For a field evolving by a Hamiltonian partial differential equation, infinite-dimensional integrability combines sufficiently rich commuting conservation laws with a mechanism such as the inverse scattering transform or a spectral coordinate system that linearizes evolution. Lax equations supply spectral invariants; Lenard-Magri recursions supply commuting Hamiltonians when their solvability conditions hold. Infinite many conserved quantities alone do not prove completeness or an infinite-dimensional action-angle theorem: the function space, boundary conditions and analytic independence all matter.

A side effect is an observable state change accompanying evaluation beyond the returned value. Writing a file, changing a shared variable or modifying an argument object are examples. A function can return the same numeric answer on repeated calls while producing different observable behaviour by appending to a log each time. Replacing the second call with a cached result then changes the program's behaviour. A pure function avoids such observable effects and returns its result from its inputs alone; languages can still provide explicit mechanisms for sequencing necessary effects.

// Target: computer-science.bigb

Online text: www.ncbi.nlm.nih.gov/books/NBK10759/?lang=en-gb

Cited passages: Immunobiology, 5th edition, Glossary: Monoclonal antibodies; Immunobiology, 5th edition, Glossary: affinity, single-site binding.

Online text: www.ncbi.nlm.nih.gov/sites/books/NBK10755/

Cited passages: Immunobiology, 5th edition, Appendix I, antiserum generation: cross-reactive antigens and absorption.

Online text: www.ncbi.nlm.nih.gov/books/NBK27160/

Cited passages: Immunobiology, 5th edition, §3-8: small recognized regions of protein and polysaccharide antigens; Immunobiology, 5th edition, §§3-6, 3-8 and 3-9: paired variable regions, discontinuous epitopes and reversible noncovalent contacts.

Online text: www.ncbi.nlm.nih.gov/books/NBK27144/

Cited passages: Immunobiology, 5th edition, §§3-1, 3-2 and 3-4, Figures 3.2 and 3.4: four chains, paired sites and hinge flexibility.

Additional cited passages: §§3-1–3-3, especially Figure 3.3’s textual caption and the surrounding explanation of papain cleavage. Official search-served section prose was inspected after direct NCBI access encountered reCAPTCHA; no textbook illustration was copied. The four credited names are whole-book authors, with no separate section byline. The year 2001 is the fifth-edition publication year, not a hosting or revision date.

PNG is a file format for raster images with lossless source coding of the encoded image data. Its first W3C Recommendation, dated 1 October 1996, described a portable replacement for earlier image formats; this situates the design in exchanging images reliably between systems.^[ref]

Two allele copies are identical by state when their observed states agree, regardless of whether they derive from one recent ancestral copy. Equal marker haplotypes can therefore reflect either identity by descent or matching founder haplotypes.

Autozygosity is homozygosity by descent: the maternal and paternal copies in one individual derive from the same ancestral copy. Unlike ordinary homozygosity, it concerns the ancestry of chromosome copies rather than only their observed states. In a pedigree with one ancestral disease-allele copy and fully penetrant recessive inheritance, every affected individual is autozygous for that copy.

In an outbred pedigree, the two full siblings receive the same paternal copy with probability $1/2$, and independently the same maternal copy with probability $1/2$. Thus the number $J$ of shared copies has a binomial distribution with probabilities $(1/4,1/2,1/4)$, and $E[J/2]=1/2$.

// Target: biology.bigb

The kinship coefficient ${\varphi }_{XY}$ is the probability that one randomly selected allele copy from each of $X,Y$ is identical by descent. For a noninbred individual, ${\varphi }_{XX}=1/2$, and more generally ${\varphi }_{XX}=(1+F_X)/2$. For distinct individuals with parents $M_X,P_X$, Mendelian segregation gives ${\varphi }_{XY}=[{\varphi }_{M_{X}Y}+{\varphi }_{P_{X}Y}]/2$. The offspring's inbreeding coefficient equals the parents' kinship coefficient.

// Target: biology.bigb

An identical-by-descent segment is a chromosomal tract inherited from the same ancestral chromosome without an intervening breakpoint in the tracked ancestry. With a specified total number $m$ of independent meioses and a uniform recombination map, the probability that a fixed tract of genetic length $d$ survives without genetic recombination is approximately $e^{-md}$. Recent common ancestry therefore tends to give longer shared tracts. Lengths of tracts selected by covering a random point are length-biased relative to tracts sampled by their boundaries.

For first cousins in an otherwise outbred pedigree, compare the copies transmitted from their related parents, who are full siblings. These transmissions trace to the same shared grandparent with probability $1/2$ and then to the same homologous copy of that grandparent with probability $1/2$, giving $k_1=1/4$. The unrelated parental contributions cannot match, so two shared copies have probability zero. Selecting one random copy from each cousin chooses those two potentially shared copies with probability $1/4$, giving kinship coefficient $k_1/4=1/16$.

For two individuals carrying labelled copies $(a,b)$ and $(c,d)$, condensed identity coefficients give the probabilities of nine identity by descent patterns, identifying states that differ only by exchanging the two homologues within either individual. They retain within-individual autozygosity, which three ordinary outbred sharing coefficients omit. One convention isEach block consists of copies identical by descent, and different blocks have distinct ancestral origins. For siblings whose parents are otherwise outbred first cousins, these coefficients are $(1,0,2,1,2,1,15,30,12)/64$. This follows by conditioning on whether the parents share one ancestral copy and enumerating the four independent Mendelian segregation choices to their two children. The kinship coefficient is ${\Delta }_1+({\Delta }_3+{\Delta }_5+{\Delta }_7)/2+{\Delta }_8/4$, here $9/32$. Weir's discussion of identity states uses this convention.

For full siblings in an outbred pedigree, condition on both being affected under multiplicative diallelic penetrance and on residual disease outcomes being independent given their genotypes. The usual Mendelian IBD sharing of full siblings probabilities $(1/4,1/2,1/4)$ are weighted by $p^2{\alpha }^4,p^2{\alpha }^2\beta ,p^2{\beta }^2$, respectively, giving the displayed conditional sharing probabilities. Since $\beta -{\alpha }^2=\varphi (1-\varphi )(\theta -1{)}^2\ge 0$, $z_1\le 1/2$, strictly for a polymorphic locus with a nontrivial effect.