For an input $x$, count the largest family of pairwise disjoint nonempty sets of coordinates such that flipping each set individually changes the Boolean function value. Maximizing over inputs gives block sensitivity. Sensitive sets need not be contiguous and need not consist of single bits. The standard bounded-error quantum query complexity lower bound is $\Omega (\sqrt{\mathrm{bs}(f)})$. A certificate for $f(x)$ meets every sensitive set, so its size bounds their disjoint count.

For independent bond percolation on ${\mathbb{Z}}^d$, let $\tau (x,y)={\mathbb{P}}_{p_c}(x\leftrightarrow y)$ be the critical percolation two-point connection probability. The percolation triangle condition istranslation invariance makes this a threefold discrete convolution evaluated at zero. The factors describe connections between three positions; their product is a diagrammatic weight, rather than the probability of three independent events in a single configuration.

The condition controls the intersection corrections that separate percolation clusters from a simple branching approximation. In the homogeneous setting, it implies the mean-field percolation exponents value $\gamma =1$ for the divergence of percolation susceptibility. It must be established for the particular model; lace expansion provides one method in suitable high-dimensional models. The condition is therefore a useful bridge between spatial connectivity estimates and critical behaviour, rather than an assumption automatically satisfied at criticality.^[ref]

If independent edges of a locally finite graph receive fair orientations, exploring outward from a root tests each fresh reached-to-unreached boundary edge with success probability $1/2$. Deferred decisions give exactly the exploration law of the root cluster in bond percolation of density $1/2$. Thus the two reachable graph vertex sets have the same distribution. An infinite out-cluster contains an infinite simple directed ray by the König infinity lemma. Geometrically biased orientations do not generally give the same uniform edge-success law.

A path in the underlying graph all of whose edges are open. Connectivity of open paths determines the percolation clusters. A ray in a graph with all edges open is an infinite ray in a graph.

For finite nonnegative measures with equal mass and first moment and finite second moments, the constant and linear terms of their Fourier transforms cancel. Taylor's remainder bounds their difference by a constant times $|\xi {|}^2$, making this distance finite. Fourier uniqueness gives definiteness. The origin is excluded from the supremum; the quotient need not have a direction-independent limit there.

The divisor contracted by a blowup of an algebraic variety to its centre. For the blowup of a smooth point on an $n$-dimensional smooth variety it is ${\mathbb{P}}^{n-1}$. Being a closed projective subvariety of positive dimension, it can obstruct affineness of a fibre containing it.

For a blowup of an algebraic variety $\pi :\tilde{X}\to X$ with centre $Z$, the strict transform of an algebraic subvariety of a subvariety $Y$ not contained in $Z$ is the closure of ${\pi }^{-1}(Y\setminus Z)$. It excludes components supported entirely in the algebraic exceptional divisor.

Constant areal velocity gives constant specific angular momentum $l=r^2\dot{\theta }$, hence zero transverse acceleration and a central force. For a focus-based ellipse $u=1/r=(1+\epsilon \cos \theta )/A$, one has $u_{\theta \theta }+u=1/A$. The Binet equation then gives the displayed radial acceleration, proving an attractive inverse-square law. An orbit's shape alone does not specify its acceleration without a time law.

For radial acceleration $-A/r^2-B/r^3$, conserved specific angular momentum $l$ and $u=1/r$ reduce the Binet equation to the displayed linear equation. If $l^2>B$, its angular oscillation frequency is $\sqrt{1-B/l^2}$; attraction proportional to $r^{-3}$ changes the apsidal angle. A zero of $u$ ends the physical positive-radius branch and corresponds to escape, not passage to a negative radius. A spatial self-intersection need not be a periodic return because velocity may differ.

An infinite path is a binary function all of whose finite prefixes belong to the tree. Its path space $[T]$ is closed in Cantor space, since failure is witnessed by one finite prefix outside $T$.

A binary tree of finite strings is computable when membership of a finite binary string is decidable. An infinite computable tree need not have any computable infinite path. Removing all nodes without infinite extensions need not preserve decidability.

A modular-addition quantum oracle for a linear function over the three-element finite field reveals its two coefficients in one query. Prepare the answer qutrit in ${\mathrm{QFT}}_3|2⟩$, whose unit shift eigenvalue is $e^{2\pi i/3}$. Quantum phase kickback produces a product of Fourier states labelled by the coefficients; inverse quantum Fourier transforms read both coefficients exactly.

The Bell basis diagonalizes the commuting Pauli operators $Z\otimes Z$ and $X\otimes X$. Two local anticommutations make the joint operators commute. Their signs encode parity and relative phase. The displayed projectors turn the signed eigenvalues into literal zero-or-one bit eigenvalues: even parity and plus phase have bit zero.

Apply a Hadamard gate to the first qubit, then a controlled-NOT gate from the first to the second. This maps a computational-basis state $|ij⟩$ to the Bell state with phase bit $i$ and parity bit $j$. Its inverse uses the same gates in reverse order because each gate is self-inverse but they do not commute. Applying the forward sequence twice is not generally a decoding operation.

If every model's likelihood is multiplied by the same positive factor independent of its parameter or model index, Bayes' theorem cancels that factor from the normalized posterior. Fixed-count Bernoulli sampling and sampling stopped at a specified positive head count give such proportional likelihoods for the same total heads and tails. This conclusion uses their actual sampling likelihoods; it is not a claim that every selection or stopping rule can be ignored.

Let $N$ have the positive-integer geometric distribution with success parameter $p\in (0,1)$, and conditionally toss $N$ independent coins with head probability $\theta \in (0,1)$. Exactly one head has conditional probability $n\theta (1-\theta {)}^{n-1}$. Combining this with the prior by Bayes' theorem gives the displayed posterior with $r=(1-p)(1-\theta )$; its normalization uses ${\sum }_{n\ge 1}n{r}^{n-1}=(1-r{)}^{-2}$. For $p=\theta =1/2$, the posterior is $9n/4^{n+1}$. It is also a shifted negative binomial distribution, with $N-1$ equal in law to the number of failures before two successes of probability $1-r$.

If $E(A)\ge \theta |A{|}^3$, a subset $B\subseteq A$ satisfies $|B|\ge c_{\theta }|A|$ and $|B-B|\le K_{\theta }|B|$. A dependent random choice argument in the popular sum graph finds many four-edge paths representing each member of the difference set. The Petridis minimal-growth lemma then bounds all higher iterated sumsets and difference sets.

There are absolute $c,C>0$ such that finite sets $A,B$ of size $n$, a bipartite graph of density at least $\delta$, and a restricted sumset of size at most $Kn$ yield subsets $A^{\prime },B^{\prime }$ with $|A^{\prime }|,|B^{\prime }|\ge c{\delta }^{C}n$ and $|A^{\prime }+B^{\prime }|\le C{\delta }^{-C}{K}^{C}n$. A graph formed by popular representations of sums turns this into the usual large-additive energy form of the Balog-Szemerédi-Gowers theorem.

For distinct increasing knots and $k\ge 2$, the explicit divided difference formula writes an order-$k$ B-spline as a finite sum of truncated power functions of degree $k-1$. It is a piecewise polynomial function and is globally $C^{k-2}$. Below its first knot the order-$k$ divided difference annihilates a degree-$k-1$ polynomial; above its last knot all values vanish. Nonzero first and last pieces give the exact closed support. Order one gives interval indicators instead of continuous splines.

A finite collection of the standard partition-normalized B-splines is nonnegative and has sum at most one on the entire real line. Extend the knot sequence in both directions. The full order-one collection sums to one, and summing the Cox-de Boor recurrence preserves that sum because the two coefficients of each lower-order function add to one. The finite collection is a subset. On the basic knot interval the finite sum equals one, also following from Marsden's identity.