Mathematics

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user193319

12:58 AM

Problem: Let $f$ be a real valued function defined on a measurable domain $E$. Suppose that $f$ is continuous except at a finite number of points. Is $f$ measurable? Proof: Let $D \subseteq E$ be set of all point at which $f$ is discontinuous. Since $D$ is finite, $m(D) = 0$, so $f$ is measurable on $D$. Since $f$ is continuous on $E \setminus D$, it must be measurable on $f$. Hence, $f$ is measurable on the union $(E \setminus D) \cup D =E$.

How does that sound?

2 hours later…

Indrasis Mitra

2:56 AM

I could use some guiding points on this problem

Q: Two fluids flowing perpendicular in thermal contact with a Wall [Help to mathematically model]

I will try to describe briefly how I am modelling the problem. (Please bear with the length). The governing equation describing temperature for a block at steady state is $$\nabla^2 T = 0$$ where $\nabla^2 T = \frac{\partial^2}{\partial x^2} + \frac{\partial^2}{\partial y^2} + \frac{\partial^2}{...

Skies burn

3:42 AM

In English, do we put a comma like in this, or leave it out:

"If R is a dedekind domain, it turns out that..."
"If R is a dedekind domain it turns out that..."

2 hours later…

Fargle

5:22 AM

@Skiesburn I think either would be acceptable, but I personally would do the first one.

1 hour later…

Noob

6:32 AM

@Fargle how are you

i want to ask , one thing its simple and will not take even 2 minutes

question is any suugestion over a book - on propostional logic ?

4 hours later…

Secret

10:45 AM

♦ $Mathematics$ Logic

This room is meant for discussion about logic, including found...

user193319

11:32 AM

Q: Open Sets in the Wedge Sum and a Homeomorphism

I am presently working through example 1.21 in Hatcher's book on wedge sums of topological spaces. He makes a few claims which I am having trouble verifying. First, let me set-up some notation. Let $\{X_i\}_{i \in I}$ be a collection of topological spaces. Then $\amalg_{i \in I} X_i := \cup_{i ...

algebraic-topology quotient-spaces

Haris Gusic

11:45 AM

Would someone be kind enough to verify my Cauchy-Riemann equations:
https://math.stackexchange.com/questions/3017630/cauchy-riemann-equations-for-z-xiy-and-fz-rx-yei-thetax-y
Thank you.

2 hours later…

abenthy

1:39 PM

@LeakyNun I got an interesting answer to a question about quaternion algebras on MathOverflow, but I don't quite understand it. Would you be up to taking a look at it maybe?

2 hours later…

Perturbative

3:34 PM

Sanity check $\operatorname{GL}_n(\mathbb{R})$ is an infinite group right?

Tobias Kildetoft

@Perturbative Yes, for $n\geq 1$.

Perturbative

Hmm, because I was in my algebra lecture today and my lecturer and my classmate said that $\operatorname{GL}_n(\mathbb{\mathbb{F}})$ was finite with $\mathbb{F}$ being some field, and I was confused. Maybe they were talking about finite fields.

Tobias Kildetoft

@Perturbative probably yes

Perturbative

But the thing is we were talking about a corollary of Cayley's theorem which said "If $G$ is a finite group of order $n$, then $G$ can be embedded into $GL_n(\mathbb{F})$" and supposedly this gave an embedding into a smaller group than the usual embedding into $S_n$ I think (but I remember thinking that wasn't correct because $GL_n(\mathbb{F})$ could be infinite)

Tobias Kildetoft

Whether it is smaller depends on the field of course

Perturbative

3:43 PM

Ahh I think I know what they meant, maybe we could choose $\mathbb{F} = \mathbb{Z}/2\mathbb{Z}$

Or some finite field like that

Tobias Kildetoft

(Actually, it will never be smaller)

Perturbative

Yeah just realized that @TobiasKildetoft

Tobias Kildetoft

Since $S_n$ always sits inside that group

(which is the way one embeds an arbitrary group)

Perturbative

@TobiasKildetoft Ohh okay, so the corollary is actually weaker. I guess the reason why we mentioned this is because we can represent elements of $G$ by matrices or something like that and work with them

Tobias Kildetoft

right, which is often useful

Perturbative

3:46 PM

Thanks @TobiasKildetoft, just talking to you helped straighten things out in my head :p

2 hours later…

Akiva Weinberger

5:21 PM

Hmm

$$a-\frac1b=(a-1)+\cfrac1{1+\cfrac1{(b-1)}}$$

I see it

Interesting

Secret

Thinking about the essence of infinity again:
hmm, a fractal with hausedoff dimension 1 < x < 2 clearly has finite area, yet you can zoom in forever and the porportions will remain the same

meaning that what I suggested a few weeks ago is indeed insufficient

besides scale invariance, what other properties does infinity have...

I guess we can start with potential infinity first:

For example, the number of naturals is larger than any fixed subcollection of naturals

So perhaps we can explosively generalise this as follows:

Let $M$ be a mathematical object of type $\text{Fin}$, $S$ be a relation that encodes some notion of size and $f$ be any relation that encodes a scaling operation. Then $M$ is potentially infinite if:

$$S(M,N)\neq S(N,M)$$

where $N$ is an object formed by any combination of $f$ and $M$ and is of type $\text{Fin}$

wait that makes no sense

1 hour later…