For two identical equally bright stars, zeros of the binary factor occur at $Y_j=(j+1/2)\lambda /\beta$, where $\beta$ is separation projected along the baseline. Their repeated spacing is $\Delta Y=\lambda /\beta$. Multiple samples separate these rapid zeros from zeros of a resolved stellar envelope. Baseline rotation is needed to recover an unknown two-dimensional separation direction; the first zero alone does not determine both diameter and separation.

A thermal change in the electroweak Higgs sector as the early Universe cools. Its order and strength depend on the particle content, couplings and effective description. A polynomial landau free energy is a simplified model and does not, by itself, establish a model-independent Higgs-mass bound.

Only single-particle states within order $k_{B}T$ of the Fermi energy can change occupation at low temperature. Their number is order $g(E_F)k_{B}T$, and their excitation energy is order $k_{B}T$, giving $U-U_0\propto T^2$ and electronic heat capacity proportional to $T$.

For an isotropic three-dimensional liquid at fixed particles per mole, the molar coefficient scales as ${\gamma }_{\mathrm{mol}}\propto m^{\ast }{V}_{\mathrm{mol}}^{2/3}$. Density and mass renormalization both affect the slope; the slope ratio is not simply the effective-mass ratio.

A feature in phonon dispersion caused by a nonanalytic wavevector dependence of electronic screening. In a quasi-one-dimensional conductor, enhanced response near $2k_F$ can soften a lattice mode and signal an approaching Peierls transition.

A second-quantized electron-phonon Hamiltonian with free electrons, harmonic phonons and terms linear in phonon displacement, $M_q{c}_{k+q,\sigma }^{†}{c}_{k,\sigma }(b_q+b_{-q}^{†})$. The name is used for both general linear electron-phonon models and the more specific long-range coupling to polar optical modes; the intended dispersion and matrix element must be stated.

For electron energy transfer $\Delta$ and phonon energy ${\Omega }_q=\hslash {\omega }_q$, the symmetrized pair amplitude is $V_{\mathrm{eff}}=2|M_q{|}^2{\Omega }_q/({\Delta }^2-{\Omega }_q^2)$. It is attractive for $|\Delta |<{\Omega }_q$. This interaction is retarded, and screened Coulomb repulsion must also be considered.

For a nonrelativistic three-dimensional ideal electron gas at zero temperature, the pressure is $p=2n{E}_F/5$ and the electronic bulk modulus is $B=2n{E}_F/3$. This is the electronic model contribution, not automatically the total bulk modulus of a real crystal.

For an Eisenstein integer $z=a+b\omega$, with $\omega =(-1+i\sqrt{3})/2$, the norm is $N(z)=z\overline{z}=a^2-ab+b^2$. It is a positive integer for $z\ne 0$, since $N(z)=(a-b/2{)}^2+3b^2/4$, and is multiplicative by the multiplicativity of the complex modulus. An element is a unit exactly when its norm is one: one direction follows from multiplicativity and integrality, and the other from the inverse $z^{-1}=\overline{z}$.

A divisibility sequence associated with multiples of a point on an elliptic curve, using suitably normalized division polynomials. With $W_0=0$ and $W_1=1$, its terms satisfy the Ward identity $W_{m+n}{W}_{m-n}=W_{m+1}{W}_{m-1}{W}_n^2-W_{n+1}{W}_{n-1}{W}_m^2$. Setting nearby indices gives doubling identities, enabling logarithmic-depth evaluation.

For a continuously differentiable vector field $v$ on a bounded sufficiently regular region $V$, apply the divergence theorem to $a\times v$ for an arbitrary constant vector $a$. The vector calculus identity $\nabla \cdot (a\times v)=-a\cdot (\nabla \times v)$ and the cyclic scalar triple product identity show that the dot products of both sides of the displayed formula with $a$ agree. Since $a$ is arbitrary, the vector integrals agree. Here $n$ is the outward unit normal vector. Reversing the order on the boundary gives the equivalent positive-sign formula $\int n\times vdS=\int \nabla \times vdV$.

A transition density with absorbing boundaries counts the joint event of survival up to time $t$ and a specified final position. Its integrated mass is survival probability, not generally one. Dividing by that mass gives the endpoint density conditional on survival, a different object.

A tensor coefficient relating an increment of electrical displacement to stress, or the corresponding converse strain response to electric field, under stated constitutive conventions. Charge-per-force units $\mathrm{C}{\mathrm{N}}^{-1}$ are equivalent to displacement-per-voltage units $\mathrm{m}{\mathrm{V}}^{-1}$ for the usual $d$ coefficients.

For electrically isolated electrodes with unchanged free charge, the increment of normal electric displacement field is approximately zero. A constitutive relation such as $D_3=d_{31}({\sigma }_1+{\sigma }_2)+ϵE_3$ then gives an induced field that balances stress-induced displacement. Finite leakage and input capacitance modify the voltage response.

For $H=c(p_x{\sigma }_x+p_y{\sigma }_y)+V(y){\sigma }_z$, $c>0$, and a mass changing from negative to positive as $y$ increases, $e^{i{k}_{x}x}\exp [-{\int }_0^{y}V(s)\mathrm{d}s/(\hslash c)](1,-1{)}^T$ has energy $-\hslash c{k}_x$. The derivative and mass terms cancel. Normalizable confinement additionally requires the envelope to be square-integrable, as for nonzero asymptotic masses. Reversing the wall reverses the propagating direction.

The dimensionless occupancy scale $n{\lambda }_T^3$, with number density $n$ and thermal de Broglie wavelength ${\lambda }_T=h/\sqrt{2\pi m{k}_{B}T}$. Increasing it approaches the regime where quantum statistics matters. This use of degeneracy must not be confused with degeneracy of an individual energy eigenspace.

For a point source $Ze$ and the Fourier transform convention $\rho (\mathbf{q})=\int e^{-i\mathbf{q}\cdot \mathbf{r}}\rho (\mathbf{r})d^{3}r$, self-consistent screening gives ${\rho }_{\mathrm{ind}}(q)=Ze[ϵ(q{)}^{-1}-1]$. Angular inversion yields ${\rho }_{\mathrm{ind}}(r)=-(Ze/r){\int }_0^{\infty }g(q)\sin (qr)dq$. The factor $1/(2{\pi }^2)$ includes the inverse-transform normalization and spherical angular integral. Perfect long-wavelength screening implies total induced charge $-Ze$.

The static independent-electron density response of a uniform three-dimensional, two-spin Fermi gas, combined with self-consistent Poisson equation screening, gives the displayed dielectric function. For bare electron mass and vacuum background, ${\kappa }^2=m_e{e}^2{k}_F/({ϵ}_0{\pi }^2{\hslash }^2)$ andHere $F(0)=1$, $F(1)=1/2$ by continuous extension, and $F(x)\sim 1/(3x^2)$ at large $x$. A host dielectric background and band effective mass require replacing the vacuum parameters appropriately. This nonlocal response contains Friedel oscillations, which its long-wavelength Thomas-Fermi screening limit misses.

A library codon with any base at the first two positions and G or T at the third. Its 32 codons represent the 20 standard amino acids unequally and include one stop codon.

A library codon with any base at the first two positions and G or C at the third. Its 32 codons represent the 20 standard amino acids unequally and include one stop codon.