For every prime number $p\ge 5$, Wolstenholme's theorem givesIts harmonic-sum form is $H_1={\sum }_{k=1}^{p-1}1/k\equiv 0(\mathrm{mod}{p}^2)$. All denominators are coprime to $p$, so fractions are interpreted using modular inverses.

Here is a proof of both assertions. Write $H_2={\sum }_{k=1}^{p-1}1/k^2$. Multiplication by $2$ permutes the nonzero residue classes modulo $p$, so $H_2\equiv H_2/4(\mathrm{mod}p)$. Since $p\ge 5$, this implies $H_2\equiv 0(\mathrm{mod}p)$. Replacing $k$ by $p-k$ in $H_1$ givesusing $1/(p-k)\equiv -1/k-p/k^2(\mathrm{mod}{p}^2)$. Thus $2H_1\equiv 0(\mathrm{mod}{p}^2)$, proving the harmonic-sum form.

Finally, expand the binomial coefficient as a product:Terms containing three or more selected factors $p/k$ vanish in this integer congruence. The two harmonic-sum estimates make both remaining correction terms zero modulo $p^3$. For $p=5$, the result reads $\left(\genfrac{}{}{0ex}{}{9}{4}\right)=126\equiv 1(\mathrm{mod}125)$.

Where a central mass dominates and radial infall is locally steady, $|v_r|\simeq \sqrt{2G{M}_{\star }/r}$. A conserved mass flux $\dot{M}=4\pi r^2\rho |v_r|$ then gives $\rho \simeq \dot{M}[4\pi \sqrt{2G{M}_{\star }}{]}^{-1}{r}^{-3/2}$. Constant accretion gives $M_{\star }\propto t$ and an inner normalization proportional to $t^{-1/2}$.

An outward collapse front and inward central accretion measure different parts of the same mass budget. In the inside-out collapse of a singular isothermal sphere, the unperturbed profile has $\rho =a_s^2/(2\pi G{r}^2)$ and the front is at $r_f=a_{s}t$. Its swept mass rate is therefore $4\pi r_f^2\rho (r_f){\dot{r}}_f=2a_s^3/G$.

The central rate is instead ${\dot{M}}_{\ast }=m_0{a}_s^3/G$, with $m_0$ determined by the dynamical solution. Material already crossed by the front can remain in the infalling envelope, so the two rates need not agree. Equating them would silently assume an empty reservoir between the front and the central object.^[ref]

A fruit-fly species used extensively to investigate inheritance, embryonic patterning and germline development.

A beetle species used to study sequential insect segmentation and its relationship to the more simultaneous mode in Drosophila melanogaster.

A solitary bee studied as the pollinator targeted by sexual deception in Ophrys sphegodes.

Apis mellifera in the cited studies of worker task development and worker policing.

The ant used in the cited test separating task specialization from individual task efficiency.

A fly genus including experimental models for cell competition and genetic conflict.

The apple maggot fly, with host-associated populations used to study partial reproductive isolation.

A butterfly genus used in studies of warning patterns, mate recognition and gene flow.

The pea aphid, including host-associated lineages used to study ecological specialization.

A bumblebee is a bee of the genus Bombus. Many species form seasonal colonies, with workers collecting floral resources. Their foraging provides examples of interactions among animal behaviour, pollination and changing food availability.

A beetle associated with dung as a resource. Brood provisioning and associated microorganisms can create maternal effects.

A butterfly is an insect in a radiation of moths and butterflies, with a life cycle passing through egg, larva, pupa and adult. The feeding and dispersal roles of caterpillar and winged adult can be very different.

Nasonia is a genus of small parasitoid wasps used in research on genetics, speciation and host interactions. Developing offspring exploit a host animal rather than merely sharing its environment. Nasonia vitripennis and Nasonia giraulti support comparisons between related species.

A water strider is an insect specialized for moving on the water surface. Water-repellent legs distribute forces at the interface, making it a biological example of the importance of surface tension and contact geometry.

A transport protein exchanging suitable dicarboxylates, including succinate and malate, with counter-substrates such as phosphate. Its relevance to respiration depends on the fuel and preparation.

A carrier exchanging adenine nucleotides across the inner mitochondrial membrane, supplying matrix ADP and exporting synthesized ATP during ordinary oxidative phosphorylation.

A transport protein supplying phosphate across the inner mitochondrial membrane for processes including ATP synthesis.