I was going through a research paper which proves the existence of infinite family of rank $6$ elliptic curve over $\mathbb{Q}$ with invariant equal to $0$. Let $k$ be a field of characteristic zero. Let $p \in k[X]$ be a unit polynomial of degree $6$. Then there exists a unique unit polynomial $g \in k[X]$, of degree $2$, such that the polynomial $r = p-g^3$ is of degree $\leq 3$. Suppose the roots $x_1,\dotsc ,x_6$ of $p$ are in $k$. The curve $E$ with equation $r(x) + y^3 = 0$ contains the $6$ $k$-rational points $P_i = (r(x_i),g(x_i))$, $1\leq i\leq 6$. In the paper the author claims that ..read more

The paper Dividing a polygon into two similar polygons provides that there are only three families of polygons that are irrep-2-tiles (can be subdivided into similar copies of the original). Right angled triangles $1:\sqrt{2}$ parallelograms The Golden Bee I wish to find examples of polyhedra that are irrep-2-tiles. The only example I have been able to find is: The $1:2^\frac{1}{3}:2^\frac{2}{3}$ parallelopipeds. Are there further examples? I would like to find as many as possible. Edit: A diagram of the Golden Bee from the linked paper. It seems plausible that there could be a $3d$ analog ..read more

Posted in MO since it is unanswered in MSE. Let $(a,b,c)$ be the side of a triangle. In its most general linear form, the triangle inequality can be expressed as: Does $ax + by \ge c$ for fixed $x,y \ge 0$ hold? Trivially the inequality holds if both $x$ and $y$ are $\ge 1$; however if one of or both of $x$ and $y$ is non-negative and $\le 1$ then $ax+y \ge c$ is not necessarily true. Assuming that vertices of a triangle are uniformly random on a circle we can ask the probability $P(ax+by \ge c)$. In this question we found a closed form for the probability $P(a+b \ge cx)$, $x \ge 1$. This ques ..read more

Let $\sigma=(1,2,3,\dots,n)$ be an odd $n-$cycle in $S_n$ (so $n$ is even). It is known that the size of its conjugacy class is $|cl_{S_n}(\sigma)|=(n-1)!$. I am interested in the size of the subset $S=cl_{cl_{S_n}(\sigma)}(\sigma)$, that is, the set of all the $n-$cycles that we can obtain by conjugating $\sigma$ only with elements in its conjugacy class. In particular, I would like to show that $S$ contains at least $|cl_{S_n}(\sigma)|/2=(n-1)!/2$ elements, which happens to seem true in the numerical experiments I run. Approaches I tried: Construct enough elements by hand. I found this tri ..read more

I came up with this while messing around with the $\gcd$ and $\operatorname{lcm}$ functions in Desmos. $$I(a)=\int\limits_{0}^{\frac{\pi}{2}}\frac{\operatorname{lcm}(a\cos x,a\sin x)}{a^2}dx$$ The function inside the integral always has a slope of $0$ but is very discontinuous. At higher values of $a$ it seems to approximate $\frac{\sin 2x}{2}$ with fewer lines approximating $\frac{\sin 2x}{4},\frac{\sin 2x}{6},$ and so on as pictured below for $a=100$. Some more things I noticed (mostly through experimentation): $I$ is undefined when $-\frac{\sqrt{2}}{2}<a<\frac{\sqrt{2}}{2}$. I can n ..read more

The below result is from Linear Algebra Done Right, 4th edition, Sheldon Axler: 5.11 linearly independent eigenvectors Suppose $T \in \mathcal{L}(V)$. Then every list of eigenvectors of $T$ corresponding to distinct eigenvalues of $T$ is linearly independent. Firstly, cool result. Secondly, I'm having trouble understanding one part of Axler's proof. He does a proof by contradiction. Since he assumes the result is false, he states: ... there exists a smallest positive integer $m$ such that there exists a linearly dependent list $v_1, \ldots, v_m$ of eigenvectors of $T$ corresponding to disti ..read more

The equation $3^x(2^{x+1}+5)=2^{x+2}$ has the solution $x=-1$. Is there an analytic way to see this solution? I have found nothing but the result seems very intriguing ..read more

The problem is to calcultate $$\int_1^2\frac{1}{x}\left(\ln(x^4+4)-\ln(x^2+4)\right)\mathrm{d}x$$ I have tried the usual substitution method. Let $y=\frac{2}{x}$, then I get $$\int_1^2\frac{1}{x}\ln\frac{x^4+4}{x^2+1}\mathrm{d}x-(\ln 2)^2,$$ but it doesn't work. I guess the result is $\frac{(\ln2)^2}{2 ..read more

Problem: Let $\Omega\subset\mathbb{R}^n$ be a domain and $u\in C^2(\Omega)\cap C(\overline{\Omega})$ a solution of $$\Delta u=u^3\text{ in } \Omega.$$ Then $u$ attains no nonnegative (local) maximum and no nonpositive (local) minimum in $\Omega$, if $u\neq 0$ holds. Attempt: Assume that $u$ has a maximum at $x_0\in\Omega$ so that $u(x_0)>0$. Since $x_0$ is a maximum $\Delta u(x_0)\leq 0$ holds. But $\Delta u(x_0)=(u(x_0))^3>0$ is a contradiction. Assume that $u$ has a minimum at $x_0\in\Omega$ so that $u(x_0)<0$. Since $x_0$ is a minimum $\Delta u(x_0)\geq 0$ holds. But $\Delta u(x_0 ..read more

For my profession I have to calculate the weighted average number of days it takes a business to review contracts. The count of each contract types reviewed is the weight, and the sum of days it takes to review is the number. The weighted average for this below data set is 35.6 days, however the raw data indicates that not a single contract took more than 25 days to review. Should my days be calculated as the average of days instead of sum? If that's the case, my weighted average is 8 days, which seems more accurate. I don't know why the weighted average using the Sum of Days is so much higher ..read more