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Multiplication by a nonzero element $a$ of an integral domain is injective, by multiplicative cancellation. When the domain is finite, this self-map is surjective, so some element $b$ satisfies $ab=1$. Commutativity gives $ba=1$, so every nonzero element has a multiplicative inverse and the domain is a field.

Even when the domain convention does not initially assume an identity element, finiteness produces one. Surjectivity supplies $e$ with $ae=a$; then $a(ex-x)=0$ forces $ex=x$ for every $x$. Commutativity makes $e$ a two-sided identity, after which the inverse argument applies.

The discriminant of an integral full-rank lattice is the field discriminant multiplied by the square of its index in the ring of integers. A squarefree lattice discriminant therefore forces index one.

If $m>1$ is a square-free integer, $3\nmid m$, and $m\not\equiv \pm 1(\mathrm{mod}9)$, then $1,\sqrt[3]{m},\sqrt[3]{m^2}$ is an integral basis of the pure cubic number field, with field discriminant $-27m^2$. The discriminant-index formula for an integral lattice restricts a possible index to primes dividing $3m$. At primes dividing $m$ the different exponent is $2$ by tame ramification. A shifted Eisenstein polynomial gives total wild ramification at $3$, so its different exponent is at least $3$. These exponents exhaust the polynomial discriminant and force index one.

A Stäckel potential allows additive separation of the Hamilton-Jacobi equation in an orthogonal coordinate system. In axisymmetric prolate spheroidal coordinates, the relative-potential form is $\psi =[G(\lambda )-G(\mu )]/(\lambda -\mu )$. It gives energy, axial angular momentum and an independent third separation integral. Here $G$ is a one-variable generating function, not the gravitational constant. Generic bounded orbits are regular invariant-torus motions; resonance and coordinate degeneracy require separate treatment.

For a square-integrable time series, let ${\mathcal{H}}_{t-1}$ be the closed linear span of its past. The innovation ${\epsilon }_t=X_t-{\mathrm{proj}}_{{\mathcal{H}}_{t-1}}{X}_t$ is orthogonal to every past linear observation. A causal invertible autoregressive moving-average model is driven by these innovations, which are weak white noise for a weakly stationary process.

For complex matrices, the inner product $⟨A,B{⟩}_F=\mathrm{tr}(A^{\ast }B)$ is conjugate-linear in its first argument. On real symmetric matrices it becomes $\mathrm{tr}(AB)$. In particular $⟨A,x{x}^T{⟩}_F=x^{T}Ax$, identifying quadratic forms with linear tests on matrices.

The Fischer inner product on polynomials is $⟨x^{\alpha },x^{\beta }{⟩}_F=\alpha !{\delta }_{\alpha \beta }$. Multiplication by $x_i$ is adjoint to differentiation by $x_i$. Therefore multiplication by $|x{|}^2$ is adjoint to the ordinary Laplacian, yielding the harmonic decomposition of homogeneous polynomials by finite-dimensional orthogonality.

Averaging squared norms of all signed sums of finitely many vectors in an inner-product space cancels every mixed term. This equates the average to the sum of the individual squared norms, distinguishing Hilbert-space geometry from other sequence-space norms.

For a Noetherian ring, injectivity of a module can be tested at all prime localizations. By the Baer criterion, test ${\mathrm{Ext}}_R^1(R/J,M)$ for every ideal $J$. Localization of Ext over a Noetherian ring identifies its localization with the corresponding ideal test over $R_P$. Every ideal of $R_P$ is extended from its contraction, and localization detects zero elements applies to the resulting Ext module even when it is not finitely generated. These observations prove both directions.

Differential entropy describes the spread of a continuous probability distribution relative to the volume scale of its coordinates. For a real random variable $X$ with probability density function $f$, use natural logarithms and writeThe convention is $0\ln 0=0$. For a finite value, require $\int f|\ln f|<\infty$; if both positive and negative contributions are infinite, the expression is undefined. Unlike a probability, a probability density can exceed one. Consequently differential entropy can be negative. Base-two logarithms give $h_2=h_e/\ln 2$, in bits instead of nats.^[ref]

This quantity does not give the exact number of bits needed to store a real number. A recording device supplies a resolution; equal-width bin entropy connects that resolution to discrete information entropy. The useful continuous quantity is what remains after removing the increasing cost of specifying finer bins. Its dependence on coordinates is visible already in differential entropy of a uniform law.

For a finite-valued random variable, an epsilon-sufficient set $S$ retains probability at least $1-ϵ$. This notion is about covering probability mass and is unrelated to a sufficient statistic. For an IID random variable source, any such set of words with fixed $ϵ<1$ requires exponentially many words at a rate approaching the Shannon entropy.

If one probability in a $k$-point distribution equals $t$, normalize the remaining $k-1$ probabilities to $q$. Then $H=h(t)+(1-t)H(q)\le h(t)+(1-t){\log }_2(k-1)$. Equality for $t<1$ means that the remaining probabilities are equal. This combines the binary entropy of the distinguished event with the largest residual Shannon entropy.

The logic $L_{{\omega }_1,\omega }$ extends first-order logic by countable logical conjunctions and logical disjunctions. Each formula has finitely many free variables, and quantifier strings are finite. In a countable first-order language, every countable structure has a Scott sentence in this logic that characterizes its isomorphism class among countable structures.

Write a small planar displacement in arclength coordinates as $X=s+X_2$ and $Y=y$. The inextensibility relation $X_s^2+Y_s^2=1$ gives $X_{2,s}=-y_s^2/2$ to quadratic order. Thus longitudinal material velocity enters one order later than the transverse velocity. For a periodic material stroke with no mean axial drift, its mean velocity vanishes, although its instantaneous contribution is needed for exact local force balance.

For an $H$-representation $V$ and $g\in G$, the conjugate representation ${}^{g}V$ of $gH{g}^{-1}$ is defined by $(gh{g}^{-1})v=hv$. This transports the acting subgroup; it is distinct from conjugating complex matrix entries. It occurs in the Mackey restriction formula.

If $n$ points and $m$ curves have the property that each pair of distinct points lies on at most $\lambda$ curves, then double counting gives ${\sum }_{\gamma }{k}_{\gamma }(k_{\gamma }-1)\le \lambda n(n-1)$, where $k_{\gamma }=|P\cap \gamma |$. The Cauchy-Schwarz inequality implies $I^2\le mI+\lambda mn(n-1)$ and hence $I\le m+\sqrt{\lambda mn(n-1)}$. The statement needs only this combinatorial multiplicity condition, not algebraicity.

An importance sampling proposal $q$ must be positive wherever the target-weighted integrand can contribute. The usual sufficient condition is that the target probability is absolutely continuous with respect to $q$. Otherwise multiplying observed draws by $f/q$ cannot recover a missing part of the integral. A strict rejection envelope supplies this condition for the rejected-proposal distribution.

For $\mu =\int hf$ and $0<A=\int |h|f<\infty$, the proposal $q_{\ast }=|h|f/A$ minimizes the variance of the ordinary unbiased importance sampling estimator. The Cauchy-Schwarz inequality gives $\int h^2{f}^2/q\ge A^2$, so the minimum variance for $n$ independent samples is $(A^2-{\mu }^2)/n$. The proposal need only cover the nonzero integrand.

If $f\le Mg$, the importance weight $w=f/g$ is bounded by $M$, and ${\mathbb{E}}_g[{\varphi }^2{w}^2]\le M{\mathbb{E}}_f{\varphi }^2$. A finite target second moment therefore guarantees finite variance and the iid central limit theorem for the unnormalized importance-sampling average. Its proposal-budget variance is ${\mathbb{E}}_g[{\varphi }^2{w}^2]-({\mathbb{E}}_f\varphi {)}^2$.

For rejection indicators with conditional acceptance $f(X_i)/(Mg(X_i))$, the conditional expectation of $(M/n){\sum }_i{I}_i\varphi (X_i)$ given all proposals is the unnormalized importance estimator. The Rao-Blackwell theorem reduces variance relative to that fixed-denominator estimator. It gives no such automatic comparison with the accepted-sample mean, whose denominator is the random count.