mathematical analysis

In mathematical analysis, a Cauchy sequence, named after Augustin Cauchy, is a sequence whose elements become close as the sequence progresses. To be more precise, by dropping a finite number of elements from the start of the sequence we can make the distance between any two remaining elements arbitrarily small. Cauchy sequences require the notion of distance so they can only be defined in a metric space. Generalizations to more abstract uniform spa ...

Including:

• Cauchy sequence - Cauchy sequence in a metric space
• Cauchy sequence - Formal definition
• Cauchy sequence - Completeness
• Cauchy sequence - Other properties
• Cauchy sequence - Cauchy sequences in topological vector spaces
• Cauchy sequence - Cauchy sequences in groups

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Avram Noam Chomsky (born December 7, 1928) is the Institute Professor Emeritus of linguistics at the Massachusetts Institute of Technology. Chomsky is credited with the creation of the theory of generative grammar, often considered the most significant contribution to the field of theoretical linguistics of the 20th century. He also helped spark the cognitive revolution in psychology through his review of B. F. Skinner's Verbal Behavior, which challenged the behaviorist approach to the study of mind and language dominant in the ...

Including:

• Noam Chomsky - Biography
• Noam Chomsky - Chomsky's name
• Noam Chomsky - Contributions to linguistics
• Noam Chomsky - Generative grammar
• Noam Chomsky - Chomsky hierarchy
• Noam Chomsky - Contributions to psychology
• Noam Chomsky - Opinion on criticism of science culture
• Noam Chomsky - Chomsky's influence in other fields
• Noam Chomsky - Political views
• Noam Chomsky - Chomsky on terrorism
• Noam Chomsky - Criticism of United States government
• Noam Chomsky - Views on globalization
• Noam Chomsky - Views on socialism
• Noam Chomsky - Mass media analysis
• Noam Chomsky - Chomsky and the Middle East
• Noam Chomsky - Criticism of intellectual communities
• Noam Chomsky - Chomsky's influence as a political activist
• Noam Chomsky - Opposition to the Vietnam War
• Noam Chomsky - Alleged marginalization in the mainstream media
• Noam Chomsky - Worldwide audience
• Noam Chomsky - Criticisms
• Noam Chomsky - Academic Achievements Awards and Honors
• Noam Chomsky - Bibliography
• Noam Chomsky - Linguistics
• Noam Chomsky - Political works
• Noam Chomsky - About Chomsky
• Noam Chomsky - Filmography
• Noam Chomsky - Political contemporaries

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The Right Honourable William Thomson, 1st Baron Kelvin, GCVO, OM, PC, PRS (26 June 1824–17 December 1907) was a Scottish-Irish mathematical physicist and engineer, an outstanding leader in the physical sciences of the 19th century. He did important work in the mathematical analysis of electricity and thermodynamics, and did much to unify the emerging discipline of physics in its modern form. He is also credited for the discovery of the atom. He also enjoyed a second career as a telegraph engineer and inventor, a career that p ...

Including:

• William Thomson 1st Baron Kelvin - Early life and work
• William Thomson 1st Baron Kelvin - Family
• William Thomson 1st Baron Kelvin - Youth
• William Thomson 1st Baron Kelvin - Cambridge
• William Thomson 1st Baron Kelvin - Thermodynamics
• William Thomson 1st Baron Kelvin - Transatlantic cable
• William Thomson 1st Baron Kelvin - Calculations on data-rate
• William Thomson 1st Baron Kelvin - Scientist to engineer
• William Thomson 1st Baron Kelvin - Disaster and triumph
• William Thomson 1st Baron Kelvin - Later expeditions
• William Thomson 1st Baron Kelvin - Other activities and contributions
• William Thomson 1st Baron Kelvin - Geology and theology
• William Thomson 1st Baron Kelvin - Honours
• William Thomson 1st Baron Kelvin - Notes
• William Thomson 1st Baron Kelvin - Bibliography
• William Thomson 1st Baron Kelvin - Kelvin's works
• William Thomson 1st Baron Kelvin - Biography

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Nicolas Bourbaki is the collective allonym under which a group of mainly French 20th-century mathematicians wrote a series of books presenting an exposition of modern advanced mathematics, beginning in 1935. With the goal of founding all of mathematics on set theory, the group strove for utmost rigour and generality, creating some new terminology and concepts along the way. While Nicolas Bourbaki is an invented personage, the Bourbaki group is officially known as the Association des collaborateurs de Nicolas Bourbaki< ...

Including:

• Nicolas Bourbaki - Books by Bourbaki
• Nicolas Bourbaki - Influence on mathematics in general
• Nicolas Bourbaki - The group
• Nicolas Bourbaki - The Bourbaki perspective and its limitations
• Nicolas Bourbaki - Dieudonné as speaker for Bourbaki
• Nicolas Bourbaki - The Bourbachique influence: education institutions trends

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In mathematical analysis, a metric space M is said to be complete (or Cauchy) if every Cauchy sequence of points in M has a limit that is also in M. Intuitively, a space is complete if it "doesn't have any holes", if there aren't any "points missing". For instance, the rational numbers are not complete, because √2 is "missing" even though you can construct a Cauchy sequence of rational numbers that converge to it. (See the examples below.) It is always possible to "fill all the holes", leading to t ...

Including:

• Complete space - Examples
• Complete space - Some theorems
• Complete space - Completion
• Complete space - Topologically complete spaces
• Complete space - Generalisations

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In measure theory (a branch of mathematical analysis), one says that a property holds almost everywhere if the set of elements for which the property does not hold is a null set, i.e. is a set with measure zero. If used for properties of the real numbers, the Lebesgue measure is assumed unless otherwise stated. It is abbreviated a. e.; in older literature one can find p. p. instead, which stands for the e ...

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(1st millennium – 2nd millennium – 3rd millennium – other millennia) 2nd millennium - Events. European crusades in Middle East Mongol Empires in Asia The Black Death The Renaissance in Europe The Protestant Reformation The agricultural and industrial revolutions The rise of nationalism and the nation state European discovery of the Americas and Australia and their colonization European colonization and decolonization in Afri ...

Including:

• 2nd millennium - Events
• 2nd millennium - Significant persons
• 2nd millennium - Inventions Discoveries Introductions
• 2nd millennium - Centuries and Decades

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Mathematics is often defined as the study of topics such as quantity, structure, space, and change. Another view, held by many mathematicians, is that mathematics is the body of knowledge justified by deductive reasoning, starting from axioms and definitions. Practical mathematics, in nearly every society, is used for such purposes as accounting, measuring land, or predicting astronomical events. Mathematical discovery or research often involves discovering and cataloging patterns, without regard for application. Today, the natural sciences, engineering, economics, and medici ...

Including:

• Mathematics - History
• Mathematics - Inspiration pure and applied mathematics and aesthetics
• Mathematics - Notation language and rigor
• Mathematics - Is mathematics a science?
• Mathematics - Overview of fields of mathematics
• Mathematics - Major themes in mathematics
• Mathematics - Quantity
• Mathematics - Structure
• Mathematics - Space
• Mathematics - Change
• Mathematics - Foundations and methods
• Mathematics - Discrete mathematics
• Mathematics - Applied mathematics
• Mathematics - Important theorems
• Mathematics - Important conjectures
• Mathematics - History and the world of mathematicians
• Mathematics - Mathematics and other fields
• Mathematics - Common misconceptions

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Adrien-Marie Legendre (September 18, 1752 - January 10, 1833) was a French mathematician. He made important contributions to statistics, number theory, abstract algebra and mathematical analysis. Most of his work was brought to perfection by others: his work on roots of polynomials inspired Galois theory; Abel's work on elliptic functions was built on Legendre's; some of Gauss' work in statistics and number theory completed that of Legendre. In 1830 he gave a proof of Fermat's last theorem for exponent n = 5, ...

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Algebra is a branch of mathematics which studies structure and quantity. It may be roughly characterized as a generalization and abstraction of arithmetic, in which operations are performed on symbols rather than numbers. It includes elementary algebra, taught to high school students, as well as abstract algebra which covers such structures as groups, rings and fields. Along with geometry and analysis, it is one of the three main branches of mathematics. Algebra - History. The origins of algebra can be trac ...

Including:

• Algebra - History
• Algebra - Classification
• Algebra - Algebras

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In mathematics, a function on a normed vector space is said to vanish at infinity if as For example, the function defined on the real line vanishes at infinity. There is a generalization of this to a locally compact setting. A function f on a locally compact space (which may not have a norm) vanishes at infinity if, given any positive number ε, there is a compact subset K such that whenever the point x

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An analysis is a critical evaluation, usually made by breaking a subject (either material or intellectual) down into its constituent parts, then describing the parts and their relationship to the whole. See also analytic and synthesis and the Scientific Method. As such, it can be applied in many different fields of study: In philosophy: philosophical analysis In mathematics: mathematical analysis real analysis - real numbers complex analysis - holo ...

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In mathematics, almost periodicity is a property of dynamical systems that appear to retrace their paths through phase space, but not exactly. An example would be a planetary system, with planets in orbits moving with periods that are not commensurable (i.e., with a period vector that is not proportional to a vector of integers). A theorem of Kronecker from diophantine approximation can be used to show that any particular configuration that occurs once, will recur to within any specified accuracy: if we wait long enough we can observe the planets all retur ...

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In mathematics, a divergent series is an infinite series that does not converge. That is, divergent series and convergent series are antonyms. If a series converges, the individual terms of the series must approach zero. Thus any series in which the individual terms do not approach zero diverges. The simplest example of a divergent series whose terms do approach zero is the harmonic series The divergence of the harmonic series w ...

Including:

• Divergent series - Theories on methods for summing divergent series

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In cybernetics and control theory, feedback is a process whereby some proportion or in general, function, of the output signal of a system is passed (fed back) to the input. Often this is done intentionally, in order to control the dynamic behaviour of the system. Feedback is observed or used in various areas dealing with complex systems, such as engineering, architecture, economics, and biology. Feedback - Feedback loop. In diagrams that depict information flow in a system, arrowed lines are usually drawn, ...

Including:

• Feedback - Feedback loop
• Feedback - Types of feedback
• Feedback - Feedback in electronic engineering
• Feedback - Feedback in mechanical engineering
• Feedback - Feedback in economics and finance
• Feedback - Feedback in nature
• Feedback - Feedback in organizations
• Feedback - Feedback in gaming
• Feedback - Sources

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This is a non-technical treatment from a historical point of view; see the article Lebesgue integration for a technical treatment from a mathematical point of view. Integration is a mathematical operation that corresponds to the informal idea of finding the area under the graph of a function. The first theory of integration was developed by Archimedes in the third century BC with his method of quadratures, but this could be applied only in limited circumstances with a high degree of geometric symmetry. In the seventeenth century, Isaa ...

See also:

Henri Lebesgue, Henri Lebesgue - Lebesgue's theory of integration, Henri Lebesgue - Lebesgue's other achievements

Read more here: » Henri Lebesgue: Encyclopedia II - Henri Lebesgue - Lebesgue's theory of integration

If a set is not closed, then it cannot be compact. If a set is not closed, then it is either an open set, or it is partially open: part of its boundary is open, by which is meant that that part of the boundary does not belong to the set. Then it is possible to come up with an infinite cover whose elements (which are all, by definition, open) are all subsets of the given open set, but whose boundaries are never tangent to the open boundary of the given set. Non-tangency implies that the elements in the cover will have to approach the boundary by decreasing both their ...

See also:

Heine–Borel theorem, Heine–Borel theorem - History and motivation, Heine–Borel theorem - Discussion of the theorem, Heine–Borel theorem - Generalizations

Read more here: » Heine–Borel theorem: Encyclopedia II - Heine–Borel theorem - Discussion of the theorem

Extended real number line - Limits. We often wish to describe the behavior of a function f(x), as either the argument x or the function value f(x) get "very big" in some sense. For example, consider the function The graph of this function has a horizontal asymptote of y = 0. Geometrically, as we move farther and farther to the right down the x-axis, the value of 1 / x2 gets closer and ...

See also:

Extended real number line, Extended real number line - Motivation, Extended real number line - Limits, Extended real number line - Measure and integration, Extended real number line - Order and topological properties, Extended real number line - Arithmetic operations, Extended real number line - Algebraic properties, Extended real number line - Miscellaneous

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The evolution of mathematics might be seen to be an ever-increasing series of abstractions, or alternatively an expansion of subject matter. The first abstraction was probably that of numbers. The realization that two apples and two oranges do have something in common, namely that they fill the hands of exactly one person, was a breakthrough in human thought. In addition to recognizing how to count concrete objects, prehistoric peoples also recognized how to count abstract quantities, like time -- days, seasons, years. Arithmetic (e.g., addition, subtraction, mul ...

See also:

Mathematics, Mathematics - History, Mathematics - Inspiration pure and applied mathematics and aesthetics, Mathematics - Notation language and rigor, Mathematics - Is mathematics a science?, Mathematics - Overview of fields of mathematics, Mathematics - Major themes in mathematics, Mathematics - Quantity, Mathematics - Structure, Mathematics - Space, Mathematics - Change, Mathematics - Foundations and methods, Mathematics - Discrete mathematics, Mathematics - Applied mathematics, Mathematics - Important theorems, Mathematics - Important conjectures, Mathematics - History and the world of mathematicians, Mathematics - Mathematics and other fields, Mathematics - Mathematical tools, Mathematics - Common misconceptions

Read more here: » Mathematics: Encyclopedia II - Mathematics - History

Jaina mathematicians in ancient India first conceived of logarithms between 200 BC and 400 CE. They performed a number of operations using logarithmic functions to base-2. From the 13th century, logarithmic tables were produced by Muslim mathematicians. Joost Bürgi, a Swiss clockmaker in the employ of the Duke of Hesse-Kassel, first discovered logarithms as a computational tool; however he did not publish his discovery until 1620. The method of logarithms was first publicly propounded in 1614, in a book entitled Mirifici Logarithm ...

See also:

Logarithm, Logarithm - Bases, Logarithm - Other notations, Logarithm - Change of base, Logarithm - Uses of logarithms, Logarithm - Science and engineering, Logarithm - Exponential functions, Logarithm - Easier computations, Logarithm - Calculus, Logarithm - Generalizations, Logarithm - History, Logarithm - Tables of logarithms, Logarithm - Trivia, Logarithm - Unicode glyph, Logarithm - Graphical interpretation, Logarithm - Irrationality, Logarithm - Relationships between binary natural and common logarithms

Read more here: » Logarithm: Encyclopedia II - Logarithm - History