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If $L_j=d{P}_j/dQ$ and the nonnegative mixture weights sum to one, the likelihood ratio of the mixture model is $L={\sum }_j{w}_j{L}_j$. Expanding ${\mathbb{E}}_Q{L}^2-1$ proves the displayed chi-squared divergence identity. For independent random variables sampled from a fixed component, one-observation cross moments multiply across observations. The component is drawn once for the whole sample; redrawing it per observation gives a different mixture model.

For an ideal gas mixture, the volume mixing ratio $x_i$ equals the mole fraction, number fraction, and ratio of partial to total pressure: $x_i=N_i/N=n_i/n=P_i/P$, with ${\sum }_i{x}_i=1$. This equivalence fails for general nonideal mixtures.

The mole fraction of species $i$ is $x_i=N_i/{\sum }_j{N}_j$, where $N_i$ is its amount in moles. It is dimensionless and equals the volume mixing ratio for an ideal gas mixture.

A line charge is a charge distribution idealized as lying on a curve, described by charge per unit length. An infinite straight line with constant density $\mu$ produces the radial electric field $E_r=\mu /(2\pi {\epsilon }_{0}r)$ by Gauss's law, and a logarithmic electrostatic potential.

Write a matrix-valued gauge field ${\mathcal{A}}_{\mu }=A_{\mu }^a{T}^a$ in the fundamental representation. Charge conjugation sends it to $-{\mathcal{A}}_{\mu }^T$ because reordering the fermions transposes both the spinor and internal matrices. The identity $[A^T,B^T]=-[A,B{]}^T$ then sends the gauge field strength to $-{\mathcal{F}}_{\mu \nu }^T$. This distinction prevents treating all color components as individually C-odd.

A field has characteristic zero when no positive integer multiple of its identity vanishes. Equivalently, its prime field is isomorphic to $\mathbb{Q}$. Every irreducible polynomial over such a field is a separable polynomial.

In a field of characteristic two, $1+1=0$. The usual splitting of a tensor product into its symmetric square and exterior square by $(I\pm \tau )/2$ is unavailable.

For a sample $X_1,\dots ,X_n$, ${\phi }_n(u)=n^{-1}{\sum }_j{e}^{iu{X}_j}$ is the characteristic function of the sample's empirical measure. For independent and identically distributed random variables, $\mathbb{E}{\phi }_n(u)=\phi (u)$ and ${\mathrm{Var}}_{\mathbb{C}}{\phi }_n(u)=(1-|\phi (u){|}^2)/n$, without any moment condition on the observations.

For a linear recurrence relation with two distinct characteristic equation of a linear recurrence roots $r_1,r_2$, the solution is $u_n=A{r}_1^n+B{r}_2^n$. If $|r_1|<1<|r_2|$, boundedness for $n\to \infty$ forces $B=0$, and $u_0$ fixes $A$. This selects the decaying discrete mode in a finite difference method for a boundary problem on an unbounded interval. Roots on the unit circle require separate treatment; a repeated root there can contribute an unbounded factor $n$.

For a finite group, the regular representation contains $d_{\rho }$ copies of each irreducible representation $\rho$. Its character of a representation is $|G|$ at the identity and zero elsewhere. Thus ${\sum }_{\rho }{d}_{\rho }{\chi }_{\rho }(x)=|G|1_{\{x=e\}}$, including the trivial representation. This formula converts independent uniform expectations into constrained configuration counts.

For proper lower semicontinuous convex functions on finite-dimensional Hilbert spaces, a nonempty saddle point set, and constant positive steps satisfying the displayed inequality, the primal-dual hybrid gradient method with extrapolation parameter one converges to a saddle point. The strict condition controls the bilinear coupling by the operator norm. It does not remove the need for existence of a saddle point. The fixed-step theorem and assumptions are recalled in Malitsky and Pock, A first-order primal-dual algorithm with linesearch, Section 1.

Informative censoring occurs when censoring carries information about the future event time beyond the variables conditioned on in the analysis. It can invalidate ordinary Kaplan–Meier estimator and survival-model residual interpretations.

The future horismos is the part of the causal future outside the chronological future. For a compact set in a globally hyperbolic spacetime, it equals the boundary of the chronological future. Its normal null geodesic generators cannot have passed a conjugate point to a spacelike surface when the initial set is that surface.

A probability distribution is causally minimal for a Directed acyclic graph when it satisfies the global Markov property for a directed acyclic graph for that graph and for no proper subgraph obtained by deleting edges on the same vertex set. It excludes redundant edges without excluding every extra conditional independence.

Continuous dependence means that convergence of the initial data in a specified topology implies convergence of the corresponding solutions in the specified solution topology. It is a stability requirement, not merely existence or uniqueness. Smooth-data instability of the Cauchy-Riemann Cauchy problem gives a counterexample even when the initial data converge uniformly with every fixed number of derivatives.

A holomorphic function on an open neighbourhood of a closed triangle has zero contour integral around its boundary. Goursat's subdivision proof uses complex differentiability alone: repeatedly choose a quarter-size triangle carrying at least one quarter of the integral, and subtract the affine expansion at the limiting point. No continuity of the derivative is assumed.

A coseparator $S$ in a category distinguishes parallel arrows by postcomposition: if $u\ne v:X\to B$, some $t:B\to S$ satisfies $tu\ne tv$. Equivalently, the contravariant functor $\mathcal{C}(-,S)$ is faithful. In a complete category that is a locally small category, the evaluation arrow $B\to {\prod }_{t\in \mathcal{C}(B,S)}S$ is a monomorphism.

An object of ${\mathcal{C}}_n$ is a set with a finite list of partial unary operations. The first is total; the domain of each later operation consists exactly of the defined fixed points of the immediately preceding operation. Morphisms preserve every defined operation. Take ${\mathcal{C}}_0$ to be the Category of sets. This tower gives examples of arbitrary finite monadic length even though its long free–forgetful composites induce the same first monad.

The category with one object and only its identity morphism. A functor from it selects an object of the target category.

The objects are the entities between which a category specifies morphisms. Each object has an identity morphism.