query: Does $(\omega^a)^b \neq \omega^{(ab)}$ for complex numbers $\omega, a, b \in \mathbb{C}$?
positive_passages.docid: 11089
positive_passages.text: For which complex $a,\,b,\,c$ does $(a^b)^c=a^{bc}$ hold?
negative_passages.docid: 127622
negative_passages.text: Does there exist a $1$-form $\alpha$ with $d\alpha = \omega$?
negative_passages.docid: 83713
negative_passages.text: If $z,w\in \mathbb{C}^{\times}$, then $|z+w|=|z|+|w|$ iff $w=tz$ for some $t\gt 0$
negative_passages.docid: 52750
negative_passages.text: Prove that $f^\ast\omega=\det f \cdot \omega$
negative_passages.docid: 115176
negative_passages.text: Can we say that $\left(a+b\right)^{\alpha}>a^{\alpha}+b^{\alpha}$ for all $a,b>0$ and $\alpha>1$?
negative_passages.docid: 63446
negative_passages.text: The existence of analytical branch of the logarithm of a holomorphic function
negative_passages.docid: 82097
negative_passages.text: If $A \succeq B$, is it true that $B^{-1} \succeq A^{-1}$?
negative_passages.docid: 59442
negative_passages.text: Are there $a,b>1$ with $a^4\equiv 1 \pmod{b^2}$ and $b^4\equiv1 \pmod{a^2}$?
negative_passages.docid: 158406
negative_passages.text: $A,B$ be Hermitian.Is this true that $tr[(AB)^2]\le tr(A^2B^2)$?
negative_passages.docid: 49330
negative_passages.text: If for two suitable matrices $B$ and $C$, $AA^TB=AA^TC$ can we conclude that $A^TB=A^TC$?
negative_passages.docid: 148516
negative_passages.text: Proof that $\omega^\omega$ is completely metrizable and second countable
negative_passages.docid: 157187
negative_passages.text: Why $\omega+1$ and $\omega^2$ are not cardinal numbers?
negative_passages.docid: 56143
negative_passages.text: Proof of a basic $AC_\omega$ equivalence
negative_passages.docid: 6777
negative_passages.text: Showing $(a^{2}+2)(b^{2}+2)(c^{2}+2)\geq 9(ab+bc+ca)$
negative_passages.docid: 100549
negative_passages.text: If $e^A$ and $e^B$ commute, do $A$ and $B$ commute?
negative_passages.docid: 90276
negative_passages.text: Is $\sqrt{2}\in{\Bbb Z}[\sqrt{2}+\sqrt{3}]$ true?
negative_passages.docid: 107450
negative_passages.text: False proof that $ρe^{iθ} = ρ$ and so complex numbers do not exist?