Hyperreal numberIn mathematics, the system of hyperreal numbers is a way of treating infinite and infinitesimal (infinitely small but non-zero) quantities. The hyperreals, or nonstandard reals, *R, are an extension of the real numbers R that contains numbers greater than anything of the form (for any finite number of terms). Such numbers are infinite, and their reciprocals are infinitesimals. The term "hyper-real" was introduced by Edwin Hewitt in 1948. The hyperreal numbers satisfy the transfer principle, a rigorous version of Leibniz's heuristic law of continuity.
Distributive latticeIn mathematics, a distributive lattice is a lattice in which the operations of join and meet distribute over each other. The prototypical examples of such structures are collections of sets for which the lattice operations can be given by set union and intersection. Indeed, these lattices of sets describe the scenery completely: every distributive lattice is—up to isomorphism—given as such a lattice of sets. As in the case of arbitrary lattices, one can choose to consider a distributive lattice L either as a structure of order theory or of universal algebra.
Formal concept analysisIn information science, formal concept analysis (FCA) is a principled way of deriving a concept hierarchy or formal ontology from a collection of objects and their properties. Each concept in the hierarchy represents the objects sharing some set of properties; and each sub-concept in the hierarchy represents a subset of the objects (as well as a superset of the properties) in the concepts above it. The term was introduced by Rudolf Wille in 1981, and builds on the mathematical theory of lattices and ordered sets that was developed by Garrett Birkhoff and others in the 1930s.
Integral curveIn mathematics, an integral curve is a parametric curve that represents a specific solution to an ordinary differential equation or system of equations. Integral curves are known by various other names, depending on the nature and interpretation of the differential equation or vector field. In physics, integral curves for an electric field or magnetic field are known as field lines, and integral curves for the velocity field of a fluid are known as streamlines.
Monomial orderIn mathematics, a monomial order (sometimes called a term order or an admissible order) is a total order on the set of all (monic) monomials in a given polynomial ring, satisfying the property of respecting multiplication, i.e., If and is any other monomial, then . Monomial orderings are most commonly used with Gröbner bases and multivariate division. In particular, the property of being a Gröbner basis is always relative to a specific monomial order.
Integer (computer science)In computer science, an integer is a datum of integral data type, a data type that represents some range of mathematical integers. Integral data types may be of different sizes and may or may not be allowed to contain negative values. Integers are commonly represented in a computer as a group of binary digits (bits). The size of the grouping varies so the set of integer sizes available varies between different types of computers. Computer hardware nearly always provides a way to represent a processor register or memory address as an integer.
Vector calculus identitiesThe following are important identities involving derivatives and integrals in vector calculus. Gradient For a function in three-dimensional Cartesian coordinate variables, the gradient is the vector field: where i, j, k are the standard unit vectors for the x, y, z-axes. More generally, for a function of n variables , also called a scalar field, the gradient is the vector field: where are orthogonal unit vectors in arbitrary directions. As the name implies, the gradient is proportional to and points in the direction of the function's most rapid (positive) change.
Arithmetic of abelian varietiesIn mathematics, the arithmetic of abelian varieties is the study of the number theory of an abelian variety, or a family of abelian varieties. It goes back to the studies of Pierre de Fermat on what are now recognized as elliptic curves; and has become a very substantial area of arithmetic geometry both in terms of results and conjectures. Most of these can be posed for an abelian variety A over a number field K; or more generally (for global fields or more general finitely-generated rings or fields).
One-parameter groupIn mathematics, a one-parameter group or one-parameter subgroup usually means a continuous group homomorphism from the real line (as an additive group) to some other topological group . If is injective then , the image, will be a subgroup of that is isomorphic to as an additive group. One-parameter groups were introduced by Sophus Lie in 1893 to define infinitesimal transformations. According to Lie, an infinitesimal transformation is an infinitely small transformation of the one-parameter group that it generates.
