I've been doing some calculations in fractions, and found this equation pop up to calculate my answer: $$\frac{1-x}{1+x}=x$$ the initial equation is $$\frac{2(x-1)}{\frac{4(x+1)}{2}}+x=4x+9(-4x-2)-2(-17x+34)+61+6$$ (I used a random number generator)I started tackling it by solving the right ...

Do we need the tag algebraic-equations? It was created recently in this question by Michael Hardy. (I have added algebra-precalculus, which I consider a good fit for that question.) This tag is similar to algebraic-identities, previously discussed here and then removed. And it is also somewhat s...

Let $L/k$ be a finite extension . $L_1,L_2 $ subfields of $L$ containing $k$ such that $L_1/k$ is separable and $L_2/k$ is normal . Then it is easy to see $L_1L_2/L_2$ is separable . But how to show that $[L_1L_2:L_2]=[L_1:L_1\cap L_2]$ ?

Show that the following statements are equivalent; $S$ is a left simple semigroup that contains an idempotent $S$ is isomorphic to the direct product of a subgroup of $S$ and a left zero semigroup $L$ In one way this is kind of obvious. If $S \cong G\times L$, then the semigroup...

Problem in 3-d space important for computer vision. We have four points: $P_0$ where we know coordinates $(0,0,0)$ and $P_1, P_2, P_3$ where coordinates are unknown. However we know distances between $P_1, P_2, P_3$ (let's name them $d_{12}, d_{23}, d_{13}$) and unit vectors $v_1, v_2, v_3$, co...

Suppose there are 'n' lines in 3D. Since each line has 4 Degrees of Freedom (DOF), total DOF as I understand should be $4n$. As per Camera Calibration Using Line Correspondences, Section 2.5: Degree of Freedom up to projectivity n lines have $4n - 15$ DOF provided $n \ge 5.^2 $. I don't q...

Given a Finsler-Minkowski norm $F$ on $R^n$, let $\mathfrak{F}=\{\Sigma_r\}_{r\geq0}$, where $\Sigma_r=\{y\in R^n: F(y)=r\}$, be a partition of $R^n$. There exists any (infinitely) smooth Morse function $f:R^n\to R$ such that $f^{-1}(c)=\Sigma_r$ (that is gave the same partition)? FYI: A func...