= Mass-normalized temporal kernels for nail clippings
{title2=$\int w(s)\,ds=1$}
\b[<Mass-normalized temporal kernels for nail clippings>] assign each earlier material time its fraction of the analyzed sample's <mass>. They convert a finite clipping into an averaging operator on a time-dependent retained <concentration>. The weighting is normalized by the assayed material amount, rather than by elapsed time alone.
For an original one-dimensional <advection> model, let $s$ be the time when a material element crossed a defined anatomical reference, and collect at time $t$. With spatially uniform positive advance speed $v(u)$,
$$
x(s)=\int_s^t v(u)\,du,\qquad \left|\frac{dx}{ds}\right|=v(s).
$$
Suppose the clipping's retained elements correspond to $s\in[s_1,s_2]$. Let $m(x,t)$ be structural <mass> per unit length at collection, and assume the marker's concentration per structural <mass> is preserved along each element. Then
$$
M=\int_{s_1}^{s_2}m(x(s),t)v(s)\,ds,\qquad
w(s)=\frac{m(x(s),t)v(s)}{M},\qquad
\overline c=\int_{s_1}^{s_2}w(s)c(s)\,ds.
$$
The kernel has units of inverse time, is nonnegative and integrates to one. It can be treated mathematically like a <probability density function> for the reference-crossing time of material selected in proportion to its <mass>. This is a material-accounting interpretation, not a claim that the biological transport is random.
For a stipulated 20-day window, take line density 1 mg/mm, advance 0.10 mm/day in the earlier ten days and 0.20 mm/day in the recent ten days. The two intervals contribute 1 and 2 mg, respectively. Their kernel heights are $1/30$ and $1/15$ per day. A retained concentration of 90 ng/mg only in the earlier interval gives $\overline c=30$ ng/mg; a uniform time average would give 45. A constant 30 ng/mg history gives the same sample result. Thus one pooled measurement does not identify a unique history.
A <centroid date of a mass-normalized nail kernel> summarizes the weighting, while the kernel's spread records its finite time resolution. Neither removes the difference between reference crossing and earlier formation in the hidden <nail matrix>. The primary nail-marking experiment measures displacement over observation intervals, not the complete kernel or marker retention law.<Growth rate of human fingernails and toenails in healthy American young adults by S. Yaemsiri and others>[Abstract, Methods]{ref} Measuring segment <mass>, geometry and growth history constrains separate parts of this model.
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