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Aug
25
comment Suitable composition of concave and convex functions is convex?
I was just about to post $f(x)=\begin{cases}\frac12\sqrt {3x},&x<\frac13\\\frac{3x+1}4,&x\ge\frac13\end{cases}$ explicitly ...
Aug
25
answered Is the Legendre sieve explicit?
Aug
25
comment does a positive/negative number cancel itself?
If you mean $x\pm 3$, this is a shorthand for two expressions $x+3$ and $x-3$. For example, the two equations $(a+b)^2=a^2+2ab+b^2$ and $(a-b)^2=a^2-2ab+b^2$ can be summarized as $(a\pm b)^2=a^2\pm 2ab+b^2$. There is no cancellation, i.e. this summary cannot be simplified to $(a+0)^2=a^2+0+b^2$.
Aug
25
comment 'Proof ' that $\ln(x)$ converges
@alexqwx It is not circular. Your "proof" consists of boldly writing "$\implies$", whereas the divergence of $\ln(x)$ or $\sqrt x$ can be easily checked.
Aug
25
comment Proving that a function is strictly monotonic knowing that $|f(x)-x^2|\le2|x|$
You cant't. $f(x)=x^2$ is a function matching the condition and is not monotonic. Or think of $f(x)=x^2+x\cdot \mathbf 1_{\mathbb Q}(x)$, which is not monotonic anywhere.
Aug
25
revised Proving that a function is strictly monotonic knowing that $|f(x)-x^2|\le2|x|$
added 3 characters in body; edited title
Aug
25
comment Question about various percentages to a 100%
Can you give a full quote of the statistics?
Aug
25
answered Equivalence of ordered field and an order relation
Aug
25
answered derivative of $x\cdot|\sin x|$
Aug
25
answered Countable Set & Formal Grammar
Aug
25
comment Countable Set & Formal Grammar
Is $\Sigma$ really arbitrary or shouldn't we assume it is countable?
Aug
25
answered Equality of cardinality of $\mathbb{N}$ and $\mathbb{N} - \{0\}$
Aug
24
comment Equivalence of ordered field and an order relation
A better statement would be: ${<}\mapsto \{\,x\in K\mid 0<x\,\}$ is a natural bijection between order relations with (a),(b),(c),(d) on $K$ and subsets $P$ of $K$ with $P+P\subseteq P$, $P\cdot P\subseteq K$, $K=P\cup\{0\}\cup {-P}$. So yes, if there are several $P$ possible, then there are also several $<$ possible and vice versa
Aug
24
answered $(A\cap B)C=AC\cap BC$ in an infinite group
Aug
22
answered Find the values of $f(0)$, $f(4)$, $f(6)$ and $f(18)$
Aug
22
comment General formula for $\frac{a^{2n+1}-a^{2n-1}}{a^n-a^{n-1}}$. Is such proof correct?
$a\ne 0$ is only needed if $n\ne 1$
Aug
22
comment General formula for $\frac{a^{2n+1}-a^{2n-1}}{a^n-a^{n-1}}$. Is such proof correct?
Compare yor result with lab's answer
Aug
22
comment General formula for $\frac{a^{2n+1}-a^{2n-1}}{a^n-a^{n-1}}$. Is such proof correct?
Why do you think that division is not a proof? -- But also: Did you notice which values of $a$ are forbidden (for the statement as such, so both kinds of proof are affected)? -- And finally, your result and proof do not match the problem statement
Aug
21
comment Notation problem in integration: write $dx$ or ${\mathrm{d}}x$?
Actually, I prefer $\int f(x)\,\mathrm dx$. :)
Aug
21
answered The number of divisors of any positive number $n$ is $\le 2\sqrt{n}$