The coherent electron-flavor mass combination entering the usual light-neutrino-exchange amplitude. Its phases can cancel contributions, so a null double-beta result does not imply vanishing individual masses.

A search in which isotope-enriched detector material is also the decay source. germanium detectors combine a contained energy deposit with energy resolution, pulse information and material-background controls. Background peaks and resolution tails must be tested before assigning a signal.

In the upper half-space $z>0$, the displayed decaying harmonic function has outward normal derivative ${\partial }_n\Phi =-{\partial }_z\Phi =z/[2\pi (x^2+y^2+z^2{)}^{3/2}]$. This is the Poisson kernel for the upper half-space, whose integral on a horizontal plane is one and whose limit as $z\downarrow 0$ is the two-dimensional Dirac delta. Thus ${\partial }_n\Phi =\delta (x)\delta (y)$ in the distributional boundary sense. The compensating flux through a large hemisphere is minus one. This illustrates why a nonzero total boundary source is possible on an unbounded domain even when the interior source is zero.

A failure changes loads or dependencies so that further components fail. Connectivity of the surviving graph alone does not determine whether a cascade occurs.

Nasonia vitripennis is a parasitoid wasp whose offspring develop using resources from a fly host. Its reproductive system and tractable laboratory breeding make it useful for investigating heredity and interactions between host biology and associated microorganisms.

Nasonia giraulti is a parasitoid wasp studied in comparison with Nasonia vitripennis. Crosses and behavioral measurements can separate inherited differences from environmental influences and help investigate barriers between species.

A method for detecting differential RNA-modification signals using Gaussian mixture distributions of nanopore current, with condition-specific component proportions. A shifted signal requires biological interpretation as well as statistical separation.^[ref]

For real even coefficients, ${\Psi }_k=(a_k,a_{-k}^{†}{)}^{\mathsf{T}}$ and a full-zone sum, this expression equals $C+{\sum }_k[A_k{a}_k^{†}{a}_k+B_k(a_k{a}_{-k}+a_k^{†}{a}_{-k}^{†})/2]$. The constant subtraction follows from $a_{-k}{a}_{-k}^{†}=1+a_{-k}^{†}{a}_{-k}$. Pairing redundancy and normal ordering determine both factors together. Switching to half a zone requires grouping both members of each pair; changing only a diagonal factor is inconsistent.

In a fixed square coil beside a long wire, neglect self-inductance compared with resistance. A changing wire current induces coil current proportional to $dI/dt$. Forces on the two perpendicular sides cancel, and the unequal parallel-side fields give magnitude ${\mu }_{0}Na|Ii|[1/s-1/(s+a)]/(2\pi )$. When $I=I_0\sin \omega t$, the signed force alternates at twice the drive frequency and has zero cycle mean, while Joule dissipation is positive with mean ${\mathcal{M}}^2{\omega }^2{I}_0^2/(2R)$. A fixed coil has no mechanical work despite its time-dependent force.

An area-based estimate treating each nanotube shell as a graphene cylinder. For graphene atomic area density $n_s$ and shell radii $r_j$, the carbon count per axial length is $N_C/\ell =2\pi n_s{\sum }_j{r}_j$. Defects and partial shells change the estimate.

A uniform thin square of side $l$ and mass $M$ has $I_z=M{l}^2/6$ about the perpendicular central axis. Any central in-plane axis and its perpendicular in-plane partner have equal moments by $90^{\circ }$ rotational symmetry. The perpendicular axis theorem says their sum is $I_z$, proving $I_{\varphi }=M{l}^2/12$ for every angle $\varphi$. A non-square rectangle generally lacks this angle independence.

The $r$th raw moment of a random variable $X$ is $\mathbb{E}[X^r]$, when it exists. A central moment instead uses $\mathbb{E}[(X-\mathbb{E}[X]{)}^r]$.

A rotation, reflection, inversion or improper rotation that carries a fixed molecular geometry into an indistinguishable configuration. Products of such operations must also be symmetries.

The four-element molecular point group with one twofold axis and two mirror planes containing that axis. Water has this symmetry at its equilibrium bent geometry.

An eight-element molecular point group with classes $E,2S_4,C_2,2C_2^{\prime },2{\sigma }_d$. The vector representation decomposes as $B_2+E$.

The twelve-element molecular point group of an ideal trigonal bipyramid. Reflection in its equatorial plane distinguishes primed and double-primed irreducible representations.

The sixteen-element molecular point group of an ideal square with a horizontal mirror plane. Inversion distinguishes gerade and ungerade irreducible representations.

The full icosahedral molecular point group, including inversion, with 120 operations. Its single-valued orbital irreducible representations have dimensions 1, 3, 3, 4 and 5 in each inversion parity.

An orbital set whose span is invariant under every molecular symmetry operation. Completeness here means closure of the chosen representation space; it does not mean that a finite orbital set spans the entire electronic Hilbert space.

For a finite molecular point group, the multiplicity of irrep $\lambda$ in a representation with character $\chi$ is $n_{\lambda }=|G{|}^{-1}{\sum }_C|C|\chi (C){\chi }_{\lambda }(C{)}^{\ast }$. The sum is over conjugacy classes.