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However, there is a non-constructive proof that shows that linkedness is decidable in polynomial time. The proof relies on the following facts: The set of graphs for which the answer is "yes" is closed under taking minors. I. e., if a graph G can be embedded linklessly in 3-d space, then every minor of G can also be embedded linklessly.
In the view of Errett Bishop, classical mathematics, which includes Robinson's approach to nonstandard analysis, was nonconstructive and therefore deficient in numerical meaning (Feferman 2000). Bishop was particularly concerned about the use of nonstandard analysis in teaching as he discussed in his essay "Crisis in mathematics" (Bishop 1975).
A proof of P ≠ NP would lack the practical computational benefits of a proof that P = NP, but would represent a great advance in computational complexity theory and guide future research. It would demonstrate that many common problems cannot be solved efficiently, so that the attention of researchers can be focused on partial solutions or ...
The following famous example of a nonconstructive proof shows that there exist two irrational numbers a and b such that is a rational number. This proof uses that 2 {\displaystyle {\sqrt {2}}} is irrational (an easy proof is known since Euclid ), but not that 2 2 {\displaystyle {\sqrt {2}}^{\sqrt {2}}} is irrational (this is true, but the proof ...
The original proof that the Hausdorff–Young inequality cannot be extended to > is probabilistic. The proof of the de Leeuw–Kahane–Katznelson theorem (which is a stronger claim) is partially probabilistic. [1] The first construction of a Salem set was probabilistic. [2] Only in 1981 did Kaufman give a deterministic construction.
Such a proof is again a refutation by contradiction. A typical example is the proof of the proposition "there is no smallest positive rational number": assume there is a smallest positive rational number q and derive a contradiction by observing that q / 2 is even smaller than q and still positive.
As is often the case with probabilistic arguments, this theorem is nonconstructive and gives no method of determining an explicit element of the probability space in which no event occurs. However, algorithmic versions of the local lemma with stronger preconditions are also known (Beck 1991; Czumaj and Scheideler 2000).
(which holds, for example, for n = 5 and r = 4), there must exist a coloring in which there are no monochromatic r-subgraphs. [a] By definition of the Ramsey number, this implies that R(r, r) must be bigger than n. In particular, R(r, r) must grow at least exponentially with r. A weakness of this argument is that it is entirely nonconstructive.