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#fourier-analysis

When the function space has an interval as the domain, the bilinear form may be the integral of the product of functions over the interval:

The functions *f* and *g* are orthogonal when this integral is zero: As with a basis of vectors in a finite-dimensional space, orthogonal functions can form an infinite basis for a function space.

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ia (Redirected from Orthogonal function) Jump to: navigation, search In mathematics, orthogonal functions belong to a function space which is a vector space (usually over R) that has a bilinear form. <span>When the function space has an interval as the domain, the bilinear form may be the integral of the product of functions over the interval: ⟨ f , g ⟩ = ∫ f ( x ) g ( x ) d x . {\displaystyle \langle f,g\rangle =\int f(x)g(x)\,dx.} The functions f and g are orthogonal when this integral is zero: ⟨ f , g ⟩ = 0. {\displaystyle \langle f,\ g\rangle =0.} As with a basis of vectors in a finite-dimensional space, orthogonal functions can form an infinite basis for a function space. Suppose {f n }, n = 0, 1, 2, … is a sequence of orthogonal functions. If f n has positive support then ⟨ f n

Tags

#sprectral-theorem #stochastics

Question

the **Karhunen–Loève theorem** represents [...] as an infinite linear combination of orthogonal functions

Answer

a stochastic process

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d> Jump to: navigation, search In the theory of stochastic processes, the Karhunen–Loève theorem (named after Kari Karhunen and Michel Loève), also known as the Kosambi–Karhunen–Loève theorem [1] [2] is a representation of a stochastic process as an infinite linear combination of orthogonal functions, analogous to a Fourier series representation of a function on a bounded interval. <html>

Karhunen–Loève theorem - Wikipedia Karhunen–Loève theorem From Wikipedia, the free encyclopedia (Redirected from Karhunen–Loeve expansion) Jump to: navigation, search In the theory of stochastic processes, the Karhunen–Loève theorem (named after Kari Karhunen and Michel Loève), also known as the Kosambi–Karhunen–Loève theorem [1] [2] is a representation of a stochastic process as an infinite linear combination of orthogonal functions, analogous to a Fourier series representation of a function on a bounded interval. The transformation is also known as Hotelling Transform and Eigenvector Transform, and is closely related to Principal Component Analysis (PCA) technique widely used in image processing

Tags

#stochastics

Question

If the parameter constant of the Poisson process is replaced with [...] , the resulting process is called an inhomogeneous or nonhomogeneous Poisson process

Answer

some non-negative integrable function of

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If the parameter constant of the Poisson process is replaced with some non-negative integrable function of , the resulting process is called an inhomogeneous or nonhomogeneous Poisson process, where the average density of points of the process is no longer constant. <

sses. [49] The homogeneous Poisson process can be defined and generalized in different ways. It can be defined such that its index set is the real line, and this stochastic process is also called the stationary Poisson process. [102] [103] <span>If the parameter constant of the Poisson process is replaced with some non-negative integrable function of t {\displaystyle t} , the resulting process is called an inhomogeneous or nonhomogeneous Poisson process, where the average density of points of the process is no longer constant. [104] Serving as a fundamental process in queueing theory, the Poisson process is an important process for mathematical models, where it finds applications for models of events randoml

Tags

#poisson-process #stochastics

Question

Answer

the number of points in a region of finite size

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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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#spanish

Question

Se informó que City [...] de la carrera después de negarse a cumplir con el precio de venta del Arsenal y las demandas salariales de Sánchez.

Answer

se retiró

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Se informó que City se retiró de la carrera después de negarse a cumplir con el precio de venta del Arsenal y las demandas salariales de Sánchez. Mourinho, sin embargo, ha desestimado los reclamos. </

el Manchester United sobre Manchester City no estuvo motivada por el dinero. United y City se enfrentaron cara a cara por el chileno este mes con el delantero moviéndose a Old Trafford y Henrikh Mkhitaryan firmando por el Arsenal a cambio. <span>Se informó que City se retiró de la carrera después de negarse a cumplir con el precio de venta del Arsenal y las demandas salariales de Sánchez. Mourinho, sin embargo, ha desestimado los reclamos. Lleva ESPN a todos lados Si quieres recibir alertas del Futbol Europeo, descarga la App ahora. espn.com/app » Él dijo en una conferencia de prensa el jueves: "Sé que si otros club

Tags

#measure-theory #stochastics

Question

In the Cantor devil's staircase, the lengths of the removed intervals add up to [...].

Answer

one

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This distribution attributes probability zero to each of the intervals removed, and the lengths of these intervals add up to one. So all of the probability is concentrated on the Cantor set C ∞ , which is what the measure-theoretic jargon calls a set of Lebesgue measure zero, Lebesgue measure being the measure-th

Tags

#measure-theory #stochastics

Question

the new real analysis started with the PhD thesis of [...] (1902)

Answer

Henri Lebesgue

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What Kolmogorov did was to say that the new real analysis that had started with the PhD thesis of Henri Lebesgue (1902) and had been rapidly generalized to integrals of real-valued functions on arbitrary spaces by Radon, Fr´echet, and others (called Lebesgue integration or abstract integration) sh

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#bayesian-network

Question

In Bayesian networks each node represents a variable with **[...]**

Answer

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>

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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#kalman-filter

Question

In Kalman filter [...] variables have Gaussian distributions.

Answer

all latent and observed

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The underlying model of Kalman filter is similar to a hidden Markov model except that the state space of the latent variables is continuous and all latent and observed variables have Gaussian distributions.

stimate in the special case that all errors are Gaussian-distributed. Extensions and generalizations to the method have also been developed, such as the extended Kalman filter and the unscented Kalman filter which work on nonlinear systems. <span>The underlying model is similar to a hidden Markov model except that the state space of the latent variables is continuous and all latent and observed variables have Gaussian distributions. Contents [hide] 1 History 2 Overview of the calculation 3 Example application 4 Technical description and context 5 Underlying dynamical system model 6 Details 6.1 Predict

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#dot-product

Question

Answer

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In mathematics, the dot product or scalar product [note 1] is an algebraic operation that takes two equal-length sequences of numbers (usually coordinate vectors) and returns a single number.

ia Jump to: navigation, search "Scalar product" redirects here. For the abstract scalar product, see Inner product space. For the product of a vector and a scalar, see Scalar multiplication. <span>In mathematics, the dot product or scalar product [note 1] is an algebraic operation that takes two equal-length sequences of numbers (usually coordinate vectors) and returns a single number. In Euclidean geometry, the dot product of the Cartesian coordinates of two vectors is widely used and often called inner product (or rarely projection product); see also inner product s

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#calculus

Question

Answer

rates of change and slopes of curves

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Calculus has two major branches, differential calculus (concerning rates of change and slopes of curves), [2] and integral calculus (concerning accumulation of quantities and the areas under and between curves). [3] These two branches are related to each other by the fundamental th

small pebble', used for counting and calculations, as on an abacus) [1] is the mathematical study of continuous change, in the same way that geometry is the study of shape and algebra is the study of generalizations of arithmetic operations. <span>It has two major branches, differential calculus (concerning rates of change and slopes of curves), [2] and integral calculus (concerning accumulation of quantities and the areas under and between curves). [3] These two branches are related to each other by the fundamental theorem of calculus. Both branches make use of the fundamental notions of convergence of infinite sequences and infinite series to a well-defined limit. Generally, modern calculus is considered to have been

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#dual-space

Question

A **dual vector space has** the vector space structure of [...].

Answer

pointwise addition and scalar multiplication by constants

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In mathematics, any vector space V has a corresponding dual vector space consisting of all linear functionals on V, together with the vector space structure of pointwise addition and scalar multiplication by constants.

Dual space - Wikipedia Dual space From Wikipedia, the free encyclopedia Jump to: navigation, search In mathematics, any vector space V has a corresponding dual vector space (or just dual space for short) consisting of all linear functionals on V, together with the vector space structure of pointwise addition and scalar multiplication by constants. The dual space as defined above is defined for all vector spaces, and to avoid ambiguity may also be called the algebraic dual space. When defined for a topological vector space, ther

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#topology

Question

A topology must satisfy axioms of **[...12...]**

Answer

inclusion, complete under infinite union and finit intersection

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A topological space is an ordered pair (X, τ), where X is a set and τ is a collection of subsets of X, satisfying the following axioms: [7] The empty set and X itself belong to τ. Any (finite or infinite) union of members of τ still belongs to τ. The intersection of any finite number of members of τ still belongs to τ.

three-point set {1,2,3}. The bottom-left example is not a topology because the union of {2} and {3} [i.e. {2,3}] is missing; the bottom-right example is not a topology because the intersection of {1,2} and {2,3} [i.e. {2}], is missing. <span>A topological space is an ordered pair (X, τ), where X is a set and τ is a collection of subsets of X, satisfying the following axioms: [7] The empty set and X itself belong to τ. Any (finite or infinite) union of members of τ still belongs to τ. The intersection of any finite number of members of τ still belongs to τ. The elements of τ are called open sets and the collection τ is called a topology on X. Examples[edit source] Given X = {1, 2, 3, 4}, the collection τ = {{}, {1, 2, 3, 4}} of only the two subsets of X required by the axioms forms a topology of X, the trivial topology (

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] The Hasse diagram of the set of all subsets of a three-element set {x, y, z}, ordered by inclusion. Sets on the same horizontal level are incomparable with each other. Some other pairs, such as {x} and {y,z}, are also incomparable. <span>In mathematics, especially order theory, a partially ordered set (also poset) formalizes and generalizes the intuitive concept of an ordering, sequencing, or arrangement of the elements of a set. A poset consists of a set together with a binary relation indicating that, for certain pairs of elements in the set, one of the elements precedes the other in the ordering. The word &qu

[unknown IMAGE 1759935859980]

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#has-images

Question

a **[...]** formalizes and generalizes the intuitive concept of an ordering, sequencing, or arrangement of the elements of a set.

Answer

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In mathematics, especially order theory, a partially ordered set (also poset) formalizes and generalizes the intuitive concept of an ordering, sequencing, or arrangement of the elements of a set.

] The Hasse diagram of the set of all subsets of a three-element set {x, y, z}, ordered by inclusion. Sets on the same horizontal level are incomparable with each other. Some other pairs, such as {x} and {y,z}, are also incomparable. <span>In mathematics, especially order theory, a partially ordered set (also poset) formalizes and generalizes the intuitive concept of an ordering, sequencing, or arrangement of the elements of a set. A poset consists of a set together with a binary relation indicating that, for certain pairs of elements in the set, one of the elements precedes the other in the ordering. The word &qu

Question

SADR is the initialism for **[...]**

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Si Neymar no quiere quedarse a la sombra de Messi debería cambiar de deporte

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orts Juegos Olímpicos Olimpismo Paralímpicos Rugby Toros Turf Volvo Ocean Race Champions League Champions LeagueAsí reaccionó el exjugador del Arsenal y del Barça ante la comparación entre los dos Thierry Henry: "<span>Si Neymar no quiere quedarse a la sombra de Messi debería cambiar de deporte" Redacción Marca Compartir en Facebook Compartir en Twitter Enviar por email 21/02/2018 10:15 CET In English 273 comentarios Comentar Thierry Henry compartió tres temporadas

Question

Si Neymar no quiere quedarse a la [...] de Messi debería cambiar de deporte

Answer

sombra

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Si Neymar no quiere quedarse a la sombra de Messi debería cambiar de deporte

orts Juegos Olímpicos Olimpismo Paralímpicos Rugby Toros Turf Volvo Ocean Race Champions League Champions LeagueAsí reaccionó el exjugador del Arsenal y del Barça ante la comparación entre los dos Thierry Henry: "<span>Si Neymar no quiere quedarse a la sombra de Messi debería cambiar de deporte" Redacción Marca Compartir en Facebook Compartir en Twitter Enviar por email 21/02/2018 10:15 CET In English 273 comentarios Comentar Thierry Henry compartió tres temporadas

#spectral-analysis

The set of all possible eigenvalues of a linear operator *D* is sometimes called its spectrum, which may be discrete, continuous, or a combination of both.^{[1]}

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he solutions to Equation (1) may also be subject to boundary conditions. Because of the boundary conditions, the possible values of λ are generally limited, for example to a discrete set λ 1 , λ 2 , ... or to a continuous set over some range. <span>The set of all possible eigenvalues of D is sometimes called its spectrum, which may be discrete, continuous, or a combination of both. [1] Each value of λ corresponds to one or more eigenfunctions. If multiple linearly independent eigenfunctions have the same eigenvalue, the eigenvalue is said to be degenerate and the ma

#spectral-analysis

Depending on whether the spectrum is discrete or continuous, the eigenfunctions can be normalized by setting the inner product of the eigenfunctions equal to either a Kronecker delta or a Dirac delta function, respectively.

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ition always holds for λ i ≠ λ j . For degenerate eigenfunctions with the same eigenvalue λ i , orthogonal eigenfunctions can always be chosen that span the eigenspace associated with λ i , for example by using the Gram-Schmidt process. [5] <span>Depending on whether the spectrum is discrete or continuous, the eigenfunctions can be normalized by setting the inner product of the eigenfunctions equal to either a Kronecker delta or a Dirac delta function, respectively. [8] [9] For many Hermitian operators, notably Sturm-Liouville operators, a third property is Its eigenfunctions form a basis of the function space on which the operator is defined [

#stochastics

In mathematics, the **Kronecker delta** is a function of two variables that equals 1 if the variables are equal, and 0 otherwise:

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pedia Kronecker delta From Wikipedia, the free encyclopedia Jump to: navigation, search Not to be confused with the Dirac delta function, nor with the Kronecker symbol. <span>In mathematics, the Kronecker delta (named after Leopold Kronecker) is a function of two variables, usually just non-negative integers. The function is 1 if the variables are equal, and 0 otherwise: δ i j = {

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In mathematics, the Kronecker delta is a function of two variables that equals 1 if the variables are equal, and 0 otherwise:

pedia Kronecker delta From Wikipedia, the free encyclopedia Jump to: navigation, search Not to be confused with the Dirac delta function, nor with the Kronecker symbol. <span>In mathematics, the Kronecker delta (named after Leopold Kronecker) is a function of two variables, usually just non-negative integers. The function is 1 if the variables are equal, and 0 otherwise: δ i j = {

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In mathematics, the Kronecker delta is a function of two variables that equals 1 if the variables are equal, and 0 otherwise:

pedia Kronecker delta From Wikipedia, the free encyclopedia Jump to: navigation, search Not to be confused with the Dirac delta function, nor with the Kronecker symbol. <span>In mathematics, the Kronecker delta (named after Leopold Kronecker) is a function of two variables, usually just non-negative integers. The function is 1 if the variables are equal, and 0 otherwise: δ i j = {

#fourier-analysis

When a function space has an interval as the domain, the integral of the product of functions as the bilinear form

The functions *f* and *g* are orthogonal when this integral is zero

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Orthogonal functions - Wikipedia Orthogonal functions From Wikipedia, the free encyclopedia Jump to: navigation, search In mathematics, orthogonal functions belong to a function space which is a vector space (usually over R) that has a bilinear form. When the function space has an interval as the domain, the bilinear form may be the integral of the product of functions over the interval: ⟨ f , g ⟩ = ∫ f ( x ) ¯ g ( x ) d x . {\displaystyle \langle f,g\rangle =\int {\overline {f(x)}}g(x)\,dx.} The functions f and g are orthogonal when this integral is zero: ⟨ f , g ⟩ = 0. {\displaystyle \langle f,\ g\rangle =0.} As with a basis of vectors in a finite-dimensional space, orthogonal functions can form an infinite basis for a function space. Suppose {f n }, n = 0, 1, 2, … is a sequence of ort

Tags

#fourier-analysis

Question

The functions *f* and *g* are orthogonal when [...inner product...]

Answer

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When a function space has an interval as the domain, the integral of the product of functions as the bilinear form The functions f and g are orthogonal when this integral is zero

Orthogonal functions - Wikipedia Orthogonal functions From Wikipedia, the free encyclopedia Jump to: navigation, search In mathematics, orthogonal functions belong to a function space which is a vector space (usually over R) that has a bilinear form. When the function space has an interval as the domain, the bilinear form may be the integral of the product of functions over the interval: ⟨ f , g ⟩ = ∫ f ( x ) ¯ g ( x ) d x . {\displaystyle \langle f,g\rangle =\int {\overline {f(x)}}g(x)\,dx.} The functions f and g are orthogonal when this integral is zero: ⟨ f , g ⟩ = 0. {\displaystyle \langle f,\ g\rangle =0.} As with a basis of vectors in a finite-dimensional space, orthogonal functions can form an infinite basis for a function space. Suppose {f n }, n = 0, 1, 2, … is a sequence of ort

#spectral-analysis

The set of all possible eigenvalues of a linear operator *D* is sometimes called its spectrum

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The set of all possible eigenvalues of a linear operator D is sometimes called its spectrum, which may be discrete, continuous, or a combination of both. [1]

he solutions to Equation (1) may also be subject to boundary conditions. Because of the boundary conditions, the possible values of λ are generally limited, for example to a discrete set λ 1 , λ 2 , ... or to a continuous set over some range. <span>The set of all possible eigenvalues of D is sometimes called its spectrum, which may be discrete, continuous, or a combination of both. [1] Each value of λ corresponds to one or more eigenfunctions. If multiple linearly independent eigenfunctions have the same eigenvalue, the eigenvalue is said to be degenerate and the ma

Tags

#spectral-analysis

Question

Answer

The set of all possible eigenvalues

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The set of all possible eigenvalues of a linear operator D is sometimes called its spectrum

he solutions to Equation (1) may also be subject to boundary conditions. Because of the boundary conditions, the possible values of λ are generally limited, for example to a discrete set λ 1 , λ 2 , ... or to a continuous set over some range. <span>The set of all possible eigenvalues of D is sometimes called its spectrum, which may be discrete, continuous, or a combination of both. [1] Each value of λ corresponds to one or more eigenfunctions. If multiple linearly independent eigenfunctions have the same eigenvalue, the eigenvalue is said to be degenerate and the ma

Tags

#spectral-analysis

Question

The set of all possible eigenvalues of a linear operator *D* is sometimes called its [...]

Answer

spectrum

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The set of all possible eigenvalues of a linear operator D is sometimes called its spectrum

he solutions to Equation (1) may also be subject to boundary conditions. Because of the boundary conditions, the possible values of λ are generally limited, for example to a discrete set λ 1 , λ 2 , ... or to a continuous set over some range. <span>The set of all possible eigenvalues of D is sometimes called its spectrum, which may be discrete, continuous, or a combination of both. [1] Each value of λ corresponds to one or more eigenfunctions. If multiple linearly independent eigenfunctions have the same eigenvalue, the eigenvalue is said to be degenerate and the ma

#english

sad or miserable in appearance: don't look so woebegone, Joanna. ternary adjective composed of three parts. • Mathematics using three as a base. a soft modelling material, used especially by children: I made a snake by rolling out plasticine an acrobatic movement in which a person turns head over heels in the air or on the ground and lands or finishes on their feet: a backward somersault from sobre ‘above’ + saut ‘leap’. castanets is small concave pieces of wood, ivory, or plastic, joined in pairs by a cord and clicked together by the fingers as a rhythmic accompaniment to Spanish dancing. principality is a state ruled by a prince. from Latin principalis ‘first, original’ a genre of crime film or fiction characterized by cynicism, fatalism, and moral ambiguity: his film proved that a Brit could do noir as darkly as any American. the husband of an adulteress, often regarded as an object of derision: jokes in literature about elderly cuckolds and misers are rife. (originally in Hinduism and Buddhism)

**separation of variables** solves ordinary and partial differential equations by allowing one to rewrite an equation so that each of two variables occurs on a different side of the equation.

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People [show] Isaac Newton Leonhard Euler Émile Picard Józef Maria Hoene-Wroński Ernst Lindelöf Rudolf Lipschitz Augustin-Louis Cauchy John Crank Phyllis Nicolson Carl David Tolmé Runge Martin Wilhelm Kutta v t e <span>In mathematics, separation of variables (also known as the Fourier method) is any of several methods for solving ordinary and partial differential equations, in which algebra allows one to rewrite an equation so that each of two variables occurs on a different side of the equation. Contents [hide] 1 Ordinary differential equations (ODE) 1.1 Alternative notation 1.2 Example 2 Partial differential equations 2.1 Example: homogeneous case 2.2 Example:

Question

**separation of variables** solves ordinary and partial differential equations by allowing one to rewrite an equation so that [...].

Answer

each of two variables occurs on a different side of the equation

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separation of variables solves ordinary and partial differential equations by allowing one to rewrite an equation so that each of two variables occurs on a different side of the equation.

People [show] Isaac Newton Leonhard Euler Émile Picard Józef Maria Hoene-Wroński Ernst Lindelöf Rudolf Lipschitz Augustin-Louis Cauchy John Crank Phyllis Nicolson Carl David Tolmé Runge Martin Wilhelm Kutta v t e <span>In mathematics, separation of variables (also known as the Fourier method) is any of several methods for solving ordinary and partial differential equations, in which algebra allows one to rewrite an equation so that each of two variables occurs on a different side of the equation. Contents [hide] 1 Ordinary differential equations (ODE) 1.1 Alternative notation 1.2 Example 2 Partial differential equations 2.1 Example: homogeneous case 2.2 Example:

Question

**[...]** solves ordinary and partial differential equations by allowing one to rewrite an equation so that each of two variables occurs on a different side of the equation.

Answer

separation of variables

status | not learned | measured difficulty | 37% [default] | last interval [days] | |||
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repetition number in this series | 0 | memorised on | scheduled repetition | ||||

scheduled repetition interval | last repetition or drill |

separation of variables solves ordinary and partial differential equations by allowing one to rewrite an equation so that each of two variables occurs on a different side of the equation. </spa

People [show] Isaac Newton Leonhard Euler Émile Picard Józef Maria Hoene-Wroński Ernst Lindelöf Rudolf Lipschitz Augustin-Louis Cauchy John Crank Phyllis Nicolson Carl David Tolmé Runge Martin Wilhelm Kutta v t e <span>In mathematics, separation of variables (also known as the Fourier method) is any of several methods for solving ordinary and partial differential equations, in which algebra allows one to rewrite an equation so that each of two variables occurs on a different side of the equation. Contents [hide] 1 Ordinary differential equations (ODE) 1.1 Alternative notation 1.2 Example 2 Partial differential equations 2.1 Example: homogeneous case 2.2 Example:

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