A tessellation by rectangular tiles in which no point is a corner of four tiles. In the common lattice version, the tiles are $1\times 2$ rectangles. The local corner restriction makes existence depend on more than the total area.

A spatially localized forcing, such as persistent tropical heating, can excite Rossby waves that propagate wave activity into distant regions. Mean winds Doppler-shift the waves, and background potential vorticity gradients and jets guide or refract the packets. A stationary pattern can thus connect remote ridges and troughs even though the underlying wave energy propagates. Group velocity, rather than phase velocity alone, diagnoses packet propagation; damping, seasonality and eddy feedbacks affect the response.

A family of mineral-analogue ceramic waste forms. Different titanate phases accept different waste ions by solid solution; phase assemblage is tailored to the waste inventory.

A tunnel-structured titanate host, represented by $\mathrm{BaA}{\mathrm{l}}_2\mathrm{T}{\mathrm{i}}_6{\mathrm{O}}_{16}$, used to incorporate large cations such as caesium with appropriate charge compensation in a crystal.

A titanate host represented by $\mathrm{CaTi}{\mathrm{O}}_3$. strontium can replace calcium; other ions require suitable charge balance.

For an ultrarelativistic circular orbit of curvature radius $\rho$, the conventional critical angular frequency is ${\omega }_c=3{\gamma }^{3}c/(2\rho )$. It sets the broad spectral scale rather than a strict maximum frequency.^[ref]

The forward radiation cone sweeps past a distant observer while the source moves closer. The difference between emission time and arrival time shortens the observed pulse to order $\rho /({\gamma }^{3}c)$ and broadens its spectrum.

Four equivalent $s$ atomic orbitals at the vertices of a regular tetrahedron carry the tetrahedral symmetry representation $A_1\oplus T_2$. The normalized equal sum is $A_1$; three coordinate-sign combinations span $T_2$. Every normalized coefficient vector has entries $\pm 1/2$ in a suitable orthonormal real basis. These functions simplify a Hamiltonian invariant under the tetrahedral operations, but equivalent symmetry copies on other sites may still mix.

An oblate symmetric top has two equal smaller principal moments of inertia and a larger moment of inertia about its symmetry axis. A planar trigonal molecule, such as boron trifluoride, is an example. Its axial rotational constant is smaller than its transverse rotational constant.

A rotor with two equal transverse moments and a smaller symmetry-axis moment, $I_a<I_b=I_c$. Its rotational constants obey $A>B=C$.

If $A^T=A$ and $B^T=B$, then $(AB{)}^T=B^T{A}^T=BA$. Hence their product is a symmetric matrix exactly when they commute. For example, $A=\mathrm{diag}(1,0)$ and $B=\left(\begin{array}{cc}0& 1\\ 1& 0\end{array}\right)$ are symmetric but have nonsymmetric product $\left(\begin{array}{cc}0& 1\\ 0& 0\end{array}\right)$. This criterion differs from asking whether an arbitrary matrix can be factored as a product of symmetric matrices.

For real $a,b$, the displayed symmetric matrix satisfies $M^2=(a^2+b^2)I$. If $s=\sqrt{a^2+b^2}>0$, choose $\theta$ with $\cos 2\theta =a/s$ and $\sin 2\theta =b/s$. The eigenvectors $(\cos \theta ,\sin \theta )$ and $(-\sin \theta ,\cos \theta )$ are orthonormal and have eigenvalues $s$ and $-s$. The half-angle description works even when $b=0$, avoiding an artificial division by an off-diagonal entry. If $s=0$, the matrix is zero and every direction is an eigenvector. The normalized matrix $M/s$ is reflection in the axis at angle $\theta$.

For a circular projected profile with a positive power-law tail, an annulus contributes $dF=2\pi I(R)RdR$. Thus the outer integral converges only when $p>2$; $p=2$ diverges logarithmically and $p<2$ as $R^{2-p}$. A shallower tail measured over a finite range must steepen or terminate before an infinite-radius extrapolation is physical. This projected criterion already includes line-of-sight integration and must not be replaced by a three-dimensional $r^{2}dr$ volume factor.

For freely propagating radiation with the usual distance duality relation, a fixed emitted bolometric surface brightness is observed smaller by $(1+z{)}^4$. One factor each comes from photon energy and arrival rate, and two from the relation of projected angular area to luminosity distance. In a fixed observed photometric passband, spectral shifts add a band-dependent correction; the bolometric power is not automatically the monochromatic one. A distortion created by scattering the local cosmic microwave background has a different redshift comparison, as in the thermal Sunyaev-Zeldovich effect.

Thermally activated reorientation of a particle's internal magnetic moment across an anisotropy barrier. It differs from rotating the whole particle in a fluid. Its timescale can determine a measurement-dependent magnetic blocking temperature.

Spectral labels classify observed features rather than directly naming the energy source. Hydrogen-rich Type II events are associated with core collapse, but hydrogen-poor stripped-envelope Ib/c events can also be core collapse. Ordinary Type Ia events are thermonuclear white-dwarf explosions. Absence of hydrogen alone does not establish a thermonuclear mechanism, and light-curve standardization does not imply identical progenitors.