Let $U,V$ be nonzero finite-dimensional complex vector spaces, and let $\varphi :U\otimes V\to W$ have injective restrictions when either nonzero factor is fixed. It induces a map from ${\mathbb{CP}}^{a-1}\times {\mathbb{CP}}^{b-1}$ to ${\mathbb{CP}}^{r-1}$, where $a=\dim U$, $b=\dim V$, and $r=\mathrm{rank}\varphi$. Pullback of the degree-two generator is $x+y$, since both factor restrictions are projective linear embeddings. Its $(a+b-2)$th power is $\left(\genfrac{}{}{0ex}{}{a+b-2}{a-1}\right)x^{a-1}{y}^{b-1}\ne 0$ in integral cohomology. The target must therefore have complex dimension at least $a+b-2$. Positivity of both factor dimensions is necessary; a zero factor can make the claimed inequality false.

A continuous bilinear map obeys $\Vert f(h,k)\Vert \le C\Vert h\Vert \Vert k\Vert$. Expanding $f(a+h,b+k)$ leaves the displayed linear term and remainder $f(h,k)$. For product norm $\rho =(\Vert h{\Vert }^2+\Vert k{\Vert }^2{)}^{1/2}$, the remainder has norm at most $C{\rho }^2/2$, hence is $o(\rho )$. This proves the Fréchet derivative formula.

Let $S:W\times W\to W$ be a symmetric bilinear map on a finite-dimensional real vector space. If $S(v,v)$ is parallel to $v$ for every $v$, then it has the displayed form for one covector $V$. Write $S(e_i,e_i)={\alpha }_i{e}_i$ in a basis. Applying the condition to $e_i+e_j$ and $e_i-e_j$ excludes every component outside their span and gives $S(e_i,e_j)=({\alpha }_j{e}_i+{\alpha }_i{e}_j)/2$. Set $V(e_i)={\alpha }_i/2$ and extend by bilinearity. Dimension one follows by the same definition. The trace identity $S^a{}_{ac}=(n+1)V_c$ proves uniqueness without depending on the chosen basis.

On a real semisimple Lie group, the nondegenerate invariant Killing form gives a bi-invariant pseudo-Riemannian metric. For left-invariant fields the Koszul formula gives ${\nabla }_{X}Y=[X,Y]/2$ and $R(X,Y)Z=-[[X,Y],Z]/4$. Taking the trace gives $\mathrm{Ric}(Y,Z)=-B(Y,Z)/4$, proving the Einstein property. On a compact semisimple group, $-B$ is positive definite and has Einstein constant $1/4$. The indefinite general construction is not a proof of existence of a positive-definite bi-invariant metric on a noncompact group.

Let $[e_1,e_2]=e_6$, $[e_1,e_3]=e_4$ and $[e_2,e_3]=e_5$, with $e_4,e_5,e_6$ central. The symmetric tensor $S=e_1\otimes e_5+e_5\otimes e_1+e_3\otimes e_6+e_6\otimes e_3-e_2\otimes e_4-e_4\otimes e_2$ is invariant under the Adjoint representation of a Lie algebra: the derivation action of each of $e_1,e_2,e_3$ cancels in pairs. For $t\ne 0$, $Q=tS+{\sum }_{j=4}^6{a}_j{e}_j\otimes e_j$ has determinant $-t^6$ and inverse metric $g=t^{-1}S(\lambda )-t^{-2}(a_5({\lambda }^1{)}^2+a_4({\lambda }^2{)}^2+a_6({\lambda }^3{)}^2)$ in the dual coframe. Its three hyperbolic planes give the displayed metric signature. On a connected Lie group this supplies a four-parameter family of bi-invariant pseudo-Riemannian metrics. Disconnected components must preserve the tensor separately; the automorphism reversing $e_1,e_4,e_6$ while fixing the other generators reverses $S$ and prevents this extension.

In the orientation-preserving affine group of the real line, translations $T_{1/n}$ converge to the identity element, but conjugation by dilation $D_n$ sends each one to $T_1$. A compatible bi-invariant group metric would give $d(e,T_{1/n})=d(e,T_1)>0$ for every $n$, contradicting convergence. The group is nonetheless metrizable in its ordinary two real coordinates.

For projective plane curves without a common irreducible component, project from a point outside both curves. In coordinates where that point is $[1:0:0]$, their defining homogeneous polynomials have constant nonzero leading coefficients in $X$. The resultant in $X$ is nonzero by Gauss's lemma, and is homogeneous of degree $\mathrm{deg}C\mathrm{deg}D$ in the remaining two coordinates. Every intersection projects to one of its zeros. Each projection line has only finitely many intersections with the curves, so their intersection is finite. Choose the center again outside all joining lines of pairs of intersection points; the projection is now injective on that finite set, proving the bound.

Two adjacent Bézier curves of degrees $n,m$ and interval lengths $h,k$ have a $C^1$ join precisely when their endpoint positions and their derivatives in the common parameter agree. Parallel endpoint polygon edges alone give tangent-direction agreement; their scaled vectors must be equal for parametric first-derivative continuity.

For a space with finite-dimensional homology over a specified field and only finitely many nonzero Betti numbers, this polynomial records their graded dimensions. Its value at one is the total Betti number, and its value at minus one is the Euler characteristic. The Künneth theorem gives $P_{M\times N}(t)=P_M(t)P_N(t)$ over a field. For integer homology ranks, use the rational Betti numbers.

For a partition padded to $l$ rows, its beta numbers are the distinct nonnegative integers ${\lambda }_i+l-i$. They are its occupied abacus of a partition positions. With exactly the positive row count, they equal the first-column hook lengths; extra zero rows change the representation of the beta set while preserving the partition. Individual beta numbers describe bead positions, whereas the beta set is the collection of all these numbers.

Parallel beta reduction permits simultaneous reduction in the body of a lambda abstraction and both sides of an application, together with $(\lambda x.M)N\Rightarrow M^{\prime }[x:=N^{\prime }]$ when $M\Rightarrow M^{\prime }$ and $N\Rightarrow N^{\prime }$. Variables reduce to themselves. Its capture-avoiding substitution lemma is $M\Rightarrow M^{\prime },N\Rightarrow N^{\prime }\Rightarrow M[x:=N]\Rightarrow M^{\prime }[x:=N^{\prime }]$. Its transitive closure equals ordinary finite beta reduction, while its one-step diamond property yields the Church-Rosser theorem.

A reduction relation $\to$ has the diamond property if $a\to b$ and $a\to c$ imply a common one-step successor $b\to d\leftarrow c$. For relations allowing an identity step, the same definition includes zero-length joins. A finite grid of these diamonds proves confluence of ${\to }^{\ast }$. Ordinary one-step beta reduction fails this property because contracting before duplication can save two separate contractions.

Commutation allows exchanging the factors at their existing arguments; it does not allow exchanging their arguments while retaining the same pole sign. With the displayed convention, differentiation produces $\partial \beta (z)\gamma (w)\sim (z-w{)}^{-2}$ and $\partial \gamma (z)\beta (w)\sim -(z-w{)}^{-2}$. These signs are essential for the conformal weights and central charge of a commuting beta-gamma system.

Using the radial-ordering sign in commuting beta-gamma contractions, single contractions give $T(z)\beta (w)\sim \lambda \beta (w)/(z-w{)}^2+\partial \beta (w)/(z-w)$ and the corresponding weight $1-\lambda$ for $\gamma$. Put $A=:\partial \beta \gamma :$ and $B=:\beta \partial \gamma :$. Their double-contraction coefficients in $AA,BB,AB,BA$ are respectively $1,1,2,2$. The fourth-order pole of $TT$ is therefore $(1-\lambda {)}^2+{\lambda }^2-4\lambda (1-\lambda )=6{\lambda }^2-6\lambda +1$, which equals $c/2$. There is no anticommuting Wick sign.

Mix a binomial distribution with a beta distribution for its success probability: $\Theta \sim \mathrm{Beta}(\alpha ,\beta )$ and $Y\mid \Theta \sim \mathrm{Binomial}(n,\Theta )$. Integrating their densities gives the displayed mass for $y=0,\dots ,n$. Its expectation is $n\alpha /(\alpha +\beta )$, and its variance is $n\alpha \beta (\alpha +\beta +n)/[(\alpha +\beta {)}^2(\alpha +\beta +1)]$. It is the posterior predictive distribution for repeated Bernoulli trials after a beta-prior update.

Couple $dX=dB+a{X}^{-1}dt$ and $dY=dB+a{Y}^{-1}dt$ from $0<x<y$, with $1/4<a<1/2$. Suppose their finite lifetimes end at zero and ${\int }_0^{{\sigma }_y}{Y}_t^{-2}dt=\infty$. Their difference is positive and has derivative $-a(Y-X)/(XY)$, so ${\sigma }_x\le {\sigma }_y$. For $\Theta =(Y-X)/Y$, the Itô formula givesThe normalized scale function of a one-dimensional diffusion $I_{\Theta }(4a-1,1-2a)$ is a bounded local martingale. Its finite quadratic variation and the divergent $Y^{-2}$ clock force $\Theta \to 0$ on simultaneous extinction; on strict extinction $\Theta \to 1$. Its terminal value is therefore the strict-extinction indicator, proving the formula by optional sampling theorem. For SLE boundary images, $a=2/\kappa$; this compares strict and simultaneous boundary swallowing times for $4<\kappa <8$.

The radial part of three-dimensional Brownian motion started away from the origin is a Bessel process of dimension three. The radial Laplacian is $\Delta |x|=2/|x|$, so the Itô formula gives the displayed equation. The radial martingale has unit quadratic variation rate and is Brownian motion by the Lévy characterization of Brownian motion. The reciprocal $1/R$ is a local martingale. Stopping it on $(\epsilon ,M)$ gives ${\mathbb{P}}_x(T_{\epsilon }<T_M)=\epsilon (M-x)/(x(M-\epsilon ))$. Letting $\epsilon \downarrow 0$ shows that zero is not hit.

For $0<\epsilon <x$, the drift $g_{\epsilon }(y)=a/(y\vee \epsilon )$ is globally Lipschitz and bounded. Solve its equation strongly for the prescribed Brownian motion, and stop at the first hit of $\epsilon$. Local uniqueness makes these stopped solutions agree up to the larger threshold; their stopping times increase as the threshold decreases. With $\nu =2a-1>0$, Itô's formula for $X^{-\nu }$ gives $\mathbb{P}(T_{\epsilon }\le t)\le (\epsilon /x{)}^{\nu }$ on every finite horizon. Hence the patched lifetime is infinite. Localization away from zero proves pathwise uniqueness of the positive global solution. Uniqueness before a stopping time refers to the stopped solution, not to arbitrary unconstrained continuations afterward.

For $0<a<1/2$, couple $dX=dB+a{X}^{-1}dt$ and $dY=-dB+a{Y}^{-1}dt$ from positive $x,y$, killing each at zero. Their sum has no Brownian term and increases, so simultaneous extinction is impossible. The ratio $U=Y/(X+Y)$ has time-changed generator $\frac{1}{2}{\partial }_u^2+a(1-2u)[u(1-u){]}^{-1}{\partial }_u$. Its scale density is $u^{-2a}(1-u{)}^{-2a}$, integrable at both endpoints. Optional stopping of its normalized scale function at the first extinction gives the displayed regularized incomplete beta function. At $a=1/3$ this is Cardy's formula.

A squared Bessel process of dimension $\delta \ge 0$ is a nonnegative continuous solution of the displayed stochastic differential equation. It extends the squared radial part of integer-dimensional Brownian motion to arbitrary nonnegative dimensions. Before hitting zero its square root solves $dR=dB+(\delta -1)(2R{)}^{-1}dt$. At dimension zero, a process started at zero remains there. The coefficients are not Lipschitz at zero, so positivity or uniqueness must not be justified by a global Lipschitz theorem.