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Graphs of functions commonly used in the analysis of algorithms, showing the number of operations versus input size for each function. The following tables list the computational complexity of various algorithms for common mathematical operations.
A mnemonic for the products of the unit octonions [9] A 3D mnemonic visualization showing the 7 triads as hyperplanes through the real (e 0) vertex of the octonion example given above [9] A convenient mnemonic for remembering the products of unit octonions is given by the diagram, which represents the multiplication table of Cayley and Graves.
In geometry, a 9-cube is a nine-dimensional hypercube with 512 vertices, 2304 edges, 4608 square faces, 5376 cubic cells, 4032 tesseract 4-faces, 2016 5-cube 5-faces, 672 6-cube 6-faces, 144 7-cube 7-faces, and 18 8-cube 8-faces. It can be named by its Schläfli symbol {4,3 7}, being composed of three 8-cubes around each 7-face.
More generally, the difference between the n th m-gonal number and the n th (m + 1)-gonal number is the (n − 1) th triangular number. For example, the sixth heptagonal number (81) minus the sixth hexagonal number (66) equals the fifth triangular number, 15.
The next number in the sequence (the smallest number of additive persistence 5) is 2 × 10 2×(10 22 − 1)/9 − 1 (that is, 1 followed by 2222222222222222222222 9's).
The definition of matrix multiplication is that if C = AB for an n × m matrix A and an m × p matrix B, then C is an n × p matrix with entries = =. From this, a simple algorithm can be constructed which loops over the indices i from 1 through n and j from 1 through p, computing the above using a nested loop:
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