# Tagged Questions

For questions about approaches and techniques for discovering a proof, as opposed to writing it down clearly (which involves (proof-writing)). Should not be used unless the focus is on the technique of the proof instead of the solution.

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### How to prove the Hubble law is the unique expansion law compatible with homogeneity and isotropy?

In the book physical foundations of cosmology, it saids that Hubble law is unique and a problem seems to be a hint of proving that. In order for a general expansion law,v=f(r,t), to be the same ...
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### From 2 to 3 dimensions: integrating a force along a contour/surface.

I am studying the following problem: Consider a closed contour $\mathcal{C}$ in $\mathbb{R}^2$ defined by $r(\theta)$ where $\theta\in[0,2\pi)$ and $r(0)=r(2\pi)$ (let the center to be zero for ...
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### How to prove that the sequence is decreasing $a_{n}=\frac{ln(n)}{n^2}$

Is my way/proof good and completely mathematically rigorous? $a_{n}=\frac{ln(n)}{n^2}$ --> $a_{n+1}=\frac{ln(n+1)}{(n+1)^2}$ $\frac{ln(n)}{n^2} > \frac{ln(n+1)}{(n+1)^2}$ ...
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### prove by induction $7 \mid 3^{3^n}+8$

Okay so ive been trying to prove this for about 5 hours... really need salvation from the geniouses around here. prove by induction $7\mid 3^{3^n}+8$ i really need some directions on what to do ...
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### Proving that, if a function f is O(g), the ceiling of f is also O(g).

I'm having a bit of trouble with this problem: $$\forall (f, g) \in F, f \in O(g) \implies \lceil{f}\rceil \in O(g)$$ Where F is the family of functions from $\mathbb{N}$ to $\mathbb{R}^+$. I know ...
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### Prove a limit with condition specified at infinity

Suppose that $$\lim_{t\rightarrow \infty}\left(\dot{x}(t)+\gamma x(t)\right)=0,\quad \gamma>0.$$ How can I prove $$\lim_{t\rightarrow \infty}x(t)=0~?$$ Please give a strict proof. Thanks!
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### Giving restriction on the value of $N$ in $\epsilon-N$ proof.

I'm still working on $\epsilon-N$ proof. If you don't mind is it possible for us to give restriction on the value of $N$ as illustrated by this example: Say after some manipulation of the limit ...
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### Showing that two maps are homotopic

Let $X$ be a topological space and let $S^2 \subset \mathbb{R^3}$ be the unit sphere with the metric $d$ inherited from $\mathbb{R^3}$. Show that if $f,g:X\to S^2$ are continuous maps such that ...
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### Choosing the right N in $\epsilon-N$ proof

I'm just a little bit confused in choosing the right $N$ when working on the rough sketch of the proof. Suppose after some algebra we have reached the point where we get this expression, say: ...
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### General lists of techniques to prove whether a set is a generator of a matrix group

It seems like a rather common problem in group theory, at least in undergraduate maths, to check whether a set is a generator of a group. The question is usually as follow: Given a group $G$, and a ...
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### A question about the proof of an obvious result

This is obviously true that a local homeomorphism is a continuous map. I tried to prove it this way : Suppose $f:X \to Y$ is a local homeomorphism, then $f$ is continuous if for each $x\in X$ and ...
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### Finding posterior of normal distributions and logistic regression.

$P(w_0 | x) = \frac{1}{1 + e^{-log\frac{P(x|w_0)}{P(x|w_1)}-log\frac{P(w_0)}{P(w_1)}}}$ Note: x = $[x_1, \dots, x_d]^T$; a $d$ dimensional vector. $w$ can take on one of two values: $w_0$ or $w_1$. ...
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### A question related to the Descartes-Frenicle-Sorli conjecture on odd perfect numbers

Good day to everyone! I apologize in advance for the somewhat long post, but I had to put in all the details into a single question to communicate what I believe to be a viable approach to odd ...
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### Coming up with short “magical” proofs

I was reading the solution to this problem: Prove that $f(n) = 2n$ is the only non-constant solution to $2f (m^2 + n^2 ) = (f (m))^2 + (f (n))^2 .$ The solution used these identities, pulled out of ...
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### Solutions depending on something continuously

Let $V$ be a a real Banach space, $K \subset V$ a closed convex set, $A: K \rightarrow V^{*}$ a (nonlinear) operator and $F \in V^{*}$. Then the variational inequality is the following problem: find ...
Suppose $a,b\in\mathbb{Z}$. Then $a = \operatorname{lcm}(a,b)$ if and only if $b\mid a$ Unsure of how to approach this problem.