Irrational numberIn mathematics, the irrational numbers (from in- prefix assimilated to ir- (negative prefix, privative) + rational) are all the real numbers that are not rational numbers. That is, irrational numbers cannot be expressed as the ratio of two integers. When the ratio of lengths of two line segments is an irrational number, the line segments are also described as being incommensurable, meaning that they share no "measure" in common, that is, there is no length ("the measure"), no matter how short, that could be used to express the lengths of both of the two given segments as integer multiples of itself.
Whittaker modelIn representation theory, a branch of mathematics, the Whittaker model is a realization of a representation of a reductive algebraic group such as GL2 over a finite or local or global field on a space of functions on the group. It is named after E. T. Whittaker even though he never worked in this area, because pointed out that for the group SL2(R) some of the functions involved in the representation are Whittaker functions. Irreducible representations without a Whittaker model are sometimes called "degenerate", and those with a Whittaker model are sometimes called "generic".
Lebesgue constantIn mathematics, the Lebesgue constants (depending on a set of nodes and of its size) give an idea of how good the interpolant of a function (at the given nodes) is in comparison with the best polynomial approximation of the function (the degree of the polynomials are fixed). The Lebesgue constant for polynomials of degree at most n and for the set of n + 1 nodes T is generally denoted by Λn(T ). These constants are named after Henri Lebesgue. We fix the interpolation nodes and an interval containing all the interpolation nodes.
Category of small categoriesIn mathematics, specifically in , the category of small categories, denoted by Cat, is the whose objects are all and whose morphisms are functors between categories. Cat may actually be regarded as a with natural transformations serving as 2-morphisms. The initial object of Cat is the empty category 0, which is the category of no objects and no morphisms. The terminal object is the terminal category or trivial category 1 with a single object and morphism. The category Cat is itself a , and therefore not an object of itself.
SinuositySinuosity, sinuosity index, or sinuosity coefficient of a continuously differentiable curve having at least one inflection point is the ratio of the curvilinear length (along the curve) and the Euclidean distance (straight line) between the end points of the curve. This dimensionless quantity can also be rephrased as the "actual path length" divided by the "shortest path length" of a curve. The value ranges from 1 (case of straight line) to infinity (case of a closed loop, where the shortest path length is zero or for an infinitely-long actual path).
Going up and going downIn commutative algebra, a branch of mathematics, going up and going down are terms which refer to certain properties of chains of prime ideals in integral extensions. The phrase going up refers to the case when a chain can be extended by "upward inclusion", while going down refers to the case when a chain can be extended by "downward inclusion". The major results are the Cohen–Seidenberg theorems, which were proved by Irvin S. Cohen and Abraham Seidenberg. These are known as the going-up and going-down theorems.
Direct sumThe direct sum is an operation between structures in abstract algebra, a branch of mathematics. It is defined differently, but analogously, for different kinds of structures. To see how the direct sum is used in abstract algebra, consider a more elementary kind of structure, the abelian group. The direct sum of two abelian groups and is another abelian group consisting of the ordered pairs where and . To add ordered pairs, we define the sum to be ; in other words addition is defined coordinate-wise.
Krull–Schmidt theoremIn mathematics, the Krull–Schmidt theorem states that a group subjected to certain finiteness conditions on chains of subgroups, can be uniquely written as a finite direct product of indecomposable subgroups. We say that a group G satisfies the ascending chain condition (ACC) on subgroups if every sequence of subgroups of G: is eventually constant, i.e., there exists N such that GN = GN+1 = GN+2 = ... . We say that G satisfies the ACC on normal subgroups if every such sequence of normal subgroups of G eventually becomes constant.
Associative algebraIn mathematics, an associative algebra A is an algebraic structure with compatible operations of addition, multiplication (assumed to be associative), and a scalar multiplication by elements in some field K. The addition and multiplication operations together give A the structure of a ring; the addition and scalar multiplication operations together give A the structure of a vector space over K. In this article we will also use the term [[algebra over a field|K-algebra]] to mean an associative algebra over the field K.
Complete fieldIn mathematics, a complete field is a field equipped with a metric and complete with respect to that metric. Basic examples include the real numbers, the complex numbers, and complete valued fields (such as the p-adic numbers). The real numbers are the field with the standard euclidean metric . Since it is constructed from the completion of with respect to this metric, it is a complete field. Extending the reals by its algebraic closure gives the field (since its absolute Galois group is ).
