|
Separant
In mathematics, a differential field ''K'' is differentially closed if every finite system of differential equations with a solution in some differential field extending ''K'' already has a solution in ''K''. This concept was introduced by . Differentially closed fields are the analogues for differential equations of algebraically closed fields for polynomial equations. The theory of differentially closed fields We recall that a differential field is a field equipped with a derivation operator. Let ''K'' be a differential field with derivation operator ∂. *A differential polynomial in ''x'' is a polynomial in the formal expressions ''x'', ∂''x'', ∂2''x'', ... with coefficients in ''K''. *The order of a non-zero differential polynomial in ''x'' is the largest ''n'' such that ∂''n''''x'' occurs in it, or −1 if the differential polynomial is a constant. *The separant ''S''''f'' of a differential polynomial of order ''n''≥0 is the derivative of ''f'' with respect to ∠... [...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] |
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 c ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Proceedings Of The American Mathematical Society
''Proceedings of the American Mathematical Society'' is a monthly peer-reviewed scientific journal of mathematics published by the American Mathematical Society. As a requirement, all articles must be at most 15 printed pages. According to the ''Journal Citation Reports'', the journal has a 2018 impact factor of 0.813. Scope ''Proceedings of the American Mathematical Society'' publishes articles from all areas of pure and applied mathematics, including topology, geometry, analysis, algebra, number theory, combinatorics, logic, probability and statistics. Abstracting and indexing This journal is indexed in the following databases: 2011. American Mathematical Society. * |
Israel Journal Of Mathematics
'' Israel Journal of Mathematics'' is a peer-reviewed mathematics journal published by the Hebrew University of Jerusalem (Magnes Press). Founded in 1963, as a continuation of the ''Bulletin of the Research Council of Israel'' (Section F), the journal publishes articles on all areas of mathematics. The journal is indexed by ''Mathematical Reviews'' and Zentralblatt MATH. Its 2009 MCQ was 0.70, and its 2009 impact factor The impact factor (IF) or journal impact factor (JIF) of an academic journal is a scientometric index calculated by Clarivate that reflects the yearly mean number of citations of articles published in the last two years in a given journal, as i ... was 0.754. External links * Mathematics journals Publications established in 1963 English-language journals Bimonthly journals Hebrew University of Jerusalem {{math-journal-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bulletin Of The American Mathematical Society
The ''Bulletin of the American Mathematical Society'' is a quarterly mathematical journal published by the American Mathematical Society. Scope It publishes surveys on contemporary research topics, written at a level accessible to non-experts. It also publishes, by invitation only, book reviews and short ''Mathematical Perspectives'' articles. History It began as the ''Bulletin of the New York Mathematical Society'' and underwent a name change when the society became national. The Bulletin's function has changed over the years; its original function was to serve as a research journal for its members. Indexing The Bulletin is indexed in Mathematical Reviews, Science Citation Index, ISI Alerting Services, CompuMath Citation Index, and Current Contents/Physical, Chemical & Earth Sciences. See also *'' Journal of the American Mathematical Society'' *''Memoirs of the American Mathematical Society'' *''Notices of the American Mathematical Society'' *'' Proceedings of the American M ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Differential Galois Theory
In mathematics, differential Galois theory studies the Galois groups of differential equations. Overview Whereas algebraic Galois theory studies extensions of algebraic fields, differential Galois theory studies extensions of differential fields, i.e. fields that are equipped with a derivation, ''D''. Much of the theory of differential Galois theory is parallel to algebraic Galois theory. One difference between the two constructions is that the Galois groups in differential Galois theory tend to be matrix Lie groups, as compared with the finite groups often encountered in algebraic Galois theory. See also *Picard–Vessiot theory References * * * * * *{{Citation , last1=van der Put , first1=Marius , last2=Singer , first2=Michael F. , title=Galois theory of linear differential equations , url=http://www4.ncsu.edu/~singer/ms_papers.html , publisher=Springer-Verlag Springer Science+Business Media, commonly known as Springer, is a German multinational publishing company ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Superstable
In the mathematical field of model theory, a complete theory is called stable if it does not have too many types. One goal of classification theory is to divide all complete theories into those whose models can be classified and those whose models are too complicated to classify, and to classify all models in the cases where this can be done. Roughly speaking, if a theory is not stable then its models are too complicated and numerous to classify, while if a theory is stable there might be some hope of classifying its models, especially if the theory is superstable or totally transcendental. Stability theory was started by , who introduced several of the fundamental concepts, such as totally transcendental theories and the Morley rank. Stable and superstable theories were first introduced by , who is responsible for much of the development of stability theory. The definitive reference for stability theory is , though it is notoriously hard even for experts to read, as mentioned, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Stable Theory
In the mathematical field of model theory, a complete theory is called stable if it does not have too many types. One goal of classification theory is to divide all complete theories into those whose models can be classified and those whose models are too complicated to classify, and to classify all models in the cases where this can be done. Roughly speaking, if a theory is not stable then its models are too complicated and numerous to classify, while if a theory is stable there might be some hope of classifying its models, especially if the theory is superstable or totally transcendental. Stability theory was started by , who introduced several of the fundamental concepts, such as totally transcendental theories and the Morley rank. Stable and superstable theories were first introduced by , who is responsible for much of the development of stability theory. The definitive reference for stability theory is , though it is notoriously hard even for experts to read, as mentioned, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Morley Rank
In mathematical logic, Morley rank, introduced by , is a means of measuring the size of a subset of a model of a theory, generalizing the notion of dimension in algebraic geometry. Definition Fix a theory ''T'' with a model ''M''. The Morley rank of a formula ''φ'' defining a definable (with parameters) subset ''S'' of ''M'' is an ordinal or −1 or ∞, defined by first recursively defining what it means for a formula to have Morley rank at least ''α'' for some ordinal ''α''. *The Morley rank is at least 0 if ''S'' is non-empty. *For ''α'' a successor ordinal, the Morley rank is at least ''α'' if in some elementary extension ''N'' of ''M'', the set ''S'' has countably infinitely many disjoint definable subsets ''Si'', each of rank at least ''α'' − 1. *For ''α'' a non-zero limit ordinal, the Morley rank is at least ''α'' if it is at least ''β'' for all ''β'' less than ''α''. The Morley rank is then defined to be ''α'' if it is at least ''α'' but ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
ω-stable
In the mathematical field of model theory, a complete theory is called stable if it does not have too many types. One goal of classification theory is to divide all complete theories into those whose models can be classified and those whose models are too complicated to classify, and to classify all models in the cases where this can be done. Roughly speaking, if a theory is not stable then its models are too complicated and numerous to classify, while if a theory is stable there might be some hope of classifying its models, especially if the theory is superstable or totally transcendental. Stability theory was started by , who introduced several of the fundamental concepts, such as totally transcendental theories and the Morley rank. Stable and superstable theories were first introduced by , who is responsible for much of the development of stability theory. The definitive reference for stability theory is , though it is notoriously hard even for experts to read, as mentioned, ... [...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 \ldots" can be viewed as a question "When is there an x such that \ldots?", 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 high school 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 He ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
