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The polar molecular field is the functional derivative of the free energy with respect to the polar order parameter. In the positive-derivative convention, dissipative dynamics is $D_t\mathbf{p}=-\Gamma \mathbf{h}$ with positive kinetic coefficient. For a quartic local term and an isotropic gradient penalty,Some authors instead define the molecular field with a minus sign, which changes the associated stress and evolution conventions.

A binary planetesimal is a gravitationally bound pair of planetesimals. Its internal two-body problem and its centre of mass orbit respond differently to external forces, including gas drag.

The geometric collision cross-section in a coplanar calculation is a length $2R_p$, rather than an area. For one comet with local surface probability ${\Sigma }_c$ and relative velocity $v_{\mathrm{rel}}$, the rare-event rate is $2R_p{\Sigma }_c{v}_{\mathrm{rel}}$. Without gravitational focusing, the resulting mean collision time scales as $Q^{3/2}{M}_p^{-1/3}{a}_p$ for deep planet-crossing orbits at fixed stellar mass and planetary mass density.

A gravitational assist ejects a weakly bound comet when the increase in its specific orbital energy exceeds its binding energy. For near-parabolic coplanar encounters, the weak-deflection ejection impact parameter scales as $(M_p/M_{\star })Q$. Combining this with the phase-mixed comet surface density gives a one-encounter ejection time proportional to $Q^{1/2}$, within the local two-body encounter approximation.

In planar bond percolation, declare a dual edge open exactly when the primal edge it crosses is closed. Independent primal parameter $p$ then gives independent dual parameter $1-p$.

Between two prescribed values, linear interpolation assigns $(1-t)f(a)+tf(b)$ at $(1-t)a+tb$, for $0\le t\le 1$. Joining successive samples produces a piecewise linear function. For a continuous function on a compact interval, interpolants along grids with mesh tending to zero converge in the uniform norm; their error is bounded by the modulus of continuity at the mesh size.

The traditional near- and thermal-infrared photometric bands in the 1–5 micrometre region are $J,H,K,L,M$, with approximate centers $1.25,1.65,2.2,3.5,4.8$ micrometres. Variants such as $K_s$, $L^{\prime }$ and $M^{\prime }$ have different detailed responses. They largely follow atmospheric windows; transmission, sky brightness, and detector response determine practical band edges. The ISAAC filter documentation records instrument-specific variants, including an $L$ filter closer to $L^{\prime }$ near 3.8 micrometres.

For ${\varphi }_K(x-X)=c\tanh (c(x-X))$ in the unit kinetic normalization, $\int ({\varphi }_K^{\prime }{)}^{2}dx=M=4c^3/3$. Substitution of $X(t)$ therefore gives the free-particle collective-coordinate effective Lagrangian $-M+M{\dot{X}}^2/2$. Its quantization has momentum $p$ and energy $M+p^2/(2M)$ to this order. The exact uniformly moving classical kink is a Lorentz boost of the static one, with energy $M/\sqrt{1-v^2}$.

Linearizing about the phi-four kink gives the displayed one-dimensional Schrödinger operator in unit kinetic normalization. Its translational zero mode in field theory is proportional to ${\mathrm{sech}}^2(cx)$. The localized shape eigenfunction $\mathrm{sech}(cx)\tanh (cx)$ has squared frequency $3c^2$, while the continuum starts at $4c^2$. Substitution verifies both bound-state eigenfunctions. The Pöschl-Teller potential permits a short completeness argument for the bound modes. In $y=cx$ let $A_{\ell }={\partial }_y+\ell \tanh y$. Then $H_K/c^2=A_2^{†}{A}_2$, its partner is $A_2{A}_2^{†}=A_1^{†}{A}_1+3$, and $A_1{A}_1^{†}=-{\partial }_y^2+1$. The last operator has no bound states. The kernels of $A_1$ and $A_2$ give the two stated modes, while the partner spectra exclude any further normalizable bound modes.

The four-point tree-level Feynman diagram has two cubic vertices joined by one propagator. The three ways to divide four labeled external legs into vertex pairs give the $s$, $t$, and $u$ channels. For interaction $+\lambda {\varphi }^3/3!$, the amplitude is $\mathcal{M}=-{\lambda }^2[(s-m^2{)}^{-1}+(t-m^2{)}^{-1}+(u-m^2{)}^{-1}]$, with the Feynman i-epsilon prescription understood. All channels interfere in its modulus squared.

A canonically normalized real scalar field in six dimensions has mass dimension two. The cubic coupling is classically a marginal coupling, whereas a local quartic coupling has mass dimension minus two. This theory is perturbatively renormalizable by power counting in quantum field theory, but a real cubic Euclidean potential is unbounded below. Its expansion about a massive Gaussian theory is therefore formal perturbation theory, not a convergent positive functional integral at nonzero real $g$.

At fixed temperature, choose a coexistence pressure $p_{\ast }$ such that the gas and liquid have equal chemical potentials. Because $d\mu =vdp$, this gives ${\int }_{v_l}^{v_g}vdp=0$, or equivalently ${\int }_{v_l}^{v_g}[p(v,T)-p_{\ast }]dv=0$. The equal areas replace the unstable part of an isotherm by a horizontal coexistence segment.

For a bulk free-energy density $f(\varphi )$, two compositions coexist across a flat interface whenThe first condition equates the chemical potentials, and the second equates the pressure. Equivalently the same straight line of slope $\mu$ is tangent to the graph of $f$ at both compositions. A stable coexistence pair uses a supporting tangent, so the resulting mixture lowers the bulk free energy.

Porosity is the fraction of a representative bulk material volume occupied by voids:It is dimensionless and measures storage space. A sample with bulk volume $1.0\mathrm{L}$ and pore volume $0.25\mathrm{L}$ has porosity $\varphi =0.25=25\%$. If all its pores are saturated by one liquid, the liquid occupies $0.25\mathrm{L}$; partial fluid saturation $s$ instead gives liquid volume $s\varphi V_{\mathrm{bulk}}$.

Porosity and permeability of a porous medium answer different questions: how much fluid fits, and how easily it flows. Isolated bubbles or cavities can contribute to total porosity while providing no throughgoing path. Even connected passages may be narrow enough to transmit very little discharge. In the simple fully participating saturated model, Darcy velocity is $\varphi$ times pore velocity; changing porosity at a stipulated fixed Darcy flux changes ideal advective travel time, but porosity alone cannot predict the Darcy flux.

For a universal abundance depending on halo peak height, the peak-background split gives a potential-response coefficient $b_{\phi }=2f_{\mathrm{NL}}{\delta }_c{b}_L$, with $b_L$ the linear Lagrangian halo bias. If $\delta =\alpha (k)\phi$, the corresponding deterministic galaxy bias correction is $\Delta b=b_{\phi }/\alpha (k)$, proportional to $k^{-2}$ on sufficiently large scales. For a physical Eulerian tracer, $b_E=1+b_L$, so $b_{\phi }=2f_{\mathrm{NL}}{\delta }_c(b_E-1)$. The deterministic tracer power spectrum is $(b_E+\Delta b{)}^2{P}_m$; it remains nonnegative even if a negative correction changes the sign of the tracer-matter correlation.

Initialize the PC skeleton search with an estimated conditional independence graph to reduce candidate pairs and conditioning sets. An exact graph contains the true Directed acyclic graph skeleton because it equals its moral graph under faithfulness. For initially absent pairs record the full complement as a separating set; an empty default can create false collider orientations. Estimation errors can affect this screening step.

With an exact conditional independence oracle and faithfulness of a directed acyclic graph, the skeleton phase of the PC algorithm deletes precisely the nonedges. A true edge cannot be D-separated by a set excluding its endpoints. For nonadjacent vertices, choose the later vertex $v$ in a topological ordering; its parents separate it from the earlier nonparent. True parent edges survive throughout, so this separator remains available among the algorithm's candidate neighbour sets. For an unshielded triple $u-v-w$, any recorded separator of $u,w$ excludes $v$ exactly when the triple is an unshielded collider. The skeleton and collider characterization of Markov equivalence therefore identifies the correct equivalence class. Directions compelled throughout that class give its completed partially directed acyclic graph.