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Local degrees multiply under composition: $e_p(S\circ R)=e_p(R)e_{R(p)}(S)$, directly by substituting the two local power-series expansions. ConsequentlyIn particular every critical point of $R$ remains critical for each positive iterate, and every critical point of an iterate reaches a critical point of $R$ in fewer than $n$ steps. This covers infinity as well as finite points. For polynomials at finite points the same conclusion follows from $(P^n{)}^{\prime }(z)={\prod }_{j=0}^{n-1}{P}^{\prime }(P^j(z))$.

A critical orbit is the forward sequence $p,R(p),R^2(p),\dots$ starting at a critical point of a rational map. The postcritical set records the forward images, usually from the first iterate onward, of all critical points; a single critical orbit and the union of all such images are distinct notions. For quadratic polynomials there is one finite critical point, zero, and its orbit controls connectedness of the filled Julia set and membership in the Mandelbrot set. A bounded orbit can be preperiodic to a repelling cycle, as $0\mapsto i\mapsto -1+i\mapsto -i\mapsto -1+i$ for $z^2+i$; boundedness then does not imply Fatou membership.

In the crystallographic convention, a Coxeter graph is admissible when it is realized by linearly independent unit vectors $v_i$ in a Euclidean space, with nonpositive mutual inner products and $4(v_i,v_j{)}^2=n_{ij}\in \{0,1,2,3\}$ for distinct vertices. The number $n_{ij}$ records a single, double or triple connection; equivalently, the displayed Gram matrix is positive-definite. In edge-label notation the nonzero connections have labels $3,4,6$. Forgetting multiplicity, the underlying graph is a forest: for any cycle of $k\ge 3$ distinct vertices, the sum of the corresponding unit vectors would have squared norm at most $k-k=0$, contradicting their linear independence. Multiple lines on one connection are labels and are not length-two cycles. This is the convention of Wehler's Lie algebra notes, section 6.3.

A five-cycle whose points are in orbit order along the real line gives four interval vertices with arrows $0\to 1\to 2\to 3$ and $3\to 0,1,2,3$. The transition-matrix traces for lengths one through four are $1,3,7,15$. Primitive closed words force at least one fixed point, one two-cycle, two three-cycles and three four-cycles. The second iterate has a two-branch horseshoe for an interval map on vertices two and three. The map itself need not have a one-step horseshoe: the unimodal affine example $F(x)=x+1$ for $x\le 3$, $F(x)=16-4x$ for $x\ge 3$ has precisely this five-cycle and has no two disjoint self-covering branches.

For cycle points ordered $x_3<x_1<x_0<x_2<x_4$, successive gap vertices have arrows $0\to 3$, $1\to 1,2$, $2\to 0$, $3\to 0,1$. Their transition-matrix traces through length four are $1,3,1,7$, forcing periods one, two and four but not three. The affine interpolation through ordered values $(4,3,1,0,2)$ has no three-cycle: its only length-three closed interval itinerary is $111$, on which the map is $5-2x$ and its third iterate has only its fixed point. Two distinct length-four return words at vertex one give a horseshoe for the fourth iterate.

With the local martingale definition requiring each stopped process itself to be a martingale, the initial value is integrable. A normalized continuous semimartingale decomposition $X=M+A$ with $A_0=0$ can therefore have a local-martingale part only if $X_0$ is integrable. A smooth transformation need not preserve this condition: take $M=0$, $A_t=G$ for an initially known standard normal random variable, and $f(x,a)=e^{a^2}$. Then $f(M_0,A_0)$ is finite almost surely but has infinite expectation. The always-valid decomposition is instead $X=X_0+N+V$ with a zero-starting continuous local martingale $N$ and a continuous finite-variation process $V$ starting at zero.

For a smooth map $F=(f,g)$ on the plane with zero Jacobian determinant and $f_y\ne 0$, set $e=g_y/f_y$. The determinant identity gives $\nabla g=e\nabla f$. Along a curve solving $y^{\prime }=-f_x/f_y$, both $f$ and $g$ have zero derivative. Local existence and uniqueness for this smooth differential equation through every nearby point produce a local foliation by curves of constant $F$.

A relative-consistency result constructs a model of a first-order theory $S$, or interprets its proofs, from the assumed consistency of a background theory $T$. It establishes $\mathrm{Con}(T)\Rightarrow \mathrm{Con}(S)$ in the stated metatheory, not an unconditional proof of consistency. For independence of an axiom $\phi$ from $S$, obtain consistent models of both $S+\phi$ and $S+\neg \phi$, relative to the background assumption.

A relative cycle is represented by a singular chain $b$ whose boundary lies in $A$. Define $\delta [\bar{b}]=[\partial b]$. If a representative changes by $\partial c+a$ with $a\in C_n(A)$, its boundary changes by $\partial a$, an $A$-boundary. Thus the map is well-defined and additive. Its image is exactly the kernel of $H_{n-1}(A)\to H_{n-1}(B)$: a cycle in $A$ dies in $B$ precisely when it is the boundary of a relative cycle lifted to $B$. This proves the corresponding exactness in the long exact sequence in relative homology directly.

Let a nonzero Jacobi field $J$ vanish at times zero and $a$. On a longer geodesic segment ending at $b>a$, extend it by zero to a continuous piecewise smooth field $Z$. Then $I(Z,Z)=0$. Choose an endpoint-vanishing field $P$ with $P(a)=-D_{t}J(a)$; integration by parts gives $I(Z,P)=-|D_{t}J(a){|}^2<0$. Thus $I(Z+\epsilon P,Z+\epsilon P)<0$ for small positive $\epsilon$. Smooth approximation preserves negativity. The second variation of geodesic energy supplies a shorter path, so a minimizing geodesic cannot extend beyond its first conjugate point.

// Target: differential-geometry.bigb

For an odd prime $p$, let $E_p:y^2=x^3-p^{2}x$. Its two-isogeny descent partner is $E_p^{\prime }:y^2=x^3+4p^{2}x$. The first two-torsion square-class homomorphism has image exactly $\{1,-1,p,-p\}$: prime-support bounds allow only these classes, and the four rational two-torsion points realize all of them. Every nonidentity point on $E_p^{\prime }$ has positive $x$, so its square-class image is contained in $\{1,2,p,2p\}$. The two-isogeny rank formula gives $2^{\mathrm{rank}{E}_p}=\#{\alpha }^{\prime }(E_p^{\prime })\le 4$, hence the rank is at most $2$.

A Lambert quadrilateral with acute angle $\pi /14$ and sides $x,y$ incident to the opposite right-angle vertex satisfies the displayed relation. Reflect four copies around that vertex to make a symmetric hyperbolic quadrilateral whose four corner angles are $\pi /14$. Identifying opposite sides yields a torus with one cone angle $2\pi /7$. The two central axis loops have lengths $2x,2y$ and meet once at right angles. Cutting along one loop gives a pair of pants with two equal geodesic boundaries and the cone point; reglue the boundaries with a Fenchel–Nielsen twist to obtain the two-parameter cone-torus family. Its seven-sheeted torsion-free covers have genus four.

For a regular two-parameter exponential family, condition on the sample mean of the nuisance canonical statistic. Exponential tilting to its observed value and the saddlepoint density approximation give a nuisance-statistic density proportional to $d_{\lambda \lambda }^{-1/2}$ times its large-deviation exponential. Dividing the joint likelihood by this density gives the profile log-likelihood plus the displayed adjustment, up to a data-only constant. The sign is positive. This construction assumes an interior saddle with positive curvature and a sufficiently regular nonlattice density; conditioning on a lattice statistic requires the corresponding mass approximation.

For the pumped Gross–Pitaevskii equation with gain term $i(\gamma -\Gamma n)\psi$, put $\psi =\sqrt{n}\exp (im\varphi /\hslash )$. The physical superfluid velocity is $\nabla \varphi$. The imaginary part gives the displayed number balance, while the real part gives $m{\varphi }_t+m|\nabla \varphi {|}^2/2+gn-{\hslash }^2{\nabla }^2\sqrt{n}/(2m\sqrt{n})=0$. These Madelung equations hold away from density zeros. The source term permits stationary density with nonzero current divergence; local gain need not balance local loss pointwise.

The tangent bundle of a complex torus is holomorphically trivial with a flat Hermitian metric. Apply the curvature formula for a holomorphic subbundle to $TV\subset TM{|}_V$: its determinant dual $K_V$ has semipositive curvature. For a closed smooth hypersurface, adjunction for a smooth submanifold identifies $K_V$ with the restriction of its divisor line bundle. Thus a positive ambient line bundle remains positive on the hypersurface after tensoring by any nonnegative power of the restricted divisor bundle.

Let $Y$ be a embedded submanifold of a complex manifold. If its tangent spaces are invariant under the almost complex structure induced by a complex atlas, choose ambient holomorphic coordinates making its tangent space at one point ${\mathbb{C}}^k\times 0$. The first $k$ coordinates restrict to a local real diffeomorphism by the inverse function theorem, so $Y$ is locally the graph of a smooth function $F$. Invariance of the tangent graph gives $dF(iv)=idF(v)$, the Cauchy-Riemann equations. Thus $F$ is holomorphic and these graph coordinates give the induced complex submanifold structure. This proves the result directly, without needing the general integrability theorem as an unproved step.

A closed analytic subset of a complex manifold is nonsingular when every point has a neighbourhood on which the subset is the zero set of a holomorphic map of full rank along that zero set. The holomorphic implicit function theorem then makes it a local coordinate subspace. This includes a closed complex submanifold, whereas an arbitrary open piece of a complex submanifold need not be globally closed in its ambient manifold.

For every real $x>0$,For a short proof, put $I(x)={\int }_x^{\infty }{e}^{-t^2}dt$. Integration by parts givesThe last integral is positive and strictly smaller than $I(x)/x^2$. Using these two inequalities gives the upper and lower bounds after multiplication by $2/\sqrt{\pi }$. Dividing both bounds by $e^{-x^2}/(\sqrt{\pi }x)$ also proves the leading asymptotic equivalence by the squeeze theorem. For small positive $x$, the bounds remain true but can be loose; they are especially useful when the Gaussian integral tail is already small.

Capacity of a disjoint union can strictly exceed the sum of the capacities. On the five-element subsets of a twenty-point set, join distinct sets when their intersection is odd. Linear polynomials over ${\mathbb{F}}_2$ bound its capacity by twenty; squarefree quadratic polynomials over ${\mathbb{F}}_3$ bound its complement's capacity by $\left(\genfrac{}{}{0ex}{}{20}{2}\right)=190$. The complement-pair lower bound is $2\sqrt{\left(\genfrac{}{}{0ex}{}{20}{5}\right)}>210$, establishing strict nonadditivity.

The displayed distance uses the oscillation length of a compactly supported Hamiltonian isotopy. On a closed manifold one may normalize each Hamiltonian to have zero mean. Additive time-dependent constants do not change the flow or its oscillation. Pullback by a symplectomorphism preserves extrema, so this distance is bi-invariant. Nondegeneracy requires an additional geometric argument; it does not follow merely from the definition as an infimum of path lengths.