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Zariski Geometry
In mathematics, a Zariski geometry consists of an abstract structure introduced by Ehud Hrushovski and Boris Zilber, in order to give a characterisation of the Zariski topology on an algebraic curve, and all its powers. The Zariski topology on a product of algebraic varieties is very rarely the product topology, but richer in closed sets defined by equations that mix two sets of variables. The result described gives that a very definite meaning, applying to projective curves and compact Riemann surfaces in particular. Definition A Zariski geometry consists of a set ''X'' and a topological structure on each of the sets :''X'', ''X''2, ''X''3, ... satisfying certain axioms. (N) Each of the ''X''''n'' is a Noetherian topological space, of dimension at most ''n''. Some standard terminology for Noetherian spaces will now be assumed. (A) In each ''X''''n'', the subsets defined by equality in an ''n''-tuple are closed. The mappings : ''X''''m'' → ''X''''n'' defined by projecting ... [...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] |
Tuple
In mathematics, a tuple is a finite sequence or ''ordered list'' of numbers or, more generally, mathematical objects, which are called the ''elements'' of the tuple. An -tuple is a tuple of elements, where is a non-negative integer. There is only one 0-tuple, called the ''empty tuple''. A 1-tuple and a 2-tuple are commonly called a singleton and an ordered pair, respectively. The term ''"infinite tuple"'' is occasionally used for ''"infinite sequences"''. Tuples are usually written by listing the elements within parentheses "" and separated by commas; for example, denotes a 5-tuple. Other types of brackets are sometimes used, although they may have a different meaning. An -tuple can be formally defined as the image of a function that has the set of the first natural numbers as its domain. Tuples may be also defined from ordered pairs by a recurrence starting from an ordered pair; indeed, an -tuple can be identified with the ordered pair of its first elements and its t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Model Theory
In mathematical logic, model theory is the study of the relationship between theory (mathematical logic), formal theories (a collection of Sentence (mathematical logic), sentences in a formal language expressing statements about a Structure (mathematical logic), mathematical structure), and their Structure (mathematical logic), models (those Structure (mathematical logic), structures in which the statements of the theory hold). The aspects investigated include the number and size of models of a theory, the relationship of different models to each other, and their interaction with the formal language itself. In particular, model theorists also investigate the sets that can be definable set, defined in a model of a theory, and the relationship of such definable sets to each other. As a separate discipline, model theory goes back to Alfred Tarski, who first used the term "Theory of Models" in publication in 1954. Since the 1970s, the subject has been shaped decisively by Saharon Shel ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Journal Of The American Mathematical Society
The ''Journal of the American Mathematical Society'' (''JAMS''), is a quarterly peer-reviewed mathematical journal published by the American Mathematical Society. It was established in January 1988. Abstracting and indexing This journal is abstracted and indexed in: 2011. American Mathematical Society. * Mathematical Reviews * Zentralblatt MATH * Science Citation Index * ISI Alerting Services * CompuMath Citation Index * [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Algebraic Curve
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 homogenization of a polynomial, 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 function, 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. If the defining polynomial of a plane algebraic curve is irreducible polynomial, irreducible, then one has an ''irreducible plane algebraic curve''. Otherwise, the algebraic curve is the union of one or several irreducible curves, called its ''Irreduc ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Algebraically Closed Field
In mathematics, a field is algebraically closed if every non-constant polynomial in (the univariate polynomial ring with coefficients in ) has a root in . In other words, a field is algebraically closed if the fundamental theorem of algebra holds for it. Every field K is contained in an algebraically closed field C, and the roots in C of the polynomials with coefficients in K form an algebraically closed field called an algebraic closure of K. Given two algebraic closures of K there are isomorphisms between them that fix the elements of K. Algebraically closed fields appear in the following chain of class inclusions: Examples As an example, the field of real numbers is not algebraically closed, because the polynomial equation x^2+1=0 has no solution in real numbers, even though all its coefficients (1 and 0) are real. The same argument proves that no subfield of the real field is algebraically closed; in particular, the field of rational numbers is not algebraically cl ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Parametric Family
In mathematics and its applications, a parametric family or a parameterized family is a indexed family, family of objects (a set of related objects) whose differences depend only on the chosen values for a set of parameters. Common examples are parametrized (families of) Function (mathematics), functions, probability distributions, curves, shapes, etc. In probability and its applications For example, the probability density function of a random variable may depend on a parameter . In that case, the function may be denoted f_X( \cdot \, ; \theta) to indicate the dependence on the parameter . is not a formal argument of the function as it is considered to be fixed. However, each different value of the parameter gives a different probability density function. Then the ''parametric family'' of densities is the set of functions \ , where denotes the parameter space, the set of all possible values that the parameter can take. As an example, the normal distribution is a family ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Very Ample Line Bundle
Very may refer to: * English's prevailing intensifier Businesses * The Very Group, a British retail/consumer finance corporation ** Very (online retailer), their main e-commerce brand * VERY TV, a Thai television channel Places * Véry, a commune in Meuse department, France * Very (lunar crater), on the Moon * Very (Martian crater), on Mars Music * ''Very'' (Pet Shop Boys album), 1993 * ''Very'' (Dreamscape album), 1999 * ''Very'', an album by Miki Furukawa, 2010 People * Edward Wilson Very (1847–1910), US Navy officer, inventor of the Very flare gun * Frank Washington Very (1852–1927), American astronomer * Jones Very (1813–1880), American poet, essayist, clergyman and mystic * Lydia Louisa Anna Very (1823–1901), American author and illustrator * Pierre Véry (1900–1960), French novelist and screenwriter * Very Idham Henyansyah (born 1978), Indonesian serial killer Other uses * Very, the most common type of flare gun See also * Vary (other) * V ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Quantifier Elimination
Quantifier elimination is a concept of simplification used in mathematical logic, model theory, and theoretical computer science. Informally, a quantified statement "\exists x such that ..." can be viewed as a question "When is there an x such that ...?", and the statement without quantifiers can be viewed as the answer to that question. One way of classifying formulas is by the amount of quantification. Formulas with less depth of quantifier alternation are thought of as being simpler, with the quantifier-free formulas as the simplest. A theory has quantifier elimination if for every formula \alpha, there exists another formula \alpha_ without quantifiers that is equivalent to it (modulo this theory). Examples An example from mathematics says that a single-variable quadratic polynomial has a real root if and only if its discriminant is non-negative: \exists x\in\mathbb. (a\neq 0 \wedge ax^2+bx+c=0)\ \ \Longleftrightarrow\ \ a\neq 0 \wedge b^2-4ac\geq 0 Here the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Irreducible Set
In the mathematical field of topology, a hyperconnected space or irreducible space is a topological space ''X'' that cannot be written as the union of two proper closed subsets (whether disjoint or non-disjoint). The name ''irreducible space'' is preferred in algebraic geometry. For a topological space ''X'' the following conditions are equivalent: * No two nonempty open sets are disjoint. * ''X'' cannot be written as the union of two proper closed subsets. * Every nonempty open set is dense in ''X''. * Every open set is connected. * The interior of every proper closed subset of ''X'' is empty. * Every subset is dense or nowhere dense in ''X''. * No two points can be separated by disjoint neighbourhoods. A space which satisfies any one of these conditions is called ''hyperconnected'' or ''irreducible''. Due to the condition about neighborhoods of distinct points being in a sense the opposite of the Hausdorff property, some authors call such spaces anti-Hausdorff. The empty s ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Noetherian Topological Space
In mathematics, a Noetherian topological space, named for Emmy Noether, is a topological space in which closed subsets satisfy the descending chain condition. Equivalently, we could say that the open subsets satisfy the ascending chain condition, since they are the complements of the closed subsets. The Noetherian property of a topological space can also be seen as a strong compactness condition, namely that every open subset of such a space is compact, and in fact it is equivalent to the seemingly stronger statement that ''every'' subset is compact. Definition A topological space X is called Noetherian if it satisfies the descending chain condition for closed subsets: for any sequence : Y_1 \supseteq Y_2 \supseteq \cdots of closed subsets Y_i of X, there is an integer m such that Y_m=Y_=\cdots. Properties * A topological space X is Noetherian if and only if every subspace of X is compact (i.e., X is hereditarily compact), and if and only if every open subset of X is ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ehud Hrushovski
Ehud Hrushovski (; born 30 September 1959) is a mathematical logician. He is a Merton Professor of Mathematical Logic at the University of Oxford and a Fellow of Merton College, Oxford. He was also Professor of Mathematics at the Hebrew University of Jerusalem. Early life and education Hrushovski's father, Benjamin Harshav (Hebrew: בנימין הרשב, né Hruszowski; 1928–2015), was a literary theorist, a Yiddish and Hebrew poet and a translator, professor at Yale University and Tel Aviv University in comparative literature. Ehud Hrushovski earned his PhD from the University of California, Berkeley in 1986 under Leo Harrington; his dissertation was titled ''Contributions to Stable Model Theory''. He was a professor of mathematics at the Massachusetts Institute of Technology until 1994, when he became a professor at the Hebrew University of Jerusalem. Hrushovski moved in 2017 to the University of Oxford, where he is the Merton Professor of Mathematical Logic. Career Hrush ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
