Generators define a surjection $R^k\to M$, and conversely images of the standard basis generate any quotient. Writing $\varphi (m_i)={\sum }_j{a}_{ij}{m}_j$ with $a_{ij}\in I$ and applying the adjugate to $XI-A$ gives a monic annihilating polynomial whose lower coefficients lie in $I$. Taking $\varphi =1$ when $IM=M$ yields $xM=0$ with $x-1\in I$; for $M=\mathbb{Z}/3$ and $I=(2)$ take $x=3$. The Jacobson radical criterion follows by placing a nonunit $1-rs$ in a maximal ideal. It yields Nakayama’s lemma. Applying the determinant trick to preimages of a finite generating set constructs a polynomial right inverse to any surjective endomorphism, proving injectivity.

The maximum-flow problem maximises source-to-sink flow subject to capacity and conservation constraints. Ford–Fulkerson repeatedly finds an augmenting path in the residual network and augments by its bottleneck capacity, stopping when none exists. Multiplying rational capacities by a common denominator makes them integers; every augmentation then raises the flow by at least one while the value is bounded by the finite source capacity, so termination is finite.

The MLE is $\hat{\mu }=\bar{X}$ with MSE $1/n$. Combining likelihood and the $N(0,{\tau }^{-2})$ prior gives posterior $N(n\bar{X}/(n+{\tau }^2),1/(n+{\tau }^2))$. The frequentist MSE of its mean is $[n+{\tau }^4{\mu }^2]/(n+{\tau }^2{)}^2$, which is below $1/n$ exactly when ${\mu }^2<2{\tau }^{-2}+1/n$.

For $z>0$, $\varphi =q(4\pi {\epsilon }_0{)}^{-1}(|x-d\hat{z}{|}^{-1}-|x+d\hat{z}{|}^{-1})$ and $E=-\nabla \varphi$. The normal field just above the plane is $-qd/[2\pi {\epsilon }_0(r^2+d^2{)}^{3/2}]$, so $\sigma ={\epsilon }_0{E}_z$ is the stated expression. Integrating $2\pi r\sigma (r)dr$ from zero to infinity gives total induced charge $-q$.

Differentiability at $p$ means $f(p+h)=f(p)+Lh+o(|h|)$ for a linear map $L=Df(p)$; partial derivatives are its values on coordinate vectors. Integrating the partial derivatives along the two coordinate segments and using continuity at $p$ proves the stated sufficient condition. Here $f(x,y)=g(x)+g(y)$ with $g(t)=t^2\sin (1/t)$ and $g(0)=0$. Since $g\prime (0)=0$, $f$ is differentiable everywhere. But $g\prime (t)=2t\sin (1/t)-\cos (1/t)$ does not approach $0$ at zero, so $f$ is $C^1$ exactly where $xy\ne 0$.

Invariant means $\pi P=\pi$. In a finite irreducible chain, $E_i{T}_i^{+}=1/{\pi }_i$ and the expected visits to $i$ between visits to $j$ are ${\pi }_i/{\pi }_j$. The first burglar has uniform invariant law on $64$ PINs, so (a) is $64$ and (b) is $16$. First-step equations by Hamming distance give the adjacent-state hitting time (c) as $63$. For the modified one-coordinate chain, destination weights are $1$ for digits $0$–$5$ and $2$ for $6,7$; its invariant weights are proportional to $w_a(10-w_a)$, namely $9$ and $16$. The product invariant law and the occupation formula give (d) ${\pi }_{(6,7)}/{\pi }_{(0,0)}=256/81$.

Odd terms vanish and direct evaluation gives $L(p)=(8/45)c_4$, so the rule is exact through degree $3$. With $n=3$, the Peano kernel isIts nonnegativity and $\int K=L(x^4)/4!=1/135$ give $|L(f)|\le \Vert f^{(4)}{\Vert }_{\infty }/135$, so $N=135$. Equality holds for a quartic with fourth derivative of constant sign.