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For a rectangular channel with stationary width $b(x)$ and constant reduced gravity $g^{\prime }$, volume conservation and momentum conservation give $h_t+u{h}_x+h{u}_x=-hu{b}^{\prime }/b$ and $u_t+u{u}_x+g^{\prime }{h}_x=0$. With $c=\sqrt{g^{\prime }h}$ the characteristic curves have speeds $u\pm c$. The combinations $u\pm 2c$ obey the displayed compatibility equations; they are Riemann invariants only when the width source term vanishes.

Oogenesis is development of female gametes; the unequal divisions of human meiosis produce polar bodies alongside the developing ovum.

In angiosperms, pollination transfers pollen from an anther to a stigma. Fertilization is a later fusion of gametes; seed development follows. Pollination may use animals, wind or water, so an insect visit is neither required for every flower nor sufficient evidence of fertilization.^[ref]

Fertilization is fusion of gametes. Fusion of two haploid gametes restores the diploid state in the inheritance model used for Mendelian segregation.

A gamete is a reproductive biological cell that can fuse with another gamete during fertilization. In the ordinary diploid inheritance model, a haploid gamete carries one copy at each genetic locus.

The Bateman equation gives the populations in a sequential radioactive decay chain. For a parent with population $N_1(0)$ and an initially absent daughter, distinct decay constants give $N_2(t)=N_1(0){\lambda }_1(e^{-{\lambda }_{1}t}-e^{-{\lambda }_{2}t})/({\lambda }_2-{\lambda }_1)$; the first population satisfies $N_1^{\prime }=-{\lambda }_1{N}_1$.

Cyclic adenosine monophosphate is a cyclic nucleotide made from ATP. It can serve as an intracellular second messenger, changing the activity of target proteins after a signal has been received. Different biological cells can respond differently because they contain different targets. ^[ref]

In signal transduction, inositol trisphosphate usually means inositol 1,4,5-trisphosphate: a soluble intracellular second messenger. An extracellular ligand, its cell-surface receptor, IP3 and the released calcium ions are distinct components. IP3 binds an IP3 receptor on the endoplasmic reticulum, promoting stored ${\mathrm{Ca}}^{2+}$ release into the cytosol.^[ref]

The useful separation is spatial: an extracellular message can control an internal store without entering it. The existing intracellular-injection experiment under metabotropic receptor tests bypassing reception; the Pancreatic IP3 store fractionation experiment below tests which internal compartment supplies the released calcium. Neither question is answered merely by observing a downstream membrane electric current.

Online text: docs.scipy.org/doc/scipy-1.16.2/reference/generated/scipy.integrate.solve_ivp.html

Cited passages: solve_ivp, method parameter: explicit versus implicit methods and stiffness; solve_ivp, rtol and atol parameters: local error estimates, additive scaling and component-specific atol.

A cathetometer measures differences of vertical height using a sighting telescope mounted on a vertical measuring column. The telescope is aligned horizontally, sighted on a first feature, translated vertically and sighted on a second feature. Their scale readings determine the height difference without placing a ruler directly against either feature.

A manometer supplies a useful application: the two liquid surfaces can be sighted separately to determine the column-height difference used to infer pressure. The optical reference, column alignment, meniscus reading and scale measurement calibration all affect the result. Greater apparent magnification alone cannot correct a tilted reference or an incorrectly identified liquid surface.^[ref]

For a nonrelativistic spin-one-half electron of charge $-e$, with $e>0$, the Pauli Hamiltonian isThe wavefunction has two spin components and $B=\nabla \times A$. If $\pi =-i\hslash \nabla +eA$, then $[{\pi }_i,{\pi }_j]=-ie\hslash {ϵ}_{ijk}{B}_k$; the Pauli matrix multiplication law gives $(\sigma \cdot \pi {)}^2={\pi }^2+e\hslash \sigma \cdot B$, explaining the spin-coupling sign. Thus $i\hslash {\partial }_t\Psi =H_P\Psi$ extends the Schrödinger equation to include the electron's magnetic moment.