Clay mathematics institute unsolved problems

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clay mathematics institute unsolved problems

The Millennium Problems by Keith J. Devlin

In the hands of Keith Devlin, the Math Guy from NPRs Weekend Edition, each Millennium Problem becomes a fascinating window onto the deepest and toughest questions in the field. For mathematicians, physicists, engineers, and everyone else with an interest in mathematics cutting edge, The Millennium Problems is the definitive account of a subject that will have a very long shelf life.
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The Millennium Prize Problems I

The prime number theorem determines the average distribution of the primes. The Riemann hypothesis tells us about the deviation from the average.
Keith J. Devlin

The Millennium Prize Problems

Experiment and computer simulations suggest the existence of a "mass gap" in the solution to the quantum versions of the Yang-Mills equations. But no proof of this property is known. The prime number theorem determines the average distribution of the primes. The Riemann hypothesis tells us about the deviation from the average. If it is easy to check that a solution to a problem is correct, is it also easy to solve the problem? This is the essence of the P vs NP question.

Libraries and resellers, please contact cust-serv ams. See our librarian page for additional eBook ordering options. Guided by the premise that solving some of the world's most important mathematical problems will advance the field, this book offers a fascinating look at the seven unsolved Millennium Prize problems. This work takes the unprecedented approach of describing these important and difficult problems at the professional level. An essay by Jeremy Gray, a well-known expert in the history of mathematics, outlines the history of prize problems in mathematics and shows how some of mathematics' most important discoveries were first revealed in papers submitted for prizes.

Suppose that you are organizing housing accommodations for a group of four hundred university students. Space is limited and only one hundred of the students will receive places in the dormitory. To complicate matters, the Dean has provided you with a list of pairs of incompatible students, and requested that no pair from this list appear in your final choice. This is an example of what computer scientists call an NP-problem, since it is easy to check if a given choice of one hundred students proposed by a coworker is satisfactory i. Indeed, the total number of ways of choosing one hundred students from the four hundred applicants is greater than the number of atoms in the known universe! Thus no future civilization could ever hope to build a supercomputer capable of solving the problem by brute force; that is, by checking every possible combination of students. However, this apparent difficulty may only reflect the lack of ingenuity of your programmer.

The Millennium Prize Problems. In order to celebrate mathematics.
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The Navier-Stokes equations

The laws of quantum physics stand to the world of elementary particles in the way that Newton's laws of classical mechanics stand to the macroscopic world. Almost half a century ago, Yang and Mills introduced a remarkable new framework to describe elementary particles using structures that also occur in geometry. Quantum Yang-Mills theory is now the foundation of most of elementary particle theory, and its predictions have been tested at many experimental laboratories, but its mathematical foundation is still unclear. The successful use of Yang-Mills theory to describe the strong interactions of elementary particles depends on a subtle quantum mechanical property called the "mass gap": the quantum particles have positive masses, even though the classical waves travel at the speed of light. This property has been discovered by physicists from experiment and confirmed by computer simulations, but it still has not been understood from a theoretical point of view. Progress in establishing the existence of the Yang-Mills theory and a mass gap will require the introduction of fundamental new ideas both in physics and in mathematics. Skip to main content.

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