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#### Flashcard 1735985859852

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#stochastics
Question
Viewed from a function analysis perspective, a single outcome of a stochastic process can be called a [...]
sample function

Again, remember a function is just a vector with infinite length, and a topology for the notion of proximity and continuity.

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A stochastic process can have many outcomes, due to its randomness, and a single outcome of a stochastic process is called, among other names, a sample function or realization

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Stochastic process - Wikipedia
r n {\displaystyle n} -dimensional Euclidean space. [1] [5] An increment is the amount that a stochastic process changes between two index values, often interpreted as two points in time. [48] [49] <span>A stochastic process can have many outcomes, due to its randomness, and a single outcome of a stochastic process is called, among other names, a sample function or realization. [28] [50] [imagelink] A single computer-simulated sample function or realization, among other terms, of a three-dimensional Wiener or Brownian motion process for time 0 ≤ t ≤ 2.

#### Flashcard 2971486260492

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#python
Question

When your script is run by passing it as a command to the Python interpreter,

 python myscript.py 

[...] gets executed.

all level 0 indented code

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When your script is run by passing it as a command to the Python interpreter, python myscript . py all of the code that is at indentation level 0 gets executed. Functions and classes that are defined are, well, defined, but none of their code gets run. Unlike other languages, there's no main() function that gets run automatical

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python - What does if __name__ == &quot;__main__&quot;: do? - Stack Overflow
add a comment | up vote 1313 down vote <span>When your script is run by passing it as a command to the Python interpreter, python myscript.py all of the code that is at indentation level 0 gets executed. Functions and classes that are defined are, well, defined, but none of their code gets run. Unlike other languages, there's no main() function that gets run automatically - the main() function is implicitly all the code at the top level. In this case, the top-level code is an if block. __name__ is a built-in variable which evaluates to the name of the current module. However, if a module is being run directly (as in myscript.py above), then __name__ instead is set to the string "__main__" . Thus, you can test whether your script is being run directly or being imported by something else by testing if __name__ == "__main__": ... If your script is being imported into another module, its various function and class definitions will be imported and its top-level code will be executed, but the code in the then-body of the if clause above won't get run as the condition is not met. As a basic example, consider the following two scripts: # file one.py def func(): print("func() in one.py") print("top-level in one.py") if __name__ == "__main__": print("one.py is being run di

#### Flashcard 2976246795532

Tags
#probability-theory
Question
Let $$(\Omega ,{\mathcal {F}},P)$$ be a probability space and $$(E,{\mathcal {E}})$$ a measurable space. Then an $$(E,{\mathcal {E}})$$-valued random variable means that [... description ...].
every subset $$B\in {\mathcal {E}}$$ has a preimage $$X^{-1}(B)\in {\mathcal {F}}$$

Pretty much like any measurable function, just with a probability measure.
where
$$X^{-1}(B)=\{\omega :X(\omega )\in B\}$$

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dy>Let $$(\Omega ,{\mathcal {F}},P)$$ be a probability space and $$(E,{\mathcal {E}})$$ a measurable space. Then an $$(E,{\mathcal {E}})$$-valued random variable is a measurable function $$X\colon \Omega \to E$$, which means that, for every subset $$B\in {\mathcal {E}}$$, its preimage $$X^{-1}(B)\in {\mathcal {F}}$$ where $$X^{-1}(B)=\{\omega :X(\omega )\in B\}$$. [5] This definition enables us to measure any subset $$B\in {\mathcal {E}}$$ in the target space by looking at its preimage, which by assumption is measurable. <body><

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Random variable - Wikipedia
fined over any sets that can be derived either directly from continuous intervals of numbers or by a finite or countably infinite number of unions and/or intersections of such intervals. [2] The measure-theoretic definition is as follows. <span>Let ( Ω , F , P ) {\displaystyle (\Omega ,{\mathcal {F}},P)} be a probability space and ( E , E ) {\displaystyle (E,{\mathcal {E}})} a measurable space. Then an ( E , E ) {\displaystyle (E,{\mathcal {E}})} -valued random variable is a measurable function X : Ω → E {\displaystyle X\colon \Omega \to E} , which means that, for every subset B ∈ E {\displaystyle B\in {\mathcal {E}}} , its preimage X − 1 ( B ) ∈ F {\displaystyle X^{-1}(B)\in {\mathcal {F}}} where X − 1 ( B ) = { ω : X ( ω ) ∈ B } {\displaystyle X^{-1}(B)=\{\omega :X(\omega )\in B\}} . [5] This definition enables us to measure any subset B ∈ E {\displaystyle B\in {\mathcal {E}}} in the target space by looking at its preimage, which by assumption is measurable. In more intuitive terms, a member of Ω {\displaystyle \Omega } is a possible outcome, a member of

#### Flashcard 2976410635532

Tags
#item-response-theory
Question
[...] is a paradigm for the design, analysis, and scoring of tests, questionnaires, and similar instruments measuring abilities, attitudes, or other variables
item response theory

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tem response theory (IRT) (also known as latent trait theory, strong true score theory, or modern mental test theory) is a paradigm for the design, analysis, and scoring of tests, questionnaires, and simil

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Item response theory - Wikipedia
>Item response theory - Wikipedia Item response theory From Wikipedia, the free encyclopedia Jump to: navigation, search In psychometrics, item response theory (IRT) (also known as latent trait theory, strong true score theory, or modern mental test theory) is a paradigm for the design, analysis, and scoring of tests, questionnaires, and similar instruments measuring abilities, attitudes, or other variables. It is a theory of testing based on the relationship between individuals' performances on a test item and the test takers' levels of performance on an overall measure of the ability tha

#### Annotation 2976501075212

 #mathematical-structures Isomorphisms (in general) are "structure-preserving" bijections, whether they be preserving group, ring, field, topological, (partial) order, or some other sort of structure. Homeomorphisms, specifically, are topology-preserving isomorphisms.

abstract algebra - What's the difference between isomorphism and homeomorphism? - Mathematics Stack Exchange
add a comment | up vote 11 down vote <span>Isomorphisms (in general) are "structure-preserving" bijections, whether they be preserving group, ring, field, topological, (partial) order, or some other sort of structure. Homeomorphisms, specifically, are topology-preserving isomorphisms. share|cite|edit|flag answered Nov 29 '13 at 15:47 [imagelink]

#### Annotation 2976502648076

 #mathematical-structures Homomorphism - an algebraical term for a function preserving some algebraic operations. For a group homomorphism ϕ ϕ \phi we have ϕ ( a b ) = ϕ ( a ) ϕ ( b ) ϕ(ab)=ϕ(a)ϕ(b) \phi(ab)=\phi(a)\phi(b) and ϕ ( 1 ) = 1 ϕ(1)=1 \phi(1)=1 , for a ring homomorphism we have additionally ϕ ( a + b ) = ϕ ( a ) + ϕ ( b ) ϕ(a+b)=ϕ(a)+ϕ(b) \phi(a+b) = \phi(a) + \phi(b) and for a vector-space homomorphism also ϕ ( r ⋅ a ) = r ⋅ ϕ ( a ) ϕ(r⋅a)=r⋅ϕ(a) \phi(r\cdot a)=r\cdot\phi(a) , where r r r is a scalar and a a a is a vector. Isomorphism (in a narrow/algebraic sense) - a homomorphism which is 1-1 and onto. In other words: a homomorphism which has an inverse. However, homEomorphism is a topological term - it is a continuous function, having a continuous inverse. In the category theory one defines a notion of a morphism (specific for each category) and then an isomorphism is defined as a morphism having an inverse, which is also a morphism. With such an approach, morphisms in the category of groups are group homomorphisms and isomorphisms in this category are just group isomorphisms. Similarly for rings, vector spaces etc. In the category of topological spaces, morphisms are continuous functions, and isomorphisms are homeomorphisms. Extra remark: A fundamental difference between algebra and topology is that in algebra any morphism (homomorphism) which is 1-1 and onto is an isomorphism - i.e., its inverse is a morphism. In topology it is not true: there are continuous and bijective functions whose inverses are not continuous. That's (one of the reasons) why we like compact Hausdorff topological spaces: for them inverses are always continuous, just like in algebra inverses of homomorphisms are homomorphisms.

#### Flashcard 2976504745228

Tags
#mathematical-structures
Question

Isomorphisms (in general) are [...], whether they be preserving group, ring, field, topological, (partial) order, or some other sort of structure.

"structure-preserving" bijections

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Isomorphisms (in general) are "structure-preserving" bijections, whether they be preserving group, ring, field, topological, (partial) order, or some other sort of structure. Homeomorphisms, specifically, are topology-preserving isomorphisms. </spa

#### Original toplevel document

abstract algebra - What's the difference between isomorphism and homeomorphism? - Mathematics Stack Exchange
add a comment | up vote 11 down vote <span>Isomorphisms (in general) are "structure-preserving" bijections, whether they be preserving group, ring, field, topological, (partial) order, or some other sort of structure. Homeomorphisms, specifically, are topology-preserving isomorphisms. share|cite|edit|flag answered Nov 29 '13 at 15:47 [imagelink]

#### Flashcard 2976507104524

Tags
#mathematical-structures
Question

[...] are topology-preserving isomorphisms.

Homeomorphisms

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Isomorphisms (in general) are "structure-preserving" bijections, whether they be preserving group, ring, field, topological, (partial) order, or some other sort of structure. Homeomorphisms, specifically, are topology-preserving isomorphisms.

#### Original toplevel document

abstract algebra - What's the difference between isomorphism and homeomorphism? - Mathematics Stack Exchange
add a comment | up vote 11 down vote <span>Isomorphisms (in general) are "structure-preserving" bijections, whether they be preserving group, ring, field, topological, (partial) order, or some other sort of structure. Homeomorphisms, specifically, are topology-preserving isomorphisms. share|cite|edit|flag answered Nov 29 '13 at 15:47 [imagelink]

#### Annotation 2976509463820

 #mathematical-structures Homomorphism - an algebraical term for a function preserving some algebraic operations.

#### Parent (intermediate) annotation

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Homomorphism - an algebraical term for a function preserving some algebraic operations. For a group homomorphism ϕ ϕ \phi we have ϕ ( a b ) = ϕ ( a ) ϕ ( b ) ϕ(ab)=ϕ(a)ϕ(b) \phi(ab)=\phi(a)\phi(b) and ϕ ( 1 ) = 1 ϕ(1)=1 \phi(1)=1 , for a ring homomorphism we have additiona

#### Flashcard 2976511560972

Tags
#mathematical-structures
Question
[...] - an algebraical term for a function preserving some algebraic operations.
Homomorphism

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Homomorphism - an algebraical term for a function preserving some algebraic operations.

#### Flashcard 2976518901004

Tags
#python
Question

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Quick Start — The Hitchhiker's Guide to Packaging 1.0 documentation
Quick Start¶ Here’s how to create a new project, which we’ll call TowelStuff . 1. Lay out your project¶ The smallest python project is two files. <span>A setup.py file which describes the metadata about your project, and a file containing Python code to implement the functionality of your project. In this example project we are going to add a little more to the project to provide the typical mini

#### Annotation 2976522833164

 #epigrams-on-programming Get into a rut early: Do the same processes the same way. Accumulate idioms. Standardize. The only difference (!) between Shakespeare and you was the size of his idiom list - not the size of his vocabulary.

Epigrams on Programming
gram than understand a correct one. A programming language is low level when its programs require attention to the irrelevant. It is better to have 100 functions operate on one data structure than 10 functions on 10 data structures. <span>Get into a rut early: Do the same processes the same way. Accumulate idioms. Standardize. The only difference (!) between Shakespeare and you was the size of his idiom list - not the size of his vocabulary. If you have a procedure with 10 parameters, you probably missed some. Recursion is the root of computation since it trades description for time. If two people write exactly t

#### Annotation 2976524406028

 #epigrams-on-programming Everything should be built top-down, except the first time.

Epigrams on Programming
computation since it trades description for time. If two people write exactly the same program, each should be put in micro-code and then they certainly won't be the same. In the long run every program becomes rococo - then rubble. <span>Everything should be built top-down, except the first time. Every program has (at least) two purposes: the one for which it was written and another for which it wasn't. If a listener nods his head when you're explaining your program, wake

#### Annotation 2976525978892

 #epigrams-on-programming Simplicity does not precede complexity, but follows it.

Epigrams on Programming
ms, the analogue of a face-lift is to add to the control graph an edge that creates a cycle, not just an additional node. In programming, everything we do is a special case of something more general - and often we know it too quickly. <span>Simplicity does not precede complexity, but follows it. Programmers are not to be measured by their ingenuity and their logic but by the completeness of their case analysis. The 11th commandment was "Thou Shalt Compute" or

#### Annotation 2976527551756

 #epigrams-on-programming Programmers are not to be measured by their ingenuity and their logic but by the completeness of their case analysis.

Epigrams on Programming
aph an edge that creates a cycle, not just an additional node. In programming, everything we do is a special case of something more general - and often we know it too quickly. Simplicity does not precede complexity, but follows it. <span>Programmers are not to be measured by their ingenuity and their logic but by the completeness of their case analysis. The 11th commandment was "Thou Shalt Compute" or "Thou Shalt Not Compute" - I forget which. The string is a stark data structure and everywhere it is passed t

#### Annotation 2976529124620

 #epigrams-on-programming Functions delay binding: data structures induce binding. Moral: Structure data late in the programming process.

Epigrams on Programming
pt to capture some of the dimensions of this traffic in imagery that sharpens, focuses, clarifies, enlarges and beclouds our view of this most remarkable of all mans' artifacts, the computer. One man's constant is another man's variable. <span>Functions delay binding: data structures induce binding. Moral: Structure data late in the programming process. Syntactic sugar causes cancer of the semi-colons. Every program is a part of some other program and rarely fits. If a program manipulates a large amount of data, it does so in

#### Flashcard 2976530697484

Tags
#epigrams-on-programming
Question
Programmers are not to be measured by their ingenuity and their logic but by the [...].
completeness of their case analysis

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Programmers are not to be measured by their ingenuity and their logic but by the completeness of their case analysis.

#### Original toplevel document

Epigrams on Programming
aph an edge that creates a cycle, not just an additional node. In programming, everything we do is a special case of something more general - and often we know it too quickly. Simplicity does not precede complexity, but follows it. <span>Programmers are not to be measured by their ingenuity and their logic but by the completeness of their case analysis. The 11th commandment was "Thou Shalt Compute" or "Thou Shalt Not Compute" - I forget which. The string is a stark data structure and everywhere it is passed t

#### Flashcard 2976532270348

Tags
#epigrams-on-programming
Question
Everything should be built [...], except the first time.
top-down

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Everything should be built top-down, except the first time.

#### Original toplevel document

Epigrams on Programming
computation since it trades description for time. If two people write exactly the same program, each should be put in micro-code and then they certainly won't be the same. In the long run every program becomes rococo - then rubble. <span>Everything should be built top-down, except the first time. Every program has (at least) two purposes: the one for which it was written and another for which it wasn't. If a listener nods his head when you're explaining your program, wake

#### Flashcard 2976533843212

Tags
#epigrams-on-programming
Question
The only difference (!) between Shakespeare and you was the size of his [...] - not the size of his vocabulary.
idiom list

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Get into a rut early: Do the same processes the same way. Accumulate idioms. Standardize. The only difference (!) between Shakespeare and you was the size of his idiom list - not the size of his vocabulary.

#### Original toplevel document

Epigrams on Programming
gram than understand a correct one. A programming language is low level when its programs require attention to the irrelevant. It is better to have 100 functions operate on one data structure than 10 functions on 10 data structures. <span>Get into a rut early: Do the same processes the same way. Accumulate idioms. Standardize. The only difference (!) between Shakespeare and you was the size of his idiom list - not the size of his vocabulary. If you have a procedure with 10 parameters, you probably missed some. Recursion is the root of computation since it trades description for time. If two people write exactly t

#### Flashcard 2976536202508

Tags
#epigrams-on-programming
Question
Functions delay binding: data structures induce binding. Moral: [...] late in the programming process.
Structure data

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Functions delay binding: data structures induce binding. Moral: Structure data late in the programming process.

#### Original toplevel document

Epigrams on Programming
pt to capture some of the dimensions of this traffic in imagery that sharpens, focuses, clarifies, enlarges and beclouds our view of this most remarkable of all mans' artifacts, the computer. One man's constant is another man's variable. <span>Functions delay binding: data structures induce binding. Moral: Structure data late in the programming process. Syntactic sugar causes cancer of the semi-colons. Every program is a part of some other program and rarely fits. If a program manipulates a large amount of data, it does so in

#### Annotation 2976540396812

 #mathematical-structures A fundamental difference between algebra and topology is that in algebra any morphism (homomorphism) which is 1-1 and onto is an isomorphism - i.e., its inverse is a morphism.

abstract algebra - What's the difference between isomorphism and homeomorphism? - Mathematics Stack Exchange
phisms and isomorphisms in this category are just group isomorphisms. Similarly for rings, vector spaces etc. In the category of topological spaces, morphisms are continuous functions, and isomorphisms are homeomorphisms. Extra remark: <span>A fundamental difference between algebra and topology is that in algebra any morphism (homomorphism) which is 1-1 and onto is an isomorphism - i.e., its inverse is a morphism. In topology it is not true: there are continuous and bijective functions whose inverses are not continuous. That's (one of the reasons) why we like compact Hausdorff topological spaces:

#### Annotation 2976541969676

 #mathematical-structures In topology it is not true: there are continuous and bijective functions whose inverses are not continuous.

abstract algebra - What's the difference between isomorphism and homeomorphism? - Mathematics Stack Exchange
functions, and isomorphisms are homeomorphisms. Extra remark: A fundamental difference between algebra and topology is that in algebra any morphism (homomorphism) which is 1-1 and onto is an isomorphism - i.e., its inverse is a morphism. <span>In topology it is not true: there are continuous and bijective functions whose inverses are not continuous. That's (one of the reasons) why we like compact Hausdorff topological spaces: for them inverses are always continuous, just like in algebra inverses of homomorphisms are homomorphisms.

#### Flashcard 2976543542540

Tags
#mathematical-structures
Question
In topology there are [...] functions whose inverses are not continuous.
continuous and bijective

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In topology it is not true: there are continuous and bijective functions whose inverses are not continuous.

#### Original toplevel document

abstract algebra - What's the difference between isomorphism and homeomorphism? - Mathematics Stack Exchange
functions, and isomorphisms are homeomorphisms. Extra remark: A fundamental difference between algebra and topology is that in algebra any morphism (homomorphism) which is 1-1 and onto is an isomorphism - i.e., its inverse is a morphism. <span>In topology it is not true: there are continuous and bijective functions whose inverses are not continuous. That's (one of the reasons) why we like compact Hausdorff topological spaces: for them inverses are always continuous, just like in algebra inverses of homomorphisms are homomorphisms.

#### Flashcard 2976545901836

Tags
#mathematical-structures
Question
in algebra any bijective morphism is an isomorphism - i.e., its [...] is a morphism.
inverse

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A fundamental difference between algebra and topology is that in algebra any morphism (homomorphism) which is 1-1 and onto is an isomorphism - i.e., its inverse is a morphism.

#### Original toplevel document

abstract algebra - What's the difference between isomorphism and homeomorphism? - Mathematics Stack Exchange
phisms and isomorphisms in this category are just group isomorphisms. Similarly for rings, vector spaces etc. In the category of topological spaces, morphisms are continuous functions, and isomorphisms are homeomorphisms. Extra remark: <span>A fundamental difference between algebra and topology is that in algebra any morphism (homomorphism) which is 1-1 and onto is an isomorphism - i.e., its inverse is a morphism. In topology it is not true: there are continuous and bijective functions whose inverses are not continuous. That's (one of the reasons) why we like compact Hausdorff topological spaces:

#### Annotation 2976557698316

 However, there seems to be no way for the user to configure these keyboard shortcuts, which is something I'd really want my extension to allow.