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 Yearling
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Apr
24
awarded  Yearling
Mar
18
comment What does it mean to have no proper non-trivial subgroup
A "proper nontrivial subgroup" is a subgroup which is both a proper subgroup and a nontrivial subgroup.
Mar
11
revised What is the difference between $\mathbb{R}$ with the $K$ topology and $\mathbb{R}_k$?
added 89 characters in body
Mar
11
answered What is the difference between $\mathbb{R}$ with the $K$ topology and $\mathbb{R}_k$?
Feb
21
comment Give a natural deduction proof of $\varphi\vdash\top$, there $\varphi$ is any formula
The way I was taught, the Natural Deduction calculus had a $\top$ formation rule. That was kind of the whole use of introducing the predicate $\top$ in the first place. It might help to reference what exposition of Natural Deduction you are following.
Feb
21
answered Question about deduction in first order logic
Feb
21
suggested rejected edit on Question about deduction in first order logic
Feb
7
comment How come $\lambda(\mathbb{Q})= \inf\left\{\sum_{n \ge 1} \lambda([a_n,b_n)) : \mathbb{Q} \subset \bigcup_{n \ge 1} [a_n,b_n)\right\} = 0$?
My intuition is game-theoretic one: I imagine a game where Player 1 has countably many points he can throw out and demand get covered, and Player 2 has $\epsilon$ much length of tape that he's allowed to cut pieces off from to tape over Player 1's challenges. Each time Player 1 demands a point get covered, Player 2 just cuts off a small portion of his tape and covers that point. At no point does Player 2 run out of tape, so in the limit, Player 2 has responded to every challenge of Player 1, and has used at most $\epsilon$ much tape.
Feb
7
revised Odd-degree polynomials have roots (Intermediate Value Theorem)
Provided a more descriptive title
Feb
7
suggested approved edit on Odd-degree polynomials have roots (Intermediate Value Theorem)
Feb
7
comment Shortest possible unreachable shape
One class of examples is to double over the snake, and then spiral it around until the ends can't be unbent. i.imgur.com/5vYPn7p.jpg I think you can at least beat Harald Hanche-Olsen's example with this approach, you might even be able to get down to 29 but I have to check more carefully.
Feb
4
comment How many times can a $4^{th}$ degree polynomial be equal to a prime number?
This problem is weaker than showing that there exists a /single/ polynomial which takes arbitrarily many prime values. For example, the corresponding problem for quadratics was proven by Sierpenski in 1964 (that there is no bound on $n$).
Feb
4
comment Extremely difficult: Polynomials $f,g,h$ such that…
At least for some $n$ there are no nontrivial solutions. Plug in $0$ for $x$. Then for the constant terms $a, b, c$ of $f, g, h$ we must have $(ab+bc+ca)/(a^2+b^2+c^2)$ must be equal to $(3n-3)/(9n+1)$ (if $a^2+b^2+c^2 \neq 0$). This is impossible, for example, for $n=3$. So $a^2+b^2+c^2=0$. So $a, b, c=0$. Divide everything through by $x$ and repeat. So $f, g, h=0$.
Feb
4
comment Model theoretic answer for having algebraic closure
If you're going to iterate the process anyway, why bother applying compactness at all? Why not just add the roots in one at a time? What actual benefit does compactness have on the proof?
Nov
16
comment Finding $\int_0^{\pi/2} \sin x\,dx$
$\lim_{x^\circ \rightarrow 0} \frac{\sin(x^\circ )}{x^\circ}$ is most certainly $1$, by any standard. Whatever your interpretation of $x^\circ$ is, it must be as a real variable, and $\lim_{y\rightarrow 0} \frac{\sin y}{y}=1$ for any real variable $y$. Typically (e.g. by Mathematica), $x^\circ$ is interpreted via $x^\circ=x\times\frac{\pi}{180}$. Perhaps you meant to write $\lim_{x\rightarrow 0} \frac{\sin (x^\circ )}{x} = \frac{\pi}{180}$ or $\lim_{x^\circ\rightarrow 0} \frac{\sin (x^\circ )}{x} = \frac{\pi}{180}$? Either would be correct. What you have is not.
Nov
16
comment Intuitive understanding of the derivatives of $\sin x$ and $\cos x$
I disagree with the cynical analysis of people's calculus education. I think the reason many people answer this question poorly is because maths students are simply not familiar with degrees, as they are abandoned immediately upon entering a calculus course. And rightly so. Reasoning that an inability to answer questions about degrees is evidence of a poor calculus education is like asking European children to convert between cups and gallons, or about the Fahrenheit scale, and then concluding when they give you puzzled looks that it is "a sign of how badly the French learn arithmetic".
Nov
16
revised Find $\int_0^2 x^2f''(2x)dx $ given $f(2)$, $f'(2)$, and $\int_0^2 f(x)dx$
typo
Nov
16
revised Find $\int_0^2 x^2f''(2x)dx $ given $f(2)$, $f'(2)$, and $\int_0^2 f(x)dx$
typo
Nov
16
answered Find $\int_0^2 x^2f''(2x)dx $ given $f(2)$, $f'(2)$, and $\int_0^2 f(x)dx$
Nov
16
comment Uncountable reals in the theory
If you want the reals to really be the reals, so in particular have theory RCF, then no. RCF is not expressive enough. If you are just asking for the sake of asking, fix a countable model of ZF and a bijection between its universe and $\mathbb{R}$. Then look at the induced structure.