A Lorentz scalar Dirac bilinear connects opposite chiralities. In the stated on-shell normalization ${\bar{u}}_R{u}_L=m_{\mathrm{spinor}}$, whereas ${\bar{u}}_L{u}_L=0$. Such an overlap verifies the allowed Lorentz structure but does not determine an independent coefficient multiplying a different mass operator.

The positive hole mobility is the magnitude of an electron hole's weak-field drift velocity per unit electric field. In an isotropic relaxation-time model, ${\mu }_h=e{\tau }_h/m_h^{\ast }$, and its contribution to electrical conductivity is $ep{\mu }_h$. The sign is opposite to an electron's drift because the electron hole has positive charge.

A transport quantity obtained by combining longitudinal electrical conductivity and a Hall coefficient. It equals single-carrier drift mobility only under the corresponding one-carrier and scattering assumptions. This does not independently determine effective mass and scattering time.

For a real skew-symmetric matrix $B$, its Cayley transform of a real skew-symmetric matrix with the opposite sign convention has inverse $C^{-1}=(I+B)(I-B{)}^{-1}$. Since $C$ is an orthogonal matrix, symmetry means $C=C^{-1}$, or $(I-B{)}^2=(I+B{)}^2$. This forces $B=0$. Skew-symmetry means $C=-C^{-1}$, or $(I-B{)}^2=-(I+B{)}^2$, which is equivalent to $B^2=-I$. In that case $C=-B$. Such examples exist in even dimension by taking two-dimensional right-angle rotation matrices in diagonal blocks; in odd real dimension the determinant equation $(\det B{)}^2=(-1{)}^n$ rules them out.

In three-dimensional Euclidean space, for a real symmetric Cartesian second-rank tensor $T$ and a nonzero vector $A$, require $B\cdot A=0$ and $C$ symmetric and traceless with $CA=0$. Put $q=A^{T}A$, $s=A^{T}TA$, $t=\mathrm{tr}T$, and $P=I-A{A}^T/q$. The unique decomposition isThe orthogonal projection $P$ isolates the plane perpendicular to $A$. Splitting $I=P+A{A}^T/q$ on both sides of $T$ gives its longitudinal, mixed and transverse blocks, proving reconstruction. The transverse trace fixes $\alpha$, the longitudinal component fixes $\beta$, and the mixed block fixes $B$, proving uniqueness. The remaining transverse block $C$ has two degrees of freedom. Together $\alpha ,\beta ,B,C$ carry $1+1+2+2=6$ components, the number in a symmetric three-by-three matrix.

A band filling constraint counts electrons per primitive unit cell of the actual translation symmetry. If an ordered state doubles the cell, a filling of one electron per original cell becomes two per new cell. The reduced Brillouin zone then permits reconstruction into different energy bands.

For spin-degenerate independent electrons, a filled band holds two electrons per primitive cell. Doubling can therefore remove the odd-filling obstruction to a band insulator, provided the reconstruction opens a global band gap. It does not guarantee a gap: folded bands can still overlap or retain symmetry-protected crossings. The original counting argument remains correct for its original symmetry assumptions; applying it after translation symmetry changes requires recounting using the new cell.

Occupation outside the macroscopically occupied condensate mode, caused by interactions or thermal excitations. A weak homogeneous three-dimensional Bose gas has zero-temperature depletion fraction $8\sqrt{n{a}_s^3}/(3\sqrt{\pi })$ in the leading dilute-gas approximation. Depletion measures single-particle mode occupation, not directly the superfluid fraction.

For a noninteracting ground state of energy zero, $N_0=(e^{-\beta \mu }-1{)}^{-1}$ gives $\mu =-k_{B}T\ln (1+1/N_0)$. Thus a finite condensate has a slightly negative chemical potential; exact zero pinning is a macroscopic limiting statement.

A one-dimensional condensate on a ring of circumference $2\pi R$ has a uniform winding state $\psi =\sqrt{N/(2\pi R)}{e}^{in\theta }$ when the order parameter is normalized to atom number. Its angular momentum per atom is $n\hslash$, while the flow contribution to its total chemical potential is ${\hslash }^2{n}^2/(2m{R}^2)$.

For a homogeneous gas, the condensate density is the macroscopic occupation $N_0$ of its condensate mode divided by volume. More generally its spatial density is $N_0|{\phi }_0(x){|}^2$ for a normalized condensate orbital. The mode and occupation are defined by a macroscopic eigenvalue of the one-body density matrix, so a finite fixed-number state need not have a nonzero field expectation. Condensate density is distinct from the superfluid density defined by response to a phase twist; interactions can deplete the former even when the latter equals the total density at zero temperature in a uniform Galilean-invariant gas.

All $N$ bosons occupy the same normalized one-particle superposition with $|C{|}^2+|S{|}^2=1$. Its occupation distribution in the original two modes is binomial, not two independent Poisson distributions. In particular, $⟨N_1⟩=N|S{|}^2$ and $⟨N_0{N}_1⟩=N(N-1)|C{|}^2|S{|}^2$. These factorial moments give finite-particle corrections to interaction energies. The state is also the two-mode realization of a spin coherent state with spin $N/2$.

At fixed irradiation, heating per grain is independent of temperature while cooling is proportional to $T^4$. With positive heat capacity and intact grains, a positive temperature perturbation increases net cooling, and a negative one increases net heating. Fixed dust properties and sufficient dust-gas thermal coupling are necessary assumptions.

A quartz retarder used to identify specimen fast and slow directions through changes of interference colour or compensation. Its own retardation and orientation must be specified before unique final colours can be predicted.

Variation among independent biological samples. Pooling reads can conceal this variation; treating within-sample technical observations as independent biological replicates causes pseudoreplication.

For a dilute-to-dense liquid coexistence problem, the dilute-phase coexistence concentration at specified conditions. It is a branch of the coexistence boundary, not generally the concentration at the critical endpoint.

Diagnostic sensitivity is the probability that a test is positive in a person who has the target condition. For past-infection assays, the target is prior infection rather than current disease. Estimate it from suitable independently characterized positive samples; using the same assay to define truth would make the assessment circular.

Diagnostic specificity is the probability that a test is negative in someone without the target condition. For an antibody test, relevant negative controls can include samples with antibodies to related pathogens to assess cross-reactivity. With true positive fraction $p$, observed positivity is $Sep+(1-Sp)(1-p)$.

Two subsets of a Euclidean space are weakly linearly separable when a hyperplane with nonzero normal vector places them in opposite closed half-spaces. Points on the separating hyperplane may belong to either subset. Strict separation instead requires strictly opposite signs, excluding all boundary ties.

A plot of true positive rate against false positive rate while a decision threshold varies over a fixed evaluation population. A random ranking has an expected diagonal curve; a perfect ranking reaches the upper-left corner.

For a binary classifier, classification precision is $TP/(TP+FP)$, the fraction of predicted positives that are correct, while classification recall is $TP/(TP+FN)$, the fraction of actual positives recovered. If a search returns ten items, six relevant, from a collection containing fifteen relevant items, precision is $6/10$ and recall is $6/15$. Raising a decision threshold often trades recall for precision, traced by a precision-recall curve, but precision need not improve at every threshold. Zero denominators require an explicit reporting convention. Both measures depend on which class is called positive; precision also depends strongly on prevalence.