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Charge conjugation exchanges particles with antiparticles. For a matrix-valued gauge potential it acts as $A_{\mu }\mapsto -A_{\mu }^T$.

A period-doubling bifurcation occurs when a fixed-point multiplier crosses $-1$ and a nearby period-two orbit is created. Under the generic nondegeneracy conditions, the orbit amplitude is proportional to the square root of the parameter displacement.

For a period-$k$ orbit $x_0,\dots ,x_{k-1}$ of a differentiable map, the multiplier isThe orbit is locally asymptotically stable when the modulus of this product is less than one.

The Jury criterion tests whether every root of a real discrete-time characteristic polynomial lies strictly inside the unit circle. For $z^2+az+b$, this is equivalent to $|b|<1$, $1+a+b>0$, and $1-a+b>0$.

An interval map is a continuous function $F:I\to I$ from a real interval to itself. Its iterates form a one-dimensional discrete dynamical system.

D-separation blocks paths through conditioned noncolliders and through unconditioned colliders without conditioned descendants.

A topological ordering is a topological order: a list containing every vertex once and placing the source of every directed edge before its destination. In the parent-and-child language of a Directed acyclic graph, every parent precedes every child, and hence every ancestor precedes every descendant along a directed path. This is an ordering of the vertices, whereas topological sorting is the task of finding it. See topological order for the precise position condition and Kahn's algorithm for a construction.

If an irrational algebraic number has degree $d$, then $|\alpha -p/q|>C(\alpha )q^{-d}$. Approximations of arbitrarily larger polynomial order therefore prove that a Liouville number is transcendental.

A simple continued fraction recursively expands a real number by taking integer parts and reciprocals. Its convergents are exceptionally good rational approximations.

Baker's theorem gives transcendence and quantitative nonvanishing results for such logarithmic forms.

A homogeneous linear form in logarithms of nonzero algebraic numbers has the form

Let $L_1,\dots ,L_n$ be linearly independent linear forms in $n$ variables with algebraic coefficients. For every $\epsilon >0$, the nonzero $\mathbf{x}\in {\mathbb{Z}}^n$ satisfyinglie in finitely many proper linear subspaces of ${\mathbb{Q}}^n$. There are variants involving a finite set of places of a number field.

A reduced fraction $p/q$ is a best rational approximation of the first kind to a real number $\alpha$ when no rational number with denominator at most $q$ is closer to $\alpha$.

For a smooth function $f(x,y)$,where the smooth remainder is

The Grassmannian $\mathrm{Gr}(k,V)$ parametrizes the $k$-dimensional linear subspaces of a finite-dimensional vector space $V$. Its Plücker embedding makes it a projective algebraic variety.

The real symplectic group consists of matrices $A$ satisfying $A^{T}JA=J$. It is an embedded matrix Lie group of dimension $2n^2+n$ in size $2n$.

For a smooth map $f:X\to Y$ between compact connected $n$-manifolds and a regular value $y$, defineThe regular-value theorem makes the preimage discrete and compact, hence finite. A transverse path between regular values produces a compact one-manifold whose boundary is the two fibres, proving independence of $y$. The degree is invariant under smooth homotopy.

A value is regular when the derivative is surjective at every point of its preimage.

Every value of a submersion is regular, but a map can have some regular values without being a submersion everywhere.

A chart is a homeomorphism from an open subset of a manifold to an open subset of Euclidean space.

A cone is intrinsically flat away from its vertex but has nontrivial angular holonomy.

A ruled surface is swept out by a smoothly varying one-parameter family of affine lines.