On the projectivization by quotients, the fibre over a one-dimensional quotient of $E_x$ is that quotient line itself. These lines glue to the universal quotient. Its restriction to a projective fibre has degree one. It is the dual of the relative tautological line after identifying quotient projectivization of $E$ with line projectivization of $E^{\ast }$.

The special linear group $S{L}_2(5)$ has $24$ nonzero first columns and five second columns giving determinant one, hence order $120$; its centre is $\{I,-I\}$. In the quotient by this centre, the nonidentity conjugacy classes have sizes $15,20,12,12$, corresponding to projective element orders $2,3,5,5$. These arise from trace zero, paired traces $\pm 1$, and paired unipotent traces $\pm 2$. No proper union of these classes with the identity has size dividing $60$, so the quotient is simple. The simple groups with order a power of two times fifteen result identifies it with $A_5$.

On the six-dimensional exterior square of ${\mathbb{F}}_2^4$, the Pfaffian $Q=x_{12}{x}_{34}+x_{13}{x}_{24}+x_{14}{x}_{23}$ is a nonsingular plus-type quadratic form. The action of $G{L}_4(2)$ preserves it and is faithful: fixing every decomposable exterior line fixes every two-space and then every one-space. The even-weight binary subset module on eight letters, modulo the all-one vector, gives another six-dimensional plus-type space with $Q=\mathrm{wt}(x)/2\mathrm{mod}2$. Coordinate permutations embed $S_8$ faithfully into its orthogonal group. That orthogonal group has order $40320$, while $G{L}_4(2)$ has order $20160$, identifying its index-two image with the alternating group $A_8$.

For a nonzero homogeneous polynomial of degree $d>0$, multiplication gives $0\to {\mathcal{O}}_{{\mathbb{P}}^2}(-d)\to {\mathcal{O}}_{{\mathbb{P}}^2}\to i_{\ast }{\mathcal{O}}_X\to 0$. Intermediate cohomology of the twisting sheaves vanishes. The top cohomology of $\mathcal{O}(-d)$ has Laurent basis $t_0^{-b_0}{t}_1^{-b_1}{t}_2^{-b_2}$ with $b_i\ge 1$ and $\sum b_i=d$. There are $(d-1)(d-2)/2$ such monomials, with zero basis for $d=1,2$. The long exact sequence gives the displayed dimensions. The calculation allows singular, reducible and nonreduced hypersurfaces; it is an arithmetic-genus calculation, not an assertion about a normalization's genus.

Translate the point to the origin of an affine chart and expand a local defining polynomial as $\sum a_{ij}{x}^i{y}^j$. Its lowest nonzero total degree is the multiplicity. A smooth point of a variety has multiplicity one; a double point has multiplicity two. On a point blowup of a smooth algebraic surface, the total transform is the strict transform of an algebraic subvariety plus this multiplicity times the algebraic exceptional divisor.

A degree-four projective plane curve. Its arithmetic genus is three by the genus-degree formula; singularities can lower the genus of its normalization of an algebraic curve.

A degree-two projective plane curve, allowing singular and nonreduced cases. In characteristic different from two, write its equation as $z^{\mathsf{T}}Mz$ with $M$ a symmetric matrix. It is smooth exactly when $M$ has matrix rank three. Over an algebraically closed field, rank two gives two distinct projective lines, while rank one gives a double line.

Over the complex numbers, an irreducible projective variety is recovered by applying projective dual variety twice. In its conormal incidence variety, pairs $(y,H)$ with $H$ tangent at $y$ become pairs $(H,y)$ under exchange of the two projective factors. At general smooth pairs, differentiating the incidence equation shows that the hyperplane defined by $y$ is tangent to the dual variety at $H$. Closure gives the same conormal variety with the factors exchanged, proving biduality. This is why the branch hypersurface of the Gauss map of a theta divisor determines a nonhyperelliptic canonical curve.

If $T^2=T$, every vector splits as $v=(v-Tv)+Tv$ in $\ker T\oplus \mathrm{im}T$. On these summands $T$ is respectively zero and the identity. Its matrix in an adapted basis is therefore $\mathrm{diag}(0,I_r)$, where $r$ is its rank of a linear map. Two idempotent linear maps on the same finite-dimensional vector space are similar matrices exactly when their ranks agree.

For an idempotent linear map $P$ on a finite-dimensional real vector space, take any positive definite inner product on that space. The displayed new inner product is positive definite since its squared norm vanishes only if both $Pu$ and $(I-P)u$ vanish. It makes the kernel of a linear map and image of a linear map orthogonal. Since $P^2=P$, one has $⟨Pu,v{⟩}_P=⟨u,Pv{⟩}_P$; thus $P$ becomes an orthogonal projection.

Every finite-dimensional vector subspace $E$ of a Banach space $X$ is the range of a bounded linear projection with the displayed norm bound. In maximal inscribed Euclidean or Hermitian ellipsoid coordinates, write the John contact decomposition as ${\sum }_r{c}_r{u}_r{u}_r^{\ast }=I_E$. Its contact functionals ${\varphi }_r(x)=u_r^{\ast }x$ have norm one on $E$. Extend them to $X$ by the Hahn-Banach theorem and set $Px={\sum }_r{c}_r{\tilde{\varphi }}_r(x)u_r$. This restricts to the identity on $E$. The contact synthesis operator has Euclidean norm one, the weights sum to $\dim E$, and the Euclidean unit ball is contained in the norm unit ball; these facts give $\Vert Px{\Vert }_E\le \sqrt{\dim E}\Vert x{\Vert }_X$.

For two independent uniform points on the surface of a sphere, rotational invariance permits fixing the first at the north pole. The surface area of a latitude band of angular width $d\theta$ is $2\pi R^2\sin \theta d\theta$, yielding the displayed density for their central angle. Equivalently their unit position vectors have a dot product uniformly distributed on $[-1,1]$. The law is independent of sphere radius.

Bi-axial extension has two positive principal strain rates and one negative rate in incompressible flow. It flattens a material volume toward a sheet while extending its two in-plane directions. A negative mean product of the three rates indicates amplitude-weighted preference for this geometry, but alone does not establish an unweighted event-frequency majority.

Choose simple root vectors with $[e_i,f_i]=h_i={\alpha }_i^{\vee }$, and let ${\rho }^{\vee }$ be the half-sum of positive coroots. It satisfies ${\alpha }_i({\rho }^{\vee })=1$. Expand $h=2{\rho }^{\vee }={\sum }_i{c}_i{h}_i$. For $i\ne j$, $[e_i,f_j]=0$, since the difference of distinct simple roots is not a root of a root system. Thus $[e,f]=h$, while $[h,e]=2e$ and $[h,f]=-2f$. These identities construct an embedded sl2 Lie algebra in any nonzero complex semisimple Lie algebra. Its raising vector is the principal nilpotent element with a nonzero component in each simple-root space.

The $n$th Postnikov invariant is the cohomology class $k_n\in H^{n+1}(P_{n-1}X;{\pi }_{n}X)$ whose representing map has homotopy fibre $P_{n}X$. It specifies how the new homotopy group is attached to the previous stage. A zero invariant gives the product $P_{n}X\simeq P_{n-1}X\times K({\pi }_{n}X,n)$. Merely tensoring homotopy groups with $\mathbb{Q}$ does not describe rationalization of a topological space unless these classes are also transported.

A score is an observation's coordinate along a principal component direction. Center an observation by the fitted sample mean, apply the fitted variable scaling when using principal component analysis on a correlation matrix, then take its inner product with the unit principal component loading vector. New observations must use the training center and scaling. The score variance equals the corresponding eigenvalue when the covariance normalization is consistent.

Assume finitely many nonempty routes, positive weights and adjustment rates, and continuous nonnegative nondecreasing resource prices. Each route must meet a resource whose price is positive somewhere. Then the primal congestion potential has compact superlevel sets inside the positive orthant and a unique maximum. Along gradient flow with diagonal mobility, $\dot{U}={\sum }_r{\kappa }_r{x}_r({\partial }_{r}U{)}^2$. The substitution $x_r={\kappa }_r{z}_r^2/4$ makes the dynamics ordinary gradient flow ascent for $V(z)=U(({\kappa }_r{z}_r^2/4{)}_r)$. This $V$ is strictly concave: the logarithmic terms are strictly concave, and each integrated price is convex and nondecreasing, composed with a convex quadratic load. Thus its continuous gradient is monotone decreasing, which proves uniqueness of trajectories even without differentiability of the prices. Compact trapping and the dissipation identity force every limit point to be the unique maximizer. If every price on some route vanishes identically, its rate instead grows linearly and there is no equilibrium.

The virial theorem and hydrostatic equilibrium relate a cluster's gas temperature to $GM/R$, while $M=(4\pi /3){\Delta }_v{\rho }_c{R}^3$. A calibrated mass-temperature relation converts the Press-Schechter halo mass function into $dn/dT=(dn/dM)|dM/dT|$. Its exponential rare-cluster tail is sensitive to the density contrast variance on cluster scales, hence to the power-spectrum amplitude, often parameterized by ${\sigma }_8$. Mass calibration, temperature scatter, nonthermal pressure, sample selection and the matter density must be included rather than treating temperature counts as an independent measurement of amplitude alone.

For a small category $\mathcal{C}$, its presheaf category is the free enlargement under small colimits. Each categorical presheaf $P$ is the colimit of the diagram sending $(C,x)\in \int P$ to the representable presheaf $yC$. The map at $(C,x)$ sends $f:A\to C$ to $P(f)x$. It is surjective at each $A$ because $z\in P(A)$ is the image of $1_A$ at $(A,z)$. If two representatives have the same image $z$, they are both identified with the representative $1_A$ at $(A,z)$ by arrows in the category of elements, proving injectivity. Thus $P\cong {\mathrm{colim}}_{(C,x)\in \int P}yC$. For a cocomplete target, a functor on $\mathcal{C}$ extends by these colimits, uniquely up to natural isomorphism among colimit-preserving extensions.

For a categorical presheaf $P$ on a small category, its category of elements has objects $(c,x)$ with $x\in P(c)$ and arrows $u:(c,x)\to (d,y)$ satisfying $P(u)y=x$. The canonical natural transformation $H_c\to P$ sends $v:a\to c$ to $P(v)x$. These maps exhibit $P$ as the displayed colimit. At $a$, a representative $(c,x,v)$ maps to $P(v)x$, with inverse $z\mapsto (a,z,1_a)$. The arrow $v:(a,P(v)x)\to (c,x)$ identifies every representative with its canonical inverse image, proving the bijection and the colimit universal property.