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At linear order in halo correlations,

The one-halo term comes from pairs of mass elements in the same halo:

In a $k$-regular bipartite graph, edge counting gives equal vertex-class sizes and $k|S|\le k|N(S)|$. Hall's theorem therefore supplies a perfect matching.

For Hall's theorem, either every proper nonempty set has one excess neighbour and one deletes an arbitrary edge's endpoints, or a tight set $S$ with $|N(S)|=|S|$ splits the problem into the induced graph on $S\cup N(S)$ and its complement.

An abelian variety is a complete connected group variety. Despite its name, commutativity follows from completeness and the rigidity lemma.

A quasi-tree is a geodesic metric space quasi-isometric to a tree.

A finitely generated group is hyperbolic when one, equivalently every, Cayley graph for a finite generating set is a Gromov-hyperbolic metric space.

If $\gcd (a,b)=1$ and $a\mid bc$ in a greatest-common-divisor domain, then $a\mid c$. Indeed, the least common multiple of $a$ and $b$ is associate to $ab$ and divides every common multiple; applying this to $bc$ and cancelling $b$ proves the claim.

The real projective plane is the quotient of $S^2$ by $x\sim -x$ and parametrizes lines through the origin in ${\mathbb{R}}^3$.

Every rank-one orthogonal projection has the form $x{x}^T$ for a unit vector, uniquely up to replacing $x$ by $-x$.

The paw graph is a triangle with one pendant edge.

For the star $K_{1,t}$,The parity distinction follows from the handshake lemma applied to a hypothetical $(t-1)$-regular colour class on $2t-1$ vertices.

The Ramsey number $r(G,H)$ is the least $n$ such that every red-blue colouring of $K_n$ contains a red copy of $G$ or a blue copy of $H$.

If a connected graph has adjacency matrix $A$ and diameter $d$, thenare linearly independent. For the largest index $k$ with a nonzero coefficient, choose vertices at distance $k$; the corresponding entries of all lower powers vanish, while $(A^k{)}_{uv}>0$ by the walk count from powers of an adjacency matrix.

For a symmetric bilinear form $B$, the same projection formulaproduces a $B$-orthogonal basis whenever the chosen pivots $B(u_j,u_j)$ are nonzero. Reordering the input vectors can avoid a zero pivot when a suitable anisotropic vector remains.

A reduced QR decomposition of an $m\times k$ matrix over the real numbers with full column rank, $m\ge k$, writes $A=QR$, where $Q$ is $m\times k$ with orthonormal columns and $R$ is a $k\times k$ upper triangular matrix with positive diagonal. Applying the Gram-Schmidt process to the columns $v_j$ of $A$ constructs $Q=(q_1\cdots q_k)$ and records the coefficientsThis last column identity is exactly the matrix multiplication $A=QR$. The upper triangular matrix records that the $j$th input uses only the first $j$ new directions, while its positive diagonal records the norms of the nonzero orthogonal projection residuals. Since $Q^{T}Q=I$, one also has $R=Q^{T}A$; $Q^T$ denotes the transpose and $I$ the identity matrix. Over the complex numbers, use the conjugate transpose $Q^{\ast }$ in these identities.

For the three-dimensional Gram-Schmidt process example, the exact matrices areFor example, the last column reconstructs asThe diagonal entries are approximately $1.414214,1.224745,1.154701$. Each is a norm of the new direction before it was scaled to length one.

For a linear least-squares problem, the orthogonal projection of $b$ onto the column space of $A$ is $Q{Q}^{T}b$. Hence the minimizing coefficients solve the linear system $Rx=Q^{T}b$ by back substitution. Let $A_2=(v_1{v}_2)$ consist of the first two columns of the example; its reduced factors are $Q_2=(q_1{q}_2)$ and the leading $2\times 2$ block $R_2$ of $R$. For $b=(0,0,1{)}^T$, the linear system $R_{2}x=Q_2^{T}b$ isThus $A_{2}x=(1/3,-1/3,2/3{)}^T$, agreeing with the orthogonal projection computed above, and the minimum Euclidean norm of $A_{2}x-b$ is $1/\sqrt{3}$. The QR decomposition separates building an orthonormal basis from recovering coefficients in the original basis.^[ref]

For a graded module $M={⨁}_{n\ge 0}{M}_n$ whose pieces have finite length, its Hilbert series is

For an ideal $I$, the associated graded ring is

Repeated Cauchy-Schwarz inequality gives

If two senders share a GHZ state with a receiver, one sender can encode two bits with $X^x{Z}^z$ and the other one bit with $X^b$. Sending both qubits to the receiver produces one of eight orthogonal GHZ-basis states, so a joint measurement recovers all three bits.