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Risch Algorithm
In symbolic computation, the Risch algorithm is a method of indefinite integration used in some computer algebra systems to find antiderivatives. It is named after the American mathematician Robert Henry Risch, a specialist in computer algebra who developed it in 1968. The algorithm transforms the problem of integration into a problem in algebra. It is based on the form of the function being integrated and on methods for integrating rational functions, radicals, logarithms, and exponential functions. Risch called it a decision procedure, because it is a method for deciding whether a function has an elementary function as an indefinite integral, and if it does, for determining that indefinite integral. However, the algorithm does not always succeed in identifying whether or not the antiderivative of a given function in fact can be expressed in terms of elementary functions. The complete description of the Risch algorithm takes over 100 pages. The Risch–Norman algorithm is ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Symbolic Computation
In mathematics and computer science, computer algebra, also called symbolic computation or algebraic computation, is a scientific area that refers to the study and development of algorithms and software for manipulating mathematical expressions and other mathematical objects. Although computer algebra could be considered a subfield of scientific computing, they are generally considered as distinct fields because scientific computing is usually based on numerical computation with approximate floating point numbers, while symbolic computation emphasizes ''exact'' computation with expressions containing variables that have no given value and are manipulated as symbols. Software applications that perform symbolic calculations are called ''computer algebra systems'', with the term ''system'' alluding to the complexity of the main applications that include, at least, a method to represent mathematical data in a computer, a user programming language (usually different from the languag ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Differentiable Function
In mathematics, a differentiable function of one real variable is a function whose derivative exists at each point in its domain. In other words, the graph of a differentiable function has a non-vertical tangent line at each interior point in its domain. A differentiable function is smooth (the function is locally well approximated as a linear function at each interior point) and does not contain any break, angle, or cusp. If is an interior point in the domain of a function , then is said to be ''differentiable at'' if the derivative f'(x_0) exists. In other words, the graph of has a non-vertical tangent line at the point . is said to be differentiable on if it is differentiable at every point of . is said to be ''continuously differentiable'' if its derivative is also a continuous function over the domain of the function f. Generally speaking, is said to be of class if its first k derivatives f^(x), f^(x), \ldots, f^(x) exist and are continuous over the domain of the func ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Constant Problem
In mathematics, the constant problem is the problem of deciding whether a given expression is equal to zero. The problem This problem is also referred to as the identity problem or the method of zero estimates. It has no formal statement as such but refers to a general problem prevalent in transcendental number theory. Often proofs in transcendence theory are proofs by contradiction. Specifically, they use some auxiliary function to create an integer ''n'' ≥ 0, which is shown to satisfy ''n'' < 1. Clearly, this means that ''n'' must have the value zero, and so a contradiction arises if one can show that in fact ''n'' is ''not'' zero. In many transcendence proofs, proving that ''n'' ≠ 0 is very difficult, and hence a lot of work has been done to develop methods that can be used to prove the non-vanishing of certain expressions. The sheer generality of the problem is what makes it difficult to prove general results or come up with general metho ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
RE (complexity)
In computability theory and computational complexity theory, RE (recursively enumerable) is the class of decision problems for which a 'yes' answer can be verified by a Turing machine in a finite amount of time. Informally, it means that if the answer to a problem instance is 'yes', then there is some procedure that takes finite time to determine this, and this procedure never falsely reports 'yes' when the true answer is 'no'. However, when the true answer is 'no', the procedure is not required to halt; it may go into an "infinite loop" for some 'no' cases. Such a procedure is sometimes called a semi-algorithm, to distinguish it from an algorithm, defined as a complete solution to a decision problem. Similarly, co-RE is the set of all languages that are complements of a language in RE. In a sense, co-RE contains languages of which membership can be disproved in a finite amount of time, but proving membership might take forever. Equivalent definition Equivalently, RE is the class o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Axiom (computer Algebra System)
Axiom is a free, general-purpose computer algebra system. It consists of an interpreter environment, a compiler and a library, which defines a strongly typed hierarchy. History Two computer algebra systems named Scratchpad were developed by IBM. The first one was started in 1965 by James Griesmer at the request of Ralph Gomory, and written in Fortran. The development of this software was stopped before any public release. The second Scratchpad, originally named Scratchpad II, was developed from 1977 on, at Thomas J. Watson Research Center, under the direction of Richard Dimick Jenks. The design is principally due to Richard D. Jenks (IBM Research), James H. Davenport (University of Bath), Barry M. Trager (IBM Research), David Y.Y. Yun (Southern Methodist University) and Victor S. Miller (IBM Research). Early consultants on the project were David Barton (University of California, Berkeley) and James W. Thatcher (IBM Research). Implementation included Robert Sutor (IBM Resear ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
James H
James is a common English language surname and given name: *James (name), the typically masculine first name James * James (surname), various people with the last name James James or James City may also refer to: People * King James (other), various kings named James * Saint James (other) * James (musician) * James, brother of Jesus Places Canada * James Bay, a large body of water * James, Ontario United Kingdom * James College, a college of the University of York United States * James, Georgia, an unincorporated community * James, Iowa, an unincorporated community * James City, North Carolina * James City County, Virginia ** James City (Virginia Company) ** James City Shire * James City, Pennsylvania * St. James City, Florida Arts, entertainment, and media * ''James'' (2005 film), a Bollywood film * ''James'' (2008 film), an Irish short film * ''James'' (2022 film), an Indian Kannada-language film * James the Red Engine, a character in ''Thomas the Tank En ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Reduce (computer Algebra System)
Reduce is a general-purpose computer algebra system geared towards applications in physics. The development of the Reduce computer algebra system was started in the 1960s by Anthony C. Hearn. Since then, many scientists from all over the world have contributed to its development under his direction. Reduce is written entirely in its own LISP dialect called Portable Standard Lisp, expressed in an ALGOL-like syntax called RLISP. The latter is used as a basis for Reduce's user-level language. Implementations of Reduce are available on most variants of Unix, Linux, Microsoft Windows, or Apple Macintosh systems by using an underlying Portable Standard Lisp or Codemist Standard LISP implementation. The Julia package Reduce.jl uses Reduce as a backend and implements its semantics in Julia style. Reduce was open sourced in December 2008 and is available for free under a modified BSD license on SourceForge. Previously it had cost $695. See also * Comparison of computer algebra ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Macsyma
Macsyma (; "Project MAC's SYmbolic MAnipulator") is one of the oldest general-purpose computer algebra systems still in wide use. It was originally developed from 1968 to 1982 at MIT's Project MAC. In 1982, Macsyma was licensed to Symbolics and became a commercial product. In 1992, Symbolics Macsyma was spun off to Macsyma, Inc., which continued to develop Macsyma until 1999. That version is still available for Microsoft's Windows XP operating system. The 1982 version of MIT Macsyma remained available to academics and US government agencies, and it is distributed by the US Department of Energy (DOE). That version, DOE Macsyma, was maintained by Bill Schelter. Under the name of Maxima, it was released under the GPL in 1999, and remains under active maintenance. Development The project was initiated in July, 1968 by Carl Engelman, William A. Martin (front end, expression display, polynomial arithmetic) and Joel Moses (simplifier, indefinite integration: heuristic/Risch). Martin ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Joel Moses
Joel Moses (24 November 1941 – 29 May 2022) was an Israeli-American mathematician, computer scientist, and Institute Professor at the Massachusetts Institute of Technology (MIT). Biography Joel Moses was born in Mandatory Palestine on 24 November 1941 and emigrated to the United States in 1954. He attended Midwood High School in Brooklyn, New York. He received his Bachelor of Science (B.Sc.) degree in mathematics from Columbia University and a Master of Science (M.Sc.) in mathematics, also from Columbia. Under the supervision of Marvin Minsky, Moses received his Doctor of Philosophy (Ph.D.) in mathematics at MIT in 1967 with a thesis entitled ''Symbolic Integration''. This laid the groundwork for the Macsyma symbolic mathematics program that was created at MIT largely under his supervision between 1969 and 1983. Macsyma was able to solve problems such as simplification, polynomial factorization, indefinite integration, solution of differential equations, and other higher-order ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Yegor Ivanovich Zolotarev
Yegor (Egor) Ivanovich Zolotarev (russian: Его́р Ива́нович Золотарёв) (31 March 1847, Saint Petersburg – 19 July 1878, Saint Petersburg) was a Russian mathematician. Biography Yegor was born as a son of Agafya Izotovna Zolotareva and the merchant Ivan Vasilevich Zolotarev in Saint Petersburg, Imperial Russia. In 1857 he began to study at the fifth St Petersburg gymnasium, a school which centred on mathematics and natural science. He finished it with the silver medal in 1863. In the same year he was allowed to be an auditor at the physico-mathematical faculty of St Petersburg university. He had not been able to become a student before 1864 because he was too young. Among his academic teachers were Somov, Chebyshev and Aleksandr Korkin, with whom he would have a tight scientific friendship. In November 1867 he defended his Kandidat thesis ''“About the Integration of Gyroscope Equations”'', after 10 months there followed his thesis pro venia lege ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pafnuty Chebyshev
Pafnuty Lvovich Chebyshev ( rus, Пафну́тий Льво́вич Чебышёв, p=pɐfˈnutʲɪj ˈlʲvovʲɪtɕ tɕɪbɨˈʂof) ( – ) was a Russian mathematician and considered to be the founding father of Russian mathematics. Chebyshev is known for his fundamental contributions to the fields of probability, statistics, mechanics, and number theory. A number of important mathematical concepts are named after him, including the Chebyshev inequality (which can be used to prove the weak law of large numbers), the Bertrand–Chebyshev theorem, Chebyshev polynomials, Chebyshev linkage, and Chebyshev bias. Transcription The surname Chebyshev has been transliterated in several different ways, like Tchebichef, Tchebychev, Tchebycheff, Tschebyschev, Tschebyschef, Tschebyscheff, Čebyčev, Čebyšev, Chebysheff, Chebychov, Chebyshov (according to native Russian speakers, this one provides the closest pronunciation in English to the correct pronunciation in old Russian), and ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
