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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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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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Measurable function - Wikipedia Measurable function From Wikipedia, the free encyclopedia Jump to: navigation, search In mathematics and in particular measure theory, a measurable function is a function between two measurable spaces such that the preimage of any measurable set is measurable, analogously to the definition that a function between topological spaces is continuous if the preimage of each open set is open. In real analysis, measurable functions are used in the

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In mathematics and in particular measure theory, a measurable function is a function between two measurable spaces such that the preimage of any measurable set is measurable

Measurable function - Wikipedia Measurable function From Wikipedia, the free encyclopedia Jump to: navigation, search In mathematics and in particular measure theory, a measurable function is a function between two measurable spaces such that the preimage of any measurable set is measurable, analogously to the definition that a function between topological spaces is continuous if the preimage of each open set is open. In real analysis, measurable functions are used in the

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In mathematics and in particular measure theory, a measurable function is a function between two measurable spaces such that the preimage of any measurable set is measurable

Measurable function - Wikipedia Measurable function From Wikipedia, the free encyclopedia Jump to: navigation, search In mathematics and in particular measure theory, a measurable function is a function between two measurable spaces such that the preimage of any measurable set is measurable, analogously to the definition that a function between topological spaces is continuous if the preimage of each open set is open. In real analysis, measurable functions are used in the

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tive · Surjective · Bijective Constructions Restriction · Composition · λ · Inverse Generalizations Partial · Multivalued · Implicit v t e In mathematics, a function space is a set of functions between two fixed sets. <span>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 natural vector space structure given by pointwise addition and scalar multiplication. In other scenarios, the function space might inherit a topological or metric structure, hence the name function space. Contents [hide] 1 In linear algebra 2 Examples 3 Functional analysis 4 Norm 5 Bibliography 6 See also 7 Footnotes In linear algebra[edit source] See also: Vector spac

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up vote 11 down vote favorite 6 One distinct difference between axioms of topology and sigma algebra is the asymmetry between union and intersection; meaning <span>topology is closed under finite intersections sigma-algebra closed under countable union. It is very clear mathematically but is there a way to think; so that we can define a geometric difference? In other words I want to have an intuitive idea in application of this object

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topology is closed under finite intersections while sigma-algebra is closed under countable intersections

up vote 11 down vote favorite 6 One distinct difference between axioms of topology and sigma algebra is the asymmetry between union and intersection; meaning <span>topology is closed under finite intersections sigma-algebra closed under countable union. It is very clear mathematically but is there a way to think; so that we can define a geometric difference? In other words I want to have an intuitive idea in application of this object

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a bijective function a one-to-one and onto (surjective) mapping of a set X to a set Y.

nction between the elements of two sets, where each element of one set is paired with exactly one element of the other set, and each element of the other set is paired with exactly one element of the first set. There are no unpaired elements. <span>In mathematical terms, a bijective function f: X → Y is a one-to-one (injective) and onto (surjective) mapping of a set X to a set Y. A bijection from the set X to the set Y has an inverse function from Y to X. If X and Y are finite sets, then the existence of a bijection means they have the same number of elements.

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a bijective function a one-to-one and onto (surjective) mapping of a set X to a set Y.

nction between the elements of two sets, where each element of one set is paired with exactly one element of the other set, and each element of the other set is paired with exactly one element of the first set. There are no unpaired elements. <span>In mathematical terms, a bijective function f: X → Y is a one-to-one (injective) and onto (surjective) mapping of a set X to a set Y. A bijection from the set X to the set Y has an inverse function from Y to X. If X and Y are finite sets, then the existence of a bijection means they have the same number of elements.

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><head>Given two vector spaces V and W over a field F, a linear transformation (also called linear map, linear mapping or linear operator) is a map \(T:V\to W\) that is compatible with addition and scalar multiplication: \(T(u+v)=T(u)+T(v),\quad T(av)=aT(v)\) for any vectors u,v ∈ V and a scalar a ∈ F. <html>

to all vector spaces. Linear transformations[edit source] Main article: Linear map Similarly as in the theory of other algebraic structures, linear algebra studies mappings between vector spaces that preserve the vector-space structure. <span>Given two vector spaces V and W over a field F, a linear transformation (also called linear map, linear mapping or linear operator) is a map T : V → W {\displaystyle T:V\to W} that is compatible with addition and scalar multiplication: T ( u + v ) = T ( u ) + T ( v ) , T ( a v ) = a T ( v ) {\displaystyle T(u+v)=T(u)+T(v),\quad T(av)=aT(v)} for any vectors u,v ∈ V and a scalar a ∈ F. Additionally for any vectors u, v ∈ V and scalars a, b ∈ F: T ( a u + b v ) =

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The repository setup that we use and that works well with this branching model, is that with a central “truth” repo.

e development model. The model that I’m going to present here is essentially no more than a set of procedures that every team member has to follow in order to come to a managed software development process. Decentralized but centralized ¶ <span>The repository setup that we use and that works well with this branching model, is that with a central “truth” repo. Note that this repo is only considered to be the central one (since Git is a DVCS, there is no such thing as a central repo at a technical level). We will refer to this repo as origin

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that there are legitimate reasons to do all of these. However, you should not attempt any of these things without understanding the potential negative effects of each and why they might be in a best practices “Don’t” list. DO NOT <span>commit anything that can be regenerated from other things that were committed. Things that can be regenerated include binaries, object files, jars, .class , flex/yacc generated code, etc. Really the only place there is room for disagreement about this

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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

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here's the list of kernels that I have on my system: $ jupyter kernelspec list Available kernels : python2 . 7 / Users / jakevdp /. ipython / kernels / python2 . 7 python3 . 3 / Users / jakevdp /. ipython / kernels / python3 . 3 python3 . 4 / Users / jakevdp /. ip

know where to look for the associated executable: that is, it needs to know which path the python sits in. These paths are specified in jupyter's kernelspec , and it's possible for the user to adjust them to their desires. For example, <span>here's the list of kernels that I have on my system: $ jupyter kernelspec list Available kernels: python2.7 /Users/jakevdp/.ipython/kernels/python2.7 python3.3 /Users/jakevdp/.ipython/kernels/python3.3 python3.4 /Users/jakevdp/.ipython/kernels/python3.4 python3.5 /Users/jakevdp/.ipython/kernels/python3.5 python2 /Users/jakevdp/Library/Jupyter/kernels/python2 python3 /Users/jakevdp/Library/Jupyter/kernels/python3 Each of these is a directory containing some metadata that specifies the kernel name, the path to the executable, and other relevant info. You can adjust kernels manually, editing the metadata inside the directories listed above. The command to install a kernel can change depending on the kernel. IPython relies on the i

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to the codomain. In mathematics, and more specifically in naive set theory, the range of a function refers to either the codomain or the image of the function, depending upon usage. Modern usage almost always uses range to mean image. <span>The codomain of a function is some arbitrary super-set of image. In real analysis, it is the real numbers. In complex analysis, it is the complex numbers. The image of a function is the set of all outputs of the function. The image is always a subs

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The codomain of a function is some arbitrary super-set of image.

to the codomain. In mathematics, and more specifically in naive set theory, the range of a function refers to either the codomain or the image of the function, depending upon usage. Modern usage almost always uses range to mean image. <span>The codomain of a function is some arbitrary super-set of image. In real analysis, it is the real numbers. In complex analysis, it is the complex numbers. The image of a function is the set of all outputs of the function. The image is always a subs

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If you understand several basic principles of the human visual system, you can eliminate inefficiencies and increase speed while improving retention.

more than 3,000 words-per-minute (wpm), or 10 pages per minute. One page every 6 seconds. By comparison, the average reading speed in the US is 200-300 wpm (1/2 to 1 page per minute), with the top 1% of the population reading over 400 wpm… <span>If you understand several basic principles of the human visual system, you can eliminate inefficiencies and increase speed while improving retention. To perform the exercises in this post and see the results, you will need: a book of 200+ pages that can lay flat when open, a pen, and a timer (a stop watch with alarm or kitchen time

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You must minimize the number and duration of fixations per line to increase speed.

when open, a pen, and a timer (a stop watch with alarm or kitchen timer is ideal). You should complete the 20 minutes of exercises in one session. First, several definitions and distinctions specific to the reading process: A) Synopsis: <span>You must minimize the number and duration of fixations per line to increase speed. You do not read in a straight line, but rather in a sequence of saccadic movements (jumps). Each of these saccades ends with a fixation, or a temporary snapshot of the text within yo

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accepted Your question is a little vague, but here is something to consider: Topology is normally discussed as its own subject while σσ-algebras are typically just used as a tool in measure theory. <span>One reason why finite intersections are needed in a topology is that it preserves what we think of as "openness" in a metric space. For instance, the finite intersection of any intervals of the form (a,b)⊆ℝ(a,b)⊆R still has the property of containing a ball around each point. This property is not shared with σσ-al

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One reason why finite intersections are needed in a topology is that it preserves what we think of as "openness" in a metric space.

accepted Your question is a little vague, but here is something to consider: Topology is normally discussed as its own subject while σσ-algebras are typically just used as a tool in measure theory. <span>One reason why finite intersections are needed in a topology is that it preserves what we think of as "openness" in a metric space. For instance, the finite intersection of any intervals of the form (a,b)⊆ℝ(a,b)⊆R still has the property of containing a ball around each point. This property is not shared with σσ-al

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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

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

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the relative complement of A in B is the set of elements in B but not in A . The relative complement of \({\displaystyle B\cap A^{\complement }=B\setminus A}\)

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{\displaystyle {\mathcal {F}}\,} is called a probability measure if P ( Ω ) = 1. {\displaystyle P(\Omega )=1.\,} <span>If F {\displaystyle {\mathcal {F}}\,} is the Borel σ-algebra on the set of real numbers, then there is a unique probability measure on F {\displaystyle {\mathcal {F}}\,} for any cdf, and vice versa. The measure corresponding to a cdf is said to be induced by the cdf. This measure coincides with the pmf for discrete variables and pdf for continuous variables, making the measure-theo

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If \({\mathcal {F}}\,\) is the Borel σ-algebra on the set of real numbers, then there is a unique probability measure on \({\mathcal {F}}\,\) for any cdf, and vice versa.

{\displaystyle {\mathcal {F}}\,} is called a probability measure if P ( Ω ) = 1. {\displaystyle P(\Omega )=1.\,} <span>If F {\displaystyle {\mathcal {F}}\,} is the Borel σ-algebra on the set of real numbers, then there is a unique probability measure on F {\displaystyle {\mathcal {F}}\,} for any cdf, and vice versa. The measure corresponding to a cdf is said to be induced by the cdf. This measure coincides with the pmf for discrete variables and pdf for continuous variables, making the measure-theo

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4 2 1 1 1 1 4 x 1 2 x 1 1 x 1 But once we have used up all the places, we need more bits , or binary digit, which stores a 0 or 1 . It turns out that <span>computers can conveniently represent a 0 or 1 with electricity, since something can either be turned on or off. And computers have lots of transistors, microscopic switches inside, that can be turned on and off to represent data. Now that we can store numbers, we need to represent words, or letters. Luckily, there is a standard mapping from numbers to letters, called ASCII . We can also similarly use ce

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out that computers can conveniently represent a 0 or 1 with electricity, since something can either be turned on or off. And computers have lots of transistors, microscopic switches inside, that can be turned on and off to represent data. <span>Now that we can store numbers, we need to represent words, or letters. Luckily, there is a standard mapping from numbers to letters, called ASCII . We can also similarly use certain standards to represent graphics and videos. A series of bits, that represent the numbers 72 73 33 might be the characters H I ! in

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out that computers can conveniently represent a 0 or 1 with electricity, since something can either be turned on or off. And computers have lots of transistors, microscopic switches inside, that can be turned on and off to represent data. <span>Now that we can store numbers, we need to represent words, or letters. Luckily, there is a standard mapping from numbers to letters, called ASCII . We can also similarly use certain standards to represent graphics and videos. A series of bits, that represent the numbers 72 73 33 might be the characters H I ! in

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istors, microscopic switches inside, that can be turned on and off to represent data. Now that we can store numbers, we need to represent words, or letters. Luckily, there is a standard mapping from numbers to letters, called ASCII . <span>We can also similarly use certain standards to represent graphics and videos. A series of bits, that represent the numbers 72 73 33 might be the characters H I ! in ASCII, but could also be interpreted by graphics programs as a color. RGB, for

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ta. Now that we can store numbers, we need to represent words, or letters. Luckily, there is a standard mapping from numbers to letters, called ASCII . We can also similarly use certain standards to represent graphics and videos. <span>A series of bits, that represent the numbers 72 73 33 might be the characters H I ! in ASCII, but could also be interpreted by graphics programs as a color. RGB, for example, is a system where a color is represented by the amount of red, green, and blue light it is composed of. By mixing the above amounts of red, green, and blue, we get

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similarly use certain standards to represent graphics and videos. A series of bits, that represent the numbers 72 73 33 might be the characters H I ! in ASCII, but could also be interpreted by graphics programs as a color. <span>RGB, for example, is a system where a color is represented by the amount of red, green, and blue light it is composed of. By mixing the above amounts of red, green, and blue, we get a color like a murky yellow. A picture on a screen, then, can be represented by lots and lots of these pixels, or single squares of color. For both ASCII and RGB, the maximum value that each character or amount of one color can be is 255, because one common standard group of bits is a byte , or 8 bits. In computer

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ere we start by taking ideas to solve simple problems, and layering these solutions until we can build more and more interesting applications. Algorithms Now that we know how to represent inputs and outputs, we can work on <span>algorithms, which is just step-by-step instructions on how to solve a problem. Computational thinking is the idea of having these precise instructions. For example, David might want to make a peanut butter and jelly sandwich from bread, peanut butter, and

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on the plate". Then "unscrew the jam", "grab the knife", and "stick the knife in the jam". We continue with these instructions that get more and more specific, until David completes his sandwich. <span>In fact, when we write algorithms to solve problems, we need to think about cases when something unexpected happens. For example, the input might not be within the range of what we expect, so our computer might freeze or come up with an incorrect solution. We can see this in action with trying to find a name in the phone book, Mike Smith. One correct algorithm might be flipping through the phone book, page by page, until we find t

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ferent shape, with time to solve growing more and more slowly as the size of the problem increases, since we are dividing the problem in half with each step. So an increase from 1000 to 2000 pages only requires one more step to solve. <span>We also need to formalize the steps we are using to solve this problem. We can write something like the following: 0 pick up phone book 1 open to middle of phone book 2 look at names 3 if Smith is among names 4 call Mike 5 else if Smith is earlier in book 6 open to middle of left half of book 7 go back to step 2 8 else if "Smith" is later in book 9 open to middle of right half of book 10 go back to step 2 11 else 12 quit We start counting at 0 because that’s the default lowest value, with all the bits off. In step 3, we have the word if , which is a fork in the road, where the next step may not be taken, so we indent it to visually separate it from the lines that are always followed. The last else , in step 11, happens if we’re on the last page and Mike isn’t in the phone book, since we can no longer divide it. These steps are pseudocode , English-like syntax that is similar in precision to code. Words like pick up , open , and look are equivalent to functions in code, like verbs or actions that allow us to do something. if , else if , and else are the keywords which represent forks in the road, or decisions based on answers to certain questions. These questions are called Boolean expressions , which have an answer of either true or false. For example, Smith among names is a question, as is Smith is earlier in book and Smith is later in book . Notice too, that with one bit, we can represent true, with on, or 1, and false, with off, or 0. Finally, go back creates loops, or series of steps that happen over and over, until we complete our algorithm. Introductions CS50 students are supported by a team of over 100 staff members, a few of whom will say hello. Doug Lloyd, who took CS50 12 years ago with no exp