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Question

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

Answer

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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#gaussian-process

Question

Answer

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>

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

Tags

#topology

Question

An affine transformation preserve [...] between points lying on a straight line.

Answer

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.

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

Tags

#sprectral-theorem #stochastics

Question

the **Karhunen–Loève theorem** represents a stochastic process as [...]

Answer

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>

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

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

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

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#logic

Question

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

Answer

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

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

Tags

#logic

Question

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

Answer

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>

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

Question

Answer

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

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

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#state-space-models

Question

state space models are Markov models with [...]

Answer

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.

Tags

#bayesian-ml #prml

Question

the conditional distributions of the observed variables given the hidden variables are known as [...]

Answer

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

Tags

#measure-theory #stochastics

Question

abstract integration gives the same result as [...] when the latter exists

Answer

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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#graphical-models

Question

A [...] is a model over an undirected graph.

Answer

Markov random field

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

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

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#bayesian-network

Question

In a Bayesian network each node takes **[...]** as input

Answer

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>

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

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#forward-backward-algorithm #hmm

Question

The foreward-backward algorithm makes use of the principle of [...] in its two passes.

Answer

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

| 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

Tags

#forward-backward-algorithm #hmm

Question

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

Answer

.

*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:

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

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.

Answer

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.

Tags

#measure-theory #stochastics

Question

A sigma-algebra for Ω must contain **[...]**

Answer

Ω

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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.

Tags

#measure-theory #stochastics

Question

A sigma-algebra for Ω is closed under [...].

Answer

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.

Tags

#measure-theory #stochastics

Question

Lebesgue measure for a set A on R corresponds to **[...]** of ordinary calculus

Answer

dx

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Tags

#poisson-process #stochastics

Question

a homogeneous Poisson point process has a parameter of the form [...]

Answer

,

*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.

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

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#topology

Question

Answer

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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.

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

Tags

#inner-product-space #vector-space

Question

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

Answer

a norm or inner product

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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.

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.

#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.

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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

Tags

#topology

Question

Answer

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.

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

#borel-algebra #measure-theory

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

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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

Tags

#borel-algebra #measure-theory

Question

[...] on *X* is the smallest σ-algebra containing all open sets

Answer

The Borel algebra

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

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

Tags

#borel-algebra #measure-theory

Question

The Borel algebra on *X* is the [...] containing all open sets

Answer

smallest σ-algebra

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

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

#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.

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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

Tags

#borel-algebra #measure-theory

Question

any measure defined on the open sets of a space, or on the closed sets of a space, must also be defined on [...].

Answer

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.

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

Tags

#metric-space #topological-space

Question

"metric" is a generalization of [...]

Answer

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.

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

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

l space. There are, however, topological spaces that are not metric spaces. Properties[edit] 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[edit] Open sets have a fundamental im

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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.

l space. There are, however, topological spaces that are not metric spaces. Properties[edit] 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[edit] Open sets have a fundamental im

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Tags

#has-images #lagrange-multiplier #optimization

Question

At the stationary point the vector ∇f(x) is orthogonal to the constraint surface because otherwise [...]

Answer

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

Tags

#calculus-of-variations

Question

the weak formulation of the necessary condition of extremum is an [...form...]

Answer

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 .

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

Tags

#functional-analysis

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Answer

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

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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Answer

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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

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

Tags

#calculus

Question

The **total derivative** is the [...description...]

Answer

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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.

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

Tags

#calculus

Question

The **total derivative** taking into account the exogenous argument's **[...]** to the function

Answer

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>

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

Tags

#calculus

Question

Answer

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

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

Tags

#incremental-reading

Question

Answer

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 .

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

Tags

#incremental-reading

Question

wording must be optimized to ensure [...].

Answer

minimum-effort recall

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

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:

Tags

#measure-theory #stochastics

Question

The Lebesgue measure for a set A on R is **[...formula...]**

Answer

\( \mu(A) = \int_A d\mu \)

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#topology

Question

A complement of an open set is always relative to **[...]**

Answer

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.

l space. There are, however, topological spaces that are not metric spaces. Properties[edit] 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[edit] Open sets have a fundamental im

Tags

#calculus

Question

Answer

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

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

Tags

#logic

Question

The Aristotelian conceptions of language and metaphysics fell out of favour in [...when...] with the rise of a new scientific paradigm.

Answer

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>

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

#has-images

羽绒服脏了怎么洗？8招教你轻松搞定！

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！

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羽绒服脏了怎么洗？8招教你轻松搞定！ - 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 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！ 标签：洗羽绒服 相关推荐 [imagelink]除了硬给钱，软银还喜欢用什么方式来投资它看中的公司？ [imagelink]图书馆艳遇 [imagelink]保卫龙脉大作战 [imagelink]一口气印度史（1） [imagelink]我家25万的装修，什么水平？ [imagelink]那本杨国强想烧掉的书，到底写了什么？ [imagel

#politics

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[imagelink] Atrocity propaganda From Wikipedia, the free encyclopedia Jump to: navigation, search <span>Atrocity propaganda is the spreading information about the crimes committed by an enemy, especially deliberate fabrications or exaggerations. [citation needed] It is a form of psychological warfare. [citation needed] The inherently violent nature of war means that exaggeration and invention of atrocities often becomes the

Tags

#politics

Question

Answer

Atrocity propaganda

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

[imagelink] Atrocity propaganda From Wikipedia, the free encyclopedia Jump to: navigation, search <span>Atrocity propaganda is the spreading information about the crimes committed by an enemy, especially deliberate fabrications or exaggerations. [citation needed] It is a form of psychological warfare. [citation needed] The inherently violent nature of war means that exaggeration and invention of atrocities often becomes the

Tags

#politics

Question

Answer

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.

[imagelink] Atrocity propaganda From Wikipedia, the free encyclopedia Jump to: navigation, search <span>Atrocity propaganda is the spreading information about the crimes committed by an enemy, especially deliberate fabrications or exaggerations. [citation needed] It is a form of psychological warfare. [citation needed] The inherently violent nature of war means that exaggeration and invention of atrocities often becomes the

#english-literature

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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

Tags

#english-literature

Question

Answer

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.

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

#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.

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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

Tags

#history

Question

The **Hundred Years' War** started in [...]

Answer

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. </

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

Tags

#history

Question

The **Hundred Years' War** ended in [...]

Answer

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

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

**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]}

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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

Question

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

Answer

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

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

#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.

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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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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

Question

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

Answer

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.

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

#germany #has-images

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#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:

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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,

Tags

#history

Question

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

Answer

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>

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,

#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.

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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

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#rome

Question

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

Answer

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

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

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#germany

Question

The **Weimar Republic** began in [...].

Answer

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.

#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.

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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

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#rome

Question

The **Roman Republic** began in [...]

Answer

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>

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

#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.

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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

Tags

#ancient-history #history #roman-empire #rome #wiki

Question

the Roman Senate formally granted Octavian overarching power and the new title *Augustus* in [...]

Answer

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.

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

#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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#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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#Biochemistry #voets

A nucleoside consists only of a base and a pentose.

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#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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#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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#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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#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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#Biochemistry #voets

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

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#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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#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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#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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#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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#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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#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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#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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#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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#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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#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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#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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#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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#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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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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#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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#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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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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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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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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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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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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#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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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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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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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