Conditionally on the observed values, bootstrap draws are iid from their empirical distribution. Each draw has variance $n^{-1}{\sum }_i(a_i-\overline{a}{)}^2$, so averaging $n$ independent draws gives the displayed variance. The variance across $B$ independent replicate means, using denominator $B-1$, is unbiased for this conditional quantity. Its expectation over the original iid data is $(n-1)\mathrm{Var}(a_1)/n^2$, which is slightly smaller than the actual sampling variance $\mathrm{Var}(a_1)/n$.

For independent identically distributed pairs $(X_i,Y_i)$, resample whole pairs using the same index for both coordinates. This preserves the empirical within-pair dependence needed to estimate a correlation coefficient or another joint statistic. Independently resampling the two margins destroys that dependence and estimates a different sampling law.

For independent mean-zero errors in a fixed-design regression, form $y_i^{\ast }=x_i^{\top }\hat{\beta }+{\hat{e}}_i{W}_i$, where the independent multipliers have mean zero and variance one, and refit. Rademacher multipliers taking the values $-1$ and $1$ with equal probability are a simple choice. Conditional on the data, the bootstrap errors have mean zero and preserve observation-specific residual variances, supporting heteroscedasticity rather than imposing a common residual distribution. Appropriate leverage corrections and regularity assumptions are required for accurate studentized inference.

A percentile bootstrap confidence interval takes the lower and upper chosen quantiles of the conditional bootstrap distribution of the estimator itself. With $B$ simulated estimates, ranks near $(B+1)\alpha /2$ and $(B+1)(1-\alpha /2)$ provide an approximate central $1-\alpha$ confidence interval. Unlike a basic bootstrap confidence interval, the endpoints are not reflected about the observed estimate. Coverage requires suitable regularity and accurate bootstrap approximation; the ranks do not establish exact finite-sample coverage.

For each bootstrap sample, form $T^{\ast }=({\hat{\theta }}^{\ast }-\hat{\theta })/{\hat{s}}^{\ast }$, using that sample's estimated standard error, possibly from a nested bootstrap. If $q_p$ are conditional quantiles of $T^{\ast }$ and $\hat{s}$ is the original estimated standard error, invert the studentized inequalities to obtain the displayed interval. The quantile order is reversed by subtraction. Its coverage is approximate and requires a nondegenerate statistic and suitable bootstrap regularity.

For $\theta ={\beta }_1{\beta }_2$, use $\hat{\theta }={\hat{\beta }}_1{\hat{\beta }}_2$. Under regular regression assumptions, the delta method gives standard error $\hat{s}=(d^{\top }\hat{V}d{)}^{1/2}$, where $d=({\hat{\beta }}_2,{\hat{\beta }}_1{)}^{\top }$ and $\hat{V}$ estimates the covariance of the regression estimate. Resample the observational units, refit, and compute both ${\hat{\theta }}^{\ast }$ and ${\hat{s}}^{\ast }$. A percentile bootstrap confidence interval uses the .025 and .975 quantiles of ${\hat{\theta }}^{\ast }$. A bootstrap-t confidence interval uses quantiles of $({\hat{\theta }}^{\ast }-\hat{\theta })/{\hat{s}}^{\ast }$ and reverses them when solving for $\theta$. If both true coefficients vanish, the first derivative is zero and the standard first-order justification is unavailable; product-specific nonregular inference is then needed.

A finitely generated group with soluble word problem for a group embeds in a countably generated simple group with a recursive presentation of a group. Start with $P_0=G\ast ⟨a⟩$, retaining decidable membership in $⟨a⟩$. At stage $n$, add a free infinite cyclic generator $z_n$. For every nonidentity base element $g$, conjugate $[g,z_n]$ to $a$ and conjugate $g{z}_n$ to $a$; also conjugate $z_n$ to $a$, using distinct stable letters. All associated subgroups are infinite cyclic, with decidable membership by effective cyclic membership in a free product. Britton's lemma preserves embeddings, decidable word problem and decidable membership in the inherited $⟨a⟩$.

In the union, any nontrivial normal subgroup contains some nonidentity $g$ and hence $a$, by the commutator-conjugation relation. The other relations force $z_n$ and then every old $g$ into the normal closure of $a$. Every stable letter becomes an old element at a later stage, so the normal closure of $a$ is the whole union. The union is nontrivial and simple. Effective word enumeration at each stage makes all relators recursively enumerable. A two-generator HNN embedding followed by the Higman embedding theorem embeds this simple envelope in a finitely presented group.

Let $S$ be a nontrivial simple group inside a finitely presented group $P$, and fix $a\ne 1$ in $S$. For an input word $w$ in $S$, enumerate products of conjugates in $P$ of $w^{\pm 1}$ and proofs in $P$ that such a product equals $a$. This search terminates exactly when $w\ne 1$: the normal closure of a nontrivial element of $S$ is all of $S$, and its conjugators are also elements of $P$, while a null word has only trivial conjugates. Run this search alongside the enumerable proofs that $w=1$. Exactly one search terminates, solving the word problem for a group on any specified finitely generated subgroup of $S$. No effective generating set for the possibly infinitely generated $S$ is needed.

For a name $a$ and formula $\phi$, form the set of pairs $(u,p)$ with $u$ in the immediate domain of $a$, $p$ a Boolean truth value, and some name $v$ satisfying $p\le a(u)\wedge [[\phi (u,v)]]$. Full separation makes this a set, and intuitionistic collection gives a set of witness names covering all its pairs. A name with these witnesses in its domain collects the entire Boolean value of every existential assertion over $a$, without selecting one witness whose value is top. No choice or maximum principle is used.

A well-founded recursively built set code with a set of immediate subnames and a Boolean coefficient for each. The coefficient records the extent to which the corresponding subname belongs. More generally one may permit repeated coded subnames, joining their coefficients. Recursive valued equality identifies different codes representing the same valued set; external equality of codes is not the interpreted equality.

In a fixed central Newtonian gravitational potential, pressureless free fall has speed $u=a{R}^{-1/2}$, and randomly moving stars have typical gas-relative speed $V=b{R}^{-1/2}$. A steady, undepleted spherical mass flux $A$ gives $\rho =A{R}^{-3/2}/(4\pi a)$. The Bondi--Hoyle--Lyttleton accretion rate therefore becomes $\dot{M}=\lambda M^2$ with $\lambda =G^{2}A/(a{b}^3)$, independent of radius. Integrating $d(1/M)/dt=-\lambda$ gives the displayed solution before $\lambda M_{0}t=1$. Its finite-time divergence signals failure of the fixed potential, undepleted gas supply and negligible feedback assumptions; it is not a physically unlimited mass supply.

For a finite positive Borel measure on the circle, averaged squared Fourier coefficients tend to the sum of squares of its atomic masses. Express each squared coefficient as a double integral and average the circle character. The averaged character is bounded by one, tends to one on the diagonal and to zero elsewhere, so the dominated convergence theorem gives the diagonal mass. In particular an atomless measure has vanishing averaged absolute coefficients by the Cauchy-Schwarz inequality.

Partition a regular factorial design by fixed signs of independent factorial contrasts. Products of those block-generating columns span the block effects. A treatment contrast that is constant on each block is therefore confounded with block differences. Select generators so that only scientifically dispensable treatment effects enter this space; randomize the assignment of blocks and the run order within each block.

For a linear regression model $Y_{bj}={\alpha }_b+f(x_{bj}{)}^T\beta +{ϵ}_{bj}$ with one unrestricted intercept per two-run experimental block, differencing the responses removes all experimental block effects. The parameter vector $\beta$ is identifiable precisely when the matrix whose rows are $f(x_{b1})-f(x_{b2})$ has full column rank. This criterion applies to arbitrary pairings, including nonregular factorial designs; requiring regular orthogonal confounding is a stronger constraint.

If an analytic family of finite normalized block sequences meets every infinite-dimensional block subspace, then after an arbitrarily prescribed coordinatewise perturbation it is strategically large in some block subspace. In the block game, one player chooses an infinite-dimensional block subspace at each turn and the other chooses a successive unit vector from it; strategically large means the vector player can force a finite initial segment into the family.

Suppose a Bloch stability operator has expansion $L(q)=L_0+iq{L}_1-q^2{L}_2+\cdots$, with a simple neutral eigenfunction $v_0$ and an adjoint neutral function $w$ normalized by $⟨w,v_0⟩=1$. Expand $v(q)=v_0+iq{v}_1+\cdots$ and $\lambda (q)=icq-D{q}^2+\cdots$. The Fredholm alternative gives $c=⟨w,L_1{v}_0⟩$ and the cell problem $L_0{v}_1=c{v}_0-L_1{v}_0$, $⟨w,v_1⟩=0$. Applying the adjoint solvability condition at second order gives the displayed phase-diffusion coefficient. Multiple translation phases or a conserved mean require a coupled matrix problem rather than this scalar formula.

A sequence in the unit disc, counted with multiplicity, is a Blaschke sequence when it satisfies the Blaschke condition. The zeros of every nonzero bounded holomorphic function form such a sequence: Jensen's formula bounds ${\sum }_{|a_n|<r}\log (r/|a_n|)$ uniformly as $r\uparrow 1$, and $1-|a|\le -\log |a|$ gives the assertion. There are only finitely many zeros at the origin, which can be factored out first.

With constant continuous stock-dividend yield $d$, the risk-neutral measure gives stock drift $r-d$ but cash claims are discounted at $r$. The same terminal payoff in a no-dividend Black-Scholes model with interest rate $r-d$ has the identical stock-transition law but a different discount factor. Their prices therefore obey the displayed identity, under the usual payoff integrability conditions.

For nonzero $J_h,J_t$ and a noncollinear initial configuration, the alignment equation integrates to $\mathrm{artanh}c(t)=\mathrm{artanh}c(0)+J_h{J}_t{\int }_0^t{K}_2(s)ds$. Hence the positive dissipative torque aligns the hole with the total vector if its integrated strength diverges. A fixed positive coefficient is sufficient. Pointwise positivity alone need not suffice for time-dependent coefficients: with $J_h=J_t=1$, $K_1=0$, $K_2=e^{-t}$ and $c(0)=0$, the limiting cosine is $\tanh 1<1$. Exactly antiparallel initial spin and total angular momentum form a stationary, nongeneric exception.

After persistent dissipative alignment of the hole with the conserved total angular momentum, the final vectors are ${\mathbf{J}}_{h,f}=J_h{\hat{\mathbf{J}}}_t$ and ${\mathbf{J}}_{d,f}=(J_t-J_h){\hat{\mathbf{J}}}_t$. The disk is therefore antiparallel to the hole if $J_t<J_h$. Squaring $J_t^2=J_h^2+J_{d,0}^2+2J_h{J}_{d,0}\cos {\theta }_0$ gives the displayed initial-angle criterion. The disk and hole need not finish aligned with each other even though the hole aligns with the total. Equality $J_t=J_h$ leaves zero final disk angular momentum rather than a defined disk spin direction.