Project a bosonic field on a ring onto angular-momentum modes zero and one. The kinetic level spacing is $\mathcal{E}={\hslash }^2/(2m{R}^2)$. The contact-interaction integral enforces angular momentum conservation, leaving two same-mode terms and four mixed terms. The displayed projected Quantum Hamiltonian follows by normal ordering. Couplings to discarded modes belong to the full model and can limit the two-mode approximation.

A low-energy contact coupling is defined through a physical scattering length. Ultraviolet pieces of zero-point energy cannot be interpreted independently of the matching or counterterms fixing that coupling.

Let $r=(\gamma /A)(n/2{)}^{1/3}$ and $P=(n_{\uparrow }-n_{\downarrow })/n$. In units $2A(n/2{)}^{5/3}$ the spin-dependent energy is $[(1+P{)}^{5/3}+(1-P{)}^{5/3}]/2-r(1+P^2)/2$. Comparing only $P=0$ and $P=1$ gives the endpoint crossing $r=2^{5/3}-2$. Expanding near $P=0$ gives a quadratic coefficient $5/9-r/2$, so the unpolarized state first loses local stability at $r=10/9$. The endpoint crossing is therefore not the actual continuous onset; intermediate polarizations matter.

A contact line can remain pinned while the observed contact angle changes between receding and advancing limits. Surface roughness and chemical heterogeneity can produce this hysteresis. Slowly adding or removing liquid from a sessile drop measures advancing and receding angles separately. An observed pinned angle therefore need not be the single equilibrium angle predicted by Young's equation for a perfectly homogeneous surface.

The planar interfacial balance ${\gamma }_{SV}-{\gamma }_{SL}={\gamma }_{LV}\cos \theta$ for the equilibrium contact angle through the liquid, with interfacial free energies defined per actual area. Roughness and retained air require additional area accounting.

Represent feasible variables using unconstrained coordinates. For example, $x=a+e^y$ enforces $x>a$, while $x=a+y^2$ includes $x=a$ but introduces a stationary point at $y=0$. A transformation can change conditioning and does not remove a nonconvex objective's multiple extrema.

For alternatives represented by projection operators at successive times, set $C_{\alpha }=P_{{\alpha }_n}(t_n)\cdots P_{{\alpha }_1}(t_1)$ in the Heisenberg picture. The displayed functional uses the initial density operator $\rho$. Combining two histories gives $D(\alpha +\beta ,\alpha +\beta )=D(\alpha ,\alpha )+D(\beta ,\beta )+2\mathrm{Re}D(\alpha ,\beta )$. Thus vanishing real cross-terms gives the probability sum rule; the stronger commonly used condition $D(\alpha ,\beta )=0$ for distinct histories is sufficient as well. Its diagonal is nonnegative because each $C_{\alpha }\rho C_{\alpha }^{†}$ is positive.

Conservative cloud scattering produces no material heating when each interaction preserves photon energy and the model includes no absorption or thermal emission. This global constraint does not imply that an approximation retaining only one scattering event predicts the correct outgoing angular distribution.

For example, a thick cloud can redirect a photon many times before escape. A single-scattering calculation can miss that redistributed intensity or violate the total energy budget if scattered light is simply removed after its first interaction. Conversely, a method can enforce total energy conservation yet still predict a poor specific intensity in a chosen direction. Conservation and directional accuracy are separate tests of a radiative transfer calculation.

A conducting conical pendulum connected to a grounded pivot rotating with signed angular velocity $\omega$ in a uniform vertical magnetic field $B$ develops a motional electric potential difference $B\omega r^2/2$ between the rotation axis and a bob at radius $r$. If the bob dominates the capacitance, its electric charge is $CB\omega r^2/2$. An additional uniform vertical electric field changes both the force on that charge and, while the grounding connection remains, the background electric potential entering the charging relation. These effects must be treated consistently when calculating a first-order mechanical response.

A geometric congruence class is an equivalence class of figures under geometric congruence: two figures represent the same class when a Euclidean isometry maps one to the other. It forgets position and orientation but retains lengths and angles. For example, every triangle with side lengths $3,4,5$ represents the same class, by side-side-side congruence. A mirror image belongs to the same class when reflections are allowed; restricting to orientation-preserving motions can distinguish the two orientations of an asymmetric labelled figure.

For a coaxial cylinder standing on the base of a right circular cone of height $h>0$ and semi-vertical angle $0<\alpha <\pi /2$, similar triangles give radius $r=(h-x)\tan \alpha$ at cylinder height $x$. Its volume is $V=\pi x(h-x{)}^2{\tan }^2\alpha$. Differentiation gives $V^{\prime }=\pi {\tan }^2\alpha (h-x)(h-3x)$, so the unique positive-volume maximum has $x=h/3$ and occupies $4/9$ of the cone's volume. Its curved surface area is instead $A=2\pi x(h-x)\tan \alpha$, maximized at $x=h/2$. Changing the optimized quantity changes the optimal shape, even though the geometric constraint is identical.

If the lowest allowed angular harmonic has a nonzero coefficient, differentiating its potential gives the displayed field scale. The field vanishes at a right-angle recess, stays finite at a smooth face, and diverges at a protruding conducting corner whose exterior vacuum angle exceeds $\pi$. A rounded tip cuts off the ideal power law at its radius scale. If the first coefficient vanishes, the first nonzero higher harmonic determines the exponent instead.

For a piston advancing slowly into a gas, the arrival rate of molecules differs slightly from that at a stationary wall. In calculating compression work from molecular impacts to first order in piston speed $u$, however, the stationary molecular arrival flux is sufficient: the energy change per reflected molecule is already proportional to $u$.

A molecule with incoming normal velocity $v_x$ acquires leading energy $2mu{v}_x$. Multiplying by the stationary arrival weighting proportional to $v_x$ and averaging over an isotropic distribution gives the pressure work rate $pSu$, where $S$ is piston area. The first-order correction to the arrival flux multiplies an order-$u$ energy change and contributes only at order $u^2$. This ordering requires piston speed small compared with the typical molecular speed; it does not justify the same approximation for rapid compression or for the force correction itself.

In a smooth fixed-composition thermodynamic region, write $a=(\partial S/\partial T{)}_V$, $b=(\partial S/\partial V{)}_T$, $c=(\partial p/\partial T{)}_V$, and $d=(\partial p/\partial V{)}_T$, with $a,d\ne 0$. The chain rule gives $(\partial S/\partial T{)}_p=a-bc/d$ and $(\partial p/\partial V{)}_S=d-bc/a$. Multiplying these expressions by $d$ and $a$, respectively, gives the same Jacobian determinant $ad-bc$.

Using heat capacity at constant pressure $C_p=T(\partial S/\partial T{)}_p$, heat capacity at constant volume $C_V=Ta$, isothermal compressibility ${\kappa }_T=-1/(Vd)$ and isentropic compressibility ${\kappa }_S=-1/[V(\partial p/\partial V{)}_S]$ proves the displayed ratio. The derivation does not require division by thermal expansion, so it still applies when that derivative vanishes. For a stable single phase, the Maxwell relation $b=c$ also explains why $C_p\ge C_V$ and ${\kappa }_T\ge {\kappa }_S$.

The static density susceptibility is the linear response of number density to a spatially varying chemical potential:Here $\rho$ denotes particle number per volume, not mass density, and $\mathbf{q}$ is a wavevector.

For a uniform weakly interacting Bose-Einstein condensate at zero temperature, the Gross–Pitaevskii equation gives a quadratic static energy for a small density perturbation:with repulsive contact interaction $g>0$, background number density ${\rho }_0$ and particle mass $m$. Stationarity givesThusThe expression without ${\hslash }^2$ uses units $\hslash =1$. The gradient term suppresses response at short wavelengths; the inverse susceptibility is the density stiffness.

At $q=0$, $(\partial \rho /\partial \mu {)}_T=1/g$ within this mean-field model. The isothermal compressibility additionally contains the density normalization:Thermal and interaction corrections beyond the stated zero-temperature approximation can change this response.

A global phase symmetry sends a complex field to $e^{i\alpha }\varphi$ with the same constant real $\alpha$ everywhere. It is distinct from allowing a spacetime-dependent phase together with a compensating gauge-field transformation. A global phase symmetry can produce a conserved Noether current and, in an appropriate ordered mean-field state, a continuum of degenerate phase choices.

Arithmetic operations on complex numbers. Writing two inputs as $a+bi$ and $c+di$ turns multiplication into $(ac-bd)+(ad+bc)i$ and addition into $(a+c)+(b+d)i$. This is distinct from the specialized theory of complex multiplication of elliptic curves.

Multiplication is defined by $(a+bi)(c+di)=(ac-bd)+(ad+bc)i$. In polar form, it multiplies magnitudes and adds arguments. This arithmetic operation is distinct from elliptic-curve complex multiplication.

For $z\ne 0$, every logarithmic value is $\log |z|+i(\arg z+2\pi k)$ with integer $k$. Multiplying by $z$ and taking the real part proves the displayed formula for the modulus of each value of complex exponentiation. For nonreal $z$, these values form a geometric progression with nonunit positive ratio $e^{-2\pi \mathrm{Im}z}$. For real nonzero $z$, changing the logarithm branch changes only the phase and leaves the modulus fixed. A fixed initial argument changes only the indexing of the set of values.