With thermal diffusivity $\xi$, a Fourier mode of angular wavenumber $k$ decays at rate $\beta =\xi k^2$. To avoid confusing this $k$ with thermal conductivity, use $\alpha =\xi$ and angular wavenumber $K$ below: the same rate is $\beta =\alpha K^2$, with units ${\mathrm{s}}^{-1}$.

Consider an insulated bar $0\le x\le L$ with constant thermal diffusivity, no sources and no heat flux through either end or the sides. An original initial temperature profile and its exact evolution areThe cosine has zero temperature gradient at both ends, satisfying the insulated Neumann boundary conditions. Its spatial mean is zero, so conservation of energy leaves the mean temperature $\overline{T}$ unchanged while the nonuniform part disappears. A short derivation substitutes $T=\overline{T}+A(t)\cos (K_{n}x)$ into the heat equation: the second spatial derivative supplies $-K_n^2$, hence $A^{\prime }=-\alpha K_n^{2}A$. The exponential decay time is ${\tau }_n=1/(\alpha K_n^2)$; doubling angular wavenumber quarters this time. If cycles-per-length spatial frequency $f_x$ is used instead, $K=2\pi f_x$ and the rate is $4{\pi }^2\alpha f_x^2$.

For the two model materials in thermal diffusivity and $L=0.010\mathrm{m}$, ${\tau }_{1,A}\approx 5.07\mathrm{s}$ and ${\tau }_{1,B}\approx 20.3\mathrm{s}$. After $10\mathrm{s}$, the remaining amplitude fractions are approximately $0.139$ and $0.610$. A narrow hot-and-cold pattern disappears faster than a broad one because its temperature curvature, and hence the imbalance in conductive heat flux, is larger.

Write ${\epsilon }_i(b)$ and ${\phi }_i(b)$ for the numbers of raising and lowering steps of color $i$ available from a vertex of a crystal basis. One convention for the tensor product of crystals isThe raising operator uses the first factor for ${\phi }_i(a)\ge {\epsilon }_i(b)$ and the second otherwise. The different strictness makes raising and lowering inverse along every edge. An edge is absent if the chosen Kashiwara operator vanishes. Stating the convention is essential when identifying the highest-weight vectors as ordered tensor vertices.

For the defining real representation of $SO(d)$, the symmetric square splits as a scalar trace plus symmetric trace-free tensors: ${\mathrm{Sym}}^2({\mathbb{R}}^d)=\mathbb{R}\oplus {\mathrm{Sym}}_0^2({\mathbb{R}}^d)$. The trace projection is $S\mapsto S-(\mathrm{tr}S)I/d$, so the second summand has dimension $d(d+1)/2-1$.

For $d\ge 3$ the trace-free summand is an irreducible real group representation. To see this, diagonalize a nonzero symmetric trace-free matrix in a nonzero invariant subspace. Two diagonal entries differ. Infinitesimal rotation in their plane yields a nonzero symmetric off-diagonal matrix; rotations carry it to every coordinate pair, and a rotation through $\pi /4$ produces a difference of two diagonal entries. These off-diagonal matrices and diagonal differences span all symmetric trace-free matrices, so the invariant subspace is the whole summand. For $d=24$, its dimension is 299, the third-level nontrivial multiplet of the lowest levels of a fully transverse ND bosonic string.

For a smooth $n$-dimensional variety $X$ over a field, its canonical line bundle is ${\omega }_X={\bigwedge }^n{\Omega }_{X/k}^1$. It is the bundle of top-degree algebraic differential forms. A nonzero rational section defines a canonical divisor of a smooth variety. This algebraic definition works in arbitrary characteristic and is distinct from restricting the analytic canonical-bundle definition to complex manifolds.

For an injective ring homomorphism between nonzero integral domains $A\to B$, the scheme-theoretic fibre over the generic point of $\mathrm{Spec}A$ is $\mathrm{Spec}(S^{-1}B)$ with $S=A\setminus \{0\}$. Localization at these nonzero elements is a nonzero integral domain, so the generic fibre is a nonempty integral scheme. A surjective morphism between these affine schemes necessarily gives an injective ring map: a prime above $(0)$ contains its kernel.

For a light scalar in the Bunch-Davies vacuum with metric perturbations neglected, the mode $e^{-ik\tau }(1-i/(k\tau ))/\sqrt{2k}$ gives the displayed equal-time spectrum. The dimensionless power tends to $H^2/(4{\pi }^2)$ on superhorizon scales. The exact massless infinite-volume correlation has an infrared divergence; a finite-volume or finite-duration infrared regulator, and smearing for coincident products, makes the correlators well defined.

In an irreducible root system with two root lengths, a short root has the smaller length with respect to a Weyl group invariant inner product.

In an irreducible root system with two root lengths, a long root has the larger length with respect to a Weyl group invariant inner product.

An automorphism of a root system $\Phi \subset E$ is an orthogonal transformation of $E$ mapping $\Phi$ onto itself. The Weyl group is a subgroup of this automorphism group; additional automorphisms can act nontrivially on a Dynkin diagram while preserving a root basis.

The $A_2$ root system has equal-length simple roots ${\alpha }_1,{\alpha }_2$ with angle $2\pi /3$, and roots $\pm {\alpha }_1$, $\pm {\alpha }_2$, $\pm ({\alpha }_1+{\alpha }_2)$. Its Cartan matrix is $\left(\begin{array}{cc}2& -1\\ -1& 2\end{array}\right)$. The corresponding complex simple Lie algebra is ${\mathfrak{sl}}_3(\mathbb{C})$, of dimension eight.

A reduced root system of rank one is $\Phi =\{\alpha ,-\alpha \}$. Its fundamental weight is $\alpha /2$, and its Weyl group has two elements. Its ordinary Dynkin diagram has one node; adjoining ${\alpha }_0=-\alpha$ gives affine Cartan matrix $\left(\begin{array}{cc}2& -2\\ -2& 2\end{array}\right)$, drawn as a double bond between equal-length roots.

A root is an individual nonzero vector in a root system. For a complex semisimple Lie algebra, it is a nonzero weight of the Adjoint representation restricted to a Cartan subalgebra; its root space consists of vectors $x$ satisfying $[h,x]=\alpha (h)x$ for all $h$ in that subalgebra.

A root vector is a nonzero element of a root space ${\mathfrak{g}}_{\alpha }$ in the root-space decomposition of a semisimple Lie algebra.

The holonomy representation is the natural action of the Riemannian holonomy group on $T_{p}M$. It induces representations on cotangent spaces and tensor powers. Irreducibility means that no nonzero proper real linear subspace of $T_{p}M$ is invariant.

For the retention parameter $p$, the Holevo capacity isEvery output has Von Neumann entropy at least the value on a pure input, $h_2((1+p)/2)$. Two equiprobable orthogonal pure inputs attain this minimum entropy individually and have maximally mixed average output, attaining the bound. Additivity of Holevo capacity makes the unassisted classical capacity of a quantum channel equal to this value. King's depolarizing-channel capacity paper proves additivity with an arbitrary other channel.

A quadratic differential with holomorphic local coefficients. If it is nonzero, its zeros have total order $4g-4$ on a compact genus-$g$ Riemann surface. Away from its zeros, a nonzero differential has flat coordinates whose changes of coordinate have the form $w\mapsto \pm w+c$.

A surface with local Euclidean charts whose changes of coordinate are $w\mapsto \pm w+b$, outside finitely many cone points. A nonzero holomorphic quadratic differential defines such a surface; a zero of order $m$ has cone angle $(m+2)\pi$.

For $q=\varphi (z)d{z}^2$, the length element is $|\varphi (z){|}^{1/2}|dz|$, locally Euclidean in flat coordinates. Zeros give cone points, not punctures. Its infimal length in a homotopy class need not determine the hyperbolic length of that class.

A fundamental wave and its second harmonic satisfy phase matching for a quadratic wave interaction when $\omega (2k)=2\omega (k)$. The square of the fundamental drives the second harmonic, while its product with the conjugate second harmonic drives the fundamental. Projecting these terms onto the resonant modes gives coupled amplitude equations rather than a uniformly valid correction at fixed amplitude.