The Pearson chi-squared test compares observed categorical counts with null expected counts using the displayed statistic. Its applications include Pearson chi-squared goodness-of-fit test for specified probabilities and Pearson chi-squared test of independence for contingency tables; the fitted restrictions determine the limiting degrees of freedom.

For paired binary observations $(C_i,M_i)$, let $b$ count pairs $(1,0)$ and $c$ pairs $(0,1)$. Under the null hypothesis of equal marginal event probabilities, these discordant outcomes have equal probabilities. Conditional on $b+c$, $b$ has the binomial distribution $\mathrm{Bin}(b+c,1/2)$, giving an exact test. For enough discordant pairs, $(b-c)/\sqrt{b+c}$ has an approximate standard normal distribution, or its square has an approximate one-degree-of-freedom chi-squared distribution. Concordant pairs provide no information about the direction of the marginal difference. If there are no discordant pairs, there is no directional information and the divided statistic is not computed. The pairing and independence between pairs are essential; the usual independent-two-sample formula is not a paired test.

A rejection region is the set of sample outcomes on which a hypothesis test rejects its null hypothesis. Its probability under the null determines the type I error rate.

For independent multivariate normal observations with positive-definite covariance, this statistic is the maximum squared studentized mean contrast. With unbiased sample covariance matrix $S$ and $n>p$, its null distribution obeys $(n-p)T^2/(p(n-1))\sim F_{p,n-p}$. This follows from independence of the Gaussian sample mean and the residual Wishart distribution.

The alternative hypothesis specifies the parameter values or distributions against which a null hypothesis is tested. If the null is $\theta \in {\Theta }_0$, the alternative is $\theta \in {\Theta }_1$ with ${\Theta }_0\cap {\Theta }_1=\varnothing$. For example an upper one-sided comparison uses $\theta >{\theta }_0$, while a two-sided comparison uses $\theta \ne {\theta }_0$. The alternative determines the relevant rejection tail. A p-value is computed under the null; it is not the probability that the alternative is false.

A one-sided test targets deviations in one specified direction, such as $\theta >{\theta }_0$. If larger values of a statistic $T$ indicate this direction, its p-value is an upper-tail null probability $P_0(T\ge t_{\mathrm{obs}})$. The direction should be chosen from the scientific question before examining the data. An upper normal test uses $1-\Phi (z)$; a lower test uses $\Phi (z)$.

A two-sided test detects departures in either direction from the null hypothesis, for example $\theta \ne {\theta }_0$. With a continuous symmetric null statistic centred at zero, its p-value is $P_0(|T|\ge |t_{\mathrm{obs}}|)=2P_0(T\ge |t_{\mathrm{obs}}|)$. This is the usual coefficient test printed by Gaussian regression summaries, unlike a directional one-sided comparison.

A result is statistically significant at a specified significance level $\alpha$ when its valid p-value is at most $\alpha$ and the corresponding null hypothesis is rejected. Significance does not measure effect size, establish practical importance, or give the probability that the null is true. Software stars encode thresholds for $p$-values; they do not supply an additional statistic.

A sequential probability ratio test accumulates the log-likelihood ratio for two specified simple statistical hypotheses until it reaches an upper or lower boundary. An upper crossing favors the alternative, a lower crossing favors the null, and observations continue between them. Error-calibrated boundaries are approximately $\log [(1-\beta )/\alpha ]$ and $\log [\beta /(1-\alpha )]$; discrete overshoot means these approximations need calibration.

A statistical hypothesis is a specified set of possible data-generating probability distributions, usually expressed as a restriction on model parameters. A simple statistical hypothesis specifies one distribution; a composite statistical hypothesis permits several. A hypothesis test compares a null hypothesis with an alternative hypothesis using a statistic calibrated under the null.

For a real symmetric positive definite matrix $A$, the error in each eigenvector direction is multiplied by $1-\tau \lambda$. Convergence from every initial vector is therefore equivalent to $0<\tau <2/{\lambda }_{\max }$. At either endpoint there is a nondecaying error direction. The optimal constant parameter is $2/({\lambda }_{\min }+{\lambda }_{\max })$, giving contraction factor $({\lambda }_{\max }-{\lambda }_{\min })/({\lambda }_{\max }+{\lambda }_{\min })$.

The periodic second-difference circulant matrix has constant null space, so solvability requires $\sum b_j=0$. Its Jacobi method has eigenvalues $\cos (2\pi j/n)$: for odd $n$ it converges to some solution, while for even $n$ an alternating mode oscillates unless absent from the error. Lexicographic Gauss-Seidel iteration converges to some solution for every compatible right-hand side and every initial vector. For $K=-A=D+L+U$, a coordinate sweep decreases the energy $e^{T}Ke/2$ by the sum of squared coordinate changes. This excludes all nonconstant unit-modulus eigenmodes. The constant unit eigenvalue is simple, since the invariant left vector $w=(0,1,\dots ,1,2{)}^T$ has $w^{T}1=n\ne 0$. Neither method is a strict contraction to a unique solution without fixing the constant nullspace.

For a primary operator, $T(z)\mathcal{O}(0)\sim h\mathcal{O}(0)/z^2+\partial \mathcal{O}(0)/z$. Taking the residue against $z^{n+1}$ gives the weight for $n=0$, the derivative descendant for $n=-1$, and zero for every $n>0$. Conversely the absence of positive-mode actions excludes higher poles in this operator product and gives the primary transformation law. The antiholomorphic argument is identical. Thus primary states and primary operators correspond under radial quantization.

A linear subalgebra closed under the involution. In bounded operators, this means closure under the adjoint operator; neither norm closure nor operator-topology closure is part of the definition. These distinctions matter in the Von Neumann double commutant theorem and the Kaplansky density theorem.

A Hermitian algebra element of a star-algebra is an element fixed by its algebra involution. The identity and zero are Hermitian algebra elements. For every $x$, both $x{x}^{\ast }$ and $x^{\ast }x$ are Hermitian algebra elements. A real spectrum for all such elements is an extra condition in a general Banach algebra, rather than a consequence of the involution identities alone.

In one Standard Model generation, the left-handed up and down quark fields form a weak $SU(2)$ doublet of hypercharge $1/6$. Each weak component has three colours. The same field therefore counts as three doublets for weak gauge-loop contributions and two colour fundamentals for strong gauge-loop contributions. Right-handed up and down quarks are separate weak singlets.

The stalk supporting an anther.

The pollen-producing part of a stamen.

The Gale-Shapley algorithm constructs a stable matching by letting one side propose and the other hold its best offer so far. The word "deferred" is essential: a held proposal is tentative and can be replaced. David Gale and Lloyd Shapley presented this construction in their 1962 paper, College Admissions and the Stability of Marriage, §3, printed pp. 12–13; §4 extends it to college quotas and unacceptable pairs.

For the one-partner bipartite graph model, designate $L$ as proposers and $R$ as receivers:

Held proposers wait; only unmatched proposers act. A receiver can replace its held partner, whereas a proposer never tries the same edge twice. Isolated vertices simply remain unmatched. These rules also describe incomplete acceptable lists and unequal sides without adding fictitious acceptable partners.

For the preference lists under stable matching, choose available proposers in queue data structure order, initially $A,B,C$, adding rejected proposers at the end. The proposals are $A\to X$, $B\to X$, $C\to Y$, $B\to Y$, $C\to Z$. At the fourth proposal, $Y$ replaces its held $C$ by $B$ because $B{\succ }_{Y}C$. This frees $C$ to try $Z$, producing $M_L$.

Reverse the proposing side. Starting in the order $X,Y,Z$, the proposals are $X\to C$, $Y\to A$, $Z\to A$, $Z\to B$. Now $A$ rejects $Z$ because it already holds $Y$. The result is $M_R$. Thus the choice of proposing side can change the output even with strict preferences and complete lists.

Why does the procedure work? Each edge receives at most one proposal, so at most $|E|$ proposals occur. At all times, the held edges form a matching in a graph, and each receiver holds its favourite proposer among all proposals received. A receiver's held partner can only improve according to its strict preference. If a proposer prefers $r$ to its final partner, or ends unmatched while $r$ is acceptable, it must already have proposed to $r$. A rejection means $r$ then held someone better, and its final partner is at least as good. Such an edge cannot be a blocking pair in stable matching. This covers every possible blocking pair in stable matching, proving stability at termination.

With a queue data structure of unmatched proposers, a next-choice pointer for each list, and constant-time receiver-rank lookups, each proposal takes constant work. Using Big O notation, the time complexity is $O(|E|+|V|)$ including initialization, provided list construction and rank preprocessing also take $O(|E|+|V|)$. Edge-indexed ranks with cross-referenced list entries, or expected constant-time hash lookups, allow sparse storage proportional to $|E|+|V|$. Repeatedly scanning a receiver's preference list to compare offers would lose this bound. For complete lists with $n$ vertices per side, $|E|=n^2$ and the bound is $O(n^2)$. Initializing full $n\times n$ rank tables itself costs $\Theta (n^2)$ time and space, even when only a sparse set of pairs is acceptable.

The Gale-Shapley algorithm does more than establish existence: with strict preferences, its output is a proposer-optimal stable matching.^[ref] It is also a receiver-pessimal stable matching, as proved below. Both qualifications compare partners across stable matchings.

If the endpoints of an edge rank one another last and one stable matching in a bipartite graph contains that edge, every stable matching contains it. In the symmetric difference of two stable matchings, preference comparisons propagate along alternating paths: if a vertex prefers its new partner, that partner must prefer its old partner to avoid blocking the old matching. An alternating path cannot end unmatched on the preferred side, so the component is an alternating cycle, on which the two parts prefer opposite matchings. A mutual-worst edge absent from the new matching would make both its endpoints prefer the new matching, contradicting this propagation.