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Conjugate Index
In mathematics, two real numbers p, q>1 are called conjugate indices (or Hölder conjugates) if : \frac + \frac = 1. Formally, we also define q = \infty as conjugate to p=1 and vice versa. Conjugate indices are used in Hölder's inequality, as well as Young's inequality for products; the latter can be used to prove the former. If p, q>1 are conjugate indices, the spaces ''L''''p'' and ''L''''q'' are dual to each other (see ''L''''p'' space). Properties The following are equivalent characterizations of Hölder conjugates: * \frac + \frac = 1, * pq = p + q, * \frac = p - 1, * \frac = q - 1. See also * Beatty's theorem In mathematics, a Beatty sequence (or homogeneous Beatty sequence) is the sequence of integers found by taking the floor of the positive multiples of a positive irrational number. Beatty sequences are named after Samuel Beatty, who wrote about th ... References * Antonevich, A. ''Linear Functional Equations'', Birkhäuser, 1999. . Functional analysi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many areas of mathematics, which include number theory (the study of numbers), algebra (the study of formulas and related structures), geometry (the study of shapes and spaces that contain them), Mathematical analysis, analysis (the study of continuous changes), and set theory (presently used as a foundation for all mathematics). Mathematics involves the description and manipulation of mathematical object, abstract objects that consist of either abstraction (mathematics), abstractions from nature orin modern mathematicspurely abstract entities that are stipulated to have certain properties, called axioms. Mathematics uses pure reason to proof (mathematics), prove properties of objects, a ''proof'' consisting of a succession of applications of in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
