For a compact $(N-1)$-dimensional link $L$ and traditional perversity $\bar{p}$, finite-chain intersection homology of its open cone is $I{H}_i^{\bar{p}}(cL)=I{H}_i^{\bar{p}}(L)$ for $i<N-1-\bar{p}(N)$ and zero for $i\ge N-1-\bar{p}(N)$. Below the cutoff both chains and possible filling chains avoid the vertex. At and above the cutoff the cone on any allowable cycle is an allowable filling: its new vertex is allowed precisely when $0\le i+1-N+\bar{p}(N)$.

A traditional perversity is a function $\bar{p}:\{2,3,\dots \}\to \mathbb{Z}$ satisfying $\bar{p}(2)=0$ and $\bar{p}(k+1)-\bar{p}(k)\in \{0,1\}$. It specifies how far chains may meet strata of codimension $k$.

An interpolant is a function satisfying the prescribed sample conditions $g(x_i)=y_i$. It agrees with the data exactly, but need not agree with an unknown underlying function between samples. The polynomial interpolation example shows that a family or a polynomial degree restriction can supply uniqueness; the McShane extension theorem shows a useful nonunique alternative.

A slowly established upward plane internal gravity wave has $\overline{u^{\prime }{w}^{\prime }}=-m|a{|}^2/(2k)$. The mean horizontal conservation of momentum equation gives ${\overline{u}}_t=-{\partial }_z\overline{u^{\prime }{w}^{\prime }}$. Using envelope transport and an initially vanishing mean gives $\overline{u}=-m|a{|}^2/(2k{c}_{gz})$. Its energy density per unit mass is $\mathcal{E}=(k^2+m^2)|a{|}^2/(2k^2)$, so $\overline{u}=(k/\omega )\mathcal{E}$. This is the mean flow left behind the advancing wave envelope, rather than a claim that an established uniform wavetrain has nonzero local stress divergence.

In a nonrotating ideal Boussinesq layer with uniform current $U>0$, constant buoyancy frequency $N$ and rigid walls, any stationary superposition of channel wave modes with $k_n^2+(n\pi /H{)}^2=N^2/U^2$ obeys the displayed relations. With $(u,w)=({\psi }_z,-{\psi }_x)$, the total streamfunction is $\Psi =Uz+{\psi }^{\prime }$ and total buoyancy is $\sigma =(N^2/U)\Psi$. Its material derivative vanishes. The perturbation vorticity is proportional to ${\psi }^{\prime }$, so its nonlinear self-advection vanishes as well. The modes are therefore exact finite-amplitude solutions, although large amplitudes may violate physical approximation assumptions or become unstable.

Incompressibility makes pressure adjust throughout the surrounding fluid. Projecting the linear Boussinesq equations perpendicular to a Fourier wavevector gives $w_t=[k_h^2/(k_h^2+m^2)]b$ and $b_t=-N^{2}w$. A localized spherical disturbance contains many wavevector directions and therefore many internal gravity wave frequencies, rather than one oscillator at $N$. Its shape evolves and energy radiates into the surroundings; a rigid-sphere added mass analogy illustrates extra inertia but does not give an exact coherent frequency for a free material blob.

For a lower uniformly stratified half-space and an upper well-mixed half-space, the vertical velocity amplitude is continuous and its derivative jumps by $[W^{\prime }]=-k^2\Delta bW/{\omega }^2$. A wave incident upwards with vertical phase factor $e^{-imz}$ therefore has upper evanescent amplitude $T/W_i=2im/[k(1-S)+im]$, where $m=k\sqrt{N^2/{\omega }^2-1}$ and $S=\Delta bk/{\omega }^2$. Its magnitude peaks at $S=1$, with amplitude twice the incident amplitude. Lossless reflection carries all incident energy back down; the evanescent upper field carries no mean outgoing vertical energy flux.

A thermodynamic force conjugate to an internal variable is the energetic driving quantity paired with that variable's rate in the dissipation. Its sign and density normalization must be specified. With specific internal energy $U(F,\eta ,\xi )$, reference mass density ${\rho }_0$ and the convention used here,Equivalently, if the specific Helmholtz free energy is $\psi (F,T,\xi )$, then $f_{\xi }=-{\rho }_0(\partial \psi /\partial \xi {)}_{F,T}$. The negative sign makes positive force-rate product correspond to free-energy decrease at fixed external variables.

For an isothermal free energy per reference volume $W=E(\epsilon -\xi {)}^2/2$, the driving force is $f_{\xi }=E(\epsilon -\xi )$. Choosing $\dot{\xi }=f_{\xi }/{\eta }_v$ with ${\eta }_v>0$ yields $f_{\xi }\dot{\xi }=f_{\xi }^2/{\eta }_v\ge 0$. The internal variable relaxes toward the imposed strain, reducing stored elastic energy. Such a law supplies the mechanical part of a Clausius-Duhem inequality check; the thermal contributions still need their own consistent description.

A dissipation potential is a function used to generate dissipative rates or their conjugate driving forces. In a force-based convention, a potential $\Phi (X)$ specifies an internal variable rate through $\dot{\alpha }={\partial }_X\Phi$. For example, $\Phi =X^2/(2\eta )$ with $\eta >0$ gives $\dot{\alpha }=X/\eta$ and dissipation $X\dot{\alpha }=X^2/\eta \ge 0$.

If $\Phi$ is differentiable and convex with a minimum at $X=0$, the convexity inequality gives $X{\partial }_X\Phi \ge \Phi (X)-\Phi (0)\ge 0$. This supplies a direct route to nonnegative dissipation. A rate-based convention instead generates forces from rates, so its variables must be stated explicitly. The potential is useful for organizing constitutive equations, but the complete model must still satisfy the Clausius-Duhem inequality, including thermal terms and the chosen energetic sign convention.

Integral-scale intermittency is modulation of the energy-containing motions and the energy supply on large lengths or long turnover times. Averaging velocity statistics over regions or realizations with differing viscous dissipation rates can invalidate universal coefficients based only on the global mean viscous dissipation.

Coarse-grained energy dissipation is a spatial average of the local rate $2\nu S_{ij}{S}_{ij}$ over a region of size $r$. Its random value ${ϵ}_r$ has mean $ϵ$ in homogeneous turbulence. Refined similarity uses this local averaged viscous dissipation to scale velocity increments.

Refined similarity writes $\Delta v=(r{ϵ}_r{)}^{1/3}V$ with universal normalized conditional statistics at high local Reynolds number. Then $S_p=C_p{r}^{p/3}⟨{ϵ}_r^{p/3}⟩$. The distribution of coarse-grained energy dissipation need not be universal. Refined similarity itself does not prescribe that distribution.

In the unprojected polynomial Neveu–Schwarz vertex sector, zero fermion insertions give the usual closed-string tachyon and the massless graviton, two-form and dilaton. Two insertions split into a left-right pair ${\psi }^i{\bar{\psi }}^j{e}^{ipX}$ of internal weights $(1/2,1/2)$ and a same-chirality current pair of weight one. The first has $M^2=-2/{\alpha }^{\prime }$ and 44-squared scalar components. The second requires a bosonic derivative on the opposite side for level matching and gives the two sets of 946 massless gauge vectors. Four-fermion scalar-current products belong to a different insertion count. A chosen global projection may retain a subset, so these operators do not by themselves specify a projected string spectrum.

Joule's paper On the Mechanical Equivalent of Heat (1850) describes a copper vessel holding water and a brass paddle with stationary vanes. Descending lead weights drove its shaft through strings and pulleys; the vanes opposed simple co-rotation. A boxwood shaft segment reduced thermal conduction. He measured fall distances and initial/final temperatures, repeating twenty descents per run and disconnecting the drive for rewinding. Calibrated thermometers read the water and room.^[ref]

His matched resting trials retained the water amount, duration, surroundings and reading procedure, omitting paddle motion to estimate environmental heat exchange. A screen reduced thermal radiation from his body. His calculation included the vessel and paddle heat capacities, external pulley/string losses, descent kinetic energy and string elasticity; it acknowledged residual vibration and sound.^[ref]

The measured observable is a corrected temperature rise, interpreted through the participating heat capacity; it is not a direct view of molecular motion. The experiment tests whether measured mechanical input accounts quantitatively for thermal storage. Frictional warming by itself would establish neither the conversion factor nor the absence of heat exchange. A modern authored demonstration at UCSB instead measures crank turns and a cylinder's temperature with an electrical thermometer: the apparatus differs, but again both mechanical input and thermal response must be measured. ^[ref]

Constant temperature does not generally mean constant internal energy. At fixed amount and composition, within one smooth thermodynamic phase of a simple compressible system, an equilibrium identity isThe bracket vanishes for the ideal gas law $pV=nRT$. It need not vanish for a real gas: changing the volume changes the average interaction potential energy, even when the random-motion contribution stays at the same temperature. The identity follows by writing $dU=TdS-pdV$ and using a Maxwell relation; it does not require the actual process to be reversible. ^[ref]

For a Van der Waals equation model with volume per mole $v=V/n$, $p=RT/(v-b)-a/v^2$ gives $(\partial U/\partial V{)}_{T,n}=a/v^2$. Thus an isothermal expansion from $v_1$ to $v_2>v_1$, at fixed amount of substance $n$, increases internal energy bySeparating attractive particles makes their interaction potential energy less negative. For stipulated $n=1\mathrm{mol}$, $a=0.10\mathrm{Pa}{\mathrm{m}}^6\mathrm{mo}{\mathrm{l}}^{-2}$, $b=10^{-5}{\mathrm{m}}^3\mathrm{mo}{\mathrm{l}}^{-1}$, $T=300\mathrm{K}$, $v_1=0.010$ and $v_2=0.020{\mathrm{m}}^3\mathrm{mo}{\mathrm{l}}^{-1}$, the calculated increase for this one-mole sample is $5\mathrm{J}$, despite $\Delta T=0$. This dilute stable-branch model example is not a measurement of a named gas. ^[ref]

A phase transition gives another reason. During equilibrium vaporization of a pure substance at fixed pressure, the temperature can stay at its phase coexistence value while the phase fraction changes. Molecular separation changes interaction potential energy.^[ref] If the closed system does only pressure-volume work, its latent heat input obeysHere $H=U+pV$ is enthalpy. A vaporization latent heat therefore is not automatically the internal energy increment: some input supplies expansion work. A fixed temperature reading alone cannot tell how much internal energy changed.

For an ordinary stationary laboratory sample, a useful convention isThe bulk kinetic energy $K_{\mathrm{bulk}}$ includes organized whole-body motion; for pure translation it is $M{v}_{\mathrm{cm}}^2/2$, with total mass $M$ and center of mass speed $v_{\mathrm{cm}}$. The external potential energy ${\Phi }_{\mathrm{ext}}$ includes elevation in an imposed gravitational field, approximately $Mgz$ in a uniform field. Accelerating a sealed bottle or lifting it without otherwise changing its state changes these mechanical terms, rather than its internal energy. Molecular kinetic energy is measured relative to the chosen bulk motion.^[ref]

State the convention rather than applying this separation blindly. If an experiment separately tracks a rotating apparatus or an organized flow, its mechanical kinetic energy must not also be counted in $U$. In Joule's paddle-wheel experiment, waiting for the apparatus and fluid to come to rest removes such an endpoint ambiguity. Including the vessel and paddle changes whose internal energy is measured; including the falling weights additionally puts their changing gravitational potential energy inside the enlarged system's total energy balance.

Interferometric visibility measures the contrast of a scanned interference fringe. For the mutually coherent, equally polarized two-beam model in wave interference, with fixed individual wave intensities and $I_1+I_2>0$, the constructive interference intensity and destructive interference intensity giveEqual nonzero wave intensities give $\mathcal{V}=1$; $I_1=1,I_2=0.25$ give $\mathcal{V}=0.8$. Unequal wave intensities can therefore reduce interferometric visibility even with perfect mutual optical coherence. Random phase differences or distinguishable wave polarizations can reduce it further; zero contrast alone does not identify the cause. Both intensities zero leave the ratio undefined.

For normalized paths with complex amplitudes $\alpha ,\beta$ and identical internal states, the output probability is $1/2+\mathrm{Re}({\alpha }^{\ast }\beta e^{i\theta })$, so $\mathcal{V}=2|\alpha \beta |$. A photon polarization fringe alone determines transverse quantum coherence but not the sign of the population difference.

// Target: optics.bigb

If the two arms have internal marker states $|a⟩,|b⟩$, an interference cross term is multiplied by $⟨a|b⟩$. Orthogonal markers remove interference from polarization-insensitive detection even if nobody reads the markers. Coherently making the two markers identical erases that distinguishing information and restores the cross term.

// Target: quantum-measurement.bigb

An interference fringe is a spatial band of enhanced or reduced wave intensity caused by a spatially varying phase difference. Uniform crossing plane waves produce straight parallel interference fringes; curved wavefronts can give curved bands. The interference fringes mark the relative wave phase, rather than the individual instantaneous wavefronts.

For the same mutually coherent, equally polarized monochromatic waves, the wave interference minimum occurs at phase difference $\delta =(2m+1)\pi$, with integer $m$. The wave amplitudes subtract, giving $I_{\min }=I_1+I_2-2\sqrt{I_1{I}_2}$. Complete cancellation requires equal individual wave intensities; $1$ and $0.25$ leave $0.25$ in their common unit. A dark band is a local minimum, and the average across an interference fringe spacing still obeys conservation of energy.