The coherent topology on a small coherent category is generated by finite families of arrows whose image subobjects have union equal to the whole target. Pullback stability of images and unions and closure under composition make these a coverage. On a coherent syntactic category, this is the site topology used to construct the classifying topos of its coherent theory.

A functor between coherent categories is coherent when it preserves finite limits, image factorizations and finite unions of subobjects, including the empty union. It consequently preserves the interpretation of every coherent formula. A coherent functor from a coherent syntactic category to a topos is equivalent to a model of the corresponding coherent theory; natural transformations correspond to ordinary model homomorphisms.

On l2 sequence space ${\ell }^2$, let $(Lu{)}_n=u_n/n$ and $f_n=1/n$. This bounded self-adjoint operator has $⟨Lu,u⟩>0$ for nonzero $u$, but the equation $Lu=f$ requires $u_n=1$, which is not in ${\ell }^2$. Taking the first $N$ coordinates equal to one makes $I(u)=-{\sum }_{n=1}^{N}1/n\to -\infty$. Thus strict positivity in isolation does not give existence in that Hilbert space; a coercive bilinear form and a continuous forcing functional in its norm are essential hypotheses for the usual minimization proof.

A real continuous function on $\mathbb{R}$ that tends to $+\infty$ as $|x|\to \infty$ has range $[m,\infty )$ for an attained minimum $m$. Choose a compact interval outside which its values exceed $f(0)$, and minimize there by the extreme value theorem. For any $y>m$, choose a far point with value larger than $y$ and apply the intermediate value theorem between it and a minimizer. Distance from a fixed point to a continuous graph is an example because it is at least the horizontal distance.

A Bogdanov–Takens bifurcation has a double zero eigenvalue with a nontrivial Jordan block and generic two-parameter unfolding. Its local diagram includes saddle-node bifurcations, Hopf bifurcations, and a curve of small homoclinic orbits. A weakly perturbed Hamiltonian scaling can locate the homoclinic curve through energy balance.

A saddle-node separatrix-loop point occurs when a saddle-node equilibrium simultaneously has a global returning separatrix at its strong stable manifold. In the local form $\dot{x}=x^2-\mu$, $\dot{y}=-\lambda y$, a global return to $y=h$ at $x=\nu$ makes the codimension-two point $\mu =\nu =0$. For $\mu >0$ a saddle loop occurs on $\nu =\sqrt{\mu }$. The saddle-node boundary $\mu =0$ has an invariant-circle segment on the $\nu <0$ side, while for $\nu >0$ a nearby periodic orbit avoids the bottleneck and survives the local equilibrium fold. The return geometry, rather than the local saddle-node normal form alone, determines the global orbit.

A fold-Hopf bifurcation combines a simple zero eigenvalue and a nonzero imaginary pair at an equilibrium point. In a three-dimensional centre manifold, a normal form of a dynamical system separates a real fold amplitude from a complex oscillatory amplitude. A rotationally symmetric normal form permits reduction to two real amplitude equations, while the oscillation phase continues to evolve. Secondary bifurcations of the amplitude flow can produce invariant tori in the full dynamical system.

A cusp is a two-parameter degeneracy at which two saddle-node bifurcation curves meet and the reduced steady equation has a triple root. In the displayed scalar normal form, simultaneous vanishing of the polynomial and its derivative gives ${\beta }_2=3x^2$, ${\beta }_1=-2x^3$. Thus the folds satisfy $4{\beta }_2^3=27{\beta }_1^2$, with ${\beta }_2\ge 0$. Between the folds there are three real equilibria, and outside them one. A transverse stable direction may be present in a higher-dimensional realization.

The codeword length is the number of alphabet symbols in a codeword. For a binary word it counts bits; in a prefix code, it is the depth of the corresponding leaf in the code tree.

// Target: algebra.bigb

A systematic encoder retains the message symbols unchanged in designated coordinates of its codeword and calculates the remaining check symbols. For a linear code, choosing an invertible set of $k$ columns of a generator matrix and changing its row basis gives this form.

The shift $E[1]$ has $E[1{]}^i=E^{i+1}$ and differential $-d_E^{i+1}$. Thus $H^i(E[1])=H^{i+1}(E)$. In general the differential of $E[r]$ is $(-1{)}^r{d}_E$.

A cochain map $f:E\to F$ consists of degreewise maps $f^i:E^i\to F^i$ satisfying $d_F^i{f}^i=f^{i+1}{d}_E^i$. This identity sends cocycles to cocycles and coboundaries to coboundaries, hence induces maps on cohomology. For complexes of sheaves the same identity holds as an identity of sheaf morphisms.

On the equatorial plane of the Kerr metric, the displayed component is negative for sufficiently small negative $r$. A curve of constant $t,r,\theta =\pi /2$, with $\varphi$ increasing through $2\pi$, is therefore a regular closed timelike curve. It avoids the ring at $r=0$. Such circles occur on the negative-radius sheet of the exact analytic extension, not in the exterior region accessible from positive-radius infinity.

The left and right transverse oscillator sectors each have generating function ${\prod }_{m\ge 1}(1-q^m{)}^{-24}$. Its coefficient satisfies $d_N\sim 2^{-1/2}(N-1{)}^{-27/4}{e}^{4\pi \sqrt{N-1}}$. Closed-string level matching requires equal levels but leaves their polarizations independent, so $g_N=d_N^2$. With ${\alpha }^{\prime }{M}^2=4(N-1)$, the exponential is $e^{{\beta }_{H}M}$ with ${\beta }_H=4\pi \sqrt{{\alpha }^{\prime }}$. The smooth rest-mass density is proportional to $M^{-26}{e}^{{\beta }_{H}M}$, since $dN/dM={\alpha }^{\prime }M/2$. This distinguishes degeneracy per level from density per unit mass. The polynomial prefactor matters at the boundary of canonical convergence.

For a complete comoving observer at the north pole of the closed slicing of de Sitter spacetime, radial null rays have $d\chi =\pm dT$. The finite past conformal interval gives the displayed particle horizon; the finite remaining future interval gives the displayed observer event horizon. The angular distances range strictly between zero and $\pi$ at every finite proper time. Complete constant-$T$ three-spheres are Cauchy hypersurfaces, so their domains of dependence cover the global spacetime and they have no Cauchy horizons. Cosmological horizons are observer-dependent and are not failures of global hyperbolicity.

The cycloidal closed matter-dominated Friedmann solution reaches its maximum scale factor at $u=\pi$ and the Big Crunch at $u=2\pi$. Thus its total lifetime is twice its time of maximum expansion. The displayed formula uses a presently expanding normalization with $H_0>0$ and $a_0=1$; the remaining time is this lifetime minus the age of a closed dust universe.

The Planck feedback is the stabilizing increase in outgoing thermal radiation when temperatures rise, with other properties held fixed. For a blackbody the local derivative is $4\sigma T^3$, so a colder emitter has a smaller absolute radiation increase per kelvin. The atmospheric response requires radiative transfer, and cannot be obtained by replacing the entire climate system with a single surface blackbody.

Lapse-rate feedback is the radiative effect of changes in the vertical temperature profile beyond a vertically uniform temperature change. In the tropics, enhanced warming aloft often increases radiative cooling and is stabilizing. In a stably stratified polar atmosphere, warming concentrated near the surface can yield less compensating cooling aloft and help produce polar amplification. The sign and magnitude depend on the profile and radiation physics.

Gamma matrices represent the spacetime Clifford algebra throughFor the Minkowski metric with signature $(+,-,-,-)$, ${\gamma }^0$ squares to $I$ and each spatial gamma matrix squares to $-I$. Distinct orthogonal generators anticommute. A four-dimensional complex representation is used for Dirac spinors in four-dimensional spacetime.

The particular matrix entries depend on the chosen basis; simultaneous similarity transformations preserve the defining relations. Their usefulness in the Dirac equation is that contracting them with derivatives produces a first-order operator whose square reduces to the second-order relativistic wave operator, apart from interactions or curvature contributions.

Clifford multiplication is the action of a Clifford algebra on a module. The displayed relation uses the negative Riemannian metric convention; in a compatible Hermitian inner product, $c(v{)}^{\ast }=-c(v)$ for real $v$. Another convention writes $C(v)=ic(v)$, giving self-adjoint generators with positive anticommutator. Switching between these conventions switches a geometric Dirac operator between self-adjoint and skew-adjoint realizations.