For a strictly increasing ordinal class function continuous at limits, start with ${\alpha }_0=0$ and iterate ${\alpha }_{n+1}=f({\alpha }_n)$. If the sequence stabilizes, its stable value is a fixed point. Otherwise its supremum $\alpha$ is a limit and continuity gives $f(\alpha )={\sup }_{n}f({\alpha }_n)={\sup }_n{\alpha }_{n+1}=\alpha$. Starting the iteration above any prescribed ordinal gives arbitrarily large fixed points.

If C is proper in both factors of $G=G_1{\ast }_C{G}_2$, every central element lies in C. In the Bass-Serre tree, a central element fixing a vertex fixes its entire orbit and then the whole tree, hence an edge. A central element acting as a translation would have an invariant axis; minimality forces the tree to be that axis, but a vertex stabilizer then contains a reflection, contradicting centrality of the translation. Thus the center fixes an edge and is contained in its stabilizer C.

// Target: algebraic-topology.bigb

For a transverse intersection, the quotient map $TM{|}_L\to {\nu }_1{|}_L\oplus {\nu }_2{|}_L$ is onto and has kernel $TL$. This gives the displayed isomorphism of actual normal bundles. Complex structures on the two normal bundles therefore induce a complex structure on the intersection's normal bundle.

// Target: geometry-and-topology.bigb

Near a projective subspace, transverse directions are linear maps from the tautological line to the complementary coordinate vector space. This describes the actual complex normal bundle. The conjugate line and the dual line are complex-isomorphic using a Hermitian metric; neither is the original tautological line as a complex bundle in general.

// Target: geometry-and-topology.bigb

Embed a smooth plane conic in ${\mathbf{P}}^3$ through a plane. The plane equation and conic equation form a regular sequence of degrees one and two, so its normal sheaf is ${\mathcal{O}}_C(1)\oplus {\mathcal{O}}_C(2)$. Under $C\cong {\mathbf{P}}^1$, hyperplane degree is two, hence this normal bundle is ${\mathcal{O}}_{{\mathbf{P}}^1}(2)\oplus {\mathcal{O}}_{{\mathbf{P}}^1}(4)$. The normal exact sequence splits as well because $H^1({\mathbf{P}}^1,\mathcal{O}(2))=0$.

For a nonzero fractional ideal of a number field, the displayed generated ideal equals the relative norm of a fractional ideal. Locally, multiplication by $x$ on the integral lattice has determinant $N_{L/K}(x)$. A Smith normal form calculation and the residue filtrations give $v_{\mathfrak{p}}(N(x))={\sum }_{\mathfrak{P}\mid \mathfrak{p}}{f}_{\mathfrak{P}/\mathfrak{p}}{v}_{\mathfrak{P}}(x)$. The Chinese remainder theorem for ideals chooses $x\in I$ simultaneously attaining the minimum valuation of $I$ at each prime above a fixed $\mathfrak{p}$. Thus the generated ideal has valuation $\sum f_{\mathfrak{P}/\mathfrak{p}}{v}_{\mathfrak{P}}(I)$. Adding these valuations proves multiplicativity, including fractional ideals after clearing denominators.

For a rational prime $p\ne 229$, representation by this norm form is equivalent to complete splitting in the Hilbert class field of Q of square root 229. Its Frobenius is a permutation of the three roots of $X^3-4X-1$. The quadratic residue condition $(p/229)=1$ makes this permutation even; a root modulo $p$ gives a fixed point, so the even permutation is the identity. Complete splitting makes the prime above $p$ principal. Its generator has norm $\pm p$, and the unit of norm minus one changes the sign to positive, giving the representation.

Kernel regression estimates a regression function using localized kernel weights rather than a fixed finite-dimensional functional form. The Nadaraya–Watson estimator is the local-constant version; local polynomial regression improves its local approximation by fitting a polynomial.

This nonlinear ordinary differential equation governs the amplitude of a complex oscillator solution. Independent real solutions of $y^{\prime \prime }+p(x)y=0$ can be combined into a nonvanishing complex solution; its modulus satisfies the equation with $c$ equal to the squared Wronskian. A real positive amplitude and a conserved phase flux give an exact representation, not merely a WKB approximation.

For a positive density $u$, logarithmic diffusion is the nonlinear diffusion equation $u_t=\kappa \Delta \log u$, equivalently $u_t=\kappa \mathrm{div}(u^{-1}\nabla u)$. Its diffusion coefficient is inversely proportional to the density. In one dimension, a mass-$M$ self-similar solution isIts integral is $M$, and substitution verifies the equation. This is $M$ times a Cauchy distribution density of scale $2\pi \kappa t/M$. It tends weakly to $M$ times the Dirac delta distribution as $t\downarrow 0$; the slow algebraic tails illustrate the enhanced diffusion where density is small.

Reachability within $2^k$ steps can be tested by enumerating a midpoint and recursively testing two half-paths. Configurations have $O(s)$ bits and recursion depth $O(s)$, yielding deterministic space $O(s^2)$. The space-bound discovery by exit reachability permits this simulation without assuming that the supplied function $s$ is constructible.

A bilinear form $b:U\times V\to K$ with zero left and right radicals induces injective linear maps $U\to V^{\ast }$ and $V\to U^{\ast }$. Finite dimensions force $\dim U=\dim V$, so both maps are isomorphisms. For any other bilinear form $c$, composing its maps to the dual spaces with these inverse isomorphisms gives unique endomorphisms $S,T$ with $c(x,y)=b(Sx,y)=b(x,Ty)$. In bases with invertible pairing matrix $B$ and coefficient matrix $C$, $S=B^{-T}{C}^T$ and $T=B^{-1}C$.

For an automorphism $\sigma$ of a left Noetherian ring $A$, the skew polynomial ring of an automorphism $A[t;\sigma ]$ is left Noetherian. For a left ideal $L$, the normalized leading-coefficient ideals $I_n=\{{\sigma }^{-n}(a_n):{\sum }_{j=0}^n{a}_j{t}^j\in L\}$ form an ascending chain of left ideals of $A$. Choose finite generating lifts through the stabilization degree. Every element of $L$ has its leading coefficient cancelled by a suitable left multiple of one of these lifts, and induction on degree proves finite generation. Passing to the skew Laurent polynomial ring preserves the property by clearing negative powers. The right-sided statement follows by applying the argument to the opposite ring.

For a right ideal $I\subseteq R[t]$, let $L_d$ be the set of degree-$d$ coefficients of its elements of degree at most $d$. These are right ideals of $R$, and multiplication by the central variable makes $L_d\subseteq L_{d+1}$. They stabilize, and every $L_d$ before stabilization is finitely generated. Choose polynomial lifts of those generators. Subtracting a suitable right linear combination of the lifts, shifted by powers of $t$, cancels the leading coefficient of any element of $I$. Induction on degree proves that the finitely many chosen lifts generate $I$.

// Destination: noncommutative-algebra.bigb

For intensity $x^{-2}(1-x{)}^{-1}$ on $(0,1)$, the cumulative-intensity maps based at $1/2$ areThe left and right images are independent unit-rate Poisson processes. Labeling the original points by $X_{-k}<1/2<X_n$, with $k\ge 1,n\ge 0$, gives $L(X_{-k})/k\to 1$ and $F(X_n)/n\to 1$. Since $xL(x)\to 1$ at zero and $F(x)=1-\log (1-x)+o(1)$ at one, $k{X}_{-k}\to 1$ and $\log (1-X_n)/n\to -1$.

If an intensity function of a point process has $\lambda (x)\sim c(b-x{)}^{-\alpha }$ at a finite endpoint, with $c>0$ and $\alpha >1$, the cumulative intensity satisfies $F(x)\sim c(b-x{)}^{1-\alpha }/(\alpha -1)$. The Poisson mapping theorem turns $F(X_n)$ into unit-rate arrivals. The strong law for Poisson arrival times gives $F(X_n)/n\to 1$ almost surely, hence the displayed accumulation rate. Reversing the coordinate gives the same argument at a left endpoint. Different singularity exponents produce different powers of the index.

For an ideal-gas heated pure plume with entrainment proportional to $w\sqrt{\rho /{\rho }_0}$, the excess heat flux implies conserved kinematic buoyancy flux $B$. Put $q=Q/{\rho }_0$, $m=M/{\rho }_0$. Their balances are $q^{\prime }=2\alpha \sqrt{m}$ and $m^{\prime }=Bq/m$, identical to the pi-suppressed Boussinesq pure-plume flux equations. With $d=6\alpha /5$ and $s=[25B/(48{\alpha }^2){]}^{1/3}$, $q=d^{2}s{z}^{5/3}$, $m=d^2{s}^2{z}^{4/3}$ and $z_b^{5/3}=B/(g{d}^{2}s)$. The radius is $b=dz\sqrt{1+(z_b/z{)}^{5/3}}$ and reduced gravity is $g/[1+(z/z_b{)}^{5/3}]$. At $z=z_b$ mass density is half ambient; far above this height the Boussinesq approximation becomes accurate.

An absolute value on a field is non-Archimedean if and only if $|n\cdot 1|\le 1$ for every positive integer $n$. Necessity follows from the ultrametric inequality. Conversely the binomial theorem gives $|x+y{|}^m\le (m+1)\max (|x|,|y|{)}^m$, since every binomial coefficient has absolute value at most one. Take $m$th roots and let $m\to \infty$. In particular an extension of an absolute value that is non-Archimedean stays non-Archimedean.

For local Kraus operators $A_i$ with ${\sum }_i{A}_i^{†}{A}_i=I$, let ${\rho }_{AB}^{\prime }={\sum }_i(A_i\otimes I){\rho }_{AB}(A_i^{†}\otimes I)$. For every remote operator $B$, cyclicity of the trace gives $\mathrm{Tr}[(I\otimes B){\rho }_{AB}^{\prime }]={\sum }_i\mathrm{Tr}[(A_i^{†}{A}_i\otimes B){\rho }_{AB}]=\mathrm{Tr}[(I\otimes B){\rho }_{AB}]$. Hence the remote reduced density matrix is unchanged. In particular every remote POVM has the same outcome probabilities, proving the no-communication theorem from the measurement rule rather than assuming it as a physical postulate.

A square matrix is nilpotent when one of its positive powers is zero. It represents a nilpotent linear map in a chosen basis; changing basis preserves nilpotence. Over an algebraically closed field, its Jordan normal form has only zero-eigenvalue blocks.