For lossless parallel Hertzian transmitting and receiving dipoles separated by distance $r$ at angle $\theta$ to their common axis, directional gains are each $G=3{\sin }^2\theta /2$. The emitted polarization matches the receiving axis's transverse projection. Matched-load power is therefore $P_L=P[3{\sin }^2\theta /2{]}^2[\lambda /(4\pi r){]}^2$. One angular factor describes transmission and the other reception. This assumes reciprocal ideal dipoles, conjugate matching, far-field separation and negligible dissipative loss.

For $y^{\prime \prime }-2x{y}^{\prime }+2\lambda y=0$, substitute a power series $y={\sum }_{n\ge 0}{a}_n{x}^n$ to obtain the displayed recurrence relation. It separates into an even solution with $a_0=1,a_1=0$ and an odd solution with $a_0=0,a_1=1$:The ratio of successive terms within each parity tends to zero at every fixed complex $x$, giving infinite radius of convergence. The two initial-value vectors are independent, so these solutions span the solution space. A nonzero polynomial solution exists precisely when $\lambda =N$ is a nonnegative integer: its highest power forces this value, and the recurrence terminates the matching parity. These terminating solutions are nonzero scalar multiples of the Hermite polynomials. Multiplication by $e^{-x^2/2}$ connects them to Hermite oscillator quantization.

The ratio of additive genetic variance to total phenotypic variance in a specified biological population and environment. It is not the fraction of one individual attributable to genes.

A helium-containing cavity in an irradiated material. helium helps stabilise small cavities and can accumulate at grain boundaries, contributing to embrittlement.

A neutral helium atom contains two electrons bound to a nucleus of charge $+2e$. Even with a fixed nucleus and nonrelativistic kinetics, the repulsion $e^2/(4\pi {ϵ}_0{r}_{12})$ prevents exact separation into two hydrogenic one-electron problems. Its ground configuration is $1s^2$ with total spin zero, while distinct occupied orbitals allow both singlet helium states and triplet helium states.

If zero conserved helical momentum reduces to $r^2\dot{\theta }+b\dot{z}=0$, a closed transverse projection $C$ has axial displacement $\Delta z=-b^{-1}{\oint }_C(xdy-ydx)=-2A/b$. Here $b\ne 0$ and $A$ is the signed area, with winding multiplicity for self-intersections. This is a path-dependent displacement even when the transverse coordinates return to their initial values. It follows from the momentum constraint and Green's theorem; it is not a prediction that every chosen closed curve is dynamically realizable.

The outflow end of the developing heart, receiving progenitor contributions on a stage-specific schedule.

The inflow end of the developing heart, with progenitors contributing to inflow and atrial regions under species- and stage-specific conditions.

Interference with detection or discrimination of a sound by other sounds. Signal frequency, amplitude, timing and receiver processing can affect masking.

At a stationary Hartree-Fock wavefunction $|{\Phi }_0⟩$, the Hamiltonian has zero matrix element to every singly excited Slater determinant $|{\Phi }_i^a⟩$ formed by replacing an occupied orbital $i$ with an unoccupied orbital $a$:An infinitesimal occupied-to-unoccupied orbital rotation adds precisely such singly excited components to the reference. Stationarity of the energy with respect to that rotation forces their linear coefficients to vanish. The theorem explains why single excitations do not provide the usual second-order reference-energy contribution in perturbation theory built on optimized Hartree-Fock orbitals; double excitations can still contribute. It does not state that the reference is an exact many-electron energy eigenstate, or that all stationary Hartree-Fock solutions are minima.

For number densities $n_{\uparrow },n_{\downarrow }$, $\zeta =(n_{\uparrow }-n_{\downarrow })/(n_{\uparrow }+n_{\downarrow })$. Uniform spin polarization raises kinetic energy by concentrating occupation in one Fermi sphere, while lowering exchange energy.

In a three-dimensional unpolarized gas, exchange energy per particle is $-3e^2{k}_F/(16{\pi }^2\epsilon )$, with Coulomb convention $e^2/(4\pi \epsilon r)$. It scales as minus the inverse dimensionless spacing radius, while kinetic energy scales as its inverse square. Exchange acts only between equal-spin occupied orbitals.

The uniform paramagnetic three-dimensional electron-gas determinant is unstable within Hartree-Fock theory to a spin-density-wave deformation, even at high density. Comparing only spatially uniform polarized and unpolarized determinants therefore does not find the unrestricted Hartree-Fock ground state.^[ref]

In a uniform three-dimensional Hartree-Fock electron gas with repulsive contact coupling $\gamma$, a spin-independent neutralizing background can cancel the Hartree energy. The remaining same-spin exchange term is negative and quadratic in each spin density. Integrating the filled single-spin Fermi sphere gives $A=3{\hslash }^2(6{\pi }^2{)}^{2/3}/(10m)$. At fixed total density the exchange term favors spin polarization, whereas the kinetic term opposes it.

The local mean Coulomb potential generated by the occupied electron density. A separate positive background or external charge contribution must be included where appropriate. Hartree-Fock exchange is nonlocal and removes equal-orbital self-interaction from the total direct-plus-exchange result.