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The first key result is the cone theorem of Shigefumi Mori, describing the structure of the cone of curves of . Briefly, the theorem shows that starting with X {\displaystyle X} , one can inductively construct a sequence of varieties X i {\displaystyle X_{i}} , each of which is "closer" than the previous one to having K X i {\displaystyle K_{X ...
Natural patterns are sometimes formed by animals, as in the Mima mounds of the Northwestern United States and some other areas, which appear to be created over many years by the burrowing activities of pocket gophers, [90] while the so-called fairy circles of Namibia appear to be created by the interaction of competing groups of sand termites ...
r-selected species are those that emphasize high growth rates, typically exploit less-crowded ecological niches, and produce many offspring, each of which has a relatively low probability of surviving to adulthood (i.e., high r, low K). [10]
This dilemma can be described as the absence or rarity of transitional varieties in habitat space. [8] Another dilemma, [9] related to the first one, is the absence or rarity of transitional varieties in time. Darwin pointed out that by the theory of natural selection "innumerable transitional forms must have existed", and wondered "why do we ...
In population genetics, directional selection is a type of natural selection in which one extreme phenotype is favored over both the other extreme and moderate phenotypes. This genetic selection causes the allele frequency to shift toward the chosen extreme over time as allele ratios change from generation to generation.
A prerequisite for natural selection to result in adaptive evolution, novel traits and speciation is the presence of heritable genetic variation that results in fitness differences. Genetic variation is the result of mutations, genetic recombinations and alterations in the karyotype (the number, shape, size and internal arrangement of the ...
Gröbner basis computation is one of the main practical tools for solving systems of polynomial equations and computing the images of algebraic varieties under projections or rational maps. Gröbner basis computation can be seen as a multivariate, non-linear generalization of both Euclid's algorithm for computing polynomial greatest common ...
The Gromov-Witten invariants of smooth projective varieties can be defined entirely within algebraic geometry. The classical enumerative geometry of plane curves and of rational curves in homogeneous spaces are both captured by GW invariants.