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 #geometry 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.

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

Mean reversion (finance) - Wikipedia
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, [...] is the assumption that a stock's price will tend to move to the average price over time.
mean reversion

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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.

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Mean reversion (finance) - Wikipedia
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

#### Flashcard 1729615760652

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isotropic process depend only on distance, not [...]
direction

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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.

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Gaussian process - Wikipedia
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 [...] may be obtained by deforming a sphere with an affine transformation .
ellipsoid

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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.

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Ellipsoid - Wikipedia
= 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

#### Flashcard 1729662684428

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An ellipsoid may be obtained by deforming a sphere with an [...].

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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.

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Ellipsoid - Wikipedia
= 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

#### Annotation 1729724550412

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

Object Oriented Programming – Quantitative Economics
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 [...] — an important determinant of productivity

well organized code

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

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Object Oriented Programming – Quantitative Economics
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

#### Annotation 1731471478028

 #reinforcement-learning 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.

Mountain car problem - Wikipedia

#### Annotation 1731505294604

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

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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

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Mountain car problem - Wikipedia

#### Flashcard 1731506867468

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

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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.

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Mountain car problem - Wikipedia

#### Annotation 1731635318028

 #linear-algebra 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.

Linear map - Wikipedia
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

#### Flashcard 1731637415180

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An important special case (of linear mapping) is when V = W , in which case the map is called a [...] ,[1] or an endomorphism of V .
linear operator

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Linear map - Wikipedia
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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a linear map is a mapping VW that preserves the operations of [...].

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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>

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Linear map - Wikipedia
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

#### Annotation 1731642658060

 #matrices #spectral-theorem 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).

Spectral theorem - Wikipedia

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

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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).

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Spectral theorem - Wikipedia

#### Flashcard 1731703737612

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The Black–Scholes model is a mathematical model of a financial market containing [...] instruments
derivative investment

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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

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Black–Scholes model - Wikipedia
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]

#### Flashcard 1731730476300

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The Wiener process starts at [...

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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 ,

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Wiener process - Wikipedia
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

#### Flashcard 1732507733260

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▸ ¡ [...] una respuesta!
exijo

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#### Annotation 1732660825356

 #matrices #spectral-theorem 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.

Spectral theorem - Wikipedia
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

#### Flashcard 1732662398220

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

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

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Spectral theorem - Wikipedia
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

#### Flashcard 1732663971084

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

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

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Spectral theorem - Wikipedia
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

#### Annotation 1732735536396

 #linear-algebra 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)

Bilinear form - Wikipedia
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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a bilinear form is [...descriptive]

linear in each argument separately

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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>

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Bilinear form - Wikipedia
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

#### Flashcard 1736018889996

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a homogeneous Poisson process is defined with a [...]
single positive constant

The constant denotes a fixed area (or length) on the domain.

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

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Stochastic process - Wikipedia
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

#### Flashcard 1736250101004

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if [...] 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.
a collection of random points

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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.

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Poisson point process - Wikipedia
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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#lists #python
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An map operation “maps” a function onto [...] in a sequence.
each of the elements

like capitalize_all

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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.

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

Fancy Indexing | Python Data Science Handbook
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)

#### Flashcard 1737345076492

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

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

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Fancy Indexing | Python Data Science Handbook
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 passes [...] to access multiple array elements at once.
an array of indices

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

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Fancy Indexing | Python Data Science Handbook
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)

#### Flashcard 1737348222220

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

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

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Fancy Indexing | Python Data Science Handbook
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)

#### Annotation 1737942502668

 #numpy 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

Fancy Indexing | Python Data Science Handbook
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

#### Flashcard 1737944075532

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

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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

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Fancy Indexing | Python Data Science Handbook
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

#### Annotation 1738454207756

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

Noticias, Estadísticas y Resultados de Premiership de Inglaterra - ESPNDEPORTES - ESPNDeportes
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

#### Flashcard 1738481208588

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

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

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Noticias, Estadísticas y Resultados de Premiership de Inglaterra - ESPNDEPORTES - ESPNDeportes
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

#### Annotation 1738484092172

 #spanish 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".

Noticias, Estadísticas y Resultados de Premiership de Inglaterra - ESPNDEPORTES - ESPNDeportes
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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#spanish
Question
El entrenador del Chelsea destacó que, considerando las [...] del City y United, el título de liga de la temporada pasada fue un "pequeño milagro".
inversiones

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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".

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Noticias, Estadísticas y Resultados de Premiership de Inglaterra - ESPNDEPORTES - ESPNDeportes
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

#### Flashcard 1738487237900

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#spanish
Question
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

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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".

#### Original toplevel document

Noticias, Estadísticas y Resultados de Premiership de Inglaterra - ESPNDEPORTES - ESPNDeportes
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

#### Annotation 1738544909580

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

Cantor set - Wikipedia
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 <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

#### Flashcard 1738561686796

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#measure-theory #stochastics
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[...] gives the same result as Riemann integration when the latter exists
Lebesgue integration

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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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#### Flashcard 1738593144076

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#measure-theory
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The [...] is created by iteratively deleting the open middle third from a set of line segments.
Cantor ternary set

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

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Cantor set - Wikipedia
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 <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

#### Flashcard 1738595765516

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#measure-theory
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The Cantor ternary set is created by iteratively deleting [...] from a set of line segments.
the open middle third

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

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Cantor set - Wikipedia
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 <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

#### Flashcard 1739058187532

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#bayesian-network
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In Bayesian networks each node represents a variable with [...]
a probability distribution

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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>

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Bayesian network - Wikipedia
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

#### Annotation 1739184540940

 #linear-algebra 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.

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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) +

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Bilinear form - Wikipedia
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

#### Flashcard 1739186113804

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#linear-algebra
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a bilinear form on a vector space V is a bilinear map [...] ,
V × VK

K is the field of scalars.
An inner product is obviously a bilinear form

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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.

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Bilinear form - Wikipedia
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

#### Flashcard 1739187686668

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#linear-algebra
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a bilinear form on a vector space V is a bilinear map V × VK , where K is [...]
the field of scalars.

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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.

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Bilinear form - Wikipedia
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

#### Flashcard 1741100027148

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#probability-measure
Question
Compared to the more general notion of measure, a probability measure must assign value 1 to [...].
the entire probability space

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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.

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Probability measure - Wikipedia
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

#### Annotation 1741818825996

 #functional-analysis 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.

Support (mathematics) - Wikipedia
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

#### Flashcard 1741820923148

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#functional-analysis
Question
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.

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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.

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Support (mathematics) - Wikipedia
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

#### Flashcard 1741822496012

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

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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.

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Support (mathematics) - Wikipedia
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

#### Flashcard 1744136703244

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#vector-space
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A norm must also satisfy certain properties pertaining to [...property...]

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

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Norm (mathematics) - Wikipedia
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

#### Annotation 1744187821324

 #spanish El entrenador de Manchester United cree que no hay "mucho entusiasmo" de la hinchada, aunque los futbolistas "les gusta jugar" en casa.

Noticias, Estadísticas y Resultados de Premiership de Inglaterra - ESPNDEPORTES - ESPNDeportes
[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

#### Flashcard 1744274590988

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#optimal-control
Question
Optimal control finds a control law for a given system such that a certain [...] is achieved.
optimality criterion

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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.

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Optimal control - Wikipedia
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

#### Annotation 1749121633548

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

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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).

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Spectral theorem - Wikipedia

#### Flashcard 1749123206412

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#matrices #spectral-theorem
Question
a spectral theorem is a result about when [...]

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

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Spectral theorem - Wikipedia

#### Annotation 1752698326284

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

Parallelogram law - Wikipedia
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 ‖

#### Flashcard 1752701472012

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#inner-product-space
Question
In a normed space, the statement of the parallelogram law is [...]:
an equation relating norms

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

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Parallelogram law - Wikipedia
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 ‖

#### Annotation 1758250274060

 #linear-algebra 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.

Linear map - Wikipedia
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

#### Flashcard 1758253419788

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

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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.

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Linear map - Wikipedia
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

#### Annotation 1759680793868

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

Spectral theorem - Wikipedia
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

#### Flashcard 1759683153164

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#matrices #spectral-theorem
Question
The spectral theorem provides a [...] of the underlying vector space on which the operator acts.
spectral decomposition

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

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Spectral theorem - Wikipedia
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

#### Annotation 1759747116300

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

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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

#### Flashcard 1759749475596

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#spanish
Question
El hijo de Drogba [...] por el Guingamp siguiendo los pasos de su padre
ficha

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

#### Original toplevel document

Unknown title
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

#### Annotation 1767465422092

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

Atrocity propaganda - Wikipedia

#### Flashcard 1767467519244

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#politics
Question
[...] is the spread of information about the crimes committed by an enemy, especially deliberate fabrications or exaggerations.
Atrocity propaganda

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

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Atrocity propaganda - Wikipedia

#### Flashcard 1767469092108

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#politics
Question
Atrocity propaganda is the spreading information about the crimes committed by an enemy, especially deliberate [...]
fabrications or exaggerations.

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

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Atrocity propaganda - Wikipedia

#### Annotation 1767514180876

 #rome 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.

Roman Kingdom - Wikipedia
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

#### Flashcard 1767516278028

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#rome
Question

The [...] was the period of the ancient Roman civilization characterized by a monarchical form of government

Roman Kingdom

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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

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Roman Kingdom - Wikipedia
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

#### Flashcard 1798944984332

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#hilbert-space
Question

The basic feature of the dot product that connects it with Euclidean geometry is that it is related to both [...] and [...]

length and angle

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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

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Hilbert space - Wikipedia
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

#### Flashcard 1802551823628

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

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

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Affine transformation - Wikipedia
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

#### Flashcard 1802553396492

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#topology
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any affine transformation is equivalent to a [...] followed by a translation.

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

#### Original toplevel document

Affine transformation - Wikipedia
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

#### Flashcard 1802554969356

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#topology
Question
any affine transformation is equivalent to a linear transformation (of position vectors) followed by a [...].
translation

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

#### Original toplevel document

Affine transformation - Wikipedia
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

#### Annotation 1802559163660

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

Functional equation - Wikipedia
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

#### Flashcard 1802561522956

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#functional-analysis
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a [...] is any equation in which the unknown represents a function.
functional equation

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

#### Original toplevel document

Functional equation - Wikipedia
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

#### Flashcard 1802563095820

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#functional-analysis
Question
a functional equation is any equation in which [...].
the unknown represents a function

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

#### Original toplevel document

Functional equation - Wikipedia
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

#### Annotation 1802567290124

 #functional-analysis A differential equation relates some function with its derivatives.

Differential equation - Wikipedia
[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

#### Flashcard 1802569649420

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#functional-analysis
Question
A [...] relates some function with its derivatives.
differential equation

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A differential equation relates some function with its derivatives.

#### Original toplevel document

Differential equation - Wikipedia
[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

#### Annotation 1802574368012

 #real-analysis 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.

Real analysis - Wikipedia
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

#### Flashcard 1802575940876

Tags
#functional-analysis
Question
[...] studies
1. vector spaces that are endowed with some kind of limit-related structure and
2. the linear functions defined on these spaces and respecting these structures in a suitable sense.
Functional analysis

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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

#### Original toplevel document

Functional analysis - Wikipedia
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

#### Flashcard 1802578300172

Tags
#real-analysis
Question
[...]
1. studies the theory of real numbers and real functions and
2. deals with the properties of limits, continuity, differentiability and integrability of these functions.
real analysis

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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

#### Original toplevel document

Real analysis - Wikipedia
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

#### Annotation 1802580659468

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

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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

#### Original toplevel document

Functional analysis - Wikipedia
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

#### Annotation 1802582232332

 #functional-analysis linear operators are transformations of functions between function spaces

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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>

#### Original toplevel document

Functional analysis - Wikipedia
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

#### Annotation 1802583805196

 #functional-analysis both differential and integral equations can be understand as linear operations on functions.

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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>

#### Original toplevel document

Functional analysis - Wikipedia
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

#### Flashcard 1802586164492

Tags
#functional-analysis
Question
The historical roots of functional analysis lie in the study of [...and...]
spaces of functions and transformations of functions

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

#### Original toplevel document

Functional analysis - Wikipedia
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

#### Flashcard 1802589310220

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#functional-analysis
Question
[...] are transformations of functions between function spaces
linear operators

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linear operators are transformations of functions between function spaces

#### Original toplevel document

Functional analysis - Wikipedia
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

#### Flashcard 1802591669516

Tags
#functional-analysis
Question
both differential and integral equations can be understand as [...] on functions.
linear operations

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

#### Original toplevel document

Functional analysis - Wikipedia
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

#### Flashcard 1802638069004

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#French
Question
On the A and U [...] usually serves to distinguish between words that would otherwise be identical
grave accent  ​​​​​​​

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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)

#### Original toplevel document

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

#### Annotation 1802640428300

 #French 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,

#### Annotation 1802642001164

 #French 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

#### Flashcard 1802643574028

Tags
#French
Question
[...] is used when two vowels are next to each other and both must be pronounced
umlaut ¨

ex: maïs.

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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.

#### Original toplevel document

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,

#### Flashcard 1802645933324

Tags
#French
Question
[...] usually indicates that an S used to follow that vowel
circumflex ˆ

e.g., forêt (forest).

status measured difficulty not learned 37% [default] 0

#### Parent (intermediate) annotation

Open it
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).