Taiwan Quiz 2014: Prove that for positive reals $a$, $b$, $c$ we have $$3(a+b+c) \ge 8\sqrt[3]{abc} + \sqrt[3]{\frac{a^3+b^3+c^3}{3}}$$ Applying Power-mean inequality (weighted) in $$a_1={abc} , a_2={\frac{a^3+b^3+c^3}{3}}$$ with $r=1$, $s=\dfrac{1}{3}$ and weights $\dfrac{8}{9}, \dfrac{1}{9}$ ...
In several papers I came across powers of the infinite radical of a module category. I don't really understand what that is and I couldn't find any definitions. Let $A$ be an Algebra and let mod$(A)$ be the category of finitely generated left $A$-modules. What is (rad$^{\infty}($mod$(A)))^2$ or ...
Let $A$ and $B$ rings together with a ring morphism $\phi : B \to A$. Note that $A$ becomes a $(B,A)$-module for $$b.x.a = \phi(b) x a$$ My notes claim that there exists a Morita context $(B,A,P,Q, \mu, \tau)$ where $P$ is the $(B,A)$-module $A$. How can I construct $Q$? I don't need a detailed a...
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