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A skew product evolves a base coordinate by a dynamical system $R$ and a fibre coordinate by a map $S_x$ depending on the current base coordinate: $F(x,y)=(R(x),S_x(y))$. The base evolution is independent of the fibre, while the fibre may depend on the base. Continuity or measurability is imposed according to the setting. An irrational skew shift is a useful affine example on a torus.

A model retaining an oscillation's phase while neglecting amplitude dynamics. Interactions can synchronize relative phases through phase locking; noise can cause phase slips. The Adler phase equation is a simple example for a coupled relative phase.

Stability theory studies the response of a dynamical system to small changes of its state. Linear stability of a planar equilibrium describes the response near equilibria using the linearized evolution.

If a successful step in action $i$ has expected reward $a_i\ge 0$ and survival probability $q_i=1-p_i$, a finite-horizon retirement problem with zero wealth after capture has threshold ${\max }_i{a}_i{q}_i/p_i$. Above it, induction makes retirement optimal; below it, a single further step already beats retirement. The threshold does not depend on the remaining positive horizon.

Orient the plane $\mathbf{n}\cdot \mathbf{r}=d$ with a unit normal $\mathbf{n}$. A point $\mathbf{p}$ then has signed distance from a point to a planeIts absolute value is the ordinary distance from a point to a plane. The closest point on a plane is $\mathbf{p}-s\mathbf{n}$, so the displacement from the point to its orthogonal projection onto a plane is $-s\mathbf{n}$. Reversing the unit normal and $d$ reverses $s$ without changing the plane or the distance from a point to a plane. A sphere of radius $q$ centred at $\mathbf{p}$ has sphere-plane intersection radius $\sqrt{q^2-s^2}$ when $|s|<q$.

For a general, not necessarily unit, normal vector ${\mathbf{n}}_0$ in ${\mathbf{n}}_0\cdot \mathbf{x}=d_0$, the oriented distance is $({\mathbf{n}}_0\cdot \mathbf{p}-d_0)/\Vert {\mathbf{n}}_0\Vert$. In the worked example of distance from a point to a plane, orientation toward $(2,-1,2)$ assigns $+8/3$ to $(4,1,2)$; reversing that unit normal assigns $-8/3$. The closest point on a plane and the unsigned distance are unaffected.

A pseudo-Riemannian manifold is a smooth manifold with a smooth metric tensor that is a nondegenerate bilinear form on each tangent space. Positivity is not required. A Riemannian manifold is the positive-definite case, while a Lorentzian manifold has exactly one temporal direction in its metric signature.

Morse theory relates the critical points of a Morse function to changes in the homotopy type of its sublevel sets. Crossing a nondegenerate critical value attaches a cell or handle of dimension equal to the Morse index. The theory also applies to geodesic energy through broken-geodesic approximations.

A submanifold is a manifold sitting inside another manifold through an immersion or embedding. An embedded submanifold has the subspace topology and local coordinates in which it is a coordinate plane. Smooth transverse intersections are governed by the tangent spaces of the participating submanifolds.

A harmonic map between Riemannian manifolds is a critical point of the Dirichlet energy $\frac{1}{2}\int |dQ{|}^2$. For a Euclidean domain and target unit sphere, the equation is $\Delta Q+|\nabla Q{|}^{2}Q=0$. Stationary wave maps are exactly harmonic maps of their spatial domain.

On a smooth manifold, ${\mathcal{A}}^k(U)$ is the space of smooth complex-valued degree-$k$ differential forms on $U$. Restriction of forms supplies the sheaf maps. Smooth partitions of unity make this a fine sheaf.

A smooth section of ${\bigwedge }^k(T^{\ast }X\otimes \mathbb{C})\otimes E$ is a differential form with values in the complex vector bundle $E$. A connection on a vector bundle extends to these forms by the graded Leibniz rule.

A classical solution is a sufficiently differentiable function that satisfies a differential equation and any prescribed initial or boundary conditions. A weak solution instead satisfies an appropriate integral or distributional identity in a specified function space. For a smooth locally Lipschitz continuous vector field, an ordinary differential equation has a unique local solution for each initial state; this does not by itself ensure global existence or boundedness.

A delay differential equation relates a derivative at the current argument to values at earlier arguments. For example, the Buchstab function obeys $\frac{d}{du}(uw(u))=w(u-1)$ for $u>2$.

The defining feature is memory: a derivative can depend on earlier values of the unknown function. Constant delays, time-dependent delays, and delays depending on the unknown function belong to this wider class. The retarded delay differential equation below isolates a single positive constant delay.^[ref]

The Malthusian delay differential equation is a simple linear member of this class. The method of steps for a delay differential equation constructs its future from a history function of a delay differential equation, and the characteristic equation of a delay differential equation studies its exponential function modes. The exact polynomial steps for a delay equation below make that construction explicit without approximating the solution of a differential equation. Applications include population dynamics and feedback control; the delayed negative feedback model and feedback delay explain why the timing of a return matters as well as its sign.

In the Born approximation for scalar wave scattering, with $n=1+\mu W$ and zero-mean stationary unit-variance $W$, define $S_W(\mathbf{q})=\int C(\mathbf{s})e^{-i\mathbf{q}\cdot \mathbf{s}}{d}^{3}s$. The leading expectation of the differential cross-section for scalar wave scattering isFor a regular volume much larger than the covariance range, the overlap is approximately $|V|$, giving ${\mu }^2{k}^4|V|S_W({\mathbf{k}}_s)/(4{\pi }^2)$. Statistical isotropy, beyond statistical homogeneity, permits the scalar argument $|{\mathbf{k}}_s|=2k\sin (\theta /2)$. If the spectrum is defined for the index fluctuation rather than $W$, it already contains ${\mu }^2$.

On a compact smooth manifold without boundary, every smooth self-map sufficiently $C^1$-close to a diffeomorphism is a diffeomorphism. Reducing to a map near the identity, invertibility of its differential follows uniformly by compactness. It is then a proper local diffeomorphism, hence a finite covering map. Sufficient $C^0$ closeness gives a homotopy to the identity on each component; its induced map on the fundamental group is surjective. The covering therefore has one sheet on each component, proving the assertion. This lets a nearby Lagrangian submanifold transverse to cotangent fibers be represented by a single global differential one-form graph.

The method of conditional probabilities turns an expected value guarantee into a deterministic construction. For a maximization objective function, fix one choice at a time, always keeping the conditional expectation of the final value at least as large as before. Once every choice is fixed, that conditional expectation is the actual value. The useful ingredient is a way to evaluate the conditional expectations efficiently.

No actual random choices are needed during this construction. The probability distribution describes hypothetical completions used in the analysis; the algorithm compares their conditional expectations and commits deterministically. For maximum cut, this converts the random cut lower bound into a cut of a graph containing at least half the edges. A maximum cut can still be larger, as the conditional expectations cut example shows.

An endpoint label attached to an open string. With several coincident branes, two endpoint labels organize the massless open-string fields as matrices and give nonabelian gauge symmetry. Strings between different stacks carry representations associated with the two endpoint groups. These labels are physical internal degrees of freedom, not extra spacetime coordinates.

A multicritical point where disordered, spatially uniform ordered and spatially modulated phases meet. The quadratic stiffness in one or more directions vanishes, so a positive higher-derivative term stabilises the Landau-Ginzburg theory. For a uniaxial case the kernel at the point has $p_{\perp }^2+p_{\parallel }^4$, with the coefficient of $p_{\parallel }^2$ tuned to zero.

The Christoffel trace identity converts the covariant derivative of a vector field into a coordinate divergence weighted by the metric volume density. This is valid for the Levi-Civita connection. Applied to the raised gradient of a scalar field, it gives $\square \varphi =|g{|}^{-1/2}{\partial }_{\mu }(\sqrt{|g|}{g}^{\mu \nu }{\partial }_{\nu }\varphi )$ in Lorentzian signature, the curved d'Alembert operator.

A coalescing coupling of probability distributions of two Markov chains keeps them identical after a meeting time $T$. The coupling inequality for total variation gives $\Vert P^t(x,\cdot )-P^t(y,\cdot ){\Vert }_{\mathrm{TV}}\le \mathbb{P}(T>t)$. A uniform bound on $\mathbb{E}T$, combined with Markov inequality, bounds the mixing time of a Markov chain.