For $h\in C_c(G)$, apply Fourier inversion on a locally compact abelian group at the identity to $h\ast h^{\ast }$, whose transform is $|\hat{h}{|}^2$. This proves the displayed norm identity on $C_c(G)$. Density and completeness give a unique linear isometric extension to all $L^2(G)$. On $L^1\cap L^2$ it agrees almost everywhere with the integral Fourier transform on a locally compact abelian group.

With the Fourier convention $\hat{f}(t)=\int f(x)e^{-itx}dx$, translation by $k$ multiplies the transform by $e^{ikt}$. The Plancherel theorem gives the displayed correlation identity for every square-integrable function, by extension from integrable square-integrable functions. Thus distinct integer translates are orthogonal exactly when the nonzero integer Fourier integrals of $|\hat{f}{|}^2$ vanish. The products in both integrals are integrable by the Cauchy-Schwarz inequality.

Every finite planar graph admits a proper vertex coloring with at most five colors. Euler formula for a connected planar graph supplies a vertex of degree at most five. Inductively color its deletion. The only obstruction is five differently colored neighbours in cyclic order. If opposite neighbours 1 and 3 are in different two-color components, interchange one component. If they are connected, planarity prevents opposite neighbours 2 and 4 from being connected in their two-color subgraph, and that interchange frees a color instead. This is a Kempe chain proof.

Every finite planar graph has chromatic number at most four. This stronger theorem is distinct from the elementary five colour theorem; the latter's Kempe chain proof does not prove this assertion.

The honeycomb lattice is the periodic degree-three planar graph consisting of the vertices and edges of a regular hexagonal tiling. It has two vertex classes; every edge joins opposite classes. Its planar dual graph is the triangular lattice.

A plane graph is a planar graph with a specified graph drawing whose edges meet only at common endpoints, also called an embedding in the Euclidean plane. Its faces of a plane graph and facial boundary walks refer to that embedding. This geometric use of embedding differs from a graph embedding, which maps one abstract graph into another.

A finite simple graph is a planar graph if and only if it contains no graph subdivision of the complete graph $K_5$ or the complete bipartite graph $K_{3,3}$ as a subgraph; equivalently, neither is a topological minor. The easy direction follows by tracing each replacement graph path as one drawn edge. The converse asserts that every failure of planarity contains one of these two patterns.^[ref]

Graph Theory by Reinhard Diestel (Planar Graphs preview, §4.4, Theorem 4.4.6, printed p. 111) dates Kuratowski's theorem to 1930 and the equivalent graph minor characterization to 1937. These dates are that textbook's historical attribution.

In the displayed pitchfork bifurcation normal form, zero is stable for $\mu <0$ and unstable for $\mu >0$. At $\mu =0$, a symmetric stable pair $x=\pm \sqrt{\mu }$ appears on the side where zero is unstable. Reversing the parameter orientation changes the side, while retaining the supercritical relationship between the new stable branches and the destabilized central branch.

The polar form of the cubic confinement model has an origin pitchfork on ${\mu }^2+{\sigma }^2=1$ when $\mu \ne 0$. Its small squared amplitude satisfies $S\sim 2\mu \delta \mu /(\mu +\sigma )$ along fixed $\sigma$. Thus the leading cubic coefficient vanishes when $\mu +\sigma =0$, at $(\sigma ,\mu )=(\pm 1/\sqrt{2},\mp 1/\sqrt{2})$. At either point the exact radius equation gives $\delta \mu =-S^2/(4{\mu }_0)+o(S^2)$, so amplitude scales as $|\delta \mu {|}^{1/4}$. The lower point has a stabilizing quintic coefficient $1/(4{\mu }_0)<0$; folds of nonzero equilibrium pairs meet the pitchfork curve there. The upper point has the opposite quintic sign and an already unstable transverse eigenvalue.

For $k>1$, the rational equilibrium point curve in the display has expansion $r=1+q+[1-q(k-1)]u+qk(k-1)u^2+\cdots$. Its cubic pitchfork bifurcation coefficient vanishes at $q=1/(k-1)$, leaving $r-(1+q)=k{u}^2+\cdots$. For larger $q$, the nonzero branch folds at $u=[\sqrt{q(k-1)}-1]/k$ and $r=[k-1+q+2\sqrt{q(k-1)}]/k$. If the other eigenvalues remain stable, the two folds bound a bistable wedge between stable zero and stable nonzero states, separated by an unstable branch. If another eigenvalue is critical, a higher-dimensional centre manifold is required instead.

The Hagen-Poiseuille equation describes steady, fully developed, laminar flow of an incompressible Newtonian fluid through a straight rigid circular pipe. With pipe radius $R$, axial coordinate $z$ and constant viscosity $\mu$, force balance reduces to $\mu r^{-1}\frac{d}{dr}(r\frac{dw}{dr})=dp/dz$.

Steady pressure-driven flow through a circular pipe has the parabolic profile

Regularity at the centre and the no-slip boundary condition at $r=R$ fix the displayed profile. Integrating over the cross-section gives $Q=-\pi R^4(dp/dz)/(8\mu )=\pi R^4\Delta p/(8\mu L)$ for length $L$ and positive pressure drop $\Delta p$. The fourth power makes volume flux strongly sensitive to radius; the assumptions exclude arbitrary turbulent or developing pipe flow. ^[ref]

For specified exposure periods with constant death hazard functions $h_k$, the integrated hazard through the endpoint is ${\sum }_k{h}_k\Delta t_k$. The displayed event probability follows from the survivor function identity. For different latent exposure schedules mixed with probabilities $w_j$, average the schedule-specific probabilities, ${\sum }_j{w}_j(1-e^{-H_j})$, rather than exponentiating the average integrated hazard. When the hazards are small, $1-e^{-H_j}\simeq H_j$ gives a person-time approximation. A comparison must specify the entire custody or treatment schedule; later changes to exposure cannot be inferred from an endpoint rate alone.

For isotropic rotational diffusion, orientation drift and probability density obey the displayed Fokker-Planck equation, normalized with respect to solid angle. Without flow, constant light $L{\mathbf{e}}_x$ gives the zero-current density $f_0=\lambda e^{\lambda p_x}/(4\pi \sinh \lambda )$, $\lambda =L/(\alpha D_r)$. Integration over the sphere gives unit mass; its mean projection is the Langevin function. The zero-light limit is the uniform density $1/(4\pi )$.

In a long channel $-a<x<a$, uniform horizontal mean swimming $V_s{K}_0{\mathbf{e}}_x$ and isotropic translational diffusion coefficient $D$ give zero transverse cell flux when $n^{\prime }=\beta n$, $\beta =V_s{K}_0/D$. Normalizing the transverse mean concentration to $n_0$ gives the displayed exponential. For $\beta a\gg 1$, cells form a layer of thickness ${\beta }^{-1}$ at the illuminated wall. This is a bulk approximation; exact closed-end cell conservation equation also requires the integrated axial cell flux to vanish.

A colour index is the difference of an object's measured apparent magnitudes in two photometric passbands, such as $B-V$. Since magnitude is logarithmic, this difference measures a flux ratio after the respective magnitude zero points are accounted for. It describes spectral shape rather than the object's overall brightness, and changes with both intrinsic radiation properties and interstellar extinction.

NIST Measurement Services: Photometric Calibrations by Yuqin Zong and others describes a concrete experiment (§§3.2–3.3.2): a silicon photodiode, multilayer photometric matching filter, precision aperture, temperature sensor and current-to-voltage converter. A monochromator supplies a small beam; scanning wavelength yields spectral responsivity, and scanning position checks response across the aperture. Finite spectral bandwidth requires correction. The average aperture response converts uniform illumination to illuminance.

On the photometric bench (§§3.4.1, 3.5.3, 3.5.6), align lamp and aperture reference planes, set the lamp's electric current, stabilize, and compare a check lamp with standard photometers. The readout is a voltmeter signal. Subtract the shutter-closed reading (§3.4.4), divide by converter gain to infer electric current, and use calibrated illuminance responsivity $S_A$ to obtain $E_v=i/S_A$. For a distant source, luminous intensity follows from $I_v=E_v{d}^2$, with $d$ between reference planes.

Controls include repeat lamp operations, distance changes to test the inverse-square law, temperature corrections and gain linearity checks. Spectral mismatch correction is source dependent (§3.4.3). Unknown optical spectra, spatial nonuniformity and incorrect reference planes remain limitations.^[ref]

A photometric matching filter shapes a photodiode's combined spectral responsivity toward the photopic luminous efficiency function. Matching is imperfect, so photometric calibration must retain spectral information. ^[ref]

For a photodiode in its linear regime, spectral responsivity is the dark-subtracted electric current per incident radiant flux in a narrow band at wavelength $\lambda$, measured in $\mathrm{A}{\mathrm{W}}^{-1}$. More generally a photodetector's readout can be electrical voltage or another quantity, with corresponding units. A broad-band readout weights the incoming optical spectrum by the detector's spectral responsivity; it is not the unweighted total incident power. ^[ref].

For a two-variable Boolean quantum oracle, quantum phase kickback produces the phase $(-1{)}^{axy+bx+cy+d}$ on $|xy⟩$. If the quadratic coefficient $a$ is known, apply the Controlled-Z gate to the power $a$: its phase $(-1{)}^{axy}$ cancels the quadratic term. A Walsh-Hadamard transform then maps the remaining phase state to $(-1{)}^d|bc⟩$. The constant term is only a global phase, and both linear coefficients are read in one quantum measurement in the computational basis.

For a known-phase contribution $L_1(\theta )\propto {\theta }^{r_1}(1-\theta {)}^{m_1-r_1}$ and an independent phase-averaged contribution $L_2(\theta )$, an interior maximum satisfiesThis follows by differentiating $\log L_1+\log L_2$ and multiplying by $\theta (1-\theta )L_2$. The constrained maximum-likelihood estimate on $[0,1/2]$ must also be compared with the endpoints. An interior root cannot be asserted without the observed pedigree counts; complete nonrecombinant data, for example, have their maximum at zero.

// Target: biology.bigb