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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.

For a smoothly parametrized curve on an embedded surface in Euclidean space, the geodesic equations hold exactly when the ambient acceleration is normal to the surface.

If two embedded surfaces are tangent along a curve, their tangent planes agree there. The curve's ambient acceleration is normal to one surface exactly when it is normal to the other, so it is a geodesic of one exactly when it is a geodesic of the other.

A line is a unit-speed geodesic $\gamma :\mathbb{R}\to M$ that minimizes between every two of its points: $d_g(\gamma (s),\gamma (t))=|s-t|$.

This is a globally minimizing geodesic, a Riemannian specialization of the general geometric line notion.

A geodesic is complete when its maximal affine parameter interval is all of $\mathbb{R}$.

For a tangent vector field $V$ along a curve on an embedded surface, the surface covariant derivative is the tangential projection of its ordinary derivative:It is the covariant derivative induced by the surface metric.

A geodesic triangle is bounded by three geodesic segments. On a surface its angle excess is the integral of the Gaussian curvature over the triangle.

It is the three-sided instance of a geodesic polygon.

If $G$ is a generic filter over a countable transitive model $M$, the generic extension $M[G]$ consists of the interpretations by $G$ of all forcing names in $M$.

A random intercept gives every group a latent additive shift, inducing dependence among observations from the same group.

For equal masses $m$, wall-spring constants $k_1,k_2$, and coupling-spring constant $k_3$, the small-oscillation mass and stiffness matrices areIf $k_1=k(1+\epsilon \delta )$, $k_2=k(1-\epsilon \delta )$, and $k_3=k\epsilon$, the squared normal frequencies are

If the mass and stiffness matrices commute with a permutation representation, each symmetry subspace is invariant under the generalized eigenvalue problem for small oscillations. On an irreducible subspace, Schur lemma forces a commuting operator to be scalar, producing a degenerate normal mode multiplet.

A weakly coupled degenerate pair can split by order $\sqrt{\mu }$ when the small parameter enters a singular mass or stiffness matrix.

For equal lengths and mass ratio $\mu =m/M$, the squared frequencies are ${\omega }_0^2(1+\mu \pm \sqrt{\mu (1+\mu )})$.

For symmetric stiffness matrix $K$ and positive-definite mass matrix $M$, eigenvectors $a_i,a_j$ ofwith distinct squared frequencies satisfyThus normal modes are orthogonal in the kinetic-energy inner product.