A resistive loop with self-inductance $L_r$ coupled by mutual inductance $M(z)$ to an imposed alternating current obeys $L_r{\dot{I}}_r+R{I}_r=-M{\dot{I}}_s$. For $I_s=I_0\cos \omega t$, averaging the magnetic force $I_s{I}_r{M}^{\prime }(z)$ gives $-I_0^2{\omega }^{2}M{L}_r{M}^{\prime }/[2(R^2+{\omega }^2{L}_r^2)]$. It is repulsive where $M{M}^{\prime }<0$. The self-inductance supplies the current component opposite to the source current; a purely resistive, zero-inductance loop has zero average force in this model.

An N-turn square coil of side a lies in the plane of a long straight wire, with nearest edge at perpendicular distance s and parallel to the wire. Integrating $B={\mu }_{0}I/(2\pi r)$ across the coil gives flux-linkage magnitude $\Lambda ={\mu }_{0}NaI\ln [(s+a)/s]/(2\pi )$. Its coefficient of I is the mutual inductance magnitude; its sign depends on the selected loop normal and current directions. Flux through one turn and N-turn flux linkage must be distinguished.

One of a repeated series of axial muscle units, useful in comparing early chordate organization.

The proportion of entries in an alignment column, or columns in a pairwise alignment, that are counted as gaps under a stated convention.

For an amino acid alignment, this relative entropy measures departure of the nongap empirical distribution in a column from the uniform distribution on the twenty standard amino acids.

Index notation describes vectors, matrices and tensors by their components relative to a chosen basis. For example, matrix multiplication reads $(AB{)}_{ij}={\sum }_k{A}_{ik}{B}_{kj}$. The free indices $i,j$ identify the resulting component, while $k$ is summed. An explicit summation sign avoids depending on an implicit Einstein summation convention.

For $L(t)=vt$ and $\alpha =3{\pi }^2\hslash /(2m{v}^2)$, first-order excitation out of the instantaneous ground state has the displayed oscillatory integral. Its lower endpoint means the limit of a ground-state preparation at positive time and positive width; no normalized state exists in a zero-width well. Substitution $u=\alpha /s$ makes convergence explicit, and $P_{1\to 2}\sim 16t^2/(9{\alpha }^2)$ as $t/\alpha \to 0$. Large excitation invalidates the first-order approximation.

For a photon emitted from the origin and arriving at receding mirror position $x(t)$, its emission time is $t_e=t-x(t)/c$. Thus $d{t}_e/dt=1-\beta$ and a steady source rate $R$ gives impact rate $R(1-\beta )$ in source-frame coordinate time. This is distinct from the emission rate and from a rate per mirror proper time.

For an n-channel MOSFET, assume a gradual channel, constant charge carrier mobility $\mu$, gate capacitance per area $C_{\mathrm{ox}}$, width $W$ and length $L$. With overdrive $V_{\mathrm{ov}}=V_{GS}-V_T>0$, the inversion-sheet charge magnitude is $C_{\mathrm{ox}}(V_{\mathrm{ov}}-V(x))$. The drift-current relation $I_D=\mu W{C}_{\mathrm{ox}}(V_{\mathrm{ov}}-V)V^{\prime }$ is constant along the channel. Integrating from $V=0$ to $V_{DS}$ over length $L$ proves the displayed expression for $0\le V_{DS}\le V_{\mathrm{ov}}$. At the drain pinch-off condition $V_{DS}=V_{\mathrm{ov}}$, it yields $I_{D,\mathrm{sat}}=\mu C_{\mathrm{ox}}W{V}_{\mathrm{ov}}^2/(2L)$. This ideal saturation neglects channel-length modulation and short-channel velocity saturation.

The Monte Carlo standard error estimates the standard deviation of a simulation-based estimate across repeated simulation runs. For an average of $N$ independent random variables with finite variance ${\sigma }_h^2$, it is ${\sigma }_h/\sqrt{N}$. Under a suitable central limit theorem for Markov chain Monte Carlo, correlated draws instead give the approximation ${\sigma }_h/\sqrt{N_{\mathrm{eff}}}$, where $N_{\mathrm{eff}}$ is the effective sample size of a Markov chain for the particular estimated function $h$. This measures numerical uncertainty in an estimate, not uncertainty in the unknown parameter itself.

For downward pre-ejection speed $v_0$, total mass $M_0$, payload mass $m_p$, and downward payload speed $v_p$ in the original inertial frame, the remaining body's upward speed is $(m_p{v}_p-M_0{v}_0)/(M_0-m_p)$. When the numerator nearly cancels, distinguishing total initial mass from final body mass is essential.

In pyramidal Ammonia, the hydrogen symmetry orbitals span $A_1\oplus E$ of the C3v molecular point group. Nitrogen $2s$ and $2p_z$ both have $A_1$ symmetry and $(2p_x,2p_y)$ span $E$. In the planar D3h molecular point group, these become $A_1^{\prime }$, $A_2^{\prime \prime }$ and $E^{\prime }$, respectively, while the hydrogen combinations span $A_1^{\prime }\oplus E^{\prime }$. Thus the planar out-of-plane lone pair is a symmetry-isolated nonbonding orbital; pyramidalization permits mixing with the other $a_1$ functions.^[ref]

A quasimodular form is a holomorphic function on the upper half-plane whose transformation under a modular substitution differs from the modular form law by a polynomial correction in $c/(c\tau +d)$, together with suitable growth conditions at cusps. The correction has finite degree, called the depth; depth zero gives an ordinary modular form.

The basic example is $E_2(\tau )=1-24{\sum }_{n\ge 1}{\sigma }_1(n)q^n$, with $q=e^{2\pi i\tau }$. It obeysfor $\left(\begin{array}{cc}a& b\\ c& d\end{array}\right)\in {\mathrm{SL}}_2(\mathbb{Z})$. The extra term is why it is quasimodular rather than modular. Quasimodular forms accommodate derivatives of modular forms, whose chain rule introduces precisely this kind of correction.

For $m,\beta >0$, set $K_m(r)=e^{-mr}/r$. Distributionally, $(\Delta -m^2)K_m=-4\pi \delta$, so the modified Helmholtz equation with unit delta source has Green function $-K_m/(4\pi )$. The difference $[K_{\beta }-K_m]/({\beta }^2-m^2)$ solves the equation with source $K_{\beta }$: the two delta terms cancel. It has no added point source at zero and decays at infinity. The Green function representation proves the displayed convolution with its positive sign.

At coincident rates, the continuous limit is $e^{-mr}/(2m)$; at $r=0$, it is $1/(m+\beta )$. The latter also follows by directly integrating $e^{-(m+\beta )r^{\prime }}d{r}^{\prime }$ after the spherical factors cancel. These limits prevent the formula's apparent singular denominators from being mistaken for physical divergences.

Let $\mu >0$ and let $u$ be a real smooth solution on a bounded regular domain with zero Dirichlet boundary conditions. Green's first identity givesThe two sides have opposite signs unless both vanish, so $u=0$. This energy method also proves uniqueness for prescribed boundary data by applying it to the difference of two solutions. For complex solutions use the complex conjugate in the integral. This argument does not apply with the same sign to the Helmholtz equation, which can have nonzero zero-boundary eigenfunctions.

A low-acceleration modification associated with Mordehai Milgrom. For an isolated spherical system, $g\mu (g/a_0)=g_N$, with $\mu (x)\to 1$ at high $x$ and $\mu (x)\to x$ at low $x$. An isolated point-like baryonic distribution then gives $v_c^4=G{M}_b{a}_0$ and a flat asymptotic galaxy rotation curve. The algebraic spherical relation is not a complete relativistic field theory and should not be assumed to hold vectorially for arbitrary mass distributions.^[ref]