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The reduced positive definite binary quadratic form bounds $|b|\le a\le c$ and $b^2-4ac=-39$ imply $1\le a\le \sqrt{13}$. Enumerating $a=1,2,3$ and using the nonnegative boundary sign convention gives the four triples $(a,b,c)=(1,1,10),(2,1,5),(2,-1,5),(3,3,4)$. Thus the number of reduced forms is four.

Local equilibrium associates the outgoing heat flux $q$ with outgoing entropy flux $q/\theta$ at positive temperature $\theta$. Its divergence appears in the Lagrangian entropy inequality, with the appropriate nominal flux in reference coordinates.

The pinching map is self-adjoint for the trace pairing, and $\log \mathcal{P}(\rho )$ is block diagonal. Thus $\mathrm{Tr}\rho \log \mathcal{P}(\rho )=\mathrm{Tr}\mathcal{P}(\rho )\log \mathcal{P}(\rho )$. The kernel of $\mathcal{P}(\rho )$ is contained in that of $\rho$, so the relative entropy is finite on its support. Expanding the quantum relative entropy gives the displayed identity. Its nonnegativity proves entropy increase, with equality exactly when the original state is already block diagonal.

Comoving entropy is the entropy density multiplied by the physical volume factor $a^3$ for a fixed comoving region in an expanding homogeneous universe. In adiabatic expansion it is conserved. Separate decoupled radiation sectors conserve their own comoving entropy, so disappearance of massive species can heat the still-equilibrated sector relative to a decoupled species.

For positive depth and reduced gravity, the characteristic speeds are $u,u\pm c$. Along the material characteristic, $D{g}^{\prime }=-2g^{\prime }{w}_e/h$. Along $D_{\pm }={\partial }_t+(u\pm c){\partial }_x$, the left-eigenvector compatibility equations areIf $w_e=0$ and $g^{\prime }$ is spatially constant, these reduce to the Riemann invariants $u\pm 4c$. Variable buoyancy or entrainment prevents that simplification.

The Picard theorem has two forms concerning the image of a holomorphic function. The Little Picard theorem permits at most one omitted value for a nonconstant entire function. The Great Picard theorem is local: near an isolated essential singularity, every complex value except possibly one occurs infinitely often. The complex exponential function $e^z$ and the function $e^{1/z}$ show that the possible exception is necessary.

Émile Picard presented the research in 1879 and published proofs in his 1880 memoir. Its opening footnote lists communications to the Académie des Sciences on 28 April, 19 May, and 20 October 1879; sections 3 and 15 contain the global and local results respectively. ^[ref]

An entire function is of exponential type if constants $A,B>0$ give the displayed bound everywhere. A genus-zero canonical product with ${\sum }_n|a_n{|}^{-1}<\infty$ obeys $|{\prod }_n(1-z/a_n)|\le \exp (|z|{\sum }_n|a_n{|}^{-1})$, since $1+u\le e^u$ for $u\ge 0$.

For a nonconstant entire function, put $M_f(R)={\max }_{|s|\le R}|f(s)|$. Its order measures the exponent in its exponential growth. Bounds $\log M_f(R)=O(R\log R)$ imply order at most one; a matching lower bound along the positive real axis proves order exactly one. This distinction is stronger than mere entire analyticity and controls the genus in Hadamard factorization.

// Target: analytic-number-theory.bigb

An exponential polynomial is a finite sum of polynomial multiples of exponentials. For distinct exponents, its terms are linearly independent over the polynomial ring: to isolate one term, apply the product of operators $(D-{\lambda }_j{)}^{\mathrm{deg}{P}_j+1}$ for all the other terms. On the remaining polynomial each operator becomes $D+{\lambda }_0-{\lambda }_j$, which is injective because its nonzero constant preserves the leading coefficient. A nonzero exponential polynomial is an entire function of exponential type. Its maximum-modulus logarithm is $O(R+\log R)$, and Jensen's formula gives at most $O(R)$ zeros, counted with multiplicity, in a disk of radius $R$.

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.