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Hard Lefschetz Theorem
In mathematics, specifically in algebraic geometry and algebraic topology, the Lefschetz hyperplane theorem is a precise statement of certain relations between the shape of an algebraic variety and the shape of its subvarieties. More precisely, the theorem says that for a variety ''X'' embedded in projective space and a hyperplane section ''Y'', the homology, cohomology, and homotopy groups of ''X'' determine those of ''Y''. A result of this kind was first stated by Solomon Lefschetz for homology groups of complex algebraic varieties. Similar results have since been found for homotopy groups, in positive characteristic, and in other homology and cohomology theories. A far-reaching generalization of the hard Lefschetz theorem is given by the decomposition theorem. The Lefschetz hyperplane theorem for complex projective varieties Let X be an n-dimensional complex projective algebraic variety in \mathbb\mathbf^N, and let Y be a hyperplane section of X such that U=X\setminus Y is smo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Hardness
In materials science, hardness (antonym: softness) is a measure of the resistance to plastic deformation, such as an indentation (over an area) or a scratch (linear), induced mechanically either by Pressing (metalworking), pressing or abrasion (mechanical), abrasion. In general, different materials differ in their hardness; for example hard metals such as titanium and beryllium are harder than soft metals such as sodium and metallic tin, or wood and common plastics. Macroscopic hardness is generally characterized by strong intermolecular bonds, but the behavior of solid materials under force is complex; therefore, hardness can be measured in different ways, such as scratch hardness, indentation hardness, and rebound hardness. Hardness is dependent on ductility, elasticity (physics), elastic stiffness, plasticity (physics), plasticity, deformation (mechanics), strain, strength of materials, strength, toughness, viscoelasticity, and viscosity. Common examples of hard matter are cer ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |