Two parallel ideal conducting plates at zero temperature have separation-dependent electromagnetic energy per area $-{\pi }^2\hslash c/(720d^3)$ and pressure $-{\pi }^2\hslash c/(240d^4)$. The negative sign denotes attraction in this geometry. Finite temperature, material response and boundary geometry change the ideal result; an absolute divergent vacuum energy is not the measured quantity.

Confining a near-critical fluctuating order parameter changes its free energy. Its range is governed by the growing correlation length, and its sign depends on the boundary preferences. This is a thermal collective effect, distinct from the zero-temperature ideal-conductor electromagnetic formula.

A fluctuation-induced force depends on how boundaries restrict fluctuating fields. Changing the boundary conditions changes the permitted normal modes, their frequencies and the separation-dependent free energy. The resulting force is $-\partial \mathcal{F}/\partial L$ at the stated control conditions; confinement alone does not determine its sign.

For a concrete ideal example, take a massless scalar field of wave speed $c$ on a one-dimensional interval of length $L$ at zero temperature. Two Dirichlet boundary conditions give mode wavenumbers $n\pi /L$, $n\ge 1$. Replacing one end by a Neumann boundary condition gives $(n+1/2)\pi /L$, $n\ge 0$. With the same bulk subtraction, their finite zero-point energies are respectivelyDifferentiation gives attraction in the first case and repulsion in the second. These scalar examples are distinct from the three-dimensional electromagnetic Casimir force. The comparison illustrates why the boundary prescription must accompany any reported fluctuation-force law.

The organization of symmetry in a flower. Whole-flower symmetry, individual organ shape and developmental identity are distinct characters; similar outlines can arise through different molecular mechanisms.

A flower containing both male and female reproductive organs, stamens and carpels. It differs from monoecy, in which male and female structures occur separately on the same individual.

The floral structures surrounding the reproductive organs, often differentiated into sepals and petals. Protection and pollinator attraction can be combined in a shared display.

Spatial separation of male and female reproductive organs that can reduce self-pollen transfer while retaining them within one flower.

Temporal separation of male and female reproductive function. It can reduce self-fertilization without requiring separate male and female flowers.

For two equal-site-volume polymers of chain length $N$, the mixing free energy per lattice site is $f/(k_{B}T)=[\varphi \ln \varphi +(1-\varphi )\ln (1-\varphi )]/N+\chi \varphi (1-\varphi )$. Each chain is one translating object, reducing the entropy of mixing per monomer by $N$ relative to the regular solution model. Its curvature is $f^{\prime \prime }/(k_{B}T)=1/[N\varphi (1-\varphi )]-2\chi$. The symmetric thermodynamic critical point has ${\varphi }_c=1/2$ and ${\chi }_c=2/N$. Long chains can therefore demix under a very small unfavorable Flory interaction parameter.

For a single-site solvent and polymer length $N$, expand $(1-\varphi )\ln (1-\varphi )=-\varphi +{\varphi }^2/2+{\varphi }^3/6+O({\varphi }^4)$ in Flory–Huggins solution theory. The displayed result follows. Its linear term changes reference chemical potentials; its quadratic term sets the effective two-segment excluded volume. The latter vanishes at the theta temperature, while the positive cubic term remains. Taking $\varphi f^{\prime }-f$ yields the osmotic pressure in Flory-Huggins solution theory, whose cubic coefficient is $1/3$ rather than the free energy's $1/6$.

An incompressible, equal-site-volume mean-field approximation for two polymers of degrees of polymerization $N_A,N_B$. Here $f$ is mixing free energy per site and $\varphi$ is the volume fraction of species $A$. For a composition-independent Flory interaction parameter, local stability requires $1/(N_A\varphi )+1/[N_B(1-\varphi )]-2\chi >0$. The thermodynamic critical point is ${\varphi }_c=\sqrt{N_B}/(\sqrt{N_A}+\sqrt{N_B})$ and ${\chi }_c=(N_A^{-1/2}+N_B^{-1/2}{)}^2/2$.

Compensated summation retains a correction for low-order information lost to roundoff error in successive floating-point arithmetic additions. Kahan summation is a common implementation; it can substantially reduce accumulated error when many terms are added.

A finite floating-point number is a value representable in a specified floating-point representation. For fixed precision and exponent bounds, only finitely many real values are available. Most real numbers therefore require rounding before storage.

In binary arithmetic, $1/2$ is exact but $1/10$ is not: a finite binary fraction has a reduced denominator that is a power of two. This is a property of the representation, even before any arithmetic operation occurs. The associated IEEE encodings additionally include infinities and NaN; NaN is a special arithmetic datum rather than a real value.^[ref]

A finite normal floating-point representation stores a sign, a significand and a bounded integer exponent, representing $(-1{)}^{s}m{\beta }^e$ in radix $\beta$. Fixing the number of significand digits gives roughly constant relative precision across a wide dynamic range; the gap between adjacent representable values grows with the exponent.

For example, $6.5=1.101_2{2}^2$ in the binary numeral system. The normal leading binary digit is one and can be implicit in storage. Subnormal numbers use a different leading-digit rule near zero. IEEE formats also reserve encodings for signed zero, infinities and NaN, so the bit pattern cannot always be interpreted by the normal finite formula.^[ref]

Floating-point underflow occurs when a nonzero numerical result is smaller in magnitude than the representable range or loses precision in the subnormal range. A long product of probabilities can become zero despite a mathematically positive result. Adding logarithms avoids forming the product; adding alternative path probabilities then requires log-sum-exp or controlled rescaling.

The flat barotropic matter-vacuum solution has $H_{\Lambda }/H_0=\sqrt{{\Omega }_{\Lambda 0}}$ and therefore the displayed age, measured from its Big Bang. At fixed $H_0$ and density fractions it is $1/(1+w)$ times the dust-plus-vacuum age. Its $w=0$ case is the age of a flat matter-Lambda universe.

For a constant-$w$ material fluid and positive vacuum energy, the Friedmann acceleration equation gives $\ddot{a}/(a{H}_0^2)={\Omega }_{\Lambda 0}-(1+3w){\Omega }_{m0}(1+z{)}^{3(1+w)}/2$. If $w>-1/3$, its zero determines the displayed transition. A past transition additionally requires the right-hand side of the displayed equation to exceed one. The equation does not imply the same transition for a fluid that already has sufficiently negative pressure.

A freshwater fish used in live embryological studies; rapid external development makes cell tracking and experimental manipulation particularly useful.

Gasterosteus aculeatus, used in experiments on personality, social feedback and collective decisions.