Order subsets first by increasing cardinality and, within each level, by lexicographic order, with the smallest differing coordinate belonging to the earlier set. This convention makes simplicial initial segments the extremizers in Harper theorem.

A face is obtained by fixing some coordinates of the Boolean hypercube and letting the remaining $d$ coordinates vary freely. It is isomorphic to $Q_d$. A two-dimensional face is a square with four vertices, and a face is contained in a set family when all its vertices belong to that family.

Let $E_q={\max }_{(a,q)=1}|\pi (x;q,a)-\mathrm{Li}(x)/\varphi (q)|$. For $Q\le x^{1-\eta }$, the Brun–Titchmarsh theorem and Cauchy-Schwarz inequality giveWrite each summand as $\sqrt{E_q}{3}^{\omega (q)}\sqrt{E_q}$, then use $E_q{\ll }_{\eta }x/(\varphi (q)\log x)$.

The Bloch sphere is the unit sphere of Bloch vectors, corresponding to pure qubit states through $\rho =(I+\mathbf{r}\cdot \mathbf{\sigma })/2$. A local unitary change of qubit basis induces a rotation of this sphere, with its common phase physically irrelevant. Mixed states occupy the interior of the unit ball.

A biregular graph is a bipartite graph whose vertices have constant degree on each part separately. The two degree constants can differ. If its parts are $X,Y$ and its edge density of a bipartite graph is $\gamma$, the two degrees are $\gamma |Y|$ and $\gamma |X|$. Thus its adjacency function has constant row and column averages $\gamma$.

Equip the function spaces of the two nonempty parts of a bipartite graph with uniform inner products. Its normalized adjacency operator is $({\theta }_{G}v)(x)={\mathbb{E}}_{y}G(x,y)v(y)$. In orthonormal coordinate bases, its matrix is $G/\sqrt{|X||Y|}$, not the unscaled adjacency matrix. For a biregular graph of edge density of a bipartite graph $\gamma$, constant functions give a largest singular value $\gamma$, and the remaining singular values belong to the centered kernel $G-\gamma$.

For a bipartite graph with ordered parts $X,Y$, its biadjacency matrix has rows indexed by $X$ and columns indexed by $Y$, with entry one for an edge and zero otherwise. It has size $|X|$ by $|Y|$, rather than the square full bipartite adjacency matrix on $X\bigsqcup Y$, which is the block matrix $\left(\begin{array}{cc}0& M\\ M^{\mathsf{T}}& 0\end{array}\right)$. Its singular values describe the normalized adjacency operator of a bipartite graph after division by $\sqrt{|X||Y|}$.

For nonnegative integers $r,n$, the alternating binomial moment satisfies $S_r(n)=0$ when $r<n$ and $S_n(n)=(-1{)}^{n}n!$. Expand the polynomial $X^r$ in falling factorials and differentiate the binomial theorem to prove these identities. Equivalently, alternating binomial coefficients implement a finite difference that annihilates polynomials of degree less than $n$.

For the binomial random graph $G(n,1/2)$, the chromatic number is $(1+o(1))n/(2{\log }_{2}n)$ with high probability. A first moment method bounds the independence number above. Edge-disjoint clique packing in the complement graph, followed by an edge-exposure martingale and a union bound over moderately large vertex sets, supplies independent sets for greedy colouring by removing independent sets.

A coeval binary has components formed at the same time. A population of such binary stars can still have a distribution of system ages. If component masses have independence and the state at age $t$ is determined separately for each component, the states have conditional independence given $t$. If $g(t)$ and $w(t)$ are the conditional probabilities of two states, the unconditional unordered pair fractions are $\mathbb{E}[g(t{)}^2]$ and $2\mathbb{E}[g(t)w(t)]$, not generally products of marginal state fractions.

A detached binary has both components lying inside their respective Roche lobes, so neither currently transfers matter by Roche-lobe overflow. A detached system may still exchange matter through a stellar wind.

A binary binary mass ratio is the quotient of its component masses. For mass-transfer stability, specify the donor-to-accretor convention $q=M_d/M_a$ explicitly, since the inverse convention also occurs. Mass-ratio reversal changes which component is more massive.

For a circular Kepler orbit, neglecting stellar spin, the orbital angular momentum of a binary star is $J=\mu a^2\Omega =\mu \sqrt{GMa}$, where $M$ is total mass, $\mu$ is the reduced mass, and $a$ is separation. Kepler's third law also gives $J=G^{2/3}{P}^{1/3}{M}_1{M}_2/[(2\pi {)}^{1/3}{M}^{1/3}]$. Distances from the center of mass are $a{M}_2/M$ and $a{M}_1/M$; adding their individual orbital angular momenta proves the formula.

For a real rectangular matrix $S$, maximize $p^{T}Sq$ over separate sign choices for $p$ and $q$. With $x=(p,q)$, the symmetric block matrix $A=\frac{1}{2}\left(\begin{array}{cc}0& S\\ S^T& 0\end{array}\right)$ satisfies $x^{T}Ax=p^{T}Sq$. Its zero diagonal blocks allow the Krivine rounding scheme to linearize the cross-block expected objective.

A bimatrix game is nondegenerate if every mixed strategy with $k$ positive coordinates has at most $k$ pure best responses for the other player. This condition prevents ties in the leaving-variable choice along the usual Lemke-Howson algorithm path and allows unique continuation once the dropped label has been selected.

A square bimatrix game is symmetric when exchanging players exchanges their payoffs: its payoff matrices are $P,Q$ with $Q=P^T$. If $(x,y)$ is a Nash equilibrium, then $(y,x)$ is also a Nash equilibrium. A symmetric bimatrix game can still have asymmetric equilibria, so the two strategy vectors must be kept separate in the Lemke-Howson algorithm.

A Type I sum in analytic number theory has the form ${\sum }_{d\le U}{a}_d{\sum }_{m\in I_d}f(dm)$, with a short weighted variable and a relatively long inner sum whose coefficients are simple. The Vaughan identity produces these sums when the Möbius function is truncated.

A Type II sum has the form ${\sum }_{d\sim D}{\sum }_{m\sim M}{a}_d{b}_{m}f(dm)$ with two nontrivially weighted variables, both long enough for a Cauchy-Schwarz inequality and a large sieve estimate. The Vaughan identity supplies this bilinear structure for sums weighted by the Von Mangoldt function.

A real symmetric bilinear form is positive-definite when $B(v,v)>0$ for every nonzero vector. It defines an inner product and the associated norm $\sqrt{B(v,v)}$. Symmetry is part of this definition: positive diagonal values by themselves do not imply symmetry.^[ref]

The identity matrix $A=\mathrm{diag}(1,1)$ gives $B(v,w)=v_1{w}_1+v_2{w}_2$ and $B(v,v)=v_1^2+v_2^2$. For $v=(3,4)$ this is $25$, giving norm $5$. A weighted example, $A=\mathrm{diag}(4,1)$, gives norm $\sqrt{4x^2+y^2}$: the first coordinate costs twice as much in length. Such inner products let geometry assign different weights to different directions.

Every positive-definite bilinear form is nondegenerate: a vector in its radical of a bilinear form has $B(v,v)=0$, so positivity forces $v=0$. The converse fails, as $\mathrm{diag}(1,-1)$ shows. Even positivity of the associated quadratic form does not force a general bilinear form to be symmetric: $A=\left(\begin{array}{cc}1& 1\\ -1& 1\end{array}\right)$ has $v^{T}Av=x^2+y^2$ but $B(e_1,e_2)=1$ and $B(e_2,e_1)=-1$.

The complex Killing form is not positive-definite on the entire complex Cartan subalgebra; its restriction to the Euclidean subspace of a Cartan subalgebra is a real positive-definite form.

For real $f$, a candidate $u\in U\subset C(K,\mathbb{R})$, and active set $E=\{x:|f(x)-u(x)|=\Vert f-u{\Vert }_{\infty }\}$, the candidate is best precisely when every $v\in U$ has some $x\in E$ with $(f(x)-u(x))v(x)\le 0$. A direction with strictly matching error signs on the whole active set gives a uniformly improving small perturbation by compactness. Conversely one oppositely signed active value already prevents improvement. This criterion yields the Chebyshev alternation theorem.