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But it is not sufficient to exclude these values, because they may have been legitimate solutions to the original equation. For example, suppose we multiply both sides of our original equation x + 2 = 0 {\displaystyle x+2=0} by x + 2. {\displaystyle x+2.}
Geometrically, we can see this problem by the following simple case where is a one-dimensional subspace: We want to find the closest approximation to the vector x {\displaystyle x} by a vector x ^ {\displaystyle {\hat {x}}} in the space W {\displaystyle W} .
These examples, one from mathematics and one from natural language, illustrate the concept of vacuous truths: "For any integer x, if x > 5 then x > 3." [11] – This statement is true non-vacuously (since some integers are indeed greater than 5), but some of its implications are only vacuously true: for example, when x is the integer 2, the statement implies the vacuous truth that "if 2 > 5 ...
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Letting be the quantity of output produced (to be chosen), () be sales revenue with a positive first derivative and with a zero value at zero output, () be production costs with a positive first derivative and with a non-negative value at zero output, and be the positive minimal acceptable level of profit, then the problem is a meaningful one ...
However, this condition is not sufficient on its own to ensure that the step size is nearly optimal, since any value of that is sufficiently small will satisfy the condition. Thus, the backtracking line search strategy starts with a relatively large step size, and repeatedly shrinks it by a factor τ ∈ ( 0 , 1 ) {\displaystyle \tau \,\in \,(0 ...
Test validity is the extent to which a test (such as a chemical, physical, or scholastic test) accurately measures what it is supposed to measure.In the fields of psychological testing and educational testing, "validity refers to the degree to which evidence and theory support the interpretations of test scores entailed by proposed uses of tests". [1]