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Cubic Surface
In mathematics, a cubic surface is a surface in 3-dimensional space defined by one polynomial equation of degree 3. Cubic surfaces are fundamental examples in algebraic geometry. The theory is simplified by working in projective space rather than affine space, and so cubic surfaces are generally considered in projective 3-space \mathbf^3. The theory also becomes more uniform by focusing on surfaces over the complex numbers rather than the real numbers; note that a complex surface has real dimension 4. A simple example is the Fermat cubic surface :x^3+y^3+z^3+w^3=0 in \mathbf^3. Many properties of cubic surfaces hold more generally for del Pezzo surfaces. Rationality of cubic surfaces A central feature of smooth scheme, smooth cubic surfaces ''X'' over an algebraically closed field is that they are all rational variety, rational, as shown by Alfred Clebsch in 1866. That is, there is a one-to-one correspondence defined by rational functions between the projective plane \mathbf^2 min ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many areas of mathematics, which include number theory (the study of numbers), algebra (the study of formulas and related structures), geometry (the study of shapes and spaces that contain them), Mathematical analysis, analysis (the study of continuous changes), and set theory (presently used as a foundation for all mathematics). Mathematics involves the description and manipulation of mathematical object, abstract objects that consist of either abstraction (mathematics), abstractions from nature orin modern mathematicspurely abstract entities that are stipulated to have certain properties, called axioms. Mathematics uses pure reason to proof (mathematics), prove properties of objects, a ''proof'' consisting of a succession of applications of in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Uniruled Variety
In algebraic geometry, a variety over a field k is ruled if it is birational to the product of the projective line with some variety over k. A variety is uniruled if it is covered by a family of rational curves. (More precisely, a variety X is uniruled if there is a variety Y and a dominant rational map Y\times \mathbf^1 \to X which does not factor through the projection to Y.) The concept arose from the ruled surfaces of 19th-century geometry, meaning surfaces in affine space or projective space which are covered by lines. Uniruled varieties can be considered to be relatively simple among all varieties, although there are many of them. Properties Every uniruled variety over a field of characteristic zero has Kodaira dimension −∞. The converse is a conjecture which is known in dimension at most 3: a variety of Kodaira dimension −∞ over a field of characteristic zero should be uniruled. A related statement is known in all dimensions: Boucksom, Demailly, Păun a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Group (mathematics)
In mathematics, a group is a Set (mathematics), set with an Binary operation, operation that combines any two elements of the set to produce a third element within the same set and the following conditions must hold: the operation is Associative property, associative, it has an identity element, and every element of the set has an inverse element. For example, the integers with the addition, addition operation form a group. The concept of a group was elaborated for handling, in a unified way, many mathematical structures such as numbers, geometric shapes and polynomial roots. Because the concept of groups is ubiquitous in numerous areas both within and outside mathematics, some authors consider it as a central organizing principle of contemporary mathematics. In geometry, groups arise naturally in the study of symmetries and geometric transformations: The symmetries of an object form a group, called the symmetry group of the object, and the transformations of a given type form a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Permutation
In mathematics, a permutation of a set can mean one of two different things: * an arrangement of its members in a sequence or linear order, or * the act or process of changing the linear order of an ordered set. An example of the first meaning is the six permutations (orderings) of the set : written as tuples, they are (1, 2, 3), (1, 3, 2), (2, 1, 3), (2, 3, 1), (3, 1, 2), and (3, 2, 1). Anagrams of a word whose letters are all different are also permutations: the letters are already ordered in the original word, and the anagram reorders them. The study of permutations of finite sets is an important topic in combinatorics and group theory. Permutations are used in almost every branch of mathematics and in many other fields of science. In computer science, they are used for analyzing sorting algorithms; in quantum physics, for describing states of particles; and in biology, for describing RNA sequences. The number of permutations of distinct objects is factorial, us ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Grassmannian
In mathematics, the Grassmannian \mathbf_k(V) (named in honour of Hermann Grassmann) is a differentiable manifold that parameterizes the set of all k-dimension (vector space), dimensional linear subspaces of an n-dimensional vector space V over a field (mathematics), field K that has a differentiable structure. For example, the Grassmannian \mathbf_1(V) is the space of lines through the origin in V, so it is the same as the projective space \mathbf(V) of one dimension lower than V. When V is a real number, real or complex number, complex vector space, Grassmannians are compact space, compact smooth manifolds, of dimension k(n-k). In general they have the structure of a nonsingular projective algebraic variety. The earliest work on a non-trivial Grassmannian is due to Julius Plücker, who studied the set of projective lines in real projective 3-space, which is equivalent to \mathbf_2(\mathbf^4), parameterizing them by what are now called Plücker coordinates. (See below.) Herma ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Schubert Calculus
In mathematics, Schubert calculus is a branch of algebraic geometry introduced in the nineteenth century by Hermann Schubert in order to solve various counting problems of projective geometry and, as such, is viewed as part of enumerative geometry. Giving it a more rigorous foundation was the aim of Hilbert's 15th problem. It is related to several more modern concepts, such as characteristic classes, and both its algorithmic aspects and applications remain of current interest. The term Schubert calculus is sometimes used to mean the enumerative geometry of linear subspaces of a vector space, which is roughly equivalent to describing the cohomology ring of Grassmannians. Sometimes it is used to mean the more general enumerative geometry of algebraic varieties that are homogenous spaces of simple Lie groups. Even more generally, Schubert calculus is sometimes understood as encompassing the study of analogous questions in generalized cohomology theories. The objects introduced ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
George Salmon
George Salmon (25 September 1819 – 22 January 1904) was a distinguished and influential Irish mathematician and Anglican theologian. After working in algebraic geometry for two decades, Salmon devoted the last forty years of his life to theology. His entire career was spent at Trinity College Dublin, having served as the 32nd Provost of the university from 1888 to 1904. Personal life Salmon was born in Cork in 1819, to Michael Salmon and Helen Weekes (the daughter of the Reverend Edward Weekes). During his boyhood in Cork, where his father Michael was a linen merchant, he attended Hamblin and Porter's School there before starting at Trinity College Dublin in 1833. In 1837, he won a scholarship and graduated from Trinity in 1839 with first-class honours in mathematics. In 1841, at the age of 21, he attained a paid fellowship and teaching position in mathematics at Trinity. In 1845, he was additionally appointed to a position in theology at the university, after having been ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Arthur Cayley
Arthur Cayley (; 16 August 1821 – 26 January 1895) was a British mathematician who worked mostly on algebra. He helped found the modern British school of pure mathematics, and was a professor at Trinity College, Cambridge for 35 years. He postulated what is now known as the Cayley–Hamilton theorem—that every square matrix is a root of its own characteristic polynomial, and verified it for matrices of order 2 and 3. He was the first to define the concept of an abstract group, a set with a binary operation satisfying certain laws, as opposed to Évariste Galois' concept of permutation groups. In group theory, Cayley tables, Cayley graphs, and Cayley's theorem are named in his honour, as well as Cayley's formula in combinatorics. Early life Arthur Cayley was born in Richmond, London, England, on 16 August 1821. His father, Henry Cayley, was a distant cousin of George Cayley, the aeronautics engineer innovator, and descended from an ancient Yorkshire family. He settled i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Algebraic Variety
Algebraic varieties are the central objects of study in algebraic geometry, a sub-field of mathematics. Classically, an algebraic variety is defined as the solution set, set of solutions of a system of polynomial equations over the real number, real or complex numbers. Modern definitions generalize this concept in several different ways, while attempting to preserve the geometric intuition behind the original definition. Conventions regarding the definition of an algebraic variety differ slightly. For example, some definitions require an algebraic variety to be Irreducible component, irreducible, which means that it is not the Union (set theory), union of two smaller Set (mathematics), sets that are Closed set, closed in the Zariski topology. Under this definition, non-irreducible algebraic varieties are called algebraic sets. Other conventions do not require irreducibility. The fundamental theorem of algebra establishes a link between algebra and geometry by showing that a mon ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Complex Manifold
In differential geometry and complex geometry, a complex manifold is a manifold with a ''complex structure'', that is an atlas (topology), atlas of chart (topology), charts to the open unit disc in the complex coordinate space \mathbb^n, such that the transition maps are Holomorphic function, holomorphic. The term "complex manifold" is variously used to mean a complex manifold in the sense above (which can be specified as an ''integrable'' complex manifold) or an almost complex manifold, ''almost'' complex manifold. Implications of complex structure Since holomorphic functions are much more rigid than smooth functions, the theories of smooth manifold, smooth and complex manifolds have very different flavors: compact space, compact complex manifolds are much closer to algebraic variety, algebraic varieties than to differentiable manifolds. For example, the Whitney embedding theorem tells us that every smooth ''n''-dimensional manifold can be Embedding, embedded as a smooth subma ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Conic
A conic section, conic or a quadratic curve is a curve obtained from a cone's surface intersecting 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, though it was sometimes considered a fourth type. The ancient Greek mathematicians studied conic sections, culminating around 200 BC with Apollonius of Perga's systematic work on their properties. The conic sections in the Euclidean plane have various distinguishing properties, many of which can be used as alternative definitions. One such property defines a non-circular conic to be the set of those points whose distances to some particular point, called a ''focus'', and some particular line, called a ''directrix'', are in a fixed ratio, called the ''eccentricity''. 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 o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Orientability
In mathematics, orientability is a property of some topological spaces such as real vector spaces, Euclidean spaces, surfaces, and more generally manifolds that allows a consistent definition of "clockwise" and "anticlockwise". A space is orientable if such a consistent definition exists. In this case, there are two possible definitions, and a choice between them is an orientation of the space. Real vector spaces, Euclidean spaces, and spheres are orientable. A space is non-orientable if "clockwise" is changed into "counterclockwise" after running through some loops in it, and coming back to the starting point. This means that a geometric shape, such as , that moves continuously along such a loop is changed into its own mirror image . A Möbius strip is an example of a non-orientable space. Various equivalent formulations of orientability can be given, depending on the desired application and level of generality. Formulations applicable to general topological manifolds oft ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
