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#### Flashcard 1735803669772

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#logic
Question
A History of Formal Logic (1961) is written by the distinguished [...]
J M Bocheński

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According to A History of Formal Logic (1961) by the distinguished J M Bocheński, the golden periods for logic were the ancient Greek period, the medieval scholastic period, and the mathematical period of the 19th and 20th centuries.

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The rise and fall and rise of logic | Aeon Essays
the hands of thinkers such as George Boole, Gottlob Frege, Bertrand Russell, Alfred Tarski and Kurt Gödel, it’s clear that Kant was dead wrong. But he was also wrong in thinking that there had been no progress since Aristotle up to his time. <span>According to A History of Formal Logic (1961) by the distinguished J M Bocheński, the golden periods for logic were the ancient Greek period, the medieval scholastic period, and the mathematical period of the 19th and 20th centuries. (Throughout this piece, the focus is on the logical traditions that emerged against the background of ancient Greek logic. So Indian and Chinese logic are not included, but medieval Ara

#### Flashcard 1738022194444

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#bayesian-ml #has-images #prml
Question
The marginal distribution for The initial latent node $$z_1$$ is [...]
[unknown IMAGE 1738019573004]

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#### Flashcard 1739069984012

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#d-separation
Question
The only twist on this simple idea of "connecting path" is that we are dealing with a system of directed arrows in which some vertices correspond to [...], whose values are known precisely.
measured variables

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or "separation". The only twist on this simple idea is to define what we mean by "connecting path", given that we are dealing with a system of directed arrows in which some vertices (those residing in Z) correspond to <span>measured variables, whose values are known precisely. To account for the orientations of the arrows we use the terms "d-separated" and "d-connected" (d connotes "directional"

#### Original toplevel document

Unknown title
d-SEPARATION WITHOUT TEARS (At the request of many readers) Introduction d-separation is a criterion for deciding, from a given a causal graph, whether a set X of variables is independent of another set Y, given a third set Z. <span>The idea is to associate "dependence" with "connectedness" (i.e., the existence of a connecting path) and "independence" with "unconnected-ness" or "separation". The only twist on this simple idea is to define what we mean by "connecting path", given that we are dealing with a system of directed arrows in which some vertices (those residing in Z) correspond to measured variables, whose values are known precisely. To account for the orientations of the arrows we use the terms "d-separated" and "d-connected" (d connotes "directional"). We start by considering separation between two singleton variables, x and y; the extension to sets of variables is straightforward (i.e., two sets are separated if and only if each el

#### Annotation 1758412803340

 know what the very best possible deal is? It’s leaving some meat in the current deal for the broker, so that he gives you the next great deal that comes along.

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#### Annotation 1758414376204

 Do What You Say You Will Do

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#### Annotation 1758415949068

 Did you say that you’d make an offer on the property by the end of the day? Then do it. If you don’t, the seller or broker will wonder if you’re interested in the deal.

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#### Annotation 1758417784076

 f you get in their mind in the first place, make sure you turn that space into positive thoughts. When you do this, you’ll get the rare reputation of a performer

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#### Annotation 1758419356940

 Don’t Be a Pain in the Butt

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#### Annotation 1758420929804

 get the very best deal possible. But after the negotiation phase is over and you’ve shaken hands, do the deal.

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#### Annotation 1758422502668

 you should renegotiate when a major unknown problem comes up during the property inspection. These items include bad roofs, boiler problems, erosion problems, and so on. They will cost some serious money to fix.

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#### Annotation 1758466805004

 #topology Being so general, topological spaces are a central unifying notion and appear in virtually every branch of modern mathematics.

Topological space - Wikipedia
tical space that allows for the definition of concepts such as continuity, connectedness, and convergence.  Other spaces, such as manifolds and metric spaces, are specializations of topological spaces with extra structures or constraints. <span>Being so general, topological spaces are a central unifying notion and appear in virtually every branch of modern mathematics. The branch of mathematics that studies topological spaces in their own right is called point-set topology or general topology. Contents [hide] 1 History 2 Definition 2.1 De

#### Flashcard 1758469688588

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#topology
Question
Being so general, [...] are a central unifying notion and appear in virtually every branch of modern mathematics.
topological spaces

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Being so general, topological spaces are a central unifying notion and appear in virtually every branch of modern mathematics.

#### Original toplevel document

Topological space - Wikipedia
tical space that allows for the definition of concepts such as continuity, connectedness, and convergence.  Other spaces, such as manifolds and metric spaces, are specializations of topological spaces with extra structures or constraints. <span>Being so general, topological spaces are a central unifying notion and appear in virtually every branch of modern mathematics. The branch of mathematics that studies topological spaces in their own right is called point-set topology or general topology. Contents [hide] 1 History 2 Definition 2.1 De

#### Annotation 1758472572172

 #topology In topology, a branch of mathematics, a topological manifold is a topological space (which may also be a separated space) which locally resembles real n-dimensional space in a sense defined below.

Topological manifold - Wikipedia
Topological manifold - Wikipedia Topological manifold From Wikipedia, the free encyclopedia Jump to: navigation, search In topology, a branch of mathematics, a topological manifold is a topological space (which may also be a separated space) which locally resembles real n-dimensional space in a sense defined below. Topological manifolds form an important class of topological spaces with applications throughout mathematics. A manifold can mean a topological manifold, or more frequently, a topolog

#### Flashcard 1758474931468

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#topology
Question
a [...] is a topological space which locally resembles real n-dimensional space in some sense
topological manifold

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In topology, a branch of mathematics, a topological manifold is a topological space (which may also be a separated space) which locally resembles real n-dimensional space in a sense defined below.

#### Original toplevel document

Topological manifold - Wikipedia
Topological manifold - Wikipedia Topological manifold From Wikipedia, the free encyclopedia Jump to: navigation, search In topology, a branch of mathematics, a topological manifold is a topological space (which may also be a separated space) which locally resembles real n-dimensional space in a sense defined below. Topological manifolds form an important class of topological spaces with applications throughout mathematics. A manifold can mean a topological manifold, or more frequently, a topolog

#### Annotation 1758480436492

 #function-space In topological spaces, a function f is a function is defined on the sets X and Y, but the continuity of f depends on the topologies used on X and Y.

Continuous function - Wikipedia
∈ X | f ( x ) ∈ V } {\displaystyle f^{-1}(V)=\{x\in X\;|\;f(x)\in V\}} is an open subset of X. That is, <span>f is a function between the sets X and Y (not on the elements of the topology T X ), but the continuity of f depends on the topologies used on X and Y. This is equivalent to the condition that the preimages of the closed sets (which are the complements of the open subsets) in Y are closed in X. An extreme example: if a set X is giv

#### Annotation 1758482533644

 #function-space A function between two topological spaces X and Y is continuous if for every open set V ⊆ Y, the inverse image is an open subset of X.

Continuous function - Wikipedia
intersections that generalize the properties of the open balls in metric spaces while still allowing to talk about the neighbourhoods of a given point. The elements of a topology are called open subsets of X (with respect to the topology). <span>A function f : X → Y {\displaystyle f\colon X\rightarrow Y} between two topological spaces X and Y is continuous if for every open set V ⊆ Y, the inverse image f − 1 ( V ) = { x ∈ X | f ( x ) ∈ V } {\displaystyle f^{-1}(V)=\{x\in X\;|\;f(x)\in V\}} is an open subset of X. That is, f is a function between the sets X and Y (not on the elements of the topology T X ), but the continuity of f depends on the topologies used on X and Y. This is equivalent to

#### Flashcard 1758485417228

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#function-space
Question

A function between two topological spaces X and Y is continuous if for every open set VY, [...].

the inverse image is an open subset of X

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A function between two topological spaces X and Y is continuous if for every open set V ⊆ Y, the inverse image is an open subset of X.

#### Original toplevel document

Continuous function - Wikipedia
intersections that generalize the properties of the open balls in metric spaces while still allowing to talk about the neighbourhoods of a given point. The elements of a topology are called open subsets of X (with respect to the topology). <span>A function f : X → Y {\displaystyle f\colon X\rightarrow Y} between two topological spaces X and Y is continuous if for every open set V ⊆ Y, the inverse image f − 1 ( V ) = { x ∈ X | f ( x ) ∈ V } {\displaystyle f^{-1}(V)=\{x\in X\;|\;f(x)\in V\}} is an open subset of X. That is, f is a function between the sets X and Y (not on the elements of the topology T X ), but the continuity of f depends on the topologies used on X and Y. This is equivalent to

#### Flashcard 1758488825100

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#function-space
Question
In topological spaces, a function f is defined on [...] , but the continuity of f depends on [...]
the sets X and Y, the topologies used on X and Y.

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In topological spaces, a function f is a function is defined on the sets X and Y, but the continuity of f depends on the topologies used on X and Y.

#### Original toplevel document

Continuous function - Wikipedia
∈ X | f ( x ) ∈ V } {\displaystyle f^{-1}(V)=\{x\in X\;|\;f(x)\in V\}} is an open subset of X. That is, <span>f is a function between the sets X and Y (not on the elements of the topology T X ), but the continuity of f depends on the topologies used on X and Y. This is equivalent to the condition that the preimages of the closed sets (which are the complements of the open subsets) in Y are closed in X. An extreme example: if a set X is giv

#### Annotation 1758491184396

 #topology A topological space is an ordered pair (X, τ), where X is a set and τ is a collection of subsets of X, satisfying the following axioms: The empty set and X itself belong to τ.Any (finite or infinite) union of members of τ still belongs to τ.The intersection of any finite number of members of τ still belongs to τ. The elements of τ are called open sets and the collection τ is called a topology on X.

Topological space - Wikipedia
three-point set {1,2,3}. The bottom-left example is not a topology because the union of {2} and {3} [i.e. {2,3}] is missing; the bottom-right example is not a topology because the intersection of {1,2} and {2,3} [i.e. {2}], is missing. <span>A topological space is an ordered pair (X, τ), where X is a set and τ is a collection of subsets of X, satisfying the following axioms:  The empty set and X itself belong to τ. Any (finite or infinite) union of members of τ still belongs to τ. The intersection of any finite number of members of τ still belongs to τ. The elements of τ are called open sets and the collection τ is called a topology on X. Examples[edit source] Given X = {1, 2, 3, 4}, the collection τ = {{}, {1, 2, 3, 4}} of only the two subsets of X required by the axioms forms a topology of X, the trivial topology (

#### Annotation 1758493281548

 #topology A topological space is an ordered pair (X, τ), where X is a set and τ is a topology of X

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A topological space is an ordered pair (X, τ), where X is a set and τ is a collection of subsets of X, satisfying the following axioms:  The empty set and X itself belong to τ. Any (finite or infinite) union of members of τ still belongs to τ. The intersection of any finite number of members of τ still belongs to τ.

#### Original toplevel document

Topological space - Wikipedia
three-point set {1,2,3}. The bottom-left example is not a topology because the union of {2} and {3} [i.e. {2,3}] is missing; the bottom-right example is not a topology because the intersection of {1,2} and {2,3} [i.e. {2}], is missing. <span>A topological space is an ordered pair (X, τ), where X is a set and τ is a collection of subsets of X, satisfying the following axioms:  The empty set and X itself belong to τ. Any (finite or infinite) union of members of τ still belongs to τ. The intersection of any finite number of members of τ still belongs to τ. The elements of τ are called open sets and the collection τ is called a topology on X. Examples[edit source] Given X = {1, 2, 3, 4}, the collection τ = {{}, {1, 2, 3, 4}} of only the two subsets of X required by the axioms forms a topology of X, the trivial topology (

#### Flashcard 1758495640844

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#topology
Question
A [...] is an ordered pair (X, τ), where X is a set and τ is a topology of X
topological space

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A topological space is an ordered pair (X, τ), where X is a set and τ is a topology of X

#### Original toplevel document

Topological space - Wikipedia
three-point set {1,2,3}. The bottom-left example is not a topology because the union of {2} and {3} [i.e. {2,3}] is missing; the bottom-right example is not a topology because the intersection of {1,2} and {2,3} [i.e. {2}], is missing. <span>A topological space is an ordered pair (X, τ), where X is a set and τ is a collection of subsets of X, satisfying the following axioms:  The empty set and X itself belong to τ. Any (finite or infinite) union of members of τ still belongs to τ. The intersection of any finite number of members of τ still belongs to τ. The elements of τ are called open sets and the collection τ is called a topology on X. Examples[edit source] Given X = {1, 2, 3, 4}, the collection τ = {{}, {1, 2, 3, 4}} of only the two subsets of X required by the axioms forms a topology of X, the trivial topology (

#### Flashcard 1758497213708

Tags
#topology
Question
The elements of [...] are called open sets
A topology

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ion of subsets of X, satisfying the following axioms:  The empty set and X itself belong to τ. Any (finite or infinite) union of members of τ still belongs to τ. The intersection of any finite number of members of τ still belongs to τ. <span>The elements of τ are called open sets and the collection τ is called a topology on X. <span><body><html>

#### Original toplevel document

Topological space - Wikipedia
three-point set {1,2,3}. The bottom-left example is not a topology because the union of {2} and {3} [i.e. {2,3}] is missing; the bottom-right example is not a topology because the intersection of {1,2} and {2,3} [i.e. {2}], is missing. <span>A topological space is an ordered pair (X, τ), where X is a set and τ is a collection of subsets of X, satisfying the following axioms:  The empty set and X itself belong to τ. Any (finite or infinite) union of members of τ still belongs to τ. The intersection of any finite number of members of τ still belongs to τ. The elements of τ are called open sets and the collection τ is called a topology on X. Examples[edit source] Given X = {1, 2, 3, 4}, the collection τ = {{}, {1, 2, 3, 4}} of only the two subsets of X required by the axioms forms a topology of X, the trivial topology (

#### Flashcard 1758499835148

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#topology
Question
A [...] is a collection of subsets of X satisfying certain axioms (inclusion, infinite union, finit intersection).
topology

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A topological space is an ordered pair (X, τ), where X is a set and τ is a collection of subsets of X, satisfying the following axioms:  The empty set and X itself belong to τ. Any (finite or infinite) union of members of τ still belongs to τ. The intersection of any finite number of members of τ still belongs to τ.

#### Original toplevel document

Topological space - Wikipedia
three-point set {1,2,3}. The bottom-left example is not a topology because the union of {2} and {3} [i.e. {2,3}] is missing; the bottom-right example is not a topology because the intersection of {1,2} and {2,3} [i.e. {2}], is missing. <span>A topological space is an ordered pair (X, τ), where X is a set and τ is a collection of subsets of X, satisfying the following axioms:  The empty set and X itself belong to τ. Any (finite or infinite) union of members of τ still belongs to τ. The intersection of any finite number of members of τ still belongs to τ. The elements of τ are called open sets and the collection τ is called a topology on X. Examples[edit source] Given X = {1, 2, 3, 4}, the collection τ = {{}, {1, 2, 3, 4}} of only the two subsets of X required by the axioms forms a topology of X, the trivial topology (

#### Annotation 1758504291596

 #topological-space A topological space X is said to be disconnected if it is the union of two disjoint nonempty open sets.

Connected space - Wikipedia
ected spaces 2 Examples 3 Path connectedness 4 Arc connectedness 5 Local connectedness 6 Set operations 7 Theorems 8 Graphs 9 Stronger forms of connectedness 10 See also 11 References 12 Further reading Formal definition[edit source] <span>A topological space X is said to be disconnected if it is the union of two disjoint nonempty open sets. Otherwise, X is said to be connected. A subset of a topological space is said to be connected if it is connected under its subspace topology. Some authors exclude the empty set (with it

#### Flashcard 1758506913036

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#topological-space
Question
A topological space X is said to be [...] if it is the union of two disjoint nonempty open sets.
disconnected

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A topological space X is said to be disconnected if it is the union of two disjoint nonempty open sets.

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Connected space - Wikipedia
ected spaces 2 Examples 3 Path connectedness 4 Arc connectedness 5 Local connectedness 6 Set operations 7 Theorems 8 Graphs 9 Stronger forms of connectedness 10 See also 11 References 12 Further reading Formal definition[edit source] <span>A topological space X is said to be disconnected if it is the union of two disjoint nonempty open sets. Otherwise, X is said to be connected. A subset of a topological space is said to be connected if it is connected under its subspace topology. Some authors exclude the empty set (with it

#### Annotation 1758512942348

 A point x of the topological space (X, τ) is the limit of the sequence (xn) if, for every neighbourhood U of x, there is an N such that, for every , .

Limit of a sequence - Wikipedia
tyle \epsilon } less than half this distance, sequence terms cannot be within a distance ϵ {\displaystyle \epsilon } of both points. Topological spaces[edit source] Definition[edit source] <span>A point x of the topological space (X, τ) is the limit of the sequence (x n ) if, for every neighbourhood U of x, there is an N such that, for every n ≥ N {\displaystyle n\geq N} , x n ∈ U {\displaystyle x_{n}\in U} . This coincides with the definition given for metric spaces if (X,d) is a metric space and τ {\displaystyle \tau } is the topology generated b

#### Flashcard 1758515825932

Question
A point x of the topological space (X, τ) is the limit of the sequence (xn) if, for every neighbourhood U of x, [...]
there is an N such that, for every , .

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A point x of the topological space (X, τ) is the limit of the sequence (x n ) if, for every neighbourhood U of x, there is an N such that, for every , .

#### Original toplevel document

Limit of a sequence - Wikipedia
tyle \epsilon } less than half this distance, sequence terms cannot be within a distance ϵ {\displaystyle \epsilon } of both points. Topological spaces[edit source] Definition[edit source] <span>A point x of the topological space (X, τ) is the limit of the sequence (x n ) if, for every neighbourhood U of x, there is an N such that, for every n ≥ N {\displaystyle n\geq N} , x n ∈ U {\displaystyle x_{n}\in U} . This coincides with the definition given for metric spaces if (X,d) is a metric space and τ {\displaystyle \tau } is the topology generated b

#### Flashcard 1758518185228

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#topology
Question
continuous deformations includes [...] , but not tearing or gluing.
streching, crumpling and bending

Mind the 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.

#### Original toplevel document

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

#### Annotation 1758527884556

 #measure-theory Any countable set of real numbers has Lebesgue measure 0.

Lebesgue measure - Wikipedia
c, d] is Lebesgue measurable, and its Lebesgue measure is (b − a)(d − c), the area of the corresponding rectangle. Moreover, every Borel set is Lebesgue measurable. However, there are Lebesgue measurable sets which are not Borel sets.   <span>Any countable set of real numbers has Lebesgue measure 0. In particular, the Lebesgue measure of the set of rational numbers is 0, although the set is dense in R. The Cantor set is an example of an uncountable set that has Lebesgue measure zer

#### Annotation 1758529981708

 #measure-theory The Lebesgue outer measure of a set E emerges as the greatest lower bound (infimum) of the lengths from among all possible such sets (unions of open intervals that include E).

Lebesgue measure - Wikipedia
, because E {\displaystyle E} is a subset of the union of the intervals, and so the intervals may include points which are not in E {\displaystyle E} . <span>The Lebesgue outer measure emerges as the greatest lower bound (infimum) of the lengths from among all possible such sets. Intuitively, it is the total length of those interval sets which fit E {\displaystyle E} most tightly and do not overlap. That characterize

#### Flashcard 1758533389580

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#measure-theory
Question
The Lebesgue outer measure of a set E emerges as [...] of the lengths from among all possible such sets (unions of open intervals that include E).

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The Lebesgue outer measure of a set E emerges as the greatest lower bound (infimum) of the lengths from among all possible such sets (unions of open intervals that include E).

#### Original toplevel document

Lebesgue measure - Wikipedia
, because E {\displaystyle E} is a subset of the union of the intervals, and so the intervals may include points which are not in E {\displaystyle E} . <span>The Lebesgue outer measure emerges as the greatest lower bound (infimum) of the lengths from among all possible such sets. Intuitively, it is the total length of those interval sets which fit E {\displaystyle E} most tightly and do not overlap. That characterize

#### Flashcard 1758534962444

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#measure-theory
Question
The Lebesgue outer measure of a set E is the greatest lower bound of the lengths from among all possible such sets that [...]
unions of open intervals that include E

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The Lebesgue outer measure of a set E emerges as the greatest lower bound (infimum) of the lengths from among all possible such sets (unions of open intervals that include E).

#### Original toplevel document

Lebesgue measure - Wikipedia
, because E {\displaystyle E} is a subset of the union of the intervals, and so the intervals may include points which are not in E {\displaystyle E} . <span>The Lebesgue outer measure emerges as the greatest lower bound (infimum) of the lengths from among all possible such sets. Intuitively, it is the total length of those interval sets which fit E {\displaystyle E} most tightly and do not overlap. That characterize

#### Flashcard 1758536535308

Tags
#measure-theory
Question
[...] of a set E emerges as the greatest lower bound (infimum) of the lengths from among all possible such sets (unions of open intervals that include E).
The Lebesgue outer measure

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The Lebesgue outer measure of a set E emerges as the greatest lower bound (infimum) of the lengths from among all possible such sets (unions of open intervals that include E).

#### Original toplevel document

Lebesgue measure - Wikipedia
, because E {\displaystyle E} is a subset of the union of the intervals, and so the intervals may include points which are not in E {\displaystyle E} . <span>The Lebesgue outer measure emerges as the greatest lower bound (infimum) of the lengths from among all possible such sets. Intuitively, it is the total length of those interval sets which fit E {\displaystyle E} most tightly and do not overlap. That characterize

#### Flashcard 1758538108172

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#measure-theory
Question
Any [...] set of real numbers has Lebesgue measure 0.

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Any countable set of real numbers has Lebesgue measure 0.

#### Original toplevel document

Lebesgue measure - Wikipedia
c, d] is Lebesgue measurable, and its Lebesgue measure is (b − a)(d − c), the area of the corresponding rectangle. Moreover, every Borel set is Lebesgue measurable. However, there are Lebesgue measurable sets which are not Borel sets.   <span>Any countable set of real numbers has Lebesgue measure 0. In particular, the Lebesgue measure of the set of rational numbers is 0, although the set is dense in R. The Cantor set is an example of an uncountable set that has Lebesgue measure zer

#### Flashcard 1758539681036

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#measure-theory
Question
As a set the rational number is [...but...] .
infinite but countable

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#### Annotation 1758542826764

 [unknown IMAGE 1758545186060] #has-images #sets In mathematics, the infimum of a subset S of a partially ordered set T is the greatest element in T that is less than or equal to all elements of S, if such an element exists.

Infimum and supremum - Wikipedia
ed sets the infimum and the minimum are equal. [imagelink] A set A of real numbers (blue balls), a set of upper bounds of A (red diamond and balls), and the smallest such upper bound, that is, the supremum of A (red diamond). <span>In mathematics, the infimum (abbreviated inf; plural infima) of a subset S of a partially ordered set T is the greatest element in T that is less than or equal to all elements of S, if such an element exists. Consequently, the term greatest lower bound (abbreviated as GLB) is also commonly used. The supremum (abbreviated sup; plural suprema) of a subset S of a partially ordered set T is th

#### Flashcard 1758547283212

Tags
#has-images #sets
[unknown IMAGE 1758545186060]
Question
the infimum of a subset S of a partially ordered set T is [...], if such an element exists.
the greatest element in T that is less than or equal to all elements of S

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In mathematics, the infimum of a subset S of a partially ordered set T is the greatest element in T that is less than or equal to all elements of S, if such an element exists.

#### Original toplevel document

Infimum and supremum - Wikipedia
ed sets the infimum and the minimum are equal. [imagelink] A set A of real numbers (blue balls), a set of upper bounds of A (red diamond and balls), and the smallest such upper bound, that is, the supremum of A (red diamond). <span>In mathematics, the infimum (abbreviated inf; plural infima) of a subset S of a partially ordered set T is the greatest element in T that is less than or equal to all elements of S, if such an element exists. Consequently, the term greatest lower bound (abbreviated as GLB) is also commonly used. The supremum (abbreviated sup; plural suprema) of a subset S of a partially ordered set T is th

#### Flashcard 1758548856076

Tags
#has-images #sets
[unknown IMAGE 1758545186060]
Question
[...] is the greatest element in T that is less than or equal to all elements of S, if such an element exists.
the infimum of a subset S of a partially ordered set T

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In mathematics, the infimum of a subset S of a partially ordered set T is the greatest element in T that is less than or equal to all elements of S, if such an element exists.

#### Original toplevel document

Infimum and supremum - Wikipedia
ed sets the infimum and the minimum are equal. [imagelink] A set A of real numbers (blue balls), a set of upper bounds of A (red diamond and balls), and the smallest such upper bound, that is, the supremum of A (red diamond). <span>In mathematics, the infimum (abbreviated inf; plural infima) of a subset S of a partially ordered set T is the greatest element in T that is less than or equal to all elements of S, if such an element exists. Consequently, the term greatest lower bound (abbreviated as GLB) is also commonly used. The supremum (abbreviated sup; plural suprema) of a subset S of a partially ordered set T is th

#### Annotation 1758554361100

 More formally, linear programming is a technique for the optimization of a linear objective function, subject to linear equality and linear inequality constraints.

Linear programming - Wikipedia
hieve the best outcome (such as maximum profit or lowest cost) in a mathematical model whose requirements are represented by linear relationships. Linear programming is a special case of mathematical programming (mathematical optimization). <span>More formally, linear programming is a technique for the optimization of a linear objective function, subject to linear equality and linear inequality constraints. Its feasible region is a convex polytope, which is a set defined as the intersection of finitely many half spaces, each of which is defined by a linear inequality. Its objective functio

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More formally, [...] is a technique for the optimization of a linear objective function, subject to linear equality and linear inequality constraints.
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linear programming

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More formally, linear programming is a technique for the optimization of a linear objective function, subject to linear equality and linear inequality constraints.

#### Original toplevel document

Linear programming - Wikipedia
hieve the best outcome (such as maximum profit or lowest cost) in a mathematical model whose requirements are represented by linear relationships. Linear programming is a special case of mathematical programming (mathematical optimization). <span>More formally, linear programming is a technique for the optimization of a linear objective function, subject to linear equality and linear inequality constraints. Its feasible region is a convex polytope, which is a set defined as the intersection of finitely many half spaces, each of which is defined by a linear inequality. Its objective functio

#### Annotation 1758561701132

 In geometry and topology, crumpling is the process whereby a sheet of paper or other two-dimensional manifold undergoes disordered deformation to yield a three-dimensional structure comprising a random network of ridges and facets with variable density.

Crumpling - Wikipedia
ikipedia Crumpling From Wikipedia, the free encyclopedia Jump to: navigation, search "Crumpled" redirects here. For the deformation feature, see Crumple zone. <span>In geometry and topology, crumpling is the process whereby a sheet of paper or other two-dimensional manifold undergoes disordered deformation to yield a three-dimensional structure comprising a random network of ridges and facets with variable density. The geometry of crumpled structures is the subject of some interest the mathematical community within the discipline of topology.  Crumpled paper balls have been studied and found t

#### Flashcard 1758563798284

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[...] is the process whereby a two-dimensional manifold undergoes disordered deformation to yield a three-dimensional structure.
crumpling

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In geometry and topology, crumpling is the process whereby a sheet of paper or other two-dimensional manifold undergoes disordered deformation to yield a three-dimensional structure comprising a random network of ridges a

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Crumpling - Wikipedia
ikipedia Crumpling From Wikipedia, the free encyclopedia Jump to: navigation, search "Crumpled" redirects here. For the deformation feature, see Crumple zone. <span>In geometry and topology, crumpling is the process whereby a sheet of paper or other two-dimensional manifold undergoes disordered deformation to yield a three-dimensional structure comprising a random network of ridges and facets with variable density. The geometry of crumpled structures is the subject of some interest the mathematical community within the discipline of topology.  Crumpled paper balls have been studied and found t

#### Annotation 1758566681868

 #calculus-of-variations the coefficient of arbitrarily small function δf in the first order term is called the functional derivative

Functional derivative - Wikipedia
integral of functions, their arguments, and their derivatives. In an integral L of a functional, if a function f is varied by adding to it another function δf that is arbitrarily small, and the resulting integrand is expanded in powers of δf, <span>the coefficient of δf in the first order term is called the functional derivative. For example, consider the functional J [ f ] = ∫ a

#### Flashcard 1758569565452

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the coefficient of arbitrarily small function δf in the first order term is called the [...]
functional derivative

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the coefficient of arbitrarily small function δf in the first order term is called the functional derivative

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Functional derivative - Wikipedia
integral of functions, their arguments, and their derivatives. In an integral L of a functional, if a function f is varied by adding to it another function δf that is arbitrarily small, and the resulting integrand is expanded in powers of δf, <span>the coefficient of δf in the first order term is called the functional derivative. For example, consider the functional J [ f ] = ∫ a

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#### Annotation 1758576643340

 The multi-family world becomes small once you get involved

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#### Annotation 1758578216204

 When you retrade, you put the broker’s commission in jeopardy.

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#### Annotation 1758579789068

 When I look at a town, I use the other end of the binoculars. I see amazing detail in the world around me. Everything is crisp and clear. And I notice things most people just pass by.

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#### Annotation 1758581361932

 Real Estate Brokers: Your Walking Gold Mines

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#### Annotation 1758582934796

 Brokers dominate the real estate market, because few sellers know of other ways to get rid of their properties.

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 It will cost you no money to pick up the phone, call a broker, and start to establish a relationship.

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#### Annotation 1758586080524

 People with valuable assets to dispense will not do business with you unless they like you first and trust you second. Brokers are no different.

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 Because they need qualified buyers to sell their listings, if you be- come qualified, they will like you.

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#### [赛雷话车]汽车喷火是为了加速？原来我们都被游戏骗了！                 #### Annotation 1758678355212

 Cox's theorem, named after the physicist Richard Threlkeld Cox, is a derivation of the laws of probability theory from a certain set of postulates. This derivation justifies the so-called "logical" interpretation of probability, as the laws of probability derived by Cox's theorem are applicable to any proposition.

Cox's theorem - Wikipedia
>Cox's theorem - Wikipedia Cox's theorem From Wikipedia, the free encyclopedia (Redirected from Probability theory as extended logic) Jump to: navigation, search Cox's theorem, named after the physicist Richard Threlkeld Cox, is a derivation of the laws of probability theory from a certain set of postulates. This derivation justifies the so-called "logical" interpretation of probability, as the laws of probability derived by Cox's theorem are applicable to any proposition. Logical (a.k.a. objective Bayesian) probability is a type of Bayesian probability. Other forms of Bayesianism, such as the subjective interpretation, are given other justifications.

#### Annotation 1758689627404

 #syntax short open tag

#### Annotation 1758691200268

 #syntax or if PHP was configured with the --enable-short-tags option).

#### Annotation 1758692773132

 #syntax If a file is pure PHP code, it is preferable to omit the PHP closing tag

#### Annotation 1758694345996

 #syntax This prevents accidental whitespace or new lines being added after the PHP closing tag

#### Annotation 1758695918860

 #syntax opening tag is "

#### Annotation 1758699588876

 Everything outside of a pair of opening and closing tags is ignored by the PHP parser which allows PHP files to have mixed content.

PHP: Escaping from HTML - Manual
ge language: English Brazilian Portuguese Chinese (Simplified) French German Japanese Romanian Russian Spanish Turkish Other Edit Report a Bug Escaping from HTML ¶ <span>Everything outside of a pair of opening and closing tags is ignored by the PHP parser which allows PHP files to have mixed content. This allows PHP to be embedded in HTML documents, for example to create templates. <span>

This is going to be ignored by PHP and displayed by the browser.<

Article 1758712171788

php.net

manual voor php

#### Flashcard 1758713482508

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ecause when the PHP interpreter hits the ?> closing tags, it simply starts outputting whatever it finds (except for an immediately following newline
hen the PHP interpreter

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PHP: Escaping from HTML - Manual
example to create templates.

This is going to be ignored by PHP and displayed by the browser.

This will also be ignored by PHP and displayed by the browser.

This works as expected, because when the PHP interpreter hits the ?> closing tags, it simply starts outputting whatever it finds (except for an immediately following newline - see instruction separation) until it hits another opening tag unless in the middle of a conditional statement in which case the interpreter will determine the outcome of t

#### Annotation 1758716890380

 For outputting large blocks of text, dropping out of PHP parsing mode is generally more efficient than sending all of the text through echo or print .

PHP: Escaping from HTML - Manual
he condition is not met, even though they are outside of the PHP open/close tags; PHP skips them according to the condition since the PHP interpreter will jump over blocks contained within a condition that is not met. <span>For outputting large blocks of text, dropping out of PHP parsing mode is generally more efficient than sending all of the text through echo or print. In PHP 5, there are up to five different pairs of opening and closing tags available in PHP, depending on how PHP is configured. Two of these, and <

#### Annotation 1758717938956

 In PHP 5, there are up to five different pairs of opening and closing tags available in PHP

PHP: Escaping from HTML - Manual
blocks contained within a condition that is not met. For outputting large blocks of text, dropping out of PHP parsing mode is generally more efficient than sending all of the text through echo or print. <span>In PHP 5, there are up to five different pairs of opening and closing tags available in PHP, depending on how PHP is configured. Two of these, and <span> , are always available. There is also the short echo tag , w

#### Annotation 1758718987532

 short echo tag , which is always available in PHP 5.4.0 and later.

#### Flashcard 1758736026892

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short echo tag <?= ?>, which is always available in PHP [...] and later.

5.4.0

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short echo tag , which is always available in PHP 5.4.0 and later.

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PHP: Escaping from HTML - Manual

#### Flashcard 1758737599756

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short echo tag [...], which is always available in PHP 5.4.0 and later.