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Vojta's Conjecture
In mathematics, Vojta's conjecture is a conjecture introduced by about heights of points on algebraic varieties over number fields. The conjecture was motivated by an analogy between diophantine approximation and Nevanlinna theory (value distribution theory) in complex analysis. It implies many other conjectures in Diophantine approximation, Diophantine equations, arithmetic geometry, and mathematical logic. Statement of the conjecture Let F be a number field, let X/F be a non-singular algebraic variety, let D be an effective divisor on X with at worst normal crossings, let H be an ample divisor on X, and let K_X be a canonical divisor on X. Choose Weil height functions h_H and h_ and, for each absolute value v on F, a local height function \lambda_. Fix a finite set of absolute values S of F, and let \epsilon>0. Then there is a constant C and a non-empty Zariski open set U\subseteq X, depending on all of the above choices, such that :: \sum_ \lambda_(P) + h_(P) \le \epsilon h_ ... [...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] |
Absolute Value (algebra)
In algebra, an absolute value is a function that generalizes the usual absolute value. More precisely, if is a field or (more generally) an integral domain, an ''absolute value'' on is a function, commonly denoted , x, , from to the real numbers satisfying: It follows from the axioms that , 1, = 1, , -1, = 1, and , -x, =, x, for every . Furthermore, for every positive integer , , n, \le n, where the leftmost denotes the sum of summands equal to the identity element of . The classical absolute value and its square root are examples of absolute values, but not the square of the classical absolute value, which does not fulfill the triangular inequality. An absolute value such that , x+y, \le \max(, x, , , y, ) is an '' ultrametric absolute value.'' An absolute value induces a metric (and thus a topology) by d(f,g) = , f - g, . Examples *The standard absolute value on the integers. *The standard absolute value on the complex numbers. *The ''p''-adic absolute val ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Conjectures
In mathematics, a conjecture is a conclusion or a proposition that is proffered on a tentative basis without proof. Some conjectures, such as the Riemann hypothesis or Fermat's conjecture (now a theorem, proven in 1995 by Andrew Wiles), have shaped much of mathematical history as new areas of mathematics are developed in order to prove them. Resolution of conjectures Proof Formal mathematics is based on ''provable'' truth. In mathematics, any number of cases supporting a universally quantified conjecture, no matter how large, is insufficient for establishing the conjecture's veracity, since a single counterexample could immediately bring down the conjecture. Mathematical journals sometimes publish the minor results of research teams having extended the search for a counterexample farther than previously done. For instance, the Collatz conjecture, which concerns whether or not certain sequences of integers terminate, has been tested for all integers up to 1.2 × 1012 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Springer-Verlag
Springer Science+Business Media, commonly known as Springer, is a German multinational publishing company of books, e-books and peer-reviewed journals in science, humanities, technical and medical (STM) publishing. Originally founded in 1842 in Berlin, it expanded internationally in the 1960s, and through mergers in the 1990s and a sale to venture capitalists it fused with Wolters Kluwer and eventually became part of Springer Nature in 2015. Springer has major offices in Berlin, Heidelberg, Dordrecht, and New York City. History Julius Springer founded Springer-Verlag in Berlin in 1842 and his son Ferdinand Springer grew it from a small firm of 4 employees into Germany's then second-largest academic publisher with 65 staff in 1872.Chronology ". Springer Science+Business Media. In 1964, Springer expanded its business internationally, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
ABC Conjecture
ABC are the first three letters of the Latin script. ABC or abc may also refer to: Arts, entertainment and media Broadcasting * Aliw Broadcasting Corporation, Philippine broadcast company * American Broadcasting Company, a commercial American TV broadcaster ** Disney–ABC Television Group, the former name of the parent organization of ABC * Australian Broadcasting Corporation, one of the national publicly funded broadcasters of Australia ** ABC Television (Australian TV network), the national television network of the Australian Broadcasting Corporation *** ABC TV (Australian TV channel), the flagship TV station of the Australian Broadcasting Corporation *** ABC Canberra (TV station), Canberra, and other ABC TV local stations in state capitals *** ABC Australia (Southeast Asian TV channel), an international pay TV channel * ABC Radio (other), several radio stations * Associated Broadcasting Corporation, the former name of TV5 Network, Inc., a Philippine media com ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bombieri–Lang Conjecture
In arithmetic geometry, the Bombieri–Lang conjecture is an unsolved problem conjectured by Enrico Bombieri and Serge Lang about the Zariski density of the set of rational points of an algebraic variety of general type. Statement The weak Bombieri–Lang conjecture for surfaces states that if X is a smooth surface of general type defined over a number field k, then the points of X do not form a dense set in the Zariski topology on X. The general form of the Bombieri–Lang conjecture states that if X is a positive-dimensional algebraic variety of general type defined over a number field k, then the points of X do not form a dense set in the Zariski topology. The refined form of the Bombieri–Lang conjecture states that if X is an algebraic variety of general type defined over a number field k, then there is a dense open subset U of X such that for all number field extensions k' over k, the set of points in U is finite. History The Bombieri–Lang conjecture was independentl ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
General Type
In algebraic geometry, the Kodaira dimension measures the size of the canonical model of a projective variety . Soviet mathematician Igor Shafarevich in a seminar introduced an important numerical invariant of surfaces with the notation . Japanese mathematician Shigeru Iitaka extended it and defined the Kodaira dimension for higher dimensional varieties (under the name of canonical dimension), and later named it after Kunihiko Kodaira. The plurigenera The canonical bundle of a smooth algebraic variety ''X'' of dimension ''n'' over a field is the line bundle of ''n''-forms, :\,\!K_X = \bigwedge^n\Omega^1_X, which is the ''n''th exterior power of the cotangent bundle of ''X''. For an integer ''d'', the ''d''th tensor power of ''K''''X'' is again a line bundle. For ''d'' ≥ 0, the vector space of global sections ''H''0(''X'',''K''''X''''d'') has the remarkable property that it is a birational invariant of smooth projective varieties ''X''. That is, this vector space i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Serge Lang
Serge Lang (; May 19, 1927 – September 12, 2005) was a French-American mathematician and activist who taught at Yale University for most of his career. He is known for his work in number theory and for his mathematics textbooks, including the influential ''Algebra''. He received the Frank Nelson Cole Prize in 1960 and was a member of the Bourbaki group. As an activist, Lang campaigned against the Vietnam War, and also successfully fought against the nomination of the political scientist Samuel P. Huntington to the National Academies of Science. Later in his life, Lang was an HIV/AIDS denialist. He claimed that HIV had not been proven to cause AIDS and protested Yale's research into HIV/AIDS. Early life Lang was born in Saint-Germain-en-Laye, close to Paris, in 1927. He had a twin brother who became a basketball coach and a sister who became an actress. Lang moved with his family to California as a teenager, where he graduated in 1943 from Beverly Hills High School. Afte ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Calabi-Yau Variety
In algebraic and differential geometry, a Calabi–Yau manifold, also known as a Calabi–Yau space, is a particular type of manifold which has certain properties, such as Ricci flatness, yielding applications in theoretical physics. Particularly in superstring theory, the extra dimensions of spacetime are sometimes conjectured to take the form of a 6-dimensional Calabi–Yau manifold, which led to the idea of mirror symmetry. Their name was coined by , after , who first conjectured that compact complex manifolds of Kähler type with vanishing first Chern class always admit Ricci-flat Kähler metrics, and , who proved the Calabi conjecture. Calabi–Yau manifolds are complex manifolds that are generalizations of K3 surfaces in any number of complex dimensions (i.e. any even number of real dimensions). They were originally defined as compact Kähler manifolds with a vanishing first Chern class and a Ricci-flat metric, though many other similar but inequivalent d ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
K3 Surface
In mathematics, a complex analytic K3 surface is a compact connected complex manifold of dimension 2 with а trivial canonical bundle and irregularity of a surface, irregularity zero. An (algebraic) K3 surface over any field (mathematics), field means a smooth scheme, smooth proper morphism, proper geometrically connected algebraic surface that satisfies the same conditions. In the Enriques–Kodaira classification of surfaces, K3 surfaces form one of the four classes of minimal surfaces of Kodaira dimension zero. A simple example is the Fermat quartic surface x^4+y^4+z^4+w^4=0 in complex projective space, complex projective 3-space. Together with two-dimensional compact complex tori, K3 surfaces are the Calabi–Yau manifolds (and also the hyperkähler manifolds) of dimension two. As such, they are at the center of the classification of algebraic surfaces, between the positively curved del Pezzo surfaces (which are easy to classify) and the negatively curved surfaces of general t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Abelian Variety
In mathematics, particularly in algebraic geometry, complex analysis and algebraic number theory, an abelian variety is a smooth Algebraic variety#Projective variety, projective algebraic variety that is also an algebraic group, i.e., has a group law that can be defined by regular functions. Abelian varieties are at the same time among the most studied objects in algebraic geometry and indispensable tools for research on other topics in algebraic geometry and number theory. An abelian variety can be defined by equations having coefficients in any Field (mathematics), field; the variety is then said to be defined ''over'' that field. Historically the first abelian varieties to be studied were those defined over the field of complex numbers. Such abelian varieties turn out to be exactly those Complex torus, complex tori that can be holomorphic, holomorphically embedded into a complex projective space. Abelian varieties defined over algebraic number fields are a special case, which ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Height Function
A height function is a function that quantifies the complexity of mathematical objects. In Diophantine geometry, height functions quantify the size of solutions to Diophantine equations and are typically functions from a set of points on algebraic varieties (or a set of algebraic varieties) to the real numbers. For instance, the ''classical'' or ''naive height'' over the rational numbers is typically defined to be the maximum of the numerators and denominators of the coordinates (e.g. for the coordinates ), but in a logarithmic scale. Significance Height functions allow mathematicians to count objects, such as rational points, that are otherwise infinite in quantity. For instance, the set of rational numbers of naive height (the maximum of the numerator and denominator when expressed in lowest terms) below any given constant is finite despite the set of rational numbers being infinite. In this sense, height functions can be used to prove asymptotic results such as Baker's ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
