# on 27-Feb-2018 (Tue)

#### Flashcard 1729617333516

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#gaussian-process
Question
a stochastic process is called [...] if it depends only on distance but not the direction
isotropic

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

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

#### Flashcard 1729628343564

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#gaussian-process
Question
Gaussian processes can be completely defined by their [...].
second-order statistics

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A key fact of Gaussian processes is that they can be completely defined by their second-order statistics.

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Gaussian process - Wikipedia
μ ℓ {\displaystyle \mu _{\ell }} can be shown to be the covariances and means of the variables in the process. [3] Covariance functions[edit source] <span>A key fact of Gaussian processes is that they can be completely defined by their second-order statistics. [4] Thus, if a Gaussian process is assumed to have mean zero, defining the covariance function completely defines the process' behaviour. Importantly the non-negative definiteness of t

#### Flashcard 1729669762316

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#multivariate-normal-distribution
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The equidensity contours of a non-singular multivariate normal distribution are [...] centered at the mean.
ellipsoids

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The equidensity contours of a non-singular multivariate normal distribution are ellipsoids (i.e. linear transformations of hyperspheres) centered at the mean.

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Multivariate normal distribution - Wikipedia
implies that the variance of the dot product must be positive. An affine transformation of X such as 2X is not the same as the sum of two independent realisations of X. Geometric interpretation[edit source] See also: Confidence region <span>The equidensity contours of a non-singular multivariate normal distribution are ellipsoids (i.e. linear transformations of hyperspheres) centered at the mean. [17] Hence the multivariate normal distribution is an example of the class of elliptical distributions. The directions of the principal axes of the ellipsoids are given by the eigenvec

#### Flashcard 1729699122444

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#multivariate-normal-distribution
Question
The multivariate normal distribution is often used to describe correlated real-valued random variables each of which [...]
clusters around a mean value

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The multivariate normal distribution is often used to describe, at least approximately, any set of (possibly) correlated real-valued random variables each of which clusters around a mean value

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Multivariate normal distribution - Wikipedia
e definition is that a random vector is said to be k-variate normally distributed if every linear combination of its k components has a univariate normal distribution. Its importance derives mainly from the multivariate central limit theorem. <span>The multivariate normal distribution is often used to describe, at least approximately, any set of (possibly) correlated real-valued random variables each of which clusters around a mean value. Contents [hide] 1 Notation and parametrization 2 Definition 3 Properties 3.1 Density function 3.1.1 Non-degenerate case 3.1.2 Degenerate case 3.2 Higher moments 3.3 Lik

#### Flashcard 1731663629580

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#dynamic-programming
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[...] speeds up computer programs by storing the results of expensive function calls and returning the cached result when the same inputs occur again.
memoization

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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 a

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Memoization - Wikipedia
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

#### Flashcard 1731666775308

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#dynamic-programming
Question
memoization speeds up computer programs by storing [...] and returning the cached result when the same inputs occur again.
results of expensive function calls

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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.

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Memoization - Wikipedia
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

#### Flashcard 1731732835596

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#stochastics
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The Wiener process has [...] increments: for every the future increments , are independent of the past values ,

independent

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

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

#### Flashcard 1731745680652

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#gauss-markov-process
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Gauss–Markov stochastic processes satisfy the requirements for [...]

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Gauss–Markov stochastic processes (named after Carl Friedrich Gauss and Andrey Markov) are stochastic processes that satisfy the requirements for both Gaussian processes and Markov processes. [1] [2] The stationary Gauss–Markov process (also known as a Ornstein–Uhlenbeck process) is a very special case because it is unique, except for some trivial exceptions. </sp

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Gauss–Markov process - Wikipedia
translations!] Gauss–Markov process From Wikipedia, the free encyclopedia Jump to: navigation, search Not to be confused with the Gauss–Markov theorem of mathematical statistics. <span>Gauss–Markov stochastic processes (named after Carl Friedrich Gauss and Andrey Markov) are stochastic processes that satisfy the requirements for both Gaussian processes and Markov processes. [1] [2] The stationary Gauss–Markov process (also known as a Ornstein–Uhlenbeck process) is a very special case because it is unique, except for some trivial exceptions. Every Gauss–Markov process X(t) possesses the three following properties: If h(t) is a non-zero scalar function of t, then Z(t) = h(t)X(t) is also a Gauss–Markov process If f(t) is

#### Flashcard 1732483091724

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#differential-equations
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[...] deals with functions of a single variable and their derivatives.

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In mathematics, a partial differential equation (PDE) is a differential equation that contains unknown multivariable functions and their partial derivatives. (A special case is ordinary differential equations (ODEs), which deal with functions of a single variable and their derivatives.)

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Partial differential equation - Wikipedia
Wikipedia, the free encyclopedia Jump to: navigation, search [imagelink] A visualisation of a solution to the two-dimensional heat equation with temperature represented by the third dimension <span>In mathematics, a partial differential equation (PDE) is a differential equation that contains unknown multivariable functions and their partial derivatives. (A special case is ordinary differential equations (ODEs), which deal with functions of a single variable and their derivatives.) PDEs are used to formulate problems involving functions of several variables, and are either solved by hand, or used to create a relevant computer model. PDEs can be used to describe

#### Flashcard 1732608658700

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#linear-algebra #matrix-decomposition
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[...] has real eigenvalues and the eigenvectors can be chosen to be orthogonal

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Eigendecomposition of a matrix - Wikipedia
{\displaystyle A=A^{*}} ), which implies that it is also complex normal, the diagonal matrix Λ has only real values, and if A is unitary, Λ takes all its values on the complex unit circle. Real symmetric matrices[edit source] <span>As a special case, for every N×N real symmetric matrix, the eigenvalues are real and the eigenvectors can be chosen such that they are orthogonal to each other. Thus a real symmetric matrix A can be decomposed as A = Q Λ Q T {\displaystyle \mathbf {A} =\mathbf {Q} \mathbf {\Lambda } \mathbf {Q} ^{T}} where Q is an orthogonal matrix, and Λ is a diagonal matrix whose entries are the eigenvalues of A. Useful facts[edit source] Useful facts regarding eigenvalues[edit source] The product of the eigenvalues is equal to the determinant of A det

#### Flashcard 1732720069900

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the domain of definition of a function is [...] for which the function is defined.
the set of "input" values

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In mathematics, and more specifically in naive set theory, the domain of definition (or simply the domain) of a function is the set of "input" or argument values for which the function is defined.

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Domain of a function - Wikipedia
main (disambiguation). [imagelink] Illustration showing f, a function from the pink domain X to the blue codomain Y. The yellow oval inside Y is the image of f. Both the image and the codomain are sometimes called the range of f. <span>In mathematics, and more specifically in naive set theory, the domain of definition (or simply the domain) of a function is the set of "input" or argument values for which the function is defined. That is, the function provides an "output" or value for each member of the domain. [1] Conversely, the set of values the function takes on as output is termed the image of th

#### Flashcard 1733059022092

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#kalman-filter
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Kalman filtering estimats a [...] over the variables for each timeframe.

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hat uses a series of measurements observed over time, containing statistical noise and other inaccuracies, and produces estimates of unknown variables that tend to be more accurate than those based on a single measurement alone, by estimating <span>a joint probability distribution over the variables for each timeframe. <span><body><html>

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Kalman filter - Wikipedia
into account; P k ∣ k − 1 {\displaystyle P_{k\mid k-1}} is the corresponding uncertainty. <span>Kalman filtering, also known as linear quadratic estimation (LQE), is an algorithm that uses a series of measurements observed over time, containing statistical noise and other inaccuracies, and produces estimates of unknown variables that tend to be more accurate than those based on a single measurement alone, by estimating a joint probability distribution over the variables for each timeframe. The filter is named after Rudolf E. Kálmán, one of the primary developers of its theory. The Kalman filter has numerous applications in technology. A common application is for guidanc

#### Flashcard 1735993986316

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#stochastics
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Random walks are usually defined as [...] of iid random variables or random vectors in Euclidean space
sums

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Random walks are stochastic processes that are usually defined as sums of iid random variables or random vectors in Euclidean space, so they are processes that change in discrete time.

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Stochastic process - Wikipedia
one, while the value of a tail is zero. [61] In other words, a Bernoulli process is a sequence of iid Bernoulli random variables, [62] where each coin flip is a Bernoulli trial. [63] Random walk[edit source] Main article: Random walk <span>Random walks are stochastic processes that are usually defined as sums of iid random variables or random vectors in Euclidean space, so they are processes that change in discrete time. [64] [65] [66] [67] [68] But some also use the term to refer to processes that change in continuous time, [69] particularly the Wiener process used in finance, which has led to some c

#### Flashcard 1735995559180

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#stochastics
Question
Random walks are usually defined as sums of [...] in Euclidean space
iid random variables or random vectors

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Random walks are stochastic processes that are usually defined as sums of iid random variables or random vectors in Euclidean space, so they are processes that change in discrete time.

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Stochastic process - Wikipedia
one, while the value of a tail is zero. [61] In other words, a Bernoulli process is a sequence of iid Bernoulli random variables, [62] where each coin flip is a Bernoulli trial. [63] Random walk[edit source] Main article: Random walk <span>Random walks are stochastic processes that are usually defined as sums of iid random variables or random vectors in Euclidean space, so they are processes that change in discrete time. [64] [65] [66] [67] [68] But some also use the term to refer to processes that change in continuous time, [69] particularly the Wiener process used in finance, which has led to some c

#### Flashcard 1735997132044

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#stochastics
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the simple random walk has [...] as the state space
the integers

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A classic example of a random walk is known as the simple random walk, which is a stochastic process in discrete time with the integers as the state space, and is based on a Bernoulli process, where each iid Bernoulli variable takes either the value positive one or negative one.

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Stochastic process - Wikipedia
ere are other various types of random walks, defined so their state spaces can be other mathematical objects, such as lattices and groups, and in general they are highly studied and have many applications in different disciplines. [69] [71] <span>A classic example of a random walk is known as the simple random walk, which is a stochastic process in discrete time with the integers as the state space, and is based on a Bernoulli process, where each iid Bernoulli variable takes either the value positive one or negative one. In other words, the simple random walk takes place on the integers, and its value increases by one with probability, say, p {\displaystyle p}

#### Flashcard 1736014171404

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#stochastics
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Almost surely, a [...] of a Wiener process is continuous everywhere but nowhere differentiable.
sample path

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Almost surely, a sample path of a Wiener process is continuous everywhere but nowhere differentiable. It can be considered a continuous version of the simple random walk.

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Stochastic process - Wikipedia
stant μ {\displaystyle \mu } , which is a real number, then the resulting stochastic process is said to have drift μ {\displaystyle \mu } . [84] [85] [86] <span>Almost surely, a sample path of a Wiener process is continuous everywhere but nowhere differentiable. It can be considered a continuous version of the simple random walk. [49] [85] The process arises as the mathematical limit of other stochastic processes such as certain random walks rescaled, [87] [88] which is the subject of Donsker's theorem or inva

#### Flashcard 1736015744268

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#stochastics
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Almost surely, a sample path of a Wiener process is [...property...].
continuous everywhere but nowhere differentiable

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Almost surely, a sample path of a Wiener process is continuous everywhere but nowhere differentiable. It can be considered a continuous version of the simple random walk.

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Stochastic process - Wikipedia
stant μ {\displaystyle \mu } , which is a real number, then the resulting stochastic process is said to have drift μ {\displaystyle \mu } . [84] [85] [86] <span>Almost surely, a sample path of a Wiener process is continuous everywhere but nowhere differentiable. It can be considered a continuous version of the simple random walk. [49] [85] The process arises as the mathematical limit of other stochastic processes such as certain random walks rescaled, [87] [88] which is the subject of Donsker's theorem or inva

#### Flashcard 1736192953612

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#abstract-algebra
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an [...] is a set with one or more operations defined on it that satisfies a list of axioms.
algebraic structure

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In mathematics, and more specifically in abstract algebra, an algebraic structure is a set (called carrier set or underlying set) with one or more operations defined on it that satisfies a list of axioms.

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Algebraic structure - Wikipedia
Module-like[show] Module Group with operators Vector space Linear algebra Algebra-like[show] Algebra Associative Non-associative Composition algebra Lie algebra Graded Bialgebra v t e <span>In mathematics, and more specifically in abstract algebra, an algebraic structure is a set (called carrier set or underlying set) with one or more operations defined on it that satisfies a list of axioms. [1] Examples of algebraic structures include groups, rings, fields, and lattices. More complex structures can be defined by introducing multiple operations, different underlying sets,

#### Flashcard 1736195050764

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#abstract-algebra
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The underlying set for an algebraic structure is called a [...]
carrier set

Think sigma-algebra and measurable space

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In mathematics, and more specifically in abstract algebra, an algebraic structure is a set (called carrier set or underlying set) with one or more operations defined on it that satisfies a list of axioms.

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Algebraic structure - Wikipedia
Module-like[show] Module Group with operators Vector space Linear algebra Algebra-like[show] Algebra Associative Non-associative Composition algebra Lie algebra Graded Bialgebra v t e <span>In mathematics, and more specifically in abstract algebra, an algebraic structure is a set (called carrier set or underlying set) with one or more operations defined on it that satisfies a list of axioms. [1] Examples of algebraic structures include groups, rings, fields, and lattices. More complex structures can be defined by introducing multiple operations, different underlying sets,

#### Flashcard 1736196885772

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#abstract-algebra
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an algebraic structure is a set with one or more [...] defined on it that satisfies a list of axioms.

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In mathematics, and more specifically in abstract algebra, an algebraic structure is a set (called carrier set or underlying set) with one or more operations defined on it that satisfies a list of axioms.

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Algebraic structure - Wikipedia
Module-like[show] Module Group with operators Vector space Linear algebra Algebra-like[show] Algebra Associative Non-associative Composition algebra Lie algebra Graded Bialgebra v t e <span>In mathematics, and more specifically in abstract algebra, an algebraic structure is a set (called carrier set or underlying set) with one or more operations defined on it that satisfies a list of axioms. [1] Examples of algebraic structures include groups, rings, fields, and lattices. More complex structures can be defined by introducing multiple operations, different underlying sets,

#### Flashcard 1736198458636

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#abstract-algebra
Question
an algebraic structure is a set with one or more operations defined on it that satisfies [...].
a list of axioms

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tml> In mathematics, and more specifically in abstract algebra, an algebraic structure is a set (called carrier set or underlying set) with one or more operations defined on it that satisfies a list of axioms. <html>

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Algebraic structure - Wikipedia
Module-like[show] Module Group with operators Vector space Linear algebra Algebra-like[show] Algebra Associative Non-associative Composition algebra Lie algebra Graded Bialgebra v t e <span>In mathematics, and more specifically in abstract algebra, an algebraic structure is a set (called carrier set or underlying set) with one or more operations defined on it that satisfies a list of axioms. [1] Examples of algebraic structures include groups, rings, fields, and lattices. More complex structures can be defined by introducing multiple operations, different underlying sets,

#### Flashcard 1737450720524

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#linear-state-space-models
Question

In the linear state-space system
\begin{aligned} x_{t+1} & = A x_t + C w_{t+1} \\ y_t & = G x_t \nonumber \\ x_0 & \sim N(\mu_0, \Sigma_0) \nonumber \end{aligned}

C is called the [...]

volatility matrix

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An n×1 vector xt denoting the state at time t=0,1,2,… An iid sequence of m×1 random vectors wt∼N(0,I) A k×1 vector yt of observations at time t=0,1,2,… An n×n matrix A called the transition matrix An n×m matrix C called the <span>volatility matrix A k×n matrix G sometimes called the output matrix Here is the linear state-space system xt+1ytx0=Axt+Cwt+1=Gxt∼N(μ0,Σ0) . .

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Linear State Space Models – Quantitative Economics

#### Flashcard 1737454914828

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#linear-state-space-models
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the [...] distribution is specialized to $$N(0,I)$$
shock

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The primitives of the model are the matrices A , C , G A,C,G A, C, G shock distribution, which we have specialized to N ( 0 , I ) N(0,I) N(0,I) the distribution of the initial condition x 0 x0 x_0 , which we have set to N ( μ 0 , Σ 0 )

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Linear State Space Models – Quantitative Economics

#### Flashcard 1738569813260

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#measure-theory #stochastics
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In [...theory...] every distribution function corresponds to a probability distribution
measure-theoretic probability theory

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Also the notion that every DF corresponds to a probability distribution (which comes from measure-theoretic probability theory) allows much more bizarre distributions than master’s level theory can handle

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cannot see any pdfs

#### Flashcard 1741100027148

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

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

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

#### Flashcard 1744159247628

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#lebesgue-integration
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Riemann integral considers the area under a curve as made out of [...shape...]
vertical rectangles

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While the Riemann integral considers the area under a curve as made out of vertical rectangles, the Lebesgue definition considers horizontal slabs that are not necessarily just rectangles, and so it is more flexible.

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Lebesgue integration - Wikipedia
es, Fourier transforms, and other topics. The Lebesgue integral is better able to describe how and when it is possible to take limits under the integral sign (via the powerful monotone convergence theorem and dominated convergence theorem). <span>While the Riemann integral considers the area under a curve as made out of vertical rectangles, the Lebesgue definition considers horizontal slabs that are not necessarily just rectangles, and so it is more flexible. For this reason, the Lebesgue definition makes it possible to calculate integrals for a broader class of functions. For example, the Dirichlet function, which is 0 where its argument is

#### Flashcard 1744162393356

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#lebesgue-integration
Question
it is actually impossible to assign a length to [...] in a way that preserves some natural additivity and translation invariance properties.
all subsets of ℝ

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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

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Lebesgue integration - Wikipedia
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

#### Flashcard 1752715889932

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#banach-space
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A normed space has a metric that allows the computation of [...]
vector length and distance between vectors

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A normed space has a metric that allows the computation of vector length and distance between vectors

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Banach space - Wikipedia
Banach space - Wikipedia Banach space From Wikipedia, the free encyclopedia Jump to: navigation, search In mathematics, more specifically in functional analysis, a Banach space (pronounced [ˈbanax]) is a complete normed vector space. Thus, a Banach space is a vector space with a metric that allows the computation of vector length and distance between vectors and is complete in the sense that a Cauchy sequence of vectors always converges to a well defined limit that is within the space. Banach spaces are named after the Polish mathematician Stefan Banach, who introduced this concept and studied it systematically in 1920–1922 along with Hans Hahn and Eduard Helly. [1]

#### Flashcard 1753276353804

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With incremental reading, you ensure high-retention of [...]
the most important pieces of text

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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.

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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

#### Flashcard 1753298636044

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#fourier-analysis
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Fourier analysis has been extended to [...].
harmonic analysis

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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.

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Fourier analysis - Wikipedia
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

#### Flashcard 1759676075276

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#spectral-analysis
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an eigenfunction of a linear operator D defined on some function space is any non-zero function f in that space that satisfies [...]

for some scalar eigenvalue λ.

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In 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.

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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 1759715134732

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#hilbert-space
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Linear operators on a Hilbert space are simply transformations that [...]
stretch the space by different factors in mutually perpendicular directions

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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.

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Hilbert space - Wikipedia
is 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

#### Annotation 1766818712844

 #cormen_2009_introductiontoalgorithms #datastructuresandalgorithms In general, an instance of a problem consists of the input (satisfying whatever constraints are imposed in the problem statement) needed to compute a solution to the problem

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#### Flashcard 1766820285708

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#_definition #cormen_2009_introductiontoalgorithms #datastructuresandalgorithms
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In general, an [...] consists of the input (satisfying whatever constraints are imposed in the problem statement) needed to compute a solution to the problem
instance of a problem

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In general, an instance of a problem consists of the input (satisfying whatever constraints are imposed in the problem statement) needed to compute a solution to the problem

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#### Flashcard 1766822907148

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#_definition #cormen_2009_introductiontoalgorithms #datastructuresandalgorithms
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In general, an instance of a problem consists of [...]
the input (satisfying whatever constraints are imposed in the problem statement) needed to compute a solution to the problem

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In general, an instance of a problem consists of the input (satisfying whatever constraints are imposed in the problem statement) needed to compute a solution to the problem

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#### Annotation 1766840208652

 You did the quick analysis and determined that the property looks good. The next step is to verify the numbers on both the income and expense side.

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#### Annotation 1766841781516

 Verify income by requesting the last two years of profit-and-loss statements, the year-to-date profit-and loss-statement, and the current rent roll. That’s a list of which tenants live in which units and how much they are paying in rent.

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#### Annotation 1766843354380

 I must be missing some- thing, because I can’t seem to reconcile the income statement with the rent roll.

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#### Annotation 1766844927244

 The last two years of profit-and-loss statements will tell you the fi- nancial story of a property. This story will emerge from the net income numbers (that is, revenue minus expenses).

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#### Annotation 1766846500108

 it will most likely arrive in the form of a trend report.In other words, it will show the last 24 months of profit and loss, all nicely laid out in front of you. Simply look at the total income line and see what it’s been doing for the last 24 months. Has it been stable? Was it a steady riser? Did it experience a period of instability? Do the same with the total expense line. You’ll soon get a good idea of the history of the property.

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#### Annotation 1766848072972

 I like to get these numbers before I make an offer. That way I really know what I’m dealing with.

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#### Annotation 1766850694412

 we want to base our offer on actual results.

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#### Annotation 1766852267276

 If the property is not running properly, get rid of the bad apples, but do keep an eye out for good employees in a bad situation

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#### Annotation 1766853840140

 get a good qualified management company to run the property for you.

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#### Annotation 1766855413004

 Have they been collecting all the rents that are due, and have they kept occupancy high? Those are the top responsibilities of a manager. If either answer is no, then dump ’em.

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#### Annotation 1766856985868

 while you’re doing your due diligence on the property, you should have a say in who will be the staff. It will be your property, after all.

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#### Annotation 1766858558732

 The main reason you should get rid of most staff on a suffering property is because once bad habits are formed, they are very hard to break. It’s much easier on both you and the tenants if you start fresh.

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#### Annotation 1766860131596

 Ask to see copies of work orders. How long did it take to complete them? The answer should be between 24 and 48 hours. Ask how long it takes to turn a vacant unit around and get it ready for the next tenant (that is, a make ready). The answer should be three days or less.

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#### Annotation 1766863015180

 #incremental-reading The key to creativity is an association of remote ideas.

ther. With incremental reading, there is virtually no limit on how many articles you can study at the same time. Only the availability of time and your memory capacity will keep massive learning in check Creativity (the association bonus) <span>The key to creativity is an association of remote ideas. By studying multiple subjects in unpredictable order, you will increase your power to associate ideas. This will immensely improve your creativity. Incremental reading may be compared t

#### Annotation 1766865112332

 #incremental-reading All written materials, depending on the reader's knowledge, pose a degree of difficulty in accurately interpreting their contents.

s. This will immensely improve your creativity. Incremental reading may be compared to brainstorming with yourself Understanding (the slot-in factor) One of the limiting factors in acquiring new knowledge is the barrier of understanding. <span>All written materials, depending on the reader's knowledge, pose a degree of difficulty in accurately interpreting their contents. This is particularly visible in highly specialist scientific papers that use a sophisticated symbol-rich language. A symbol-rich language is a language that gains conciseness by the use

#### Annotation 1766866685196

 #incremental-reading For an average reader, symbol-rich language may exponentially raise the bar of lexical competence

s is particularly visible in highly specialist scientific papers that use a sophisticated symbol-rich language. A symbol-rich language is a language that gains conciseness by the use of highly specialist vocabulary and notational conventions. <span>For an average reader, symbol-rich language may exponentially raise the bar of lexical competence (i.e. knowledge of vocabulary required to gain understanding). Incremental reading makes it possible to delay the processing of those articles, paragraphs or sentences that require prio

#### Annotation 1766868520204

 #incremental-reading Incremental reading stochastically juxtaposes pieces of information coming from various sources and uses the associative qualities of human memory to emphasize and then resolve contradiction

tantly faced with a chaos of disparate and often contradictory statements. Incremental reading makes it possible to resolve contradictions and build harmonious models of knowledge on the basis of the information chaos drawn from the Internet. <span>Incremental reading stochastically juxtaposes pieces of information coming from various sources and uses the associative qualities of human memory to emphasize and then resolve contradiction Stresslessness The information era tends to overwhelm us with the amount of information we feel compelled to process. Incremental reading does not require all-or-nothing choices on

#### Annotation 1766870093068

reading 3% of an article may provide 50% of its reading value. Reading of articles may be delayed transparently, i.e. not by stressful procrastination but by a sheer competition with other pieces of information on the basis of their priority. <span>In incremental reading, instead of hesitating or procrastinating, you simply prioritize Attention Incremental reading widely stretches the span of your attention. You will notice that a single paragraph in an article may greatly reduce your enthusiasm for reading. If

#### Annotation 1766871665932

 #incremental-reading This slow process of jelling out knowledge provides you with an enhanced understanding and applicability of individual pieces of information.

ave already been established in a favorable context (i.e. context that makes remembering easier). This comes from the need to extract a given piece of information from a larger body of knowledge that provides your items with relevant context. <span>This slow process of jelling out knowledge provides you with an enhanced sense of meaning and applicability of individual pieces of information. In addition, semantically equivalent pieces of information may be consolidated in varying contexts adding additional angles to their associative power. In other words, not only will you

#### Flashcard 1766909676812

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#calculus
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The tangent line is a limit of secant lines just as the derivative is a limit of [...].
difference quotients

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The tangent line is a limit of secant lines just as the derivative is a limit of difference quotients.

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Calculus - Wikipedia
h . {\displaystyle \lim _{h\to 0}{f(a+h)-f(a) \over {h}}.} Geometrically, the derivative is the slope of the tangent line to the graph of f at a. <span>The tangent line is a limit of secant lines just as the derivative is a limit of difference quotients. For this reason, the derivative is sometimes called the slope of the function f. Here is a particular example, the derivative of the squaring function at the input 3. Let f(x) = x 2

#### Flashcard 1766921473292

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#stochastics
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Random walks change in [...] time.
discrete

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Random walks are stochastic processes that are usually defined as sums of iid random variables or random vectors in Euclidean space, so they are processes that change in discrete time.

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Stochastic process - Wikipedia
one, while the value of a tail is zero. [61] In other words, a Bernoulli process is a sequence of iid Bernoulli random variables, [62] where each coin flip is a Bernoulli trial. [63] Random walk[edit source] Main article: Random walk <span>Random walks are stochastic processes that are usually defined as sums of iid random variables or random vectors in Euclidean space, so they are processes that change in discrete time. [64] [65] [66] [67] [68] But some also use the term to refer to processes that change in continuous time, [69] particularly the Wiener process used in finance, which has led to some c

#### Flashcard 1766930124044

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#linear-algebra #matrix-decomposition
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The eigendecomposition of a real symmetric matrix can be represented as [...]

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Eigendecomposition of a matrix - Wikipedia
{\displaystyle A=A^{*}} ), which implies that it is also complex normal, the diagonal matrix Λ has only real values, and if A is unitary, Λ takes all its values on the complex unit circle. Real symmetric matrices[edit source] <span>As a special case, for every N×N real symmetric matrix, the eigenvalues are real and the eigenvectors can be chosen such that they are orthogonal to each other. Thus a real symmetric matrix A can be decomposed as A = Q Λ Q T {\displaystyle \mathbf {A} =\mathbf {Q} \mathbf {\Lambda } \mathbf {Q} ^{T}} where Q is an orthogonal matrix, and Λ is a diagonal matrix whose entries are the eigenvalues of A. Useful facts[edit source] Useful facts regarding eigenvalues[edit source] The product of the eigenvalues is equal to the determinant of A det

#### Flashcard 1766939823372

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The key to creativity is an [...].
association of remote ideas

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The key to creativity is an association of remote ideas.

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ther. With incremental reading, there is virtually no limit on how many articles you can study at the same time. Only the availability of time and your memory capacity will keep massive learning in check Creativity (the association bonus) <span>The key to creativity is an association of remote ideas. By studying multiple subjects in unpredictable order, you will increase your power to associate ideas. This will immensely improve your creativity. Incremental reading may be compared t

#### Flashcard 1766941396236

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All written materials, depending on the reader's knowledge, pose a degree of difficulty in accurately [...].
interpreting their contents

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All written materials, depending on the reader's knowledge, pose a degree of difficulty in accurately interpreting their contents.

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s. This will immensely improve your creativity. Incremental reading may be compared to brainstorming with yourself Understanding (the slot-in factor) One of the limiting factors in acquiring new knowledge is the barrier of understanding. <span>All written materials, depending on the reader's knowledge, pose a degree of difficulty in accurately interpreting their contents. This is particularly visible in highly specialist scientific papers that use a sophisticated symbol-rich language. A symbol-rich language is a language that gains conciseness by the use

#### Flashcard 1766942969100

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All written materials, depending on [...], pose a degree of difficulty in accurately interpreting their contents.

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All written materials, depending on the reader's knowledge, pose a degree of difficulty in accurately interpreting their contents.

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s. This will immensely improve your creativity. Incremental reading may be compared to brainstorming with yourself Understanding (the slot-in factor) One of the limiting factors in acquiring new knowledge is the barrier of understanding. <span>All written materials, depending on the reader's knowledge, pose a degree of difficulty in accurately interpreting their contents. This is particularly visible in highly specialist scientific papers that use a sophisticated symbol-rich language. A symbol-rich language is a language that gains conciseness by the use

#### Flashcard 1766944541964

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hesitate or procrastinate

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reading 3% of an article may provide 50% of its reading value. Reading of articles may be delayed transparently, i.e. not by stressful procrastination but by a sheer competition with other pieces of information on the basis of their priority. <span>In incremental reading, instead of hesitating or procrastinating, you simply prioritize Attention Incremental reading widely stretches the span of your attention. You will notice that a single paragraph in an article may greatly reduce your enthusiasm for reading. If

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reading 3% of an article may provide 50% of its reading value. Reading of articles may be delayed transparently, i.e. not by stressful procrastination but by a sheer competition with other pieces of information on the basis of their priority. <span>In incremental reading, instead of hesitating or procrastinating, you simply prioritize Attention Incremental reading widely stretches the span of your attention. You will notice that a single paragraph in an article may greatly reduce your enthusiasm for reading. If

#### Flashcard 1766950046988

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This slow process of jelling out knowledge provides you with an enhanced [...] of individual pieces of information.
understanding and applicability

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This slow process of jelling out knowledge provides you with an enhanced understanding and applicability of individual pieces of information.

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ave already been established in a favorable context (i.e. context that makes remembering easier). This comes from the need to extract a given piece of information from a larger body of knowledge that provides your items with relevant context. <span>This slow process of jelling out knowledge provides you with an enhanced sense of meaning and applicability of individual pieces of information. In addition, semantically equivalent pieces of information may be consolidated in varying contexts adding additional angles to their associative power. In other words, not only will you

#### Annotation 1766961057036

 #has-images 极简唐朝史！ 梁萧 发布于 4小时前分类：文摘 未经允许不得转载：博海拾贝 » 极简唐朝史！

#### Annotation 1766978620684

 #Biochemistry For example, three- to seven- membered rings of hydrogen-bonded molecules commonly occur in liquid water (Fig. 2-4), in contrast to the six-membered rings characteristic of ice

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#### Annotation 1766980979980

 #Biochemistry Interactions among permanent dipoles such as carbonyl groups (Fig. 2-5a) are much weaker than ionic interactions. A permanent dipole also induces a dipole moment in a neighboring group by electrostatically distort- ing its electron distribution (Fig. 2-5b). Such dipole–induced dipole interac- tions are generally much weaker than dipole–dipole interactions. At any instant, nonpolar molecules have a small, randomly oriented di- pole moment resulting from the rapid fluctuating motion of their electrons. This transient dipole moment can polarize the electrons in a neighboring group (Fig. 2-5c), so that the groups are attracted to each other. These so- called London dispersion forces are extremely weak and fall off so rapidly with distance that they are significant only for groups in close contact

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#### Annotation 1766983339276

 #Biochemistry The strength of association of ionic groups of opposite charge depends on the chemical nature of the ions, the distance between them, and the polarity of the medium. In general, the strength of the interaction between two charged groups (i.e., the energy required to completely separate them in the medium of interest) is less than the energy of a covalent bond but greater than the en- ergy of a hydrogen bond (Table 2-1). The noncovalent associations between neutral molecules, collectively known as van der Waals forces, arise from electrostatic interactions among

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#### Annotation 1766985698572

 #Biochemistry Why do salts such as NaCl dissolve in water? Polar solvents, such as wa- ter, weaken the attractive forces between oppositely charged ions (such as Na ⫹ and Cl ⫺ ) and can therefore hold the ions apart. (In nonpolar solvents, ions of opposite charge attract each other so strongly that they coalesce to form a solid salt.) An ion immersed in a polar solvent such as water attracts the op- positely charged ends of the solvent dipoles (Fig. 2-6). The ion is thereby sur- rounded by one or more concentric shells of oriented solvent molecules. Such ions are said to be solvated or, when water is the solvent, to be hydrated.

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#### Annotation 1766988057868

 #Biochemistry The aggregation of the non- polar groups thereby minimizes the surface area of the cavity and therefore max- imizes the entropy of the entire system.

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#### Annotation 1766990417164

 #Biochemistry Despite the temptation to attribute some mutual attraction to a collection of nonpolar groups excluded from water, their exclusion is largely a function of the entropy of the surround- ing water molecules, not some “hydrophobic force” among them (the London dispersion forces between the nonpolar groups are relatively weak).

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#### Annotation 1766992776460

 #Biochemistry Osmotic pressure also depends on the solute concentration. When a solu- tion is separated from pure water by a semipermeable membrane that permits the passage of water molecules but not solutes, water tends to move into the solution in order to equalize its concentration on both sides of the membrane. Osmosis is the movement of solvent across the membrane from a region of high concentration (here, pure water) to a region of relatively low concentra- tion (water containing dissolved solute). The osmotic pressure of a solution is the pressure that must be applied to the solution to prevent the inward flow of water; it is proportional to the concentration of the solute (Fig. 2-13). For a 1 M solution, the osmotic pressure is 22.4 atm.

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#### Annotation 1766995922188

 #Biochemistry Diffusion of solutes is the basis for the laboratory technique of dialysis. In this process, solutes smaller than the pore size of the dialysis membrane freely exchange between the sample and the bulk solution until equilibrium is reached (Fig. 2-14). Larger substances cannot cross the membrane and remain where they are. Dialysis is particularly useful for separating large molecules, such as proteins or nucleic acids, from smaller molecules.

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#### Annotation 1766998281484

 #Biochemistry Individuals with kidney failure can undergo a dialysis procedure in which the blood is pumped through a machine containing a semipermeable mem- brane. As blood flows along one side of the membrane, a fluid called the dialysate flows in the opposite direction on the other side. This countercur- rent arrangement maximizes the concentration differences between the two so- lutions so that waste materials such as urea and creatinine (present at high concentration in the blood) will efficiently diffuse through the membrane into the dialysate (where their concentrations are low). Excess water can also be eliminated, as it moves into the dialysate by osmosis. The “cleansed” blood is then returned to the patient.

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#### Annotation 1767000640780

 #Biochemistry There is actually no such thing as a free proton (H ⫹ ) in solution. Rather, the proton is associated with a water molecule as a hydronium ion, H 3 O ⫹

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#### Annotation 1767003000076

 #Biochemistry The proton of a hydronium ion can jump rapidly to another water mol- ecule and then to another (Fig. 2-15). For this reason, the mobilities of H ⫹ and OH ⫺ ions in solution are much higher than for other ions, which must move through the bulk water carrying their waters of hydration. Proton jumping is also responsible for the observation that acid–base reactions are among the fastest reactions that take place in aqueous solution

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#### Annotation 1767005359372

 #Biochemistry The ionization (dissociation) of water is described by an equilibrium ex- pression in which the concentration of the parent substance is in the denom- inator and the concentrations of the dissociated products are in the numerator: [2-1] K is the dissociation constant (here and throughout the text, quantities in square brackets symbolize the molar concentrations of the indicated sub- stances, which in many cases are only negligibly different from their activities; Section 1-3D). Because the concentration of the undissociated H 2 O ([H 2 O]) is so much larger than the concentrations of its component ions, it can be considered constant and incorporated into K to yield an expression for the ionization of water, [2-2] The value of K w , the ionization constant of water, is 10 ⫺14 at 25°C

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#### Annotation 1767007718668

 #Biochemistry Pure water must contain equimolar amounts of H ⫹ and OH ⫺ , so [H ⫹ ] ⫽ [OH ⫺ ] ⫽ (K w ) 1兾2 ⫽ 10 ⫺7 M. Since [H ⫹ ] and [OH ⫺ ] are reciprocally related by Eq. 2-2, when [H ⫹ ] is greater than 10 ⫺7 M, [OH ⫺ ] must be cor- respondingly less and vice versa. Solutions with [H ⫹ ] ⫽ 10 ⫺7 M are said to be neutral, those with [H ⫹ ] ⬎ 10 ⫺7 M are said to be acidic, and those with [H ⫹ ] ⬍ 10 ⫺7 M are said to be basic. Most physiological solutions have hydrogen ion concentrations near neutrality. For example, human blood is normally slightly basic with [H ⫹ ] ⫽ 4.0 ⫻ 10 ⫺8 M.

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#### Annotation 1767010077964

 #Biochemistry The values of [H ⫹ ] for most solutions are inconveniently small and thus impractical to compare. A more practical quantity, which was devised in 1909 by Søren Sørenson, is known as the pH: [2-3] The higher the pH, the lower is the H ⫹ concentration; the lower the pH, the higher is the H ⫹ concentration

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#### Annotation 1767012437260

 #Biochemistry Note that solutions that differ by one pH unit differ in [H ⫹ ] by a factor of 10.

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#### Annotation 1767014796556

 #Biochemistry According to a definition formulated in 1923 by Johannes Brønsted and Thomas Lowry, an acid is a substance that can do- nate a proton, and a base is a substance that can accept a proton. Under the Brønsted–Lowry definition, an acid–base reaction can be written as An acid (HA) reacts with a base (H 2 O) to form the conjugate base of the acid (A ⫺ ) and the conjugate acid of the base (H 3 O ⫹ ). Accordingly, the ac- etate ion (CH 3 COO ⫺ ) is the conjugate base of acetic acid (CH 3 COOH), and the ammonium ion is the conjugate acid of ammonia (NH 3 ). The acid–base reaction is frequently abbreviated HA Δ H ⫹ ⫹ A ⫺

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#### Annotation 1767017155852

 #Biochemistry The Strength of an Acid Is Specified by Its Dissociation Constant. The equilibrium constant for an acid–base reaction is expressed as a dissociation constant with the concentrations of the “reactants” in the denominator and the concentrations of the “products” in the numerator: [2-4] In dilute solutions, the water concentration is essentially constant, 55.5 M (1000 g ⴢ L ⫺1 兾18.015 g ⴢ mol ⫺1 ⫽ 55.5 M). Therefore, the term [H 2 O] is custom- arily combined with the dissociation constant, which then takes the form [2-5] For brevity, however, we will henceforth omit the subscript “a.” The dissociation constants of some common acids are listed in Table 2-4. Because acid dissociation constants, like [H ⫹ ] values, can be cumbersome to work with, they are transformed to pK values by the formula [2-6] which is analogous to Eq. 2-3. pK ⫽⫺log K K a ⫽ K 3H 2 O4⫽ 3H ⫹ 43A ⫺ 4 3HA4 K ⫽ 3H 3 O ⫹ 43A ⫺ 4

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#### Annotation 1767019515148

 #Biochemistry Since strong acids rapidly transfer all their protons to H 2 O, the strongest acid that can stably exist in aqueous solutions is H 3 O ⫹ . Likewise, there can be no stronger base in aqueous solutions than OH ⫺ . Virtually all the acid–base reactions that occur in biological systems involve H 3 O ⫹ (and OH ⫺ ) and weak acids (and their conjugate bases).

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#### Annotation 1767021874444

 #Biochemistry The acids listed in Table 2-4 are known as weak acids because they are only partially ionized in aqueous solution (K ⬍ 1). Many of the so-called mineral acids, such as HClO 4 , HNO 3 , and HCl, are strong acids (K ⬎⬎ 1).

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#### Annotation 1767024495884

 #Biochemistry he pH of a Solution Is Determined by the Relative Concentrations of Acids and Bases. The relationship between the pH of a solution and the concentrations of an acid and its conjugate base is easily derived. Equation 2-5 can be rearranged to [2-7] Taking the negative log of each term (and letting pH ⫽⫺log[H ⫹ ]; Eq. 2-3) gives [2-8] Substituting pK for ⫺log K (Eq. 2-6) yields [2-9] This relationship is known as the Henderson–Hasselbalch equation. When the molar concentrations of an acid (HA) and its conjugate base (A ⫺ ) are equal, log ([A ⫺ ]兾[HA]) ⫽ log 1 ⫽ 0, and the pH of the solution is numerically equiv- alent to the pK of the acid. The Henderson–Hasselbalch equation is invaluable for calculating, for example, the pH of a solution containing known quanti- ties of a weak acid and its conjugate base (see Sample Calculation 2-2). However, since the Henderson–Hasselbalch equation does not account for the ionization of water itself, it is not useful for calculating the pH of solu- tions of strong acids or bases. For example, in a 1 M solution of a strong acid, [H ⫹ ] ⫽ 1 M and the pH is 0. In a 1 M solution of a strong base, [OH ⫺ ] ⫽ 1 M, so [H ⫹ ] ⫽ [OH ⫺ ]兾K w ⫽ 1 ⫻ 10 ⫺14 M and the pH is 14.

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#### Annotation 1767026855180

 #Biochemistry Several details about the titration curves in Fig. 2-17 should be noted: 1. The curves have similar shapes but are shifted vertically along the pH axis. 2. The pH at the midpoint of each titration is numerically equivalent to the pK of its corresponding acid; at this point, [HA] ⫽ [A ⫺ ]. 3. The slope of each titration curve is much lower near its midpoint than near its wings. This indicates that when [HA] ⬇ [A ⫺ ], the pH of the so- lution is relatively insensitive to the addition of strong base or strong acid. Such a solution, which is known as an acid–base buffer, resists pH changes because small amounts of added H ⫹ or OH ⫺ react with A ⫺ or HA, respectively, without greatly changing the value of log([A ⫺ ]兾[HA]).

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#### Annotation 1767029214476

 #Biochemistry In a biomolecule that contains numerous ionizable groups with different pK values, the many dissociation events may yield a titration curve without any clear “plateaus.”

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#### Annotation 1767031573772

 #Biochemistry Bicarbonate is the most significant buffer compound in human blood; other buffering agents, including proteins and organic acids, are present at much lower concentrations. The buffering capacity of blood depends primarily on two equilibria: (1) between gaseous CO 2 dissolved in the blood and carbonic acid formed by the reaction and (2) between carbonic acid and bicarbonate formed by the dissociation of H ⫹ The overall pK for these two sequential reactions is 6.35. (The further dissociation of to pK ⫽ 10.33, is not signifi- cant at physiological pH.) When the pH of the blood falls due to metabolic production of H ⫹ , the bicarbonate–carbonic acid equilibrium shifts toward more carbonic acid. At the same time, carbonic acid loses water to become CO 2 , which is then expired in the lungs as gaseous CO 2 . Conversely, when the blood pH rises, relatively more forms. Breathing is adjusted so that increased amounts of CO 2 in the lungs HCO ⫺ 3 CO 2⫺ 3 ,HCO ⫺ 3 H 2 CO 3 Δ H ⫹ ⫹ HCO ⫺ 3 CO 2 ⫹ H 2 O Δ H 2 CO 3 can be reintroduced into the blood for conversion to carbonic acid. In this manner, a near-constant hydrogen ion concentration can be maintained. The kidneys also play a role in acid–base balance by excreting and

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#### Annotation 1767036554508

 #cormen_2009_introductiontoalgorithms #datastructuresandalgorithms An algorithm is said to be correct if, for every input instance, it halts with the correct output.

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#### Flashcard 1767038913804

Tags
#cormen_2009_introductiontoalgorithms #datastructuresandalgorithms
Question
An algorithm is said to be [...] if, for every input instance, it halts with the correct output.