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Let $S_n={\sum }_{j=1}^n{X}_j$, where the increments are nonconstant independent and identically distributed random variables with zero expected value and $|X_j|\le c$. Then the random walk oscillates between arbitrarily large positive and negative levels almost surely. For $T_a=\inf \{n:S_n\ge a\}$, the stopped martingale $a+c-S_{n\wedge T_a}$ is nonnegative. The martingale convergence theorem implies that $S_n$ converges finitely on $\{T_a=\infty \}$. However, some $\delta >0$ has $\mathbb{P}(|X_1|>\delta )>0$, and the Borel-Cantelli lemmas imply $|X_n|>\delta$ infinitely often. Hence finite convergence is impossible. Apply this to every positive integer $a$ and to $-S_n$.

The bounded increments also imply infinitely many visits to $[-c/2,c/2]$: a path cannot cross from above $c/2$ to below $-c/2$ without entering this interval. Identical distribution cannot simply be dropped. For independent symmetric signs ${\epsilon }_j$, the increments $X_j=c{4}^{-(j-1)}{\epsilon }_j$ satisfy $|S_n|\ge 2c/3$ for every $n$, even though they are nonzero, independent, symmetric and uniformly bounded.

If $X_1,\dots ,X_n$ are independent random variables and $f$ has bounded differences constants $c_i$, thenand the same bound holds for the lower tail.

A meromorphic function on the unit disc has bounded characteristic if and only if it has this representation. Its pole Blaschke product $B$ makes $g=Bf$ analytic with bounded positive logarithmic means. The least harmonic majorant by expanding disk lifts gives $H\ge {\log }^{+}|g|$. A harmonic conjugate gives an analytic $A$ with $\mathrm{Re}A=H$, so $u=g{e}^{-A}$ and $v=B{e}^{-A}$ are bounded. Conversely the Nevanlinna first main theorem bounds the characteristic of $u/v$ by that of $u$ and $1/v$, hence by a constant.

For $0<\kappa <4$ and $x>0$, stop the simple chordal SLE trace on first entering the radius-$r$ half-disc about $x$, with $r<x/4$. A crosscut from its tip to $x$ separates the positive-side boundary arc containing $x/2$ from infinity. Harmonic measure comparison with the half-disc bounds the length of that arc's mapped interval, $g_t(x/2)-{\xi }_t$, by a universal constant times $r$. This centered boundary image is a positive Bessel process of dimension $1+4/\kappa >2$ and has a strictly positive all-time minimum. Hence the full trace stays a positive distance from each fixed nonzero real point. Reflection gives the negative-side version.

For a drop with internal viscosity $\lambda \mu$, exterior viscosity $\mu$, continuous velocity and a decaying exterior Stokes flow, use the drop-outward normal $n$ and curvature $\kappa ={\nabla }_s\cdot n$, twice the mean curvature. Constant surface tension gives $({\sigma }^{+}-{\sigma }^{-})n=\gamma \kappa n$. Multiply the interior boundary representation by $\lambda$ and add it to the exterior one: the traction difference yields the single-layer term shown, while the viscosity-independent stress kernel yields $(\lambda -1)D[u]$. Here $S,D$ denote the kernels in boundary integral representation of Stokes flow. For equal viscosities the double layer vanishes.

For a homogeneous ideal gas in $d$ dimensions with $A,\alpha >0$, the density of states scales as ${ϵ}^{d/\alpha -1}$. At limiting chemical potential zero, the Bose-Einstein distribution is proportional to $1/ϵ$ near zero energy. The excited-particle integral is finite exactly when $d/\alpha >1$. Thus positive-temperature condensation at fixed nonzero density is possible precisely for $0<\alpha <d$. A nonconserved particle number, as for ordinary equilibrium photons, removes the fixed-number saturation mechanism even when the integral is finite.

For real $h\in C_c^{\infty }(D)$, the displayed real compactly supported contrast has Fourier transform $\hat{V}(\mathbf{\kappa })=[|\mathbf{\kappa }{|}^4-4({\mathbf{k}}_i\cdot \mathbf{\kappa }{)}^2]\hat{h}(\mathbf{\kappa })$. This vanishes on the shifted Ewald sphere because $|\mathbf{\kappa }{|}^2+2{\mathbf{k}}_i\cdot \mathbf{\kappa }=0$ there. Thus nonzero contrasts can have zero fixed-frequency, fixed-incidence Born approximation data. A small amplitude preserves weak contrast and a positive refractive index.

For a topological space, ${\Sigma }_1^0$ is the class of open sets and ${\Pi }_{\alpha }^0$ consists of complements of ${\Sigma }_{\alpha }^0$ sets. For countable $\alpha >1$, ${\Sigma }_{\alpha }^0$ consists of countable unions of sets in ${\bigcup }_{\beta <\alpha }{\Pi }_{\beta }^0$. These countable levels contain every Borel set. Continuous inverse images preserve each level, by induction using openness, complementation and countable unions.

A bounded normal operator admits a normal square root even when no continuous scalar square root exists on its Banach algebra spectrum. Choose a bounded Borel branch $s(z)=|z{|}^{1/2}\exp (i\mathrm{Arg}(z)/2)$, with $s(0)=0$. The Borel functional calculus for a normal operator gives $S=s(T)$; multiplicativity gives $S^2=T$ and preservation of the adjoint gives $S^{\ast }S=S{S}^{\ast }=|s{|}^2(T)$. Unlike a positive square root, this square root need not be unique.

For a bounded Borel measurable function $f$ and the projection-valued measure $E$ of a normal operator, the displayed set is the spectrum of an element of $f(T)$. Outside it, a bounded reciprocal away from a spectral-null set supplies the inverse. Inside it, unit vectors supported by the indicated arbitrarily small spectral sets make $f(T)-\lambda$ fail to be bounded below. The kernel of the calculus consists of functions satisfying $E(|f|>\epsilon )=0$ for every $\epsilon >0$, and its norm is the corresponding essential supremum. Ordinary pointwise range is insufficient: for multiplication by $t$ on $L^2[0,1]$, $1_{\{0\}}(T)=0$ even though the function takes the value $1$.

An unordered nonparametric bootstrap sample from $n$ distinct observations is specified by its $n$ nonnegative multiplicities. Stars and bars gives $\left(\genfrac{}{}{0ex}{}{2n-1}{n}\right)$ such vectors. They are not equiprobable: a vector has probability $n!/(n^n{\prod }_i{N}_i!)$, because this counts its ordered sequences of sampled indices.

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.