An injective holomorphic function cannot have an essential singularity: Casorati-Weierstrass theorem would give values from a small punctured neighbourhood inside the open image of a disjoint neighbourhood, contradicting injectivity. Apply this at infinity to an injective entire function. A removable singularity there would make the function constant by Liouville's theorem. A pole makes it a polynomial by the Cauchy coefficient formula; its derivative has no zero, so the fundamental theorem of algebra forces degree one.

For a bipartite pure state, entanglement entropy is the Von Neumann entropy of either reduced density matrix. When Alice makes a local measurement with pure conditional branches, Bob's reductions average to his original reduction. Concavity of Von Neumann entropy therefore gives ${\sum }_r{p}_{r}S({\rho }_{B,r})\le S({\rho }_B)$. When Bob measures, apply concavity to Alice's reductions. In each pure branch the two reduced entropies coincide by the Schmidt decomposition. Refining measurement outcomes to individual Kraus operators and iterating proves the inequality for arbitrary finite-round LOCC. Discarded local systems can be measured and their outcomes refined as well, so discarding does not evade the bound.

An entanglement monotone quantifies entanglement without increasing on average under local operations and classical communication. The average is over recorded measurement in quantum mechanics outcomes: individual successful branches can have more entanglement, provided their probabilities compensate for that increase. This distinction makes an entanglement monotone useful for bounding probabilistic state conversion. A pure-state entanglement monotone need only satisfy this inequality on pure input states and pure conditional branches, with local measurement outcomes refined to individual Kraus operators.

Entanglement dilution uses local operations and classical communication to prepare partially entangled states from maximally entangled states, such as Bell pairs. For many identical bipartite pure targets, the asymptotic cost in Bell pairs per target, allowing a vanishing approximation error, is their entanglement entropy in bits. The quantum typical subspace of the target's Schmidt coefficients has almost all its weight and dimension at most $2^{m(S+\delta )}$. A maximally entangled state of this dimension can be converted to the normalized truncated target by Nielsen's pure-state conversion theorem. The resulting state approaches the full target as the retained weight tends to one. Average monotonicity of pure-state entanglement entropy and entropy continuity give the converse lower bound. This is the reverse asymptotic task to entanglement concentration.

Entanglement distillation uses local operations and classical communication on many noisy entangled states to produce fewer high-fidelity Bell pairs. It is also called entanglement purification. The protocol may fail or abort; naming a purification step does not guarantee successful output for arbitrary inputs. If the output is an exact pure state of Bell pairs, every purification of a density operator factors between those pairs and the environment, so measuring each pair in the computational basis gives a shared uniform bit independent of an eavesdropper. Approximate privacy follows when the entire retained output is sufficiently close to that pure target. A separable quantum state cannot be distilled into a Bell pair by LOCC, because each local branch preserves separability.

A reaction energy barrier is the height of a specified peak relative to the starting state in an energy profile. State which energy is used and which direction is being considered. A schematic barrier is not automatically the empirical activation energy obtained from the temperature dependence of a rate constant.

A potential-energy profile plots the potential energy of the reacting system along a specified path from reactants through a transition state to reaction products. It is a particular kind of energy profile. For an elementary reaction, a suitable reaction coordinate parametrizes the path; a multi-step reaction mechanism needs the coordinates of its successive steps. See the IUPAC Gold Book: potential-energy profile.

For a normalized state in a unitary representation of the ordinary four-dimensional Super-Poincaré algebra, define $M_{ab}=⟨\{Q_a,Q_b\}⟩/2$ in a real Hermitian charge basis. Then $\Vert u\cdot Q\psi {\Vert }^2=u^{T}Mu$, so its annihilator is precisely the kernel of a linear map represented by the positive matrix $M$. In a suitable real basis its eigenvalues are the two eigenvalues of $⟨P^0⟩I+⟨\mathbf{P}⟩\cdot \mathbf{\sigma }$, each repeated twice. A nonzero kernel therefore has dimension two at nonzero null momentum expectation or four at zero expectation. Complex non-Hermitian lowering operators and algebras with domain-wall extensions are not covered by this argument.

A fixed-endpoint variation of a curve $\gamma :[a,b]\to S$ is a smooth family $\Gamma (s,t)$ with $\Gamma (0,t)=\gamma (t)$, $\Gamma (s,a)=\gamma (a)$ and $\Gamma (s,b)=\gamma (b)$. Its variation vector field $V(t)={\partial }_s\Gamma (0,t)$ vanishes at the endpoints. Consequently integration by parts in the first variation of energy of a curve has no boundary contribution, and stationarity is characterized by the geodesic equation.

A field configuration is critical for an energy functional $E$ if its first variation vanishes for every admissible infinitesimal field variation. For $E[\varphi ]=\int (\frac{1}{2}|\nabla \varphi {|}^2+U(\varphi ))$, integration by parts against compactly supported variations gives $\delta E=\int (-\Delta \varphi +U^{\prime }(\varphi ))\delta \varphi$, so stationary field configurations satisfy the Euler-Lagrange equation $\Delta \varphi =U^{\prime }(\varphi )$. A constrained target instead allows only tangent variations, changing that field equation. Criticality is stationarity, not necessarily energetic stability or a local minimum.

A spatially localized electronic level produced by a defect or foreign atom in a crystalline potential. Such levels can lie in a host band gap, where no extended host state has the same energy, while impurity resonances can also occur inside an energy band. Dilute localized levels are nearly independent; increasing overlap can broaden them into an impurity band. The location and existence of a bound level depend on the impurity potential.

For independent event times with survivor function $S(t)=\mathbb{P}(T>t)$, an exact event at $t$ contributes mass $S(t-)-S(t)$, a right-censored observation at $c$ contributes $S(c)$, and an interval-censored observation $l<T\le u$ contributes $S(l)-S(u)$. Multiplying these factors gives the event-time likelihood when the observation mechanism is noninformative. Maximizing over nonincreasing right-continuous survivor functions gives a nonparametric maximum-likelihood estimator rather than requiring a parametric density.

If $u\in H^1(V)$ and $\Delta u=f\in L^2(V)$ weakly, then $u\in H_{\mathrm{loc}}^2(V)$. Test the weak equation with $-{\delta }_{-h}({\eta }^2{\delta }_{h}u)$, move the first difference quotient onto $Du$, and bound the remaining quotient on the forcing side by the first derivative of ${\eta }^2{\delta }_{h}u$. Young inequality absorbs the localized second derivatives. Uniform bounds in $h$, together with the Sobolev characterization by bounded difference quotients, give the assertion without differentiating $f$. The constant depends only on dimension and the positive distance from $U$ to the boundary of $V$.

// Target: geometry-and-topology.bigb

Choose the holomorphic square root equal to one at zero on the unit disc. The resulting elliptic integral maps that disc conformally onto the square with vertices $\pm B,\pm iB$, where $B={\int }_0^{1}dt/\sqrt{1-t^4}$. For the inverse Cayley transform between the half-plane and disk $h(z)=i(1+z)/(1-z)$, the square of the derivative of $F_1\circ h$, with $F_1^{\prime }(w)=1/\sqrt{w(1-w^2)}$, is $2i/(1-z^4)$. Thus the two square maps differ by an affine change of coordinate.

For an ellipse with semiaxes $1,x$, the perimeter integral has integrand $\sqrt{1-t{\cos }^2\theta }$. The uniformly convergent binomial series near $t=0$ permits termwise integration. Integrals of ${\cos }^2\theta$ and ${\cos }^4\theta$ over a full period give the displayed first corrections. The same coefficients follow from the Frobenius method for the transformed perimeter equation.

The semi-major and semi-minor axes are the larger and smaller half-axis lengths of an ellipse. In principal-axis coordinates the ellipse has equation $x^2/a^2+y^2/b^2=1$ with $a\ge b>0$. The same terminology is used for a hyperbola $x^2/a^2-y^2/b^2=1$, where $a$ is half its transverse axis and $b$ half its conjugate axis. Orbital semiaxes apply these geometric quantities to conic sections.

In the space of infinite subsets of the natural numbers, the families $C_n=\{X:n\notin X\}$ are clopen sets in the ordinary topology on infinite subsets and thus in the finer Ellentuck topology. Their union is $[\mathbb{N}{]}^{\omega }\setminus \{\mathbb{N}\}$. Every basic Ellentuck topology neighborhood of $\mathbb{N}$ contains $\mathbb{N}\setminus \{n\}$ for some sufficiently large $n$, so this union is not a closed set.

For an infinite coinfinite set $A\subseteq \mathbb{N}$, let $E_A=[A{]}^{\omega }$ in the space of infinite subsets of the natural numbers. This cone is a nonempty clopen set in the Ellentuck topology, because it is the basic neighbourhood $[\varnothing ,A]$, and any $X\notin E_A$ has a finite initial segment containing a point outside $A$ and hence a basic neighbourhood disjoint from $E_A$.

The cone is a closed set with empty interior in the ordinary topology on infinite subsets: membership of a point outside $A$ witnesses its complement as an open set, while every ordinary finite-stem neighbourhood can be extended by an arbitrarily large point outside $A$. Thus $E_A$ is an ordinary nowhere dense set, but it is neither ordinary open nor Ellentuck nowhere dense. It provides both distinctions between these two topologies.

Every Borel set for the Ellentuck topology is completely Ramsey. First, an acceptance-and-rejection fusion makes every open set homogeneous after thinning the infinite reservoir while keeping its finite stem. Every nowhere-dense set is consequently completely Ramsey-null, and a fusion handling all subsets of each finite selected prefix shows that countable unions of such null sets remain null. Finally, sets differing from an open set by a meagre set form a sigma-algebra containing the open sets. Thus every Borel set has the required stem-preserving homogeneous refinement.

For zonal averages, $u^{\prime }=-{\psi }_y^{\prime }$, $v^{\prime }={\psi }_x^{\prime }$, and $Q^{\prime }={\nabla }_H^2{\psi }^{\prime }+{\partial }_z(S{\psi }_z^{\prime })$, with $S=f_0^2/N^2$ depending only on height, define $F=-\overline{u^{\prime }{v}^{\prime }}$ and $G=S\overline{{\psi }_x^{\prime }{\psi }_z^{\prime }}$. Integrating zonal derivatives by parts gives $\overline{v^{\prime }{Q}^{\prime }}={\partial }_y\overline{{\psi }_x^{\prime }{\psi }_y^{\prime }}+{\partial }_z(S\overline{{\psi }_x^{\prime }{\psi }_z^{\prime }})$. This identity is exact within quasi-geostrophic potential vorticity, without assuming small disturbance amplitudes. Periodicity or appropriate vanishing zonal boundary terms is essential.