The Laplace contour-integral method seeks a solution in the formReplacing multiplication by $z$ with differentiation of $e^{zt}$ in $t$ and integrating by parts turns the differential equation for $w$ into a first-order equation for $f$. The contour $\gamma$ must make the resulting endpoint term vanish; finite endpoints are often branch points of $f$, while infinite ends must enter sectors where the exponential decays.

The past light cone of a spacetime event $O$ is the past-directed part of its light cone. In Minkowski spacetime, its spacetime events satisfy $|\Delta \mathbf{x}|=-c\Delta t$ with $\Delta t<0$. They are the possible emission events of light rays arriving at $O$. The interior also contains events that can send slower signals to $O$. The retarded potential of a moving point electric charge, given by the Liénard–Wiechert potential, uses the emission event where the source's world line intersects this boundary.

A connected Lie group is simple when its Lie algebra is nonabelian and simple. Some conventions additionally quotient out or exclude a discrete center; the infinitesimal gauge fields depend only on the Lie algebra.

A matrix Lie group is a subgroup of a general linear group that is also a Lie group for its matrix topology. Near the identity, the matrix exponential and matrix logarithm give inverse coordinate charts whenever they are restricted to sufficiently small neighborhoods in the group and its tangent space.

A Lie group action is a smooth group action of a Lie group on a smooth manifold. Differentiating the action gives an infinitesimal action of its Lie algebra by vector fields.

The direct sum $\mathfrak{g}\oplus \mathfrak{h}$ has componentwise bracketThe two summands are commuting ideals.

A Lie algebra isomorphism is a bijective linear map $\phi :\mathfrak{g}\to \mathfrak{h}$ satisfying $\phi ([x,y])=[\phi (x),\phi (y)]$. Its inverse automatically preserves the Lie bracket.

For a Matrix Lie group $G$, its tangent space $\mathfrak{g}=T_{I}G$ is closed under the matrix commutator. Indeed, for $X,Y\in \mathfrak{g}$, the group commutatorlies in $G$, and the mixed derivative at $(0,0)$ of its matrix logarithm is $XY-YX$. Thus $[X,Y]=XY-YX$ belongs to $\mathfrak{g}$.

A Lie algebra generator is an element of a chosen spanning or generating set for a Lie algebra. In a matrix representation, an infinitesimal group transformation has the form $1+i{\alpha }_a{t}^a+O({\alpha }^2)$ in terms of generator matrices $t^a$.

A compound Poisson process has the formwhere $N$ is a Poisson process and the independent identically distributed marks $Y_n$ are independent of $N$. It is a Lévy process with a finite Lévy measure.

The Lévy measure $\nu$ of a Lévy process records the intensity of its jumps: for every measurable set $A$ bounded away from zero, $\nu (A)$ is the expected number per unit time of jumps whose sizes lie in $A$. It satisfies

The expectation-maximization algorithm fits a latent-variable model by alternating a conditional expectation over the latent variables with a maximization over the statistical parameters. Its target is the observed-data likelihood function, although its updates use the simpler complete-data log-likelihood.^[ref]

Write $y$ for the observations, $z$ for the latent variables, and $p_{\theta }(y,z)$ for their joint probability density function or probability mass function. The observed-data log-likelihood iswhere the integral becomes a sum for discrete $z$. Directly maximizing a logarithm of this sum can be awkward. The E-step instead computes a conditional distribution at the current statistical parameters ${\theta }^{(t)}$; the M-step uses it to construct a new fit. The attractive idea is to retain uncertainty about the hidden labels while recovering simple weighted means and other complete-data updates. The Known-variance Gaussian mixture EM example shows the mechanism without estimating a variance or mixture weight.

Arthur Pentland Dempster, Nan M. Laird, and Donald B. Rubin gave the general framework its name in their 1977 paper, which also discusses earlier special-case algorithms.^[ref]

Let $\pi :G\to G/[G,G]$ be the abelianization map and let $A$ be a finite $K$-approximate group. Suppose a coset progression $HP(x_1,\dots ,x_r;L_1,\dots ,L_r)$ lies in $\pi (A^4)$ and has size at least $\delta |\pi (A)|$. ThenA section of $\pi$ is multiplicative up to a bounded power of $A$ inside the commutator subgroup; successively separating the subgroup and progression coordinates proves the inclusion needed for this estimate.

A metastable phase is a local but nonglobal minimum of the thermodynamic potential. A finite nucleation barrier can delay conversion to the globally stable phase until fluctuations create a sufficiently large nucleus or the metastable minimum reaches a spinodal point.

For the Landau free energy $G(\varphi )={\varphi }^6/6+g{\varphi }^4/4+\epsilon {\varphi }^2/2$, positive $g$ gives a continuous phase transition at $\epsilon =0$. For negative $g$, phase coexistence between the disordered global minimum $\varphi =0$ and the ordered global minima occurs on $\epsilon =3g^2/16$, with ${\varphi }^2=-3g/4$. Indeed, on that locus the polynomial is $G(\varphi )={\varphi }^2({\varphi }^2+3g/4{)}^2/6\ge 0$, so these three zeros are global minima. For $g<0$, the disordered and ordered spinodal points are respectively $\epsilon =0$ and $\epsilon =g^2/4$; equality of stationary free energies determines phase coexistence, while loss of a local minimum determines a spinodal point.

Taxicab geometry equips Euclidean coordinate space with the distance induced by the L1 norm, so the distance between two points is the sum of their coordinatewise absolute differences.

The completion ${\mathbb{C}}_p$ of an algebraic closure of ${\mathbb{Q}}_p$ is an algebraically closed field. Approximate a polynomial over ${\mathbb{C}}_p$ by one over ${\overline{\mathbb{Q}}}_p$, use continuity of roots over a non-Archimedean field to obtain a nearby algebraic root, and then use Krasner's lemma to show that the original root already belongs to ${\mathbb{C}}_p$.

The integral cohomology groups of the Klein bottle are $H^0(K;\mathbb{Z})=\mathbb{Z}$, $H^1(K;\mathbb{Z})=\mathbb{Z}$, $H^2(K;\mathbb{Z})=\mathbb{Z}/2$, and zero in higher degrees. The degree-one generator is pulled back from the base circle of the circle-bundle presentation, so its square is zero; hence every product of positive-degree classes vanishes.

A top-hat planar jet has one constant streamwise velocity in a layer and another constant velocity outside it. In the inviscid model, the two layer boundaries are parallel vortex sheets.

The fixed-point set of a map $f:M\to M$ is $M^f=\{x:f(x)=x\}$. For an isometry of a Riemannian manifold, each smooth component of the fixed-point set is totally geodesic.