Win a Million Dollars with Mathematics
Published by There are six mathematical problems which attract a prize of a million dollars for anyone who can solve them.
In 2000 the Clay Mathematics Institute declared seven problems worthy of a reward of $1,000,000 (US dollars). So far only one has been solved, leaving six million dollars up for grabs for any maths genius able to solve them.
The solved one is called the Poincaré Conjecture which was a topological problem concerning the nature of a sphere’s surface. The others yet unproven are:
1. P v. NP
This concerns whether it’s true or not that a solution is soluble quickly given that it is verifiable and is part of a branch of computer science.
2. Hodge’s Conjecture
Hodge’s Conjecture states that linear cycles can be expressed as rational combinations of Hodge Cycles.
3. Riemann Hypothesis
This determines the distribution of primes and will help solve other unsolved problems.
4. Yang-Mills Existence
This is a mathematical treatment of particles called gluons which are said to hold quarks together inside an atom’s nucleus.
5. Navier-Stokes Existence
The Navier-Stokes Existence is based on equations describing fluid motion. This should provide important advances in the field of the nature of fluids.
6. Birch & Swinnerton-Dyer Conjecture
This states that it is possible to determine whether equations regarding ellipses have an infinite or finite number of solutions.
The official website concerning the Millennium Prize Problems is http://www.claymath.org/millennium
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http://scienceray.com/technology/the-synthesis-of-petrol-gasoline/
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On September 15, 2010 at 5:40 am
The Navier-Stokes problem, as it is stated by Clay, is solved in the peer-reviewed reputable journal Electon J. Diff. Equ. http://ejde.math.txstate.edu Vol 2010(2010), No.93, pp 1-14.
The proof of Statement D is very simple even for a beginner:
The solutions for zero force are not unique since there are no conditions on pressure. Lemma 2.1 shows an explicit example with a free function g(t). In Theorem 2.3 the function g(t) is selected to have a singularity at a finite value a. Thus, for zero force there exists this blowup solution, but also other solutions. In Theorem 2.4 the force is selected as a feedback force that steers the solution to a unique solution, which is the one in Theorem 2.3. The official problem statement does not exclude feedback forces, so they are allowed. Indeed, they are the only allowed forces unless the problem statement has errors: it states that solutions can be uniquely continued from t00 to some finite t>0. This is not true as pressure has no constraints, unless what is means is feedback forces, which do give unique solutions. The only statement of the external force is that it is a given external force, like gravitation. However, gravitation is not space-periodic, so in Statement D, the external force cannot be just like gravitation, only more or less like gravitation. feefback forces are more or less that, since they are external forces to the controlled system. This article has lasted for two years in arxiv without anybody finding errors, and was published in a journal after a long review 7. July 2010. Many people have read it, reddit.com tried to attack it, Robert Coulter and Terence tao tried to attack it. So far, the proof is considered valid. Either the official problem statement has three errors (false claim of local uniqueness, explicitely stating in Statement B that no restrictions are set to pressure, not excluding feedback forces), or Theorem 3.4 is a proof of Statement D. The corrections, if Clay so chooses to make, are not minor. They show that the posers of the problem did not understand the problem. The errors were caused by a faulty theorem “proven” in Temam (see the paper refs).