mathjacks
Reputation
1,470
Next privilege 2,000 Rep.
 8h comment Suppose $u,v \in \mathbb{C}$ are in the open unit disk. Is $|u|^n - |v|^n \leq |u - v|^n$? Thanks for this. I will have to do my proof another way. 8h comment Suppose $u,v \in \mathbb{C}$ are in the open unit disk. Is $|u|^n - |v|^n \leq |u - v|^n$? Thank you for pointing out this easy counter-example. 8h accepted Suppose $u,v \in \mathbb{C}$ are in the open unit disk. Is $|u|^n - |v|^n \leq |u - v|^n$? 9h asked Suppose $u,v \in \mathbb{C}$ are in the open unit disk. Is $|u|^n - |v|^n \leq |u - v|^n$? 2d comment What is the Laurent series of $\frac{2}{z-1} - z$ in $1<|z|<2$? @C.Dubussy I see it now. So is it true then that this Laurent series converges as long as $1 < |z| <\infty$ holds? Hence, it converges within the upper bound of the annulus in the question. 2d awarded Mortarboard 2d asked What is the Laurent series of $\frac{2}{z-1} - z$ in $1<|z|<2$? Apr 23 revised Let $A$ be an $n \times n$ matrix over $\mathbb{C}$ or $\mathbb{R}$. Does $\det(e^A) = e^{\mathrm{tr}(A)}$ always hold? deleted 3 characters in body; edited title Apr 23 asked Let $A$ be an $n \times n$ matrix over $\mathbb{C}$ or $\mathbb{R}$. Does $\det(e^A) = e^{\mathrm{tr}(A)}$ always hold? Apr 18 comment Is there a diagonalizable matrix $A \neq B$ such that $e^A = e^B$? Helpful. This could also be generalized to any finite dimensional space by taking any diagonal matrix with diagonal entries $2 \pi i$, I believe. Apr 18 asked Is there a diagonalizable matrix $A \neq B$ such that $e^A = e^B$? Apr 16 accepted Let $z=x+iy$. What is $\frac{\partial}{\partial x} \log|z-w|$ and $\frac{\partial}{\partial y} \log|z-w|$? Apr 16 asked Let $z=x+iy$. What is $\frac{\partial}{\partial x} \log|z-w|$ and $\frac{\partial}{\partial y} \log|z-w|$? Apr 4 comment All solutions of $A^2+I=0$ in $M(n,\mathbb{C})$ are similar How did you deduce the minimum polynomial from $A^2+I=0$? Is this a variant of Cayley-Hamilton? I calculated it as $p_A(x) = (i-x)^m (-i-x)^p$ which obviously does have repeated roots. Apr 4 accepted All solutions of $A^2+I=0$ in $M(n,\mathbb{C})$ are similar Apr 4 comment All solutions of $A^2+I=0$ in $M(n,\mathbb{C})$ are similar In practice, we should be able to deduce the "appropriate" $m$ and $n$ from the characteristic polynomial of $A$, right? Apr 4 comment All solutions of $A^2+I=0$ in $M(n,\mathbb{C})$ are similar Shouldn't we mention that $A$ is diagonalizable over $\mathbb{C}$? I believe the diagonalizability of $A$ is why there are no $1$'s on the super-diagonal over the Jordan canonical form. Apr 4 accepted Let $V$ be the space of complex polynomials on $[0,1]$. Is the differentiation operator self-adjoint? Apr 3 asked When does the the eigenspace $E_\lambda(T)$ equal the generalized eigenspace $M_\lambda(T)$? Apr 3 asked All solutions of $A^2+I=0$ in $M(n,\mathbb{C})$ are similar