For a differential one-form $Pdx+Qdy$, seek a nonzero integrating factor $\mu (x)$. Expanding $(\mu P{)}_y=(\mu Q{)}_x$ gives the displayed equation where $Q\ne 0$. Such a factor exists locally when $(P_y-Q_x)/Q$ depends only on $x$, and is $\mu =\exp \int [(P_y-Q_x)/Q]dx$. Likewise, a factor depending only on $y$ obeys ${\mu }^{\prime }/\mu =(Q_x-P_y)/P$ where $P\ne 0$. The resulting product is an exact differential form on a sufficiently small regular domain; singular lines of the factor need separate treatment.

A collective coordinate labels a family of physically distinct saddle configurations, such as the center ${\tau }_0$ of an instanton. Differentiating the saddle with respect to that coordinate gives a zero mode of the quadratic fluctuation operator. An ordinary Gaussian integral over its coefficient would be divergent, because moving along the family costs no quadratic action.

Instead, change variables from the zero-mode coefficient to the collective coordinate and include the associated Jacobian. For a one-dimensional instanton with $m\int {\dot{q}}_I^{2}d\tau =S_I$, the standard normalized measure is $\sqrt{S_I/(2\pi \hslash )}d{\tau }_0$. The remaining fluctuations contribute a primed functional determinant, excluding this zero mode. Integrating over the allowed center positions supplies the physical observation-time factor rather than deleting the mode and losing that factor.

With $y_0=Y\sin (\pi x/L)$ and $F_c={\pi }^{2}EI/L^2$, the equilibrium branch connected to zero load has $y=Y\sin (\pi x/L)/(1-F/F_c)$. The pole as $F$ approaches $F_c$ marks breakdown of the small-deflection model, not a physically infinite beam deflection.

For positive loop resistance, coupled mechanical and circuit equations relax toward constant sliding speed on sufficiently long rails. The steady current balances the downhill gravitational force, and motional electromotive force balances the resistive voltage. The undamped zero-resistance limit has no relaxation, so its late-time limit is singular.

For $-V_0(\cos Qx+\cos Qy)$, two-dimensional bands are sums of one-dimensional bands. At first order, the first-band maximum is $E_Q/2-V_0$ at the corner, while the second-band minimum is $E_Q/4+V_0/2$ at an edge centre. Hence ${\Delta }_{\mathrm{ind}}\simeq 3V_0/2-E_Q/4$, giving a rough onset $V_0\sim E_Q/6$. Higher plane waves shift this estimate; insulating filling must be checked separately.

Matrices $\left(\begin{array}{cc}\alpha & \beta \\ {\beta }^{\ast }& {\alpha }^{\ast }\end{array}\right)$ with the displayed constraint form a real three-dimensional noncompact Lie group. A real Lie algebra basis is $m_1={\sigma }_x/2$, $m_2={\sigma }_y/2$, $m_3=i{\sigma }_z/2$, with $[m_1,m_2]=m_3$, $[m_2,m_3]=-m_1$, and $[m_3,m_1]=-m_2$.

For a direct-gap semiconductor with equal parabolic conduction-electron effective masses, restrict the final valence hole to zero crystal momentum and consider a normal process. Momentum conservation gives ${\mathbf{k}}_1={\mathbf{k}}_2+{\mathbf{k}}_3$, while energy conservation gives $E_1=E_g+\alpha (k_2^2+k_3^2)$ with $\alpha ={\hslash }^2/(2m^{\ast })$. The Cauchy-Schwarz inequality gives $k_2^2+k_3^2\ge k_1^2/2$, so $E_1\ge E_g+E_1/2$ and the threshold is $2E_g$. Equal final wavevectors attain the bound; opposite spins allow the two electrons to occupy that wavevector. This is the threshold under the stated hole constraint, not the general threshold when the final hole can carry momentum.

Magnetic hysteresis means that a material's magnetization depends on its field history as well as the present applied field. A closed quasistatic magnetic cycle dissipates energy per unit volume equal to its $H$–$B$ loop area. It is relevant to remanence in permanent magnets and core losses in electrical transformers.

For real $|a|<\pi$, integrate $g(z)=e^{az}/\cosh \pi z$ around a rectangle of height $i$ and width $2R$. Only the simple pole $i/2$ lies inside, with residue $e^{ia/2}/(\pi i)$. Since $g(x+i)=-e^{ia}g(x)$, the reversed top edge adds $e^{ia}$ times the bottom integral. The two vertical edges vanish as $R\to \infty$, using $|a|<\pi$. The residue theorem gives $(1+e^{ia})J=2e^{ia/2}$, proving the formula. The condition is also the exact convergence range: the two tails decay with exponents $\pi -a$ and $\pi +a$.

At fixed temperature, the continuum excited population of a three-dimensional ideal Bose gas reaches a finite maximum as the chemical potential approaches the ground energy. Adding particles beyond that maximum creates macroscopic ground occupation. Thus $N^{\prime }=\min (N,N_c)$ and $N_0=\max (0,N-N_c)$ to leading thermodynamic-limit order. Finite systems retain nonmacroscopic ground occupation below the nominal threshold.

A hard-wall cube has positive one-particle ground energy $E_0=3{\pi }^2{\hslash }^2/(2m{L}^2)$. Condensation corresponds to the absolute chemical potential approaching $E_0$, not zero. Shifting the energy reference makes the limiting chemical potential zero, while the exact shifted excited-state density of states also shifts. Keeping its bulk square-root form is a large-system approximation valid when $k_{B}T\gg E_0$. A periodic cube instead has zero ground energy.

For quadratic three-dimensional dispersion below condensation, the excited-state thermal energy is proportional to $T^{5/2}$. Differentiating at fixed particle number and volume gives the displayed heat capacity, with limiting value approximately $1.926N{k}_B$ at $T_c^{-}$. The macroscopic ground occupation carries no excitation energy in the shifted reference. The exponent differs from the linear-phonon result of an interacting gas.

A scale for the combined gravitational influence of two small orbiting bodies relative to their common host star. It is often used to express the separation of neighboring planetary embryos. For equal masses and nearly equal orbital radii it exceeds the individual Hill radius by $2^{1/3}$.

For $x=m_W^2/m_H^2<1/4$, the on-shell longitudinal pair contributes the displayed tree-level partial decay width. In the Higgs boson rest frame, its polarization vectors have contraction $(m_H^2/2-m_W^2)/m_W^2$. Multiplying by the Higgs boson coupling to W bosons gives $|{\mathcal{M}}_{LL}{|}^2=m_H^4(1-2x{)}^2/v^2$; the two-body Lorentz-invariant phase space then gives the width. The sum over transverse polarizations adds $8x^2$ to the longitudinal polynomial. Below threshold, physical decays through virtual W bosons require a different multiparticle calculation.

At leading order, neglecting phase-space suppression and radiative corrections, Higgs decay to a fermion pair has partial decay width proportional to $N_c^f{m}_f^2$, with $N_c=3$ for a quark and one for a lepton. A known sum of branching fractions therefore fixes the proportionality between these weights and individual branching fractions without requiring the Higgs vacuum expectation value. The same normalization estimates a small additional fermionic channel if its weight is not included in the known sum.

A nonzero Hermitian rank-one positive semidefinite matrix has zero determinant and positive trace, representing a future-directed null vector. It factors as $\chi {\chi }^{†}$, with an irrelevant spinor phase. A past-directed null vector instead requires a minus sign; a positive outer product alone does not cover both time directions.

The relative envelope delay accumulated by waves with different group velocities over length $L$. In nonlinear pulse gating this can blur the temporal overlap and limits the useful crystal thickness.

For a differentiable positive phase speed $c$ at positive angular frequency, assume $dk/d\omega \ne 0$. Since $k=\omega /c(\omega )$, differentiation gives $u=(dk/d\omega {)}^{-1}=c/[1-(\omega /c)dc/d\omega ]$. Using the period $P=2\pi /\omega$ gives the displayed formula. A positive period slope makes the group velocity smaller than the phase velocity on this branch. A slope read from a graph produces an approximate local result, not a globally fitted dispersion law.

The density-gradient contribution to the real stationary condensate equation after writing $\psi =\sqrt{\rho }{e}^{i\varphi }$. Neglecting it gives the bulk Thomas–Fermi density relation, but fails near a depleted interface or a vortex core.

The incremental radiative forcing per increment of a specified atmospheric gas burden. State whether the abundance unit is emitted mass, mole fraction or ppb, and specify the radiation-adjustment convention. For equal emitted masses in a fixed atmospheric air inventory, a per-ppb coefficient must be divided by the gas molar mass before comparing efficiencies. Spectral overlap with existing absorption makes efficiency depend on the background state.