Phenylalanine is an amino acid, represented by Phe or F for an incorporated amino acid residue. It was identified at the other insulin N-terminus in Sanger's insulin sequencing experiment. See IUPAC Blue Book, P-103.1: amino-acid names and symbols.

Glycine is an amino acid, represented by Gly or G for an incorporated amino acid residue. Its amino acid side chain is a hydrogen chemical atom. It was identified at one insulin N-terminus in Sanger's insulin sequencing experiment. See IUPAC Blue Book, P-103.1: amino-acid names, symbols and structures.

An alpha-amino acid has amino functionality on the carbon next to its carboxyl carbon. The skeleton in amino acid shows this adjacency explicitly. Numbering the carboxyl carbon as C1 makes the adjacent position C2, also called the alpha position. ^[ref]

Alanine is an amino acid represented by Ala or A in protein sequence notation. GCU is one of its codons in the standard genetic code, and supplies the second residue in the worked gene expression example. ^[ref]

Methionine is an amino acid represented by Met or M in protein sequence notation. In the standard genetic code, AUG encodes it, as in the worked gene expression example. ^[ref]

In a discrete-time binomial market with $B_t=(1+r{)}^t$ and one-step stock factors $1\pm \epsilon$, use the risk-neutral probability $p=(1+r/\epsilon )/2$, where $0\le r<\epsilon <1$. For the exercise payoff $S_t^2$, the discounted one-step reward multiplier has meanThe Snell envelope of a multiplicative process gives the value $V_t=S_t^2{\lambda }^{T-t}$. Continuation strictly exceeds exercise before maturity, so the optimal exercise time is $T$. The replicating portfolio in a binomial market holds $2S_t{\lambda }^{T-t-1}$ shares on the next step, found by subtracting the two next-state values and dividing by the stock-price difference. At time zero the price is $S_0^2{\lambda }^T$.

An American call option permits buying the underlying at strike price $X$ at any chosen stopping time from the current date $t$ through maturity $T$. Under standard complete-market pricing and integrability hypotheses,The conditional expectation uses information available at time $t$, and the optimization allows only future exercise times. A European call option restricts exercise to $T$. The no early exercise of a call without dividends result applies with nonnegative interest under its stated assumptions; dividends or negative rates can change that conclusion. The discounted value is the Snell envelope of discounted exercise rewards, as in American option.

For a countable discrete group, the Reiter condition requires a sequence of probability measures in ${\ell }^1(G)$ asymptotically invariant under every left translation. It is equivalent to the existence of an invariant mean and a Følner sequence. Separation by bounded functions shows that amenability gives approximate invariance simultaneously on each finite set of translations; countability makes the successive finite tests into a sequence.

An inclinometer measures tilt relative to the vertical established by gravity, usually reporting an angle from the perpendicular horizontal plane. A weighted-string inclinometer compares a sighting tube with a freely hanging reference; its scale must be oriented so a level sight gives zero altitude. A raw protractor may instead indicate the complementary angle, so check the markings rather than assuming that every instrument has the same scale. ^[ref].

A finite alphabet is an alphabet whose symbols form a finite set. The binary alphabet $\{0,1\}$ and a finite vocabulary of possible messages are examples.

A group variety is an algebraic variety whose multiplication and inversion are morphisms.

On an integral algebraic variety, a rational function is an equivalence class of regular functions on nonempty open subsets, with equality on a common dense open subset. These form the function field $k(X)$. On an affine open $\mathrm{Spec}A$, this field is the fraction field of $A$.

// Target: geometry-and-topology.bigb

For a possibly reducible algebraic variety $X$, take pairs $(U,f)$ with $U$ a dense open subset and $f$ a regular function on $U$. Identify pairs agreeing on a common dense open subset. Addition and multiplication on intersections make these classes a ring. For an affine variety with reduced coordinate ring $A$, this ring is its total quotient ring. For an irreducible variety, it is the function field. An isomorphism of dense open subsets induces an isomorphism of these rings.

A rational map is a morphism of algebraic varieties defined on a dense open subset, with representatives identified when they agree on a smaller dense open subset. For an integral source and a separated variety as target, agreement on a dense open determines a morphism uniquely. This definition also permits nonprojective sources.

An exceptional curve of a surface resolution is an irreducible curve on the smooth resolving surface that maps to a point on the original normal surface. For an affine toric surface resolved by fan subdivision, the interior inserted rays give these proper curves. Each has a complete one-dimensional star and is isomorphic to the projective line.

A cubic surface is a degree-three projective hypersurface in the projective space ${\mathbb{P}}^3$. Over an algebraically closed field every cubic surface contains a projective line. The smooth case has finitely many lines, although the incidence proof of existence does not require smoothness.

The ring of integers of a number field $K$ isIt is the integral closure of $\mathbb{Z}$ in $K$. Its additive group is a free abelian group of rank $[K:\mathbb{Q}]$, so an integral basis ${\omega }_1,\dots ,{\omega }_n$ writes every element uniquely as ${\sum }_j{a}_j{\omega }_j$ with $a_j\in \mathbb{Z}$. Its field of fractions is $K$.^[ref]

For $K=\mathbb{Q}$, this gives ${\mathcal{O}}_K=\mathbb{Z}$; for $K=\mathbb{Q}(\sqrt{2})$, it gives $\mathbb{Z}[\sqrt{2}]$. The Gaussian integers give the corresponding example inside $\mathbb{Q}(i)$. The ring of integers is determined by the field, and need not equal $\mathbb{Z}[\theta ]$ for a chosen algebraic integer generator: in $\mathbb{Q}(\sqrt{5})$, the larger ring $\mathbb{Z}[(1+\sqrt{5})/2]$ contains $\mathbb{Z}[\sqrt{5}]$ with index two.

This is the setting for divisibility, global units, and prime ideal factorization. A ring of integers is a Dedekind domain, so every nonzero proper ideal factors uniquely into nonzero prime ideals, even when unique factorization of elements fails.^[ref]

There is a sequence of real numbers for which every finite initial segment consists of algebraically independent elements over the rationals. Given an independent initial segment, specialize each nonzero integer polynomial in one extra variable at those values. At least one coefficient remains nonzero, so each resulting polynomial has finitely many roots. There are countably many such polynomials by countability of integer polynomial rings. Choosing the next real outside the countable union of their roots continues the construction. Clearing denominators makes independence over the integers equivalent to independence over the rationals.

An $R$-point of a unital algebra over a field $A$ is a unital algebra homomorphism over a field $A\to R$. For the quantum plane over the complex numbers, scalar points lie on the two coordinate axes when $q\ne 1$, while matrix-valued points are pairs $X,Y$ with $YX=qXY$.

For $L=-d^2/d{x}^2+x$ on $(0,1)$, with homogeneous conditions $v(0)=0$, $v^{\prime }(1)=0$, integration by parts gives $⟨v,Lv⟩={\int }_0^1(v^{\prime 2}+x{v}^2)dx>0$ for nonzero $v$. The affine energy $J[u]=\frac{1}{2}{\int }_0^1(u^{\prime 2}+x{u}^2)dx$, minimized over $H^1$ functions with $u(0)=a$, yields $Lu=0$ and the natural boundary condition $u^{\prime }(1)=0$. The inequality $\Vert v{\Vert }_2^2\le \frac{1}{2}\Vert v^{\prime }{\Vert }_2^2$ on $v(0)=0$ proves that the form is a coercive bilinear form; Lax-Milgram theorem supplies a unique critical point, and $J[u+v]-J[u]=\frac{1}{2}\int (v^{\prime 2}+x{v}^2)$ proves that it is the unique minimum.