A locally integrable function $f$ defines a distribution by $⟨u_f,\phi ⟩=\int f\phi$. The pairing is bilinear if distributions are taken complex-linear. This identifies ordinary functions with a subset of distributions without claiming that every distribution is a function.

For distributions $u$ on ${\mathbb{R}}^m$ and $v$ on ${\mathbb{R}}^n$, their tensor product acts on a joint test function by $⟨u\otimes v,\Phi ⟩=⟨u_x,⟨v_y,\Phi (x,y)⟩⟩$. The inner function is smooth with compact support in the projection of $\mathrm{supp}\Phi$, and finite order of a distribution estimates prove continuity. Reversing the pairings gives the same value: finite sums of product kernels are dense in the required smooth seminorms, and the two orders agree on product kernels. Its support is contained in the Cartesian product of the two supports. This provides a rigorous replacement for formal integration in separate distribution variables.

For every positive distance, draw equal-radius circles centred at the points of $P$. Their incidences between points and curves count the ordered pairs at that distance. The Szemerédi–Trotter theorem for unit circles bounds each distance class by $O(|P{|}^{4/3})$. Summing over all classes accounts for $|P|(|P|-1)$ pairs, proving $|\Delta (P)|\gtrsim |P{|}^{2/3}$.

Suppose $n$ independent axisymmetric pure plumes share total buoyancy flux $B$ equally, have negligible source volume flux, and are supplied with a separate ambient ventilation flux $F$ at floor level. Their combined flux is $P(h)=\lambda n^{2/3}{B}^{1/3}{h}^{5/3}$. Volume conservation at a steady displacement-ventilation interface height and buoyancy flux conservation giveA positive two-layer state needs $0<h_{\ast }<H$, sources small relative to $h_{\ast }$, and plumes sufficiently separated to remain independent. Splitting the sources increases total fluid entrainment and reduces the interface height by $n^{-2/5}$. A finite source flux cannot simultaneously be neglected and supply all the replacement air; no steady displacement layer without ambient supply explains this restriction.

Consider entraining plumes supplied with total source volume flux $S$, a separate lower-layer ambient inflow $L$, and ceiling exhaust $F=S+L$. Write $P(h)=S+E(h)$, where $E(h)$ is total fluid entrainment from the lower layer. Volume conservation gives $A\dot{h}=L-E(h)$. A steady positive lower layer therefore requires $L=E(h)>0$. If $L=0$ and $E(h)>0$ for every $h>0$, no positive steady displacement-ventilation interface height exists. In a shifted pure plume approximation $P_i(h)=\lambda B_i^{1/3}(h+z_i{)}^{5/3}$ matched to source flux $S_i$ at $h=0$, this follows immediately from $P_i(h)>S_i$ for $h>0$. This is a statement about the ideal two-layer model, not a prohibition of all possible stratification in a real ventilated room.

For total plume flux $P(h)$, fixed ceiling exhaust $F$, zero ambient buoyancy, and uniform upper-layer reduced gravity $b$, the ideal two-layer balances areAt $P(h_{\ast })=F$ and $b_{\ast }=B/F$, linearization yields the decay times ${\tau }_h=A/P^{\prime }(h_{\ast })$ and ${\tau }_b=A(H-h_{\ast })/F$. For $P(h)\propto h^{5/3}$, ${\tau }_h=3A{h}_{\ast }/(5F)$. Upper-layer buoyancy relaxation usually dominates when $h_{\ast }\ll H$. Exact equilibrium is approached asymptotically; a fixed small relative tolerance takes order $\max ({\tau }_h,{\tau }_b)\log (1/\epsilon )$ after the initial transient.

A finite-alphabet erasure channel transmits the input symbol unchanged with probability $1-f$ and replaces it by a distinct erasure flag with probability $f$, independently of the symbol. Given the flag, the posterior remains the input distribution; otherwise the input is known. Thus its mutual information is $(1-f)H(X)$.

For a record of $N$ samples separated by time $\Delta t$, the $j$th Fourier frequency has angular frequencyThe total record duration is $T=N\Delta t$; writing $2\pi j/T$ uses that duration rather than the sample count. Frequencies differing by $2\pi /\Delta t$ give the same discrete Fourier mode. For a real record, the distinct positive interior modes have $1\le j<N/2$; the zero mode and, for even $N$, the Nyquist frequency are separate real modes.

Their sine and cosine sample vectors are mutually orthogonal. If the samples are independent centered Gaussian random variables with a common variance, projections on these orthogonal vectors are independent Gaussian random variables. Negative-frequency coefficients are instead fixed by discrete Fourier conjugate symmetry; they are not additional independent measurements.

For fixed $A,C>0$,uniformly for $q\le {\log }^{C}x$ and $(a,q)=1$. The implied constant may be ineffective. This small-modulus input complements the large sieve in the proof of the Bombieri–Vinogradov theorem.

A Dirichlet polynomial is a finite Dirichlet series, $A(s)={\sum }_{n\le X}{a}_n{n}^{-s}$. Vertical integration separates equal product indices from oscillatory unequal ones.

For a full-rank Euclidean lattice in ${\mathbb{R}}^n$, its Epstein zeta function is $E_{\Lambda }(s)={\sum }_{0\ne x\in \Lambda }|x{|}^{-2s}$, initially convergent for $\mathrm{Re}s>n/2$. A Mellin transform of its theta function gives its completed Epstein zeta function and a meromorphic continuation. Its sole pole has residue ${\pi }^{n/2}/[m(\Lambda )\Gamma (n/2)]$ at $s=n/2$.

For the positive Laplacian convention $\Delta G=\delta$, the method of images in the first quadrant reflects an interior pole $(a,b)$ across each axis with negative sign and across both with positive sign:The Dirichlet Green function vanishes on both axes. Its outward normal derivative on the horizontal axis isand the vertical-axis kernel follows by exchange of coordinates. Green second identity converts these derivatives into the quadrant Poisson integral for boundary data.

The density of expected Brownian occupation time before exit from a planar domain. For planar Brownian motion with infinitesimal generator $\frac{1}{2}\Delta$, it obeys $-\frac{1}{2}{\Delta }_w{G}_D(z,w)={\delta }_z(w)$ and has singularity $-{\pi }^{-1}\log |w-z|$. A conformal bijection preserves this Dirichlet Green function: the squared derivative in the conformal Brownian clock cancels the area Jacobian determinant.

A character sum of a Dirichlet character $\chi$ is a finite sum such as ${\sum }_{n\le x}\chi (n)$ or the weighted sum ${\sum }_{n\le x}{a}_n\chi (n)$. Orthogonality of Dirichlet characters converts residue class restrictions to these sums.

A rooted directed graph has a distinguished root. Root-connectedness means each vertex can be reached by a finite directed path from that root. This is directed reachability, rather than connectedness of the underlying undirected graph.

The restricted direct sum of a family of groups $G_i$ is the subgroup of their direct product of groups consisting of tuples with only finitely many nonidentity coordinates. Multiplication is coordinatewise. If every factor is a finite group, each subgroup that is a finitely generated group lies in a finite subproduct, so the restricted direct sum is a locally finite group. This gives concrete infinite torsion groups without a uniform bound on element orders.

For real $f\in L^2({\mathbb{R}}^3)$, this energy functional on $H^1({\mathbb{R}}^3)$ is coercive and attains its minimum by the direct method in the calculus of variations. The nonnegative potential term is weakly lower semicontinuous by local Sobolev compactness gives lower semicontinuity of a nonnegative integral. Its Euler-Lagrange equation is $-\Delta u+u+\sin u=f$ in the weak solution sense. The extra linear restoring term screens the Sine-Gordon equation nonlinearity. The scalar potential $V(s)=s^2/2+1-\cos s$ is convex because $V^{\prime \prime }(s)=1+\cos s\ge 0$; hence any weak critical point is a minimizer. The elliptic regularity estimate for $-\Delta +1$, together with $|\sin u|\le |u|$, gives $u\in H^2$. The Sobolev inequality then puts $u$ in $W^{1,6}$, so Morrey's inequality and uniformly continuous integrable functions vanish at infinity give a continuous representative tending to zero.

For a differentiable objective on an admissible real Hilbert space $H$, with $\Vert {\mathbf{u}}_0{\Vert }^2/2=E_0>0$, stationarity on the sphere in a normed vector space of fixed kinetic energy means that the projected gradient is normal to the sphere in a normed vector space. Thus $P_H\nabla J=\lambda {\mathbf{u}}_0$ for a real Lagrange multiplier. It follows by testing all tangent variations orthogonal to ${\mathbf{u}}_0$. The multiplier need not be positive; second-order conditions are still needed for a local minimum.

A Dirac field theory invariant under $\psi \mapsto e^{i\alpha }\psi$, $\bar{\psi }\mapsto e^{-i\alpha }\bar{\psi }$ has a conserved global charge, subject to the quantum theory preserving the symmetry. In particle language this charge counts particles minus antiparticles. When the S-matrix commutes with it, a scattering amplitude between states of different charge vanishes. A neutral scalar and a Yukawa interaction preserve the charge, so a process with one incoming Dirac particle and only one outgoing Dirac antiparticle plus neutral particles is forbidden. This net charge is distinct from a sum of positive occupation numbers for all particle and antiparticle modes.

For a smooth hypersurface $S$, its surface delta is the distribution $⟨{\delta }_S,\phi ⟩={\int }_S\phi dS$. Its normalization uses geometric surface measure rather than a defining function. On the unit circle, ${\delta }_{S^1}$ pairs by integrating $\phi (\cos t,\sin t)$ over $0\le t\le 2\pi$.