Category of modulesIn algebra, given a ring R, the category of left modules over R is the whose are all left modules over R and whose morphisms are all module homomorphisms between left R-modules. For example, when R is the ring of integers Z, it is the same thing as the . The category of right modules is defined in a similar way. One can also define the category of bimodules over a ring R but that category is equivalent to the category of left (or right) modules over the enveloping algebra of R (or over the opposite of that).
Diagonalizable matrixIn linear algebra, a square matrix is called diagonalizable or non-defective if it is similar to a diagonal matrix, i.e., if there exists an invertible matrix and a diagonal matrix such that , or equivalently . (Such , are not unique.) For a finite-dimensional vector space , a linear map is called diagonalizable if there exists an ordered basis of consisting of eigenvectors of .
SoilSoil, also commonly referred to as earth, is a mixture of organic matter, minerals, gases, liquids, and organisms that together support life of plants and soil organisms. Some scientific definitions distinguish dirt from soil by restricting the former term specifically to displaced soil. Soil consists of a solid phase of minerals and organic matter (the soil matrix), as well as a porous phase that holds gases (the soil atmosphere) and water (the soil solution). Accordingly, soil is a three-state system of solids, liquids, and gases.
Hamiltonian mechanicsHamiltonian mechanics emerged in 1833 as a reformulation of Lagrangian mechanics. Introduced by Sir William Rowan Hamilton, Hamiltonian mechanics replaces (generalized) velocities used in Lagrangian mechanics with (generalized) momenta. Both theories provide interpretations of classical mechanics and describe the same physical phenomena. Hamiltonian mechanics has a close relationship with geometry (notably, symplectic geometry and Poisson structures) and serves as a link between classical and quantum mechanics.
Soil scienceSoil science is the study of soil as a natural resource on the surface of the Earth including soil formation, classification and mapping; physical, chemical, biological, and fertility properties of soils; and these properties in relation to the use and management of soils. Sometimes terms which refer to branches of soil science, such as pedology (formation, chemistry, morphology, and classification of soil) and edaphology (how soils interact with living things, especially plants), are used as if synonymous with soil science.
