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The Möbius transformations commuting with $J(z)=1/\overline{z}$ are exactlyThey preserve the open unit disc exactly when $|a|>|b|$. The identityproves the latter assertion.

For determinant-one real representatives, trace magnitude below, equal to, or above two gives elliptic, parabolic, or hyperbolic type.

A condition $\arg ((z-z_1)/(z-z_2))=\theta$ fixes the oriented angle subtended by $z_1,z_2$. Its locus is an arc of a circle through those two points, with the endpoints excluded.

A fixed point of a Möbius transformation satisfies a quadratic equation on the Riemann sphere.

The transformations that take every Euclidean circle to a Euclidean circle are preciselyThey form the stabilizer of infinity in the Möbius group and are not a normal subgroup.

A generalized circle is either a Euclidean circle or a straight line together with the point at infinity. Every Möbius transformation maps generalized circles to generalized circles. A Euclidean circle becomes a line exactly when it contains the pole, the preimage of infinity.

For four distinct points of the Riemann sphere, their cross-ratio iswith the evident limiting conventions at infinity. Every Möbius transformation preserves it.

If adjacent states admit one-step couplings that contract a path metric by a common factor, then the induced Wasserstein distance contracts by the same factor for arbitrary starting distributions.

A sequence of chains exhibits cutoff when its distance from stationarity drops from near one to near zero in a time window negligible compared with its mixing-time scale.

For a finite transitive reversible chain with nontrivial eigenvalues ${\lambda }_i$,

The field has ring of integers $\mathbb{Z}[\sqrt{-33}]$, discriminant $-132$, and Minkowski bound below eight. The nonprincipal ramified ideals above two and three generate four classes, and the split primes above seven introduce no new class. Hence

For $K=\mathbb{Q}(\sqrt[3]{3})$ with ${\mathcal{O}}_K=\mathbb{Z}[\sqrt[3]{3}]$, the field discriminant is $-243$ and the Minkowski bound is $8\sqrt{3}/\pi <5$. The prime ideals above $2$ and $3$ are principal, so every ideal class has a principal representative and

If a rule has constant risk $r$ and a sequence of priors has Bayes risks tending to $r$, then the rule is minimax. Every competing rule has worst-case risk at least every one of those Bayes risks.

An equalizer rule has the same risk at every parameter value. A Bayes equalizer rule is minimax when its constant risk bounds from below the worst-case risk of every competing rule.

If an upper-triangular $n$ by $n$ matrix has diagonal entries ${\lambda }_1,\dots ,{\lambda }_n$, its standard invariant flag satisfiesThe commuting product ${\prod }_k(A-{\lambda }_{k}I)$ therefore vanishes, proving $\mathrm{deg}{m}_A\le n$ without invoking the full Cayley-Hamilton theorem.

If $p_1,\dots ,p_r$ are pairwise coprime, thenBezout identities between the factors construct the corresponding projections.

An information projection chooses the probability distribution that satisfies a constraint while departing least from a reference law in relative entropy. The constraint might specify an expected value, several marginal probability distributions, or another convex set of permissible laws. The reference is held fixed: the variable is the first argument of $D(q\Vert p)$. Reversing the arguments gives a different minimization problem.

For a finite alphabet $\mathcal{A}$, take a positive reference probability mass function $p(a)>0$ and a nonempty closed convex set $\mathcal{C}$ in the probability simplex. ThenThe minimum exists and its minimizing probability distribution is unique. In this article, natural logarithms give nats. Using base-two logarithms rescales the objective without changing the minimizer.^[ref]

If $p$ already meets the constraint, its information projection is $p$ itself by relative entropy nonnegativity. If not, a constraint can change the relative weights of outcomes without specifying every weight individually. Relative entropy minimization under a mean constraint gives an especially useful example: it tilts the reference law rather than choosing an arbitrary distribution with the right average.

If $f$ is locally Lebesgue integrable on ${\mathbb{R}}^n$, then for Lebesgue almost every $x$,

A measure is sigma-finite when its space is a countable union of measurable sets of finite measure.

A measure $\mu$ on $X$ is finite when $\mu (X)<\infty$.