help-49

[midnight plank]

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[pearl hull]

.reopen

[midnight plank]

✅

[pearl hull]

@sleek marsh Please reply to depression when you’re back. He wrote a whole page of explanation for you.

[finite wharf]

my bad

[midnight plank]

@sleek marsh Has your question been resolved?

[midnight plank]

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[molten bay]

19

[civic gazelle]

find its derivative

[molten bay]

f'(x)=3(a+2)x^2-6ax+9a

My doubt is for decreases monotonically is <=0 or <0?

@civic gazelle

,rotate

[civic gazelle]

we should do 2 cases

[molten bay]

Which one?

[civic gazelle]

-9ax is wrong

[molten bay]

-6ax

Yeah wrong multiplication in hurryness

[civic gazelle]

alright so what if a = -2

the x^2 term is gone right?

[molten bay]

it will be one order

An equation of line

12x-18

[civic gazelle]

and because it is not a constant

it is not <= 0 for ALL x

[molten bay]

What do you mean?

[civic gazelle]

we wanted to find a so that f'(x) <= 0 for all values of x

[molten bay]

Why<=0?

Why not <0?

[civic gazelle]

this is a quadratic

The root of f'(x)=0 can be a double root

[sudden yacht]

Actually, it's this because they say monotonically

[molten bay]

I see

[civic gazelle]

hmm?

[sudden yacht]

Wait

No, probably both are ok

[molten bay]

[sudden yacht]

Monotonically means that there must not be intervals where y' = 0, but single points are acceptable

[molten bay]

Single points?

[civic gazelle]

here f'(x) = 0 at x = -1 but the function monotonically increases

[midnight plank]

@molten bay Has your question been resolved?

[fallen sparrow]

monotonic includes equal to case as well

[midnight plank]

@molten bay Has your question been resolved?

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[vernal dagger]

How the hell do I explain this without the Pythagorean theorem

[vernal dagger]

Ik the process and answer

But how do I teach this

To kids

I don't wanna use phytagorean theorem

They're 3rd and 4th graders

Wait

Where

Answer is supposed to be D

Bruh

<@&286206848099549185>

Bro

Can someone help me

[gaunt nimbus]

Area of square = EDH + BFG + EFGH + EAF + GHC?

@vernal dagger

Ye this works

[vernal dagger]

Thank you bro

W

Now I don't need to scare my students

.close

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[midnight plank]

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[last slate]

Not sure where im going wrong, i know i have made a mistake but idk where or when

[raven zenith]

,rccw

[raven zenith]

what was the original problem?

!original

[midnight plank]

Please show the original problem, exactly as it was stated to you, with the entire original context. A picture or screenshot is best. If the original problem is not in English, then post it anyway! The additional context might still be helpful. Do your best to provide a translation.

[dawn dagger]

It looks like a system of two equations

[last slate]

it is 3/x + 2/y = 14 and 5/x - 3/y = -2

yea

[dawn dagger]

How did -3/y become +3b

[last slate]

because i stated let 1/x = a, and 1/y = b, so then 3/y = 3b

so i can eliminate/substitute variables

[dawn dagger]

but there is a minus

[last slate]

wait where are you seeing this?

[dawn dagger]

or also there is a y somehow

-3/y should be -3b not 3y

[last slate]

ohhhhh

so ive flipped signs and assigned incorrectly?

[dawn dagger]

Yes

[last slate]

kk

also once ive got to the point where

[dawn dagger]

y was probably a silly but changing signs was the crucial mistake, I think.

[last slate]

i have an expression for what a is

do i then plug it into the let 1/x = a and 1/y = b

[last slate]

kk

[dawn dagger]

Yes, you can do that

(keep in mind a and b must not be 0)

[last slate]

kk so i just undo the reassignment at that stage?

[last slate]

what would that do?

[dawn dagger]

Yeah, and I would try to do it a bit cleaner

[dawn dagger]

undefined

[last slate]

k

[dawn dagger]

1/x = 0 has no solutions

[last slate]

ohhhhhhhh

yeahh true

[last slate]

awesome tysm bro ill redo it

.close

[midnight plank]

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[midnight plank]

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[gilded tiger]

Can someone tell me where im going wrong

[gaunt nimbus]

Given ${f'(3) + f'(2) = 0}$
[ \lim_{x \to 0} \left( \frac{1 + f(3-x) - f(3)}{1 + f(2-x)-f(2)}\right)^{\frac{1}{x}}]

[grand pond]

k

[gaunt nimbus]

hmm

[gilded tiger]

....

[gaunt nimbus]

i would take ln of both sides

any apply l'hopital

[gilded tiger]

After the e^
Or before tht

[thin crater]

after the e^ i think

did you learn about the 1^infinity form?

[gaunt nimbus]

oh

[gilded tiger]

Yes thats why i used the e^(f(x)-1) 1/gx

[thin crater]

oh yeah i forgot to look at your solution mb 😭

[gilded tiger]

👍ooooh

[gilded tiger]

So what would be better
Ln directly or after the 1 power indity form

[thin crater]

hold on i dont think you need to use l'hopital

[gilded tiger]

Ok

[gaunt nimbus]

it makes thing simpler imo

[thin crater]

it just becomes e^0?

the limit becomes 1

is that not the answer

[gaunt nimbus]

thats what i got too

[gilded tiger]

1 is the answer but

[thin crater]

over here, shouldnt this be +f'(2)?

[gaunt nimbus]

ye

[thin crater]

f(2-x)-f(2)/x = -f'(2)

multiply that by a negative sign and it becomes f'(2)

[gilded tiger]

I mean it's it e^1/0

[gilded tiger]

Ohhh yea noted

[thin crater]

noooo its e^0

the 1/x was consumed when you made f'(3) and f'(2)

so it becomes

[gilded tiger]

OH

Yea

[thin crater]

e^(f'(3)+f'(2))/(1 + f(2-h) - f(2))

[gilded tiger]

mb i looked at the above expression

Okay so now we have no x

[thin crater]

you can assume f(2-h) - f(2) = 0 cause its not making any indeterminate form

[gilded tiger]

But f'3+f'2 =0

[junior jewel]

hello

[mortal mirage]

!occupied

[gilded tiger]

? Confused

[midnight plank]

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).

[junior jewel]

Can anyone explain to me what reciprocal means?

[mortal mirage]

^^

[thin crater]

the denominator: 1 + f(2-x) - f(2) can be written as simply 1

[gilded tiger]

Ohhh okay when x=0

Got it

Yea so it will jus be

Got it thanks

[thin crater]

nw all the best fellow jee aspirant 👍

[gilded tiger]

😅

You too

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[novel flax]

I have a doubt in the conditions of rolles theorem

[novel flax]

why cant f(x) just be continous on (a,b)?

instead of [a,b]

[surreal moon]

Because then condition 3 is meaningless

[novel flax]

y?

[proper kernel]

Try to visualize it

[surreal moon]

Imagine f(x)=x everywhere except f(a)=f(b)=0

[novel flax]

oh fuck right

[vast loom]

you can be non continuous but defined

[novel flax]

but if domain is [a,b] but i state that it should be cont on (a,b) then?

[vast loom]

try to sketch this function

on a piece of paper

[nova compass]

✅

[novel flax]

wont a=b be the only possible thing?

[quartz hornet]

.

[vast loom]

look at this

its continuous on (a,b) but rolle's is wrong here

get it now?

[novel flax]

nope

could you tell from start sry

[vast loom]

this function verifies all conditions, except its only continuous on (a,b) instead of [a,b], and because of that, f' never = 0 on (a,b)

so rolles theorem cant be applied if its only continuous on (a,b) and this is a counter example to that

[novel flax]

ohhh

[vast loom]

(the point I made for f(b) should be on the x-axis to verify the third condition but i moved it up so you could see it)

[novel flax]

wb this tho, if i make it a piecewise typa function

[vast loom]

well f' = 0 on (a.b) somewhere sure, but it doesnt mean it will always be true

it doesnt come from rolles theorem here

[blissful pier]

indeed while that specific f(x) may satisfy Rolle's Theorem if we don't have continuity on [a,b] we cannot assert it for general f(x)

[vast loom]

You cant say this function verifies the theorem so the theorem is correct, counter examples with specific functions can only show that something doesnt work, you cant use it to show that a theorem will always work

(except if your theorem is like it will work for that specific function exactly but you get the point)

[solid iris]

its possible for the conditions of a theorem to fail but its result still holds

[novel flax]

yeah understood

[solid iris]

all it means is that you cant apply the theorem to get the result

[novel flax]

took time for brain to process it 😭

[vast loom]

no worries

as long as you understand in the end

[novel flax]

thanks a lot!

[fallow scarab]

happens to everyone

[novel flax]

Hey, i have another doubt

what is the geometrical meaning of line integral

ik that normal definite integral shows area under curve

[blissful pier]

area under the curve still :)

nothing's changed

[novel flax]

yo damn

so its 3d shit

or not nexessarily

[blissful pier]

tis

would not be equipped to answer that but ive only ever done them in 3d

[steel crest]

not necessarily

if the curve represents something that's accumulating in the wire in some sense

the integral adds up what's being accumulated in that sense, for examkple

Ampere's law

[novel flax]

whats the upper and the lower line? are they separate curves?

[steel crest]

it's the projection of the upper cure onto the xy plane

know about projections?

[novel flax]

oo ye

kinda learnt in vectors

[steel crest]

it can also add up work done along a path

int F dot dl

[blissful pier]

if flattened/smooshed the curve C onto the xy plane, then that's the lower line

[steel crest]

and of course

it doesn't have to be interpreted geometrically at all

many line integrals can be a contour integral

in the complex plane

[novel flax]

idk contour stuff, just started uni

[steel crest]

which don't really have a geometric interpretation like area

[steel crest]

know complex number basics? Euler's formula?

[novel flax]

what if the function is f(x,y,z)?

[novel flax]

yeah

[steel crest]

so you can have something like

$$\oint_{\mathcal C} f(e^{i \theta}) , \mathrm d(e^{i \theta})$$ for a circle $\mathcal C$

[grand pond]

gfauxpas

[novel flax]

ohk

[steel crest]

in this case a circle of radius 1

for a simple example

so instead of f(x) you have f(z), where z is a complex number on the unit circle

[novel flax]

what if the function is f(x,y,z)
where would the projection be?

[steel crest]

i think the analogy would be

the projection is onto a 3d space, and (x,y,z,w) is a 4d space not easy to graph

[novel flax]

ohk ic

[steel crest]

however

if a function f is real-valued, regardless of how many variables it takes in

let's introduce a notation

f=f^+ - f^-

where

$$f^+ = \max{f,0}$$
$$f^- = \min{f,0}$$

[grand pond]

gfauxpas

[steel crest]

IF both f^+ and f^- are integrable

then int f is a measure on f

and it can be interpreted as a generalization of the idea of length/area/volume , all those concepts are genralizable and the generalization is called "measure"

so in that sense you can find: the length of something, the area of something, the volume of something, and in general, the "measure" of something

[novel flax]

ohh

i have understood somewhat somewhat

i have another doubt, not related to this, but surface area typa

our physics teacher gave us a homework, on a satelite problem
if the satelite is at a distance R_e from the surface of earth, find the max arc lenght it covers on the earth

basically 2R from centre of earth

i found the arc length

but if i want to find the total surface area it covers, then how do i do so?

[quartz hornet]

solid angle?

[novel flax]

ye but idk how to use that

my sir told that for 4pi its 4 pi r^2
then he told similarly for a theta, there will be a x

i didnt understand that properly..

[novel flax]

or maybe im a little wrong in telling what my teacher explained

[midnight plank]

@novel flax Has your question been resolved?

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[lyric charm]

$y = \frac{x}{\sqrt{a^2-1}} - \frac{2}{\sqrt{a^2-1}} \arctan\paren{\frac{\sin(x)}{a+\sqrt{a^2-1}+\cos(x)}}$

[grand pond]

Ann

[gaunt nimbus]

Why do all ur questions look cursed

[woeful turret]

😔

[lyric charm]

this is one of those other bullshit jee problems where you either know the trick and do it in 5 seconds or you dont and spend 17 years

for sure

[woeful turret]

yeah is there a trick?

or do we have to use chain rule

and differentiate that

[lyric charm]

but my thinking is that we might want to write a := 1/cos(alpha) or something to at least make the bullshit look a bit more presentable

where alpha lies in... (0, pi/2) \cup (pi/2, pi) i guess

what's a^2 - 1 gonna be then, tan^2(alpha)?

[woeful turret]

yeah

[heady badger]

Isn't "a" a constant?

[lyric charm]

it is but i wanna make the formula have less roots in it

and sqrt(a^2-1) is obviously a recurring element

[woeful turret]

yeah but just to make it more presentable

i was thinking we have to use some trig identity

to remove the tan inverse function

[heady badger]

Oh

[feral sedge]

honestly this does not even take that long to differentiate normally and plug in x = pi/2

[heady badger]

If a is cosectheta then lots of things would go easier right??the range would be (-pi,-pi/2)U(pi/2,pi)

[woeful turret]

im getting a lot of ugly terms while doing that tho

[feral sedge]

you can write the arctan(sinx / (C + cosx)) or something

just condense the a into a single constant and a lot of the ugliness goes away

everything is fine because you do not actually have to simplify the derivative. once you differentiate you can plug in x = pi/2

[woeful turret]

ok

ill let u know what i get one minute

[plain finch]

that looks miserable to solve

[woeful turret]

i do have to simplify it

cause im getting an ugly ass expression

that somehow simplifies into 1/a

(1/k1) - (2/k1)(1/(1+k2^2))(1/(k2+1)^2)

could someone use texit to type this pls

also k1= sqrt(a^2-1)

k2 = (a+sqrt(a^2-1))

which makes it more ugly

[midnight plank]

@woeful turret Has your question been resolved?

[midnight plank]

@woeful turret Has your question been resolved?

[summer geode]

$\frac{1}{k_1}-\frac{2}{k_1}\frac{1}{1+k_2^2}\frac{1}{(k_2+1)^2}$

[grand pond]

Yeatte

[summer geode]

this?

[quartz hornet]

^ yeah because you don't really have to write f prime , you can just directly write fprime(pi/2)
Which means you can substitute sin x as 1 and cos x as zero as they form ( notice how quickly the other terms vanishes )

[quartz hornet]

So real

[midnight plank]

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[summer geode]

@woeful turret it does end up simplifying into 1/a yeah(your answer with the k's was correct btw), tedious but if you combine the 2 fractions into 1 they do simplify

[summer geode]

.close

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[woeful turret]

Ohh ok ty

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[prime hornet]

a) does this proof seem okay? did I miss anything or overlook anything?
b) if it's okay, could I have made this more concise? I feel like I spent a lot of effort in the 2nd paragraph checking carefully that Phi is a global trivialization; could I have somehow made it quicker?

[blissful pier]

I cannot help with any math but I see that down there

[runic hamlet]

why are you even still in the help channels

[frail carbon]

This is obviously a pre-university homework-type question, wdym?

[civic lynx]

math makes sense: diff geo

[runic hamlet]

well its so obvious, you should go and ask the next toddler on the street

instead of being here

[prime hornet]

I'm crossposting, admittedly

[civic lynx]

literally what i was thinking about ty

[prime hornet]

I've also posted in my thread

[frail carbon]

Even worse 🗿

[prime hornet]

one of the issues I'm finding now is that asking somebody to review a proof is quite a lot of work because there're so many definitions

I'm pretty confident that my work here is sufficient, but I'm not confident at all that it couldn't be made significantly more concise

[frail carbon]

I have not studied diff geo but I do want to ask why in the first paragraph you mention something from a definition, but then don't use it

and instead just refer to a different part of the same definition

[prime hornet]

which part?

[frail carbon]

the one with projection

[prime hornet]

oh, in the parentheses?

[frail carbon]

are both needed pi_m o Phi = pi and the restrictions being isomorphisms needed to show Phi is a smooth bundle isomorphism

[prime hornet]

yeah

not that the restrictions are isos, but that they are linear

that's why I explicitly called out their linearity

pi_M serves the role of pi' here

[midnight plover]

😭

Rip

[fallow scarab]

stop spamming help channels

[midnight plover]

Kk who wants help

[fallow scarab]

#discussion

[midnight plover]

Stop spamming

[fallow scarab]

<@&268886789983436800> troll

[midnight plover]

Understod

[prime hornet]

hm, I have to go now for a bit

I guess I'll close this for now and check it again later

.close

[midnight plank]

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[merry glacier]

Boa noite quanto é dois mais dois mesmo?

[midnight plank]

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[crude kindle]

hey can someone help me understand this example of the Bellman-Ford Alg. ?

[midnight plank]

@crude kindle Has your question been resolved?

[fathom onyx]

Willst du's auf Deutsch oder auf Englisch? [bin Englischsprechender aber kanns auf Deutsch]
-# do you want it in German or English?

[crude kindle]

dann gerne auf deutsch wenn es geht

wenns schwer wird dann englisch

[fathom onyx]

Cool

(hehe, geht auf beide Sprache lol)

[crude kindle]

perfekt

ich check alles bis zum zyklus hin ab da wirds ganz komisch

[fathom onyx]

Nennen wir die Knoten, from the first row rightwards, A bis D, dann E bis H

Auf A starten

[crude kindle]

ok

[fathom onyx]

B und E direkt rechnen, indem wir die verbindende Pfeile nutzen

Also is B 14, und E 6

[crude kindle]

yes

B wird dann aber zu 9 da man über E abkürzen kann

[fathom onyx]

ja sicher

Bin der Meinung dass man C, dann D rechnen

Dennoch wird E für dir Abkürzing benutzt

Der Grund dafür? ist nicht mit diesem Bild klar erklärt

soweit dass ichs sehe

[crude kindle]

stimmt eigentlich haben die sachen ja ne rheinfolge

[midnight plank]

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[crude kindle]

ich verstehe aber besonders nicht wie sie das letzte ding erreich haben like why C 7 and not 12

.reopen

[midnight plank]

✅

[crude kindle]

or is the example simply wron

[fathom onyx]

Yh that's what's confusing me as well

Bis zum dritten Beispiel solls klar genug

Der letzte aber...

[crude kindle]

jaja bis zum dritten check ich auch noch aber das vierte ist falsch vielleicht schreib ich einfach ne mail an die prof

[fathom onyx]

ja vielleicht

Dennoch: 9 + (-12) + 2 gilt aber -1?

dh. go around in circles and get maximum efficiency?

Hier stimmt was definitiv nicht

[crude kindle]

ich glaube das alg wird abgebrochen nach einem durchlauf

und wird dann nochmal durchgeführt um auf negative zyklen zu untersuchen

[fathom onyx]

yeah...

[crude kindle]

wenn es sich dann nochmal verringert weiß man das ein negativer zyklus drin ist

[fathom onyx]

Beim englischen Artikel von https://en.wikipedia.org/wiki/Bellman–Ford_algorithm gibts einen guten Beispiel

ooh it's animated in the embed

[crude kindle]

uiiii

[fathom onyx]

OH

Ist das, was hier gemeint wird?

[crude kindle]

ohhh das könnte sein

damn die hat das aber nicht in ihrer vorlesung dahin geschrieben haha

[fathom onyx]

If these are lecture slides, then this might be what was explained in the lecture

[crude kindle]

Nah she just put it in without context haha

[fathom onyx]

Mal suchen obs notes dafür gibt

Oder frag sie, ob sie sie schicken kann

[crude kindle]

mach ich

vielen dank für deine hilfe

[fathom onyx]

kein problem

[crude kindle]

schönen tag noch

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[fierce silo]

So, I'm probably missing something so unbelievably simple but I cannot seem to figure this one out. My graph does not match with this graph given in the answers part in my textbook, if anyone else could see how they got this graph as the answer, I only need to know if it's actually correct or maybe there's a mistake?

If it's correct, I'll eventually figure it out on my own, sorry for the dumb question. But I'm asking in terms of question 43.

I'm not enrolled in school or classes or anything so I don't have a teacher I can go and ask but if anyone might be able to just let me know if they also think this graph is wrong, I'd appreciate it. AI isn't helping me much and I don't want to trust it because it also said the graph was wrong but I don't trust myself to be right.

I have the same values as them, and I plug in other values that match the rules for X but I do not get that same graph ever

[grim forge]

Ye its wrong because at f(-1), it should be at (-1, 0) not (-1,1)

[fierce silo]

That's exactly what I'm thinking too and I'm trying to find any documentation online that might confirm it's wrong? It's James Stewart Calculus single variable and early Transcendentals if anyone else has that book.

[grim forge]

I mean just use your own reasoning

[dapper slate]

Yes, this is incorrect.

[fierce silo]

Ok, you're right. Just always sketchy trusting myself since I'm only self studying so far, so I'm extra cautious about trusting myself over a textbook

Thank you though guys, this does help me, I've been stuck for like an hour on it

[dapper slate]

It makes sense. Usually you can trust these books.

I would definitely not trust AI to resolve the debate because it will probably answer based on the cues in your question

[fierce silo]

Yeah, it gave me a graph completely wrong and way out of left field so I tried to think of another place I might be able to ask. Anyway thank you all

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[languid mica]

is there a general method to find a formula for series?

[sudden yacht]

Not that I know

[languid mica]

im studying series

and i can see we can use proving that a formula is true for n=1 then suppose its true for n and prove its true for n+1

[next bramble]

No. Because the function could be infinitely piecewise

[languid mica]

but does someone someone come up with the formula

for example

[next bramble]

$$f(n) = \begin{cases} 1 & \mbox{if}; n = 0 \ 5 & \mbox{if}; n = 1 \ 39 & \mbox{if}; n = 2 \ ... \end{cases}$$

[grand pond]

Shuba

[next bramble]

There is no "general" way to determine this function.

[languid mica]

for example the sum from 1 to n of a_i^2

[languid mica]

is there some logic behind it

[sudden yacht]

No

[languid mica]

for this

[next bramble]

nope. randomise everything

[sudden yacht]

Don't even try to find it 😅

[next bramble]

We have some tricks to help though. You're probably studing some of those.
For example, a sequence could be constant, polynomial, exponential, arithmetic, geometric, etc...

But, there is no "general" way.

[languid mica]

$$ \sum_{k=1}^{n} k^2 &= \frac{n(n+1)(2n+1)}{6} $$

[grand pond]

taebek
Compile Error! Click the reaction for more information.
(You may edit your message to recompile.)

[languid mica]

like this

how did the guy think of the formula

[sudden yacht]

Search on the Internet

[languid mica]

cant be pure luck

[next bramble]

For example ${ 1, 2, 4, 8, 16 }$ looks exponential, but then I tell you the next number is $31$. This is a well known sequence, but very hard to determine what it is.

[languid mica]

surely there is some logic

[grand pond]

Shuba

[languid mica]

thats something different

thats finding the sequence of which the output is these

[vast loom]

You can do cubic interpolation to find the formula

[languid mica]

thats a completely different question

[next bramble]

Nope. Finding a function for a sequence is trying to find every next number. But for any sequence, the next number could be random and indeterminable.

[vast loom]

the formula for the sum of squares i mean

[languid mica]

what i asked is if you know the sequence like the function inside the sum how can you construct a formula that finds the value

[languid mica]

no thats not what i asked

i asked if we have a sequence how do we find formula of the sum

so basically

[vast loom]

there isnt a general method\

there are many ways depending on the series itself

[languid mica]

solving s_n = Sum from 1 to n of f(i)

[vast loom]

well that you cant simplify

its just = sum

[languid mica]

yes in general case you cant

[vast loom]

doesnt get any better if you dont know anything about f

[languid mica]

but whats the logic in specific case

now

if you know f

here

how would you guess the formula is n(n+1)(2n+1)/6

and then prove it by mathematical induction

is there some logic in this case ?

[sudden yacht]

https://math.stackexchange.com/questions/183316/how-to-get-to-the-formula-for-the-sum-of-squares-of-first-n-numbers

[midnight plank]

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[chilly cobalt]

For a positive integer $N>1$ with unique factorization $N=p_1^{\alpha_1}p_2^{\alpha_2}\dotsb p_k^{\alpha_k}$, we define
[\Omega(N)=\alpha_1+\alpha_2+\dotsb+\alpha_k.]
Let $a_1,a_2,\dotsc, a_n$ be positive integers and $p(x)=(x+a_1)(x+a_2)\dotsb (x+a_n)$ such that for all positive integers $k$, $\Omega(P(k))$ is even. Show that $n$ is an even number.

[grand pond]

Copter

[chilly cobalt]

could someone hint me, i cant really begin

well i know that omega is additive

omega(ab) = omega(a) + omega(b)

[steel crest]

I assume you mean P(x) is the polynomial not p(x)

[chilly cobalt]

that isnt even the same p lol

[steel crest]

but... this is difficult

[chilly cobalt]

yeah

i mean

p(0) => omega(a1) + ... omega(an) is even

thats something

[steel crest]

what do you mean by "P(0) implies"

let me see if this Omega has a name so i can look up properties

[chilly cobalt]

omega(p(0)) = omega(a1a2...an) is even

[steel crest]

oh so "=>" was just a typo , got it

[chilly cobalt]

yeah😭

my bad

[steel crest]

lol wait until you hear this name

it's very surprising

[chilly cobalt]

?

[steel crest]

ready for it? you sitting down?

dont want to shock you

it's called the "big Omega function"

[chilly cobalt]

there is no way

,w big omega function

[grand pond]

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[steel crest]

lmao

[chilly cobalt]

are there any properties you found??

[steel crest]

looiking

have an asymptotic that seems not useful at all for us

$$\sum_{n \le x} \Omega(n) = x \log \log x + B_2 x + o(x)$$
where
$$B_2 \approx 1.0345$$

[grand pond]

gfauxpas

[steel crest]

yeah that's u seless

[chilly cobalt]

what the hell

what do mathematicians be doign in their free time

[steel crest]

lol

completely additive as you found

[chilly cobalt]

whats "completely" mean

[steel crest]

regardless of whether the numbers are coprime

oooh found somethign

the Liouville function is defined as:

working on it

let P_E be the property "has an even number of prime factors" and P_O be the property "has an odd number of prime factors", counting multiplicity

$$\lambda(n) = \begin{cases} +1 , \text{ if $n$ is $P_{\mathcal E}$} \ -1 , \text{ if $n$ is $P_{\mathcal O}$} \end{cases}$$

[grand pond]

gfauxpas

[steel crest]

then:

$$\lambda(n) = (-1)^{\Omega(n)}$$

[grand pond]

gfauxpas

[steel crest]

is that helpful

no its not unless we know interesting propeties of lambda

[chilly cobalt]

theres definetly a better solution for this using only omega 😭

[steel crest]

yeah definitely, but i dont know how to approach this, sorry man

<@&286206848099549185>

[chilly cobalt]

its okay bro

For a positive integer $N>1$ with unique factorization $N=p_1^{\alpha_1}p_2^{\alpha_2}\dotsb p_k^{\alpha_k}$, we define
[\Omega(N)=\alpha_1+\alpha_2+\dotsb+\alpha_k.]
Let $a_1,a_2,\dotsc, a_n$ be positive integers and $p(x)=(x+a_1)(x+a_2)\dotsb (x+a_n)$ such that for all positive integers $k$, $\Omega(P(k))$ is even. Show that $n$ is an even number.

back to this

[grand pond]

Copter

[chilly cobalt]

im thinking we plug in the roots of this polynomial

[chilly cobalt]

from this uhh

$\prod_{i=1}^{n}p(a_i + 2) = 2^n$ something something

[grand pond]

Copter

[chilly cobalt]

might be useful?

[steel crest]

oh, roots, that's an idea

we can use roots of polynomial theorems

Vieta's formulas

[chilly cobalt]

p(-ai) = 0

[chilly cobalt]

i dont see how that would relate to n though,,

wait

the last term?

if n is even then the products should be positive

[steel crest]

yes

[chilly cobalt]

copter processing

[steel crest]

we also know that Omega(N) is negative, but that might not be helpful

wait no

[chilly cobalt]

wait why

[steel crest]

I take it back

[chilly cobalt]

omega is defined for positive integers only

[steel crest]

I confused alpha and a

[chilly cobalt]

sob

[chilly cobalt]

oh yeah and omega(2^n) = n

this is getting somewhere

[steel crest]

so we knmow that n has to be an even number otherwise their product would be a negative number

because 1 is positive and x^n is positive

that notation means "the coefficient of"

wait is that right

then maybe Omega has nothing to do with it

[chilly cobalt]

we're being trolled

sob

[steel crest]

oh nooo

we did a stupid mistake

a_1, a_2, ... , a_n are the NEGATIVES of the roots of the polynomial

[fresh sparrow]

yeah i was about to say that lol

[steel crest]

alright that changes a lot

[fresh sparrow]

there are no roots after >0

[chilly cobalt]

oh right yeah

[steel crest]

I feel more confident now that cuttlefish in 3d glasses is here

[chilly cobalt]

fr

[fresh sparrow]

thank you, i don't know what i am doing

xd

[steel crest]

(-a1)(-a2)...(-an)=(-1)^n a0/an

positive only if n is even

which you can derive without vietas at all

because it's (-1)^n a1a2a3...an

[chilly cobalt]

so we got trolled

cries

[steel crest]

I give up sorry

discrete math is not my forte

[chilly cobalt]

its alright bro

i shall think harder

[chilly cobalt]

i run back to this idea cause i think its promising

[chilly cobalt]

we make this a square ,so when we cover it in omega its even and omega(2^n) = n so we might be able to argue by modulo 2

think think

oh wait

multiply by p(1) and the like term becomes square

and the rest are all squares too because we can swap the order of addition

how do i notate this 😭

$p(1) \cdot \prod_{i=1}^n p(a_i + 2) = 2^n \cdot \prod_{i=1}^n (a_i + 1)^2$ product of something something someting squared

[grand pond]

Copter

[chilly cobalt]

take omega of both sides we get rhs even

and lhs is n + even terms

so n even

amazing

[chilly cobalt]

<@&286206848099549185> someone check this peak solution 🙏

math gods i summon thee

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@chilly cobalt Has your question been resolved?

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@chilly cobalt Has your question been resolved?

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[grave kernel]

p is a prime number and im proving that sqrtp is irrational
doing the process, assuming sqrtp = a/b and gcd(a,b) = 1 and then p = a^2/b^2
then pb^2 = a^2. now, in the solution it says, "hence, p|a^2 and so from euclids lemma p|a"
but euclids lemma says that gcd(a,p) should be 1 no? how is gcd(a,p) = 1 i dont get it, can it not be p itself or idk

[lyric charm]

but euclids lemma says that gcd(a,p) should be 1 no?
can you state euclid's lemma

[grave kernel]

If gcd(a,m) = 1 and m|ab then m|b

thats what i learned

[feral sedge]

i don’t like this solution

[grave kernel]

like, wdym

[feral sedge]

it very weirdly invokes a completely unnecessary lemma (and apparently it “uses” gcd(a, p) = 1 to conclude p | a, which is nonsense)

[runic hamlet]

the other form of euclids lemma is: if p is prime and p|ab, then p|a or p|b

[feral sedge]

that’s better

[grave kernel]

right

[runic hamlet]

use that version

[grave kernel]

can you prove that

[runic hamlet]

it follows from the other one pretty quickly

either the gcd is 1 or its not...

what happens in either case?

[grave kernel]

it divides the other number?

oooohhhhh wait

right

if gcd(a,p) = p

then p|a

if gcd(a,p) = 1

then p|b

is that correct?

given that p|ab obv

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[wary epoch]

Experience has convinced me that it is pedagogically unsound (though logically correct) to start off with the construction of the real numbers from the rational ones
Wait, why is it pedagogically bad to define real number via Dedekind cut? Quote is from Baby Rudin third edition.

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@wary epoch Has your question been resolved?

[prime hornet]

you can try asking in #math-pedagogy too, maybe

my (unrefined) thoughts might be that Rudin is saying that Dedekind cuts are the sort of construction you only ever see once and are thus not super important

nobody thinks about the real numbers in terms of cuts afterwards

if you define them using Cauchy sequences instead though, then that's probably more pedagogically useful, because students will be seeing those sequences again in the future

[hoary pulsar]

At the beginning, most students simply fail to appreciate the need for doing this. Accordingly, the real number system is introduced as an ordered field with the least-upper-bound property, and a few interesting applications of this property are quickly made.

These are the sentences following that one

Like higher! said, Dedekind cuts are correct but they're not really the best way to approach it with new students

Though rather than Cauchy sequences, they're using the LUB property. Which is actually how it was defined for me as well via Stephen Abbott's RA book

[wary epoch]

To be fair, I used to hate Dedekind cuts, because it's not intuitive, but it's grown on me because of how simple the construction is compared to Cauchy sequences.

Anyway, thanks

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[molten bay]

I can do it with expansion but looking for some ideas

[feral sedge]

i believe none of those answers are correct

[molten bay]

And how do you believe?

[feral sedge]

by plugging in theta = pi

all choices of lambda fail

[molten bay]

How can you say lambda fail

Left side we got -1

-2a^3-3a^2+1=-1

,w -2a^3-3a^2+1=-1

[grand pond]

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[lyric charm]

,w det {{0, cos(t), -sin(t)}, {sin(t), 0, cos(t)}, {cos(t), sin(t), 0}}

[grand pond]

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[molten bay]

Yes i got same thing ann

[lyric charm]

.w det {{1, -λ, λ}, {-λ, 1, λ}, {λ, λ, 1}}

[molten bay]

Cubic polynomial

[lyric charm]

,w det {{1, -λ, λ}, {-λ, 1, λ}, {λ, λ, 1}}

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[molten bay]

.

[lyric charm]

why =-1 ?

[molten bay]

He said put theta=pi

[sudden yacht]

I think it's better to start from the answers, which say that λ can be only 0 or 1 (when theta is π)

[feral sedge]

lambda is dependent on theta

you are given four answer choices, functions in terms of theta

all four satisfy lambda(pi) = 1 or lambda(pi) = 0, and evaluating the right determinant gives 1 or -4

[lyric charm]

put theta=pi, the options for lambda become:
A) 0, B) 1, C) 0, D) 0
LHS det gives -1, RHS det gives either 1 or -4 based on WA's results

[feral sedge]

but you found the left to evaluate as -1

this is a clear contradiction. therefore no answer is correct

[molten bay]

Where does it give 1,-4?

[lyric charm]

put λ=0 or λ=1 into RHS value according to wolfrram

[molten bay]

cos^3(t)-sin^3(t)=-2a^3-3a^2+1

t=theta, a=lambda

What are you guys doing, i am confused

Okay if i put left side t=pi then it becomes

-1=-2a^3-3a^2+1

what next?

And option become what ann wrote

if I put lambda=a=0 then equal become -1=1 not possible

But why are we doing such weird thing

[feral sedge]

if lambda(0) = 0 then there is no contradiction

[molten bay]

I'm not getting it

[feral sedge]

cos(0)^3 - sin(0)^3 = 1

det(lambda(0)) = 1

it aligns

so i told you theta = pi gives you a contradiction

ann explained it very well. i suggest rereading her explanation

[molten bay]

I am not getting properly you guys put random values RHS then LHS

And contradiction!!

Which is totally confusion i have wrote my whole work

So here lambda=0 is satisfying

cos^3(t)-sin^3(t)=-2a^3-3a^2+1

Lambda=0, theta=0

So what conclusion?

[empty cove]

do you agree that when theta = pi, the lhs and rhs should be equal?

[molten bay]

For theta=pi

lambda will have different value

[empty cove]

what is the value of lambda when theta = pi?

[molten bay]

,w -2a^3-3a^2+2=0

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[molten bay]

Here is a value

[empty cove]

ok

so now let's look at the answer choices

[molten bay]

Hang on

Ohh got it

No option perfect!!

[empty cove]

:)

[molten bay]

SO theta=0 is not a good thing

Only theta=pi

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[molten bay]

Thanks all brilliant guys

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[viral dagger]

i genuenly dont understand where the solution is going

it just made a diffrent polynomial Q*(x) such that it has roots 21 and 32, and we want to find N such that Q*(N)=4

[runic hamlet]

why the fuck is remainder theorem censored

[feral sedge]

this book censors a bunch of random words, apparently...

[viral dagger]

i think it incentivizes readers to think about what theorems would be useful for the problem... i dont think its doing a good job at that though

[runic hamlet]

Q*(N)=4 is a product of three integers

so there arent many options for what those integers could be

that should probably finish it

[viral dagger]

ohg

[empty cove]

hi

[viral dagger]

i see

[empty cove]

for integer polynomials we have (a-b) | (P(a) - P(b))

this is a very useful property

[runic hamlet]

wait, Q*(N)=4?

is that correct?

not =55?

[viral dagger]

oop its 55

your point should still holds though

[runic hamlet]

yeah

[viral dagger]

alr ty

.solved

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[grave kernel]

i didnt get this solution at all, its also missing a few of the lcm and gcd texts

[fathom onyx]

We often shorthand, for some dumb reason, the GCD of two numbers as follows

(x, y) = gcd(x,y)

[grave kernel]

oh really

[fathom onyx]

yh

It should be contextually obviously not coordinates

[mortal mirage]

its really stupid imo

[fathom onyx]

But it does need explaining, yh - usually this SHOULD be stated at the beginning of the notes you're reading

[grave kernel]

okay but still tho, the solution is pretty vague to me

nah, this just randomly started, i just thought they are missing the gcd part

[fathom onyx]

Grab the gcd of a and b, call it d; then a = dm and b = dn for some m and n whose gcd is 1

[mortal mirage]

i think theres a typo in the second sentence

[fathom onyx]

Then a+b = (as explained)

[grave kernel]

right

[mortal mirage]

it should be (a,b)

[fathom onyx]

Then LCM is (as explained);
Then the rest of this solution involves hunting for d, m and n

[grave kernel]

how are they getting the 4 x 13

[grave kernel]

right

i dont get the last part

[mortal mirage]

do you get the two equations?

these two

[lethal path]

well it's by process of elimination

d can't be 13 or 52, because that would imply lcm(a, b) would be divisible by 13

then figure out for yourself why d can't be 1 either

[grave kernel]

right

[grave kernel]

how is lcm(a,b) = dmn

[mortal mirage]

by definition

but if you think about it because d is a common divisor, it only should appear once in the lcm

[grave kernel]

oooooohhhh

right

right

[lethal path]

this reasoning is actually a bit tricky, so I'll start you off with the fact that if d = 1, then lmc(a, b) = ab, so ab = 168

but then since a, b are again coprime, exactly one of a, b must be divisible by 4, and the other one must not be

but then a + b = 52 ||which is a multiple of 4....||

[grave kernel]

okay

right

i got it

[lethal path]

no worries!

[grave kernel]

right

that makes sense

thank you

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[hybrid crow]

I just need help showing the only if part

Suppose null S is a subset of null T

how do we show there exists an E with that property

Intuitively I get why E must exist but I'm not sure how to formally say it

My non-rigorous explanation is that S is gonna take basis vectors v1,...,vn to either 0 or to some w's

and then we construct E to take some more w's to 0, (so that all v's taken to 0 in ET make up T's null space) and E takes the remaining w's to where T would take the v that they came from

[outer girder]

Try to construct E from the given property

[hybrid crow]

yeah idk how

S(a1v1 + ... + anvn) = a1w1 + ... + anwn

lets just say as an example that v1,v2,v3 all go to 0 under S

but that T(a1v1 + ... + anvn) takes v1,v2,v3, and v4 to 0

[outer girder]

V is not given to be finite-dimensional so you can't just take a finite basis like that

[hybrid crow]

oh

right

wait

if V is infinite-dimensional

then doesn't that mean the null space is infinite

[outer girder]

it could be

actually yes it is

Here's an idea :
given a basis {b1, ... bn} of range(S), find v1, ..., vn from V with S(vi) = bi
then define E(bi) = T(vi) for i=1,...,n, and define E to be zero wherever the input is not in range(S)

it is a good exercise to show that given any vector v in V we may write v = a + b where a is in span(v1,...,vn) and b is in null(S) [-- in fact, this representation is unique]

by using the fact that null(S) is contained in null(T), you can easily show that E is well-defined

[midnight plank]

@hybrid crow Has your question been resolved?

[hybrid crow]

alr

lemme think about this

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[icy herald]

can i have help

[icy herald]

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[floral apex]

I have a question.

[floral apex]

This book says that the system:

[grand pond]

jan Niku

jan Niku

jan Niku

[floral apex]

that this system has a center at p_j = q_j = 0

oh wait

oh

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[chilly cobalt]

A positive integer $n$ is known as an interesting number if $n$ satisfies
[{\ {\frac{n}{10^k}} } > \frac{n}{10^{10}} ]
for all $k=1,2,\ldots 9$.
Find the number of interesting numbers.

[grand pond]

Copter

[lusty python]

i mean

doesn't all $n$ satisfy that?

[grand pond]

1 divided by 0 equals Infinity

[chilly cobalt]

i was looking at the solution for this, and they said we can establish a bijection between interesting numbers and aperiodic necklaces $10$ beads of $10$ possible colors.

[lusty python]

$10^{10} > 10^k$ for $1 \leq k \leq 9$

[grand pond]

Copter

[chilly cobalt]

no idea what that means tho

[grand pond]

1 divided by 0 equals Infinity

[chilly cobalt]

fractional part

i think?

[lusty python]

and the numerator is $n$ for both fractions

[grand pond]

1 divided by 0 equals Infinity

[lusty python]

so isn't it true for all $n > 0$?

[grand pond]

1 divided by 0 equals Infinity

[chilly cobalt]

n = 10^i is false

cause for some k the fractional part will be 0

also the anwer is $\frac{1}{10}(10^{10}-10-(10^5-10)-(10^2-10))=10^9-10^4-9=999989991$?

[grand pond]

Copter

[chilly cobalt]

wth 😭

[chilly cobalt]

anyways could someone explain this?

[chilly cobalt]

also n cant have more than 10 digits

thats a restriction

<@&286206848099549185>

[midnight plank]

@chilly cobalt Has your question been resolved?

[wheat nymph]

Do u know what a bijection or period is?

[chilly cobalt]

one to one correspondence?

i dont know what "beads" and "necklaces" are unfortunately

[wheat nymph]

That's fine

[chilly cobalt]

can you fix it 💔

[wheat nymph]

Essentially the analogy with the circular beads is that for every 10-digit number (leading zeroes allowed), we can always shift the digits

e.g. abc...hij can be shifted 1 digit to bc...hija

Hence the necklace

Basically u can think of the digits as arranged in a circle

Does that make sense?

[chilly cobalt]

yeah

[wheat nymph]

More formally, we r partitioning the space of 10-digit numbers into these classes

The solution claims that interesting numbers must not be periodic (in the sense that their class is of size <10), and each non-periodic class contains exactly one interesting number

That's the bijection

Is that clear?

[chilly cobalt]

do "classes" mean like, all the possible shiftings of a number

[wheat nymph]

Yeah the set of these shifts

The formal term I should be using is equivalence class btw

[chilly cobalt]

spooky ;-;

how does the solution come from this though?

[wheat nymph]

It's counting the number of non-periodic equivalence classes

[wheat nymph]

^

We can do this since there's a bijection between the set of equivalence classes and the set of interesting numbers

[chilly cobalt]

this?

[wheat nymph]

Um almost

The solution says that u only need to count the non-periodic necklaces

[wheat nymph]

These r quite general formulas
In ur case u alr have a nice case where k=n

[chilly cobalt]

oh

[wheat nymph]

It's reasonable to figure out why this is the case without plugging it into the formula

[chilly cobalt]

hmmm

i dont really get how, sorry😭

[wheat nymph]

On a second look at this formula, yes actually it's exactly the right thing

[wheat nymph]

So when I say this, I mean it's not too difficult to derive

[chilly cobalt]

oh dear

[wheat nymph]

I think we should first get some definitions clear

Do u understand what it means for a necklace to be periodic

[chilly cobalt]

if its periodic does that mean there are some "shifts" that are equal?

[wheat nymph]

Yeah

Let's call a necklace q-periodic if it equals itself by a shift of q digits

Since the necklace is of length 10, what possible periods can it be (note that a necklace can have multiple periods)?

[chilly cobalt]

divisors of 10 right?

[wheat nymph]

Yeah and I should've said smallest periods it can be

Anyways

We're trying to count the number of necklaces whose smallest period is 10, so the ones that are not 1, 2, or 5

[chilly cobalt]

uh huh..

[wheat nymph]

Can u see what the 1,2 and 5- periodic necklaces should look like

And use that to count them

[chilly cobalt]

for 1 period theres 10 right

for 2 its 10^2..?

[wheat nymph]

Yes

But when u count the 2-periodic necklaces u also count the 1-perioidic ones

[chilly cobalt]

so you subtract?

[wheat nymph]

Yes

[chilly cobalt]

ooooh

[wheat nymph]

In general the number of q-periodic necklaces is 10^q

[wheat nymph]

Hence the inner term

[chilly cobalt]

well, where does the 1/n come from?

[wheat nymph]

So now we're left with the number of aperiodic necklaces

We haven't partitioned them into their equivalence classes yet

Rmb that we're trying to count the number of equivalence classes of aperiodic necklaces

[chilly cobalt]

oh

[wheat nymph]

Btw this general formula is using the Mobius function, which is what u need when u replace 10 with k
This is arising from the inclusion exclusion stuff that's going on in our proof

But for ur question the remaining bit is to establish why u have a bijection

[chilly cobalt]

hmm

[wheat nymph]

O and also u need to show why aperiodic necklaces r not interesting

[chilly cobalt]

brain is loading

each interesting number is atleast a 10 digit number which are each beads or something? each color is a digit and each number is different among its shifting

sob

is this a bijection

[wheat nymph]

um almost

interesting number satisfies the inequalities in the question, and we can deduce the this means its 10-digit (leading zeros allowed) and aperiodic

so on one side we have the set of interesting numbers

on the other side we have 10 bead aperiodic necklaces where each bead is labelled by some digit 1 to 9

sry abt all that

[chilly cobalt]

its alright

[wheat nymph]

maybe its best to figure out this bijection by means of an example

so just take an arbitary 10-digit aperiodic number, write down all 10 of its shifts and figure out which one is interesting

then figure out the common rule for all these necklaces

[chilly cobalt]

uhh

im guessing its the least one?

ahh

im kinda lost :<

sob

[wheat nymph]

Yeah so u might have alr noticed the leading digit of the interesting number must be the smallest on the necklace

So now u have to consider what happens when the necklace has multiple beads with the smallest digit

[chilly cobalt]

hm

i dont really know

sorry

[wheat nymph]

E.g. what's the interesting number arising from the necklace 0911103111

[chilly cobalt]

is it 311109111?

sob

[wheat nymph]

Yeah

The general rule, if u can't see it yet, is that u ||shift the necklace to the smallest possible value||; ||there exists a unique minimum because of the aperiodicity||

[chilly cobalt]

ohh