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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.

For finite-dimensional complex linear operators with $C=[A,B]$ commuting with both $A$ and $B$, differentiating $e^{-vB}A{e}^{vB}$ gives the constant derivative $C$. Hence $e^{-B}A{e}^B=A+C$, and conjugating the matrix exponential gives $e^{-B}{e}^{-A}{e}^B=e^{-A-C}$. Since $[A,C]=0$, multiplication by $e^A$ proves the identity. The commutator order determines the minus sign.

The free commutative monoid on $d$ generators consists of the nonnegative integral sums of those generators, with unique coefficients. Its semigroup algebra over a commutative ring is the polynomial ring in $d$ variables. In a smooth toric chart, the dual-cone lattice points form such a monoid, yielding affine-space coordinates.