An ideal dilution estimate for a random copolymer whose comonomer units are excluded from the crystal. Let $x$ be the crystallizable-unit fraction in the melt and $\Delta h_f$ the molar enthalpy of fusion per such unit. Mixing contributes $RT\ln x$ to its liquid chemical potential. Approximate the unmixed liquid-minus-solid chemical-potential difference by $\Delta h_f(1-T/T_m^0)$. Equating the sum to zero gives the displayed relation. For $x<1$, the predicted melting temperature decreases. Nonideal mixing, finite crystallizable sequences, lamellar surface energy and comonomer inclusion can change the result.

On $\mathbb{C}\setminus ((-\infty ,-1]\cup [1,\infty ))$, choose the holomorphic branch $f(z)=\sqrt{1-z}\sqrt{1+z}$, taking each factor's principal square root. This gives $f(x)>0$ for $-1<x<1$. Across the branch cuts, its limits are $f(x+i0)=-i\sqrt{x^2-1}$ and $f(x-i0)=+i\sqrt{x^2-1}$ for $x>1$, and $f(x+i0)=+i\sqrt{x^2-1}$ and $f(x-i0)=-i\sqrt{x^2-1}$ for $x<-1$. The endpoints are branch points; analytic continuation around a single endpoint changes the square-root sign.

For a nonnegative integer $n$, its integer square root is the unique nonnegative integer $r$ satisfying $r^2\le n<(r+1{)}^2$, equivalently $r=\lfloor \sqrt{n}\rfloor$. Testing $r^2=n$ decides whether $n$ is a perfect square using exact integer arithmetic.

For a nonnegative integer $n$, its square root $\sqrt{n}$ is the unique nonnegative real number whose square is $n$. It is an integer exactly when $n$ is a perfect square; otherwise it is an irrational number. Indeed, if $\sqrt{n}=a/b$ with coprime positive integers $a,b$, then $a^2=n{b}^2$ forces every prime divisor of $b$ to divide $a$, so $b=1$.

For example, $\sqrt{49}=7$, whereas $\sqrt{12}=2\sqrt{3}$ is irrational. The integer square root is a different operation: it rounds $\sqrt{n}$ down, so the integer square root of $12$ is $3$.

Movement of spores away from their source. Resistant walls and elevated release can improve dispersal without guaranteeing germination or successful fertilization in the next generation.

A resistant wall material in spores and pollen. It helps protect dispersal stages without eliminating the environmental requirements of later development.

Production of one principal spore type, as in many ferns. The resulting gametophytes may produce both kinds of gametes, with their actual sexual development depending on the species and conditions.

Production of distinct microspores and megaspores that give rise to male and female gametophytes. Retention of developmental stages within spore walls changes protection and provisioning without itself removing the need for water-dependent sperm movement in heterosporous ferns.

An even double-well Landau free energy can have stable homogeneous phases at $M=\pm M_0$. In a selected phase, for example the thermodynamic limit with an infinitesimal symmetry-breaking field, mean magnetisation approaches one of those values. An exactly symmetry-preserving finite-system ensemble instead averages both signs and can have mean zero. This ensemble average does not make $M=0$ a stable homogeneous minimum. Distinguish a free-energy minimum, a domain mixture and an unrestricted ensemble average when discussing spontaneous magnetisation.

A spirocyclic framework containing a dienone ring. An acid-promoted dienone-phenol rearrangement can convert a suitably substituted spiro framework into fused-ring connectivity.

The bipartite GTPase-activating complex that inhibits Tem1 in budding yeast. Its localization and phosphorylation connect the spindle position checkpoint to mitotic exit.

The overlapping spindle microtubule array assembled between segregating chromosome masses during anaphase, contributing to cytokinetic organisation.

The microtubule-organizing structure at a yeast spindle pole. Its location relative to mother and bud contributes to spatial regulation of mitotic exit.

A ceramide contains a long-chain amino alcohol carrying a fatty acid through an amide linkage; sphingosine is a common backbone. Ceramides are central building blocks of more complex sphingolipids and also participate in signal transduction.

Changing the fatty-acid chain changes packing within a cell membrane without changing the identity of the polar linkage. Thus the name ceramide denotes a lipid class, rather than one fixed molecular formula.^[ref]

The angular parts $x/r$, $y/r$ and $z/r$ span the $\ell =1$ spherical harmonic basis. A radial function times $x+y+z$ therefore has definite $L^2=2{\hslash }^2$ and equal probabilities for the three $L_z$ outcomes.

A polar solvent lacking an acidic hydrogen-bond-donor group. Relative to strongly protic solvation, it can leave anionic nucleophiles more reactive, while counterion and solubility effects still matter.

A solvent containing an appropriate proton-donating group, often an O–H or N–H group. It can hydrogen bond, shuttle protons and quench strongly basic organometallic reagents.

A short-range solvent-mediated interaction associated with water organization and hydration near surfaces. It is distinct from an ideal point-ion double layer and can contribute repulsion at separations where continuum far-field models fail.