ideals

One also speaks of polynomials in several variables, obtained by taking the ring of polynomials of a ring of polynomials: R[X,Y] = (R[X])[Y] = (R[Y])[X]. These are of fundamental importance in algebraic geometry which studies the simultaneous zero sets of several such multivariate polynomials. Polynomials are frequently used to encode information about some other object. The characteristic polynomial of a matrix or linear operator contains information about the operator's eigenvalues. The minimal polynomial of an algebraic element records the simplest ...

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Polynomial, Polynomial - Elementary properties of polynomials, Polynomial - More advanced examples of polynomials, Polynomial - History, Polynomial - Polynomial functions, Polynomial - Graphs, Polynomial - End behavior, Polynomial - Number of x-intercepts, Polynomial - Number of turning points, Polynomial - Examples, Polynomial - Notes, Polynomial - Roots, Polynomial - Numerical analysis, Polynomial - Polynomials and calculus, Polynomial - Evaluation of polynomials, Polynomial - Finding roots, Polynomial - Several variables, Polynomial - Abstract algebra, Polynomial - Divisibility, Polynomial - More variables

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In commutative algebra, one major focus of study is divisibility among polynomials. If R is an integral domain and f and g are polynomials in R[X], it is said that f divides g if there exists a polynomial q in R[X] such that f q = g. One can then show that "every zero gives rise to a linear factor", or more formally: if f is a polynomial in R[X] and r is an element of R such that f(r) = 0, then the polynomial (X − r) divides f. The converse is also true. The quotie ...

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Polynomial, Polynomial - Elementary properties of polynomials, Polynomial - More advanced examples of polynomials, Polynomial - History, Polynomial - Polynomial functions, Polynomial - Graphs, Polynomial - End behavior, Polynomial - Number of x-intercepts, Polynomial - Number of turning points, Polynomial - Examples, Polynomial - Notes, Polynomial - Roots, Polynomial - Numerical analysis, Polynomial - Polynomials and calculus, Polynomial - Evaluation of polynomials, Polynomial - Finding roots, Polynomial - Several variables, Polynomial - Abstract algebra, Polynomial - Divisibility, Polynomial - More variables

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In abstract algebra, one must take care to distinguish between polynomials and polynomial functions. A polynomial f is defined to be a formal expression of the form where the coefficients a0, ..., an are elements of some ring R and X is considered to be a formal symbol. Two polynomials are considered to be equal if and only if the sequences of their coefficients are equal. Polynomials with coefficients in R can be added by simply ...

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Polynomial, Polynomial - Elementary properties of polynomials, Polynomial - More advanced examples of polynomials, Polynomial - History, Polynomial - Polynomial functions, Polynomial - Graphs, Polynomial - End behavior, Polynomial - Number of x-intercepts, Polynomial - Number of turning points, Polynomial - Examples, Polynomial - Notes, Polynomial - Roots, Polynomial - Numerical analysis, Polynomial - Polynomials and calculus, Polynomial - Evaluation of polynomials, Polynomial - Finding roots, Polynomial - Several variables, Polynomial - Abstract algebra, Polynomial - Divisibility, Polynomial - More variables

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In multivariate calculus, polynomials in several variables play an important role. These are the simplest multivariate functions and can be defined using addition and multiplication alone. An example of a polynomial in the variables x, y, and z is The total degree of such a multivariate polynomial can be found by adding the exponents of the variables in every term, and taking the maximum. The above polynomial f(x,&# ...

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Polynomial, Polynomial - Elementary properties of polynomials, Polynomial - More advanced examples of polynomials, Polynomial - History, Polynomial - Polynomial functions, Polynomial - Graphs, Polynomial - End behavior, Polynomial - Number of x-intercepts, Polynomial - Number of turning points, Polynomial - Examples, Polynomial - Notes, Polynomial - Roots, Polynomial - Numerical analysis, Polynomial - Polynomials and calculus, Polynomial - Evaluation of polynomials, Polynomial - Finding roots, Polynomial - Several variables, Polynomial - Abstract algebra, Polynomial - Divisibility, Polynomial - More variables

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As explained before, orders are ubiquitous in mathematics. However, earliest explicit mentionings of partial orders are probably to be found not before the 19th century. In this context the works of George Boole are of great importance. Moreover, works of Charles S. Peirce, Richard Dedekind, and Ernst Schröder also consider concepts of order theory. Certainly, there are others to be named in this context and surely there exists more detailed material on the history of order theory. Please contribute if any fu ...

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Order theory, Order theory - Background and motivation, Order theory - Introduction to the basic definitions, Order theory - Partially ordered sets, Order theory - Visualizing orders, Order theory - Special elements within an order, Order theory - Duality, Order theory - Constructing new orders, Order theory - Functions between orders, Order theory - Special types of orders, Order theory - Subsets of ordered sets, Order theory - Related mathematical areas, Order theory - Universal algebra, Order theory - Topology, Order theory - Category theory, Order theory - History, Order theory - Literature

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Although most mathematical areas use orders in one or the other way, there are also a few theories that have relationships which go far beyond mere application. Together with their major touching points with order theory, some of these are to be presented below. Order theory - Universal algebra. As already mentioned, the methods and formalisms of universal algebra are an important tool for many order theoretic considerations. Beside formalizing orders in terms of algebraic structures that satisfy ce ...

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Order theory, Order theory - Background and motivation, Order theory - Introduction to the basic definitions, Order theory - Partially ordered sets, Order theory - Visualizing orders, Order theory - Special elements within an order, Order theory - Duality, Order theory - Constructing new orders, Order theory - Functions between orders, Order theory - Special types of orders, Order theory - Subsets of ordered sets, Order theory - Related mathematical areas, Order theory - Universal algebra, Order theory - Topology, Order theory - Category theory, Order theory - History, Order theory - Literature

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In public discussions, some ideas seem to arise more commonly than others. Indeed, often completely separate people may be found to think alike in startling ways. For social scientists, one way of explaining such instances of common opinion is the presence of an ideology. Every society has an ideology that forms the basis of the "public opinion" or common sense, a basis that usually remains invisible to most people within the society. This dominant ideology appears as "neutral", holding to assumptions that are largely unchallenged. Me ...

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Ideology, Ideology - Ideology in everyday society, Ideology - History of the concept of ideology, Ideology - The analysis of ideology, Ideology - Ideology as an instrument of social reproduction, Ideology - Feminism as critique of ideology, Ideology - Political ideologies, Ideology - List of political ideologies, Ideology - Epistemological ideologies

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In general parlance, "idealism" or "idealist" is also used to describe a person having high ideals, sometimes with the connotation that those ideals are unrealisable or at odds with "practical" life. The word "ideal" is commonly used as an adjective to designate qualities of perfection, desirability, and excellence. This is foreign to the epistemological use of the word "idealism" which pertains to internal mental representations. These internal ideas represent objects that ar ...

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Idealism, Idealism - History, Idealism - Plato, Idealism - Plotinus, Idealism - Malebranche, Idealism - George Berkeley, Idealism - Arthur Collier, Idealism - Jonathan Edwards, Idealism - Immanuel Kant, Idealism - Fichte, Idealism - Hegel, Idealism - Schopenhauer, Idealism - British idealism, Idealism - Karl Pearson, Idealism - Critique of Idealism, Idealism - G. E. Moore, Idealism - David Stove, Idealism - John Searle, Idealism - Idealism in religious thought, Idealism - Other uses

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Not all religion and belief in the supernatural is, strictly speaking, anti-materialist in nature. While many types of religious belief are indeed specifically idealist, for example, Hindu beliefs about the nature of the Brahman, Zen Buddhism stands in the middle way of dialectics between idealism and materialism, and mainstream Christian doctrine affirms the importance of the materiality of Christ's human body and the necessi ...

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Idealism, Idealism - History, Idealism - Plato, Idealism - Plotinus, Idealism - Malebranche, Idealism - George Berkeley, Idealism - Arthur Collier, Idealism - Jonathan Edwards, Idealism - Immanuel Kant, Idealism - Fichte, Idealism - Hegel, Idealism - Schopenhauer, Idealism - British idealism, Idealism - Karl Pearson, Idealism - Critique of Idealism, Idealism - G. E. Moore, Idealism - David Stove, Idealism - John Searle, Idealism - Idealism in religious thought, Idealism - Other uses

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Idealism - G. E. Moore. The most influential criticism of Idealism is Moore's The Refutation of Idealism. This was the first application of Moore's analytic philosophical method, which greatly influenced Analytic philosophy. Moore proceeds by examining the Berkeleian aphorism esse is percipi: "to be is to be perceived". He examines in detail each of the three terms in the aphorism, finding that it must mean that the object and the subject are necessarily connected. So, he argues, for t ...

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Idealism, Idealism - History, Idealism - Plato, Idealism - Plotinus, Idealism - Malebranche, Idealism - George Berkeley, Idealism - Arthur Collier, Idealism - Jonathan Edwards, Idealism - Immanuel Kant, Idealism - Fichte, Idealism - Hegel, Idealism - Schopenhauer, Idealism - British idealism, Idealism - Karl Pearson, Idealism - Critique of Idealism, Idealism - G. E. Moore, Idealism - David Stove, Idealism - John Searle, Idealism - Idealism in religious thought, Idealism - Other uses

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Perhaps the most accessible source for the original meaning of "ideology" is Hippolyte Taine's work on the Ancien Regime (first volume of "Origins of Contemporary France"). He describes ideology as rather like teaching philosophy by the Socratic method, but without extending the vocabulary beyond what the general reader already possessed, and without the examples from observation which practical science would require. Taine identifies it not just with Destutt de Tracy, but wi ...

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Ideology, Ideology - Ideology in everyday society, Ideology - History of the concept of ideology, Ideology - The analysis of ideology, Ideology - Ideology as an instrument of social reproduction, Ideology - Feminism as critique of ideology, Ideology - Political ideologies, Ideology - List of political ideologies, Ideology - Epistemological ideologies

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In social studies, a political ideology is a certain ethical, set of ideals, principles, doctrines, myths or symbols of a social movement, institution, class, or large group that explain how society should work, and offer some political and cultural blueprint for a certain social order. A political ideology largely concerns itself with how to allocate power and to what ends it should be used. It can be a construct of political thought, often defining political parties and their policy. Studies of the concept of ideology itself (rather than specific ideologies) have been c ...

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Ideology, Ideology - Ideology in everyday society, Ideology - History of the concept of ideology, Ideology - The analysis of ideology, Ideology - Ideology as an instrument of social reproduction, Ideology - Feminism as critique of ideology, Ideology - Political ideologies, Ideology - List of political ideologies, Ideology - Epistemological ideologies

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In an ordered set, one can define many types of special subsets based on the given order. A simple example are upper sets; i.e. sets that contain all elements that are above them in the order. Formally, the upper closure of a set S in a poset P is given by the set {x in P | there is some y in S with y ≤ x}. A set that is equal to its upper closure is ca ...

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Order theory, Order theory - Background and motivation, Order theory - Introduction to the basic definitions, Order theory - Partially ordered sets, Order theory - Visualizing orders, Order theory - Special elements within an order, Order theory - Duality, Order theory - Constructing new orders, Order theory - Functions between orders, Order theory - Special types of orders, Order theory - Subsets of ordered sets, Order theory - Related mathematical areas, Order theory - Universal algebra, Order theory - Topology, Order theory - Category theory, Order theory - History, Order theory - Literature

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Many of the structures that are studied in order theory employ order relations with further properties. In fact, even some relations that are not partial orders are of special interest. Mainly the concept of a preorder has to be mentioned. A preorder is a relation that is reflexive and transitive, but not necessarily antisymmetric. Each preorder induces an equivalence relation between elements, where a is equivalent to b, if a ≤ b and a ≥ b. Preorders can be turned into orders by identifying all ele ...

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Order theory, Order theory - Background and motivation, Order theory - Introduction to the basic definitions, Order theory - Partially ordered sets, Order theory - Visualizing orders, Order theory - Special elements within an order, Order theory - Duality, Order theory - Constructing new orders, Order theory - Functions between orders, Order theory - Special types of orders, Order theory - Subsets of ordered sets, Order theory - Related mathematical areas, Order theory - Universal algebra, Order theory - Topology, Order theory - Category theory, Order theory - History, Order theory - Literature

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It is reasonable to consider functions between partially ordered sets having certain additional properties, that are related to the ordering relations of the two sets. The most fundamental condition that occurs in this context is monotonicity. A function f from a poset P to a poset Q is monotone, or order-preserving, if a ≤ b in P implies f(a) ≤ f(b) in Q (Noting that, strictly, the two relations here are different since they apply to different set ...

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Order theory, Order theory - Background and motivation, Order theory - Introduction to the basic definitions, Order theory - Partially ordered sets, Order theory - Visualizing orders, Order theory - Special elements within an order, Order theory - Duality, Order theory - Constructing new orders, Order theory - Functions between orders, Order theory - Special types of orders, Order theory - Subsets of ordered sets, Order theory - Related mathematical areas, Order theory - Universal algebra, Order theory - Topology, Order theory - Category theory, Order theory - History, Order theory - Literature

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Although written Chinese remained the official language of the Heian period imperial court, the introduction and wide use of kana saw a boom in Japanese literature. Despite the establishment of several new literary genre such as the novel and narrative monogatari (物語) and essays, literacy was only common among the court and Buddhist clergy. The lyrics of the modern Japanese national anthem, "Kimi Ga Yo," were written in the Heian period, as was The Tale of Genji by Murasaki Shikibu, one of the first novels in Japanes ...

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Heian period, Heian period - History, Heian period - Developments in Buddhism, Heian period - Heian period literature, Heian period - Heian period economics, Heian period - The Fujiwara Regency, Heian period - The Rise of the military class

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Orders appear everywhere - at least as far as mathematics and related areas, such as computer science, are concerned. The first order that one typically meets in primary school mathematical education is the order ≤ of natural numbers. This intuitive concept is easily extended to orderings of other sets of numbers, such as the integers and the reals. Indeed the idea of being greater or smaller than another number is one of the basic intuitions of number systems in general (although one usually is also interested in the actual difference of two numbers, which is not given by the order). Another familiar example of an ord ...

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Order theory, Order theory - Background and motivation, Order theory - Introduction to the basic definitions, Order theory - Partially ordered sets, Order theory - Visualizing orders, Order theory - Special elements within an order, Order theory - Duality, Order theory - Constructing new orders, Order theory - Functions between orders, Order theory - Special types of orders, Order theory - Subsets of ordered sets, Order theory - Related mathematical areas, Order theory - Universal algebra, Order theory - Topology, Order theory - Category theory, Order theory - History, Order theory - Literature

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This is a sitemap for Popular Pages IX - I . Click on a link and you will find multiple definitions and articles related to the word.

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