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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.

Starting from $|000⟩$, apply a Hadamard gate to the first qubit and then controlled-NOT gates from the first qubit to each of the other two qubits.

For $|\alpha ⟩=a|0⟩+b|1⟩$, a Bell measurement on $|\alpha ⟩$ and the first leg of a GHZ state leaves the other two legs in $a|00⟩\pm b|11⟩$, after applying $X$ to both legs for the two $\Psi$ outcomes.

The space $\mathcal{G}$ of germs over a domain $D$ has basic open sheetsfor holomorphic $f$ on open $U\subseteq D$. The projection $\pi ([f{]}_z)=z$ restricts to a homeomorphism on each sheet, and these projections form holomorphic coordinate charts.

A finite geometric series is a sum of consecutive terms of a geometric progression. With first term $a$ and $N$ terms, including $N=0$ for the empty sum,Unlike an infinite geometric series, this identity does not need $|r|<1$. For example, $3+6+12+24+48=3(2^5-1)=93$. The number of terms is five; the largest exponent is four.

The cancellation behind the formula is especially useful to remember: multiplying $S_N$ by the common ratio shifts all terms one place. In $S_N-r{S}_N$, every interior term cancels, leaving $a-a{r}^N$. Division by $1-r$ gives the identity when $r\ne 1$; at $r=1$ the sum is directly $Na$. The same argument works for complex numbers.

For integers $m\le n$ and $r\ne 1$, the shifted version isThere are $n-m+1$ terms. All powers must be defined; if a negative exponent is present, $r$ must be nonzero. At $r=1$ this sum is simply $n-m+1$. The finite trigonometric sum uses this identity with a complex exponential as the common ratio.

Finite sums of positive integers in a geometric progression already appear in Euclid's Elements, Book IX, Proposition 35. The argument there compares differences of proportional numbers; Joyce's guide rewrites its conclusion as the modern finite-sum formula. This historical result concerns finite sums, without asserting an infinite limit. ^[ref]

A genus one curve is a smooth projective curve of geometric genus one. Choosing a rational point makes it an elliptic curve.

An affine parameter on a geodesic is one for which the coordinate equation has the form ${\ddot{x}}^k+{\Gamma }_{ij}^k{\dot{x}}^i{\dot{x}}^j=0$. Its affine changes $\lambda \mapsto A\lambda +B$, with $A\ne 0$, preserve this form.

A constant-speed intersection with a plane containing every surface normal along the curve is a geodesic.