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For every state on ${\mathcal{H}}_A\otimes {\mathcal{H}}_B$,The upper bound follows from Subadditivity of Von Neumann entropy and the lower bound from the Araki–Lieb inequality.

If $\epsilon =\Vert {\rho }_{AB}-{\sigma }_{AB}{\Vert }_1/2$ and $d_A=\dim {\mathcal{H}}_A$, thenThe proof couples the two states through their positive and negative differences, then combines concavity of quantum conditional entropy with the entropy bound for a binary mixture.

For bipartite density operators,This is a consequence of Strong subadditivity of Von Neumann entropy.

If $F_n\to F$ at every continuity point of $F$, then the generalized inverses satisfyat every continuity point of $F^{-1}$. Since a monotone function has only countably many discontinuities, evaluating all quantiles at one uniform random variable gives an almost-sure coupling of the corresponding convergence in distribution.

If $U$ has a uniform distribution on $(0,1)$ and $Q$ is the quantile function of a cumulative distribution function $F$, then $Q(U)$ has the desired probability distribution. Indeed, right continuity givesso $\mathbb{P}(Q(U)\le x)=\mathbb{P}(U\le F(x))=F(x)$. This short argument works for both jumps and smooth cumulative distribution functions. MIT's continuous-probability notes introduce the inverse-transformation construction. ^[ref].

For the three-value example, send $U\le 0.2$ to $0$, $0.2<U\le 0.7$ to $1$, and $U>0.7$ to $3$. These intervals have exactly the desired widths $0.2$, $0.5$, $0.3$. For the density example, output $\sqrt{U}$: for $0\le y\le 1$, $\mathbb{P}(\sqrt{U}\le y)=\mathbb{P}(U\le y^2)=y^2$. Independent uniform draws give independent random variables after this deterministic transformation. This direction of construction works even when $F$ has jumps; applying the probability integral transform in the other direction requires its own continuity hypotheses.

A Majorana fermion operator is self-adjoint and obeys $\{c_j,c_k\}=2{\delta }_{jk}$.

A Bogoliubov--de Gennes Hamiltonian acts on a Nambu particle-hole spinor and describes quadratic fermion hopping and pairing.

An element is purely inseparable over $K$ exactly when its minimal polynomial has the formTo prove the nontrivial direction, repeatedly factor a zero formal derivative through $T\mapsto T^p$ until the remaining irreducible polynomial has nonzero derivative. Since $T^{p^r}-x^{p^r}$ has only the root $x$, that remaining separable polynomial must be linear.

The punctured plane ${\mathbb{C}}^{\ast }$ is a connected Riemann surface homeomorphic to a cylinder.

It is the complex plane with a point removed, a special punctured Riemann surface.

A path in a surface can be perturbed inside coordinate discs to avoid finitely many prescribed points.

With atoms $p_{x,y}$, finitary propositions can enforce that the true pairs form an injective partial function. If the codomain is infinite, they cannot enforce that every $x$ has a value: adjoining every $\neg p_{x_0,y}$ remains finitely satisfiable, so compactness produces a model omitting $x_0$.

For $V={\xi }^i{\partial }_{x_i}+\eta {\partial }_u$, the second prolongation adds coefficients for $u_i$ and $u_{ij}$ obtained by total differentiation:

A projective variety is a closed algebraic subvariety of some projective space.

An invertible linear map of the underlying vector space induces a projective linear transformation of ${\mathbb{P}}^n$; scalar multiples induce the same transformation.

The projective plane is the two-dimensional projective space ${\mathbb{P}}^2$.

A projective point is an equivalence class $[x_0:\cdots :x_n]$ of nonzero coordinate vectors under multiplication by a nonzero scalar.

For a field $k$,This nonvanishing obstructs affine covers of ${\mathbb{P}}_k^n$ having only $n$ members.

For $n>k\ge 0$, take an $(n-k)$-dimensional irreducible projective variety with one singular point and form its product with ${\mathbb{P}}^k$. The Segre embedding makes the product projective, and its singular locus is the singular point times ${\mathbb{P}}^k$, of dimension $k$. A quadric cone works in dimension at least two and a cuspidal cubic works in dimension one.

In the standard graded homogeneous coordinate ring $S=k[x_0,\dots ,x_n]$ of ${\mathbb{P}}^n$, the irrelevant ideal isIt vanishes only at the origin of the corresponding affine cone, which does not represent a projective point.

It is the polynomial-ring instance of the irrelevant ideal of a graded ring.

Lift the image of each basis element independently through the given surjection, then extend the chosen lifts linearly.