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George Dantzig

From Wikipedia, the free encyclopedia

George Bernard Dantzig (/ˈdæntsɪɡ/; November 8, 1914 – May 13, 2005) was an American mathematical scientist who made contributions to industrial engineering, operations research, computer science, economics, and statistics.

Dantzig is known for his development of the simplex algorithm,[1] an algorithm for solving linear programming problems, and for his other work with linear programming. In statistics, Dantzig solved two open problems in statistical theory, which he had mistaken for homework after arriving late to a lecture by Jerzy Neyman.[2]

Dantzig was the Professor Emeritus of Transportation Sciences and Professor of Operations Research and of Computer Science at Stanford.

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  • ✪ The Simplest Impossible Problem
  • ✪ George Dantzig Interview
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  • ✪ George Dantzig
  • ✪ How to solve the maths problem from Good Will Hunting

Transcription

Pure mathematics, that is, math for its own sake, has produced fascinating patterns, such as erie strange attractors, or tables of knots. Applied mathematics has been used in many areas, such as heat flow or turbulence. There's one problem, however, which leaves mathematicians utterly defeated. And it only involves simple arithmetic that a seven-year-old can follow. This is definitely the simplest impossible problem. Starting with a positive whole number n, let's produce a new number according to the following rule: if n is even, divide it by 2. If it's odd, multiply by 3, then add 1. For example, let's start with 10. Since 10 is even, divide it by 2 to get 5. 5 is odd, so multiply it by 3 and add 1 to get 16. Keep going to produce 8, 4, 2, 1, 4, 2, 1, etc. This pattern of 4,2,1 repeats forever. Well, this isn't hard. So what's the problem? Try other starting numbers, like 11, 23, or 29. They all eventually reach one. This is the challenge; show that no matter which starting number you choose, the numbers will always reach one. This problem drives mathematicians crazy because there don't seem to be any clear patterns. Sure, some special numbers, such as 8192, which is a power of 2, collapse down to 1 pretty quickly; it takes only 13 steps to get there. However, if you start with 27, it takes 110 steps to reach the number 1. A graph of the points when we start at 27 shows the erratic nature of these numbers. The graph reaches its peak at 9232. Of course researchers have used computers to help out. You can click on this box to enter your own starting number and explore what happens. To date, all starting numbers less than 5x2^60 have been checked and they all eventually reach one. Of course this doesn't prove the conjecture for larger starting numbers, but it does mean that working by hand is not a good idea. This impossible problem is usually called the 3x+1 problem, but it's also known as the Collatz Conjecture, named after Lothar Collatz who invented the problem back in the 1930s. Other mathematicians who were intrigued by the problem mentioned it in their lectures, so this conjecture also became known as Hasse's problem, Kakutani’s Problem, and Ulam's problem. With all this interest, it was joked in 1960 that the 3x+1 problem was part of a conspiracy to slow down mathematical research in the U.S. But getting back to the problem, what could happen if a starting number doesn't reach the cycle {4,2,1}? One possiblity is that it approaches some other cycle. Advanced theory shows that any cycle besides {4,2,1} must have at least 10 billion numbers. The only other possibility is that the numbers would get arbitrarily large and approach infinity. But both of these scenarios are highly unlikely. Over time, mathematicians have built complex theories to try to understand the 3x+1 problem, but they've made little progress. Even the 20th century genius Paul Erdos said about this challenge, "Mathematics is not yet ready for such problems". But hey, but don't let me or Paul discourage you. What can you see in this problem?

Contents

Life

Born in Portland, Oregon, George Bernard Dantzig was named after George Bernard Shaw, the Irish writer.[3][4] Born to Jewish parents, his father, Tobias Dantzig, was a mathematician and linguist, and his mother, Anja Dantzig (née Ourisson), was a linguist of French Jewish origin. Dantzig's parents met during their study at the University of Paris, where Tobias studied mathematics under Henri Poincaré, after whom Dantzig's brother was named.[4] The Dantzigs immigrated to the United States, where they settled in Portland, Oregon.

Early in the 1920s the Dantzig family moved from Baltimore to Washington. His mother became a linguist at the Library of Congress, and his father became a math tutor at the University of Maryland, College Park. Dantzig attended Powell Junior High School and Central High School; one of his friends there was Abraham Seidenberg, who also became a professional mathematician.[4] By the time he reached high school he was already fascinated by geometry, and this interest was further nurtured by his father, challenging him with complicated problems, particularly in projective geometry.[2][4]

George Dantzig received his B.S. from University of Maryland in 1936 in mathematics and physics, which is part of the University of Maryland College of Computer, Mathematical, and Natural Sciences. He earned his master's degree in mathematics from the University of Michigan in 1938. After a two-year period at the Bureau of Labor Statistics, he enrolled in the doctoral program in mathematics at the University of California, Berkeley, where he studied statistics under Jerzy Neyman.

With the outbreak of World War II, Dantzig took a leave of absence from the doctoral program at Berkeley to join the U.S. Air Force Office of Statistical Control. In 1946, he returned to Berkeley to complete the requirements of his program and received his Ph.D. that year.[3] Although he had a faculty offer from Berkeley, he returned to the Air Force as mathematical advisor to the comptroller.[4]

In 1952 Dantzig joined the mathematics division of the RAND Corporation. By 1960 he became a professor in the Department of Industrial Engineering at UC Berkeley, where he founded and directed the Operations Research Center. In 1966 he joined the Stanford faculty as Professor of Operations Research and of Computer Science. A year later, the Program in Operations Research became a full-fledged department. In 1973 he founded the Systems Optimization Laboratory (SOL) there. On a sabbatical leave that year, he headed the Methodology Group at the International Institute for Applied Systems Analysis (IIASA) in Laxenburg, Austria. Later he became the C. A. Criley Professor of Transportation Sciences at Stanford, and kept going, well beyond his mandatory retirement in 1985.[3]

He was a member of the National Academy of Sciences, the National Academy of Engineering, and the American Academy of Arts and Sciences. Dantzig was the recipient of many honors, including the first John von Neumann Theory Prize in 1974, the National Medal of Science in 1975,[5] an honorary doctorate from the University of Maryland, College Park in 1976. The Mathematical Programming Society honored Dantzig by creating the George B. Dantzig Prize, bestowed every three years since 1982 on one or two people who have made a significant impact in the field of mathematical programming.

Dantzig died on May 13, 2005, in his home in Stanford, California, of complications from diabetes and cardiovascular disease. He was 90 years old.[2]

Work

Freund wrote further that "through his research in mathematical theory, computation, economic analysis, and applications to industrial problems, Dantzig contributed more than any other researcher to the remarkable development of linear programming".[6]

Dantzig's work allows the airline industry, for example, to schedule crews and make fleet assignments. Based on his work tools are developed "that shipping companies use to determine how many planes they need and where their delivery trucks should be deployed. The oil industry long has used linear programming in refinery planning, as it determines how much of its raw product should become different grades of gasoline and how much should be used for petroleum-based byproducts. It is used in manufacturing, revenue management, telecommunications, advertising, architecture, circuit design and countless other areas".[2]

Mathematical statistics

An event in Dantzig's life became the origin of a famous story in 1939, while he was a graduate student at UC Berkeley. Near the beginning of a class for which Dantzig was late, professor Jerzy Neyman wrote two examples of famously unsolved statistics problems on the blackboard. When Dantzig arrived, he assumed that the two problems were a homework assignment and wrote them down. According to Dantzig, the problems "seemed to be a little harder than usual", but a few days later he handed in completed solutions for the two problems, still believing that they were an assignment that was overdue.[4][7]

Six weeks later, Dantzig received a visit from an excited professor Neyman, who was eager to tell him that the homework problems he had solved were two of the most famous unsolved problems in statistics.[2][4] He had prepared one of Dantzig's solutions for publication in a mathematical journal.[8] As Dantzig told it in a 1986 interview in the College Mathematics Journal:[9]

A year later, when I began to worry about a thesis topic, Neyman just shrugged and told me to wrap the two problems in a binder and he would accept them as my thesis.

Years later another researcher, Abraham Wald, was preparing to publish an article that arrived at a conclusion for the second problem, and included Dantzig as its co-author when he learned of the earlier solution.[4][10]

This story began to spread and was used as a motivational lesson demonstrating the power of positive thinking. Over time Dantzig's name was removed, and facts were altered, but the basic story persisted in the form of an urban legend and as an introductory scene in the movie Good Will Hunting.[7]

Linear programming

Linear programming is a mathematical method for determining a way to achieve the best outcome (such as maximum profit or lowest cost) in a given mathematical model for some list of requirements represented as linear relationships. Linear programming arose as a mathematical model developed during World War II to plan expenditures and returns in order to reduce costs to the army and increase losses to the enemy. It was kept secret until 1947. Postwar, many industries found its use in their daily planning.

The founders of this subject are Leonid Kantorovich, a Russian mathematician who developed linear programming problems in 1939, Dantzig, who published the simplex method in 1947, and John von Neumann, who developed the theory of the duality in the same year.

Dantzig's original example of finding the best assignment of 70 people to 70 jobs exemplifies the usefulness of linear programming. The computing power required to test all the permutations to select the best assignment is vast; the number of possible configurations exceeds the number of particles in the universe. However, it takes only a moment to find the optimum solution by posing the problem as a linear program and applying the Simplex algorithm. The theory behind linear programming drastically reduces the number of possible optimal solutions that must be checked.

In 1963, Dantzig's Linear Programming and Extensions was published by Princeton University Press. Rich in insight and coverage of significant topics, the book quickly became "the bible" of linear programming.

Publications

Books by George Dantzig:

  • 1953. Notes on linear programming. RAND Corporation.
  • 1956. Linear inequalities and related systems. With others. Edited by H.W. Kuhn and A.W. Tucker. Princeton University Press.
  • 1963. Linear programming and extensions. Princeton University Press and the RAND Corporation. pdf from RAND
  • 1966. On the continuity of the minimum set of a continuous function. With Jon H. Folkman and Norman Shapiro.
  • 1968. Mathematics of the decision sciences. With Arthur F. Veinott, Jr. Summer Seminar on Applied Mathematics 5th : 1967 : Stanford University. American Mathematical Society.
  • 1969. Lectures in differential equations. A. K. Aziz, general editor. Contributors: George B. Dantzig and others.
  • 1970. Natural gas transmission system optimization. With others.
  • 1973. Compact city; a plan for a liveable urban environment. With Thomas L. Saaty.
  • 1974. Studies in optimization. Edited with B.C. Eaves. Mathematical Association of America.
  • 1985. Mathematical programming : essays in honor of George B. Dantzig. Edited by R.W. Cottle. Mathematical Programming Society.
  • 1997. Linear programming 1: Introduction. G.B.D. and Mukund N. Thapa. Springer-Verlag.
  • 2003. Linear programming 2: Theory and Extensions. G.B.D. and Mukund N. Thapa. Springer-Verlag.
  • 2003. The Basic George B. Dantzig. Edited by Richard W. Cottle. Stanford Business Books, Stanford University Press, Stanford, California.[11]

Book chapters:

  • Dantzig, George B. (1960), "General convex objective forms", in Arrow, Kenneth J.; Karlin, Samuel; Suppes, Patrick (eds.), Mathematical models in the social sciences, 1959: Proceedings of the first Stanford symposium, Stanford mathematical studies in the social sciences, IV, Stanford, California: Stanford University Press, pp. 151–158, ISBN 9780804700214.

Articles, a selection:

  • Dantzig, George B. (June 1940). "On the Non-Existence of Tests of 'Student's' Hypothesis Having Power Functions Independent of σ". The Annals of Mathematical Statistics. 11 (2): 186–92. doi:10.1214/aoms/1177731912. JSTOR 2235875.
  • Wood, Marshall K.; Dantzig, George B. (1949). "Programming of Interdependent Activities: I General Discussion". Econometrica. 17 (3/4): 193–9. doi:10.2307/1905522. JSTOR 1905522.
  • Dantzig, George B. (1949). "Programming of Interdependent Activities: II Mathematical Model". Econometrica. 17 (3): 200–211. doi:10.2307/1905523. JSTOR 1905523.
  • Dantzig, George B. (1955). "Optimal Solution of a Dynamic Leontief Model with Substitution". Econometrica. 23 (3): 295–302. doi:10.2307/1910385. JSTOR 1910385.

See also

Notes

  1. ^ Gass, Saul I. (2011). "George B. Dantzig". Profiles in Operations Research. International Series in Operations Research & Management Science. 147. pp. 217–240. doi:10.1007/978-1-4419-6281-2_13. ISBN 978-1-4419-6280-5.
  2. ^ a b c d e Joe Holley (2005). "Obituaries of George Dantzig". In: Washington Post, May 19, 2005; B06
  3. ^ a b c Richard W. Cottle, B. Curtis Eaves and Michael A. Saunders (2006). "Memorial Resolution: George Bernard Dantzig". Stanford Report, June 7, 2006.
  4. ^ a b c d e f g h Albers, Donald J.; Alexanderson, Gerald L.; Reid, Constance, eds. (1990). "George B. Dantzig". More Mathematical People. Harcourt Brace Jovanovich. pp. 60–79. ISBN 978-0-15-158175-7.
  5. ^ National Science Foundation – The President's National Medal of Science
  6. ^ Robert Freund (1994). "Professor George Dantzig: Linear Programming Founder Turns 80". In: SIAM News, November 1994.
  7. ^ a b "The Unsolvable Math Problem". Snopes. June 28, 2011.
  8. ^ Dantzig, George (1940). "On the non-existence of tests of "Student's" hypothesis having power functions independent of σ". The Annals of Mathematical Statistics. 11 (2): 186–192. doi:10.1214/aoms/1177731912.
  9. ^ Allende, Sira M.; Bouza, Carlos N. (2005). "Professor George Bernard Dantzig,  Life & Legend" (PDF). Revista Investigación Operacional. 26 (3): 205–11.
  10. ^ Dantzig, George; Wald, Abraham (1951). "On the Fundamental Lemma of Neyman and Pearson". The Annals of Mathematical Statistics. 22: 87–93. doi:10.1214/aoms/1177729695. Retrieved 14 October 2014.
  11. ^ Todd, Michael J. (2011). "Review: The Basic George B. Dantzig, by Richard W. Cottle". Bull. Amer. Math. Soc. (N.S.). 48 (1): 123–129. doi:10.1090/S0273-0979-2010-01303-3.

Further reading

External links

This page was last edited on 30 April 2019, at 00:38
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