Without rotational diffusion, $x=p_z$ follows $\dot{x}=(1-x^2)/B$, hence $x=(x_0+\tanh (t/B))/(1+x_0\tanh (t/B))$. Pushing forward the initial uniform orientation distribution gives $f=\alpha /[\pi (1+\alpha -(1-\alpha )x{)}^2]$, $\alpha =e^{-2t/B}$, per unit solid angle. Integrating $2\pi xf$ yields the displayed mean, interpreted continuously as zero at $t=0$. Its late deficit is $(4t/B-2)e^{-2t/B}+O((t/B)e^{-4t/B})$. At each finite time the density is smooth and normalized; the infinitely late distribution is a point mass at the upward pole.

This convention defines the gyrotactic reorientation time by an aligning rate $1/(2B)$. Multiplying the torque balance by $2B$ and perturbing about the upward direction gives ${\mathbf{p}}_{\perp }=B\mathbf{\omega }\times {\mathbf{e}}_z$. For an incompressible flow, its horizontal divergence is $-B{\nabla }^{2}w$, because $\nabla \times \mathbf{\omega }=-{\nabla }^2\mathbf{u}$. The parameter $B$ is half the reorientation parameter used in the convention with aligning rate $1/B$; writing the governing equation prevents a factor-of-two ambiguity.

A matter vertex with left and right oscillator levels $N,\tilde{N}$ and momentum $k$ has weights $N+{\alpha }^{\prime }{k}^2/4,\tilde{N}+{\alpha }^{\prime }{k}^2/4$. An integrated string vertex operator must have weights $(1,1)$, giving the displayed mass shell and level matching with $M^2=-k^2$. Physical states are further identified modulo null string states, equivalently through BRST cohomology; merely counting all weight-one primaries would include gauge redundancies.

With group commutator convention $[a,b]=a^{-1}{b}^{-1}ab$, the displayed group presentation describes the fundamental group of the Borromean rings complement. Its abelianization is ${\mathbb{Z}}^3$, since each relator is a group commutator and hence imposes no abelian relation. It embeds discretely in the Gaussian Bianchi group by sending the meridians to $\left(\begin{array}{cc}1& 1\\ 0& 1\end{array}\right)$, $\left(\begin{array}{cc}1& 0\\ 2i& 1\end{array}\right)$ and $\left(\begin{array}{cc}i& 1\\ 2i& 2-i\end{array}\right)$.

Every bijection of pure-state rays that preserves transition probabilities is induced by either a unitary or an antiunitary operator on the Hilbert space.

For two orthonormal energy eigenstates ${\psi }_1,{\psi }_2$, let an observable have distinct nondegenerate eigenstates ${\varphi }_{\pm }=({\psi }_1\pm {\psi }_2)/\sqrt{2}$. Starting in ${\varphi }_{-}$, unitary time evolution gives $(e^{-i{E}_{1}t/\hslash }{\psi }_1-e^{-i{E}_{2}t/\hslash }{\psi }_2)/\sqrt{2}$. Its probability amplitude along ${\varphi }_{+}$ is $(e^{-i{E}_{1}t/\hslash }-e^{-i{E}_{2}t/\hslash })/2$. The Born rule therefore gives the stated transition probability. The energy difference determines the oscillation frequency, while a common energy shift has no effect.

A normalized quantum state $\chi =\sqrt{q}{\psi }_1+e^{i\eta }\sqrt{1-q}{\psi }_2$ in two orthonormal energy eigenstates returns to its original one-dimensional measurement subspace with probabilityThis follows by applying unitary time evolution to each energy eigenstate, then the Born rule. The relative phase $\eta$ cancels from the return amplitude. Degenerate energies make the probability identically one.

For a static D3-brane with one scalar and no magnetic field, taking the Legendre transform of the DBI action gives the displayed identity. It bounds energy by the electric scalar charge and is saturated at $\mathbf{d}=\nabla X$. The opposite orientation uses the analogous identity with reversed signs. On saturation, the constitutive relation gives $\mathbf{E}=\mathbf{d}$ and the Gauss law constraint in gauge theory reduces to the Laplace equation for $X$.

For one constant charge angle, the DBI Hamiltonian bound for an electric spike and its dyonic extension give the displayed equations away from sources. Each aligned branch preserves eight of the sixteen linearly realized worldvolume supersymmetries. The corresponding spacetime string–D3 configuration preserves eight of thirty-two. The bound and charge interpretation are derived in BPS Bounds for Worldvolume Branes.

For $X-X_{\infty }=c/r$, subtracting the planar D3-brane energy leaves $T_3\int |\nabla X{|}^2{d}^{3}x=4\pi T_3{c}^2/r_0$ outside a cutoff. Since $X_{\mathrm{cut}}-X_{\infty }=c/r_0$, its coefficient is the carried string tension. Electric charge quantization gives $c=\pi g_s|N|{\alpha }^{\prime }$ and coefficient $|N|T_{F1}$; magnetic charge gives $c=\pi |N|{\alpha }^{\prime }$ and coefficient $|N|T_{D1}$.

The DBI action resums field strengths but truncates their derivative expansion. A slowly varying large-charge throat may therefore be controlled even when its field strengths are not small. Large derivatives, large embedding curvature, string loops and gravitational backreaction restrict this description, and its ideal singular core is not uniformly controlled. The electric throat window $|N|g_s\gg 1$, $g_s^2|N|\ll 1$ and magnetic window $|N|\gg 1$, $g_s|N|\ll 1$ separate derivative and backreaction requirements.

Choose one random variable $U$ with the continuous uniform distribution on $[0,1]$, and reuse it in every event:Each probability is $1/(n+1)$, so their harmonic series diverges, just as in summable and harmonic event probabilities. But the events are decreasing: $A_{n+1}\subseteq A_n$. For every outcome $U=u>0$, all indices with $n+1>1/u$ fail. Only finitely many events occur on that outcome. The limsup of events is exactly $\{U=0\}$, a zero-probability event. This is a concrete closed-interval version of the shrinking-interval dependence example.^[ref]

The missing assumption is visible already in two events. For $i<j$, their intersection is $A_j$, henceMoreover, the whole tail union is just $A_m$, with probability $1/(m+1)\to 0$. This calculation explains directly why divergence alone fails: overlapping events keep spending their individual probabilities on the same outcomes.

Let $N={\sum }_{n\ge 1}{1}_{A_n}$ be the total number of occurrences, a sum of indicator random variables. Although $N<\infty$ almost surely, Tonelli theorem gives $\mathbb{E}[N]={\sum }_{n\ge 1}1/(n+1)=\infty$. More explicitly, $\mathbb{P}(N\ge k)=1/(k+1)$ for $k\ge 1$: very small positive values of $U$ give very long, but finite, initial runs. Infinite expected value does not imply a positive probability of an infinite count.

Replace the unknown sampling distribution by the empirical distribution, generate independent bootstrap samples, and recompute the statistic. The sample standard deviation of those recomputed values estimates the conditional bootstrap standard deviation, which approximates the statistic's sampling standard error when the bootstrap is consistent. It is different from the Monte Carlo standard error of the estimated bootstrap mean.

For a continuous uniform distribution on $[0,\theta ]$, the scaled maximum gap $n(\theta -X_{(n)})/\theta$ tends to a unit-rate exponential distribution. An ordinary bootstrap sample includes the unique observed maximum with probability $1-(1-1/n{)}^n\to 1-e^{-1}$. The bootstrap maximum gap therefore retains an atom at zero, whereas the target limit is continuous. This proves failure of the ordinary empirical bootstrap for this endpoint functional; consistency of the endpoint estimate alone does not justify resampling its extreme-value limit.

A family in the Boolean lattice whose pairwise Hamming distances are at most $d<n$ has size at most $D(n,d)$. Downward compression followed by elementary set shifts preserves the distance bound. In the resulting down-set, the section containing the last coordinate has diameter at most $d-2$: two members leave a coordinate free, and shifting the last coordinate to it creates a pair with two additional disagreements. This gives the recurrence $D(n,d)=D(n-1,d)+D(n-1,d-2)$. The boundary case $d=n-1$ follows by pairing complementary sets. Hamming balls attain the even case; the union of two radius-$q$ balls with adjacent centers attains the odd case.

A tribes function is an OR of AND blocks: partition $mw$ coordinates into $m$ blocks of size $w$, and let $f=1$ when at least one block consists entirely of ones. For independent random variables with each bit uniform, $\mathbb{P}(f=0)=(1-2^{-w}{)}^m$. The influence of a variable is $2^{-(w-1)}(1-2^{-w}{)}^{m-1}$, because every other bit in its block must be one and every other block must fail. Taking $m$ approximately $(\log 2)2^w$ keeps both output probabilities bounded away from zero and gives influences of order $\log (mw)/(mw)$. This shows the order in the Kahn-Kalai-Linial theorem is sharp.

A $p$-random restriction independently leaves each coordinate live with probability $p$, otherwise fixing it to zero or one with probability $(1-p)/2$ each. Composing independent restrictions with live probabilities $p_1,\dots ,p_d$ produces the same distribution with live probability ${\prod }_j{p}_j$. Fixed values stay unbiased.

Every Boolean function of distributed inputs is an exclusive or of products of locally computable bits. The displayed identity follows because exactly one indicator is one. It uses at most $2^m$ terms, each a function of $x$ times a function of $y$; exchanging the two input roles gives at most $2^n$ terms. This elementary truth-table factorization is useful for distributed computation with PR boxes and does not depend on a probabilistic approximation.

The algebraic normal form is the unique multilinear polynomial representation of a Boolean function over the field with two elements, using exclusive or for addition and ordinary bit multiplication for products. The empty product is one. Existence and uniqueness follow by induction from $f(z^{\prime },z_d)=f(z^{\prime },0)\oplus z_d(f(z^{\prime },0)\oplus f(z^{\prime },1))$: the first polynomial supplies all coefficients not containing $z_d$, and the difference supplies exactly those that do. For split inputs $(x,y)$, each monomial factors into one locally computed product of Alice's bits and one of Bob's bits, yielding a separated Boolean decomposition.

Under uniform independent input bits, a zero-one Boolean function is quite fair when its mean lies between one quarter and three quarters. This convention excludes functions whose output is nearly constant; it does not require exact balance or equal coordinate influences.