If solutions exist but " A " does not have full column rank, then we have an indeterminate system, all of whose infinitude of solutions are given by this last equation.
42.
As with the Nyquist Shannon sampling theorem, this theorem also assumes an idealization of any real-world situation, as it only applies to functions that are sampled over an infinitude of points.
43.
This inequality restriction on the exponents has the effect of precluding consideration of the known infinitude of solutions of ( 1 ) in which two of the exponents are 2 ( such as Pythagorean triples ).
44.
Other researchers have characterized the literature as having " yielded an infinitude of publications with very few consistent replications " and called for a phase out of candidate-gene studies in favor of polygenic scores.
45.
In other cases, though, \ bold { w } remains and hence an infinitude of potential values of the free parameter vector \ bold { w } give an infinitude of solutions of the equation.
46.
In other cases, though, \ bold { w } remains and hence an infinitude of potential values of the free parameter vector \ bold { w } give an infinitude of solutions of the equation.
47.
The analog of a weighted average in this context, in which there are an infinitude of possibilities for the precise value of the variable, is called the " mean of the probability distribution ".
48.
Their proof reduced the infinitude of possible maps to 1, 936 reducible configurations ( later reduced to 1, 476 ) which had to be checked one by one by computer and took over a thousand hours.
49.
And as the sky darkened in each story _ or rather, failed to brighten _ I found myself trying to second-guess Trevor's outcomes, which are almost never without the brutal infinitudes of suffering.
50.
Using \ Phi _ n, one can give an elementary proof for the infinitude of primes congruent to 1 modulo " n ", which is a special case of Dirichlet's theorem on arithmetic progressions.
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