Suppose $$\lambda_{1} = \frac{1}{2} (x^2z - \sqrt{4wy^2 + x^4z^2}) , \lambda_2 =\frac{1}{2} (x^2z + \sqrt{4wy^2 + x^4z^2})$$ $$\lambda_3 = \frac{1}{2} (w^2 + y+xz^2 - \sqrt{-4w^2y + (-w^2-y-xz^2)^2}) $$ $$\lambda_4 = \frac{1}{2} (w^2 + y+xz^2 + \sqrt{-4w^2y + (-w^2-y-xz^2)^2})$$ $S=\{(x,y,z,w)| |...

Let $G_i,i=1,2$ be two topological groups. Is it true that the product space $G_1\times G_2$ is a topological group too? Suppose that $G:=G_1\times G_2$. Let $o_i$ denote the group operation of $G_i$ and let $i_j$ denote the map $x\to x^{-1}$ on $G_j$. $G$ is Hausdorff as product of two Hausdorff...

$\DeclareMathOperator\Var{Var}\DeclareMathOperator\Bern{Bern}\DeclareMathOperator\Pois{Pois}$Question: What not-trivial mistakes do students often make when solving problems in probability theory, mathematical statistics and random processes? Some examples of wrong solutions: Problem 1: Find dist...

Consider the map$$\begin{array}{rccc}\Pi\colon&\Bbb R&\longrightarrow&\Bbb T\\&x&\mapsto&\bigl(\cos(x),\sin(x)\bigr),\end{array}$$which is a surjective continuous group homomorphism. If $G$ is a subgroup of $\Bbb T$, $\Pi^{-1}(G)$ is a subgroup of $(\Bbb R,+)$. But every subgroup of $(\Bbb R,+)$ ...