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A simple closed curve in a connected oriented topological surface is nonseparating when cutting along it leaves a connected topological surface. In that case an arc joining the two new boundary components closes to a simple closed curve meeting the original once. Their algebraic intersection number of curves on an oriented surface is $\pm 1$. Consequently the original curve represents a nonzero primitive homology class.

On a closed oriented topological surface, a separating simple closed curve bounds a compact subsurface, whose fundamental class exhibits it as a boundary in homology. Thus on a closed oriented topological surface, separation is equivalent to vanishing of its integral first homology class.

The closed-surface condition cannot be omitted. In the open annulus $S^1\times \mathbb{R}$, the core $S^1\times \{0\}$ separates the two ends, yet generates $H_1(S^1\times \mathbb{R};\mathbb{Z})\cong \mathbb{Z}$. Either side is noncompact and supplies no finite singular chain bounding that core.

For an unknown nonzero period, a collision $f(x)=f(y)$ reveals the period $x\oplus y$. Birthday problem sampling finds one using $O(2^{n/2})$ queries with bounded error. A matching lower bound follows by choosing the period and output labels randomly: without a collision, $q$ queries rule out at most $\left(\genfrac{}{}{0ex}{}{q}{2}\right)$ candidate periods. For $q=o(2^{n/2})$, almost all periods remain consistent with the transcript. The exponential separation is in query complexity, compared with Simon's algorithm's linear quantum query count.

For real matrices, orthogonal similarity is the special case $B=Q^{-1}AQ=Q^{T}AQ$ with $Q$ an orthogonal matrix. Then a symmetric matrix remains symmetric, because $B^T=Q^T{A}^{T}Q=B$. The expression $QA{Q}^T$ describes the reverse conversion and has the same property. An orthonormal basis retains the ordinary dot product formula for its coordinate vectors.

An arbitrary change of basis need not preserve entrywise symmetry. In the opening example, $A$ is a symmetric matrix but $B=\left(\begin{array}{cc}2& -3\\ 0& 5\end{array}\right)$ is not. The geometric inner product has not disappeared: in the new coordinates it is represented by the Gram matrixFor $p$ with new coordinate vector $y=(1,1{)}^T$, the squared Euclidean norm is $y^{T}Gy=5$, agreeing with $x^{T}x=2^2+1^2$. The naive coordinate sum $y^{T}y=2$ would use the wrong inner product.

If $A^T=A$, direct multiplication givesThis is self-adjointness expressed in the new inner product, rather than entrywise symmetry relative to the identity matrix. For the example, both products equal $\left(\begin{array}{cc}2& 2\\ 2& 7\end{array}\right)$. Thus arbitrary matrix similarity preserves the linear operator, while its geometric interpretation requires transforming the Gram matrix as well. Matrix congruence supplies the corresponding rule for a bilinear form and should be distinguished from matrix similarity.

When $c=e$ in the signed cyclic three-mode amplitude equations, $\dot{X}=2X(\mu -aX)$ and $\dot{V}=3V(\mu -aX)$ for $V=xyz$. Thus $V/X^{3/2}$ is conserved wherever $X>0$, precluding asymptotically stable isolated equilibria at nonzero radius. This does not exclude attraction to the origin: for $\mu <0$ and $a>0$, $X$ tends to zero. When $\mu /a>0$, on the invariant sphere $X=\mu /a$, each octant contains closed contours of $V$ around the equal-magnitude equilibrium, bounded by heteroclinic arcs between coordinate axes if $c\ne 0$. For $c=0$ the sphere consists entirely of equilibria.

The unit tangent of a simple regular closed plane curve traversed anticlockwise turns once. Consequently its integrated signed curvature is $2\pi$; clockwise orientation gives $-2\pi$. Simplicity and one traversal are essential to this value.

For two real primitive nonprincipal Dirichlet characters with nonprincipal product, the real zeros in this interval are counted with multiplicity. The logarithmic-derivative coefficients of $\zeta L({\chi }_1)L({\chi }_2)L({\chi }_1{\chi }_2)$ are $\Lambda (n)(1+{\chi }_1(n))(1+{\chi }_2(n))\ge 0$. The global partial-fraction expansion of a Dirichlet L-function logarithmic derivative therefore gives $0\le 1/(\sigma -1)-{\sum }_{j=1}^2(\sigma -{\beta }_j{)}^{-1}+C\log (q_1{q}_2)$ if two nearby real zeros exist. Put $\sigma =1+a/\log (q_1{q}_2)$, choose $a$ small, and test $1-{\beta }_j<a/(4\log (q_1{q}_2))$. The right side is negative, a contradiction.

// Target: analytic-number-theory.bigb

For $(a,q)=1$ and $q\le {\log }^{A}N$, apply the Siegel–Walfisz theorem to every reduced residue class with error exponent $A+B+2$. The main terms sum to $N{c}_q(a)/\phi (q)$, where the Ramanujan sum satisfies $c_q(a)=\mu (q)$ for coprime $a,q$. Nonunits with nonzero Von Mangoldt function are powers of primes dividing $q$ and contribute at most $\omega (q)\log N$, which is absorbed by the error. For arbitrary $a$, the correct main term is $N{c}_q(a)/\phi (q)$; the reduced numerator hypothesis cannot be omitted.

A spatial sideband is a Fourier perturbation with wavenumber close to a pattern's carrier wavenumber, such as $k_c+p$ or $k_c-p$. A real physical field also includes the conjugate Fourier component. Linearizing a nonlinear complex amplitude equation can couple the two displaced components, so testing only a single uniform amplitude misses their joint linear stability problem.

For a spatially periodic nonlinear pattern, a linear stability disturbance is written $e^{\lambda t+iq\cdot x}v(x)$ with $v$ periodic on the pattern cell. The conjugated operator $L(q)$ acts on that fixed cell and determines the sideband eigenvalues. Translation invariance supplies zero eigenfunctions at $q=0$, namely spatial derivatives of the base pattern. Their small-$q$ eigenvalues determine long-wavelength phase stability; other parts of the spectrum can give independent instabilities.

Near a saddle-focus equilibrium, linearized coordinates satisfy $\dot{z}={\lambda }_{+}z$, $\dot{r}={\lambda }_{-}r$ and $\dot{\theta }=\omega$. A trajectory entering at a small positive coordinate $z$ takesto reach a fixed outgoing section. Hence $r_{\mathrm{out}}\propto z^{\delta }$ with $\delta =-{\lambda }_{-}/{\lambda }_{+}$, while the accumulated angle is logarithmic in $z$. Composing this passage with a smooth global return gives the leading scalar formwhere $|\Omega |=|\omega |/{\lambda }_{+}$ and the phase convention absorbs constant angular shifts. The map applies only on branches whose trajectories return to the section.^[ref]

Differentiation givesFor $\delta <1$, the envelope becomes unbounded as $z\downarrow 0$, but the derivative also has infinitely many zeros. The map therefore has strongly expanding branches, rather than uniform expansion everywhere.

The full two-dimensional return has Jacobian determinant of order $z^{2\delta -1}$ in regular section coordinates: radial scaling contributes $z^{\delta -1}$ and conversion of polar coordinates contributes $z^{\delta }$. For $\delta >1/2$ this contracts area near the singular limit, but area contraction alone does not prove that a scalar reduction is valid. A one-passage periodic trajectory has period $T_g+T_{\mathrm{loc}}$, explaining the logarithmically diverging time near the homoclinic connection.

For an invertible matrix $A$, the rank-one matrix update $B=A+u{v}^T$ is a singular matrix exactly when $1+v^T{A}^{-1}u=0$. This follows directly from the matrix determinant lemma. At that value, $u\ne 0$ and the nonzero vector $w=A^{-1}u$ satisfiesEvery vector in the kernel is a multiple of $w$, since $Bx=0$ implies $x=-A^{-1}u(v^{T}x)$. The matrix rank therefore drops by exactly one.

For $B(t)=I+tu{v}^T$ with $u=(1,2{)}^T$ and $v=(3,-1{)}^T$, the update becomes a singular matrix at $t=-1$, and $B(t)u=(1+t)u$. At $t=-0.99$, this direction is compressed by a factor $0.01$ and the inverse amplifies it by a factor 100. The actual spectral norm of the matrix inverse, which maximizes amplification over every direction, is about $700.17853409$.

The exact formula can expose a large condition number in the updated linear system even when the original one had a small condition number. It does not guarantee numerical stability merely because its evaluation uses fewer arithmetic operations. The denominator condition certifies existence; the condition number and floating-point behavior address different questions about using the resulting answer.

A shearing box solves the shearing sheet equations in a finite domain with periodic azimuthal and vertical faces and a shearing-periodic boundary condition on its radial faces. It follows a small region rotating at a reference orbital frequency. The model can sustain stress and exchange energy with its imposed differential rotation, but its identified radial faces do not represent a global disk mass sink.

A sheaf of topological rings takes values in topological rings, with continuous restriction maps. For an open cover, the sheaf condition is an equalizer of the product of sections on the cover and the product on pairwise overlaps, in the category of topological rings. The structure sheaf of a formal scheme is an inverse limit of discrete structure sheaves with this topology.

// Target: algebra.bigb

A rational section is a section on a dense open subset, with two representatives identified when they agree on a common dense open subset. Intersecting representatives' domains defines addition. Multiplication by a similarly represented rational function on an algebraic variety makes these classes a module over the ring $\mathrm{Rat}(X)$. For an irreducible variety, this ring is its function field.

A local system is a locally constant sheaf of modules, for example the integral cohomology of fibres of a smooth proper family of compact complex manifolds. Parallel continuation around loops gives its monodromy representation.

For sheaves of modules $\mathcal{F},\mathcal{G}$ on a ringed space, the internal Hom sheaf assigns to an open $U$ the module of sheaf morphisms $\mathcal{F}{|}_U\to \mathcal{G}{|}_U$ respecting ${\mathcal{O}}_U$. Compatible morphisms glue, so this is a sheaf. Its global sections are precisely the global module-sheaf morphisms. For a locally free sheaf $\mathcal{F}$ of finite rank it is ${\mathcal{F}}^{\vee }\otimes \mathcal{G}$.

For a normal vector $\xi$, this identity defines the corresponding shape operator using the vector-valued second fundamental form. It is self-adjoint because of the symmetry of the second fundamental form. Differentiating $⟨\xi ,Y⟩=0$ shows that $A_{\xi }X=-(D_X\xi {)}^{\top }$. In higher codimension there is a different operator for each normal direction.

For a steady, slowly varying rotating channel, negligible transverse velocity and zero shallow-water potential vorticity give $u_y=f$. Cross-channel geostrophic balance gives $g{h}_y=-fu$, producing the displayed profiles. The specific energy $E=h+u^2/(2g)$ is constant across the channel, and $E+H$ is constant along it. Zero potential vorticity is not the value in an ordinary finite-depth reservoir at rest in the rotating frame; it requires vanishing absolute vorticity or a deep-reservoir approximation.

For an upstream irrotational current on an f-plane in a rigid layer of height $h$, a circular region of height $h(1-ϵ)$ has the displayed relative vorticity if its fluid retains upstream potential vorticity $f/h$. To leading order in the height contrast, take $u=-{\Psi }_y$, $v={\Psi }_x$ and $\Psi =-Uy+\psi$. Solving the Poisson equation for $\psi$ gives $\psi =-C{r}^2/2$ inside and $\psi =C{a}^2[\log (a/r)-1/2]$ outside, where $C=fϵ/2$. The disturbance is clockwise for positive $fϵ$. Variable-depth transport and the transition across the rim introduce higher-order corrections to this leading columnar inversion.