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Intermediate Value Theorem
In mathematical analysis, the intermediate value theorem states that if f is a continuous function whose domain contains the interval , then it takes on any given value between f(a) and f(b) at some point within the interval. This has two important corollaries: # If a continuous function has values of opposite sign inside an interval, then it has a root in that interval (Bolzano's theorem). # The image of a continuous function over an interval is itself an interval. Motivation This captures an intuitive property of continuous functions over the real numbers: given ''f'' continuous on ,2/math> with the known values f(1) = 3 and f(2) = 5, then the graph of y = f(x) must pass through the horizontal line y = 4 while x moves from 1 to 2. It represents the idea that the graph of a continuous function on a closed interval can be drawn without lifting a pencil from the paper. Theorem The intermediate value theorem states the following: Consider an interval I = ,b/math> of real ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Illustration For The Intermediate Value Theorem
An illustration is a decoration, interpretation or visual explanation of a text, concept or process, designed for integration in print and digital published media, such as posters, flyers, magazines, books, teaching materials, animations, video games and films. An illustration is typically created by an illustrator. Digital illustrations are often used to make websites and apps more user-friendly, such as the use of emojis to accompany digital type. llustration also means providing an example; either in writing or in picture form. The origin of the word "illustration" is late Middle English (in the sense ‘illumination; spiritual or intellectual enlightenment’): via Old French from Latin ''illustratio''(n-), from the verb ''illustrare''. Illustration styles Contemporary illustration uses a wide range of styles and techniques, including drawing, painting, printmaking, collage, montage, digital design, multimedia, 3D modelling. Depending on the purpose, illustration ma ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Completeness (order Theory)
In the mathematical area of order theory, completeness properties assert the existence of certain infima or suprema of a given partially ordered set (poset). The most familiar example is the completeness of the real numbers. A special use of the term refers to complete partial orders or complete lattices. However, many other interesting notions of completeness exist. The motivation for considering completeness properties derives from the great importance of suprema (least upper bounds, joins, "\vee") and infima (greatest lower bounds, meets, "\wedge") to the theory of partial orders. Finding a supremum means to single out one distinguished least element from the set of upper bounds. On the one hand, these special elements often embody certain concrete properties that are interesting for the given application (such as being the least common multiple of a set of numbers or the union of a collection of sets). On the other hand, the knowledge that certain types of subsets are guarant ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Louis Arbogast
Louis may refer to: * Louis (coin) * Louis (given name), origin and several individuals with this name * Louis (surname) * Louis (singer), Serbian singer * HMS Louis, HMS ''Louis'', two ships of the Royal Navy See also Derived or associated terms * Lewis (other) * Louie (other) * Luis (other) * Louise (other) * Louisville (other) * Louis Cruise Lines * Louis dressing, for salad * Louis Quinze, design style Associated names * * Chlodwig, the origin of the name Ludwig, which is translated to English as "Louis" * Ladislav and László - names sometimes erroneously associated with "Louis" * Ludovic, Ludwig (other), Ludwig, Ludwick, Ludwik, names sometimes translated to English as "Louis" {{disambiguation ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Foundations Of Science
''Foundations of Science'' is a peer-reviewed interdisciplinary academic journal focussing on methodological and philosophical topics concerning the structure and the growth of science. It is the official journal of the Association for Foundations of Science, Language and Cognition and is published quarterly by Springer Science+Business Media. The journal was established in 1995. The editor in chief is Diederik Aerts. Abstracting and indexing The journal is abstracted and indexed in Arts and Humanities Citation Index, Cengage, EBSCO Databases, FRANCIS, Google Scholar, Mathematical Reviews, PASCAL, Science Citation Index Expanded, Scopus, and Zentralblatt MATH. External links * Journal pageat the Free University of Brussels University of Brussels may refer to several institutions in Brussels, Belgium: Current institutions * Université libre de Bruxelles (ULB), a French-speaking university established as a separate entity in 1970 *Vrije Universiteit Brussel (VUB) ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mikhail Katz
Mikhail "Mischa" Gershevich Katz (born 1958, in Chișinău)Curriculum vitae retrieved 2011-05-23. is an Israeli , a professor of mathematics at . His main interests are , and |
Cubic Function
In mathematics, a cubic function is a function of the form f(x)=ax^3+bx^2+cx+d where the coefficients , , , and are complex numbers, and the variable takes real values, and a\neq 0. In other words, it is both a polynomial function of degree three, and a real function. In particular, the domain and the codomain are the set of the real numbers. Setting produces a cubic equation of the form :ax^3+bx^2+cx+d=0, whose solutions are called roots of the function. A cubic function has either one or three real roots (which may not be distinct); all odd-degree polynomials have at least one real root. The graph of a cubic function always has a single inflection point. It may have two critical points, a local minimum and a local maximum. Otherwise, a cubic function is monotonic. The graph of a cubic function is symmetric with respect to its inflection point; that is, it is invariant under a rotation of a half turn around this point. Up to an affine transformation, there are only thre ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Polynomial
In mathematics, a polynomial is an expression consisting of indeterminates (also called variables) and coefficients, that involves only the operations of addition, subtraction, multiplication, and positive-integer powers of variables. An example of a polynomial of a single indeterminate is . An example with three indeterminates is . Polynomials appear in many areas of mathematics and science. For example, they are used to form polynomial equations, which encode a wide range of problems, from elementary word problems to complicated scientific problems; they are used to define polynomial functions, which appear in settings ranging from basic chemistry and physics to economics and social science; they are used in calculus and numerical analysis to approximate other functions. In advanced mathematics, polynomials are used to construct polynomial rings and algebraic varieties, which are central concepts in algebra and algebraic geometry. Etymology The word ''polynomial'' joins tw ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Simon Stevin
Simon Stevin (; 1548–1620), sometimes called Stevinus, was a Flemish mathematician, scientist and music theorist. He made various contributions in many areas of science and engineering, both theoretical and practical. He also translated various mathematical terms into Dutch, making it one of the few European languages in which the word for mathematics, '' wiskunde'' ('' wis'' and '' kunde'', i.e., "the knowledge of what is certain"), was not a loanword from Greek but a calque via Latin. He also replaced the word ''chemie'', the Dutch for chemistry, by '' scheikunde'' ("the art of separating"), made in analogy with '' wiskunde''. Biography Very little is known with certainty about Simon Stevin's life, and what we know is mostly inferred from other recorded facts. E. J. Dijksterhuis (1970) ''Simon Stevin: Science in the Netherlands around 1600'', The Hague: Martinus Nijhoff Publishers, Dutch original 1943, 's-Gravenhage The exact birth date and the date and place of his death ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Joseph-Louis Lagrange
Joseph-Louis Lagrange (born Giuseppe Luigi LagrangiaJoseph-Louis Lagrange, comte de l’Empire ''Encyclopædia Britannica'' or Giuseppe Ludovico De la Grange Tournier; 25 January 1736 – 10 April 1813), also reported as Giuseppe Luigi Lagrange or Lagrangia, was an Italian and , later naturalized |
Augustin-Louis Cauchy
Baron Augustin-Louis Cauchy (, ; ; 21 August 178923 May 1857) was a French mathematician, engineer, and physicist who made pioneering contributions to several branches of mathematics, including mathematical analysis and continuum mechanics. He was one of the first to state and rigorously prove theorems of calculus, rejecting the heuristic principle of the generality of algebra of earlier authors. He almost singlehandedly founded complex analysis and the study of permutation groups in abstract algebra. A profound mathematician, Cauchy had a great influence over his contemporaries and successors; Hans Freudenthal stated: "More concepts and theorems have been named for Cauchy than for any other mathematician (in elasticity alone there are sixteen concepts and theorems named for Cauchy)." Cauchy was a prolific writer; he wrote approximately eight hundred research articles and five complete textbooks on a variety of topics in the fields of mathematics and mathematical physics. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bernard Bolzano
Bernard Bolzano (, ; ; ; born Bernardus Placidus Johann Gonzal Nepomuk Bolzano; 5 October 1781 – 18 December 1848) was a Bohemian mathematician, logician, philosopher, theologian and Catholic priest of Italian extraction, also known for his liberal views. Bolzano wrote in German, his native language. For the most part, his work came to prominence posthumously. Family Bolzano was the son of two pious Catholics. His father, Bernard Pompeius Bolzano, was an Italian who had moved to Prague, where he married Maria Cecilia Maurer who came from Prague's German-speaking family Maurer. Only two of their twelve children lived to adulthood. Career Bolzano entered the University of Prague in 1796 and studied mathematics, philosophy and physics. In 1796 Bolzano enrolled in the Faculty of Philosophy at the University of Prague. During his studies he wrote: "My special predilection for Mathematics is based in a particular way on its speculative aspects, in other words, I greatly appreci ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Squaring The Circle
Squaring the circle is a problem in geometry first proposed in Greek mathematics. It is the challenge of constructing a square with the area of a circle by using only a finite number of steps with a compass and straightedge. The difficulty of the problem raised the question of whether specified axioms of Euclidean geometry concerning the existence of lines and circles implied the existence of such a square. In 1882, the task was proven to be impossible, as a consequence of the Lindemann–Weierstrass theorem, which proves that pi (\pi) is a transcendental number. That is, \pi is not the root of any polynomial with rational coefficients. It had been known for decades that the construction would be impossible if \pi were transcendental, but that fact was not proven until 1882. Approximate constructions with any given non-perfect accuracy exist, and many such constructions have been found. Despite the proof that it is impossible, attempts to square the circle have be ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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