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Clash Royale CLAN TAG #URR8PPP up vote 0 down vote favorite Let $A$ be a finite dimensional algebra. $P$ is an idecomposable projective $A$-module such that its radical $rad(P)$ is non-projective. Is it right that every non-zero proper submodule of $P$ is not projective? modules representation-theory share | cite | improve this question asked Aug 11 at 3:05 Xiaosong Peng 639 4 14 add a comment  |  up vote 0 down vote favorite Let $A$ be a finite dimensional algebra. $P$ is an idecomposable projective $A$-module such that its radical $rad(P)$ is non-projective. Is it right that every non-zero proper submodule of $P$ is not projective? modules representation-theory share | cite | improve this question asked Aug 11 at 3:05 Xiaosong Peng 639 4 14 add a comment  |  up vote 0 down vote favorite up vote 0 down vote favorite Let $A$ be a fini...

Clash Royale CLAN TAG #URR8PPP up vote 8 down vote favorite 4 I'm finding some bounds for the Si function defined as $$ operatornameSi(x) := int_0^xfracsin ttdt. $$ I observed from WolframAlpha that the inequality $$ operatornameSi(x)>arctan(x) $$ holds for $x>0$. I tried to show this analytically but failed and could not find any references regarding this. Could someone help me with this? calculus inequality share | cite | improve this question asked Aug 11 at 3:20 Ramanasa 56 4 I don't know if this helps but maybe try to express $arctan(x)$ as the integral from $0$ to $x$ of its derivative $frac1t^2+1$ and then form one integral. – zzuussee Aug 11 at 3:26 I tried to but could not proceed more. It seems there needs some more manipulations. – Ramanasa Aug 11 at 3:34 I'm currently writing a sketch of some thoughts. – zzuuss...