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NIP (model Theory)
In model theory, a branch of mathematical logic, a complete theory ''T'' is said to satisfy NIP ("not the independence property") if none of its formulae satisfy the independence property—that is, if none of its formulae can pick out any given subset of an arbitrarily large finite set. Definition Let ''T'' be a complete ''L''-theory. An ''L''-formula φ(''x'',''y'') is said to have the independence property (with respect to ''x'', ''y'') if in every model ''M'' of ''T'' there is, for each ''n'' = < ω, a family of s ''b''0,...,''b''''n''−1 such that for each of the 2''n'' subsets ''X'' of ''n'' there is a tuple ''a'' in ''M'' for which : The theory ''T'' is said to have the independence property if some formula has the in ... [...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] |
O-minimal Theory
In mathematical logic, and more specifically in model theory, an infinite structure (''M'',<,...) that is totally ordered by < is called an o-minimal structure if and only if every definable subset ''X'' ⊆ ''M'' (with parameters taken from ''M'') is a finite union of intervals and points. O-minimality can be regarded as a weak form of . A structure ''M'' is o-minimal if and only if every |
Cambridge University Press
Cambridge University Press was the university press of the University of Cambridge. Granted a letters patent by King Henry VIII in 1534, it was the oldest university press in the world. Cambridge University Press merged with Cambridge Assessment to form Cambridge University Press and Assessment under Queen Elizabeth II's approval in August 2021. With a global sales presence, publishing hubs, and offices in more than 40 countries, it published over 50,000 titles by authors from over 100 countries. Its publications include more than 420 academic journals, monographs, reference works, school and university textbooks, and English language teaching and learning publications. It also published Bibles, runs a bookshop in Cambridge, sells through Amazon, and has a conference venues business in Cambridge at the Pitt Building and the Sir Geoffrey Cass Sports and Social Centre. It also served as the King's Printer. Cambridge University Press, as part of the University of Cambridge, was a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
P-adic Numbers
In number theory, given a prime number , the -adic numbers form an extension of the rational numbers which is distinct from the real numbers, though with some similar properties; -adic numbers can be written in a form similar to (possibly infinite) decimals, but with digits based on a prime number rather than ten, and extending to the left rather than to the right. For example, comparing the expansion of the rational number \tfrac15 in base vs. the -adic expansion, \begin \tfrac15 &= 0.01210121\ldots \ (\text 3) &&= 0\cdot 3^0 + 0\cdot 3^ + 1\cdot 3^ + 2\cdot 3^ + \cdots \\ mu\tfrac15 &= \dots 121012102 \ \ (\text) &&= \cdots + 2\cdot 3^3 + 1 \cdot 3^2 + 0\cdot3^1 + 2 \cdot 3^0. \end Formally, given a prime number , a -adic number can be defined as a series s=\sum_^\infty a_i p^i = a_k p^k + a_ p^ + a_ p^ + \cdots where is an integer (possibly negative), and each a_i is an integer such that 0\le a_i < p. A -adic integer is a -adic number such that < ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Valued Field
In algebra (in particular in algebraic geometry or algebraic number theory), a valuation is a function on a field that provides a measure of the size or multiplicity of elements of the field. It generalizes to commutative algebra the notion of size inherent in consideration of the degree of a pole or multiplicity of a zero in complex analysis, the degree of divisibility of a number by a prime number in number theory, and the geometrical concept of contact between two algebraic or analytic varieties in algebraic geometry. A field with a valuation on it is called a valued field. Definition One starts with the following objects: *a field and its multiplicative group ''K''×, *an abelian totally ordered group . The ordering and group law on are extended to the set by the rules * for all ∈ , * for all ∈ . Then a valuation of is any map : that satisfies the following properties for all ''a'', ''b'' in ''K'': * if and only if , *, *, with equality if ''v''(' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Linearly Ordered Group
In mathematics, specifically abstract algebra, a linearly ordered or totally ordered group is a group (mathematics), group ''G'' equipped with a total order "≤" that is ''translation-invariant''. This may have different meanings. We say that (''G'', ≤) is a: * left-ordered group if ≤ is left-invariant, that is ''a'' ≤ ''b'' implies ''ca'' ≤ ''cb'' for all ''a'', ''b'', ''c'' in ''G'', * right-ordered group if ≤ is right-invariant, that is ''a'' ≤ ''b'' implies ''ac'' ≤ ''bc'' for all ''a'', ''b'', ''c'' in ''G'', * bi-ordered group if ≤ is bi-invariant, that is it is both left- and right-invariant. A group ''G'' is said to be left-orderable (or right-orderable, or bi-orderable) if there exists a left- (or right-, or bi-) invariant order on ''G''. A simple necessary condition for a group to be left-orderable is to have no elements of finite order; however this is not a sufficient condition. It is equival ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Tree (set Theory)
In botany, a tree is a perennial plant with an elongated stem, or trunk, usually supporting branches and leaves. In some usages, the definition of a tree may be narrower, e.g., including only woody plants with secondary growth, only plants that are usable as lumber, or only plants above a specified height. But wider definitions include taller palms, tree ferns, bananas, and bamboos. Trees are not a monophyletic taxonomic group but consist of a wide variety of plant species that have independently evolved a trunk and branches as a way to tower above other plants to compete for sunlight. The majority of tree species are angiosperms or hardwoods; of the rest, many are gymnosperms or softwoods. Trees tend to be long-lived, some trees reaching several thousand years old. Trees evolved around 400 million years ago, and it is estimated that there are around three trillion mature trees in the world currently. A tree typically has many secondary branches supported clear of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Total Order
In mathematics, a total order or linear order is a partial order in which any two elements are comparable. That is, a total order is a binary relation \leq on some set X, which satisfies the following for all a, b and c in X: # a \leq a ( reflexive). # If a \leq b and b \leq c then a \leq c ( transitive). # If a \leq b and b \leq a then a = b ( antisymmetric). # a \leq b or b \leq a ( strongly connected, formerly called totality). Requirements 1. to 3. just make up the definition of a partial order. Reflexivity (1.) already follows from strong connectedness (4.), but is required explicitly by many authors nevertheless, to indicate the kinship to partial orders. Total orders are sometimes also called simple, connex, or full orders. A set equipped with a total order is a totally ordered set; the terms simply ordered set, linearly ordered set, toset and loset are also used. The term ''chain'' is sometimes defined as a synonym of ''totally ordered set'', but generally refers to ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Prime Number
A prime number (or a prime) is a natural number greater than 1 that is not a Product (mathematics), product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime because the only ways of writing it as a product, or , involve 5 itself. However, 4 is composite because it is a product (2 × 2) in which both numbers are smaller than 4. Primes are central in number theory because of the fundamental theorem of arithmetic: every natural number greater than 1 is either a prime itself or can be factorization, factorized as a product of primes that is unique up to their order. The property of being prime is called primality. A simple but slow primality test, method of checking the primality of a given number , called trial division, tests whether is a multiple of any integer between 2 and . Faster algorithms include the Miller–Rabin primality test, which is fast but has a small chance of error ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematical Logic
Mathematical logic is the study of Logic#Formal logic, formal logic within mathematics. Major subareas include model theory, proof theory, set theory, and recursion theory (also known as computability theory). Research in mathematical logic commonly addresses the mathematical properties of formal systems of logic such as their expressive or deductive power. However, it can also include uses of logic to characterize correct mathematical reasoning or to establish foundations of mathematics. Since its inception, mathematical logic has both contributed to and been motivated by the study of foundations of mathematics. This study began in the late 19th century with the development of axiomatic frameworks for geometry, arithmetic, and Mathematical analysis, analysis. In the early 20th century it was shaped by David Hilbert's Hilbert's program, program to prove the consistency of foundational theories. Results of Kurt Gödel, Gerhard Gentzen, and others provided partial resolution to th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Stable Theory
In the mathematical field of model theory, a theory is called stable if it satisfies certain combinatorial restrictions on its complexity. Stable theories are rooted in the proof of Morley's categoricity theorem and were extensively studied as part of Saharon Shelah's classification theory, which showed a dichotomy that either the models of a theory admit a nice classification or the models are too numerous to have any hope of a reasonable classification. A first step of this program was showing that if a theory is not stable then its models are too numerous to classify. Stable theories were the predominant subject of pure model theory from the 1970s through the 1990s, so their study shaped modern model theory and there is a rich framework and set of tools to analyze them. A major direction in model theory is "neostability theory," which tries to generalize the concepts of stability theory to broader contexts, such as simple and NIP theories. Motivation and history A common ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
VC Dimension
VC may refer to: Military decorations * Victoria Cross, a military decoration awarded by the United Kingdom and other Commonwealth nations ** Victoria Cross for Australia ** Victoria Cross (Canada) ** Victoria Cross for New Zealand * Victorious Cross, Idi Amin's self-bestowed military decoration Organisations * Ocean Airlines (IATA airline designator 2003-2008), Italian cargo airline * Voyageur Airways (IATA airline designator since 1968), Canadian charter airline * Visual Communications, an Asian-Pacific-American media arts organization in Los Angeles, California * Viet Cong, a political and military organization during the Vietnam War (1959–1975) Education * Vanier College, Canada * Vassar College, US * Velez College, Philippines * Virginia College, US * Ventura College, US Places * Saint Vincent and the Grenadines (ISO country code) * Sri Lanka (ICAO airport prefix code) * Watsonian vice-counties, subdivisions of Great Britain or Ireland * Ventura County, in S ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
