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Suppose a unitary matrix $U$ is symmetric under ordinary transpose, and write its entrywise real and imaginary parts as $A$ and $B$. They are real symmetric matrices. Expanding $U^{†}U=I$ gives $A^2+B^2+i(AB-BA)=I$, proving the displayed identities. Commuting real symmetric matrices admit simultaneous diagonalization: diagonalize $A$, then diagonalize the restriction of $B$ to each invariant eigenspace of $A$. A common real orthonormal eigenbasis givesIf $A$ has simple eigenvalues, every eigenvector of $A$ is already an eigenvector of $B$, because its one-dimensional eigenspace is invariant. The eigenvalues of $A$ and $B$ are paired as the real and imaginary parts of the same eigenvalue of $U$.

For orthonormal orbitals $a,b$, the symmetric spatial state $C=(|ab⟩+|ba⟩)/\sqrt{2}$ multiplied by the antisymmetric spin singlet has one electron on each center and no double occupancy. The full state remains antisymmetric under fermion exchange.

A configuration with both electrons on one center, such as $aa$ or $bb$, in a two-center electronic basis. A coherent superposition of such configurations in a neutral molecule represents charge fluctuations; it does not imply a permanent classical ionic charge assignment.

For real hopping $t$ and repulsion $U>0$, the covalent singlet and symmetric ionic state have relative Hamiltonian $\left(\begin{array}{cc}0& 2t\\ 2t& U\end{array}\right)$. The lower energy is $(U-\sqrt{U^2+16t^2})/2$, relative to the common one-electron energy. It approaches $-4t^2/U$ at small hopping, and its state continuously mixes covalent and ionic configurations.

At one electron per site and $U\gg |t|$, the low-energy singlet of a repulsive two-site Hubbard dimer mixes virtually with doubly occupied configurations. Its energy is $-4t^2/U$ to leading order, while the three triplets have no such shift. With dimensionless spin-half operators this yields the displayed effective antiferromagnetic Hamiltonian. Restoring angular-momentum operators with units of $\hslash$ requires the corresponding $1/{\hslash }^2$ in the spin product.

For independent exponential escape, equal prior probabilities and no state relaxation, the mean classification error is $P_e(\tau )=[e^{-{\Gamma }_f\tau }+1-e^{-{\Gamma }_s\tau }]/2$. Setting its derivative to zero gives the displayed interval. The error increases both when the fast state is not given enough time to escape and when the slow state is given too much. Detector bandwidth, relaxation and reloading require an expanded measurement model.

A larvacean tunicate with dispersed Hox genes whose expression still shows aspects of anteroposterior ordering. The derived dispersed state is a counterexample to obligatory linkage for all spatial Hox regulation.

The trivial representation of a molecular point group: every symmetry operation is represented by $1$. Labels such as $A_1$ or $A_{1g}$ depend on the group and its labelling convention.

Fine-grained occupation is conserved along trajectories of a collisionless distribution. Coarse graining cannot increase its maximum. For fermions the primordial occupation is bounded, so a measured halo density and velocity scale restrict the particle mass required to produce that halo.

For normalized oscillator states and $x=\sqrt{\hslash /(2m\omega )}(a+a^{†})$, define $\eta =k\sqrt{\hslash /(2m\omega )}$. Normal ordering the displacement operator gives the displayed matrix element. A sudden momentum kick therefore creates a Poisson distribution of oscillator occupations with mean ${\eta }^2$. Its mean added motional energy is ${\eta }^2\hslash \omega ={\hslash }^2{k}^2/(2m)$, although most events retain $n=0$ when $\eta \ll 1$.

Below the lowest waveguide cutoff frequency, the axial wave number of the ${\mathrm{TE}}_{10}$ mode is $i\kappa$, with $\kappa >0$. For a real peak scale $E_0$ and time convention $e^{-i\omega t}$, $E_y=E_0\sin (\pi x/a)e^{-\kappa z}$, $B_x=-i\kappa E_y/\omega$ and $B_z=-i{E}_0(\pi /a)\cos (\pi x/a)e^{-\kappa z}/\omega$. The electric field and magnetic field are in quadrature. A single decaying mode in a lossless semi-infinite guide consequently has zero time-averaged Poynting vector, although its instantaneous field stores energy.

Bypass of a potential initiation site by a scanning small-subunit complex, allowing initiation farther downstream. Local context and factor availability affect the probability of selection.

Initiation after a preceding biological translation event on the same RNA, often involving retention of a small subunit and reacquisition of required factors. Upstream reading-frame effects depend on these kinetics.

A biological translation inhibitor used to enrich ribosomes near initiation sites while much ongoing elongation runs off. Exposure time and trapping kinetics affect inferred initiation positions.

A biological translation inhibitor with preferential effects on early ribosomes, used in initiation-site profiling. Peak size need not equal initiation flux.

A transition state theory expression relating a rate constant to the standard free energy of activation. For a bimolecular reaction expressed in concentration units, $k_2=(k_{\mathrm{B}}T/h)(c^{\circ }{)}^{-1}\exp [-\Delta G^{‡\circ }/(RT)]$ in the unit-transmission approximation.

The standard enthalpy difference between the activated-complex reference state and the reactants at the same temperature and pressure. Under the usual Eyring assumptions it controls the slope of $\ln (k/T)$ against $1/T$.