Mathematics - 2018-02-01 (page 2 of 4)

William Oliver

04:00

I also can't tell if I do actually get it, but there is just a communication barrier between me and everyone who really understands it

Leaky Nun

04:30

2 hours ago, by Ted Shifrin

Tensor product is not a map.

@WilliamOliver kindly follow this

47 mins ago, by 0celo7

@WilliamOliver honestly unless you're trying to be very algebraic just think of tensors as multilinear maps

so this is wrong

William Oliver

Whats the difference between a map and an operation

Leaky Nun

the name

@WilliamOliver it makes multilinear maps into linear maps

William Oliver

Okay so, I found something, a few lines down from en.wikipedia.org/wiki/… here it says that it is an operation

A bilinear one

that takes two vector spaces to the tensor product space

"This product operation $ \otimes :V\times W\rightarrow V\otimes W$ is quickly verified to be bilinear.

Leaky Nun

1 min ago, by William Oliver

Okay so, I found something, a few lines down from https://en.wikipedia.org/wiki/Tensor_product#Tensor_product_of_vector_spaces here it says that it is an operation

no, it doesn't

ok

William Oliver

Then what does that quote mean

Leaky Nun

04:35

you're confusing the process of generating the product and the object itself that we call the tensor product

also, the quote is talking about the small tensor symbol

i.e. $v \otimes w$

William Oliver

Right

Leaky Nun

it does not say that $V \otimes W$ is an operation

William Oliver

Oh right okay

I think I am just having communication issues

I just communicate poorly

I think I fully understand now

Leaky Nun

ok

William Oliver

Although, actually, I think it is still a bilinear operation right. If we take a vector space to itself be a vector in the vector space of vector spaces with the operation $\oplus$

But that is irrelevant

0celo7

04:41

@LeakyNun ted's statement and mine are not incompatible

@WilliamOliver read Chapter 12 of Lee's smooth manifolds if you can get it

pages 305 - 307

Daminark

That's a rather short chapter

Leaky Nun

@Daminark o.O

0celo7

@Daminark that's the part he needs

Leaky Nun

>.>

Secret

[Random]

(The following message is delayed)

0celo7

04:49

@WilliamOliver and then go to 311

there's really no need for free vector spaces or the universal property

William Oliver

@0celo7 Thanks!

Clarinetist

@MichaelHardy I'm back on. Thanks for pinging me

I'm going to open up a room

@MichaelHardy Please see chat.stackexchange.com/rooms/72526/…

WDUK

05:42

Any one about ? Need some help with an integral

ManishKumar Singh

Hi, how would you prove $\int_{0}^{2\pi} u(e^{i\theta}(x+iy)) sin(\theta)d\theta$ is a polynomial if $u$ is harmonic

polynomila in indeterminates x,y

Secret

07:41

@Daminark That is what happens when I wrote some of my proofs in my linear algebra course back in the past

Secret

[Random]

Annelise t.

Hello, I have a yes or no question:) Does this set $S=\{x:x_1+2x_2\le 4\}$ has extreme points?

Tobias Kildetoft

There, I made a plot of the grades given as determined by my own scores and after meeting with the external evaluator. For a large enough sample size, the grading scale was designed so that it should be normally distributed with an average of 7

It is not quite like that here imgur.com/a/bjYQd

Daminark

Huh

Secret

3

Q: Why, in terms of the structure of proofs and proof strategy, is this proof of mine said to be backwards in logic?

Last year when I was doing the linear algebra and proof writing course, I was often said by my friends and my professors that the logical flow of my proofs are weird or even backwards. Recently, I accidentally stumbled upon this site. After analysing it and comparing with the standard approaches...

Daminark: for example here

I unawarely (mis)proved the converse instead of the intended

Vrouvrou

07:55

hello

`someone know this theorem

Let $u\in L^1_{loc}(\Omega)$ such that $$\int_{\Omega} u(x)\xi(x) dx=0,~\xi\in C_0(\Omega).$$ Then, $$u\equiv0~\text{a.e. in}~ \Omega.$$

Tobias Kildetoft

Just got a Google Scholar update, because apparently the place I uploaded my lecture notes for representation theory is indexed there.

Vrouvrou

?

what i must surch ?

Tobias Kildetoft

@Vrouvrou depends on what you want to find

Vrouvrou

this theorem:

Let $u\in L^1_{loc}(\Omega)$ such that $$\int_{\Omega} u(x)\xi(x) dx=0,~\xi\in C_0(\Omega).$$ Then, $$u\equiv0~\text{a.e. in}~ \Omega.$$

i founded in a thesis as the " Du Bois Reymond"

Secret

Tobias, vrou. It seems to be a L^1 special case of this:

11

Q: Integral vanishes on all intervals implies the function is a.e. zero

I am having trouble with the following problem: $f:\mathbb{R}\to \mathbb{R}$ is a measurable function such that for all $a$: $$\int_{[0,a]}f\,dm=0.$$ Prove that $f=0$ for $m$ almost every $x$ (here $m$ is the Lebesgue measure). I have no problem proving this for $f$ non-negative, or u...

real-analysis measure-theory lebesgue-integral

Vrouvrou

08:10

@Secret but in the version i have $\xi$ under the integral

Secret

$\xi$ has something to do with the integration region, I suspect it might be a measure or something

I recall seeing something similar in my electrodynamics 3rd year undergrad crouse, but they have gave no name

Vrouvrou

$\xi\in C_0$

Secret

08:27

@Anneliset. I think no as every point in $S$ can be covered by open line segments that are parallel to x+2y=4

Secret

physicsforums.com/threads/…

vrou: all similar looking theorems don't seemed to have a name

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[Random]

Vrouvrou

there is no book where i can found it ?

Secret

I have no idea, but that results (and its relatives) are often tied to measure zero functions, so I guess some books on lebesgue integration and measure theory might have it, though I strongly doubt it even has a name

Vrouvrou

ok thank you

Secret

[Random]

①
\begin{align}
\textbf{x}+\textbf{0} & = \textbf{x}\\
\textbf{x}1 & = \textbf{x}\\
a(b\textbf{x}) & = (ab)\textbf{x}\\
(\textbf{x}+ \textbf{y}) + \textbf{z} & = \textbf{x} + (\textbf{y} + \textbf{z})\\
\textbf{x}+ \textbf{y} & = \textbf{y} + \textbf{x}\\
\textbf{x}+ (-\textbf{x}) & = \textbf{0}\\
a(\textbf{x}+ \textbf{y}) & = a\textbf{x}+ a\textbf{y}\\
(a + b)\textbf{x} & = a\textbf{x}+ b\textbf{x}\\
\end{align}

②
\begin{align}
\textbf{x} : V \mapsto V^*\\
\tilde{w} : V^* \mapsto V\\
\end{align}

③
\begin{align}
\text{f}_i \in\mathcal{B}\\
\text{v} & = \sum_{i \in I} a_i\textbf{f}^i\\
\text{g}_i \in\mathcal{B}\\
\tilde{w} & = \sum_{i \in J} b^i\textbf{g}_i
\end{align}

typo

②
\begin{align}
\textbf{x} : V \mapsto \Bbb{F}\\
\tilde{w} : V^* \mapsto \Bbb{F}\\
\end{align}

o great ths is all wrong

going to do this in one go to prevent flooding

Mithlesh Upadhyay

10:48

1

Q: $G’$ be the graph constructed by squaring the weights of edges in $G$

Let $G$ be a weighted graph with edge weights greater than one and $G’$ be the graph constructed by squaring the weights of edges in $G$. Let $T$ and $T’$ be the minimum spanning trees of $G$ and $G’$, respectively, with total weights $t$ and $t’$. Which of the following statements is TRUE? $T’...

discrete-mathematics graph-theory trees

Tuki

11:01

hello

Shobhit

hi

Secret

[Random cont.]

Example of utterly wrong and totally incomprehensible stuff because I have no idea what I am doing and suffer from the Dunning–Kruger effect:

①
\begin{align}
V & := \{ \textbf{x,y,z},a,b \in \Bbb{F} :\\
\textbf{x}+\textbf{0} & = \textbf{x}\\
\textbf{x}1 & = \textbf{x}\\
a(b\textbf{x}) & = (ab)\textbf{x}\\
(\textbf{x}+ \textbf{y}) + \textbf{z} & = \textbf{x} + (\textbf{y} + \textbf{z})\\
\textbf{x}+ \textbf{y} & = \textbf{y} + \textbf{x}\\
\textbf{x}+ (-\textbf{x}) & = \textbf{0}\\
a(\textbf{x}+ \textbf{y}) & = a\textbf{x}+ a\textbf{y}\\
(a + b)\textbf{x} & = a\textbf{x}+ b\textbf{x}\}\\
\end{align}
②
\begin{align}
\textbf{x} : V \mapsto \Bbb{F}\\

Tuki

if i have $f(x,y)=x+y+4\sin(x)\sin(y)$ what can i say when $f(x,y)\ge 0$ ??

Secret

Tuki: I don't see what can be said about the function if it is nonegative, bumps?

Tuki

how large is your image ?

plot from both $x,y$ in $]-2,2[$

Secret

11:16

ok, so mathpix does not even align its graph at the origin...

Tuki

The exact question is: What can be said from following statements when dot (x,y) is close enough to origin ?

then the exact statement is $$ x+y+4\sin(x)\sin(y)\ge 0 $$

something like matlab is great. I can have very specific plots

Secret

(0,0) seemed to be a saddle point, thus your f can be +ve or -ve depending on how you approach it

Tuki

yes

however if you compute gradient at point $(0,0)$ you get vector $$\begin{bmatrix} 1 \\ 1 \end{bmatrix}$$

but shouldn't gradient be zero in saddle point ?

now this is confusing

Tuki

11:37

@Secret what you mean with ve ?

vector ?

Secret

negative and positive

Tuki

vector right ?

or gradient ?

so direction $\begin{bmatrix} 1 \\ 1\end{bmatrix} $ is increasing

$\begin{bmatrix} -1 \\ -1 \end{bmatrix}$ is also increasing ?

$\begin{bmatrix} 1 \\ 0 \end{bmatrix}$ and $\begin{bmatrix} 0 \\ 1 \end{bmatrix}$ are decreasing directions ?

if we look from $(0,0)$ ?

now this does makes sense but how to prove this ?

Shobhit

hi

Can someone help me prove $E = F - int(F)$ is a nowhere dense closed set if $F$ is a closed set

closed part is easy

i am not able to show N.D

I know two definitions of N.D set

$Int(Cl(A)) = \phi$

and, for any open set $U$ there exists an open proper subset $V$ of $U$ such that $A \cap V = \phi$

Joás Fellipe

11:57

I hope one day to be smart enough to make difference

Shobhit

me too

Unknown123

12:11

3 days unanswered, please help guys math.stackexchange.com/questions/2625222/…

Shobhit

I can start a bounty on your question to attract attention @Unknown123

Alessandro Codenotti

You already know it's closed, so show it has empty interior

Shobhit

yes, i have tried to no avail. I am using the first definition, but cant show it.

Alessandro Codenotti

Suppose $F-\text{int}(F)$ has an interior point for contradiction

What does that mean?

Leaky Nun

@AlessandroCodenotti contradiction :'(

Shobhit

12:22

then there exists a ball contained in that set around that point

@AlessandroCodenotti

Alessandro Codenotti

Right, but if that ball is in $F-\text{int}(F)$ then it's also in $F$

Shobhit

yes

Alessandro Codenotti

So that point is an interior point of $F$

Shobhit

yes

Alessandro Codenotti

So you have an interior point of $F$ in $F-\text{int}(F)$

Shobhit

12:28

but B(x,r) is not in Int(F), so x is not in interior of F, contradiction?

noooo

sorry

so $Int(F -Int(F))$ is a subset of $Int(F)$

@AlessandroCodenotti

Alessandro Codenotti

Yes, but that's not a problem, the contradiction is the one in my last message

Shobhit

oh, got it

an interior point of $F$ cannot be contained in $F - int(F)$

correct?

@AlessandroCodenotti

Alessandro Codenotti

Yes, you just removed them

Shobhit

yeah, thank you :)

nathdwek

12:47

Hello stupid question

what is the covariance matrix of (x, y) where x and y are the coordinates of a point randomly chosen from a circle with uniform angular probability?

Tuki

so does anyone know about this problem ?

How do you prove a point is saddle point ?

nathdwek

ok I am dumb

@Tuki Hessian matrix

Tuki

even though the point isn't considered as critical point since $\nabla f(0,0)\neq 0$ does hessian matrix work in this case @nathdwek ?

well i can always try

nathdwek

My understanding is that a saddle point is always a critical point

Tuki

i agree but isn't the definition that critical point exist when $\nabla f(x,y)=0$, when $\nabla f(x,y)$ is not defined (example: division with 0), when the point belongs to border of the domain ?

domain in this case would be $(x,y)\in \mathbb{R}$ ?

this wouldn't fall into any of these categories

Shobhit

13:02

I need to prove that every complete metric space is a baire space.

i am thinking of showing that if $A_{n}$ is a sequence of open dense sets then $A = \cap_{n \in N} A_{n}$ is dense in $X$.

Can someone help me in how to start this

I am thinking of starting with contradiction that there exists an open set $O$ in $X$ s.t $A \cap O = \phi$

but how can i use here that X is complete?

where to use the complete property?

Tuki

what is baire space @Shobhit ?

Shobhit

have you studied topology? @Tuki

Tuki

nope

Shobhit

then it will be hard to understand now, try googling it if you wish to know

Tuki

I am familiar with what topology means in terms of 3d-modeling but this doesn't probably translate to mathematics

PVAL-inactive

13:10

@tuki It probably does, but it might not translate to the undergraduate curriculum.

Tuki

I agree

Shobhit

yes, your 3D model should be like you can stretch but cannot break, along those ines

*lines

@PVAL-inactive any ideas? how can i use complete property here?

PVAL-inactive

You are going to have to use the metric in some way.

Shobhit

the metric is not given

PVAL-inactive

I'm unsure how to give a hint without giving it away.

Shobhit

13:16

:(

ok, i'll continue thinking

user777

13:27

Hi I have this simple question in graph theory: math.stackexchange.com/questions/2630532/…

I want to understand this statement:
Observe that a graph is a cluster graph if and only if it does not contain a path on three vertices (P3) as an induced subgraph

suppose I have several connected component, each are complete graph. Then, by fact above, it is not cluster graph; since it does have path with 3 vertices.
cluster graph is the collection of disjoint cliques.

Perturbative

Hey everyone

Yo @Daminark

Alessandro Codenotti

@Sobhit to show that the intersection is dense you want to show that it hits every open set, in particular every open ball

So call $A=\bigcap A_i$ where $\{A_i\}$ is your countable family of dense open sets and let $U$ be an open ball, you want to show $U\cap A\neq\varnothing$ knowing $U\cap A_i\neq\varnothing$ for all $i$

But you actually know more about $U\cap A_i$ than just the fact that the intersection is nonempty

And I've given too many hints already :P

Trey

14:10

is there any trig identity which can be used for $\int tan^5 x sec^3 xdx$?

Xander Henderson

@Trey The typical approach for those kinds of problems is to use power reduction formulae

and to know the derivatives of $\tan$ and $\sec$

Perturbative

Why is $\phi[M]$ an open submanifold of $\widehat{M}$?

$\phi[M]$ need not necessarily be an open subset of $\widehat{M}$

Semiclassical

14:38

@Trey You might want to check via WolframAlpha whether that has a nice antiderivative in the first place

If it was $\sec^2 x$ I'd be sure that it does. But with $\sec^3 x$ I'm not nearly as confident

Trey

@Semiclassical Indeed it's not that pretty, I managed to solve it by rewriting it as $\int(-1 sec^2(x))^2 tan(x) sec^3 x$$ and setting $u = sec(x)$

Semiclassical

hmm

0celo7

@Perturbative inverse function theorem

Semiclassical

it might be easier with the substitution $v=\cos x$

But that's just a variation on the same theme

0celo7

@Perturbative an embedding of equidimensional manifolds is an open map

PVAL-inactive

14:43

sec^3 has an antiderivative

its actually a problem in spivak

Semiclassical

sure. it's just not as obvious to me what it is, whereas I've got $\frac{d}{dx}\tan x=\sec^2 x$ in my memory banks

PVAL-inactive

He calls it "very important"

it reduces to sec

Semiclassical

hmm

PVAL-inactive

so you need to know how to integrate that

Semiclassical

yeah, $\int \sec x\,dx$ isn't very obvious iirc

not absurdly hard, by any means

but not obvious

0celo7

14:45

@PVAL-inactive is he being sarcastic?

PVAL-inactive

I don't think so.

0celo7

why would the antiderivative of secant be important?

Balarka Sen

iirc its log|sec x + tan x|

PVAL-inactive

Lots of trignometric functions show up naturally in physics, geometry, pdes and odes.

I don't know where sec^3 shows up only that Spivak seemed to be hinting at something important.

Semiclassical

$$\frac{d}{dx}\sec x=\frac{\sin x}{(\cos x)^2}=\tan x\sec x$$

that's cute, I guess

Balarka Sen

14:48

what? no cube

Semiclassical

durrrrrr brain

Balarka Sen

its sec(x) tan(x)

philmcole

Hey guys

Semiclassical

yeah, not sure wtf I was thinking

Balarka Sen

thats why the thing i said works; log(sec(x) + tan(x)) differentiates to (sec(x)tan(x)
+ sec^2(x))/(sec(x) + tan(x))

which is sec(x), conveniently

Semiclassical

14:50

right

i'm trying to think how $\int \sec^3 x\,dx$ would relate to $\int \sec x\,dx$ in terms of integration by parts

Balarka Sen

well it's sec(x) d(tan(x))

Semiclassical

actually, though, I'm probably making things too complicated

Balarka Sen

So that is convenient

Semiclassical

oh, nice

yeah

philmcole

I'm trying to find the value of the series $\sum _{n=1}^\infty \frac {1}{(2n+1)^{2}}$ by using Fourier. I have this equation and need to find a suitable $x$ to get a value for the above series

$$x(1-x) = \frac{1}{6} - \sum_{n=1}^\infty \frac{\cos(2\pi n x)}{\pi^2 n^2}$$

Does anybody have an idea which $x$ to choose? (already tried $0,\frac{1}{2}$, but those yield different series...)

Balarka Sen

14:51

Hm, I have to integrate tan(x) d(sec(x)) later on

Which is sec(x) tan^2(x) dx

Semiclassical

@philmcole well, my first instinct would be to choose $x$ so that $\cos(2\pi n x)=0$ when $n$ is even

So in particular $\cos(4\pi x)=0$

What values of $x\in [0,1]$ will work for that?

philmcole

This could work!

$[0,1)$

Semiclassical

right

same difference since the series is 1-periodic, though

Xander Henderson

Yeah, but $1=0$

because you are on the torus!

philmcole

Hmm I need to choose an $x$ so that the inside is something like $\frac{\pi}{2},\frac{3\pi}{2},\frac{5\pi}{2}$... so that the cosine is zero

Semiclassical

14:58

Right. So you want $4\pi x = \pi (m+1/2)$ for integer $m$

though writing it like that a bit overkill here.

So what $x$ is that?

philmcole

$x = \frac{m + 1/2}{4}$

Semiclassical

yeah. so 1/8,3/8,5/8,7/8

I'm a little unsure this is the right route, though

philmcole

So $\frac{2m+1}{8}$

for $m$ an integer

let's choose $m=1$ then

then this is in the interval

or $m=0$ should work too I guess?

Semiclassical

the problem I see is that, while cos will vanish on the even n, the odd terms won't all have the same sign

and you definitely don't want that

I think I see the 'right' approach, though

philmcole

ok?

Semiclassical

15:04

You noted that the series for x=0 and x=1/2 don't give you what you want

just for the sake of argument, can you write out those two cases here?

I'm fine with you just writing down the first few terms for each of them

philmcole

Like this?

Case $x=0$:

$$0 = x(1-x) = \frac{1}{6} - \sum_{n=1}^\infty \frac{\cos(2\pi n x)}{\pi^2 n^2} = \frac{1}{6} - \frac{1}{\pi^2}\sum_{n=1}^\infty \frac{1}{n^2}$$

therefore $\sum_{n=1}^\infty \frac{1}{n^2} = \frac{\pi^2}{6}$.

Case $x=\frac{1}{2}$

$$\frac{1}{4} = x(1-x) = \frac{1}{6} - \sum_{n=1}^\infty \frac{\cos(2\pi n x)}{\pi^2 n^2} = \frac{1}{6} - \frac{1}{\pi^2}\sum_{n=1}^\infty \frac{\cos(\pi n)}{n^2} = \frac{1}{6} - \frac{1}{\pi^2}\sum_{n=1}^\infty \frac{(-1)^n}{n^2}$$

therefore $\sum_{n=1}^\infty \frac{(-1)^n}{n^2} = -\frac{\pi^2}{12}$.

Semiclassical

Right. Or, to put it more concretely:

\begin{align}
1+\frac{1}{4}+\frac{1}{9}+\frac{1}{16}+\cdots &= \frac{\pi^2}{6},\\
-1+\frac{1}{4}-\frac{1}{9}+\frac{1}{16}-\cdots &= -\frac{\pi^2}{12},\\
\end{align}

Agreed?

philmcole

yeah

Semiclassical

What happens if you take the difference of those two series?

philmcole

Ahhhh

Semiclassical

15:08

yuuup

philmcole

haha I thought way to complicated

Semiclassical

Equivalently, you can look at this as $f(1/2)-f(0)$

philmcole

yeah

Semiclassical

That's probably the shortest way to write this out, since it makes it unnecessary to verify those two series separately

philmcole

That's the easier solution of the three indeed

Thanks man

Semiclassical

15:11

np

I should note that you see variations on this trick a lot

For example, suppose $F(z)=\sum_{n=0}^\infty a_n z^n$

philmcole

Yes

Semiclassical

Then $F(z)+F(-z)=\sum_{n=0}^\infty a_n [z^n+(-z)^n]=\sum_{n=0}^\infty a_n z^n(1+(-1)^n)=\sum_{k=0}^\infty 2a_{2k}z^{2k}$

Which is really not much more than saying $\frac{1}{2}(F(z)+F(-z))$ is the even part of $F(z)$

philmcole

Got it

Semiclassical

But it means that, if we know how to compute $F(z)$, then we can not only get $F(1)=\sum_{n=0}^\infty a_n$ but we can actually get the sum of just the even/odd terms

philmcole

I get the feeling to be good at math you need to have really a lot of those tricks up your sleeve all the time

Semiclassical

15:16

depends what you do, really

I've used this trick in actual research, but it was natural in context

a more advanced version of this is to do something like $F(z)+F(\omega z)+F(\omega^2 z)$ with $\omega = e^{2\pi i/3}$

philmcole

yikes

Semiclassical

It's not bad, actually: $1+\omega^n+\omega^{2n}=0$ if $n$ isn't a multiple of 3

and if it is, it just equals 3

So that sum will 'filter out' those coefficients which aren't multiples of 3

philmcole

And with something like that you can get the sum of only the $n$-th terms?

Shobhit

@Semiclassicali have used this method many times in calculating sum of binomial coefficient

Semiclassical

^

@philmcole right

philmcole

15:19

Will get more complicated for $n \gt 3$ then I suppose

Semiclassical

it do

philmcole

nice trick

Semiclassical

yeah. the name you'll see for it is the 'root of unity filter'

philmcole

oh yeah the first result shows the binomial coefficient immediately

Semiclassical

right

Shobhit

15:24

@AlessandroCodenotti i also know that every intersection $A_{n} \cap U$, i used that and formed a ball inside one another, had to use cantor intersection theorom (advice of a friend) to conclude. thank you.

Secret

15:45

muhahahaha, functionally beaten

Daniel Cortild

16:03

Hello

Shobhit

hi

Daniel Cortild

Do you know something about probability?

Shobhit

nope

but ask away

someone might

Sha Vuklia

guys, is this true:
$$
\int_{-\infty}^\infty\delta(x-\lambda)^2dx=\delta(0)
$$

Daniel Cortild

So you throw an infinite number of coins, equal probabilty to land on each side

Sha Vuklia

16:06

Slereah

@ShaVuklia probably not since that's not a well defined quantity

Sha Vuklia

I'm trying to see why this is true

Daniel Cortild

Call the sides $1$ and $2$...

Sha Vuklia

@Slereah right, well can you explain the screen shot then to me?

Slereah

What you might be interested in is the wavefront set

Sha Vuklia

16:07

because what I suggested was my attempt at an explanation

Daniel Cortild

Find the probability that you get a sequence of $5$ "0"'s before 3 "1"'s

Slereah

Which allows you to multiply distributions if their singularities aren't in the "same place"

Sha Vuklia

@Slereah that's directed at me?

Slereah

So this quantity is well defined if $\lambda \neq \mu$

yes

Sha Vuklia

the context here is just some algebraic manipulations in an introductory course in quantum

wait, I'll give you a bit more

Slereah

16:08

If it's a QFT book then it's probably not meant to be rigorous

Sha Vuklia

Slereah

The rigorous version is gonna be using the wavefront set, yes

Sha Vuklia

quantum mechanics

I just want to have some intuition on why it's true

we are basically showing that the position eigenfunctions satisfy some sort of orthogonality condition

Slereah

well if $\lambda \neq \mu$, the product of those two is gonna be $0$ outside of $\lambda$ and $\mu$

Sha Vuklia

I am interested in the case $\lambda=\mu$

Slereah

16:09

Well here's the secret : there's no position eigenfunction :p

They're not part of the Hilbert space

Sha Vuklia

that's not the point griffith's is making

Slereah

the case $\lambda = \mu$ isn't defined

Sha Vuklia

right

Slereah

if you want an intuitive version of this, try defining the delta function as the limit of a gaussian

that should give you the reason why this work if the singular supports aren't overlapping

Sha Vuklia

with that last sentence you mean when $\mu\neq\lambda$?

Slereah

16:14

yes

(the singular support is the domain on which the distribution can't be defined by a function, roughly)

$\text{sing supp}(\delta) = 0$

Semiclassical

I'll note that this is a point in standard QM treatments that @0celo7 takes issue with

I think it's the same, anyways (that there's no such thing as position eigenfunctions in Hilbert space as such)

Balarka Sen

16:57

@Daminark eyo

do you want to hear a small rant on continued fractions

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