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with the bilinear functions of the and is possible only for n ∈ {1, 2, 4, 8} . However, the more general Pfister's theorem (1965) shows that if the z i {\displaystyle z_{i}} are rational functions of one set of variables, hence has a denominator , then it is possible for all n = 2 m {\displaystyle n=2^{m}} . [ 3 ]
One consequence of Germain's identity is that the numbers of the form + cannot be prime for >. (For =, the result is the prime number 5.)They are obviously not prime if is even, and if is odd they have a factorization given by the identity with = and = /.
The system + =, + = has exactly one solution: x = 1, y = 2 The nonlinear system + =, + = has the two solutions (x, y) = (1, 0) and (x, y) = (0, 1), while + + =, + + =, + + = has an infinite number of solutions because the third equation is the first equation plus twice the second one and hence contains no independent information; thus any value of z can be chosen and values of x and y can be ...
An illustration of the five-point stencil in one and two dimensions (top, and bottom, respectively). In numerical analysis, given a square grid in one or two dimensions, the five-point stencil of a point in the grid is a stencil made up of the point itself together with its four "neighbors".
There are many names for interaction information, including amount of information, [1] information correlation, [2] co-information, [3] and simply mutual information. [4] Interaction information expresses the amount of information (redundancy or synergy) bound up in a set of variables, beyond that which is present in any subset of those variables.
Brahmagupta polynomials are a class of polynomials associated with the Brahmagupa matrix which in turn is associated with the Brahmagupta's identity.The concept and terminology were introduced by E. R. Suryanarayan, University of Rhode Island, Kingston in a paper published in 1996.
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For this purpose, we consider for each x i the y i which is in the same residue class modulo m and between (–m + 1)/2 and m/2 (possibly included). It follows that y 1 2 + y 2 2 + y 3 2 + y 4 2 = mr, for some strictly positive integer r less than m. Finally, another appeal to Euler's four-square identity shows that mpmr = z 1 2 + z 2 2 + z 3 2 ...