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For a one-dimensional shear elastic wave, the seismic impedance is the magnitude of the ratio of stress to particle velocity in a progressive wave. With the positive coordinate aligned with elastic wave propagation and tension-positive stress, $\sigma =-Zv$ for the forward wave and $\sigma =Zv$ for the backward wave. Its instantaneous elastic-wave energy flux is $-v\sigma$.

The difference of two solutions with the same body force, boundary traction and initial data satisfies a homogeneous elastic equation with zero boundary work. Multiplication by its velocity and integration by parts conserve the displayed positive energy. Zero initial energy forces zero velocity for all times, and zero initial displacement field then forces zero displacement field. Rigid zero-strain motions do not cause nonuniqueness after initial data are fixed. This assumes positive mass density, stable real elastic moduli and sufficient regularity for the energy identity.

A localized zero-resultant seismic moment tensor represents the leading force-dipole part of an elastic wave source through the displayed equivalent body force. For a prescribed displacement jump $b_i$ across a surface with normal $n_j$, its moment density is $c_{ijkl}{b}_k{n}_l$, where $c$ is the elastic stiffness tensor. Spatial integration gives the point-source moment when the source is small compared with the wavelengths.

The causal Green function for isotropic linear elasticity is tensor-valued: column $k$ is the displacement caused by a point impulsive body force in direction $k$. Its radiative terms are longitudinal P waves at $t=R/\alpha$ and transverse S waves at $t=R/\beta$, proportional to $R^{-1}$. A term supported between the arrivals carries the intermediate/near-field response. Convolution in time and space constructs the displacement for a general source.

For homogeneous stable linear elasticity, the three positive eigenvalues of the acoustic tensor give three phase velocities for each wavefront normal. Plotting the corresponding three vectors $\mathbf{p}=\mathbf{n}/v$ gives the three sheets of the elastic slowness surface. Sheets can meet at degeneracies. The elastic-wave energy flux is normal to each regular sheet: differentiating its eigenvalue equation along a sheet tangent $d\mathbf{p}$ gives $c_{ijkl}{a}_i{a}_k{p}_{l}d{p}_j=0$.

In a right-going simple wave, the state emitted at boundary time $s$ travels along $X=c_b(s)(t-s)$. The mapping from emission time to position first becomes singular when ${\dot{c}}_b(t-s)-c_b=0$, giving the displayed shock time and position $X=c_b^2/{\dot{c}}_b$. A smooth increase of traction produces steepening only if it increases the propagation speed. This describes the first gradient singularity; continuation as a shock wave needs admissible jump conditions.

The elastic polarization is the excess stress relative to a homogeneous comparison elastic stiffness tensor. In heterogeneous linear elasticity, $\tau =(C-C^0)\epsilon$. The comparison static elastic equilibrium equation gives the integral equation $\epsilon ={\epsilon }^0-\Gamma \tau$, where ${\epsilon }^0$ is the reference strain under the same external loading.

Let $B_{n}a=\mathrm{sym}(a\otimes n)$; its adjoint on symmetric stress tensors is $B_n^T\tau =\tau n$. Then $B_n^{T}C{B}_n$ is the acoustic tensor. The directional elastic strain Green operator satisfies the resolvent identity ${\tilde{\Gamma }}_r(C^r-C^s){\tilde{\Gamma }}_s={\tilde{\Gamma }}_s-{\tilde{\Gamma }}_r$, by the elementary inverse identity $K_r^{-1}(K_r-K_s)K_s^{-1}=K_s^{-1}-K_r^{-1}$.

For a three-dimensional isotropic comparison material, the contraction of its elastic strain Green operator is the displayed distributional derivative. Its double trace is ${\Gamma }_{iikk}=\delta /L$, because $\Delta |x{|}^{-1}=-4\pi \delta$. Away from zero the contraction is $({\delta }_{ij}|x{|}^2-3x_i{x}_j)/(4\pi L|x{|}^5)$; the full distribution also includes ${\delta }_{ij}\delta /(3L)$. This contact term must not be discarded when computing hydrostatic strain.

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.