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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.

Repeatedly take the center of a group after quotienting by the preceding term. A group is a nilpotent group exactly when a finite term is the whole group. This yields the normalizer condition for nilpotent groups.

Choose the first term of the upper central series not contained in $H$. The preceding term lies in $H$, so an element of the new term outside $H$ commutes with $H$ modulo that preceding term and normalizes $H$.

A finite group is nilpotent exactly when it is the direct product of groups of its Sylow subgroups. The normalizer condition for nilpotent groups and Sylow conjugacy force each Sylow normalizer to be the whole group. Distinct normal Sylow subgroups commute because their commutators lie in their trivial intersection. Conversely finite p-groups are nilpotent by the nontrivial center and induction; direct products preserve nilpotence.

If a nilpotent group is generated by $a_1,\dots ,a_d$, every element of its commutator subgroup is a product $[x_1,a_1]\cdots [x_d,a_d]$. The sets in the displayed formula need not be subgroups. Induct on the nilpotency class $c$. Collection in $G/{\gamma }_c(G)$ gives the desired product with a central error. Since ${\gamma }_c(G)=[{\gamma }_{c-1}(G),G]$, that error is a product $[y_1,a_1]\cdots [y_d,a_d]$ with $y_i\in {\gamma }_{c-1}(G)$. These factors are central, and ${\gamma }_{c-1}(G)$ centralizes $G\prime$, so replacing $x_i$ by $x_i{y}_i$ absorbs the error. For a finitely generated pro-p group, applying this formula in all finite quotients and using compactness makes its algebraically generated commutator subgroup closed.

If a nilpotent group is generated by $a_1,\dots ,a_d$, every element of its commutator subgroup is a product $[x_1,a_1]\cdots [x_d,a_d]$. The sets in the displayed formula need not be subgroups. Induct on the nilpotency class $c$. Collection in $G/{\gamma }_c(G)$ gives the desired product with a central error. Since ${\gamma }_c(G)=[{\gamma }_{c-1}(G),G]$, that error is a product $[y_1,a_1]\cdots [y_d,a_d]$ with $y_i\in {\gamma }_{c-1}(G)$. These factors are central, and ${\gamma }_{c-1}(G)$ centralizes $G\prime$, so replacing $x_i$ by $x_i{y}_i$ absorbs the error. For a finitely generated pro-p group, applying this formula in all finite quotients and using compactness makes its algebraically generated commutator subgroup closed.

Every finitely generated torsion-free group which is a nilpotent group embeds as a discrete cocompact subgroup of a unique simply connected nilpotent real Lie group. Its dimension equals the group's integral cohomological dimension of a group. In dimension three the Lie algebra is either abelian or has a basis with a single nonzero bracket $[Y,T]=X$, giving the real Heisenberg group.

For $tv{t}^{-1}=Bv$, the group commutator convention $[x,y]=x^{-1}{y}^{-1}xy$ gives $[t,v]=(I-B^{-1})v$ in additive notation. Its image is $(B-I){\mathbb{Z}}^r$, since $B$ is an integral automorphism. The lower central series from its second term is $(B-I){\mathbb{Z}}^r,(B-I{)}^2{\mathbb{Z}}^r,\dots$. It terminates exactly when $B-I$ is a nilpotent matrix. For rank two this means $(B-I{)}^2=0$, equivalently $\det B=1$ and $\mathrm{tr}B=2$. Nilpotence of a finite-index subgroup alone gives virtual nilpotence of the whole group.

A pseudo-Anosov homeomorphism preserves two transverse measured foliations and stretches one while contracting the other by the same factor greater than one. Unlike the Anosov homeomorphism case, the foliations may have singularities.

For finite-index subgroups of a rank-$n$ free group, the Nielsen–Schreier formula gives the displayed quotient. Along any sequence whose finite indices tend to infinity, the quotient tends to $n-1$. Subtracting one from the rank before dividing gives the constant rank gradient $n-1$.

// Target: geometry-and-topology.bigb