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For vectors $x,y$ in a Hilbert space, the rank-one operator $(x\otimes y)z=⟨z,y⟩x$ has $\Vert x\otimes y{\Vert }_1=\Vert x\Vert \Vert y\Vert$ and $\mathrm{tr}(x\otimes y)=⟨x,y⟩$.

For a trace-class operator, the series $\mathrm{tr}T={\sum }_n⟨T{e}_n,e_n⟩$ converges absolutely and is independent of the orthonormal basis. It satisfies $|\mathrm{tr}T|\le \Vert T{\Vert }_1$.

The trace-class operator condition makes this operator trace well-defined; the class of eligible operators and the trace functional are distinct concepts.

An operator is trace class exactly when it factors as $T=AB$ with $A$ and $B$ Hilbert--Schmidt. The factors can be chosen so that $\Vert T{\Vert }_1=\Vert A{\Vert }_{\mathrm{HS}}\Vert B{\Vert }_{\mathrm{HS}}$ by using the polar decomposition of a bounded operator.

This operation is a fan subdivision. A star subdivision inserts a ray through a lattice point in a cone and subdivides every cone containing that point. It induces a proper birational toric morphism.

A divisor is basepoint-free when its global sections have no common zero. Its complete linear system therefore defines a Kodaira map everywhere.

For an invariant divisor $D={\sum }_{\rho }{a}_{\rho }{D}_{\rho }$, its lattice polytope isIts lattice points index torus-character sections of the associated line bundle.

For a toric variety whose rays span $N_{\mathbb{R}}$, the divisor class group is computed bywhere $m\mapsto (⟨m,v_{\rho }⟩{)}_{\rho }$.

For a character ${\chi }^m$ of the dense torus,

Dynamical-friction stalling by tidal stripping occurs when inward motion shrinks a satellite's tidal radius so rapidly that its remaining mass and frictional drag tend to zero before it reaches the host center.

The Euler class of an oriented rank-$d$ vector bundle is the pullback $e(E)=s^{\ast }{u}_E\in H^d(B;R)$ of its Thom class along the zero section. Over ${\mathbb{F}}_2$ it equals the top Stiefel-Whitney class.

For an $R$-oriented rank-$d$ vector bundle, multiplication by the Thom class gives isomorphisms

A circular helix on a circular cylinder has radial ambient acceleration and is therefore a geodesic. Any surface tangent to the cylinder along that helix has the same curve as a geodesic.

If $X^{\ast }$ is separable and $K$ is a nonempty weak-star closed subset of $B_{X^{\ast }}$, then $K_{\epsilon }^{\prime }$ is a proper subset of $K$ for every $\epsilon >0$. Metrizability produces sequences witnessing membership in the derivative, while a Baire-category argument applied to a universal weakly null sequence yields the strict inclusion.

A symplectic basis organizes a nondegenerate alternating bilinear form into independent pairs. Let $(V,\omega )$ be a finite-dimensional vector space over a field $K$, with $\omega$ a nondegenerate bilinear form satisfying $\omega (v,v)=0$ for every vector $v$. Its dimension is even, say $2n$, and a symplectic basis is an ordered basis $\mathcal{B}=(e_1,\dots ,e_n,f_1,\dots ,f_n)$ such thatHere ${\delta }_{ij}$ is the Kronecker delta. Each $(e_i,f_i)$ is a hyperbolic pair; different pairs have zero mutual pairing. These conventions apply in every characteristic. In characteristic two, $-1=1$, but the condition $\omega (v,v)=0$ is still essential: a symmetric bilinear form need not be an alternating bilinear form. The existence theorem and the characteristic caveat are given in Bilinear Forms by Keith Conrad (Keith Conrad, Bilinear Forms, §1, Theorem 1.6; §5, Theorems 5.1 and 5.4 and Definition 5.3).

The benefit is a common coordinate language for every symplectic vector space of the same finite dimension over the same field. The span of all $e_i$, and separately of all $f_i$, is a Lagrangian subspace; each paired span is a symplectic subspace. In Hamiltonian mechanics, the corresponding pairing of position and momentum coordinates describes phase space and canonical transformations. On a symplectic manifold, a symplectic basis describes one tangent space; Darboux coordinates additionally describe a whole local neighborhood.^[ref]

Forboth players optimally choose cards $1,2,3$ with probabilities $41/50,2/25,1/10$, and the row player's expected payoff is $57/25$.

A support vector is a training observation on or inside the fitted margin; it has a nonzero dual coefficient and can affect the fitted boundary.

For a linear support vector machine, the geometric margin lies between $f(x)=-1$ and $f(x)=1$ and has width $2/\Vert \beta \Vert$.

For a linear score $f(x)=\alpha +x^T\beta$, the decision boundary is $f(x)=0$.

The slack variables satisfy ${\xi }_i\ge \max \{0,1-y_{i}f(x_i)\}$ and quantify margin violations.

When several eigenvalues share the largest modulus, normalized power iterates need not converge individually, but every limit lies in their joint invariant subspace.