Over a field of characteristic zero, let $w(x)$ be the unique formal power series with constant coefficient zero satisfying $w=x\varphi (w)$, where $\varphi (0)\ne 0$. Lagrange inversion gives, for $n\ge 1$,Here the brackets denote coefficient extraction. The theorem obtains coefficients of an implicitly defined series without solving for it explicitly.

For $w=x(1+w{)}^2$ and $F(t)=t$, it gives $[x^n]w=\left(\genfrac{}{}{0ex}{}{2n}{n-1}\right)/n=\left(\genfrac{}{}{0ex}{}{2n}{n}\right)/(n+1)$, the Catalan numbers for $n\ge 1$. This is useful when a recursive counting construction naturally gives an implicit generating function.

For a formal power series $f(x)={\sum }_{n\ge 0}{a}_n{x}^n$ over a commutative ring $R$, a reciprocal is a series $g$ with $fg=1$. It exists exactly when $a_0$ is a unit. The constant coefficient forces $b_0=a_0^{-1}$, and the higher coefficients are determined successively byThus the reciprocal is unique and uses no analytic convergence assumption. For example, $(1-x{)}^{-1}=1+x+x^2+\cdots$ in $R[[x]]$. Newton inversion of a formal power series accelerates the computation when many coefficients are needed.

The elbow is the joint region between upper arm and forearm, involving the humerus, radius and ulna. Its geometry supports flexion and extension while interacting with forearm rotation. A named joint region can therefore contain more than one articular relation.

The shoulder is the region connecting the proximal forelimb to the trunk. In human anatomy, movement involves several linked articulations and the shoulder girdle, rather than only the joint between humerus and scapula.

Motion of a previously pinned Abrikosov vortex when drive, thermal activation or other fluctuations overcome its trapping. The resulting flux motion can generate an electric field, so a transport critical current depends on the measurement's voltage and time criteria.

Slow motion of pinned superconducting vortices, commonly through thermally activated escape over energy barriers. It permits magnetic relaxation and a small dissipative voltage even below an ideal static depinning force. The barrier depends on current and pinning structure; a universal sharp transport threshold should not be assumed.

Live-cell imaging follows living biological cells over time. Labels, illumination and immobilization must be checked for effects on the process being measured; an apparently sharp image is not sufficient evidence that the cell remains physiologically unperturbed.

Total internal reflection fluorescence microscopy excites labels close to a boundary using the evanescent wave accompanying total internal reflection. Illumination travels through the higher-refractive index side, typically a coverslip, at an angle above the critical angle for total internal reflection. The field penetrates the specimen side but decays with distance from the interface.

A prism or a sufficiently high-numerical aperture microscope objective supplies the required incidence. This confines excitation near the surface and reduces background from deeper fluorescent material, making it useful for surface-associated molecular events. Changing incidence angle changes the depth weighting; an intensity change can therefore reflect vertical motion as well as molecule number.^[ref]

A floor sum adds values of a floor function, often in the form $F(n,a,b,m)={\sum }_{i=0}^{n-1}\lfloor (ai+b)/m\rfloor$ for integers $n\ge 0$, $m>0$, $a$, and $b$. If $a,b\ge 0$, it counts lattice points $(i,j)$ with $0\le i<n$ and $1\le j\le (ai+b)/m$. Splitting $a$ and $b$ into quotients and remainders isolates an arithmetic progression and a constant contribution; swapping the remaining counting axes leads to an Euclidean algorithm reduction.

An algorithm for evaluating sums that depend on $\lfloor N/i\rfloor$. If $q=\lfloor N/l\rfloor$, then all indices from $l$ through $r=\lfloor N/q\rfloor$ have quotient $q$. Jumping to $r+1$ visits only $O(\sqrt{N})$ blocks rather than all $N$ indices.

The operation returning $\lfloor a/b\rfloor$ for $b\ne 0$. It rounds the exact quotient toward negative infinity; for nonnegative integers and positive $b$, it is the quotient in Euclidean division.

For a positive integer $n$, indices with the same quotient $q=\lfloor n/i\rfloor$ form intervals. If an interval begins at $l$, it ends at $r=\lfloor n/q\rfloor$. There are $O(\sqrt{n})$ distinct quotients: either $i\le \sqrt{n}$, or the quotient is below $\sqrt{n}$. This permits summing constant-quotient blocks instead of visiting every index.

The temperature derivative of the Fermi occupation has opposite signs below and above $\mu$. Pairing energies $\mu \pm x$ leaves a positive contribution because the density of states increases with energy. The unpaired higher-energy tail is also positive. Hence the density increases at fixed $\mu$ for an unbounded three-dimensional parabolic band.

For $a,c>0$, $b\le 0$ and the molecular bound $Q\in [-1/2,1]$, coexistence with $Q=0$ occurs at $b_t=-{\min }_{0<Q\le 1}(a/Q+3cQ)$. Negative $Q$ has positive free energy and cannot be the stable phase in this coefficient range. Thus $a\le 3c$ gives $b_t=-\sqrt{12ac}$ and jump $0\to \sqrt{a/(3c)}$, whereas $a>3c$ gives $b_t=-(a+3c)$ and jump $0\to 1$. The interior case is the usual small-order-parameter Landau theory result. A formal unconstrained quartic minimization can violate the molecular bounds when extrapolated too far.

At coexistence, the landau free energy can factor as $F(\varphi )=\lambda {\varphi }^2(\varphi -v{)}^2/4$. The minima at $0,v$ have equal free energy. Differentiating shows the intermediate maximum at $v/2$, with barrier height $\lambda v^4/64$. The barrier is a local free-energy barrier; predicting spatial nucleation also requires interfacial or gradient information.

A thermal time constant characterizes exponential relaxation in a first-order lumped thermal model. For heat capacity $C$ and thermal conductance $G$ to a fixed bath, a small temperature deviation obeys $C\dot{\delta T}=-G\delta T$, givingAfter one $\tau$, the initial deviation has fallen to $1/e$ of its original value. The ratio has units of time because $C$ is measured in $\mathrm{J}{\mathrm{K}}^{-1}$ and $G$ in $\mathrm{W}{\mathrm{K}}^{-1}$.

A heating step provides an experimental check: the body approaches its final temperature rise with the same $\tau$ in this linear model. Greater heat capacity slows changes, while greater thermal conductance speeds exchange. Several coupled regions generally produce several relaxation modes, so fitting one exponential over a short interval does not establish a unique time constant for the entire apparatus.

The coefficient $k$ of a first-order concentration term in a rate law. Its units are inverse time; two reversible first-order directions contribute additively to the relaxation rate.

A first-order electric dipole transition connects opposite parities, but this does not prohibit all transitions between states of the same parity. In a symmetric infinite square well, use the centered position operator $X=x-L/2$. Each factor of $X$ reverses parity, so a second-order transition with two factors can connect the ground state to an excited state of the same parity through opposite-parity intermediate states.

For two-photon absorption, the second-order amplitude contains sums of products $⟨f|X|m⟩⟨m|X|i⟩$ divided by detuning-dependent energy denominators. Equal initial and final parity is therefore allowed, provided appropriate intermediate states and drive frequencies exist. “Allowed” does not guarantee a nonzero amplitude at every parameter value: different intermediate contributions can cancel. The distinction separates a symmetry-forbidden first-order matrix element from an all-orders prohibition.

Let $0<a<b$ and $y^{\prime }-\lambda y=A[H(t-a)-H(t-b)]$, with zero initial data. For $\lambda \ne 0$, the Laplace transform time-shift rule gives the displayed response, since the transform of the input is $A(e^{-as}-e^{-bs})/s$ and the equation supplies the factor $(s-\lambda {)}^{-1}$. Equivalently, integrate the impulse response $e^{\lambda t}H(t)$ over the active pulse. Before $a$ the solution is zero; during the pulse it is $A(e^{\lambda (t-a)}-1)/\lambda$; after $b$ it is $A{e}^{\lambda t}(e^{-\lambda a}-e^{-\lambda b})/\lambda$. The solution is continuous at both switching times. At $\lambda =0$ take the limit, giving linear growth during the pulse and the constant $A(b-a)$ afterward. Positive $\lambda$ gives growing memory, while negative $\lambda$ gives decay after forcing stops.

For flat expansion with a canonical scalar and no additional fluid, differentiating the Friedmann equation gives $\dot{H}=-4\pi G{\dot{\varphi }}^2\le 0$. Expansion can still accelerate if $-\dot{H}/H^2<1$. Equality at an isolated turning point does not imply eternal de Sitter expansion.