The fixed-time regular perturbation is $(1-ϵt)\sin t+O({ϵ}^2)$. Retaining the exponential decay and the first accumulated phase correction gives $e^{-ϵt}[\sin t-{ϵ}^{2}t\cos t/2]+O({ϵ}^2)$ on $ϵt=O(1)$. When ${ϵ}^{2}t=O(1)$ its phase series is unordered. For positive damping the absolute error is nevertheless exponentially small on that later scale, so phase failure and failure in absolute displacement must be distinguished.

The Dirac-Born-Infeld action describes the leading derivative dynamics of one D-brane, including nonlinear dependence on slowly varying worldvolume field strengths. Its induced tensors are D-brane worldvolume pullbacks. The combination $P[B]+2\pi {\alpha }^{\prime }F$ is invariant under the combined two-form and boundary gauge transformation. The dilaton factor reflects a disk's Euler characteristic. Expanding about flat space with small field strengths and embedding gradients gives the worldvolume gauge and scalar kinetic terms. A coincident stack has a non-Abelian low-energy Yang-Mills action; this does not imply that a naive traced determinant is exact at every order.

A D0-brane is a point-like D-brane in space, with a one-dimensional worldvolume consisting only of time. It belongs to type IIA superstring theory and couples electrically to its one-form Ramond–Ramond potential. “Zero” counts spatial dimensions, rather than asserting zero mass or absence of dynamics.

Its mass scales as $1/(g_s\sqrt{{\alpha }^{\prime }})$, so it is heavy at weak string coupling. Unlike an ordinary perturbative string excitation, its open-string endpoints are fixed in every spatial direction for a static brane, while the brane's own position is a dynamical collective coordinate. The low-energy description of several nearby D0-branes is quantum mechanics of their worldvolume fields; it is not a field theory with an additional extended spatial direction.^[ref]

A D1-brane has one spatial dimension and couples to the two-form Ramond–Ramond potential. Its tension differs from the fundamental string tension by a factor $g_s^{-1}$ at zero axion.

A D2-brane is a membrane-like D-brane with two spatial directions and a three-dimensional worldvolume including time. It is a charged extended object of type IIA superstring theory, coupling electrically to a three-form Ramond–Ramond potential.

The three-form can be integrated over its worldvolume to give the charge coupling, just as a one-form electromagnetic potential is integrated along a charged particle's trajectory. Open strings ending on the membrane supply a worldvolume gauge field and fields describing transverse motion. The relation between this three-dimensional gauge field and a scalar is useful in connecting the D2 description with an M2-brane, while the bulk and worldvolume descriptions address different degrees of freedom.

A D3-brane has three spatial dimensions. The low-energy fields of an isolated flat D3-brane form an abelian theory with sixteen linearly realized supersymmetry generators. Its transverse scalar fields describe its geometric position, and its gauge field can carry the charges of strings ending on it.

A D4-brane is a D-brane with four spatial directions and a five-dimensional worldvolume, belonging to type IIA superstring theory. It supports the low-energy modes of open strings ending on it, while also coupling to bulk closed-string fields.

Its electric coupling uses a five-form Ramond–Ramond potential, dual to the three-form potential coupled electrically to a D2-brane. Thus brane dimension determines which differential-form field measures its charge, not merely the size of a geometric object. At long wavelengths, an isolated brane's slowly varying fields are described by a Dirac-Born-Infeld action, with transverse scalar fields encoding its position and fluctuations.

A D6-brane is a D-brane with six spatial dimensions. Its seven-dimensional worldvolume includes time, leaving three transverse directions in ten-dimensional type IIA superstring theory. Open-string endpoints can move along the brane while satisfying fixed-position Dirichlet boundary conditions in the transverse directions.

It couples electrically to the seven-form description of a Ramond–Ramond potential, equivalently carrying magnetic charge for the IIA one-form potential. This electric–magnetic distinction relates the D6-brane to the D0-brane: their relevant field strengths are dual in ten dimensions. The open-string description supplies worldvolume fields, whereas the closed-string description treats the same brane as a source for bulk fields.

A D8-brane is a D-brane with eight spatial directions and a nine-dimensional worldvolume. In ten-dimensional type IIA superstring theory, it has one transverse spatial direction, so it acts as a domain wall. Its charge is associated with a nine-form Ramond–Ramond potential.

The dual zero-form field strength is the Romans mass; its value can jump across a charged D8-brane. Consequently a D8-brane cannot simply be inserted into the elementary massless M-theory circle duality dictionary used for lower-dimensional IIA branes. Its gravitational matching and scalar-field profiles must be treated consistently on both sides of the wall.^[ref]

Identical, parallel, equally oriented type-II D-branes preserve the same sixteen spacetime supercharges. Their annulus oscillator sum has the displayed numerator, which vanishes by the Jacobi abstruse identity. In the closed-string channel this is cancellation of attractive NS-NS exchange and repulsive RR exchange. The force vanishes in the flat supersymmetric configuration; anti-oriented branes, relative angles or supersymmetry-breaking backgrounds need not have this cancellation.

For the asymptotically flat extremal D$p$ solution with $p<7$, $H=1+Q/r^{7-p}$ is harmonic away from the source and $e^{\Phi }=g_s{H}^{(3-p)/4}$. The electric potential can be chosen as $C_{0\cdots p}=g_s^{-1}(H^{-1}-1)$; magnetic descriptions use the dual flux, and a D3-brane has self-dual five-form flux. An equally oriented probe has DBI density $-{\mu }_p{g}_s^{-1}{H}^{-1}$ and Wess-Zumino density $+{\mu }_p{g}_s^{-1}(H^{-1}-1)$, whose sum is position independent. This proves the corresponding parallel D-brane no-force identity in the classical background. The supergravity approximation requires controlled curvature and coupling; it does not resolve every brane core.

A type-IIB D-brane with seven spatial dimensions. Its RR charge is magnetic with respect to $C_0$, or electric with respect to the dual eight-form potential. Its transverse geometry is codimension two, so the simple asymptotically flat $r^{p-7}$ harmonic formula for lower-dimensional D-branes does not apply unchanged.

A space-filling type-IIB D-brane. The associated ten-form potential is nonpropagating; its charge is not an additional local massless polarization. A consistent compact configuration must cancel any net space-filling RR charge, for example through appropriate other charged objects or projections.

A Euclidean type-IIB D-brane localized in all spacetime directions, coupling electrically to the RR scalar $C_0$. It is an instanton, not a Lorentzian particle with negative spatial dimension. The magnetic partner of that scalar charge is a D7-brane.

A type-IIB D-brane with five spatial dimensions. Its RR charge is magnetic with respect to the two-form potential $C_2$, or electric with respect to the dual six-form potential. It is distinct from the NS5-brane, whose charge is magnetic under the NS-NS two-form and whose string-frame tension has a different power of the coupling.

A standard positive-total form of the Dvoretzky-Motzkin cycle lemma uses a nonempty sequence $a=(a_1,\dots ,a_n)$ of integers satisfyingExactly $k$ starting positions have every nonempty partial sum strictly positive, using the same rotation convention as the sum-one cycle lemma. Negative entries can be arbitrarily large in magnitude, and zero entries are allowed. Addario-Berry's proposition 2.2.1 gives an equivalent form with increments at least $-1$ and negative total; reversing the sequence and changing all signs gives the form here. ^[ref]

Four starts, two successes. For $a=(1,1,1,-1)$, the sum is $k=2$. Starts $0$ and $1$ give partial sums $(1,2,3,2)$ and $(1,2,1,2)$, respectively. Start $2$ gives $(1,0,1,2)$ and touches zero; start $3$ gives $(-1,0,1,2)$ and begins below zero. Thus exactly two starts are good. The rotated sequence $(-1,1,1,1)$ has the same positive total, but its first start is bad: a positive final lead alone does not control earlier partial sums.

Positions can give identical words. The sequence $(1,1,-1,1,1,-1)$ repeats a block of length three and has sum two. Its good starts are $0$ and $3$, both giving the same word $(1,1,-1,1,1,-1)$ with partial sums $(1,2,1,2,3,2)$. Starts $1$ and $4$ touch zero; starts $2$ and $5$ begin negatively. The cycle lemma with increments at most one counts two starting positions even though there is only one distinct good rotated word. Counting distinct words would lose the multiplicity that makes the statement true.

The upper bound matters. The one-entry sequence $(2)$ has positive integer sum two, but only one starting position. Thus the sum-$k$ conclusion fails for arbitrary positive increments when $k>1$. The sum-one cycle lemma above needs no such upper bound. Strict positivity also matters: counting paths that merely stay at or above zero would include additional starts in both diagrams.

From rotation to enumeration. For a fixed admissible sequence, a start chosen from a uniform distribution on a finite set of its $n$ positions is good with probability $k/n$. More generally, let $\mathcal{F}$ be any finite set of length-$n$ sequences satisfying the increment bound and total $k$, closed under rotation. Double counting pairs consisting of a sequence and a good starting position givesFor each good output sequence and each of the $n$ positions, inverse rotation gives one such pair, even when several rotated words coincide. For example, the four sequences containing three $+1$ votes and one $-1$ vote have exactly $\frac{2}{4}\left(\genfrac{}{}{0ex}{}{4}{3}\right)=2$ vote orders with a strictly positive running lead: $(1,1,1,-1)$ and $(1,1,-1,1)$. This is how the cycle lemma with increments at most one converts many prefix inequalities into a simple counting factor.

Model a spatially uniform cell membrane as a capacitor of membrane capacitance $C_m$ in parallel with membrane electrical resistance $R_m$ and a fixed leak reversal potential $E_L$. Exclude active ion channels, seal leak current and pipette capacitance. For electric current $I$ injected into the biological cell, electric charge conservation yieldsStarting at $V(0)=E_L$ and applying constant $I$ gives the ordinary differential equation solutionThe steady state rise is $I{R}_m$; the time constant is $R_m{C}_m$. With $I=50\mathrm{pA}$, $R_m=200\mathrm{M\Omega }$, $C_m=50\mathrm{pF}$ and $E_L=-70\mathrm{mV}$, the rise is $10\mathrm{mV}$ and ${\tau }_m=10\mathrm{ms}$. At one time constant the rise is $10(1-e^{-1})\mathrm{mV}\approx 6.321\mathrm{mV}$. The figure's uncompensated pipette voltage adds $I{R}_s=1\mathrm{mV}$ for $R_s=20\mathrm{M\Omega }$. The membrane voltage is continuous at step onset in this model, while the pipette voltage immediately acquires that access drop.

Checking the units is instructive: electrical resistance times capacitance has units $\Omega \mathrm{F}=\mathrm{s}$, and electric current times electrical resistance has units of voltage. Unlike the passive voltage clamp transient under patch-clamp series resistance, this ideal current clamp membrane time constant is set by $R_m{C}_m$, not $R_s{C}_m$. Strong current steps can recruit voltage-gated ion channels; the passive model then fails and cannot generate an action potential. Small pulses of both signs, stable access and repeated baseline measurements help test whether the passive approximation is appropriate.

A saddle point of a scalar function is a stationary point with nearby values both greater and less than its value. An indefinite Hessian matrix proves this by choosing directions with opposite signs of the quadratic term. A degenerate Hessian matrix may require higher-order terms: $x^2-y^4$ has a saddle point of a scalar function at the origin despite a zero quadratic coefficient in the $y$ direction. This is distinct from a saddle equilibrium of a dynamical system.

For $b>0$, the origin is a strict local minimum. The four axis points at distance one have Hessian eigenvalues $-4$ and $2(1-b)$. The four diagonal points with $x^2=y^2=1/(1+b)$ have Hessian eigenvalues $-4$ and $-4(1-b)/(1+b)$. Thus for $b<1$ the diagonal points are the global maxima and the axis points are saddle points of a scalar function; for $b>1$ these roles reverse. In polar coordinates, $f=r^2-\frac{1}{2}q(\theta )r^4$, where $q=1+\frac{1}{2}(b-1){\sin }^2(2\theta )>0$. Maximizing over $r$ gives $1/(2q)$ and proves the global statements. At $b=1$, $f=\frac{1}{2}-\frac{1}{2}(r^2-1{)}^2$, so the entire unit circle is the maximum set.

For $F(x,y)=(y-c{x}^2{)}^2+\alpha x^2-\beta x^4$ with $\beta >0$, its stationary points are the origin and, when $\alpha >0$, $x=\pm \sqrt{\alpha /(2\beta )}$, $y=c{x}^2$. The origin is a strict local minimum for $\alpha >0$ and a saddle point of a scalar function for $\alpha <0$. At $\alpha =0$ its Hessian matrix is singular, but the vertical line gives positive values and the parabola $y=c{x}^2$ negative values, so it remains a degenerate saddle. The other stationary points have Hessian determinant $-8\alpha$ and are saddles.