An electron annihilation operator restricted to the no-double-occupancy sector of the t-J model. It removes a spin-$\sigma$ electron only when the opposite-spin site occupation vanishes. The corresponding creation operator cannot create a doubly occupied site. These projected operators do not obey the unrestricted canonical on-site fermion algebra on all states.

The system-equivalent flux density is the unpolarized spectral flux density that would produce an antenna temperature equal to $T_{\mathrm{sys}}$. With effective collecting area $A_e$, $\mathrm{SEFD}=2k_B{T}_{\mathrm{sys}}/A_e$. The factor two follows from one polarization of waves measurement of an unpolarized source. State effective area and polarization of waves conventions when comparing sensitivities.^[ref]

A multiphase Synroc formulation with zirconolite, hollandite, perovskite and titanium-oxide phases. The mineral hosts distribute different radionuclides rather than requiring one lattice to accept the entire inventory.

The permutation group of a three-element set has six elements: the identity, three transpositions, and two three-cycles. A three-cycle and any transposition generate it, with relations $x^3=h^2=e$, $hx{h}^{-1}=x^{-1}$. Its alternating group is the normal cyclic subgroup of order three. It is the unique nonabelian isomorphism type of a group of order six.

For a strongly segregated, symmetric long-chain polymer blend, take the square-gradient functional $F/(k_{B}T{\rho }_0)=\int [\chi \varphi (1-\varphi )+a^2({\varphi }^{\prime }{)}^2/(24\varphi (1-\varphi ))]dz$. Its first integral equates the two terms, giving ${\varphi }^{\prime }=\sqrt{24\chi }\varphi (1-\varphi )/a$. Thus $\varphi =[1+\tanh (z/\ell )]/2$ with $\ell =a/\sqrt{6\chi }$. The inverse-central-slope width $w=1/{\varphi }^{\prime }(0)=2\ell$ differs by a factor two from the tanh-argument scale. The continuum approximation is marginal when these lengths approach the statistical segment size. An observed width can additionally include roughness and instrument broadening.

For two spin-one-half qubits, the SWAP gate acts as one on the triplet sector and minus one on the singlet sector. The displayed unitary squares to SWAP. An isotropic exchange interaction with integrated angle $\pi /2$ implements it up to a common phase. There are other square-root branches; a pulse decomposition must use the declared branch consistently.

Jamming is the loss of the ability of a dense disordered material to flow as its particle fraction or applied confinement reaches a critical state.

Stationary one-dimensional flow has constant particle current $j=\rho v$. A lower-density segment must flow faster at the same current and also has smaller local speed of sound $\sqrt{g\rho /m}$. It can therefore lose local phonon energetic stability before a denser segment.

In a first-order condensate action, density and phase are canonically paired variables. Linear density and phase oscillations are therefore coupled components of a single Bogoliubov branch, rather than two independent positive-frequency particles. For $i\dot{\psi }=-{\psi }^{\prime \prime }+\beta (|\psi {|}^2-1)\psi$, writing $\psi \simeq 1+\rho /2+i\theta$ gives $\dot{\rho }=-2{\theta }^{\prime \prime }$ and $\dot{\theta }={\rho }^{\prime \prime }/2-\beta \rho$, hence ${\omega }^2=k^4+2\beta k^2$.

In a minimal hole description of a metal–ligand–metal path, two holes initially occupy separate metal orbitals and the ligand has no hole. Creating a virtual ligand hole costs a charge-transfer energy $\Delta$. Two hops generate an effective metal–metal hopping of order $t_{pd}^2/\Delta$; two more hops through metal double occupancy generate the antiferromagnetic scale in the title. Ligand double occupancy adds a channel with denominator $2\Delta +U_p$. Orbital geometry and intra-atomic exchange determine when the antiferromagnetic channel dominates.

A complex momentum-dependent pairing amplitude. Its crystal-symmetry transformation, nodes and relative signs distinguish candidate superconducting states. Magnitude-sensitive spectra alone generally do not establish all relative signs; phase-sensitive superconducting gap experiments test them through interference.

The boundary of a stated superconducting transport criterion in temperature, magnetic field and current-density space. A value measured under one combination does not specify a winding's usable current at another field or temperature. Strain and sample geometry can require further parameters.

For an irrational number $0<\alpha <1$, the characteristic Sturmian word of slope $\alpha$ is the binary sequenceEach term is zero or one because the arguments differ by less than one. This fixes an intercept convention: it is the lower mechanical word with intercept $\alpha$.

For $\alpha =(3-\sqrt{5})/2$, the beginning is $0100101001\dots$, the Fibonacci word. Like every Sturmian word, it has exactly $n+1$ distinct factors of length $n$. Its relation to continued fractions makes its recursive block structure reflect the partial quotients of its slope.

Reduction of stomatal aperture through changes in guard cell mechanics and turgor pressure. In ABA responses, regulated solute efflux promotes water loss from guard cells.

An epidemic reaches an absorbing state when no infected individuals remain and there is no external source of infection. Extinction time differs from a simulation horizon at which infected individuals are still present.

A steady outward radial flow with negligible pressure force conserves $v^2/2-GM/r$. If its speed at a reference radius equals the mechanical escape velocity, this constant is zero and $v\propto r^{-1/2}$. Conservation of mass gives $\rho \propto r^{-3/2}$. An adiabatically evolving ideal gas carried along that leading flow has $p\propto r^{-3\gamma /2}$ and $T\propto r^{-3(\gamma -1)/2}$.

Finite specific enthalpy changes the dynamical problem: the full conserved Bernoulli equation is $v^2/2+h-GM/r=B$. At the same boundary mechanical escape speed, $B=h_e$ rather than zero. A warm wind therefore needs its enthalpy and equation of state specified before the explicit ballistic velocity can be replaced by a consistent pressure-driven solution.

A dense stellar-scale object such as a white dwarf, neutron star or stellar-mass black hole. accretion releases a large gravitational binding energy per unit mass because the accreting surface or inner orbit is small. A black hole has no material surface, so its radiative yield depends on the accretion flow rather than surface impact.

The midpoint of the two wings of a stellar absorption line at each depth. A changing line bisector can reveal profile distortions caused by active regions or blends rather than a rigid Doppler shift of the whole spectrum.

An activity-period alias in radial-velocity sampling is an apparent period created when a stellar activity signal is measured at a restricted set of observing times. For observations exactly separated by $1/f_s$, a sinusoid of frequency $f$ is indistinguishable at those times from appropriate sinusoids at $|f\pm n{f}_s|$, with integer $n$. Thus a genuine stellar rotation signal can produce peaks at periods different from the rotation period.

Real observing times are irregular, so the aliases have different amplitudes and phases rather than being perfectly equivalent. Simulating a candidate signal at those same times helps test which observed peaks it predicts. The method applies to planetary signals as well as stellar noise; a periodogram peak alone does not identify a new exoplanet.^[ref]

Before local expulsion, a linear star-formation law $\psi =\nu {\rho }_g$ without return gives ${\rho }_g={\rho }_0{e}^{-\nu t}$. If $q=\gamma E_{\mathrm{SN}}$ is the energy supplied per unit stellar mass, accumulated energy is $q{\rho }_0(1-e^{-\nu t})$. A fixed binding energy per unit remaining gas mass $B(r)$ gives the threshold $q(e^{\nu t}-1)=B(r)$. This assumes a specified retained gravitational profile and neglects exchange between volumes.