If $r=dQ/dP$, independence gives $E_{P^{\otimes n}}{\prod }_{i}r(X_i{)}^2=(E_P{r}^2{)}^n$. Since $E_{P}r=1$, this proves the displayed identity. The Cauchy-Schwarz inequality also gives $\Vert Q^{\otimes n}-P^{\otimes n}{\Vert }_1^2\le {\chi }^2(Q^{\otimes n}\Vert P^{\otimes n})$. Thus perturbations of size ${\chi }^2(Q\Vert P)=O(1/n)$ have a uniformly bounded joint total variation distance, a useful input to a metric squared-loss two-point bound.

For positive summable coefficients, the Weierstrass M-test gives a continuous uniform sum. The partial sum over $3^k\le n$ is the unique best uniform approximation of degree at most $n$. If $3^K$ is the first omitted frequency, every tail term equals $(-1{)}^j$ at $x_j=\cos (j\pi /3^K)$ because its frequency ratio is odd. The entire tail therefore attains alternating extrema equal in magnitude to its coefficient sum at more than $n+1$ points. The Chebyshev alternation theorem proves the assertion, including the empty partial sum at $n=0$.

The local form of conservation of electric charge relates charge density to current density. Taking the divergence of the Ampère-Maxwell equation and substituting Gauss's law proves the equation, using that the divergence of a curl is zero.

For a real random variable $Y$ and a sigma-algebra $\mathcal{G}$, the conditional characteristic function is $\mathbb{E}[e^{i\theta Y}\mid \mathcal{G}]$. If it equals the nonrandom characteristic function of a probability law for every $\theta$, then $Y$ has that law and is independent of $\mathcal{G}$. To verify the independence assertion, multiply the identity by any bounded $\mathcal{G}$-measurable test variable and use the uniqueness theorem for characteristic functions on the resulting finite measures. Equality for rational $\theta$ extends by continuity, so simultaneous null sets can be arranged when needed.

For a primitive nonprincipal Dirichlet character modulo $q$, finite Fourier inversion theorem and the primitive Gauss magnitude give this bound uniformly in the interval. Bound the exponential sums by $C\Vert a/q{\Vert }^{-1}$ and sum the resulting harmonic series. The general Gauss identity follows from the prime-power identity by the Chinese remainder theorem.

A causal time-series representation expresses a time series using present and past driving white noise. Its coefficients vanish at negative lags. Square summability ensures mean-square convergence for a white-noise input; absolute summability is the stronger stable-filter convention. Causality refers to the particular driving sequence, not merely to stationary existence.

Integrating the smooth positive-time impulse kernel gives the zero-data response to the Heaviside step function. For nonzero $\omega$,At zero frequency and nonzero $k$ the limit is $[1-e^{-kt}(1+kt)]/k^2$; when both parameters vanish it is $t^2/2$. The integral representation avoids ambiguous divisions in these limiting cases.

Objects are small categories and morphisms are functors. Its products pair objects and arrows componentwise. Its exponential objects are functor categories, as shown by the currying adjunction for small categories.

Given functions $f_i:Y\to X_i$, there is a unique function $g:Y\to X_1\times X_2$ whose coordinate projection maps are $f_1,f_2$. It is the displayed pointwise pairing. Equality of ordered pairs is equality of both coordinates, which proves uniqueness and makes the Cartesian product a categorical product in the Category of sets.

Radial capillary imbibition into a half-space has hemispherical flow area $2\pi r^2$. Its radial hydraulic resistance is proportional to $1/R_s-1/R$, so its inlet volume flux approaches $2\pi k\Delta P{R}_s/\mu$. The growing front area gives late-time $R\propto t^{1/3}$, unlike the planar $t^{1/2}$ law. Both results assume capillary-dominated geometry with negligible directional gravity effects.

For a quasistatic change of capacitance at fixed terminal electric potential difference, the capacitor energy is $U=C{V}^2/2$ and plate electric charge is $Q=CV$. Electrical work supplied to the electric field by the battery is $VdQ$, so mechanical work done on the capacitor is $d{W}_{\mathrm{mech}}=dU-VdQ=-V^{2}dC/2$. Increasing the separation of a parallel-plate capacitor makes $dC<0$: mechanical work is positive even though electrostatic energy decreases, because electrical energy returns to the source. Conversely, when $dC>0$, the battery supplies $V^{2}dC$: half increases capacitor energy and half is available as mechanical work done by the electric field. Differentiating capacitor energy alone at fixed terminal electric potential difference omits this exchange with the battery.

A continuous function on Cantor space is uniformly continuous. Replacing it by one constant on each length-$n$ prefix cylinder set changes it by at most its oscillation on sets of diameter $2^{-n}$. This tends uniformly to zero. Each approximant is a finite linear combination of continuous indicator functions, proving density in the supremum norm.

Every open set in Cantor space is a countable disjoint union of prefix cylinder sets. Select all finite words whose cylinder lies in the open set but whose immediate predecessor's cylinder does not. These words are prefix-free, so their cylinders are disjoint. Every point of the open set has a shortest qualifying prefix. The whole space is represented by the empty prefix.

A binary tree of finite strings is a set of finite binary strings closed under taking prefixes. The empty tree is allowed; a nonempty such tree contains the empty string. Finite nodes may have no infinite extensions. Such presentations describe closed subsets of Cantor space through their infinite paths.

Independent functions $Q_1,\dots ,Q_n$ on a $2n$-dimensional symplectic manifold can locally form the position half of canonical coordinates exactly when $\{Q_i,Q_j\}=0$ throughout a neighbourhood. Their commuting Hamiltonian vector fields span the fibres of $Q$. Flow from a transverse section gives coordinates $(Q,t)$ with $\omega =\sum d{t}_i\wedge d{Q}_i+\beta (Q)$. Write $\beta =d\alpha$ locally and take $P_i=-t_i-{\alpha }_i(Q)$. Then $\omega =\sum d{Q}_i\wedge d{P}_i$.

For $H=\omega \cdot y+\frac{1}{2}|y{|}^2+ϵV(x)$ with a Diophantine frequency vector, solve ${\mathcal{D}}_{\omega }u=⟨V⟩-V$ and apply translated conjugacy of a Diophantine vector field to $w=ϵ\nabla u$. Set $\eta =ϵ\nabla u+a$ and use the symplectic cotangent lift with a closed momentum shift. The transformed Hamiltonian is exactly $E+\omega \cdot y^{\prime }+\frac{1}{2}{y}^{\prime T}D{\chi }^{-1}D{\chi }^{-T}{y}^{\prime }+\frac{1}{2}|\eta \circ \chi {|}^2$, where $E=ϵ⟨V⟩+\omega \cdot a$. Since $a=O({ϵ}^2)$, the last term is an angular potential of order ${ϵ}^2$ and the quadratic coefficient is identity plus $O(ϵ)$. The construction is on a smaller angular strip and momentum ball.

The trace of imaginary-time evolution is the canonical partition function of a quantum system. In a coordinate basis it is the integral of the diagonal Euclidean path integral kernel. Inserting a symmetry operator instead gives a weighted thermal trace.

For a complex function on $X_1\times X_2\times X_3$, its eighth power is the average of the conjugated product over the eight vertices of a coordinate cube. Three successive applications of the Cauchy-Schwarz inequality show that it bounds correlation with a product of three bounded functions, each omitting one coordinate. This is the analytic estimate behind the tetrahedron counting lemma.

Given a finite acyclic Boolean circuit and its input assignment, determine the output bit. Topological gate evaluation places the problem in P; it is a standard P-complete problem under logspace many-one reductions. Boolean circuit evaluation is different from Circuit satisfiability problem, which asks whether some assignment succeeds.

Put $A=f^{\prime }{f}_{\ast }^{\prime }$ and $B_0=f{f}_{\ast }$. The weak Boltzmann collision identity applied to $\log f$ gives $H^{\prime }(t)=-\int D(f)dx$ when transport boundary fluxes vanish. Monotonicity of the logarithm makes every integrand nonnegative. For positive sufficiently regular distributions and nondegenerate angular scattering, zero dissipation forces $\log f$ to be a collision invariant, hence $f$ to be a local Maxwellian. This equality statement alone does not establish convergence rates.