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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 $\le 2\sqrt{n}$
Aug
21
revised What's time complexity of algorithm for “Word Break”?
added 204 characters in body
Aug
21
answered What's time complexity of algorithm for “Word Break”?
Aug
21
answered Is it true that a field is a vector space over a field?
Aug
21
comment The process of solving the inequality $\frac{8}{19} x\ge -1$
If you follow the next steps, you should see that by choosing $\frac{19}8$, the coefficient $\frac 8{19}$ gets happily cancelled. This would not happen with $\frac8{19}$ or any other multiplier.
Aug
21
answered For a cubic equation, prove that two critical points of the same sign imply one root
Aug
21
comment Set of points of $[0,1)$ that have a unique binary expansion
Assume a number $x$ has two distinct expansions, first differing at the $n$th place. Then my multiplying with $2^n$ and subtracting an integer, you get two representations $0.a_1a_2\ldots$ and $1.b_1b_2\ldots$ of the same number. The latter is $\ge 1$, the former is $\le 1$, hence both must equal $1$, so they must be $0.1111\ldots$ and $1.000\ldots$. Thus one of the original representations for $x$ ends in all zeroes, i.e. is finite. But a finite 0-1-string (ending in $1$) can be viewed as a reversed representation of an integer, hence there are only countably many
Aug
21
answered Estimate bias of a coin
Aug
21
comment Do journals that published a proof of an important theorem $T$ publish another proof of $T$?
It's less about $T$ than about $P'$, you might say: A substantially new proof may use different (new?) methods, from which we can possibly learn more than from $T$ itself. Also, $P'$ may be so differnt that it may suggest different generalizations of $T$. You might even publish a new(!) proof of Pythagoras, adding to an already long collection.
Aug
21
comment Is there a general way to parameterize all implicit functions?
Depends on what you precisely mean by parametrizing. What about disconnected sets like $xy=1$?
Aug
21
answered Path Connectedness argument for $SO(n, \mathbb{R})$