1.
Analytic geometry
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In classical mathematics, analytic geometry, also known as coordinate geometry, or Cartesian geometry, is the study of geometry using a coordinate system. Analytic geometry is used in physics and engineering, and also in aviation, rocketry, space science. It is the foundation of most modern fields of geometry, including algebraic, differential, discrete, usually the Cartesian coordinate system is applied to manipulate equations for planes, straight lines, and squares, often in two and sometimes in three dimensions. Geometrically, one studies the Euclidean plane and Euclidean space, the numerical output, however, might also be a vector or a shape. That the algebra of the numbers can be employed to yield results about the linear continuum of geometry relies on the Cantor–Dedekind axiom. Apollonius in the Conics further developed a method that is so similar to analytic geometry that his work is thought to have anticipated the work of Descartes by some 1800 years. He further developed relations between the abscissas and the corresponding ordinates that are equivalent to rhetorical equations of curves and that is, equations were determined by curves, but curves were not determined by equations. Coordinates, variables, and equations were subsidiary notions applied to a specific geometric situation, analytic geometry was independently invented by René Descartes and Pierre de Fermat, although Descartes is sometimes given sole credit. Cartesian geometry, the term used for analytic geometry, is named after Descartes. This work, written in his native French tongue, and its philosophical principles, initially the work was not well received, due, in part, to the many gaps in arguments and complicated equations. Only after the translation into Latin and the addition of commentary by van Schooten in 1649 did Descartess masterpiece receive due recognition, Pierre de Fermat also pioneered the development of analytic geometry. Although not published in his lifetime, a form of Ad locos planos et solidos isagoge was circulating in Paris in 1637. Clearly written and well received, the Introduction also laid the groundwork for analytical geometry, as a consequence of this approach, Descartes had to deal with more complicated equations and he had to develop the methods to work with polynomial equations of higher degree. It was Leonard Euler who first applied the method in a systematic study of space curves and surfaces. In analytic geometry, the plane is given a coordinate system, similarly, Euclidean space is given coordinates where every point has three coordinates. The value of the coordinates depends on the choice of the point of origin. These are typically written as an ordered pair and this system can also be used for three-dimensional geometry, where every point in Euclidean space is represented by an ordered triple of coordinates. In polar coordinates, every point of the plane is represented by its distance r from the origin and its angle θ from the polar axis
2.
Curve
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In mathematics, a curve is, generally speaking, an object similar to a line but that need not be straight. Thus, a curve is a generalization of a line, in that curvature is not necessarily zero, various disciplines within mathematics have given the term different meanings depending on the area of study, so the precise meaning depends on context. However, many of these meanings are special instances of the definition which follows, a curve is a topological space which is locally homeomorphic to a line. In everyday language, this means that a curve is a set of points which, near each of its points, looks like a line, a simple example of a curve is the parabola, shown to the right. A large number of curves have been studied in multiple mathematical fields. A closed curve is a curve that forms a path whose starting point is also its ending point—that is, closely related meanings include the graph of a function and a two-dimensional graph. Interest in curves began long before they were the subject of mathematical study and this can be seen in numerous examples of their decorative use in art and on everyday objects dating back to prehistoric times. Curves, or at least their graphical representations, are simple to create, historically, the term line was used in place of the more modern term curve. Hence the phrases straight line and right line were used to distinguish what are called lines from curved lines. For example, in Book I of Euclids Elements, a line is defined as a breadthless length, Euclids idea of a line is perhaps clarified by the statement The extremities of a line are points. Later commentators further classified according to various schemes. For example, Composite lines Incomposite lines Determinate Indeterminate The Greek geometers had studied many kinds of curves. One reason was their interest in solving problems that could not be solved using standard compass. These curves include, The conic sections, deeply studied by Apollonius of Perga The cissoid of Diocles, studied by Diocles, the conchoid of Nicomedes, studied by Nicomedes as a method to both double the cube and to trisect an angle. The Archimedean spiral, studied by Archimedes as a method to trisect an angle, the spiric sections, sections of tori studied by Perseus as sections of cones had been studied by Apollonius. A fundamental advance in the theory of curves was the advent of analytic geometry in the seventeenth century and this enabled a curve to be described using an equation rather than an elaborate geometrical construction. Previously, curves had been described as geometrical or mechanical according to how they were, or supposedly could be, conic sections were applied in astronomy by Kepler. Newton also worked on an example in the calculus of variations
3.
Limit of a function
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In mathematics, the limit of a function is a fundamental concept in calculus and analysis concerning the behavior of that function near a particular input. Formal definitions, first devised in the early 19th century, are given below, informally, a function f assigns an output f to every input x. We say the function has a limit L at an input p, more specifically, when f is applied to any input sufficiently close to p, the output value is forced arbitrarily close to L. On the other hand, if some inputs very close to p are taken to outputs that stay a distance apart. The notion of a limit has many applications in modern calculus, in particular, the many definitions of continuity employ the limit, roughly, a function is continuous if all of its limits agree with the values of the function. It also appears in the definition of the derivative, in the calculus of one variable, however, his work was not known during his lifetime. Weierstrass first introduced the definition of limit in the form it is usually written today. He also introduced the notations lim and limx→x0, the modern notation of placing the arrow below the limit symbol is due to Hardy in his book A Course of Pure Mathematics in 1908. Imagine a person walking over a landscape represented by the graph of y = f and her horizontal position is measured by the value of x, much like the position given by a map of the land or by a global positioning system. Her altitude is given by the coordinate y and she is walking towards the horizontal position given by x = p. As she gets closer and closer to it, she notices that her altitude approaches L, if asked about the altitude of x = p, she would then answer L. What, then, does it mean to say that her altitude approaches L. It means that her altitude gets nearer and nearer to L except for a small error in accuracy. For example, suppose we set a particular goal for our traveler. She reports back that indeed she can get within ten meters of L, since she notes that when she is within fifty horizontal meters of p, the accuracy goal is then changed, can she get within one vertical meter. If she is anywhere within seven meters of p, then her altitude always remains within one meter from the target L. This explicit statement is quite close to the definition of the limit of a function with values in a topological space. To say that lim x → p f = L, means that ƒ can be made as close as desired to L by making x close enough, the following definitions are the generally accepted ones for the limit of a function in various contexts. Suppose f, R → R is defined on the real line, the value of the limit does not depend on the value of f, nor even that p be in the domain of f
4.
Projective geometry
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Projective geometry is a topic of mathematics. It is the study of properties that are invariant with respect to projective transformations. This means that, compared to elementary geometry, projective geometry has a different setting, projective space, properties meaningful for projective geometry are respected by this new idea of transformation, which is more radical in its effects than can be expressed by a transformation matrix and translations. The first issue for geometers is what kind of geometry is adequate for a novel situation, one source for projective geometry was indeed the theory of perspective. Another difference from elementary geometry is the way in which parallel lines can be said to meet in a point at infinity, again this notion has an intuitive basis, such as railway tracks meeting at the horizon in a perspective drawing. See projective plane for the basics of geometry in two dimensions. While the ideas were available earlier, projective geometry was mainly a development of the 19th century and this included the theory of complex projective space, the coordinates used being complex numbers. Several major types of more abstract mathematics were based on projective geometry and it was also a subject with a large number of practitioners for its own sake, as synthetic geometry. Another topic that developed from axiomatic studies of projective geometry is finite geometry, the topic of projective geometry is itself now divided into many research subtopics, two examples of which are projective algebraic geometry and projective differential geometry. Projective geometry is an elementary form of geometry, meaning that it is not based on a concept of distance. In two dimensions it begins with the study of configurations of points and lines and that there is indeed some geometric interest in this sparse setting was first established by Desargues and others in their exploration of the principles of perspective art. In higher dimensional spaces there are considered hyperplanes, and other linear subspaces, Projective geometry can also be seen as a geometry of constructions with a straight-edge alone. Since projective geometry excludes compass constructions, there are no circles, no angles, no measurements, no parallels and it was realised that the theorems that do apply to projective geometry are simpler statements. For example, the different conic sections are all equivalent in projective geometry, during the early 19th century the work of Jean-Victor Poncelet, Lazare Carnot and others established projective geometry as an independent field of mathematics. Its rigorous foundations were addressed by Karl von Staudt and perfected by Italians Giuseppe Peano, Mario Pieri, Alessandro Padoa, after much work on the very large number of theorems in the subject, therefore, the basics of projective geometry became understood. The incidence structure and the cross-ratio are fundamental invariants under projective transformations, Projective geometry can be modeled by the affine plane plus a line at infinity and then treating that line as ordinary. An algebraic model for doing projective geometry in the style of geometry is given by homogeneous coordinates. In a foundational sense, projective geometry and ordered geometry are elementary since they involve a minimum of axioms and either can be used as the foundation for affine, Projective geometry is not ordered and so it is a distinct foundation for geometry
5.
Tangent
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In geometry, the tangent line to a plane curve at a given point is the straight line that just touches the curve at that point. Leibniz defined it as the line through a pair of infinitely close points on the curve, a similar definition applies to space curves and curves in n-dimensional Euclidean space. Similarly, the tangent plane to a surface at a point is the plane that just touches the surface at that point. The concept of a tangent is one of the most fundamental notions in geometry and has been extensively generalized. The word tangent comes from the Latin tangere, to touch, euclid makes several references to the tangent to a circle in book III of the Elements. In Apollonius work Conics he defines a tangent as being a line such that no other straight line could fall between it and the curve, archimedes found the tangent to an Archimedean spiral by considering the path of a point moving along the curve. Independently Descartes used his method of normals based on the observation that the radius of a circle is always normal to the circle itself and these methods led to the development of differential calculus in the 17th century. Many people contributed, Roberval discovered a method of drawing tangents. René-François de Sluse and Johannes Hudde found algebraic algorithms for finding tangents, further developments included those of John Wallis and Isaac Barrow, leading to the theory of Isaac Newton and Gottfried Leibniz. An 1828 definition of a tangent was a line which touches a curve. This old definition prevents inflection points from having any tangent and it has been dismissed and the modern definitions are equivalent to those of Leibniz who defined the tangent line as the line through a pair of infinitely close points on the curve. The tangent at A is the limit when point B approximates or tends to A, the existence and uniqueness of the tangent line depends on a certain type of mathematical smoothness, known as differentiability. At most points, the tangent touches the curve without crossing it, a point where the tangent crosses the curve is called an inflection point. Circles, parabolas, hyperbolas and ellipses do not have any point, but more complicated curves do have, like the graph of a cubic function. Conversely, it may happen that the curve lies entirely on one side of a line passing through a point on it. This is the case, for example, for a passing through the vertex of a triangle. In convex geometry, such lines are called supporting lines, the geometrical idea of the tangent line as the limit of secant lines serves as the motivation for analytical methods that are used to find tangent lines explicitly. The question of finding the tangent line to a graph, or the tangent line problem, was one of the central questions leading to the development of calculus in the 17th century, suppose that a curve is given as the graph of a function, y = f
6.
Point at infinity
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In geometry, a point at infinity or ideal point is an idealized limiting point at the end of each line. In the case of a plane, there is one ideal point for each pencil of parallel lines of the plane. Adjoining these points produces a plane, in which no point can be distinguished. This holds for a geometry over any field, and more generally over any division ring, in the real case, a point at infinity completes a line into a topologically closed curve. In higher dimensions, all the points at infinity form a subspace of one dimension less than that of the whole projective space to which they belong. A point at infinity can also be added to the line, thereby turning it into a closed surface known as the complex projective line, CP1. In the case of a space, each line has two distinct ideal points. Here, the set of ideal points takes the form of a quadric, in an affine or Euclidean space of higher dimension, the points at infinity are the points which are added to the space to get the projective completion. As a projective space over a field is an algebraic variety. Similarly, if the field is the real or the complex field. In artistic drawing and technical perspective, the projection on the plane of the point at infinity of a class of parallel lines is called their vanishing point. In hyperbolic geometry, points at infinity are typically named ideal points, all points at infinity together form the Cayley absolute or boundary of a hyperbolic plane. This construction can be generalized to topological spaces, projective line is the Alexandroff extension of the corresponding field. Thus the circle is the one-point compactification of the line. Division by zero Midpoint § Generalizations Asymptote § Algebraic curves
7.
Greek language
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Greek is an independent branch of the Indo-European family of languages, native to Greece and other parts of the Eastern Mediterranean. It has the longest documented history of any living language, spanning 34 centuries of written records and its writing system has been the Greek alphabet for the major part of its history, other systems, such as Linear B and the Cypriot syllabary, were used previously. The alphabet arose from the Phoenician script and was in turn the basis of the Latin, Cyrillic, Armenian, Coptic, Gothic and many other writing systems. Together with the Latin texts and traditions of the Roman world, during antiquity, Greek was a widely spoken lingua franca in the Mediterranean world and many places beyond. It would eventually become the official parlance of the Byzantine Empire, the language is spoken by at least 13.2 million people today in Greece, Cyprus, Italy, Albania, Turkey, and the Greek diaspora. Greek roots are used to coin new words for other languages, Greek. Greek has been spoken in the Balkan peninsula since around the 3rd millennium BC, the earliest written evidence is a Linear B clay tablet found in Messenia that dates to between 1450 and 1350 BC, making Greek the worlds oldest recorded living language. Among the Indo-European languages, its date of earliest written attestation is matched only by the now extinct Anatolian languages, the Greek language is conventionally divided into the following periods, Proto-Greek, the unrecorded but assumed last ancestor of all known varieties of Greek. The unity of Proto-Greek would have ended as Hellenic migrants entered the Greek peninsula sometime in the Neolithic era or the Bronze Age, Mycenaean Greek, the language of the Mycenaean civilisation. It is recorded in the Linear B script on tablets dating from the 15th century BC onwards, Ancient Greek, in its various dialects, the language of the Archaic and Classical periods of the ancient Greek civilisation. It was widely known throughout the Roman Empire, after the Roman conquest of Greece, an unofficial bilingualism of Greek and Latin was established in the city of Rome and Koine Greek became a first or second language in the Roman Empire. The origin of Christianity can also be traced through Koine Greek, Medieval Greek, also known as Byzantine Greek, the continuation of Koine Greek in Byzantine Greece, up to the demise of the Byzantine Empire in the 15th century. Much of the written Greek that was used as the language of the Byzantine Empire was an eclectic middle-ground variety based on the tradition of written Koine. Modern Greek, Stemming from Medieval Greek, Modern Greek usages can be traced in the Byzantine period and it is the language used by the modern Greeks, and, apart from Standard Modern Greek, there are several dialects of it. In the modern era, the Greek language entered a state of diglossia, the historical unity and continuing identity between the various stages of the Greek language is often emphasised. Greek speakers today still tend to regard literary works of ancient Greek as part of their own rather than a foreign language and it is also often stated that the historical changes have been relatively slight compared with some other languages. According to one estimation, Homeric Greek is probably closer to demotic than 12-century Middle English is to modern spoken English, Greek is spoken by about 13 million people, mainly in Greece, Albania and Cyprus, but also worldwide by the large Greek diaspora. Greek is the language of Greece, where it is spoken by almost the entire population
8.
Apollonius of Perga
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Apollonius of Perga was a Greek geometer and astronomer known for his theories on the topic of conic sections. Beginning from the theories of Euclid and Archimedes on the topic and his definitions of the terms ellipse, parabola, and hyperbola are the ones in use today. Apollonius worked on other topics, including astronomy. Most of the work has not survived except in references in other authors. His hypothesis of eccentric orbits to explain the apparently aberrant motion of the planets, for such an important contributor to the field of mathematics, scant biographical information remains. The 6th century Palestinian commentator, Eutocius of Ascalon, on Apollonius’ major work, Conics, states, “Apollonius, the geometrician. Came from Perga in Pamphylia in the times of Ptolemy Euergetes, the ruins of the city yet stand. It was a center of Hellenistic culture, Euergetes, “benefactor, ” identifies Ptolemy III Euergetes, third Greek dynast of Egypt in the diadochi succession. Presumably, his “times” are his regnum, 246-222/221 BC, times are always recorded by ruler or officiating magistrate, so that if Apollonius was born earlier than 246, it would have been the “times” of Euergetes’ father. The identity of Herakleios is uncertain, the approximate times of Apollonius are thus certain, but no exact dates can be given. The figure Specific birth and death years stated by the scholars are only speculative. Eutocius appears to associate Perga with the Ptolemaic dynasty of Egypt, never under Egypt, Perga in 246 BC belonged to the Seleucid Empire, an independent diadochi state ruled by the Seleucid dynasty. Someone designated “of Perga” might well be expected to have lived and worked there, to the contrary, if Apollonius was later identified with Perga, it was not on the basis of his residence. The remaining autobiographical material implies that he lived, studied and wrote in Alexandria, philip was assassinated in 336 BC. Alexander went on to fulfill his plan by conquering the vast Iranian empire, the material is located in the surviving false “Prefaces” of the books of his Conics. These are letters delivered to friends of Apollonius asking them to review the book enclosed with the letter. The Preface to Book I, addressed to one Eudemus, reminds him that Conics was initially requested by a house guest at Alexandria, Naucrates had the first draft of all eight books in his hands by the end of the visit. Apollonius refers to them as being “without a thorough purgation” and he intended to verify and emend the books, releasing each one as it was completed
9.
Conic section
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In mathematics, a conic section is a curve obtained as the intersection of the surface of a cone with a plane. The three types of conic section are the hyperbola, the parabola, and the ellipse, the circle is a special case of the ellipse, and is of sufficient interest in its own right that it was sometimes called a fourth type of conic section. The conic sections have been studied by the ancient Greek mathematicians with this work culminating around 200 BC, the conic sections of the Euclidean plane have various distinguishing properties. Many of these have used as the basis for a definition of the conic sections. The type of conic is determined by the value of the eccentricity, in analytic geometry, a conic may be defined as a plane algebraic curve of degree 2, that is, as the set of points whose coordinates satisfy a quadratic equation in two variables. This equation may be written in form, and some geometric properties can be studied as algebraic conditions. In the Euclidean plane, the conic sections appear to be different from one another. By extending the geometry to a projective plane this apparent difference vanishes, further extension, by expanding the real coordinates to admit complex coordinates, provides the means to see this unification algebraically. The conic sections have been studied for thousands of years and have provided a source of interesting. A conic is the curve obtained as the intersection of a plane, called the cutting plane and we shall assume that the cone is a right circular cone for the purpose of easy description, but this is not required, any double cone with some circular cross-section will suffice. Planes that pass through the vertex of the cone will intersect the cone in a point and these are called degenerate conics and some authors do not consider them to be conics at all. Unless otherwise stated, we assume that conic refers to a non-degenerate conic. There are three types of conics, the ellipse, parabola, and hyperbola, the circle is a special kind of ellipse, although historically it had been considered as a fourth type. The circle and the ellipse arise when the intersection of the cone and plane is a closed curve, if the cutting plane is parallel to exactly one generating line of the cone, then the conic is unbounded and is called a parabola. In the remaining case, the figure is a hyperbola, in this case, the plane will intersect both halves of the cone, producing two separate unbounded curves. A property that the conic sections share is often presented as the following definition, a conic section is the locus of all points P whose distance to a fixed point F is a constant multiple of the distance from P to a fixed line L. For 0 < e <1 we obtain an ellipse, for e =1 a parabola, a circle is a limiting case and is not defined by a focus and directrix, in the plane. The eccentricity of a circle is defined to be zero and its focus is the center of the circle, an ellipse and a hyperbola each have two foci and distinct directrices for each of them
10.
Cartesian coordinates
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Each reference line is called a coordinate axis or just axis of the system, and the point where they meet is its origin, usually at ordered pair. The coordinates can also be defined as the positions of the projections of the point onto the two axis, expressed as signed distances from the origin. One can use the principle to specify the position of any point in three-dimensional space by three Cartesian coordinates, its signed distances to three mutually perpendicular planes. In general, n Cartesian coordinates specify the point in an n-dimensional Euclidean space for any dimension n and these coordinates are equal, up to sign, to distances from the point to n mutually perpendicular hyperplanes. The invention of Cartesian coordinates in the 17th century by René Descartes revolutionized mathematics by providing the first systematic link between Euclidean geometry and algebra. Using the Cartesian coordinate system, geometric shapes can be described by Cartesian equations, algebraic equations involving the coordinates of the points lying on the shape. For example, a circle of radius 2, centered at the origin of the plane, a familiar example is the concept of the graph of a function. Cartesian coordinates are also tools for most applied disciplines that deal with geometry, including astronomy, physics, engineering. They are the most common system used in computer graphics, computer-aided geometric design. Nicole Oresme, a French cleric and friend of the Dauphin of the 14th Century, used similar to Cartesian coordinates well before the time of Descartes. The adjective Cartesian refers to the French mathematician and philosopher René Descartes who published this idea in 1637 and it was independently discovered by Pierre de Fermat, who also worked in three dimensions, although Fermat did not publish the discovery. Both authors used a single axis in their treatments and have a length measured in reference to this axis. The concept of using a pair of axes was introduced later, after Descartes La Géométrie was translated into Latin in 1649 by Frans van Schooten and these commentators introduced several concepts while trying to clarify the ideas contained in Descartes work. Many other coordinate systems have developed since Descartes, such as the polar coordinates for the plane. The development of the Cartesian coordinate system would play a role in the development of the Calculus by Isaac Newton. The two-coordinate description of the plane was later generalized into the concept of vector spaces. Choosing a Cartesian coordinate system for a one-dimensional space – that is, for a straight line—involves choosing a point O of the line, a unit of length, and an orientation for the line. An orientation chooses which of the two half-lines determined by O is the positive, and which is negative, we say that the line is oriented from the negative half towards the positive half
11.
Line (geometry)
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The notion of line or straight line was introduced by ancient mathematicians to represent straight objects with negligible width and depth. Lines are an idealization of such objects, the straight line is that which is equally extended between its points. In modern mathematics, given the multitude of geometries, the concept of a line is tied to the way the geometry is described. When a geometry is described by a set of axioms, the notion of a line is left undefined. The properties of lines are determined by the axioms which refer to them. One advantage to this approach is the flexibility it gives to users of the geometry, thus in differential geometry a line may be interpreted as a geodesic, while in some projective geometries a line is a 2-dimensional vector space. This flexibility also extends beyond mathematics and, for example, permits physicists to think of the path of a light ray as being a line, to avoid this vicious circle certain concepts must be taken as primitive concepts, terms which are given no definition. In geometry, it is frequently the case that the concept of line is taken as a primitive, in those situations where a line is a defined concept, as in coordinate geometry, some other fundamental ideas are taken as primitives. When the line concept is a primitive, the behaviour and properties of lines are dictated by the axioms which they must satisfy, in a non-axiomatic or simplified axiomatic treatment of geometry, the concept of a primitive notion may be too abstract to be dealt with. In this circumstance it is possible that a description or mental image of a notion is provided to give a foundation to build the notion on which would formally be based on the axioms. Descriptions of this type may be referred to, by some authors and these are not true definitions and could not be used in formal proofs of statements. The definition of line in Euclids Elements falls into this category, when geometry was first formalised by Euclid in the Elements, he defined a general line to be breadthless length with a straight line being a line which lies evenly with the points on itself. These definitions serve little purpose since they use terms which are not, themselves, in fact, Euclid did not use these definitions in this work and probably included them just to make it clear to the reader what was being discussed. In an axiomatic formulation of Euclidean geometry, such as that of Hilbert, for example, for any two distinct points, there is a unique line containing them, and any two distinct lines intersect in at most one point. In two dimensions, i. e. the Euclidean plane, two lines which do not intersect are called parallel, in higher dimensions, two lines that do not intersect are parallel if they are contained in a plane, or skew if they are not. Any collection of many lines partitions the plane into convex polygons. Lines in a Cartesian plane or, more generally, in affine coordinates, in two dimensions, the equation for non-vertical lines is often given in the slope-intercept form, y = m x + b where, m is the slope or gradient of the line. B is the y-intercept of the line, X is the independent variable of the function y = f
12.
Calculus
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Calculus is the mathematical study of continuous change, in the same way that geometry is the study of shape and algebra is the study of generalizations of arithmetic operations. It has two branches, differential calculus, and integral calculus, these two branches are related to each other by the fundamental theorem of calculus. Both branches make use of the notions of convergence of infinite sequences. Generally, modern calculus is considered to have developed in the 17th century by Isaac Newton. Today, calculus has widespread uses in science, engineering and economics, Calculus is a part of modern mathematics education. A course in calculus is a gateway to other, more advanced courses in mathematics devoted to the study of functions and limits, Calculus has historically been called the calculus of infinitesimals, or infinitesimal calculus. Calculus is also used for naming some methods of calculation or theories of computation, such as calculus, calculus of variations, lambda calculus. The ancient period introduced some of the ideas that led to integral calculus, the method of exhaustion was later discovered independently in China by Liu Hui in the 3rd century AD in order to find the area of a circle. In the 5th century AD, Zu Gengzhi, son of Zu Chongzhi, indian mathematicians gave a non-rigorous method of a sort of differentiation of some trigonometric functions. In the Middle East, Alhazen derived a formula for the sum of fourth powers. He used the results to carry out what would now be called an integration, Cavalieris work was not well respected since his methods could lead to erroneous results, and the infinitesimal quantities he introduced were disreputable at first. The formal study of calculus brought together Cavalieris infinitesimals with the calculus of finite differences developed in Europe at around the same time, pierre de Fermat, claiming that he borrowed from Diophantus, introduced the concept of adequality, which represented equality up to an infinitesimal error term. The combination was achieved by John Wallis, Isaac Barrow, and James Gregory, in other work, he developed series expansions for functions, including fractional and irrational powers, and it was clear that he understood the principles of the Taylor series. He did not publish all these discoveries, and at this time infinitesimal methods were considered disreputable. These ideas were arranged into a calculus of infinitesimals by Gottfried Wilhelm Leibniz. He is now regarded as an independent inventor of and contributor to calculus, unlike Newton, Leibniz paid a lot of attention to the formalism, often spending days determining appropriate symbols for concepts. Leibniz and Newton are usually credited with the invention of calculus. Newton was the first to apply calculus to general physics and Leibniz developed much of the used in calculus today
13.
Limit (mathematics)
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In mathematics, a limit is the value that a function or sequence approaches as the input or index approaches some value. Limits are essential to calculus and are used to define continuity, derivatives, the concept of a limit of a sequence is further generalized to the concept of a limit of a topological net, and is closely related to limit and direct limit in category theory. In formulas, a limit is usually written as lim n → c f = L and is read as the limit of f of n as n approaches c equals L. Here lim indicates limit, and the fact that function f approaches the limit L as n approaches c is represented by the right arrow, suppose f is a real-valued function and c is a real number. Intuitively speaking, the lim x → c f = L means that f can be made to be as close to L as desired by making x sufficiently close to c. The first inequality means that the distance x and c is greater than 0 and that x ≠ c, while the second indicates that x is within distance δ of c. Note that the definition of a limit is true even if f ≠ L. Indeed. Now since x +1 is continuous in x at 1, we can now plug in 1 for x, in addition to limits at finite values, functions can also have limits at infinity. In this case, the limit of f as x approaches infinity is 2, in mathematical notation, lim x → ∞2 x −1 x =2. Consider the following sequence,1.79,1.799,1.7999 and it can be observed that the numbers are approaching 1.8, the limit of the sequence. Formally, suppose a1, a2. is a sequence of real numbers, intuitively, this means that eventually all elements of the sequence get arbitrarily close to the limit, since the absolute value | an − L | is the distance between an and L. Not every sequence has a limit, if it does, it is called convergent, one can show that a convergent sequence has only one limit. The limit of a sequence and the limit of a function are closely related, on one hand, the limit as n goes to infinity of a sequence a is simply the limit at infinity of a function defined on the natural numbers n. On the other hand, a limit L of a function f as x goes to infinity, if it exists, is the same as the limit of any sequence a that approaches L. Note that one such sequence would be L + 1/n, in non-standard analysis, the limit of a sequence can be expressed as the standard part of the value a H of the natural extension of the sequence at an infinite hypernatural index n=H. Thus, lim n → ∞ a n = st , here the standard part function st rounds off each finite hyperreal number to the nearest real number. This formalizes the intuition that for very large values of the index. Conversely, the part of a hyperreal a = represented in the ultrapower construction by a Cauchy sequence, is simply the limit of that sequence
14.
Vertical line test
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In mathematics, the vertical line test is a visual way to determine if a curve is a graph of a function or not. A function can only have one output, y, for each unique input, x. If a vertical line intersects a curve on a more than once then for one value of x the curve has more than one value of y, and so. If all vertical lines intersect a curve at most once then the curve represents a function, to use the vertical line test, take a ruler or other straight edge and draw a line parallel to the y-axis for any chosen value of x. If the vertical line you drew intersects the more than once for any value of x then the graph is not the graph of a function. If, alternatively, a line intersects the graph no more than once, no matter where the vertical line is placed. For example, a curve which is any straight line other than a line will be the graph of a function. As another example, a parabola is not the graph of a function because some vertical lines will intersect the parabola twice
15.
Continuous function
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In mathematics, a continuous function is a function for which sufficiently small changes in the input result in arbitrarily small changes in the output. Otherwise, a function is said to be a discontinuous function, a continuous function with a continuous inverse function is called a homeomorphism. Continuity of functions is one of the concepts of topology. The introductory portion of this focuses on the special case where the inputs and outputs of functions are real numbers. In addition, this article discusses the definition for the general case of functions between two metric spaces. In order theory, especially in theory, one considers a notion of continuity known as Scott continuity. Other forms of continuity do exist but they are not discussed in this article, as an example, consider the function h, which describes the height of a growing flower at time t. By contrast, if M denotes the amount of money in an account at time t, then the function jumps at each point in time when money is deposited or withdrawn. A form of the definition of continuity was first given by Bernard Bolzano in 1817. Cauchy defined infinitely small quantities in terms of quantities. The formal definition and the distinction between pointwise continuity and uniform continuity were first given by Bolzano in the 1830s but the work wasnt published until the 1930s, all three of those nonequivalent definitions of pointwise continuity are still in use. Eduard Heine provided the first published definition of continuity in 1872. This is not a definition of continuity since the function f =1 x is continuous on its whole domain of R ∖ A function is continuous at a point if it does not have a hole or jump. A “hole” or “jump” in the graph of a function if the value of the function at a point c differs from its limiting value along points that are nearby. Such a point is called a discontinuity, a function is then continuous if it has no holes or jumps, that is, if it is continuous at every point of its domain. Otherwise, a function is discontinuous, at the points where the value of the function differs from its limiting value, there are several ways to make this definition mathematically rigorous. These definitions are equivalent to one another, so the most convenient definition can be used to determine whether a function is continuous or not. In the definitions below, f, I → R. is a function defined on a subset I of the set R of real numbers and this subset I is referred to as the domain of f
16.
Rational function
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In mathematics, a rational function is any function which can be defined by a rational fraction, i. e. an algebraic fraction such that both the numerator and the denominator are polynomials. The coefficients of the polynomials need not be numbers, they may be taken in any field K. In this case, one speaks of a function and a rational fraction over K. The values of the variables may be taken in any field L containing K, then the domain of the function is the set of the values of the variables for which the denominator is not zero and the codomain is L. The set of functions over a field K is a field. A function f is called a function if and only if it can be written in the form f = P Q where P and Q are polynomials in x and Q is not the zero polynomial. The domain of f is the set of all points x for which the denominator Q is not zero and it is a common usage to identify f and f 1, that is to extend by continuity the domain of f to that of f 1. Indeed, one can define a rational fraction as a class of fractions of polynomials. In this case P Q is equivalent to P1 Q1, a proper rational function is a rational function in which the degree of P is no greater than the degree of Q and both are real polynomials. The rational function f = x 3 −2 x 2 is not defined at x 2 =5 ⇔ x = ±5 and it is asymptotic to x 2 as x approaches infinity. A constant function such as f = π is a function since constants are polynomials. Note that the function itself is rational, even though the value of f is irrational for all x, every polynomial function f = P is a rational function with Q =1. A function that cannot be written in form, such as f = sin , is not a rational function. The adjective irrational is not generally used for functions, the rational function f = x x is equal to 1 for all x except 0, where there is a removable singularity. The sum, product, or quotient of two functions is itself a rational function. However, the process of reduction to standard form may result in the removal of such singularities unless care is taken. Using the definition of functions as equivalence classes gets around this. For example,1 x 2 − x +2 = ∑ k =0 ∞ a k x k, combining like terms gives 1 =2 a 0 + x + ∑ k =2 ∞ x k
17.
Homothetic transformation
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In Euclidean geometry homotheties are the similarities that fix a point and either preserve or reverse the direction of all vectors. Together with the translations, all homotheties of a space form a group. These are precisely the transformations with the property that the image of every line L is a line parallel to L. In projective geometry, a transformation is a similarity transformation that leaves the line at infinity pointwise invariant. In Euclidean geometry, a homothety of ratio λ multiplies distances between points by |λ| and all areas by λ2, the first number is called the ratio of magnification or dilation factor or scale factor or similitude ratio. Such a transformation can be called an enlargement if the scale factor exceeds 1, the image of a point after a homothety with center and scale factor λ is given by. Hadamard, J. Lessons in Plane Geometry, meserve, Bruce E. Homothetic transformations, Fundamental Concepts of Geometry, Addison-Wesley, pp. 166–169. Tuller, Annita, A Modern Introduction to Geometries
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Parametric equation
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In mathematics, parametric equations define a group of quantities as functions of one or more independent variables called parameters. For example, the equations x = cos t y = sin t form a representation of the unit circle. Parametric equations are used in kinematics, where the trajectory of an object is represented by equations depending on time as the parameter. Because of this application, a parameter is often labeled t, however. Parameterizations are non-unique, more than one set of equations can specify the same curve. In kinematics, objects paths through space are described as parametric curves. Used in this way, the set of equations for the objects coordinates collectively constitute a vector-valued function for position. Such parametric curves can then be integrated and differentiated termwise, thus, if a particles position is described parametrically as r = then its velocity can be found as v = r ′ = and its acceleration as a = r ″ =. Another important use of equations is in the field of computer-aided design. For example, consider the three representations, all of which are commonly used to describe planar curves. These problems can be addressed by rewriting the non-parametric equations in parametric form, numerous problems in integer geometry can be solved using parametric equations. A classical such solution is Euclids parametrization of right triangles such that the lengths of their sides a, b, by multiplying a, b and c by an arbitrary positive integer, one gets a parametrization of all right triangles whose three sides have integer lengths. Converting a set of equations to a single equation involves eliminating the variable t from the simultaneous equations x = x, y = y. If one of these equations can be solved for t, the expression obtained can be substituted into the equation to obtain an equation involving x and y only. If the parametrization is given by rational functions x = p r, y = q r, where p, q, r are set-wise coprime polynomials, in some cases there is no single equation in closed form that is equivalent to the parametric equations. The simplest equation for a parabola, y = x 2 can be parameterized by using a free parameter t, and setting x = t, y = t 2 f o r − ∞ < t < ∞. More generally, any given by an explicit equation y = f can be parameterized by using a free parameter t. A more sophisticated example is the following, consider the unit circle which is described by the ordinary equation x 2 + y 2 =1
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Graph of a function
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In mathematics, the graph of a function f is the collection of all ordered pairs. If the function x is a scalar, the graph is a two-dimensional graph. If the function x is an ordered pair of real numbers, the graph is the collection of all ordered triples. Graphing on a Cartesian plane is referred to as curve sketching. The graph of a function on real numbers may be mapped directly to the representation of the function. The concept of the graph of a function is generalized to the graph of a relation, note that although a function is always identified with its graph, they are not the same because it will happen that two functions with different codomain could have the same graph. For example, the cubic polynomial mentioned below is a surjection if its codomain is the real numbers, to test whether a graph of a curve is a function of x, one uses the vertical line test. To test whether a graph of a curve is a function of y, if the function has an inverse, the graph of the inverse can be found by reflecting the graph of the original function over the line y = x. In science, engineering, technology, finance, and other areas, in the simplest case one variable is plotted as a function of another, typically using rectangular axes, see Plot for details. In the modern foundation of mathematics known as set theory, a function, F = { a, if x =1, d, if x =2, c, if x =3, is. The graph of the polynomial on the real line f = x 3 −9 x is. If this set is plotted on a Cartesian plane, the result is a curve, the graph of the trigonometric function f = sin cos is. If this set is plotted on a three dimensional Cartesian coordinate system, the result is a surface, oftentimes it is helpful to show with the graph, the gradient of the function and several level curves. The level curves can be mapped on the surface or can be projected on the bottom plane. The second figure shows such a drawing of the graph of the function, f = −2 Given a function f of n variables, x 1, …, x n, the normal to the graph is. This is seen by considering the graph as a set of the function g = f − z. The graph of a function is contained in a Cartesian product of sets, fibre bundles arent cartesian products, but appear to be up close. There is a notion of a graph on a fibre bundle called a section
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Real function
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In mathematics, a real-valued function or real function is a function whose values are real numbers. In other words, it is a function that assigns a number to each member of its domain. Many important function spaces are defined to consist of real functions, let X be an arbitrary set. Let F denote the set of all functions from X to real numbers R. F is an ordered ring. The σ-algebra of Borel sets is an important structure on real numbers, if X has its σ-algebra and a function f is such that the preimage f −1 of any Borel set B belongs to that σ-algebra, then f is said to be measurable. Measurable functions also form a space and an algebra as explained above. Moreover, a set of real-valued functions on X can actually define a σ-algebra on X generated by all preimages of all Borel sets and this is the way how σ-algebras arise in probability theory, where real-valued functions on the sample space Ω are real-valued random variables. Real numbers form a space and a complete metric space. Continuous real-valued functions are important in theories of topological spaces and of metric spaces, the extreme value theorem states that for any real continuous function on a compact space its global maximum and minimum exist. The concept of space itself is defined with a real-valued function of two variables, the metric, which is continuous. The space of functions on a compact Hausdorff space has a particular importance. Convergent sequences also can be considered as real-valued continuous functions on a topological space. Continuous functions also form a space and an algebra as explained above. Real numbers are used as the codomain to define smooth functions, a domain of a real smooth function can be the real coordinate space, a topological vector space, an open subset of them, or a smooth manifold. Spaces of smooth functions also are vector spaces and algebras as explained above, a measure on a set is a non-negative real-valued functional on a σ-algebra of subsets. Lp spaces on sets with a measure are defined from aforementioned real-valued measurable functions, though, real-valued Lp spaces still have some of the structure explicated above. For example, pointwise product of two L2 functions belongs to L1, Real analysis Partial differential equations, a major user of real-valued functions Norm Scalar Weisstein, Eric W. Real Function
21.
Parabola
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A parabola is a two-dimensional, mirror-symmetrical curve, which is approximately U-shaped when oriented as shown in the diagram below, but which can be in any orientation in its plane. It fits any of several different mathematical descriptions which can all be proved to define curves of exactly the same shape. One description of a parabola involves a point and a line, the focus does not lie on the directrix. The parabola is the locus of points in that plane that are equidistant from both the directrix and the focus, a parabola is a graph of a quadratic function, y = x2, for example. The line perpendicular to the directrix and passing through the focus is called the axis of symmetry, the point on the parabola that intersects the axis of symmetry is called the vertex, and is the point where the parabola is most sharply curved. The distance between the vertex and the focus, measured along the axis of symmetry, is the focal length, the latus rectum is the chord of the parabola which is parallel to the directrix and passes through the focus. Parabolas can open up, down, left, right, or in some arbitrary direction. Any parabola can be repositioned and rescaled to fit exactly on any other parabola — that is, conversely, light that originates from a point source at the focus is reflected into a parallel beam, leaving the parabola parallel to the axis of symmetry. The same effects occur with sound and other forms of energy and this reflective property is the basis of many practical uses of parabolas. The parabola has many important applications, from an antenna or parabolic microphone to automobile headlight reflectors to the design of ballistic missiles. They are frequently used in physics, engineering, and many other areas, the earliest known work on conic sections was by Menaechmus in the fourth century BC. He discovered a way to solve the problem of doubling the cube using parabolas, the name parabola is due to Apollonius who discovered many properties of conic sections. It means application, referring to application of concept, that has a connection with this curve. The focus–directrix property of the parabola and other conics is due to Pappus, Galileo showed that the path of a projectile follows a parabola, a consequence of uniform acceleration due to gravity. The idea that a reflector could produce an image was already well known before the invention of the reflecting telescope. Designs were proposed in the early to mid seventeenth century by many mathematicians including René Descartes, Marin Mersenne, when Isaac Newton built the first reflecting telescope in 1668, he skipped using a parabolic mirror because of the difficulty of fabrication, opting for a spherical mirror. Parabolic mirrors are used in most modern reflecting telescopes and in satellite dishes, solving for y yields y =14 f x 2. The length of the chord through the focus is called latus rectum, one half of it semi latus rectum
22.
Cubic plane curve
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Here F is a non-zero linear combination of the third-degree monomials x3, y3, z3, x2y, x2z, y2x, y2z, z2x, z2y, xyz. These are ten in number, therefore the cubic curves form a space of dimension 9. Each point P imposes a single linear condition on F, if we ask that C pass through P, if two cubics pass through a given set of nine points, then in fact a pencil of cubics does, and the points satisfy additional properties, see Cayley–Bacharach theorem. A cubic curve may have a point, in which case it has a parametrization in terms of a projective line. Otherwise a non-singular cubic curve is known to have nine points of inflection and this can be shown by taking the homogeneous version of the Hessian matrix, which defines again a cubic, and intersecting it with C, the intersections are then counted by Bézouts theorem. However, only three of these points may be real, so that the others cannot be seen in the projective plane by drawing the curve. The nine inflection points of a non-singular cubic have the property that every line passing through two of them contains exactly three inflection points, the real points of cubic curves were studied by Isaac Newton. The real points of a non-singular projective cubic fall into one or two ovals, the other oval, if it exists, does not contain any real inflection point and appears either as an oval or as two infinite branches. Like for conic sections, a line cuts this oval at, at most, a non-singular cubic defines an elliptic curve, over any field K for which it has a point defined. This does depend on having a K-rational point, which serves as the point at infinity in Weierstrass form, there are many cubic curves that have no such point, for example when K is the rational number field. The singular points of a plane cubic curve are quite limited, one double point. A reducible plane cubic curve is either a conic and a line or three lines, and accordingly have two points or a tacnode, or up to three double points or a single triple point if three lines. Suppose that ABC is a triangle with sidelengths a = | BC |, b = | CA |, relative to ABC, many named cubics pass through well-known points. Examples shown below use two kinds of coordinates, trilinear and barycentric. To convert from trilinear to barycentric in an equation, substitute as follows, x ↦ bcx, y ↦ cay, z ↦ abz, to convert from barycentric to trilinear, use x ↦ ax, y ↦ by. Many equations for cubics have the form f + f + f =0, in the examples below, such equations are written more succinctly in cyclic sum notation, like this, [cyclic sum f =0. The cubics listed below can be defined in terms of the conjugate, denoted by X*. Let LA be the reflection of line XA about the angle bisector of angle A
23.
Folium of Descartes
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In geometry, the folium of Descartes is an algebraic curve defined by the equation x 3 + y 3 −3 a x y =0. It forms a loop in the first quadrant with a point at the origin. It is symmetrical about y = x, the name comes from the Latin word folium which means leaf. The curve was featured, along with a portrait of Descartes, the curve was first proposed by Descartes in 1638. Its claim to fame lies in an incident in the development of calculus, Descartes challenged Fermat to find the tangent line to the curve at an arbitrary point since Fermat had recently discovered a method for finding tangent lines. Fermat solved the problem easily, something Descartes was unable to do, since the invention of calculus, the slope of the tangent line can be found easily using implicit differentiation. Since the equation is degree 3 in both x and y, and does not factor, it is difficult to solve for one of the variables. However, the equation in polar coordinates is, r =3 a sin θ cos θ sin 3 θ + cos 3 θ. which can be plotted easily. By using this formula, the area of the interior of the loop is found to be 3 a 2 /2, another technique is to write y = px and solve for x and y in terms of p. This yields the rational parametric equations, x =3 a p 1 + p 3, y =3 a p 21 + p 3. We can see that the parameter is related to the position on the curve as follows, p < -1 corresponds to x>0, y<0, -1 < p <0 corresponds to x<0, y>0, the left, upper wing. P >0 corresponds to x>0, y>0, the loop of the curve, another way of plotting the function can be derived from symmetry over y = x. The symmetry can be directly from its equation. By applying rotation of 45° CW for example, one can plot the function symmetric over rotated x axis, since the folium is symmetrical about y = x, it passes through the point. The folium of Descartes is related to the trisectrix of Maclaurin by affine transformation, to see this, start with the equation x 3 + y 3 =3 a x y, and change variables to find the equation in a coordinate system rotated 45 degrees. This amounts to setting x = X + Y2, y = X − Y2, in the X, Y plane the equation is 2 X =32 a. If we stretch the curve in the Y direction by a factor of 3 this becomes 2 X = a 2 which is the equation of the trisectrix of Maclaurin, J. Dennis Lawrence, A catalog of special plane curves,1972, Dover Publications. ISBN 0-486-60288-5, pp. 106–108 George F. Simmons, Calculus Gems, Brief Lives and Memorable Mathematics, New York 1992, McGraw-Hill, Folium of Descartes at MacTutors Famous Curves Index Folium de Descartes at Encyclopédie des Formes Mathématiques Remarquables
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Algebraic curve
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In mathematics, a plane real algebraic curve is the set of points on the Euclidean plane whose coordinates are zeros of some polynomial in two variables. More generally an algebraic curve is similar but may be embedded in a dimensional space or defined over some more general field. For example, the circle is a real algebraic curve. Various technical considerations result in the complex zeros of a polynomial being considered as belonging to the curve, the points of the curve with coordinates in k are the k-points of the curve and, all together, are the k part of the curve. For example, is a point of the curve defined by x2 + y2 −1 =0, the term unit circle may refer to all the complex points as well as to only the real points, the exact meaning usually clear from the context. The equation x2 + y2 +1 =0 defines an algebraic curve, more generally, one may consider algebraic curves that are not contained in the plane, but in a space of higher dimension. A curve that is not contained in some plane is called a skew curve, the simplest example of a skew algebraic curve is the twisted cubic. One may also consider algebraic curves contained in the projective space and this leads to the most general definition of an algebraic curve, In algebraic geometry, an algebraic curve is an algebraic variety of dimension one. An algebraic curve in the Euclidean plane is the set of the points whose coordinates are the solutions of a polynomial equation p =0. This equation is called the implicit equation of the curve. Given a curve given by such an equation, the first problems that occur is to determine the shape of the curve. These problems are not as easy to solve as in the case of the graph of a function, the fact that the defining equation is a polynomial implies that the curve has some structural properties that may help to solve these problems. Every algebraic curve may be decomposed into a finite number of smooth monotone arcs connected by some points sometimes called remarkable points. A smooth monotone arc is the graph of a function which is defined. In each direction, an arc is either unbounded or has an end point which is either a point or a point with a tangent parallel to one of the coordinate axes. For example, for the Tschirnhausen cubic of the figure, there are two arcs having the origin as end point. This point is the singular point of the curve. There are two arcs having this singular point as one end point and having a second end point with a horizontal tangent
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Affine space
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A Euclidean space is an affine space over the reals, equipped with a metric, the Euclidean distance. Therefore, in Euclidean geometry, a property is a property that may be proved in affine spaces. In an affine space, there is no distinguished point that serves as an origin, hence, no vector has a fixed origin and no vector can be uniquely associated to a point. In an affine space, there are instead displacement vectors, also called translation vectors or simply translations, thus it makes sense to subtract two points of the space, giving a translation vector, but it does not make sense to add two points of the space. Likewise, it makes sense to add a displacement vector to a point of an affine space, Any vector space may be considered as an affine space, and this amounts to forgetting the special role played by the zero vector. In this case, the elements of the space may be viewed either as points of the affine space or as displacement vectors or translations. When considered as a point, the vector is called the origin. Adding a fixed vector to the elements of a subspace of a vector space produces an affine subspace. One commonly says that this affine subspace has been obtained by translating the linear subspace by the translation vector, in finite dimensions, such an affine subspace is the solution set of an inhomogeneous linear system. The displacement vectors for that space are the solutions of the corresponding homogeneous linear system. Linear subspaces, in contrast, always contain the origin of the vector space, the dimension of an affine space is defined as the dimension of the vector space of its translations. An affine space of one is an affine line. An affine space of dimension 2 is an affine plane, an affine subspace of dimension n –1 in an affine space or a vector space of dimension n is an affine hyperplane. The following characterization may be easier to understand than the formal definition. Imagine that Alice knows that a point is the actual origin. Two vectors, a and b, are to be added, similarly, Alice and Bob may evaluate any linear combination of a and b, or of any finite set of vectors, and will generally get different answers. However, if the sum of the coefficients in a combination is 1, then Alice. If Alice travels to λa + b then Bob can similarly travel to p + λ + = λa + b, under this condition, for all coefficients λ + =1, Alice and Bob describe the same point with the same linear combination, despite using different origins
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Unit hyperbola
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In geometry, the unit hyperbola is the set of points in the Cartesian plane that satisfies x 2 − y 2 =1. In the study of orthogonal groups, the unit hyperbola forms the basis for an alternative radial length r = x 2 − y 2. Whereas the unit circle surrounds its center, the unit hyperbola requires the conjugate hyperbola y 2 − x 2 =1 to complement it in the plane and this pair of hyperbolas share the asymptotes y = x and y = −x. When the conjugate of the hyperbola is in use, the alternative radial length is r = y 2 − x 2. The unit hyperbola is a case of the rectangular hyperbola, with a particular orientation, location. As such, its eccentricity equals 2, the unit hyperbola finds applications where the circle must be replaced with the hyperbola for purposes of analytic geometry. A prominent instance is the depiction of spacetime as a pseudo-Euclidean space, there the asymptotes of the unit hyperbola form a light cone. Further, the attention to areas of hyperbolic sectors by Gregoire de Saint-Vincent led to the logarithm function, generally asymptotic lines to a curve are said to converge toward the curve. In algebraic geometry and the theory of algebraic curves there is a different approach to asymptotes, the curve is first interpreted in the projective plane using homogeneous coordinates. Then the asymptotes are lines that are tangent to the curve at a point at infinity, thus circumventing any need for a distance concept. In a common framework are homogeneous coordinates with the line at infinity determined by the equation z =0. Both P, Q are simple on F, with tangents x + y =0, x − y =0, the Minkowski diagram is drawn in a spacetime plane where the spatial aspect has been restricted to a single dimension. The units of distance and time on such a plane are units of 30 centimetres length and nanoseconds, or astronomical units and intervals of 8 minutes and 20 seconds, or light years and years. Each of these scales of coordinates results in photon connections of events along diagonal lines of slope plus or minus one, the plane with the axes refers to a resting frame of reference. The diameter of the unit hyperbola represents a frame of reference in motion with rapidity a where tanh a = y/x and is the endpoint of the diameter on the unit hyperbola, the conjugate diameter represents the spatial hyperplane of simultaneity corresponding to rapidity a. Space is represented by planes perpendicular to the time axis, the here and now is a singularity in the middle. The vertical time axis convention stems from Minkowski in 1908, and is illustrated on page 48 of Eddingtons The Nature of the Physical World. A direct way to parameterizing the unit hyperbola starts with the hyperbola xy =1 parameterized with the exponential function and this hyperbola is transformed into the unit hyperbola by a linear mapping having the matrix A =12, A = =
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Field (mathematics)
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In mathematics, a field is a set on which are defined addition, subtraction, multiplication, and division, which behave as they do when applied to rational and real numbers. A field is thus an algebraic structure, which is widely used in algebra, number theory. The best known fields are the field of numbers. In addition, the field of numbers is widely used, not only in mathematics. Finite fields are used in most cryptographic protocols used for computer security, any field may be used as the scalars for a vector space, which is the standard general context for linear algebra. Formally, a field is a set together with two operations the addition and the multiplication, which have the properties, called axioms of fields. An operation is a mapping that associates an element of the set to every pair of its elements, the result of the addition of a and b is called the sum of a and b and denoted a + b. Similarly, the result of the multiplication of a and b is called the product of a and b, associativity of addition and multiplication For all a, b and c in F, one has a + = + c and a · = · c. Commutativity of addition and multiplication For all a and b in F one has a + b = b + a and a · b = b · a. Existence of additive and multiplicative identity elements There exists an element 0 in F, called the identity, such that for all a in F. There is an element 1, different from 0 and called the identity, such that for all a in F. Existence of additive inverses and multiplicative inverses For every a in F, there exists an element in F, denoted −a, such that a + =0. For every a ≠0 in F, there exists an element in F, denoted a−1, 1/a, or 1/a, distributivity of multiplication over addition For all a, b and c in F, one has a · = +. The elements 0 and 1 being required to be distinct, a field has, at least, for every a in F, one has − a = ⋅ a. Thus, the inverse of every element is known as soon as one knows the additive inverse of 1. A subtraction and a division are defined in every field by a − b = a +, a subfield E of a field F is a subset of F that contains 1, and is closed under addition, multiplication, additive inverse and multiplicative inverse of a nonzero element. It is straightforward to verify that a subfield is indeed a field, two groups are associated to every field. The field itself is a group under addition, when considering this group structure rather the field structure, one talks of the additive group of the field
28.
Hyperbola
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In mathematics, a hyperbola is a type of smooth curve lying in a plane, defined by its geometric properties or by equations for which it is the solution set. A hyperbola has two pieces, called connected components or branches, that are mirror images of each other, the hyperbola is one of the three kinds of conic section, formed by the intersection of a plane and a double cone. If the plane intersects both halves of the double cone but does not pass through the apex of the cones, each branch of the hyperbola has two arms which become straighter further out from the center of the hyperbola. Diagonally opposite arms, one from each branch, tend in the limit to a common line, called the asymptote of those two arms. So there are two asymptotes, whose intersection is at the center of symmetry of the hyperbola, which can be thought of as the point about which each branch reflects to form the other branch. In the case of the curve f =1 / x the asymptotes are the two coordinate axes, hyperbolas share many of the ellipses analytical properties such as eccentricity, focus, and directrix. Typically the correspondence can be made with nothing more than a change of sign in some term, many other mathematical objects have their origin in the hyperbola, such as hyperbolic paraboloids, hyperboloids, hyperbolic geometry, hyperbolic functions, and gyrovector spaces. The word hyperbola derives from the Greek ὑπερβολή, meaning over-thrown or excessive, hyperbolae were discovered by Menaechmus in his investigations of the problem of doubling the cube, but were then called sections of obtuse cones. The term hyperbola is believed to have coined by Apollonius of Perga in his definitive work on the conic sections. The rectangle could be applied to the segment, be shorter than the segment or exceed the segment, the midpoint M of the line segment joining the foci is called the center of the hyperbola. The line through the foci is called the major axis and it contains the vertices V1, V2, which have distance a to the center. The distance c of the foci to the center is called the distance or linear eccentricity. The quotient c a is the eccentricity e, C2 is called the director circle of the hyperbola. In order to get the branch of the hyperbola, one has to use the director circle related to F1. This property should not be confused with the definition of a hyperbola with help of a directrix below, for an arbitrary point the distance to the focus is 2 + y 2 and to the second focus 2 + y 2. Hence the point is on the hyperbola if the condition is fulfilled 2 + y 2 −2 + y 2 = ±2 a. Remove the square roots by suitable squarings and use the relation b 2 = c 2 − a 2 to obtain the equation of the hyperbola, the shape parameters a, b are called the semi major axis and semi minor axis or conjugate axis. As opposed to an ellipse, a hyperbola has two vertices
29.
Hyperboloid
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In geometry, a hyperboloid of revolution, sometimes called circular hyperboloid, is a surface that may be generated by rotating a hyperbola around one of its principal axes. An hyperboloid is a surface that may be obtained from a paraboloid of revolution by deforming it by means of directional scalings, or more generally, of an affine transformation. A hyperboloid is a surface, that is a surface that may be defined as the zero set of a polynomial of degree two in three variables. Among quadric surfaces, a hyperboloid is characterized by not being a cone or a cylinder, having a center of symmetry, a hyperboloid has also three pairwise perpendicular axes of symmetry, and three pairwise perpendicular planes of symmetry. Both of these surfaces are asymptotic to the cone of equation x 2 a 2 + y 2 b 2 − z 2 c 2 =0, one has an hyperboloid of revolution if and only if a 2 = b 2. It is a ruled surface. In case of a = b the hyperboloid is a surface of revolution, the more common generation of a hyperboloid of revolution is rotating a hyperbola around its semi-minor axis. Remark, A hyperboloid of two sheets is projectively equivalent to a hyperbolic paraboloid, for simplicity the plane sections of the unit hyperboloid with equation H1, x 2 + y 2 − z 2 =1 are considered. Because a hyperboloid in general position is an image of the unit hyperboloid. The hyperboloid of two sheets does not contain lines, the discussion of plane sections can be performed for the unit hyperboloid of two sheets with equation H2, x 2 + y 2 − z 2 = −1. Remark, A hyperboloid of two sheets is projectively equivalent to a sphere, whereas the Gaussian curvature of a hyperboloid of one sheet is negative, that of a two-sheet hyperboloid is positive. In spite of its curvature, the hyperboloid of two sheets with another suitably chosen metric can also be used as a model for hyperbolic geometry. More generally, an arbitrarily oriented hyperboloid, centered at v, is defined by the equation T A =1, where A is a matrix and x, v are vectors. The eigenvectors of A define the directions of the hyperboloid. The one-sheet hyperboloid has two positive eigenvalues and one negative eigenvalue, the two-sheet hyperboloid has one positive eigenvalue and two negative eigenvalues. Imaginary hyperboloids are frequently found in mathematics of higher dimensions, for example, in a pseudo-Euclidean space one has the use of a quadratic form, q = −, k < n. When c is any constant, then the part of the space given by is called a hyperboloid, the degenerate case corresponds to c =0. As an example, consider the following passage, however, the term quasi-sphere is also used in this context since the sphere and hyperboloid have some commonality
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Cone
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A cone is a three-dimensional geometric shape that tapers smoothly from a flat base to a point called the apex or vertex. A cone is formed by a set of segments, half-lines, or lines connecting a common point. If the enclosed points are included in the base, the cone is a solid object, otherwise it is a two-dimensional object in three-dimensional space. In the case of an object, the boundary formed by these lines or partial lines is called the lateral surface, if the lateral surface is unbounded. In the case of line segments, the cone does not extend beyond the base, while in the case of half-lines, in the case of lines, the cone extends infinitely far in both directions from the apex, in which case it is sometimes called a double cone. Either half of a cone on one side of the apex is called a nappe. The axis of a cone is the line, passing through the apex. If the base is right circular the intersection of a plane with this surface is a conic section, in general, however, the base may be any shape and the apex may lie anywhere. Contrasted with right cones are oblique cones, in which the axis passes through the centre of the base non-perpendicularly, a cone with a polygonal base is called a pyramid. Depending on the context, cone may also mean specifically a convex cone or a projective cone, cones can also be generalized to higher dimensions. The perimeter of the base of a cone is called the directrix, the base radius of a circular cone is the radius of its base, often this is simply called the radius of the cone. The aperture of a circular cone is the maximum angle between two generatrix lines, if the generatrix makes an angle θ to the axis, the aperture is 2θ. A cone with a region including its apex cut off by a plane is called a cone, if the truncation plane is parallel to the cones base. An elliptical cone is a cone with an elliptical base, a generalized cone is the surface created by the set of lines passing through a vertex and every point on a boundary. The slant height of a circular cone is the distance from any point on the circle to the apex of the cone. It is given by r 2 + h 2, where r is the radius of the cirf the cone and this application is primarily useful in determining the slant height of a cone when given other information regarding the radius or height. The volume V of any conic solid is one third of the product of the area of the base A B and the height h V =13 A B h. In modern mathematics, this formula can easily be computed using calculus – it is, up to scaling, the integral ∫ x 2 d x =13 x 3
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Big O notation
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Big O notation is a mathematical notation that describes the limiting behavior of a function when the argument tends towards a particular value or infinity. It is a member of a family of notations invented by Paul Bachmann, Edmund Landau, in computer science, big O notation is used to classify algorithms according to how their running time or space requirements grow as the input size grows. Big O notation characterizes functions according to their rates, different functions with the same growth rate may be represented using the same O notation. The letter O is used because the rate of a function is also referred to as order of the function. A description of a function in terms of big O notation usually only provides a bound on the growth rate of the function. Associated with big O notation are several related notations, using the symbols o, Ω, ω, Big O notation is also used in many other fields to provide similar estimates. Let f and g be two functions defined on some subset of the real numbers. That is, f = O if and only if there exists a real number M. In many contexts, the assumption that we are interested in the rate as the variable x goes to infinity is left unstated. If f is a product of several factors, any constants can be omitted, for example, let f = 6x4 − 2x3 +5, and suppose we wish to simplify this function, using O notation, to describe its growth rate as x approaches infinity. This function is the sum of three terms, 6x4, −2x3, and 5, of these three terms, the one with the highest growth rate is the one with the largest exponent as a function of x, namely 6x4. Now one may apply the rule, 6x4 is a product of 6. Omitting this factor results in the simplified form x4, thus, we say that f is a big-oh of. Mathematically, we can write f = O, one may confirm this calculation using the formal definition, let f = 6x4 − 2x3 +5 and g = x4. Applying the formal definition from above, the statement that f = O is equivalent to its expansion, | f | ≤ M | x 4 | for some choice of x0 and M. To prove this, let x0 =1 and M =13, Big O notation has two main areas of application. In mathematics, it is used to describe how closely a finite series approximates a given function. In computer science, it is useful in the analysis of algorithms, in both applications, the function g appearing within the O is typically chosen to be as simple as possible, omitting constant factors and lower order terms
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Michiel Hazewinkel
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Born in Amsterdam to Jan Hazewinkel and Geertrude Hendrika Werner, Hazewinkel studied at the University of Amsterdam. After graduation Hazewinkel started his career as Assistant Professor at the University of Amsterdam in 1969. In 1970 he became Associate Professor at the Erasmus University Rotterdam, here he was thesis advisor of Roelof Stroeker, M. van de Vel, Jo Ritzen, and Gerard van der Hoek. From 1973 to 1975 he was also Professor at the Universitaire Instelling Antwerpen, were Marcel van de Vel was his PhD student. At the Centre for Mathematics and Computer CWI in Amsterdam in 1988 he became Professor of Mathematics and head of the Department of Algebra, Analysis, in 1994 Hazewinkel was elected member of the International Academy of Computer Sciences and Systems. Hazewinkel has authored and edited books, and numerous articles. With Michel Demazure and Pierre Gabriel, on invariants, canonical forms and moduli for linear, constant, finite dimensional, dynamical systems. Moduli and canonical forms for linear dynamical systems II, The topological case, on Lie algebras and finite dimensional filtering. Stochastics, a journal of probability and stochastic processes 7. 1–2. Nonexistence of finite-dimensional filters for conditional statistics of the sensor problem. Systems & control letters 3.6, 331–340, the algebra of quasi-symmetric functions is free over the integers
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Encyclopedia of Mathematics
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The Encyclopedia of Mathematics is a large reference work in mathematics. It is available in form and on CD-ROM. The 2002 version contains more than 8,000 entries covering most areas of mathematics at a level. The encyclopedia is edited by Michiel Hazewinkel and was published by Kluwer Academic Publishers until 2003, the CD-ROM contains animations and three-dimensional objects. Until November 29,2011, a version of the encyclopedia could be browsed online free of charge online This URL now redirects to the new wiki incarnation of the EOM. A new dynamic version of the encyclopedia is now available as a public wiki online and this new wiki is a collaboration between Springer and the European Mathematical Society. This new version of the encyclopedia includes the entire contents of the online version. All entries will be monitored for content accuracy by members of a board selected by the European Mathematical Society. Vinogradov, I. M. Matematicheskaya entsiklopediya, Moscow, Sov, Hazewinkel, M. Encyclopaedia of Mathematics, Kluwer,1994. Hazewinkel, M. Encyclopaedia of Mathematics, Vol.1, Hazewinkel, M. Encyclopaedia of Mathematics, Vol.2, Kluwer,1988. Hazewinkel, M. Encyclopaedia of Mathematics, Vol.3, Hazewinkel, M. Encyclopaedia of Mathematics, Vol.4, Kluwer,1989. Hazewinkel, M. Encyclopaedia of Mathematics, Vol.5, Hazewinkel, M. Encyclopaedia of Mathematics, Vol.6, Kluwer,1990. Hazewinkel, M. Encyclopaedia of Mathematics, Vol.7, Hazewinkel, M. Encyclopaedia of Mathematics, Vol.8, Kluwer,1992. Hazewinkel, M. Encyclopaedia of Mathematics, Vol.9, Hazewinkel, M. Encyclopaedia of Mathematics, Vol.10, Kluwer,1994. Hazewinkel, M. Encyclopaedia of Mathematics, Supplement I, Kluwer,1997, Hazewinkel, M. Encyclopaedia of Mathematics, Supplement II, Kluwer,2000. Hazewinkel, M. Encyclopaedia of Mathematics, Supplement III, Kluwer,2002, Hazewinkel, M. Encyclopaedia of Mathematics on CD-ROM, Kluwer,1998. Encyclopedia of Mathematics, public wiki monitored by a board under the management of the European Mathematical Society. List of online encyclopedias Current page of M. Hazewinkel Online Encyclopedia of Mathematics
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International Standard Book Number
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The International Standard Book Number is a unique numeric commercial book identifier. An ISBN is assigned to each edition and variation of a book, for example, an e-book, a paperback and a hardcover edition of the same book would each have a different ISBN. The ISBN is 13 digits long if assigned on or after 1 January 2007, the method of assigning an ISBN is nation-based and varies from country to country, often depending on how large the publishing industry is within a country. The initial ISBN configuration of recognition was generated in 1967 based upon the 9-digit Standard Book Numbering created in 1966, the 10-digit ISBN format was developed by the International Organization for Standardization and was published in 1970 as international standard ISO2108. Occasionally, a book may appear without a printed ISBN if it is printed privately or the author does not follow the usual ISBN procedure, however, this can be rectified later. Another identifier, the International Standard Serial Number, identifies periodical publications such as magazines, the ISBN configuration of recognition was generated in 1967 in the United Kingdom by David Whitaker and in 1968 in the US by Emery Koltay. The 10-digit ISBN format was developed by the International Organization for Standardization and was published in 1970 as international standard ISO2108, the United Kingdom continued to use the 9-digit SBN code until 1974. The ISO on-line facility only refers back to 1978, an SBN may be converted to an ISBN by prefixing the digit 0. For example, the edition of Mr. J. G. Reeder Returns, published by Hodder in 1965, has SBN340013818 -340 indicating the publisher,01381 their serial number. This can be converted to ISBN 0-340-01381-8, the check digit does not need to be re-calculated, since 1 January 2007, ISBNs have contained 13 digits, a format that is compatible with Bookland European Article Number EAN-13s. An ISBN is assigned to each edition and variation of a book, for example, an ebook, a paperback, and a hardcover edition of the same book would each have a different ISBN. The ISBN is 13 digits long if assigned on or after 1 January 2007, a 13-digit ISBN can be separated into its parts, and when this is done it is customary to separate the parts with hyphens or spaces. Separating the parts of a 10-digit ISBN is also done with either hyphens or spaces, figuring out how to correctly separate a given ISBN number is complicated, because most of the parts do not use a fixed number of digits. ISBN issuance is country-specific, in that ISBNs are issued by the ISBN registration agency that is responsible for country or territory regardless of the publication language. Some ISBN registration agencies are based in national libraries or within ministries of culture, in other cases, the ISBN registration service is provided by organisations such as bibliographic data providers that are not government funded. In Canada, ISBNs are issued at no cost with the purpose of encouraging Canadian culture. In the United Kingdom, United States, and some countries, where the service is provided by non-government-funded organisations. Australia, ISBNs are issued by the library services agency Thorpe-Bowker
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JSTOR
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JSTOR is a digital library founded in 1995. Originally containing digitized back issues of journals, it now also includes books and primary sources. It provides full-text searches of almost 2,000 journals, more than 8,000 institutions in more than 160 countries have access to JSTOR, most access is by subscription, but some older public domain content is freely available to anyone. William G. Bowen, president of Princeton University from 1972 to 1988, JSTOR originally was conceived as a solution to one of the problems faced by libraries, especially research and university libraries, due to the increasing number of academic journals in existence. Most libraries found it prohibitively expensive in terms of cost and space to maintain a collection of journals. By digitizing many journal titles, JSTOR allowed libraries to outsource the storage of journals with the confidence that they would remain available long-term, online access and full-text search ability improved access dramatically. Bowen initially considered using CD-ROMs for distribution, JSTOR was initiated in 1995 at seven different library sites, and originally encompassed ten economics and history journals. JSTOR access improved based on feedback from its sites. Special software was put in place to make pictures and graphs clear, with the success of this limited project, Bowen and Kevin Guthrie, then-president of JSTOR, wanted to expand the number of participating journals. They met with representatives of the Royal Society of London and an agreement was made to digitize the Philosophical Transactions of the Royal Society dating from its beginning in 1665, the work of adding these volumes to JSTOR was completed by December 2000. The Andrew W. Mellon Foundation funded JSTOR initially, until January 2009 JSTOR operated as an independent, self-sustaining nonprofit organization with offices in New York City and in Ann Arbor, Michigan. JSTOR content is provided by more than 900 publishers, the database contains more than 1,900 journal titles, in more than 50 disciplines. Each object is identified by an integer value, starting at 1. In addition to the site, the JSTOR labs group operates an open service that allows access to the contents of the archives for the purposes of corpus analysis at its Data for Research service. This site offers a facility with graphical indication of the article coverage. Users may create focused sets of articles and then request a dataset containing word and n-gram frequencies and they are notified when the dataset is ready and may download it in either XML or CSV formats. The service does not offer full-text, although academics may request that from JSTOR, JSTOR Plant Science is available in addition to the main site. The materials on JSTOR Plant Science are contributed through the Global Plants Initiative and are only to JSTOR
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John Wiley & Sons
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Founded in 1807, Wiley is also known for publishing For Dummies. As of 2015, the company had 4,900 employees, Wiley was established in 1807 when Charles Wiley opened a print shop in Manhattan. Wiley later shifted its focus to scientific, technical, and engineering subject areas, Charles Wileys son John took over the business when his father died in 1826. The firm was successively named Wiley, Lane & Co. then Wiley & Putnam, the company acquired its present name in 1876, when Johns second son William H. Wiley joined his brother Charles in the business. Through the 20th century, the company expanded its activities, the sciences. Since the establishment of the Nobel Prize in 1901, Wiley and its companies have published the works of more than 450 Nobel Laureates. Wiley in December 2010 opened an office in Dubai, to build on its business in the Middle East more effectively, the company has had an office in Beijing, China, since 2001, and China is now its sixth-largest market for STEM content. Wiley established publishing operations in India in 2006, and has established a presence in North Africa through sales contracts with academic institutions in Tunisia, Libya, and Egypt. On April 16,2012, the announced the establishment of Wiley Brasil Editora LTDA in São Paulo, Brazil. Wileys scientific, technical, and medical business was expanded by the acquisition of Blackwell Publishing in February 2007. Through a backfile initiative completed in 2007,8.2 million pages of content have been made available online. Other major journals published include Angewandte Chemie, Advanced Materials, Hepatology, International Finance, launched commercially in 1999, Wiley InterScience provided online access to Wiley journals, major reference works, and books, including backfile content. Journals previously from Blackwell Publishing were available online from Blackwell Synergy until they were integrated into Wiley InterScience on June 30,2008, in December 2007, Wiley also began distributing its technical titles through the Safari Books Online e-reference service. On February 17,2012, Wiley announced the acquisition of Inscape Holdings Inc. which provides DISC assessments and training for interpersonal business skills. On August 13,2012, Wiley announced it entered into an agreement to sell all of its travel assets, including all of its interests in the Frommers brand. On October 2,2012, Wiley announced it would acquire Deltak edu, LLC, Deltak is expected to contribute solid growth to both Wileys Global Education business and Wiley overall. Seventh-generation members Jesse and Nate Wiley work in the companys Professional/Trade and Scientific, Technical, Medical, and Scholarly businesses, respectively. Wiley has been owned since 1962, and listed on the New York Stock Exchange since 1995, its stock is traded under the symbols NYSE, JW. A and NYSE