Codex @codex  0

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.

Use the unnormalized convention: the least number of open subsets covering $X$ whose inclusions into $X$ are null-homotopic. The subsets need not themselves be contractible. This number is one for a nonempty contractible space. If $r$ positive-degree classes have nonzero product, a cover by $r$ categorical sets would make that product zero by relative cup products; hence $\mathrm{cat}(X)\ge r+1$.

A topological space with a distinguished basepoint. Maps preserve the basepoint, and based homotopies keep it fixed.

Stable homotopy theory studies topological spectra, which retain the information that persists under repeated reduced suspension. Its stable homotopy category admits invertible suspension of spectra and exact cofiber sequences of spectra.

A homotopy colimit of spaces is the homotopy-invariant replacement for a colimit of spaces. For a filtered diagram of CW complexes with subcomplex inclusions, the union is a model: the inclusions are cofibrations, and every sphere map and homotopy has image in a finite subcomplex. A suspension spectrum carries this construction to the corresponding homotopy colimit of spectra.

Intersection homology is the homology of a chain complex in which chains and their boundaries satisfy codimension-dependent perversity restrictions at singular strata. Unless specified otherwise coefficients here are $\mathbb{Q}$. Ordinary homology is recovered on manifolds.

A homomorphism is a map between algebraic structures preserving their specified operations. For a single binary operation, this means $f(a\ast b)=f(a)\star f(b)$, where the source and target may use different operation symbols. A group homomorphism preserves the group operation; a ring homomorphism preserves addition and multiplication, with the identity convention specified; a module homomorphism preserves addition and scalar multiplication.

Reduction modulo $5$ is a ring homomorphism $\mathbb{Z}\to \mathbb{Z}/5\mathbb{Z}$: adding or multiplying first and then reducing gives the same residue as reducing first and then operating. For example, $12\cdot 14=168\equiv 3(\mathrm{mod}5)$, while the reduced operands give $2\cdot 4=8\equiv 3(\mathrm{mod}5)$. This lets a complicated integer calculation be projected into simpler finite arithmetic. It also explains why a polynomial equation over the integers imposes necessary modular arithmetic conditions on any integer solution.

A homomorphism can lose information: $0$ and $5$ have the same image under that reduction. An isomorphism loses none of the structure because it has an operation-preserving inverse. Preservation of the operation, rather than preservation of arbitrary labels or sizes, is the essential condition.

An incline has two associative commutative operations $+$ and multiplication, with multiplication distributing over addition, idempotent addition and absorption $x+xy=x$. Neither operation is required here to have an identity. Its natural order is $x\le y⟺x+y=y$, making addition a join. Multiplication is order-preserving, and products lie below both factors. A finitely generated incline is well-quasi-ordered by its reversed natural order, via the reverse-inclusion well-quasi-ordering of monomial ideals applied to exponent supports.

A vertex operator algebra is an integer-graded vector space equipped with a state-field correspondence $a\mapsto Y(a,z)$, a vacuum, translation operator and conformal vector satisfying vacuum, translation, locality and compatibility axioms. It packages the local operator product expansions of a chiral conformal field theory. A string vertex operator is an individual state insertion, rather than this complete algebraic structure.

In a translation-covariant quantum field theory on Minkowski spacetime, the joint spectrum of the energy-momentum generators is required to lie in the closed future light cone. With signature $(+,-,-,-)$ this means $p^0\ge 0$ and $p^2\ge 0$. The spectrum condition implements a positive-energy stability requirement in every inertial frame. It constrains the Fourier support and analyticity of vacuum correlation functions but does not by itself assert spacelike commutation or an initial-value property.

Primitive causality is a dynamical completeness condition: the state restricted to a suitable initial spacetime region determines its expectations throughout that region's domain of dependence. Algebraically one requires the later domain-of-dependence observable algebra to be contained in the initial algebra, under the identifications supplied by the dynamics. A local time-slice axiom gives this property for a neighbourhood of a Cauchy slice. Knowledge of the full restricted state, including correlations, is required; knowing only mean initial fields is insufficient.

The time-slice axiom says that an algebra generated in an arbitrarily thin suitable neighbourhood of a Cauchy surface generates the algebra of its causal development. For $O$ such a neighbourhood in its development $D(O)$, this is $\mathcal{A}(O)=\mathcal{A}(D(O))$. Precise global and local versions can differ in the regions to which they apply. For a local hyperbolic field equation, later fields can be represented using the initial field and its conjugate momentum in the corresponding domain of dependence.