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The infimum of $E(u)$ over $\Vert u{\Vert }_2=1$ is attained. A minimizing sequence is bounded in $H_0^1$ because $V$ is bounded below; weak compactness, Rellich strong $L^2$ convergence, and weak lower semicontinuity complete the direct-method argument.

If $u:[0,T]\to H$ is weakly continuous and an energy equality makes $t\mapsto \Vert u(t){\Vert }_H$ continuous, then $u$ is strongly continuous. This follows from the Radon-Riesz theorem applied whenever $t_n\to t$.

Let $E/K$ be an elliptic curve and $n>0$ an integer prime to the characteristic. The Weil pairing is the canonical perfect pairingwhere the arguments are geometric $n$-torsion points of an elliptic curve and ${\mu }_n$ is the group of $n$th roots of unity. It is bilinear and alternating, so $e_n(P,P)=1$ and $e_n(P,Q)=e_n(Q,P{)}^{-1}$. Perfection means that every nonzero first argument pairs nontrivially with some second argument.^[ref]

It is compatible with the absolute Galois group: $e_n(\sigma P,\sigma Q)=\sigma (e_n(P,Q))$. For an isogeny $\varphi :E\to E^{\prime }$ and its dual isogeny, $e_n^{E^{\prime }}(\varphi P,Q)=e_n^E(P,\hat{\varphi }Q)$. In a torsion basis, alternation makes the action on the pairing equal to the determinant; hence the determinant of the torsion Galois representation is the cyclotomic character. In characteristic $p$, the pairing on geometric $p$-torsion points cannot have this same prime-to-characteristic interpretation: ${\mu }_p$ has only one geometric point.

The $n$-division field is obtained by adjoining to $K$ the coordinates of every point of $E[n]$. Over a finite field its degree is the order of Frobenius acting on $E[n]$.

In an orthonormal basis $({\epsilon }_i)$, the $C_n$ roots are $\pm {\epsilon }_i\pm {\epsilon }_j$ for $i<j$ and $\pm 2{\epsilon }_i$.

For a vector field $A=A_r{e}_r+A_{\theta }{e}_{\theta }+A_{\varphi }{e}_{\varphi }$ in the local orthonormal basis of spherical coordinates, take $\theta$ as colatitude and $\varphi$ as azimuth. For $r>0$ and $0<\theta <\pi$, its curl isThese are physical components along unit vectors, not components along unnormalized coordinate tangent vectors. The scale factors of orthogonal coordinates are $1,r,r\sin \theta$; applying the Stokes theorem to a small coordinate rectangle gives each component as circulation divided by its area.

As a check, rigid rotation about the $z$ axis has $A=\Omega r\sin \theta e_{\varphi }$. The formula gives $\nabla \times A=2\Omega \cos \theta e_r-2\Omega \sin \theta e_{\theta }=2\Omega e_z$, agreeing with the Cartesian calculation. Apparent singularities at the poles are coordinate singularities; regularity there must be checked for the original vector field.

If orthogonal coordinates $q_i$ satisfythen $h_i=|\partial \mathbf{x}/\partial q_i|$ are their scale factors.

These factors determine the diagonal metric and volume element of orthogonal coordinates.

A normal operator $A$ on a separable complex Hilbert space has a unique normalized projection-valued measure $E$ such thatFor a bounded normal operator, $E$ is supported on its compact spectrum and the integral defines an everywhere-defined bounded operator. For a closed densely defined normal operator, its operator domain isThe scalar measure in this expression is positive and has total mass $\Vert x{\Vert }^2$.

A spectral integral can exist without a single eigenvector. On $L^2(0,1)$, $(Af)(t)=tf(t)$ is bounded and self-adjoint, and $E(B)f=1_{B}f$. An eigenvector for $\lambda$ would have to be supported on $\{t=\lambda \}$, a set of Lebesgue measure zero, so it would vanish in the L2 space. The spectral measure therefore extends orthonormal diagonalization to continuous spectral parameters rather than claiming an eigenbasis always exists.

For a finite irreducible Reversible Markov chain with Stationary distribution $\pi$, let $Q(x,y)=\pi (x)P(x,y)$ and define its conductance profile byFor ${\pi }_{\ast }:={\min }_x\pi (x)\le r\le 1/2$ and the variance-normalized Spectral profile,The star here names the small-set profile on this restricted domain; some sources reserve a star for its extension truncated at $1/2$.^[ref]

For the upper bound, test the spectral-profile quotient with $f=1_A$. Its energy is $Q(A,A^c)$ and its variance is $\pi (A)(1-\pi (A))$. For the lower bound, take a nonnegative function supported on a set of mass at most $r$. Applying the level-set integral identity to $f^2$ bounds ${\Phi }_{\ast }(r)\Vert f{\Vert }_2^2$ by $\frac{1}{2}{\sum }_{x,y}Q(x,y)|f(x{)}^2-f(y{)}^2|$. Factoring the difference of squares and applying the Cauchy-Schwarz inequality bounds this last expression by $\sqrt{2\mathcal{E}(f,f)}\Vert f{\Vert }_2$. Squaring and using ${\mathrm{Var}}_{\pi }(f)\le \Vert f{\Vert }_2^2$ proves the lower bound. Thus a geometric obstruction to boundary flow produces an analytic obstruction to relaxation.

The $B_2$ root system consists of $\pm {\epsilon }_1$, $\pm {\epsilon }_2$, and $\pm {\epsilon }_1\pm {\epsilon }_2$. Its vector representation has weights ${\epsilon }_1,{\epsilon }_2,0,-{\epsilon }_2,-{\epsilon }_1$.

The Lie algebra ${\mathfrak{so}}_5$ has dimension ten and root system $B_2$.

The Lorentz algebra has rotation generators $J_i$ and boost generators $K_i$ satisfying

The Lie algebra ${\mathfrak{sl}}_2$ has generators $e,h,f$ with $[h,e]=2e$, $[h,f]=-2f$, and $[e,f]=h$.

This is the $n=2$ instance of the special linear Lie algebra. Classifying its modules is a separate representation-theory problem.

The two fundamental representations of ${\mathfrak{sl}}_3$ are the defining representation $V({\omega }_1)={\mathbb{C}}^3$ and its dual

Under $Q\cong {\mathbb{P}}^1\times {\mathbb{P}}^1$, the fibres of the two coordinate projections are the two rulings of the smooth quadric surface. Two distinct fibres in one ruling are disjoint projective lines, while one fibre from each ruling meets in one point.

A horn ${\Lambda }_i^n$ is inner when $0<i<n$ and outer when $i=0$ or $i=n$.

For a finite-type morphism of Noetherian schemes, separatedness is equivalent to uniqueness in every lifting problem $\mathrm{Spec}K\to X$ over $\mathrm{Spec}R\to Y$, where $R$ is a valuation ring with fraction field $K$.

This is the uniqueness part of the valuative criterion.

In the general reflection convention, a root system in a finite-dimensional real inner product space $E$ is a finite spanning set $\Phi \subset E\setminus \{0\}$ such that every root reflectionpermutes $\Phi$. In particular $-\alpha \in \Phi$ whenever $\alpha \in \Phi$. A reduced root system additionally requires $\Phi \cap \mathbb{R}\alpha =\{\alpha ,-\alpha \}$. A crystallographic root system additionally requires all Cartan integers $2(\beta ,\alpha )/(\alpha ,\alpha )$ to be integers. Some authors include these extra conditions in the term root system; they are stated separately here.

For example, the A2 root system is $\{e_i-e_j:i\ne j\}$ in the plane $x_1+x_2+x_3=0$. Its six roots form a regular hexagon, and their reflections generate a Weyl group isomorphic to $S_3$. The roots of a complex semisimple Lie algebra form a reduced crystallographic root system: its root-space decomposition converts infinitesimal Lie brackets into this finite Euclidean geometry. The Dynkin diagram then records enough simple-root data to classify its simple components.^[ref]

This is a Riemannian splitting theorem whose hypotheses include a globally minimizing geodesic line.

A complete connected Riemannian manifold with nonnegative Ricci curvature that contains a line splits isometrically as $N\times \mathbb{R}$.

This is the positive-definite metric case of a Ricci-flat manifold.

A Ricci-flat Riemannian manifold has identically zero Ricci tensor. Every flat Riemannian manifold is Ricci-flat, while the converse can fail in dimension at least four.