High Quality Content by WIKIPEDIA articles! In probability theory, a probability mass function (pmf) is a function that gives the probability that a discrete random variable is exactly equal to some value. A pmf differs from a probability density function (pdf) in that the values of a pdf, defined only for continuous random variables, are not probabilities as such. Instead, the integral of a pdf over a range of possible values (a,b] gives the probability of the random variable falling within that range. See interval notation (mathematics) for the meaning of the notation (a,b]. Since the image of X is countable, the probability mass function fX(x) is zero for all but a countable number of values of x. The discontinuity of probability mass...

ISBN: 978-6-1336-7319-9
Издательство: Книга по требованию
Дата выхода: июль 2011

High Quality Content by WIKIPEDIA articles! In probability theory, the probability P of some event E, denoted P(E), is usually defined in such a way that P satisfies the Kolmogorov axioms, named after Andrey Kolmogorov, which are described below. These assumptions can be summarised as: Let ( , F, P) be a measure space with P( )=1. Then ( , F, P) is a probability space, with sample space , event space F and probability measure P. An alternative approach to formalising probability, favoured by some Bayesians, is given by Cox's theorem. This is the assumption of unit measure: that the probability that some elementary event in the entire sample space will occur is 1. More specifically, there are no elementary events outside the sample space....

ISBN: 978-6-1336-7253-6
Издательство: Книга по требованию
Дата выхода: июль 2011

High Quality Content by WIKIPEDIA articles! In probability theory and statistics, the probit function is the inverse cumulative distribution function (CDF), or quantile function associated with the standard normal distribution. It has applications in exploratory statistical graphics and specialized regression modeling of binary response variables. For the standard normal distribution (often denoted N(0,1)), the CDF is commonly denoted ?(z). ?(z) is a continuous, monotone increasing sigmoid function whose domain is the real line and range is (0,1). As an example, consider the familiar fact that the N(0,1) distribution places 95% of probability between -1.96 and 1.96, and is symmetric around zero. It follows that Данное...

ISBN: 978-6-1303-4810-6
Издательство: Книга по требованию
Дата выхода: июль 2011

High Quality Content by WIKIPEDIA articles! In probability theory and statistics, the probit function is the inverse cumulative distribution function, or quantile function associated with the standard normal distribution. It has applications in exploratory statistical graphics and specialized regression modeling of binary response variables. The idea of probit was published in 1934 by Chester Ittner Bliss in an article in Science on how to treat data such as the percentage of a pest killed by a pesticide. Bliss proposed transforming the percentage killed into a "probability unit" which was linearly related to the modern definition. He included a table to aid other researchers to convert their kill percentages to his probit, which they...

ISBN: 978-6-1303-4313-2
Издательство: Книга по требованию
Дата выхода: июль 2011

High Quality Content by WIKIPEDIA articles! A problem domain is the area of expertise or application that needs to be examined to solve a problem. A problem domain is simply looking at only the topics you are interested in, and excluding everything else. For example, if you were developing a system trying to measure good practice in medicine, you wouldn't include carpet drawings at hospitals in your problem domain. In this example the domain refers to relevant topics solely within your interest: medicine. This points to one of the limitations of overly specific and bounded problem domains, one may think they are interested in medicine and not interior design, but a better solution exists outside of the problem domain as it was initially...

ISBN: 978-6-1336-7589-6
Издательство: Книга по требованию
Дата выхода: июль 2011

High Quality Content by WIKIPEDIA articles! Problem solving is a mental process and is part of the larger problem process that includes problem finding and problem shaping. Considered the most complex of all intellectual functions, problem solving has been defined as higher-order cognitive process that requires the modulation and control of more routine or fundamental skills. Problem solving occurs when an organism or an artificial intelligence system needs to move from a given state to a desired goal state. Данное издание не является оригинальным. Книга печатается по технологии принт-он-деманд после получения заказа.

ISBN: 978-6-1303-1754-6
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Дата выхода: июль 2011

High Quality Content by WIKIPEDIA articles! The problem of points, also called the problem of division of the stakes, is a classical problem in probability theory. One of the famous problems that motivated the beginnings of modern probability theory in the 17th century, it led Blaise Pascal to the first explicit reasoning about what today is known as an expectation value. The problem concerns a game of chance with two players who have equal chances of winning each round. The players contribute equally to a prize pot, and agree in advance that the first player to have won a certain number of rounds will collect the entire prize. Now suppose that the game is interrupted by external circumstances before either player has achieved victory....

ISBN: 978-6-1336-8253-5
Издательство: Книга по требованию
Дата выхода: июль 2011

High Quality Content by WIKIPEDIA articles! Problems involving arithmetic progressions are of interest in number theory, combinatorics, and computer science, both from theoretical and applied points of view. Find the cardinality (denoted by Ak(m)) of the largest subset of [0, 1, ..., m – 1] which contains no progression of k distinct terms. The elements of the forbidden progressions are not required to be consecutive. For example, A4(10) = 8, because [0, 1, 2, 4, 5, 7, 8, 9] has no arithmetic progressions of length 4, while all 9-element subsets of [0, 1, ..., 9] have one. Paul Erdos set a $1000 prize for a question related to this number, collected by Endre Szemeredi for what has become known as Szemeredi's theorem. Szemeredi's...

ISBN: 978-6-1336-8260-3
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Дата выхода: июль 2011

High Quality Content by WIKIPEDIA articles! Lors du deuxieme congres international des mathematiciens tenu a Paris en 1900, David Hilbert presenta une liste de problemes qui tenaient jusqu'alors les mathematiciens en echec. Ces problemes devaient, selon Hilbert, marquer le cours des mathematiques du XXe siecle, et l'on peut dire aujourd'hui que cela a ete grandement le cas. La liste definitive fut publiee apres la tenue du congres et est aujourd'hui familierement appelee problemes de Hilbert. Les sections suivantes presentent brievement chaque probleme. Данное издание не является оригинальным. Книга печатается по технологии принт-он-деманд после...

ISBN: 978-6-1340-5276-4
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Дата выхода: июль 2011

High Quality Content by WIKIPEDIA articles! A procept is an amalgam of three components: a process which produces a mathematical object and a symbol which is used to represent either process or object. It derives from the work of Eddie Gray and David O. Tall, and is a much used construct in mathematics education research. The notion was first published in a paper in the Journal for Research in Mathematics Education in 1994, and is part of the process-object literature. This body of literature suggests that mathematical objects are formed by encapsulating processes, that is to say that the mathematical object 3 is formed by an encapsulation of the process of counting: 1,2,3... Данное издание не является...

ISBN: 978-6-1311-4244-4
Издательство: Книга по требованию
Дата выхода: июль 2011

High Quality Content by WIKIPEDIA articles! En mathematiques, un processus de Markov est un processus stochastique possedant la propriete de Markov. Dans un tel processus, la prediction du futur a partir du present n'est pas rendue plus precise par des element d'information concernant le passe. Les processus de Markov portent le nom de leur decouvreur, Andrei Markov. Данное издание не является оригинальным. Книга печатается по технологии принт-он-деманд после получения заказа.

ISBN: 978-6-1340-5230-6
Издательство: Книга по требованию
Дата выхода: июль 2011

High Quality Content by WIKIPEDIA articles! In statistics, Procrustes analysis is a form of statistical shape analysis used to analyse the distribution of a set of shapes. The name Procrustes refers to a bandit from Greek mythology who made his victims fit his bed either by stretching their limbs or cutting them off. Here we just consider objects made up from a finite number k of points in n dimensions, these points are called landmark points. The shape of object can be considered as a member of an equivalence class formed by removing the translational, rotational and scaling components. For example, translational components can be removed from an object by translating the object so that the mean of all the points lies at the origin....

ISBN: 978-6-1303-4342-2
Издательство: Книга по требованию
Дата выхода: июль 2011

High Quality Content by WIKIPEDIA articles! In statistics, Procrustes analysis is a form of statistical shape analysis used to analyse the distribution of a set of shapes. The name Procrustes refers to a bandit from Greek mythology who made his victims fit his bed either by stretching their limbs or cutting them off. Here we just consider objects made up from a finite number k of points in n dimensions, these points are called landmark points. The shape of object can be considered as a member of an equivalence class formed by removing the translational, rotational and scaling components. Данное издание не является оригинальным. Книга печатается по технологии...

ISBN: 978-6-1303-4833-5
Издательство: Книга по требованию
Дата выхода: июль 2011

High Quality Content by WIKIPEDIA articles! In category theory, the product of two (or more) objects in a category is a notion designed to capture the essence behind constructions in other areas of mathematics such as the cartesian product of sets, the direct product of groups, the direct product of rings and the product of topological spaces. Essentially, the product of a family of objects is the "most general" object which admits a morphism to each of the given objects. Данное издание не является оригинальным. Книга печатается по технологии принт-он-деманд после получения заказа.

ISBN: 978-6-1303-1896-3
Издательство: Книга по требованию
Дата выхода: июль 2011

Please note that the content of this book primarily consists of articles available from Wikipedia or other free sources online. In mathematics, it is possible to combine several rings into one large product ring. This is done as follows: if I is some index set and Ri is a ring for every i in I, then the cartesian product i in I Ri can be turned into a ring by defining the operations coordinatewise, i.e. (ai) + (bi) = (ai + bi) / (ai) · (bi) = (ai · bi). The resulting ring is called a direct product of the rings Ri. The direct product of finitely many rings R1,...,Rk is also written as R1 x R2 x ... x Rk or R1 R2 ... Rk, and can also be called the direct sum (and sometimes the complete direct sum) of the rings Ri. If R = i in I Ri is a...

ISBN: 978-6-1345-2383-7
Издательство: Книга по требованию
Дата выхода: июль 2011

High Quality Content by WIKIPEDIA articles! In the mathematical field of category theory, the product of two categories C and D, denoted C x D and called a product category, is a straightforward extension of the concept of the Cartesian product of two sets. For small categories, this is the same as the action on objects of the categorical product in the category Cat. A functor whose domain is a product category is known as a bifunctor. Just as the binary Cartesian product is readily generalized to an n-ary Cartesian product, binary product of two categories can be generalized, completely analogously, to a product of n categories. The product operation on categories is commutative and associative, up to isomorphism, and so this...

ISBN: 978-6-1345-0829-2
Издательство: Книга по требованию
Дата выхода: июль 2011

High Quality Content by WIKIPEDIA articles! It is a common error, when studying calculus, to suppose that the derivative of (uv) equals (u ?)(v ?) (Leibniz himself made this error initially); however, there are clear counterexamples to this. For a (x) whose derivative is '(x), the function can also be written as (x) · 1, since 1 is the identity element for multiplication. If the above-mentioned misconception were true, (u?)(v?) would equal zero. This is true because the derivative of a constant (such as 1) is zero and the product of '(x) · 0 is also zero. Данное издание не является оригинальным. Книга печатается по технологии принт-он-деманд после...

ISBN: 978-6-1303-2558-9
Издательство: Книга по требованию
Дата выхода: июль 2011

High Quality Content by WIKIPEDIA articles! In category theory, the product of two objects in a category is a notion designed to capture the essence behind constructions in other areas of mathematics such as the cartesian product of sets, the direct product of groups, the direct product of rings and the product of topological spaces. Essentially, the product of a family of objects is the "most general" object which admits a morphism to each of the given objects. In mathematics, category theory deals in an abstract way with mathematical structures and relationships between them: it abstracts from sets and functions to objects linked in diagrams by morphisms or arrows. Данное издание не является...

ISBN: 978-6-1303-4353-8
Издательство: Книга по требованию
Дата выхода: июль 2011

High Quality Content by WIKIPEDIA articles! In topology and related areas of mathematics, a product space is the cartesian product of a family of topological spaces equipped with a natural topology called the product topology. This topology differs from another, perhaps more obvious, topology called the box topology, which can also be given to a product space and which agrees with the product topology when the product is over only finitely many spaces. However, the product topology is "correct" in that it makes the product space a categorical product of its factors, whereas the box topology is too fine; this is the sense in which the product topology is "natural". Данное издание не является...

ISBN: 978-6-1303-4846-5
Издательство: Книга по требованию
Дата выхода: июль 2011

Please note that the content of this book primarily consists of articles available from Wikipedia or other free sources online. In mathematics, given two measurable spaces and measures on them, one can obtain the product measurable space and the product measure on that space. Conceptually, this is similar to defining the Cartesian product of sets and the product topology of two topological spaces. The existence of this measure is guaranteed by the Hahn–Kolmogorov theorem. The uniqueness of product measure is guaranteed only in case that both (X1, 1, 1) and (X2, 2, 2) are -finite. The Borel measure on the Euclidean space Rn can be obtained as the product of n copies of the Borel measure on the real line R. Even if the two factors of the...

ISBN: 978-6-1345-2110-9
Издательство: Книга по требованию
Дата выхода: июль 2011

High Quality Content by WIKIPEDIA articles! In mathematics, a product is the result of multiplying, or an expression that identifies factors to be multiplied. The order in real or complex numbers are multiplied has no bearing on the product; this is known as the commutative law of multiplication. When matrices or members of various other associative algebras are multiplied the product usually depends on the order of the factors; in other words, matrix multiplication, and the multiplications in those other algebras, are non-commutative. The product operator for the product of a sequence is denoted by the capital Greek letter Pi ? (in analogy to the use of the capital Sigma ? as summation symbol). Данное издание не...

ISBN: 978-6-1305-1257-6
Издательство: Книга по требованию
Дата выхода: июль 2011

High Quality Content by WIKIPEDIA articles! En mathematiques, le produit de convolution de deux fonctions f et g, qui se note generalement « », est une autre fonction, obtenue a l'aide d'un calcul d'integrales, et generalisant l'idee de moyenne glissante. En statistique, on utilise une formule tres voisine pour definir la correlation croisee. Данное издание не является оригинальным. Книга печатается по технологии принт-он-деманд после получения заказа.

ISBN: 978-6-1340-3871-3
Издательство: Книга по требованию
Дата выхода: июль 2011

Please note that the content of this book primarily consists of articles available from Wikipedia or other free sources online. In programming languages and type theory, the product of two types is the type that characterizes the expressions which behaves, with respect to the evaluation mechanism, as pairs whose first component is an expression of the first type and whose second component is an expression of the second type. It is written A x B, where A and B are its component types. Elements of a product type are called tuples, and product types are thus also known as tuple types. In many languages, product types take the form of a record type, for which the components of a tuple can be accessed by label. In languages that have...

ISBN: 978-6-1345-2147-5
Издательство: Книга по требованию
Дата выхода: июль 2011

High Quality Content by WIKIPEDIA articles! In topology and related areas of mathematics, a product space is the cartesian product of a family of topological spaces equipped with a natural topology called the product topology. This topology differs from another, perhaps more obvious, topology called the box topology, which can also be given to a product space and which agrees with the product topology when the product is over only finitely many spaces. However, the product topology is "correct" in that it makes the product space a categorical product of its factors, whereas the box topology is too fine; this is the sense in which the product topology is "natural". Данное издание не является...

ISBN: 978-6-1303-4361-3
Издательство: Книга по требованию
Дата выхода: июль 2011

This volume on professional competence in mathematics education would make an invaluable addition to the libraries of educational researchers, teacher educators, school administrators, and graduate students. This book advances the theoretical understanding about the knowledge, skills, attitudes, and practices that distinguish competent mathematics teachers. The unique contribution of the book is that the theory of professional competence that it advances is grounded in the specifics of classroom practices in two countries, the United States and India. The use of grounded theory methodology is another unique feature of the book, and the painstaking attention to detail in the use of the methodology makes this a useful book for graduate...

ISBN: 978-3-6392-5162-3
Издательство: Книга по требованию
Дата выхода: июль 2011