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An element $m$ of a partially ordered set is minimal when $x\le m$ implies $x=m$.

An element $m$ of a partially ordered set is maximal when $m\le x$ implies $x=m$. A poset can have several maximal elements and need not have a greatest element.

For a parabolic trajectory with periapsis radius $r_p$, put $D=\tan (f/2)$, where $f$ is true anomaly. The Barker equation is $t-t_p=\sqrt{2r_p^3/(GM)}(D+D^3/3)$. Since $r=r_p(1+D^2)$, it is equivalent to the radius-time relation for a parabolic Kepler orbit. The sign of $D$ distinguishes approach from recession; $t_p$ is the periapsis time.

Multiplying a solution by $(z-z_0{)}^{\lambda }$ adds $\lambda$ to both exponents at the finite point $z_0$ and subtracts $\lambda$ from both exponents at infinity, under the convention that an exponent $\rho$ at infinity corresponds to behavior $z^{-\rho }$.

A Möbius change of independent variable permutes the singular points while carrying their characteristic exponent pairs with them.

For a second-order equation with three regular singular points, the sum of all six characteristic exponents is one.

For an order-$n$ Fuchsian differential equation with $r$ finite singular points and infinity, the sum of all characteristic exponents is $n(n-1)(r-1)/2$. The three-singularity second-order case has $n=2$ and $r=2$.

An unordered pair $\{a,b\}$ is a set containing the specified elements, with no distinction between first and second positions. Hence $\{a,b\}=\{b,a\}$. If $a=b$, the notation denotes the singleton $\{a\}$, rather than a two-element set.

For distinct $a,b$, an unordered pair represents an edge in a simple undirected graph. An ordered pair instead represents a directed edge because reversing its entries changes the direction. If multiplicity must be retained even for equal elements, a multiset is needed instead of a set.

An ordered pair $(a,b)$ records two elements together with their positions. Its defining equality rule is $(a,b)=(c,d)$ exactly when $a=c$ and $b=d$. Thus $(1,2)\ne (2,1)$, and repeated entries such as $(a,a)$ are allowed.

The Cartesian product $A\times B$ consists of the ordered pairs with first entry in $A$ and second entry in $B$. A function can be represented by a set of such pairs with exactly one output for each input. In set theory, a Kuratowski ordered pair realizes the equality rule using ordinary sets.

For a non-Archimedean local field $K$ with maximal ideal ${\mathfrak{m}}_K$, a principal unit is an element of $1+{\mathfrak{m}}_K$. The higher principal-unit groups are $1+{\mathfrak{m}}_K^n$.

At $r=a$,For $\omega a/c_0\gg 1$, kinetic and compressional energies are equal and the field is locally radiative. For $\omega a/c_0\ll 1$, reactive kinetic energy dominates and the mean radiated flux is small.

The mean power radiated by the pulsating sphere is

Choosing an intermediate radius uniformly and exploring the open cluster meeting that sphere gives a decision tree for a one-arm event. Translation and a union bound control each revealment by a constant times the average of the one-arm probabilities over the possible radii.

A decision tree represents a deterministic computation of a known Boolean function $f:\{0,1{\}}^n\to \{0,1\}$ by inspecting input bits. Each internal vertex of its binary tree names a coordinate $i$ to query, its two outgoing edges carry answers 0 and 1, and each leaf carries the output 0 or 1. Starting at the root of a tree, read the named bit, follow its answer edge, and repeat until a leaf is reached. Correctness requires the leaf's label to equal $f(x)$ for every input $x$.^[ref]

The next coordinate, or the decision to stop, can depend on earlier answers: this is adaptive querying. The same input always follows the same graph path; adaptivity does not require randomness. A correct leaf must have the same output for every completion of the observed bits. Thus a decision tree can stop as soon as the observations form a query certificate. Querying an already known coordinate adds no information, so we use decision trees without repeated coordinates on a graph path; an efficient tree also removes queries made after the output is already forced.

The motivation is exact computation with shortcuts. If inspecting an input entry is expensive, the question is how much of the input must actually be read. The decision-tree query cost counts these inspections, while local calculations on known values are free in this model; this cost need not equal an implementation's time complexity. Here the Boolean function is specified in advance and must be computed exactly on every input. Decision tree learning instead fits a predictor from data, with a different objective.

A Hilbertian basis is a complete orthonormal family: its closed linear span is the whole Hilbert space, equivalently every vector is the norm-convergent sum of its Fourier coefficients against the basis.

For orthogonal polynomials $Q_0,\dots ,Q_n$ under a positive weighted integral inner product, the least-squares approximation to $f$ of degree at most $n$ isIts residual is orthogonal to every polynomial of degree at most $n$.

A matrix represents a linear projection when applying it twice has the same effect as applying it once. For an orthogonal projection with the standard Hermitian inner product on ${\mathbb{C}}^m$, the additional condition is that the conjugate transpose equals the original matrix:Thus $P$ is both idempotent and a Hermitian matrix. Over the real numbers this says $P^2=P=P^T$. Its image of a linear map is the retained linear subspace, while its kernel of a linear map is the orthogonal complement: vectors in the image are unchanged, and vectors in the kernel become zero.

For a matrix $A$ with full column rank, projecting onto its column space givesThe invertibility of $A^{†}A$ follows from linear independence of the columns. If those columns are an orthonormal set, $A^{†}A=I$ and $P=A{A}^{†}$. These formulas and the conditions $P^2=P=P^T$ in the real case are developed in MIT 18.06SC, Projections onto Subspaces, pp. 1–3.

For the line spanned by $e=(1,2{)}^T$,In particular, $Pe=e$ and $P(2,-1{)}^T=0$. The eigenvalues are $1$ on the retained direction and $0$ on the removed direction. This exposes the meaning of an orthogonal projection as retaining selected components, rather than stretching them.

Symmetry refers to the specified inner product. In the real weighted setting, the formula becomes $P_M=A(A^{T}MA{)}^{-1}{A}^{T}M$ and satisfies $P_M^2=P_M$ and $P_M^{T}M=M{P}_M$. For the preceding weighted line example,It is not symmetric under the ordinary transpose, but it is an orthogonal projection for the weighted Euclidean inner product. The matrix criterion $P=P^{†}$ should therefore be used with the standard Hermitian inner product, not silently transferred to a different geometry.

Two unitary gates are perfectly distinguishable in one use exactly when

A Thom class of an $R$-oriented rank-$d$ vector bundle is a class $u_E\in H^d(D(E),S(E);R)$ restricting to the chosen generator on every fiber pair $(D^d,S^{d-1})$.

An orientation-reversing diffeomorphism reverses the chosen orientation. In oriented local coordinates its derivative has negative determinant.

For a nonconstant predictor, putThe minimized residual sum of squares isEach centered sum can be recovered in constant time from the five raw sums $\sum x_i$, $\sum y_i$, $\sum x_i^2$, $\sum x_i{y}_i$, and $\sum y_i^2$.