In algebraic geometry, divisors are a generalization of codimension-1 subvarieties of algebraic varieties.Two different generalizations are in common use, Cartier divisors and Weil divisors (named for Pierre Cartier and André Weil by David Mumford).Both are derived from the notion of divisibility in the integers and algebraic number fields.. Globally, every codimension-1 …

Algebraic geometry is a branch of mathematics, classically studying zeros of multivariate polynomials.Modern algebraic geometry is based on the use of abstract algebraic techniques, mainly from commutative algebra, for solving geometrical problems about these sets of zeros.. The fundamental objects of study in algebraic geometry are algebraic varieties, which are …

In algebraic geometry, universal algebraic geometry generalizes the geometry of rings to geometries of arbitrary varieties of algebras, so that every variety of algebras has its own algebraic geometry.The two terms algebraic variety and variety of algebras should not be confused.. See also. Algebraic geometry; Universal algebra; References. Seven Lectures on …

In classical Euclidean geometry, a point is a primitive notion that models an exact location in space, and has no length, width, or thickness. In modern mathematics, a point refers more generally to an element of some set called a space.. Being a primitive notion means that a point cannot be defined in terms of previously defined objects. That is, a point is defined only by …

Background. Euclidean geometry, named after the Greek mathematician Euclid, includes some of the oldest known mathematics, and geometries that deviated from this were not widely accepted as legitimate until the 19th century.. The debate that eventually led to the discovery of the non-Euclidean geometries began almost as soon as Euclid wrote Elements.In the …

In algebra, ring theory is the study of rings —algebraic structures in which addition and multiplication are defined and have similar properties to those operations defined for the integers.Ring theory studies the structure of rings, their representations, or, in different language, modules, special classes of rings (group rings, division rings, universal enveloping algebras), …

In geometry, the tangent line (or simply tangent) to a plane curve at a given point is the straight line that "just touches" the curve at that point. Leibniz defined it as the line through a pair of infinitely close points on the curve. More precisely, a straight line is said to be a tangent of a curve y = f(x) at a point x = c if the line passes through the point (c, f(c)) on the curve and ...

In mathematics, a cubic function is a function of the form () = + + + where the coefficients a, b, c, and d are complex numbers, and the variable x takes real values, and .In other words, it is both a polynomial function of degree three, and a real function.In particular, the domain and the codomain are the set of the real numbers.. Setting f(x) = 0 produces a cubic equation of the form

In applied mathematics, topological based data analysis (TDA) is an approach to the analysis of datasets using techniques from topology.Extraction of information from datasets that are high-dimensional, incomplete and noisy is generally challenging. TDA provides a general framework to analyze such data in a manner that is insensitive to the particular metric chosen and provides …

Differential geometry is a mathematical discipline that studies the geometry of smooth shapes and smooth spaces, otherwise known as smooth manifolds.It uses the techniques of differential calculus, integral calculus, linear algebra and multilinear algebra.The field has its origins in the study of spherical geometry as far back as antiquity.It also relates to astronomy, the geodesy …