Do you want BuboFlash to help you learning these things? Click here to log in or create user.

Tags

#gaussian-process

Question

a stochastic process is called [...] if it depends only on distance but not the direction

Answer

isotropic

status | not learned | measured difficulty | 37% [default] | last interval [days] | |||
---|---|---|---|---|---|---|---|

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

scheduled repetition interval | last repetition or drill |

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

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

Tags

#gaussian-process

Question

Gaussian processes can be completely defined by their [...].

Answer

second-order statistics

status | not learned | measured difficulty | 37% [default] | last interval [days] | |||
---|---|---|---|---|---|---|---|

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

scheduled repetition interval | last repetition or drill |

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

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

Tags

#multivariate-normal-distribution

Question

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

Answer

ellipsoids

status | not learned | measured difficulty | 37% [default] | last interval [days] | |||
---|---|---|---|---|---|---|---|

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

scheduled repetition interval | last repetition or drill |

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

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

Tags

#multivariate-normal-distribution

Question

The multivariate normal distribution is often used to describe correlated real-valued random variables each of which [...]

Answer

clusters around a mean value

status | not learned | measured difficulty | 37% [default] | last interval [days] | |||
---|---|---|---|---|---|---|---|

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

scheduled repetition interval | last repetition or drill |

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

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

Tags

#dynamic-programming

Question

Answer

status | not learned | measured difficulty | 37% [default] | last interval [days] | |||
---|---|---|---|---|---|---|---|

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

scheduled repetition interval | last repetition or drill |

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

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

Tags

#dynamic-programming

Question

Answer

results of expensive function calls

status | not learned | measured difficulty | 37% [default] | last interval [days] | |||
---|---|---|---|---|---|---|---|

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

scheduled repetition interval | last repetition or drill |

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

Tags

#stochastics

Question

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

Answer

independent

status | not learned | measured difficulty | 37% [default] | last interval [days] | |||
---|---|---|---|---|---|---|---|

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

scheduled repetition interval | last repetition or drill |

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

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

Tags

#gauss-markov-process

Question

**Gauss–Markov stochastic processes** satisfy the requirements for [...]

Answer

both Gaussian processes and Markov processes.

status | not learned | measured difficulty | 37% [default] | last interval [days] | |||
---|---|---|---|---|---|---|---|

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

scheduled repetition interval | last repetition or drill |

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

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

Tags

#differential-equations

Question

status | not learned | measured difficulty | 37% [default] | last interval [days] | |||
---|---|---|---|---|---|---|---|

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

scheduled repetition interval | last repetition or drill |

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.)

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

Tags

#linear-algebra #matrix-decomposition

Question

**[...]** has real eigenvalues and the eigenvectors can be chosen to be orthogonal

Answer

real symmetric matrix

status | not learned | measured difficulty | 37% [default] | last interval [days] | |||
---|---|---|---|---|---|---|---|

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

scheduled repetition interval | last repetition or drill |

{\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

Question

Answer

the set of "input" values

status | not learned | measured difficulty | 37% [default] | last interval [days] | |||
---|---|---|---|---|---|---|---|

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

scheduled repetition interval | last repetition or drill |

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.

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

Tags

#kalman-filter

Question

status | not learned | measured difficulty | 37% [default] | last interval [days] | |||
---|---|---|---|---|---|---|---|

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

scheduled repetition interval | last repetition or drill |

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>

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

Tags

#stochastics

Question

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

Answer

sums

status | not learned | measured difficulty | 37% [default] | last interval [days] | |||
---|---|---|---|---|---|---|---|

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

scheduled repetition interval | last repetition or drill |

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.

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

Tags

#stochastics

Question

Random walks are usually defined as sums of [...] in Euclidean space

Answer

iid random variables or random vectors

status | not learned | measured difficulty | 37% [default] | last interval [days] | |||
---|---|---|---|---|---|---|---|

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

scheduled repetition interval | last repetition or drill |

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.

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

Tags

#stochastics

Question

the *simple random walk has* [...] as the state space

Answer

the integers

status | not learned | measured difficulty | 37% [default] | last interval [days] | |||
---|---|---|---|---|---|---|---|

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

scheduled repetition interval | last repetition or drill |

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.

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}

Tags

#stochastics

Question

Answer

sample path

status | not learned | measured difficulty | 37% [default] | last interval [days] | |||
---|---|---|---|---|---|---|---|

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

scheduled repetition interval | last repetition or drill |

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.

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

status | not learned | measured difficulty | 37% [default] | last interval [days] | |||
---|---|---|---|---|---|---|---|

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

scheduled repetition interval | last repetition or drill |

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.

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

Tags

#abstract-algebra

Question

Answer

algebraic structure

status | not learned | measured difficulty | 37% [default] | last interval [days] | |||
---|---|---|---|---|---|---|---|

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

scheduled repetition interval | last repetition or drill |

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.

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,

Tags

#abstract-algebra

Question

The **underlying set **for** **an **algebraic structure** is called a **[...]**

Answer

status | not learned | measured difficulty | 37% [default] | last interval [days] | |||
---|---|---|---|---|---|---|---|

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

scheduled repetition interval | last repetition or drill |

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.

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,

Tags

#abstract-algebra

Question

Answer

status | not learned | measured difficulty | 37% [default] | last interval [days] | |||
---|---|---|---|---|---|---|---|

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

scheduled repetition interval | last repetition or drill |

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.

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,

Tags

#abstract-algebra

Question

Answer

a list of axioms

status | not learned | measured difficulty | 37% [default] | last interval [days] | |||
---|---|---|---|---|---|---|---|

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

scheduled repetition interval | last repetition or drill |

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>

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,

Tags

#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 **[...]**

Answer

volatility matrix

status | not learned | measured difficulty | 37% [default] | last interval [days] | |||
---|---|---|---|---|---|---|---|

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

scheduled repetition interval | last repetition or drill |

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) . .

Tags

#linear-state-space-models

Question

the **[...]** distribution is specialized to \(N(0,I)\)

Answer

shock

status | not learned | measured difficulty | 37% [default] | last interval [days] | |||
---|---|---|---|---|---|---|---|

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

scheduled repetition interval | last repetition or drill |

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 )

Tags

#measure-theory #stochastics

Question

In **[...theory...]** every distribution function corresponds to a probability distribution

Answer

measure-theoretic probability theory

status | not learned | measured difficulty | 37% [default] | last interval [days] | |||
---|---|---|---|---|---|---|---|

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

scheduled repetition interval | last repetition or drill |

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

Tags

#probability-measure

Question

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

Answer

the entire probability space

status | not learned | measured difficulty | 37% [default] | last interval [days] | |||
---|---|---|---|---|---|---|---|

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

scheduled repetition interval | last repetition or drill |

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.

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

Tags

#lebesgue-integration

Question

Riemann integral considers the area under a curve as made out of [...shape...]

Answer

vertical rectangles

status | not learned | measured difficulty | 37% [default] | last interval [days] | |||
---|---|---|---|---|---|---|---|

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

scheduled repetition interval | last repetition or drill |

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.

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

Tags

#lebesgue-integration

Question

it is actually impossible to assign a length to [...] in a way that preserves some natural additivity and translation invariance properties.

Answer

all subsets of ℝ

status | not learned | measured difficulty | 37% [default] | last interval [days] | |||
---|---|---|---|---|---|---|---|

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

scheduled repetition interval | last repetition or drill |

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

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

Tags

#banach-space

Question

A normed space has a metric that allows the computation of **[...]**

Answer

vector length and distance between vectors

status | not learned | measured difficulty | 37% [default] | last interval [days] | |||
---|---|---|---|---|---|---|---|

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

scheduled repetition interval | last repetition or drill |

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

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]

Tags

#incremental-reading

Question

With incremental reading, you ensure high-retention of [...]

Answer

the most important pieces of text

status | not learned | measured difficulty | 37% [default] | last interval [days] | |||
---|---|---|---|---|---|---|---|

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

Tags

#fourier-analysis

Question

Fourier analysis has been extended to [...].

Answer

harmonic analysis

status | not learned | measured difficulty | 37% [default] | last interval [days] | |||
---|---|---|---|---|---|---|---|

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

Tags

#spectral-analysis

Question

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

Answer

for some scalar eigenvalue λ.

status | not learned | measured difficulty | 37% [default] | last interval [days] | |||
---|---|---|---|---|---|---|---|

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

scheduled repetition interval | last repetition or drill |

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.

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

Tags

#hilbert-space

Question

Linear operators on a Hilbert space are simply transformations that **[...]**

Answer

stretch the space by different factors in mutually perpendicular directions

status | not learned | measured difficulty | 37% [default] | last interval [days] | |||
---|---|---|---|---|---|---|---|

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

scheduled repetition interval | last repetition or drill |

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.

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

#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

status | not read | reprioritisations | ||
---|---|---|---|---|

last reprioritisation on | reading queue position [%] | |||

started reading on | finished reading on |

Tags

#_definition #cormen_2009_introductiontoalgorithms #datastructuresandalgorithms

Question

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

Answer

instance of a problem

status | not learned | measured difficulty | 37% [default] | last interval [days] | |||
---|---|---|---|---|---|---|---|

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

scheduled repetition interval | last repetition or drill |

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

Tags

#_definition #cormen_2009_introductiontoalgorithms #datastructuresandalgorithms

Question

In general, an instance of a problem consists of [...]

Answer

the input (satisfying whatever constraints are imposed in the problem statement) needed to compute a solution to the problem

status | not learned | measured difficulty | 37% [default] | last interval [days] | |||
---|---|---|---|---|---|---|---|

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

scheduled repetition interval | last repetition or drill |

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

status | not read | reprioritisations | ||
---|---|---|---|---|

last reprioritisation on | reading queue position [%] | |||

started reading on | finished reading on |

status | not read | reprioritisations | ||
---|---|---|---|---|

last reprioritisation on | reading queue position [%] | |||

started reading on | finished reading on |

status | not read | reprioritisations | ||
---|---|---|---|---|

last reprioritisation on | reading queue position [%] | |||

started reading on | finished reading on |

status | not read | reprioritisations | ||
---|---|---|---|---|

last reprioritisation on | reading queue position [%] | |||

started reading on | finished reading on |

status | not read | reprioritisations | ||
---|---|---|---|---|

last reprioritisation on | reading queue position [%] | |||

started reading on | finished reading on |

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

status | not read | reprioritisations | ||
---|---|---|---|---|

last reprioritisation on | reading queue position [%] | |||

started reading on | finished reading on |

we want to base our offer on actual results.

status | not read | reprioritisations | ||
---|---|---|---|---|

last reprioritisation on | reading queue position [%] | |||

started reading on | finished reading on |

status | not read | reprioritisations | ||
---|---|---|---|---|

last reprioritisation on | reading queue position [%] | |||

started reading on | finished reading on |

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

status | not read | reprioritisations | ||
---|---|---|---|---|

last reprioritisation on | reading queue position [%] | |||

started reading on | finished reading on |

status | not read | reprioritisations | ||
---|---|---|---|---|

last reprioritisation on | reading queue position [%] | |||

started reading on | finished reading on |

status | not read | reprioritisations | ||
---|---|---|---|---|

last reprioritisation on | reading queue position [%] | |||

started reading on | finished reading on |

status | not read | reprioritisations | ||
---|---|---|---|---|

last reprioritisation on | reading queue position [%] | |||

started reading on | finished reading on |

status | not read | reprioritisations | ||
---|---|---|---|---|

last reprioritisation on | reading queue position [%] | |||

started reading on | finished reading on |

#incremental-reading

The key to creativity is an association of remote ideas.

status | not read | reprioritisations | ||
---|---|---|---|---|

last reprioritisation on | reading queue position [%] | |||

started reading on | finished reading on |

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

#incremental-reading

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

status | not read | reprioritisations | ||
---|---|---|---|---|

last reprioritisation on | reading queue position [%] | |||

started reading on | finished reading on |

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

#incremental-reading

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

status | not read | reprioritisations | ||
---|---|---|---|---|

last reprioritisation on | reading queue position [%] | |||

started reading on | finished reading on |

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

#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

status | not read | reprioritisations | ||
---|---|---|---|---|

last reprioritisation on | reading queue position [%] | |||

started reading on | finished reading on |

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

#incremental-reading

In incremental reading, instead of hesitating or procrastinating, you simply prioritize

status | not read | reprioritisations | ||
---|---|---|---|---|

last reprioritisation on | reading queue position [%] | |||

started reading on | finished reading on |

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

#incremental-reading

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

status | not read | reprioritisations | ||
---|---|---|---|---|

last reprioritisation on | reading queue position [%] | |||

started reading on | finished reading on |

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

Tags

#calculus

Question

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

Answer

difference quotients

status | not learned | measured difficulty | 37% [default] | last interval [days] | |||
---|---|---|---|---|---|---|---|

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

scheduled repetition interval | last repetition or drill |

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

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

status | not learned | measured difficulty | 37% [default] | last interval [days] | |||
---|---|---|---|---|---|---|---|

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

scheduled repetition interval | last repetition or drill |

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.

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

Tags

#linear-algebra #matrix-decomposition

Question

The eigendecomposition of a real symmetric matrix can be represented as **[...]**

Answer

status | not learned | measured difficulty | 37% [default] | last interval [days] | |||
---|---|---|---|---|---|---|---|

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

scheduled repetition interval | last repetition or drill |

{\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

Tags

#incremental-reading

Question

The key to creativity is an [...].

Answer

association of remote ideas

status | not learned | measured difficulty | 37% [default] | last interval [days] | |||
---|---|---|---|---|---|---|---|

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

scheduled repetition interval | last repetition or drill |

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

Tags

#incremental-reading

Question

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

Answer

interpreting their contents

status | not learned | measured difficulty | 37% [default] | last interval [days] | |||
---|---|---|---|---|---|---|---|

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

scheduled repetition interval | last repetition or drill |

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

Tags

#incremental-reading

Question

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

Answer

the reader's knowledge

status | not learned | measured difficulty | 37% [default] | last interval [days] | |||
---|---|---|---|---|---|---|---|

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

scheduled repetition interval | last repetition or drill |

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

Tags

#incremental-reading

Question

In incremental reading, instead of [...], you simply prioritize

Answer

hesitate or procrastinate

status | not learned | measured difficulty | 37% [default] | last interval [days] | |||
---|---|---|---|---|---|---|---|

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

scheduled repetition interval | last repetition or drill |

In incremental reading, instead of hesitating or procrastinating, you simply prioritize

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

Tags

#incremental-reading

Question

In incremental reading, instead of hesitate or procrastinate, you simply [...]

Answer

prioritize

status | not learned | measured difficulty | 37% [default] | last interval [days] | |||
---|---|---|---|---|---|---|---|

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

scheduled repetition interval | last repetition or drill |

In incremental reading, instead of hesitating or procrastinating, you simply prioritize

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

Tags

#incremental-reading

Question

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

Answer

understanding and applicability

status | not learned | measured difficulty | 37% [default] | last interval [days] | |||
---|---|---|---|---|---|---|---|

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

scheduled repetition interval | last repetition or drill |

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

status | not read | reprioritisations | ||
---|---|---|---|---|

last reprioritisation on | reading queue position [%] | |||

started reading on | finished reading on |

极简唐朝史！ - 博海拾贝 - 萝卜网 博海拾贝 关于 联系 每日博海拾贝 萝卜网关闭公告 订阅 微博 腾讯微博 微信 诸暨 | 最优购| 烧饼博客 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！ 标签：唐朝史 相关推荐 [imagelink]你经历过把私聊内容发到群里的绝望吗？ [imagelink]高铁的一等座和商务座有什么区别？ [imagelink]饿了么卖身阿里 张旭豪还能否与王兴一战？ [imagelink]20个简短而深刻的精彩回复（第十五期） [imagelink]微语录精选0227：脱离自拍看长相 [imagelin

#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

status | not read | reprioritisations | ||
---|---|---|---|---|

last reprioritisation on | reading queue position [%] | |||

started reading on | finished reading on |

#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

status | not read | reprioritisations | ||
---|---|---|---|---|

last reprioritisation on | reading queue position [%] | |||

started reading on | finished reading on |

#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

status | not read | reprioritisations | ||
---|---|---|---|---|

last reprioritisation on | reading queue position [%] | |||

started reading on | finished reading on |

#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.

status | not read | reprioritisations | ||
---|---|---|---|---|

last reprioritisation on | reading queue position [%] | |||

started reading on | finished reading on |

#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.

status | not read | reprioritisations | ||
---|---|---|---|---|

last reprioritisation on | reading queue position [%] | |||

started reading on | finished reading on |

#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).

status | not read | reprioritisations | ||
---|---|---|---|---|

last reprioritisation on | reading queue position [%] | |||

started reading on | finished reading on |

#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.

status | not read | reprioritisations | ||
---|---|---|---|---|

last reprioritisation on | reading queue position [%] | |||

started reading on | finished reading on |

#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.

status | not read | reprioritisations | ||
---|---|---|---|---|

last reprioritisation on | reading queue position [%] | |||

started reading on | finished reading on |

#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.

status | not read | reprioritisations | ||
---|---|---|---|---|

last reprioritisation on | reading queue position [%] | |||

started reading on | finished reading on |

#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 ⫹

status | not read | reprioritisations | ||
---|---|---|---|---|

last reprioritisation on | reading queue position [%] | |||

started reading on | finished reading on |

#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

status | not read | reprioritisations | ||
---|---|---|---|---|

last reprioritisation on | reading queue position [%] | |||

started reading on | finished reading on |

#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

status | not read | reprioritisations | ||
---|---|---|---|---|

last reprioritisation on | reading queue position [%] | |||

started reading on | finished reading on |

#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.

status | not read | reprioritisations | ||
---|---|---|---|---|

last reprioritisation on | reading queue position [%] | |||

started reading on | finished reading on |

#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

status | not read | reprioritisations | ||
---|---|---|---|---|

last reprioritisation on | reading queue position [%] | |||

started reading on | finished reading on |

#Biochemistry

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

status | not read | reprioritisations | ||
---|---|---|---|---|

last reprioritisation on | reading queue position [%] | |||

started reading on | finished reading on |

#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 ⫺

status | not read | reprioritisations | ||
---|---|---|---|---|

last reprioritisation on | reading queue position [%] | |||

started reading on | finished reading on |

#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

status | not read | reprioritisations | ||
---|---|---|---|---|

last reprioritisation on | reading queue position [%] | |||

started reading on | finished reading on |

#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).

status | not read | reprioritisations | ||
---|---|---|---|---|

last reprioritisation on | reading queue position [%] | |||

started reading on | finished reading on |

#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).

status | not read | reprioritisations | ||
---|---|---|---|---|

last reprioritisation on | reading queue position [%] | |||

started reading on | finished reading on |

#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.

status | not read | reprioritisations | ||
---|---|---|---|---|

last reprioritisation on | reading queue position [%] | |||

started reading on | finished reading on |

#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]).

status | not read | reprioritisations | ||
---|---|---|---|---|

last reprioritisation on | reading queue position [%] | |||

started reading on | finished reading on |

#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.”

status | not read | reprioritisations | ||
---|---|---|---|---|

last reprioritisation on | reading queue position [%] | |||

started reading on | finished reading on |

#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

status | not read | reprioritisations | ||
---|---|---|---|---|

last reprioritisation on | reading queue position [%] | |||

started reading on | finished reading on |

#cormen_2009_introductiontoalgorithms #datastructuresandalgorithms

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

status | not read | reprioritisations | ||
---|---|---|---|---|

last reprioritisation on | reading queue position [%] | |||

started reading on | finished reading on |

Tags

#cormen_2009_introductiontoalgorithms #datastructuresandalgorithms

Question

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

Answer

correct

status | not learned | measured difficulty | 37% [default] | last interval [days] | |||
---|---|---|---|---|---|---|---|

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

scheduled repetition interval | last repetition or drill |

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