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A bootstrap confidence interval uses the conditional resampling distribution of an estimator to choose its endpoints. Basic intervals invert bootstrap error quantiles; percentile intervals use quantiles of the resampled estimates themselves. Their asymptotic coverage requires conditions appropriate to the statistic and interval construction.

If $n\ge 2d+1$ and the field characteristic is zero or greater than $d+1$, the matrix recording inclusion of $d$-sets in $(d+1)$-sets has full row rank. Subtracting equations on sets differing in one coordinate reduces the kernel problem to degree $d-1$ on $n-2$ coordinates. Mathematical induction then makes all coefficients equal, and the factor $d+1$ forces them to vanish. This gives the auxiliary-polynomial independence used in the uniform Frankl-Wilson theorem.

For nearest-neighbour bond percolation with $0<p<1$ on the strip $T_k=\mathbb{Z}\times \{-k,\dots ,k\}$, put $q_k(n)={\mathbb{P}}_p((0,0)\leftrightarrow (n,0)\text{ in }{T}_k)$. The Harris-FKG inequality and translation invariance imply $q_k(n+m)\ge q_k(n)q_k(m)$. Thus $-\log q_k(n)$ is a subadditive sequence, and the Fekete lemma givesThe direct horizontal graph path gives $0\le f_k(p)\le -\log p$. Enlarging the strip increases every percolation two-point connection probability, so $f_k(p)$ is nonincreasing in $k$ and has a nonnegative limit of a sequence.

The two-point percolation two-point connection probability ${\tau }_p(x,y)={\mathbb{P}}_p(x\leftrightarrow y)$ is the probability that two fixed graph vertices belong to the same percolation cluster. On a translation-invariant lattice it depends on their displacement. Restricting the permitted connecting graph paths to a subgraph gives the corresponding confined percolation two-point connection probability. The Harris-FKG inequality implies ${\tau }_p(x,z)\ge {\tau }_p(x,y){\tau }_p(y,z)$.

A nonrelativistic species with $g$ states and negligible chemical potential has equilibrium density $g(mT/(2\pi ){)}^{3/2}{e}^{-m/T}$. If particle-antiparticle annihilation maintains equilibrium after the species becomes massive, its eventual freeze-out population can be much smaller than a relativistically decoupled population. The nonrelativistic annihilation rate and any asymmetry determine the final abundance.

For real even coefficients with $A_k>|B_k|$, the quadratic Hamiltonianis diagonalized by the Bogoliubov transformation $a_k=\cosh {\theta }_k{\alpha }_k-\sinh {\theta }_k{\alpha }_{-k}^{†}$, with $\tanh 2{\theta }_k=B_k/A_k$. The canonical commutation relations follow from ${\cosh }^2{\theta }_k-{\sinh }^2{\theta }_k=1$. The diagonal form is ${\sum }_k[{\omega }_k{\alpha }_k^{†}{\alpha }_k+({\omega }_k-A_k)/2]$, with ${\omega }_k=\sqrt{A_k^2-B_k^2}$. The original-mode vacuum depletion is ${\sinh }^2{\theta }_k=(A_k/{\omega }_k-1)/2$. At $A_k=|B_k|$, the exact zero mode requires separate infrared treatment.

For ferromagnetic exchange $J$ and coordination $z$, the mean-field approximation has $f=zJ{M}^2/2-k_{B}T\log [1+2e^{-g/(k_{B}T)}\cosh (zJM/(k_{B}T))]$. With $K=zJ/(k_{B}T)$ and $q=2e^{-g/(k_{B}T)}/[1+2e^{-g/(k_{B}T)}]$, its quadratic and quartic coefficients vanish at $q=1/3$, $K=3$. The sextic coefficient of $f/(k_{B}T)$ is $9/40>0$. Hence $T_{\mathrm{TCP}}=zJ/(3k_B)$ and $g_{\mathrm{TCP}}=k_B{T}_{\mathrm{TCP}}\log 4$; these are not exact finite-dimensional lattice values.

For $v,m\ge 0$, define $F(v,m)=\mathbb{E}[(e^{-v/2+\sqrt{v}Y}-m{)}^{+}]$, where $Y$ has the standard normal distribution. For $v,m>0$, completing the square in the standard normal density giveswhere $\Phi$ is the standard normal distribution function. The boundary values are $F(0,m)=(1-m{)}^{+}$ and $F(v,0)=1$. Multiplying by the current stock price produces a zero-interest Black-Scholes formula with total variance $v$ and relative strike $m$.

For a biregular graph of edge density of a bipartite graph $\gamma$, small rectangular discrepancy, a box norm fourth power close to ${\gamma }^4$, and a small second singular value of the normalized adjacency operator of a bipartite graph are equivalent notions of quasirandomness, with quantitative bounds independent of the part sizes. Centering the adjacency function removes the constant singular component.

Conditional iid accepted values have the ordinary target-sample CLT at scale $\sqrt{N}$. Since $N/n\to 1/M$, the same estimator has asymptotic variance $M{\mathrm{Var}}_f\varphi$ at scale $\sqrt{n}$. The centered count has limiting variance $(1/M)(1-1/M)$. Any fixed convention on an empty output is asymptotically negligible.

An attractive inverse-square force together with radial drag $-k\dot{r}/r^2$ preserves angular momentum $h=r^2\dot{\theta }$ but modifies the Binet equation for $u=1/r$ by a first derivative. For $k/|h|<2$ and $h\ne 0$, the reciprocal radius is a decaying oscillation about $p/h^2$. Forward angular distance $\mathrm{sgn}(h)(\theta -{\theta }_0)$ makes its damping sign explicit for either orbital orientation.

Conjugation by a Controlled-Z gate leaves the $a$ bits unchanged and updates the displayed $b$ bits and phase, using the old bits. The phase occurs when the extra $Z$ from one line is moved past $X$ on the other line. These constant-size updates support Heisenberg propagation of a Pauli observable through a Clifford circuit.

For $d(a,b)=a\log (a/b)+(1-a)\log ((1-a)/(1-b))$, the second derivative with respect to $a$ is $1/[a(1-a)]\ge 4$. Its value and first derivative at $a=b$ vanish, so $d(a,b)\ge 2(a-b{)}^2$. Boundary values follow by continuity or the infinite-divergence convention. Together with the binary partition bound for relative entropy, this gives Pinsker's inequality with the natural-logarithm constant.

An initial segment of the binary order on the discrete cube contains the vertices whose integer encodings are $0,\dots ,m-1$. It is a down-set, so its internal edges and two-dimensional cube faces can be counted at their unique upper vertices by ${\sum }_{j<m}\mathrm{wt}(j)$ and ${\sum }_{j<m}\left(\genfrac{}{}{0ex}{}{\mathrm{wt}(j)}{2}\right)$.

Fix a coordinate of the Boolean hypercube, delete it in each of the two sections, and replace each section by a binary initial segment of the same size. This preserves the total size. Mathematical induction on dimension proves that it does not decrease contained edge counts; the crossing contribution is the size of the section intersection, maximized by nested initial segments.

Incompressibility and periodicity imply that the spatial mean of the normal velocity is constant with height. Zero normal flow at infinity therefore requires zero mean normal boundary velocity. In the bounded biharmonic stream function for planar Stokes flow solution, the mean streamfunction is a constant plus a term linear in height.

A bounded linear operator between Banach spaces is weakly compact exactly when its Banach-space adjoint is weakly compact. Equivalently, that adjoint is continuous from its domain's weak-star topology to its range's weak topology. For the reverse direction, the bidual condition applied to $T^{\ast }$ forces $T^{\ast \ast \ast }$ to annihilate every functional on $Y^{\ast \ast }$ vanishing on $J_{Y}Y$; the Hahn-Banach theorem then places $T^{\ast \ast }(X^{\ast \ast })$ inside $J_{Y}Y$.

The tensor of left modules over a bialgebra has action $h(u\otimes v)=\sum (h_{(1)}u)\otimes (h_{(2)}v)$. The unit module $k$ has action through the counit.

A normalized scalar map $\gamma :H\otimes H\to k$ satisfying $\sum \gamma (a_{(1)},b_{(1)})\gamma (a_{(2)}{b}_{(2)},c)=\sum \gamma (b_{(1)},c_{(1)})\gamma (a,b_{(2)}{c}_{(2)})$, and $\gamma (1,a)=\epsilon (a)=\gamma (a,1)$. Some conventions require convolution invertibility; that extra hypothesis distinguishes strong from lax tensor transformations.

For a Boolean quantum oracle whose function is $g(x,y)=a_y\cdot x$, place its target in $|-⟩$. Conjugating the oracle by a Walsh-Hadamard transform on the $x$ register translates that register by $a_y$. This is Bernstein-Vazirani phase kickback without measuring its output, so the index $y$ may remain in an arbitrary quantum superposition. A second application undoes the translation. A phase evaluation between the two calls therefore transfers a hidden-label-dependent phase to the index register while uncomputation returns the decoding register to zero.