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Structural Ramsey Theory
In mathematics, structural Ramsey theory is a categorical generalisation of Ramsey theory, rooted in the idea that many important results of Ramsey theory have "similar" logical structure. The key observation is noting that these Ramsey-type theorems can be expressed as the assertion that a certain category (or class of finite structures) has the Ramsey property (defined below). Structural Ramsey theory began in the 1970s with the work of Nešetřil and Rödl, and is intimately connected to Fraïssé theory. It received some renewed interest in the mid-2000s due to the discovery of the Kechris–Pestov–Todorčević correspondence, which connected structural Ramsey theory to topological dynamics. History is given credit for inventing the idea of a Ramsey property in the early 70s, and the first publication of this idea appears to be Graham, Leeb and Rothschild's 1972 paper on the subject. Key development of these ideas was done by Nešetřil and Rödl in their series of 19 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Stevo Todorčević
Stevo Todorčević ( sr-Cyrl, Стево Тодорчевић; born February 9, 1955), is a Yugoslavian mathematician specializing in mathematical logic and set theory. He holds a Canada Research Chair in mathematics at the University of Toronto, and a director of research position at the Centre national de la recherche scientifique in Paris. Early life and education Todorčević was born in Ubovića Brdo. As a child he moved to Banatsko Novo Selo, and went to school in Pančevo. At Belgrade University, he studied pure mathematics, attending lectures by Đuro Kurepa. He began graduate studies in 1978, and wrote his doctoral thesis in 1979 with Kurepa as his advisor. Research Todorčević's work involves mathematical logic, set theory, and their applications to pure mathematics. In Todorčević's 1978 master’s thesis, he constructed a model of MA + ¬wKH in a way to allow him to make the continuum any regular cardinal, and so derived a variety of topological consequences. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
:Category Theory
Category theory is a general theory of mathematical structures and their relations that was introduced by Samuel Eilenberg and Saunders Mac Lane in the middle of the 20th century in their foundational work on algebraic topology. Category theory is used in almost all areas of mathematics. In particular, many constructions of new mathematical objects from previous ones that appear similarly in several contexts are conveniently expressed and unified in terms of categories. Examples include quotient spaces, direct products, completion, and duality. Many areas of computer science also rely on category theory, such as functional programming and semantics. A category is formed by two sorts of objects: the objects of the category, and the morphisms, which relate two objects called the ''source'' and the ''target'' of the morphism. One often says that a morphism is an ''arrow'' that ''maps'' its source to its target. Morphisms can be ''composed'' if the target of the first morph ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Formal Language
In logic, mathematics, computer science, and linguistics, a formal language consists of words whose letters are taken from an alphabet and are well-formed according to a specific set of rules. The alphabet of a formal language consists of symbols, letters, or tokens that concatenate into strings of the language. Each string concatenated from symbols of this alphabet is called a word, and the words that belong to a particular formal language are sometimes called ''well-formed words'' or ''well-formed formulas''. A formal language is often defined by means of a formal grammar such as a regular grammar or context-free grammar, which consists of its formation rules. In computer science, formal languages are used among others as the basis for defining the grammar of programming languages and formalized versions of subsets of natural languages in which the words of the language represent concepts that are associated with particular meanings or semantics. In computational complexity ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fixed Point (mathematics)
A fixed point (sometimes shortened to fixpoint, also known as an invariant point) is a value that does not change under a given transformation. Specifically, in mathematics, a fixed point of a function is an element that is mapped to itself by the function. In physics, the term fixed point can refer to a temperature that can be used as a reproducible reference point, usually defined by a phase change or triple point. Fixed point of a function Formally, is a fixed point of a function if belongs to both the domain and the codomain of , and . For example, if is defined on the real numbers by f(x) = x^2 - 3 x + 4, then 2 is a fixed point of , because . Not all functions have fixed points: for example, , has no fixed points, since is never equal to for any real number. In graphical terms, a fixed point means the point is on the line , or in other words the graph of has a point in common with that line. Fixed-point iteration In numerical analysis, ''fixed-point iter ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Continuous Group Action
In topology, a continuous group action on a topological space ''X'' is a group action of a topological group ''G'' that is continuous: i.e., :G \times X \to X, \quad (g, x) \mapsto g \cdot x is a continuous map. Together with the group action, ''X'' is called a ''G''-space. If f: H \to G is a continuous group homomorphism of topological groups and if ''X'' is a ''G''-space, then ''H'' can act on ''X'' ''by restriction'': h \cdot x = f(h) x, making ''X'' a ''H''-space. Often ''f'' is either an inclusion or a quotient map. In particular, any topological space may be thought of as a ''G''-space via G \to 1 (and ''G'' would act trivially.) Two basic operations are that of taking the space of points fixed by a subgroup ''H'' and that of forming a quotient by ''H''. We write X^H for the set of all ''x'' in ''X'' such that hx = x. For example, if we write F(X, Y) for the set of continuous maps from a ''G''-space ''X'' to another ''G''-space ''Y'', then, with the action (g \cdot f)(x) = g f ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Topological Group
In mathematics, topological groups are logically the combination of groups and topological spaces, i.e. they are groups and topological spaces at the same time, such that the continuity condition for the group operations connects these two structures together and consequently they are not independent from each other. Topological groups have been studied extensively in the period of 1925 to 1940. Haar and Weil (respectively in 1933 and 1940) showed that the integrals and Fourier series are special cases of a very wide class of topological groups. Topological groups, along with continuous group actions, are used to study continuous symmetries, which have many applications, for example, in physics. In functional analysis, every topological vector space is an additive topological group with the additional property that scalar multiplication is continuous; consequently, many results from the theory of topological groups can be applied to functional analysis. Formal definition ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Alexander S
Alexander is a male given name. The most prominent bearer of the name is Alexander the Great, the king of the Ancient Greek kingdom of Macedonia who created one of the largest empires in ancient history. Variants listed here are Aleksandar, Aleksander and Aleksandr. Related names and diminutives include Iskandar, Alec, Alek, Alex, Alexandre, Aleks, Aleksa and Sander; feminine forms include Alexandra, Alexandria, and Sasha. Etymology The name ''Alexander'' originates from the (; 'defending men' or 'protector of men'). It is a compound of the verb (; 'to ward off, avert, defend') and the noun (, genitive: , ; meaning 'man'). It is an example of the widespread motif of Greek names expressing "battle-prowess", in this case the ability to withstand or push back an enemy battle line. The earliest attested form of the name, is the Mycenaean Greek feminine anthroponym , , (/ Alexandra/), written in the Linear B syllabic script. Alaksandu, alternatively called ''Alakasandu' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Surjective Function
In mathematics, a surjective function (also known as surjection, or onto function) is a function that every element can be mapped from element so that . In other words, every element of the function's codomain is the image of one element of its domain. It is not required that be unique; the function may map one or more elements of to the same element of . The term ''surjective'' and the related terms ''injective'' and ''bijective'' were introduced by Nicolas Bourbaki, a group of mainly French 20th-century mathematicians who, under this pseudonym, wrote a series of books presenting an exposition of modern advanced mathematics, beginning in 1935. The French word '' sur'' means ''over'' or ''above'', and relates to the fact that the image of the domain of a surjective function completely covers the function's codomain. Any function induces a surjection by restricting its codomain to the image of its domain. Every surjective function has a right inverse assuming the axiom ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Rigidity (mathematics)
In mathematics, a rigid collection ''C'' of mathematical objects (for instance sets or functions) is one in which every ''c'' ∈ ''C'' is uniquely determined by less information about ''c'' than one would expect. The above statement does not define a mathematical property. Instead, it describes in what sense the adjective rigid is typically used in mathematics, by mathematicians. __FORCETOC__ Examples Some examples include: #Harmonic functions on the unit disk are rigid in the sense that they are uniquely determined by their boundary values. #Holomorphic functions are determined by the set of all derivatives at a single point. A smooth function from the real line to the complex plane is not, in general, determined by all its derivatives at a single point, but it is if we require additionally that it be possible to extend the function to one on a neighbourhood of the real line in the complex plane. The Schwarz lemma is an example of such a rigidity theorem. #By the fu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hales–Jewett Theorem
In mathematics, the Hales–Jewett theorem is a fundamental combinatorial result of Ramsey theory named after Alfred W. Hales and Robert I. Jewett, concerning the degree to which high-dimensional objects must necessarily exhibit some combinatorial structure; it is impossible for such objects to be "completely random". An informal geometric statement of the theorem is that for any positive integers ''n'' and ''c'' there is a number ''H'' such that if the cells of a ''H''-dimensional ''n''×''n''×''n''×...×''n'' cube are colored with ''c'' colors, there must be one row, column, or certain diagonal (more details below) of length ''n'' all of whose cells are the same color. In other words, the higher-dimensional, multi-player, ''n''-in-a-row generalization of a game of tic-tac-toe cannot end in a draw, no matter how large ''n'' is, no matter how many people ''c'' are playing, and no matter which player plays each turn, provided only that it is played on a board of sufficiently high ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Affine Transformation
In Euclidean geometry, an affine transformation or affinity (from the Latin, ''affinis'', "connected with") is a geometric transformation that preserves lines and parallelism, but not necessarily Euclidean distances and angles. More generally, an affine transformation is an automorphism of an affine space (Euclidean spaces are specific affine spaces), that is, a function which maps an affine space onto itself while preserving both the dimension of any affine subspaces (meaning that it sends points to points, lines to lines, planes to planes, and so on) and the ratios of the lengths of parallel line segments. Consequently, sets of parallel affine subspaces remain parallel after an affine transformation. An affine transformation does not necessarily preserve angles between lines or distances between points, though it does preserve ratios of distances between points lying on a straight line. If is the point set of an affine space, then every affine transformation on can be repre ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Natural Number
In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and ordering (as in "this is the ''third'' largest city in the country"). Numbers used for counting are called ''Cardinal number, cardinal numbers'', and numbers used for ordering are called ''Ordinal number, ordinal numbers''. Natural numbers are sometimes used as labels, known as ''nominal numbers'', having none of the properties of numbers in a mathematical sense (e.g. sports Number (sports), jersey numbers). Some definitions, including the standard ISO/IEC 80000, ISO 80000-2, begin the natural numbers with , corresponding to the non-negative integers , whereas others start with , corresponding to the positive integers Texts that exclude zero from the natural numbers sometimes refer to the natural numbers together with zero as the whole numbers, while in other writings, that term is used instead for the integers (including negative integers). The natural ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
