In the published chlorine experiment, negative ions were accelerated and mass-selected, then overlapped with a tunable laser beam. A downstream electric field deflected charged ions; neutral chemical atoms continued to a plate, whose secondary electrons were counted. Scanning the laser frequency produced an onset in this neutral-atom signal. The apparatus therefore inferred an electron affinity from a threshold, rather than measuring the heat of attachment.^[ref]

Reversing the laser direction and averaging the two thresholds corrected the first-order Doppler shift. A Fabry–Pérot interferometer and reference spectral lines calibrated the frequency. Finite laser bandwidth, unresolved quantum state splitting and shifts of the calibration lines limited the inference; the authors included these effects in fitting and the error estimate. The level diagram shows only the extracted gap, not a measured signal curve.^[ref]

For the real scalar field with metric $(+,-)$ and potential $U={\kappa }^2({\varphi }^2-1{)}^2/2$, the static phi-four kink is ${\varphi }_0(x)=\tanh (\kappa x)$ and satisfies ${\varphi }_0^{\prime 2}=2U({\varphi }_0)$. Its mass is $M=\int {\varphi }_0^{\prime 2}dx=4\kappa /3$. A Lorentz boost gives $\varphi (t,x)={\varphi }_0(\gamma (x-vt))$. The stress-energy tensor gives $P=-\int {\varphi }_t{\varphi }_{x}dx=\gamma v\int {\varphi }_0^{\prime 2}dx=\gamma Mv$. Similarly, $E=\frac{1}{2}[\gamma (1+v^2)+{\gamma }^{-1}]M=\gamma M$, proving the energy–momentum relation $E^2-P^2=M^2$.

For $E(0,z)=e^{i\varphi (z)}$ and sufficiently smooth real phase, expand free paraxial propagation as $E=E_0+ix{E}_0^{\prime \prime }/(2k)-x^2{E}_0^{\prime \prime \prime \prime }/(8k^2)+O(x^3)$. Taking the squared modulus gives the displayed intensity. Negative phase curvature initially focuses the field. With $\varphi =\cos z$, $I=1+(x/k)\cos z+(x/k{)}^2\cos 2z+O(x^3)$, with transverse length scaled so the modulation wavenumber is one. Period-averaged intensity stays one.

A zigzag instability is a long-wave transverse phase modulation of parallel rolls. In the modela small Fourier mode has growth rate $\sigma (k)=-\lambda k^2-\gamma k^4$. Thus the straight pattern is linearly stable for $\lambda >0$ and unstable to sufficiently long transverse waves for $\lambda <0$.

When $\lambda <0$, the unstable band is $0<k^2<-\lambda /\gamma$. Differentiating $\sigma$ with respect to $k^2$ gives the fastest growing scale $k^2=-\lambda /(2\gamma )$ and maximum growth ${\lambda }^2/(4\gamma )$. The fourth derivative stabilizes short waves rather than allowing arbitrarily fine structure to grow fastest.

Writing the slope as $p={\varphi }_Y$ givesa conserved Cahn--Hilliard equation for the local orientation. When $\lambda <0$ and $\alpha <0$, its homogeneous bulk potential $W(p)=\lambda p^2/2-\alpha p^4/12$ has two minima at $p=\pm \sqrt{3\lambda /\alpha }$. These are preferred local slopes; boundary conditions and the conserved mean slope still constrain the complete pattern.

Let $p={\varphi }_Y$, $r={\varphi }_{YY}$ and $w_2=C_1(\xi )r+C_2(\xi )p^2$. Two applications of the chain rule give ${\mathcal{D}}_Y^2{w}_2=C_1{\varphi }_{YYYY}+2(C_1^{\prime }+C_2)p{\varphi }_{YYY}+(C_1^{\prime }+2C_2)r^2+(C_1^{\prime \prime }+5C_2^{\prime })p^{2}r+C_2^{\prime \prime }{p}^4$. In a Fredholm solvability condition, the fourth term generally contributes to the coefficient of ${\varphi }_Y^2{\varphi }_{YY}$. It cannot be omitted merely because the fast-coordinate correction functions have been denoted by $C_j(x)$.

Insert Hadamard gates on every data qubit immediately after encoding by a bit-flip repetition code and immediately before its bit-flip detection and recovery. The intervening physical error is thereby conjugated by $H^{\otimes n}$, and $HZH=X$ turns a single Pauli Z gate error into a single bit flip for the original correction circuit. The encoded logical codewords are now $|+{⟩}^{\otimes n}$ and $|-{⟩}^{\otimes n}$. The syndrome ancillas and the final inverse encoding stay unchanged. This protects against phase flips in the new basis; it does not simultaneously correct arbitrary bit flips in the physical basis.

The phase-flip repetition code with codewords $|+++⟩,|---⟩$ corrects any one physical Pauli Z gate error. Its two adjacent-pair error syndromes cannot distinguish an error pattern from its three-bit complement. Minimum-weight recovery therefore leaves no residual error at weight zero or one, and leaves $Z_1{Z}_2{Z}_3$ at weight two or three. This residual operator exchanges the codewords, so inverse encoding gives a logical Pauli X gate. For independent physical phase errors of probability $ϵ$, the decoded quantum channel is $(1-p_L)\rho +p_{L}X\rho X$, where $p_L=3{ϵ}^2(1-ϵ)+{ϵ}^3$. For $0<ϵ<1/2$, $ϵ-p_L=ϵ(1-ϵ)(1-2ϵ)>0$.

The codewords $|+++⟩,|---⟩$ have residual logical Pauli X gate after an uncorrectable two- or three-site phase error. Precede their encoding by a logical Hadamard gate and follow decoding by its inverse. Since $HXH=Z$, the same error now acts as a logical Pauli Z gate. Explicitly the encoded coefficients are $(\alpha +\beta )/\sqrt{2}$ and $(\alpha -\beta )/\sqrt{2}$ for an input $\alpha |0⟩+\beta |1⟩$. This convention matters when translating physical phase errors into a claimed decoded phase-error channel.

A phase-contrast microscope makes small specimen-induced wave phase changes visible through wave interference. An illumination annulus is imaged onto a phase ring in the microscope objective's rear focal plane. The ring changes the relative phase and amplitude of largely undeviated illumination compared with specimen-diffracted light, producing intensity differences in the image.^[ref]

Correctly matching and centering the annulus and ring is an apparatus requirement. A transparent unstained specimen can then acquire strong image contrast, but halos and image contrast reversals can occur. Comparing ordinary illumination with the aligned phase system demonstrates that image brightness is not simply a map of optical absorption.

A system can split into macroscopic regions with distinct local order parameters, densities or compositions. In a short-range system at phase coexistence, regions of two phases with means $M_1,M_2$ and volume fractions $q,1-q$ have spatial mean $M=q{M}_1+(1-q)M_2$. Their interface contribution is subextensive, so their thermodynamic free-energy density is the corresponding convex combination of the phase densities. This can lie below the homogeneous-branch Landau free energy at the same $M$. The equilibrium constrained order-parameter free energy is consequently convexified, with a straight coexistence segment described by the Maxwell construction. A nonconvex local potential remains useful for homogeneous branches, metastability and interfaces, but must not be identified with the full equilibrium constrained density throughout a coexistence interval.

The solid-liquid segregation coefficient is the ratio of the solute concentration in newly formed solid to that in the adjacent equilibrium liquid. A coefficient of zero means complete solute rejection: newly formed crystals are pure solvent. It does not mean the bulk solute in a mushy layer vanishes, because its pore liquid can remain concentrated.

A planar freezing front can push a spherical particle steadily when repulsive sphere-plane dispersion force balances lubrication resistance. The thin parabolic gap gives hydraulic drag $6\pi \mu V{a}^2/d$, and balancing it against $Aa/(6d^2)$ gives the displayed speed. This balance assumes a nonretarded interaction, small gap and negligible inertial and other particle forces.

For a compact group $G$, closed subgroup $K$ and finite-dimensional unitary $K$-representation $W$, induced sections are functions satisfying $f(gk)=k^{-1}f(g)$. A $K$-intertwiner $\varphi :E\to W$ produces the $G$-intertwiner $v\mapsto (g\mapsto \varphi (g^{-1}v))$. Evaluation at the identity reverses this construction on each finite-dimensional continuous representation summand. Thus the multiplicity of an irreducible $E$ in the induced section space is $\dim {\mathrm{Hom}}_K(E{|}_K,W)$.

A continuous symmetric approximate identity on a compact group gives a compact self-adjoint convolution operator $T$ on $L^2(G)$ which maps into $C(G)$ and commutes with left translations. Its nonzero eigenspaces are finite-dimensional invariant spaces of continuous functions, hence consist of matrix coefficients of finite-dimensional unitary representations. If $S_N$ projects onto increasing finite sums of these eigenspaces, $S_{N}Tf\to Tf$ in $L^2$, and the bound $\Vert Th{\Vert }_{\infty }\le \Vert k{\Vert }_2\Vert h{\Vert }_2$ gives $T{S}_{N}Tf\to T^{2}f$ uniformly. Choosing the approximate identity narrow makes $T^{2}f$ uniformly close to $f$. This supplies uniform approximation without confusing an $L^2$ limit with a uniform limit.

Suppose finite-dimensional continuous group representations separate points of a compact Lie group. Choose an identity neighborhood $U$ containing no nontrivial subgroup, using Lie groups have no small subgroups. For each $g\notin U$, choose a representation whose matrix at $g$ is not the identity. The corresponding open nonidentity sets cover the compact complement of $U$, so finitely many representations suffice there. Their direct sum has kernel inside $U$ and therefore trivial kernel. Unitarization of a compact-group representation makes it a finite-dimensional faithful unitary representation.

The lower middle perversity is $\bar{m}(k)=\lfloor (k-2)/2\rfloor$. The complementary upper middle perversity is $\bar{n}(k)=\lceil (k-2)/2\rceil$, and $\bar{m}(k)+\bar{n}(k)=k-2$. They agree at even codimensions.

The upper middle perversity is $\bar{n}(k)=\lceil (k-2)/2\rceil$. It differs from the lower middle perversity by one at odd codimensions, and is complementary to it.

An $i$-dimensional singular simplex $\sigma :{\Delta }^i\to X$ is $\bar{p}$-allowable if ${\sigma }^{-1}(X_{N-k})$ is contained in the $(i-k+\bar{p}(k))$-skeleton of ${\Delta }^i$ for every $k\ge 2$. A negative skeleton is empty. Individual faces need not be allowable; this is why the definition of intersection chains separately imposes allowability on the total boundary.

If the Dirichlet series $F(s)={\sum }_n{a}_n{n}^{-s}$ is absolutely convergent at $\sigma >0$, thenThe scalar kernel is $(2\pi {)}^{-1}{\int }_{\mathbb{R}}{e}^{v(\sigma +it)}(\sigma +it{)}^{-2}dt=v_{+}$. This follows by Fourier inversion applied to $u{e}^{-\sigma u}{1}_{u\ge 0}$, whose Fourier transform is $(\sigma +it{)}^{-2}$. Absolute convergence of the series and of $\int |\sigma +it{|}^{-2}dt$ justifies interchange. The factor $ds=idt$ is essential.

A nonconstant meromorphic function on the complex plane is a simply periodic function when its period group of a function is generated by one nonzero complex number: $\mathrm{Per}(f)=\omega \mathbb{Z}$. It repeats in one independent complex direction.

For example, $f(z)=e^z$ satisfies $e^{z+\omega }=e^z$ exactly when $e^{\omega }=1$, so its periods are $2\pi i\mathbb{Z}$. A doubly periodic elliptic function instead has a period lattice generated by two real-linearly independent complex numbers. Single periodicity alone imposes no algebraic addition law.