An angle in radians is the ratio of its subtended arc length to the radius of a circle. A full turn is $2\pi$ radians and a right angle is $\pi /2$ radians.

Partial summation is the discrete analogue of integration by parts. If $A(x)={\sum }_{n\le x}{a}_n$ and $g$ is continuously differentiable, then

Dirichlet's theorem on arithmetic progressions states that for positive integer $m$ and integer $a$ with $\gcd (a,m)=1$, infinitely many primes satisfy $p\equiv a(\mathrm{mod}m)$.

For example, each of the arithmetic progressions $4n+1$ and $4n+3$ contains infinitely many primes. The theorem establishes infinitude separately in every coprime residue class, rather than only in their union.

The coprimality condition is necessary. If $d=\gcd (a,m)>1$, every integer in the progression is divisible by $d$. A member can then be prime only when it equals $d$ and $d$ itself is prime. This leaves at most one prime value in that residue class. The difficult part of the theorem is sufficiency; its analytic treatment uses Dirichlet L-functions.

The circle method represents a weighted additive count as an integral of products of exponential sums against a character. Near rationals with small denominator, major arcs produce a structured main term involving a singular series and an archimedean singular integral. Away from those rationals, minor arcs require cancellation bounds. The partition and approximation estimates depend on the problem; a formal integral alone is not a positivity proof.

For $0<M<a$, $\Delta =(r-M{)}^2+a^2-M^2>0$, so no horizon blocks need adjoining. The oblate Cartesian coordinates of the Kerr metric satisfy the displayed relation, where ${\rho }^2=x^2+y^2$. Keeping only the positive root $r$ leaves a cut along the disk $z=0$, $\rho <a$. Continue $r$ through zero away from its boundary, the Kerr ring singularity, onto a negative-radius sheet. Equivalently cut two Cartesian sheets along the disk and join opposite banks, so $r$ changes sign continuously. The metric is analytic at the disk interior in signed oblate coordinates since $\Sigma =a^2{\cos }^2\theta \ne 0$ there. The extension has asymptotically flat ends at $r\to +\infty$ and $r\to -\infty$; only the singular ring remains inextendible by an analytic Lorentz metric.

Arrow's theorem says that a rank-order voting system with at least three alternatives cannot simultaneously satisfy unrestricted preferences, unanimity, independence of irrelevant alternatives, and absence of a dictator.

For a Boolean function $f:\{0,1{\}}^n\to \{0,1\}$ under the uniform probability measure, some coordinate has influence of a variable at least $c\mathrm{Var}(f)\log (n)/n$, where $c>0$ is an absolute constant and $I_i(f)$ is the probability that flipping coordinate $i$ changes $f$. For variance bounded below, this is order $\log (n)/n$. The tribes function shows that this order cannot be improved for general balanced Boolean functions.

There is an absolute $C>0$ such that any nontrivial transitive increasing event on $n\ge 2$ independent bits satisfies ${\mu }_q(A)>1-\epsilon$ if ${\mu }_p(A)>\epsilon$, $0<\epsilon <1/2$, and the displayed bound holds with $p<q<1$. A uniform weighted Kahn-Kalai-Linial theorem gives one pivotal probability at least $c{\mu }_r(A)(1-{\mu }_r(A))\log (n)/n$. Transitivity makes the total influence at least $c{\mu }_r(A)(1-{\mu }_r(A))\log n$. The Margulis–Russo formula identifies this with the derivative of ${\mu }_r(A)$. Integration of the derivative of its log odds proves the threshold bound.

If the state-preparation unitary sends $|0⟩$ to $|\psi ⟩=\sin \chi |g⟩+\cos \chi |b⟩$, use the orthonormal good/bad basis. The marked-space phase is diagonal there, and conjugating a phase on $|0⟩$ gives the rank-one phase in the displayed formula. Their product is unitary. An arbitrary preparation unitary not sending zero to this state need not preserve this particular two-dimensional subspace. Matrices in unnormalized good/bad vectors must not be described as unitary in the ordinary Euclidean inner product.

For known initial success probability $p\in (0,1)$, this choice of iteration count ensures $|(2r+1)\theta -\pi /2|\le \theta$. Hence the success probability after amplitude amplification is at least ${\cos }^2\theta =1-p$. Counting initial state preparation and the forward and inverse preparation in each iterate gives $2r+1$ preparation-unitary applications, asymptotic to $\pi /(2\sqrt{p})$ as $p\to 0$.

A glutamic acid residue is an amino acid residue represented by Glu or E in polypeptide sequence notation. GAA and GAG specify glutamic acid residues in the standard genetic code; the ribosome example uses GAA as its third codon. See Elzanowski and Ostell, NCBI: standard genetic code, table 1. The name denotes an incorporated chain position, as distinguished from a free amino acid molecule.

The amino acid alpha carbon is the backbone carbon carrying the amino acid side chain in the usual protein amino acid residue. Marking successive amino acid alpha carbons is a useful way to trace a polypeptide chain, but omits intervening backbone chemical atoms. See Clark, Douglas and Choi: the central alpha carbon.

An amino acid side chain is the variable group attached to an amino acid alpha carbon, conventionally denoted $R$. For alanine it is $\mathrm{C}{\mathrm{H}}_3$: a carbon bonded to three hydrogen chemical atoms. In a beta strand, successive amino acid side chains alternate between the two faces in the idealized arrangement. See Clark, Douglas and Choi: amino-acid R groups.

A tyrosine residue is an amino acid residue whose amino acid side chain can accept a phosphate group during receptor tyrosine kinase signaling. The conventional one-letter sequence symbol is $Y$.

Capture of an alpha particle by carbon-12 makes oxygen-16. Competition between this reaction and the Triple-alpha process helps determine the carbon-to-oxygen ratio after core helium burning.

An allele fixes when all copies at the relevant locus in the biological population are that allele. Fixation can occur through selection or genetic drift. Spatial polymorphism means that fixation within an individual habitat need not imply fixation across a metapopulation.

For a based finite acyclic chain complex, Reidemeister torsion is an alternating product of determinants comparing each given chain basis with bases formed from boundaries and lifts of boundaries. With suitable local coefficients this gives an invariant of a space finer than its homology. Analytic torsion constructs a corresponding invariant from spectra of Laplacian operators, while Turaev torsion refines combinatorial sign and normalization choices.

The suspension is the union of two cones on a space along their common base.

The cone of a topological space $X$ is $CX=(X\times [0,1])/(X\times \{0\})$, with one end collapsed to a vertex. Realization of a simplicial cone gives this construction for the realized base complex.

A loop in a topological space is a continuous map $\gamma :[0,1]\to X$ with $\gamma (0)=\gamma (1)$. Equivalently it is a map from the circle. A based loop also specifies the common endpoint as a chosen base point.