A total binary-valued function is diagonally noncomputable if it differs from each defined diagonal value in an effective enumeration of partial computable functions. Nonbinary diagonal outputs are automatically different. No such function is computable: its own program index would contradict the defining condition. Choosing the opposite bit at every defined binary diagonal value proves existence without supplying a computable procedure.

A function in intension is a finite program or formal construction description. Its associated function in extension is the partial or total input-output map it denotes. A Gödel number records the description rather than identifying all descriptions with the same extension.

For a charged Dirac particle, incoming momenta $p,k$ and outgoing momenta $p^{\prime },k^{\prime }$ obey $p+k=p^{\prime }+k^{\prime }$. The two QED diagrams have internal fermion momenta $p+k$ and $p-k^{\prime }$, giving $s$ and $u$ channels. The amplitude is proportional to $\bar{u}(p^{\prime })[/{ϵ}^{\prime \ast }R(p+k)/ϵ+/ϵR(p-k^{\prime })/{ϵ}^{\prime \ast }]u(p)$, where $R(q)=(/q+m)/(q^2-m^2+i0)$. It is related to particle-antiparticle two-photon annihilation by crossing symmetry, with appropriate changes to external wavefunctions and conjugations.

A self-similar blast wave is an expanding shock wave whose post-shock profiles retain their shape in a rescaled radius or distance. A fixed explosion energy, ambient mass density and geometry set the front scaling through conservation of total energy. The dimensionless coefficient requires the profiles and an energy-normalization convention.

A unital commutative ring generated by finitely many elements under addition and multiplication, starting from the image of the integers. Equivalently it is a quotient of a finite-variable polynomial ring over $\mathbb{Z}$. localization of a ring at one element remains finite type, since $R[1/r]\cong R[T]/(rT-1)$. Generation here is as an algebra, not merely as a field extension when the ring happens to be a field.

An oriented knot cobordism is a compact oriented smooth surface properly embedded in $S^3\times [0,1]$ with prescribed knots at its incoming and outgoing boundary. A connected genus-zero knot cobordism between knots is a knot concordance. Composing knot cobordisms adds their genera.

Eligibility criteria specify which individuals may enter a clinical trial, defining its enrolled population and excluding individuals for whom the intervention or measurements are unsuitable. Screening applies those criteria before randomization. The treatment comparison's external validity depends on how this population relates to the population of interest.

A climate feedback is a response to climate change that alters the initial warming or cooling. Positive feedback amplifies the change; negative feedback opposes it. Positive feedback alone does not prove an irreversible transition.

A global symmetry has transformation parameters constant across spacetime and preserves the action. It can relate physically distinct states. Continuous global symmetries give Noether currents and conserved charges when boundary fluxes vanish; discrete symmetries do not receive currents from the same infinitesimal theorem. A gauge redundancy instead uses local parameters to relate descriptions of the same physical state, except that nontrivial boundary transformations may carry physical charges. The global phase symmetry of a complex scalar field is an important internal example.

Languages computable by polynomial-size Boolean circuit families of bounded fan-in and depth $O(\log n)$. One can specify the nonuniform or logspace-uniform convention explicitly. Balanced constant-size operations give examples such as the MOD3 binary divisibility problem without nonuniform advice.

For $X\sim N(m,v)$ with $v>0$ and $0<\epsilon <1$, $P(X\ge C)\le \epsilon$ is equivalent to the displayed chance constraint, because the standard normal distribution function is continuous and strictly increasing. The coefficient is a standard normal quantile, which can be negative. If $v=0$, equality is an atom and the exact condition becomes $C>m$ instead.

A deterministic demand has probability one of meeting or exceeding its mean. Thus replacing a positive-variance Gaussian chance constraint by its naive zero-variance limit with a non-strict capacity inequality loses the boundary condition. The example $X=0$, $C=0$, $\epsilon =e^{-1}$ satisfies $m+\varphi \sqrt{v}\le C$ for finite $\varphi$, but violates the chance requirement.

A coreflexive pair $f,g:B\rightrightarrows A$ has a common retraction $r:A\to B$ with $rf=rg=1_B$. It is the dual of a reflexive pair. Both $f$ and $g$ are split monomorphisms.

A sequence of functionals Gamma-converges in a specified topology when every convergent $u_n\to u$ satisfies $F(u)\le {\mathrm{liminf}}_n{F}_n(u_n)$, and each $u$ has a recovery sequence $u_n\to u$ with $F_n(u_n)\to F(u)$. With appropriate compactness of energy sublevels, limits of approximate global minimizers minimize $F$, and the minimum values converge. This is suited to variational approximation such as the Ambrosio–Tortorelli approximation; it does not describe arbitrary local minima.

The worldvolume is the history of a brane. An embedding $X:\Sigma \to \mathcal{M}$ maps its local coordinates into target spacetime. A string worldvolume is a worldsheet.

The brane tension is the energy per unit spatial brane volume and multiplies its geometric action. In units $\hslash =c=1$, a $p$-brane tension has mass dimension $p+1$. The $p=1$ case is the string tension.

For the normalized isotropic cutoff model, collisions redistribute the direction of relative velocity uniformly while keeping its length. With center $C=(v+v_{\ast })/2$ and relative speed $r=|v-v_{\ast }|$, outgoing velocities are $C\pm r\sigma /2$. Momentum and kinetic energy are preserved, and the collision rate has no relative-speed factor. The displayed operator uses unit total angular rate. General Maxwell molecule kernels can have nonconstant angular dependence.

The integral uses $0\log 0=0$ when it is defined. For the Boltzmann equation, the Boltzmann H theorem makes it nonincreasing under suitable regularity and integrability. This functional is distinct from the microstate-counting Boltzmann entropy; the physical kinetic entropy has the opposite sign.

A Gaussian velocity profile at each position, with coefficients encoding local density, mean velocity and temperature. Its logarithm is a linear combination of collision invariants, so its collision term vanishes pointwise in position. Transport need not vanish: the coefficients must also satisfy their transport equations for a local Maxwellian to solve the full Boltzmann equation.

For equal-mass elastic collisions, one reflection parametrization is $v^{\prime }=v-((v-v_{\ast })\cdot n)n$ and $v_{\ast }^{\prime }=v_{\ast }+((v-v_{\ast })\cdot n)n$. The fixed-$n$ transformation preserves pairwise momentum, kinetic energy and velocity-pair measure. A nonnegative kernel invariant under particle interchange and collision reversal yields a gain-loss operator. Its weak Boltzmann collision identity proves conservation and its entropy identity proves the Boltzmann H theorem. Unlike the Maxwell molecule collision operator, this general kernel may depend on relative speed.