For the prescribed sine-mode model $\omega =k\sqrt{c_s^2+\alpha k^2}$, positive small stiffness shifts higher harmonics more strongly. Writing $\eta =\alpha k_1^2/c_s^2$ and $k_2=2k_1$ gives ${\omega }_2/{\omega }_1=2+3\eta +O({\eta }^2)$. At the original fundamental quarter-period, the fundamental displacement changes to first order, but the second-mode cosine is at an extremum and changes only to second order. If the quarter-period is redefined with the shifted fundamental frequency, that fundamental displacement is exactly zero. Comparing profiles requires specifying which clock convention is held fixed.

A junction produced when suitable partial dislocations on intersecting FCC slip planes react to form a low-energy sessile stair-rod segment. It obstructs further slip and can promote work hardening. Frank energetic favorability alone is not a complete calculation of junction geometry or mobility.

A crystallographic plane together with an allowed dislocation glide direction in that plane. Geometry alone does not determine its activation stress.

A lattice-friction stress for dislocation glide, associated with Rudolf Peierls and Frank Nabarro. A simple model gives ${\tau }_P\sim 3G\exp (-2\pi w/b)$, where $w$ is core width and $b$ the Burgers vector magnitude. Spreading slip over many atomic rows reduces the periodic energy barrier; narrower cores are harder to move in this model.

A screw dislocation is a crystal line defect whose Burgers vector is parallel to the dislocation line. Atomic planes around it have a helical arrangement, rather than the extra-half-plane geometry associated with an edge dislocation.

Its motion permits crystal slip. Because several planes can contain the same line and Burgers vector, screw segments can sometimes change their glide plane by cross-slip, with the actual possibility depending on the crystal and defect structure.

An edge dislocation is a crystal line defect whose Burgers vector is perpendicular to the dislocation line. It is often pictured as the termination of an extra half-plane of atoms inside the crystal.

Motion of the line allows a slip displacement to advance through the material without every bond across a whole plane changing simultaneously. Its surrounding elastic distortion includes compressive and tensile regions, connecting an atomic-scale defect with macroscopic plastic deformation.

A mixed dislocation has a line direction neither parallel nor perpendicular to its Burgers vector, combining screw and edge character. Along a curved dislocation line, the local character can change continuously while the Burgers vector remains fixed.

A dislocation loop therefore need not be assigned one character everywhere. Decomposing a local segment into edge and screw contributions helps analyze its stress field, mobility, and interaction with obstacles.

For arrays $a,b$ indexed by integers modulo $k$, their cyclic convolution isWhen $a_i$ and $b_j$ count choices with respective residue classes $i$ and $j$, the cyclic convolution counts pairs whose sum has residue class $r$. Repeated cyclic convolution therefore counts sums of longer sequences. The discrete Fourier transform converts cyclic convolution into pointwise multiplication.

For arrays indexed by $b$-bit nonnegative integers, their XOR convolution is $(a{\ast }_{\oplus }c{)}_r={\sum }_i{a}_i{c}_{i\oplus r}$, where $\oplus$ is bitwise exclusive or. If entries count choices of Grundy numbers, its repeated application counts sums of impartial games with a specified combined Grundy number. The Hadamard transform turns this operation into pointwise multiplication.

A restricted passage in a casting mould that progressively excludes competing crystal grains so a single surviving orientation can grow into the component.

With the generators $c,a$ of the direct product of cyclic groups of orders four and two, the subgroups are the trivial group, the whole group, the three order-two groups $⟨c^2⟩$, $⟨a⟩$, $⟨c^{2}a⟩$, and the three order-four groups $⟨c⟩$, $⟨ca⟩$, $⟨c^2,a⟩$. These are exhaustive by Lagrange's theorem: a proper nontrivial subgroup has order two or four; an order-four subgroup either contains an element of order four and is one of the two cyclic examples, or is precisely the four-element set killed by squaring. The latter is a Klein four-group.

A direct intersite Coulomb exchange integral acts without an intermediate doubly occupied state. In a repulsive one-band Hubbard model, by contrast, virtual hopping into a doubly occupied state lowers the singlet energy by $4t^2/U$ relative to the triplet. The latter is antiferromagnetic kinetic exchange, even when hopping is directly between neighbouring magnetic orbitals. Ligand-mediated superexchange introduces an intermediate ligand orbital as well.

Radiation polarized along the static field drives the zero-projection change, while transverse polarization supplies the two circular components with projection changes plus or minus one. The opposite signs of the excited magnetic quantum number remain degenerate in a pure static electric field; they need not be separate spectral lines.

Take $r$ as a counterclockwise quarter-turn and $s$ as reflection across the horizontal axis. The eight symmetries of a square are $e,r,r^2,r^3,s,rs,r^{2}s,r^{3}s$. The last four are reflections across the horizontal axis, the diagonal $y=x$, the vertical axis and the diagonal $y=-x$, respectively. This is a dihedral group of order eight; the subscript convention $D_4$ counts polygon vertices, while some authors use $D_8$ for the same group.

Dihedral symmetry consists of the rotations and reflections preserving a regular polygon. For an $n$-gon with $n\ge 3$, its dihedral group has $2n$ elements generated by a rotation $r$ through $2\pi /n$ and a reflection $s$, with $r^n=s^2=1$ and $srs=r^{-1}$.

Every element is uniquely $r^k$ or $r^{k}s$, for $0\le k<n$. For a square this gives four rotations and four reflections. The relation $srs=r^{-1}$ explains why changing orientation reverses the direction of rotation and why the group is generally noncommutative. Notation varies: $D_n$ and $D_{2n}$ are both used for the group of order $2n$.

A diffusion approximation for a scalar neutron flux, with one prescribed energy group. Its terms balance current loss, absorption, nuclear fission production and a source. A solution also requires appropriate boundary conditions.

Ewald sphere curvature is the curvature of the reciprocal-space surface sampled by elastic scattering. Accounting for its departure from a plane can improve very high-resolution electron microscopy reconstructions.

Diffuse scattering generated by fluctuating lattice displacements. Resolved one-phonon creation and annihilation transfer both energy and crystal momentum, with Bose factors $n_B+1$ and $n_B$. An energy-integrating diffraction measurement can record their summed intensity without resolving the small energy transfer. Phonon polarization, soft-mode frequencies and atomic scattering weights determine the pattern; static disorder supplies a distinct possible diffuse contribution.

For an ideal uniform dielectric between large parallel plates, $Q=CV=(ϵA/d)(Ed)=ϵAE$. A prescribed breakdown field $E_{\mathrm{bd}}$ therefore limits charge independently of plate separation. Voltage capacity still grows with separation. This ideal result neglects field enhancement near edges and any dependence of the breakdown field on gap geometry or environmental conditions.

For $A=B=I_2$, $\det (A+B)=\det (2I_2)=4$, while $\det A+\det B=2$. The determinant is multilinear in its separate columns, rather than a linear function of the matrix as a whole. Expanding the columns of a sum introduces mixed terms that the proposed additive identity omits.