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In mathematics, the polar decomposition of a square real or complex matrix is a factorization of the form =, where is a unitary matrix and is a positive semi-definite Hermitian matrix (is an orthogonal matrix and is a positive semi-definite symmetric matrix in the real case), both square and of the same size.
In set theory, a branch of mathematics, a rank-into-rank embedding is a large cardinal property defined by one of the following four axioms given in order of increasing consistency strength. (A set of rank < λ {\displaystyle <\lambda } is one of the elements of the set V λ {\displaystyle V_{\lambda }} of the von Neumann hierarchy .)
The Grandmother chord is an eleven-interval, twelve-note, invertible chord with all of the properties of the Mother chord. Additionally, the intervals are so arranged that they alternate odd and even intervals (counted by semitones) and that the odd intervals successively decrease by one whole-tone while the even intervals successively increase by one whole-tone. [13]
An m × n (read as m by n) order matrix is a set of numbers arranged in m rows and n columns. Matrices of the same order can be added by adding the corresponding elements. Two matrices can be multiplied, the condition being that the number of columns of the first matrix is equal to the number of rows of the second matrix.
The Z-ordering can be used to efficiently build a quadtree (2D) or octree (3D) for a set of points. [4] [5] The basic idea is to sort the input set according to Z-order.Once sorted, the points can either be stored in a binary search tree and used directly, which is called a linear quadtree, [6] or they can be used to build a pointer based quadtree.
In computability theory, computational complexity theory and proof theory, a fast-growing hierarchy (also called an extended Grzegorczyk hierarchy, or a Schwichtenberg-Wainer hierarchy) [1] is an ordinal-indexed family of rapidly increasing functions f α: N → N (where N is the set of natural numbers {0, 1, ...}, and α ranges up to some large countable ordinal).
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Numbers of the form pq where p and q are distinct primes congruent to 3 (mod 4). A016105: Magic numbers: 2, 8, 20, 28, 50, 82, 126, ... A number of nucleons (either protons or neutrons) such that they are arranged into complete shells within the atomic nucleus. A018226: Superperfect numbers