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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 ...
This is a rational number, the Gromov–Witten invariant for the given classes. This number gives a "virtual" count of the number of pseudoholomorphic curves (in the class A {\displaystyle A} , of genus g {\displaystyle g} , with domain in the β {\displaystyle \beta } -part of the Deligne–Mumford space) whose n {\displaystyle n} marked ...
Gneiss, a foliated metamorphic rock. Quartzite, a non-foliated metamorphic rock. Foliation in geology refers to repetitive layering in metamorphic rocks. [1] Each layer can be as thin as a sheet of paper, or over a meter in thickness. [1] The word comes from the Latin folium, meaning "leaf", and refers to the sheet-like planar structure. [1]
In ecology, pressure-volume curves describe the relationship between total water potential (Ψt) and relative water content (R) of living organisms. These values are widely used in research on plant-water relations, and provide valuable information on the turgor , osmotic and elastic properties of plant tissues .
The species–area relationship or species–area curve describes the relationship between the area of a habitat, or of part of a habitat, and the number of species found within that area. Larger areas tend to contain larger numbers of species, and empirically, the relative numbers seem to follow systematic mathematical relationships. [ 1 ]
The modularity of an elliptic curve E of conductor N can be expressed also by saying that there is a non-constant rational map defined over ℚ, from the modular curve X 0 (N) to E. In particular, the points of E can be parametrized by modular functions. For example, a modular parametrization of the curve y 2 − y = x 3 − x is given by [18]
Water retention curve is the relationship between the water content, θ, and the soil water potential, ψ. The soil moisture curve is characteristic for different types of soil, and is also called the soil moisture characteristic. It is used to predict the soil water storage, water supply to the plants (field capacity) and soil aggregate stability.
Faltings's theorem is a result in arithmetic geometry, according to which a curve of genus greater than 1 over the field of rational numbers has only finitely many rational points. This was conjectured in 1922 by Louis Mordell, [1] and known as the Mordell conjecture until its 1983 proof by Gerd Faltings. [2]