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The image of the diagonal map in $X\times X$. For a manifold, the diagonal subset is an embedded submanifold diffeomorphic to $X$.

If the input qubit of the Deutsch algorithm acquires conditional environment states with overlap $⟨e_0|e_1⟩=v{e}^{i\alpha }$ between its two Hadamard gates, its fixed zero-for-constant, one-for-balanced rule succeeds with probability $(1+v\cos \alpha )/2$. A known phase can be compensated before the last Hadamard gate, raising the success probability to $(1+v)/2$. An orthogonal pair of environment states gives $v=0$ and destroys all information in the input qubit about the answer; unit overlap modulus preserves coherence up to a known phase.

For smooth fields into ${\mathbb{R}}^l$ with nonnegative differentiable potential and finite energy, Derrick scaling in two spatial dimensions forces the potential integral to vanish. Thus the field takes values where $U=0$ and $\nabla U=0$, so its static field equation makes every component harmonic. Each first derivative is an entire harmonic function in L2 space; the mean-value inequality on arbitrarily large balls forces it to vanish. Hence the field is constant. The scale identity alone does not exclude a pure-gradient model; this additional argument uses the flat unconstrained target. Curved-target harmonic maps, for example the O3 nonlinear sigma model, can instead have finite-energy nonconstant solutions.

The category $D^{+}(X)$ is obtained from bounded below cochain complexes of sheaves by passing to cochain-homotopy classes of maps and then inverting quasi-isomorphisms. It can equally be modeled by injective resolutions. The cohomology-sheaf functors descend to this localization, and an object is zero exactly when all its cohomology sheaves vanish.

A bounded complex is constructible for a given stratification when each of its cohomology sheaves restricts to a finite-rank local system on every stratum. Constructibility concerns cohomology sheaves, not an arbitrary choice of resolution.

On bounded constructible complexes over locally compact stratified spaces, Verdier duality is $D_{X}E=R\mathcal{H}om(E,{\omega }_X^{\bullet })$, where ${\omega }_X^{\bullet }$ is the dualizing complex. For an oriented real $N$-manifold, $D_X{\mathbb{Q}}_X\cong {\mathbb{Q}}_X[N]$. Proper direct image commutes with this duality, and $i_x^{!}{D}_{X}E\cong D_{\{x\}}{i}_x^{\ast }E$. These properties exchange stalk support conditions with costalk conditions.

A ghost is a morphism inducing zero on every cohomology sheaf, but it need not itself be zero. For the equator $j:S^1\hookrightarrow S^2$ and complement $i:U\hookrightarrow S^2$, the connecting morphism $j_{\ast }{\mathbb{Q}}_{S^1}\to i_{!}{\mathbb{Q}}_U[1]$ is a ghost. Its source and target have cohomology sheaves in different degrees, but its induced map $H^1(S^1)\to H_c^2(U)$ is nonzero, proving that the derived morphism is nonzero.

For a perfect fluid with $P=w\rho$, in Newtonian gauge in cosmology the density perturbation on fluid-comoving slices is the displayed combination of the fractional density perturbation and velocity potential. The Einstein energy and momentum constraints combine into ${\nabla }^2\varphi =4\pi G{a}^2\bar{\rho }\Delta$ when scalar anisotropic stress vanishes. This gives a relativistic Poisson constraint valid beyond the subhorizon limit.

Revealed preference infers rankings from choices: a chosen bundle is directly revealed at least as desirable as any bundle affordable in the same budget. Indirect comparisons use chains of such observations.

A demagnetizing factor relates a component of demagnetizing field to magnetization. The three principal factors of a uniformly magnetized ellipsoid sum to one in the International System of Units. Factors quoted in a different unit convention need conversion; state the convention and measurement geometry before applying them.

Couple the nucleon isospin $1/2$ to the pion isospin $1$. The highest state $|3/2,3/2⟩=|p{\pi }^{+}⟩$ is normalized. One application of the total isospin lowering operator gives $|3/2,1/2⟩=\sqrt{2/3}|p{\pi }^0⟩+\sqrt{1/3}|n{\pi }^{+}⟩$. For a common isospin-invariant two-body decay amplitude, the partial decay rates relative to the ${\Delta }^{++}$ channel are $2/3$ and $1/3$. The lowest state gives ${\Delta }^{-}\to n{\pi }^{-}$. Exact relations assume isospin-degenerate masses and common kinematic factors; electromagnetic and quark-mass differences cause corrections.

A delayed feedback characteristic root solves the characteristic equation of a delay differential equation and supplies an exponential function mode. Its real part determines decay or growth of that mode.

The degree sequence of a labelled graph records the degree of a vertex at each label. The handshaking lemma gives ${\sum }_i{d}_i=2|E|$. A nonincreasing list forgets the assignment of degrees to labels unless a particular ordering is specified.

The reference configuration is a chosen arrangement used to label a body's material particles and measure their subsequent deformation. A particle at $\mathbf{X}$ in the chosen reference has position $\mathbf{x}=\mathbf{\chi }(\mathbf{X},t)$ in the current configuration. The reference coordinates are Lagrangian coordinates and remain attached to the particles.

The initial shape of a specimen is a convenient reference, but a reference configuration need not be stress-free. A preloaded or residually stressed specimen can also be chosen as the reference. For example, engineering strain compares length with the chosen original gauge length, while the deformation gradient compares infinitesimal material line elements. Current and reference mass densities satisfy $\rho J={\rho }_0$ for conserved mass, with $J$ the local volume ratio. Keeping this choice explicit prevents mixing stress or strain measures based on different areas and lengths.

The current configuration is the spatial arrangement of a material body at the time being studied. A deformation map $\mathbf{x}=\mathbf{\chi }(\mathbf{X},t)$ gives the current position $\mathbf{x}$ of the particle labelled by reference position $\mathbf{X}$. Fields expressed as functions of $\mathbf{x}$ use Eulerian coordinates.

A marked rubber strip illustrates the distinction: its original marks identify material particles, while their observed positions after stretching form the current configuration. A current length or area can differ from the corresponding quantity in the reference configuration. The Cauchy stress tensor measures force per current area, and mass density per current volume changes under compression even though each marked material region retains its mass. Specifying the configuration is therefore part of specifying a measurement, rather than a change in the underlying physical body.

Let $E$ be a finite normal closure of the number field $K$. The permutation representation of its Galois group on the embeddings of $K$ into $E$ has one trivial constituent, because the action is transitive. At a prime, the inertia group invariant basis consists of sums over inertia orbits of embeddings; Frobenius automorphism permutes them in cycles of lengths equal to the residue degrees of the primes of $K$. A cycle of length $f$ contributes $\det (1-T\rho (\mathrm{Frob}))=1-T^f$. Hence the local factors equal those of the Dedekind zeta function, including at ramified primes. If nontrivial irreducible Artin L-functions are holomorphic and nonzero at one, this factorization and the simple pole of the Riemann zeta function show that ${\zeta }_K$ has a simple pole there.

The invariant decay amplitude is the transition matrix element for one incoming particle and a specified on-shell final state, with the overall momentum delta function factored out. For covariantly normalized states, write the connected S-matrix element as $i(2\pi {)}^4{\delta }^4(p_i-p_f)\mathcal{M}(i\to f)$. Its overall phase does not affect the decay width. The rate is $\Gamma =(2M_i{)}^{-1}{\sum }_{\mathrm{spins}}\int d{\Phi }_f|\mathcal{M}{|}^2$, with the initial spin average included when the initial state is unpolarized. For a spin-zero parent and an angle-independent two-body amplitude, the two-body Lorentz-invariant phase space gives $\Gamma =|\mathbf{k}|\sum |\mathcal{M}{|}^2/(8\pi M_i^2)$. In leptonic pion decay, contracting the axial hadronic matrix element with the chiral weak charged current makes this amplitude proportional to the charged-lepton mass, proving helicity suppression.

A branching fraction is the probability that a decay occurs through a particular mutually exclusive channel, equal to its partial decay width divided by the total decay width. Inclusive channel fractions sum to one. Rounding displayed fractions can make their numerical sum differ from one.

A streamfunction for an incompressible zonal-mean flow in the meridional-vertical plane. Its derivatives give meridional and vertical velocity, and differences in its values measure volume transport per zonal span. Constant values on closed impermeable boundaries encode zero net normal flow.

The effective meridional circulation associated with eddy buoyancy transport. In a stratified Southern Ocean, mesoscale eddies tend to flatten the isopycnals tilted by the wind-driven mean circulation, so their contribution can largely oppose the mean Deacon circulation. The residual combines mean and eddy contributions with the diabatic water-mass budget.