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Cramer's Paradox
In mathematics, Cramer's paradox or the Cramer–Euler paradoxWeisstein, Eric W. "Cramér-Euler Paradox." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/Cramer-EulerParadox.html is the statement that the number of points of intersection of two higher-order curves in the plane can be greater than the number of arbitrary points that are usually needed to define one such curve. It is named after the Genevan mathematician Gabriel Cramer. This phenomenon appears paradoxical because the points of intersection fail to uniquely define any curve (they belong to at least two different curves) despite their large number. It is the result of a naive understanding or a misapplication of two theorems: * Bézout's theorem states that the number of points of intersection of two algebraic curves is equal to the product of their degrees, provided that certain necessary conditions are met. In particular, two curves of degree n generally have n^2 points of intersection. * Crame ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Two Cubic Curves
2 (two) is a number, numeral and digit. It is the natural number following 1 and preceding 3. It is the smallest and only even prime number. Because it forms the basis of a duality, it has religious and spiritual significance in many cultures. Evolution Arabic digit The digit used in the modern Western world to represent the number 2 traces its roots back to the Indic Brahmic script, where "2" was written as two horizontal lines. The modern Chinese and Japanese languages (and Korean Hanja) still use this method. The Gupta script rotated the two lines 45 degrees, making them diagonal. The top line was sometimes also shortened and had its bottom end curve towards the center of the bottom line. In the Nagari script, the top line was written more like a curve connecting to the bottom line. In the Arabic Ghubar writing, the bottom line was completely vertical, and the digit looked like a dotless closing question mark. Restoring the bottom line to its original horizontal ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Colin Maclaurin
Colin Maclaurin (; gd, Cailean MacLabhruinn; February 1698 – 14 June 1746) was a Scottish mathematician who made important contributions to geometry and algebra. He is also known for being a child prodigy and holding the record for being the youngest professor. The Maclaurin series, a special case of the Taylor series, is named after him. Owing to changes in orthography since that time (his name was originally rendered as M'Laurine), his surname is alternatively written MacLaurin. Early life Maclaurin was born in Kilmodan, Argyll. His father, John Maclaurin, minister of Glendaruel, died when Maclaurin was in infancy, and his mother died before he reached nine years of age. He was then educated under the care of his uncle, Daniel Maclaurin, minister of Kilfinan. A child prodigy, he entered university at age 11. Academic career At eleven, Maclaurin, a child prodigy at the time, entered the University of Glasgow. He graduated Master of Arts three years later by defending ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Algebraic Curves
In mathematics, an affine algebraic plane curve is the zero set of a polynomial in two variables. A projective algebraic plane curve is the zero set in a projective plane of a homogeneous polynomial in three variables. An affine algebraic plane curve can be completed in a projective algebraic plane curve by homogenizing its defining polynomial. Conversely, a projective algebraic plane curve of homogeneous equation can be restricted to the affine algebraic plane curve of equation . These two operations are each inverse to the other; therefore, the phrase algebraic plane curve is often used without specifying explicitly whether it is the affine or the projective case that is considered. More generally, an algebraic curve is an algebraic variety of dimension one. Equivalently, an algebraic curve is an algebraic variety that is birationally equivalent to an algebraic plane curve. If the curve is contained in an affine space or a projective space, one can take a projection for such a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Determinant (mathematics)
In mathematics, the determinant is a scalar value that is a function of the entries of a square matrix. It characterizes some properties of the matrix and the linear map represented by the matrix. In particular, the determinant is nonzero if and only if the matrix is invertible and the linear map represented by the matrix is an isomorphism. The determinant of a product of matrices is the product of their determinants (the preceding property is a corollary of this one). The determinant of a matrix is denoted , , or . The determinant of a matrix is :\begin a & b\\c & d \end=ad-bc, and the determinant of a matrix is : \begin a & b & c \\ d & e & f \\ g & h & i \end= aei + bfg + cdh - ceg - bdi - afh. The determinant of a matrix can be defined in several equivalent ways. Leibniz formula expresses the determinant as a sum of signed products of matrix entries such that each summand is the product of different entries, and the number of these summands is n!, the factorial of (the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Line (geometry)
In geometry, a line is an infinitely long object with no width, depth, or curvature. Thus, lines are one-dimensional objects, though they may exist in two, three, or higher dimension spaces. The word ''line'' may also refer to a line segment in everyday life, which has two points to denote its ends. Lines can be referred by two points that lay on it (e.g., \overleftrightarrow) or by a single letter (e.g., \ell). Euclid described a line as "breadthless length" which "lies evenly with respect to the points on itself"; he introduced several postulates as basic unprovable properties from which he constructed all of geometry, which is now called Euclidean geometry to avoid confusion with other geometries which have been introduced since the end of the 19th century (such as non-Euclidean, projective and affine geometry). In modern mathematics, given the multitude of geometries, the concept of a line is closely tied to the way the geometry is described. For instance, in analytic ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Julius Plücker
Julius Plücker (16 June 1801 – 22 May 1868) was a German mathematician and physicist. He made fundamental contributions to the field of analytical geometry and was a pioneer in the investigations of cathode rays that led eventually to the discovery of the electron. He also vastly extended the study of Lamé curves. Biography Early years Plücker was born at Elberfeld (now part of Wuppertal). After being educated at Düsseldorf and at the universities of Bonn, Heidelberg and Berlin he went to Paris in 1823, where he came under the influence of the great school of French geometers, whose founder, Gaspard Monge, had only recently died. In 1825 he returned to Bonn, and in 1828 was made professor of mathematics. In the same year he published the first volume of his ''Analytisch-geometrische Entwicklungen'', which introduced the method of "abridged notation". In 1831 he published the second volume, in which he clearly established on a firm and independent basis projective ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
James Stirling (mathematician)
James Stirling (11 May O.S. 1692, Garden, Stirlingshire – 5 December 1770, Edinburgh) was a Scottish mathematician. He was nicknamed "The Venetian". The Stirling numbers, Stirling permutations, and Stirling's approximation are named after him. He also proved the correctness of Isaac Newton's classification of cubics. Biography Stirling was born on 11 May 1692 O.S. at Garden House near Stirling, the third son of Archibald Stirling, Lord Garden. At 18 years of age he went to Balliol College, Oxford, where, chiefly through the influence of the Earl of Mar, he was nominated in 1711 to be one of Bishop Warner's exhibitioners (or Snell exhibitioner) at Balliol. In 1715 he was expelled on account of his correspondence with his cousins, who were members of the Keir and Garden families, who were noted Jacobites, and had been accessory to the " Gathering of the Brig o' Turk" in 1708. From Oxford he made his way to Venice, where he occupied himself as a professor of mathematics. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Introductio In Analysin Infinitorum
''Introductio in analysin infinitorum'' (Latin: ''Introduction to the Analysis of the Infinite'') is a two-volume work by Leonhard Euler which lays the foundations of mathematical analysis. Written in Latin and published in 1748, the ''Introductio'' contains 18 chapters in the first part and 22 chapters in the second. It has Eneström index, Eneström numbers E101 and E102. Carl Benjamin Boyer, Carl Boyer's lectures at the 1950 International Congress of Mathematicians compared the influence of Euler's ''Introductio'' to that of Euclid's ''Euclid's Elements, Elements'', calling the ''Elements'' the foremost textbook of ancient times, and the ''Introductio'' "the foremost textbook of modern times". Boyer also wrote: :The analysis of Euler comes close to the modern orthodox discipline, the study of functions by means of infinite processes, especially through infinite series. :It is doubtful that any other essentially didactic work includes as large a portion of original material tha ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Leonhard Euler
Leonhard Euler ( , ; 15 April 170718 September 1783) was a Swiss mathematician, physicist, astronomer, geographer, logician and engineer who founded the studies of graph theory and topology and made pioneering and influential discoveries in many other branches of mathematics such as analytic number theory, complex analysis, and infinitesimal calculus. He introduced much of modern mathematical terminology and notation, including the notion of a mathematical function. He is also known for his work in mechanics, fluid dynamics, optics, astronomy and music theory. Euler is held to be one of the greatest mathematicians in history and the greatest of the 18th century. A statement attributed to Pierre-Simon Laplace expresses Euler's influence on mathematics: "Read Euler, read Euler, he is the master of us all." Carl Friedrich Gauss remarked: "The study of Euler's works will remain the best school for the different fields of mathematics, and nothing else can replace it." Euler is a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Degeneracy (mathematics)
In mathematics, a degenerate case is a limiting case of a class of objects which appears to be qualitatively different from (and usually simpler than) the rest of the class, and the term degeneracy is the condition of being a degenerate case. The definitions of many classes of composite or structured objects often implicitly include inequalities. For example, the angles and the side lengths of a triangle are supposed to be positive. The limiting cases, where one or several of these inequalities become equalities, are degeneracies. In the case of triangles, one has a ''degenerate triangle'' if at least one side length or angle is zero. Equivalently, it becomes a "line segment". Often, the degenerate cases are the exceptional cases where changes to the usual dimension or the cardinality of the object (or of some part of it) occur. For example, a triangle is an object of dimension two, and a degenerate triangle is contained in a line, which makes its dimension one. This is similar ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics with the major subdisciplines of number theory, algebra, geometry, and analysis, respectively. There is no general consensus among mathematicians about a common definition for their academic discipline. Most mathematical activity involves the discovery of properties of abstract objects and the use of pure reason to prove them. These objects consist of either abstractions from nature orin modern mathematicsentities that are stipulated to have certain properties, called axioms. A ''proof'' consists of a succession of applications of deductive rules to already established results. These results include previously proved theorems, axioms, andin case of abstraction from naturesome basic properties that are considered true starting points of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
General Position
In algebraic geometry and computational geometry, general position is a notion of genericity for a set of points, or other geometric objects. It means the ''general case'' situation, as opposed to some more special or coincidental cases that are possible, which is referred to as special position. Its precise meaning differs in different settings. For example, generically, two lines in the plane intersect in a single point (they are not parallel or coincident). One also says "two generic lines intersect in a point", which is formalized by the notion of a generic point. Similarly, three generic points in the plane are not collinear; if three points are collinear (even stronger, if two coincide), this is a degenerate case. This notion is important in mathematics and its applications, because degenerate cases may require an exceptional treatment; for example, when stating general theorems or giving precise statements thereof, and when writing computer programs (see '' generic compl ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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