Efficient solutions to the vehicle routing problem require tools from combinatorial optimization and integer programming.

Applied mathematics is a branch of mathematics that concerns itself with mathematical methods that are typically used in science, engineering, business, and industry. Thus, "applied mathematics" is a mathematical science with specialized knowledge. The term "applied mathematics" also describes the professional specialty in which mathematicians work on practical problems; as a profession focused on practical problems, applied mathematics focuses on the formulation and study of mathematical models. In the past, practical applications have motivated the development of mathematical theories, which then became the subject of study in pure mathematics, where mathematics is developed primarily for its own sake. Thus, the activity of applied mathematics is vitally connected with research in pure mathematics.

History[link]

A numerical solution to the heat equation on a pump casing model using the finite element method.

Historically, applied mathematics consisted principally of applied analysis, most notably differential equations; approximation theory (broadly construed, to include representations, asymptotic methods, variational methods, and numerical analysis); and applied probability. These areas of mathematics were intimately tied to the development of Newtonian physics, and in fact the distinction between mathematicians and physicists was not sharply drawn before the mid-19th century. This history left a legacy as well: until the early 20th century subjects such as classical mechanics were often taught in applied mathematics departments at American universities rather than in physics departments, and fluid mechanics may still be taught in applied mathematics departments.^[1] Engineering and computer science departments have traditionally made use of applied mathematics.

Divisions[link]

Fluid mechanics is often considered a branch of applied mathematics.

Today, the term applied mathematics is used in a broader sense. It includes the classical areas above, as well as other areas that have become increasingly important in applications. Even fields such as number theory that are part of pure mathematics are now important in applications (such as cryptography), though they are not generally considered to be part of the field of applied mathematics per se. Sometimes the term applicable mathematics is used to distinguish between the traditional applied mathematics that developed alongside physics and the many areas of mathematics that are applicable to real-world problems today.

There is no consensus as to what the various branches of applied mathematics are. Such categorizations are made difficult by the way mathematics and science change over time, and also by the way universities organize departments, courses, and degrees.

Many mathematicians distinguish between applied mathematics, which is concerned with mathematical methods, and the applications of mathematics within science and engineering. A biologist using a population model and applying known mathematics would not be doing applied mathematics, but rather using it; however mathematical biologists have posed problems that have stimulated the growth of pure mathematics. However, mathematicians like Poincaré and Arnold deny the existence of "applied mathematics" and claim that there are only "applications of mathematics"; similarly, nonmathematicians blend applied mathematics and applications of mathematics. The use and development of mathematics to solve industrial problems is also called industrial mathematics.^[2]

The success of modern numerical mathematical methods and software has led to the emergence of computational mathematics, computational science, and computational engineering, which use high performance computing for the simulation of phenomena and the solution of problems in the sciences and engineering. These are often considered interdisciplinary disciplines.

Utility[link]

Mathematical finance is concerned with the modelling of financial markets.

Historically, mathematics was most important in the natural sciences and engineering. However, since World War II, fields outside of the physical sciences have spawned the creation of new areas of mathematics, such as game theory and social choice theory, which grew out of economic considerations, or neural networks, which arose out of the study of the brain in neuroscience.

The advent of the computer has created new applications: studying and using the new computer technology itself (computer science), using computers to study problems arising in other areas of science (computational science), and studying the mathematics of computation (for example, theoretical computer science, computer algebra, numerical analysis). Statistics is probably the most widespread mathematical science used in the social sciences, but other areas of mathematics are proving increasingly useful in these disciplines, most notably in economics.

Status in academic departments[link]

This unreferenced section requires citations to ensure verifiability.

Academic institutions are not consistent in the way they group and label courses, programs, and degrees in applied mathematics. At some schools, there is a single mathematics department, whereas others have separate departments for Applied Mathematics and (Pure) Mathematics. It is very common for Statistics departments to be separate at schools with graduate programs, but many undergraduate-only institutions include statistics under the mathematics department.

Many applied mathematics programs (as opposed to departments) consist of primarily cross-listed courses and jointly-appointed faculty in departments representing applications. Some Ph.D. programs in applied mathematics require little or no coursework outside of mathematics, while others require substantial coursework in a specific area of application. In some respects this difference reflects the distinction between "application of mathematics" and "applied mathematics".

Some universities in the UK host departments of Applied Mathematics and Theoretical Physics, but it is now much less common to have separate departments of pure and applied mathematics. A notable exception to this is the Department of Applied Mathematics and Theoretical Physics at the University of Cambridge, housing the Lucasian Professor of Mathematics whose past holders include Isaac Newton, Charles Babbage, James Lighthill, Paul Dirac and Stephen Hawking.

Schools with separate applied mathematics departments range from Brown University, which has a well-known and large Division of Applied Mathematics that offers degrees through the doctorate, to Santa Clara University, which offers only the M.S. in applied mathematics.^[3] Research universities dividing their mathematics department into pure and applied sections include Harvard and MIT.

Other associated mathematical sciences[link]

Applied mathematics has substantial overlap with statistics.

Applied mathematics is closely related to other mathematical sciences.

Scientific computing[link]

Scientific computing includes applied mathematics (especially numerical analysis), computing science (especially high-performance computing), and mathematical modelling in a scientific discipline.

Computer science[link]

Computer science relies on logic, algebra, and combinatorics.

Operations research and management science[link]

Operations research and management science are often taught in faculties of engineering, business, public policy.

Statistics[link]

Applied mathematics has substantial overlap with the discipline of statistics. Statistical theorists study and improve statistical procedures with mathematics, and statistical research often raises mathematical questions. Statistical theory relies on probability and decision theory, and makes extensive use of scientific computing, analysis, and optimization; for the design of experiments, statisticians use algebra and combinatorial design. Applied mathematicians and statisticians often work in a department of mathematical sciences (particularly at colleges and small universities).

Actuarial science[link]

Actuarial science uses probability, statistics, and economic theory.

Economics[link]

A standard classification system in mathematics is the Mathematics Subject Classification (MSC). It is used by many mathematics journals to allow authors to classify their articles. In the MSC Applied mathematics/other classification there is category 91:

Game theory, economics, social and behavioral sciences.

MSC2010 classifications for 'Game theory' are at 91A here. Classifications for 'Mathematical economics' are at 91B here. In the latter, subjects in economics are listed, as possibly distinguished from mathematical methods that are used to develop them. For example 'Social choice' (91B14) is a subject in economics, but it is developed using methods of mathematical logic.

The Handbook of Mathematical Economics series (Elsevier), currently 4 volumes, distinguishes between mathematical methods in economics per 1st-page chapter links in the accompanying footnote^[4] and mathematical approaches to different subjects in in economic theory.^[5]

Another source with a similar distinction is The New Palgrave: A Dictionary of Economics (1987, 4 vols., 1,300 subject entries). In it, a "Subject Index" includes mathematical entries under 2 headings (vol. IV, pp. 982–3):

Mathematical Economics (24 listed, such as "acyclicity", "aggregation problem", "comparative statics", "lexicographic orderings", "linear models", "orderings^{[disambiguation needed ]}", and "qualitative economics")

Mathematical Methods (42 listed, such as "calculus of variations", "catastrophe theory", "combinatorics," "computation of general equilibrium", "convexity", "convex programming", and "stochastic optimal control").

A widely-used classification system in economics that includes mathematical methods on the subject is the JEL classification codes. Articles in journals are usually classified according to the system. It was originated by the Journal of Economic Literature for classifying new books and articles. The relevant categories are listed below (simplified below to omit "Miscellaneous" and "Other" JEL codes) with the same Wikipedia links at JEL classification codes#Mathematical and quantitative methods JEL: C Subcategories. The New Palgrave Dictionary of Economics (2008, 2nd ed.) also uses the JEL codes to classify its entries. The corresponding footnotes below have links to abstracts of The New Palgrave Online for each JEL category (10 or fewer per page, similar to Google searches).

JEL: C02 - Mathematical Methods (following JEL: C00 - General and JEL: C01 - Econometrics)

JEL: C6 - Mathematical Methods; Programming Models; Mathematical and Simulation Modeling ^[6]

JEL: C60 - General

JEL: C61 - Optimization techniques; Programming models; Dynamic analysis^[7]

JEL: C62 - Existence and stability conditions of equilibrium^[8]

JEL: C63 - Computational techniques; Simulation modeling^[9]

JEL: C67 - Input–output models

JEL: C68 - Computable General Equilibrium models^[10]

JEL: C7 - Game theory and Bargaining theory^[11]

JEL: C70 - General^[12]

JEL: C71 - Cooperative games^[13]

JEL: C72 - Noncooperative games^[14]

JEL: C73 - Stochastic and Dynamic games; Evolutionary games; Repeated Games^[15]

JEL: C78 - Bargaining theory; Matching theory^[16]

Mathematical methods listed in economics texts include:

• Alpha C. Chiang and Kevin Wainwright, 2005. Fundamental Methods of Mathematical Economics, McGraw-Hill Irwin. Contents.
• E. Roy Weintraub, 1982. Mathematics for Economists, Cambridge. Contents.
• Stephen Glaister, 1984. Mathematical Methods for Economists, 3rd ed., Blackwell. Contents.
• Akira Takayama, 1985. Mathematical Economics, 2nd ed. Cambridge. Contents.
• Nancy L. Stokey and Robert E. Lucas with Edward Prescott, 1989. Recursive Methods in Economic Dynamics, Harvard University Press. Desecription and chapter-preview links.
• A. K. Dixit, [1976] 1990. Optimization in Economic Theory, 2nd ed., Oxford. Description and contents preview.
• Kenneth L. Judd, 1998. Numerical Methods in Economics, MIT Press. Description and chapter-preview links.
• Michael Carter, 2001. Foundations of Mathematical Economics, MIT Press. Contents.
• D. Wade Hands, 2004. Introductory Mathematical Economics, 2nd ed. Oxford. Contents.
• Giancarlo Gandolfo, [1997] 2009. Economic Dynamics, 4th ed., Springer. Description and preview.
• John Stachurski, 2009. Economic Dynamics: Theory and Computation, MIT Press. Description and preview.

Other disciplines[link]

The line between applied mathematics and specific areas of application is often blurred. Many universities teach mathematical and statistical courses outside of the respective departments, in departments and areas including business, engineering, physics, chemistry, psychology, biology, computer science, and mathematical physics.

See also[link]

Mathematics portal

	Book: Applied science
Wikipedia books are collections of articles that can be downloaded or ordered in print.

• Society for Industrial and Applied Mathematics

References[link]

1. ^ Stolz, M. (2002), "The History Of Applied Mathematics And The History Of Society", Synthese 133 (1): 43–57, DOI:10.1023/A:1020823608217, http://www.springerlink.com/index/T152575218M865W4.pdf, retrieved 2009-07-07 
2. ^ University of Strathclyde (17 January 2008), Industrial Mathematics, http://www.maths.strath.ac.uk/applying/postgraduate/research_topics/industrial_mathematics, retrieved 8 January 2009 
3. ^ Santa Clara University Dept of Applied Mathematics, http://www.scu.edu/academics/bulletins/undergraduate/Department-of-Applied-Mathematics.cfm, retrieved 2011-03-05 
4. ^ Kenneth J. Arrow and Michael D. Intriligator, ed. (1981), Handbook of Mathematical Economics, v. 1.
5. ^ • Kenneth J. Arrow and Michael D. Intriligator, ed. (1982). Handbook of Mathematical Economics, v. 2.
       • _____ (1986). Handbook of Mathematical Economics, v. 3.
       • Hildenbrand, Werner, and Hugo Sonnenschein, ed. (1991). Handbook of Mathematical Economics, v. 4.
6. ^ http://www.dictionaryofeconomics.com/search_results?q=&field=content&edition=all&topicid=C6. The JEL Classification Codes Guide for JEL: 6 has this comment: "Covers studies about general issues related to mathematical methods that are of interest to economists."
7. ^ http://www.dictionaryofeconomics.com/search_results?q=&field=content&edition=all&topicid=C61
8. ^ http://www.dictionaryofeconomics.com/search_results?q=&field=content&edition=all&topicid=C62
9. ^ http://www.dictionaryofeconomics.com/search_results?q=&field=content&edition=all&topicid=C63
10. ^ http://www.dictionaryofeconomics.com/search_results?q=&field=content&edition=all&topicid=C68
11. ^ http://www.dictionaryofeconomics.com/search_results?q=&field=content&edition=all&topicid=C7
12. ^ http://www.dictionaryofeconomics.com/search_results?q=&field=content&edition=all&topicid=C70
13. ^ http://www.dictionaryofeconomics.com/search_results?q=&field=content&edition=all&topicid=C71
14. ^ http://www.dictionaryofeconomics.com/search_results?q=&field=content&edition=all&topicid=C72
15. ^ http://www.dictionaryofeconomics.com/search_results?q=&field=content&edition=all&topicid=C73
16. ^ http://www.dictionaryofeconomics.com/search_results?q=&field=content&edition=all&topicid=C78

External links[link]

Wikiversity has learning materials about School of Mathematics

• The Society for Industrial and Applied Mathematics (SIAM) is a professional society dedicated to promoting the interaction between mathematics and other scientific and technical communities. Aside from organizing and sponsoring numerous conferences, SIAM is a major publisher of research journals and books in applied mathematics.

This article needs additional citations for verification. Please help improve this article by adding citations to reliable sources. Unsourced material may be challenged and removed. (March 2011)

Donald O'Connor
in I Love Melvin
Born	Donald David Dixon Ronald O’Connor (1925-08-28)August 28, 1925 Chicago, Illinois
Died	September 27, 2003(2003-09-27) (aged 78) Calabasas, California
Occupation	dancer, singer, actor
Years active	1937-1997
Spouse	Gwen Carter (m.1944-1954; divorced) Gloria Noble (m.1956-2003; his death)

Donald David Dixon Ronald O’Connor (August 28, 1925 - September 27, 2003) was an American dancer, singer, and actor who came to fame in a series of movies in which he co-starred alternately with Gloria Jean, Peggy Ryan, and Francis the Talking Mule. He is best known today for his role as Gene Kelly's friend and colleague Cosmo Brown in Singin' in the Rain (1952).

Early years[link]

Though he considered Danville, Illinois to be his home town, O’Connor was born in St. Elizabeth Hospital in Chicago. His parents, Effie Irene (née Crane) and John Edward "Chuck" O'Connor, were vaudeville entertainers.^[1][2] His father's family was from County Cork, Ireland.^[3] When O'Connor was only a few years old, he and his sister Arlene were in a car crash outside a theater in Hartford, Connecticut; O'Connor survived, but his sister was killed. Several weeks later, his father died of a heart attack while dancing on stage in Brockton, Massachusetts.^[4] O'Connor at the time was being held in the arms of the theater manager Mr. Maurice Sims.

Career[link]

O'Connor began performing in movies in 1937. He appeared opposite Bing Crosby in Sing, You Sinners at age 12. Paramount Pictures used him in both A and B films, including Tom Sawyer, Detective and Beau Geste.^[4] In 1940, when he had outgrown child roles, he returned to vaudeville.

In 1942 O'Connor joined Universal Pictures. He played roles in four of the Gloria Jean musicals, and achieved stardom with Mister Big (1943).

In 1944, O'Connor was drafted into the Army. Before he reported for induction, Universal Pictures rushed him through three feature films, done simultaneously and released when he was overseas. After his discharge, Universal (now reorganized as Universal-International) cast him in lightweight musicals and comedies.

In 1949, he played the lead role in Francis, the story of a soldier befriended by a talking mule. The film was a huge success. However, his musical career was constantly interrupted by his making one Francis film a year until 1955. It was because of Francis that O'Connor missed out on playing Bing Crosby's companion in White Christmas. O'Connor was unavailable because he contracted an illness transmitted by the mule, and was replaced in the film by Danny Kaye. O'Connor's role as Cosmo the piano player in Singin' in the Rain earned him a Golden Globe Award for Best Performance by an Actor in a Comedy or Musical.

Donald O'Connor was a regular host of NBC's Colgate Comedy Hour.^[4] He hosted a color television special on NBC in 1957, one of the earliest color programs to be preserved on a color kinescope; an excerpt of the telecast was included in NBC's 50th anniversary special in 1976. He also had a television series in the late 1960s.

After overcoming alcoholism in the 1970s,^[4] he had a career boost when he hosted the Academy Awards, which earned him two Primetime Emmy nominations. He appeared as a gaslight-era entertainer in the 1981 film Ragtime, notable for similar encore performances by James Cagney and Pat O'Brien. O'Connor appeared in the short-lived Bring Back Birdie on Broadway in 1981, and continued to make film and television appearances into the 1990s. Donald O'Connor's last feature film was the Jack Lemmon-Walter Matthau comedy Out to Sea, in which he played a dance host on a cruise ship. O’Connor was still making public appearances well into 2003.

Death[link]

O'Connor died from congestive heart failure in 2003 at age 78. His last words were reported to have been a tongue-in-cheek thanks to the Academy of Motion Picture Arts and Sciences for a Lifetime Achievement award that he expected to receive at a "future date". His remains were cremated and buried at the Forest Lawn – Hollywood Hills Cemetery in Los Angeles.

O'Connor was survived by his wife, Gloria, and four children.^[4]

References in popular culture[link]

Donald O'Connor's computer-generated likeness appeared, alongside that of Gene Kelly, in a 2011 television commercial, for the Volkswagen Jetta, in which they performed a seated dance in the backseat of a Jetta to illustrate the large amount of leg room available.^[5] O'Connor's signature "Make 'Em Laugh" number has often been repeated by other entertainers. In 1995, Tim Curry, Kathy Najimy, and Mara Wilson reprised it onstage at the Academy Awards.^{[citation needed]} Versions of the routine have also been performed on other television programs, including Glee^[6] and Saturday Night Live.^[7]

Filmography[link]

Film[link]

Television[link]

• The Milton Berle Show (producer) – 1948
• Colgate Comedy Hour – 1951-54
• The Donald O'Connor Show – 19 episodes on NBC, 1954–55
• Petticoat Junction (director) – 1964
• Bell Telephone Hour – 1964-66
• The Donald O'Connor Show - 1968
• Ellery Queen – episode "The Comic Book Crusader" – October 2, 1975
• The Bionic Woman – episode "A Thing of the Past" (February 18, 1976)
• The Love Boat – 1981-84
• Alice – "Guinness on Tap", as himself, 1982
• The Littlest Hobo – episode "The Clown" as Freddie the Clown (1982)^[8]
• Alice in Wonderland (1985 film) as The Lory Bird, 1985
• Tales from the Crypt - 1992
• Frasier – episode "Crane vs. Crane" as Harlow Safford, 1996
• The Nanny – episode "Freida Needa Man" as Fred (1996)

References[link]

External links[link]

Biography portal

Wikimedia Commons has media related to: Donald O'Connor

• Donald O'Connor at Find a Grave
• Donald O'Connor at the Internet Broadway Database
• Donald O'Connor at the Internet Movie Database
• Essay on Donald O'Connor
• Mindy Alloff's 1979 interview with O'Connor
• Donald O'Connor on "The Colgate Comedy Hour" (1951-54) at Classic TV Info.
• Donald O'Connor on "Texaco Star Theater" (1954-55) at Classic TV Info.
• Film-shots