The two minor symmetries of the elastic stiffness tensor make it a map between six-dimensional spaces of symmetric strain and stress. An elastic energy gives major symmetry, so the matrix of this map is symmetric and has 21 independent entries. For a symmetry operation acting on strains with parity spaces of dimensions $m$ and $6-m$, invariance eliminates the mixed block and leaves $m(m+1)/2+(6-m)(7-m)/2$ independent coefficients.

An elastic plate coupled to one compressible fluid half-space has an outgoing Fourier transform kernel $D$, where ${\gamma }^2=k^2-{\omega }^2/c_0^2$. A harmonic line force $F_0$ produces displacement transform $F_0\gamma /D$. Propagating acoustic components have $\gamma =-i\sqrt{{\omega }^2/c_0^2-k^2}$, whereas guided components decay normally to the plate. The outgoing prescription separates acoustic energy flux from guided-wave energy transport.

For a period lattice $\Lambda \subset \mathbb{C}$ and an integer $k>2$, this sum converges absolutely. Pairing $\omega$ and $-\omega$ makes it zero when $k$ is odd. For even weights it supplies the Laurent coefficients of the Weierstrass elliptic function. Scaling the period lattice gives $G_k(a\Lambda )=a^{-k}{G}_k(\Lambda )$.

Use $K_{abc}=e_a{}^{\mu }(\omega -\omega (e){)}_{\mu bc}$ and $T_{abc}=K_{bac}-K_{cab}$. For the symmetrized Dirac action $-e\bar{\chi }{\gamma }^{\mu }{\overleftrightarrow{\nabla }}_{\mu }\chi /2-em\bar{\chi }\chi$, varying the spin connection gives $-e\bar{\chi }{\gamma }^{\mu ab}\chi \delta {\omega }_{\mu ab}/4$. Only totally antisymmetric contorsion is sourced. Its action is $-e{K}^2/(2{\kappa }^2)-e{K}_{abc}\bar{\chi }{\gamma }^{abc}\chi /4$, hence the stated solution. Substitution generates $e{\kappa }^2(\bar{\chi }{\gamma }^{abc}\chi )(\bar{\chi }{\gamma }_{abc}\chi )/32$. The torsion is an auxiliary field, not an independent propagating degree of freedom.

Solve the algebraic connection equation for $\hat{\omega }(e,\chi )$, but perform the remaining variations as though the connection were independent. The chain-rule contribution $(\delta S/\delta \omega )\delta \hat{\omega }$ vanishes on that solution. The resulting equations are those of the connection-eliminated second-order action. This is especially useful for local supersymmetry calculations, where the gravitino-induced contorsion is quadratic in fermions and the omitted induced connection variation would otherwise be lengthy.

Given a diagram of algebras for a monad, take any chosen limit $L$ of its underlying diagram. The displayed equations uniquely determine $a:TL\to L$, because their right-hand sides form a cone. Joint monicity of the limit projections proves $a{\eta }_L=1$ and $aT(a)=a{\mu }_L$. The same projection argument proves that every induced mediating map preserves the algebra action. Thus the forgetful functor strictly creates this limit with the prescribed underlying cone, without assuming the monad preserves limits.

For algebra maps $f,g:(A,a)\rightrightarrows (B,b)$ whose underlying maps have a split coequalizer $q:B\to Q$, every functor preserves the split diagram, including $T$ and $T^2$. Since $qb$ equalizes $Tf,Tg$, it factors uniquely to the displayed action $c:TQ\to Q$. Cancellation of the epimorphisms $q,Tq,T^{2}q$ proves the unit and multiplication laws. Any algebra-valued map equalizing the pair factors through $q$, and cancelling $Tq$ proves that its factor preserves the action. This establishes creation of that coequalizer.

An eigenvalue collision occurs when distinct branches of eigenvalues coincide in a parameter-dependent family of matrices or linear operators. The resulting repeated eigenvalue need not retain a full eigenbasis; its geometric and algebraic multiplicities must be compared to decide diagonalizability.

When velocity consists of impulses $U_1$ spaced by $T$, the periodic corrector jumps by $U_1$ and decays between kicks. With $a=\kappa m^2$, $g(t)=U_1{e}^{-at}/(1-e^{-aT})$ for $0<t<T$. Integrating $g^2$ in the effective diffusivity of a periodic sinusoidal shear formula yields the displayed result. For $aT\gg 1$, independent transverse phases give longitudinal variance $U_1^2/2$ per kick and enhancement $U_1^2/(4T)$. For $aT\ll 1$, the enhancement is $U_1^2/(2\kappa m^2{T}^2)$ to leading order, the steady-shear result for average speed $U_1/T$. The zero-diffusivity limit does not commute with the long-time limit: without transverse diffusion the displacement is ballistic.

A relative effective divisor of degree $m$ on a smooth projective curve is an effective Cartier divisor in $C\times T$ that is finite and flat of degree $m$ over $T$. Its associated line bundle has degree $m$ on every fibre. The symmetric product of a curve $C^{(m)}$ represents these families and carries their universal divisor. Locally on a smooth coordinate chart, the divisor is described by a monic polynomial, whose coefficients are elementary symmetric functions of its points; this description includes collisions and works in positive characteristic.

A proper three-color edge coloring of a cubic graph. Label its colors by the nonzero elements of ${\mathbb{F}}_2^2$: the incident colors at each vertex then sum to zero. On a planar map without bridges in a graph, integrating these labels on the dual graph gives a four-color face coloring. On a torus, a nonzero period can obstruct this integration.

The chromatic index $\chi \prime (G)$, also called the edge chromatic number, is the least number of labels in a proper edge colouring of a finite undirected graph without self-loops. Equivalently, it is the smallest number of matchings in a graph whose set partition covers all the edges. We take $\chi \prime (G)=0$ when there are no edges. For an arbitrary edge colouring, one colour always suffices whenever an edge exists, so properness is essential to this definition. The chromatic number $\chi (G)$ instead colours vertices.^[ref]

To establish an exact chromatic index, supply both a proper edge colouring and a reason that fewer colours cannot work. Every edge at a fixed vertex must have its own colour, giving the lower bound $\chi \prime (G)\ge \Delta (G)$, where $\Delta (G)$ is the maximum degree of a graph. The bound need not be attained: it counts a single resource's workload, while conflicts can also accumulate around a graph cycle.

For the triangle in a graph $K_3$ with edges $ab,bc,ca$, every pair of edges shares a vertex, so all three require different labels. The assignment $c(ab)=1,c(bc)=2,c(ca)=3$ works, proving $\chi \prime (K_3)=3$ although $\Delta (K_3)=2$. In its line graph, those three edges become a three-vertex clique. Treating this as three pair-resource jobs makes the obstruction tangible: each resource has only two jobs, but every pair of jobs conflicts, so no slot can contain two jobs.

For the square cycle graph $C_4$ with edges $ab,bc,cd,da$, use $c(ab)=c(cd)=1$ and $c(bc)=c(da)=2$. Each colour class is a matching in a graph, and each vertex has one edge of each colour. Since $\Delta (C_4)=2$, this proves $\chi \prime (C_4)=2$. More generally, alternation closes on any even cycle graph $C_{2m}$, $m\ge 2$. On an odd cycle $C_{2m+1}$, $m\ge 1$, two-colour alternation would make the last and first edges share a colour at their common vertex; a third colour on the last edge suffices. Thus even cycle graphs have chromatic index two and odd cycle graphs have chromatic index three. These exact examples show why maximum degree of a graph alone does not determine the answer.

Each edge has a list of allowed colors; a list edge coloring chooses an allowed color on each edge, giving different colors to incident edges. The list-chromatic index is the uniform list-size threshold guaranteeing success.

A proper edge colouring gives different labels to any two distinct edges sharing a vertex. We work with finite undirected graphs without self-loops; distinct parallel edges in a multigraph must also have different colours. For a simple graph, whose edges are two-element sets, the condition isEvery colour class $M_i=\{e\in E(G):c(e)=i\}$ is a matching in a graph. Conversely, a set partition of the edges into matchings in a graph supplies a proper edge colouring. Thus minimizing colours means grouping all the edges into as few matchings in a graph as possible.

The line graph makes the relationship with proper vertex colouring explicit: replace each original edge by a vertex, joining two new vertices exactly when their original edges share an endpoint. A proper edge colouring of $G$ becomes a proper vertex colouring of its line graph $L(G)$, so $\chi \prime (G)=\chi (L(G))$. For a multigraph, retain a distinct new vertex for each parallel edge.^[ref]

Every finite simple graph of maximum degree $\Delta$ has edge chromatic number $\Delta$ or $\Delta +1$.

For a finite graph with at least two vertices, the displayed minimum measures the fewest outgoing edges per vertex in a nonempty set occupying at most half the graph. It uses the edge boundary, rather than the external vertex boundary. For the hypercube graph $Q_n$, $n\ge 1$, it equals one: edge-isoperimetry gives the lower bound and a coordinate half-cube attains it.

For the affine angular ansatz $I=A+C\cos \theta$, the radiative flux is $F=4\pi C/3$. Setting the inward hemispheric flux to zero gives $A(0)=2C/3$, but then the formal intensity at $\cos \theta =-1$ is negative when $F>0$. Exact no-incoming radiation and nonnegative intensity cannot both be represented by a nonzero affine polynomial on all directions. The Eddington surface boundary condition therefore specifies approximate moments, rather than an exact emergent angular radiation field.

Earth's outer core is a liquid, electrically conducting region whose buoyant motion and rotation sustain the geodynamo. Its magnetic field both responds to the flow through induction and modifies that flow through the Lorentz force. A prescribed-velocity kinematic model describes onset rather than the entire nonlinear core dynamics.

Let a vector of E-functions satisfy $f^{\prime }=A(z)f$ with $A\in {\mathrm{Mat}}_s(\overline{\mathbb{Q}}(z))$. At a nonzero algebraic point $\xi$ which is not a pole of $A$, the transcendence degree of the values over $\overline{\mathbb{Q}}$ equals the transcendence degree of the functions over $\overline{\mathbb{Q}}(z)$. In particular algebraically independent functions give algebraically independent values. The exponential system $f_j=e^{{\gamma }_{j}z}$, for rationally independent algebraic ${\gamma }_j$, recovers the Lindemann–Weierstrass theorem by evaluating at one.

Work in the compositum of the two coefficient fields. Conjugate coefficient bounds add, while multiplying the common denominators clears both coefficient lists. For Siegel's original bounds $O_{ϵ}(n^{ϵn})$, use $ϵ/2$ in each denominator estimate. A nonzero linear differential equation for the sum follows because its derivatives lie in the sum of the two finite-dimensional derivative spaces over the rational function field. The stricter exponential arithmetic bounds are preserved as well.