Mar 26, 2026 · Note: $24n+9 = x^2+2y^2+6z^2$ implies $$8n+3 = \Bigl (\frac {x+2y} {3}\Bigr)^2 + 2\Bigl (\frac {x-y} {3}\Bigr)^2 + 2z^2$$ which may be less restrictive than Eureka’s requirement $8n+3 = …

Aug 4, 2021 · Whenever a problem or question of isomorphism comes up I am clueless as to what they mean. From my understanding an isomorphism is in terms of graph theory where it is the same thing …

For basic questions about limits, continuity, derivatives, differentiation, integrals, and their applications, mainly of one-variable functions.

Mar 27, 2026 · As shown, on pg 259 of Ramanujan's 2nd notebook he states a few identities: I am easily able to decipher the second half, which just consists of some class equations for the theory of …

Suppose I am trying to prove a statement in the form A if and only if B. I know I need to prove that If A, then B If B, then A I know that 1 is equivalent to proving "If not B, then not A". My

Oct 8, 2020 · $ABCD$ is a parallelogram, $P$ is any point on $AC$. Through $P$, $MN$ is drawn parallel to $BA$ cutting $BC$ in $M$ and $AD$ in $N$. $SR$ is drawn parallel to $BC ...

Oct 29, 2011 · What is difference between implicit and explicit solution of an initial value problem? Please explain with example both solutions (implicit and explicit)of same initial value problem? Or …

Seeing your answer, I tried to figure out how to guarantee that it was general, and couldn't easily do so. That's what led me to my answer, where I started by moving the x-axis to the point where the light …

Sep 16, 2025 · @L.R.Stanis I understand you are looking only for the proof that the Karoubi envelope is semiadditive, but as Naïm points out the whole idea of the construction (also called the idempotent …

Apr 29, 2024 · A proof of each property follows from the definition of variance and linearity of expectation: $$\operatorname {Var} [X] = \operatorname {E} [ (X - \operatorname {E ...