A group of predatory fishes including the Red Sea fish studied in coordinated hunting with moray eels.

Gymnothorax javanicus, the moray in the cited interspecific hunting study.

A member of the fish family Cichlidae, with radiations informing studies of sensory evolution and speciation.

A fish species with surface and cave populations used in studies of eye development and degeneration.

A cleaner wrasse is a fish that obtains food by removing material, including external parasites, from other fish. Such interactions make benefits, conflicts and partner choice observable in a natural social setting. Labroides dimidiatus is a widely studied example.

The guppy, Poecilia reticulata, is a small live-bearing fish used in research on sexual selection, predation and life-history evolution. Differences among populations make it possible to connect environmental conditions with inherited and plastic traits.

For a quasistatic simple compressible system with only pressure-volume work, the direction of heat transfer is determined by $dU+pdV$. A reversal occurs where this vanishes and changes sign. It need not coincide with a maximum or minimum of temperature, because the substance can still do work when its internal energy is stationary.

Every group of order six is isomorphic to the cyclic group $C_6$ or the symmetric group on three points $S_3$. Lagrange's theorem restricts element orders to $1,2,3,6$. If there is an element of order six, it generates the group. If every nonidentity element had order two, the group of exponent two is abelian result would produce a subgroup $\{e,a,b,ab\}$ of order four, contradicting Lagrange's theorem. Thus the noncyclic case has an element $x$ of order three.

There cannot be two distinct subgroups of order three: their intersection would be trivial and their nine pairwise products would be distinct elements of a six-element group. Hence exactly two elements have order three, and the other three nonidentity elements have order two. The subgroup $⟨x⟩$ has index two and is normal. Choose an involution $h$ outside it. Conjugation sends $x$ to $x$ or $x^{-1}$. The first case would make $xh$ have order six, so $hx{h}^{-1}=x^{-1}$. The generators therefore obey $x^3=h^2=e$, $hx{h}^{-1}=x^{-1}$, the relations realized by $(123)$ and $(12)$ in $S_3$.

The first difference of a sequence records the change between consecutive terms. Its partial sums telescope: $u_n=u_0+{\sum }_{j=0}^{n-1}\Delta u_j$. Thus a compact description of the differences can give an efficient way to reconstruct distant terms of the original sequence.

The finite increment ${\Delta }_{h}f(x)=f(x+h)-f(x)$ for a specified step $h$. Higher forward differences are obtained by iterating this operation. For a polynomial of degree $d$, its $(d+1)$st forward difference is zero.

A Newton series uses the binomial-polynomial expansionFor data on nonnegative integers, the value at $x=n$ is always the finite identity $f(n)={\sum }_{k=0}^n\left(\genfrac{}{}{0ex}{}{n}{k}\right){\Delta }^{k}f(0)$. For a polynomial of degree $d$, higher finite differences vanish, and the expansion terminates and equals that polynomial for every real or complex $x$. For example, $x^2=\left(\genfrac{}{}{0ex}{}{x}{1}\right)+2\left(\genfrac{}{}{0ex}{}{x}{2}\right)$.

For a general nonpolynomial function, convergence and recovery away from integer sites need additional hypotheses. Integer samples do not determine an arbitrary continuous function or even an entire function: $\sin (\pi x)$ vanishes at every integer, so all coefficients computed from its integer samples are zero, although the function is not zero. A formal Newton series must therefore be distinguished from an asserted expansion valid on a continuous domain.

The thumb is the first digit of the human hand. Its basal joint and muscular control permit opposition toward other fingers, supporting precision grasp. Mobility, orientation and sensory control together explain its function more fully than digit length alone.

A vertex written in terms of composite hadron fields, with a coupling or form factor that summarizes unresolved Quantum chromodynamics. For a strong Delta baryon decay, a $\Delta p\pi$ vertex describes the external channel but does not assert that the decay is perturbative at constituent-quark scales.

A schematic tracking of valence quark and antiquark flavor through a hadronic process. It can explain flavor conservation and required pair creation without computing the nonperturbative hadronization amplitude. A complete elementary perturbative diagram contains additional interaction and color information.

For an isotropic three-dimensional free electron gas at zero temperature, occupied wavevectors fill a sphere of radius $k_F$. Including two spin states, $n=k_F^3/(3{\pi }^2)$. The volume in reciprocal space counts occupied states; folding an extended-zone sphere into a first Brillouin zone does not remove states outside that first zone.

An electron pocket surrounds occupied states near a conduction-band minimum. In two dimensions its carrier density is $gA/(2\pi {)}^2$, where $A$ is its total reciprocal-space area and $g$ the unresolved degeneracy. Portions drawn at opposite zone faces can belong to the same pocket after periodic identification.

A hole pocket surrounds unoccupied states near a band maximum. Its hole carrier density counts missing states in the nearly filled band, not all occupied electrons. In two dimensions the count is $g{A}_h/(2\pi {)}^2$. Four corner pieces of a square-zone plot can be one pocket on the periodically identified zone.

For a free circle of radius $R$ and square reciprocal period $Q$ with $\sqrt{2}R<Q<2R$, the circle cuts the faces but not the corners of the first zone. Weak Bragg gaps reconnect the folded branches into two inequivalent edge-centred electron pockets and one corner-centred hole pocket. The zero-potential limit retains band crossings instead of avoided crossings.

For box-normalized plane waves in volume $\mathcal{V}$, a potential has matrix element $\tilde{W}({\mathbf{k}}_f-{\mathbf{k}}_i)/\mathcal{V}$. The final-state density per energy and solid angle is $\mathcal{V}m{k}_f/[(2\pi {)}^3{\hslash }^2]$. Dividing the golden-rule rate by incident flux $\hslash k_i/(m\mathcal{V})$ gives $(m^2/4{\pi }^2{\hslash }^4)(k_f/k_i)|\tilde{W}{|}^2$. Elastic scattering has $k_f=k_i$; the arbitrary box volume cancels.

When $E-\mu \gg kT$, the unit term in the denominator of the Fermi-Dirac distribution is negligible. The resulting exponentially small occupation is the Boltzmann approximation. It justifies the usual nondegenerate semiconductor carrier formulas; it fails near strongly occupied band states.