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A computation is any type of arithmetic or non-arithmetic calculation that is well-defined. Common examples of computation are mathematical equation solving and the execution of computer algorithms.
The P versus NP problem is a major unsolved problem in theoretical computer science. Informally, it asks whether every problem whose solution can be quickly verified can also be quickly solved.
In mathematics, computable numbers are the real numbers that can be computed to within any desired precision by a finite, terminating algorithm. They are also known as the recursive numbers, effective numbers or the computable reals or recursive reals. The concept of a computable real number was introduced by Emile Borel in 1912, using the intuitive notion of computability available at the time.
In computability theory, the Church–Turing thesis is a thesis about the nature of computable functions. It states that a function on the natural numbers can be calculated by an effective method if and only if it is computable by a Turing machine. The thesis is named after American mathematician Alonzo Church and the British mathematician Alan Turing. Before the precise definition of computable function, mathematicians often used the informal term effectively calculable to describe functions that are computable by paper-and-pencil methods. In the 1930s, several independent attempts were made to formalize the notion of computability:
Herbert Alexander Simon was an American political scientist whose work also influenced the fields of computer science, economics, and cognitive psychology. His primary research interest was decision-making within organizations and he is best known for the theories of "bounded rationality" and "satisficing". He received the Nobel Memorial Prize in Economic Sciences in 1978 and the Turing Award in computer science in 1975. His research was noted for its interdisciplinary nature, spanning the fields of cognitive science, computer science, public administration, management, and political science. He was at Carnegie Mellon University for most of his career, from 1949 to 2001, where he helped found the Carnegie Mellon School of Computer Science, one of the first such departments in the world.
Mathematical logic is the study of formal logic within mathematics. Major subareas include model theory, proof theory, set theory, and recursion theory. Research in mathematical logic commonly addresses the mathematical properties of formal systems of logic such as their expressive or deductive power. However, it can also include uses of logic to characterize correct mathematical reasoning or to establish foundations of mathematics.
In theoretical computer science and mathematics, the theory of computation is the branch that deals with what problems can be solved on a model of computation, using an algorithm, how efficiently they can be solved or to what degree. The field is divided into three major branches: automata theory and formal languages, computability theory, and computational complexity theory, which are linked by the question: "What are the fundamental capabilities and limitations of computers?".
Stephen Arthur Cook is an American-Canadian computer scientist and mathematician who has made significant contributions to the fields of complexity theory and proof complexity. He is a university professor emeritus at the University of Toronto, Department of Computer Science and Department of Mathematics.
Computability theory, also known as recursion theory, is a branch of mathematical logic, computer science, and the theory of computation that originated in the 1930s with the study of computable functions and Turing degrees. The field has since expanded to include the study of generalized computability and definability. In these areas, computability theory overlaps with proof theory and effective descriptive set theory.
"The Unreasonable Effectiveness of Mathematics in the Natural Sciences" is a 1960 article written by the physicist Eugene Wigner, published in Communication in Pure and Applied Mathematics. In it, Wigner observes that a theoretical physics's mathematical structure often points the way to further advances in that theory and to empirical predictions. Mathematical theories often have predictive power in describing nature.
Hypercomputation or super-Turing computation is a set of hypothetical models of computation that can provide outputs that are not Turing-computable. For example, a machine that could solve the halting problem would be a hypercomputer; so too would one that could correctly evaluate every statement in Peano arithmetic.
Intuitively, an algorithmically random sequence is a sequence of binary digits that appears random to any algorithm running on a universal Turing machine. The notion can be applied analogously to sequences on any finite alphabet. Random sequences are key objects of study in algorithmic information theory.
S. Barry Cooper was an English mathematician and computability theorist. He was a professor of Pure Mathematics at the University of Leeds.
The unreasonable ineffectiveness of mathematics is a phrase that alludes to the article by physicist Eugene Wigner, "The Unreasonable Effectiveness of Mathematics in the Natural Sciences". This phrase is meant to suggest that mathematical analysis has not proved as valuable in other fields as it has in physics.
The history of the Church–Turing thesis ("thesis") involves the history of the development of the study of the nature of functions whose values are effectively calculable; or, in more modern terms, functions whose values are algorithmically computable. It is an important topic in modern mathematical theory and computer science, particularly associated with the work of Alonzo Church and Alan Turing.
Mathematical economics is the application of mathematical methods to represent theories and analyze problems in economics. Often, these applied methods are beyond simple geometry, and may include differential and integral calculus, difference and differential equations, matrix algebra, mathematical programming, or other computational methods. Proponents of this approach claim that it allows the formulation of theoretical relationships with rigor, generality, and simplicity.
Richard M. Goodwin was an American mathematician and economist.
Applied mathematics is the application of mathematical methods by different fields such as physics, engineering, medicine, biology, finance, business, computer science, and industry. Thus, applied mathematics is a combination of mathematical science and specialized knowledge. The term "applied mathematics" also describes the professional specialty in which mathematicians work on practical problems by formulating and studying mathematical models.
Jean-Paul Fitoussi was a French economist and sociologist of Sephardi Jewish descent.
Alain A. Lewis is an American mathematician. A student of the mathematical economist Kenneth Arrow, Lewis is credited by the historian of economics Philip Mirowski with making Arrow aware of computational limits to economic agency.
References
- 1 2 "CURRICULUM VITAE : Kumaraswamy (Vela) Velupillai" (PDF). Assru.org. Retrieved 2017-02-23.
- ↑ "CURRICULUM VITAE : Kumaraswamy (Vela) Velupillai" (PDF). Assru.org. Retrieved 2017-02-23.
- ↑ "Computable, Constructive and Behavioural Economic Dynamics: Essays in Honour of Kumaraswamy (Vela) Velupillai (Hardback)". Routledge . Retrieved 2017-02-23.
- ↑ "Origins and Pioneers of Behavioural Economics", "Interdisciplinary Journal of Economics and Business Law", Vol.1, Issue.3, 47-73, 2012. (with Ying-Fang Kao)
- ↑ "The Alan Turing Year - 2012 Turing Centenary". Mathcomp.leeds.ac.uk. Retrieved 2017-02-23.
- ↑ "Premio NORDSUD". www.fondazionepescarabruzzo.it. Archived from the original on 9 October 2011. Retrieved 17 January 2022.
