Search results
Results From The WOW.Com Content Network
A condition can be both necessary and sufficient. For example, at present, "today is the Fourth of July" is a necessary and sufficient condition for "today is Independence Day in the United States". Similarly, a necessary and sufficient condition for invertibility of a matrix M is that M has a nonzero determinant.
Wherever logic is applied, especially in mathematical discussions, it has the same meaning as above: it is an abbreviation for if and only if, indicating that one statement is both necessary and sufficient for the other. This is an example of mathematical jargon (although, as noted above, if is more often used than iff in statements of definition).
Thus, tr AA* = tr BB* is a necessary condition for unitary equivalence, but it is not sufficient. Specht's theorem gives infinitely many necessary conditions which together are also sufficient. The formulation of the theorem uses the following definition. A word in two variables, say x and y, is an expression of the form
An example traditionally used by logicians contrasting sufficient and necessary conditions is the statement "If there is fire, then oxygen is present". An oxygenated environment is necessary for fire or combustion, but simply because there is an oxygenated environment does not necessarily mean that fire or combustion is occurring.
The Arzelà–Ascoli theorem is a fundamental result of mathematical analysis giving necessary and sufficient conditions to decide whether every sequence of a given family of real-valued continuous functions defined on a closed and bounded interval has a uniformly convergent subsequence. The main condition is the equicontinuity of
The necessary conditions are sufficient for optimality if the objective function of a maximization problem is a differentiable concave function, the inequality constraints are differentiable convex functions, the equality constraints are affine functions, and Slater's condition holds. [11]
In the field of complex analysis in mathematics, the Cauchy–Riemann equations, named after Augustin Cauchy and Bernhard Riemann, consist of a system of two partial differential equations which form a necessary and sufficient condition for a complex function of a complex variable to be complex differentiable. These equations are
Necessary and sufficient conditions for contiguity and entire asymptotic separation of probability measures R Sh Liptser et al 1982 Russ. Math. Surv. 37 107–136; The unconscious as infinite sets By Ignacio Matte Blanco, Eric (FRW) Rayner "Contiguity of Probability Measures", David J. Scott, La Trobe University