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Every partial computable function is represented on Church numerals by a closed untyped lambda calculus term. First represent all primitive recursive functions using zero, successor, projections, composition and lambda definition of primitive recursion by pair iteration. Apply the Kleene normal form theorem and search successive history codes with a fixed-point combinator. Test the represented total predicate using a Church numeral zero test; return the decoded output only on the true branch. If no history passes, normal-order head reduction continues the search indefinitely and the term has no head normal form, so cannot be in beta equivalence with a Church numeral. This avoids the incorrect assumption that a lazy outer function must evaluate a divergent argument in an unguarded composition.

For standard probability systems, a factor of a measure-preserving system with at most $q$ points in every fibre preserves Kolmogorov-Sinai entropy. Conditional on the complete factor point, every finite orbit name has at most $q$ possibilities, so its conditional entropy is at most $\log q$. Dividing by the orbit length gives zero relative entropy.

Let $p,q\ge 2$ be relatively prime integers. If $\nu$ is invariant and ergodic for the integer multiplication map on the circle $T_p$ and $h_{\nu }(T_p)>0$, then for $\nu$-almost every $x$ the sequence $(T_q^{n}x)$ is an equidistributed sequence for Lebesgue measure. For a non-ergodic $T_p$ invariant measure, the same conclusion holds if almost every measure in its ergodic decomposition has positive entropy. Positive entropy of the whole measure alone does not suffice.

For an invertible measure-preserving system, a generating measurable partition $\xi$ has $\sigma ({\bigvee }_{j\in \mathbb{Z}}{T}^{-j}\xi )=\mathcal{B}$ modulo null sets. Thus knowing the entire two-sided itinerary determines all measurable observations. A finite or countable finite-entropy generating partition computes the system entropy through the Kolmogorov-Sinai generator theorem.

A probability measure-preserving system has completely positive entropy if $h_{\mu }(T,\xi )>0$ for every finite measurable partition with $H_{\mu }(\xi )>0$. Equivalently, its Pinsker sigma-algebra is trivial. Since finite partitions measurable in a partition tail have zero entropy rate, this property forces every finite-partition tail sigma-algebra of a measurable partition to be trivial.

For a conformal bijection $f:D\to \tilde{D}$ between proper planar domains, put $d=\mathrm{dist}(z,\partial D)$ and $\tilde{d}=\mathrm{dist}(f(z),\partial \tilde{D})$. Apply the Koebe quarter theorem to $(f(z+dw)-f(z))/(d{f}^{\prime }(z))$ on the unit disc to obtain $d|f^{\prime }(z)|/4\le \tilde{d}$. Apply it again to $f^{-1}$ to get the opposite bound. Thus $\boxed{\tilde{d}/(4d)\le |f^{\prime }(z)|\le 4\tilde{d}/d}$. The domains need not be simply connected domains. For the whole plane, the boundary distances are infinite and these ratios are undefined.

If a meridian of a knot component in a surgery link links only one other component, filling it with coefficient $b\ne 0$ removes it and replaces the other coefficient $a$ by $a-1/b$. This gives the negative continued-fraction chain description of rational Dehn filling.

For $t>0$, the number $B_t$ of blocks is finite almost surely. On a finite restriction, the expected time to reach $m$ blocks is at most ${\sum }_{j>m}2/[j(j-1)]=2/m$. Markov inequality, followed by increasing the restriction, gives $\mathbb{P}(B_t>m)\le 2/(mt)$. Let $m\to \infty$, then use rational times and monotonicity to obtain the assertion simultaneously for every $t>0$.

The Kähler cone of a compact Kähler manifold is the open cone of real $(1,1)$ cohomology classes represented by positive closed $(1,1)$ forms. If every real degree-two cohomology class has type $(1,1)$, density of rational classes supplies a rational Kähler class. An integral multiple is the First Chern class of a positive holomorphic line bundle, and the Kodaira embedding theorem implies projectivity.

Here a Calabi-Yau threefold is a compact complex three-dimensional Kähler manifold with trivial canonical bundle and $H^1(M,\mathbb{C})=0$. The Hodge decomposition theorem for compact Kähler manifolds gives $H^1(M,{\mathcal{O}}_M)=0$, and Serre duality for compact complex manifolds then gives $H^2(M,{\mathcal{O}}_M)=0$. This supplies a positive integral Kähler class and hence a projective embedding.

A square nonnegative matrix $A$ is primitive if some positive integer power $A^k$ has strictly positive entries. It is therefore an irreducible nonnegative matrix. The Perron–Frobenius theorem gives a leading eigenvalue whose modulus is strictly larger than that of every other eigenvalue. Every strictly positive matrix is primitive; the two-cycle permutation matrix is irreducible but not primitive.

For a continuously differentiable nonzero vector field $F$, the curvature of a space curve following its direction isTake the unit tangent vector $t=\pm F/|F|$ and differentiate with respect to arc length. The derivative of $|F{|}^{-1}$ is parallel to $F$ and disappears from $t\times t^{\prime }$. Since $t\cdot t^{\prime }=0$, we have $|t\times t^{\prime }|=|t^{\prime }|=\kappa$, proving the formula. It remains valid with zero curvature, even where the usual Frenet frame is undefined.

Write the Integer Heisenberg group with central generator $c=[a,b]$, so $ab=cba$. Moving $a$ successively past powers of $b$ givesFor positive exponents this follows by two inductions; inverses give the remaining signs. The representation $a=I+E_{12}$, $b=I+E_{23}$, $c=I+E_{13}$ checks the identity and shows $c$ has infinite order.

A functor $F:I\to J$ is initial if restriction preserves categorical limits for diagrams in a category with codomain ${\mathbf{Set}}^{\mathrm{op}}$. To test necessity, view $J(-,j)$ as such a diagram. Its limit is a singleton, since the colimit of the corresponding representable presheaf is the connected-component set of the slice $J\downarrow j$, which has a terminal object. The restricted colimit is the connected-component set of $(F\downarrow j)$. Thus that comma category is nonempty and connected. Sufficiency follows from cone restriction along an initial functor.

For ${\mathcal{F}}_L=\sigma ({\bigvee }_{j\ge L}{T}^{-j}\xi )$ and a finite measurable partition $\xi$, $H_{\mu }({\xi }_0^{L-1}\mid {\mathcal{F}}_L)=L{h}_{\mu }(T,\xi )$. The backward conditional entropy chain rule expresses the left side as ${\sum }_{j=0}^{L-1}{H}_{\mu }(T^{-j}\xi \mid {\mathcal{F}}_{j+1})$. Each term equals $h_{\mu }(T,\xi )$ by measure preservation and the infinite-future formula for partition entropy rate. No invertibility is needed.

An event is decreasing when closing additional edges cannot destroy it. Its complement is an increasing event.

For a connected Riemannian manifold, the full holonomy group consists of parallel transport maps around all piecewise smooth loops based at $p$. Metric compatibility makes them orthogonal. Basepoint changes conjugate this group by parallel transport. This is full holonomy, not just the subgroup using contractible loops.

For tracers sampled uniformly on a Galactocentric shell of radius $r$, with the Sun at radius $R_{\odot }$, the correction in the heliocentric projection of a spherical velocity dispersion isThe limiting value at $r=R_{\odot }$ is $1/2$; it tends to $2/3$ as $r/R_{\odot }\to 0$ and to $2R_{\odot }^2/(3r^2)$ at large radius. Uniform heliocentric solid angle at fixed $r>R_{\odot }$ instead gives $A=2R_{\odot }^2/(3r^2)$ exactly. Specify the selection measure when applying the correction.

Averaging conjugation by all Heisenberg-Weyl operators completely depolarizes a system:Summing over phases $Z^m$ kills off-diagonal matrix elements, and averaging shifts $X^k$ makes all diagonal elements equal. Acting on subsystem $B$ of a bipartite operator gives $({\mathrm{Tr}}_{B}A)\otimes I_B/d$. This identity turns joint convexity of quantum relative entropy into monotonicity under partial trace.

Thicken the planar graph underlying a knot diagram. Its boundary supports a Heegaard diagram with face disks on one side and crossing-tunnel disks on the other. The face generators and crossing curves give a Dehn presentation of a knot group.