cactus314
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 5h comment Show $\int_0^\infty \frac{\sin ax \sin bx}{x^2} \, dx = \frac{\pi a}{2}$ if $a < b$ my question is a duplicate: math.stackexchange.com/questions/441106/… Apr 16 comment Show that $\mathrm{SO}_3(\mathbb{Q}_p) \simeq \mathrm{SL}_2(\mathbb{Q}_p)$ this did not depend on $p = 4k\pm 1$ ? Apr 8 comment show that $\sum P(x,y) e^{x^2 + y^2}$ is a modular form over $\Gamma_0(4)$ where $P(x,y) = x^4 - 6 x^2 y^2 + y^4$ My best guess this comes from $f(x,y) = P(x,y) e^{-(x^2+y^2)}$ and saying $$\sum_{(x,y) \in \mathbb{Z}^2} f(x,y) = \sum_{(x,y) \in \mathbb{Z}^2} \hat{f}(x,y)$$ which is Poisson summation in two variables and $\mathbb{Z}^2$ is a lattice dual to itself. Apr 8 comment Show that $\mathrm{SO}_3(\mathbb{Q}_p) \simeq \mathrm{SL}_2(\mathbb{Q}_p)$ @anomaly covering map is fine. I reasoned for example that $\mathrm{Spin}(2) \to \mathrm{SO}_3$ is a 2-1 covering map for $\mathbb{R}$. And we have $\mathrm{Spin}_2 \simeq \mathrm{SU}_2$. Apr 6 comment Show the Dedekind $\eta(\tau)$ function is a modular form of weight $\frac{1}{2}$ for $\Gamma_0(6)$ I have read that $\eta(\tau)^24$ is a modular form of weight 12 over $\mathrm{SL}_2(\mathbb{Z})$, so maybe $\eta$ is modular of weight $\frac{1}{2}$ ? Apr 4 comment A set such that $A$, $A+A$ have density zero but $A+A+A$ has positive density. Is that called log density? Not even. Like $\sqrt{\log}$ density. Apr 4 comment A set such that $A$, $A+A$ have density zero but $A+A+A$ has positive density. Finally $d(\square + \square + \square + \square) = 1$ as all positive integers can be expressed as the sum of 4 squares. So what is happening here? Mar 24 comment $S=\frac{-d^2}{dx^2}$ self-adjoint operator or not? I am a little bit confused by symmetric but not self-adjoint as this could never happen in finite dimensional space Mar 22 comment equivalent to $A \to (C \leftrightarrow D)$ If $A$ is not true, then maybe $C$ and $D$ are not equivalent. In number theory, there are rings where "prime" and "irreducible" are not equivalent. Mar 21 comment Ruler and compass question how is this Galois theory? Mar 20 comment sum of divisors function $\sum \tau(n) = \frac{1}{4}$ @Gerben Divergent series get hated on in math courses but I trudge on since I see them so often in physics courses. Mar 18 comment How to visualize the region $\mathbb{H}/\Gamma_0(4)$ and its cusps? math.stackexchange.com/questions/1288478/… Mar 4 comment Why is $1 - \frac{1}{1 - \frac{1}{1 - \ldots}}$ not real? Notice that $n \geq 2$. In this guy's example $n \equiv 1$. Feb 7 comment Show that $J_n(x)$ satisfies Bessel equation $x^2 \frac{d^2 y}{dx^2} + x \frac{dy}{dx} + (x^2 - n^2)y = 0$ Jan 29 comment Does $\overline{ \sqrt{1 + i}} = \sqrt{1-i} \$? The paper that I am reading is unusual as they are cutting along $\mathbb{R}^+$ instead of $\mathbb{R}^-$ I should have mentioned that thanks. Jan 29 comment Does $\overline{ \sqrt{1 + i}} = \sqrt{1-i} \$? what if you cut at positive reals? Jan 26 comment express as contour integral $f(x) = \int_0^\infty dt \; e^{-t/g} \; \frac{1}{\sqrt{1 - 2 t x}}$ Yeah this could be the one Jan 25 comment express as contour integral $f(x) = \int_0^\infty dt \; e^{-t/g} \; \frac{1}{\sqrt{1 - 2 t x}}$ @tired yes. the $e^{\frac{1}{2g}}$ is an instanton correction Jan 25 comment express as contour integral $f(x) = \int_0^\infty dt \; e^{-t/g} \; \frac{1}{\sqrt{1 - 2 t x}}$ @tired you get the error function if you don't add any correcting terms wolframalpha.com/input/… that's why I started making stuff up. it comes from a physics paper Jan 24 comment How to prove that $f$ is integrable if $\forall \epsilon, \ \exists$ partition $M\in [a,b]$ such that $U_f(M) - L_f(M)\lt\epsilon$? In practice these partitions might get pretty exotic. What partition makes the upper and lower Riemann sums close when $f(x)=\sum_{n=0}^{100}\frac{1}{n}\sin nx$ ? This converges to a sawtooth wave