For a thermodynamic cycle, with heat positive into the system and each transfer divided by the absolute temperature of the relevant thermal boundary or reservoir, the cyclic integral is nonpositive. Equality holds for a reversible cycle. Closing an arbitrary process by a reversible return path yields $\Delta S\ge \int \delta Q/T$, because the reversible integral defines the entropy difference. The difference is the nonnegative entropy production; entropy transferred with heat need not itself be positive.

With constant heat capacities $C_1,C_2$, a reversible heat engine taking two bodies to a common temperature has $T_f=T_1^{C_1/(C_1+C_2)}{T}_2^{C_2/(C_1+C_2)}$. The maximum work is $C_1{T}_1+C_2{T}_2-(C_1+C_2)T_f$. heat isolation does not prohibit transferring work to a store included in the total isolated system.

When a system at instantaneous temperature $T$ absorbs heat $dQ$ from a reservoir at $T_R$, with no other entropy production in the system, its entropy increases by $dQ/T$ while the reservoir changes by $-dQ/T_R$. The total is positive for spontaneous heat transfer across a finite temperature difference. A quasistatic mechanical path can therefore be thermally irreversible.

Assign values from specified domains to variables so that all stated constraints hold simultaneously. A partial assignment can be rejected when it violates a constraint or leaves an unassigned variable with no permissible value. Backtracking applies this test during a depth-first search.

A rooted tree whose nodes are partial search states and whose children are possible next choices. Leaves represent completed candidates or terminated branches. Backtracking traverses this tree while storing the current path, and search pruning removes branches that cannot yield a useful completion.

Effective exchange of internal states through virtual boson hopping into energetically costly occupation sectors. For two bosons initially one per site and charging energy $U{\Delta }^2/4$, the second-order operator is $-(t^2/U)(3+4{\mathbf{s}}_a\cdot {\mathbf{s}}_b)$ for dimensionless spins. The triplet is lowered and the singlet is unchanged in this example; signs for other energy functions require their actual denominators.

For a radial timelike geodesic of specific Killing energy $k=fdt/d\tau$, the static observer measures $v_{\mathrm{loc}}=(dr/dt)/f=\dot{r}/k$. Its relative Lorentz factor is $k/\sqrt{f}$, giving $v_{\mathrm{loc}}^2=c^2(1-f/k^2)$.

For an integer partition $\lambda$, $s_{\lambda }(1^m)={\prod }_{(i,j)\in \lambda }(m+j-i)/h_{ij}$, where $h_{ij}$ is the cell's hook length. It counts Semistandard Young tableaux of shape $\lambda$ whose entries belong to $\{1,\dots ,m\}$.

The normalized sum $N_{+}(|\alpha ⟩+|-\alpha ⟩)$ contains only even photon numbers, since the two coherent-state expansions cancel every odd term. It tends to vacuum at small amplitude. With real $\alpha$, its second quadrature can be squeezed below the vacuum variance.

The difference $N_{-}(|\alpha ⟩-|-\alpha ⟩)$ contains only odd photon numbers and tends to a one-photon state as $\alpha \to 0$. Exactly at zero amplitude the unnormalized difference vanishes; the normalized limiting state is obtained only by taking a limit.

For real $\alpha$, put $s=e^{-2{\alpha }^2}$ in $N(|\alpha ⟩+e^{i\varphi }|-\alpha ⟩)$. Its mean second quadrature is $-\alpha s\sin \varphi /(1+s\cos \varphi )$, which is generally nonzero. Subtracting its square from the second moment gives the displayed variance and squeezing precisely when $\cos \varphi >-s$, for nonzero finite amplitude. A useful counterexample to confusing moment and variance is $\varphi =\pi /2$.

A pure-loss channel of transmissivity $\eta$ maps the two branches to system amplitudes $\pm \sqrt{\eta }\alpha$ and environment amplitudes $\pm \sqrt{1-\eta }\alpha$. Tracing over the environment multiplies their cross terms by its branch overlap, the factor displayed. Large cat states can therefore lose interference rapidly even while their mean intensities change only moderately.

If site-transfer energies are $\mathcal{E}(\Delta )$, the left-ordered raising-hop generator has coefficient $f(\Delta )=t/[\mathcal{E}(\Delta )-\mathcal{E}(\Delta -2)]$. Nonlinear occupation energies generally make the effective second-order shifts and spin-exchange amplitudes depend on the occupation sector.

A perturbative generator dividing by an unperturbed energy difference cannot eliminate a nonzero coupling between exactly degenerate states. Those states must remain in the retained degenerate block. Small denominators also invalidate an expansion when the actual hopping matrix element, including occupation enhancement, is not small in comparison.

The plane spanned by incoming and outgoing propagation directions when those directions are not parallel or antiparallel. Polarization perpendicular or parallel to it is useful when expressing a differential scattering cross-section. The plane is not unique at exact forward or backward scattering.