Mathematics - 2019-05-07

Ultradark

00:02

How do you translate a function diagonally?

Konformist Liberal

00:45

Hi, (possibly silly) quick question: Is Ho(Top) an abelian category?

Ultradark

01:01

I don't know^

jonem

01:34

Stupid question: Is the set of all infinitely long binary sequences compact? That is, the set containing all sequences of the form: {1,0,0,0,...}, {0,1,0,0,...}, {0,0,1,0,...}, ..., {1,1,1,1,...}, ...

Karl Kronenfeld

What's your topology?

jonem

I'm not sure...not so strong in topology. Does there exist one?

Karl Kronenfeld

Yeah. You can give an arbitrary product of spaces a topology. Give it a basis by declaring products of open subsets of the factors, all but finitely of which are the whole factors, to be open.

Then, Tychonoff's Theorem implies that products of compact spaces are compact.

This is famously equivalent to the axiom of choice.

jonem

Thanks!

Newbie

03:06

hi @TedShifrin

@TedShifrin You know what I find weird in North America. The emphasis on people's skin colour. One friend introduced me as his "white" Egyptian friend. In Egypt we have huge issues with religions and education, but we don't have this skin colour designation like north america.

I mean most Egyptian's are tan olive colour, but I didn't even notice this stuff until I came to North America.

haha

@TedShifrin I am thinking about doing videos on grad math on youtube. I will probably start working on that next week.

Secret

03:41

@Ultradark shear operator?

user398843

04:36

Hi, a quick question: is SO(2, \mathbb{C}) = SU(2)? I think it is, right? There is a question in Artin's Algebra that asks if SO(2, \mathbb{C}) is a bounded subset of \mathbb{C}^{2\times 2}

I think it is bounded, since I know there is a proof showing that $SU(2) is isomorphic to Sp(1), where Sp(1) = {(a+bi+cj+dk)| a^2 + b^2 + c^2 +d^2 = 1}

but there is an answer on Math SE that says SO(2, \mathbb{C}) is not a bounded subset of \mathbb{C}^{2\times 2}

5

A: Is ${\rm SO}(n, \mathbb{C})$ a bounded subset of $\mathbb{C}^{n\times n}$?

No, it is a non-empty affine variety in $\mathbb C^{n^2}$, and so is unbounded. Of course, one can see this explicitly in this case. For example, we can change basis so that the quadratic forms is $x_1x_2 + x_3x_4 + \cdots + x_{2m-1}x_{2m}$ (if $n = 2m$ is even) or $x_1x_2 + x_3x_4 + \cdots + x...

@loch what do you think?

Rudi_Birnbaum

04:54

@Newbie you are right. It also irritates me as a European a lot. Similar with this "race" buisness, they seem to think or to want to suggest that the tone of the skin has something to do with the biological subspecies concept...

Also the use of the word "Caucasian" in this context, because some either do not want to notice that they are just European, nor do they want to be put in one box with Spanish or Italian. Of course no such a distinction has any meaning other the xenophobia and racism...

ÍgjøgnumMeg

07:52

Morning all

Andrews

08:33

Is there a post as a big list conclude proofs of analysis/geometry/algebra theorems in probability methods?

Oh I find this

user123

09:27

hi ! does anyone familiar with tensor/symmetric/exterior products?

Andrews

09:41

@user123 so what's your problem?

user123

@Andrews im trying to prove this. but i'm not sure even why both sides has the same dimension..

Secret

11:17

[Random]

Concepts that cannot describe itself:

e.g. in Kleene's 3-value logic, the logical truth value known as null has this property:

"This sentence is null" is neither true nor false. Thus it has to be null.

It goes further, that it is unprovable that "this sentence is null"

14

Q: Can unprovability unprovable? Is there an $\omega$-fold unprovability?

I was just thinking about unprovability. I just wanted to know if it is possible to make a concrete boundary between provable problems and unprovable problems in a certain axiomatic system. We know that there is a statement that is true yet unprovable. Then is it possible that a statement is tr...

It will be cool if there is a topological net counterpart to this

some kind of constant net $\phi_c$ such that there exists $c$ such that $\phi_c$ diverges

christmas_light

11:55

Hi chat, I have some problems with the definition of reductive homogeneous space: we say that a homogeneous space $G/K$ is reductive if there exists a subspace $\mathcal{m}$ of $\mathcal{g}$ such that $\mathcal{g}=\mathcal{m}\oplus \mathcal{k}$, and $Ad(k)\mathcal{m} subset \mathcal{m}$ $\forall k \in \mathcal{m}$ (where $\mathcal{g}$ is the Lie algebra of $G$, \mathcal{k} of K)$:
I surely didn't study enough, but I can't make "$Ad(k)\mathcal{m} \subset \mathcal{m}$" make sense
Is it right to read this as "Pick the quotient algebra $\mathcal{g}/\mathcal{m}$, consider all the images of the a…

sorry, I made a lot of mistakes... now I fix

Hi chat, I have some problems with the definition of reductive homogeneous space: we say that a homogeneous space $G/K$ is reductive if there exists a subspace $\mathcal{m}$ of $\mathcal{g}$ such that $\mathcal{g}=\mathcal{m}\oplus \mathcal{k}$, and $Ad(k)\mathcal{m} \subset \mathcal{m}$ $\forall k \in K$ (where $\mathcal{m}$ is the Lie algebra of G, $\mathcal{k}$ of K).
I surely didn′t study enough, but I can′t make" "$Ad(k)\mathcal{m} \subset \mathcal{m}$" make sense
Is it right to read this as "Pick the quotient algebra $\mathcal{g}/\mathcal{m}$ , consider all the images of the adjoint m…

(obv other errors, I apologize)

Semiclassical

12:11

@Rudi_Birnbaum the really weird example for me is the notion that Jesus was white, which I mostly associate with US evangelicals

Rudi_Birnbaum

@Semiclassical lol!

user193319

If $d$ is a translation invariant metric on the vector space $X$, is it true that $d(x-y,z) = d(y-x,z)$?

Semiclassical

i mean, I get people who depict him as a certain race for the purpose of relating him to their community

Most famous example of this is probably: politico.com/blogs/media/2013/12/…

That last statement is in reference to Jesus being white, not the disclaimer in the message immediately prior(

Depicting Jesus as white, not necessarily racist. Insisting that Jesus is white...

PranshuKhandal

12:28

hey

can i ask something?

user193319

Don't ask to ask, just ask.

PranshuKhandal

2

Q: Eight points are in/on the circle of radius 1cm. Show that distance between some two points is less than 1cm.

Original problem If 8 points in a plane are chosen to lie on or inside a circle of diameter 2cm then show that the distance between some two points will be less than 1cm. My proof Let the points be $p_1,p_2,\dots,p_8$ Placing point $p_8$ at the center of the circle and placing other seven poi...

proof-verification

accepted answer

case 2

why is the person telling to exclude something?

he has excluded one point from a peripheral hexagon, and the middle hexa has at most 1 point, so how he concluded for 7 points

christmas_light

hi chat

Mathphile

are there a finite number of happy primes of the form 111111...

so far the only one's i can find are: 1111111111111111111 (19 digits) ,11111111111111111111111 (23 digits)

Note that 19 and 23 are happy primes themselves which explains why 1111111111111111111,11111111111111111111111 are also happy primes.

PranshuKhandal

12:46

also 317

(10^317-1)/9 is prime

11111111...(317)...1

317 itself is also a prime, and it has to be

the last one i know

there is no other such number b/w 23 and 317

Mathphile

but 317 is not happy prime

Happy number

A happy number is defined by the following process: Starting with any positive integer, replace the number by the sum of the squares of its digits in base-ten, and repeat the process until the number either equals 1 (where it will stay), or it loops endlessly in a cycle that does not include 1. Those numbers for which this process ends in 1 are happy numbers, while those that do not end in 1 are unhappy numbers (or sad numbers). == Origin == The origin of happy numbers is not clear. Happy numbers were brought to the attention of Reg Allenby (a British author and senior lecturer in pure ...

PranshuKhandal

i solved for it on SageMatg

*SageMath

sorry

Adam

@Mathphile howdy old chap that sounds like the type of fun that's right up my ally unfortunately im too fatigued at this point but thankyou for the interesting contribution to the room, my only contribution at this hour is pretty dark in some respects but please don't take it so, one of the straight forward conclusions of general relativity is that there exists no such thing as a pair of events that are "simultaneous" in space time from a fairly lay man appraisal of the theory, but...

we all share a common fear as human beings, of dying alone, something which is an inevitability according to the conclusions made in the former statement

Mathphile

13:05

this question stemmed from another question i asked on mse

2

Q: Circular Happy Palindromic Primes

$(1)$ A circular prime is a prime number with the property that the number generated at each intermediate step when cyclically permuting its (base 10) digits will be prime. For example, 1193 is a circular prime, since 1931, 9311 and 3119 all are also prime. $(2)$ A happy number is defined by th...

elementary-number-theory prime-numbers conjectures palindrome

Adam

oh right well ill certainly place both of these on my favorite list for when I eventually wake but on my former note, it's of the most ridiculous things about the human condition really, that the one the most beautiful things about our existence, being given moments of isolation, our own thought space, complete freedom, would be plagued by things like depression and mental illness, its the sheer illogic of such things that cheers me up to be honest

Mathphile

@Adam thanks

abenthy

13:28

Hi, I have a covering theory question: Suppose $\Gamma$ is a group acting by covering transformations on a manifold $M$ and $A \subset M$ is a submanifold such that the map $\Gamma_A \backslash A \to \Gamma \backslash M$ is an embedding, where $\Gamma_A = \{ \gamma \in \Gamma : \gamma A = A\}$. Does it follow that $\Gamma_A$ acts by covering transformations on $A$?

Newbie

13:49

@Rudi_Birnbaum I have been to Germany, but I don't know how people think do they think like North America as well or is it different ?

Rudi_Birnbaum

@Newbie There are many Germans but what concerns me and my close environment I can say its totally different. Race based on skin color is in my social environment a non subject, except for curiosity things. In general the to apply concept of race to humans in Germany is considered a major fauxpaus on the transition to a crime. Of course xenophobia and similar exists but its not focusing on anything biological rather than social and cultural.

Newbie

@Rudi_Birnbaum That sucks I think the reason people think like that is based on tribalism. They see who looks similar to them and get attached to them. This sucks because I was hoping humanity to reach at some point one government ruling the world.

Rudi_Birnbaum

usually (as you will find of course lots among neo nazis and extreme right wingers or otherwise uncultivated persons being racists and telling their steroetyps of "races")

@Newbie same here. I think xenophobia is a kind of biological human instinct, that in our world brings nothing good.

Newbie

Yeah. Human instinct itself sucks. I think it inhibits our ability to use our full capacity of our brain. Instead we it trains like lower functioning brain.

Semiclassical

good for whom, I guess

Secret

13:58

1 hour ago, by Adam

@Mathphile howdy old chap that sounds like the type of fun that's right up my ally unfortunately im too fatigued at this point but thankyou for the interesting contribution to the room, my only contribution at this hour is pretty dark in some respects but please don't take it so, one of the straight forward conclusions of general relativity is that there exists no such thing as a pair of events that are "simultaneous" in space time from a fairly lay man appraisal of the theory, but...

Semiclassical

it is an adaptive strategy, and---despite it hardly being an optimal strategy---it is unfortunately a rather consistent and effective strategy

Secret

Why does this not seemed to be a response to mathphile?

Newbie

I don't know how things are taught in North America as I educated myself while I was in Egypt, but I think one good thing in school is to teach kids to know when they have emotions that rely more on instinct and rationalize through it.

Secret

> it's of the most ridiculous things about the human condition really, that the one the most beautiful things about our existence, being given moments of isolation, our own thought space, complete freedom, would be plagued by things like depression and mental illness, its the sheer illogic of such things that cheers me up to be honest

What does this have to do with prime numbers?

Mathphile

kinda what i was wondering but okay

Secret

14:01

Some day I might turn Adam into an infinite Dedekind finite set

Joe Shmo

14:46

@TedShifrin, suppose $A: X \rightarrow X$ is an $n \times k$ (suppose $n > k$), orthogonal matrix, where $X$ is an (arbitrary) $n-$dimensional Euclidean space. Then $A^T A = I_k$. Now the (half-assed) claim is (by a very dubious source) that $A A^T = I$ where $I$ is an identity element on X, *and may not even be a matrix*. This is an interesting claim, especially the latter part, since the product of two matrices somehow gave me an identity element in an entirely brand new ring. I claim
$(AA^T)^2 = A(A^TA)A^T = A \cdot I_k \cdot A^T = AA^T$. That is, $AA^T$ is it's own right/left identity. …

N. Maneesh

15:19

Let $P$ be a $4\times 4$ matrices from $M_4(\mathbb Q)$. If $\sqrt 2+i$ be an eigenvalue of $P$.. Then
(A) $P^4=4P^2+9I$ (B)$P^4=4P^2-9I$ (C) $P^4=2P^2+9I$ (D) $P^4=2P^2-9I$

Since $\sqrt 2+i$ root, $\sqrt 2-i$ is also a root of characteristic polynomial

$(x-(\sqrt 2+i))(x-(\sqrt 2-i))$ is a factor of characteristic polynomial

$x^2-2\sqrt 2x +3$ is a factor of characteristic polynomial.

Ted Shifrin

@JoeShmo: First of all, only square matrices can be called orthogonal. eye rolls The equation says that the $k$ columns of $A$ form an orthonormal set. OK. If $n>k$, then the $n\times n$ matrix $AA^\top$ cannot possibly be the identity, because it has rank $k$. Of course it is a matrix. WTF. You don't say an element is its own "identity": the equation $B^2=B$ characterizes projections.

N. Maneesh

I can equate $x^2-2\sqrt 2x +3=0$ and removing the radical I get $x^4-2x^2+9=0$ mechanically. I got similar answer there. But what am I doing here?

Ted Shifrin

@N.Maneesh: You're saying that $-\sqrt2+i$ and $-\sqrt2-i$ are also roots.

Since you're working over $\Bbb Q$, you can conjugate the $\sqrt2$ just like you conjugate the $i$.

N. Maneesh

okay. Thank you.

Joe Shmo

@TedShifrin, i think im complaining to the department at the end of this semester. this is the most bizzare class i have ever sat in. is that gonna do any good?

Ted Shifrin

15:33

I have no idea. But you should complain regardless.

Joe Shmo

the lecturer talks to himself, giggles to himself, sort of knows the motions of how these proofs go, but as soon as you ask the most basic of questions about something bizzare that he puts on the board, he starts laughing hysterically and proceeds to avoid your question. because he knows that he doesn't know

prefixes every other statement with "this is trivial" (because it isn't trivial to him)

Ted Shifrin

He has a math Ph.D.?

Joe Shmo

that's what i want to know. who gave him a phd? i think his phd may be in computer science

and how did he become a professor at this institution

Ted Shifrin

Is he actually?

Joe Shmo

yes. and his phd is from a very fine american university. i am truly at a loss of words

Ted Shifrin

15:40

How old?

Joe Shmo

middle aged

he peddles conspiracy theories in the middle of lecture, completely out of the blue

Ted Shifrin

Hmmm, so presumably it's not senility — just someone who's gotten away with being incompetent at least in the classroom.

Joe Shmo

no i think he is demented

Ted Shifrin

Definitely should talk to the department head.

Joe Shmo

i saw GFauxPas (so he can corroborate) in the hall yesterday, and stopped by to chat after lecture, and i told him aobut this class. this (i think undergrad) overheard us, and turns out he had the same guy for lin alg last semester, and had a very similar experience.

Semiclassical

16:01

about 80 minutes away from a talk in the math department

Joe Shmo

just checked, phd in mathematics apparently. his papers seem to concentrate on applied mathematics / computational problems.

good luck, semi!

Secret

what kind of conspiracy theory does he sprew?

I am in need of sufficiently wild ideas

Joe Shmo

the ruling class set up mathematics such that its inherently abstract, and you might call modern science stupid.

Secret

lol is he a marxist?

Joe Shmo

yes

its obvious

Secret

16:08

no wonder

Joe Shmo

newton was an idiot (hinging on the fact that most of his work was actually in alchemy)

its hard to actually sift through what he is saying because he doesn't speak in complete sentences his accent is very thick, and his english is lacking.

so that's just the pebbles that i managed to recover

Secret

> the ruling class set up mathematics such that its inherently abstract

To be honest, I never heard of that kind of accusation in all the socialist and anarchist circles I am involved with

Joe Shmo

that's a gem, isn't it

Secret

though there are some folks there do have a anti modernist view, particularly the identity politics sector

Joe Shmo

well the thing is that he invokes one of those every time he can't answer a question

so his answer to most questions he doesn't know how to answer is a monologue in something or another

Semiclassical

16:12

@JoeShmo thx

Joe Shmo

littered with giggles, and comments under his breath

Secret

he should probably start some kind of mass movement that take back mathematics if he is not actually using that to excuse himself for failing to answer questions

For me, maths is not that abstract and concentrated, else we don't have nerds on the internet that can solve an age old maths problem every now and then

e.g. that crazy number theory problem which is solved by binge reading mangas back in 2017

Joe Shmo

yeah thats pretty cool..

i think there was one in number theory, and another in combinatorics

the latter is the binge reading manga you might be referring to

he was a computer scientist i think answering a question in combinatorics that remained unanswered until someone noticed/remembered that this guys' answer exists

and i think it properly showed one direction of the bidirectional implication in the problem statement

Secret

Superpermutation

In combinatorial mathematics, a superpermutation on n symbols is a string that contains each permutation of n symbols as a substring. While trivial superpermutations can simply be made up of every permutation listed together, superpermutations can also be shorter (except for the trivial case of n = 1) because overlap is allowed. For instance, in the case of n = 2, the superpermutation 1221 contains all possible permutations (12 and 21), but the shorter string 121 also contains both permutations. It has been shown that for 1 ≤ n ≤ 5, the smallest superpermutation on n symbols has length 1! + 2!...

Yeah

Joe Shmo

and then the internet went gaga over how to site this unnamed source

that was entertaining

Secret

16:17

It sometimes makes you wonder. Perhaps all fictions are real in that it can change a person so much that it can produce a solution to a real world problem

And technically, even science itself is fiction as otherwise you can bring me this object called a derivative and I can touch it

Joe Shmo

logically, you can

to me, mathematics is the construction of proper abstractions to reason about problems (in mathematics :-) )

so in particular, brute forcing through any problem isn't the kind of mathematics any of us would enjoy

Secret

The mathematics problem I usually like to study, besides the nature of infinite objects, is the symmetries and invariances that governs all problem solving methods and problems in mathematics

Joe Shmo

yes, i enjoy that too.

are you an algebraist by chance?

Secret

Nah I am really just a computational chemist PhD working on some metal containing catalyst molecules

Joe Shmo

sounds exciting

whats your background in mathematics?

Secret

16:22

I however have interests spanning through most of the human knowledge spectrum, which is why sometimes my knowledge to a given problem can be dense or nowhere dense depending on how big the filter corresponding to the problem is

I did my linear algebra 2nd year undergrad as well solving some 1st and 2nd order ODEs, all my other solid maths background came from my physics part of my undergrad double major

Anything else I basically self learnt by mingling with maths people and maths conferences and seminars

as well this chat and the internet in general

Joe Shmo

gowtcha

Secret

So linear algebra is really the only subject I have solid authority on. Anything else I just happened to have a knowledge set sufficiently dense to answer other users questions without them suspecting I am not a maths major

Joe Shmo

how much linear algebra?

up to.. what?

Secret

let's see... things like rank nullity, adjoint matrices, inner products, Jordan normal forms, generalised eigenspaces, solutions to exponential equations involving vectors is all I can remember form my brain

Currently trying to expand that to functional analysis, because infinite dimensional operators are more fun and I tend to bump into them a lot when trying to make sense of integrals

Joe Shmo

very nice

im into functional analysis lately

i think i might take it either next semester

or $next(next(next))$ semester

gotta run

cheers!

Secret

16:30

The other topic which I do read a textbook is general topology. I read munkres but I never managed to go beyond Ch.3 before chemistry PhD starts

see ya

Joe Shmo

munkres is one of my favorite books!

you would really enjoy the chapters following chapter 3

N. Maneesh

I studied from Foundations of Topology by C. W Patty, It is better than Munkres. Lots of historical notes, motivations etc are there.

0

Q: $Dim(W_{1}\cap W_{2}\cap W_{3})$, for given $9$ dimension sub spaces $W_{1},W_{2}$ and $W_{3}$ of $\mathbb{R}^{10}.$

Let $W_{1},W_{2}$ and $W_{3}$ be three distict subspaces of $\mathbb{R}^{10}$ such that each of $W_{i}$ has dimension $9.$ Let $W=W_{1}\cap W_{2}\cap W_{3}.$ Then we can conclude that $1.$ $W$ may not a subspace of $\mathbb{R}^{10}.$ $2.$ $dim(W)\leq8.$ $3.$ $dim(W)\geq7.$ $4.$ $dim(W)\leq3....

linear-algebra

If $W_1$ and $W_2$ are distinct subspaces, how $W_1+W_2=\mathbb R^{10}$?

Secret

hmm.. that book starts with metric spaces, looks like they mostly motivate their examples under metric spaces and hence does not put as strong focus on non metrisisable topologies

@N.Maneesh distinct does not mean disjoint

N. Maneesh

16:46

@Secret Yes. But How do I prove it rigorously?

Secret

Suppose they only intersect at the origin (as they must since they are subspaces), i.e. $W_1 \cap W_2 = \{0\}$. Then we have $\text{dim}(W_1 + W_2) = \text{dim} (W_1) + \text{dim}(W_2) - \text{dim} (W_1 \cap W_2) \implies \text{dim}(W_1 + W_2) = 9 +9 - 0 = 18 > \text{dim}(\Bbb{R}^{10})$ which is a contradiction since sum of subspaces can never be larger than the full vector space

N. Maneesh

Yes, similarly, I can eliminate the cases $\dim(W_1\cap W_2)\leq 7$

jacobian

if they are distinct, there's a vector v in W2 not in W1, adding v to W1 gives dimension 10

N. Maneesh

$\dim(W_1 \cap W_2)$ can not be $10$.

@jacobian yes. Thank you.

@Secret Thank you for you help also.

Secret

np

Adam

19:05

I'm sorry this isn't mathematics specifically related, but the only exception to me telling anyone of any profession that they must never rest on their laurels is the entire cast and crew, production of the series

Seinfeld. honestly how old is this show and to this day, they have me laughing so hard my sides hurt despite having the exact same episode so many times I cant even count

Ultradark

21:16

Is it ever useful to take a sum of all the solutions of a family of functions

Derso

22:27

What does it mean a sequence of surfaces $\Sigma_n$ in a compact riemannian manifold to "$\mathcal{C}^{2,\alpha}$-converge to a surface $\Sigma$ in $M$"?

Transcript for

$Mathematics$ Mathematics