Mathematics

2:45 AM

Hello, I'm trying to prove Hamming distance is a metric distance. The way to do it is to prove a) $ Positive case d(x) \geq 0$. b) Symmetric case $d(x,y) = d(y,x)$, and d) Triangle inequality $d(x,y) + d(y,z) \geq d(x,z) $. I think I'm able to prove a) and b) but d) is difficult

0celo7

What's Hamming distance?

Axoren

It's about how far you can ham.

Serious note, it's an edit distance.

$d(cat, dog) = 3$ because you need to replace $3$ letters in "cat" to get "dog".

YOUSEFY

Hamming distance is a function which gives you the different between two numbers. for example: x=100 and y=111, then H(x,y) = 2

Semiclassical

Given two strings of binaries, it's the number of bits that need to be flipped to make them agree

Axoren

I don't think it's a good idea to denote it as $H(x, y)$, seeing as how that already has an important meaning in each setting you'd see Hamming distance discussed in.

Most of those settings, it's Joint Shannon Entropy.

0celo7

2:52 AM

how do you people know this

@Semiclassical long time no see

I bombed the QM exam

Jessy Cat

1

Q: For $x,y \in \mathbb{Q}\setminus\{ 0\}$, prove that $\mathbb{Q}/\langle x \rangle \simeq \mathbb{Q}/ \langle y \rangle$.

For $x$, $y$ nonzero rationals, I need to prove that $\mathbb{Q}/\langle x \rangle \simeq \mathbb{Q}/ \langle y \rangle$. To do this, I was thinking about using the Third Isomorphism Theorem: Let $M, N \trianglelefteq G$ and $M \leq N$. Then, $(G/M)/(N/M)\simeq G/N$. However, this would...

Semiclassical

Hamming shows up in abstract algebra courses a lot

@0celo7 oof

0celo7

but I'm hoping everyone did

Axoren

@0celo7 Am computer scientist. Generally ham on a regular basis.

Jessy Cat

I just need somebody to explain the kernel of this thing to me.

0celo7

2:53 AM

it was extremely hard -- two people finished

Semiclassical

yikes

0celo7

out of 40

arctic tern

@JessyCat kernel of what thing?

Axoren

They posted a link above. Scary notation.

Jessy Cat

Remember that problem we were talking about earlier, @arctictern?

arctic tern

2:54 AM

yes. kernel of what?

Jessy Cat

I don't understand why the kernel of $f: x \mapsto \frac{x}{a} + \langle 1 \rangle$ is $\langle a \rangle$

Semiclassical

@YOUSEFY actually, wouldn't (d) just be: if it takes $a$ changes to turn $x$ into $y$, and $b$ to turn $y$ into $z$, then I can turn $x$ into $z$ with $a+b$ changes

Jessy Cat

wait a sec. I wasn't finished typing.

YOUSEFY

@Axoren You're definitely right, but just in our context we want to explain what is it. Moreover, in the book they use small letter of d, that is: d(x,y)

Jessy Cat

Okay, now I am.

Axoren

2:55 AM

@YOUSEFY $d$ is the notation used for most dissimilarities, distances metrics, etc.

Semiclassical

there may be a shorter sequence of changes, but all you need is the upper bound

Axoren

It's a nice general one which doesn't tend to overlap with other meanings

Jessy Cat

Because isn't the identity element in $\mathbb{Q}/\langle 1 \rangle $ 0?

arctic tern

@JessyCat $x/a+\langle 1\rangle=\langle 1\rangle\iff x/a\in\langle 1\rangle\iff x\in\langle a\rangle$

the identity element is $0+\langle 1\rangle$, or in other words $\langle 1\rangle$

Jessy Cat

Oooh!

It's great how I don't see these things, isn't it?

arctic tern

2:58 AM

heh

Jessy Cat

Seriously though, I feel like such a loser.

arctic tern

everything's relative. you're in a chatroom full of geniuses. but learning abstract algebras puts you above 99%+ of other people.

Axoren

This place is literally where all the smart people come to be smart at each other.

Jessy Cat

I'm actually doing better than everybody else in my class, too.

Axoren

So how are you a loser, then?

Jessy Cat

3:00 AM

@Axoren I've noticed that ;)

Because @Axoren when I ask things here sometimes I feel like I should already know what I'm asking about.

0celo7

@Semiclassical one of the "smart" grad students asked the prof after the exam

"I thought you said it was easy"

and the prof answered "oh I thought it was"

Semiclassical

lol

Jessy Cat

They always do.

Our prof is pretty awful, though. He's extremely incoherent. The only thing that saves things is that he types up a set of lecture notes for each week and gives them to us.

YOUSEFY

@Semiclassical you mean a: x ---> y and b: y ---> z, then (a+b): x,y--->y,z

0celo7

no, this prof is very good

very clear

but that exam, oh my god

Axoren

3:02 AM

@JessyCat Generally, you should be given the tools to understand everything that builds up from your foundations. If you lack some knowledge, there's a few things that could have happened: (1) you weren't given the tools, (2) you didn't use the tools, or (3) the tools you had need to be used in a different way than you've used them before.

A lot of things fall into the third case, which is where a lot of people aren't prepared even when they have every opportunity.

Semiclassical

hmm. that's a bit of a weird way to write it---is a the number of substitutions, or the act of making those substitutions

All I think one needs, though, is that you can get to z by first going through y

Jessy Cat

@Axoren we also don't use some of the more technical terminology that the experts are inclined to use. Like for example, I have no idea what "killing" an element is, because I've never heard anything in any of the classes I've taken referred to that way.

Semiclassical

and that can be done in d(x,y)+d(y,z) steps

so whatever d(x,z) actually is, it can't be bigger than that

Axoren

@JessyCat I don't think that's standard terminology but then again, it's not my field.

Jessy Cat

@arctictern thanks again! :)

arctic tern

3:05 AM

mmhmm

Jessy Cat

@Axoren it's not mine either. I'm taking it as a foundations course.

Axoren

Killing an element could just be the figurative killing of it.

Semiclassical

I think I've seen the word 'annihilates' used in a similar way

Jessy Cat

Math is so violent.

LOL

Semiclassical

e.g. that differentiating a constant function annihilates it

arctic tern

3:05 AM

> When I was at the University of Oklahoma in the early '80s, we were all required to write a brief description of our research for the (rather conservative, this being Oklahoma) Board of Regents of the University. An colleague in algebra, perhaps hoping for more state support, wrote that he was studying "annihilating radical left ideals."

Axoren

If you kill an element on the imaginary axis, is it a crime?

2

Jessy Cat

In the early 80s, I'm sure that he got it.

@Axoren not if there isn't a body.

0celo7

QFT in curved spacetime

Killing fields

annihilation operators

black holes that destroy everything

curvature blowups

smashing wave packets together

Axoren

Radiation coming out of black holes. In my day, we used to call that a fart.

4

Semiclassical

scattering theory oh me oh my

arctic tern

3:07 AM

blowing up eight points on a plane

0celo7

lol

Jessy Cat

@arctictern I didn't realize you were old enough to have been at university in the 80s.

Axoren

Now, we call it Hawking Radiation.

arctic tern

@JessyCat I was born in 91. That's a quote from the MO thread mathematical urban legends.

Axoren

Oh damn, you're my age, @arctictern

I thought you were older.

Jessy Cat

3:08 AM

Oh damn, you're all young'uns.

0celo7

91? I thought you were 12

Jessy Cat

No, that's Balarka who's 12

Axoren

If you flip it 180 degrees, he was born this year.

91 $\to$ 16

Jessy Cat

I was actually born in the early 80s

Semiclassical

i'm an 87 birthday myself

Jessy Cat

3:10 AM

That's not early, LOL

i didn't say it was?

I know ;P

anyways

I'm on the starred list for a fart joke.

Oh jeez.

Jessy Cat

@Axoren congratulations. This is the place where smart people come to be smart to each other...and to tell fart jokes.

Semiclassical

3:11 AM

@YOUSEFY Did that help?

0celo7

> As a young postdoc, Misha was giving a talk at a prestigious US university about his new diagrammatic formula for a certain finite type invariant, which had 158 terms. A famous (but unnamed) mathematician was sitting, sleeping, in the front row. "Oh dear, he doesn't like my talk," thought Misha.
But then, just as Misha's talk was coming to a close, the famous professor wakes with a start. Like a man possessed, the famous professor leaps up out of his chair, and cries, "By golly! That looks exactly like the Grothendieck-Riemann-Roch Theorem!!!"

lol

YOUSEFY

3:27 AM

@Semiclassical I'm trying to use your technique. I'm working on it right now

Semiclassical

okay.

0celo7

I hear a similar story a few years ago, where a student proved exciting theorems about holomorphic functions with compact support. — Orbicular May 23 '11 at 22:12

haha

Adeek

3:45 AM

hey @0celo7

I would like to discuss something super trivial

I think

so I don't understand 2

does 2 says that for each $\epsilon_k$ we have that there exists an increasing sequence such that the distance between the subsequences are all closer than $\epsilon_k$ for all k !! ?

that doesn't seem believable

arctic tern

@Adeek it's saying the convergence of any sequence can be sped up to an arbitrarily high degree by picking subsequences

Adeek

we are given it is cauchy not convergent no ?

0celo7

what is $p$?

Adeek

metric

0celo7

why no love for $d$?

what is the $\nearrow$ doing?

Adeek

3:51 AM

increasing

haha you cracked me up @0celo7

0celo7

it's believable.

Adeek

but it is cauchy not convergent !!?

maybe he forgot to say convergent ?

0celo7

what is the word in front of sequence in the second line

looks like Luud.

Adeek

fund

fundmental sequence

0celo7

what is that?

Adeek

3:54 AM

cauchy sequence

0celo7

why not say that then

Adeek

he likes saying fundmental

0celo7

him and no one else, but ok

so we have a sequence $(\epsilon_k)$ of reals

Adeek

I don't think this is true for cauchy sequence actually

0celo7

take it to be decreasing, otherwise this is trivial

so we have some $\epsilon_1$

Since the sequence is Cauchy we can find $N_1,N_2$ such that $d(x_{N_1},x_{N_2})<\epsilon_1$, right?

Adeek

3:57 AM

yes but we need to satisfy for all $\epsilon_k$ no ?

0celo7

no, that's certainly false

not even convergent will save you then

Adeek

I think I am not parsing the statement correctly ?

usukidoll

Is that real analysis?

Adeek

maybe it is exactly

what your saying that is

0celo7

you need $d(x_{n_k},x_{n_{k+1}})<\epsilon_k$ to hold $\forall k$.

that means

$d(x_{n_1},x_{n_{2}})<\epsilon_1$

$d(x_{n_2},x_{n_{3}})<\epsilon_2$

etc.

Adeek

3:58 AM

yeah yeah I see

I thought this would work for all k.

yeah that is trivial.

You can construct them by cauchyness at each step

0celo7

sure

Adeek

yeah ok.

Ok I see it is not that bad

He used it to prove that X is banach iff every absolutely convergent sequence is convergent.

@usukidoll functional analysis I guess

0celo7

what does absolutely convergent mean again?

Mussulini

the sum of the absolute values converges

Adeek

yeah

Balarka Sen

4:01 AM

@Semiclassical Hi.

0celo7

@Mussulini that's a series

he said sequence.

the sum of the norms you mean?

usukidoll

I did Cauchy Sequences last semester

Adeek

convergent series I meant to say

@0celo7

0celo7

hm, ok

usukidoll

4:17 AM

back

YOUSEFY

@Semiclassical I think I proved it. Check it. We want to show that d(x,z) steps is larger than d(x,y)+d(y,z). Now, Case d(x,z) = 0 iff x = z: then in this case no matter whatever value is assigned to d(x,y) or d(y,z), then they will always are bigger than or equal to d(x,z). Case $d(x,z) > 0$ iff $x \inq z$: Suppose that d(x,y) + d(y,z) = r. then in this case at least r = 1 or at most r= 2 while d(x,z) at most equal to 1. Thus, d(x,y) + d(y,z) is always greater than or equal d(x,z).

dm__

hi, anyone here?

usukidoll

I am

dm__

interested in a quick physics dilemma? I am confused.

usukidoll

I am a horrible science person

I am so sorry T_T

Semiclassical

4:24 AM

just ask, don't ask to ask

dm__

oil sits on top of water. a block is immersed, floating in the boundary between the two. now, how in the world does the oil, which is on top, supply an UPWARD buoyant force? the sides and top of the block contact oil; side pressures cancel, and wouldn't the top pressure be down?

resolved in physics chat. oil pressure is transmitted through the water to the bottom of the block.

Balarka Sen

4:41 AM

Maybe I can finally do some math today

Jessy Cat

Whodat?

Who just pinged me?

Weirdest thing - I was just closing some tabs in my browser, and I heard the ping noise

YOUSEFY

5:33 AM

What is different between showing a function or variable to be a "metric distance" and "metric space"?

They have the same definition: Positivity, Symmetric, and Triangle Inequality