Mathematics - 2023-02-26 (page 1 of 2)

Sine of the Time

00:04

Any suggestions for a book about ODE's? I've Arnold's book but I don't know if it's appropriate for a basic course :/

monoidaltransform

@TedShifrin When you say "take an inner tubular nbhd of $\partial{U}$ in $U$, this means to me to take an open subset $Z$ such that $\partial{U} \subseteq Z \subseteq U$. Which implies $\partial{U}\subseteq U$, but $U$ here is open.

Ted Shifrin

00:20

No, $Z\subset U\cup \partial U$. Get the picture in your mind and then make it work.

@SineoftheTime one dimension or systems, @Sine?

Sine of the Time

@TedShifrin let me check the programm of the course

monoidaltransform

Ah, so just take $Z\subseteq U\cup \partial{U}$ that misses $K$?

冥王 Hades

Is hospital/doctor anxiety a thing?

I was shaking having to see a doctor for a normal routine checkup. Not sure why

Sine of the Time

@ted -Motivation and examples. Basic definitions (ordinary differential equations, Cauchy problem). Basic general theory: local existence and uniqueness of the solution under Lipschitz hypothesis by the method of Picard's iterations.
- Gronwall's Lemma. Maximal solutions and extension of solutions. Continuous dependence on the initial datum.Maximal solutions and extension of solutions.
- Basic resolution methods for differential equations in normal form: separable variables equations, first order linear differential equation, 1-forms and differential equations. Bernoulli's equation. Differ…

冥王 Hades

Even scarier when I get asked questions about medical history

Sine of the Time

00:23

-Qualitative study of an equation: comparison theorem, monotonicity theorem and asymptote theorem.
- Existence for the Cauchy problem in the case of continuity. Ascoli-Arzelà Ascoli theorem. Peano's theorem.
- First order linear systems: exponential of a matrix and Jordan normal form. -Linear differential equations of order n. Characteristic polynomial. Euler equation. Method of variation of constants. D'Alembert method for the reduction of the order.

user 85795

00:35

Anything can become an anxiety if you think about it too much.

leslie townes

hades, totally a thing

Sine of the Time

@leslie any suggestion for a good book about ODE?

user 85795

Test anxiety has its roots in fear of failure.

leslie townes

arnold's book is pretty good. what do you want? in a lot of universities, the first course with "ODE" in the title is just a mishmash of recipes to memorize with no theory. and you actually need quite a lot of prerequisites to work out meaningful theory.

Sine of the Time

I've written the program

Ted Shifrin

00:45

@SineoftheTime This is a single-variable ODE course, not systems. Birkhoff & Rota is a classic text that's more theoretical than the typical books used in the US. These days, many US ODE courses are more centered on systems of ODEs (and the interplay with linear algebra).

leslie townes

yeah, birkhoff and rota, that's the other good one.

Sine of the Time

thank you both :)

Ted Shifrin

The end of your course is systems, but it's not the bulk of the course. For that material, there's very little better than Hirsch and Smale (and new edition with Devaney, and I don't know that version).

冥王 Hades

@leslietownes Part of it could be because I'm just not good at explaining/talking but another part of it felt like I was asked by the professor to solve a problem on the board, which I can do quite well if its mathematics but I have almost 0 medical knowledge

leslie townes

no, anxiety around medical care is a huge and normal thing. my mom was a nurse. they take 'vital signs' like pulse and blood pressure, and write down whatever those are, but the numbers they get are often discounted mentally because people are generally somewhat worked up from being in a situation where a nurse is involved.

Ted Shifrin

00:53

My BP in any doctor’s office is always too high.

leslie townes

they definitely know that. one of the weird things where a measurement is taken where everybody knows it might be off by 50%.

Ted Shifrin

And I don’t have anxiety other than about my BP.

leslie townes

in our wonderful USA you can also pile economic anxiety on top of whatever else might be going on in a medical situation.

user 85795

19 mins ago, by user 85795

Anything can become an anxiety if you think about it too much.

Isn't that how the brain works.

Astyx

I have a rank 16 integral lattice that I know is isomorphic to $E_8\oplus E_8$ (in particular it's definite), is there an efficient way to compute an explicit isomorphism? The Gram matrix is pretty sparse, but PARI/GP is struggling quite a lot

user 85795

00:59

~~think~~ obsessively worry about it, actually

冥王 Hades

@TedShifrin my doctor measured my heart rate, looked at me and laughed slightly before she told me to calm down. I haven't felt more embarrassed ever since I stepped on a banana peel, and landed face first in mud

monoidaltransform

@TedShifrin I really am stuck and confused. So, just to repeat. $f=g$ on $U-K$, where $f:W\rightarrow X$ and $g:W\rightarrow X$. Define $F$ by $f(x)$ on $W-U$, $g(x)$ on $U$. We want to show $F$ is continuous.

Where $U\subseteq closure(U)\subseteq W$.

You claim that $f=g$ on a neighborhood of $\partial{U}$ in $W$ and therefore that implies continuity of $F$.

D Left Adjoint to U

@Koro

Proof this:

-1

Q: A factorial modular equation form of the twin prime conjecture.

Conjecture. Let $f_q(X) = X(X - 2)(X - 3)\cdots (X - (q - 2)) = X \dfrac{ (X - 2)!}{(X - q + 1)!}$ be a function from $I = [p_n + 2, p_{n+1}^2 - 2]$ into $\Bbb{Z}$ as $q$ ranges over $2..p_n$. Then there exists a solution $X \in I$ such that: $f_q(X) = 0 \pmod q, \forall q \in\{2, .., p_n\}$ Th...

monoidaltransform

Sorry, not $g:W\rightarrow X$. I meant $g:U\rightarrow X$

冥王 Hades

@leslietownes when I was told I'm gonna need a laser surgery to correct my vision, I was visibly shaking and started feeling nauseous, the idea of a surgery is like a nightmare for me. The specialist actually had to give me a mini-medical lesson on how the surgery works in order to get me to calm down before I head home

monoidaltransform

01:10

I don't see (1). Why $f=g$ in a neighborhood of $\partial{U}$ in $W$ and (2). Why (1). implies continuity

of all of $F$.

Ted Shifrin

I’ve been through 3 major surgeries and am headed for a 4th. They were all superbly done. No worry.

The two functions agree on the overlap because they agree off $K$. @monoidal

leslie townes

the laser eye stuff is extremely non invasive.

冥王 Hades

@TedShifrin well, not everyone can be Ted you know. And I mean that in a good way btw

@leslietownes that's precisely what the specialist explained and got me to calm down

monoidaltransform

Yes, ofcourse. But what overlap? W-U and U are disjoint @TedShifrin

Ted Shifrin

I meant that to encourage you, Hades, not to be an ass.

monoidaltransform

01:13

Ofcourse they agree on $U-K$. But why does that imply $F$ is continuous

leslie townes

like, 30 years ago i'd think it was definitely something to be worried about. my uncle got it almost the minute after they authorized some version of it in the USA (and was fine, but, it was in a context of there not being decades of people having gotten it).

and most of the advances in it have only been to do more with less invasion.

Ted Shifrin

Reread what we did. We thickened up $W-U$.

monoidaltransform

Yes. Could you just elaborate on that please? I keep rereading it but i'm confused for some reason

I think that's where i'm confused

Ted Shifrin

We made $W-U$ into an open set going into $U$. Whence the overlap.

monoidaltransform

Okay. So we take an open set $Z$ such that $W-U \subseteq Z\subseteq U$. Right? okay that is clear to me

冥王 Hades

01:16

@leslietownes You know what I like about doctors and medical professionals in general? They can be really good at calming you down. Both then and today, when I was having trouble getting back to sanity, the doctor started asking about my interests, what I do, what I like to work on and so on.

Ted Shifrin

Not right, of course.

monoidaltransform

Not right not right. ofcourse notright

冥王 Hades

She asked about my academic path, which I informed her of, and we got into a discussion of how I tend to approach solving problems as opposed to her as a medical professional

monoidaltransform

I got my $U$ and $W$ other way around.

In my head

Ted Shifrin

Are you drawing a picture?

monoidaltransform

01:17

Yes.

Ted Shifrin

So we know $f$ and $g$ agree on the overlap. The union of the two open sets is $W$.

monoidaltransform

Just a minute. 'made $W-U$ into an open set going into $U$'. in terms for $Z$ and $W-U$ and $U$, what does that mean?

I've drawn $W$, the 'largest' open set which contains $\overline{U}$ which inside contains $U$. There is some 'space' between boundary of $\overline{U}$ and boundary of $U$.

So they're nested

$W-U$ is just everything except $U$.

Ted Shifrin

And we added to $W-U$ the collar around $\partial U$. Call that union $Z$.

The two open sets are $U$ and $Z$.

monoidaltransform

Okay. So by 'collar', you mean we thickened up the boundary of $U$ to an open set $C$. Then, put $Z=(W-U)\cup C$. Right?

So, the 'collar' intersects the complement of $U$, but also the interior

Aha. Now that's where i'm confused. Why is $Z$ open?

Ted Shifrin

01:32

Just check.

monoidaltransform

Just curious. Are there known proofs with a similar flavor to this problem?

D.C. the III

Let's talk Jordan canonical forms.............I don't understand what Thm 7.1 (b) is getting at. And the idea has been used in two proofs since that moment in the text.

leslie townes

dc3, the screenshot is missing a definition of K_lambda but as far as i can tell i'm with you, who cares

maybe it's just some boring lemma they use to establish something more important

D.C. the III

Sorry I thought it was a general notion

leslie townes

if that's the case, focus on that instead of this

D.C. the III

01:42

$K_\lambda$ are the generalized eignespaces

leslie townes

OK. at the moment, i honestly can't think of why you would use 7.1(b) in anything, except as some step in some other result.

so some author arranged their treatment this way and in another book you maybe never see 7.1(b). that is what i would take out of it

D.C. the III

let me look at the proofs where it came up

Ted Shifrin

@monoidaltransform This problem was cluttered with obscure red herrings. Techniques with thickening and taking tubular nbhds are all over differential topology, geometric topology, and differential geometry.

monoidaltransform

Yeah. I wonder why he chose to write that problem that way

D.C. the III

If you want to look at their uses @leslietownes but I'll just carry on without getting bogged down by it

Ted Shifrin

01:48

He seems to thrive on obscurity.

leslie townes

i don't know why anyone would want to set out 7.3 as something worthy of attention. 7.4 seems useful but i would regard it as a black box, who cares how you prove it

D.C. the III

Yea that's how I'm treating 7.4 as a black box of sorts

leslie townes

it looks like an author made some expository choices that leaned on this lemma or whatever being used in multiple places. which, OK. but i would not give it independent importance

Ted Shifrin

I disagree. 7.3 is great.

D.C. the III

Why wouldn't it just be a corollary to 7.4 Ted?

Ted Shifrin

01:51

7.4 is harder to parse. Did they not use 7.3 to prove 7.4?

D.C. the III

that's how I interpreted it before I had read 7.4 the first time....Yes 7.3 was used for 7.4

Thorgott

why not just talk about direct sum decompositions, easier to parse than both of these

D.C. the III

They summarize it up with the direct sum decomposition THor

Thorgott

that's nice

monoidaltransform

Okay so just to make sure I got the idea. So, We thicken up $W-U$ to $Z$ as above. For suitable collar $C$, we have $Z\cap U= C\cap U$. On $C\cap U$, $f=g$. Now $F=f$ on $W-U$ and $g$ on $U$. Define $\tilde{F}$ by $F$ on $Z$ and $g$ on $U$. Then, $\tilde{F}$ is continuous extension of $F$. So, $F$ is continuous?

it's 7:37 Am here, and I really want to get this

Ted Shifrin

02:09

Yup.

Unless I missed some typographical issue.

Somewhere in there you should make sure that $C\cap K=\emptyset$.

monoidaltransform

Ah okay. Yeah. Thanks Ted. I'll make sure I got all the details down after I rest. Thanks ocne again. So much

Ted Shifrin

G’morning!

monoidaltransform

Goodmorning to you too :)

冥王 Hades

All of my medical reports seem to be clear, or so the doctor claims. That's definitely reassuring

Ted Shifrin

Yippee!

robjohn

02:33

My post operative test was supposed to come back 0.10 or less. It came back 0.39. I talk to the doctor on Wednesday.

Not Tfue

Does anyone know how to prove this f bijective?

I really couldn't see what else information I can get from set about the input x.

Ted Shifrin

@robjohn Damn, sorry to hear that. Good thoughts.

@NotTfue Sure. If a digit isn’t $2$, what is it?

Not Tfue

@TedShifrin 1 and if it is 1 then f(x) skips the the natural number j.

I was curious about the injection using

Ted Shifrin

So, if you know $f(x)$ you know the decimal uniquely,

Not Tfue

definition

Ted Shifrin

02:44

We just argued that.

Not Tfue

@TedShifrin how? Decimal expansion is unique of course but this is what I don't get. How did you know it is unique using function.

Ted Shifrin

You know exactly what every decimal digit is if you know $f(x)$.

monoidaltransform

I can't rest. I need to figure this out. I understand the collar construction really well now. Now just the issue of continuity. Why is $F$ continuous? ($\tilde{F}$ is defined on $W$ as well, so not necessarily continuous extension).

Ted Shifrin

If $j\in f(xj$, then the $j$th digit is …. And if $j\notin f(x)$, then ….

monoidaltransform

I also understand why $(W-U)\cup C$ is open. It's because the boundary points of $U$ have neighborhoods living in the collar

Not Tfue

02:49

I will try to use pc. Can't use chat Jax.

Ted Shifrin

We’re not extending, monoidal. We’re patching.

monoidaltransform

Okay, so now I have set $Z=(W-U)\cup C$. Which is clearly open. Then, $W=Z\cup U$. So the function $F$ coincides with the function defined by $f$ on $Z$ and $g$ on $U$?

Not Tfue

If j∈f(x)
, then the jth digit is 2 And if j∉f(x), then 1.

This step is enough right?

I also made a function $g(A)=0.111...+∑_{n∈A}10^{−n}$ that extract the decimal from set produced by $f(x)$. As this step didn't look rigorous enough.

monoidaltransform

Ah right. Yes. $F$ IS the function that is defined by $f$ on $Z$ and $g$ on $U$ right @TedShifrin ?

Ted Shifrin

03:10

I believe so.

@NotTfue That’s fine, too.

Not Tfue

@TedShifrin but really how exactly does it say that it is injective though?

monoidaltransform

So if I define $\tilde{F}:W\rightarrow X$ by $\tilde{F} =f$ on $Z$ and $g$ on $U$. Then, I claim $F=\tilde{F}$: If $x\in W-U$ or $x\in U$ then clearly done. So assume $x\in C$. If $x\in C$ and $x\in U$ then we are done. If $x\in C$ and $x\notin U$ then since we are in $W$, we still are done.

And $F$ is continuous because $\tilde{F}$ is. $f$ and $g$ coincide on $U\cap Z=U\cap C$.

Is that reasoning correct @TedShifrin? or is there another way to see this?

Not Tfue

This is the reasoning I can come up with if j∈f(x), then the jth digit is 2 And if j∉f(x), then jth digit is 1. So because every decimal expansion is unique so this implies the set f(x) is also unique. Is it the correct reasoning?

Not Tfue

03:54

This is not quite satisfying.

D Left Adjoint to U

04:08

@Koro I'm onto some deep NT shet:
https://math.stackexchange.com/questions/4646639/for-all-n-geq-1-there-exists-6x-inp-n2-dots-p-n12-2-st-6x-2-q

Obliv

ever notice i j and k (unit vectors in the x,y,z) are aligned on the keyboard together

weird

D Left Adjoint to U

Yes, that is weird

abstract nonsense

orthonormality?

:D

Obliv

04:24

how does one construct a number using only symbols of first order set theory

and how is such a language of logic defined such that it isn't just shorthand for other languages

D.C. the III

I believe the idea is using sets of empty sets

Obliv

@D.C.theIII Would you say a countable set must exist within an uncountable set?

for infinite sets, we say one is countable if there is 1-1 correspondence with the natural numbers. For those that do not have this, can we not make further analysis on the "size" or countability of the sets?

D.C. the III

04:41

Not sure Obliv, that is getting above my level at this moment. I'm sure one of the veterans will be able to expand on that

Not Tfue

I guess I am not capable of doing real analysis. It is impossible for me to prove this theorem.

Real analysis require certain level of intelligence that I don't have.

Obliv

What's the distinguishing feature of a computable/uncomputable algorithm for a turing machine :\

Koro

Suppose that E is a real Banach space, U is a bijection isometry. Fix a, b in E. Define D such that D(U)=$\| U(a)+U(b)-\frac{U(a)+U(b)}2\|$. Define $s_U(e)= U(a)+U(b)-e$ for every e in E. Define U’= $U^{-1} \circ s_U \circ U$. Then it is to be shown that $D(U’)=2D(U)$.

I have shown that.

Now, from here how do I conclude that D(U)=0?

Obliv

@NotTfue whats the theorem

Koro

It can also be seen that $D(U)\le \|a-b\|/2$.

Not Tfue

04:45

Koro

@DLeftAdjointtoU

Not Tfue

Been thinking for 2 days. Can't come up with rigorous injection proof.

With this definition.

I don't know what I can do with the basic set stuff I know to build the proof.

Koro

@NotTfue hint: think about complement of f(x) in $\mathbb N$.

Obliv

I don't even know what a proof is.

Are proofs subjective?

Not Tfue

@Obliv For physics guy like us I think it doesn't matter at all lol.

Obliv

04:50

I mean i've tried to form proofs but I got frustrated and gave up

I think proof writing is something you need a mentor/teacher for

Not Tfue

Computation is what matters. I was trying out analysis because my professor said it might be used as graduate student.

Obliv

so you're self studying it?

Not Tfue

Yup. And I get really frustrated this time I just think I should give up. I wonder how math student deal with it.

Obliv

Math is pretty abstract, I'm taking only math courses this semester to catch up on requisites, I miss physics lol

Actually I take it back, I think proof writing scales with your knowledge. I'm sure you could prove very simple things, but the material you're going over might be too foreign

Not Tfue

@Koro You mean x that is in P(N) but not f(x)?

Obliv

04:59

@Koro what is this called?

Not Tfue

@Obliv I think the calculus course they(math student) take is different from ours(physics student). Also they take introduction to proof as freshman here so may be you are right.

Obliv

what is that symbol before $\mathbb{N}$ in that theorem

like that function

is this a proof of countability

Not Tfue

@Obliv power set of natural number

It's proof of cardinality.

Obliv

Have you shown such a proof before @notTfue

What's your strategy

Not Tfue

05:15

I want to prove it is bijection which means it is both subjective and invective so first I want to prove it using invective definition that I know. I know that each decimal expansion is unique but can't come up with rigorous description of what does the set say about the input x. I just know definition.

@Obliv You mean seen or shown? I have seen this kind of proof but never seen a function that maps to a set so this is the problem.

The set is the only problem.

copper.hat

dammit, what does invective mean?

Not Tfue

injective

Obliv

what does the hint even mean $f(x) = \{j \in \mathbb{N} : d_j = 2\}$

what does the colon mean again :

Not Tfue

auto-correct is making me sound too smart lol

@Obliv it is stating a condition

Obliv

maybe it's saying $f(x)$ is $0.2,0.22,0.222,...$

then you show $g(x) = \{j \in \mathbb{N} : d_j = 1\}$ also is bijective then you add f(x) and g(x) or something

oh well i gotta sleep good luck lol

Not Tfue

05:25

@Obliv the element of E is not finite

finite decimal

Koro

05:38

@NotTfue ohh. I mean think of $\mathbb N-f(x)$.

@Obliv what?

nvm, I got D(U)=0.

Not Tfue

@Koro I am not sure what this step will lead to.

Koro

This will tell you that if f(x)= f(y), then there in x and y 1 occurs in exactly the same slots after decimal.

Is uniform boundedness principal applicable in case of uncontountably many linear operators on a Banach space?

Not Tfue

@Koro I guess this is the missing step. We already know abut 2 now we need to know about 1. But does that really prove it is injective? It makes sense. I will think about it.

Koro

05:56

yeah, I think you’re overthinking this.

@Koro yes, it is.

Not Tfue

@Koro Yeah I think so. I spend my whole holiday looping around this question not giving fresh view. I am convinced it is injective now :)

Koro

08:05

why is (1,1,...) an extreme point of a closed unit ball in c (the space of all convergent sequences with sup norm.)?

Suppose that $(1,1,...)= t (x_n)+(1-t) (y_n)$ for some t in (0,1) and $(x_n), (y_n)\in c$.

Then, $tx_n+(1-t)y_n=1$ for all n, so t x+(1-t)y=1, where $x_n\to x, y_n\to y$.

nickbros123

08:27

In this answer, this person considers the range of numbers after a given index in a sequence. He argues this range is bounded so has a supermum. I agree with that, but aren't sets and sequences fundamentally different? One allows for repetition while sets should have distinct elements: math.stackexchange.com/a/2708043/1118236

Question is multiple indices can have the same value hence the many indices can act as a supremum

leslie townes

10:13

koro it makes sense to look entrywise although i'm not sure you specifically need to consider the limit. from the definition of the norm, any particular x_n and y_n will be in the closed unit ball of your field of scalars. is 1 an extreme point of that?

Koro

11:52

I understood what I missing.

:-)

leslie townes

we're having "cold" weather now. 40F, may not go beyond the mid 50s today

Xander Henderson

13:43

@leslietownes I hear that it snowed on the Hollywood sign yesterday.

And my daughter says that they got a little snow in Redlands.

user123234

0

Q: How can I show that this set is not dense in $S^1\times S^1$?

Let me define by $R_\alpha: S^1\rightarrow S^2$ to be the rotation on the circle by the angle $\alpha$. Now consider $\alpha, \beta$ and $R_\alpha, R_\beta$ and define $$M:S^1\times S^1\rightarrow S^1\times S^1;~~(x,y)\mapsto (R_\alpha x, R_\beta y)$$ Assume that $\alpha=2\beta$ I need to show t...

functional-analysis dynamical-systems rotations dense-subspaces

Can someone take a look at this question here?

Koro

14:03

suppose that H is a Hilbert space. For c in H, define H_c:= span(c). For some a,b in H, if $\|a-b\|=\inf \{\|a-x\|: x\in H_b\}=\inf\{\|x- b\|: x\in H_a\|$, then a=b.

How do I prove this?

$H_b$ is finite dimensional, hence closed. So there exists a unique element in H_b closest to a so the question makes sense.

If a lies in $H_b$, then we are done.

Aira Thunberg

Guys help here please math.stackexchange.com/questions/4646752/…

Astyx

15:05

@AiraThunberg The $m$-th term of the $n$-th line of that "triangle" is $\sum_{k=0}^n (m+k)\choose{n}{k} = 2^{n-1}(2m+n)$

Then it's just writing $m$ and $n$ in terms of the $k$-th number of the series

I messed up the latex, here is the correct version $\sum_{k=0}^n (m+k){n \choose k} = 2^{n-1}(2m+n)$

Aira Thunberg