For electronic $J=1/2$, nuclear spin $I$ and conserved angular momentum component $m$, define ${\Delta }_{\mathrm{hf}}=A(I+1/2)$ and the dimensionless field $x$ shown in the title. The hyperfine-plus-magnetic energies areThe factors $g_I,g_J$ use the same Hamiltonian convention and Bohr-magneton units. The two stretched states require the branch consistent with their unique spin configuration. The formula follows by diagonalizing each two-dimensional fixed-$m$ block; it connects weak-field coupled-spin levels to decoupled high-field states. The name refers to Gregory Breit and Isidor Isaac Rabi.^[ref]

Reduction using hydrogen and a catalyst. Selectivity depends on the catalyst, pressure, solvent and functional groups: reducing an alkene while retaining a ketone is a distinct requirement from reducing a nitrile while retaining an ester.

A porous nickel catalyst prepared by selective removal of aluminium from a nickel–aluminium alloy. It can catalyse hydrogenations; its use does not guarantee that every other reducible functional group will remain unchanged.

Approximating a dilute polymer coil as a hydrodynamic object of size $R$ gives $D\sim k_{B}T/(\eta R)$. If a stipulated model uses the theta-chain end-to-end distance $R=2a\sqrt{N}$ as a literal solid-sphere radius, Stokes law gives $D=k_{B}T/(12\pi \eta a\sqrt{N})$. The scaling matches theta-solvent hydrodynamic coupling, but a permeable coil's hydrodynamic radius is not generally equal to its end-to-end distance.

The energy cost $U=\iint |a({\mathbf{r}}_1){|}^2|a({\mathbf{r}}_2){|}^2{e}^2/[4\pi {ϵ}_0|{\mathbf{r}}_1-{\mathbf{r}}_2|]{\mathrm{d}}^3{r}_1{\mathrm{d}}^3{r}_2$ for double occupancy of a localized orbital. It is positive for an ordinary nonzero orbital. A model retaining only this integral also neglects intersite and mixed Coulomb terms; orbital orthogonality alone does not make those terms vanish.

Two localized orbitals coupled by hopping, with an on-site penalty for double occupancy. At two-electron filling the competition between hopping and local repulsion can be solved exactly within the finite basis. It gives a compact example of a molecular bonding crossover and electron correlation; it is not by itself a complete real-molecule binding curve.

Two electrons of opposite spin occupying the same spatial orbital. The mean occupation product measures its probability at a specified site. In the two-electron dimer, the sum over sites is also the probability of any doubly occupied site, since two sites cannot both be doubly occupied.

For spin imbalance $s=n_{\uparrow }-n_{\downarrow }$ per site, the uniform mean-field interaction energy is $U(n^2-s^2)/4$. With noninteracting density of states $D$ per spin per site, the kinetic cost is $s^2/(4D)$. The quadratic coefficient changes sign at $UD=1$, producing the Stoner mean-field instability. This does not describe every strong-coupling Hubbard phase.

An effective low-energy lattice model obtained from a strongly repulsive Hubbard model near one electron per site. The retained Hilbert space excludes double occupancy. Its usual nearest-neighbor Quantum Hamiltonian has projected hopping $-t{\sum }_{⟨ij⟩,\sigma }({\tilde{c}}_{i\sigma }^{†}{\tilde{c}}_{j\sigma }+h.c.)$ and antiferromagnetic superexchange $J{\sum }_{⟨ij⟩}({\mathbf{S}}_i\cdot {\mathbf{S}}_j-n_i{n}_j/4)$. Virtual intermediate states with one double occupancy cost energy $U$, giving $J=4t^2/U$ to leading order. Three-site hopping terms of the same expansion order are omitted in this simplified version.

At half filling the projected hopping is blocked. Doped holes restore charge motion coupled to spin correlations. Nearest-neighbor singlet pairing can support a d-wave bond order ansatz on a square lattice, but the Quantum Hamiltonian alone is not a proof of a unique superconducting ground state for all parameters.

Correspondence between gene order and the anterior boundaries of Hox expression domains along a body axis. It must be distinguished from temporal collinearity.

Sequential onset of Hox expression following genomic order in relevant developmental contexts. Rearrangements can alter onset timing without equivalently changing every later spatial boundary.

An association between genomic position and expression level or prevalence in a developmental domain. It is a separate property from spatial and temporal order.

The cis-regulatory domains flanking the mouse HoxD cluster. Developmental changes in enhancer contacts support distinct proximal and distal limb expression programmes.

For real unit vectors $a,b$, the Householder reflections $H_a=I-2a{a}^T$ and $H_b=I-2b{b}^T$ commute exactly when their normals are parallel or orthogonal. Expanding their products proves the formula. If $a\cdot b\ne 0$, a zero commutator requires $b{a}^T=a{b}^T$; applying this identity to $a$ gives $b=(a\cdot b)a$, hence parallelism. Orthogonal normals make the prefactor zero.

Equality on one selected vector is weaker than equality of the operators. For nonparallel, nonorthogonal normals in three dimensions, the commutator's kernel is the line perpendicular to both normals, spanned by $a\times b$. A vector on that line is fixed by both reflections even though the operators do not commute.

The maximum primordial occupation of a collisionless fermion distribution limits the coarse-grained phase-space density attainable in a halo. Combining that bound with an observed velocity dispersion and mass density gives a lower candidate mass. This tests light-fermion halo models independently of whether the candidate is optically visible.^[ref]