Let $T_{ij}$ count transmitted modes from injector $j$ to collector $i$, excluding reflection $R_{ii}$. Scattering conservation gives ${\sum }_{i\ne j}{T}_{ij}=N_j-R_{jj}$; unitarity also gives the corresponding row balance. The conductance matrix has zero row sum, making equal lead potentials carry no current, and zero column sum, conserving total current.

For a symmetric interaction kernel, $\delta E/\delta n_a={ϵ}_a+V^{-1}{\sum }_b{f}_{ab}\delta n_b$. Smooth Fermi-surface kernels and fixed-density chemical-potential shifts give an $O(T^2)$ thermal correction by the Sommerfeld expansion. The reference dispersion already includes zero-temperature interactions.

For $f=f_0+a(T-T_c)|\mathbf{m}{|}^2+b|\mathbf{m}{|}^4+c|\nabla \mathbf{m}{|}^2$ with positive coefficients and uniform free boundaries, the minimum is $\mathbf{m}=0$ at $T\ge T_c$. Below $T_c$ it is any constant vector with $|\mathbf{m}{|}^2=a(T_c-T)/(2b)$. A chosen direction leaves an $O(2)$ subgroup of $O(3)$ invariant; for proper rotations the pattern is $SO(3)\to SO(2)$. The mean-field transition is continuous, with square-root onset. Long-wavelength fluctuations can invalidate the ordered state in low dimensions.

For $\varphi \ge 0$ and positive $c,B,\lambda ,T_0$, define $\gamma =2B^2/(\lambda c)$. Completing the square givesFor $0<\gamma <1$, phase coexistence occurs at $T^{\ast }=T_0/\sqrt{1-\gamma }$ with a discontinuous minimum at ${\varphi }^{\ast }=2B{T}^{\ast }/\lambda$. For $\gamma \ge 1$, a nonzero global minimum has lower free energy at every finite temperature. Following the global equilibrium branch then does not force a restoration transition.

A stationary point on the domain $\varphi \ge 0$ must be tested using allowed one-sided variations. For $F=A{\varphi }^2/2-B{\varphi }^3+\lambda {\varphi }^4/4$ with $B>0$, the boundary is a local minimum when $A>0$ and a local maximum when $A<0$. At $A=0$, the leading negative cubic term makes it a one-sided local maximum even though $F^{\prime \prime }(0)=0$; on an unrestricted real line it is instead a stationary inflection point.

For $f=a|\varphi {|}^2+|\varphi {|}^4/2+c|{\partial }_x\varphi {|}^2+|{\partial }_x^2\varphi {|}^2$, a plane-wave order parameter has quadratic coefficient $a+c{k}^2+k^4$. If $c<0$, this is minimized at $k^2=-c/2$, and a nonzero amplitude appears when $a<c^2/4$. If $c\ge 0$, the minimizing wave number is zero and the ordered mean-field phase requires $a<0$. The arbitrary global phase is a degeneracy, not a missing minimization condition. These are functional-minimization results; finite-temperature fluctuations can invalidate long-range order in low dimensions.

For a repulsive homogeneous condensate with Bogoliubov energy $E(p)=\sqrt{(p^2/2m)(p^2/2m+2\mu )}$, the infimum of $E(p)/p$ is the speed of sound $\sqrt{\mu /m}$. This is energetic stability against the given bulk modes, not a universal measured dissipation threshold.

The contact-gas spectrum $E_k=\sqrt{{ϵ}_k({ϵ}_k+2g{n}_0)}$ gives $\min E_k/(\hslash k)=\sqrt{g{n}_0/m}$. This energetic homogeneous-flow bound need not equal a measured critical speed when vortices or boundaries enable other dissipation.

The excitation-based kinematic velocity bound in the Landau criterion. For a heavy weak obstacle and an isotropic excitation spectrum, emitting momentum parallel to the obstacle's motion costs $E(p)-vp$; requiring this to remain positive for every excitation gives the displayed infimum. Vortex nucleation or other dynamical processes can make a measured critical velocity lower than this bound.

In a parabolic condensate in the Thomas–Fermi approximation for a condensate, the local sound speed is $c(D)={\omega }_r\sqrt{R^2-D^2}/\sqrt{2}$. Equating it to the obstacle speed $\Omega D$ gives the displayed threshold. This is the stipulated local phonon estimate; a finite strongly depleting obstacle can modify flow and nucleate vortices, so the inequality is not a universal exact onset law for every stirring experiment.

A DNA recognition sequence bound by LacI. Repeated copies are used in fluorescent operator-repressor chromosome labelling.

For fixed bond length $b$, fixed forward bond angle $\alpha$, and independently uniform torsion angles, the bond correlations are $⟨{\mathbf{b}}_i\cdot {\mathbf{b}}_{i+s}⟩=b^2{c}^s$. Thus for $N$ bondsFor fixed $|c|<1$ and a sufficiently long chain, divide by contour length $Nb$ and sum the geometric series to obtain the title formula. Positive forward persistence raises the Kuhn length, and negative correlation lowers it relative to $b$. A chemical internal angle is supplementary to the forward-vector angle. The long-chain formula cannot be used uniformly as $c\to 1$ at fixed $N$: the exact chain obeys $R\le Nb$ and hence $l_K\le Nb$, with equality in the rigid-rod limit.

A Kondo lattice develops coherent low-energy quasiparticle bands below a material-dependent coherence scale. This lattice scale need not equal the single-impurity Kondo screening scale. Coherence can produce a heavy Fermi liquid with narrow bands, large electronic heat capacity and low-temperature quadratic resistivity.

A diffraction-loss ratio is normalized to the unobstructed intensity at the same receiver position. With a point source that unobstructed intensity changes as inverse source–receiver distance squared. Moving the receiver can therefore decrease the transmission ratio while increasing its absolute received intensity. The two curves require separate normalization; a Cornu-chord plot alone omits geometric spreading.

A canonical spatially homogeneous field with $V=m^2{\varphi }^2/2$ obeys $\ddot{\varphi }+3H\dot{\varphi }+m^2\varphi =0$. The $3H\dot{\varphi }$ term expresses dilution of field momentum by expansion. Conservation of its stress energy gives this equation wherever $\dot{\varphi }\ne 0$; the canonical field equation is needed to fix an identically stationary interval, while continuity extends a regular solution across isolated turning points.

A local scalar nonlinearity preserves Lorentz covariance of the wave operator when the wave speed is one. If $f=F^{\prime }$, the equation follows from $\mathcal{L}=({\partial }_t\varphi {)}^2/2-({\partial }_x\varphi {)}^2/2+F(\varphi )$. The physical potential is $-F$, so its sign must be distinguished from the sign written in the Lagrangian.

In a sufficiently strong electrostatic step, the positive-energy incident channel overlaps a negative-kinetic-energy channel on the far side. For a charged scalar field the outgoing antiparticle branch has negative charge current, allowing reflected particle current larger than the incident current while signed charge is conserved. The external source supplies energy for particle-antiparticle creation. This requires a multiparticle interpretation; negative transmitted charge current is not a negative probability.

Write the Klein four-group as $\{1,a,b,ab\}$. Its four one-dimensional irreducible representations independently assign signs to $a,b$, so the displayed characters have $p,q\in \{0,1\}$. The character table isEvery conjugacy class is a singleton because the group is abelian. The four rows exhaust the complex irreducibles by the equality between their number and the number of classes. If $H$ is this table, $H{H}^T=4I$ proves character orthogonality. The diagonal three-dimensional representation by half-turns about the coordinate axes is the direct sum of the three nontrivial characters, with total character $(3,-1,-1,-1)$.

For charges $+e,-e$ and relative coordinate $\mathbf{r}={\mathbf{r}}_2-{\mathbf{r}}_1$, total magnetic pseudomomentum is conserved in a uniform magnetic field. In an appropriate symmetric-gauge Lagrangian differing by a total derivative, it is the canonical centre-of-mass momentum. Mechanical centre-of-mass momentum instead equals $\mathbf{K}+e\mathbf{B}\times \mathbf{r}$ and need not be constant. The sector $\mathbf{K}=0$ is not generally a stationary-centre-of-mass frame.

With metric signature $(+---)$ and a future timelike stationary Killing vector $K$, positive energy is $E=g(P,K)$ for future physical momentum $P$. A common $(-+++)$ convention instead uses $-g(P,K)$. The normalization of $K$ fixes the observer or asymptotic time scale.