# on 01-Mar-2018 (Thu)

#### Flashcard 1729406831884

Question
the probabilistic framework to machine learning infers [...] to explain observed data
plausible models

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The key idea behind the probabilistic framework to machine learning is that learning can be thought of as inferring plausible models to explain observed data

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#### Flashcard 1729622052108

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#gaussian-process
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Stationarity process' behaviour depends on the distance between points, not their [...].
the actual positions

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y> Stationarity refers to the process' behaviour regarding the separation of any two points x and x' . If the process is stationary, it depends on their separation, x − x', while if non-stationary it depends on the actual position of the points x and x'. <body><html>

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Gaussian process - Wikipedia
initeness of this function enables its spectral decomposition using the Karhunen–Loeve expansion. Basic aspects that can be defined through the covariance function are the process' stationarity, isotropy, smoothness and periodicity. [5] [6] <span>Stationarity refers to the process' behaviour regarding the separation of any two points x and x' . If the process is 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. If the process depends only on |x − x'|, the Euclidean distance (not the dire

#### Flashcard 1729659538700

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#topology
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An affine transformation preserve [...] between points lying on a straight line.
ratios of distances

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An affine transformation does not necessarily preserve angles between lines or distances between points, though it does preserve ratios of distances between points lying on a straight line.

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Affine transformation - Wikipedia
s related to each other leaf by an affine transformation. For instance, the red leaf can be transformed into both the small dark blue leaf and the large light blue leaf by a combination of reflection, rotation, scaling, and translation. <span>In geometry, an affine transformation, affine map [1] or an affinity (from the Latin, affinis, "connected with") is a function between affine spaces which preserves points, straight lines and planes. Also, sets of parallel lines remain parallel after an affine transformation. An affine transformation does not necessarily preserve angles between lines or distances between points, though it does preserve ratios of distances between points lying on a straight line. Examples of affine transformations include translation, scaling, homothety, similarity transformation, reflection, rotation, shear mapping, and compositions of them in any combination

#### Flashcard 1732487023884

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#sprectral-theorem #stochastics
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the Karhunen–Loève theorem represents a stochastic process as [...]

an infinite linear combination of orthogonal functions

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

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

#### Flashcard 1732726099212

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#functional-analysis
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a [...] is a set of functions between two fixed sets.
function space

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n mathematics, a function space is a set of functions between two fixed sets.

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Function space - Wikipedia
tional · Algebraic · Analytic · Smooth · Continuous · Measurable · Injective · Surjective · Bijective   Constructions   Restriction · Composition · λ · Inverse   Generalizations   Partial · Multivalued · Implicit v t e I<span>n mathematics, a function space is a set of functions between two fixed sets. Often, the domain and/or codomain will have additional structure which is inherited by the function space. For example, the set of functions from any set X into a vector space have a

#### Flashcard 1735816514828

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#logic
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The fall of [...] culture wasn’t the only cause for the demise of scholastic logic
disputational

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The fall of disputational culture wasn’t the only cause for the demise of scholastic logic, however. Scholastic logic was also viewed – rightly or wrongly – as being tied to broadly Aristotelian conceptions of language

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The rise and fall and rise of logic | Aeon Essays
s Diafoirus resorts to disputational vocabulary to make a point about love: Distinguo, Mademoiselle; in all that does not concern the possession of the loved one, concedo, I grant it; but in what does regard that possession, nego, I deny it. <span>The fall of disputational culture wasn’t the only cause for the demise of scholastic logic, however. Scholastic logic was also viewed – rightly or wrongly – as being tied to broadly Aristotelian conceptions of language and metaphysics, which themselves fell out of favour in the dawn of the modern era with the rise of a new scientific paradigm. Despite all this, disputations continued to be practised in certain university contexts for some time – indeed, they live on in the ceremonial character of PhD defences. The point, thou

#### Flashcard 1735818087692

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#logic
Question
Scholastic logic was also viewed as being tied to Aristotelian conceptions of [...and...]
language and metaphysics

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The fall of disputational culture wasn’t the only cause for the demise of scholastic logic, however. Scholastic logic was also viewed – rightly or wrongly – as being tied to broadly Aristotelian conceptions of language and metaphysics, which themselves fell out of favour in the dawn of the modern era with the rise of a new scientific paradigm. <body><html>

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The rise and fall and rise of logic | Aeon Essays
s Diafoirus resorts to disputational vocabulary to make a point about love: Distinguo, Mademoiselle; in all that does not concern the possession of the loved one, concedo, I grant it; but in what does regard that possession, nego, I deny it. <span>The fall of disputational culture wasn’t the only cause for the demise of scholastic logic, however. Scholastic logic was also viewed – rightly or wrongly – as being tied to broadly Aristotelian conceptions of language and metaphysics, which themselves fell out of favour in the dawn of the modern era with the rise of a new scientific paradigm. Despite all this, disputations continued to be practised in certain university contexts for some time – indeed, they live on in the ceremonial character of PhD defences. The point, thou

#### Flashcard 1737361067276

Question
functional programming is a style of building [...] of computer programs
the structure and elements

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

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

#### Flashcard 1737957444876

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#state-space-models
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state space models are Markov models with [...]
latent variables

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Although such models are tractable, they are also severely limited. We can ob- tain a more general framework, while still retaining tractability, by the introduction of latent variables, leading to state space models.

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#### Flashcard 1738029272332

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#bayesian-ml #prml
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the conditional distributions of the observed variables given the hidden variables are known as [...]
emission probabilities

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the conditional distributions of the observed variables is , where φ is a set of parameters governing the distribution. These are known as emission probabilities

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#### Flashcard 1738563259660

Tags
#measure-theory #stochastics
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abstract integration gives the same result as [...] when the latter exists
Riemann integration

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Lebesgue integration or abstract integration gives the same result as Riemann integration when the latter exists, so nothing you know from calculus changes, but a lot more functions are integrable

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#### Flashcard 1739048488204

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#graphical-models
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A [...] is a model over an undirected graph.
Markov random field

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A Markov random field, also known as a Markov network, is a model over an undirected graph.

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Graphical model - Wikipedia
ne learning models like hidden Markov models, neural networks and newer models such as variable-order Markov models can be considered special cases of Bayesian networks. Markov random field[edit source] Main article: Markov random field <span>A Markov random field, also known as a Markov network, is a model over an undirected graph. A graphical model with many repeated subunits can be represented with plate notation. Other types[edit source] A factor graph is an undirected bipartite graph connecting variables a

#### Flashcard 1739056614668

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#bayesian-network
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In a Bayesian network each node takes [...] as input
a set of values from parent nodes

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re not connected (there is no path from one of the variables to the other in the Bayesian network) represent variables that are conditionally independent of each other. Each node is associated with a probability function that takes, as input, <span>a particular set of values for the node's parent variables, and gives (as output) the probability (or probability distribution, if applicable) of the variable represented by the node. <span><body><html>

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Bayesian network - Wikipedia
ed acyclic graph (DAG). For example, a Bayesian network could represent the probabilistic relationships between diseases and symptoms. Given symptoms, the network can be used to compute the probabilities of the presence of various diseases. <span>Formally, Bayesian networks are DAGs whose nodes represent variables in the Bayesian sense: they may be observable quantities, latent variables, unknown parameters or hypotheses. Edges represent conditional dependencies; nodes that are not connected (there is no path from one of the variables to the other in the Bayesian network) represent variables that are conditionally independent of each other. Each node is associated with a probability function that takes, as input, a particular set of values for the node's parent variables, and gives (as output) the probability (or probability distribution, if applicable) of the variable represented by the node. For example, if m {\displaystyle m} parent nodes represent m {\displaystyle m} Boolean variables

#### Flashcard 1739080994060

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#forward-backward-algorithm #hmm
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The foreward-backward algorithm makes use of the principle of [...] in its two passes.
dynamic programming

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The foreward-backward algorithm makes use of the principle of dynamic programming to compute efficiently the values that are required to obtain the posterior marginal distributions in two passes. The first pass goes forward in time while the second goes backward in t

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Forward–backward algorithm - Wikipedia
| o 1 : t ) {\displaystyle P(X_{k}\ |\ o_{1:t})} . This inference task is usually called smoothing. <span>The algorithm makes use of the principle of dynamic programming to compute efficiently the values that are required to obtain the posterior marginal distributions in two passes. The first pass goes forward in time while the second goes backward in time; hence the name forward–backward algorithm. The term forward–backward algorithm is also used to refer to any algorithm belonging to the general class of algorithms that operate on sequence models in a forward–backward manner. I

#### Flashcard 1739930864908

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#forward-backward-algorithm #hmm
Question

In the second pass, the forward-backward algorithm computes [...]

.

the probability of observing the remaining observations given any starting point

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rward probabilities which provide, for all , the probability of ending up in any particular state given the first observations in the sequence, i.e. . In the second pass, the algorithm computes a set of backward probabilities which provide <span>the probability of observing the remaining observations given any starting point , i.e. . These two sets of probability distributions can then be combined to obtain the distribution over states at any specific point in time given the entire observation sequence:

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Forward–backward algorithm - Wikipedia
cific instance of this class. Contents [hide] 1 Overview 2 Forward probabilities 3 Backward probabilities 4 Example 5 Performance 6 Pseudocode 7 Python example 8 See also 9 References 10 External links Overview[edit source] <span>In the first pass, the forward–backward algorithm computes a set of forward probabilities which provide, for all k ∈ { 1 , … , t } {\displaystyle k\in \{1,\dots ,t\}} , the probability of ending up in any particular state given the first k {\displaystyle k} observations in the sequence, i.e. P ( X k | o 1 : k ) {\displaystyle P(X_{k}\ |\ o_{1:k})} . In the second pass, the algorithm computes a set of backward probabilities which provide the probability of observing the remaining observations given any starting point k {\displaystyle k} , i.e. P ( o k + 1 : t | X k ) {\displaystyle P(o_{k+1:t}\ |\ X_{k})} . These two sets of probability distributions can then be combined to obtain the distribution over states at any specific point in time given the entire observation sequence: P ( X k | o 1 : t ) = P ( X k | o 1 : k , o k + 1 : t ) ∝ P ( o k + 1 : t | X k ) P ( X k | o 1 : k ) {\displaystyle P(X_{k}\ |\ o_{1:t})=P(X_{k}\ |\ o_{1:k},o_{k+1:t})\propto P(o_{k+1:t}\ |\ X_{k})P(X_{k}|o_{1:k})} The last step follows from an application of the Bayes' rule and the conditional independence of o k + 1 : t {\displaystyle o_{k+1:t}} and o 1 : k {\displaystyle o_{1:k}} given X k {\displaystyle X_{k}} . As outlined above, the algorithm involves three steps: computing forward probabilities computing backward probabilities computing smoothed values. The forward and backward steps m

#### Flashcard 1741136989452

Tags
#measure-theory #stochastics
Question
A [...] for set Ω is a family $$\mathcal{A}$$ of subsets of Ω that contains Ω and is closed under complements and countable unions and intersections.
sigma-algebra

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Let Ω be an arbitrary set. A sigma-algebra for Ω is a family  of subsets of Ω that contains Ω and is closed under complements and countable unions and intersections.

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#### Flashcard 1741139348748

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#measure-theory #stochastics
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A sigma-algebra for Ω must contain [...]

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Let Ω be an arbitrary set. A sigma-algebra for Ω is a family  of subsets of Ω that contains Ω and is closed under complements and countable unions and intersections.

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#### Flashcard 1741140921612

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#measure-theory #stochastics
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A sigma-algebra for Ω is closed under [...].
complements and countable unions and intersections

Is this property stopping the Banach-Tarski paradox?

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Let Ω be an arbitrary set. A sigma-algebra for Ω is a family of subsets of Ω that contains Ω and is closed under complements and countable unions and intersections.

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#### Flashcard 1741164514572

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#measure-theory #stochastics
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Lebesgue measure for a set A on R corresponds to [...] of ordinary calculus
dx

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#### Flashcard 1741238177036

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#poisson-process #stochastics
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a homogeneous Poisson point process has a parameter of the form [...]
,

where is Lebegues measure, and is a constant

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

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

#### Flashcard 1741389171980

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#topology
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topology concerns the properties of space that are preserved under [...]
continuous deformations

i.e. homeomorphisms. Remeber scrub

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In mathematics, topology (from the Greek τόπος, place, and λόγος, study) is concerned with the properties of space that are preserved under continuous deformations, such as stretching, crumpling and bending, but not tearing or gluing.

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Topology - Wikipedia
ogy (disambiguation). For a topology of a topos or category, see Lawvere–Tierney topology and Grothendieck topology. [imagelink] Möbius strips, which have only one surface and one edge, are a kind of object studied in topology. <span>In mathematics, topology (from the Greek τόπος, place, and λόγος, study) is concerned with the properties of space that are preserved under continuous deformations, such as stretching, crumpling and bending, but not tearing or gluing. This can be studied by considering a collection of subsets, called open sets, that satisfy certain properties, turning the given set into what is known as a topological space. Important

#### Flashcard 1744145353996

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#inner-product-space #vector-space
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Among the topologies of vector spaces, those that are defined by [...] are more commonly used, as having a notion of distance between two vectors.

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Among the topologies of vector spaces, those that are defined by a norm or inner product are more commonly used, as having a notion of distance between two vectors.

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Vector space - Wikipedia
roperties, which in some cases can be visualized as arrows. Vector spaces are the subject of linear algebra and are well characterized by their dimension, which, roughly speaking, specifies the number of independent directions in the space. <span>Infinite-dimensional vector spaces arise naturally in mathematical analysis, as function spaces, whose vectors are functions. These vector spaces are generally endowed with additional structure, which may be a topology, allowing the consideration of issues of proximity and continuity. Among these topologies, those that are defined by a norm or inner product are more commonly used, as having a notion of distance between two vectors. This is particularly the case of Banach spaces and Hilbert spaces, which are fundamental in mathematical analysis. Historically, the first ideas leading to vector spaces can be traced back as far as the 17th century's analytic geometry, matrices, systems of linear equations, and Euclidean vectors.

#### Annotation 1748725009676

 #borel-algebra #measure-theory In mathematics, a Borel set is any set in a topological space that can be formed from open sets (or, equivalently, from closed sets) through the operations of countable union, countable intersection, and relative complement.

Borel set - Wikipedia
Borel set - Wikipedia Borel set From Wikipedia, the free encyclopedia Jump to: navigation, search In mathematics, a Borel set is any set in a topological space that can be formed from open sets (or, equivalently, from closed sets) through the operations of countable union, countable intersection, and relative complement. Borel sets are named after Émile Borel. For a topological space X, the collection of all Borel sets on X forms a σ-algebra, known as the Borel algebra or Borel σ-algebra. The Borel al

#### Flashcard 1748733660428

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#topology
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a [...] may be defined by a set of of points, neighbourhoods, and axioms.
topological space

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a topological space may be defined as a set of points, along with a set of neighbourhoods for each point, satisfying a set of axioms relating points and neighbourhoods.

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Topological space - Wikipedia
n>Topological space - Wikipedia Topological space From Wikipedia, the free encyclopedia Jump to: navigation, search In topology and related branches of mathematics, a topological space may be defined as a set of points, along with a set of neighbourhoods for each point, satisfying a set of axioms relating points and neighbourhoods. The definition of a topological space relies only upon set theory and is the most general notion of a mathematical space that allows for the definition of concepts such as continuity, c

#### Annotation 1748735233292

 #borel-algebra #measure-theory The Borel algebra on X is the smallest σ-algebra containing all open sets

Borel set - Wikipedia
of countable union, countable intersection, and relative complement. Borel sets are named after Émile Borel. For a topological space X, the collection of all Borel sets on X forms a σ-algebra, known as the Borel algebra or Borel σ-algebra. <span>The Borel algebra on X is the smallest σ-algebra containing all open sets (or, equivalently, all closed sets). Borel sets are important in measure theory, since any measure defined on the open sets of a space, or on the closed sets of a space, must also be

#### Flashcard 1748736806156

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#borel-algebra #measure-theory
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[...] on X is the smallest σ-algebra containing all open sets
The Borel algebra

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The Borel algebra on X is the smallest σ-algebra containing all open sets

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Borel set - Wikipedia
of countable union, countable intersection, and relative complement. Borel sets are named after Émile Borel. For a topological space X, the collection of all Borel sets on X forms a σ-algebra, known as the Borel algebra or Borel σ-algebra. <span>The Borel algebra on X is the smallest σ-algebra containing all open sets (or, equivalently, all closed sets). Borel sets are important in measure theory, since any measure defined on the open sets of a space, or on the closed sets of a space, must also be

#### Flashcard 1748738379020

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#borel-algebra #measure-theory
Question
The Borel algebra on X is the [...] containing all open sets
smallest σ-algebra

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The Borel algebra on X is the smallest σ-algebra containing all open sets

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Borel set - Wikipedia
of countable union, countable intersection, and relative complement. Borel sets are named after Émile Borel. For a topological space X, the collection of all Borel sets on X forms a σ-algebra, known as the Borel algebra or Borel σ-algebra. <span>The Borel algebra on X is the smallest σ-algebra containing all open sets (or, equivalently, all closed sets). Borel sets are important in measure theory, since any measure defined on the open sets of a space, or on the closed sets of a space, must also be

#### Annotation 1748744932620

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

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

#### Flashcard 1748756204812

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#borel-algebra #measure-theory
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any measure defined on the open sets of a space, or on the closed sets of a space, must also be defined on [...].
all Borel sets of that space

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

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

#### Flashcard 1753310694668

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#metric-space #topological-space
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"metric" is a generalization of [...]
the Euclidean distance

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

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

#### Flashcard 1755466304780

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#topology
Question
[...] is called a closed set.
A complement of an open set

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A complement of an open set (relative to the space that the topology is defined on) is called a closed set.

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Open set - Wikipedia
l space. There are, however, topological spaces that are not metric spaces. Properties The union of any number of open sets, or infinitely many open sets, is open. [2] The intersection of a finite number of open sets is open. [2] <span>A complement of an open set (relative to the space that the topology is defined on) is called a closed set. A set may be both open and closed (a clopen set). The empty set and the full space are examples of sets that are both open and closed. [3] Uses Open sets have a fundamental im

#### Flashcard 1755467877644

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#topology
Question
A complement of an open set is called a [...].
closed set

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A complement of an open set (relative to the space that the topology is defined on) is called a closed set.

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Open set - Wikipedia
l space. There are, however, topological spaces that are not metric spaces. Properties The union of any number of open sets, or infinitely many open sets, is open. [2] The intersection of a finite number of open sets is open. [2] <span>A complement of an open set (relative to the space that the topology is defined on) is called a closed set. A set may be both open and closed (a clopen set). The empty set and the full space are examples of sets that are both open and closed. [3] Uses Open sets have a fundamental im

#### Flashcard 1756791180556

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#has-images #lagrange-multiplier #optimization
Question
At the stationary point the vector ∇f(x) is orthogonal to the constraint surface because otherwise [...]
[unknown IMAGE 1756479753484]
move along the constraint surface and increase f(x)

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At the stationary point the vector ∇f(x) is orthogonal to the constraint surface because otherwise we could increase the value of f(x) by moving a short distance along the constraint surface

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#### Flashcard 1758284614924

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#calculus-of-variations
Question
the weak formulation of the necessary condition of extremum is an [...form...]
integral with an arbitrary function δf .

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the weak formulation of the necessary condition of extremum is an integral with an arbitrary function δf .

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Fundamental lemma of calculus of variations - Wikipedia
pedia Jump to: navigation, search In mathematics, specifically in the calculus of variations, a variation δf of a function f can be concentrated on an arbitrarily small interval, but not a single point. <span>Accordingly, the necessary condition of extremum (functional derivative equal zero) appears in a weak formulation (variational form) integrated with an arbitrary function δf. The fundamental lemma of the calculus of variations is typically used to transform this weak formulation into the strong formulation (differential equation), free of the integration with arbitrary function. The proof usually exploits the possibility to choose δf concentrated on an interval on which f keeps sign (positive or negative). Several versions of the lemma are in use. Basic version

#### Flashcard 1759666375948

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#functional-analysis
Question
a [...] is a vector space whose elements are infinite sequences
sequence space

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In functional analysis and related areas of mathematics, a sequence space is a vector space whose elements are infinite sequences of real or complex numbers

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Sequence space - Wikipedia
a, the free encyclopedia Jump to: navigation, search For usage in evolutionary biology, see Sequence space (evolution). For mathematical operations on sequence numbers, see Serial number arithmetic. <span>In functional analysis and related areas of mathematics, a sequence space is a vector space whose elements are infinite sequences of real or complex numbers. Equivalently, it is a function space whose elements are functions from the natural numbers to the field K of real or complex numbers. The set of all such functions is naturally identif

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#functional-analysis
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a sequence space is a vector space whose elements are [...]
infinite sequences of real or complex numbers

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In functional analysis and related areas of mathematics, a sequence space is a vector space whose elements are infinite sequences of real or complex numbers

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Sequence space - Wikipedia
a, the free encyclopedia Jump to: navigation, search For usage in evolutionary biology, see Sequence space (evolution). For mathematical operations on sequence numbers, see Serial number arithmetic. <span>In functional analysis and related areas of mathematics, a sequence space is a vector space whose elements are infinite sequences of real or complex numbers. Equivalently, it is a function space whose elements are functions from the natural numbers to the field K of real or complex numbers. The set of all such functions is naturally identif

#### Flashcard 1760858606860

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#calculus
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The total derivative is the [...description...]
limiting ratio of ∆ƒ/∆t

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The total derivative of a function of several variables, e.g., , , , with respect to an exogenous argument , is the limiting ratio of the change in the function's value to the change in the exogenous argument's value, taking into account the exogenous argument's direct effect as well as indirect effects via the other arguments of the function.

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Total derivative - Wikipedia
integral Line integral Surface integral Volume integral Jacobian Hessian Specialized[show] Fractional Malliavin Stochastic Variations Glossary of calculus[show] Glossary of calculus v t e <span>In the mathematical field of differential calculus, a total derivative or full derivative of a function f {\displaystyle f} of several variables, e.g., t {\displaystyle t} , x {\displaystyle x} , y {\displaystyle y} , etc., with respect to an exogenous argument, e.g., t {\displaystyle t} , is the limiting ratio of the change in the function's value to the change in the exogenous argument's value (for arbitrarily small changes), taking into account the exogenous argument's direct effect as well as indirect effects via the other arguments of the function. The total derivative of a function is different from its corresponding partial derivative ( ∂ {\displaystyle \partial } ). Calculation of the

#### Flashcard 1760860179724

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#calculus
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The total derivative taking into account the exogenous argument's [...] to the function
direct and indirect effects

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a function of several variables, e.g., , , , with respect to an exogenous argument , is the limiting ratio of the change in the function's value to the change in the exogenous argument's value, taking into account the exogenous argument's <span>direct effect as well as indirect effects via the other arguments of the function. <span><body><html>

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Total derivative - Wikipedia
integral Line integral Surface integral Volume integral Jacobian Hessian Specialized[show] Fractional Malliavin Stochastic Variations Glossary of calculus[show] Glossary of calculus v t e <span>In the mathematical field of differential calculus, a total derivative or full derivative of a function f {\displaystyle f} of several variables, e.g., t {\displaystyle t} , x {\displaystyle x} , y {\displaystyle y} , etc., with respect to an exogenous argument, e.g., t {\displaystyle t} , is the limiting ratio of the change in the function's value to the change in the exogenous argument's value (for arbitrarily small changes), taking into account the exogenous argument's direct effect as well as indirect effects via the other arguments of the function. The total derivative of a function is different from its corresponding partial derivative ( ∂ {\displaystyle \partial } ). Calculation of the

#### Flashcard 1760891374860

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#calculus
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derivative defined by limit considers [...] and extracts a consistent value for the exact point
the behavior of f at nearby inputs

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derivative defined by limit considers the behavior of f for all small values of h and extracts a consistent value for the case when h equals zero

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Calculus - Wikipedia
e behavior of the function at the point a because it does not account for what happens between a and a + h. It is not possible to discover the behavior at a by setting h to zero because this would require dividing by zero, which is undefined. <span>The derivative is defined by taking the limit as h tends to zero, meaning that it considers the behavior of f for all small values of h and extracts a consistent value for the case when h equals zero: lim h → 0 f ( a + h ) − f ( a ) 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. The tangent line is a limit of secant lines just as the derivative is a limit of difference

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you should always try to make sure your brain [...] at each repetition.
works in the exactly same way

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you should always try to make sure your brain works in the exactly same way at each repetition .

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20 rules of formulating knowledge in learning
he set of countries that can be listed in any order. Paradoxically, despite containing more information, enumerations are easier to remember. The reason for this has been discussed earlier in the context of the minimum information principle: <span>you should always try to make sure your brain works in the exactly same way at each repetition . In the case of sets, listing members in varying order at each repetition has a disastrous effect on memory. It is nearly impossible to memorize sets containing more than five members wi

#### Flashcard 1767191481612

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wording must be optimized to ensure [...].
minimum-effort recall

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wording must be optimized to make sure that the brain can recall it with minimum effort.

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20 rules of formulating knowledge in learning
apt but I will never know which is which!") in SuperMemo use View : Other browsers : Leeches (Shift+F3) to regularly review and eliminate most difficult items read more: Memory interference Optimize wording <span>The wording of your items must be optimized to make sure that in minimum time the right bulb in your brain lights up. This will reduce error rates, increase specificity, reduce response time, and help your concentration. Less optimum item: cloze deletion that is too wordy Q:

#### Flashcard 1767377079564

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#measure-theory #stochastics
Question
The Lebesgue measure for a set A on R is [...formula...]
$$\mu(A) = \int_A d\mu$$

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#### Flashcard 1767389662476

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#topology
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A complement of an open set is always relative to [...]
a certain topology in a certain space

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A complement of an open set (relative to the space that the topology is defined on) is called a closed set.

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Open set - Wikipedia
l space. There are, however, topological spaces that are not metric spaces. Properties The union of any number of open sets, or infinitely many open sets, is open. [2] The intersection of a finite number of open sets is open. [2] <span>A complement of an open set (relative to the space that the topology is defined on) is called a closed set. A set may be both open and closed (a clopen set). The empty set and the full space are examples of sets that are both open and closed. [3] Uses Open sets have a fundamental im

#### Flashcard 1767393856780

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#calculus
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derivative defined by limit considers the behavior of f at nearby inputs and extracts [...]
a consistent value for the exact point

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derivative defined by limit considers the behavior of f for all small values of h and extracts a consistent value for the case when h equals zero

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Calculus - Wikipedia
e behavior of the function at the point a because it does not account for what happens between a and a + h. It is not possible to discover the behavior at a by setting h to zero because this would require dividing by zero, which is undefined. <span>The derivative is defined by taking the limit as h tends to zero, meaning that it considers the behavior of f for all small values of h and extracts a consistent value for the case when h equals zero: lim h → 0 f ( a + h ) − f ( a ) 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. The tangent line is a limit of secant lines just as the derivative is a limit of difference

#### Flashcard 1767406963980

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#logic
Question
The Aristotelian conceptions of language and metaphysics fell out of favour in [...when...] with the rise of a new scientific paradigm.
the dawn of the modern era

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ture wasn’t the only cause for the demise of scholastic logic, however. Scholastic logic was also viewed – rightly or wrongly – as being tied to broadly Aristotelian conceptions of language and metaphysics, which themselves fell out of favour <span>in the dawn of the modern era with the rise of a new scientific paradigm. <span><body><html>

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The rise and fall and rise of logic | Aeon Essays
s Diafoirus resorts to disputational vocabulary to make a point about love: Distinguo, Mademoiselle; in all that does not concern the possession of the loved one, concedo, I grant it; but in what does regard that possession, nego, I deny it. <span>The fall of disputational culture wasn’t the only cause for the demise of scholastic logic, however. Scholastic logic was also viewed – rightly or wrongly – as being tied to broadly Aristotelian conceptions of language and metaphysics, which themselves fell out of favour in the dawn of the modern era with the rise of a new scientific paradigm. Despite all this, disputations continued to be practised in certain university contexts for some time – indeed, they live on in the ceremonial character of PhD defences. The point, thou

#### Annotation 1767447596300

 #has-images 羽绒服脏了怎么洗？8招教你轻松搞定！ 梁萧 发布于 2小时前分类：文摘 寒冬即将结束，厚重的羽绒服也该清洗干净收起来了，如何清洗？戳图↓↓，学会这8招，教你轻松清洁羽绒服，洁净舒适如新！ 未经允许不得转载：博海拾贝 » 羽绒服脏了怎么洗？8招教你轻松搞定！

ç¾½ç»æèäºæä¹æ´ï¼8ææä½ è½»æ¾æå®ï¼ - åæµ·æ¾è´ - èåç½

#### Annotation 1767465422092

 #politics Atrocity propaganda is the spreading information about the crimes committed by an enemy, especially deliberate fabrications or exaggerations.

Atrocity propaganda - Wikipedia

#### Flashcard 1767467519244

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#politics
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[...] is the spread of information about the crimes committed by an enemy, especially deliberate fabrications or exaggerations.
Atrocity propaganda

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Atrocity propaganda is the spreading information about the crimes committed by an enemy, especially deliberate fabrications or exaggerations.

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Atrocity propaganda - Wikipedia

#### Flashcard 1767469092108

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#politics
Question
Atrocity propaganda is the spreading information about the crimes committed by an enemy, especially deliberate [...]
fabrications or exaggerations.

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Atrocity propaganda is the spreading information about the crimes committed by an enemy, especially deliberate fabrications or exaggerations.

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Atrocity propaganda - Wikipedia

#### Annotation 1767472237836

 #english-literature William Shakespeare ( / ˈ ʃ eɪ k s p ɪər / ; 26 April 1564 (baptised) – 23 April 1616)[a] was an English poet, playwright and actor, widely regarded as the greatest writer in the English language and the world's pre-eminent dramatist.

William Shakespeare - Wikipedia
Era Elizabethan era Jacobean era Movement English Renaissance Spouse(s) Anne Hathaway ( m. 1582) Children Susanna Hall Hamnet Shakespeare Judith Quiney Parent(s) John Shakespeare Mary Arden Signature [imagelink] <span>William Shakespeare (/ˈʃeɪkspɪər/; 26 April 1564 (baptised) – 23 April 1616) [a] was an English poet, playwright and actor, widely regarded as the greatest writer in the English language and the world's pre-eminent dramatist. [2] [3] [4] He is often called England's national poet and the "Bard of Avon". [5] [b] His extant works, including collaborations, consist of approximately 39 plays, [c] 15

#### Flashcard 1767476432140

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#english-literature
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William Shakespeare was born in [...]
1564

Look, here comes the tall shrew!

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William Shakespeare ( / ˈ ʃ eɪ k s p ɪər / ; 26 April 1564 (baptised) – 23 April 1616) [a] was an English poet, playwright and actor, widely regarded as the greatest writer in the English language and the world's pre-eminent dramatist.

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William Shakespeare - Wikipedia
Era Elizabethan era Jacobean era Movement English Renaissance Spouse(s) Anne Hathaway ( m. 1582) Children Susanna Hall Hamnet Shakespeare Judith Quiney Parent(s) John Shakespeare Mary Arden Signature [imagelink] <span>William Shakespeare (/ˈʃeɪkspɪər/; 26 April 1564 (baptised) – 23 April 1616) [a] was an English poet, playwright and actor, widely regarded as the greatest writer in the English language and the world's pre-eminent dramatist. [2] [3] [4] He is often called England's national poet and the "Bard of Avon". [5] [b] His extant works, including collaborations, consist of approximately 39 plays, [c] 15

#### Annotation 1767480102156

 #history The Hundred Years' War was a series of conflicts waged from 1337 to 1453 by the House of Plantagenet, rulers of the Kingdom of England, against the House of Valois, rulers of the Kingdom of France, over the succession to the French throne.

Hundred Years' War - Wikipedia
Anglo-French wars 1202–04 1213–14 1215–17 1242–43 1294–1303 1337–1453 (1337–60, 1369–89, 1415–53) 1496-98 1512–14 1522–26 1542–46 1557–59 1627–29 1666–67 1689–97 1702–13 1744–48 1744–1763 1754–63 1778–83 1793–1802 1803–14 1815 <span>The Hundred Years' War was a series of conflicts waged from 1337 to 1453 by the House of Plantagenet, rulers of the Kingdom of England, against the House of Valois, rulers of the Kingdom of France, over the succession to the French throne. Each side drew many allies into the war. It was one of the most notable conflicts of the Middle Ages, in which five generations of kings from two rival dynasties fought for the throne o

#### Flashcard 1767482199308

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#history
Question
The Hundred Years' War started in [...]
1337

The war between the tame and the meek (Christians).

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The Hundred Years' War was a series of conflicts waged from 1337 to 1453 by the House of Plantagenet, rulers of the Kingdom of England, against the House of Valois, rulers of the Kingdom of France, over the succession to the French throne. </

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Hundred Years' War - Wikipedia
Anglo-French wars 1202–04 1213–14 1215–17 1242–43 1294–1303 1337–1453 (1337–60, 1369–89, 1415–53) 1496-98 1512–14 1522–26 1542–46 1557–59 1627–29 1666–67 1689–97 1702–13 1744–48 1744–1763 1754–63 1778–83 1793–1802 1803–14 1815 <span>The Hundred Years' War was a series of conflicts waged from 1337 to 1453 by the House of Plantagenet, rulers of the Kingdom of England, against the House of Valois, rulers of the Kingdom of France, over the succession to the French throne. Each side drew many allies into the war. It was one of the most notable conflicts of the Middle Ages, in which five generations of kings from two rival dynasties fought for the throne o

#### Flashcard 1767484558604

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#history
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The Hundred Years' War ended in [...]
1453

England would hate losing its heirloom.

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The Hundred Years' War was a series of conflicts waged from 1337 to 1453 by the House of Plantagenet, rulers of the Kingdom of England, against the House of Valois, rulers of the Kingdom of France, over the succession to the French throne. </b

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Hundred Years' War - Wikipedia
Anglo-French wars 1202–04 1213–14 1215–17 1242–43 1294–1303 1337–1453 (1337–60, 1369–89, 1415–53) 1496-98 1512–14 1522–26 1542–46 1557–59 1627–29 1666–67 1689–97 1702–13 1744–48 1744–1763 1754–63 1778–83 1793–1802 1803–14 1815 <span>The Hundred Years' War was a series of conflicts waged from 1337 to 1453 by the House of Plantagenet, rulers of the Kingdom of England, against the House of Valois, rulers of the Kingdom of France, over the succession to the French throne. Each side drew many allies into the war. It was one of the most notable conflicts of the Middle Ages, in which five generations of kings from two rival dynasties fought for the throne o

#### Annotation 1767488228620

 A lingua franca ( / ˌ l ɪ ŋ ɡ w ə ˈ f r æ ŋ k ə / ; lit.  Frankish tongue ),[1] also known as a bridge language, common language, trade language or vehicular language, is a language or dialect systematically used to make communication possible between people who do not share a native language or dialect, particularly when it is a third language that is distinct from both native languages.[2]

Lingua franca - Wikipedia
span>Lingua franca - Wikipedia Lingua franca From Wikipedia, the free encyclopedia Jump to: navigation, search For other uses, see Lingua franca (disambiguation). A lingua franca (/ˌlɪŋɡwə ˈfræŋkə/; lit.  Frankish tongue), [1] also known as a bridge language, common language, trade language or vehicular language, is a language or dialect systematically used to make communication possible between people who do not share a native language or dialect, particularly when it is a third language that is distinct from both native languages. [2] Lingua francas have developed around the world throughout human history, sometimes for commercial reasons (so-called "trade languages") but also for cultural, religious, dip

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A [...] is a language systematically used to make communication possible between people who do not share a native language

lingua franca

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A lingua franca ( / ˌ l ɪ ŋ ɡ w ə ˈ f r æ ŋ k ə / ; lit.  Frankish tongue ), [1] also known as a bridge language, common language, trade language or vehicular language, is a language or dialect system

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Lingua franca - Wikipedia
span>Lingua franca - Wikipedia Lingua franca From Wikipedia, the free encyclopedia Jump to: navigation, search For other uses, see Lingua franca (disambiguation). A lingua franca (/ˌlɪŋɡwə ˈfræŋkə/; lit.  Frankish tongue), [1] also known as a bridge language, common language, trade language or vehicular language, is a language or dialect systematically used to make communication possible between people who do not share a native language or dialect, particularly when it is a third language that is distinct from both native languages. [2] Lingua francas have developed around the world throughout human history, sometimes for commercial reasons (so-called "trade languages") but also for cultural, religious, dip

#### Annotation 1767493995788

 #germany #has-images The Weimar Republic (German: Weimarer Republik [ˈvaɪmaʁɐ ʁepuˈbliːk] ( [imagelink] listen ) ) is an unofficial, historical designation for the German state as it existed between 1919 and 1933.

#### Annotation 1767498976524

 Enlightened absolutism refers to the conduct and policies of European absolute monarchs during the 18th and 19th centuries who were influenced by the ideas of the Enlightenment.

Enlightened absolutism - Wikipedia
ated topics[show] Aristocracy Autocracy Crowned republic Conservatism Thomas Hobbes Legitimists Oligarchy Philosopher king Primogeniture Royalism Regicide Regnal number Royal family Ultra-royalist Politics portal v t e <span>Enlightened absolutism refers to the conduct and policies of European absolute monarchs during the 18th and 19th centuries who were influenced by the ideas of the Enlightenment. Contents [hide] 1 History 2 Political reforms 3 Major nations 4 Associated rulers 5 Chinese Legalism 6 See also 7 References 8 Further reading History[edit source] En

#### Flashcard 1767501073676

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[...] refers to the conduct and policies of European absolute monarchs during the 18th and 19th centuries

Enlightened absolutism

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Enlightened absolutism refers to the conduct and policies of European absolute monarchs during the 18th and 19th centuries who were influenced by the ideas of the Enlightenment.

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Enlightened absolutism - Wikipedia
ated topics[show] Aristocracy Autocracy Crowned republic Conservatism Thomas Hobbes Legitimists Oligarchy Philosopher king Primogeniture Royalism Regicide Regnal number Royal family Ultra-royalist Politics portal v t e <span>Enlightened absolutism refers to the conduct and policies of European absolute monarchs during the 18th and 19th centuries who were influenced by the ideas of the Enlightenment. Contents [hide] 1 History 2 Political reforms 3 Major nations 4 Associated rulers 5 Chinese Legalism 6 See also 7 References 8 Further reading History[edit source] En

#### Annotation 1767503957260

 #germany #has-images The Weimar Republic (German: Weimarer Republik [ˈvaɪmaʁɐ ʁepuˈbliːk] ( [imagelink] listen ) ) is an unofficial, historical designation for the German state as it existed between 1919 and 1933.

#### Annotation 1767508413708

 #history The rulers commonly known as the "Five Good Emperors" were Nerva, Trajan, Hadrian, Antoninus Pius and Marcus Aurelius.[4] The term was coined based on what the political philosopher Niccolò Machiavelli said in 1503:

Nerva–Antonine dynasty - Wikipedia
y his biological son, Commodus. [imagelink] Antoninus Pius [imagelink] Marcus Aurelius [imagelink] Lucius Verus [imagelink] Commodus Five Good Emperors[edit source] <span>The rulers commonly known as the "Five Good Emperors" were Nerva, Trajan, Hadrian, Antoninus Pius and Marcus Aurelius. [4] The term was coined based on what the political philosopher Niccolò Machiavelli said in 1503: From the study of this history we may also learn how a good government is to be established; for while all the emperors who succeeded to the throne by birth, except Titus, were bad,

#### Flashcard 1767510510860

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#history
Question

The term "Five Good Emperors" was coined based on what [...] said

Niccolò Machiavelli

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ad><head> The rulers commonly known as the "Five Good Emperors" were Nerva, Trajan, Hadrian, Antoninus Pius and Marcus Aurelius. [4] The term was coined based on what the political philosopher Niccolò Machiavelli said in 1503: <html>

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Nerva–Antonine dynasty - Wikipedia
y his biological son, Commodus. [imagelink] Antoninus Pius [imagelink] Marcus Aurelius [imagelink] Lucius Verus [imagelink] Commodus Five Good Emperors[edit source] <span>The rulers commonly known as the "Five Good Emperors" were Nerva, Trajan, Hadrian, Antoninus Pius and Marcus Aurelius. [4] The term was coined based on what the political philosopher Niccolò Machiavelli said in 1503: From the study of this history we may also learn how a good government is to be established; for while all the emperors who succeeded to the throne by birth, except Titus, were bad,

#### Annotation 1767514180876

 #rome The Roman Kingdom, or regal period, was the period of the ancient Roman civilization characterized by a monarchical form of government of the city of Rome and its territories.

Roman Kingdom - Wikipedia
Roman law Ius Imperium Mos maiorum Collegiality Auctoritas Roman citizenship Cursus honorum Senatus consultum Senatus consultum ultimum Assemblies Centuriate Curiate Plebeian Tribal Other countries Atlas v t e <span>The Roman Kingdom, or regal period, was the period of the ancient Roman civilization characterized by a monarchical form of government of the city of Rome and its territories. Little is certain about the history of the kingdom, as nearly no written records from that time survive, and the histories about it that were written during the Republic and Empire ar

#### Flashcard 1767516278028

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The [...] was the period of the ancient Roman civilization characterized by a monarchical form of government

Roman Kingdom

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The Roman Kingdom, or regal period, was the period of the ancient Roman civilization characterized by a monarchical form of government of the city of Rome and its territories. </h

#### Original toplevel document

Roman Kingdom - Wikipedia
Roman law Ius Imperium Mos maiorum Collegiality Auctoritas Roman citizenship Cursus honorum Senatus consultum Senatus consultum ultimum Assemblies Centuriate Curiate Plebeian Tribal Other countries Atlas v t e <span>The Roman Kingdom, or regal period, was the period of the ancient Roman civilization characterized by a monarchical form of government of the city of Rome and its territories. Little is certain about the history of the kingdom, as nearly no written records from that time survive, and the histories about it that were written during the Republic and Empire ar

#### Flashcard 1767518637324

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#germany
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The Weimar Republic began in [...].
1919

I'd say the government is tip-top!

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The Weimar Republic (German: Weimarer Republik [ˈvaɪmaʁɐ ʁepuˈbliːk] ( [imagelink] listen ) ) is an unofficial, historical designation for the German state as it existed between 1919 and 1933.

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Weimar Republic - Wikipedia

#### Annotation 1767522307340

 #rome The Roman Republic (Latin: Res publica Romana ; Classical Latin: [ˈreːs ˈpuːb.lɪ.ka roːˈmaː.na] ) was the era of ancient Roman civilization beginning with the overthrow of the Roman Kingdom, traditionally dated to 509 BC, and ending in 27 BC with the establishment of the Roman Empire.

Roman Republic - Wikipedia
art of a series on Ancient Rome and the fall of the Republic Mark Antony Cleopatra VII Assassination of Julius Caesar Pompey Theatre of Pompey Cicero First Triumvirate Roman Forum Comitium Rostra Curia Julia Curia Hostilia v t e <span>The Roman Republic (Latin: Res publica Romana; Classical Latin: [ˈreːs ˈpuːb.lɪ.ka roːˈmaː.na]) was the era of ancient Roman civilization beginning with the overthrow of the Roman Kingdom, traditionally dated to 509 BC, and ending in 27 BC with the establishment of the Roman Empire. It was during this period that Rome's control expanded from the city's immediate surroundings to hegemony over the entire Mediterranean world. Roman government was headed by two consu

#### Flashcard 1767524404492

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#rome
Question
The Roman Republic began in [...]
509 BC

Apparently the Romans are beginning to lace up.

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> The Roman Republic (Latin: Res publica Romana ; Classical Latin: [ˈreːs ˈpuːb.lɪ.ka roːˈmaː.na] ) was the era of ancient Roman civilization beginning with the overthrow of the Roman Kingdom, traditionally dated to 509 BC, and ending in 27 BC with the establishment of the Roman Empire. <span><body><html>

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Roman Republic - Wikipedia
art of a series on Ancient Rome and the fall of the Republic Mark Antony Cleopatra VII Assassination of Julius Caesar Pompey Theatre of Pompey Cicero First Triumvirate Roman Forum Comitium Rostra Curia Julia Curia Hostilia v t e <span>The Roman Republic (Latin: Res publica Romana; Classical Latin: [ˈreːs ˈpuːb.lɪ.ka roːˈmaː.na]) was the era of ancient Roman civilization beginning with the overthrow of the Roman Kingdom, traditionally dated to 509 BC, and ending in 27 BC with the establishment of the Roman Empire. It was during this period that Rome's control expanded from the city's immediate surroundings to hegemony over the entire Mediterranean world. Roman government was headed by two consu

#### Annotation 1767526763788

 #ancient-history #history #roman-empire #rome #wiki Octavian's power was then unassailable and in 27 BC the Roman Senate formally granted him overarching power and the new title Augustus, effectively marking the end of the Roman Republic.

Roman Empire - Wikipedia
perpetual dictator and then assassinated in 44 BC. Civil wars and executions continued, culminating in the victory of Octavian, Caesar's adopted son, over Mark Antony and Cleopatra at the Battle of Actium in 31 BC and the annexation of Egypt. <span>Octavian's power was then unassailable and in 27 BC the Roman Senate formally granted him overarching power and the new title Augustus, effectively marking the end of the Roman Republic. The imperial period of Rome lasted approximately 1,500 years compared to the 500 years of the Republican era. The first two centuries of the empire's existence were a period of unprec

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#ancient-history #history #roman-empire #rome #wiki
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the Roman Senate formally granted Octavian overarching power and the new title Augustus in [...]

27 BC

Octavian had the Senate at its neck.

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Octavian's power was then unassailable and in 27 BC the Roman Senate formally granted him overarching power and the new title Augustus, effectively marking the end of the Roman Republic.

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Roman Empire - Wikipedia
perpetual dictator and then assassinated in 44 BC. Civil wars and executions continued, culminating in the victory of Octavian, Caesar's adopted son, over Mark Antony and Cleopatra at the Battle of Actium in 31 BC and the annexation of Egypt. <span>Octavian's power was then unassailable and in 27 BC the Roman Senate formally granted him overarching power and the new title Augustus, effectively marking the end of the Roman Republic. The imperial period of Rome lasted approximately 1,500 years compared to the 500 years of the Republican era. The first two centuries of the empire's existence were a period of unprec

#### Annotation 1767533841676

 #Biochemistry #voets • The nitrogenous bases of nucleotides include two types of purines and three types of pyrimidines. • A nucleotide consists of a nitrogenous base, a ribose or deoxyribose sugar, and one or more phosphate groups. • DNA contains adenine, guanine, cytosine, and thymine deoxyribonucleotides, whereas RNA contains adenine, guanine, cytosine, and uracil ribonucleotides.

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#### Annotation 1767538560268

 #Biochemistry #voets The most common purines are adenine (A) and guanine (G), and the major pyrimidines are cytosine (C), uracil (U), and thymine (T). The purines form bonds to a five-carbon sugar (a pentose) via their N9 atoms, whereas pyrimidines do so through their N1 atoms (Table 3-1). In ribonucleotides, the pentose is ribose, while in deoxyribonucleotides (or just deoxynucleotides), the sugar is 2ⴕ-deoxyribose (i.e., the carbon at position 2¿ lacks a hydroxyl group)

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#### Annotation 1767540919564

 #Biochemistry #voets A nucleoside consists only of a base and a pentose.

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#### Annotation 1767543278860

 #Biochemistry #voets In a ribonucleotide or a deoxyribonucleotide, one or more phosphate groups are bonded to atom C3¿ or atom C5¿ of the pentose to form a 3¿-nucleotide or a 5¿-nucleotide, respectively (Fig. 3-1). When the phosphate group is absent, the compound is known as a nucleoside. A 5¿-nucleotide can therefore be called a nucleoside-5¿-phosphate. Nucleotides most commonly

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#### Annotation 1767545638156

 #Biochemistry #voets contain one to three phosphate groups at the C5¿ position and are called nu- cleoside monophosphates, diphosphates, and triphosphates.

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#### Annotation 1767547997452

 #Biochemistry #voets Perhaps the best known nucleotide is adenosine triphosphate (ATP), a nucleotide containing adenine, ribose, and a triphosphate group. ATP is often mistakenly referred to as an energy-storage molecule, but it is more accurately termed an energy carrier or energy transfer agent.

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#### Annotation 1767550618892

 #Biochemistry #voets Identify the purines and pyrimidines commonly found in nucleic acids. • Practice drawing the structures of adenine, adenosine, and adenylate. • Describe the chemical differences between a ribonucleoside triphosphate and a deoxyribonucleoside monophosphate

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#### Annotation 1767552978188

 #Biochemistry #voets starch synthesis in plants proceeds by repeated additions of glu- cose units donated by ADP–glucose

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#### Annotation 1767555337484

 #Biochemistry #voets The phosphates of these polynucleotides are acidic, so at physiological pH, nucleic acids are polyanions. The linkage between individual nucleotides is known as a phos- phodiester bond, so named because the phosphate is esterified to two ribose units.

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#### Annotation 1767557696780

 #Biochemistry #voets The terminal residue whose C5¿ is not linked to another nucleotide is called the 5¿ end, and the terminal residue whose C3¿ is not linked to another nucleotide is called the 3¿ end. By convention, the sequence of nucleotide residues in a nucleic acid is written, left to right, from the 5¿ end to the 3¿ end.

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#### Annotation 1767560056076

 #Biochemistry #voets Although there appear to be no rules governing the nucleotide composition of typical RNA molecules, DNA has equal numbers of adenine and thymine residues (A ⫽ T) and equal numbers of guanine and cytosine residues (G ⫽ C). These relation- ships, known as Chargaff’s rules, were discovered in the late 1940s by Erwin Chargaff, who devised the first reliable quantitative methods for the composi- tional analysis of DNA.

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#### Annotation 1767562415372

 #Biochemistry #voets Watson and Crick’s accomplishment, which is ranked as one of science’s major intellectual achievements, was based in part on two pieces of evidence in addition to Chargaff’s rules: the correct tautomeric forms of the bases and indications that DNA is a helical molecule. The purine and pyrimidine bases of nucleic acids can assume different tautomeric forms (tautomers are easily converted isomers that differ only in hydrogen positions; Fig. 3-4). X-Ray, nuclear magnetic resonance (NMR), and spectroscopic investigations have firmly established that the nucleic acid bases are overwhelmingly in the keto tautomeric forms

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#### Annotation 1767564774668

 #Biochemistry #voets Many bacteria are able to resist infection by bacteriophages (viruses that are specific for bacteria) by virtue of a restriction–modification system. The bac- terium modifies certain nucleotides in specific sequences of its own DNA by adding a methyl (¬CH 3 ) group in a reaction catalyzed by a modification methylase. A restriction endonuclease, which recognizes the same nucleotide sequence as does the methylase, cleaves any DNA that has not been modified on at least one of its two strands. (An endonuclease cleaves a nucleic acid within the polynucleotide strand; an exonuclease cleaves a nucleic acid by re- moving one of its terminal residues.) This system destroys foreign (phage) DNA containing a recognition site that has not been modified by methyla- tion. The host DNA is always at least half methylated, because although the daughter strand is not methylated until shortly after it is synthesized, the parental strand to which it is paired is already modified (and thus protects both strands of the DNA from cleavage by the restriction enzyme).

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#### Annotation 1767567133964

 #Biochemistry #voets Negatively charged molecules such as DNA migrate through the gel matrix toward the anode in response to an applied electric field. Because smaller molecules move faster, the molecules in each lane are separated according to size. Following electrophoresis, the separated molecules may be visualized by staining or fluorescence

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#### Annotation 1767569493260

 #Biochemistry #voets Until recent years, the most widely used technique for sequencing DNA was the chain-terminator method, which was devised by Frederick Sanger. The first step in this procedure is to obtain single polynucleotide strands. Complementary DNA strands can be separated by heating, which breaks the hydrogen bonds between bases. Next, polynucleotide fragments that terminate at positions corresponding to each of the four nucleotides are generated. Finally, the fragments are separated and detected.

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#### Annotation 1767571852556

 #Biochemistry #voets The chain-terminator method (also called the dideoxy method) uses an E. coli enzyme to make complementary copies of the single-stranded DNA being sequenced. The en- zyme is a fragment of DNA polymerase I, one of the enzymes that partic- ipates in replication of bacterial DNA (Section 25-2A). Using the single DNA strand as a template, DNA polymerase I assembles the four deoxynu- cleoside triphosphates (dNTPs), dATP, dCTP, dGTP, and dTTP, into a com- plementary polynucleotide chain that it elongates in the 5¿S3¿ direction (Fig. 3-18). DNA polymerase I can sequentially add deoxynucleotides only to the 3¿ end of a polynucleotide. Hence, replication is initiated in the presence of a short polynucleotide (a primer) that is complementary to the 3¿ end of the template DNA and thus becomes the 5¿ end of the new strand. The primer base-pairs with the template strand, and nucleotides are sequentially added to the 3¿ end of the primer. If the DNA being sequenced is a restriction frag- ment, as it usually is, it begins and ends with a restriction site.

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#### Annotation 1767574211852

 #Biochemistry #voets In the chain-terminator technique (Fig. 3-19), the DNA to be sequenced is incubated with DNA poly- merase I, a suitable primer, and the four dNTP substrates (reactants in enzy- matic reactions) for the polymerization reaction. The key component of the reaction mixture is a small amount of a 2ⴕ,3ⴕ-dideoxynucleoside triphos- phate (ddNTP), which lacks the 3¿-OH group of deoxynucleotides.When the dideoxy analog is incorporated into the growing polynucleotide in place of the corresponding normal nucleotide, chain growth is terminated because addition of the next nucleotide re- quires a free 3¿-OH.

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#### Annotation 1767576571148

 #Biochemistry #voets By using only a small amount of the ddNTP, a series of truncated chains is generated, each of which ends with the dideoxy analog at one of the positions occupied by the corresponding base. Each ddNTP bears a different fluorescent “tag” so that the products of the polymerase reaction can be readily detected. Gel electrophoresis separates the newly synthesized DNA segments, which differ in size by one nucleotide. Thus, the sequence of the replicated strand can be di- rectly read from the gel. Note that the sequence obtained by the chain-termi- nator method is complementary to the DNA strand being sequenced

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#### Annotation 1767578930444

 #Biochemistry #voets In pyrosequencing, molecules of the template DNA are immobilized on the surfaces of microscopic plastic beads that are deposited in small wells in a fiber-optic slide with one bead per well. A primer and DNA polymerase are added, and then a dNTP substrate is introduced. If DNA polymerase adds that nucleotide to the new DNA strand, pyrophosphate is released and trig- gers a chemical reaction involving the firefly enzyme luciferase, which gener- ates a flash of light. Solutions of each of the four dNTPs are successively washed across the immobilized DNA template, and a detector records whether light is produced in the presence of a particular dNTP. In this way, the se- quence of nucleotides complementary to the template strand can be deduced, and no electrophoretic separation is needed.

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#### Annotation 1767581289740

 #Biochemistry #voets In metagenomic sequencing, the DNA sequences of multiple organisms are analyzed as a single data set. This approach is used to characterize com- plex microbial communities, such as those in marine environments, where in- dividual species—including many not yet identified—cannot be cultured and sequenced one by one. Metagenomic sequence data reveal the overall gene number and an estimate of the collective metabolic capabilities of the com- munity. Over 3 million genes have been identified in a metagenomic analysis of the microorganisms that inhabit the human gut, representing some 1000 bacterial species. While most humans share a common core set of about 60 gut microorganisms, significant differences appear to correlate with metabolic variables such as body mass.

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#### Annotation 1767583649036

 #Biochemistry #voets Rather, it appears that vertebrate proteins them- selves are more complex than those of invertebrates; that is, vertebrate proteins tend to have more domains (modules) than invertebrate proteins, and these modules are more often selectively expressed through alternative gene splic- ing (a phenomenon in which a given gene transcript can be processed in multiple ways so as to yield different proteins when translated; Section 26-3B). In fact, most vertebrate genes encode several different although similar proteins.

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#### Annotation 1767586008332

 #Biochemistry #voets Phylogenetic relationships can be revealed by comparing the sequences of similar genes in different organisms. The number of nucleotide differences be- tween the corresponding genes in two species roughly indicates the degree to which the species have diverged through evolution. The regrouping of prokary- otes into archaea and bacteria (Section 1-2C) according to rRNA sequences present in all organisms illustrates the impact of sequence analysis.

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#### Annotation 1767588367628

 #Biochemistry #voets Recombinant DNA technology, also called molecular cloning or genetic engineering, makes it possible to isolate, amplify, and mod- ify specific DNA sequences.

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#### Annotation 1767593086220

 #Biochemistry #voets Ligase Joins Two DNA Segments. A DNA segment to be cloned is often obtained through the action of restriction endonucleases. Most restriction en- zymes cleave DNA to yield sticky ends (Section 3-4A). Therefore, as Janet Mertz and Ron Davis first demonstrated in 1972, a restriction fragment can be inserted into a cut made in a cloning vector by the same restriction enzyme (Fig. 3-24). The complementary ends of the two DNAs form base pairs (an- neal) and the sugar–phosphate backbones are covalently ligated, or spliced together, through the action of an enzyme named DNA ligase. (A ligase pro- duced by a bacteriophage can also join blunt-ended restriction fragments.) A great advantage of using a restriction enzyme to construct a recombinant DNA molecule is that the DNA insert can later be precisely excised from the cloned vector by cleaving it with the same restriction enzyme.

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#### Annotation 1767595969804

 #Biochemistry #voets It is essential to select only those host organisms that have been transformed and that contain a properly constructed vector. In the case of plasmid transfor- mation, selection can be accomplished through the use of antibiotics and/or chromogenic (color-producing) substances. For example, the lacZ gene in the pUC18 plasmid (see Fig. 3-22) encodes the enzyme ␤-galactosidase, which cleaves the colorless compound X-gal to a blue product:

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#### Annotation 1767598329100

 #Biochemistry #voets Cells of E. coli that have been transformed by an unmodified pUC18 plasmid form blue colonies. However, if the plasmid contains a foreign DNA insert in its polylinker region, the colonies are colorless because the insert interrupts the protein-coding sequence of the lacZ gene and no functional ␤-galactosidase is produced. Bacteria that have failed to take up any plasmid are also colorless due to the absence of ␤-galactosidase, but these cells can be excluded by adding the antibiotic ampicillin to the growth medium (the plasmid includes the gene amp R , which confers ampicillin resistance). Thus, successfully transformed cells

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#### Annotation 1767600688396

 #Biochemistry #voets The cloned set of all DNA fragments from a particular organism is known as its genomic library. Genomic libraries are generated by a procedure known as shotgun cloning. The chromosomal DNA of the organism is isolated, cleaved to frag- ments of cloneable size, and inserted into a cloning vector. The DNA is usu- ally fragmented by partial rather than exhaustive restriction digestion so that the genomic library contains intact representatives of all the organism’s genes, including those that contain restriction sites. DNA in solution can also be me- chanically fragmented (sheared) by rapid stirring.

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#### Annotation 1767603047692

 #Biochemistry #voets A different type of DNA library contains only the expressed sequences from a particular cell type. Such a cDNA library is constructed by isolating all the cell’s mRNAs and then copying them to DNA using a specialized type of DNA polymerase known as reverse transcriptase because it synthesizes DNA using RNA templates (Box 25-2). The complementary DNA (cDNA) molecules are then inserted into cloning vectors to form a cDNA library. A cDNA library can also be used to construct a DNA microarray (DNA chip), in which each different cDNA is immobilized at a specific position on a slide. A DNA chip can be used for detecting the presence of mRNA in a biological sample (the mRNA, if pres- ent, will bind to its complementary cDNA

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#### Annotation 1767605406988

 #Biochemistry #voets Colonies are transferred from a “master” culture plate by replica plating. Clones containing the DNA of interest are identified by the ability to bind a specific probe.

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#### Annotation 1767607766284

 #Biochemistry #voets In PCR, a DNA sample is separated into single strands and incubated with DNA polymerase, dNTPs, and two oligonucleotide primers whose sequences flank the DNA segment of interest. The primers direct the DNA polymerase to synthesize com- plementary strands of the target DNA (Fig. 3-27). Multiple cycles of this process, each doubling the amount of the target DNA, geometrically amplify the DNA starting from as little as a single gene copy. In each cycle, the two strands of the duplex DNA are separated by heating, then the reaction mixture is cooled to allow the primers to anneal to their complementary segments on the DNA. Next, the DNA polymerase directs the synthesis of the complementary strands. The use of a heat-stable DNA polymerase, such as Taq polymerase isolated from Thermus aquaticus, a bacterium that thrives at 75°C, eliminates the need to add fresh enzyme after each round of heating (heat inactivates most en- zymes). Hence, in the presence of sufficient quantities of primers and dNTPs, PCR is carried out simply by cyclically varying the temperature.

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#### Annotation 1767610125580

 #Biochemistry #voets Bacteria can produce eukaryotic proteins only if the recombinant DNA that carries the protein-coding sequence also includes bacterial transcriptional and translational control sequences. Synthesis of eukaryotic proteins in bacte- ria also presents other problems. For example, many eukaryotic genes are large and contain stretches of nucleotides (introns) that are transcribed and excised before translation (Section 26-3A); bacteria lack the machinery to excise the introns. In addition, many eukaryotic proteins are posttranslationally modi- fied by the addition of carbohydrates or by other reactions. These problems can be overcome by using expression vectors that propagate in eukaryotic hosts, such as yeast or cultured insect or animal cells.

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#### Annotation 1767612484876

 #Biochemistry #voets process of evolution and allows predictions about the structural and functional roles of particular amino acids in a protein to be rigorously tested in the laboratory

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#### Annotation 1767614844172

 An open source package is usually distributed as a compressed archive containing all source code and header files plus build scripts and documentation. A dev package , sometimes referred to as a -devel package by Linux package managers, is usually dis- tributed as an archive containing a lib plus its associated header files

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#### Annotation 1767616417036

 This device driver understands the device controller and provides the rest of the operating system with a uniform interface to the device

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of special-purpose registers. The device controller is responsible for moving the data between the peripheral devices that it controls and its local buffer storage. Typically, operating systems have a device driver for each device controller. <span>This device driver understands the device controller and provides the rest of the operating system with a uniform interface to the device. To start an I/O operation, the device driver loads the appropriate registers within the device controller. The device controller, in turn, examines the contents of these registers to d

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