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With an intercept present, a categorical predictor having $m$ represented levels adds $m-1$ linearly independent indicator columns and therefore $m-1$ model degrees of freedom.

For $w=-2/3$, a quintessence-dominated scale factor grows quadratically and $tH$ approaches two.

For $w=-2/3$, zero cosmological constant, and negative curvature, $a(t)=\sqrt{\beta }t+\gamma t^2/4$ with density and curvature parameters $\gamma ,\beta >0$.

In a Gromov-hyperbolic geodesic metric space, every quasigeodesic segment stays within a uniformly bounded Hausdorff distance of a geodesic segment with the same endpoints. The bound depends only on the hyperbolicity and quasigeodesic constants.

The bounded-distance conclusion is a property of a Gromov-hyperbolic metric space, under the geodesic and quasigeodesic hypotheses stated here.

When the shared resource is $|{\Psi }^{+}⟩$, Bell outcomes ${\Phi }^{+},{\Phi }^{-},{\Psi }^{+},{\Psi }^{-}$ leave the receiver with $X|\alpha ⟩,XZ|\alpha ⟩,|\alpha ⟩,Z|\alpha ⟩$, respectively, up to global phases.

A deterministic finite-dimensional universal programmable quantum gate cannot implement every unitary exactly: programs for physically distinct unitaries would have to be mutually orthogonal, but a finite-dimensional program register contains only finitely many mutually orthogonal states.

This is a restriction on programmable quantum circuits and their unitary operators, rather than a statement about the existence of classical program descriptions.

The quantum conditional entropy has the relative-entropy representationwhere the minimum ranges over density operators on $B$.

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 a self-adjoint operator in a fermionic algebra satisfying $\{{\gamma }_i,{\gamma }_j\}=2{\delta }_{ij}I$. A single complex fermionic mode $c$ can be written using two such operators:The canonical anticommutation relations verify their Hermiticity and normalization, and the occupation is $c^{†}c=(I+i{\gamma }_1{\gamma }_2)/2$.

This algebraic rewriting is always possible for a complex mode. A spatially separated Majorana zero mode requires additional physical properties, including localization and near-zero commutator with the Hamiltonian. The existence of two Majorana operators in a formula alone therefore does not establish protected boundary states or a topological phase.

A Bogoliubov--de Gennes Hamiltonian is the matrix representation of a quadratic fermion Hamiltonian in a Nambu spinor combining annihilation operators and creation operators. It treats ordinary one-particle motion and fermionic pairing in one eigenvalue problem.

For a column $c$ of $N$ fermionic annihilation operators, let $h=h^{†}$ and $\Delta =-{\Delta }^T$. With $\Psi =(c,c^{†}{}^T{)}^T$, defineThe canonical anticommutation relations give the identityThe factor $1/2$ compensates for the full particle-hole doubling; the trace constant follows when the hole block is normal-ordered.

A two-by-two opposite-spin pairing blockhas eigenvalues $E_{\pm }=\pm \sqrt{{\xi }^2+|\Delta {|}^2}$, as follows from its characteristic polynomial. Pairing opens a spectral gap at $\xi =0$. Negative and positive partners in a fully doubled representation are related by Particle-hole symmetry of a Bogoliubov--de Gennes Hamiltonian; they are not independent species of physical quasiparticles. A reduced block and a full doubled basis therefore require consistent counting conventions.

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.

Let $X\ne \varnothing$ and let $Y$ be infinite, with an injection $X\to Y$ available. In the propositional logic with variables $p_{x,y}$, no finitary theory has as its models exactly all total injections $X\to Y$, represented by their true pairs. Fix $x_0\in X$ and suppose $T$ axiomatizes that entire class. Adjoin $\neg p_{x_0,y}$ for every $y\in Y$.

Every finite subset of these extra requirements is satisfiable together with $T$: start with any total injection and permute $Y$ so that the value at $x_0$ avoids the finitely many forbidden values. The propositional compactness theorem then supplies a model of $T$ in which $x_0$ has no value, a contradiction.

The requirement that all total injections be models is essential. A theory can certainly force totality by specifying a particular injection, with one axiom $p_{x,f(x)}$ for each $x$. What fails is a finitary axiomatization of the full unrestricted class. Conditions saying at most one value per input and at most one input per value are expressible by pairwise prohibitions.

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.