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The geometric mean of a data set {,, …,} is given by: (=) =. [3]The above figure uses capital pi notation to show a series of multiplications. Each side of the equal sign shows that a set of values is multiplied in succession (the number of values is represented by "n") to give a total product of the set, and then the nth root of the total product is taken to give the geometric mean of the ...
If g is a primitive root modulo p, then g is also a primitive root modulo all powers p k unless g p −1 ≡ 1 (mod p 2); in that case, g + p is. [15] If g is a primitive root modulo p k, then g is also a primitive root modulo all smaller powers of p. If g is a primitive root modulo p k, then either g or g + p k (whichever one is odd) is a ...
The trigonometric definition gives an explicit formula ... Chebyshev polynomials of even order have even symmetry and therefore ... Chebyshev cube root; Dickson ...
To determine whether isometry or allometry is present, an expected relationship between variables needs to be determined to compare data to. This is important in determining if the scaling relationship in a dataset deviates from an expected relationship (such as those that follow isometry).
The order of a tensor is the sum of these two numbers. The order (also degree or rank) of a tensor is thus the sum of the orders of its arguments plus the order of the resulting tensor. This is also the dimensionality of the array of numbers needed to represent the tensor with respect to a specific basis, or equivalently, the number of indices ...
Its volume would be multiplied by the cube of 2 and become 8 m 3. The original cube (1 m sides) has a surface area to volume ratio of 6:1. The larger (2 m sides) cube has a surface area to volume ratio of (24/8) 3:1. As the dimensions increase, the volume will continue to grow faster than the surface area. Thus the square–cube law.
Z[ω] (where ω is a primitive (non-real) cube root of unity), the ring of Eisenstein integers. Define f (a + bω) = a 2 − ab + b 2, the norm of the Eisenstein integer a + bω. K[X], the ring of polynomials over a field K. For each nonzero polynomial P, define f (P) to be the degree of P. [4] K[[X]], the ring of formal power series over the ...
Adjoining the real cube root of 2 to the rational numbers gives the cubic field (). This is an example of a pure cubic field, and hence of a complex cubic field. This is an example of a pure cubic field, and hence of a complex cubic field.