Mathematics - 2017-07-17 (page 1 of 5)

Semiclassical

00:00

e.g. $\int_{\partial D} z\frac{f'(z)}{f(z)}dz=2\pi i z_0$ if $f(z)$ is an analytic function with exactly one zero $z_0$ in $D$.

(assuming I'm remembering that right, which I may not be)

On that note, I should be doing some contour integration arguments

Once I figure out which integral I should be doing :P

(I'm basically trying to invent an example of a contour integral to serve as a toy model of some stuff)

Daminark

Hmm

Semiclassical

Something of the form $g(p)=\int_C e^{\beta (pz-f(z))}\,dz$ where $\beta\gg 0$

If $pz-f(z)$ has a global max at $z^*$, then the saddle-point approximation gives $g(p)\sim e^{\beta (pz^*-f(z^*))}$

But $f^*(p)=pz^*-f(z^*)$ will be the Legendre transform of $f(z)$.

On the other hand, I've got a case where $pz-f(z)$ only has a global max when $p>0$.

blah blah blah

Akiva Weinberger

01:22

@MaryStar Looks like it

Typhon

01:52

hi guys

how is it going?

skullpatrol

02:32

They say that an n-digit number is curious if the last n digits of its square are the same as the original number. For example, 25^2 = 625 and 76^2 = 5776 (Curious numbers are also known as automorphic numbers.)

Kirill

03:17

Is there any good reason to say that the inverse of an upper triangular matrix is an upper triangular matrix, too?

Semiclassical

I think that's true, but I don't remember why.

Secret

Mar 21 '15 at 21:55, by 0xFFF1

yes. Magic, being a sufficiently advanced technology or concept too complex to understand as of yet.

The only known way to violate Clark's 3rd Law is to have some magic unreachable by any technology. Otherwise magic is isomorphic to a finite limit of technology making the term meaningless and redundant

arctic tern

@Kirill unitriangular = invertible diagonal + strictly upper triangular

Semiclassical

Ah, there we go.

arctic tern

(D+N)^-1 = (I+D^-1 N)^-1 D^-1, expand with geo series

skullpatrol

03:26

@arctictern would you like to put a drop down feed of meta posts in this room?

arctic tern

can't say I do

Semiclassical

You're thinking something similar to what's in the h Bar?

skullpatrol

yup

Semiclassical

Yeah, I find that more distracting than helpful tbh

skullpatrol

I see.

Kirill

03:29

@arctictern wow! thanks

skullpatrol

How, in those terms, would you define "black magic" @Secret?

ALannister

WOW!!

😍😍😍😍

Hey @arctictern

arctic tern

heya

ALannister

I posted a question in here earlier about showing that the ring of fractions of a sub ring of a field is embedded in the field.

0

Q: If $R$ is a subring of a field $F$, then the ring of fractions is embedded in $F$

Let $R$ be a subring of a field $F$ and $D$ a multiplicative subgroup of $R$. I need to prove that the ring of fractions $D^{-1}R$ is embedded in $F$. To do so, I began by supposing that $R$ is such a subring of $F$ and $D$ a multiplicative semigroup of $R$. Then, I noted that since the subring ...

Leaky Nun

03:46

@skullpatrol they are also truncations of the 2 non-trivial solutions to the 10-adic equation $x^2=x$.

skullpatrol

cool @LeakyNun

Lucas Henrique

04:02

Hey there guys.

I was looking at the curriculum of various universities on the mathematics major

Why don't universities teach things as number theory or euclidean geometry in-depth?

All I can see is "calculus, algebra and differential stuff"

skullpatrol

Because that's how math is taught.

AreaMan

they do teach these classes, though maybe not so much Euclidean geometry

Daminark

I mean, I think number theory often comes up

@LucasHenrique

skullpatrol

Applications and problem solving come first.

Daminark

So they might have a course on the side about elementary number theory, most of it is likely to come up in discrete math and algebra

But once you get through algebra, you have algebraic number theory

skullpatrol

04:11

^

Daminark

@skullpatrol I mean not necessarily, it kinda depends on which place you're at

Like, the math major tends to be a bit too small to offer too many pathways into the major

And there tend to be more engineering majors than math majors

skullpatrol

Yup.

Daminark

This sorta forces math departments to do the first few courses with engineering and applications as the end instead of theory

But I don't think there's a pedagogical reason why this ought be the case

AreaMan

@LucasHenrique somtimes when you study Euclidean geometry in college level math it is in comparison with hyperbolic and elliptic geometries, an it comes in as a discussion of angles in a triangle (leading to Gauss-Bonnet).... or Euclidean geometry style stuff comes up in algebraic geometry when you talk about projective geometry ... Maybe the point is that hyperbolic and projective geometry just ended up being richer than Euclidean geometry, so Euclidean geometry tends to get sidelined? Idk.

skullpatrol

We've had enough of Euclid and his elements :P

AreaMan

04:17

@LucasHenrique When you study linear algebra (correctly) you see some ideas from Euclidean geometry also (lines and planes, etc).

Daminark

notes that what AreaMan is saying is in a bit of contrast with how I did it

just a bit tho

I mean later we did do inner product spaces so yeah, Euclidean geometry was definitely floating around

AreaMan

Yeah. I don't really know what the "big ideas" are in Euclidean geometry. I guess they probably got absorbed into other branches of math.

I mean, all this stuff about axioms, and what is a point and so on...

Daminark

Yeah I'm pretty sure a lot of it is absorbed especially into stuff like calculus

Because you can imagine a lot of things analytically

AreaMan

Yeah -- Descarte made a lot of hard geometry problems easy

Oh yeah and Euclidean geometry esque ideas also come up as a motivation for Galois theory

when you talk about trisections and doubling the cube

Daminark

Oh really?

That's neat

AreaMan

04:22

well, I may be conflating what "Greek geometry" with Euclidean geometry.

Daminark

Oh I mean I guess I can sorta see stuff, like radicals or something

AreaMan

Yeah

they end up being statements about non-existence of certain field extensions, iirc

I hope I'm not totally making this up

quick google*

Daminark

I don't know any algebra yet so I'm not sure but I do think that's at least kinda right

AreaMan

I think the point is you can describe what points on the complex plane you can build by the standard compass and straight edge moves...

If you are given two points

0 and 1

cut-the-knot.org/arithmetic/rational.shtml

It's seems like its about what lengths of segments you can construct, and theorem is the set of such lengths is a field and that it is the field obtained by adjoining any square roots starting with Q. Which really kind of makes sense because you are allowed to work with a line and a circle (quadratic).

(The first idea about points in the complex plane is wrong, or anyway not what is done.)

Daminark

(removed)

skullpatrol

04:35

Better than "moved to the trash."

AreaMan

I guess it's oriented lengths though because otherwise where would negative numbers come from...

skullpatrol

^exactly

Direction as well as size.

AreaMan

Constructible number

In geometry and algebra, a real number r is constructible if and only if, given a line segment of unit length, a line segment of length |r| can be constructed with compass and straightedge in a finite number of steps. Not all real numbers are constructible and to describe those that are, algebraic techniques are usually employed. However, in order to employ those techniques, it is useful to first associate points with constructible numbers. A point in the Euclidean plane is a constructible point if it is either endpoint of the given unit segment, or the point of intersection of two lines determined...

I don't know why I don't just go to wikipedia first every time.

Daminark

@skullpatrol [DATA EXPUNGED]

skullpatrol

Yup

AreaMan

04:40

yesssss SCP foundation fond memories

Daminark

I actually haven't seen SCP yet, I just saw someone who did that at one point, told a friend who was like "SCP reference that"

AreaMan

well if you have anything to do don't start reading it

Eric Silva

hi chat

AreaMan

Hi eric :)

I am chat

skullpatrol

Hi pal

Eric Silva

04:42

Don't ever read SCP if you want to have a productive life

Daminark

I've got a few things to do so yeah, I'll hold off for now

AreaMan

yeah agree

or tvtropes

or stackexchange ?

Daminark

@AreaMan shh shh shhhhhhh

skullpatrol

-_-

AreaMan

I mean or [DATA EXPUNGED]

speaking of which, bye bye

Daminark

04:46

bai

skullpatrol

cya

Lucas Henrique

@AreaMan So... there's no focus when talking about graduation courses?

Those are very interesting topics and IMO it's sad that college math doesn't talk about them

I wonder when I'm going to learn these in depth after going to uni

Secret

05:02

@skullpatrol If you are talking about an inherently neutral magic that is used for bad purposes, then the same idea on object applies. If you are talking about intrinsically malicious magic, then depending on whether it is sentient, it is more like an animal or an individual with a personality or even supernatural entities like demons and angels. Either way, the point is that you cannot have any technology to approximate it

Daminark

@Lucas I think part of it is that Euclidean geometry is, in some respect, "done" or something

skullpatrol

I see @Secret

Lucas Henrique

@Daminark Agreed.

Daminark

Like, I think there's an algorithm to decide every statement

Lucas Henrique

Most of "open problems" in Euclidean geometry aren't even about geometry itself

Are mostly about constructions (compass and straighedge) or constructibility

skullpatrol

05:07

Ala hilberts formalization

Lucas Henrique

Now, when talking about number theory, there's no excuse

Daminark

Yeah, so for that reason I think the subject just gets distributed in various places, while mainstream stuff is more algebraic geometry, differential geometry, and the like

Well number theory is totally a thing

Elementary stuff might be subsumed by discrete math and algebra, but like, algebraic number theory

Semiclassical

Analytic number theory is basically complex analysis.

Daminark

True

Semiclassical

That points to another reason why calculus is such a thing by comparison with number theory: It's an essential component of a lot of different subjects.

skullpatrol

05:11

Not to mention a great vocabulary builder :P

Lucas Henrique

I'm a bit afraid of uni

It seems so... boring

skullpatrol

You won't have time to be bored.

3

Daminark

@Semi I mean, I don't know if that'd really explain much, like number theory seems to be one of the dominant motivators of, say, algebra, so you'd think it wouldn't be nearly as far behind as it is

Lucas Henrique

"Algebra, analysis, calculus, topology... differential stuff"

Looks like pretty much everything in the course from various, different universities

Hey there @Ted

Daminark

Hey @Ted

skullpatrol

05:16

How did you do in high school algebra? @LucasHenrique

Ted Shifrin

hi @Lucas, Demonark

I don't think my students ever complained of being ... bored.

Lucas Henrique

@skullpatrol I'm in highschool. I find it pretty easy and pretty boring

I find Brazil's curriculum on mathematics poor

skullpatrol

I see.

Daminark

Yeah skull raises a good point, algebra is a very different thing

In college it becomes groups, rings, ideals, modules, etc

Lucas Henrique

Ironically, Brazil's curriculum on chemistry is the biggest/hardest of the whole world

Daminark

05:17

Even what's called "linear algebra"

Lucas Henrique

@Daminark yup, those are cool

(even if I do not understand them xD)

I can imagine

But what about discrete mathematics?

Daminark

You'd think it's all that $y=mx + b$ stuff, but you reach college and then they start talking about vector spaces, linear transformations, minimal polynomials, all that jazz. Also topology can be neat, especially stuff like algebraic topology and the like

Lucas Henrique

All this continuity makes me uncomfortable

Probably analysis will kill that pain

Daminark

Here, discrete math is actually a compsci class

Many math majors take it since we've got this requirement to do like, 4 classes in sister departments

And many people do compsci classes since a lot of them are theory

Lucas Henrique

@LucasHenrique I tried to construct the reals once, ended up constructing the rationals

Daminark

05:21

But from what I know of it, it tends to cover some number theory, counting, graph theory, finite probability spaces, and linear algebra

Lucas Henrique

@Daminark Oh my, that's what I'm looking for

Here in Brazil, some Majors in Mathematics also have Physics. That makes me so sad

I hate Physics when it comes to rigorosity

Daminark

Do you mean that physicists are sometimes a bit sketchy with their mathematical manipulations?

Lucas Henrique

Obviously I speak as a highschooler. Superior-level Physics must be totally different but the whole cientific method is a bit suspicious

@Daminark depending on the physicist, yes

Daminark

Oh you mean the scientific method. I mean, it feels less "firm" than math in a sense, true. Though that's by necessity, if you're studying the natural world, you're bound by observational techniques, which is a hard hit

Alessandro Codenotti

Hi chat

Daminark

05:25

Though I mean, labs and whatnot aren't really my style either, I had thought I was into physics for quite some time when I realized that I was more into it for the math, and working physics didn't resonate (kek) with me all too well

Lucas Henrique

My physics teacher, former physicist, while solving mechanics problems, finds negative "vector modulus" and says that "intuitively, this happens because the force sense is opposite, so there's no problem"

Daminark

Hey @Alessandro!

-chat

Lucas Henrique

You know... dude, just don't. You're basically saing that norm in a vector space can be negative.

@AlessandroCodenotti Hey!

Daminark

Oh wait no that just nope

Semiclassical

eh. what would be fine is to say that the force components are negative.

(and in a 1D problem one tends to be sloppy about that distinction)

Lucas Henrique

05:28

@Semiclassical If considering 1D, then you'd have to make different vectors and make different transformations

This usually happens with pulley problems with indeterminate masses

Semiclassical

eh. pulley problems are basically 1D.

Lucas Henrique

like, find tension given masses

@Semiclassical the "same" force can have two different senses and equal modulus

so one of the solutions "has" negative modulus

that, according to my teacher's rigorosity

IMO, that's just no

Semiclassical

I really don't see the issue. The relevant forces are all along a 1D curve. Hence one can meaningfully talk about the components of force along that curve.

Lucas Henrique

Semiclassical

It's that which can be positive/negative.

BAYMAX

05:34

I dont understand the solution to this problem >< math.stackexchange.com/questions/2361210/…

Leaky Nun

@BAYMAX the eleven roots of $z^{11}=2$ form a regular polygon

BAYMAX

yes

Leaky Nun

and $|P|=3$

well, it would really help if you can kindly point out where you don't understand

so as not to waste my time

BAYMAX

$|p - z_{0}| = 3$
where $z_{0}$ is the center of the circle!

Leaky Nun

let $z_0$ be the origin.

Lucas Henrique

05:37

tbh, you can just change the coordinates

The Raiders of Las Vegas

How would you explain a + (-a) = 0 without negative numbers? @LucasHenrique

Leaky Nun

@TheRaidersofLasVegas what is -a?

Lucas Henrique

idk, aren't negative numbers extensions of rings?

inverse additive

Daminark

@TheRaiders the point isn't that numbers can be negative, the teacher presented things such that $\|x\| < 0$, which is kinda nonsense

Leaky Nun

@LucasHenrique negative numbers are defined quite formally

BAYMAX

05:38

why coordinates of $A_{i}$ taken as $2\zeta$?

Daminark

Perhaps had it been explained better it would've worked, but that statement on its own is just like "m8 no"

Leaky Nun

@BAYMAX because the roots of unity form a regular polygon

double that and you have a regular polygon inscribed in a circle with radius 2

Lucas Henrique

@BAYMAX Use Euler's formula to find them

Use the center of the circle as (0,0) (that is, 0 + 0i)

BAYMAX

gotcha

Lucas Henrique

and then use that every complex number can be written as $|p|\mathrm{cis}\theta$

BAYMAX

05:41

$|3u -2\zeta|^2$

The Raiders of Las Vegas

@Daminark that is nonsense :-/

BAYMAX

$= (3u - 2\zeta)(3\bar{u} - 2\bar{\zeta})$

Leaky Nun

@BAYMAX $|z|^2 = z \bar{z}$

Lucas Henrique

Every $\mathrm{A}_i$ is, in fact, the extremity of a rotation of a line by 360º/11

Leaky Nun

or, $|a+bi|^2 = a^2+b^2 = (a+bi)(a-bi)$

The Raiders of Las Vegas

05:42

@LeakyNun the opposite or additive inverse of a.

Leaky Nun

@TheRaidersofLasVegas then why would it not be 0?

BAYMAX

yes

next what upon addition of $\zeta$ and $u$?

of the last expression?.

Leaky Nun

@BAYMAX $\sum \zeta = \sum \bar\zeta = 0$

The Raiders of Las Vegas

Sorry @LeakyNun mix communications

Leaky Nun

because $\zeta$ is the roots of $z^{11} - 1 = 0$

$\sum \zeta$ corresponds to the negative of the cofficient of the $z^{10}$ (Vieta's formulas)

which is $0$

and then $\sum$ commutes with $x \mapsto \bar x$

Lucas Henrique

05:45

I didn't understand you question @TheRaidersofLasVegas

BAYMAX

nice1 leaky nun!

The Raiders of Las Vegas

Where you not disputing negative numbers as being the opposite of positives? @LucasHenrique with your physics teacher

Lucas Henrique

So, after all, it equals 13?

Leaky Nun

@LucasHenrique yes

Lucas Henrique

@TheRaidersofLasVegas define "disputing", hahaha

Leaky Nun

05:48

22 mins ago, by Lucas Henrique

My physics teacher, former physicist, while solving mechanics problems, finds negative "vector modulus" and says that "intuitively, this happens because the force sense is opposite, so there's no problem"

Don't call it vector modulus then

just call it vector

Daminark

Sniped @Leaky

Leaky Nun

interpret the force as a 1D vector

The Raiders of Las Vegas

^@LucasHenrique

Lucas Henrique

@LeakyNun exactly

@LeakyNun he won't

Leaky Nun

@LucasHenrique it's really a pointless dispute

Lucas Henrique

05:53

yes, that's why I won't even bother

Daminark

Okay @Lucas hold on one second, when you said that some math majors have physics, do you mean that some people who do math as a major choose to take physics? Or do you mean some math majors at some places require physics?

Lucas Henrique

I'm just a student; he's teaching Physics and people are understanding the concepts, if his math is a bit wrong (at least in concepts), that's his choice. He has a diploma, I don't. :p

Daminark

The former is pretty common in general, the latter is... I mean I guess it probably happens sometimes

Lucas Henrique

@Daminark There are math Majors with emphasis. You can have a Math Major with emphasis on Computational Math or emphasis in Physics

Even tho you will take Physics at your Math course

Daminark

Here we have a core physical science requirement, for which there are a lot of classes that aren't at all major classes, but the math department wants you to use the first year of chemistry or physics, and in fact finish the whole year. Why? I dunno

Lucas Henrique

05:56

Its purpose is, mainly, to apply concepts of vectorial calculus

Daminark

I mean, that's just a thing that people want to do. Though requiring specifically physics for all math majors seems iffy to me

Lucas Henrique

That's why you need to take Mechanics at the Math course. Optionally, you can get more in depth with other parts of Physics like Electrodynamics or Quantum Physics

Daminark

Meh

Lucas Henrique

But as a required discipline... for me, it's just dumb

I don't like Physics

Daminark

Yeah merp

Lucas Henrique

06:06

Oh my, I'm so anxious with uni...

Any of you guys has done a postgraduate course on mathematics?

Daminark

I've sat in on one? If that counts?

Next year I may do one outright

Kane

my lecturer said that the zero vector is not counted as a vector in the kernel of a matrix. So when vectors are in the kernel of the identity matrix?

Daminark

Wait hold on that's not true though. The kernel of a matrix always contains 0

You discount 0 if you're trying to do eigenvector stuff

Kane

wait i think im mixed up. He said eigenvector cant be zero vector

so does that mean that any vector is an eigenvector of the identity matrix?

Daminark

Well, no

Okay so wait here's the rundown

You say $\lambda$ is an eigenvalue of $A$ if $U_{\lambda} := \ker(\lambda I - A)$ is not trivial

Kane

06:13

yep

Daminark

Non-trivial meaning, it doesn't contain /just/ the 0 vector

Now, $v$ is an eigenvector of $A$ with eigenvalue $\lambda$ if $v \in U_{\lambda}\setminus\{0\}$

Kane

yep

Daminark

Does this make sense?

Kane

yeah it does

Daminark

So you have that the kernel contains 0, just that the eigenvectors are the non-zero vectors in the kernel

Okay nice

BORN TO LEARN

06:45

brothers from where i can learn the writing mathematics symbols in stack exchange

Astyx

@BORNTOLEARN Here

BORN TO LEARN

thankx buddy

brother can i do not write mathematics symbols without learning these symbols or there is any shortcut of it?

Leaky Nun

@BORNTOLEARN there's no shortcut to learning, brother

Astyx

You don't need to learn them all, just those you need right now, and the rest will naturally follow

The Raiders of Las Vegas

There are no royal roads.

Daminark

06:54

Also wow complex is dank

The Raiders of Las Vegas

dank?

Daminark

It means cool

The Raiders of Las Vegas

oh

BORN TO LEARN

07:20

haha yeah thats true .

brother shortcuts make learning easy

Balarka Sen

Hi chat

what'd i what'd i what'd i missed?

Tobias Kildetoft

@BalarkaSen You mean miss, not missed

Balarka Sen

when you readed James Joyce for longer enough, English grammer starts not mattering

Daminark

@Tobias Balarka's statements are pretty grammerful though

skullpatrol

Things that are easy to learn @BORNTOLEARN are easy to forget.

Transcript for

$Mathematics$ Mathematics