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Oskar Schlömilch
Oskar Xavier Schlömilch (13 April 1823 – 7 February 1901) was a Germans, German mathematician, born in Weimar, working in mathematical analysis. He took a doctorate at the University of Jena in 1842, and became a professor at Dresden University of Technology, Dresden Polytechnic in 1849. He is now known as the eponym of the Polygamma function, Schlömilch function, a kind of Bessel function. He was also an important textbook writer, and editor of the journal '' Zeitschrift für Mathematik und Physik'', of which he was a founder in 1856. He published in 1868 for the first time the Missing square puzzle, dissection paradox, earlier invented by Sam Loyd. In 1862, he was elected a foreign member of the Royal Swedish Academy of Sciences. See also *Glasser's master theorem#A special case: the Cauchy–Schlömilch transformation, Cauchy–Schlömilch transformation *Schlömilch's series *Taylor's theorem#Explicit formulas for the remainder, Schömilch form of the remainder * Cau ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Weimar
Weimar is a city in the state of Thuringia, Germany. It is located in Central Germany between Erfurt in the west and Jena in the east, approximately southwest of Leipzig, north of Nuremberg and west of Dresden. Together with the neighbouring cities of Erfurt and Jena, it forms the central metropolitan area of Thuringia, with approximately 500,000 inhabitants. The city itself has a population of 65,000. Weimar is well known because of its large cultural heritage and its importance in German history. The city was a focal point of the German Enlightenment and home of the leading figures of the literary genre of Weimar Classicism, writers Johann Wolfgang von Goethe and Friedrich Schiller. In the 19th century, noted composers such as Franz Liszt made Weimar a music centre. Later, artists and architects such as Henry van de Velde, Wassily Kandinsky, Paul Klee, Lyonel Feininger, and Walter Gropius came to the city and founded the Bauhaus movement, the most important German de ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Sam Loyd
Samuel Loyd (January 30, 1841 – April 10, 1911), was an American chess player, chess composer, puzzle author, and recreational mathematics, recreational mathematician. Loyd was born in Philadelphia but raised in New York City. As a chess composer, he authored a number of chess problems, often with interesting themes. At his peak, Loyd was one of the best chess players in the US, and was ranked 15th in the world, according to chessmetrics.com. He played in the strong Paris 1867 chess tournament (won by Ignatz von Kolisch) with little success, placing near the bottom of the field. Following his death, his book ''Cyclopedia of 5000 Puzzles'' was published (1914) by his son. His son, named after his father, dropped the "Jr" from his name and started publishing reprints of his father's puzzles. Loyd (senior) was inducted into the US Chess Hall of Fame in 1987. Reputation Loyd is widely acknowledged as one of America's great puzzle writers and popularizers, often mentioned as '' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Members Of The Royal Swedish Academy Of Sciences
Member may refer to: * Military jury, referred to as "Members" in military jargon * Element (mathematics), an object that belongs to a mathematical set * In object-oriented programming, a member of a class ** Field (computer science), entries in a database ** Member variable, a variable that is associated with a specific object * Limb (anatomy), an appendage of the human or animal body ** Euphemism for penis * Structural component of a truss, connected by nodes * User (computing), a person making use of a computing service, especially on the Internet * Member (geology), a component of a geological formation * Member of parliament * The Members, a British punk rock band * Meronymy, a semantic relationship in linguistics * Church membership, belonging to a local Christian congregation, a Christian denomination and the universal Church * Member, a participant in a club or learned society A learned society (; also learned academy, scholarly society, or academic association) is an ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Paradox Of Loyd And Schlömilch
The chessboard paradoxGreg N. Frederickson: ''Dissections: Plane and Fancy''. Cambridge University Press, 2003, , chapter 23, pp. 268–277 in particular pp. 271–274 Colin Foster: "Slippery Slopes". In: ''Mathematics in School'', vol. 34, no. 3 (May, 2005), pp. 33–34JSTOR or paradox of Loyd and SchlömilchFranz Lemmermeyer: ''Mathematik à la Carte: Elementargeometrie an Quadratwurzeln mit einigen geschichtlichen Bemerkungen''. Springer 2014, , pp95–96(German) is a falsidical paradox based on an optical illusion. A chessboard or a square with a side length of 8 units is cut into four pieces. Those four pieces are used to form a rectangle with side lengths of 13 and 5 units. Hence the combined area of all four pieces is 64 area units in the square but 65 area units in the rectangle, this seeming contradiction is due an optical illusion as the four pieces don't fit exactly in the rectangle, but leave a small barely visible gap around the rectangle's diagonal. The paradox is ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cauchy Condensation Test
In mathematics, the Cauchy condensation test, named after Augustin-Louis Cauchy, is a standard convergence test for infinite series. For a non-increasing sequence f(n) of non-negative real numbers, the series \sum\limits_^ f(n) converges if and only if the "condensed" series \sum\limits_^ 2^ f(2^) converges. Moreover, if they converge, the sum of the condensed series is no more than twice as large as the sum of the original. Estimate The Cauchy condensation test follows from the stronger estimate, \sum_^ f(n) \leq \sum_^ 2^n f(2^n) \leq\ 2\sum_^ f(n), which should be understood as an inequality of extended real numbers. The essential thrust of a proof follows, patterned after Oresme's proof of the divergence of the harmonic series. To see the first inequality, the terms of the original series are rebracketed into runs whose lengths are powers of two, and then each run is bounded above by replacing each term by the largest term in that run. That term is always the first on ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Taylor's Theorem
In calculus, Taylor's theorem gives an approximation of a ''k''-times differentiable function around a given point by a polynomial of degree ''k'', called the ''k''th-order Taylor polynomial. For a smooth function, the Taylor polynomial is the truncation at the order ''k'' of the Taylor series of the function. The first-order Taylor polynomial is the linear approximation of the function, and the second-order Taylor polynomial is often referred to as the quadratic approximation. There are several versions of Taylor's theorem, some giving explicit estimates of the approximation error of the function by its Taylor polynomial. Taylor's theorem is named after the mathematician Brook Taylor, who stated a version of it in 1715, although an earlier version of the result was already mentioned in 1671 by James Gregory. Taylor's theorem is taught in introductory-level calculus courses and is one of the central elementary tools in mathematical analysis. It gives simple arithmetic formula ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Schlömilch's Series
Schlömilch's series is a Fourier series type expansion of twice continuously differentiable function in the interval (0,\pi) in terms of the Bessel function of the first kind, named after the German mathematician Oskar Schlömilch, who derived the series in 1857. The real-valued function f(x) has the following expansion: :f(x) = a_0 + \sum_^\infty a_n J_0(nx), where :\begin a_0 &= f(0) + \frac \int_0^\pi \int_0^ u f'(u\sin\theta)\ d\theta\ du, \\ a_n &= \frac \int_0^\pi \int_0^ u\cos nu \ f'(u\sin\theta)\ d\theta\ du. \end Examples Some examples of Schlömilch's series are the following: *Null functions in the interval (0,\pi) can be expressed by Schlömilch's Series, 0 = \frac+\sum_^\infty (-1)^n J_0(nx), which cannot be obtained by Fourier Series. This is particularly interesting because the null function is represented by a series expansion in which not all the coefficients are zero. The series converges only when 0 [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
The Cauchy–Schlömilch Transformation
''The'' () is a grammatical article in English, denoting persons or things that are already or about to be mentioned, under discussion, implied or otherwise presumed familiar to listeners, readers, or speakers. It is the definite article in English. ''The'' is the most frequently used word in the English language; studies and analyses of texts have found it to account for seven percent of all printed English-language words. It is derived from gendered articles in Old English which combined in Middle English and now has a single form used with nouns of any gender. The word can be used with both singular and plural nouns, and with a noun that starts with any letter. This is different from many other languages, which have different forms of the definite article for different genders or numbers. Pronunciation In most dialects, "the" is pronounced as (with the voiced dental fricative followed by a schwa) when followed by a consonant sound, and as (homophone of the archai ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Royal Swedish Academy Of Sciences
The Royal Swedish Academy of Sciences ( sv, Kungliga Vetenskapsakademien) is one of the Swedish Royal Academies, royal academies of Sweden. Founded on 2 June 1739, it is an independent, non-governmental scientific organization that takes special responsibility for promoting natural sciences and mathematics and strengthening their influence in society, whilst endeavouring to promote the exchange of ideas between various disciplines. The goals of the academy are: * to be a forum where researchers meet across subject boundaries, * to offer a unique environment for research, * to provide support to younger researchers, * to reward outstanding research efforts, * to communicate internationally among scientists, * to advance the case for science within society and to influence research policy priorities * to stimulate interest in mathematics and science in school, and * to disseminate and popularize scientific information in various forms. Every year, the academy awards the Nobel Priz ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Missing Square Puzzle
The missing square puzzle is an optical illusion used in mathematics classes to help students reason about geometrical figures; or rather to teach them not to reason using figures, but to use only textual descriptions and the axioms of geometry. It depicts two arrangements made of similar shapes in slightly different configurations. Each apparently forms a 13×5 right-angled triangle, but one has a 1×1 hole in it. Solution The key to the puzzle is the fact that neither of the 13×5 "triangles" is truly a triangle, nor would either truly be 13x5 if it were, because what appears to be the hypotenuse is bent. In other words, the "hypotenuse" does not maintain a consistent slope, even though it may appear that way to the human eye. A true 13×5 triangle cannot be created from the given component parts. The four figures (the yellow, red, blue and green shapes) total 32 units of area. The apparent triangles formed from the figures are 13 units wide and 5 units tall, so it appears ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics with the major subdisciplines of number theory, algebra, geometry, and analysis, respectively. There is no general consensus among mathematicians about a common definition for their academic discipline. Most mathematical activity involves the discovery of properties of abstract objects and the use of pure reason to prove them. These objects consist of either abstractions from nature orin modern mathematicsentities that are stipulated to have certain properties, called axioms. A ''proof'' consists of a succession of applications of deductive rules to already established results. These results include previously proved theorems, axioms, andin case of abstraction from naturesome basic properties that are considered true starting points of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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