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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.
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.
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:
Necessary and sufficient conditions can be explained by analogy in terms of the concepts and the rules of immediate inference of traditional logic. In the categorical proposition "All S is P ", the subject term S is said to be distributed, that is, all members of its class are exhausted in its expression.
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]
In mathematics, a Hurwitz polynomial (named after German mathematician Adolf Hurwitz) is a polynomial whose roots (zeros) are located in the left half-plane of the complex plane or on the imaginary axis, that is, the real part of every root is zero or negative. [1] Such a polynomial must have coefficients that are positive real numbers.
To prove that this condition is sufficient to guarantee existence of a compatible second-order tensor field, we start with the assumption that a field exists such that =. We will integrate this field to find the vector field v {\displaystyle \mathbf {v} } along a line between points A {\displaystyle A} and B {\displaystyle B} (see Figure 2), i.e.,
Hierholzer proved this is a sufficient condition in a paper published in 1873. This leads to the following necessary and sufficient statement for what a finite graph must have to be Eulerian: An undirected connected finite graph is Eulerian if and only if every vertex of G has even degree. [20]