A theorem giving sufficient conditions for a functor to have an adjoint functor. The general adjoint functor theorem uses a complete category, locally small categories, preservation of categorical limits and a solution-set condition. The Special adjoint functor theorem obtains that condition from a well-powered source and a small cogenerating family. Precise hypotheses matter in both forms.

An adjunction $L⊣R$ is idempotent when its induced monad has invertible multiplication. This is equivalent to its induced comonad having invertible comultiplication. For an idempotent monad, the two maps ${\eta }_{TA}$ and $T{\eta }_A$ agree, since both invert the multiplication. If $a:TA\to A$ is an algebra, naturality then gives ${\eta }_{A}a=Ta{\eta }_{TA}=T(a{\eta }_A)=1$. Thus every algebra unit is invertible. Apply this to $R{\epsilon }_B$ to obtain the self-dual adjunction condition.

A Boolean algebra homomorphism $B\to \mathcal{P}(A)$ is equivalent to a function assigning to every $a\in A$ an ultrafilter on $B$: test membership of $a$ in every image subset. This gives a left adjoint valued in the opposite category of sets. Its unit sends an algebra element $b$ to the set of ultrafilters containing $b$. On an infinite power-set algebra this map is not surjective, because the subset consisting of the principal ultrafilters cannot arise this way in the presence of a nonprincipal ultrafilter.

Discretization $D$ and indiscretization $J$ retain underlying sets. Every function $DX\to Y$ is continuous, and every function $X\to JY$ is continuous, giving the adjunction. Its induced monad and comonad are idempotent, since repeated discrete topology or indiscrete topology construction changes nothing.

For small categories, the object functor $O$ has left adjoint the discrete category functor and right adjoint the indiscrete category functor. Connected components, defined by zigzags of morphisms, give a left adjoint ${\pi }_0$ to the discrete functor. There is no further left adjoint: ${\pi }_0$ fails the equalizer of two functors selecting distinct objects of the indiscrete two-object category. There is no further right adjoint: the indiscrete functor fails the coproduct of two singleton sets.

A functor admitting a further left adjoint must itself be a right adjoint and preserve all existing categorical limits. A functor admitting a further right adjoint must preserve all existing colimits. Failure to preserve a single terminal object, equalizer, initial object or coproduct therefore rules out the corresponding extension. In the adjoint chain for the objects of a category, connected components fail an equalizer and the indiscrete category functor fails a binary coproduct.

For a resonant three-level chain with couplings ${\Omega }_1,{\Omega }_2$, the zero-energy dark state of a driven Hamiltonian is $|D⟩=({\Omega }_2|1⟩-{\Omega }_1|3⟩)/\sqrt{{\Omega }_1^2+{\Omega }_2^2}$. Turning on the target-to-intermediate coupling first and sweeping ${\Omega }_1/{\Omega }_2$ from zero to infinity rotates this state from $|1⟩$ to $-|3⟩$. The other eigenvalues are $\pm \sqrt{{\Omega }_1^2+{\Omega }_2^2}$, so $\hslash |\dot{\theta }|\ll \sqrt{{\Omega }_1^2+{\Omega }_2^2}$ controls nonadiabatic leakage. Ideally the intermediate state remains unoccupied, suppressing spontaneous-emission loss. The gap must not close while the dark state is rotating.

For separately conserved barotropic components, an adiabatic cosmological perturbation has equal ${\delta }_N/(1+w_N)$. Pressureless matter and a $w=-1/3$ string component therefore have ${\delta }_s=2{\delta }_c/3$ on superhorizon scales. In the supplied perfect-fluid synchronous equations, neglecting spatial gradients gives $S^{\prime \prime }+2\mathcal{H}{S}^{\prime }=0$ for $S={\delta }_s-2{\delta }_c/3$. Both $S$ and its independent decaying-mode amplitude must vanish to select the growing adiabatic solution. At finite wavenumber, pressure gradients generate departures from this leading long-wavelength relation.

A real additive function $f:\mathbb{R}\to \mathbb{R}$ that is continuous at one point $x_0$ is continuous everywhere and has form $f(x)=xf(1)$. Indeed $f(0)=0$, and $f(h)=f(x_0+h)-f(x_0)\to 0$ as $h\to 0$. Then $f(x+h)-f(x)=f(h)\to 0$ for every $x$. Additivity gives $f(q)=qf(1)$ for every rational number; continuity and the density of the rational numbers extend this identity to every real argument.

The step from the rational numbers to the real numbers can be made explicit. Put $c=f(1)$. For an arbitrary real number $x$, the density of the rational numbers supplies a sequence of rational numbers $q_n\to x$. Because continuity at the original point already implied continuity everywhere,This is where continuity enters: knowing all the rational values alone does not justify the first equality.

For an original numerical example, use the floor function to define the rational lower approximants $q_n=\lfloor 10^n\sqrt{2}\rfloor /10^n$. They obey $0\le \sqrt{2}-q_n<10^{-n}$. When $f(1)=3$ and $f$ is continuous at one point, rational homogeneity of an additive function and the preceding argument giveThe error bound here follows from the resulting linear map, rather than from a quantitative bound on the initially assumed continuity.

For centered-space diffusion, a Fourier mode gives $z^2-(1-3a/2)z-a/2=0$, with $a=4\mu {\sin }^2(\theta /2)$. For $a>0$ the roots have opposite signs, the positive root lies in $(0,1)$, and the negative root lies in $[-1,0)$ exactly when $P(-1)=2-2a\ge 0$. At $a=0,1$ the unit-modulus roots are simple. Hence the root condition for a multistep method holds for every Fourier frequency exactly when $\mu \le 1/4$. Under fixed $\mu$, the unscaled one-step defect is $-\Delta t^2{u}_{xxxx}/(12\mu )+O(\Delta t^3)$; dividing by the step changes its order to $O(\Delta t)$.

At an extreme point of $\{x:Ax\ge b\}$, the active rows span the ambient dimension: otherwise a nonzero direction orthogonal to all active rows permits small feasible moves of both signs. Select $n$ independent active rows to form an integer matrix $B$. Its nonzero determinant has absolute value at least one. Cramer's rule and the determinant expansion bound each coordinate by $n!U^n$, where $U$ bounds the entries of $A$ and $b$. The vector $b$ may be real; only the denominator matrix needs to be integer.

For a bound noncollision Kepler orbit with gravitational parameter $\mu$, the canonical actions per unit mass are $\mathcal{L}=\sqrt{\mu a}$, $\mathcal{G}=\mathcal{L}\sqrt{1-e^2}$ and $\mathcal{H}=\mathcal{G}\cos i$. Their angles are mean anomaly $\ell$, argument of periapsis $\omega$ and longitude of ascending node $\Omega$. The Hamiltonian is $-{\mu }^2/(2{\mathcal{L}}^2)$, so only $\ell$ advances in the unperturbed problem. In the prograde spherical-action chart, $J_r=\mathcal{L}-\mathcal{G}$, $J_{\theta }=\mathcal{G}-\mathcal{H}$, $J_{\varphi }=\mathcal{H}$; the type-two canonical transformation is generated by $F_2=(\mathcal{L}-\mathcal{G})w_r+(\mathcal{G}-\mathcal{H})w_{\theta }+\mathcal{H}{w}_{\varphi }$. Retrograde spherical actions have $J_{\theta }=\mathcal{G}-|\mathcal{H}|$, so the absolute value cannot be dropped globally. Circular and coplanar limits are singular angle charts, not singular orbits.

A point source moving at angular speed $\Omega$ on a circle of radius $a$ has observer-direction Mach number $m=\Omega a\sin \iota /c_0$. In the far field its retarded phase satisfies $\Theta +\theta -\pi /2=\Omega (t-r/c_0)+m\sin \Theta$. For $m<1$ there is one retarded root, and a constant source strength gives ${\rho }^{\prime }=q\dot{\Theta }/(4\pi c_0^2\Omega r)$. For $m>1$, multiple retarded roots can occur; their contributions require $|1-m\cos \Theta |$ and a sum over roots. Coalescing roots produce a caustic in the ideal point-source model.

For a small body of size $l$ in flow of speed $u$, force of order ${\rho }_0{u}^2{l}^2$ varying on time $l/u$ produces an acoustic dipole density amplitude of the displayed order, where $M=u/c_0$ is the Mach number. Its sound power scales as ${\rho }_0{u}^6{l}^2/c_0^3$. The corresponding compact acoustic quadrupole Mach-number scaling has density amplitude of order $(l/r)M^4$ and power ${\rho }_0{u}^8{l}^2/c_0^5$.

For a closed operator $L$ on a Banach space $H$, a positive-time classical solution belongs to $C([0,T],H)\cap C^1((0,T],H)$, takes values in $D(L)$ for positive time, and has $Lu\in C((0,T],H)$. It satisfies the differential equation and the initial condition in $H$. Classical regularity also at time zero requires the corresponding regularity of the initial datum, in particular $u_0\in D(L)$. For an analytic semigroup and Hölder continuous forcing, the variation-of-constants formula gives such a solution: subtracting the forcing at the current time turns the singular estimate $\Vert LS(r)\Vert \le C/r$ into the integrable bound $C{r}^{\theta -1}$.

Absorption distance is a dimensionless path variable chosen so a non-evolving comoving absorber population with fixed proper cross-section has constant $d\mathcal{N}/dX$. It absorbs the geometrical factor in absorber incidence and comoving number density. A distribution per $dNdX$ and one per $dNdz$ therefore differ by $dX/dz$.

Let the locations $a_n$ be distinct and bounded, and let $c_n>0$ with ${\sum }_n{c}_n<\infty$. The Weierstrass M-test gives locally uniform convergence of $G$, so the uniform limit theorem makes it continuous. The reverse triangle inequality bounds each term's difference quotient by $c_n$. Consequently a uniformly small tail permits passing each one-sided difference-quotient limit through the series. At every point outside the locations, $G^{\prime }(x)={\sum }_n{c}_n\mathrm{sgn}(x-a_n)$; at $a_m$, the right derivative minus the left derivative is $2c_m>0$. Thus the function has precisely the prescribed cusp locations, even when they accumulate at another point where the function remains differentiable.

Suppose a real function $g$ is differentiable at $a$, with $g(a)=0$ and $g^{\prime }(a)\ne 0$. The derivative definition gives $g(a+h)=g^{\prime }(a)h+o(|h|)$. The reverse triangle inequality then impliesThus $|g|$ has unequal one-sided derivatives, $|g^{\prime }(a)|$ on the right and $-|g^{\prime }(a)|$ on the left. More generally, multiplication by a function $w$ continuous at $a$ with $w(a)\ne 0$ gives right and left slopes $w(a)|g^{\prime }(a)|$ and $-w(a)|g^{\prime }(a)|$. This elementary expansion explains how a smooth oscillation can acquire infinitely many corners under the absolute value function, even after multiplication by a factor that makes it differentiable at an accumulation point.

This loss assigns different positive costs to underestimation and overestimation of a scalar parameter. Here $\gamma$ penalizes estimates below the parameter and $\delta$ estimates above it. Equal costs give a multiple of absolute-error loss.

Let $D_8$ denote the dihedral group of group order eight acting on a square. Here the subscript specifies group order; the same group is called $D_4$ in the convention that counts vertices of a polygon, as in the group action example. Let $r$ be a counterclockwise quarter-turn rotation and $s$ a reflection in a diagonal. ThenProducts act rightmost first. The group commutator of the generators of a group isIt is a half-turn rotation, so forcing $r$ and $s$ to commute must identify a half-turn with $e$.

More concretely, write each element uniquely as $r^i{s}^j$, with $0\le i<4$ and $j\in \{0,1\}$. The functionis a surjective group homomorphism. Indeed, $srs=r^{-1}$ giveswhere the exponents are reduced modulo four and two. Modulo two, $-k$ and $k$ agree, so $\varphi$ adds the two coordinates separately. Its group kernel is $N=\{e,r^2\}$. Since the target is an abelian group, $D_8^{\prime }\subseteq N$; since $[r,s]=r^2$, also $N\subseteq D_8^{\prime }$. ThusThe four cosets are $\{e,r^2\}$, $\{r,r^3\}$, $\{s,r^{2}s\}$, and $\{rs,r^{3}s\}$. Each coset forgets the difference of a half-turn. The quotient group still distinguishes rotations from reflections, and even from odd numbers of quarter-turns; the two surviving choices are independent. This is the Klein four-group, rather than the cyclic group $C_4$, because each nonidentity element squares to the identity element. For the general dihedral group calculation, see Conrad, Dihedral groups I, theorem A.1; that source uses the vertex-count convention and $xy{x}^{-1}{y}^{-1}$ for individual group commutators.