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Affine Geometry
In mathematics, affine geometry is what remains of Euclidean geometry when ignoring (mathematicians often say "forgetting") the metric notions of distance and angle. As the notion of '' parallel lines'' is one of the main properties that is independent of any metric, affine geometry is often considered as the study of parallel lines. Therefore, Playfair's axiom (Given a line L and a point P not on L, there is exactly one line parallel to L that passes through P.) is fundamental in affine geometry. Comparisons of figures in affine geometry are made with affine transformations, which are mappings that preserve alignment of points and parallelism of lines. Affine geometry can be developed in two ways that are essentially equivalent. In synthetic geometry, an affine space is a set of ''points'' to which is associated a set of lines, which satisfy some axioms (such as Playfair's axiom). Affine geometry can also be developed on the basis of linear algebra. In this context an a ...
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Translation Parallelogram
Translation is the communication of the Meaning (linguistic), meaning of a #Source and target languages, source-language text by means of an Dynamic and formal equivalence, equivalent #Source and target languages, target-language text. The English language draws a terminology, terminological distinction (which does not exist in every language) between ''translating'' (a written text) and ''Language interpretation, interpreting'' (oral or Sign language, signed communication between users of different languages); under this distinction, translation can begin only after the appearance of writing within a language community. A translator always risks inadvertently introducing source-language words, grammar, or syntax into the target-language rendering. On the other hand, such "spill-overs" have sometimes imported useful source-language calques and loanwords that have enriched target languages. Translators, including early translators of sacred texts, have helped shape the very l ...
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Function Composition
In mathematics, function composition is an operation that takes two functions and , and produces a function such that . In this operation, the function is applied to the result of applying the function to . That is, the functions and are composed to yield a function that maps in domain to in codomain . Intuitively, if is a function of , and is a function of , then is a function of . The resulting ''composite'' function is denoted , defined by for all in . The notation is read as " of ", " after ", " circle ", " round ", " about ", " composed with ", " following ", " then ", or " on ", or "the composition of and ". Intuitively, composing functions is a chaining process in which the output of function feeds the input of function . The composition of functions is a special case of the composition of relations, sometimes also denoted by \circ. As a result, all properties of composition of relations are true of composition of functions, such as t ...
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Worldline
The world line (or worldline) of an object is the path that an object traces in 4-dimensional spacetime. It is an important concept in modern physics, and particularly theoretical physics. The concept of a "world line" is distinguished from concepts such as an "orbit" or a "trajectory" (e.g., a planet's ''orbit in space'' or the ''trajectory'' of a car on a road) by the ''time'' dimension, and typically encompasses a large area of spacetime wherein perceptually straight paths are recalculated to show their ( relatively) more absolute position states—to reveal the nature of special relativity or gravitational interactions. The idea of world lines originates in physics and was pioneered by Hermann Minkowski. The term is now most often used in relativity theories (i.e., special relativity and general relativity). Usage in physics In physics, a world line of an object (approximated as a point in space, e.g., a particle or observer) is the sequence of spacetime events corresp ...
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Parallel Transport
In geometry, parallel transport (or parallel translation) is a way of transporting geometrical data along smooth curves in a manifold. If the manifold is equipped with an affine connection (a covariant derivative or connection on the tangent bundle), then this connection allows one to transport vectors of the manifold along curves so that they stay ''parallel'' with respect to the connection. The parallel transport for a connection thus supplies a way of, in some sense, moving the local geometry of a manifold along a curve: that is, of ''connecting'' the geometries of nearby points. There may be many notions of parallel transport available, but a specification of one — one way of connecting up the geometries of points on a curve — is tantamount to providing a ''connection''. In fact, the usual notion of connection is the infinitesimal analog of parallel transport. Or, ''vice versa'', parallel transport is the local realization of a connection. As parallel transp ...
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Dover Publications
Dover Publications, also known as Dover Books, is an American book publisher founded in 1941 by Hayward and Blanche Cirker. It primarily reissues books that are out of print from their original publishers. These are often, but not always, books in the public domain. The original published editions may be scarce or historically significant. Dover republishes these books, making them available at a significantly reduced cost. Classic reprints Dover reprints classic works of literature, classical sheet music, and public-domain images from the 18th and 19th centuries. Dover also publishes an extensive collection of mathematical, scientific, and engineering texts. It often targets its reprints at a niche market, such as woodworking. Starting in 2015, the company branched out into graphic novel reprints, overseen by Dover acquisitions editor and former comics writer and editor Drew Ford. Most Dover reprints are photo facsimiles of the originals, retaining the original paginatio ...
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Mathematical Physics
Mathematical physics refers to the development of mathematical methods for application to problems in physics. The '' Journal of Mathematical Physics'' defines the field as "the application of mathematics to problems in physics and the development of mathematical methods suitable for such applications and for the formulation of physical theories". An alternative definition would also include those mathematics that are inspired by physics (also known as physical mathematics). Scope There are several distinct branches of mathematical physics, and these roughly correspond to particular historical periods. Classical mechanics The rigorous, abstract and advanced reformulation of Newtonian mechanics adopting the Lagrangian mechanics and the Hamiltonian mechanics even in the presence of constraints. Both formulations are embodied in analytical mechanics and lead to understanding the deep interplay of the notions of symmetry and conserved quantities during the dynamical evoluti ...
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Hermann Weyl
Hermann Klaus Hugo Weyl, (; 9 November 1885 – 8 December 1955) was a German mathematician, theoretical physicist and philosopher. Although much of his working life was spent in Zürich, Switzerland, and then Princeton, New Jersey, he is associated with the University of Göttingen tradition of mathematics, represented by Carl Friedrich Gauss, David Hilbert and Hermann Minkowski. His research has had major significance for theoretical physics as well as purely mathematical disciplines such as number theory. He was one of the most influential mathematicians of the twentieth century, and an important member of the Institute for Advanced Study during its early years. Weyl contributed to an exceptionally wide range of mathematical fields, including works on space, time, matter, philosophy, logic, symmetry and the history of mathematics. He was one of the first to conceive of combining general relativity with the laws of electromagnetism. Freeman Dyson wrote that Weyl alone b ...
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Erlangen Program
In mathematics, the Erlangen program is a method of characterizing geometries based on group theory and projective geometry. It was published by Felix Klein in 1872 as ''Vergleichende Betrachtungen über neuere geometrische Forschungen.'' It is named after the University Erlangen-Nürnberg, where Klein worked. By 1872, non-Euclidean geometries had emerged, but without a way to determine their hierarchy and relationships. Klein's method was fundamentally innovative in three ways: :* Projective geometry was emphasized as the unifying frame for all other geometries considered by him. In particular, Euclidean geometry was more restrictive than affine geometry, which in turn is more restrictive than projective geometry. :* Klein proposed that group theory, a branch of mathematics that uses algebraic methods to abstract the idea of symmetry, was the most useful way of organizing geometrical knowledge; at the time it had already been introduced into the theory of equations in the form ...
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Felix Klein
Christian Felix Klein (; 25 April 1849 – 22 June 1925) was a German mathematician and mathematics educator, known for his work with group theory, complex analysis, non-Euclidean geometry, and on the associations between geometry and group theory. His 1872 Erlangen program, classifying geometries by their basic symmetry groups, was an influential synthesis of much of the mathematics of the time. Life Felix Klein was born on 25 April 1849 in Düsseldorf, to Prussian parents. His father, Caspar Klein (1809–1889), was a Prussian government official's secretary stationed in the Rhine Province. His mother was Sophie Elise Klein (1819–1890, née Kayser). He attended the Gymnasium in Düsseldorf, then studied mathematics and physics at the University of Bonn, 1865–1866, intending to become a physicist. At that time, Julius Plücker had Bonn's professorship of mathematics and experimental physics, but by the time Klein became his assistant, in 1866, Plücker's interest ...
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August Möbius
August is the eighth month of the year in the Julian and Gregorian calendars, and the fifth of seven months to have a length of 31 days. Its zodiac sign is Leo and was originally named ''Sextilis'' in Latin because it was the 6th month in the original ten-month Roman calendar under Romulus in 753 BC, with March being the first month of the year. About 700 BC, it became the eighth month when January and February were added to the year before March by King Numa Pompilius, who also gave it 29 days. Julius Caesar added two days when he created the Julian calendar in 46 BC (708 AUC), giving it its modern length of 31 days. In 8 BC, it was renamed in honor of Emperor Augustus. According to a Senatus consultum quoted by Macrobius, he chose this month because it was the time of several of his great triumphs, including the conquest of Egypt. Commonly repeated lore has it that August has 31 days because Augustus wanted his month to match the length of Julius Caesar's July, ...
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Introductio In Analysin Infinitorum
''Introductio in analysin infinitorum'' (Latin: ''Introduction to the Analysis of the Infinite'') is a two-volume work by Leonhard Euler which lays the foundations of mathematical analysis. Written in Latin and published in 1748, the ''Introductio'' contains 18 chapters in the first part and 22 chapters in the second. It has Eneström numbers E101 and E102. Carl Boyer's lectures at the 1950 International Congress of Mathematicians compared the influence of Euler's ''Introductio'' to that of Euclid's '' Elements'', calling the ''Elements'' the foremost textbook of ancient times, and the ''Introductio'' "the foremost textbook of modern times". Boyer also wrote: :The analysis of Euler comes close to the modern orthodox discipline, the study of functions by means of infinite processes, especially through infinite series. :It is doubtful that any other essentially didactic work includes as large a portion of original material that survives in the college courses today...Can be read ...
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Leonhard Euler
Leonhard Euler ( , ; 15 April 170718 September 1783) was a Swiss mathematician, physicist, astronomer, geographer, logician and engineer who founded the studies of graph theory and topology and made pioneering and influential discoveries in many other branches of mathematics such as analytic number theory, complex analysis, and infinitesimal calculus. He introduced much of modern mathematical terminology and notation, including the notion of a mathematical function. He is also known for his work in mechanics, fluid dynamics, optics, astronomy and music theory. Euler is held to be one of the greatest mathematicians in history and the greatest of the 18th century. A statement attributed to Pierre-Simon Laplace expresses Euler's influence on mathematics: "Read Euler, read Euler, he is the master of us all." Carl Friedrich Gauss remarked: "The study of Euler's works will remain the best school for the different fields of mathematics, and nothing else can replace it." Euler i ...
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