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For an element of ${\mathrm{PSL}}_2(\mathbb{R})$ represented by $A\in {\mathrm{SL}}_2(\mathbb{R})$, the absolute trace classifies nonidentity Möbius transformations: $|\mathrm{tr}A|<2$ is elliptic, $|\mathrm{tr}A|=2$ is parabolic, and $|\mathrm{tr}A|>2$ is hyperbolic. The scalar representatives $A=\pm I$ give the identity and must be excluded from the parabolic case.

The distinction follows from the quadratic $t^2-(\mathrm{tr}A)t+1$ for the eigenvalues. Elliptic elements have nonreal conjugate eigenvalues, hyperbolic ones have distinct real eigenlines, and nonidentity parabolic ones have a repeated eigenvalue with only one eigenline. These correspond to different fixed-point patterns on the boundary of the hyperbolic plane.

For distinct complex points $z_1,z_2$, a conditionfixes the oriented angle subtended by those two points, with $z=z_1,z_2$ excluded. Under the Möbius transformation $w=(z-z_1)/(z-z_2)$, it is the positive ray $w=t{e}^{i\theta }$, $t>0$. Its inverse is $z=(w{z}_2-z_1)/(w-1)$.

If $\theta$ is not zero modulo $\pi$, the ray pulls back to an open circular arc through $z_1,z_2$. At the exceptional angles the generalized circle is a straight line: $\theta =\pi$ gives the open segment between $z_1,z_2$, whereas $\theta =0$ gives the two exterior rays on that line. Thus the unrestricted statement must allow line loci, not just arcs of ordinary circles.

A fixed point of a Möbius transformation $f$ is a point $z$ of the Riemann sphere satisfying $f(z)=z$. For $f(z)=(az+b)/(cz+d)$, a finite fixed point satisfiesThe homogeneous equation $c{u}^2+(d-a)uv-b{v}^2=0$ in homogeneous coordinates $[u:v]$ also handles infinity. In particular infinity is fixed exactly when $c=0$.

If $f$ is the identity map, all three coefficients vanish and every point is fixed. Otherwise the homogeneous quadratic polynomial has two polynomial roots counted with multiplicity, so there are either two distinct fixed points or one repeated fixed point. The translation $z\mapsto z+1$ has only infinity, counted twice; $z\mapsto 2z$ fixes zero and infinity; $z\mapsto 1/z$ fixes $1,-1$. These alternatives underlie the Möbius conjugacy normal forms.

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.

The Path coupling theorem reduces a global contraction calculation to pairs joined by one edge in a connected state graph. Let $d$ be the shortest-path metric from positive edge lengths, and let $P$ be a transition matrix on this finite state space. Suppose every adjacent pair $x,y$ admits a coupling $(X^{\prime },Y^{\prime })$ of $P(x,\cdot )$ and $P(y,\cdot )$ such thatThen the Wasserstein distance associated with $d$ satisfiesfor all initial probability distributions $\mu ,\nu$. Along a shortest path between two states, the triangle inequality for Wasserstein distance sums the adjacent-pair bounds to give the same bound for that pair. Averaging over a coupling of the initial states gives the distribution-level inequality. Iteration yields contraction by ${\alpha }^t$.^[ref]

If $\pi$ is stationary, $d_{\min }={\min }_{x\ne y}d(x,y)>0$, and $D={\max }_{x,y}d(x,y)$, the coupling bound givesFor example, on $m$ bits choose one coordinate uniformly and replace it by a fair bit. Couple two configurations by choosing the same coordinate and replacement. A one-bit disagreement disappears with Probability $1/m$, so the Hamming distance contracts by $1-1/m$. Here $D=m$, $d_{\min }=1$, giving the explicit bound $m(1-1/m{)}^t$. The strength is that only one-bit differences had to be analyzed.

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 $\mu (X)<\infty$.

For pairwise disjoint measurable sets $E_n$, countable additivity means

The infinite series is the limit of its nonnegative partial sums. This controls an entire countable union, not merely each finite stage. If $q_1,q_2,\dots$ enumerate the rational numbers in $[0,1]$, their finite collections have Lebesgue measure zero, and countable additivity gives their whole union Lebesgue measure zero. Their being a dense set does not change this calculation.

The axiom makes no assertion about an arbitrary uncountable set of disjoint pieces. In particular, $[0,1]$ is the union of its singletons, each of Lebesgue measure zero, yet its Lebesgue measure is one. Extending additivity to all such unions would contradict interval length.