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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.
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 writing, phrases commonly used as alternatives to P "if and only if" Q include: Q is necessary and sufficient for P, for P it is necessary and sufficient that Q, P is equivalent (or materially equivalent) to Q (compare with material implication), P precisely if Q, P precisely (or exactly) when Q, P exactly in case Q, and P just in case Q. [3]
In mathematics, Hall's marriage theorem, proved by Philip Hall (), is a theorem with two equivalent formulations.In each case, the theorem gives a necessary and sufficient condition for an object to exist:
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
It is relatively rare for such sufficient conditions to be also necessary, so that a sharper piece of analysis may extend the domain of validity of formal results. Professionally speaking, therefore, analysts push the envelope of techniques, and expand the meaning of well-behaved for a given context. G. H.
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
In mathematics, Specht's theorem gives a necessary and sufficient condition for two complex matrices to be unitarily equivalent. It is named after Wilhelm Specht, who proved the theorem in 1940. [1] Two matrices A and B with complex number entries are said to be unitarily equivalent if there exists a unitary matrix U such that B = U *AU. [2]