Mathematics - 2018-01-30 (page 1 of 5)

Semiclassical

00:00

my favorite bit of combinatorics is generating functions, and I don't think I ever saw them in a math class

Fawad

$\Sigma n^2 -n = \dfrac{n(n+1)(4n-1)}{12}$

Semiclassical

they show up in physics esp. statistical mechanics, but not in a way that's fully spelled out

Fawad

Nvm

Akiva Weinberger

I heard of them through generatingfunctionology (the book). I don't remember where I got the book from.

Semiclassical

yeah, same

Akiva Weinberger

00:04

(Er, I don't remember where I got the PDF from)

Semiclassical

the other book I got them from was Flajolet and Sedgewick's Analytic Combinatorics

which was waaaay overkill for what I needed, and I still can't get through most of it

Akiva Weinberger

I ask because my sister asked me my opinion on combinatorics

Apparently the reason she was asking was because "[her] friend was freaking out about his combinatorics hw and I was curious"

(I'm pretty sure she wouldn't mind me sharing that piece of our conversation here, it's pretty innocuous)

Semiclassical

combinatorics is a bit weird compared to other stuff

The Great Duck

@AkivaWeinberger i think your sis just wanted to know what you thought about it so she could use the response to embarass you.

XD

Akiva Weinberger

Nah I trust her

The Great Duck

00:15

glares

gullible, are we? never trust a sibling to not play a prank.

Semiclassical

depends on the sibling, really

Xander Henderson

combinatorics is f'n' hard

it is even worse than algebraic number theory

Akiva Weinberger

I don't even know what sort of prank that would be

The Great Duck

:|

fair point

Semiclassical

the usual challenge of combinatorics, I think, is to figure out how to relate the counting problem you're interested in to one you already know how to do

Though stuff like inclusion-exclusion usually wrecks me :/

Akiva Weinberger

00:19

I honestly have no idea what sort of problems you'd be doing in that class

I mean, obviously there's "place a billion boys and a jillion girls in a row of chairs so that no two boys are next to each other" but that doesn't feel like an undergraduate problem

Or at least it would show up fairly early on in the course

The Great Duck

@AkivaWeinberger youd be surprised

Semiclassical

As a full course, yeah I dunno

I had combinatorics as part of a discrete math course

Akiva Weinberger

Would you be counting different ways of drawing graphs and stuff?

Xander Henderson

stars and bars arguments show up in combinatorics classes quite often

Akiva Weinberger

As it does in US History

Xander Henderson

00:22

relations involving binomial expressions

The Great Duck

ba dum ch

Xander Henderson

like, proving various identities

Akiva Weinberger

Pascal's triangle then

Xander Henderson

yarp

Akiva Weinberger

Big pile o' binomial coefficients

Xander Henderson

00:23

depending on the instructor, maybe some elementary graph theory

there are some fun tricks with Fibonacci and Lucas numbers, too

Daminark

Combinatorics is best subject

Eric Silva

Sizzling take: combinatorics is worst subject

Akiva Weinberger

Like how $\sum F_n^2=F_nF_{n+1}$?

As proven by the above image

Leaky Nun

"proven"

Daminark

@EricSilva it's not a sizzling take so much as pure crankhood

Oooohhhhh

Akiva Weinberger

00:27

Or more like, I have dominoes and squares and I want to make a thing $n$ long

Eric Silva

I mean it's not my actual opinion but not liking combinatorics isn't cranky

Akiva Weinberger

which ends up being Fibonacc-y

Daminark

All people who don't work in combo are cranks tbh

Akiva Weinberger

@Daminark I'm asking what sort of problems you'd find in an undergrad combinatorics class

Daminark

Not even a question of disliking, if it isn't your favorite wyd?

Eric Silva

00:28

@Daminark hot take: all people are cranks

Daminark

@Akiva I can send materials for our class from 2 years ago

Eric Silva

Except me

Akiva Weinberger

Sure why not

Or a sample homework sheet

Daminark

@EricSilva that's colder than the weather outside

Oh shit

Eric Silva

the mind of a crank

The Great Duck

00:29

@Daminark so every engineer is a crank?

Akiva Weinberger

Isn't Eric in the southern hemisphere?

Oh wait no he's at uni in the States

Or something I dunno

Eric Silva

I live in Chicago lol

And previously lived in Florida

Daminark

@TheGreatDuck unless you're engineering stars and bars

Akiva Weinberger

Wait so who's the Brazilian guy

Is that the other Eric

Daminark

Eric is originally from Brazil but we're in the same school

Eric Silva

00:30

I happen to be a Brazilian-American

Akiva Weinberger

Oh OK

Oh, but not like actually born there

Annelise t.

Hi:) In this system $$\frac{dA}{dt}=A(2-\frac{A}{5000}-\frac{L}{100})$$ $$\frac{dL}{dt}=L(-\frac{1}{2}+\frac{A}{10000})$$ suppose A=0, then $L(t)=ce^{-t/2}$ which means that if $t\to\infty$ then $L\to\infty$.Suppose L represent some specie population, then $L\to\infty$ means that the L population will be extinct in future or means that L population will decrease exponentially but won't disappear in future time?

Akiva Weinberger

You don't need to answer that if you don't want to

The Great Duck

@Daminark are you drunk?

Eric Silva

Nope I was born stateside

Daminark

00:31

I'm drunk on binomial coefficients

Akiva Weinberger

@Anneliset. Doesn't $L(t)=ce^{-t/2}$ mean that as $t$ goes to infinity, $L$ goes to zero?

Because of the minus sign in the exponent

Annelise t.

Oh sorry yes of course, $L\to 0$

The Great Duck

@Daminark real analysis is best subject period

not debatable

Annelise t.

@AkivaWeinberger But I am asking about the meaning of $L\to 0$

Eric Silva

Hottest take: math sucks

Daminark

00:33

Real analysis is for NERDS

Eric Silva

@Daminark agreed

Daminark

"Muh deltas"

Akiva Weinberger

I guess that means $L$ will decrease exponentially... although, technically speaking, population is discrete, isn't it? Like, if $ce^{-t/2}$ is one-half, that can't mean the population is one-half

The Great Duck

combinatorics is for dumb JOCKS

Akiva Weinberger

'cause you can't have half of a person

so I guess practically speaking it would mean the population eventually becomes extinct @Anneliset.

Daminark

00:34

Wombo combo is the chad of math along with number theory

Leaky Nun

o.O what is going on here

Xander Henderson

MATH IS GOING ON HERE!

The Great Duck

@Daminark amen to that brudduh

WE

ARE

MATH

THIS

IS

Akiva Weinberger

Sparta?

The Great Duck

MSE!!!!!

Daminark

00:35

@XanderHenderson impossible!

Xander Henderson

@Daminark Not only is it not impossible, it is unimpossible!

Daminark

Thonk

Eric Silva

This chat is 0/10 tbh

The Great Duck

@AkivaWeinberger even better than Sparta

@EricSilva no. this chat is 0\10.

Akiva Weinberger

In machine learning lingo, the chat is "overfitting" and "producing nonsense results"

Annelise t.

00:37

@AkivaWeinberger and when will this happen? I tried to solve for t in L but I got Ln(0), so approximately when L is going to be in extinction ?

The Great Duck

@AkivaWeinberger we arent machines. we arent learning. no contradiction occurred.

@LeakyNun demonark is drunk on combinatorics.

Leaky Nun

@Anneliset. when the population falls below 1, you know

modelling is just an approximation

Akiva Weinberger

@Anneliset. $L(t)=1$ when $ce^{-t/2}=1$, so when $t=2\ln c$. When $t$ is greater than that, $L$ is less than one. The only interpretation that makes sense to me is that $L$ is extinct then

Leaky Nun

they "tend to" go like a smooth exponential function when viewed on the big scheme of things

Akiva Weinberger

just because if this is a species' population, $L$ has to be a whole number at all times

Leaky Nun

00:39

@AkivaWeinberger sniped :P

The Great Duck

@LeakyNun no you wont. if it falls below one you cease to exist and therefore know nothing.

Akiva Weinberger

Can you guys try to make sense for one minute

4

> For almost all graphs $G$ with $n$ vertices, $\alpha(G) ≤ 1 + 2 \log_2 n$.

Do you know what $\alpha$ is here? @Daminark

The Great Duck

i can make scents if youd like?

but i gotta go

Annelise t.

:)

Leaky Nun

@AkivaWeinberger "almost all"

in the lingo of theorem provers, unification

Akiva Weinberger

00:42

Meaning, as $n$ goes to infinity, the probability that a graph satisfies that inequality goes to $1$

Leaky Nun

maybe average length of loops?

I'm making strange connections to proof theory / model theory, when I was proving that the ideal generated by S and T is S+T

Akiva Weinberger

I think it's the size of the largest independent set?

Leaky Nun

i.e. if you can prove, from the rules of ideals, starting from "the ideal contains S and T", to "x is an element of the ideal", then x is in S+T

Annelise t.

@AkivaWeinberger Should not be $t=-2\ln c$?

Leaky Nun

@Anneliset. sign errors, lol

Semiclassical

00:46

@EricSilva how goes physics

Akiva Weinberger

@Anneliset. I don't think so, because of the minus sign in the exponent

If $t=2\ln c$ then $ce^{-t/2}=ce^{-(2\ln c)/2}=ce^{-\ln c}=c\frac1c=1$

$e^{-\ln c}=\frac1c$ because it's $(e^{\ln c})^{-1}=c^{-1}$. Alternatively, it's because it's $e^{\ln(1/c)}$.

Eric Silva

@Semiclassical I have lab tomorrow and I'm going to die

The stuff that isn't lab is great though

Semiclassical

i'd say "now you know what being a TA is like"

except lab stuff isn't bad as a TA. it's grading lab reports that's the arse

Eric Silva

Yeah I feel bad for my ta having to read through this shit

Akiva Weinberger

Labs are always less fun than lab dogs

Semiclassical

00:50

what stuff are you up to in lab?

Akiva Weinberger

How To Create Dog

Eric Silva

Last week was just learning how to use an oscilloscope

Semiclassical

ah

Are you doing CRT stuff?

Eric Silva

We didn't last time

Idk what the next one is

Semiclassical

(i mean, that's what an oscilloscope is based on. but like, applying an electric or magnetic field to a CRT and seeing the beam bend)

Eric Silva

00:52

Isn't that an analog oscilloscope

The Great Duck

im back

Semiclassical

yeah

if you're doing a digital oscilloscope then that's different

Eric Silva

Yeah we had these big digital ones

It was just a bunch of boopin and boppin buttons for four hours

The Great Duck

im having trouble determining the conditions by which N functions are linearly independent using the Wronskian

Eric Silva

@Semi according to the lab guide the next lab is $e/m$ of an electron whatever that mwans

Semiclassical

00:56

oh, nice

yeah, that's fun

The Great Duck

@Semiclassical last semester we had to measure the speed of light using cables

and a computer

Leaky Nun

til having all products implies having terminal object

Eric Silva

In other physics related news I just found out about a mathematical GR summer school which I just applied to

Akiva Weinberger

My external battery pack for my phone came

(It's a replacement for my old one which broke when someone else was borrowing it)

(hmmmmm)

Semiclassical

@TheGreatDuck I think we did something similar: We pulsed a laser light down at a mirror down the hall and bounced it back to the initial setup

Akiva Weinberger

00:58

(This one is from a different company)

Semiclassical

so if you compared the time at which the pulse was send out to the time the signal returned, you could figure out the time of flight

The Great Duck

@Semiclassical yeah except the software we used was stupid and messed up the results

well sorta

Semiclassical

ours was okay. nothing stellar

The Great Duck

you know how when you open windows task manager you can see CPU percentage?

it's labelled "CPU"

Semiclassical

we could directly compare the voltage signals (pulse sent compared to pulse return) in our digital oscilloscope and read it from there

sure

The Great Duck

01:00

well the program's time axis was using its own runtime, not the operating system

Semiclassical

ooof

The Great Duck

so the time axis was off by a factor of 1/2

Semiclassical

the digital oscilloscope runs independently of a PC, so we didn't have that as an issue in our case

The Great Duck

good

needless to say the students unlucky enough to have the antivirus running a scan got really weird results

Semiclassical

@EricSilva one way that can show up is this. You apply an electric field E vertically and a magnetic field B horizontally, then send a beam of electron perpendicular to both

@TheGreatDuck yeah

Eric Silva

01:02

Lol @Semi we had a problem displaying some curve on our oscilloscope and the TA came over and was like "wtf" and pressed some buttons and it was fixed. I asked what he did and his response was just "I have no idea"

Semiclassical

lol

The Great Duck

apparently we somehow violated Einstein's core principles. XD

Eric Silva

Wait what does e/m of an electron even mean

Semiclassical

getting there

The Great Duck

energy per meter?

Eric Silva

01:02

Ah ok gotcha

Semiclassical

oh, drat. the example I'm giving isn't going to help for that.

Eric Silva

He says with the lab guide open on his desk

Semiclassical

it's the charge-to-mass ratio

Eric Silva

Makes sense

The Great Duck

ah

e as in electron charge

Semiclassical

01:03

right.

what I was going on about was a velocity selector. That's usually part of the following setup, but it's not where e/m shows up so it's a distraction

Eric Silva

Hmmk

Semiclassical

Suppose you have a vertical uniform magnetic field B, and you shoot an electron at a velocity v in the horizontal direction

then the Lorentz force on the electron will have a magnitude evB where e is the charge of the electron

Eric Silva

Agree

The Great Duck

ah I see

Semiclassical

since the field is uniform and the Lorentz force is perpendicular to the motion, the electron will go in a circular orbit

The Great Duck

01:05

the wronskian tells you if things are linearly dependent

i remember now

Eric Silva

Also agree

The Great Duck

now the question is whether a wronskian with implied derivatives allows one to tell if functions are... piecewise linear independent?

ugh

Semiclassical

the orbit's radius is determined from circular motion principles: to go in a circle of radius r, you need an acceleration a=v^2/r perpendicular to the motion

so from F=ma you get mv^2/r = evB

or mv=eBr for a pithy expression

So if you know the velocity the electron starts with, the radius of its orbit, and the field B

you can work out the charge-to-mass ratio of the electron experimentally

what you can't determine from this is the charge or the mass by themselves, though

Eric Silva

Ahh

I see

Semiclassical

one setup like this would be to apply a magnetic field to an electron beam and see how much it makes the beam deflect

from the deflection, you can work out the radius of curvature

and the initial velocity of each electron is determined by the accelerating potential applied to the beam to start with

the tricky part is getting a good magnetic field (both good in the sense of being uniform and of knowing what its strength is)

best setup for that is a Helmholtz coil setup

Eric Silva

01:11

Yup that's what it is according to the guide

Semiclassical

neat

The Great Duck

0

Q: Are two functions $f$ and $g$ fitting these qualities neccessarily linearly independent?

Suppose that there exists a functions $f$ and $g$ defined on the real numbers and differentiable everywhere. If there does not exist real numbers $c \neq 0$ and $d \neq 0$ such that $(cf(x) + dg(x))' = 0$ on some nonzero interval then are $f$ and $g$ linearly independent? I am trying to determin...

Semiclassical

lol

Eric Silva

Oh this lab seems significantly less annoying than previous ones

The Great Duck

dont laugh at my inability to copy hyperlinks properly

i gotta go

Semiclassical

01:12

nah, i assumed you were making a joke about how we're talking lab physics in the math chat room

The Great Duck

no

Eric Silva

Let's go talk math in a physics chatroom

The Great Duck

nah i have to go

Eric Silva

I was talking to semi

Semiclassical

@EricSilva when you do this with a CRT, you can use a bar magnet to see the electron beam on the screen move perpendicular to the magnet

The Great Duck

01:13

drive.google.com/file/d/1kKcMF80ZTdKFyvwTrFIGWgEwhdmlCNAp/view

im trying to prove that for my paper

it might be useful in the part about piecewise trivial solutions

Eric Silva

That is p cool

Semiclassical

if you put the same bar magnet on the other side, it'll deflect in the opposite direction of the original setup

what's really fun is to do two bar magnets, one on each side

based on what I've said, you'd expect the two effects to cancel out. And in one respect, that's true

the center of the beam is just going to be where it was without the two magnets

but the beam shape goes from being a little circle to a little ellipse :)

Eric Silva

Wait why

Semiclassical

heheheh

i actually asked this on PSE to get a good answer, lemme find it

Eric Silva

What fresh spook is this

Semiclassical

01:16

1

Q: Diagonal squeezing of an electron beam by a pair of bar magnets

While coordinating an introductory physics lab on the Lorentz force, I came across a behavior which I hadn't seen before and for which I didn't have a ready explanation. The experiment consisted of putting a CRT tube in a uniform magnetic field causing the electron beam to curve and therefore mov...

electromagnetism magnetic-fields electrons

Eric Silva

Whoa

Semiclassical

yeah

p cool

Eric Silva

That explanation is gud

Thats pretty surprising

Semiclassical

one fun bit of that is that the system has a reflection symmetry (i.e. flip everything along the horizontal axis and it remains the same)

and yet, the beam ellipse is oriented at a 45 degree angle to this axis. so the ellipse doesn't share this symmetry!

Eric Silva

Yes agree it is indeed flippy doo

Too weird my dude

Semiclassical

01:22

if memory serves it comes down to the fact that B doesn't transform as a vector but as a pseudovector

so a reflection can send B->-B

and that leads to weirdness

Eric Silva

What do you mean pseudovector

Semiclassical

"In physics and mathematics, a pseudovector (or axial vector) is a quantity that transforms like a vector under a proper rotation, but in three dimensions gains an additional sign flip under an improper rotation such as a reflection."

Eric Silva

Oh is it like a bivector

That has the right symmetry then I think

Semiclassical

probably related, yeah

Eric Silva

In 3 dim I guess

Semiclassical

01:25

hmm, wikipedia says this:

"One way to formalize pseudovectors is as follows: if $V$ is an $n$-dimensional vector space, then a pseudovector of $V$ is an element of the $(n − 1)$-th exterior power of $V$: $Λ^{n−1}(V)$. The pseudovectors of $V$ form a vector space with the same dimension as $V$."

Eric Silva

Ah so yes then in our case it's a bivector

Cool

Semiclassical

ya

I think the point in this case is that the Lorentz force is given by $\mathbf{v}\times \mathbf{B}$

the velocity $\mathbf{v}$ is a regular vector, but B is a psuedo-vector

and if you take a cross product of vector - pseudovector, you get a pseudovector

Eric Silva

Hmm

Semiclassical

So the force experienced by the electron along its trajectory is a pseudovector

wait, no

vector x pseudovector should be vector

Eric Silva

So vector x vector is pseudovector right

Semiclassical

01:30

right

Eric Silva

Bc the cross product does a weird when you flippy doo

Semiclassical

ya

it's also suggested by the fact that in index form one has $(A\times B)_i = \epsilon_{ijk} A_j B_k$

so one is running into how the Levi-Civita tensor behaves under rotations

Eric Silva

Ok I see

I want to write out what all this means formally

Semiclassical

yeah

Daminark

Ugh we had a fire alarm go off

Eric Silva

01:35

@Semiclassical I guess you need hodge star

Semiclassical

ya

Eric Silva

So then the cross product is gonna be like $\ast(v \wedge w)$

And so doing it with three boys you get $\ast(u \wedge \ast(v \wedge w))$

Horrible

Ted Shifrin

What is the world coming to when @Akiva asks people to make sense?!!

This is a good one for @EricSilva and @Balarka.

Eric Silva

Hirschhhh

That reminds me I need to a study for my diff top exam

And do my e&m pset

Ted Shifrin

Good that things happen in the world to remind you of such matters.

Eric Silva

01:46

Jury is still out on good or bad

Prototank

Can someone help me understand the red underlined sentence?

Ted Shifrin

I can't, offhand.

And I need to leave. Sorry.

Prototank

Have a good evening!

Meow Mix

i missed ted

Adeek

@MatheinBoulomenos can you tell me a good book that talks about algebraic number theory in order to talk about go into Arakelov geometry

I will wait for your recommendation.

Xander Henderson

02:59

Alexander's Theorem? Someone gave me a theorem! Yay!

Annelise t.

Hello. How to determine if time $t$ used in some mathematic model is defined in days or months?

Xander Henderson

@Anneliset. Context.

Annelise t.

@XanderHenderson This is the model $$\frac{dA}{dt}=A(2-\frac{A}{5000}-\frac{L}{100})$$ $$\frac{dL}{dt}=L(-\frac{1}{2}+\frac{A}{10000})$$ where A=number of aphids, L=number or lady-bugs.

I also have the phase portrait, do you want to see it?

Prototank

Funny enough, a googling of Alexander's Theorem will not lead one to the theorem being quoted

thought I know what they are talking about, so I'm not upset

Secret

in Logic, Jan 27 at 3:06, by user21820

For any pure first-order logic proof that you write down and empirically verify that it satisfies the rules, we can be empirically sure that the theorem it proves is true of the real world under any permitted interpretation of the symbols.

@Dattier

may be related

but if i recall, you want empirical logic to not have inference rules, i.e. that the rules themselves are empirically checked

in order to not fall into the dogma

Metalogic

Metalogic is the study of the metatheory of logic. Whereas logic studies how logical systems can be used to construct valid and sound arguments, metalogic studies the properties of logical systems. Logic concerns the truths that may be derived using a logical system; metalogic concerns the truths that may be derived about the languages and systems that are used to express truths. The basic objects of metalogical study are formal languages, formal systems, and their interpretations. The study of interpretation of formal systems is the branch of mathematical logic that is known as model theory, and...

this might be relevant

Xander Henderson

03:17

@Anneliset. You can't just look at a bunch of equations and know what the units are

you have to have some context from the statement of the problem to have any idea what is going on

according to THE GOOGLE, aphids live for about a month, and ladybugs live for a couple of years

as such, it seems unlikely that your model assumes that $t$ is measured in anything so long as a century or millennium, nor so short as a minute or second

however, anything from days to years could be reasonable

and, to be frank, the model doesn't care; turn the crank, and interpret the result as best you can

Annelise t.

@XanderHenderson I think months fit well then. The model doesn't care? what do you mean?

Xander Henderson

Mathematically, a change of units is just multiplication or division by a constant. If you solve the problem with the assumption that the units were seconds, but it turns out that they were months, you get the same basic function, rescaled by a constant to fix the units.

This constant doesn't have any deep mathematical meaning, but it is quite important if you want your model to fit your empirical observations

Again, I would suggest that the appropriate units should either (a) be clear from the statement of the problem or (b) not matter.

Annelise t.

03:33

@XanderHenderson It's not clear from the statement to determine the units, this model was just given like I gave it before in comments (it's from a book BioCalculus) however I used it as a real life model and I also google for information about both species and other things. But as you mention, it's important to know the time to fit your empirical observations.

I think I am going to take units as months :)

Silent

10:14

This is from Rudin PMA: "Theorem: If $p>0$, then $\lim\limits_{n \to \infty}\frac 1{n^p}=0$. Proof: take $n>(1/\varepsilon)^{1/p}$.(Note that the Archimedean property of the real number system is used here)." I can't see where Archimedean property is used, and what happens if Archimedean property does not hold here.

Tobias Kildetoft

@Silent You have a real and you take a larger natural number. That is precisely the property

Silent

And what if archimedean property does not hold in this case? Just curious. @TobiasKildetoft

Tobias Kildetoft

You mean what would happen in a field without that property but enough other structure to make sense of the statement?

Kasmir Khaan

@TobiasKildetoft Hey Tobias :D

Tobias Kildetoft

@KasmirKhaan Hi

Kasmir Khaan

10:21

I had a question that I could not make sense of, can you check if it is well written ? :D

no answer just if it can be solved

:D

pleaseeee :D

Tobias Kildetoft

sure

Kasmir Khaan

yeey:D

@TobiasKildetoft did you get the invitation ?><

Silent

10:51

@TobiasKildetoft So, my question is useless, right? Thank you!

Tobias Kildetoft

@Silent Not entirely, but it takes some work to get to a setting in which it makes sense.

Silent

Ok

jublikon

Hola :)

I have to draw a complex number. Can you help me with that?
This is what I have done so far:

let $z \in \mathbb{C}: \quad z = a+bi, \quad a, b \in \mathbb{R}$

$$|z+2|^2> |z-2i|^2+1$$

$\begin{align}
|(a+bi)+2|^2 > |(a+bi)-2i|^2 + 1 &\equiv (\sqrt{(a+2)^2 + b^2 } )^2 > ( \sqrt{a^2+(b-2)^2})^2 + 1\\
&\equiv (a+2)^2 + b^2 > a^2 + (b-2)^2 + 1 \\
&\equiv a^2+4a+4+b^2 > a^2+b^2-4b+4+1 \\
&\equiv 4a > -4b + 1 \\
&\equiv 4a+4b > 1
\end{align}$

jublikon

11:22

never mind

I got it now :)

user777

12:04

Hi I have a question regarding definition of Edge Clique cover problem. It defined as: we are given a graph G and a nonnegative integer k, and the goal is to decide whether the edges of G can be covered by at most k cliques.
Now, suppose you have the following graph: (v1,v2), (v1,v3), (v3,v2), (v3,v4), (v4,v2). Then if we say k=3, then it would return yes; because we have triangle v2v3v4 and two left edges: (v1,v3) and (v1,v2). Now, if k=4 then it would return no; because we cannot cover the whole graph by any clique with k=4. Now, k=5, then it would return yes; because number of edges are …

user8469759

hi guys, if $v \in \mathbb{R}^3$, and has unit length how can I formalize the idea that $v \in \Omega$, where $\Omega$ is a solid angle?

the thing is if $v \in \mathbb{R}^2$ I can get the idea that $v$ spans an angle between $\alpha$ and $\beta$

not entirely sure though what can be the idea that $v$ spans a solid angle between $\Omega_0$ and $\Omega_1$

if $v \in \mathbb{R}^3$

mick

0

Consider the sequence $f$ : $$f(1) = 1$$ $$ f(n) = f(n-1) + \frac{1}{n f(n-1)}$$ Now we have for large $n$ : $$ f(n) = \sqrt {2 \ln(n)} + \frac{1}{19} + C + eps(n) $$ Where $C$ is a constant and $eps(n)$ Goes to $0$ for large $n$. Main question : Is $C = 0 $ ? If not What is Its value and d...

asymptotics dynamical-systems conjectures constants

Hi all. Any ideas ?

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$Mathematics$ Mathematics