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A single-category model represents local ice by one thickness $h$ and one surface temperature $T_s$. With basal freezing temperature $T_m$, atmospheric temperature $T_A$, thermal conductivity $k$ and atmospheric heat transfer coefficient ${\lambda }_A$, quasistatic thermal conduction giveswhile $T_s<T_m$. If $F_O$ is upward ocean heat flux, the Stefan condition gives $\rho L\dot{h}={\lambda }_A(T_m-T_A)/(1+{\lambda }_{A}h/k)-F_O$. During surface melting, $T_s=T_m$ and $\rho L\dot{h}=-{\lambda }_A(T_A-T_m)-F_O$ when $T_A\ge T_m$. Thickness is constrained to be nonnegative, and the open-water energy balance applies after complete melt.

The integral $I(x)={\int }_0^{\infty }{e}^{xt-t^3/3}dt$ is a rescaled Scorer Hi function. It has the three useful limitsFor the negative-argument limit, scale $t=s/M$ and use the dominated convergence theorem. At zero, substitute $s=t^3/3$ and use the Gamma integral. For the positive-argument limit, the exponent has its maximum at $t=\sqrt{M}$, with second derivative $-2\sqrt{M}$; Laplace's method gives the displayed factor. These estimates describe a cubic endpoint-to-saddle transition.

A quantum source-compression scheme has encoding and decoding quantum channels with compressed space ${\mathcal{K}}_n$ and rate ${\mathrm{limsup}}_n{n}^{-1}{\log }_2\dim {\mathcal{K}}_n$. For a pure-state source ensemble, average reliability means that the mean squared quantum fidelity after decoding tends to one. A stronger coherent criterion is $F_e({\pi }^{\otimes n},{\mathcal{D}}_n\circ {\mathcal{E}}_n)\to 1$, where entanglement fidelity also tests a purification reference. Encoding the quantum typical subspace achieves both criteria above $S(\pi )$. Schumacher's original quantum coding paper establishes quantum source coding.

For a density operator $\pi ={\sum }_i{q}_i|{\varphi }_i⟩⟨{\varphi }_i|$, the quantum typical subspace is spanned by product eigenvectors of ${\pi }^{\otimes n}$ whose eigenvalues $q_{i_1}\cdots q_{i_n}$ satisfyIt applies the classical weakly typical sequence definition to the spectrum. If $P_{\epsilon }^{(n)}$ is its orthogonal projection, thenThe first statement follows from the typical-set cardinality bounds, and the second from the weak law of large numbers. This subspace holds nearly all the source probability while using exponentially fewer dimensions than the whole space when $S(\pi )<{\log }_{2}d$.

With relativistically normalized external states, a two-particle initial state has differential cross-section $d\sigma =|\mathcal{M}{|}^{2}d{\Phi }_n/\mathcal{F}$, where $d{\Phi }_n$ is the Lorentz-invariant phase-space measure and $\mathcal{F}$ is the invariant flux factor. For identical unobserved final particles, divide the full labeled phase-space integral by their permutation multiplicity.

Crossing symmetry relates scattering amplitudes by analytic continuation exchanging incoming and outgoing particles. A symmetric four-point scalar amplitude has equivalent descriptions under permutations of the Mandelstam variables.

Near a one-particle pole, a scattering amplitude factors into amplitudes coupling each side to the intermediate state, summed over its physical polarizations. The pole position gives its squared mass, while its residue encodes its spin and couplings.

A charged scalar field is a complex scalar field transforming nontrivially under a gauge symmetry. For an electromagnetic potential $A_a$, its gauge-covariant derivative is ${\mathcal{D}}_a={\nabla }_a-iq{A}_a$.

A scalar field with no spacetime coordinates reduces its path integral to an ordinary integral. It preserves Wick contractions and Feynman-diagram symmetry factors, making it a useful model of perturbative quantum field theory.

A scalar field transforming in the Adjoint representation of a Lie algebra transforms by a Lie bracket with the infinitesimal gauge parameter.

Let $K\subset \mathbb{C}$ be compact and permit polynomial terms and finite poles in $S\subset \mathbb{C}\setminus K$, meeting every bounded complementary component. In the uniform closure $A$ of these rational functions, the coordinate $u(z)=z$ has spectrum exactly $K$. Indeed the set of $\lambda \notin K$ with $(u-\lambda {)}^{-1}\in A$ is relatively open by invertibility and relatively closed by uniform continuity of the scalar resolvent; it meets each component by a prescribed pole, or by a Neumann expansion at infinity. It is therefore all of $\mathbb{C}\setminus K$. Holomorphic functional calculus then places every function holomorphic near $K$ in $A$, proving prescribed-pole Runge approximation theorem.

The classical four-stage explicit Runge-Kutta method usesIt satisfies the fourth-order conditions for a Runge-Kutta method. Its polynomial stability function excludes A-stability, while $|R(iq){|}^2=1-q^6/72+q^8/576$ gives the imaginary-axis stability interval $|q|\le 2\sqrt{2}$.

For the regular $C^2$ ruled surface $\sigma (x,y)=\mathbf{a}(x)+y\mathbf{b}(x)$, put $W=|({\mathbf{a}}^{\prime }+y{\mathbf{b}}^{\prime })\times \mathbf{b}|>0$. The second-form coefficient in the ruling direction is zero, and the mixed coefficient is ${\mathbf{a}}^{\prime }\cdot (\mathbf{b}\times {\mathbf{b}}^{\prime })/W$. The determinant formula for Gaussian curvature therefore gives the displayed nonpositive expression. Its vanishing is equivalent to the zero scalar triple product, only at regular points.

The one-qubit rotation gate is $R_z(\theta )=\mathrm{diag}(e^{-i\theta /2},e^{i\theta /2})$. It implements the two relative phases needed in simulation of a computable diagonal Hamiltonian.

The one-qubit rotation gatesatisfies $R_y(2\theta )|0⟩=\cos \theta |0⟩+\sin \theta |1⟩$ and $R_y(\alpha )R_y(\beta )=R_y(\alpha +\beta )$. The latter identity enables a binary-angle implementation of a quantum variable rotation.

For a label set $D$, define $\mathcal{T}(D)$ inductively: choose a root label $d\in D$ and a finite repetition-free sequence of rooted trees in $\mathcal{T}(D\setminus \{d\})$ as its children. Each object is a finite ordered rooted tree. Labels are distinct along each root-to-leaf path, and child rooted trees at a vertex are distinct as whole rooted trees. Labels may repeat across different branches; children need not have distinct root labels. For a finite pool of size $m$, the exact number $t_m$ of rooted trees satisfiesThis follows by choosing the root and then an ordered repetition-free list from the finite pool of smaller rooted trees.

For zero initial velocity, body masses $kM,(1-k)M$ and fuel masses $k{m}_0,(1-k)m_0$, the rocket equation gives the displayed final second-stage speed after burning the first-stage fuel, discarding its body without impulse, and burning the second-stage fuel. It assumes $M>0$ and $0\le k<1$. The speed exceeds the single-stage value when $0<k<1$ and fuel is positive. Its limit as $k\to 1^{-}$ concerns a surviving second stage of vanishing mass; at $k=1$ no such stage exists.

A polyhedral uncertainty set is specified by finitely many linear inequalities. The displayed centered example is the inverse image of a box under a linear map. It is nonempty because it contains $r_0$, and contains every line $r_0+td$ with $d\in \ker P$. Consequently it need not be bounded when $P$ is rank deficient. Its supremum norm constraint is equivalent to $-1\le P(r-r_0)\le 1$.

Worst-case linear revenue over a polyhedral uncertainty set is optimized by the linear programHere ${\Delta }_n$ is the probability simplex. Finite decisions are exactly its intersection with $\mathrm{range}{P}^T$. If this intersection is empty, all decisions have worst-case value $-\infty$ and the displayed linear program is infeasible; its supremum over the empty feasible set is also $-\infty$. Full column rank of $P$ suffices to make all simplex decisions finite.