Ur gloatng now 😑
bro what
desert siren
<@&268886789983436800>
fallen sparrow
so i can equate PQ + OP = OQ right?
✅
last slate
Wth?
mortal heart
i got OP
so now since i got all in the powers of n, i can equate em to find n
ig
last slate
Thats right
mortal heart
dont quit yet cuz the 2^198 is pretty scary 😔
dude wtf
thats actually monstrous
huh wha
yea no i quit this q
thx for the help ig
.close
midnight plankBOT
mortal heart
(i got 199)
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Available help channel!
humble loom
where the hell did this 16r come from?
midnight plankBOT
where the hell did this 16r come from?
snow ledge
(r+8)^2
echo wharf
,w expand (r+8)^2
grand pondBOT
humble loom
,w expand (r+8)^2
as soon as i saw the "expand" i was like what is wrong with me
omfg
.close
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Available help channel!
tidal tide
can somone help me
midnight plankBOT
can somone help me
silent zodiac
With?
oak nymph
Determine if the parabola is upward or downward, determine the roots, and critical point
silent zodiac
And plot
flat spire
if they dont know what a vertex is they dont know what extremas are or max or min
silent zodiac
if they dont know what a vertex is they dont know what extremas are or max or mi...
Why they plotting stuff 
flat spire
idk
silent zodiac
Should know concavity
flat spire
but this is a quadratic you dont need to use calculus
exotic coral
Hello
silent zodiac
but this is a quadratic you dont need to use calculus
Yea but if u want
A good idea
flat spire
yes but they dont know any
exotic coral
Can u guys help me idk wgat im looking at
flat spire
you dont need to do all derivative stuff for concavity when you just need to look at the sign Lol
fallen sparrow
basically this is high school maths
Do you know how a general quadratic looks like
tidal tide
yes
fallen sparrow
Can u guys help me idk wgat im looking at
!occupied
midnight plankBOT
Can u guys help me idk wgat im looking at
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).
silent zodiac
you dont need to do all derivative stuff for concavity when you just need to loo...
On quadratics yes
silent zodiac
Can u guys help me idk wgat im looking at
Get a new channel
exotic coral
Get a new channel
Okk ty
tidal tide
can you just tell me the basic how are equations like these related to graphs
pleasee
silent zodiac
Quadratics are parabolic
tidal tide
i mean ik -2 and 6 are zeroes then if i plot these on graph they are on x
then
what's y
how does y come in the equation
tidal tide
like in ax^2+bx+c = 0
alr
i understand
so y is 0 or can be any value acc to x
so y=ax^2+bx +c
then
?
should we insert the zeroes ?
to get the value of y ?
nah
sorry
zeroes make 0
mb
odd path
can you just tell me the basic how are equations like these related to graphs
For finding the vertex this: \frac{-b}{2a}
\frac{-b}{2a}
odd path
and if you need y, replace the x with the result of the vertex formula
tidal tide
what's even vertex can you please tell
odd path
what's even vertex can you please tell
its a point where the parabole starts rising(or falling) from
odd path
if yes should the vertex be in the middle ?
should be
tidal tide
so if -2 and 6 are on the x axis then the vertex should be on their mean right?
i mean
not saying that on the x
tidal tide
But u don’t need to know now
alr
@odd path i mean if 2 is mean of the two points on the x axis then how do i now find the y axis
2- x axis
?- y axis
odd path
<@670304337762975745> i mean if 2 is mean of the two points on the x axis then h...
Just replace x in the quadratic with what you got
In this case it would be this= 2²-8-12
odd path
Should be
tidal tide
@odd path graph is this ?
one of my friend sent this to me
why's there (0,12)
i mean how do we determine 0,12
alr
sir
im so dumb
sorry
i understand
it now
thx
sacred gate
help
lyric charm
- lie down
- try not to cry
- cry a lot
- after wiping your tears, try the rational root theorem
- if that gives you nothing, resume crying
sacred gate
if a cubic comes
sacred gate
what if i have to fynd the asyntote of
2x/(x^3-3x-2)
like how the duck teacher expects me to know that without explaining that to us
should i just skip this exercice and hope it doesnt appear on exam
autumn canopy
Pretty much always, it will factor nicely if this was given as an exercise
sharp coral
and you test the roots by plugging in or synthetic division
silent zodiac
2x/(x^3-3x-2)
Bro
Have u seen the roots
-1 satisfies
2 also does
And 1 also
lyric charm
wat
if there are any rational roots at all they will take the form
±(divisor of constant term)/(divisor of leading coeff)
sacred gate
wait so the candidates of rationalizeng is +-number of nx^3 and the +-without the x?
lyric charm
And 1 also
since when is 1-3-2=0
lyric charm
wait so the candidates of rationalizeng is +-number of nx^3 and the +-without th...
no. look up videos on YouTube maybe.
rational root theorem
midnight plankBOT
sacred gate
.reopen
midnight plankBOT
✅
sacred gate
update
i understood it
do i just then pop every single one trying to find which numbers give 0?
when i find p and q
autumn canopy
After you found the candidates using RRT, you plug them in to check
sacred gate
each one?
autumn canopy
You can use symmetries of the polynomial if there are some to save you of some computations, otherwise yes
midnight plankBOT
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Available help channel!
sage glen
hey need some help with maths
midnight plankBOT
hey need some help with maths
fathom jetty
with what exactly
midnight plankBOT
@sage glen Has your question been resolved?
midnight plankBOT
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Available help channel!
sand flume
midnight plankBOT
sand flume
For these two things I need to proof that they are true. What would be the proper way to include the proof by induction. In the proof itself but that would require me two include two nested proofs in this proof or should I just create two different proofs after. If I do that how do I mention that these results are proofed and valid in my original proof
Oh there is a mistake when I wrote this out for the greater than part it should be 1/sqrt(n+1)
midnight plankBOT
@sand flume Has your question been resolved?
small jasper
Just say that they’re lemmas - you can prove them at the start, name them Lemma 1 and Lemma 2, and invoke them later
sand flume
Just say that they’re lemmas - you can prove them at the start, name them Lemma ...
So before I start the base case I should say lemme 1 and then I have a whole proof for lemma 1 and then same for lemma 2. Wouldn't this be too long for one proof? And then what is the proper way of involving the lemma in my work sections?
small jasper
So before I start the base case I should say lemme 1 and then I have a whole pro...
So before I start the base case I should say lemme 1 and then I have a whole proof for lemma 1 and then same for lemma 2. Wouldn't this be too long for one proof?
idk I wouldn’t even prove these things and chalk them up to being trivial but whatever floats your boat for style (though all you’re doing is putting the same stuff in a different order, so idk how much length is going to change)
And then what is the proper way of involving the lemma in my work sections?
“By Lemma 1 and Lemma 2, it is obvious that”
sand flume
So before I start the base case I should say lemme 1 and then I have a whole pro...
Ok so two things. How can I decide what is trivial or not. And by placement I mean like under the work or to the side
small jasper
Ok so two things. How can I decide what is trivial or not. And by placement I me...
Ok so two things. How can I decide what is trivial or not.
context dependent
And by placement I mean like under the work or to the side
I feel like you just answered this
We begin by proving two lemmas:
Lemma 1: ….
Proof:
Lemma 2: …
Proof:
We now proceed by induction.
Base case: …
Inductive hypothesis: …
Inductive step: We need to prove that …
Invoking Lemmas 1 and 2 yields …
and hence the claim is proven by induction.
night gyro
is my box plot correctly drawn and for b) I think Ben is more consistent as he has lower inter quartile range
midnight plankBOT
is my box plot correctly drawn and for b) I think Ben is more consistent as he h...
small jasper
is my box plot correctly drawn and for b) I think Ben is more consistent as he h...
is my box plot correctly drawn
yes
and for b) I think Ben is more consistent as he has lower inter quartile range
Yes. You can talk about the range being smaller.
night gyro
tysm
u know the way i drew the box plot
i started at 17
i was worried if its wrong
night gyro
is my box plot correctly drawn
> yes
and for b) I think Ben is more consistent ...
kk
.close
meager ore
Prove that ( f : \mathbb{R} \to \mathbb{R} ) is continuous if and only if for every ( X \subset \mathbb{R} ), it holds that ( f(\overline{X}) \subset \overline{f(X)} ), where ( \overline{X} ) is the closure of ( X ).
midnight plankBOT
Prove that \( f : \mathbb{R} \to \mathbb{R} \) is continuous if and only if for ...
grand pondBOT
Halex
meager ore
I find myself struggling with the other direction, any hint?
it would work to show that for every sequence x_n in X converging to a, f(x_n) converges to f(a)
But still unsure how
$$ a \in X \implies a \in \overline{X} \implies f(a) \in f(\overline{X}) \implies f(a) \in \overline{f(X)}$$
grand pondBOT
Halex
meager ore
There exists $(y_n) \in f(X)$ such that $y_n \to f(a)$
grand pondBOT
Halex
meager ore
Any hint from there?
brisk iris
Let m a value in R, consider ]m-epsilon, m + épsilon [
meager ore
what
midnight plankBOT
@meager ore Has your question been resolved?
brisk iris
Sorry I Can’t write (phone) but here’s the idea
noble forge
**Halex**
Since $a\in \overline{X}$, there exists a sequence ${a_n}$ such that ${a_n} \to a $.
By the continuity of f, $f(a_n) \to f(a) = y$
Since $f(a_n) \subseteq f(X)$, y is a limit point of f(X), so $y \in \overline{f(X)}$. Therefore, $f(\overline{X}) \subseteq\overline{f(X)}$.
brisk iris
Use the fact that it works for every interval X, assume xn converges toward x and choose the right interval X centered around x
This would allow you to prove that for every x in R, f(xn) converges towards f(x)
brisk iris
Since $a\in \overline{X}$, there exists a sequence ${a_n}$ such that ${a_n} \to ...
I thinks he’s struggling with the other way around
grand pondBOT
a guy with no beard
noble forge
Should be fairly straight forward. Just use the sequential definition of continuity
half heron
$D(x, y) = f_{xx}f_{yy} - f_{xy}f_{yx}$
midnight plankBOT
$D(x, y) = f_{xx}f_{yy} - f_{xy}f_{yx}$
grand pondBOT
Vague Disbeliever
half heron
this looks like a determinant of a matrix
\begin{bmatrix}
f_{xx} & f_{xy} \\
f_{yx} & f_{yy} \\
\end{bmatrix}
Vague Disbeliever
Compile Error! Click the reaction for more information.
(You may edit your message to recompile.)
desert siren
I mean, it can be a matrix determinant, but there's no specific name for what this is
half heron
it's just coincidentally in form ad - bc?
astral canyon
```latex
\begin{bmatrix}
f_{xx} & f_{xy} \\
f_{yx} & f_{yy} \\
\end{bmatrix}...
That's a hessian matrix https://en.wikipedia.org/wiki/Hessian_matrix
desert siren
Oops, idk what that is lol
half heron
That's a hessian matrix https://en.wikipedia.org/wiki/Hessian_matrix
what's the difference between this and a jacobian matrix
they're both matrices of partial derivatives ?
(im not fully sure what a jacobian is yet i havent gotten to triple integrals and change of variables)
subtle blaze
The jacobian is the first derivative
The hessian is the second derivative
Since f_xy means $\frac{\partial^2 f}{\partial x\partialy}$
desert siren
{xy}
half heron
The jacobian is the first derivative
ah ok
desert siren
$f_{xy}$
grand pondBOT
Executor (ask on server b4 DM)
half heron
$\pdv[2]{f}{x}{y}$
grand pondBOT
Vague Disbeliever
half heron
if that's what u meant
grand pondBOT
frosst
Compile Error! Click the reaction for more information.
(You may edit your message to recompile.)
half heron
which i'm assuming iti s
half heron
Since $f_{xy}$ means $\pdv[2]{f}{x}{y}$
subtle blaze
**Vague Disbeliever**
This
grand pondBOT
Vague Disbeliever
subtle blaze
Thank you!
subtle blaze
So anyway that’s a second derivative
subtle blaze
nth order?
There is but it’s very unweildy to write down
half heron
like maybe a function that takes n and spits out a matrix 🧌
matrices can be in the codomain of a function right
half heron
$f: \mathbb{R} \mapsto \mathbb{R}^{m \times n}$
subtle blaze
The next order derivative is like a matrix of matrices
grand pondBOT
Vague Disbeliever
half heron
The next order derivative is like a matrix of matrices
ew
so a tensor?
alright wlel wtv thx
.close
subtle blaze
Yeah
midnight plankBOT
subtle blaze
so $J_f(x) = \partial_ifx^i$, and $H_f(x) = \partial_i\partial_jf(x^i)(x^j)$
grand pondBOT
frosst
subtle blaze
if i call the next term $K$ then it's $K_f(x) = (\partial_i\partial_j\partial_kf)x^ix^jx^k$
grand pondBOT
frosst
subtle blaze
Then it keeps going
I’m using Einstein summation convention here as well
At least this is for the Taylor series
desert siren
.reopen
midnight plankBOT
✅
midnight plankBOT
@half heron Has your question been resolved?
sharp cairn
hey
sharp cairn
so
f(x) < 0 and f(x) > 0
but THIS IS NOT CONTINUOUS (Checking at x=1)
How can it be n=1?
it is n= 0
fallow scarab
Translate the question if you want help faster