$\mathbf{co}\text{-}\mathbf{RP}$ consists of complements of languages in RP. Its algorithms always accept positive instances and reject negative instances with probability at least one half.

$\mathbf{RP}$ contains decision problems with a polynomial-time randomized algorithm that rejects every negative instance and accepts every positive instance with probability at least one half.

Gaussian null coordinates adapt one coordinate to the generators of a null hypersurface, one affine coordinate to transverse null geodesics, and the remaining coordinates to a spacelike cross-section. If the generator parameter is affine, the metric can locally be written

A Killing horizon is a null hypersurface whose null generators are orbits of a Killing vector field $K$. On the horizon, $K^b{\nabla }_b{K}^a=\kappa K^a$ defines its surface gravity.

An $R$-module is Noetherian when every submodule is finitely generated, equivalently when every ascending chain of submodules stabilizes. Quotients and finite direct sums of Noetherian modules are Noetherian.

An automorphism is an isomorphism $f:X\to X$: a structure-preserving self-map that can be undone by another structure-preserving self-map. Equivalently, it is an invertible endomorphism. Its inverse satisfieswhere $1_X$ is the identity morphism. Both maps must be morphisms in the chosen category. For ordinary groups, rings, and vector spaces over a fixed field, a bijective morphism automatically has a compatible inverse. For topological spaces, an automorphism is a homeomorphism from the space to itself: a continuous bijection needs a continuous inverse as well. See Riehl, Category Theory in Context, Definition 1.1.10 and Example 1.1.11.

For example, $T(x,y)=(x+y,y)$ is an automorphism of the vector space ${\mathbb{R}}^2$ over the real numbers, with inverse $T^{-1}(u,v)=(u-v,v)$. Both are linear maps. The linear endomorphism $P(x,y)=(x,0)$ from endomorphism preserves addition and scalar multiplication too, but discards a coordinate and cannot be undone. Preserving the structure and preserving enough information for reversal are separate requirements.

The automorphisms of a fixed object in a locally small category form a group under composition in a category: the identity morphism is the identity element, and the inverse element of a composite reverses the order of its factors. This makes reversible internal changes calculable. For a group, the resulting automorphism group records which relabelings preserve all products; the examples in group automorphism show how restrictive that condition can be.

Monte Carlo error is the random difference between a simulation estimate and the exact quantity it approximates. For an average of independent finite-variance draws, its standard deviation is proportional to the inverse square root of the number of draws.

Momentum conservation states that total momentum stays constant when the net force exerted from outside the chosen collection is zero. Use a closed collection, meaning the same fixed-mass point masses are followed throughout, and an inertial frame. ThenTo see why, sum Newton's second law over the members. If ${\mathbf{F}}_{ij}$ is the force on member $i$ from member $j$, Newton's third law assumes ${\mathbf{F}}_{ij}=-{\mathbf{F}}_{ji}$. Each pair cancels in the total:The internal forces act on different bodies and can change their individual momenta; their contributions cancel only in the collection's sum. An isolated system has no external interaction, but balanced external forces also suffice for momentum conservation. A closed system merely excluding matter exchange need not have conserved momentum. ^[ref]

The usefulness is that the detailed internal forces during a collision need not be known. Both an elastic collision and a perfectly inelastic collision obey the same isolated momentum balance; kinetic energy conservation is an additional condition for the elastic collision. If the external impulse is small but nonzero, the total momentum change is correspondingly small. For a variable-mass system exchanging matter, the entering or leaving momentum must also be included.

In a translation-invariant field theory, integrating over the spacetime zero mode enforces momentum conservation at interactions and scattering amplitudes.

Let $\mathcal{L}$ be the set of complex lattices. A weight-$k$ lattice function satisfies $F(\lambda \Lambda )={\lambda }^{-k}F(\Lambda )$. Evaluating at ${\Lambda }_{\tau }=\mathbb{Z}\tau +\mathbb{Z}$ identifies such functions with weight-$k$ modular-invariant functions on the upper half-plane.

A theta function is a holomorphic function formed by summing an exponential quadratic form over a lattice.

A meromorphic modular form obeys the modular transformation law and is meromorphic on the upper half-plane and at the cusps.

An Eisenstein series is a modular form constructed by summing a weight factor over a parabolic coset space. For even $k>2$, the level-one holomorphic Eisenstein series is a scalar multiple of ${\sum }_{(m,n)\ne (0,0)}(m\tau +n{)}^{-k}$.

Translation invariance at a cusp gives a Fourier expansion in its local parameter $q$. Holomorphy at the cusp excludes negative powers; vanishing of the constant term defines a cusp form.

For $\gamma =\left(\begin{array}{cc}a& b\\ c& d\end{array}\right)$, the weight-$k$ slash operator isA modular form of weight $k$ and level $\Gamma$ satisfies $f{|}_k\gamma =f$ for every $\gamma \in \Gamma$.

The modular group is $S{L}_2(\mathbb{Z})$ acting on the complex upper half-plane. Its central element $-I$ acts trivially on points but contributes the factor $(-1{)}^k$ in weight $k$.

A prime model of a complete theory elementarily embeds into every model of that theory.

An Ehrenfeucht-Mostowski model is generated, often as a Skolem hull, by a sequence of order indiscernibles whose finite order types determine their types.

A theory is model-complete when every embedding between its models is elementary. Every model-complete theory is inductive and hence admits a forall-exists axiomatization.

A first-order theory is inductive when unions of chains of its models under embeddings remain models. A theory is inductive exactly when it has an axiomatization by sentences of the form $\forall \bar{x}\exists \bar{y}\phi (\bar{x},\bar{y})$ with quantifier-free $\phi$.

In the MSSM, superpartners change the three Standard Model gauge beta functions so that the inverse gauge couplings meet accurately near $2\times 10^{16}$ GeV, up to threshold and higher-loop corrections. The corresponding Standard Model one-loop lines miss a common intersection.