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Apr
24
revised Prove $\forall u,v,x,y,z,w \in \mathbf{R}^+, \frac{u}{v} < \frac{x}{y} \wedge \frac{x}{y} < \frac{z}{w} \implies \frac{u + z}{v+w} < \frac{z}{w}$
$$ in titles takes up too much space on the main page
Apr
22
answered Odds in Pascal's Triangle
Apr
21
comment Let $n$ be a positive integer. Show that if $2^n -1$ is a prime number, then $n$ is a prime number.
I think you mean "contrapositive," not "converse" (the converse statement is actually false).
Apr
20
reviewed Reopen How many equilateral triangles can be inscribed in a triangle?
Apr
19
comment How to understand “Union of balls centered at rational numbers is way less than $\mathbb{R}$
You might think of $f$ as giving you the rationals sorted by complexity. Then an irrational number not in $S$ would be one that doesn't have any "good, simple" rational approximations. That is, as your rational approximations get better, they also get more complicated, faster than they get better. The actual identities of these irrationals are going to depend on your choices of $f$ and $\epsilon$, but good places to look would be numbers with small continued fraction coefficients (like the golden ratio) as they are hard to rationally approximate.
Apr
16
revised Definite Integral $\int_{-\infty}^{\infty}\frac{x\sin x}{(x^2+a^2)(x^2+b^2)}\,\mathrm{d}x$
\displaystyle in title takes up too much vertical space on the main page
Apr
15
answered Is the reasoning/algebra for my proof correct? (musical tuning theory proof)
Apr
15
comment Is the reasoning/algebra for my proof correct? (musical tuning theory proof)
If you want both $m$ and $n$ to be positive integers, you should probably be looking at the equation $(3/2)^m=2^n$ rather than $(3/2)^m=(1/2)^n$...
Apr
10
comment Whats the differences between the real-entire functions on $\mathbb R^{2}$ and complex entire functions on $\mathbb C$?
Related: math.stackexchange.com/questions/189366/…
Apr
10
comment Coefficients of the expansion of $(x+a)^2$ makes a perfect square?!
Notice that it doesn't work when $a=4$: $1816$ is not a perfect square. Based on Trogdor's answer, you should be able to figure out why...
Apr
3
reviewed Reject suggested edit on Is there a way to solve this problem
Apr
3
reviewed Reopen Solving the recurrence relation $T(n) = T(n-\sqrt n) + 1$
Apr
2
comment Shouldn't the yellow marked $a_0$ be $a_0+\langle p(x)\rangle?$
That's right. If you wanted to do a lot of arithmetic in this field, you'd probably go even further and say "I'm going to write $x$ as shorthand for $x+\left<p(x)\right>$," but in this proof that would obscure exactly why the whole thing vanishes.
Apr
2
answered Shouldn't the yellow marked $a_0$ be $a_0+\langle p(x)\rangle?$
Apr
1
revised Proving an inequality on $\sum_{1\leq i,j \leq n} \langle c_i ,c_j \rangle \times \langle l_i ,l_j \rangle$
$$ in titles takes up too much space on the main page
Mar
31
comment Is 91 the only number which is both a centred cube number and a centred hexagon number?
There are definitely finitely many such numbers (by Siegel's theorem...)
Mar
29
comment $v$ is Conjugate Harmonic to $u$ $\implies$ $f = u + iv$ is Analytic (Proof Verification from Ahlfors)
That sounds right (though I don't have a copy of Ahlfors handy to check). You might want to go back and check exactly what Ahlfors is using as a definition of "harmonic conjugate." (I'm actually used to defining $u$ and $v$ to be harmonic conjugate if $u+iv$ is analytic, which would render this whole thing trivial, but the proof you've given suggests that "$u$ and $v$ satisfy the CR equations" is the definition he's using.)
Mar
29
comment $v$ is Conjugate Harmonic to $u$ $\implies$ $f = u + iv$ is Analytic (Proof Verification from Ahlfors)
$v$ is a conjugate harmonic function of $u$ because they satisfy the Cauchy-Riemann equations. Any pair of functions that satisfy the Cauchy-Riemann equations are guaranteed to both be harmonic, so in a sense you're making use of the fact that $u$ and $v$ are harmonic in your question 2 step.
Mar
29
answered $v$ is Conjugate Harmonic to $u$ $\implies$ $f = u + iv$ is Analytic (Proof Verification from Ahlfors)
Mar
29
revised How to prove this integral $I=\int_{0}^{\frac{\pi}{2}}\frac{dx}{1+\sin^2{(\tan{x})}}$
\dfrac in titles takes up too much vertical space