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Flashcard 1729450085644
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Parent (intermediate) annotation
Open itImpersonal: Nunca se debe llegar tarde a una entrevista de trabajo (One should never arrive late to a job interview).
Original toplevel document
Open itgift --> I give it to her). Reflexive: Sentarse (to sit) --> Ellos se sentaron en primera fila (They sat on the first row). Reciprocal: Amarse (to love each other) --> Esa pareja se ama con locura(That couple love each other madly). <span>Impersonal: Nunca se debe llegar tarde a una entrevista de trabajo (One should never arrive late to a job interview). Passive: Desde mi ventana se ve la playa (I can see the beach from my window; literal: "The beach is seen from my window). Pure pronominal: Pedro siempre se baña después del trabaj
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Hermitian matrix - Wikipedia
A = A T ¯ {\displaystyle A={\overline {A^{\text{T}}}}} , in matrix form. <span>Hermitian matrices can be understood as the complex extension of real symmetric matrices. If the conjugate transpose of a matrix A {\displaystyle A} is denoted by A H
Flashcard 1730982317324
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Parent (intermediate) annotation
Open itHermitian matrices can be understood as the complex extension of real symmetric matrices.
Original toplevel document
Hermitian matrix - WikipediaA = A T ¯ {\displaystyle A={\overline {A^{\text{T}}}}} , in matrix form. <span>Hermitian matrices can be understood as the complex extension of real symmetric matrices. If the conjugate transpose of a matrix A {\displaystyle A} is denoted by A H
Flashcard 1730991230220
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Parent (intermediate) annotation
Open itHermitian matrices can be understood as the complex extension of real symmetric matrices.
Original toplevel document
Hermitian matrix - WikipediaA = A T ¯ {\displaystyle A={\overline {A^{\text{T}}}}} , in matrix form. <span>Hermitian matrices can be understood as the complex extension of real symmetric matrices. If the conjugate transpose of a matrix A {\displaystyle A} is denoted by A H
Flashcard 1731706096908
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Parent (intermediate) annotation
Open itThe 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 European-style options and shows that the option has a
Original toplevel document
Black–Scholes model - WikipediaBlack–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 1732503538956
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Flashcard 1732529229068
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Open it- Ella rechazó su ayuda pues él no tenía buenas intenciones. (Ella no quiso aceptar la ayuda que él le ofrecía)
Original toplevel document
Unknown titleCancel 0 comment(s) Show previous comments Please enter between 2 and 2000 characters. Characters remaining: 2000 Submit Cancel Answers Time: oldest to newest Time: newest to oldest Votes: highest to lowest [imagelink] Hola Svetlana,<span>- Ella rechazó su ayuda pues él no tenía buenas intenciones. (Ella no quiso aceptar la ayuda que él le ofrecía)- El Ministro de Economía renunció a su cargo. (Él no quiere seguir trabajando como Ministro)- Aunque él le explicó sus razones, ella le negó su ayuda. (Ella no quiso ayudarlo).Espero sea de ayuda! Please enter between 2 and 2000 characters. If you copy an answer from another italki page, please include the URL of the original page. Characters remaining: 1673 U
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Measure (mathematics) - Wikipedia
[imagelink] Informally, a measure has the property of being monotone in the sense that if A is a subset of B, the measure of A is less than or equal to the measure of B. Furthermore, the measure of the empty set is required to be 0. <span>In mathematical analysis, a measure on a set is a systematic way to assign a number to each suitable subset of that set, intuitively interpreted as its size. In this sense, a measure is a generalization of the concepts of length, area, and volume. A particularly important example is the Lebesgue measure on a Euclidean space, which assigns the conventional length, area, and volume of Euclidean geometry to suitable subsets of the n-dimensional Euclidean space R n . For instance, the Lebesgue measure of the interval [0, 1] in the real numbers is its length in the everyday sense of the word – specifically, 1. Technically, a measure is a function that assigns a non-negative real number or +∞ to (certain) subsets of a set X (see Definition below). It must further be countably additive: the measure of a 'large' subset that can be decomposed into a finite (or countably infinite) number of 'smaller' disjoint subsets, is the sum of the measures of the "smaller" subsets. In general, if one wants to associate a consistent size to each subset of a given set while satisfying the other axioms of a measure, one only finds trivial examples like the counting measure. This problem was resolved by defining measure only on a sub-collection of all subsets; the so-called measurable subsets, which are required to form a σ-algebra. This means that countable unions, countable intersections and complements of measurable subsets are measurable. Non-measurable sets in a Euclidean space, on which the Lebesgue measure cannot be defined consistently, are necessarily complicated in the sense of being badly mixed up with their complement. [1] Indeed, their existence is a non-trivial consequence of the axiom of choice. Measure theory was developed in successive stages during the late 19th and early 20th centuries by Émile Borel, Henri Lebesgue, Johann Radon, and Maurice Fréchet, among others. The ma
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Parent (intermediate) annotation
Open itIn mathematical analysis, a measure on a set is a systematic way to assign a number to each suitable subset of that set, intuitively interpreted as its size. In this sense, a measure is a generalization of the concepts of length, area, and volume. A particularly important example is the Lebesgue measure on a Euclidean space, which assigns th
Original toplevel document
Measure (mathematics) - Wikipedia[imagelink] Informally, a measure has the property of being monotone in the sense that if A is a subset of B, the measure of A is less than or equal to the measure of B. Furthermore, the measure of the empty set is required to be 0. <span>In mathematical analysis, a measure on a set is a systematic way to assign a number to each suitable subset of that set, intuitively interpreted as its size. In this sense, a measure is a generalization of the concepts of length, area, and volume. A particularly important example is the Lebesgue measure on a Euclidean space, which assigns the conventional length, area, and volume of Euclidean geometry to suitable subsets of the n-dimensional Euclidean space R n . For instance, the Lebesgue measure of the interval [0, 1] in the real numbers is its length in the everyday sense of the word – specifically, 1. Technically, a measure is a function that assigns a non-negative real number or +∞ to (certain) subsets of a set X (see Definition below). It must further be countably additive: the measure of a 'large' subset that can be decomposed into a finite (or countably infinite) number of 'smaller' disjoint subsets, is the sum of the measures of the "smaller" subsets. In general, if one wants to associate a consistent size to each subset of a given set while satisfying the other axioms of a measure, one only finds trivial examples like the counting measure. This problem was resolved by defining measure only on a sub-collection of all subsets; the so-called measurable subsets, which are required to form a σ-algebra. This means that countable unions, countable intersections and complements of measurable subsets are measurable. Non-measurable sets in a Euclidean space, on which the Lebesgue measure cannot be defined consistently, are necessarily complicated in the sense of being badly mixed up with their complement. [1] Indeed, their existence is a non-trivial consequence of the axiom of choice. Measure theory was developed in successive stages during the late 19th and early 20th centuries by Émile Borel, Henri Lebesgue, Johann Radon, and Maurice Fréchet, among others. The ma
Flashcard 1732833053964
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Parent (intermediate) annotation
Open itIn mathematical analysis, a measure on a set is a systematic way to assign a number to each suitable subset of that set, intuitively interpreted as its size.
Original toplevel document
Measure (mathematics) - Wikipedia[imagelink] Informally, a measure has the property of being monotone in the sense that if A is a subset of B, the measure of A is less than or equal to the measure of B. Furthermore, the measure of the empty set is required to be 0. <span>In mathematical analysis, a measure on a set is a systematic way to assign a number to each suitable subset of that set, intuitively interpreted as its size. In this sense, a measure is a generalization of the concepts of length, area, and volume. A particularly important example is the Lebesgue measure on a Euclidean space, which assigns the conventional length, area, and volume of Euclidean geometry to suitable subsets of the n-dimensional Euclidean space R n . For instance, the Lebesgue measure of the interval [0, 1] in the real numbers is its length in the everyday sense of the word – specifically, 1. Technically, a measure is a function that assigns a non-negative real number or +∞ to (certain) subsets of a set X (see Definition below). It must further be countably additive: the measure of a 'large' subset that can be decomposed into a finite (or countably infinite) number of 'smaller' disjoint subsets, is the sum of the measures of the "smaller" subsets. In general, if one wants to associate a consistent size to each subset of a given set while satisfying the other axioms of a measure, one only finds trivial examples like the counting measure. This problem was resolved by defining measure only on a sub-collection of all subsets; the so-called measurable subsets, which are required to form a σ-algebra. This means that countable unions, countable intersections and complements of measurable subsets are measurable. Non-measurable sets in a Euclidean space, on which the Lebesgue measure cannot be defined consistently, are necessarily complicated in the sense of being badly mixed up with their complement. [1] Indeed, their existence is a non-trivial consequence of the axiom of choice. Measure theory was developed in successive stages during the late 19th and early 20th centuries by Émile Borel, Henri Lebesgue, Johann Radon, and Maurice Fréchet, among others. The ma
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Parent (intermediate) annotation
Open itd volume of Euclidean geometry to suitable subsets of the n -dimensional Euclidean space R n . For instance, the Lebesgue measure of the interval [0, 1] in the real numbers is its length in the everyday sense of the word – specifically, 1. <span>Technically, a measure is a function that assigns a non-negative real number or +∞ to (certain) subsets of a set X (see Definition below). It must further be countably additive: the measure of a 'large' subset that can be decomposed into a finite (or countably infinite) number of 'smaller' disjoint subsets, is the sum of the measures of the "smaller"
Original toplevel document
Measure (mathematics) - Wikipedia[imagelink] Informally, a measure has the property of being monotone in the sense that if A is a subset of B, the measure of A is less than or equal to the measure of B. Furthermore, the measure of the empty set is required to be 0. <span>In mathematical analysis, a measure on a set is a systematic way to assign a number to each suitable subset of that set, intuitively interpreted as its size. In this sense, a measure is a generalization of the concepts of length, area, and volume. A particularly important example is the Lebesgue measure on a Euclidean space, which assigns the conventional length, area, and volume of Euclidean geometry to suitable subsets of the n-dimensional Euclidean space R n . For instance, the Lebesgue measure of the interval [0, 1] in the real numbers is its length in the everyday sense of the word – specifically, 1. Technically, a measure is a function that assigns a non-negative real number or +∞ to (certain) subsets of a set X (see Definition below). It must further be countably additive: the measure of a 'large' subset that can be decomposed into a finite (or countably infinite) number of 'smaller' disjoint subsets, is the sum of the measures of the "smaller" subsets. In general, if one wants to associate a consistent size to each subset of a given set while satisfying the other axioms of a measure, one only finds trivial examples like the counting measure. This problem was resolved by defining measure only on a sub-collection of all subsets; the so-called measurable subsets, which are required to form a σ-algebra. This means that countable unions, countable intersections and complements of measurable subsets are measurable. Non-measurable sets in a Euclidean space, on which the Lebesgue measure cannot be defined consistently, are necessarily complicated in the sense of being badly mixed up with their complement. [1] Indeed, their existence is a non-trivial consequence of the axiom of choice. Measure theory was developed in successive stages during the late 19th and early 20th centuries by Émile Borel, Henri Lebesgue, Johann Radon, and Maurice Fréchet, among others. The ma
Flashcard 1741091114252
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Parent (intermediate) annotation
Open itTechnically, a measure is a function that assigns a non-negative real number or +∞ to (certain) subsets of a set X (see Definition below). It must further be countably additive:
Original toplevel document
Measure (mathematics) - Wikipedia[imagelink] Informally, a measure has the property of being monotone in the sense that if A is a subset of B, the measure of A is less than or equal to the measure of B. Furthermore, the measure of the empty set is required to be 0. <span>In mathematical analysis, a measure on a set is a systematic way to assign a number to each suitable subset of that set, intuitively interpreted as its size. In this sense, a measure is a generalization of the concepts of length, area, and volume. A particularly important example is the Lebesgue measure on a Euclidean space, which assigns the conventional length, area, and volume of Euclidean geometry to suitable subsets of the n-dimensional Euclidean space R n . For instance, the Lebesgue measure of the interval [0, 1] in the real numbers is its length in the everyday sense of the word – specifically, 1. Technically, a measure is a function that assigns a non-negative real number or +∞ to (certain) subsets of a set X (see Definition below). It must further be countably additive: the measure of a 'large' subset that can be decomposed into a finite (or countably infinite) number of 'smaller' disjoint subsets, is the sum of the measures of the "smaller" subsets. In general, if one wants to associate a consistent size to each subset of a given set while satisfying the other axioms of a measure, one only finds trivial examples like the counting measure. This problem was resolved by defining measure only on a sub-collection of all subsets; the so-called measurable subsets, which are required to form a σ-algebra. This means that countable unions, countable intersections and complements of measurable subsets are measurable. Non-measurable sets in a Euclidean space, on which the Lebesgue measure cannot be defined consistently, are necessarily complicated in the sense of being badly mixed up with their complement. [1] Indeed, their existence is a non-trivial consequence of the axiom of choice. Measure theory was developed in successive stages during the late 19th and early 20th centuries by Émile Borel, Henri Lebesgue, Johann Radon, and Maurice Fréchet, among others. The ma
Flashcard 1741092687116
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Parent (intermediate) annotation
Open itTechnically, a measure is a function that assigns a non-negative real number or +∞ to (certain) subsets of a set X (see Definition below). It must further be countably additive:
Original toplevel document
Measure (mathematics) - Wikipedia[imagelink] Informally, a measure has the property of being monotone in the sense that if A is a subset of B, the measure of A is less than or equal to the measure of B. Furthermore, the measure of the empty set is required to be 0. <span>In mathematical analysis, a measure on a set is a systematic way to assign a number to each suitable subset of that set, intuitively interpreted as its size. In this sense, a measure is a generalization of the concepts of length, area, and volume. A particularly important example is the Lebesgue measure on a Euclidean space, which assigns the conventional length, area, and volume of Euclidean geometry to suitable subsets of the n-dimensional Euclidean space R n . For instance, the Lebesgue measure of the interval [0, 1] in the real numbers is its length in the everyday sense of the word – specifically, 1. Technically, a measure is a function that assigns a non-negative real number or +∞ to (certain) subsets of a set X (see Definition below). It must further be countably additive: the measure of a 'large' subset that can be decomposed into a finite (or countably infinite) number of 'smaller' disjoint subsets, is the sum of the measures of the "smaller" subsets. In general, if one wants to associate a consistent size to each subset of a given set while satisfying the other axioms of a measure, one only finds trivial examples like the counting measure. This problem was resolved by defining measure only on a sub-collection of all subsets; the so-called measurable subsets, which are required to form a σ-algebra. This means that countable unions, countable intersections and complements of measurable subsets are measurable. Non-measurable sets in a Euclidean space, on which the Lebesgue measure cannot be defined consistently, are necessarily complicated in the sense of being badly mixed up with their complement. [1] Indeed, their existence is a non-trivial consequence of the axiom of choice. Measure theory was developed in successive stages during the late 19th and early 20th centuries by Émile Borel, Henri Lebesgue, Johann Radon, and Maurice Fréchet, among others. The ma
Flashcard 1741094259980
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Parent (intermediate) annotation
Open itTechnically, a measure is a function that assigns a non-negative real number or +∞ to (certain) subsets of a set X (see Definition below). It must further be countably additive:
Original toplevel document
Measure (mathematics) - Wikipedia[imagelink] Informally, a measure has the property of being monotone in the sense that if A is a subset of B, the measure of A is less than or equal to the measure of B. Furthermore, the measure of the empty set is required to be 0. <span>In mathematical analysis, a measure on a set is a systematic way to assign a number to each suitable subset of that set, intuitively interpreted as its size. In this sense, a measure is a generalization of the concepts of length, area, and volume. A particularly important example is the Lebesgue measure on a Euclidean space, which assigns the conventional length, area, and volume of Euclidean geometry to suitable subsets of the n-dimensional Euclidean space R n . For instance, the Lebesgue measure of the interval [0, 1] in the real numbers is its length in the everyday sense of the word – specifically, 1. Technically, a measure is a function that assigns a non-negative real number or +∞ to (certain) subsets of a set X (see Definition below). It must further be countably additive: the measure of a 'large' subset that can be decomposed into a finite (or countably infinite) number of 'smaller' disjoint subsets, is the sum of the measures of the "smaller" subsets. In general, if one wants to associate a consistent size to each subset of a given set while satisfying the other axioms of a measure, one only finds trivial examples like the counting measure. This problem was resolved by defining measure only on a sub-collection of all subsets; the so-called measurable subsets, which are required to form a σ-algebra. This means that countable unions, countable intersections and complements of measurable subsets are measurable. Non-measurable sets in a Euclidean space, on which the Lebesgue measure cannot be defined consistently, are necessarily complicated in the sense of being badly mixed up with their complement. [1] Indeed, their existence is a non-trivial consequence of the axiom of choice. Measure theory was developed in successive stages during the late 19th and early 20th centuries by Émile Borel, Henri Lebesgue, Johann Radon, and Maurice Fréchet, among others. The ma
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Closed set - Wikipedia
en set. For a set closed under an operation, see closure (mathematics). For other uses, see Closed (disambiguation). In geometry, topology, and related branches of mathematics, a closed set is a set whose complement is an open set. [1] [2] <span>In a topological space, a closed set can be defined as a set which contains all its limit points. In a complete metric space, a closed set is a set which is closed under the limit operation. Contents [hide] 1 Equivalent definitions of a closed set 2 Properties of closed
Flashcard 1744285601036
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Parent (intermediate) annotation
Open itIn a topological space, a closed set can be defined as a set which contains all its limit points.
Original toplevel document
Closed set - Wikipediaen set. For a set closed under an operation, see closure (mathematics). For other uses, see Closed (disambiguation). In geometry, topology, and related branches of mathematics, a closed set is a set whose complement is an open set. [1] [2] <span>In a topological space, a closed set can be defined as a set which contains all its limit points. In a complete metric space, a closed set is a set which is closed under the limit operation. Contents [hide] 1 Equivalent definitions of a closed set 2 Properties of closed
Flashcard 1744287173900
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Parent (intermediate) annotation
Open itIn a topological space, a closed set can be defined as a set which contains all its limit points.
Original toplevel document
Closed set - Wikipediaen set. For a set closed under an operation, see closure (mathematics). For other uses, see Closed (disambiguation). In geometry, topology, and related branches of mathematics, a closed set is a set whose complement is an open set. [1] [2] <span>In a topological space, a closed set can be defined as a set which contains all its limit points. In a complete metric space, a closed set is a set which is closed under the limit operation. Contents [hide] 1 Equivalent definitions of a closed set 2 Properties of closed
Flashcard 1749111147788
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Open itHilbert spaces are complete: there are enough limits in the space to allow the techniques of calculus to be used.
Original toplevel document
Hilbert space - Wikipediae state of a vibrating string can be modeled as a point in a Hilbert space. The decomposition of a vibrating string into its vibrations in distinct overtones is given by the projection of the point onto the coordinate axes in the space. <span>The mathematical concept of a Hilbert space, named after David Hilbert, generalizes the notion of Euclidean space. It extends the methods of vector algebra and calculus from the two-dimensional Euclidean plane and three-dimensional space to spaces with any finite or infinite number of dimensions. A Hilbert space is an abstract vector space possessing the structure of an inner product that allows length and angle to be measured. Furthermore, Hilbert spaces are complete: there are enough limits in the space to allow the techniques of calculus to be used. Hilbert spaces arise naturally and frequently in mathematics and physics, typically as infinite-dimensional function spaces. The earliest Hilbert spaces were studied from this point o
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Spectral analysis - Wikipedia
power spectrum) of a time-domain signal. This may also be called frequency domain analysis Spectrum analyzer, a hardware device that measures the magnitude of an input signal versus frequency within the full frequency range of the instrument <span>Spectral theory, in mathematics, a theory that extends eigenvalues and eigenvectors to linear operators on Hilbert space, and more generally to the elements of a Banach algebra In nuclear and particle physics, gamma spectroscopy, and high-energy astronomy, the analysis of the output of a pulse height anal
Flashcard 1758001237260
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Open itSpectral theory extends eigenvalues and eigenvectors to linear operators on Hilbert space
Original toplevel document
Spectral analysis - Wikipediapower spectrum) of a time-domain signal. This may also be called frequency domain analysis Spectrum analyzer, a hardware device that measures the magnitude of an input signal versus frequency within the full frequency range of the instrument <span>Spectral theory, in mathematics, a theory that extends eigenvalues and eigenvectors to linear operators on Hilbert space, and more generally to the elements of a Banach algebra In nuclear and particle physics, gamma spectroscopy, and high-energy astronomy, the analysis of the output of a pulse height anal
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Absolute convergence - Wikipedia
of references, but its sources remain unclear because it has insufficient inline citations. Please help to improve this article by introducing more precise citations. (February 2013) (Learn how and when to remove this template message) <span>In mathematics, an infinite series of numbers is said to converge absolutely (or to be absolutely convergent) if the sum of the absolute values of the summands is finite. More precisely, a real or complex series ∑ n = 0
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Eigenfunction - Wikipedia
ected from Eigenfunction expansion) Jump to: navigation, search [imagelink] This solution of the vibrating drum problem is, at any point in time, an eigenfunction of the Laplace operator on a disk. <span>In mathematics, an eigenfunction of a linear operator D defined on some function space is any non-zero function f in that space that, when acted upon by D, is only multiplied by some scaling factor called an eigenvalue. As an equation, this condition can be written as D f = λ f {\displaystyle Df=\lambda f} for some scalar eigenvalue λ. [1] [2] [3] The solutions to this equation may also be subject to boundary conditions that limit the allowable eigenvalues and eigenfunctions. An eigenfunction is a type of eigenvect
Flashcard 1759676075276
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Parent (intermediate) annotation
Open itIn mathematics, an eigenfunction of a linear operator D defined on some function space is any non-zero function f in that space that for some scalar eigenvalue λ.
Original toplevel document
Eigenfunction - Wikipediaected from Eigenfunction expansion) Jump to: navigation, search [imagelink] This solution of the vibrating drum problem is, at any point in time, an eigenfunction of the Laplace operator on a disk. <span>In mathematics, an eigenfunction of a linear operator D defined on some function space is any non-zero function f in that space that, when acted upon by D, is only multiplied by some scaling factor called an eigenvalue. As an equation, this condition can be written as D f = λ f {\displaystyle Df=\lambda f} for some scalar eigenvalue λ. [1] [2] [3] The solutions to this equation may also be subject to boundary conditions that limit the allowable eigenvalues and eigenfunctions. An eigenfunction is a type of eigenvect
Flashcard 1759678434572
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Parent (intermediate) annotation
Open itIn mathematics, an eigenfunction of a linear operator D defined on some function space is any non-zero function f in that space that for some scalar eigenvalue λ.
Original toplevel document
Eigenfunction - Wikipediaected from Eigenfunction expansion) Jump to: navigation, search [imagelink] This solution of the vibrating drum problem is, at any point in time, an eigenfunction of the Laplace operator on a disk. <span>In mathematics, an eigenfunction of a linear operator D defined on some function space is any non-zero function f in that space that, when acted upon by D, is only multiplied by some scaling factor called an eigenvalue. As an equation, this condition can be written as D f = λ f {\displaystyle Df=\lambda f} for some scalar eigenvalue λ. [1] [2] [3] The solutions to this equation may also be subject to boundary conditions that limit the allowable eigenvalues and eigenfunctions. An eigenfunction is a type of eigenvect
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Open itIn mathematics, an infinite series of numbers is said to converge absolutely if the sum of the absolute values of the summands is finite.
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Absolute convergence - Wikipediaof references, but its sources remain unclear because it has insufficient inline citations. Please help to improve this article by introducing more precise citations. (February 2013) (Learn how and when to remove this template message) <span>In mathematics, an infinite series of numbers is said to converge absolutely (or to be absolutely convergent) if the sum of the absolute values of the summands is finite. More precisely, a real or complex series ∑ n = 0
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Open itIn mathematics, an infinite series of numbers is said to converge absolutely if the sum of the absolute values of the summands is finite.
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Absolute convergence - Wikipediaof references, but its sources remain unclear because it has insufficient inline citations. Please help to improve this article by introducing more precise citations. (February 2013) (Learn how and when to remove this template message) <span>In mathematics, an infinite series of numbers is said to converge absolutely (or to be absolutely convergent) if the sum of the absolute values of the summands is finite. More precisely, a real or complex series ∑ n = 0
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Open itLinear operators on a Hilbert space are likewise fairly concrete objects: in good cases, they are simply transformations that stretch the space by different factors in mutually perpendicular directions in a sense that is made precise by the study of their spectrum.
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Hilbert space - Wikipediais countably infinite, this means that the Hilbert space can also usefully be thought of in terms of the space of infinite sequences that are square-summable. The latter space is often in the older literature referred to as the Hilbert space. <span>Linear operators on a Hilbert space are likewise fairly concrete objects: in good cases, they are simply transformations that stretch the space by different factors in mutually perpendicular directions in a sense that is made precise by the study of their spectrum. Contents [hide] 1 Definition and illustration 1.1 Motivating example: Euclidean space 1.2 Definition 1.3 Second example: sequence spaces 2 History 3 Examples 3.1 Lebesgu
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Eigenfunction - Wikipedia
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
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Eigenfunction - Wikipedia
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 [
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Open itThe 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]
Original toplevel document
Eigenfunction - Wikipediahe 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
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Open itThe set of all possible eigenvalues of a linear operator D is sometimes called its spectrum
Original toplevel document
Eigenfunction - Wikipediahe 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
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Open itThe set of all possible eigenvalues of a linear operator D is sometimes called its spectrum
Original toplevel document
Eigenfunction - Wikipediahe 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
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Hilbert space - Wikipedia
ble spaces 9 Orthogonal complements and projections 10 Spectral theory 11 See also 12 Remarks 13 Notes 14 References 15 External links Definition and illustration[edit source] Motivating example: Euclidean space[edit source] One of <span>the most familiar examples of a Hilbert space is the Euclidean space consisting of three-dimensional vectors, denoted by ℝ 3 , and equipped with the dot product. The dot product takes two vectors x and y, and produces a real number x · y. If x and y are represented in Cartesian coordinates, then the dot product is defined by
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Hilbert space - Wikipedia
x · y = y · x. It is linear in its first argument: (ax 1 + bx 2 ) · y = ax 1 · y + bx 2 · y for any scalars a, b, and vectors x 1 , x 2 , and y. It is positive definite: for all vectors x, x · x ≥ 0 , with equality if and only if x = 0. <span>An operation on pairs of vectors that, like the dot 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. The basic feature of t
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Hilbert space - Wikipedia
to 0\quad {\text{as }}N\to \infty \,.} This property expresses the completeness of Euclidean space: that a series that converges absolutely also converges in the ordinary sense. Hilbert spaces are often taken over the complex numbers. <span>The complex plane denoted by ℂ is equipped with a notion of magnitude, the complex modulus |z| which is defined as the square root of the product of z with its complex conjugate: | z | 2 = z z ¯ . {\displaystyle |z|^{2}=z{\overline {z}}\,.} If z = x + iy is a decomposition of z into its real and imaginary parts, then the modulus is the usual Euclidean two-dimensional length: |
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Hilbert space - Wikipedia
w ¯ . {\displaystyle \langle z,w\rangle =z{\overline {w}}\,.} This is complex-valued. The real part of ⟨z,w⟩ gives the usual two-dimensional Euclidean dot product. <span>A second example is the space ℂ 2 whose elements are pairs of complex numbers z = (z 1 , z 2 ). Then the inner product of z with another such vector w = (w 1 ,w 2 ) is given by ⟨ z , w ⟩ = z 1 w ¯ 1 + z 2 w ¯ 2 . {\displaystyle \langle z,w\rangle =z_{1}{\overline {w}}_{1}+z_{2}{\overline {w}}_{2}\,.} The real part of ⟨z,w⟩ is then the four-dimensional Euclidean dot product. This inner product is Hermitian symmetric, which means that the result of interchanging z and w is the complex conjugate: ⟨ w , z
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Hilbert space - Wikipedia
3 . {\displaystyle {\begin{pmatrix}x_{1}\\x_{2}\\x_{3}\end{pmatrix}}\cdot {\begin{pmatrix}y_{1}\\y_{2}\\y_{3}\end{pmatrix}}=x_{1}y_{1}+x_{2}y_{2}+x_{3}y_{3}\,.} <span>The dot product satisfies the properties: It is symmetric in x and y: x · y = y · x. It is linear in its first argument: (ax 1 + bx 2 ) · y = ax 1 · y + bx 2 · y for any scalars a, b, and vectors x 1 , x 2 , and y. It is positive definite: for all vectors x, x · x ≥ 0 , with equality if and only if x = 0. An operation on pairs of vectors that, like the dot product, satisfies these three properties is known as a (real) inner product. A vector space equipped with such an inner product is
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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
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\|\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 displacement (in orange). <span>Multivariable calculus in Euclidean space relies on the ability to compute limits, and to have useful criteria for concluding that limits exist. A mathematical series ∑ n = 0 ∞
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→ 0 as N → ∞ . {\displaystyle \left\|\mathbf {L} -\sum _{k=0}^{N}\mathbf {x} _{k}\right\|\to 0\quad {\text{as }}N\to \infty \,.} <span>This property expresses the completeness of Euclidean space: that a series that converges absolutely also converges in the ordinary sense. Hilbert spaces are often taken over the complex numbers. The complex plane denoted by ℂ is equipped with a notion of magnitude, the complex modulus |z| which is defined as the square
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Open itthe most familiar examples of a Hilbert space is the Euclidean space consisting of three-dimensional vectors, denoted by ℝ 3 , and equipped with the dot product.
Original toplevel document
Hilbert space - Wikipediable spaces 9 Orthogonal complements and projections 10 Spectral theory 11 See also 12 Remarks 13 Notes 14 References 15 External links Definition and illustration[edit source] Motivating example: Euclidean space[edit source] One of <span>the most familiar examples of a Hilbert space is the Euclidean space consisting of three-dimensional vectors, denoted by ℝ 3 , and equipped with the dot product. The dot product takes two vectors x and y, and produces a real number x · y. If x and y are represented in Cartesian coordinates, then the dot product is defined by
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Open itthe most familiar examples of a Hilbert space is the Euclidean space consisting of three-dimensional vectors, denoted by ℝ 3 , and equipped with the dot product.
Original toplevel document
Hilbert space - Wikipediable spaces 9 Orthogonal complements and projections 10 Spectral theory 11 See also 12 Remarks 13 Notes 14 References 15 External links Definition and illustration[edit source] Motivating example: Euclidean space[edit source] One of <span>the most familiar examples of a Hilbert space is the Euclidean space consisting of three-dimensional vectors, denoted by ℝ 3 , and equipped with the dot product. The dot product takes two vectors x and y, and produces a real number x · y. If x and y are represented in Cartesian coordinates, then the dot product is defined by
Flashcard 1796946136332
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Open itthe most familiar examples of a Hilbert space is the Euclidean space consisting of three-dimensional vectors, denoted by ℝ 3 , and equipped with the dot product.
Original toplevel document
Hilbert space - Wikipediable spaces 9 Orthogonal complements and projections 10 Spectral theory 11 See also 12 Remarks 13 Notes 14 References 15 External links Definition and illustration[edit source] Motivating example: Euclidean space[edit source] One of <span>the most familiar examples of a Hilbert space is the Euclidean space consisting of three-dimensional vectors, denoted by ℝ 3 , and equipped with the dot product. The dot product takes two vectors x and y, and produces a real number x · y. If x and y are represented in Cartesian coordinates, then the dot product is defined by
Flashcard 1796951641356
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Open itThe dot product satisfies three properties: symmetric in its arguments; linear in its first argument; and positive definite for all non-zero vectors
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Hilbert space - Wikipedia3 . {\displaystyle {\begin{pmatrix}x_{1}\\x_{2}\\x_{3}\end{pmatrix}}\cdot {\begin{pmatrix}y_{1}\\y_{2}\\y_{3}\end{pmatrix}}=x_{1}y_{1}+x_{2}y_{2}+x_{3}y_{3}\,.} <span>The dot product satisfies the properties: It is symmetric in x and y: x · y = y · x. It is linear in its first argument: (ax 1 + bx 2 ) · y = ax 1 · y + bx 2 · y for any scalars a, b, and vectors x 1 , x 2 , and y. It is positive definite: for all vectors x, x · x ≥ 0 , with equality if and only if x = 0. An operation on pairs of vectors that, like the dot product, satisfies these three properties is known as a (real) inner product. A vector space equipped with such an inner product is
Flashcard 1796953214220
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Open itThe dot product satisfies three properties: symmetric in its arguments; linear in its first argument; and positive definite for all non-zero vectors
Original toplevel document
Hilbert space - Wikipedia3 . {\displaystyle {\begin{pmatrix}x_{1}\\x_{2}\\x_{3}\end{pmatrix}}\cdot {\begin{pmatrix}y_{1}\\y_{2}\\y_{3}\end{pmatrix}}=x_{1}y_{1}+x_{2}y_{2}+x_{3}y_{3}\,.} <span>The dot product satisfies the properties: It is symmetric in x and y: x · y = y · x. It is linear in its first argument: (ax 1 + bx 2 ) · y = ax 1 · y + bx 2 · y for any scalars a, b, and vectors x 1 , x 2 , and y. It is positive definite: for all vectors x, x · x ≥ 0 , with equality if and only if x = 0. An operation on pairs of vectors that, like the dot product, satisfies these three properties is known as a (real) inner product. A vector space equipped with such an inner product is
Flashcard 1796955573516
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Open itThe dot product satisfies three properties: symmetric in its arguments; linear in its first argument; and positive definite for all non-zero vectors
Original toplevel document
Hilbert space - Wikipedia3 . {\displaystyle {\begin{pmatrix}x_{1}\\x_{2}\\x_{3}\end{pmatrix}}\cdot {\begin{pmatrix}y_{1}\\y_{2}\\y_{3}\end{pmatrix}}=x_{1}y_{1}+x_{2}y_{2}+x_{3}y_{3}\,.} <span>The dot product satisfies the properties: It is symmetric in x and y: x · y = y · x. It is linear in its first argument: (ax 1 + bx 2 ) · y = ax 1 · y + bx 2 · y for any scalars a, b, and vectors x 1 , x 2 , and y. It is positive definite: for all vectors x, x · x ≥ 0 , with equality if and only if x = 0. An operation on pairs of vectors that, like the dot product, satisfies these three properties is known as a (real) inner product. A vector space equipped with such an inner product is
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Open itPrimeira Geração: são os direitos que buscam restringir a ação do Estado sobre o indivíduo, impedindo que este se intrometa de forma abusiva na vida privada das pessoas
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Open itPrimeira Geração: são os direitos que buscam restringir a ação do Estado sobre o indivíduo, impedindo que este se intrometa de forma abusiva na vida privada das pessoas
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cannot see any pdfsFlashcard 1798721637644
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