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magelink] [emptylink] The first flight of an airplane, the Wright Flyer on December 17, 1903 [imagelink] [emptylink] An All Nippon Airways Boeing 777 -300 taking off from New York JFK Airport . <span>An airplane or aeroplane (informally plane) is a powered , fixed-wing aircraft that is propelled forward by thrust from a jet engine , propeller or rocket engine . Airplanes come in a variety of sizes, shapes, and wing configurations . The broad spectrum of uses for airplanes includes recreation , transportation of goods and people, military , and

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An airplane or aeroplane A (informally plane) is a powered, fixed-wing aircraft that is propelled forward by thrust from a jet engine or propeller.

her uses, see Airplane (disambiguation) and Aeroplane (disambiguation). [imagelink] North American P-51 Mustang, a World War II fighter [imagelink] The first flight of an airplane, the Wright Flyer on December 17, 1903 <span>An airplane or aeroplane A (informally plane) is a powered, fixed-wing aircraft that is propelled forward by thrust from a jet engine or propeller. Airplanes come in a variety of sizes, shapes, and wing configurations. The broad spectrum of uses for airplanes includes recreation, transportation of goods and people, military, and re

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An airplane or aeroplane A (informally plane) is a powered, fixed-wing aircraft that is propelled forward by thrust from a jet engine or propeller.

her uses, see Airplane (disambiguation) and Aeroplane (disambiguation). [imagelink] North American P-51 Mustang, a World War II fighter [imagelink] The first flight of an airplane, the Wright Flyer on December 17, 1903 <span>An airplane or aeroplane A (informally plane) is a powered, fixed-wing aircraft that is propelled forward by thrust from a jet engine or propeller. Airplanes come in a variety of sizes, shapes, and wing configurations. The broad spectrum of uses for airplanes includes recreation, transportation of goods and people, military, and re

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El camino recorrido a la derecha o hacia arriba de un punto se desig na con el signo + y el camino recorrido a la izquierda o hacia abajo de un punto se representa con el signo —.

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is on hand does not mean it's free. See below for the definition of opportunity cost. Expected future cash flows must be measured on an after-tax basis. The firm's wealth depends on its usable after-tax funds. <span>Ignore how the project is financed. Interest payments should not be included in the estimated cash flows since the effects of debt financing are reflected in the cost of capital used to discount the cash flows. The existe

Capital budgeting decisions are based on incremental after-tax cash flows discounted at the opportunity cost of capital. Assumptions of capital budgeting are: Capital budgeting decisions must be based on cash flows, not accounting income. Accounting profits only measure the return on the invested capital. Accounting income calculations reflect non-cash items and ignore the time value of money. They are important for some purposes, but for capital budgeting, cash flows are what are relevant. Economic income is an investment's after-tax cash flow plus the change in the market value. Financing costs are ignored in computing economic income. Cash flow timing is critical because money is worth more the sooner you get it. Also, firms must have adequate cash flow to meet maturing obligations. The opportunity cost should be charged against a project. Remember that just because something is on hand does not mean it's free. See below for the definition of opportunity cost. Expected future cash flows must be measured on an after-tax basis. The firm's wealth depends on its usable after-tax funds. Ignore how the project is financed. Interest payments should not be included in the estimated cash flows since the effects of debt financing are reflected in the cost of capital used to discount the cash flows. The existence of a project depends on business factors, not financing. Important capital budgeting concepts: A sunk cost is a cash outlay that has already been incurred and which

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An airplane or aeroplane (informally plane) is a powered , fixed-wing aircraft that is propelled forward by thrust from a jet engine or propeller .

sambiguation) and Aeroplane (disambiguation). [imagelink] [emptylink] North American P-51 Mustang, a World War II fighter [imagelink] [emptylink] The first flight of an airplane, the Wright Flyer on December 17, 1903 <span>An airplane or aeroplane (informally plane) is a powered, fixed-wing aircraft that is propelled forward by thrust from a jet engine or propeller. Airplanes come in a variety of sizes, shapes, and wing configurations. The broad spectrum of uses for airplanes includes recreation, transportation of goods and people, military, and re

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EP finds approximations to a probability distribution. It uses an iterative approach that leverages the factorization structure of the target distribution.

Expectation propagation From Wikipedia, the free encyclopedia Jump to: navigation, search Expectation propagation (EP) is a technique in Bayesian machine learning. <span>EP finds approximations to a probability distribution. It uses an iterative approach that leverages the factorization structure of the target distribution. It differs from other Bayesian approximation approaches such as Variational Bayesian methods. References[edit source] Thomas Minka (August 2–5, 2001). "Expectation Propagation

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In linear algebra, a square matrix A is called diagonalizable if it is similar to a diagonal matrix, i.e., if there exists an invertible matrix P such that P −1 AP is a diagonal matrix.

Diagonalizable matrix From Wikipedia, the free encyclopedia Jump to: navigation, search This article is about matrix diagonalisation in linear algebra. For other uses, see Diagonalisation. <span>In linear algebra, a square matrix A is called diagonalizable if it is similar to a diagonal matrix, i.e., if there exists an invertible matrix P such that P −1 AP is a diagonal matrix. If V is a finite-dimensional vector space, then a linear map T : V → V is called diagonalizable if there exists an ordered basis of V with respect to which T is represented by a diagona

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

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

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In computing, memoization or memoisation is an optimization technique used primarily to speed up computer programs by storing the results of expensive function calls and returning the cached result when the same inputs occur again.

dia, the free encyclopedia Jump to: navigation, search Not to be confused with Memorization. "Tabling" redirects here. For the parliamentary procedure, see Table (parliamentary procedure). <span>In computing, memoization or memoisation is an optimization technique used primarily to speed up computer programs by storing the results of expensive function calls and returning the cached result when the same inputs occur again. Memoization has also been used in other contexts (and for purposes other than speed gains), such as in simple mutually recursive descent parsing [1] . Although related to caching, memoi

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ter science, mathematics, management science, economics and bioinformatics, dynamic programming (also known as dynamic optimization) is a method for solving a complex problem by breaking it down into a collection of simpler subproblems, <span>solving each of those subproblems just once, and storing their solutions. The next time the same subproblem occurs, instead of recomputing its solution, one simply looks up the previously computed solution, thereby saving c

This article's factual accuracy is disputed. Relevant discussion may be found on the talk page. Please help to ensure that disputed statements are reliably sourced. (November 2015) (Learn how and when to remove this template message) <span>In computer science, mathematics, management science, economics and bioinformatics, dynamic programming (also known as dynamic optimization) is a method for solving a complex problem by breaking it down into a collection of simpler subproblems, solving each of those subproblems just once, and storing their solutions. The next time the same subproblem occurs, instead of recomputing its solution, one simply looks up the previously computed solution, thereby saving computation time at the expense of a (hopefully) modest expenditure in storage space. (Each of the subproblem solutions is indexed in some way, typically based on the values of its input parameters, so as to facilitate its lookup.) The technique of storing solutions to subproblems instead of recomputing them is called "memoization". Dynamic programming algorithms are often used for optimization. A dyna

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In probability theory, a martingale is a sequence of random variables (i.e., a stochastic process) for which, at a particular time in the realized sequence, the expectation of the next value in the sequence is equal to the present observed value even given knowledge of all prior observed values. <body><html>

h For the martingale betting strategy, see martingale (betting system). [imagelink] Stopped Brownian motion is an example of a martingale. It can model an even coin-toss betting game with the possibility of bankruptcy. <span>In probability theory, a martingale is a sequence of random variables (i.e., a stochastic process) for which, at a particular time in the realized sequence, the expectation of the next value in the sequence is equal to the present observed value even given knowledge of all prior observed values. Contents [hide] 1 History 2 Definitions 2.1 Martingale sequences with respect to another sequence 2.2 General definition 3 Examples of martingales 4 Submartingales, super

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The term random function is also used to refer to a stochastic or random process, [25] [26] because a stochastic process can also be interpreted as a random element in a function space. <

are considered the most important and central in the theory of stochastic processes, [1] [4] [23] and were discovered repeatedly and independently, both before and after Bachelier and Erlang, in different settings and countries. [21] [24] <span>The term random function is also used to refer to a stochastic or random process, [25] [26] because a stochastic process can also be interpreted as a random element in a function space. [27] [28] The terms stochastic process and random process are used interchangeably, often with no specific mathematical space for the set that indexes the random variables. [27] [29]

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span> In computer science, functional programming is a programming paradigm—a style of building the structure and elements of computer programs—that treats computation as the evaluation of mathematical functions and avoids <span>changing-state and mutable data <span><body><html>

ithic) Object-oriented Actor-based Class-based Concurrent Prototype-based By separation of concerns: Aspect-oriented Role-oriented Subject-oriented Recursive Value-level (contrast: Function-level) Quantum programming v t e <span>In computer science, functional programming is a programming paradigm—a style of building the structure and elements of computer programs—that treats computation as the evaluation of mathematical functions and avoids changing-state and mutable data. It is a declarative programming paradigm, which means programming is done with expressions [1] or declarations [2] instead of statements. In functional code, the output value of a fu

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The smallest sigma-algebra is {∅, Ω}. It must contain Ω by definition, and it must contain ∅ because it is Ω c . Unions and intersections of Ω and ∅ give us the same sets back, no new sets.

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A set Ω equipped with a sigma-algebra is called a measurable space and usually denoted as a pair . In this context, the elements of are called measurable sets.

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If a Poisson point process has a parameter of the form , where is Lebegues measure, and is a constant, then the point process is called a homogeneous or stationary Poisson point process.

edit source] For all the different settings of the Poisson point process, the two key properties [b] of the Poisson distribution and complete independence play an important role. [25] [45] Homogeneous Poisson point process[edit source] <span>If a Poisson point process has a parameter of the form Λ = ν λ {\displaystyle \textstyle \Lambda =\nu \lambda } , where ν {\displaystyle \textstyle \nu } is Lebegues measure, which assigns length, area, or volume to sets, and λ {\displaystyle \textstyle \lambda } is a constant, then the point process is called a homogeneous or stationary Poisson point process. The parameter, called rate or intensity, is related to the expected (or average) number of Poisson points existing in some bounded region, [49] [50] where rate is usually used when the

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er set theory developments showed (see non-measurable set), it is actually impossible to assign a length to all subsets of ℝ in a way that preserves some natural additivity and translation invariance properties. This suggests that picking out <span>a suitable class of measurable subsets is an essential prerequisite. <span><body><html>

a useful abstraction of the notion of length of subsets of the real line—and, more generally, area and volume of subsets of Euclidean spaces. In particular, it provided a systematic answer to the question of which subsets of ℝ have a length. <span>As later set theory developments showed (see non-measurable set), it is actually impossible to assign a length to all subsets of ℝ in a way that preserves some natural additivity and translation invariance properties. This suggests that picking out a suitable class of measurable subsets is an essential prerequisite. The Riemann integral uses the notion of length explicitly. Indeed, the element of calculation for the Riemann integral is the rectangle [a, b] × [c, d], whose area is calculated to be

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A technical issue in Lebesgue integration is that the domain of integration is defined as a set (a subset of a measure space), with no notion of orientation.

1 ] ) = 0 , {\displaystyle \int _{[0,1]}1_{\mathbf {Q} }\,\mathrm {d} \mu =\mu (\mathbf {Q} \cap [0,1])=0,} because Q is countable. Domain of integration[edit source] <span>A technical issue in Lebesgue integration is that the domain of integration is defined as a set (a subset of a measure space), with no notion of orientation. In elementary calculus, one defines integration with respect to an orientation: ∫ b a

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A technical issue in Lebesgue integration is that the domain of integration is defined as a set (a subset of a measure space), with no notion of orientation.

1 ] ) = 0 , {\displaystyle \int _{[0,1]}1_{\mathbf {Q} }\,\mathrm {d} \mu =\mu (\mathbf {Q} \cap [0,1])=0,} because Q is countable. Domain of integration[edit source] <span>A technical issue in Lebesgue integration is that the domain of integration is defined as a set (a subset of a measure space), with no notion of orientation. In elementary calculus, one defines integration with respect to an orientation: ∫ b a

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In mathematics and vector algebra, the cross product or vector product (occasionally directed area product to emphasize the geometric significance) is a binary operation on two vectors in three-dimensional space (R 3 ) and is denoted by the symbol ×. Given two linearly independent vectors a and b, the cross product, a × b, is a vector that is perpendicular to both a and b and thus normal to the plane

edia, the free encyclopedia Jump to: navigation, search This article is about the cross product of two vectors in three-dimensional Euclidean space. For other uses, see Cross product (disambiguation). <span>In mathematics and vector algebra, the cross product or vector product (occasionally directed area product to emphasize the geometric significance) is a binary operation on two vectors in three-dimensional space (R 3 ) and is denoted by the symbol ×. Given two linearly independent vectors a and b, the cross product, a × b, is a vector that is perpendicular to both a and b and thus normal to the plane containing them. It has many applications in mathematics, physics, engineering, and computer programming. It should not be confused with dot product (projection product). If two vectors have the same

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To assign a value to the integral of the indicator function 1 S of a measurable set S consistent with the given measure μ, the only reasonable choice is to set: Notice that the result may be equal to +∞ , unless μ is a finite measure.

x ) {\displaystyle \int _{E}f\,\mathrm {d} \mu =\int _{E}f\left(x\right)\,\mathrm {d} \mu \left(x\right)} for measurable real-valued functions f defined on E in stages: Indicator functions: <span>To assign a value to the integral of the indicator function 1 S of a measurable set S consistent with the given measure μ, the only reasonable choice is to set: ∫ 1 S d μ = μ ( S ) . {\displaystyle \int 1_{S}\,\mathrm {d} \mu =\mu (S).} Notice that the result may be equal to +∞, unless μ is a finite measure. Simple functions: A finite linear combination of indicator functions ∑ k a

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Borel sets are important in measure theory, since any measure defined on the open sets of a space, or on the closed sets of a space, must also be defined on all Borel sets of that space.

or a topological space X, the collection of all Borel sets on X forms a σ-algebra, known as the Borel algebra or Borel σ-algebra. The Borel algebra on X is the smallest σ-algebra containing all open sets (or, equivalently, all closed sets). <span>Borel sets are important in measure theory, since any measure defined on the open sets of a space, or on the closed sets of a space, must also be defined on all Borel sets of that space. Any measure defined on the Borel sets is called a Borel measure. Borel sets and the associated Borel hierarchy also play a fundamental role in descriptive set theory. In some contexts

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One thing that makes the Borel sets so powerful is that if you know what a probability measure does on every interval, then you know what it does on all the Borel sets.

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One thing that makes the Borel sets so powerful is that if you know what a probability measure does on every interval, then you know what it does on all the Borel sets.

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A Hilbert space is an abstract vector space possessing the structure of an inner product that allows length and angle to be measured.

e 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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By the spectral theorem, real symmetric matrices and complex Hermitian matrices have an eigenbasis

x. In complex matrices, symmetry is often replaced by the concept of Hermitian matrices, which satisfy A ∗ = A, where the star or asterisk denotes the conjugate transpose of the matrix, that is, the transpose of the complex conjugate of A. <span>By the spectral theorem, real symmetric matrices and complex Hermitian matrices have an eigenbasis; that is, every vector is expressible as a linear combination of eigenvectors. In both cases, all eigenvalues are real. [29] This theorem can be generalized to infinite-dimensional situations related to matrices with infinitely many rows and columns, see below. Invertible matrix and its inverse[edit s

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

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

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

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analysis. For field theory, see Field norm. For ideals, see Ideal norm. For group theory, see Norm (group). For norms in descriptive set theory, see prewellordering. In linear algebra, functional analysis, and related areas of mathematics, <span>a norm is a function that assigns a strictly positive 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).

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a norm is a function that assigns a strictly positive length or size to each vector in a vector space

analysis. For field theory, see Field norm. For ideals, see Ideal norm. For group theory, see Norm (group). For norms in descriptive set theory, see prewellordering. In linear algebra, functional analysis, and related areas of mathematics, <span>a norm is a function that assigns a strictly positive 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).

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a norm is a function that assigns a strictly positive length or size to each vector (bar zero vector) in a vector space

analysis. For field theory, see Field norm. For ideals, see Ideal norm. For group theory, see Norm (group). For norms in descriptive set theory, see prewellordering. In linear algebra, functional analysis, and related areas of mathematics, <span>a norm is a function that assigns a strictly positive 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).

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ng and 10% of your time on adding most important findings to SuperMemo, your reading speed will actually decline only by some 10%, while the retention of the most important pieces will be as high as programmed in SuperMemo (up to 99%). <span>The concept of incremental reading introduced in SuperMemo 2000 provides you with a precise tool for finding the optimum balance between speed and retention. You will ensure high-retention of the most important pieces of text, while a large proportion of time will be spent reading at speeds comparable or higher than those typical of traditional book reading. It is worth noting that the learning speed limit in high-retention learning is imposed by your memory. If one-book-per-year sounds like a major disappointment, the roots of this lay

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The concept of incremental reading introduced in SuperMemo 2000 provides you with a precise tool for finding the optimum balance between speed and retention. You will ensure high-retention of the most important pieces of text, while a large proportion of time will be spent reading at speeds comparable or higher than those typical of traditio

ng and 10% of your time on adding most important findings to SuperMemo, your reading speed will actually decline only by some 10%, while the retention of the most important pieces will be as high as programmed in SuperMemo (up to 99%). <span>The concept of incremental reading introduced in SuperMemo 2000 provides you with a precise tool for finding the optimum balance between speed and retention. You will ensure high-retention of the most important pieces of text, while a large proportion of time will be spent reading at speeds comparable or higher than those typical of traditional book reading. It is worth noting that the learning speed limit in high-retention learning is imposed by your memory. If one-book-per-year sounds like a major disappointment, the roots of this lay

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The concept of incremental reading introduced in SuperMemo 2000 provides you with a precise tool for finding the optimum balance between speed and retention. You will ensure high-retention of the most important pieces of text, while a large proportion of time will be spent reading at speeds comparable or higher than those typical of traditional book reading.

ng and 10% of your time on adding most important findings to SuperMemo, your reading speed will actually decline only by some 10%, while the retention of the most important pieces will be as high as programmed in SuperMemo (up to 99%). <span>The concept of incremental reading introduced in SuperMemo 2000 provides you with a precise tool for finding the optimum balance between speed and retention. You will ensure high-retention of the most important pieces of text, while a large proportion of time will be spent reading at speeds comparable or higher than those typical of traditional book reading. It is worth noting that the learning speed limit in high-retention learning is imposed by your memory. If one-book-per-year sounds like a major disappointment, the roots of this lay

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 |

The concept of incremental reading introduced in SuperMemo 2000 provides you with a precise tool for finding the optimum balance between speed and retention.

ng and 10% of your time on adding most important findings to SuperMemo, your reading speed will actually decline only by some 10%, while the retention of the most important pieces will be as high as programmed in SuperMemo (up to 99%). <span>The concept of incremental reading introduced in SuperMemo 2000 provides you with a precise tool for finding the optimum balance between speed and retention. You will ensure high-retention of the most important pieces of text, while a large proportion of time will be spent reading at speeds comparable or higher than those typical of traditional book reading. It is worth noting that the learning speed limit in high-retention learning is imposed by your memory. If one-book-per-year sounds like a major disappointment, the roots of this lay

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repetition number in this series | 0 | memorised on | scheduled repetition | ||||

scheduled repetition interval | last repetition or drill |

With incremental reading, you ensure high-retention of the most important pieces of text, while a large proportion of time will be spent reading at speeds comparable to typical of traditional book reading.

ng and 10% of your time on adding most important findings to SuperMemo, your reading speed will actually decline only by some 10%, while the retention of the most important pieces will be as high as programmed in SuperMemo (up to 99%). <span>The concept of incremental reading introduced in SuperMemo 2000 provides you with a precise tool for finding the optimum balance between speed and retention. You will ensure high-retention of the most important pieces of text, while a large proportion of time will be spent reading at speeds comparable or higher than those typical of traditional book reading. It is worth noting that the learning speed limit in high-retention learning is imposed by your memory. If one-book-per-year sounds like a major disappointment, the roots of this lay

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 |

With incremental reading, you ensure high-retention of the most important pieces of text, while a large proportion of time will be spent reading at speeds comparable to traditional book reading.

ng and 10% of your time on adding most important findings to SuperMemo, your reading speed will actually decline only by some 10%, while the retention of the most important pieces will be as high as programmed in SuperMemo (up to 99%). <span>The concept of incremental reading introduced in SuperMemo 2000 provides you with a precise tool for finding the optimum balance between speed and retention. You will ensure high-retention of the most important pieces of text, while a large proportion of time will be spent reading at speeds comparable or higher than those typical of traditional book reading. It is worth noting that the learning speed limit in high-retention learning is imposed by your memory. If one-book-per-year sounds like a major disappointment, the roots of this lay

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repetition number in this series | 0 | memorised on | scheduled repetition | ||||

scheduled repetition interval | last repetition or drill |

With incremental reading, you ensure high-retention of the most important pieces of text, while a large proportion of time will be spent reading at speeds comparable to traditional book reading.

ng and 10% of your time on adding most important findings to SuperMemo, your reading speed will actually decline only by some 10%, while the retention of the most important pieces will be as high as programmed in SuperMemo (up to 99%). <span>The concept of incremental reading introduced in SuperMemo 2000 provides you with a precise tool for finding the optimum balance between speed and retention. You will ensure high-retention of the most important pieces of text, while a large proportion of time will be spent reading at speeds comparable or higher than those typical of traditional book reading. It is worth noting that the learning speed limit in high-retention learning is imposed by your memory. If one-book-per-year sounds like a major disappointment, the roots of this lay

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Euler's formula, named after Leonhard Euler, is a mathematical formula in complex analysis that establishes the fundamental relationship between the trigonometric functions and the complex exponential function. Euler's formula states that for any real number x e i x = cos x + i sin x , {\displaystyle e^{ix}=\cos x+i\sin x,} where e is the base of the natural logarithm, i is the imagina

s formula half-lives exponential growth and decay Defining e proof that e is irrational representations of e Lindemann–Weierstrass theorem People John Napier Leonhard Euler Related topics Schanuel's conjecture v t e <span>Euler's formula, named after Leonhard Euler, is a mathematical formula in complex analysis that establishes the fundamental relationship between the trigonometric functions and the complex exponential function. Euler's formula states that for any real number x e i x = cos x + i sin x , {\displaystyle e^{ix}=\cos x+i\sin x,} where e is the base of the natural logarithm, i is the imaginary unit, and cos and sin are the trigonometric functions cosine and sine respectively, with the argument x given in radians. This complex exponential function is sometimes denoted cis x ("cosine plus i sine"). The formula is still valid if x is a complex number, and so some authors refer to the more

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span> Euler's formula, named after Leonhard Euler, is a mathematical formula in complex analysis that establishes the fundamental relationship between the trigonometric functions and the complex exponential function. Euler's formula states that for any real number x e i x = cos x + i sin x , {\displaystyle e^{ix}=\cos x+i\sin x,} where e is the base of the natural logarithm, i is the imaginary unit, and cos and sin are the trigonometric functions cosine and sine respectively, with the argument x given in radians. <span><body><html>

s formula half-lives exponential growth and decay Defining e proof that e is irrational representations of e Lindemann–Weierstrass theorem People John Napier Leonhard Euler Related topics Schanuel's conjecture v t e <span>Euler's formula, named after Leonhard Euler, is a mathematical formula in complex analysis that establishes the fundamental relationship between the trigonometric functions and the complex exponential function. Euler's formula states that for any real number x e i x = cos x + i sin x , {\displaystyle e^{ix}=\cos x+i\sin x,} where e is the base of the natural logarithm, i is the imaginary unit, and cos and sin are the trigonometric functions cosine and sine respectively, with the argument x given in radians. This complex exponential function is sometimes denoted cis x ("cosine plus i sine"). The formula is still valid if x is a complex number, and so some authors refer to the more

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repetition number in this series | 0 | memorised on | scheduled repetition | ||||

scheduled repetition interval | last repetition or drill |

Euler's formula establishes the fundamental relationship between the trigonometric functions and the complex exponential function.

s formula half-lives exponential growth and decay Defining e proof that e is irrational representations of e Lindemann–Weierstrass theorem People John Napier Leonhard Euler Related topics Schanuel's conjecture v t e <span>Euler's formula, named after Leonhard Euler, is a mathematical formula in complex analysis that establishes the fundamental relationship between the trigonometric functions and the complex exponential function. Euler's formula states that for any real number x e i x = cos x + i sin x , {\displaystyle e^{ix}=\cos x+i\sin x,} where e is the base of the natural logarithm, i is the imaginary unit, and cos and sin are the trigonometric functions cosine and sine respectively, with the argument x given in radians. This complex exponential function is sometimes denoted cis x ("cosine plus i sine"). The formula is still valid if x is a complex number, and so some authors refer to the more

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repetition number in this series | 0 | memorised on | scheduled repetition | ||||

scheduled repetition interval | last repetition or drill |

Euler's formula states that with the argument x given in radians.

s formula half-lives exponential growth and decay Defining e proof that e is irrational representations of e Lindemann–Weierstrass theorem People John Napier Leonhard Euler Related topics Schanuel's conjecture v t e <span>Euler's formula, named after Leonhard Euler, is a mathematical formula in complex analysis that establishes the fundamental relationship between the trigonometric functions and the complex exponential function. Euler's formula states that for any real number x e i x = cos x + i sin x , {\displaystyle e^{ix}=\cos x+i\sin x,} where e is the base of the natural logarithm, i is the imaginary unit, and cos and sin are the trigonometric functions cosine and sine respectively, with the argument x given in radians. This complex exponential function is sometimes denoted cis x ("cosine plus i sine"). The formula is still valid if x is a complex number, and so some authors refer to the more

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, who showed that representing a function as a sum of trigonometric functions greatly simplifies the study of heat transfer. Today, the subject of Fourier analysis encompasses a vast spectrum of mathematics. In the sciences and engineering, <span>the process of decomposing a function into oscillatory components is often called Fourier analysis, while the operation of rebuilding the function from these pieces is known as Fourier synthesis. For example, determining what component frequencies are present in a musical note would involve computing the Fourier transform of a sampled musical note. One could then re-synthesize t

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ions. The decomposition process itself is called a Fourier transformation. Its output, the Fourier transform, is often given a more specific name, which depends on the domain and other properties of the function being transformed. Moreover, <span>the original concept of Fourier analysis has been extended over time to apply to more and more abstract and general situations, and the general field is often known as harmonic analysis. Each transform used for analysis (see list of Fourier-related transforms) has a corresponding inverse transform that can be used for synthesis. Contents [hide] 1 Application

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the process of decomposing a function into oscillatory components is often called Fourier analysis, while the operation of rebuilding the function from these pieces is known as Fourier synthesis.

, who showed that representing a function as a sum of trigonometric functions greatly simplifies the study of heat transfer. Today, the subject of Fourier analysis encompasses a vast spectrum of mathematics. In the sciences and engineering, <span>the process of decomposing a function into oscillatory components is often called Fourier analysis, while the operation of rebuilding the function from these pieces is known as Fourier synthesis. For example, determining what component frequencies are present in a musical note would involve computing the Fourier transform of a sampled musical note. One could then re-synthesize t

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 |

the original concept of Fourier analysis has been extended to apply to more and more abstract and general situations, and is often known as harmonic analysis.

ions. The decomposition process itself is called a Fourier transformation. Its output, the Fourier transform, is often given a more specific name, which depends on the domain and other properties of the function being transformed. Moreover, <span>the original concept of Fourier analysis has been extended over time to apply to more and more abstract and general situations, and the general field is often known as harmonic analysis. Each transform used for analysis (see list of Fourier-related transforms) has a corresponding inverse transform that can be used for synthesis. Contents [hide] 1 Application

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her, are called a metric on the set. A metric on a space induces topological properties like open and closed sets, which lead to the study of more abstract topological spaces. The most familiar metric space is 3-dimensional Euclidean space. <span>In fact, a "metric" is the generalization of the Euclidean metric arising from the four long-known properties of the Euclidean distance. The Euclidean metric defines the distance between two points as the length of the straight line segment connecting them. Other metric spaces occur for example in elliptic geometry and h

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Parallel postulate - Wikipedia Parallel postulate From Wikipedia, the free encyclopedia Jump to: navigation, search [imagelink] If the sum of the interior angles α and β is less than 180°, the two straight lines, produced indefinitely, meet on that side. In geometry, the parallel postulate, also called Euclid's fifth postulate because it is the fifth postulate in Euclid's Elements, is a distinctive axiom in Euclidean geometry. It

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Euclid's parallel postulate: If the sum of the interior angles α and β is less than 180°, the two straight lines, produced indefinitely, meet on that side.

Parallel postulate - Wikipedia Parallel postulate From Wikipedia, the free encyclopedia Jump to: navigation, search [imagelink] If the sum of the interior angles α and β is less than 180°, the two straight lines, produced indefinitely, meet on that side. In geometry, the parallel postulate, also called Euclid's fifth postulate because it is the fifth postulate in Euclid's Elements, is a distinctive axiom in Euclidean geometry. It

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In fact, a "metric" is the generalization of the Euclidean metric arising from the four long-known properties of the Euclidean distance.

her, are called a metric on the set. A metric on a space induces topological properties like open and closed sets, which lead to the study of more abstract topological spaces. The most familiar metric space is 3-dimensional Euclidean space. <span>In fact, a "metric" is the generalization of the Euclidean metric arising from the four long-known properties of the Euclidean distance. The Euclidean metric defines the distance between two points as the length of the straight line segment connecting them. Other metric spaces occur for example in elliptic geometry and h

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Jyeṣṭhadeva Descartes Pascal Minggatu Euler Sakabe Aida 1700s–1900s Gauss Lobachevsky Bolyai Riemann Klein Poincaré Hilbert Minkowski Cartan Veblen Coxeter Present day Atiyah Gromov [imagelink] Geometry portal v t e <span>Elliptic geometry is a geometry in which Euclid's parallel postulate does not hold. Elliptic geometry is studied in two, three, or more dimensions. The appearance of this geometry in the nineteenth century stimulated the development of non-Euclidean geometry generally,

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Elliptic geometry is a geometry in which Euclid's parallel postulate does not hold.

Jyeṣṭhadeva Descartes Pascal Minggatu Euler Sakabe Aida 1700s–1900s Gauss Lobachevsky Bolyai Riemann Klein Poincaré Hilbert Minkowski Cartan Veblen Coxeter Present day Atiyah Gromov [imagelink] Geometry portal v t e <span>Elliptic geometry is a geometry in which Euclid's parallel postulate does not hold. Elliptic geometry is studied in two, three, or more dimensions. The appearance of this geometry in the nineteenth century stimulated the development of non-Euclidean geometry generally,

status | not learned | measured difficulty | 37% [default] | last interval [days] | |||
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Elliptic geometry is a geometry in which Euclid's parallel postulate does not hold.

Jyeṣṭhadeva Descartes Pascal Minggatu Euler Sakabe Aida 1700s–1900s Gauss Lobachevsky Bolyai Riemann Klein Poincaré Hilbert Minkowski Cartan Veblen Coxeter Present day Atiyah Gromov [imagelink] Geometry portal v t e <span>Elliptic geometry is a geometry in which Euclid's parallel postulate does not hold. Elliptic geometry is studied in two, three, or more dimensions. The appearance of this geometry in the nineteenth century stimulated the development of non-Euclidean geometry generally,

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In mathematics, a metric space is a set for which distances between all members of the set are defined. Those distances, taken together, are called a metric on the set. A metric on a space induces topological properties like open and closed sets, which lead to the study of more abstract topological spaces.

Metric space - Wikipedia Metric space From Wikipedia, the free encyclopedia Jump to: navigation, search In mathematics, a metric space is a set for which distances between all members of the set are defined. Those distances, taken together, are called a metric on the set. A metric on a space induces topological properties like open and closed sets, which lead to the study of more abstract topological spaces. The most familiar metric space is 3-dimensional Euclidean space. In fact, a "metric" is the generalization of the Euclidean metric arising from the four long-known propertie

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A metric on a space induces topological properties like open and closed sets, which lead to the study of more abstract topological spaces.

Metric space - Wikipedia Metric space From Wikipedia, the free encyclopedia Jump to: navigation, search In mathematics, a metric space is a set for which distances between all members of the set are defined. Those distances, taken together, are called a metric on the set. A metric on a space induces topological properties like open and closed sets, which lead to the study of more abstract topological spaces. The most familiar metric space is 3-dimensional Euclidean space. In fact, a "metric" is the generalization of the Euclidean metric arising from the four long-known propertie

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set can be open (called the discrete topology), or no set can be open but the space itself and the empty set (the indiscrete topology). In practice, however, open sets are usually chosen to be similar to the open intervals of the real line. <span>The notion of an open set provides a fundamental way to speak of nearness of points in a topological space, without explicitly having a concept of distance defined. Once a choice of open sets is made, the properties of continuity, connectedness, and compactness, which use notions of nearness, can be defined using these open sets. Each choice of o

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The notion of an open set provides a fundamental way to speak of nearness of points in a topological space, without explicitly having a concept of distance defined.

set can be open (called the discrete topology), or no set can be open but the space itself and the empty set (the indiscrete topology). In practice, however, open sets are usually chosen to be similar to the open intervals of the real line. <span>The notion of an open set provides a fundamental way to speak of nearness of points in a topological space, without explicitly having a concept of distance defined. Once a choice of open sets is made, the properties of continuity, connectedness, and compactness, which use notions of nearness, can be defined using these open sets. Each choice of o

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scheduled repetition interval | last repetition or drill |

The notion of an open set provides a fundamental way to speak of nearness of points in a topological space, without explicitly having a concept of distance defined.

set can be open (called the discrete topology), or no set can be open but the space itself and the empty set (the indiscrete topology). In practice, however, open sets are usually chosen to be similar to the open intervals of the real line. <span>The notion of an open set provides a fundamental way to speak of nearness of points in a topological space, without explicitly having a concept of distance defined. Once a choice of open sets is made, the properties of continuity, connectedness, and compactness, which use notions of nearness, can be defined using these open sets. Each choice of o

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scheduled repetition interval | last repetition or drill |

The notion of an open set provides a fundamental way to speak of nearness of points in a topological space, without explicitly having a concept of distance defined.

set can be open (called the discrete topology), or no set can be open but the space itself and the empty set (the indiscrete topology). In practice, however, open sets are usually chosen to be similar to the open intervals of the real line. <span>The notion of an open set provides a fundamental way to speak of nearness of points in a topological space, without explicitly having a concept of distance defined. Once a choice of open sets is made, the properties of continuity, connectedness, and compactness, which use notions of nearness, can be defined using these open sets. Each choice of o

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In the mathematical field of topology, a homeomorphism or topological isomorphism or bi continuous function is a continuous function between topological spaces that has a continuous inverse function. Homeomorphisms are the isomorphisms in the category of topological spaces—that is, they are the mappings that preserve all the topological properties of a given space. Two spaces with a

formation between a coffee mug and a donut (torus) illustrating that they are homeomorphic. But there need not be a continuous deformation for two spaces to be homeomorphic — only a continuous mapping with a continuous inverse function. <span>In the mathematical field of topology, a homeomorphism or topological isomorphism or bi continuous function is a continuous function between topological spaces that has a continuous inverse function. Homeomorphisms are the isomorphisms in the category of topological spaces—that is, they are the mappings that preserve all the topological properties of a given space. Two spaces with a homeomorphism between them are called homeomorphic, and from a topological viewpoint they are the same. The word homeomorphism comes from the Greek words ὅμοιος (homoios) = similar or same (French pareil) and μορφή (morphē) = shape, form, introduced to mathematics by Henri Poincaré in 189

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In topology, a homeomorphism is a continuous function between topological spaces that has a continuous inverse function.

formation between a coffee mug and a donut (torus) illustrating that they are homeomorphic. But there need not be a continuous deformation for two spaces to be homeomorphic — only a continuous mapping with a continuous inverse function. <span>In the mathematical field of topology, a homeomorphism or topological isomorphism or bi continuous function is a continuous function between topological spaces that has a continuous inverse function. Homeomorphisms are the isomorphisms in the category of topological spaces—that is, they are the mappings that preserve all the topological properties of a given space. Two spaces with a homeomorphism between them are called homeomorphic, and from a topological viewpoint they are the same. The word homeomorphism comes from the Greek words ὅμοιος (homoios) = similar or same (French pareil) and μορφή (morphē) = shape, form, introduced to mathematics by Henri Poincaré in 189

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The definition of a topological space relies only upon set theory and is the most general notion of a mathematical space that allows for the definition of concepts such as continuity, connectedness, and convergence.

, search In topology and related branches of mathematics, a topological space may be defined as a set of points, along with a set of neighbourhoods for each point, satisfying a set of axioms relating points and neighbourhoods. <span>The definition of a topological space relies only upon set theory and is the most general notion of a mathematical space that allows for the definition of concepts such as continuity, connectedness, and convergence. [1] Other spaces, such as manifolds and metric spaces, are specializations of topological spaces with extra structures or constraints. Being so general, topological spaces are a centra

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The definition of a topological space relies only upon set theory

, search In topology and related branches of mathematics, a topological space may be defined as a set of points, along with a set of neighbourhoods for each point, satisfying a set of axioms relating points and neighbourhoods. <span>The definition of a topological space relies only upon set theory and is the most general notion of a mathematical space that allows for the definition of concepts such as continuity, connectedness, and convergence. [1] Other spaces, such as manifolds and metric spaces, are specializations of topological spaces with extra structures or constraints. Being so general, topological spaces are a centra

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A proxy statement is a document containing the information the Securities and Exchange Commission (SEC) requires companies to provide to shareholders so shareholders can make informed decisions about matters that will be brought up at an annual or special stockholder meeting. Issues covered in a proxy statement can include proposals for new additions to the board of directors, information on directors' salaries, information on bonus and options plans for di

[imagelink] <span>What is a 'Proxy Statement' A proxy statement is a document containing the information the Securities and Exchange Commission (SEC) requires companies to provide to shareholders so shareholders can make informed decisions about matters that will be brought up at an annual or special stockholder meeting. Issues covered in a proxy statement can include proposals for new additions to the board of directors, information on directors' salaries, information on bonus and options plans for directors, and any declarations made by the company's management. BREAKING DOWN 'Proxy Statement' A proxy statement must be filed by a publicly traded company before shareholder meetings, and it discloses material

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ntaining the information the Securities and Exchange Commission (SEC) requires companies to provide to shareholders so shareholders can make informed decisions about matters that will be brought up at an annual or special stockholder meeting. <span>Issues covered in a proxy statement can include proposals for new additions to the board of directors, information on directors' salaries, information on bonus and options plans for directors, and any declarations made by the company's management. <span><body><html>

[imagelink] <span>What is a 'Proxy Statement' A proxy statement is a document containing the information the Securities and Exchange Commission (SEC) requires companies to provide to shareholders so shareholders can make informed decisions about matters that will be brought up at an annual or special stockholder meeting. Issues covered in a proxy statement can include proposals for new additions to the board of directors, information on directors' salaries, information on bonus and options plans for directors, and any declarations made by the company's management. BREAKING DOWN 'Proxy Statement' A proxy statement must be filed by a publicly traded company before shareholder meetings, and it discloses material

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Under IFRS , the income statement may be presented as a separate statement followed by a statement of comprehensive income that begins with the profit or loss from the income statement or as a section of a single statement of comprehensive income. US GAAP permit the same alternative presentation formats.2 This reading focuses on the income statement, but also discusses comprehensive income (profit or loss from the income statem

The income statement presents information on the financial results of a company’s business activities over a period of time. The income statement communicates how much revenue the company generated during a period and what costs it incurred in connection with generating that revenue. The basic equation underlying the income statement, ignoring gains and losses, is Revenue minus Expenses equals Net income. The income statement is also sometimes referred to as the “statement of operations,” “statement of earnings,” or “profit and loss (P&L) statement.” Under International Financial Reporting Standards (IFRS), the income statement may be presented as a separate statement followed by a statement of comprehensive income that begins with the profit or loss from the income statement or as a section of a single statement of comprehensive income.1 US generally accepted accounting principles (US GAAP) permit the same alternative presentation formats.2 This reading focuses on the income statement, but also discusses comprehensive income (profit or loss from the income statement plus other comprehensive income). Investment analysts intensely scrutinize companies’ income statements.3 Equity analysts are interested in them because equity markets often reward relatively high- or low-earnings growth companies with above-average or below-average valuations, respectively, and because inputs into valuation models often include estimates of earnings. Fixed-income analysts examine the components of income statements, past and projected, for information on companies’ abilities to make promised payments on their debt over the course of the business cycle. Corporate financial announcements frequently emphasize information reported in income statements, particularly earnings, more than information reported in the other financial statements. This reading is organized as follows: Section 2 describes the components of the income statement and its format. Section 3 describes basic principles and selected applicati

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IFRS , the income statement may be presented as a separate statement followed by a statement of comprehensive income that begins with the profit or loss from the income statement or as a section of a single statement of comprehensive income. <span>US GAAP permit the same alternative presentation formats.2 This reading focuses on the income statement, but also discusses comprehensive income (profit or loss from the income statement plus other comprehensive income). <span><body><html>

The income statement presents information on the financial results of a company’s business activities over a period of time. The income statement communicates how much revenue the company generated during a period and what costs it incurred in connection with generating that revenue. The basic equation underlying the income statement, ignoring gains and losses, is Revenue minus Expenses equals Net income. The income statement is also sometimes referred to as the “statement of operations,” “statement of earnings,” or “profit and loss (P&L) statement.” Under International Financial Reporting Standards (IFRS), the income statement may be presented as a separate statement followed by a statement of comprehensive income that begins with the profit or loss from the income statement or as a section of a single statement of comprehensive income.1 US generally accepted accounting principles (US GAAP) permit the same alternative presentation formats.2 This reading focuses on the income statement, but also discusses comprehensive income (profit or loss from the income statement plus other comprehensive income). Investment analysts intensely scrutinize companies’ income statements.3 Equity analysts are interested in them because equity markets often reward relatively high- or low-earnings growth companies with above-average or below-average valuations, respectively, and because inputs into valuation models often include estimates of earnings. Fixed-income analysts examine the components of income statements, past and projected, for information on companies’ abilities to make promised payments on their debt over the course of the business cycle. Corporate financial announcements frequently emphasize information reported in income statements, particularly earnings, more than information reported in the other financial statements. This reading is organized as follows: Section 2 describes the components of the income statement and its format. Section 3 describes basic principles and selected applicati

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