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An element $a$ is a non-zero-divisor when multiplication by $a$ is injective, equivalently when $ab=0$ implies $b=0$.

In a chain $N_1\to N_2\to N_3$ with distinct decay constants, the Bateman equation expresses each population as a linear combination of the exponential functions $e^{-{\lambda }_{i}t}$. When two consecutive constants coalesce to $\lambda$, the repeated Laplace transform pole produces a term proportional to $t{e}^{-\lambda t}$.

A standing wave is the superposition of equal-frequency waves travelling in opposite directions. Its spatial nodes remain fixed while its amplitude oscillates in time.

A conservation law equates local time change of a density with the divergence of its flux.

The Schrödinger equation governs unitary quantum evolution under a Hamiltonian.

Proper time is the time measured by a clock following a timelike worldline. With metric signature $(-,+,+,+)$ and units $c=1$,along that worldline.

Restoring the speed of light, the same metric signature gives $d\tau =\sqrt{-d{s}^2}/c=dt\sqrt{1-|\mathbf{v}{|}^2/c^2}$ for a future-directed timelike curve. Integrating along a clock's world line gives its elapsed proper time, even when its velocity varies. For uniform motion this becomes the time dilation formula; it is not generally the square root of the interval between endpoints of a curved world line.

A uniformly charged spherical shell produces zero field inside and the field of a point charge outside.

An ideal superconducting loop supports persistent current that preserves linked magnetic flux when resistance vanishes.