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For example, a Fourier series of sine and cosine functions, all continuous, may converge pointwise to a discontinuous function such as a step function. Carmichael's totient function conjecture was stated as a theorem by Robert Daniel Carmichael in 1907, but in 1922 he pointed out that his proof was incomplete. As of 2016 the problem is still open.
In logic a counterexample disproves the generalization, and does so rigorously in the fields of mathematics and philosophy. [1] For example, the fact that "student John Smith is not lazy" is a counterexample to the generalization "students are lazy", and both a counterexample to, and disproof of, the universal quantification "all students are ...
Charles Akemann and Nik Weaver showed in 2003 that the statement "there exists a counterexample to Naimark's problem which is generated by ℵ 1, elements" is independent of ZFC. Miroslav Bačák and Petr Hájek proved in 2008 that the statement "every Asplund space of density character ω 1 has a renorming with the Mazur intersection property ...
For instance, an example of a first-countable space which is not second-countable is counterexample #3, the discrete topology on an uncountable set. This particular counterexample shows that second-countability does not follow from first-countability. Several other "Counterexamples in ..." books and papers have followed, with similar motivations.
Nagata (1960) gave the following counterexample to Hilbert's problem. The field k is a field containing 48 elements a 1i, ...,a 16i, for i=1, 2, 3 that are algebraically independent over the prime field. The ring R is the polynomial ring k[x 1,...,x 16, t 1,...,t 16] in 32 variables.
The third of Hilbert's list of mathematical problems, presented in 1900, was the first to be solved. The problem is related to the following question: given any two polyhedra of equal volume , is it always possible to cut the first into finitely many polyhedral pieces which can be reassembled to yield the second?
Euclid's Elements was read by anyone who was considered educated in the West until the middle of the 20th century. [10] In addition to theorems of geometry, such as the Pythagorean theorem, the Elements also covers number theory, including a proof that the square root of two is irrational and a proof that there are infinitely many prime numbers.
Many mathematical problems have been stated but not yet solved. These problems come from many areas of mathematics, such as theoretical physics, computer science, algebra, analysis, combinatorics, algebraic, differential, discrete and Euclidean geometries, graph theory, group theory, model theory, number theory, set theory, Ramsey theory, dynamical systems, and partial differential equations.
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