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= 1 : {\displaystyle {x^{2} \over a^{2}}+{y^{2} \over b^{2}}+{z^{2} \over c^{2}}=1:} sphere (top, a=b=c=4), spheroid (bottom left, a=b=5, c=3), tri-axial ellipsoid (bottom right, a=4.5, b=6, c=3) <span>An ellipsoid is a surface that may be obtained from a sphere by deforming it by means of directional scalings, or more generally, of an affine transformation. An ellipsoid is a quadric surface, that is a surface that may be defined as the zero set of a polynomial of degree two in three variables. Among quadric surfaces, an ellipsoid is char

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ocultar ahora Mean reversion (finance) From Wikipedia, the free encyclopedia Jump to: navigation, search For other uses, see Mean reversion (disambiguation). <span>In finance, mean reversion is the assumption that a stock's price will tend to move to the average price over time. [1] [2] Using mean reversion in stock price analysis involves both identifying the trading range for a stock and computing the average price using analytical techniques taking into ac

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In finance, mean reversion is the assumption that a stock's price will tend to move to the average price over time.

ocultar ahora Mean reversion (finance) From Wikipedia, the free encyclopedia Jump to: navigation, search For other uses, see Mean reversion (disambiguation). <span>In finance, mean reversion is the assumption that a stock's price will tend to move to the average price over time. [1] [2] Using mean reversion in stock price analysis involves both identifying the trading range for a stock and computing the average price using analytical techniques taking into ac

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If the process depends only on |x − x'|, the Euclidean distance (not the direction) between x and x', then the process is considered isotropic.

stationary, it depends on their separation, x − x', while if non-stationary it depends on the actual position of the points x and x'. For example, the special case of an Ornstein–Uhlenbeck process, a Brownian motion process, is stationary. <span>If the process depends only on |x − x'|, the Euclidean distance (not the direction) between x and x', then the process is considered isotropic. A process that is concurrently stationary and isotropic is considered to be homogeneous; [7] in practice these properties reflect the differences (or rather the lack of them) in the behaviour of the process given the location of the observer. Ultimately Gaussian processes translate as taking priors on functions and the smoothness of these priors can be induced by the covariance function. [5] If we expect that for "ne

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An ellipsoid is a surface that may be obtained from a sphere by deforming it by means of directional scalings, or more generally, of an affine transformation.

= 1 : {\displaystyle {x^{2} \over a^{2}}+{y^{2} \over b^{2}}+{z^{2} \over c^{2}}=1:} sphere (top, a=b=c=4), spheroid (bottom left, a=b=5, c=3), tri-axial ellipsoid (bottom right, a=4.5, b=6, c=3) <span>An ellipsoid is a surface that may be obtained from a sphere by deforming it by means of directional scalings, or more generally, of an affine transformation. An ellipsoid is a quadric surface, that is a surface that may be defined as the zero set of a polynomial of degree two in three variables. Among quadric surfaces, an ellipsoid is char

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An ellipsoid is a surface that may be obtained from a sphere by deforming it by means of directional scalings, or more generally, of an affine transformation.

= 1 : {\displaystyle {x^{2} \over a^{2}}+{y^{2} \over b^{2}}+{z^{2} \over c^{2}}=1:} sphere (top, a=b=c=4), spheroid (bottom left, a=b=5, c=3), tri-axial ellipsoid (bottom right, a=4.5, b=6, c=3) <span>An ellipsoid is a surface that may be obtained from a sphere by deforming it by means of directional scalings, or more generally, of an affine transformation. An ellipsoid is a quadric surface, that is a surface that may be defined as the zero set of a polynomial of degree two in three variables. Among quadric surfaces, an ellipsoid is char

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ramming, and nicely supported in Python OOP has become an important concept in modern software engineering because It can help facilitate clean, efficient code (if used well) The OOP design pattern fits well with many computing problems <span>OOP is about producing well organized code — an important determinant of productivity Moreover, OOP is a part of Python, and to progress further it’s necessary to understand the basics About OOP¶ OOP is supported in many languages: JAVA and Ruby are relativel

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OOP is about producing well organized code — an important determinant of productivity

ramming, and nicely supported in Python OOP has become an important concept in modern software engineering because It can help facilitate clean, efficient code (if used well) The OOP design pattern fits well with many computing problems <span>OOP is about producing well organized code — an important determinant of productivity Moreover, OOP is a part of Python, and to progress further it’s necessary to understand the basics About OOP¶ OOP is supported in many languages: JAVA and Ruby are relativel

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[imagelink] This article is an orphan, as no other articles link to it. Please introduce links to this page from related articles; try the Find link tool for suggestions. (July 2012) [imagelink] The mountain car problem <span>Mountain Car, a standard testing domain in Reinforcement Learning, is a problem in which an under-powered car must drive up a steep hill. Since gravity is stronger than the car's engine, even at full throttle, the car cannot simply accelerate up the steep slope. The car is situated in a valley and must learn to leverage potential energy by driving up the opposite hill before the car is able to make it to the goal at the top of the rightmost hill. The domain has been used as a test bed in various Reinforcement Learning papers. Contents [hide] 1 Introduction 2 History 3 Techniques used to solve mountain car 3.1 Discretization 3.2 Function approximation 3.3 Traces 4 Technical details 4.1 State v

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Mountain Car, a standard testing domain in Reinforcement Learning, is a problem in which an under-powered car must drive up a steep hill. Since gravity is stronger than the car's engine, even at full throttle, the car cannot simply accelerate up the steep slope. The car is situated in a valley and must learn to leverage p

[imagelink] This article is an orphan, as no other articles link to it. Please introduce links to this page from related articles; try the Find link tool for suggestions. (July 2012) [imagelink] The mountain car problem <span>Mountain Car, a standard testing domain in Reinforcement Learning, is a problem in which an under-powered car must drive up a steep hill. Since gravity is stronger than the car's engine, even at full throttle, the car cannot simply accelerate up the steep slope. The car is situated in a valley and must learn to leverage potential energy by driving up the opposite hill before the car is able to make it to the goal at the top of the rightmost hill. The domain has been used as a test bed in various Reinforcement Learning papers. Contents [hide] 1 Introduction 2 History 3 Techniques used to solve mountain car 3.1 Discretization 3.2 Function approximation 3.3 Traces 4 Technical details 4.1 State v

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Mountain Car, a standard testing domain in Reinforcement Learning, is a problem in which an under-powered car must drive up a steep hill.

[imagelink] This article is an orphan, as no other articles link to it. Please introduce links to this page from related articles; try the Find link tool for suggestions. (July 2012) [imagelink] The mountain car problem <span>Mountain Car, a standard testing domain in Reinforcement Learning, is a problem in which an under-powered car must drive up a steep hill. Since gravity is stronger than the car's engine, even at full throttle, the car cannot simply accelerate up the steep slope. The car is situated in a valley and must learn to leverage potential energy by driving up the opposite hill before the car is able to make it to the goal at the top of the rightmost hill. The domain has been used as a test bed in various Reinforcement Learning papers. Contents [hide] 1 Introduction 2 History 3 Techniques used to solve mountain car 3.1 Discretization 3.2 Function approximation 3.3 Traces 4 Technical details 4.1 State v

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Linear operator) Jump to: navigation, search "Linear transformation" redirects here. For fractional linear transformations, see Möbius transformation. Not to be confused with linear function. <span>In mathematics, a linear map (also called a linear mapping, linear transformation or, in some contexts, linear function) is a mapping V → W between two modules (including vector spaces) that preserves (in the sense defined below) the operations of addition and scalar multiplication. An important special case is when V = W, in which case the map is called a linear operator, [1] or an endomorphism of V. Sometimes the term linear function has the same meaning as li

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linear mapping, linear transformation or, in some contexts, linear function) is a mapping V → W between two modules (including vector spaces) that preserves (in the sense defined below) the operations of addition and scalar multiplication. <span>An important special case is when V = W, in which case the map is called a linear operator, [1] or an endomorphism of V. Sometimes the term linear function has the same meaning as linear map, while in analytic geometry it does not. A linear map always maps linear subspaces onto linear subspaces (possibl

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thematics, a linear map (also called a linear mapping, linear transformation or, in some contexts, linear function) is a mapping V → W between two modules (including vector spaces) that preserves (in the sense defined below) the operations of <span>addition and scalar multiplication. <span><body><html>

Linear operator) Jump to: navigation, search "Linear transformation" redirects here. For fractional linear transformations, see Möbius transformation. Not to be confused with linear function. <span>In mathematics, a linear map (also called a linear mapping, linear transformation or, in some contexts, linear function) is a mapping V → W between two modules (including vector spaces) that preserves (in the sense defined below) the operations of addition and scalar multiplication. An important special case is when V = W, in which case the map is called a linear operator, [1] or an endomorphism of V. Sometimes the term linear function has the same meaning as li

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for more information. [imagelink] [Help with translations!] Spectral theorem From Wikipedia, the free encyclopedia Jump to: navigation, search <span>In mathematics, particularly linear algebra and functional analysis, a spectral theorem is a result about when a linear operator or matrix can be diagonalized (that is, represented as a diagonal matrix in some basis). This is extremely useful because computations involving a diagonalizable matrix can often be reduced to much simpler computations involving the corresponding diagonal matrix. The concep

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In mathematics, particularly linear algebra and functional analysis, a spectral theorem is a result about when a linear operator or matrix can be diagonalized (that is, represented as a diagonal matrix in some basis).

for more information. [imagelink] [Help with translations!] Spectral theorem From Wikipedia, the free encyclopedia Jump to: navigation, search <span>In mathematics, particularly linear algebra and functional analysis, a spectral theorem is a result about when a linear operator or matrix can be diagonalized (that is, represented as a diagonal matrix in some basis). This is extremely useful because computations involving a diagonalizable matrix can often be reduced to much simpler computations involving the corresponding diagonal matrix. The concep

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The Black–Scholes / ˌ b l æ k ˈ ʃ oʊ l z / [1] or Black–Scholes–Merton model is a mathematical model of a financial market containing derivative investment instruments. From the partial differential equation in the model, known as the Black–Scholes equation, one can deduce the Black–Scholes formula, which gives a theoretical estimate of the price of E

Black–Scholes model - Wikipedia Black–Scholes model From Wikipedia, the free encyclopedia (Redirected from Black–Scholes) Jump to: navigation, search The Black–Scholes /ˌblæk ˈʃoʊlz/ [1] or Black–Scholes–Merton model is a mathematical model of a financial market containing derivative investment instruments. From the partial differential equation in the model, known as the Black–Scholes equation, one can deduce the Black–Scholes formula, which gives a theoretical estimate of the price of European-style options and shows that the option has a unique price regardless of the risk of the security and its expected return (instead replacing the security's expected return with the risk-neutral rate). The formula led to a boom in options trading and provided mathematical legitimacy to the activities of the Chicago Board Options Exchange and other options markets around the world. [2]

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The Wiener process is characterised by the following properties: [1] a.s. has independent increments: for every the future increments , are independent of the past values , has Gaussian increments: is normally distributed with mean and variance ,

Brownian motion 4.3 Time change 4.4 Change of measure 4.5 Complex-valued Wiener process 4.5.1 Self-similarity 4.5.2 Time change 5 See also 6 Notes 7 References 8 External links Characterisations of the Wiener process[edit source] <span>The Wiener process W t {\displaystyle W_{t}} is characterised by the following properties: [1] W 0 = 0 {\displaystyle W_{0}=0} a.s. W {\displaystyle W} has independent increments: for every t > 0 , {\displaystyle t>0,} the future increments W t + u − W t , {\displaystyle W_{t+u}-W_{t},} u ≥ 0 , {\displaystyle u\geq 0,} , are independent of the past values W s {\displaystyle W_{s}} , s ≤ t . {\displaystyle s\leq t.} W {\displaystyle W} has Gaussian increments: W t + u − W t {\displaystyle W_{t+u}-W_{t}} is normally distributed with mean 0 {\displaystyle 0} and variance u {\displaystyle u} , W t + u − W t ∼ N ( 0 , u ) . {\displaystyle W_{t+u}-W_{t}\sim {\mathcal {N}}(0,u).} W {\displaystyle W} has continuous paths: With probability 1 {\displaystyle 1} , W t {\displaystyle W_{t}} is continuous in t {\displaystyle t} . The independent increments means that if 0 ≤ s 1 < t 1 ≤ s 2 < t 2 then W t 1 −W s 1 and W t 2 −W s 2 are independent random variables, and the similar condition holds for

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putations involving the corresponding diagonal matrix. The concept of diagonalization is relatively straightforward for operators on finite-dimensional vector spaces but requires some modification for operators on infinite-dimensional spaces. <span>In general, the spectral theorem identifies a class of linear operators that can be modeled by multiplication operators, which are as simple as one can hope to find. In more abstract language, the spectral theorem is a statement about commutative C*-algebras. See also spectral theory for a historical per

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In general, the spectral theorem identifies a class of linear operators that can be modeled by multiplication operators

putations involving the corresponding diagonal matrix. The concept of diagonalization is relatively straightforward for operators on finite-dimensional vector spaces but requires some modification for operators on infinite-dimensional spaces. <span>In general, the spectral theorem identifies a class of linear operators that can be modeled by multiplication operators, which are as simple as one can hope to find. In more abstract language, the spectral theorem is a statement about commutative C*-algebras. See also spectral theory for a historical per

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In general, the spectral theorem identifies a class of linear operators that can be modeled by multiplication operators

putations involving the corresponding diagonal matrix. The concept of diagonalization is relatively straightforward for operators on finite-dimensional vector spaces but requires some modification for operators on infinite-dimensional spaces. <span>In general, the spectral theorem identifies a class of linear operators that can be modeled by multiplication operators, which are as simple as one can hope to find. In more abstract language, the spectral theorem is a statement about commutative C*-algebras. See also spectral theory for a historical per

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Bilinear form - Wikipedia Bilinear form From Wikipedia, the free encyclopedia Jump to: navigation, search In mathematics, more specifically in abstract algebra and linear algebra, a bilinear form on a vector space V is a bilinear map V × V → K, where K is the field of scalars. In other words, a bilinear form is a function B : V × V → K that is linear in each argument separately: B(u + v, w) = B(u, w) + B(v, w) and B(λu, v) = λB(u, v) B(u, v + w) = B(u, v) + B(u, w) and B(u, λv) = λB(u, v) The definition of a bilinear form can be extended to include modules over a ring, with linear maps replaced by module homomorphisms. When K is the field of complex numbers C, one

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In mathematics, more specifically in abstract algebra and linear algebra, a bilinear form on a vector space V is a bilinear map V × V → K , where K is the field of scalars. In other words, a bilinear form is a function B : V × V → K that is <span>linear in each argument separately: B(u + v, w) = B(u, w) + B(v, w) and B(λu, v) = λB(u, v) B(u, v + w) = B(u, v) + B(u, w) and B(u, λv) = λB(u, v) <span><body><html>

Bilinear form - Wikipedia Bilinear form From Wikipedia, the free encyclopedia Jump to: navigation, search In mathematics, more specifically in abstract algebra and linear algebra, a bilinear form on a vector space V is a bilinear map V × V → K, where K is the field of scalars. In other words, a bilinear form is a function B : V × V → K that is linear in each argument separately: B(u + v, w) = B(u, w) + B(v, w) and B(λu, v) = λB(u, v) B(u, v + w) = B(u, v) + B(u, w) and B(u, λv) = λB(u, v) The definition of a bilinear form can be extended to include modules over a ring, with linear maps replaced by module homomorphisms. When K is the field of complex numbers C, one

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If a Poisson process is defined with a single positive constant, then the process is called a homogeneous Poisson process.

arameter. This process has the natural numbers as its state space and the non-negative numbers as its index set. This process is also called the Poisson counting process, since it can be interpreted as an example of a counting process. [99] <span>If a Poisson process is defined with a single positive constant, then the process is called a homogeneous Poisson process. [99] [101] The homogeneous Poisson process (in continuous time) is a member of important classes of stochastic processes such as Markov processes and Lévy processes. [49] The homogen

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Its name (Poisson Process) derives from the fact that if a collection of random points in some space forms a Poisson process, then the number of points in a region of finite size is a random variable with a Poisson distribution.

oint processes, some of which are constructed with the Poisson point process, that seek to capture such interaction. [22] The process is named after French mathematician Siméon Denis Poisson despite Poisson never having studied the process. <span>Its name derives from the fact that if a collection of random points in some space forms a Poisson process, then the number of points in a region of finite size is a random variable with a Poisson distribution. The process was discovered independently and repeatedly in several settings, including experiments on radioactive decay, telephone call arrivals and insurance mathematics. [23] [24] T

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An operation like capitalize_all is sometimes called a map because it “maps” a function (in this case the method capitalize ) onto each of the elements in a sequence.

ngs: def capitalize_all(t): res = [] for s in t: res.append(s.capitalize()) return res res is initialized with an empty list; each time through the loop, we append the next element. So res is another kind of accumulator. <span>An operation like capitalize_all is sometimes called a map because it “maps” a function (in this case the method capitalize) onto each of the elements in a sequence. Another common operation is to select some of the elements from a list and return a sublist. For example, the following function takes a list of strings and returns a list that cont

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is like the simple indexing we've already seen, but we pass arrays of indices in place of single scalars. This allows us to very quickly access and modify complicated subsets of an array's values. Exploring Fancy Indexing¶ <span>Fancy indexing is conceptually simple: it means passing an array of indices to access multiple array elements at once. For example, consider the following array: In [1]: import numpy as np rand = np.random.RandomState(42) x = rand.randint(100, size=10) print(x)

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Fancy indexing is conceptually simple: it means passing an array of indices to access multiple array elements at once.

is like the simple indexing we've already seen, but we pass arrays of indices in place of single scalars. This allows us to very quickly access and modify complicated subsets of an array's values. Exploring Fancy Indexing¶ <span>Fancy indexing is conceptually simple: it means passing an array of indices to access multiple array elements at once. For example, consider the following array: In [1]: import numpy as np rand = np.random.RandomState(42) x = rand.randint(100, size=10) print(x)

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Fancy indexing is conceptually simple: it means passing an array of indices to access multiple array elements at once.

is like the simple indexing we've already seen, but we pass arrays of indices in place of single scalars. This allows us to very quickly access and modify complicated subsets of an array's values. Exploring Fancy Indexing¶ <span>Fancy indexing is conceptually simple: it means passing an array of indices to access multiple array elements at once. For example, consider the following array: In [1]: import numpy as np rand = np.random.RandomState(42) x = rand.randint(100, size=10) print(x)

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Fancy indexing is conceptually simple: it means passing an array of indices to access multiple array elements at once.

is like the simple indexing we've already seen, but we pass arrays of indices in place of single scalars. This allows us to very quickly access and modify complicated subsets of an array's values. Exploring Fancy Indexing¶ <span>Fancy indexing is conceptually simple: it means passing an array of indices to access multiple array elements at once. For example, consider the following array: In [1]: import numpy as np rand = np.random.RandomState(42) x = rand.randint(100, size=10) print(x)

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exactly as we saw in broadcasting of arithmetic operations. For example: In [8]: row[:, np.newaxis] * col Out[8]: array([[0, 0, 0], [2, 1, 3], [4, 2, 6]]) <span>It is always important to remember with fancy indexing that the return value reflects the broadcasted shape of the indices, rather than the shape of the array being indexed. Combined Indexing¶ For even more powerful operations, fancy indexing can be combined with the other indexing schemes we've seen: In [9]: print

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It is always important to remember with fancy indexing that the return value reflects the broadcasted shape of the indices, rather than the shape of the array being indexed

exactly as we saw in broadcasting of arithmetic operations. For example: In [8]: row[:, np.newaxis] * col Out[8]: array([[0, 0, 0], [2, 1, 3], [4, 2, 6]]) <span>It is always important to remember with fancy indexing that the return value reflects the broadcasted shape of the indices, rather than the shape of the array being indexed. Combined Indexing¶ For even more powerful operations, fancy indexing can be combined with the other indexing schemes we've seen: In [9]: print

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icas Estadísticas [imagelink] [imagelink] Manchester United/Man Utd via Getty Images 2hRob Dawson, ESPN Mourinho: Alexis no se mudó al United por dinero <span>El entrenador portugués desestimó los argumentos del City y Guardiola por los que desistieron de contratar al jugador chileno. [imagelink]play Guido Carrillo, nuevo jugador de Southampton (0:24) [imagelink]play0:24 9h Carrillo pasó al Southampton de Pellegrino El mediocampista ex-Estudiantes ya fue oficializado

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El entrenador portugués desestimó los argumentos del City y Guardiola por los que desistieron de contratar al jugador chileno.

icas Estadísticas [imagelink] [imagelink] Manchester United/Man Utd via Getty Images 2hRob Dawson, ESPN Mourinho: Alexis no se mudó al United por dinero <span>El entrenador portugués desestimó los argumentos del City y Guardiola por los que desistieron de contratar al jugador chileno. [imagelink]play Guido Carrillo, nuevo jugador de Southampton (0:24) [imagelink]play0:24 9h Carrillo pasó al Southampton de Pellegrino El mediocampista ex-Estudiantes ya fue oficializado

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Arsenal La leyenda gunner salió al cruce de los rumores que afirmaban que había influido en el pase del chileno al United. [imagelink] Darren Walsh/Chelsea FC via Getty Images 2dLiam Twomey, ESPN Conte: Chelsea no puede gastar como Manchester <span>El entrenador del Chelsea destacó que, considerando la inversiones del City y United, el título de liga de la temporada pasada fue un "pequeño milagro". [imagelink]play João Castelo-Branco traz todas as novidades da movimentação do mercado na Inglaterra (6:34) [imagelink]play Liverpool se estrelló ante Swansea City (1:25) [imagelink]pla

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El entrenador del Chelsea destacó que, considerando la inversiones del City y United, el título de liga de la temporada pasada fue un "pequeño milagro".

Arsenal La leyenda gunner salió al cruce de los rumores que afirmaban que había influido en el pase del chileno al United. [imagelink] Darren Walsh/Chelsea FC via Getty Images 2dLiam Twomey, ESPN Conte: Chelsea no puede gastar como Manchester <span>El entrenador del Chelsea destacó que, considerando la inversiones del City y United, el título de liga de la temporada pasada fue un "pequeño milagro". [imagelink]play João Castelo-Branco traz todas as novidades da movimentação do mercado na Inglaterra (6:34) [imagelink]play Liverpool se estrelló ante Swansea City (1:25) [imagelink]pla

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El entrenador del Chelsea destacó que, considerando la inversiones del City y United, el título de liga de la temporada pasada fue un "pequeño milagro".

Arsenal La leyenda gunner salió al cruce de los rumores que afirmaban que había influido en el pase del chileno al United. [imagelink] Darren Walsh/Chelsea FC via Getty Images 2dLiam Twomey, ESPN Conte: Chelsea no puede gastar como Manchester <span>El entrenador del Chelsea destacó que, considerando la inversiones del City y United, el título de liga de la temporada pasada fue un "pequeño milagro". [imagelink]play João Castelo-Branco traz todas as novidades da movimentação do mercado na Inglaterra (6:34) [imagelink]play Liverpool se estrelló ante Swansea City (1:25) [imagelink]pla

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bability 3.7 Cantor numbers 4 Variants 4.1 Smith–Volterra–Cantor set 4.2 Stochastic Cantor set 4.3 Cantor dust 5 Historical remarks 6 See also 7 Notes 8 References 9 External links Construction and formula of the ternary set[edit] <span>The Cantor ternary set C {\displaystyle {\mathcal {C}}} is created by iteratively deleting the open middle third from a set of line segments. One starts by deleting the open middle third (1/3, 2/3) from the interval [0, 1], leaving two line segments: [0, 1/3] ∪ [2/3, 1]. Next, the open middle third of each of these remaining

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Lebesgue integration or abstract integration gives the same result as Riemann integration when the latter exists, so nothing you know from calculus changes, but a lot more functions are integrable <

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The Cantor ternary set is created by iteratively deleting the open middle third from a set of line segments.

bability 3.7 Cantor numbers 4 Variants 4.1 Smith–Volterra–Cantor set 4.2 Stochastic Cantor set 4.3 Cantor dust 5 Historical remarks 6 See also 7 Notes 8 References 9 External links Construction and formula of the ternary set[edit] <span>The Cantor ternary set C {\displaystyle {\mathcal {C}}} is created by iteratively deleting the open middle third from a set of line segments. One starts by deleting the open middle third (1/3, 2/3) from the interval [0, 1], leaving two line segments: [0, 1/3] ∪ [2/3, 1]. Next, the open middle third of each of these remaining

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The Cantor ternary set is created by iteratively deleting the open middle third from a set of line segments.

bability 3.7 Cantor numbers 4 Variants 4.1 Smith–Volterra–Cantor set 4.2 Stochastic Cantor set 4.3 Cantor dust 5 Historical remarks 6 See also 7 Notes 8 References 9 External links Construction and formula of the ternary set[edit] <span>The Cantor ternary set C {\displaystyle {\mathcal {C}}} is created by iteratively deleting the open middle third from a set of line segments. One starts by deleting the open middle third (1/3, 2/3) from the interval [0, 1], leaving two line segments: [0, 1/3] ∪ [2/3, 1]. Next, the open middle third of each of these remaining

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ayesian network) represent variables that are conditionally independent of each other. Each node is associated with a probability function that takes, as input, a particular set of values for the node's parent variables, and gives (as output) <span>the probability (or probability distribution, if applicable) of the variable represented by the node. <span><body><html>

ed acyclic graph (DAG). For example, a Bayesian network could represent the probabilistic relationships between diseases and symptoms. Given symptoms, the network can be used to compute the probabilities of the presence of various diseases. <span>Formally, Bayesian networks are DAGs whose nodes represent variables in the Bayesian sense: they may be observable quantities, latent variables, unknown parameters or hypotheses. Edges represent conditional dependencies; nodes that are not connected (there is no path from one of the variables to the other in the Bayesian network) represent variables that are conditionally independent of each other. Each node is associated with a probability function that takes, as input, a particular set of values for the node's parent variables, and gives (as output) the probability (or probability distribution, if applicable) of the variable represented by the node. For example, if m {\displaystyle m} parent nodes represent m {\displaystyle m} Boolean variables

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In mathematics, more specifically in abstract algebra and linear algebra, a bilinear form on a vector space V is a bilinear map V × V → K , where K is the field of scalars. In other words, a bilinear form is a function B : V × V → K that is linear in each argument separately: B(u + v, w) = B(u, w) + B(v, w) and B(λu, v) = λB(u, v) B(u, v + w) = B(u, v) +

Bilinear form - Wikipedia Bilinear form From Wikipedia, the free encyclopedia Jump to: navigation, search In mathematics, more specifically in abstract algebra and linear algebra, a bilinear form on a vector space V is a bilinear map V × V → K, where K is the field of scalars. In other words, a bilinear form is a function B : V × V → K that is linear in each argument separately: B(u + v, w) = B(u, w) + B(v, w) and B(λu, v) = λB(u, v) B(u, v + w) = B(u, v) + B(u, w) and B(u, λv) = λB(u, v) The definition of a bilinear form can be extended to include modules over a ring, with linear maps replaced by module homomorphisms. When K is the field of complex numbers C, one

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In mathematics, more specifically in abstract algebra and linear algebra, a bilinear form on a vector space V is a bilinear map V × V → K , where K is the field of scalars.

Bilinear form - Wikipedia Bilinear form From Wikipedia, the free encyclopedia Jump to: navigation, search In mathematics, more specifically in abstract algebra and linear algebra, a bilinear form on a vector space V is a bilinear map V × V → K, where K is the field of scalars. In other words, a bilinear form is a function B : V × V → K that is linear in each argument separately: B(u + v, w) = B(u, w) + B(v, w) and B(λu, v) = λB(u, v) B(u, v + w) = B(u, v) + B(u, w) and B(u, λv) = λB(u, v) The definition of a bilinear form can be extended to include modules over a ring, with linear maps replaced by module homomorphisms. When K is the field of complex numbers C, one

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In mathematics, more specifically in abstract algebra and linear algebra, a bilinear form on a vector space V is a bilinear map V × V → K , where K is the field of scalars.

Bilinear form - Wikipedia Bilinear form From Wikipedia, the free encyclopedia Jump to: navigation, search In mathematics, more specifically in abstract algebra and linear algebra, a bilinear form on a vector space V is a bilinear map V × V → K, where K is the field of scalars. In other words, a bilinear form is a function B : V × V → K that is linear in each argument separately: B(u + v, w) = B(u, w) + B(v, w) and B(λu, v) = λB(u, v) B(u, v + w) = B(u, v) + B(u, w) and B(u, λv) = λB(u, v) The definition of a bilinear form can be extended to include modules over a ring, with linear maps replaced by module homomorphisms. When K is the field of complex numbers C, one

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The difference between a probability measure and the more general notion of measure (which includes concepts like area or volume) is that a probability measure must assign value 1 to the entire probability space.

inequality Venn diagram Tree diagram v t e In mathematics, a probability measure is a real-valued function defined on a set of events in a probability space that satisfies measure properties such as countable additivity. [3] <span>The difference between a probability measure and the more general notion of measure (which includes concepts like area or volume) is that a probability measure must assign value 1 to the entire probability space. Intuitively, the additivity property says that the probability assigned to the union of two disjoint events by the measure should be the sum of the probabilities of the events, e.g. t

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eds additional citations for verification. Please help improve this article by adding citations to reliable sources. Unsourced material may be challenged and removed. (November 2009) (Learn how and when to remove this template message) <span>In mathematics, the support of a real-valued function f is the subset of the domain containing those elements which are not mapped to zero. If the domain of f is a topological space, the support of f is instead defined as the smallest closed set containing all points not mapped to zero. This concept is used very widely in m

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In mathematics, the support of a real-valued function f is the subset of the domain containing those elements which are not mapped to zero.

eds additional citations for verification. Please help improve this article by adding citations to reliable sources. Unsourced material may be challenged and removed. (November 2009) (Learn how and when to remove this template message) <span>In mathematics, the support of a real-valued function f is the subset of the domain containing those elements which are not mapped to zero. If the domain of f is a topological space, the support of f is instead defined as the smallest closed set containing all points not mapped to zero. This concept is used very widely in m

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In mathematics, the support of a real-valued function f is the subset of the domain containing those elements which are not mapped to zero.

eds additional citations for verification. Please help improve this article by adding citations to reliable sources. Unsourced material may be challenged and removed. (November 2009) (Learn how and when to remove this template message) <span>In mathematics, the support of a real-valued function f is the subset of the domain containing those elements which are not mapped to zero. If the domain of f is a topological space, the support of f is instead defined as the smallest closed set containing all points not mapped to zero. This concept is used very widely in m

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A norm must also satisfy certain properties pertaining to scalability and additivity which are given in the formal definition below.

ositive length or size to each vector in a vector space—save for the zero vector, which is assigned a length of zero. A seminorm, on the other hand, is allowed to assign zero length to some non-zero vectors (in addition to the zero vector). <span>A norm must also satisfy certain properties pertaining to scalability and additivity which are given in the formal definition below. A simple example is two dimensional Euclidean space R 2 equipped with the "Euclidean norm" (see below) Elements in this vector space (e.g., (3, 7)) are usually drawn as arr

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[imagelink] 1dESPN Estadísticas e Información Kane llega a 100 goles en Premier League El goleador del Tottenham se reivindicó con ese tanto luego de fallar un penalti. [imagelink]play0:17 17h Mou criticó el ambiente tranquilo de Old Trafford <span>El entrenador de Manchester United cree que no hay "mucho entusiasmo" de la hinchada, aunque los futbolistas "les gusta jugar" en casa. [imagelink]play José Mourinho y el gol 'no soñado' de Alexis Sánchez (0:17) [imagelink]play Pep Guardiola insiste: 'No quiero a nadie en Manchester' (0:45) Prem Posiciones EQUIPO PJ

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Optimal control deals with the problem of finding a control law for a given system such that a certain optimality criterion is achieved.

1 General method 2 Linear quadratic control 3 Numerical methods for optimal control 4 Discrete-time optimal control 5 Examples 5.1 Finite time 6 See also 7 References 8 Further reading 9 External links General method[edit source] <span>Optimal control deals with the problem of finding a control law for a given system such that a certain optimality criterion is achieved. A control problem includes a cost functional that is a function of state and control variables. An optimal control is a set of differential equations describing the paths of the control

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In mathematics, particularly linear algebra and functional analysis, a spectral theorem is a result about when a linear operator or matrix can be diagonalized (that is, represented as a diagonal matrix in some basis).

for more information. [imagelink] [Help with translations!] Spectral theorem From Wikipedia, the free encyclopedia Jump to: navigation, search <span>In mathematics, particularly linear algebra and functional analysis, a spectral theorem is a result about when a linear operator or matrix can be diagonalized (that is, represented as a diagonal matrix in some basis). This is extremely useful because computations involving a diagonalizable matrix can often be reduced to much simpler computations involving the corresponding diagonal matrix. The concep

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a spectral theorem is a result about when a linear operator or matrix can be diagonalized

for more information. [imagelink] [Help with translations!] Spectral theorem From Wikipedia, the free encyclopedia Jump to: navigation, search <span>In mathematics, particularly linear algebra and functional analysis, a spectral theorem is a result about when a linear operator or matrix can be diagonalized (that is, represented as a diagonal matrix in some basis). This is extremely useful because computations involving a diagonalizable matrix can often be reduced to much simpler computations involving the corresponding diagonal matrix. The concep

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2 {\displaystyle BD^{2}+AC^{2}=2a^{2}+2b^{2}} Q.E.D. The parallelogram law in inner product spaces[edit source] [imagelink] Vectors involved in the parallelogram law. <span>In a normed space, the statement of the parallelogram law is an equation relating norms: 2 ‖ x ‖ 2 + 2 ‖ y ‖

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In a normed space, the statement of the parallelogram law is an equation relating norms:

2 {\displaystyle BD^{2}+AC^{2}=2a^{2}+2b^{2}} Q.E.D. The parallelogram law in inner product spaces[edit source] [imagelink] Vectors involved in the parallelogram law. <span>In a normed space, the statement of the parallelogram law is an equation relating norms: 2 ‖ x ‖ 2 + 2 ‖ y ‖

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u ) = c f ( u ) {\displaystyle f(c\mathbf {u} )=cf(\mathbf {u} )} homogeneity of degree 1 / operation of scalar multiplication Thus, <span>a linear map is said to be operation preserving. In other words, it does not matter whether you apply the linear map before or after the operations of addition and scalar multiplication. This is equivalent to requiring the same for any linear combination of vectors, i.e. that for any vectors u

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a linear map is operation preserving: it does not matter whether you apply the linear map before or after the operations of addition and scalar multiplication.

u ) = c f ( u ) {\displaystyle f(c\mathbf {u} )=cf(\mathbf {u} )} homogeneity of degree 1 / operation of scalar multiplication Thus, <span>a linear map is said to be operation preserving. In other words, it does not matter whether you apply the linear map before or after the operations of addition and scalar multiplication. This is equivalent to requiring the same for any linear combination of vectors, i.e. that for any vectors u

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a statement about commutative C*-algebras. See also spectral theory for a historical perspective. Examples of operators to which the spectral theorem applies are self-adjoint operators or more generally normal operators on Hilbert spaces. <span>The spectral theorem also provides a canonical decomposition, called the spectral decomposition, eigenvalue decomposition, or eigendecomposition, of the underlying vector space on which the operator acts. Augustin-Louis Cauchy proved the spectral theorem for self-adjoint matrices, i.e., that every real, symmetric matrix is diagonalizable. In addition, Cauchy was the first to be systema

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The spectral theorem provides a spectral decomposition of the underlying vector space on which the operator acts.

a statement about commutative C*-algebras. See also spectral theory for a historical perspective. Examples of operators to which the spectral theorem applies are self-adjoint operators or more generally normal operators on Hilbert spaces. <span>The spectral theorem also provides a canonical decomposition, called the spectral decomposition, eigenvalue decomposition, or eigendecomposition, of the underlying vector space on which the operator acts. Augustin-Louis Cauchy proved the spectral theorem for self-adjoint matrices, i.e., that every real, symmetric matrix is diagonalizable. In addition, Cauchy was the first to be systema

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ombates UFC Deportes Aventura eSports Juegos Olímpicos Olimpismo Paralímpicos Rugby Toros Turf Volvo Ocean Race Ligue 1 - Francia Liga FrancesaIsaac se formó en el Chelsea y ahora ficha por otro exequipo de Didier <span>El hijo de Drogba ficha por el Guingamp siguiendo los pasos de su padre Compartir en Facebook Compartir en Twitter Enviar por email 20/02/2018 16:06 CET Isaac Drogba, durante su presentación con el Guingamp. 3 comentarios Comentar Guardiola vuelve a sa

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El hijo de Drogba ficha por el Guingamp siguiendo los pasos de su padre

ombates UFC Deportes Aventura eSports Juegos Olímpicos Olimpismo Paralímpicos Rugby Toros Turf Volvo Ocean Race Ligue 1 - Francia Liga FrancesaIsaac se formó en el Chelsea y ahora ficha por otro exequipo de Didier <span>El hijo de Drogba ficha por el Guingamp siguiendo los pasos de su padre Compartir en Facebook Compartir en Twitter Enviar por email 20/02/2018 16:06 CET Isaac Drogba, durante su presentación con el Guingamp. 3 comentarios Comentar Guardiola vuelve a sa

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[imagelink] Atrocity propaganda From Wikipedia, the free encyclopedia Jump to: navigation, search <span>Atrocity propaganda is the spreading information about the crimes committed by an enemy, especially deliberate fabrications or exaggerations. [citation needed] It is a form of psychological warfare. [citation needed] The inherently violent nature of war means that exaggeration and invention of atrocities often becomes the

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Atrocity propaganda is the spreading information about the crimes committed by an enemy, especially deliberate fabrications or exaggerations.

[imagelink] Atrocity propaganda From Wikipedia, the free encyclopedia Jump to: navigation, search <span>Atrocity propaganda is the spreading information about the crimes committed by an enemy, especially deliberate fabrications or exaggerations. [citation needed] It is a form of psychological warfare. [citation needed] The inherently violent nature of war means that exaggeration and invention of atrocities often becomes the

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Atrocity propaganda is the spreading information about the crimes committed by an enemy, especially deliberate fabrications or exaggerations.

[imagelink] Atrocity propaganda From Wikipedia, the free encyclopedia Jump to: navigation, search <span>Atrocity propaganda is the spreading information about the crimes committed by an enemy, especially deliberate fabrications or exaggerations. [citation needed] It is a form of psychological warfare. [citation needed] The inherently violent nature of war means that exaggeration and invention of atrocities often becomes the

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Roman law Ius Imperium Mos maiorum Collegiality Auctoritas Roman citizenship Cursus honorum Senatus consultum Senatus consultum ultimum Assemblies Centuriate Curiate Plebeian Tribal Other countries Atlas v t e <span>The Roman Kingdom, or regal period, was the period of the ancient Roman civilization characterized by a monarchical form of government of the city of Rome and its territories. Little is certain about the history of the kingdom, as nearly no written records from that time survive, and the histories about it that were written during the Republic and Empire ar

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The Roman Kingdom, or regal period, was the period of the ancient Roman civilization characterized by a monarchical form of government of the city of Rome and its territories. </h

Roman law Ius Imperium Mos maiorum Collegiality Auctoritas Roman citizenship Cursus honorum Senatus consultum Senatus consultum ultimum Assemblies Centuriate Curiate Plebeian Tribal Other countries Atlas v t e <span>The Roman Kingdom, or regal period, was the period of the ancient Roman civilization characterized by a monarchical form of government of the city of Rome and its territories. Little is certain about the history of the kingdom, as nearly no written records from that time survive, and the histories about it that were written during the Republic and Empire ar

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The basic feature of the dot product that connects it with Euclidean geometry is that it is related to both the length of a vector, denoted || x || , and to the angle θ between two vectors x and y by means of the formula

t product, satisfies these three properties is known as a (real) inner product. A vector space equipped with such an inner product is known as a (real) inner product space. Every finite-dimensional inner product space is also a Hilbert space. <span>The basic feature of the dot product that connects it with Euclidean geometry is that it is related to both the length (or norm) of a vector, denoted ||x||, and to the angle θ between two vectors x and y by means of the formula x ⋅ y = ‖ x ‖ ‖ y ‖ cos θ . {\displaystyle \mathbf {x} \cdot \mathbf {y} =\|\mathbf {x} \|\,\|\mathbf {y} \|\,\cos \theta \,.} [imagelink] Completeness means that if a particle moves along the broken path (in blue) travelling a finite total distance, then the particle has a well-defined net displacem

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any affine transformation is equivalent to a linear transformation (of position vectors) followed by a translation.

es in Euclidean spaces, each output coordinate of an affine map is a linear function (in the sense of calculus) of all input coordinates. Another way to deal with affine transformations systematically is to select a point as the origin; then, <span>any affine transformation is equivalent to a linear transformation (of position vectors) followed by a translation. Contents [hide] 1 Mathematical definition 1.1 Alternative definition 2 Representation 2.1 Augmented matrix 2.1.1 Example augmented matrix 3 Properties 3.1 Properti

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any affine transformation is equivalent to a linear transformation (of position vectors) followed by a translation.

es in Euclidean spaces, each output coordinate of an affine map is a linear function (in the sense of calculus) of all input coordinates. Another way to deal with affine transformations systematically is to select a point as the origin; then, <span>any affine transformation is equivalent to a linear transformation (of position vectors) followed by a translation. Contents [hide] 1 Mathematical definition 1.1 Alternative definition 2 Representation 2.1 Augmented matrix 2.1.1 Example augmented matrix 3 Properties 3.1 Properti

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any affine transformation is equivalent to a linear transformation (of position vectors) followed by a translation.

es in Euclidean spaces, each output coordinate of an affine map is a linear function (in the sense of calculus) of all input coordinates. Another way to deal with affine transformations systematically is to select a point as the origin; then, <span>any affine transformation is equivalent to a linear transformation (of position vectors) followed by a translation. Contents [hide] 1 Mathematical definition 1.1 Alternative definition 2 Representation 2.1 Augmented matrix 2.1.1 Example augmented matrix 3 Properties 3.1 Properti

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Functional equation - Wikipedia Functional equation From Wikipedia, the free encyclopedia Jump to: navigation, search In mathematics, a functional equation [1] [2] [3] [4] is any equation in which the unknown represents a function. Often, the equation relates the value of a function (or functions) at some point with its values at other points. For instance, properties of functions can be determined by considering

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a functional equation is any equation in which the unknown represents a function.

Functional equation - Wikipedia Functional equation From Wikipedia, the free encyclopedia Jump to: navigation, search In mathematics, a functional equation [1] [2] [3] [4] is any equation in which the unknown represents a function. Often, the equation relates the value of a function (or functions) at some point with its values at other points. For instance, properties of functions can be determined by considering

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a functional equation is any equation in which the unknown represents a function.

Functional equation - Wikipedia Functional equation From Wikipedia, the free encyclopedia Jump to: navigation, search In mathematics, a functional equation [1] [2] [3] [4] is any equation in which the unknown represents a function. Often, the equation relates the value of a function (or functions) at some point with its values at other points. For instance, properties of functions can be determined by considering

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[imagelink] Visualization of heat transfer in a pump casing, created by solving the heat equation. Heat is being generated internally in the casing and being cooled at the boundary, providing a steady state temperature distribution. <span>A differential equation is a mathematical equation that relates some function with its derivatives. In applications, the functions usually represent physical quantities, the derivatives represent their rates of change, and the equation defines a relationship between the two. Because s

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A differential equation relates some function with its derivatives.

[imagelink] Visualization of heat transfer in a pump casing, created by solving the heat equation. Heat is being generated internally in the casing and being cooled at the boundary, providing a steady state temperature distribution. <span>A differential equation is a mathematical equation that relates some function with its derivatives. In applications, the functions usually represent physical quantities, the derivatives represent their rates of change, and the equation defines a relationship between the two. Because s

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ncyclopedia Jump to: navigation, search [imagelink] The first four partial sums of the Fourier series for a square wave. Fourier series are an important tool in real analysis. In mathematics, <span>real analysis is the theory of real numbers and real functions, which are real-valued functions of a real variable. It is thus a branch of mathematical analysis, and deals, in particular, with the properties of limits, continuity, differentiability and integrability of these functions. Contents [hide] 1 Scope 1.1 Construction of the real numbers 1.2 Order properties of the real numbers 1.3 Sequences 1.4 Limits and convergence 1.5 Continuity 1.5.1 Uniform

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Functional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure (e.g. inner product, norm, topology, et

analysis (psychology). [imagelink] One of the possible modes of vibration of an idealized circular drum head. These modes are eigenfunctions of a linear operator on a function space, a common construction in functional analysis. <span>Functional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure (e.g. inner product, norm, topology, etc.) and the linear functions defined on these spaces and respecting these structures in a suitable sense. The historical roots of functional analysis lie in the study of spaces of functions and the formulation of properties of transformations of functions such as the Fourier transform as transformations defining continuous, unitary etc. operators between function spaces. This point of view turned out to be particularly useful for the study of differential and integral equations. The usage of the word functional goes back to the calculus of variations, implying a function whose argument is a function and the name was first used in Hadamard's 1910 book on that

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real analysis is the theory of real numbers and real functions, which are real-valued functions of a real variable. It is thus a branch of mathematical analysis, and deals, in particular, with the pr

ncyclopedia Jump to: navigation, search [imagelink] The first four partial sums of the Fourier series for a square wave. Fourier series are an important tool in real analysis. In mathematics, <span>real analysis is the theory of real numbers and real functions, which are real-valued functions of a real variable. It is thus a branch of mathematical analysis, and deals, in particular, with the properties of limits, continuity, differentiability and integrability of these functions. Contents [hide] 1 Scope 1.1 Construction of the real numbers 1.2 Order properties of the real numbers 1.3 Sequences 1.4 Limits and convergence 1.5 Continuity 1.5.1 Uniform

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f which is formed by the study of vector spaces endowed with some kind of limit-related structure (e.g. inner product, norm, topology, etc.) and the linear functions defined on these spaces and respecting these structures in a suitable sense. <span>The historical roots of functional analysis lie in the study of spaces of functions and the formulation of properties of transformations of functions such as the Fourier transform as transformations defining continuous, unitary etc. operators between function spaces. This point of view turned out to be particularly useful for the stu

analysis (psychology). [imagelink] One of the possible modes of vibration of an idealized circular drum head. These modes are eigenfunctions of a linear operator on a function space, a common construction in functional analysis. <span>Functional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure (e.g. inner product, norm, topology, etc.) and the linear functions defined on these spaces and respecting these structures in a suitable sense. The historical roots of functional analysis lie in the study of spaces of functions and the formulation of properties of transformations of functions such as the Fourier transform as transformations defining continuous, unitary etc. operators between function spaces. This point of view turned out to be particularly useful for the study of differential and integral equations. The usage of the word functional goes back to the calculus of variations, implying a function whose argument is a function and the name was first used in Hadamard's 1910 book on that

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spaces and respecting these structures in a suitable sense. The historical roots of functional analysis lie in the study of spaces of functions and the formulation of properties of transformations of functions such as the Fourier transform as <span>transformations defining continuous, unitary etc. operators between function spaces. This point of view turned out to be particularly useful for the study of differential and integral equations. <span><body><html>

analysis (psychology). [imagelink] One of the possible modes of vibration of an idealized circular drum head. These modes are eigenfunctions of a linear operator on a function space, a common construction in functional analysis. <span>Functional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure (e.g. inner product, norm, topology, etc.) and the linear functions defined on these spaces and respecting these structures in a suitable sense. The historical roots of functional analysis lie in the study of spaces of functions and the formulation of properties of transformations of functions such as the Fourier transform as transformations defining continuous, unitary etc. operators between function spaces. This point of view turned out to be particularly useful for the study of differential and integral equations. The usage of the word functional goes back to the calculus of variations, implying a function whose argument is a function and the name was first used in Hadamard's 1910 book on that

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unctional analysis lie in the study of spaces of functions and the formulation of properties of transformations of functions such as the Fourier transform as transformations defining continuous, unitary etc. operators between function spaces. <span>This point of view turned out to be particularly useful for the study of differential and integral equations. <span><body><html>

analysis (psychology). [imagelink] One of the possible modes of vibration of an idealized circular drum head. These modes are eigenfunctions of a linear operator on a function space, a common construction in functional analysis. <span>Functional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure (e.g. inner product, norm, topology, etc.) and the linear functions defined on these spaces and respecting these structures in a suitable sense. The historical roots of functional analysis lie in the study of spaces of functions and the formulation of properties of transformations of functions such as the Fourier transform as transformations defining continuous, unitary etc. operators between function spaces. This point of view turned out to be particularly useful for the study of differential and integral equations. The usage of the word functional goes back to the calculus of variations, implying a function whose argument is a function and the name was first used in Hadamard's 1910 book on that

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The historical roots of functional analysis lie in the study of spaces of functions and transformations of functions

analysis (psychology). [imagelink] One of the possible modes of vibration of an idealized circular drum head. These modes are eigenfunctions of a linear operator on a function space, a common construction in functional analysis. <span>Functional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure (e.g. inner product, norm, topology, etc.) and the linear functions defined on these spaces and respecting these structures in a suitable sense. The historical roots of functional analysis lie in the study of spaces of functions and the formulation of properties of transformations of functions such as the Fourier transform as transformations defining continuous, unitary etc. operators between function spaces. This point of view turned out to be particularly useful for the study of differential and integral equations. The usage of the word functional goes back to the calculus of variations, implying a function whose argument is a function and the name was first used in Hadamard's 1910 book on that

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linear operators are transformations of functions between function spaces

analysis (psychology). [imagelink] One of the possible modes of vibration of an idealized circular drum head. These modes are eigenfunctions of a linear operator on a function space, a common construction in functional analysis. <span>Functional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure (e.g. inner product, norm, topology, etc.) and the linear functions defined on these spaces and respecting these structures in a suitable sense. The historical roots of functional analysis lie in the study of spaces of functions and the formulation of properties of transformations of functions such as the Fourier transform as transformations defining continuous, unitary etc. operators between function spaces. This point of view turned out to be particularly useful for the study of differential and integral equations. The usage of the word functional goes back to the calculus of variations, implying a function whose argument is a function and the name was first used in Hadamard's 1910 book on that

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both differential and integral equations can be understand as linear operations on functions.

analysis (psychology). [imagelink] One of the possible modes of vibration of an idealized circular drum head. These modes are eigenfunctions of a linear operator on a function space, a common construction in functional analysis. <span>Functional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure (e.g. inner product, norm, topology, etc.) and the linear functions defined on these spaces and respecting these structures in a suitable sense. The historical roots of functional analysis lie in the study of spaces of functions and the formulation of properties of transformations of functions such as the Fourier transform as transformations defining continuous, unitary etc. operators between function spaces. This point of view turned out to be particularly useful for the study of differential and integral equations. The usage of the word functional goes back to the calculus of variations, implying a function whose argument is a function and the name was first used in Hadamard's 1910 book on that

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The "accent grave" ` (grave accent) can be on an A, E, or U. On the A and U, it usually serves to distinguish between words that would otherwise be written the same, ex: ou (or) vs où (where), "a" (has) vs "à" (to, at)

low discussion Topic: French [imagelink] Accents in French Remy 19 14 13 11 11 6 4 665 In French, there are 4 accents for vowels and 1 accent for a consonant. The "accent aigu" ´ (acute accent) can only be on an E. <span>The "accent grave" ` (grave accent) can be on an A, E, or U. On the A and U, it usually serves to distinguish between words that would otherwise be written the same, ex: ou (or) vs où (where), "a" (has) vs "à" (to, at) The "accent circonflexe" ˆ (circumflex) can be on an A, E, I, O, or U. In general, it indicates that an S used to follow that vowel, e.g., forêt (forest). The "accent

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ex: ou (or) vs où (where), "a" (has) vs "à" (to, at) The "accent circonflexe" ˆ (circumflex) can be on an A, E, I, O, or U. In general, it indicates that an S used to follow that vowel, e.g., forêt (forest). <span>The "accent tréma" ¨ (dieresis or umlaut) can be on an E, I, or U. It is used when two vowels are next to each other and both must be pronounced, ex: maïs. The "cédille" ¸ (cedilla) is found only on the letter C. It changes a hard C sound (like K) into a soft C sound (like S), e.g., garçon. It never appears in front of E or I,

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t;accent grave" ` (grave accent) can be on an A, E, or U. On the A and U, it usually serves to distinguish between words that would otherwise be written the same, ex: ou (or) vs où (where), "a" (has) vs "à" (to, at) <span>The "accent circonflexe" ˆ (circumflex) can be on an A, E, I, O, or U. In general, it indicates that an S used to follow that vowel, e.g., forêt (forest). The "accent tréma" ¨ (dieresis or umlaut) can be on an E, I, or U. It is used when two vowels are next to each other and both must be pronounced, ex: maïs. The "cédil

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The "accent tréma" ¨ (dieresis or umlaut) can be on an E, I, or U. It is used when two vowels are next to each other and both must be pronounced, ex: maïs.

ex: ou (or) vs où (where), "a" (has) vs "à" (to, at) The "accent circonflexe" ˆ (circumflex) can be on an A, E, I, O, or U. In general, it indicates that an S used to follow that vowel, e.g., forêt (forest). <span>The "accent tréma" ¨ (dieresis or umlaut) can be on an E, I, or U. It is used when two vowels are next to each other and both must be pronounced, ex: maïs. The "cédille" ¸ (cedilla) is found only on the letter C. It changes a hard C sound (like K) into a soft C sound (like S), e.g., garçon. It never appears in front of E or I,

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The "accent circonflexe" ˆ (circumflex) can be on an A, E, I, O, or U. In general, it indicates that an S used to follow that vowel, e.g., forêt (forest).

t;accent grave" ` (grave accent) can be on an A, E, or U. On the A and U, it usually serves to distinguish between words that would otherwise be written the same, ex: ou (or) vs où (where), "a" (has) vs "à" (to, at) <span>The "accent circonflexe" ˆ (circumflex) can be on an A, E, I, O, or U. In general, it indicates that an S used to follow that vowel, e.g., forêt (forest). The "accent tréma" ¨ (dieresis or umlaut) can be on an E, I, or U. It is used when two vowels are next to each other and both must be pronounced, ex: maïs. The "cédil