A comonad is idempotent when its comultiplication $G\to G^2$ is invertible. Its coalgebra category identifies with the coreflective subcategory of objects on which the counit is invertible. A coreflective inclusion produces such a comonad.

For a transformation from the tensor built using $n$ to the tensor built using $m$, colinearity is $m\ast \gamma =\gamma \ast n$: $\sum m(a_{(1)},b_{(1)})\gamma (a_{(2)},b_{(2)})=\sum \gamma (a_{(1)},b_{(1)})n(a_{(2)},b_{(2)})$. Scalar maps enter this convolution product for coalgebra maps via the common algebra unit.

If $CD=DC$ and $Dx=\lambda x$, then $D(Cx)=C(Dx)=\lambda Cx$. Therefore each eigenspace of $D$ is invariant under $C$. The vector $Cx$ may be zero; invariance does not require it to be a nonzero eigenvector. This observation is the starting point for simultaneous diagonalization.

The matrix trace of $[A,B]=AB-BA$ is zero, since $\mathrm{tr}(AB)={\sum }_{i,j}{A}_{ij}{B}_{ji}=\mathrm{tr}(BA)$. This obstructs a matrix with nonzero trace from being a single commutator.

The isotropic three-dimensional Cauchy velocity distribution $f_0={\rho }_{0}b/[{\pi }^2(v^2+b^2{)}^2]$, $b>0$, has density ${\rho }_0$ and one-dimensional marginal $F(u)={\rho }_{0}b/[\pi (u^2+b^2)]$. Its second velocity moment diverges, so $b$ is a scale rather than a finite velocity dispersion. At zero phase speed, $\int F^{\prime }(u)/udu=-{\rho }_0/b^2$, giving $k_J=\sqrt{4\pi G{\rho }_0}/b$. In the upper-half frequency plane, the marginal response is ${\rho }_0/(\omega /k+ib{)}^2$; the growing branch has $\omega =i(\sqrt{4\pi G{\rho }_0}-kb)$ for $0<k<k_J$.

The collinear Lagrange point beyond the smaller primary, away from the larger one, is $L_2$. For a small mass ratio it lies approximately one Hill radius from the secondary and has a growing linear mode.

The Lagrange point lying between the components of a circular restricted three-body problem. The critical effective potential through this saddle defines their Roche lobes; gas passing through it can initiate Roche-lobe overflow.

Map $X_{d,f}$ to $K(\mathbb{Z},2)={\mathbb{CP}}^{\infty }$ using its degree-two generator. The homotopy fibre is the total space of the corresponding circle bundle, is 2-connected, and has ${\pi }_3$ equal to that of $X_{d,f}$. Over $S^2\vee S^3$ its total space is $P=S^3{\cup }_{S^1}(S^1\times S^3)$, with $H_3(P)=\mathbb{Z}A\oplus \mathbb{Z}B$. The four-cell bundle is trivial; its relative degree-four generator has boundary $dA+fB$, the lifted attaching map. The long exact sequence in relative homology and Hurewicz theorem therefore give ${\pi }_3(X_{d,f})={\mathbb{Z}}^2/⟨(d,f)⟩$. Smith normal form gives the displayed expression when $d>0$, including $f=0$.

The canonical model interprets each sort as its truth-context object, each operation by its term graph and each relation by its atomic subobject. A coherent formula is interpreted by its own formula-in-context object. Consequently a coherent sequent holds in this model exactly when it is derivable in the theory.

A coherent sheaf on ${\mathbb{P}}_k^n$ is the sheaf associated with a graded module for a finite graded $k[t_0,\dots ,t_n]$-module. Choose a finite graded free resolution by the Hilbert syzygy theorem and sheafify it; exactness of localization preserves exactness. All terms are finite sums of twisting sheaves on projective space. Twisting sufficiently positively makes every term acyclic in positive degree, and dimension shifting in long exact sequences in sheaf cohomology proves Serre vanishing. The graded-module description is supplied by Stacks Project, Section 30.15.

An ideal sheaf of a closed subscheme that is a coherent sheaf as a module over the structure sheaf. On a Noetherian scheme, every quasi-coherent ideal sheaf is coherent.

For a stationary $S\subseteq {\omega }_1$, there are cofinal ladders $A_{\delta }\subset \delta$ of order type $\omega$, indexed by limit ordinals in $S$, such that every uncountable $X\subseteq {\omega }_1$ contains one of these ladders. This predicts a contained ladder, whereas stationary diamond principle predicts a whole initial segment.

With constant atmospheric scale height $H$, a molecular band at height $\Delta z$ above the measured continuum satisfies $P_{\mathrm{band}}=P_{\mathrm{cont}}{e}^{-\Delta z/H}$. If an opaque exoplanet cloud deck sets that continuum, then $P_c\simeq P_{\mathrm{cont}}$; the band pressure is not the cloud pressure. If the clear band height would be $NH$ above a clear continuum at $P_0$ and the cloud reduces the visible feature to $mH$, thenWithout the clear-atmosphere band opacity or $N$, the measured feature amplitude alone cannot locate the cloud top. This follows directly by subtracting the two hydrostatic height relations.

For a finitely presented ring $A$, its sheafified representable is initial exactly when $A$ is the zero ring. The zero ring has a generating empty cover. Every nonzero ring maps to a field by quotienting by a maximal ideal, and that set-based domain gives a point at which the representable has a section, precluding initiality.

In the generic domain, negation of simultaneous invertibility of finitely many elements implies that one is zero. At a ring stage, localize at their product $t$. All become units, so a tuple satisfying the negation makes the localized stage empty. The empty-cover criterion for the domain-classifying site then says $A[1/t]=0$, hence $t$ is nilpotent. Repeated zero-product covers force one factor to vanish locally. The two-variable property conversely forces the integral-domain axiom in any nontrivial internal ring.

For a finite group with trivial centre of a group and a prime divisor $p$ of its order, some nonidentity conjugacy class has size coprime to $p$. Otherwise the class equation would give $|G|\equiv 1(\mathrm{mod}p)$. Primality is essential: in the symmetric group $S_3$, neither nonidentity class size is coprime to the composite divisor $6$.

Maximize the integral of a continuous strictly convex function among probability measures with a fixed barycenter. The supporting measure lemma for affine upper envelopes implies that the maximizing measure has zero integral of the nonnegative envelope gap. Strict convexity makes this gap positive at every nonextreme point, so the measure is concentrated on the extreme boundary. Metrizability makes that boundary Borel and supplies the continuous strictly convex function.

Work with the real Lp space $L^{\infty }((0,1))$ for Lebesgue measure, so its predual $L^1((0,1))$ is separable. Choose a measurable representative $f$ of a unit-ball element with $-1\le f\le 1$ and putFor every $t$, $H_t$ has modulus one almost everywhere, so it is an extreme point of the real $L^{\infty }$ closed unit ball. The map $t\mapsto H_t$ is Borel for the weak-star topology: pairing with any $g\in L^1$ is measurable, and separability makes the weak-star unit ball metrizable. Push forward uniform probability measure on $[0,1]$ through this map.

For almost every $s$,Fubini's theorem then gives ${\int }_0^1\int g(s)H_t(s)dsdt=\int g(s)f(s)ds$ for every $g\in L^1$. Thus the resulting Borel probability measure is concentrated on the extreme points and has barycenter $f$.

For $f(s)=2s-1$, the profiles are the threshold steps $H_t(s)=21_{\{t\le s\}}-1$. Averaging these sign-valued extreme points produces the linear profile. This is an explicit Choquet theorem representation in a setting where the required metrizability is actually available.

For an Abelian vector superfield, the chiral field-strength superfield is $W_{\alpha }=-{\bar{D}}^2{D}_{\alpha }V/4$. Three barred supersymmetric covariant derivatives vanish, proving ${\bar{D}}_{\dot{\beta }}{W}_{\alpha }=0$. This is a chiral spinor superfield with a gaugino as its lowest component. Phase and component signs depend on the stated Wess-Zumino gauge convention; the reality of $V$ alone does not fix those phase conventions.

For dimensionless nearest-neighbour coupling $K$ and diagonal coupling $L$, the second cumulant in checkerboard decimation of the square-lattice Ising model generates $K^2$ for each pair of retained neighbours of an eliminated centre. A retained diagonal pair shares two centres, and an axial pair at distance $2a$ shares one. The displayed two-coupling recursion neglects $K^4$, $K^{2}L$, and $L^2$, treating $L$ as order $K^2$. Its finite nontrivial renormalization-group fixed point is $(K,L)=(1/3,1/9)$. Higher orders generate additional operators, so the closure is a property of the truncation.