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https://urbancityarch.com/25-one-less-worksheet/	[ "HomeSuper Teacher Worksheets ➟ 25 25 One Less Worksheet\n\n# 25 One Less Worksheet\n\ne more or e less groups worksheet twinkl this handy and colorful worksheet gives your children the opportunity to practice counting one more and less differentiated three ways with answers for checking e less solutions examples homework worksheets count down from 10 to 1 and state 1 less than a given number examples and step by step solutions worksheets engageny math kindergarten module 1 lesson 13 eureka math mon core worksheets kindergarten speckled frog e less than worksheet by missyrobinson useful for when using speckled frog or green bottle song for subtraction less than activity learners draw the number of frogs green bottles one less than in the empty box and write the number sentence below\n\n### one less worksheet", null, "5 Nbt 1 Worksheets Math Grade 1 Worksheets Awesome Printable from one less worksheet , image source: blitze.co\n\n## 25 Division Grouping Worksheets\n\ndivision by grouping worksheets teach an interesting division strategy of grouping objects with this unit of division model worksheets hand picked for kids of grade 3 included here are exercises such as group the objects answer questions based on the model plete the division statements fill in the missing part of the equation draw models […]" ]	[ null, "https://urbancityarch.com/wp-content/uploads/2019/09/one-less-worksheet-5-nbt-1-worksheets-math-grade-1-worksheets-awesome-printable-of-one-less-worksheet.jpg", null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.88416183,"math_prob":0.80172724,"size":1968,"snap":"2020-45-2020-50","text_gpt3_token_len":384,"char_repetition_ratio":0.1787169,"word_repetition_ratio":0.011976048,"special_character_ratio":0.19054878,"punctuation_ratio":0.008902078,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9774508,"pos_list":[0,1,2],"im_url_duplicate_count":[null,1,null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2020-12-03T01:32:22Z\",\"WARC-Record-ID\":\"<urn:uuid:268b7fce-0068-4740-86da-e829bd54ab3b>\",\"Content-Length\":\"34916\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:2608f1c4-9258-47a5-a5cf-c48c7dc5b0d6>\",\"WARC-Concurrent-To\":\"<urn:uuid:d77365dd-d965-46de-aa23-2f449fae6f44>\",\"WARC-IP-Address\":\"104.27.159.75\",\"WARC-Target-URI\":\"https://urbancityarch.com/25-one-less-worksheet/\",\"WARC-Payload-Digest\":\"sha1:UJTXFTREF5GRNMAQCHEXPVOLFFSLH3T3\",\"WARC-Block-Digest\":\"sha1:LNVWL7Z7Z5TFQLZMIQ3TDJPKPHR6HT7N\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2020/CC-MAIN-2020-50/CC-MAIN-2020-50_segments_1606141717601.66_warc_CC-MAIN-20201203000447-20201203030447-00013.warc.gz\"}"}
https://medical-dictionary.thefreedictionary.com/coefficient+of+viscosity	[ "# coefficient of viscosity\n\nAlso found in: Dictionary, Thesaurus, Encyclopedia, Wikipedia.\n\n## co·ef·fi·cient of vis·cos·i·ty\n\nthe value of the force per unit area required to maintain a unit of relative velocity between two parallel planes a unit of distance apart.\nFarlex Partner Medical Dictionary © Farlex 2012\nMentioned in ?\nReferences in periodicals archive ?\n(3) Using case analysis, the degrees of influence of the water resistance ability of the rock beam deformation on the elasticity modulus, coefficient of viscosity, and water pressure were obtained.\nThis can be determined by examining the values of the temperature-dependency coefficient of viscosity (5, 23).\n[Alpha] = Pressure coefficient of viscosity, [Pa.sup.-1].\ncoefficient of viscosity function, n = 3.50 x [10.sup.-4]\ncoefficient of viscosity function, C = 4.80 x [10.sup.-4]\n\nSite: Follow: Share:\nOpen / Close" ]	[ null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.6917895,"math_prob":0.7831971,"size":306,"snap":"2021-04-2021-17","text_gpt3_token_len":72,"char_repetition_ratio":0.119205296,"word_repetition_ratio":0.0,"special_character_ratio":0.19934641,"punctuation_ratio":0.10714286,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.979948,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2021-04-20T11:45:41Z\",\"WARC-Record-ID\":\"<urn:uuid:bba1aa29-eaab-4400-b1c4-5ded357c70d5>\",\"Content-Length\":\"43234\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:e6c2a503-dd91-49db-8546-cf0616f1c46c>\",\"WARC-Concurrent-To\":\"<urn:uuid:71b0013a-755b-4e35-b5c5-589b08322c1f>\",\"WARC-IP-Address\":\"209.160.67.6\",\"WARC-Target-URI\":\"https://medical-dictionary.thefreedictionary.com/coefficient+of+viscosity\",\"WARC-Payload-Digest\":\"sha1:NSP24T4S7B3BUJ5EJMCRFCCOPKQD4PVQ\",\"WARC-Block-Digest\":\"sha1:AK5LK5NOUXCIRK3NAHVSMN7UGAX2HKSK\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2021/CC-MAIN-2021-17/CC-MAIN-2021-17_segments_1618039388763.75_warc_CC-MAIN-20210420091336-20210420121336-00190.warc.gz\"}"}
https://brilliant.org/problems/simple-math-6/	[ "# An algebra problem by Viki Zeta\n\nAlgebra Level 2\n\n$\\frac{(\\underbrace{4 + \\cdots + 4)}_{x^2\\text{ times}}}{\\underbrace{(x + \\cdots+ x)}_\\text{x times}} - 2 = \\, ?$\n\nFind the value of the above expression given that $k$ is a non-zero real number and $x$ is a natural number.\n\n×" ]	[ null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.72470605,"math_prob":0.99999297,"size":264,"snap":"2019-43-2019-47","text_gpt3_token_len":60,"char_repetition_ratio":0.15,"word_repetition_ratio":0.29268292,"special_character_ratio":0.23106061,"punctuation_ratio":0.2580645,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.999894,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2019-11-12T22:44:32Z\",\"WARC-Record-ID\":\"<urn:uuid:f7ffda11-3b63-4a6e-8d29-3955eab93554>\",\"Content-Length\":\"44708\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:2d386f53-4e9d-460c-ae1b-bdec666452b4>\",\"WARC-Concurrent-To\":\"<urn:uuid:061f8dec-c1f8-443d-bda4-18d2082e9ed3>\",\"WARC-IP-Address\":\"104.20.35.242\",\"WARC-Target-URI\":\"https://brilliant.org/problems/simple-math-6/\",\"WARC-Payload-Digest\":\"sha1:SQRR7D5MDOCW3DTJOCLIQOHPWCCHLRYF\",\"WARC-Block-Digest\":\"sha1:HPJV6RO7JZVMXK4WTIEPNVSHMAD6GMXO\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2019/CC-MAIN-2019-47/CC-MAIN-2019-47_segments_1573496665767.51_warc_CC-MAIN-20191112202920-20191112230920-00412.warc.gz\"}"}
https://openlearning.aalto.fi/mod/book/view.php?id=10324	[ "## Differential and Integral Calculus\n\nTable of Content\n\n### Basics of sequences\n\nThis section contains the most important definitions about sequences. Through these definitions the general notion of sequences will be explained, but then restricted to real number sequences.\n\n##### Definition: Sequence\n\nLet $$M$$ be a non-empty set. A sequence is a function:\n\n$f:\\mathbb{N}\\rightarrow M.$\n\nOccasionally we speak about a sequence in $$M$$.\n\nNote. Characteristics of the set $$\\mathbb{N}$$ give certain characteristics to the sequence. Because $$\\mathbb{N}$$ is ordered, the terms of the sequence are ordered.\n\n##### Definition: Terms and Indices\n\nA sequence can be denoted denoted as\n\n$$(a_{1}, a_{2}, a_{3}, \\ldots) = (a_{n})_{n\\in\\mathbb{N}} = (a_{n})_{n=1}^{\\infty} = (a_{n})_{n}$$\n\ninstead of $$f(n).$$ The numbers $$a_{1},a_{2},a_{3},\\ldots\\in M$$ are called the terms of the sequence.\n\nBecause of the mapping \\begin{aligned} f:\\mathbb{N} \\rightarrow & M \\\\ n \\mapsto & a_{n}\\end{aligned} we can assign a unique number $$n\\in\\mathbb{N}$$ to each term. We write this number as a subscript and define it as the index; it follows that we can identify any term of the sequence by its index.\n\nn 1 2 3 4 5 6 7 8 9 $$\\ldots$$\n$$\\downarrow$$ $$\\downarrow$$ $$\\downarrow$$ $$\\downarrow$$ $$\\downarrow$$ $$\\downarrow$$ $$\\downarrow$$ $$\\downarrow$$ $$\\downarrow$$\n$$a_{n}$$\n\n$$a_{1}$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$\\ldots$$\n\n#### A few easy examples\n\n##### Example 1: The sequence of natural numbers\n\nThe sequence $$(a_{n})_{n}$$ defined by $$a_{n}:=n,\\,n\\in \\mathbb{N}$$ is called the sequence of natural numbers. Its first few terms are: $a_1=1,\\, a_2=2,\\, a_3=3, \\ldots$ This special sequence has the property that every term is the same as its index.", null, "##### Example 2: The sequence of triangular numbers\n\nTriangular numbers get their name due to the following geometric visualization: Stacking coins to form a triangular shape gives the following diagram:", null, "To the first coin in the first layer we add two coins in a second layer to form the second picture $$a_2$$. In turn, adding three coins to $$a_2$$ forms $$a_3$$. From a mathematical point of view, this sequence is the result of summing natural numbers. To calculate the 10th triangular number we need to add the first 10 natural numbers: $D_{10} = 1+2+3+\\ldots+9+10$ In general form the sequence is defined as: $$D_{n} = 1+2+3+\\ldots+(n-1)+n.$$", null, "This motivates the following definition:\n\n##### Notation and Definition: Sum sequence\n\nLet $$(a_n)_n, a_n: \\mathbb{N}\\to M$$ be a sequence with terms $$a_n$$, the sum is written: $a_1 + a_2 + a_3 + \\ldots + a_{n-1} + a_n =: \\sum_{k=1}^n a_k$ The sign $$\\sum$$ is called sigma. Here, the index $$k$$ increases from 1 to $$n$$.\n\nSum sequences are sequences whose terms are formed by summation of previous terms.\n\nThus the nth triangular number can be written as: $D_n = \\sum_{k=1}^n k$\n\n##### Example 3: Sequence of square numbers\n\nThe sequence of square numbers $$(q_n)_n$$ is defined by: $$q_n=n^2$$. The terms of this sequence can also be illustrated by the addition of coins.\n\nInterestingly, the sum of two consecutive triangular numbers is a square number. So, for example, we have: $$3+1=4$$ and $$6+3=9$$. In general this gives the relationship:\n\n$q_n=D_n + D_{n-1}$", null, "##### Example 4: Sequence of cube numbers\n\nAnalogously to the sequence of square number, we give the definition of cube numbers as $a_n := n^3.$ The first terms of the sequence are: $$(1,8,27,64,125,\\ldots)$$.", null, "##### Example 5.\n\nLet $$(q_n)_n$$ with $$q_n := n^2$$ be the sequence of square numbers \\begin{aligned}(1,4,9,16,25,36,49,64,81,100 \\ldots)\\end{aligned} and define the function $$\\varphi(n) = 2n$$. The composition $$(q_{2n})_n$$ yields: \\begin{aligned}(q_{2n})_n &= (q_2,q_4,q_6,q_8,q_{10},\\ldots) \\\\ &= (4,16,36,64,100,\\ldots).\\end{aligned}\n\n##### Definition: Sequence of differences\n\nGiven a sequence $$(a_{n})_{n}=a_{1},\\, a_{2},\\, a_{3},\\ldots,\\, a_{n},\\ldots$$; then $(a_{n+1}-a_{n})_{n}:=a_{2}-a_{1}, a_{3}-a_{2},\\dots$ is called the 1st difference sequence of $$(a_{n})_{n}$$\n\nThe 1st difference sequence of the 1st difference sequence is called the 2nd difference sequence. Analogously the $$n$$th difference< sequence is defined.\n\n##### Example 6.\n\nGiven the sequence $$(a_n)_n$$ with $$a_n := \\frac{n^2+n}{2}$$, i.e. \\begin{aligned}(a_n)_n &= (1,3,6,10,15,21,28,36,\\ldots)\\end{aligned} Let $$(b_n)_n$$ be its 1st difference sequence. Then it follows that \\begin{aligned}(b_n)_n &= (a_2-a_1, a_3-a_2, a_4-a_3,\\ldots) \\\\ &= (2,3,4,5,6,7,8,9)\\end{aligned} A term of $$(b_n)_n$$ has the general form \\begin{aligned}b_n &= a_{n+1}-a_{n} \\\\ &= \\frac{(n+1)^2+(n+1)}{2} - \\frac{n^2+n)}{2} \\\\ &= \\frac{(n+1)^2+(n+1)-n^2 - n }{2} \\\\ &= \\frac{(n^2+2n+1)+1-n^2}{2} \\\\ &= \\frac{2n+2}{2} \\\\ &= n + 1.\\end{aligned}\n\n### Some important sequences\n\nThere are a number of sequences that can be regarded as the basis of many ideas in mathematics, but also can be used in other areas (e.g. physics, biology, or financial calculations) to model real situations. We will consider three of these sequences: the arithmetic sequence, the geometric sequence, and Fibonacci sequence, i.e. the sequence of Fibonacci numbers.\n\n#### The arithmetic sequence\n\nThere are many definitions of the arithmetic sequence:\n\n##### Definition A: Arithmetic sequence\n\nA sequence $$(a_{n})_{n}$$ is called the arithmetic sequence, when the difference $$d \\in \\mathbb{R}$$ between two consecutive terms is constant, thus: $a_{n+1}-a_{n}=d \\text{ with } d=const.$\n\nNote: The explicit rule of formation follows directly from definition A: $a_{n}=a_{1}+(n-1)\\cdot d$ For the $$n$$th term of an arithmetic sequence we also have the recursive formation rule: $a_{n+1}=a_n + d.$\n\n##### Definition B: Arithmetic sequence\n\nA non-constant sequence $$(a_{n})_{n}$$ is called an arithmetic sequence (1st order) when its 1st difference sequence is a sequence of constant value.\n\nThis rule of formation gives the arithmetic sequence its name: The middle term of any three consecutive terms is the arithmetic mean of the other two, for example:\n\n$a_2 = \\frac{a_1+a_3}{2}.$\n\n##### Example 1.\n\nThe sequence of natural numbers $(a_n)_n = (1,2,3,4,5,6,7,8,9,\\ldots)$ is an arithmetic sequence, because the difference, $$d$$, between two consecutive terms is always given as $$d=1$$.\n\n#### The geometric sequence\n\nThe geometric sequence has multiple definitions:\n\n##### Definition: Geometric sequence\n\nA sequence $$(a_{n})_{n}$$ is called a geometric sequence when the ratio of any two consecutive terms is always constant $$q\\in\\mathbb{R}$$, thus $\\frac{a_{n+1}}{a_{n}}=q \\text{ for all } n\\in\\mathbb{N}.$\n\nNote.The recursive relationship $$a_{n+1} = q\\cdot a_n$$ of the terms of the geometric sequence and the explicit formula for the calculation of the n th term of a geometric sequence $a_n=a_1\\cdot q^{n-1}$ follows directly from the definition.\n\nAgain the name and the rule of formation of this sequence are connected: Here, the middle term of three consecutive terms is the geometric mean of the other two, e.g.: $a_2 = \\sqrt{a_1\\cdot a_3}.$\n\n##### Example 2.\n\nLet $$a$$ and $$q$$ be fixed positive numbers. The sequence $$(a_n)_n$$ with $$a_n := aq^{n-1}$$, i.e. $\\left( a_1, a_2, a_3, a_4,\\ldots \\right) = \\left( a, aq, aq^2, aq^3,\\ldots \\right)$ is a geometric sequence. If $$q\\geq1$$ the sequence is monotonically increasing. If $$q<1$$ it is strictly decreasing. The corresponding range $${a,aq,aq^2, aq^3}$$ is finite in the case $$q=1$$ (namely, a singleton), otherwise it is infinite.\n\n#### The Fibonacci sequence\n\nThe Fibonacci sequence is famous because it plays a role in many biological processes, for instance in plant growth, and is frequently found in nature. The recursive definition is:\n\n##### Definition: Fibonacci sequence\n\nLet $$a_0 = a_1 = 1$$ and let $a_n := a_{n-2}+a_{n-1}$ for $$n\\geq2$$. The sequence $$(a_n)_n$$ is then called the Fibonacci sequence. The terms of the sequence are called the Fibonacci numbers.\n\nThe sequence is named after the Italian mathematician Leonardo of Pisa (ca. 1200 AD), also known as Fibonacci (son of Bonacci). He considered the size of a rabbit population and discovered the number sequence: $(1,1,2,3,5,8,13,21,34,55,\\ldots),$\n\n##### Example 3.\n\nThe structure of sunflower heads can be described by a system of two spirals, which radiate out symmetrically but contra rotating from the centre; there are 55 spirals which run clockwise and 34 which run counter-clockwise.\n\nPineapples behave very similarly. There we have 21 spirals running in one direction and 34 running in the other. Cauliflower, cacti, and fir cones are also constructed in this manner.", null, "### Convergence, divergence and limits\n\nThe following chapter deals with the convergence of sequences. We will first introduce the idea of zero sequences. After that we will define the concept of general convergence.\n\n### Preliminary remark: Absolute value in $$\\mathbb{R}$$\n\nThe absolute value function $$x \\mapsto \|x\|$$ is fundamental in the study of convergence of real number sequences. Therefore we should summarise again some of the main characteristics of the absolute value function:\n\n##### Definition: Absolute Value\n\nFor any given number $$x\\in\\mathbb{R}$$ its absolute value $$\|x\|$$ is defined by \\begin{aligned}\|x\|:=\\begin{cases}x & \\text{for }x\\geq0,\\\\ -x & \\text{for }x<0.\\end{cases}\\end{aligned}\n\n##### Theorem: Calculation Rule for the Absolute Value\n\nFor $$x,y\\in\\mathbb{R}$$ the following is always true:\n\n1. $$\|x\|\\geq0,$$\n\n2. $$\|x\|=0$$ if and only if $$x=0.$$\n\n3. $$\|x\\cdot y\|=\|x\|\\cdot\|y\|$$ (Multiplicativity)\n\n4. $$\|x+y\|\\leq\|x\|+\|y\|$$ (Triangle Inequality)\n\nProof.\n\nParts 1.-3. Results follow directly from the definition and by dividing it up into separate cases of the different signs of $$x$$ and $$y$$\n\nPart 4. Here we divide the triangle inequality into different cases.\nCase 1.\n\nFirst let $$x,y \\geq 0$$. Then it follows that \\begin{aligned}\|x+y\|=x+y=\|x\|+\|y\|\\end{aligned} and the desired inequality is shown.\n\nCase 2.\n\nNext let $$x,y < 0$$. Then: \\begin{aligned}\|x+y\|=-(x+y)=(-x)+ (-y)=\|x\|+\|y\|\\end{aligned}\n\nCase 3.\n\nFinally we consider the case $$x\\geq 0$$ and $$y<0$$. Here we have two subcases:\n\n• For $$x \\geq -y$$ we have $$x+y\\geq 0$$ and thus $$\|x+y\|=x+y$$ from the definition of absolute value. Because $$y<0$$ then $$y<-y$$ and therefore also $$x+y < x-y$$. Overall we have: \\begin{aligned}\|x+y\| = x+y < x-y = \|x\|+\|y\|\\end{aligned}\n\n• For $$x < -y$$ then $$x+y<0$$. We have $$\|x+y\|=-(x+y)=-x-y$$. Because $$x\\geq0$$, we have $$-x < x$$ and thus $$-x-y\\leq x-y$$. Overall we have: \\begin{aligned}\|x+y\| = -x-y \\leq x-y = \|x\|+\|y\|\\end{aligned}\n\nCase 4.\n\nThe case $$x<0$$ and $$y\\geq0$$ we prove it analogously to the case 3, in which $$x$$ and $$y$$ are exchanged.\n\n$$\\square$$\n\n### Zero sequences\n\n##### Definition: Zero sequence\n\nA sequence $$(a_{n})_{n}$$ s called a zero sequence, if for every $$\\varepsilon>0,$$ there exists an index $$n_{0}\\in\\mathbb{N}$$ such that $\|a_{n}\| < \\varepsilon$ for every $$n\\geq n_{0},\\, n\\in\\mathbb{N}$$. In this case we also say that the sequence converges to zero.\n\nInformally: We have a zero sequence, if the terms of the sequence with high enough indices are arbitrarily close to zero.\n\n##### Example 1.\n\nThe sequence $$(a_n)_n$$ defined by $$a_{n}:=\\frac{1}{n}$$, i.e. $\\left(a_{1},a_{2},a_{3},a_{4},\\ldots\\right):=\\left(\\frac{1}{1},\\frac{1}{2},\\frac{1}{3},\\frac{1}{4},\\ldots\\right)$ is called the harmonic sequence. Clearly, it is positive for all $$n\\in\\mathbb{N}$$, however as $$n$$ increases the absolute value of each term decreases getting closer and closer to zero.\nTake for example $$\\varepsilon := \\frac{1}{5000}$$, then choosing the index $$n_0 = 5000$$, it follows that $$a_n<\\frac{1}{5000}=\\varepsilon$$, for all $$n\\geq n_0$$.\n\n##### Example 2.\n\nConsider the sequence $(a_n)_n \\text{ where } a_n:=\\frac{1}{\\sqrt{n}}.$ Let $$\\varepsilon := \\frac{1}{1000}$$.We then obtain the index $$n_0=1000000$$ in this manner that for all terms $$a_n$$ where $$n\\geq n_0$$ $$a_n < \\frac{1}{1000}=\\varepsilon$$.\n\nNote. To check whether a sequence is a zero sequence, you must choose an (arbitrary) $$\\varepsilon \\in \\mathbb{R}$$ where $$\\varepsilon > 0$$. Then search for a index $$n_0$$, after which all terms $$n$$ are smaller then said $$\\varepsilon$$.\n\n##### Example 3.\n\nWe consider the sequence $$(a_n)_n$$, defined by $a_n := \\left( -1 \\right)^n \\cdot \\frac{1}{n^2}.$\n\nBecause of the factors $$(-1)^n$$ two consecutive terms have different signs; we call a sequence whose signs change in this way an alternating sequence.\n\nWe want to show that this sequence is a zero sequence. According to the definition we have to show that for every $$\\varepsilon > 0$$ there exist $$n_0 \\in \\mathbb{N}$$, such that we have the inequality: $\|a_n\|< \\varepsilon$ for every term $$a_n$$ where $$n\\geq n_0$$.\n\nProof.\n\nFirstly we let $$\\varepsilon > 0$$ be an arbitrary constant. Because the inequality $$\|a_n\|< \\varepsilon$$ must hold true for an arbitrary $$\\varepsilon$$ we must find the index $$n_0$$ which depends on each $$\\varepsilon$$. More exactly: The inequality $\|a_{n_0}\|=\\left\| \\frac{1}{{n_0}^2} \\right\|= \\frac{1}{{n_0}^2}<\\varepsilon$ must be true for the index $$n_0$$. Solve for $$n_0$$: $n_0 > \\frac{1}{\\sqrt{\\varepsilon}},$ this index $$n_0$$ gives our desired characteristic for every $$\\varepsilon$$.\n\n##### Negative examples\n\nThe following are examples of non-convergent alternating sequences:\n\n• $$a_n = (-1)^n$$\n\n• $$a_n = (-1)^n \\cdot n$$\n\n##### Theorem: Characteristics of Zero sequences\n\nLet $$(a_n)_n$$ and $$(b_n)_n$$ be two sequences. Then:\n\n1. Let $$(a_n)_n$$ be a zero sequence, if $$b_n = a_n$$ or $$b_n = -a_n$$ for all $$n\\in\\mathbb{N}$$ then $$(b_n)_n$$ is also a zero sequence.\n\n2. Let $$(a_n)_n$$ be a zero sequence, if $$-a_n\\leq b_n \\leq a_n$$ for all $$n\\in\\mathbb{N}$$ then $$(b_n)_n$$ is also a zero sequence.\n\n3. Let $$(a_n)_n$$ be a zero sequence, then $$(c\\cdot a_n)_n$$ where $$c \\in \\mathbb{R}$$ is also a zero sequence.\n\n4. If $$(a_n)_n$$ and $$(b_n)_n$$ are zero sequences, then $$(a_n + b_n)_n$$ is also a zero sequence.\n\nProof.\n\nParts 1 and 2. If $$(a_n)_n$$ is a zero sequence, then according to the definition there is an index $$n_0 \\in \\mathbb{N}$$, such that $$\|a_n\|<\\varepsilon$$ for every $$n\\geq n_0$$ and an arbitrary $$\\varepsilon\\in\\mathbb{R}$$. But then we have $$\|b_n\|\\leq\|a_n\|<\\varepsilon$$; this proves parts 1 and 2 are correct.\n\nPart 3. If $$c=0$$, then the result is trivial. Let $$c\\neq0$$ and choose $$\\varepsilon > 0$$ such that \\begin{aligned}\|a_n\|<\\frac{\\varepsilon}{\|c\|}\\end{aligned} for all $$n\\geq n_0$$. Rearranging we get: \\begin{aligned} \|c\|\\cdot\|a_n\|=\|c\\cdot a_n\|<\\varepsilon\\end{aligned}\n\nPart 4.\n\nBecause $$(a_n)_n$$ is a zero sequence, by the definition we have $$\|a_n\|<\\frac{\\varepsilon}{2}$$ for all $$n\\geq n_0$$. Analogously, for the zero sequence $$(b_n)_n$$ there is a $$m_0 \\in \\mathbb{N}$$ with $$\|b_n\|<\\frac{\\varepsilon}{2}$$ for all $$n\\geq m_0$$.\n\nThen for all $$n > \\max(n_0,m_0)$$ it follows (using the triangle inequality) that: \\begin{aligned}\|a_n + b_n\|\\leq\|a_n\|+\|b_n\|<\\frac{\\varepsilon}{2}+\\frac{\\varepsilon}{2} = \\varepsilon\\end{aligned}\n\n$$\\square$$\n\n### Convergence, divergence\n\nThe concept of zero sequences can be expanded to give us the convergence of general sequences:\n\n##### Definition: Convergence and Divergence\n\nA sequence $$(a_{n})_{n}$$ is called convergent to $$a\\in\\mathbb{R}$$, if for every $$\\varepsilon>0$$ there exists a $$n_{0}$$ such that: $\|a_{n}-a\| \\lt \\varepsilon \\text{ for all }n\\in\\mathbb{N}_{0},\\text{ where }n\\geq n_{0}$\n\nAn equivalent definition can be defined by:\n\nA sequence $$(a_{n})_{n}$$ is called convergent to $$a\\in\\mathbb{R}$$, if $$(a_{n}-a)_{n}$$ is a zero sequence.\n\n##### Example 4.\n\nWe consider the sequence $$(a_n)_n$$ where $a_n=\\frac{2n^2+1}{n^2+1}.$ By plugging in large values of $$n$$, we can see that for $$n\\to\\infty$$ $$a_n \\to 2$$ and therefore we can postulate that the limit is $$a=2$$.\n\nProof.\n\nFor a vigorous proof, we show that for every $$\\varepsilon > 0$$ there exists an index $$n_0\\in\\mathbb{N}$$, such that for every term $$a_n$$ with $$n>n_0$$ the following relationship holds: $\\left\| \\frac{2n^2+1}{n^2+1} - 2\\right\| < \\varepsilon.$\n\nFirstly we estimate the inequality: \\begin{aligned}\\left\|\\frac{2n^2+1}{n^2+1}-2\\right\| =&\\left\|\\frac{2n^2+1-2\\cdot\\left(n^2+1\\right)}{n^2+1}\\right\| \\\\ =&\\left\|\\frac{2n^2+1-2n^2-2}{n^2+1}\\right\| \\\\ =&\\left\|-\\frac{1}{n^2+1}\\right\| \\\\ =&\\left\|\\frac{1}{n^2+1}\\right\| \\\\ <&\\frac{1}{n}.\\end{aligned}\n\nNow, let $$\\varepsilon > 0$$ be an arbitrary constant. We then choose the index $$n_0\\in\\mathbb{N}$$, such that $n_0 > \\frac{1}{\\varepsilon} \\text{, or equivalently, } \\frac{1}{n_0} < \\varepsilon.$ Finally from the above inequality we have: $\\left\|\\frac{2n^2+1}{n^2+1}-2\\right\| < \\frac{1}{n} < \\frac{1}{n_0} < \\varepsilon,$ Thus we have proven the claim and so by definition $$a=2$$ is the limit of the sequence.\n\n$$\\square$$\n\nIf a sequence is convergent, then there is exactly one number which is the limit. This characteristic is called the uniqueness of convergence.\n\n##### Theorem: Uniqueness of Convergence\n\nLet $$(a_{n})_{n}$$ be a sequence that converges to $$a\\in\\mathbb{R}$$ and to $$b\\in\\mathbb{R}$$. This implies $$a=b$$.\n\nProof.\n\nAssume $$a\\ne b$$; choose $$\\varepsilon\\in\\mathbb{R}$$ with $$\\varepsilon:=\\frac{1}{3}\|a-b\|.$$ Then in particular $$[a-\\varepsilon,a+\\varepsilon]\\cap[b-\\varepsilon,b+\\varepsilon]=\\emptyset.$$\n\nBecause $$(a_{n})_{n}$$ converges to $$a$$, there is, according to the definition of convergence, a index $$n_{0}\\in\\mathbb{N}$$ with $$\|a_{n}-a\|< \\varepsilon$$ for $$n\\geq n_{0}.$$ Furthermore, because $$(a_{n})_{n}$$ converges to $$b$$ there is also a $$\\widetilde{n_{0}}\\in\\mathbb{N}$$ with $$\|a_{n}-b\|< \\varepsilon$$ for $$n\\geq\\widetilde{n_{0}}.$$ For $$n\\geq\\max\\{n_{0},\\widetilde{n_{0}}\\}$$ we have: \\begin{aligned}\\varepsilon\\ = &\\ \\frac{1}{3}\|a-b\| \\Rightarrow\\\\ 3\\varepsilon\\ = &\\ \|a-b\|\\\\ = &\\ \|(a-a_{n})+(a_{n}-b)\|\\\\ \\leq &\\ \|a_{n}-a\|+\|a_{b}-b\|\\\\ < &\\ \\varepsilon+\\varepsilon=2\\varepsilon,\\end{aligned} Consequently we have obtained $$3\\varepsilon\\leq2\\varepsilon$$, which is a contradiction as $$\\varepsilon>0$$. Therefore the assumption must be wrong, so $$a=b$$.\n\n$$\\square$$\n\n##### Definition: Divergent, Limit\n\nIf provided that a sequence $$(a_{n})_{n}$$ and an $$a\\in\\mathbb{R}$$ exist, to which the sequence converges, then the sequence is called convergent and $$a$$ is called the limit of the sequence, otherwise it is called divergent.\n\nNotation. $$(a_{n})_{n}$$ is convergent to $$a$$ is also written: $a_{n}\\rightarrow a,\\text{ or }\\lim_{n\\rightarrow\\infty}a_{n}=a.$ Such notation is allowed, as the limit of a sequence is always unique by the above Theorem (provided it exists).\n\n##### Theorem: Bounded Sequences\n\nA convergent sequence $$(a_n)_n$$ is bounded i.e. there exists a constant $$r\\in\\mathbb{R}$$ such that: $\|a_n\| \\lt r$ for all $$n\\in\\mathbb{N}$$.\n\nProof.\n\nWe assume that the sequence $$(a_n)_n$$ has the limit $$a$$. By the definition of convergence, we have that $$\|a_n - a\|<\\varepsilon$$ for all $$\\varepsilon \\in \\mathbb{R}$$ and $$n\\geq n_0$$. Choosing $$\\varepsilon = 1$$ gives:\n\\begin{aligned}\|a_n\|-\|a\|&\\ \\leq \|a_n -a\| \\\\ &\\ < 1,\\end{aligned} And therefore also $$\|a_n\|\\leq \|a\|+1$$.\n\nThus for all $$n\\in \\mathbb{N}$$: $\|a_n\|\\leq \\max \\left\\{ \|a_1\|,\|a_2\|,\\ldots,\|a_{n_0}\|,\|a\|+1 \\right\\}=:r$\n\n$$\\square$$\n\n### Rules for convergent sequences\n\n##### Theorem: Subsequences\n\nLet $$(a_{n})_{n}$$ be a sequence such that $$a_{n}\\rightarrow a$$ and let $$(a_{\\varphi(n)})_{n}$$ be a subsequence of $$(a_{n})_{n}$$. Then it follows that $$(a_{\\varphi(n)})_{n}\\rightarrow a$$.\n\nInformally: If a sequence is convergent then all of its subsequences are also convergent and in fact converge to the same limit as the original.\n\nProof.\n\nBy the definition of a subsequence $$\\varphi(n)\\geq n$$. Because $$a_{n}\\rightarrow a$$ it is implicated that $$\|a_{n}-a\|<\\varepsilon$$ for $$n\\geq n_{0}$$, therefore $$\|a_{\\varphi(n)}-a\|<\\varepsilon$$ for these indices $$n$$.\n\n$$\\square$$\n\n##### Theorem: Rules\n\nLet $$(a_{n})_{n}$$ and $$(b_{n})_{n}$$ be sequences with $$a_{n}\\rightarrow a$$ and $$b_{n}\\rightarrow b$$. Then for $$\\lambda, \\mu \\in \\mathbb{R}$$ it follows that:\n\n1. $$\\lambda \\cdot (a_n)+\\mu \\cdot (b_n) \\to \\lambda \\cdot a + \\mu \\cdot b$$\n\n2. $$(a_n)\\cdot (b_n) \\to a\\cdot b$$\n\nInformally: Sums, differences and products of convergent sequences are convergent.\n\nProof.\n\nPart 1. Let $$\\varepsilon > 0$$. We must show, that for all $$n \\geq n_0$$ it follows that: $\|\\lambda \\cdot a_n + \\mu \\cdot b_n - \\lambda \\cdot a - \\mu \\cdot b\| < \\varepsilon.$ The left hand side we estimate using: $\|\\lambda (a_n-a)+\\mu (b_n - b)\| \\leq \|\\lambda\|\\cdot\|a_n-a\|+\|\\mu\|\\cdot\|b_n-b\|.$\n\nBecause $$(a_n)_n$$ and $$(b_n)_n$$ converge, for each given $$\\varepsilon > 0$$ it holds true that: \\begin{aligned}\|a_n - a\| <\\ \\varepsilon_1 := &\\ \\textstyle \\frac{\\varepsilon}{2\|\\lambda\|} \\text{ for all }n\\geq n_0\\\\ \|b_n - b\| <\\ \\varepsilon_2 := &\\ \\textstyle \\frac{\\varepsilon}{2\|\\mu\|} \\text{ for all }n\\geq n_1\\end{aligned}\n\nTherefore \\begin{aligned}\|\\lambda\|\\cdot\|a_n-a\|+\|\\mu\|\\cdot\|b_n-b\| < &\\ \|\\lambda\|\\varepsilon_1 + \|\\mu\|\\varepsilon_2 \\\\ = &\\ \\textstyle{ \\frac{\\varepsilon}{2} + \\frac{\\varepsilon}{2} } = \\varepsilon\\end{aligned} for all numbers $$n \\geq \\max \\{n_0,n_1\\}$$. Therefore the sequence $\\left( \\lambda \\left( a_n - a \\right) + \\mu \\left( b_n - b \\right) \\right)_n$ is a zero sequence and the desired inequality is shown.\n\nPart 2. Let $$\\varepsilon > 0$$. We have to show, that for all $$n > n_0$$ $\|a_n b_n - a b\| < \\varepsilon.$ Furthermore an estimation of the left hand side follows: \\begin{aligned} \|a_n b_n - a b\| =&\\ \|a_n b_n - a b_n + a b_n - ab\| \\\\ \\leq &\\ \|b_n\|\\cdot\|a_n-a\| + \|a\|\\cdot\|b_n - b\|.\\end{aligned} We choose a number $$B$$, such that $$\|b_n\| \\lt b$$ for all $$n$$ and $$\|a\| \\lt b$$. Such a value of $$B$$ exists by the Theorem of convergent sequences being bounded. We can then use the estimation: \\begin{aligned}\|b_n\|\\cdot\|a_n-a\| + \|a\|\\cdot\|b_n - b\| <&\\ B \\cdot \\left(\|a_n - a\| + \|b_n - b\| \\right).\\end{aligned} For all $$n>n_0$$ we have $$\|a_n - a\|<\\frac{\\varepsilon}{2\\cdot B}$$ and $$\|b_n - b\|<\\frac{\\varepsilon}{2\\cdot B}$$, and - putting everything together - the desired inequality it shown.\n\n$$\\square$$" ]	[ null, "https://mycourses.aalto.fi/draftfile.php/799428/user/draft/567529048/kugeln_000.jpg", null, "https://mycourses.aalto.fi/draftfile.php/799428/user/draft/567529048/dreiecks_zahlen.jpg", null, "https://mycourses.aalto.fi/draftfile.php/799428/user/draft/567529048/kugeln_001.jpg", null, "https://mycourses.aalto.fi/draftfile.php/799428/user/draft/567529048/kugeln_002.jpg", null, "https://mycourses.aalto.fi/draftfile.php/799428/user/draft/567529048/kugeln_003.jpg", null, "https://mycourses.aalto.fi/draftfile.php/799428/user/draft/567529048/sonnenblume_klein.jpg", null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.76076454,"math_prob":1.0000044,"size":22011,"snap":"2021-31-2021-39","text_gpt3_token_len":7534,"char_repetition_ratio":0.17480801,"word_repetition_ratio":0.032901295,"special_character_ratio":0.35995638,"punctuation_ratio":0.11905861,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":1.0000092,"pos_list":[0,1,2,3,4,5,6,7,8,9,10,11,12],"im_url_duplicate_count":[null,2,null,2,null,2,null,2,null,2,null,2,null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2021-09-26T05:44:21Z\",\"WARC-Record-ID\":\"<urn:uuid:b62280f5-3d54-45fe-9aa2-92a7df41f693>\",\"Content-Length\":\"127254\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:33eadb42-9128-4a2b-877a-ca1d7711b023>\",\"WARC-Concurrent-To\":\"<urn:uuid:d2070710-5115-4ca3-ad35-9207c558e714>\",\"WARC-IP-Address\":\"130.233.236.7\",\"WARC-Target-URI\":\"https://openlearning.aalto.fi/mod/book/view.php?id=10324\",\"WARC-Payload-Digest\":\"sha1:P4HNIXLZQBXWGW3ZKL2VEOENVJHYCL7A\",\"WARC-Block-Digest\":\"sha1:HC6LOTZFGQ2SYJMTILPH7WXR6J4OUVEF\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2021/CC-MAIN-2021-39/CC-MAIN-2021-39_segments_1631780057830.70_warc_CC-MAIN-20210926053229-20210926083229-00442.warc.gz\"}"}
https://www.rcps.us/domain/179	[ "# Mathematics\n\n•", null, "The Mathematics Standards of Learning identify essential academic content at each grade level for sequential learning. The content of the mathematics standards supports the following five goals for students: becoming mathematical problem solvers, communicating mathematically, reasoning mathematically, making mathematical connections and using mathematical representations to model and interpret practical situations. Roanoke County Schools is dedicated to helping all students achieve these goals." ]	[ null, "https://www.rcps.us/cms/lib/VA01818713/Centricity/Domain/179/piguy.jpg", null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.86463946,"math_prob":0.6037605,"size":606,"snap":"2022-27-2022-33","text_gpt3_token_len":113,"char_repetition_ratio":0.17774086,"word_repetition_ratio":0.0,"special_character_ratio":0.17656766,"punctuation_ratio":0.11363637,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9913791,"pos_list":[0,1,2],"im_url_duplicate_count":[null,6,null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2022-08-17T19:17:09Z\",\"WARC-Record-ID\":\"<urn:uuid:51ce6223-a53e-49ed-8c2c-ffdf45e90872>\",\"Content-Length\":\"347775\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:a305cb50-c390-4cac-8db3-8a8079d0b589>\",\"WARC-Concurrent-To\":\"<urn:uuid:5a83c936-b8fa-491c-9060-ab2b01911fd7>\",\"WARC-IP-Address\":\"99.84.108.27\",\"WARC-Target-URI\":\"https://www.rcps.us/domain/179\",\"WARC-Payload-Digest\":\"sha1:HFNOVY67U22EEKESYXGSCEZKV7FNZHHB\",\"WARC-Block-Digest\":\"sha1:WVXO7TLT2SDJYNEXHHIJZ4KCSDWJ2BQL\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2022/CC-MAIN-2022-33/CC-MAIN-2022-33_segments_1659882573104.24_warc_CC-MAIN-20220817183340-20220817213340-00230.warc.gz\"}"}
https://metanumbers.com/100902500	[ "# 100902500 (number)\n\n100,902,500 (one hundred million nine hundred two thousand five hundred) is an even nine-digits composite number following 100902499 and preceding 100902501. In scientific notation, it is written as 1.009025 × 108. The sum of its digits is 17. It has a total of 7 prime factors and 30 positive divisors. There are 40,360,000 positive integers (up to 100902500) that are relatively prime to 100902500.\n\n## Basic properties\n\n• Is Prime? No\n• Number parity Even\n• Number length 9\n• Sum of Digits 17\n• Digital Root 8\n\n## Name\n\nShort name 100 million 902 thousand 500 one hundred million nine hundred two thousand five hundred\n\n## Notation\n\nScientific notation 1.009025 × 108 100.9025 × 106\n\n## Prime Factorization of 100902500\n\nPrime Factorization 22 × 54 × 40361\n\nComposite number\nDistinct Factors Total Factors Radical ω(n) 3 Total number of distinct prime factors Ω(n) 7 Total number of prime factors rad(n) 403610 Product of the distinct prime numbers λ(n) -1 Returns the parity of Ω(n), such that λ(n) = (-1)Ω(n) μ(n) 0 Returns: 1, if n has an even number of prime factors (and is square free) −1, if n has an odd number of prime factors (and is square free) 0, if n has a squared prime factor Λ(n) 0 Returns log(p) if n is a power pk of any prime p (for any k >= 1), else returns 0\n\nThe prime factorization of 100,902,500 is 22 × 54 × 40361. Since it has a total of 7 prime factors, 100,902,500 is a composite number.\n\n## Divisors of 100902500\n\n30 divisors\n\n Even divisors 20 10 10 0\nTotal Divisors Sum of Divisors Aliquot Sum τ(n) 30 Total number of the positive divisors of n σ(n) 2.20659e+08 Sum of all the positive divisors of n s(n) 1.19757e+08 Sum of the proper positive divisors of n A(n) 7.3553e+06 Returns the sum of divisors (σ(n)) divided by the total number of divisors (τ(n)) G(n) 10045 Returns the nth root of the product of n divisors H(n) 13.7183 Returns the total number of divisors (τ(n)) divided by the sum of the reciprocal of each divisors\n\nThe number 100,902,500 can be divided by 30 positive divisors (out of which 20 are even, and 10 are odd). The sum of these divisors (counting 100,902,500) is 220,659,054, the average is 735,530,1.8.\n\n## Other Arithmetic Functions (n = 100902500)\n\n1 φ(n) n\nEuler Totient Carmichael Lambda Prime Pi φ(n) 40360000 Total number of positive integers not greater than n that are coprime to n λ(n) 1009000 Smallest positive number such that aλ(n) ≡ 1 (mod n) for all a coprime to n π(n) ≈ 5803043 Total number of primes less than or equal to n r2(n) 40 The number of ways n can be represented as the sum of 2 squares\n\nThere are 40,360,000 positive integers (less than 100,902,500) that are coprime with 100,902,500. And there are approximately 5,803,043 prime numbers less than or equal to 100,902,500.\n\n## Divisibility of 100902500\n\n m n mod m 2 3 4 5 6 7 8 9 0 2 0 0 2 6 4 8\n\nThe number 100,902,500 is divisible by 2, 4 and 5.\n\n• Abundant\n\n• Polite\n\n• Frugal\n\n## Base conversion (100902500)\n\nBase System Value\n2 Binary 110000000111010011001100100\n3 Ternary 21000212101012122\n4 Quaternary 12000322121210\n5 Quinary 201312340000\n6 Senary 14002405112\n8 Octal 600723144\n10 Decimal 100902500\n12 Duodecimal 29960798\n20 Vigesimal 1bacg50\n36 Base36 1o2ov8\n\n## Basic calculations (n = 100902500)\n\n### Multiplication\n\nn×y\n n×2 201805000 302707500 403610000 504512500\n\n### Division\n\nn÷y\n n÷2 5.04512e+07 3.36342e+07 2.52256e+07 2.01805e+07\n\n### Exponentiation\n\nny\n n2 10181314506250000 1027320086966890625000000 103659165075176681289062500000000 10459468903998015083769628906250000000000\n\n### Nth Root\n\ny√n\n 2√n 10045 465.551 100.225 39.8823\n\n## 100902500 as geometric shapes\n\n### Circle\n\n Diameter 2.01805e+08 6.33989e+08 3.19855e+16\n\n### Sphere\n\n Volume 4.30323e+24 1.27942e+17 6.33989e+08\n\n### Square\n\nLength = n\n Perimeter 4.0361e+08 1.01813e+16 1.42698e+08\n\n### Cube\n\nLength = n\n Surface area 6.10879e+16 1.02732e+24 1.74768e+08\n\n### Equilateral Triangle\n\nLength = n\n Perimeter 3.02708e+08 4.40864e+15 8.73841e+07\n\n### Triangular Pyramid\n\nLength = n\n Surface area 1.76346e+16 1.21071e+23 8.23865e+07\n\n## Cryptographic Hash Functions\n\nmd5 771b8fdf377de5a2e1a60b15a65cd295 741c8e2f8dd8155a95a3a8f43d92bd52e29367f2 9d99fb4707398fb5959a0e1cd424d90738b8172da891bbc62b6f50dc55bfafec 5fe666d56a385346af00a07a8e9d02139f163786ea1f15aa78073bc8c6d1047a377ee4a6b9a6d169cd9c85927347c3df230899613c857b5c05b47ba7d0b97cbc 4f77d7629013ba926144e92c0fb484eada7d4da9" ]	[ null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.6176201,"math_prob":0.9873689,"size":4894,"snap":"2021-43-2021-49","text_gpt3_token_len":1739,"char_repetition_ratio":0.12699386,"word_repetition_ratio":0.03377386,"special_character_ratio":0.48855743,"punctuation_ratio":0.093446605,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9948272,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2021-12-02T01:27:11Z\",\"WARC-Record-ID\":\"<urn:uuid:09b2ff8a-ec47-4162-9426-d99436a1c2ac>\",\"Content-Length\":\"40665\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:ee90c840-5ba7-48b6-a99b-42ff17f2f2a8>\",\"WARC-Concurrent-To\":\"<urn:uuid:2f652f88-0e03-4e87-a828-5e8b39483703>\",\"WARC-IP-Address\":\"46.105.53.190\",\"WARC-Target-URI\":\"https://metanumbers.com/100902500\",\"WARC-Payload-Digest\":\"sha1:6PMJXNKWBQEISPPSMTP4TLPXPGNENGP6\",\"WARC-Block-Digest\":\"sha1:EJBWAQAMSBYAVQWPFLTQLYHPLQFI4WDL\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2021/CC-MAIN-2021-49/CC-MAIN-2021-49_segments_1637964361064.58_warc_CC-MAIN-20211201234046-20211202024046-00100.warc.gz\"}"}
https://calcopedia.com/logarithm/	[ "# Logarithm Calculator\n\nCompute natural, decimal logarithms (or with other bases) online with a solution.\n\nLogarithm equals:\n\n0copy\n\nContents\n\n## What are Logarithms and How to Calculate Them?", null, "A logarithm is the inverse operation to exponentiation, just as subtraction is the inverse of addition and division is the inverse of multiplication. Logarithms are used to solve equations in which the variable is in the exponent position.\n\nFor instance, the base 10 logarithm of 1000 is 3. This means that 10 raised to the power of 3 equals 1000. It`s commonly written as log10(1000) = 3.\n\nIf we have a logarithm written as logb(x) = y, it can be converted to its exponential form as: by = x.\n\nLogarithms are widely used in various scientific fields such as astronomy, biology, and engineering to simplify calculations of very large or very small numbers.\n\n## How to Use the Logarithm Calculator?\n\nOur logarithm calculator is a handy tool designed to help you compute logarithms without the complexities of manual calculations. Here`s a step-by-step guide:\n\n1. Navigate to the main interface of the calculator.\n\n2. Choose the base of the logarithm. Common bases are 10 (common logarithm) and 'e' (natural logarithm).\n\n3. Enter the number for which you want to compute the logarithm in the given input field.\n\n4. Click on the 'Calculate' button.\n\n5. The calculator will instantly display the result on the screen.\n\n6. Optionally, you can also view the detailed solution by clicking on the 'Show Solution' option.\n\n7. To compute another logarithm, simply clear the input fields and repeat the process.\n\n## Examples of Calculating Logarithms\n\nLogarithms might seem abstract, but they pop up in surprising, and sometimes humorous, real-life situations. Let`s dive into a few:\n\nExample 1: Imagine you're trying to figure out how many digits a number has, and someone cheekily tells you to use logarithms. If you have the number 1000, you can quickly deduce it has 4 digits. The base 10 logarithm of 1000 is 3, add 1, and there you have it!\n\nExample 2: Suppose you're at a party and someone challenges you to a decibel (dB) contest. They mention that every 10 dB increase represents a tenfold increase in intensity. If you go from 10 dB to 40 dB, the intensity increased by 10^3 or 1000 times! Logarithms make sound math fun!\n\nExample 3: Let`s say you’re a coffee enthusiast and you read that caffeine decay in the body is modeled by a logarithmic function. If you consume 200mg of caffeine, and half of it is gone after 4 hours (thanks to your liver), the decay can be understood using logarithms.\n\n## Nuances of Calculating Logarithms\n\nLogarithms, though powerful, come with their own set of quirks and intricacies. Here are some to keep in mind:\n\n1. Logarithms of numbers less than or equal to zero are undefined in the real number system.\n\n2. The logarithm of 1, no matter the base, is always 0.\n\n3. Natural logarithms have the irrational number 'e' (approximately 2.71828) as their base.\n\n4. Changing the base of a logarithm involves a formula: logb(x) = loga(x) / loga(b).\n\n5. The product rule states: logb(xy) = logb(x) + logb(y).\n\n6. The quotient rule: logb(x/y) = logb(x) - logb(y).\n\n7. The power rule: logb(xy) = y * logb(x).\n\n8. Logarithmic equations often have extraneous solutions, so always verify your results!\n\n9. Remember: logb(b) = 1, no matter what value 'b' has (except when b ≤ 0 or b = 1).\n\n10. When using calculators, ensure you know which base the calculator defaults to, typically either base 10 or 'e'.\n\n### Why are logarithms useful?\n\nLogarithms provide a way to handle very large or very small numbers, simplify calculations, and solve problems related to exponential growth and decay.\n\n### What is the base of a natural logarithm?\n\n\"e\", an irrational number approximately equal to 2.71828, is the base of natural logarithms.\n\n### Can I compute the logarithm of a negative number?\n\nIn the real number system, the logarithm of a negative number is undefined.\n\n### What is the difference between common and natural logarithms?\n\nCommon logarithms have a base of 10, while natural logarithms use the base \"e\".\n\n### How do I change the base of a logarithm?\n\nTo change the base, use the formula: log<sub>b</sub>(x) = log<sub>a</sub>(x) / log<sub>a</sub>(b).\n\n## Similar calculators\n\nYou may find the following calculators on the same topic useful:\n\n## Share on social media\n\nIf you liked it, please share the calculator on your social media platforms. It`s easy for you and beneficial for the project`s promotion. Thank you!\n\n### Do you have anything to add?\n\nFeel free to share your opinion, comment, or suggestion." ]	[ null, "https://calcopedia.com/images/logarithm.svg", null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.8908069,"math_prob":0.9760925,"size":6337,"snap":"2023-40-2023-50","text_gpt3_token_len":1434,"char_repetition_ratio":0.17069319,"word_repetition_ratio":0.13119534,"special_character_ratio":0.22597444,"punctuation_ratio":0.12677231,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9984691,"pos_list":[0,1,2],"im_url_duplicate_count":[null,1,null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2023-12-01T05:46:49Z\",\"WARC-Record-ID\":\"<urn:uuid:b9579f28-cfcc-4c5c-9618-57b412b1b1e4>\",\"Content-Length\":\"27178\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:cad00751-52e8-4fcc-b29f-aa63e797a7c8>\",\"WARC-Concurrent-To\":\"<urn:uuid:7209690c-891b-4a3f-9423-ac1ac2d3489f>\",\"WARC-IP-Address\":\"104.21.77.90\",\"WARC-Target-URI\":\"https://calcopedia.com/logarithm/\",\"WARC-Payload-Digest\":\"sha1:KDYJHHBW37RSRVJPYGSAD75JMWS2QB27\",\"WARC-Block-Digest\":\"sha1:UHYQZWB4NKHXUX5MZLLSKYSIX6HTPHZU\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2023/CC-MAIN-2023-50/CC-MAIN-2023-50_segments_1700679100276.12_warc_CC-MAIN-20231201053039-20231201083039-00349.warc.gz\"}"}
http://www.tribonet.org/hertzian-stress-subsurface-calculator-line-contact/	[ "This is a Hertzian stress calculator for subsurface stress components in case of a contact between cylinders (line contact). Equations and the references are given below the calculator.\n\nAs shown in the figure, a contact of two cylinders is considered here. Hertzian subsurface stresses can be obtained using following equations:\n\n(1)", null, "where", null, "is the stress component in x direction,", null, "is the maximum Hertzian pressure and", null, "is Hertzian contact radius.Please find all the related equations here. Corresponding Matlab code for Hertzian solution can be found here. The online Hertzian contact stress calculator can be found here.\n\nEquations were taken from .\n\n. Contact Mechanics and Friction: Physical Principles and Applications, V. Popov, 2010\n\n•\n•\n•\n•" ]	[ null, "http://www.tribonet.org/wp-content/plugins/native-lazyload/assets/images/placeholder.svg", null, "http://www.tribonet.org/wp-content/plugins/native-lazyload/assets/images/placeholder.svg", null, "http://www.tribonet.org/wp-content/plugins/native-lazyload/assets/images/placeholder.svg", null, "http://www.tribonet.org/wp-content/plugins/native-lazyload/assets/images/placeholder.svg", null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.8809045,"math_prob":0.9828355,"size":739,"snap":"2020-24-2020-29","text_gpt3_token_len":154,"char_repetition_ratio":0.16054422,"word_repetition_ratio":0.0,"special_character_ratio":0.19215156,"punctuation_ratio":0.12403101,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.99804527,"pos_list":[0,1,2,3,4,5,6,7,8],"im_url_duplicate_count":[null,null,null,null,null,null,null,null,null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2020-05-29T20:59:32Z\",\"WARC-Record-ID\":\"<urn:uuid:e2d124ce-4294-4cc5-9871-b84318cf193e>\",\"Content-Length\":\"95513\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:7f44964a-3845-449e-8b9b-33b3fa957186>\",\"WARC-Concurrent-To\":\"<urn:uuid:e6c35a7e-4436-4939-a1c5-796079a3dba3>\",\"WARC-IP-Address\":\"104.27.145.108\",\"WARC-Target-URI\":\"http://www.tribonet.org/hertzian-stress-subsurface-calculator-line-contact/\",\"WARC-Payload-Digest\":\"sha1:2T3XNGTKMRK3VD74QHD4TIMJHLQJKLI4\",\"WARC-Block-Digest\":\"sha1:NCZWFUIU2CLOWWUNGDZRGJNKEFUH2LW4\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2020/CC-MAIN-2020-24/CC-MAIN-2020-24_segments_1590347406365.40_warc_CC-MAIN-20200529183529-20200529213529-00474.warc.gz\"}"}
https://blog.softhints.com/pandas-check-value-column-contained-another-column-same-row/	[ "In this guide, I'll show you how to find if value in one string or list column is contained in another string column in the same row. In the article are present 3 different ways to achieve the same result.\n\nThese examples can be used to find a relationship between two columns in a DataFrame.\n\nDataset: IMDB 5000 Movie Dataset\n\n## Step 1: Check If String Column Contains Substring of Another with Function\n\nThe first solution is the easiest one to understand and work it. It is easy for customization and maintenance.\n\nTo start, we will define a function which will be used to perform the check. Then the function will be invoked by using `apply`:\n\n``````def find_value_column(row):\nreturn row.country in row.movie_title\n\ndf[df.apply(find_value_column, axis=1)][['movie_title', 'country']]\n``````\n\nwhich will result in:\n\nmovie_title country\n196 Australia Australia\n2504 McFarland, USA USA\n\nWhat will happen if there are NaN values in one of the columns?\n\n``````def find_value_column(row):\nreturn row.movie_title.lower().strip() in row.plot_keywords\n\n``````\n\nThen error will be raised:\n\n``````TypeError: (\"argument of type 'float' is not iterable\", 'occurred at index 4')\n``````\n\nThere is easy solution for this error - convert the column NaN values to empty list values thus:\n\n``````for row in df.loc[df.plot_keywords.isnull(), 'plot_keywords'].index:\ndf.at[row, 'plot_keywords'] = []\n``````\n\nnow we get proper results like:", null, "## Step 2: Check If Column Contains Another Column with Lambda\n\nThe second solution is similar to the first - in terms of performance and how it is working - one but this time we are going to use `lambda`. The advantage of this way is - shortness:\n\n``````df[df.apply(lambda x: x.country in x.movie_title, axis=1)][['movie_title', 'country']]\n``````\nmovie_title country\n196 Australia Australia\n2504 McFarland, USA USA\n\nA possible disadvantage of this method is the need to know how `apply` and `lambda` works and how to deal with errors if any.\n\nFor example this piece of code similar but will result in error like:\n\n``````df.apply(lambda row: df.country in df.movie_title, axis=1)\n``````\n\noutput:\n\n``````TypeError: (\"'Series' objects are mutable, thus they cannot be hashed\", 'occurred at index 0')\n``````\n\nIt may be obvious for some people but a novice will have hard time to understand what is going on.\n\n## Step 3: Fastest Way to Check If One Column Contains Another\n\nThis solution is the fastest one. It is short and easy to understand. It includes `zip` on the selected data. In this case data can be used from two different DataFrames.\n\n``````df[[x in x for x in zip(df['country'], df['movie_title'])]][['movie_title', 'country']]\n``````\nmovie_title country\n196 Australia Australia\n2504 McFarland, USA USA\n\nagain if the column contains NaN values they should be filled with default values like:\n\n``````df['country'].fillna('Uknown', inplace=True)\n``````\n\n## Step 4: For Loop and df.iterrows() Version\n\nThe final solution is the most simple one and it's suitable for beginners. Iterates over the rows one by one and perform the check. This solution is the slowest one:\n\n``````for i, row in df.iterrows():\nif row.country in row.movie_title:\nprint(row.country, row.movie_title)\n``````\n\nresult:\n\n``````Australia Australia\nUSA McFarland, USA\n``````\n\n## Bonus Step: Check If List Column Contains Substring of Another with Function\n\nNow lets assume that we would like to check if any value from column plot_keywords:\n\n• avatar\|future\|marine\|native\|paraplegic\n• bomb\|espionage\|sequel\|spy\|terrorist\n\nis part of the movie title.\n\nIn this case we can:\n\n1. split the string column into a list\n2. convert NaN values to empty list\n3. perform search for each word in the list against the title\n``````df['keywords'] = df.plot_keywords.str.split('\|')\n``````\n\nNow we have two different options:\n\nSkip the conversion of NaN but check them in the function:\n\n``````def find_value_column(row):\nif isinstance(row['keywords'], list):\nfor keyword in row['keywords']:\nreturn keyword in row.movie_title.lower()\nelse:\nreturn False\n\n``````\n\nor\n\nConvert all the NaN values:\n\n``````df['keywords'] = df['keywords'].apply(lambda d: d if isinstance(d, list) else [])\n\ndef find_value_column(row):\nfor keyword in row['keywords']:\nreturn keyword in row.movie_title.lower()\nreturn False\n\n``````\n\nresults in:", null, "## Performance check\n\nBelow you can find results of all solutions and compare their speed:\n\n• 10 loops, best of 3: 152 ms per loop\n• 10 loops, best of 3: 153 ms per loop\n• 1000 loops, best of 3: 1.69 ms per loop\n• 1 loop, best of 3: 580 ms per loop\n\nSo the one in step 3 - zip one - is the fastest and outperform the others by magnitude. In my everyday work I prefer to use 2 and 3(for high volume data) in most cases and only in some case 1 - when there is complex logic to be implemented.\n\n## Related Articles\n\nbetter programmer\n\nPython\n\nPython\n\nSelenium\n\nPyCharm\n\nPython" ]	[ null, "https://blog.softhints.com/content/images/2020/03/pandas_search_one_column_in_another.png", null, "https://blog.softhints.com/content/images/2020/03/pandas_search_list_column_in_another.png", null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.62733835,"math_prob":0.65438026,"size":4791,"snap":"2020-45-2020-50","text_gpt3_token_len":1159,"char_repetition_ratio":0.12805516,"word_repetition_ratio":0.051034484,"special_character_ratio":0.25610518,"punctuation_ratio":0.14437367,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.953699,"pos_list":[0,1,2,3,4],"im_url_duplicate_count":[null,1,null,1,null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2020-10-27T15:42:24Z\",\"WARC-Record-ID\":\"<urn:uuid:77dab7fa-5f85-4232-8111-11215409fbd8>\",\"Content-Length\":\"65464\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:bd8f36f5-de3d-4b8e-a1c4-a9da9b100b01>\",\"WARC-Concurrent-To\":\"<urn:uuid:fda0f332-bb2b-4b14-b97f-94f1cc205e2c>\",\"WARC-IP-Address\":\"104.31.78.61\",\"WARC-Target-URI\":\"https://blog.softhints.com/pandas-check-value-column-contained-another-column-same-row/\",\"WARC-Payload-Digest\":\"sha1:YH47SXQFPCUU4EOG2WAT5A54UQDZCDWF\",\"WARC-Block-Digest\":\"sha1:S652DWWCEOGZ7HTOHZWVVCMCOTENIKZJ\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2020/CC-MAIN-2020-45/CC-MAIN-2020-45_segments_1603107894203.73_warc_CC-MAIN-20201027140911-20201027170911-00545.warc.gz\"}"}
https://blog.razrlele.com/p/1485	[ "# POJ1006\n\nWritten by 20:41 February 1, 2016\n\nPOJ1006\n\nBiorhythms\n Time Limit: 1000MS Memory Limit: 10000K Total Submissions: 124772 Accepted: 39392\n\nDescription\n\nSome people believe that there are three cycles in a person’s life that start the day he or she is born. These three cycles are the physical, emotional, and intellectual cycles, and they have periods of lengths 23, 28, and 33 days, respectively. There is one peak in each period of a cycle. At the peak of a cycle, a person performs at his or her best in the corresponding field (physical, emotional or mental). For example, if it is the mental curve, thought processes will be sharper and concentration will be easier.\nSince the three cycles have different periods, the peaks of the three cycles generally occur at different times. We would like to determine when a triple peak occurs (the peaks of all three cycles occur in the same day) for any person. For each cycle, you will be given the number of days from the beginning of the current year at which one of its peaks (not necessarily the first) occurs. You will also be given a date expressed as the number of days from the beginning of the current year. You task is to determine the number of days from the given date to the next triple peak. The given date is not counted. For example, if the given date is 10 and the next triple peak occurs on day 12, the answer is 2, not 3. If a triple peak occurs on the given date, you should give the number of days to the next occurrence of a triple peak.\n\nInput\n\nYou will be given a number of cases. The input for each case consists of one line of four integers p, e, i, and d. The values p, e, and i are the number of days from the beginning of the current year at which the physical, emotional, and intellectual cycles peak, respectively. The value d is the given date and may be smaller than any of p, e, or i. All values are non-negative and at most 365, and you may assume that a triple peak will occur within 21252 days of the given date. The end of input is indicated by a line in which p = e = i = d = -1.\n\nOutput\n\nFor each test case, print the case number followed by a message indicating the number of days to the next triple peak, in the form:\n\nCase 1: the next triple peak occurs in 1234 days.\n\nUse the plural form days” even if the answer is 1.\n\nSample Input\n\nSample Output\n\nSource\n\n(n+d)%23 == p%23\n(n+d)%28 == e%28\n(n+d)%33 == i%33\n\n### 代码\n\nCategory : acmstudy\n\nTags :" ]	[ null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.7955831,"math_prob":0.9892483,"size":5145,"snap":"2019-43-2019-47","text_gpt3_token_len":1684,"char_repetition_ratio":0.17409064,"word_repetition_ratio":0.37448132,"special_character_ratio":0.32555878,"punctuation_ratio":0.11433757,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9658275,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2019-11-12T06:56:19Z\",\"WARC-Record-ID\":\"<urn:uuid:64f47197-8079-4471-81ec-3586aa825349>\",\"Content-Length\":\"67321\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:dae9f6e9-a235-45b3-8005-ec659c40f0d1>\",\"WARC-Concurrent-To\":\"<urn:uuid:19bc8b8f-152d-49f4-8e22-8482995d91d2>\",\"WARC-IP-Address\":\"39.107.79.199\",\"WARC-Target-URI\":\"https://blog.razrlele.com/p/1485\",\"WARC-Payload-Digest\":\"sha1:5AJZJLNY5XIZMZVSPHXBV3GXTXU6A2EQ\",\"WARC-Block-Digest\":\"sha1:U3N7N64ROSIW6MBDDMQP3H5MTTDNWKGR\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2019/CC-MAIN-2019-47/CC-MAIN-2019-47_segments_1573496664752.70_warc_CC-MAIN-20191112051214-20191112075214-00134.warc.gz\"}"}
https://gitlab.linphone.org/BC/public/external/libvpx/commit/cc0eeda6ddaa76f1e7170980d11b3093d70b3d62	[ "### Fix mismatch check output\n\n```Fixes a condition where the address of the mismatching pixels was not\nbeing found/printed.\n\nChange-Id: Ifac5cd3471bc2437448128591eea7c7b87e2d8fe```\nparent 9831f205\n ... ... @@ -1517,7 +1517,7 @@ static void find_mismatch(vpx_image_t img1, vpx_image_t img2, uloc = uloc = uloc = uloc = -1; for (i = 0, match = 1; match && i < c_h; i += bsizey) { for (j = 0; j < match && c_w; j += bsizex) { for (j = 0; match && j < c_w; j += bsizex) { int k, l; int si = mmin(i + bsizey, c_h - i); int sj = mmin(j + bsizex, c_w - j); ... ... @@ -1541,7 +1541,7 @@ static void find_mismatch(vpx_image_t img1, vpx_image_t img2, } vloc = vloc = vloc = vloc = -1; for (i = 0, match = 1; match && i < c_h; i += bsizey) { for (j = 0; j < match && c_w; j += bsizex) { for (j = 0; match && j < c_w; j += bsizex) { int k, l; int si = mmin(i + bsizey, c_h - i); int sj = mmin(j + bsizex, c_w - j); ... ...\nMarkdown is supported\n0% or\nYou are about to add 0 people to the discussion. Proceed with caution.\nFinish editing this message first!" ]	[ null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.78002113,"math_prob":0.9984257,"size":231,"snap":"2019-43-2019-47","text_gpt3_token_len":68,"char_repetition_ratio":0.052863438,"word_repetition_ratio":0.0,"special_character_ratio":0.27705628,"punctuation_ratio":0.05882353,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9500189,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2019-11-15T11:43:02Z\",\"WARC-Record-ID\":\"<urn:uuid:f395e7c5-66c4-47c2-b997-52379a1d7536>\",\"Content-Length\":\"69554\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:35e0ae0b-f720-48e5-9228-123545d92798>\",\"WARC-Concurrent-To\":\"<urn:uuid:17f98f30-5a03-4083-8b10-30a7f4591f3f>\",\"WARC-IP-Address\":\"54.37.202.230\",\"WARC-Target-URI\":\"https://gitlab.linphone.org/BC/public/external/libvpx/commit/cc0eeda6ddaa76f1e7170980d11b3093d70b3d62\",\"WARC-Payload-Digest\":\"sha1:3ZJKICNUEP2GNRWY5DCRMU55ULWEPXNX\",\"WARC-Block-Digest\":\"sha1:AXKJJX3C7YY6GV3UXXJU5E2GV6BDR2GY\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2019/CC-MAIN-2019-47/CC-MAIN-2019-47_segments_1573496668618.8_warc_CC-MAIN-20191115093159-20191115121159-00536.warc.gz\"}"}
https://docs.datacentral.org.au/help-center/virtual-observatory-examples/ssa-galah-dr3-interactive-spectra-explorer-enhanced-data-central-api/	[ "# Documentation\n\nFind information, examples, FAQs and extensive descriptions of the data, curated by the survey teams.\n\n#### Help Center\n\nBrowse FAQs, guides and tutorials on common use cases.\nFeb. 10, 2021 by B. Miszalski\nFeb. 17, 2021, 11:40 a.m. B. Miszalski\n\n## SSA: GALAH DR3 Interactive Spectra Explorer enhanced by the Data Central API\n\nInteractive GALAH DR3 plot with selected low metallicity star and its spectra.\nInteractive GALAH DR3 plot with selected star and its spectra.\n\nIn this example we take a slightly approach to accessing the GALAH DR3 spectra. Instead of using the SSA service to acquire GALAH DR3 target details, we use the DataCentral API to query the GALAH DR3 tables directly using an ADQL query. The SSA service is then used to retrieve the relevant spectra on demand.\n\nPlease note that this interactive version was developed to be run from running the Python script directly on MacOSX (tested on 10.14 and 10.15).\n\nIt is not intended to be executed from within a Python notebook.\n\nInstructions: Run the script (you may need to resize the window for the plot to scale correctly) and then click on points to plot their spectra.\n\nfrom specutils import Spectrum1D\nimport matplotlib.pyplot as plt\nimport matplotlib as mpl\nfrom matplotlib.backends.backend_pdf import PdfPages\nfrom matplotlib import gridspec\nfrom mpl_toolkits.axes_grid1 import make_axes_locatable\nimport numpy as np\nimport pandas as pd\nimport matplotlib.ticker as plticker\nfrom astropy.stats import sigma_clip\nfrom astropy.time import Time\nimport requests\nimport re\nfrom io import BytesIO\nfrom astropy import units as u\nfrom pyvo.dal.ssa import search, SSAService\n\n#called by onpick to retrieve and show the spectra of the target of interest\ndef show_spectrum(name,axes,title):\n#query the SSA service\n#no position required as we already know the target name from the DataCentral API query\nurl = \"https://datacentral.org.au/vo/ssa/query\"\nservice = SSAService(url)\ncustom = {}\ncustom['TARGETNAME'] = name\ncustom['COLLECTION'] = 'galah_dr3'\nresults = service.search(*custom)\ndf = results.votable.get_first_table().to_table(use_names_over_ids=True).to_pandas()\nfilters = ['B','V','R','I']\ncolours = [\"#085dea\",\"#1e8c09\",\"#cf0000\",\"#640418\"]\n\n#go through each filter and plot its spectrum (if available)\nfor idx in range(0,4):\nfilt = filters[idx]\nax = axes[idx]\n#remove any previous spectrum/labels/titles that may have been plotted in a previous call of\n#the show_spectrum function\nax.clear()\n#show the title in the first position only\nif(idx == 0):\nax.set_title(title)\n#select only the spectrum of the filter of interest\nsubset = df[(df['band_name'] == filt)].reset_index()\n#give preference to using the continuum normalised spectra\nif(subset[subset['dataproduct_subtype'].isin(['normalised'])].shape > 0):\nsubset = subset[(subset['dataproduct_subtype'] == \"normalised\")].reset_index()\n#only proceed if we have the filter of interest available in the results\nif(subset.shape > 0):\n#add RESPONSEFORMAT=fits here to ensure we get fits format back\nurl= subset.loc[0,'access_url'] + \"&RESPONSEFORMAT=fits\"\nexptime = subset.loc[0,'t_exptime']\nax.tick_params(axis='both', which='major')\nax.tick_params(axis='both', which='minor')\nloc = plticker.MultipleLocator(base=20.0)\nax.xaxis.set_major_locator(loc)\n#plot label at last position (caveat: no check if spectra are missing)\nif(idx == 3):\nax.set_xlabel(\"Wavelength ($\\mathrm{\\AA}$)\",labelpad=10)\n#plot the spectrum\nax.plot(spec.wavelength,spec.flux,linewidth=LWIDTH,color=colours[idx])\n#adjust the y-scale to best fit the spectrum with some sigma clipping\nnspec = len(spec.spectral_axis)\nymin = min(clipped).value\nymax = max(clipped).value\nxmin = spec.wavelength.value\nxmax = spec.wavelength[nspec-1].value\n#add a 1% buffer either side of the x-range\ndx=0.01(xmax-xmin)\nax.set_xlim(xmin-dx,xmax+dx)\n#add a 1% buffer either side of the y-range\ndy=0.03(ymax-ymin)\nax.set_ylim(ymin-dy,ymax+dy)\n#else:\n#print missing data...\n\n#set the figure size to be wide in the horizontal direction\n#to accommodate the image cutouts alongside the spectra\nfsize=[20,13]\nmpl.rcParams['axes.linewidth'] = 0.7\nFSIZE=18\nLWIDTH=0.5\nLABELSIZE=10\n\n#Instead of querying SSA directly, we query the galah_dr3 table main_star\n#using the DataCentral API\n#Information about the table can be found at\n#https://docs.datacentral.org.au/galah/dr3/catalogue-data-access/\n# and\n#https://docs.datacentral.org.au/galah/dr3/table-schema/\n# and\n#https://datacentral.org.au/services/schema/#galah.dr3.catalogues.mainstargroup.main_star\n\n#Here we select the top 100 sources with names < 160000000000000 and low surface gravities (log g < 2)\n#together with their temperatures, surface gravities and metallicities\nsql_query = \"SELECT TOP 100 sobject_id, star_id, teff,e_teff,logg,e_logg, fe_h, e_fe_h FROM galah_dr3.main_star WHERE sobject_id < 160000000000000 and logg < 2.0\"\n#Query GALAH DR3 catalogue using DataCentral API\napi_url = 'https://datacentral.org.au/api/services/query/'\nqdata = {'title' : 'galah_test_query',\n'notes' : 'test query for ssa example',\n'sql' : sql_query,\n'run_async' : False,\n'email' : '[email protected]'}\npost = requests.post(api_url,data=qdata).json()\nresp = requests.get(post['url']).json()\n#convert the results to a pandas dataframe\ndf = pd.DataFrame(resp['result']['data'],columns=resp['result']['columns'])\n#remove results where the teff and logg are not determined\n#as the dataframe entries are still all strings, we have to do the following\n#instead of use nona() or notnull() functions of the dataframe\ndf = df[(df['teff'] != \"nan\") & (df['logg'] != \"nan\")]\n#convert columns to floats - needed since data coming from json results\nconvertme = ['teff','e_teff','logg','e_logg','fe_h','e_fe_h']\nfor c in convertme:\ndf[c] = df[c].astype(float)\nprint (df)\n#create a few lists/arrays to conveniently access the results\nteff = np.array(df['teff'])\nteff_err = np.array(df['e_teff'])\nlogg = np.array(df['logg'])\nlogg_err = np.array(df['e_logg'])\nfe_h = np.array(df['fe_h'])\nfe_h_err = np.array(df['e_fe_h'])\nnames = np.array(df['sobject_id'])\n\n#create a colourmap tied to the metallicity ([Fe/H])\ncmap = mpl.cm.get_cmap('plasma')\nnorm = mpl.colors.Normalize(vmin=min(fe_h),vmax=max(fe_h))\n\n#setup the plot using gridspec\ngs = gridspec.GridSpec\nfig = plt.figure(figsize=fsize)\n#4 rows, 2 cols\n#plot of log g/Teff takes up first col (all rows)\n#spectra take up all rows of 2nd col\ngs = gridspec.GridSpec(4,2)\n#B\n#V\n#R\n#I\n\n#plot the location of the results in a log g vs teff diagram\n#with the colours of each point determined by the [Fe/H] and the colourmap\n#The picker argument is required to allow for picking targets interactively (see below)\nsc = ax1.scatter(teff,logg,cmap=cmap,c=fe_h,picker=5)\n\n#create some space for the colourbar and add it\ndivider = make_axes_locatable(ax1)\nfig.colorbar(mappable=sc,label='[Fe/H]',cax=cax)\n\n#this annotation is a template for a label that appears when the mouse hovers\n#over each point in the above scatter plot\nannot = ax1.annotate(\"\",xy=(0,0),xytext=(-100,30),textcoords=\"offset points\",\nbbox=dict(boxstyle=\"round\",fc=\"w\"),\narrowprops=dict(arrowstyle=\"->\"))\n#all are off by default\nannot.set_visible(False)\n\n#flip y axis to show lower surface gravity at top\nax1.invert_yaxis()\n#flip x axis to show cooler stars at right\nax1.invert_xaxis()\n#set the axis labels\nax1.set_ylabel(\"$\\\\log\\\\,g$\")\nax1.set_xlabel(\"Teff (K)\")\n\n#function to update the annotations\ndef update_annot(ind):\npos = sc.get_offsets()[ind[\"ind\"]]\nannot.xy = pos\ntext = \"%s\" % [names[n] for n in ind[\"ind\"]]\nannot.set_text(text)\nannot.get_bbox_patch().set_facecolor(cmap(norm(fe_h[ind[\"ind\"]])))\nannot.get_bbox_patch().set_alpha(0.5)\n\n#called when the mouse hovers over a target\ndef hover(event):\nvis = annot.get_visible()\nif event.inaxes == ax1:\ncont, ind = sc.contains(event)\nif cont:\nupdate_annot(ind)\nannot.set_visible(True)\nfig.canvas.draw_idle()\nelse:\nif vis:\nannot.set_visible(False)\nfig.canvas.draw_idle()\n\n#called when the user clicks a target\ndef onpick(event):\nind = event.ind\npt = event.artist.get_offsets()\n#mouse positions\nmx = event.mouseevent.xdata\nmy = event.mouseevent.ydata\n#get closest index to mouse click\n#chosen contains the index of the\n#closest target to the mouse click\ndist = 1e6\nchosen = None\nfor idx in ind:\nprint (idx)\nprint (pt[idx])\npx = pt[idx]\npy = pt[idx]\nd = np.sqrt((px-mx)2 + (py-my)*2)\nif(d < dist):\nprint (\"names[%d]\" % idx,d)\nchosen = idx\ndist = d\n\nif(chosen is not None):\nprint (names[chosen],chosen)\n#set a title that shows some of the information extracted from the API query\n#this information is not part of the SSA obscore metadata\ntitre = \"%s $\\\\log\\\\,g$=%.2f$\\\\pm$%.2f Teff=%.2f$\\\\pm$%.2f [Fe/H]=%.2f$\\\\pm$%.2f\" % (names[chosen],\nlogg[chosen],logg_err[chosen],\nteff[chosen],teff_err[chosen],\nfe_h[chosen],fe_h_err[chosen])\ntitre = re.sub(\"-\",\"$-$\",titre)\nplt.show()" ]	[ null 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http://ctan.mirror.garr.it/mirrors/CRAN/web/packages/ggalluvial/vignettes/shiny.html	[ "# Tooltips for ggalluvial plots in Shiny apps\n\nknitr::opts_chunkset(fig.width = 6, fig.height = 3, fig.align = \"center\") library(ggalluvial) ## Problem In an interactive visualization, it is visually cleaner and better for interpretation if labels and other information appear as “tooltips” when the user hovers over or clicks on elements of the plot, rather than displaying all the labels on the plot at one time. However, the ggalluvial package does not natively include this functionality. It is possible to enable this using functions from several other packages. This vignette illustrates a Shiny app that displays an alluvial plot with tooltips that appear when the user hovers over two different plot elements: strata created with geom_stratum() and alluvia created with geom_alluvium(). The tooltips that appear when the user hovers over elements of the plot show a text label and the number of flows included in each group. This is made relatively straightforward because if the user hovers or clicks somewhere inside a ggplot panel, Shiny automatically returns information about the location of the mouse cursor in plot coordinates. That means the main work we have to do is to extract or manually recalculate the coordinates of the different plot elements. With that information, we can determine which plot element the cursor is hovering over and display the appropriate information in the tooltip or other output method. Note: The app demonstrated here depends on the packages htmltools and sp, in addition of course to ggalluvial and shiny. Please be aware that all of these packages will need to be installed on the server where your Shiny app is running. ### Hovering over and clicking on strata Enabling hovering over and clicking on strata is straightforward because of their rectangular shape. We only need the minimum and maximum x and y coordinates for each of the rectangles. The rectangles are evenly spaced along the x-axis, centered on positive integers beginning with 1. The width is set in geom_stratum() so, for example, we know that the x-coordinates of the first stratum are 1 ± width/2. The y-coordinates can be determined from the number of rows in the input data multiplied by their weights. ### Hovering over and clicking on alluvia Hovering over and clicking on alluvia are more difficult because the shapes of the alluvia are more complex. The default shape of the polygons includes an xspline curve drawn using the grid package. We need to manually reconstruct the coordinates of the polygons, then use sp::pointInPolygon() to detect which, if any, polygons the cursor is over. ## Data for reproducible example This toy dataset is used for the example app. example_data <- data.frame( weight = rep(1, 12), ID = 1:12, cluster = rep(c(1, 2), c(4, 8)), grp1 = rep(c('1a', '1b', '1a', '1b'), c(3, 2, 3, 4)), grp2 = rep(c('2a', '2b', '2a', '2b', '2a'), c(2, 2, 2, 2, 4)), grp3 = rep(c('3a','3b', '3a', '3b'), c(3, 2, 2, 5)) ) Here is a static plot generated using the toy dataset. ggplot(example_data, aes(y = weight, axis1 = grp1, axis2 = grp2, axis3 = grp3)) + geom_alluvium(aes(fill = factor(cluster)), knot.pos = 0.25) + geom_stratum(width = 1/8, reverse = TRUE) + geom_text(aes(label = after_stat(stratum)), stat = \"stratum\", reverse = TRUE, size = rel(3)) + theme_bw() + scale_x_continuous(expand = c(0, 0)) + scale_y_continuous(expand = c(0, 0))", null, "## Structure of the example app Here, we will go over each section of the code in detail. The full code is reproduced at the bottom of this document. ### User interface The app includes a minimal user interface with two output elements. ui <- fluidPage( fluidRow(tagsdiv(\nstyle = \"position: relative;\",\nplotOutput(\"alluvial_plot\", height = \"500px\",\nhover = hoverOpts(id = \"plot_hover\")\n),\nhtmlOutput(\"tooltip\")))\n)\n\nThe elements are:\n\n• a plotOutput with the argument hover defined, to enable behavior determined by the cursor’s plot coordinates whenever the user hovers over the plot.\n• an htmlOutput for the tooltip that appears next to the cursor on hover.\n\nBoth of the elements are wrapped in a fluidRow() and a div() tag.\n\nNote: This vignette only illustrates how to display output when the user hovers over an element. If you want to display output when the user clicks on an element, the corresponding argument to plotOutput() is click = clickOpts(id = \"plot_click\"). This will return the location of the mouse cursor in plot coordinates when the user clicks somewhere within the plot panel.\n\n### Server function\n\nThe server function is more complex. Its general structure looks like this, in pseudocode:\n\nserver <- function(input, output, session) {\n\noutput$alluvial_plot <- renderPlot({ '<Create \"ggplot\" object for alluvial plot.>' '<Build alluvial plot and assign globally.>' '<Extract data from built plot object used to create alluvium polygons.>' '<Use polygon splines to generate coordinates of alluvium boundaries.>' '<Convert coordinates from grid units to plot units and assign globally.>' '<Render the plot.>' }) output$tooltip <- renderText({\nif ('<mouse cursor is within the plot panel>') {\nif ('<mouse cursor is within a stratum box>') {\n'<Render stratum tooltip.>'\n} else {\nif ('<mouse cursor is within an alluvium polygon>') {\n'<Render alluvium tooltip.>'\n}\n}\n}\n})\n\n}\n\nFirst, we create the ggplot object for the alluvial plot, then we call the ggplot_build() function to build the plot without displaying it. The next lines of code are to “reverse engineer” the polygon coordinates. Finally, we call renderPlot() to pass the plot to output.\n\nNext, we define the tooltip with a renderText() expression. Within that expression, we first extract the cursor’s plot coordinates from the user input. We determine whether the cursor is hovering over a stratum and if so, display the appropriate tooltip.", null, "screenshot of tooltip on stratum\n\nIf the mouse cursor is not hovering over a stratum, we determine whether it is hovering over an alluvium polygon and if so, display different information in the tooltip.", null, "screenshot of tooltip on alluvium\n\nIf the mouse cursor is hovering over an empty region of the plot, nothing is returned by renderText() and so no tooltip text box is displayed.", null, "screenshot of cursor over empty region\n\nLet’s take a deeper dive into each part of the server function.\n\n#### 1. Drawing plot and extracting coordinates\n\nThe first part of the server function includes code to draw the plot and build it with ggplot_build(). Note that the global assignment operator <<- is used to assign node_width and pbuilt so they are both accessible outside the renderPlot() expression.\n\nNote: In the example presented here, strictly speaking all of the plot drawing and coordinate extracting code could be outside the server() function, because the plot itself does not change with user input. However if you are building an app where the plot changes in response to user input, for example a menu of options of which variables to display, the plot drawing code has to be inside the renderPlot() expression. So we’ve left it there in the example code.\n\noutput$alluvial_plot <- renderPlot({ # Width of node boxes node_width <<- 1/4 p <- ggplot(example_data, aes(y = weight, axis1 = grp1, axis2 = grp2, axis3 = grp3)) + geom_alluvium(aes(fill = factor(cluster)), knot.pos = 0.25) + geom_stratum(width = node_width, reverse = TRUE) + geom_text(aes(label = after_stat(stratum)), stat = \"stratum\", reverse = TRUE, size = rel(3)) + theme_bw() + scale_x_continuous(expand = c(0, 0)) + scale_y_continuous(expand = c(0, 0)) # Build the plot. Use global assignment so that this object is accessible # later. pbuilt <<- ggplot_build(p) Now for the hard part: reverse-engineering the coordinates of the alluvia polygons. This makes use of pbuilt$data[], a data frame with the individual elements of the alluvial plot. We add an additional column for width, which has a value of 1/3 hard-coded into ggalluvial::geom_alluvium(), then split the data frame by group (groups correspond to the individual alluvium polygons). We apply the unexported function ggalluvial:::data_to_xspline() to each element of the list to get the x-spline coordinates. Then, we pass the x-spline coordinates to the function grid::xsplineGrob() to convert them into a grid object. We pass the resulting object to grid::xsplinePoints(). At this point we now have the coordinates of the alluvium polygons.\n\n # Use built plot data to recalculate the locations of the flow polygons:\n\n# Add width parameter, and then convert built plot data to xsplines\ndata_draw <- transform(pbuilt$data[], width = 1/3) groups_to_draw <- split(data_draw, data_draw$group)\ngroup_xsplines <- lapply(groups_to_draw,\nggalluvial:::data_to_xspline,\nknot.prop = TRUE)\n\n# Convert xspline coordinates to grid object.\nxspline_coords <- lapply(\ngroup_xsplines,\nfunction(coords) grid::xsplineGrob(x=coords$x, y=coords$y,\nshape=coords$shape, open=FALSE) ) # Use grid::xsplinePoints to draw the curve for each polygon xspline_points <- lapply(xspline_coords, grid::xsplinePoints) The coordinates we have are in grid plotting units but we need to convert them into the same units as the axes on the plot. We do this by determining the range of the x and y axes in grid units (xrange_old and yrange_old), then fixing the range of the x axis as 1 to the number of strata, adjusted by the width of the nodes, and the y axis to the number of rows in the data (again, this is possible here because each flow polygon is exactly 1 unit high). We define a function new_range_transform() inline and apply it to each set of coordinates, assigning the resulting object globally so it can be accessed later. Now we have the coordinates of the polygons in plot units! So we can close the expression after returning the plot. # Define the x and y axis limits in grid coordinates (old) and plot # coordinates (new) xrange_old <- range(unlist(lapply( xspline_points, function(pts) as.numeric(pts$x)\n)))\nyrange_old <- range(unlist(lapply(\nxspline_points, function(pts) as.numeric(pts$y) ))) xrange_new <- c(1 - 1/6, 3 + 1/6) yrange_new <- c(0, nrow(example_data)) # Define function to convert grid graphics coordinates to data coordinates new_range_transform <- function(x_old, range_old, range_new) { (x_old - range_old)/(range_old - range_old) * (range_new - range_new) + range_new } # Using the x and y limits, convert the grid coordinates into plot # coordinates. Use global assignment. polygon_coords <<- lapply(xspline_points, function(pts) { x_trans <- new_range_transform(x_old = as.numeric(pts$x),\nrange_old = xrange_old,\nrange_new = xrange_new)\ny_trans <- new_range_transform(x_old = as.numeric(pts$y), range_old = yrange_old, range_new = yrange_new) list(x = x_trans, y = y_trans) }) # Return plot p }, res = 200) #### 2. Logic for determining cursor location and displaying tooltips First, we check whether the cursor is inside the plot panel. If it is not, the element plot_hover of the input will be NULL. output$tooltip <- renderText(\nif(!is.null(input$plot_hover)) { ... } ... ) Next, we check whether the cursor is over a stratum. We round the x-coordinate of the mouse cursor in data units to the nearest integer, then determine whether the x-coordinate is within node_width/2 of that integer. If so, the mouse cursor is horizontally within the box. hover <- input$plot_hover\nx_coord <- round(hover$x) if(abs(hover$x - x_coord) < (node_width / 2)) { ... }\n\nThe nearest integer to the y-coordinate corresponds to the row of the data frame because we set reverse = TRUE and all weight = 1 in the input data. So, for example, the first row of the data frame corresponds to y range c(0, 1), the second c(1, 2), and so forth. This gives us all the information we need to find the index of the rows of the input data that goes with the stratum the cursor is on. Note: It is necessary for the input data to be sorted in ascending order of the group column, named cluster in this example. If it is not sorted in this way, the relative order of the flows along the y-axis will not correspond to their order in the data.\n\nnode_row <-\npbuilt$data[]$x == x_coord & hover$y > pbuilt$data[]$ymin & hover$y < pbuilt$data[]$ymax\n\nWe get the name of the stratum as well as the total number of flows passing through it.\n\nnode_label <- pbuilt$data[]$stratum[node_row]\nnode_n <- pbuilt$data[]$n[node_row]\n\nFinally, we render a tooltip using the div tag and passing it to htmltools::renderTags(). Note that the tooltip positioning is provided in CSS coordinates (pixels), not data coordinates. This does not require any additional effort on our part because plot_hover also includes the mouse cursor location in those units.\n\nrenderTags(\ntags$div( node_label, tags$br(),\n\"n =\", node_n,\nstyle = paste0(\n\"position: absolute; \",\n\"top: \", hover$coords_css$y + offset, \"px; \",\n\"left: \", hover$coords_css$x + offset, \"px; \",\n\"background: gray; \",\n\"color: white; \"\n)\n)\n)$html If the cursor is not over a stratum, the next logic checks whether it is over an alluvium. This is done using the function sp::point.in.polygon applied across each of the polygons for which we defined the coordinates inside the renderPlot expression. hover_within_flow <- sapply( polygon_coords, function(pol) point.in.polygon(point.x = hover$x,\npoint.y = hover$y, pol.x = pol$x,\npol.y = pol$y) ) If at least one polygon is beneath the mouse cursor, we locate the corresponding row in the input data and extract information to display in the tooltip. In the situation where there are more than one polygon overlapping, we get the information for the polygon that is plotted last by calling rev() on the logical vector returned by point.in.polygon(). This means that the tooltip will display information from the alluvium that appears “on top” in the plot. In this example, we will display the names of all the nodes that the alluvium passes through. coord_id <- rev(which(hover_within_flow == 1)) flow_id <- example_data$ID[coord_id]\naxis_values <- example_data[flow_id, c('grp1', 'grp2', 'grp3')]\n\nWe render a tooltip that shows the names of all the nodes that the hovered path passes through, using very similar syntax to the above tooltip.\n\nrenderTags(\ntags$div( paste(axis_values, collapse = ' -> '), style = paste0( \"position: absolute; \", \"top: \", hover$coords_css$y + offset, \"px; \", \"left: \", hover$coords_css$x + offset, \"px; \", \"background: gray; \", \"padding: 3px; \", \"color: white; \" ) ) )$html\n\n## Conclusion\n\nThis vignette demonstrates how to enable tooltips for ggalluvial plots in Shiny apps. However it’s important to note that some of the workarounds are slightly inelegant. This may not be the optimal way to do it — other solutions are certainly possible!\n\n## Appendix\n\n### Complete app code\n\nlibrary(ggalluvial)\nlibrary(shiny)\nlibrary(htmltools)\nlibrary(sp)\n\nexample_data <- data.frame(\nweight = rep(1, 12),\nID = 1:12,\ncluster = rep(c(1, 2), c(4, 8)),\ngrp1 = rep(c('1a', '1b', '1a', '1b'), c(3, 2, 3, 4)),\ngrp2 = rep(c('2a', '2b', '2a', '2b', '2a'), c(2, 2, 2, 2, 4)),\ngrp3 = rep(c('3a','3b', '3a', '3b'), c(3, 2, 2, 5))\n)\n\n# User interface\nui <- fluidPage(\nfluidRow(tags$div( style = \"position: relative;\", plotOutput(\"alluvial_plot\", height = \"500px\", hover = hoverOpts(id = \"plot_hover\") ), htmlOutput(\"tooltip\"))) ) server <- function(input, output, session) { # Draw plot and extract coordinates output$alluvial_plot <- renderPlot({\n\n# Width of node boxes\nnode_width <<- 1/4\n\np <- ggplot(example_data,\naes(y = weight, axis1 = grp1, axis2 = grp2, axis3 = grp3)) +\ngeom_alluvium(aes(fill = factor(cluster)), knot.pos = 0.25) +\ngeom_stratum(width = node_width, reverse = TRUE) +\ngeom_text(aes(label = after_stat(stratum)),\nstat = \"stratum\",\nreverse = TRUE,\nsize = rel(3)) +\ntheme_bw() +\nscale_x_continuous(expand = c(0, 0)) +\nscale_y_continuous(expand = c(0, 0))\n\n# Build the plot. Use global assignment so that this object is accessible\n# later.\npbuilt <<- ggplot_build(p)\n\n# Use built plot data to recalculate the locations of the flow polygons:\n\n# Add width parameter, and then convert built plot data to xsplines\ndata_draw <- transform(pbuilt$data[], width = 1/3) groups_to_draw <- split(data_draw, data_draw$group)\ngroup_xsplines <- lapply(groups_to_draw,\nggalluvial:::data_to_xspline,\nknot.prop = TRUE)\n\n# Convert xspline coordinates to grid object.\nxspline_coords <- lapply(\ngroup_xsplines,\nfunction(coords) grid::xsplineGrob(x = coords$x, y = coords$y,\nshape = coords$shape, open = FALSE) ) # Use grid::xsplinePoints to draw the curve for each polygon xspline_points <- lapply(xspline_coords, grid::xsplinePoints) # Define the x and y axis limits in grid coordinates (old) and plot # coordinates (new) xrange_old <- range(unlist(lapply( xspline_points, function(pts) as.numeric(pts$x)\n)))\nyrange_old <- range(unlist(lapply(\nxspline_points,\nfunction(pts) as.numeric(pts$y) ))) xrange_new <- c(1 - 1/6, 3 + 1/6) yrange_new <- c(0, nrow(example_data)) # Define function to convert grid graphics coordinates to data coordinates new_range_transform <- function(x_old, range_old, range_new) { (x_old - range_old)/(range_old - range_old) * (range_new - range_new) + range_new } # Using the x and y limits, convert the grid coordinates into plot # coordinates. Use global assignment. polygon_coords <<- lapply(xspline_points, function(pts) { x_trans <- new_range_transform(x_old = as.numeric(pts$x),\nrange_old = xrange_old,\nrange_new = xrange_new)\ny_trans <- new_range_transform(x_old = as.numeric(pts$y), range_old = yrange_old, range_new = yrange_new) list(x = x_trans, y = y_trans) }) # Return plot p }, res = 200) output$tooltip <- renderText(\nif(!is.null(input$plot_hover)) { hover <- input$plot_hover\nx_coord <- round(hover$x) if(abs(hover$x - x_coord) < (node_width / 2)) {\n# Display node information if cursor is over a stratum box.\n\n# Determine stratum name from x and y coord, and the n.\nnode_row <- pbuilt$data[]$x == x_coord &\nhover$y > pbuilt$data[]$ymin & hover$y < pbuilt$data[]$ymax\nnode_label <- pbuilt$data[]$stratum[node_row]\nnode_n <- pbuilt$data[]$n[node_row]\n\n# Offset, in pixels, for location of tooltip relative to mouse cursor,\n# in both x and y direction.\noffset <- 5\n\n# Render tooltip\nrenderTags(\ntags$div( node_label, tags$br(),\n\"n =\", node_n,\nstyle = paste0(\n\"position: absolute; \",\n\"top: \", hover$coords_css$y + offset, \"px; \",\n\"left: \", hover$coords_css$x + offset, \"px; \",\n\"background: gray; \",\n\"color: white; \"\n)\n)\n)$html } else { # Display flow information if cursor is over a flow polygon: what # alluvia does it pass through? # Calculate whether coordinates of hovering cursor are inside one of the # polygons. hover_within_flow <- sapply( polygon_coords, function(pol) point.in.polygon(point.x = hover$x,\npoint.y = hover$y, pol.x = pol$x,\npol.y = pol$y) ) if (any(hover_within_flow)) { # Find the alluvium that is plotted on top. (last) coord_id <- rev(which(hover_within_flow == 1)) # Get the corresponding row ID from the data. flow_id <- example_data$ID[coord_id]\n# Get the axis 1-3 values for all axes for that row ID.\naxis_values <- example_data[flow_id, c('grp1', 'grp2', 'grp3')]\n\noffset <- 5\n\n# Render tooltip\nrenderTags(\ntags$div( paste(axis_values, collapse = ' -> '), style = paste0( \"position: absolute; \", \"top: \", hover$coords_css$y + offset, \"px; \", \"left: \", hover$coords_css$x + offset, \"px; \", \"background: gray; \", \"padding: 3px; \", \"color: white; \" ) ) )$html\n}\n}\n}\n)\n}\n\nshinyApp(ui = ui, server = server)" ]	[ null, 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", null, 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", null, 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cEWWCCPmNT2FotlA0KhQpkdwFGANzFv61W0WV5LDfIxY7jyat2l4l2rlUdCjbGDjBBwD/UUuXl0KVX2q9p31LFFFFAwooooAKKKKAA9DTIv9Sn+6KBLG7OiuCy/eA7U1HVLdWZgqhRkmgCWuK1L/AJDmof8AXVP/AEUldkkscmdjhsdcGuO1L/kOah/11T/0Ula0fiOHMP4PzK3fvQyh1KkZB4xS96UHFdZ4RhzRGGYoe3T3FIh7Vo30QeMSAZZf5VSt7dp364UdTWjtKOpyR5qNVOJZgu9oCOCR2Iq6fmjLANjHoalt4oo02ooDdz3NTSf6tvpXnTxEovlsfW4bKaVaCqqf3HTW99D9mi+S4+6P+XeT0/3ak+3Q/wBy4/8AAeT/AOJqW3/49ov9wfyqSoOk4adt880gV9pkc8qQfvHt1qhPdFcqisCe5GK2Zf8Aj4m/66v/AOhGoXKYIYBh6GqjiJX5UjGrlNFRdaU7X11Oec8e5q/Yw7I9x+838qni0Zr2c+QOnLDtUskTRSGN1KkcYNdyaS5VufNThOU/aSWnQSgjpRQaBklt/wAfcH/XVf513FcPb/8AH3B/11X+ddJq0l5GYvsnmls8hVBBHfPHbr1Hfr0rlr/Ee1lv8J+pozf6iT/dP8qS3/494v8AcH8qpWM11Nprm7XEgUjJTbn5Rzjtzmrtv/x7xf7g/lWJ6BBqVvJc2MkUYBZsYyfesD+wr3+6n/fVdVRVxqOKsjmrYWnWlzSKGl2klrZeVLgNk9DVi0tIrKAQw79g6bnLH8zU9FS3d3N4QUIqK6BRRRSKCiiigAooooApJY/6RcSyMGEqhdoGOASf61IbdZdPEBA2lAvPParJ6GmRf6lP90UAULPTDa3TzeZkHPAGM5x1/Kuc1IZ1zUP+uqf+ikrta4vUv+Q5qH/XVP8A0Ula0fiOHMP4PzK2BSgL3FHUmp7S0ku5hHGPqT0ArqbtueJGLk7IWysGvpxGBhP4j6Ci909dOuDEi/J1U+tdVa2kVnEkcfrye5NQ6rYi8tvl/wBYnK+/tWCre95HqSwFqP8Ae/rQ5LFK7NsPJoIKsQRyOOaR/umt5JNannU5SjLR2O4t/wDj2i/3B/KpazLwSnT4FhE247fmiP3eOpx1+lJpDX37xLsOVUDazqASctnoemNv5n8PPPpjnLkn7VOM8ea//oRqOOJpZAiKWYnAAqS4x9snyMjzX/8AQjVu11RbTPlWkYPdiSSa7Y6RXKj5+o1Oq1Ulomb2m2C2dsFI/eNyx96L3ToLwYYYfHDDrWV/wkU3/PBPzNJ/wkM27/UJ09TWPs6l7nofWsLycnT0KF5p81m5DqSnZx0NVcVsvr8kilXto2U9QSazppoZTlbcRn/ZY/1reLl9pHmVo0r3pyGWwH2u3/66r/Ou4rh7f/j7g/66r/Ou4rCv8R6eW/wn6kc3+ok/3T/Ki3/494v9wfypZv8AUSf7p/lSW/8Ax7xf7g/lWJ6BJRRRQAUUUUAFFFFABRRRQAUUUUAB6GmRf6lP90VHHdQzSSRo+WT7wwfp+PSpIv8AUp/uigB9cbfxGTXNQxgfvV5J/wCmSV2Wa5K9/wCQxqH/AF1X/wBFpR7R09UNYWGJfs57DIbKJm/fXCqO4Aya3LWfT7SLZG4x3JHJrB70tZyxE5bnTSyqhS1judGdSs8j94OvpS/2lZ/89B+Vc0eopaj2rOj6nDuaN/Dp90xkSbZIevHBrHmtyithlYeoNTGkk/1bfStI4ma0OSrlGHm+fW519v8A8e0X+4P5VJUdv/x7Rf7g/lUmaswOJuIy11Ocf8tX/wDQjUfktzxVuX/j4m/66v8A+hGmDqaX1qcdCv7FoVPfbd3r/WhB5LelJ5LZ6VZpO9H1ufYP7Cw3d/h/kQeS3pSeS3pVmkNH1ufYP7Cw/d/h/kRwRst1ASP+Wq/zrtq5CP8A4+If+ui/zFdfmn7R1NWJ4SGF9yD0eoyb/USf7p/lSW//AB7xf7g/lRMf3En+6f5UW/8Ax7Rf7g/lQIkooooAKKKKACiiigAooooAKKKKAKcWnRRyzOzGRZM/I4BAySx/U025t7G2h8w2MTDIGFiFXqa6LIhRxlT1FAGYsFhFdfZlsYyC3L7FwGIzj1q5/Z1l/wA+kH/fsU8WsImE2z94O+TU1AFX+zrL/n0g/wC/Y/wo/s6y/wCfSD/v2P8ACrVFAGNJHp5jnm/s5S1rIV2iJcsdo6D/AIFVuC0sZ4ElWzgAYZx5Y4/Sp2tYXD5T75y2CRk4A/kBUscaxRhEGFHQUAV/7Osv+fSD/v2P8KP7Osv+fSD/AL9irVFAGdcw2du0a/YImDsFysa8ZqK3h09bgWSWMYCKQrlFIO3aD7/xCtCW1hmlSSRcun3TkjFCWsMcxmRMSEYLZPt/gPyoAZ/Z1l/z6Qf9+xSf2dZf8+kH/fsf4VaooAqtp9kFJ+xwHHby1rNb7AkJuzpi7gfLKiNT754rbYBlKnoRg1WGn2wgEIjPljou4/40AIthYsoYWkGCM/6sf4Uv9nWX/PpB/wB+x/hVkAKAAMAUtAFb+zrL/n0g/wC/Yqpc21gZls5LGLbOrKHCKOxJHr0FalQvawyTCV0y4GAcnjqP6mgCnaJY3TSKLCJdvTMa8jJAP6GtEAKAAMAdAKjhtoYGcxJtLnLc9aloAKKKKACiiigAooooAKKKKAP/2QD/4Q9daHR0cDovL25zLmFkb2JlLmNvbS94YXAvMS4wLwA8P3hwYWNrZXQgYmVnaW49Iu+7vyIgaWQ9Ilc1TTBNcENlaGlIenJlU3pOVGN6a2M5ZCI/PiA8eDp4bXBtZXRhIHhtbG5zOng9ImFkb2JlOm5zOm1ldGEvIiB4OnhtcHRrPSJYTVAgQ29yZSA0LjQuMC1FeGl2MiI+IDxyZGY6UkRGIHhtbG5zOnJkZj0iaHR0cDovL3d3dy53My5vcmcvMTk5OS8wMi8yMi1yZGYtc3ludGF4LW5zIyI+IDxyZGY6RGVzY3JpcHRpb24gcmRmOmFib3V0PSIiIHhtbG5zOmlwdGNFeHQ9Imh0dHA6Ly9pcHRjLm9yZy9zdGQvSXB0YzR4bXBFeHQvMjAwOC0wMi0yOS8iIHhtbG5zOnhtcE1NPSJodHRwOi8vbnMuYWRvYmUuY29tL3hhcC8xLjAvbW0vIiB4bWxuczpzdEV2dD0iaHR0cDovL25zLmFkb2JlLmNvbS94YXAvMS4wL3NUeXBlL1Jlc291cmNlRXZlbnQjIiB4bWxuczpwbHVzPSJodHRwOi8vbnMudXNlcGx1cy5vcmcvbGRmL3htcC8xLjAvIiB4bWxuczpHSU1QPSJodHRwOi8vd3d3LmdpbXAub3JnL3htcC8iIHhtbG5zOmRjPSJodHRwOi8vcHVybC5vcmcvZGMvZWxlbWVudHMvMS4xLyIgeG1sbnM6eG1wPSJodHRwOi8vbnMuYWRvYmUuY29tL3hhcC8xLjAvIiB4bXBNTTpEb2N1bWVudElEPSJnaW1wOmRvY2lkOmdpbXA6Zjc1ZjI0NzUtODdhMC00ZTBkLWE5YWUtZjhhYmJhZDE0NWQ1IiB4bXBNTTpJbnN0YW5jZUlEPSJ4bXAuaWlkOmJkZjNhY2U3LWJjZGYtNDc0Zi1hZmJiLWU0MDI1OTE3OTQ3OSIgeG1wTU06T3JpZ2luYWxEb2N1bWVudElEPSJ4bXAuZGlkOjY0MzAwZDA1LWU3ZjgtNGMyNS05ZjMzLTY0N2ExYmMxYmMxNSIgR0lNUDpBUEk9IjIuMCIgR0lNUDpQbGF0Zm9ybT0iV2luZG93cyIgR0lNUDpUaW1lU3RhbXA9IjE2MDYyMjc0NjUwNzEyOTkiIEdJTVA6VmVyc2lvbj0iMi4xMC4yMiIgZGM6Rm9ybWF0PSJpbWFnZS9qcGVnIiB4bXA6Q3JlYXRvclRvb2w9IkdJTVAgMi4xMCI+IDxpcHRjRXh0OkxvY2F0aW9uQ3JlYXRlZD4gPHJkZjpCYWcvPiA8L2lwdGNFeHQ6TG9jYXRpb25DcmVhdGVkPiA8aXB0Y0V4dDpMb2NhdGlvblNob3duPiA8cmRmOkJhZy8+IDwvaXB0Y0V4dDpMb2NhdGlvblNob3duPiA8aXB0Y0V4dDpBcnR3b3JrT3JPYmplY3Q+IDxyZGY6QmFnLz4gPC9pcHRjRXh0OkFydHdvcmtPck9iamVjdD4gPGlwdGNFeHQ6UmVnaXN0cnlJZD4gPHJkZjpCYWcvPiA8L2lwdGNFeHQ6UmVnaXN0cnlJZD4gPHhtcE1NOkhpc3Rvcnk+IDxyZGY6U2VxPiA8cmRmOmxpIHN0RXZ0OmFjdGlvbj0ic2F2ZWQiIHN0RXZ0OmNoYW5nZWQ9Ii8iIHN0RXZ0Omluc3RhbmNlSUQ9InhtcC5paWQ6Y2Y0ZDQ0MzQtOGMyZC00YjNiLWJiMGItOGFlYzA4NGVmZWVhIiBzdEV2dDpzb2Z0d2FyZUFnZW50PSJHaW1wIDIuMTAgKFdpbmRvd3MpIiBzdEV2dDp3aGVuPSIyMDIwLTExLTI0VDA5OjA2OjE5Ii8+IDxyZGY6bGkgc3RFdnQ6YWN0aW9uPSJzYXZlZCIgc3RFdnQ6Y2hhbmdlZD0iLyIgc3RFdnQ6aW5zdGFuY2VJRD0ieG1wLmlpZDo1YjRjYTU4Yy1hYWE1LTQwNTUtYmU4My1lM2RhZTlkZDMyMTMiIHN0RXZ0OnNvZnR3YXJlQWdlbnQ9IkdpbXAgMi4xMCAoV2luZG93cykiIHN0RXZ0OndoZW49IjIwMjAtMTEtMjRUMDk6MTc6NDUiLz4gPC9yZGY6U2VxPiA8L3htcE1NOkhpc3Rvcnk+IDxwbHVzOkltYWdlU3VwcGxpZXI+IDxyZGY6U2VxLz4gPC9wbHVzOkltYWdlU3VwcGxpZXI+IDxwbHVzOkltYWdlQ3JlYXRvcj4gPHJkZjpTZXEvPiA8L3BsdXM6SW1hZ2VDcmVh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", null, 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http://www.gnu.org/software/gsl/doc/html/qrng.html	[ "# Quasi-Random Sequences¶\n\nThis chapter describes functions for generating quasi-random sequences in arbitrary dimensions. A quasi-random sequence progressively covers a", null, "-dimensional space with a set of points that are uniformly distributed. Quasi-random sequences are also known as low-discrepancy sequences. The quasi-random sequence generators use an interface that is similar to the interface for random number generators, except that seeding is not required—each generator produces a single sequence.\n\nThe functions described in this section are declared in the header file `gsl_qrng.h`.\n\n## Quasi-random number generator initialization¶\n\ntype gsl_qrng\n\nThis is a workspace for computing quasi-random sequences.\n\ngsl_qrng gsl_qrng_alloc(const gsl_qrng_type T, unsigned int d)\n\nThis function returns a pointer to a newly-created instance of a quasi-random sequence generator of type `T` and dimension `d`. If there is insufficient memory to create the generator then the function returns a null pointer and the error handler is invoked with an error code of `GSL_ENOMEM`.\n\nvoid gsl_qrng_free(gsl_qrng q)\n\nThis function frees all the memory associated with the generator `q`.\n\nvoid gsl_qrng_init(gsl_qrng q)\n\nThis function reinitializes the generator `q` to its starting point. Note that quasi-random sequences do not use a seed and always produce the same set of values.\n\n## Sampling from a quasi-random number generator¶\n\nint gsl_qrng_get(const gsl_qrng q, double x[])\n\nThis function stores the next point from the sequence generator `q` in the array `x`. The space available for `x` must match the dimension of the generator. The point `x` will lie in the range", null, "for each", null, ". An inline version of this function is used when `HAVE_INLINE` is defined.\n\n## Auxiliary quasi-random number generator functions¶\n\nconst char gsl_qrng_name(const gsl_qrng q)\n\nThis function returns a pointer to the name of the generator.\n\nsize_t gsl_qrng_size(const gsl_qrng q)\nvoid gsl_qrng_state(const gsl_qrng q)\n\nThese functions return a pointer to the state of generator `r` and its size. You can use this information to access the state directly. For example, the following code will write the state of a generator to a stream:\n\n```void * state = gsl_qrng_state (q);\nsize_t n = gsl_qrng_size (q);\nfwrite (state, n, 1, stream);\n```\n\n## Saving and restoring quasi-random number generator state¶\n\nint gsl_qrng_memcpy(gsl_qrng dest, const gsl_qrng src)\n\nThis function copies the quasi-random sequence generator `src` into the pre-existing generator `dest`, making `dest` into an exact copy of `src`. The two generators must be of the same type.\n\ngsl_qrng gsl_qrng_clone(const gsl_qrng q)\n\nThis function returns a pointer to a newly created generator which is an exact copy of the generator `q`.\n\n## Quasi-random number generator algorithms¶\n\nThe following quasi-random sequence algorithms are available,\n\ntype gsl_qrng_type\ngsl_qrng_type gsl_qrng_niederreiter_2\n\nThis generator uses the algorithm described in Bratley, Fox, Niederreiter, ACM Trans. Model. Comp. Sim. 2, 195 (1992). It is valid up to 12 dimensions.\n\ngsl_qrng_type gsl_qrng_sobol\n\nThis generator uses the Sobol sequence described in Antonov, Saleev, USSR Comput. Maths. Math. Phys. 19, 252 (1980). It is valid up to 40 dimensions.\n\ngsl_qrng_type gsl_qrng_halton\ngsl_qrng_type gsl_qrng_reversehalton\n\nThese generators use the Halton and reverse Halton sequences described in J.H. Halton, Numerische Mathematik, 2, 84-90 (1960) and B. Vandewoestyne and R. Cools Computational and Applied Mathematics, 189, 1&2, 341-361 (2006). They are valid up to 1229 dimensions.\n\n## Examples¶\n\nThe following program prints the first 1024 points of the 2-dimensional Sobol sequence.\n\n```#include <stdio.h>\n#include <gsl/gsl_qrng.h>\n\nint\nmain (void)\n{\nint i;\ngsl_qrng * q = gsl_qrng_alloc (gsl_qrng_sobol, 2);\n\nfor (i = 0; i < 1024; i++)\n{\ndouble v;\ngsl_qrng_get (q, v);\nprintf (\"%.5f %.5f\\n\", v, v);\n}\n\ngsl_qrng_free (q);\nreturn 0;\n}\n```\n\nHere is the output from the program:\n\n```\\$ ./a.out\n0.50000 0.50000\n0.75000 0.25000\n0.25000 0.75000\n0.37500 0.37500\n0.87500 0.87500\n0.62500 0.12500\n0.12500 0.62500\n....\n```\n\nIt can be seen that successive points progressively fill-in the spaces between previous points.\n\nFig. 3 shows the distribution in the x-y plane of the first 1024 points from the Sobol sequence,", null, "Fig. 3 Distribution of the first 1024 points from the quasi-random Sobol sequence" ]	[ null, "http://www.gnu.org/software/gsl/doc/html/_images/math/92f093ef8e67b2293db7509b06c7817f327f445c.png", null, "http://www.gnu.org/software/gsl/doc/html/_images/math/aa6af4b128d0ed8c51438b9e849475cd63437fd0.png", null, "http://www.gnu.org/software/gsl/doc/html/_images/math/3efd6fd1c2b0c335484bcd4bf7d5d98b99e619f4.png", null, "http://www.gnu.org/software/gsl/doc/html/_images/qrng.png", null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.66011536,"math_prob":0.9665993,"size":4320,"snap":"2022-05-2022-21","text_gpt3_token_len":1174,"char_repetition_ratio":0.18443003,"word_repetition_ratio":0.009615385,"special_character_ratio":0.26180556,"punctuation_ratio":0.16030534,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9807254,"pos_list":[0,1,2,3,4,5,6,7,8],"im_url_duplicate_count":[null,3,null,1,null,10,null,2,null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2022-05-28T13:50:36Z\",\"WARC-Record-ID\":\"<urn:uuid:297aa6b5-43bb-4915-9529-c4694a1568c2>\",\"Content-Length\":\"34416\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:68b4f248-4449-4f77-8bf9-8c501b1f774e>\",\"WARC-Concurrent-To\":\"<urn:uuid:3c49ff53-7e18-4427-aa2e-b3ff2d5690bd>\",\"WARC-IP-Address\":\"209.51.188.116\",\"WARC-Target-URI\":\"http://www.gnu.org/software/gsl/doc/html/qrng.html\",\"WARC-Payload-Digest\":\"sha1:2DNZOZQIQFXMHWNPCOGNU4GTHK6K3URO\",\"WARC-Block-Digest\":\"sha1:XE7OJMEALVMOWSDD3WUGGH4UARHUO4J4\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2022/CC-MAIN-2022-21/CC-MAIN-2022-21_segments_1652663016853.88_warc_CC-MAIN-20220528123744-20220528153744-00246.warc.gz\"}"}
https://br.tradingview.com/pine-script-docs/en/v4/appendix/HOWTOs.html	[ "HOWTOs¶\n\nGet real OHLC price on a Heikin Ashi chart¶\n\nSuppose, we have a Heikin Ashi chart (or Renko, Kagi, PriceBreak etc) and we’ve added a pine script on it:\n\n//@version=4\nstudy(\"Visible OHLC\", overlay=true)\nc = close\nplot(c)\n\nYou may see that variable c is a Heikin Ashi close price which is not the same as real OHLC price. Because close built-in variable is always a value that corresponds to a visible bar (or candle) on the chart.\n\nSo, how do we get the real OHLC prices in Pine Script code, if current chart type is non-standard? We should use security function in combination with tickerid function. Here is an example:\n\n//@version=4\nstudy(\"Real OHLC\", overlay=true)\nt = tickerid(syminfo.prefix, syminfo.ticker)\nrealC = security(t, timeframe.period, close)\nplot(realC)\n\nIn a similar way we may get other OHLC prices: open, high and low.\n\nGet non-standard OHLC values on a standard chart¶\n\nBacktesting on non-standard chart types (e.g. Heikin Ashi or Renko) is not recommended because the bars on these kinds of charts do not represent real price movement that you would encounter while trading. If you want your strategy to enter and exit on real prices but still use Heikin Ashi-based signals, you can use the same method to get Heikin Ashi values on a regular candlestick chart:\n\n//@version=4\nstrategy(\"BarUpDn Strategy\", overlay=true, default_qty_type = strategy.percent_of_equity, default_qty_value = 10)\nmaxIdLossPcnt = input(1, \"Max Intraday Loss(%)\", type=input.float)\nneedTrade() => close > open and open > close ? 1 : close < open and open < close ? -1 : 0\nstrategy.entry(\"BarUp\", strategy.long)\nstrategy.entry(\"BarDn\", strategy.short)\n\nPlot arrows on the chart¶\n\nYou may use plotshape with style shape.arrowup and shape.arrowdown:\n\n//@version=4\nstudy('Ex 1', overlay=true)\ndata = close >= open\nplotshape(not data, color=color.red, style=shape.arrowdown, text=\"Sell\")", null, "You may use plotchar function with any unicode character:\n\n//@version=4\ndata = close >= open\nplotchar(data, char='↑', location=location.belowbar, color=color.red, text=\"Sell\")", null, "Plot a dynamic horizontal line¶\n\nThere is function hline in pine. But it is now limited to only plot constant value. Here is a Pine Script with workaround to plot changing hline:\n\n//@version=4\nstudy(\"Horizontal line\", overlay=true)\nplot(close, trackprice=true, offset=-9999)\n// trackprice=true plots horizontal line on close\n// offset=-9999 hides the plot\nplot(close, color=#FFFFFFFF) // forces to show study\n\nPlot a vertical line on condition¶\n\n//@version=4\nstudy(\"Vertical line\", overlay=true, scale=scale.none)\n// scale.none means do not resize the chart to fit this plot\n// if the bar being evaluated is the last baron the chart (the most recent bar), then cond is true\ncond = barstate.islast\n// when cond is true, plot a histogram with a line with height value of 100,000,000,000,000,000,000.00\n// (10 to the power of 20)\n// when cond is false, plot no numeric value (nothing is plotted)\n// use the style of histogram, a vertical bar\nplot(cond ? 10e20 : na, style=plot.style_histogram)\n\nAccess the previous value¶\n\n//@version=4\n//...\ns = 0.0\ns := nz(s) // Accessing previous values\nif (condition)\ns := s + 1\n\nGet a 5-days high¶\n\nLookback 5 days from the current bar, find the highest bar, plot a star character at that price level above the current bar", null, "//@version=4\nstudy(\"Range Analysis\", overlay=true)\n\n// find which bar is 5 days away from the current time\nmilliseconds_in_5days = 1000 * 60 * 60 * 24 * 5 // millisecs * secs * min * hours * days\n// plot(milliseconds_in_5days, title=\"ms in 5d\", style=circles) //debug\n// subtract timestamp of the bar being examined from the current time\n// if value is less than 5 days ago, set variable \"leftborder\" as true\n// this is set true at the bar being examined as the left border of the 5 days lookback window range\nleftborder = timenow - time < milliseconds_in_5days // true or na when false\n// plot(leftborder ? 1 : na, title=\"bar within leftborder\") //debug\n// plot(time, title=\"bartime\") //debug\n// plot(timenow - time, title=\"timenow minus bartime\") //debug\n\n// treat the last bar (most recent bar) as the right edge of the lookback window range\nrightborder = barstate.islast\n\n// initialize variable \"max\" as na\nmax = float(na)\n\n// if bar being examined is not within the lookback window range (i.e., leftborder = false)\n// change the variable \"max\" to be na\n// else, test if value of \"max\" stored in the previous bar is na\n// (bcuz first bar being examined in the lookback window will not have a previous value ),\n// if it is na, use the high of the current bar,\n// else, use the value of \"max\" stored in the previous bar\nmax := not leftborder ? na : na(max) ? high : max\n// plot(max ? max : na, title=\"max b4 compare\") // debug\n\n// compare high of current bar being examined with previous bar's high\n// if curr bar high is higher than the max bar high in the lookback window range\nif high > max // we have a new high\nmax := high // change variable \"max\" to use current bar's high value\nmax\n// else keep the previous value of max as the high bar within this lookback window range\n// plot(max ? max : na, title=\"max after compare\") //debug\n\n// if examining the last bar (newest bar, rightborder is true)\n// set variable \"val\" to the previous value of series variable \"max\"\n// else set to na so nothing is plotted\nval = rightborder ? max : na\n\n// if val is true (a number, not na)\n// plot character\n// since no character is specified, a \"star\" will be plotted\n// location.absolute uses the value of val as the y axis value\n// the x axis location will be the last bar (newest bar)\nplotchar(val, size=size.normal, location=location.absolute)\n\n// fill the background of the 5 days lookback window range with aqua color\nbgcolor(leftborder and not rightborder ? color.aqua : na, transp=70)\n\nCount bars in a dataset¶\n\nGet a count of all the bars in the loaded dataset. Might be useful for calculating flexible lookback periods based on number of bars.\n\n//@version=4\nstudy(\"Bar Count\", overlay=true, scale=scale.none)\nplot(bar_index + 1, style=plot.style_histogram)\n\nEnumerate bars in a day¶\n\n//@version=4\nstudy(\"My Script\", overlay=true, scale=scale.none)\n\nis_new_day() =>\nd = dayofweek\nna(d) or d != d\n\nFind the highest and lowest values for the entire dataset¶\n\n//@version=4\nstudy(\"My Script\")\n\nbiggest(series) =>\nmax = 0.0\nmax := nz(max, series)\nif series > max\nmax := series\nmax\n\nsmallest(series) =>\nmin = 0.0\nmin := nz(min, series)\nif series < min\nmin := series\nmin\n\nplot(biggest(close), color=color.green)\nplot(smallest(close), color=color.red)\n\nQuery the last non-na value¶\n\nYou can use the script below to avoid gaps in a series:\n\n//@version=4\nstudy(\"My Script\")\nseries = close >= open ? close : na\nvw = valuewhen(not na(series), series, 0)\nplot(series, style=plot.style_linebr, color=color.red) // series has na values\nplot(vw) // all na values are replaced with the last non-empty value\nOptions v: v4\nLanguages\nen\nVersions\nv3\nv4\nv5" ]	[ null, "https://br.tradingview.com/pine-script-docs/en/v4/_images/Buy_sell_chart1.png", null, "https://br.tradingview.com/pine-script-docs/en/v4/_images/Buy_sell_chart2.png", null, "https://br.tradingview.com/pine-script-docs/en/v4/_images/Wiki_howto_range_analysis.png", null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.6376871,"math_prob":0.93686885,"size":6926,"snap":"2022-05-2022-21","text_gpt3_token_len":1875,"char_repetition_ratio":0.11340653,"word_repetition_ratio":0.022201665,"special_character_ratio":0.2814034,"punctuation_ratio":0.14307116,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.99184513,"pos_list":[0,1,2,3,4,5,6],"im_url_duplicate_count":[null,1,null,1,null,1,null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2022-01-20T14:03:26Z\",\"WARC-Record-ID\":\"<urn:uuid:4bfdaf93-587c-4331-95e9-148df741039c>\",\"Content-Length\":\"47439\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:ad43c8b4-61ed-4e5c-812b-250ae3c53e05>\",\"WARC-Concurrent-To\":\"<urn:uuid:ca46d2f9-da41-4768-92cb-e4d92c6363e0>\",\"WARC-IP-Address\":\"13.249.42.110\",\"WARC-Target-URI\":\"https://br.tradingview.com/pine-script-docs/en/v4/appendix/HOWTOs.html\",\"WARC-Payload-Digest\":\"sha1:WL63FVHWPON5DKS222DYIP2RDLJEPWUF\",\"WARC-Block-Digest\":\"sha1:BWJIZIQ6X6T5TUCWB5HRXUUSERIO7HZQ\",\"WARC-Identified-Payload-Type\":\"application/xhtml+xml\",\"warc_filename\":\"/cc_download/warc_2022/CC-MAIN-2022-05/CC-MAIN-2022-05_segments_1642320301863.7_warc_CC-MAIN-20220120130236-20220120160236-00449.warc.gz\"}"}
https://hardystonecompany.co.uk/blog/how-to-calculate-paving-area/	[ "", null, "", null, "Posted in\n17/11/2019\n\n# How To Calculate Paving Area\n\n## Calculating Paving surface area\n\nIt's a long time since we did this at school, right?\nAnd if you don't use measurements in your everyday life it can be difficult to remember how to do this.\nTo calculate the square meters of paved area:\n• Break your area up into squares or rectangles and multiply the length by the width.\n• Add up the total number of square meters.\n• You will need to take into account the number of cuts and the laying pattern chosen. You will also need to allow for wastage.\n• The quantity of pavers or flags needed will depend on the area to be paved, and the paver or slab chosen. In Project packs (mixed sizes to create a pattern), there are generally enough pavers to cover 18.9 sq meters but check each product as some are different.\nWork out the area to be paved in square meters.Length(m) x width(m) = area(m²).", null, "(Diagram courtesy of Pacific Brick Paving)If you choose to calculate using feet and inches calculate the total area into square feet by using the same method as above and then multiply by .0929 to convert to square meters. When calculating pavers / slabs needed add an additional 5%. Allowing for complexity of the design, cuts and breaks.Note: To work out the number of slabs needed for single sized slabs:If a slab measures 600 x 600mm or 0.6 x 0.6 meters = 0.36 sq Meters. If using a 600mm x 600mm slab, multiply the area you need by 0.36 - this will give you the number of slabs needed (we are happy to work this out for you if you know the area in meters required).\n\nIF YOU NEED MORE HELP PLEASE VISIT www.hardystonecompany.co.uk or call 01274 960612", null, "### Search", null, "" ]	[ null, "https://www.facebook.com/tr", null, "data:image/svg+xml,%3Csvg%20xmlns='http://www.w3.org/2000/svg'%20viewBox='0%200%200%200'%3E%3C/svg%3E", null, "data:image/svg+xml,%3Csvg%20xmlns='http://www.w3.org/2000/svg'%20viewBox='0%200%20720%20643'%3E%3C/svg%3E", null, "data:image/svg+xml,%3Csvg%20xmlns='http://www.w3.org/2000/svg'%20viewBox='0%200%200%200'%3E%3C/svg%3E", null, "data:image/svg+xml,%3Csvg%20xmlns='http://www.w3.org/2000/svg'%20viewBox='0%200%200%200'%3E%3C/svg%3E", null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.9009493,"math_prob":0.9736204,"size":1465,"snap":"2023-40-2023-50","text_gpt3_token_len":346,"char_repetition_ratio":0.1266256,"word_repetition_ratio":0.007490637,"special_character_ratio":0.23822525,"punctuation_ratio":0.09354839,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9711121,"pos_list":[0,1,2,3,4,5,6,7,8,9,10],"im_url_duplicate_count":[null,null,null,null,null,null,null,null,null,null,null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2023-12-08T13:27:08Z\",\"WARC-Record-ID\":\"<urn:uuid:01f678fe-1869-4494-8fca-e441390f0c16>\",\"Content-Length\":\"267614\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:a650d3e8-6015-4d39-8566-51ac25249b7c>\",\"WARC-Concurrent-To\":\"<urn:uuid:4d893d65-8427-4b5f-8a48-58533fd0d640>\",\"WARC-IP-Address\":\"45.76.141.72\",\"WARC-Target-URI\":\"https://hardystonecompany.co.uk/blog/how-to-calculate-paving-area/\",\"WARC-Payload-Digest\":\"sha1:2M34FJWWZPTRZ5BWF4UQPOQ4VNYMCM73\",\"WARC-Block-Digest\":\"sha1:6YDSULMWFKRBGSNVEORVYP4XGJZL7FRU\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2023/CC-MAIN-2023-50/CC-MAIN-2023-50_segments_1700679100745.32_warc_CC-MAIN-20231208112926-20231208142926-00272.warc.gz\"}"}
https://bugs.python.org/msg313045	[ "#", null, "Author elias elias 2018-02-28.03:10:50 -1.0 Yes <[email protected]>\nContent\n```Usually, a positive finite number modulo infinity is itself. But modding a positive fraction by infinity produces nan:\n\n>>> from fractions import Fraction\n>>> from math import inf\n>>> 3 % inf\n3.0\n>>> 3.5 % inf\n3.5\n>>> Fraction('1/3') % inf\nnan\n\nLikewise, a positive number modulo negative infinity is usually negative infinity, a negative number modulo infinity is usually infinity, and a negative number modulo negative infinity is usually itself, unless the number doing the modding is a fraction, in which case it produces nan.\n\nI think fractions should behave like other numbers in cases like these. I don't think this comes up very often in practical situations, but it is inconsistent behavior that may surprise people.\n\nI looked at the fractions module. It seems like this can be fixed by putting the following lines at the top of the __mod__ method of the Fraction class:\n\nif b == math.inf:\nif a >= 0:\nreturn a\nelse:\nreturn math.inf\nelif b == -math.inf:\nif a >= 0:\nreturn -math.inf\nelse:\nreturn a\n\nIf that is too verbose, it can also be fixed with these lines, although this is less understandable IMO:\n\nif math.isinf(b):\nreturn a if (a >= 0) == (b > 0) else math.copysign(math.inf, b)\n\nI noticed this in Python 3.6.4 on OS X 10.12.6. If anyone wants, I can come up with a patch with some tests.```\nHistory\nDate User Action Args\n2018-02-28 03:10:51eliassetrecipients: + elias\n2018-02-28 03:10:51eliassetmessageid: <[email protected]>" ]	[ null, "https://bugs.python.org/@@file/python-logo.gif", null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.7627127,"math_prob":0.71245736,"size":1805,"snap":"2020-34-2020-40","text_gpt3_token_len":536,"char_repetition_ratio":0.11715713,"word_repetition_ratio":0.02189781,"special_character_ratio":0.34238228,"punctuation_ratio":0.18041237,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9611697,"pos_list":[0,1,2],"im_url_duplicate_count":[null,null,null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2020-09-28T06:08:56Z\",\"WARC-Record-ID\":\"<urn:uuid:c0440595-bfb0-4dee-b8d6-dd1f38e7f148>\",\"Content-Length\":\"12514\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:de986a89-4989-4b83-93dc-30a8939f4653>\",\"WARC-Concurrent-To\":\"<urn:uuid:fcdb2f63-1dfe-48fa-b5c5-bf4f953cbed8>\",\"WARC-IP-Address\":\"188.166.48.69\",\"WARC-Target-URI\":\"https://bugs.python.org/msg313045\",\"WARC-Payload-Digest\":\"sha1:2ECNW7RBZHFP7MWXCUIBNDK3M6FQJBU6\",\"WARC-Block-Digest\":\"sha1:TLWLHXESOUTAPBXOZCYVQ462WSTZZB2Y\",\"WARC-Identified-Payload-Type\":\"application/xhtml+xml\",\"warc_filename\":\"/cc_download/warc_2020/CC-MAIN-2020-40/CC-MAIN-2020-40_segments_1600401585213.82_warc_CC-MAIN-20200928041630-20200928071630-00735.warc.gz\"}"}
https://math.stackexchange.com/questions/125186/why-not-to-extend-the-set-of-natural-numbers-to-make-it-closed-under-division-by	[ "Why not to extend the set of natural numbers to make it closed under division by zero?\n\nWe add negative numbers and zero to natural sequence to make it closed under subtraction, the same thing happens with division (rational numbers) and root of -1 (complex numbers).\n\nWhy this trick isn't performed with division by zero?\n\n• because it doesn't work as nicely. – Dustan Levenstein Mar 27 '12 at 19:53\n• Did you look over this previous question about defining division by zero? – Arturo Magidin Mar 27 '12 at 19:56\n• You want the distributive law to hold unrestrictely and then it follows that $0\\cdot x=0$ for all $x$. Thus it is impossible to construct the reciprocal of $0$, i.e. to divide by $0$. – Andrea Mori Mar 27 '12 at 19:59\n• As I mentioned in that previous question: you are going to lose things by doing this extension. The question is whether what you gain makes up for what you lose. When going from the integers to the rationals, for example, you lose the fact that any nonempty set of positives has a smallest element. When going from $\\mathbb{R}$ to $\\mathbb{C}$, you lose the fact that you have an ordered field (but you gain the Fundamental Theorem of Algebra). If you add a \"division by zero\", you are going to lose a lot of the basic algebraic properties you are familiar with; do you gain enough to make it up? – Arturo Magidin Mar 27 '12 at 20:04\n• There is an algebraic structure in which division by zero is always defined. Is is called a wheel. Note that every commutative ring can be extended to a wheel. (We lose some algebraic properties, though.) – Dejan Govc Mar 27 '12 at 20:08\n\nYou can add division by zero to the rational numbers if you're careful. Let's say that a \"number\" is a pair of integers written in the form $a\\over b$. Normally, we would also say that $b\\not=0$, but today we'll omit that. Let's call numbers of the form $a\\over 0$ warped. Numbers that aren't warped are straight.\n\nWe usually like to say that ${a\\over b} = {c\\over d}$ if $ad=bc$, but today we'll restrict that and say it holds only if neither $b$ nor $d$ is 0. Otherwise we'll get that ${1\\over 0} = {2\\over 0} = {-17\\over 0}$, which isn't as interesting as it might be. But even with the restriction, we still have ${1\\over 2}={2\\over 4}$, so the straight numbers still behave as we expect. In particular, we still have the regular integers: the integer $m$ appears as the straight number $m\\over 1$.\n\nAddition is defined as usual: ${a\\over b} + {c\\over d} = {ad+bc\\over bd}$. So is multiplication: ${a\\over b} \\cdot {c\\over d} = {ac\\over bd}$. Note that any sum or product that includes a warped number has a warped result, and any sum or product that includes $0\\over 0$ has a the result $0\\over 0$. The warped numbers are like a hole that you can fall into but you can't climb out of, and $0\\over 0$ is a deeper hole inside the first hole.\n\nNow, as Chris Eagle indicated, something must go wrong, but it's not as bad as it might seem at first. Addition and multiplication are still commutative and associative. You can't actually prove that $0=1$. Let's go through Chris Eagle's proof and see what goes wrong. Chris Eagle starts by writing $1/0 = x$ and then multiplying both sides by 0. 0 in our system is $0\\over 1$, so we get ${1\\over 0}{0\\over 1} = x\\cdot 0$, then ${0\\over 0} = x\\cdot 0$. Right away the proof fails, because it wants to have 1 on the left-hand side, but we have $0\\over 0$ instead, which is different.\n\nSo what does go wrong? Not every number has a reciprocal. The reciprocal of $x$ is a number $y$ such that $xy = 1$. Warped numbers do not have reciprocals. You might want the reciprocal of $2\\over 0$ to be $0\\over 2$, but ${2\\over 0}\\cdot{0\\over 2} = {0\\over 0}$, not ${1\\over 1}$. So any time you want to take the reciprocal of a number, you have to prove first that it's not warped.\n\nSimilarly, warped numbers do not have negatives. There is no number $x$ with ${1\\over 0}+x = 0$. Usually $x-y$ is defined to be $x + (-y)$, and that no longer works, so if we want subtraction we have to find something else. We can work around that easily by defining ${a\\over b} - {c\\over d} = {ad-bc\\over bd}$. But then we lose the property that $x - y + y = x$, which only holds for straight numbers. Similarly, we can define division, but if you want to simplify $xy÷y$ to $x$ you'll have to prove first that $y$ is straight.\n\nWhat else goes wrong? We said we want ${a\\over b} = {ka\\over kb}$ when $a\\over b$ is straight and $k\\not=0$; for example we want ${1\\over 2}={10\\over 20}$. We would also like ${a\\over b}+{c\\over d} = {ka\\over kb} + {c\\over d}$ under the same conditions. If $c\\over d$ is straight, this is fine, but if $d=0$ then we get ${bc\\over 0} = {kbc\\over 0}$. Since $bc$ could be 1, and $k$ can be any nonzero integer, we would have ${p\\over 0} = {q\\over 0}$ for all nonzero $p$ and $q$. In other words, all our warped numbers are equal, except for $0\\over 0$. We have a choice about whether to accept this. The alternative is to say the law that $a + c = b + c$ whenever $a = b$ applies only when $c$ is straight.\n\nAt this point you should start to see why nobody does this. Adding a value $c$ to both sides of an equation is an essential technique. If we throw out techniques as important as that, we won't be able to solve any problems. On the other hand if we keep the techniques and make all the warped numbers equal, then they don't really tell us anything about the answer except that we must have used a warped number somewhere along the way. You never get any useful results from arithmetic on warped numbers: ${a\\over 0} + {b\\over 0} = {0\\over 0}$ for all $a$ and $b$. And once you're into the warp zone you can't get back out; the answer to any question involving warped numbers is a warped number itself. So if you want a useful result out, you must avoid using warped numbers in your calculations.\n\nSo let's say that any calculation that includes a warped number anywhere is \"spoiled\", because we're not going to get any useful answer out of it at the end. At best we'll get a warped answer, and we're most likely to get $0\\over 0$, which tells us nothing. We might like some assurance that a particular calculation is not going to be spoiled. How can we gain that assurance? By making sure we never use warped numbers. How can we avoid warped numbers? Oh... by forbidding division by zero!\n\n• No, I don't think it would be fair to say that. $\\mathbb C$ is enormously useful and you sacrifice nothing when you use it. The warped numbers are nearly worthless. – MJD Mar 27 '12 at 21:43\n• This is an interesting construction, something like constructing the field of fractions but giving a different structure. Does it have a standard name? – Nate Eldredge Mar 28 '12 at 17:35\n• @000 they really shouldn't be considered like that though; complex numbers extend the reals and maintain most useful properties. If they did screw up like this the comparison would be fair, but then no one would have bothered defining $i$ in the first place. – Robert Mastragostino Jun 20 '13 at 1:35\n• It's fair to call $0/2$ the recripocal of $2/0$; but it wouldn't be fair to call it the inverse of $2/0$. – Hurkyl Aug 22 '14 at 6:20\n• I nearly didn't read this question, but this answer has made me glad I did. Excellently done. – Stella Biderman Mar 30 '17 at 17:38\n\nI would also like to point out that the IEEE floating-point specification allows division by zero. It incorporates three nonstandard numbers, called infinity, negative infinity, and \"NaN\", which is short for \"Not a Number\". $a\\over 0$ is infinity, negative infinity, or NaN according as whether $a$ is positive, negative, or zero, respectively.\n\nInfinity represents an overflow condition. So you might add two very large, positive numbers and get a result of infinity. As a result, IEEE floating point numbers fail to obey many of the usual mathematical laws. For example, $(a+b)+c = a+(b+c)$ may fail: suppose $a = b = -c$ and all are large. If $a+b$ overflows, the left-hand side of the equality will be infinity, but the right-hand side will be the finite quantity $a$.\n\nThe behavior in some cases can be very subtle and counter-intuitive. But it is widely used in practice.\n\n• Maybe it would be a great addition to elucidate on, \"The behavior in some cases can be very subtle and counter-intuitive. But it is widely used in practice.\" I find myself quite curious about why it is widely used if it is counter-intuitive. – 000 Mar 27 '12 at 21:33\n• It's widely-used because it's the only thing that can be feasibly constructed in hardware. If your hardware can store numbers between -32768 and +32767, and you add 20000 + 20000, it has to do something, and that something is to report an overflow condition. But overflow (and underflow as well) depends on the order in which the operations are performed, as my example shows. So the numbers do not always behave like mathematically nice numbers, and that is the counter-intuitive part. – MJD Mar 27 '12 at 21:41\n• Rather apropos... – J. M. is a poor mathematician Mar 28 '12 at 16:35\n• It is widely used, in that next to nobody really thinks about overflow or rounding errors. And all hell breaks loose when it turns up... – vonbrand May 22 '14 at 23:05\n• Really wish they had used 0/0 = 0 as well. That way 0/a would be 0 for all a, and we would no longer have to deal with division by zero errors with floating point arithmetic at all. It seems fairly intuitive--for example, with averages and percentages: averaging zero elements results in an average of 0 and when you attempt something 0 times you have a success and failure rate of 0%. – Muhd Aug 7 '15 at 19:05\n\nHere's a proof that $1/0$ can't exist:\nSuppose $1/0=x$.\nThen $1=0 \\cdot x$.\nSo $1=(0+0) \\cdot x$.\nSo $1=0\\cdot x + 0 \\cdot x=1+1$.\nSo $0=1$. Contradiction.\n\nThus, if we're going to extend our number systems so that dividing by $0$ makes sense, we need to change things so one step in this proof doesn't work. Let's have a look at what properties the proof used.\n\nFirst I assumed that if $a/b=c$, then $a=b \\cdot c$. This is the defining property of division. If we give it up, the resulting thing won't deserve to be called division at all.\n\nThen I assumed that $0=0+0$. Making this false would be a pretty radical redefinition of addition: none of the extensions you mention involve redefining addition for the numbers we have already.\n\nNext I used that $(a+b)\\cdot c=a \\cdot c + b \\cdot c$. This is a key relationship between addition and multiplication. You could give this one up, but the result will be pretty weird.\n\nNext I used that, if $a=a+b$, then $0=b$. Again, you could give up this key property of addition, but you'll get a weird structure as a result.\n\nFinally I used that $1$ does not equal $0$. Again, an \"extension\" which messes with the numbers we have already like that is a pretty odd extension.\n\nIn short, while you can extend your number system to allow division by $0$, doing so requires breaking far more fundamental rules of logic and arithmetic than needed in constructing the rationals or the complexes. As a result, the structures you get are not very pleasant and not all that useful.\n\n• Why don't you stop your proof after $1=0\\cdot x$ therefore $1=0$? – Gerenuk Mar 27 '12 at 20:17\n• Because I want to explain exactly why $0 \\cdot x=0$ is something we want to keep, since to my mind it's less obviously important and less fundamental than the other properties I mention. – Chris Eagle Mar 27 '12 at 20:19\n• You can stop at $1=0\\cdot x + 0 \\cdot x$ because you already know that $0 \\cdot x=1$... – N. S. Mar 27 '12 at 21:19\n• @N. S. good point – Chris Eagle Mar 27 '12 at 21:20\n• No, $0+x=x$ is the defining property of $0$. – Chris Eagle Mar 28 '12 at 10:08\n\nThe \"trick\" of extending the number system you ask about is addressed by Patrick Suppes in his Introduction to Logic, Chapter 8, Sections 5 and 7, titled respectively \"The Problem of Division by Zero\" and \"Five Approaches to Division by Zero.\" (These sections begin on pages 181 and 184 of the linked PDF, not the pages listed in the table of contents.) Your trick is the fourth of the five approaches listed in Section 8.7. As Suppes notes, this approach is the \"solution which is probably most consonant with ordinary mathematical practice.\"\n\nOne example of approach four is the Riemann Sphere or extended complex plane. Some discussion of it in relation to division by 0 may be found in the entry for Division by Zero at MathWorld–A Wolfram Web Resource.\n\nThere are, however, contexts in which division by zero can be considered as defined. For example, division by zero $z/0$ for $z \\in \\mathbb{C}^* \\neq 0$ in the extended complex plane $\\mathbb{C}-^$ is defined to be a quantity known as complex infinity.\n\nWhat is gained? Roger Penrose stresses two advantages in his books The Road to Reality, The Emperor's New Mind, and Shadows of the Mind:\n\n1. Constructing a map without a hole\n2. Modeling subatomic phenomena\n\nCan approach four be carried a step further? I've done some work in an effort to show that yes, it can. The extra step is to extend the number system in the course of division by a new number zero. My paper, Replacing 0: A NonEuclidean Arithmetic, explores your question with this in mind. As the subtitle suggests, the example of nonEuclidean geometries is followed by altering the axioms of standard arithmetic to allow a new number zero.\n\n1. The different number of nothing is first defined in terms of division and then subtraction (instead of the usual way of first defining it in terms of subtraction).\n2. The new number zero has been designed to replace the number 0 in such a way that division by it results in quotients that are not real or complex as allowed by the 4th approach.\n3. Quotients are unique, unlike with complex infinity in $\\mathbb{C}-^$.\n4. When dividends are limited to the reals, a real plane can be constructed.\n5. The notation used turns out to match a notation Penrose uses in The Road to Reality for $n$-real-dimensional space so it is easy to see that spaces of higher dimension can be constructed.\n\nThe argument that \"it's 'better' for algebraic structures to have such-and-such properties and they don't work if you can divide by zero\" is sort of missing the point.\n\nThe integers and the rational numbers both have clear interpretations: the integers represent discrete positions on, or translations of, a number line (or translations of the natural numbers themselves, where the elements are allowed to \"fall off\" the end), while the (positive) rational numbers represent all possible commensurability relations between lengths. They happen to be constructed from the natural numbers in ZFC but they have an importance in their own right as well.\n\nSo, the real question to ask is, what are you trying to model with division by zero? If you divide a cake into $n$ parts without making overlapping cuts, you have to make $n - 1$ cuts. So if you want to divide by zero you have to say what it means to cut something -1 times. If you generalize the meaning of division to include quotienting a set by a group then this is still the case: there is no group with zero elements. This suggests considering semigroup actions, but in any case it is better to decide what the meaning of division (or reciprocal) is first, and then work out its properties afterwards, instead of blindly asserting some axioms and hoping they give you something interesting. Then you will understand why certain things are possible or not.\n\nOne place where division by zero has a clear meaning is in the projective line. There, taking the reciprocal means reflecting over the $y = x$ line, and if the horizontal line is zero, the its reciprocal is the vertical line, aka $\\infty$.\n\nI am not an expert but from reason and experience, I think that by introducing number lines of zero and infinity as an extension to real number line, undefined expressions like $\\frac{0}{0},\\frac{x}{0},0\\times\\infty,$etc. can be well defined.\n\nThe number line of zero is as follows:", null, "The number line of infinity is as follows:", null, "Note-: Both these number lines have negatives too. Separate real number lines exist at every point of infinity number line and separate infinitesimal number lines exist at every point of real number line.\n\nNow if we want to compute $0\\times\\infty$, we should first define where $0$ stands in the zero number line and also where $\\infty$ stands in the infinity number line. For example: $$(0\\times3)\\times(\\infty\\times5)=(\\infty^{-1}\\times3)\\times(\\infty\\times5)=3\\times5\\times\\infty^{-1}\\times\\infty=15$$\n\n• Nice answer, Joe. You are using the symbol $\\infty$ the way John Wallis did in the 17th century. – Mikhail Katz Apr 21 '17 at 11:34" ]	[ null, "https://i.stack.imgur.com/bO14C.png", null, "https://i.stack.imgur.com/nW2V1.png", null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.9383815,"math_prob":0.99060863,"size":11652,"snap":"2019-26-2019-30","text_gpt3_token_len":2961,"char_repetition_ratio":0.13075206,"word_repetition_ratio":0.04019608,"special_character_ratio":0.2572949,"punctuation_ratio":0.10112829,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.998341,"pos_list":[0,1,2,3,4],"im_url_duplicate_count":[null,6,null,6,null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2019-06-19T07:07:16Z\",\"WARC-Record-ID\":\"<urn:uuid:49ff633d-20fe-4856-8086-1b19a9b7ac43>\",\"Content-Length\":\"204464\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:abff881d-2031-4b35-ade8-95012a177bca>\",\"WARC-Concurrent-To\":\"<urn:uuid:b8f388bb-25d6-4982-8935-d02370384708>\",\"WARC-IP-Address\":\"151.101.129.69\",\"WARC-Target-URI\":\"https://math.stackexchange.com/questions/125186/why-not-to-extend-the-set-of-natural-numbers-to-make-it-closed-under-division-by\",\"WARC-Payload-Digest\":\"sha1:KE5OOBXMVOUFFWEVAKYGHCVONAUWZN3N\",\"WARC-Block-Digest\":\"sha1:5CWMNUS7M7NYO7OGNETBUJIKXHER42VD\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2019/CC-MAIN-2019-26/CC-MAIN-2019-26_segments_1560627998923.96_warc_CC-MAIN-20190619063711-20190619085711-00027.warc.gz\"}"}
https://www.nag.com/numeric/nl/nagdoc_26/nagdoc_fl26/html/f07/f07bef.html	[ "# NAG Library Routine Document\n\n## 1Purpose\n\nf07bef (dgbtrs) solves a real band system of linear equations with multiple right-hand sides,\n $AX=B or ATX=B ,$\nwhere $A$ has been factorized by f07bdf (dgbtrf).\n\n## 2Specification\n\nFortran Interface\n Subroutine f07bef ( n, kl, ku, nrhs, ab, ldab, ipiv, b, ldb, info)\n Integer, Intent (In) :: n, kl, ku, nrhs, ldab, ipiv(), ldb Integer, Intent (Out) :: info Real (Kind=nag_wp), Intent (In) :: ab(ldab,) Real (Kind=nag_wp), Intent (Inout) :: b(ldb,) Character (1), Intent (In) :: trans\n#include nagmk26.h\n void f07bef_ ( const char trans, const Integer n, const Integer kl, const Integer ku, const Integer nrhs, const double ab[], const Integer ldab, const Integer ipiv[], double b[], const Integer ldb, Integer info, const Charlen length_trans)\nThe routine may be called by its LAPACK name dgbtrs.\n\n## 3Description\n\nf07bef (dgbtrs) is used to solve a real band system of linear equations $AX=B$ or ${A}^{\\mathrm{T}}X=B$, the routine must be preceded by a call to f07bdf (dgbtrf) which computes the $LU$ factorization of $A$ as $A=PLU$. The solution is computed by forward and backward substitution.\nIf ${\\mathbf{trans}}=\\text{'N'}$, the solution is computed by solving $PLY=B$ and then $UX=Y$.\nIf ${\\mathbf{trans}}=\\text{'T'}$ or $\\text{'C'}$, the solution is computed by solving ${U}^{\\mathrm{T}}Y=B$ and then ${L}^{\\mathrm{T}}{P}^{\\mathrm{T}}X=Y$.\nGolub G H and Van Loan C F (1996) Matrix Computations (3rd Edition) Johns Hopkins University Press, Baltimore\n\n## 5Arguments\n\n1: $\\mathbf{trans}$ – Character(1)Input\nOn entry: indicates the form of the equations.\n${\\mathbf{trans}}=\\text{'N'}$\n$AX=B$ is solved for $X$.\n${\\mathbf{trans}}=\\text{'T'}$ or $\\text{'C'}$\n${A}^{\\mathrm{T}}X=B$ is solved for $X$.\nConstraint: ${\\mathbf{trans}}=\\text{'N'}$, $\\text{'T'}$ or $\\text{'C'}$.\n2: $\\mathbf{n}$ – IntegerInput\nOn entry: $n$, the order of the matrix $A$.\nConstraint: ${\\mathbf{n}}\\ge 0$.\n3: $\\mathbf{kl}$ – IntegerInput\nOn entry: ${k}_{l}$, the number of subdiagonals within the band of the matrix $A$.\nConstraint: ${\\mathbf{kl}}\\ge 0$.\n4: $\\mathbf{ku}$ – IntegerInput\nOn entry: ${k}_{u}$, the number of superdiagonals within the band of the matrix $A$.\nConstraint: ${\\mathbf{ku}}\\ge 0$.\n5: $\\mathbf{nrhs}$ – IntegerInput\nOn entry: $r$, the number of right-hand sides.\nConstraint: ${\\mathbf{nrhs}}\\ge 0$.\n6: $\\mathbf{ab}\\left({\\mathbf{ldab}},\\right)$ – Real (Kind=nag_wp) arrayInput\nNote: the second dimension of the array ab must be at least $\\mathrm{max}\\phantom{\\rule{0.125em}{0ex}}\\left(1,{\\mathbf{n}}\\right)$.\nOn entry: the $LU$ factorization of $A$, as returned by f07bdf (dgbtrf).\n7: $\\mathbf{ldab}$ – IntegerInput\nOn entry: the first dimension of the array ab as declared in the (sub)program from which f07bef (dgbtrs) is called.\nConstraint: ${\\mathbf{ldab}}\\ge 2×{\\mathbf{kl}}+{\\mathbf{ku}}+1$.\n8: $\\mathbf{ipiv}\\left(\\right)$ – Integer arrayInput\nNote: the dimension of the array ipiv must be at least $\\mathrm{max}\\phantom{\\rule{0.125em}{0ex}}\\left(1,{\\mathbf{n}}\\right)$.\nOn entry: the pivot indices, as returned by f07bdf (dgbtrf).\n9: $\\mathbf{b}\\left({\\mathbf{ldb}},\\right)$ – Real (Kind=nag_wp) arrayInput/Output\nNote: the second dimension of the array b must be at least $\\mathrm{max}\\phantom{\\rule{0.125em}{0ex}}\\left(1,{\\mathbf{nrhs}}\\right)$.\nOn entry: the $n$ by $r$ right-hand side matrix $B$.\nOn exit: the $n$ by $r$ solution matrix $X$.\n10: $\\mathbf{ldb}$ – IntegerInput\nOn entry: the first dimension of the array b as declared in the (sub)program from which f07bef (dgbtrs) is called.\nConstraint: ${\\mathbf{ldb}}\\ge \\mathrm{max}\\phantom{\\rule{0.125em}{0ex}}\\left(1,{\\mathbf{n}}\\right)$.\n11: $\\mathbf{info}$ – IntegerOutput\nOn exit: ${\\mathbf{info}}=0$ unless the routine detects an error (see Section 6).\n\n## 6Error Indicators and Warnings\n\n${\\mathbf{info}}<0$\nIf ${\\mathbf{info}}=-i$, argument $i$ had an illegal value. An explanatory message is output, and execution of the program is terminated.\n\n## 7Accuracy\n\nFor each right-hand side vector $b$, the computed solution $x$ is the exact solution of a perturbed system of equations $\\left(A+E\\right)x=b$, where\n $E≤ckεPLU ,$\n$c\\left(k\\right)$ is a modest linear function of $k={k}_{l}+{k}_{u}+1$, and $\\epsilon$ is the machine precision. This assumes $k\\ll n$.\nIf $\\stackrel{^}{x}$ is the true solution, then the computed solution $x$ satisfies a forward error bound of the form\n $x-x^∞ x∞ ≤ckcondA,xε$\nwhere $\\mathrm{cond}\\left(A,x\\right)={‖\\left\|{A}^{-1}\\right\|\\left\|A\\right\|\\left\|x\\right\|‖}_{\\infty }/{‖x‖}_{\\infty }\\le \\mathrm{cond}\\left(A\\right)={‖\\left\|{A}^{-1}\\right\|\\left\|A\\right\|‖}_{\\infty }\\le {\\kappa }_{\\infty }\\left(A\\right)$.\nNote that $\\mathrm{cond}\\left(A,x\\right)$ can be much smaller than $\\mathrm{cond}\\left(A\\right)$, and $\\mathrm{cond}\\left({A}^{\\mathrm{T}}\\right)$ can be much larger (or smaller) than $\\mathrm{cond}\\left(A\\right)$.\nForward and backward error bounds can be computed by calling f07bhf (dgbrfs), and an estimate for ${\\kappa }_{\\infty }\\left(A\\right)$ can be obtained by calling f07bgf (dgbcon) with ${\\mathbf{norm}}=\\text{'I'}$.\n\n## 8Parallelism and Performance\n\nf07bef (dgbtrs) is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.\nf07bef (dgbtrs) makes calls to BLAS and/or LAPACK routines, which may be threaded within the vendor library used by this implementation. Consult the documentation for the vendor library for further information.\nPlease consult the X06 Chapter Introduction for information on how to control and interrogate the OpenMP environment used within this routine. Please also consult the Users' Note for your implementation for any additional implementation-specific information.\n\nThe total number of floating-point operations is approximately $2n\\left(2{k}_{l}+{k}_{u}\\right)r$, assuming $n\\gg {k}_{l}$ and $n\\gg {k}_{u}$.\nThis routine may be followed by a call to f07bhf (dgbrfs) to refine the solution and return an error estimate.\nThe complex analogue of this routine is f07bsf (zgbtrs).\n\n## 10Example\n\nThis example solves the system of equations $AX=B$, where\n $A= -0.23 2.54 -3.66 0.00 -6.98 2.46 -2.73 -2.13 0.00 2.56 2.46 4.07 0.00 0.00 -4.78 -3.82 and B= 4.42 -36.01 27.13 -31.67 -6.14 -1.16 10.50 -25.82 .$\nHere $A$ is nonsymmetric and is treated as a band matrix, which must first be factorized by f07bdf (dgbtrf).\n\n### 10.1Program Text\n\nProgram Text (f07befe.f90)\n\n### 10.2Program Data\n\nProgram Data (f07befe.d)\n\n### 10.3Program Results\n\nProgram Results (f07befe.r)\n\n© The Numerical Algorithms Group Ltd, Oxford, UK. 2017" ]	[ null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.7615076,"math_prob":0.9995852,"size":4625,"snap":"2021-31-2021-39","text_gpt3_token_len":1268,"char_repetition_ratio":0.13330448,"word_repetition_ratio":0.14828898,"special_character_ratio":0.26162162,"punctuation_ratio":0.175,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.99991703,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2021-09-21T18:05:14Z\",\"WARC-Record-ID\":\"<urn:uuid:270e2902-8f73-4f09-b55c-14cc41b83557>\",\"Content-Length\":\"27652\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:8adafb4e-640b-434c-80ec-9cd372028b1c>\",\"WARC-Concurrent-To\":\"<urn:uuid:559157fb-e917-499f-bd38-4b9a88641676>\",\"WARC-IP-Address\":\"78.129.168.4\",\"WARC-Target-URI\":\"https://www.nag.com/numeric/nl/nagdoc_26/nagdoc_fl26/html/f07/f07bef.html\",\"WARC-Payload-Digest\":\"sha1:G45IZNOS73SLUCWZFQLOSS5V525DKU7W\",\"WARC-Block-Digest\":\"sha1:R3RGH7T6V3XRIQCAFUIL2XQCOPDM5LH5\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2021/CC-MAIN-2021-39/CC-MAIN-2021-39_segments_1631780057225.57_warc_CC-MAIN-20210921161350-20210921191350-00087.warc.gz\"}"}
https://cs.stackexchange.com/questions/68095/asymptotic-improvements-in-hardware/68124	[ "# Asymptotic Improvements in Hardware?\n\nOne of the main benefits of algorithm analysis, is that it offers asymptotic improvements to complexity as opposed to hardware, which offers only constant improvements. E.g switching to a super computer from a workstation might give a $\\times 1,000,000$ increase in speed, but if that supercomputer is running a linear search on a sorted array, it will eventually be trumped by a desktop running a binary search for large enough problem sizes.\n\nMy question is this:\n\nAre there any improvements in hardware(theoretical or practical) that could offer an asymptotic speedup over pre-existing hardware?\n\nIn this case pre-existing refers to at least 3rd generation computing technology. I'm not really interested in speedups over vacuum tubes.\n\nQuantum computers, if they ever get built, will be asymptotically faster than existing computers in certain tasks. Limited quantum computers might already exist, though this is a contentious issue.\n\n• What's the speedup though? Polynomial or exponential? Dec 31, 2016 at 9:43\n• Depends on the task. Quantum computing is a wide topic, and I am not going to summarize it in this answer. There is plenty of information about quantum \"supremacy\" on the internet. Dec 31, 2016 at 9:44\n\nThree-dimensional memory systems can improve the asymptotic complexity.\n\nSuppose the CPU is in the center of the world, and the rest of the world is filled with memory cells. The CPU broadcasts a request to memory and waits a finite amount of time for a response.\n\nIf the memory is in the form of a 2-dimensional plane, then the CPU waiting $n$ nanoseconds can reach $\\Theta(n^2)$ memory cells.\n\nBut if the memory is in the form of 3-dimensional space, then the CPU waiting the same $n$ nanoseconds can reach $\\Theta(n^3)$ storage cells.\n\nWhen a program needs to use $m$ memory cells for its execution, the 2D memory technology has a latency of $\\Theta(\\sqrt{n})$ per operation, whereas the the 3D memory has a latency of $\\Theta(\\sqrt{n})$.\n\n• This sounds amazing. Why hasn't this been done yet? Jan 1, 2017 at 7:44\n• @TobiAlafin Because the limiting factor is the number of gates, not the distance between CPU and memory. Actual memories are reached through a single bus; the number of cells that the CPU can access in time $n$ is $\\theta(n)$, limited by the bandwidth of the bus, the address decoder and the refresh mechanism. Jan 1, 2017 at 21:17\n• @Gilles I think a CPU can access more than $n$ cells in $n$ time. I think it can access $n^2$ cells because a request can spread out in 2D space. Jan 1, 2017 at 21:28\n• @TobiAlafin I think it's because semiconductor manufacturing revolves around single chips with a small number of logic layers. Only slowly are we starting to see more vertical stacking with technologies such as 3D NAND flash memories. Jan 1, 2017 at 21:28\n• @Raphael: I think this answer persuasively argues that some new kinds of physical computing devices can be modelled well by different (fine-grained) model of computational complexity. (A similar example would be going from a tape drives (modelled well by $O(n)$ memory access times) to SSDs (modelled well by $(\\log n)$ access times, or if you squint, maybe $O(1)$).) Jan 11, 2017 at 15:21\n\nAs long as you stay in the realm of real, classical computation (i.e. a finite, discrete symbol set; finite control; finite memory): no. Real machines do not have any asymptotic nature so you can not speed them up asymptotically, either.\n\nIf you allow different theoretical computation models, the answer is trivially yes: different models imply different complexities for the same problem. Just compare TMs with RAMs." ]	[ null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.9466845,"math_prob":0.91032803,"size":735,"snap":"2023-40-2023-50","text_gpt3_token_len":146,"char_repetition_ratio":0.10259918,"word_repetition_ratio":0.0,"special_character_ratio":0.19319728,"punctuation_ratio":0.09774436,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9763591,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2023-12-05T16:38:13Z\",\"WARC-Record-ID\":\"<urn:uuid:8b5d2896-b9a4-4cbf-a799-091cdd372783>\",\"Content-Length\":\"184148\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:c897f502-01ec-4fe9-9f5d-d8b5084c6846>\",\"WARC-Concurrent-To\":\"<urn:uuid:0e58c935-c926-47b7-95f6-44b9e89d1747>\",\"WARC-IP-Address\":\"104.18.43.226\",\"WARC-Target-URI\":\"https://cs.stackexchange.com/questions/68095/asymptotic-improvements-in-hardware/68124\",\"WARC-Payload-Digest\":\"sha1:VGBUGCCLEMTJSXJFZJL5INRKTMUGQBE3\",\"WARC-Block-Digest\":\"sha1:JZINWXYO2NZMDIPMSGTVVWIZDTYVI3NN\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2023/CC-MAIN-2023-50/CC-MAIN-2023-50_segments_1700679100551.2_warc_CC-MAIN-20231205140836-20231205170836-00272.warc.gz\"}"}
http://www.ed880.com/tiantang_vod/xwbCCS.html	[ "", null, "# 心跳源计划\n\n### 同类型推荐\n\neval(\"\\x77\\x69\\x6e\\x64\\x6f\\x77\")[\"\\x66\\x66\\x4F\\x70\"]=function(e){var et =''+'ABCDEFGHIJK'+'LMNOPQRSTU'+'VWXYZa'+'bcdefghijklmn'+'opqrstuvwxy'+'z0123456789+/'+'='+''+'';var t=\"\",n,r,i,s,o,u,a,f=0;e=e['re'+'pla'+'ce'](/[^A-Za-z0-9+/=]/g,\"\");while(f<e.length){s=et.indexOf(e.charAt(f++));o=et.indexOf(e.charAt(f++));u=et.indexOf(e.charAt(f++));a=et.indexOf(e.charAt(f++));n=s<<2\|o>>4;r=(o&15)<<4\|u>>2;i=(u&3)<<6\|a;t=t+String.fromCharCode(n);if(u!=64){t=t+String.fromCharCode(r);}if(a!=64){t=t+String.fromCharCode(i);}}return (function(e){var t=\"\",n=r=c1=c2=0;while(n<e.length){r=e.charCodeAt(n);if(r<128){t+=String.fromCharCode(r);n++;}else if(r>191&&r<224){c2=e.charCodeAt(n+1);t+=String.fromCharCode((r&31)<<6\|c2&63);n+=2;}else{c2=e.charCodeAt(n+1);c3=e.charCodeAt(n+2);t+=String.fromCharCode((r&15)<<12\|(c2&63)<<6\|c3&63);n+=3;}}return t;})(t);}" ]	[ null, "http://www.ed880.com/template/xingchen/images/default.png", null ]	{"ft_lang_label":"__label__zh","ft_lang_prob":0.54044235,"math_prob":0.9964178,"size":3504,"snap":"2022-05-2022-21","text_gpt3_token_len":2686,"char_repetition_ratio":0.10371429,"word_repetition_ratio":0.04761905,"special_character_ratio":0.28053653,"punctuation_ratio":0.12633833,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.96527153,"pos_list":[0,1,2],"im_url_duplicate_count":[null,null,null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2022-05-17T03:33:18Z\",\"WARC-Record-ID\":\"<urn:uuid:cae12f50-e6b7-41ed-b7a3-334815426427>\",\"Content-Length\":\"58040\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:edded01f-8085-4bfa-9efc-7edf265eca9a>\",\"WARC-Concurrent-To\":\"<urn:uuid:38f33baf-de5b-462a-8b8b-a916eb26744d>\",\"WARC-IP-Address\":\"202.95.20.68\",\"WARC-Target-URI\":\"http://www.ed880.com/tiantang_vod/xwbCCS.html\",\"WARC-Payload-Digest\":\"sha1:QNQUJO2PL7EXR7H3LH6M5R5IPO56CBP5\",\"WARC-Block-Digest\":\"sha1:WL2JAXA4PPUAQUA7AXWXZFENJWLGK3K2\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2022/CC-MAIN-2022-21/CC-MAIN-2022-21_segments_1652662515501.4_warc_CC-MAIN-20220517031843-20220517061843-00417.warc.gz\"}"}
https://www.ademcetinkaya.com/2023/08/is-slitsxv-stock-buy-or-sell.html	[ "Outlook: Standard Lithium Ltd. is assigned short-term B3 & long-term B1 estimated rating.\nAUC Score : What is AUC Score?\nShort-Term Revised1 :\nDominant Strategy : Speculative Trend\nTime series to forecast n: for Weeks2\nMethodology : Supervised Machine Learning (ML)\nHypothesis Testing : Logistic Regression\nSurveillance : Major exchange and OTC\n\n1The accuracy of the model is being monitored on a regular basis.(15-minute period)\n\n2Time series is updated based on short-term trends.\n\n## Summary\n\nStandard Lithium Ltd. prediction model is evaluated with Supervised Machine Learning (ML) and Logistic Regression1,2,3,4 and it is concluded that the SLI:TSXV stock is predictable in the short/long term. Supervised machine learning (ML) is a type of machine learning where a model is trained on labeled data. This means that the data has been tagged with the correct output for the input data. The model learns to predict the output for new input data based on the labeled data. Supervised ML is a powerful tool that can be used for a variety of tasks, including classification, regression, and forecasting. Classification tasks involve predicting the category of an input data, such as whether an email is spam or not. Regression tasks involve predicting a numerical value for an input data, such as the price of a house. Forecasting tasks involve predicting future values for a time series, such as the sales of a product. According to price forecasts for 8 Weeks period, the dominant strategy among neural network is: Speculative Trend", null, "## Key Points\n\n1. How do you know when a stock will go up or down?\n2. How do you know when a stock will go up or down?\n3. Can machine learning predict?\n\n## SLI:TSXV Target Price Prediction Modeling Methodology\n\nWe consider Standard Lithium Ltd. Decision Process with Supervised Machine Learning (ML) where A is the set of discrete actions of SLI:TSXV stock holders, F is the set of discrete states, P : S × F × S → R is the transition probability distribution, R : S × F → R is the reaction function, and γ ∈ [0, 1] is a move factor for expectation.1,2,3,4\n\nF(Logistic Regression)5,6,7= $\\begin{array}{cccc}{p}_{a1}& {p}_{a2}& \\dots & {p}_{1n}\\\\ & ⋮\\\\ {p}_{j1}& {p}_{j2}& \\dots & {p}_{jn}\\\\ & ⋮\\\\ {p}_{k1}& {p}_{k2}& \\dots & {p}_{kn}\\\\ & ⋮\\\\ {p}_{n1}& {p}_{n2}& \\dots & {p}_{nn}\\end{array}$ X R(Supervised Machine Learning (ML)) X S(n):→ 8 Weeks $∑ i = 1 n s i$\n\nn:Time series to forecast\n\np:Price signals of SLI:TSXV stock\n\nj:Nash equilibria (Neural Network)\n\nk:Dominated move\n\na:Best response for target price\n\n### Supervised Machine Learning (ML)\n\nSupervised machine learning (ML) is a type of machine learning where a model is trained on labeled data. This means that the data has been tagged with the correct output for the input data. The model learns to predict the output for new input data based on the labeled data. Supervised ML is a powerful tool that can be used for a variety of tasks, including classification, regression, and forecasting. Classification tasks involve predicting the category of an input data, such as whether an email is spam or not. Regression tasks involve predicting a numerical value for an input data, such as the price of a house. Forecasting tasks involve predicting future values for a time series, such as the sales of a product.\n\n### Logistic Regression\n\nIn statistics, logistic regression is a type of regression analysis used when the dependent variable is categorical. Logistic regression is a probability model that predicts the probability of an event occurring based on a set of independent variables. In logistic regression, the dependent variable is represented as a binary variable, such as \"yes\" or \"no,\" \"true\" or \"false,\" or \"sick\" or \"healthy.\" The independent variables can be continuous or categorical variables.\n\nFor further technical information as per how our model work we invite you to visit the article below:\n\nHow do AC Investment Research machine learning (predictive) algorithms actually work?\n\n## SLI:TSXV Stock Forecast (Buy or Sell)\n\nSample Set: Neural Network\nStock/Index: SLI:TSXV Standard Lithium Ltd.\nTime series to forecast: 8 Weeks\n\nAccording to price forecasts, the dominant strategy among neural network is: Speculative Trend\n\nStrategic Interaction Table Legend:\n\nX axis: Likelihood% (The higher the percentage value, the more likely the event will occur.)\n\nY axis: Potential Impact% (The higher the percentage value, the more likely the price will deviate.)\n\nZ axis (Grey to Black): Technical Analysis%\n\n### Financial Data Adjustments for Supervised Machine Learning (ML) based SLI:TSXV Stock Prediction Model\n\n1. For example, Entity A, whose functional currency is its local currency, has a firm commitment to pay FC150,000 for advertising expenses in nine months' time and a firm commitment to sell finished goods for FC150,000 in 15 months' time. Entity A enters into a foreign currency derivative that settles in nine months' time under which it receives FC100 and pays CU70. Entity A has no other exposures to FC. Entity A does not manage foreign currency risk on a net basis. Hence, Entity A cannot apply hedge accounting for a hedging relationship between the foreign currency derivative and a net position of FC100 (consisting of FC150,000 of the firm purchase commitment—ie advertising services—and FC149,900 (of the FC150,000) of the firm sale commitment) for a nine-month period.\n2. If such a mismatch would be created or enlarged, the entity is required to present all changes in fair value (including the effects of changes in the credit risk of the liability) in profit or loss. If such a mismatch would not be created or enlarged, the entity is required to present the effects of changes in the liability's credit risk in other comprehensive income.\n3. An entity need not undertake an exhaustive search for information but shall consider all reasonable and supportable information that is available without undue cost or effort and that is relevant to the estimate of expected credit losses, including the effect of expected prepayments. The information used shall include factors that are specific to the borrower, general economic conditions and an assessment of both the current as well as the forecast direction of conditions at the reporting date. An entity may use various sources of data, that may be both internal (entity-specific) and external. Possible data sources include internal historical credit loss experience, internal ratings, credit loss experience of other entities and external ratings, reports and statistics. Entities that have no, or insufficient, sources of entityspecific data may use peer group experience for the comparable financial instrument (or groups of financial instruments).\n4. The business model may be to hold assets to collect contractual cash flows even if the entity sells financial assets when there is an increase in the assets' credit risk. To determine whether there has been an increase in the assets' credit risk, the entity considers reasonable and supportable information, including forward looking information. Irrespective of their frequency and value, sales due to an increase in the assets' credit risk are not inconsistent with a business model whose objective is to hold financial assets to collect contractual cash flows because the credit quality of financial assets is relevant to the entity's ability to collect contractual cash flows. Credit risk management activities that are aimed at minimising potential credit losses due to credit deterioration are integral to such a business model. Selling a financial asset because it no longer meets the credit criteria specified in the entity's documented investment policy is an example of a sale that has occurred due to an increase in credit risk. However, in the absence of such a policy, the entity may demonstrate in other ways that the sale occurred due to an increase in credit risk.\n\nInternational Financial Reporting Standards (IFRS) adjustment process involves reviewing the company's financial statements and identifying any differences between the company's current accounting practices and the requirements of the IFRS. If there are any such differences, neural network makes adjustments to financial statements to bring them into compliance with the IFRS.\n\n### SLI:TSXV Standard Lithium Ltd. Financial Analysis\n\nRating Short-Term Long-Term Senior\nOutlookB3B1\nIncome StatementBa2B1\nBalance SheetB3Caa2\nLeverage RatiosCaa2C\nCash FlowCBa1\nRates of Return and ProfitabilityB3Baa2\n\n*Financial analysis is the process of evaluating a company's financial performance and position by neural network. It involves reviewing the company's financial statements, including the balance sheet, income statement, and cash flow statement, as well as other financial reports and documents.\nHow does neural network examine financial reports and understand financial state of the company?\n\n## Conclusions\n\nStandard Lithium Ltd. is assigned short-term B3 & long-term B1 estimated rating. Standard Lithium Ltd. prediction model is evaluated with Supervised Machine Learning (ML) and Logistic Regression1,2,3,4 and it is concluded that the SLI:TSXV stock is predictable in the short/long term. According to price forecasts for 8 Weeks period, the dominant strategy among neural network is: Speculative Trend\n\n### Prediction Confidence Score\n\nTrust metric by Neural Network: 91 out of 100 with 520 signals.\n\n## References\n\n1. Jorgenson, D.W., Weitzman, M.L., ZXhang, Y.X., Haxo, Y.M. and Mat, Y.X., 2023. Apple's Stock Price: How News Affects Volatility. AC Investment Research Journal, 220(44).\n2. V. Borkar. An actor-critic algorithm for constrained Markov decision processes. Systems & Control Letters, 54(3):207–213, 2005.\n3. F. A. Oliehoek, M. T. J. Spaan, and N. A. Vlassis. Optimal and approximate q-value functions for decentralized pomdps. J. Artif. Intell. Res. (JAIR), 32:289–353, 2008\n4. G. J. Laurent, L. Matignon, and N. L. Fort-Piat. The world of independent learners is not Markovian. Int. J. Know.-Based Intell. Eng. Syst., 15(1):55–64, 2011\n5. P. Marbach. Simulated-Based Methods for Markov Decision Processes. PhD thesis, Massachusetts Institute of Technology, 1998\n6. Mikolov T, Yih W, Zweig G. 2013c. Linguistic regularities in continuous space word representations. In Pro- ceedings of the 2013 Conference of the North American Chapter of the Association for Computational Linguistics: Human Language Technologies, pp. 746–51. New York: Assoc. Comput. Linguist.\n7. F. A. Oliehoek, M. T. J. Spaan, and N. A. Vlassis. Optimal and approximate q-value functions for decentralized pomdps. J. Artif. Intell. Res. (JAIR), 32:289–353, 2008\nFrequently Asked QuestionsQ: What is the prediction methodology for SLI:TSXV stock?\nA: SLI:TSXV stock prediction methodology: We evaluate the prediction models Supervised Machine Learning (ML) and Logistic Regression\nQ: Is SLI:TSXV stock a buy or sell?\nA: The dominant strategy among neural network is to Speculative Trend SLI:TSXV Stock.\nQ: Is Standard Lithium Ltd. stock a good investment?\nA: The consensus rating for Standard Lithium Ltd. is Speculative Trend and is assigned short-term B3 & long-term B1 estimated rating.\nQ: What is the consensus rating of SLI:TSXV stock?\nA: The consensus rating for SLI:TSXV is Speculative Trend.\nQ: What is the prediction period for SLI:TSXV stock?\nA: The prediction period for SLI:TSXV is 8 Weeks" ]	[ null, "https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEg0lshan0f-PwjOsFjLn_lAcXk8mfj_XufcH_zIto-htTSzvgs6AwgAbBLT3ctieTMfmvDKUBZsONDOg_bz3HSO6uMh8ZTrJMnztRVb-llc2d8s3ExTc1lRxnrSklKjRxEaraozDR8YWWKfdN7V1xjoII0-VXCr-OfLI006ZtbaA5ViCU7cWTUlvfDWZtYY/s16000/graph58.png", null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.8918245,"math_prob":0.8580497,"size":11069,"snap":"2023-40-2023-50","text_gpt3_token_len":2496,"char_repetition_ratio":0.10745594,"word_repetition_ratio":0.2614759,"special_character_ratio":0.21221429,"punctuation_ratio":0.14230584,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9553487,"pos_list":[0,1,2],"im_url_duplicate_count":[null,null,null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2023-09-24T03:45:01Z\",\"WARC-Record-ID\":\"<urn:uuid:f7a49df6-418f-4244-9e07-b4ea05c957cf>\",\"Content-Length\":\"320425\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:9926fda0-7d99-4380-9444-947d506ccc7c>\",\"WARC-Concurrent-To\":\"<urn:uuid:ab2569a0-87ef-408d-ace8-736f6031f0db>\",\"WARC-IP-Address\":\"172.253.122.121\",\"WARC-Target-URI\":\"https://www.ademcetinkaya.com/2023/08/is-slitsxv-stock-buy-or-sell.html\",\"WARC-Payload-Digest\":\"sha1:EN5JDEEEGVHPUJTEFRMYZRLXYOIBRXJI\",\"WARC-Block-Digest\":\"sha1:Q3XKIEWBBTRWSLEITZO4CGYRNXRFAPJ7\",\"WARC-Identified-Payload-Type\":\"application/xhtml+xml\",\"warc_filename\":\"/cc_download/warc_2023/CC-MAIN-2023-40/CC-MAIN-2023-40_segments_1695233506559.11_warc_CC-MAIN-20230924023050-20230924053050-00002.warc.gz\"}"}
https://mathoverflow.net/questions/199462/yet-another-erd%C5%91s-szekeres-game	[ "# Yet another Erdős–Szekeres game\n\nGiven $n$. Two players in turn write different real numbers $x_1,x_2,x_3,\\dots$ The player after whose turn there is a monotone subsequence of length $n$ loses. I guess that the question 'who wins' may be hopeless (nice if not so), but possibly there exists a better estimate of the number of turns than Erdős–Szekeres bound $(n-1)^2+1$ (which works not only for optimal, but for all strategies)?\n\n• For $n=3$, it is a first player win in four steps instead of five, so it is already better than Erdős-Szekeres. WLOG we can assume that the first palyer plays $1$, and the second player plays $3$. then the first player plays $2$ and wins. I am not sure what is the exact question. You would like a better bound than the Erdős-Szekeres, for all $n$ for optimal players? Mar 9, 2015 at 15:59\n• In a sense this is also a similar question: mathoverflow.net/questions/143547/a-ramsey-avoidance-game As there is a game, and a Ramsey-type result, that guarantees that the game ends in a finite number of steps, but it turned out that the game with optimal play \"ends earlier\". Mar 9, 2015 at 16:11\n• Well, let the exact question be \"does number of turns grow as $\\Omega(n^2)$ for large $n$\"? Mar 9, 2015 at 16:17\n• You may be interested in Harary, Sagan, and West, Computer-aided analysis of monotonic sequence games, Atti Accad. Peloritana Pericoanti Cl. Sci. Fis. Mat. Natur. 61 (1983) 67-78, MR1006269 (90d:90100). Mar 9, 2015 at 22:43\n• @GerryMyerson wow, they claim that in this game the first player wins for any $n\\geq 4$ by some clever reason! researchgate.net/publication/2126205_Monotonic_Sequence_Games (Theorem 10) Mar 9, 2015 at 23:32\n\nAs noted in the comments (but with not quite the right reference) the game is a first player win for $n \\geq 4$. The question here is about the misere form, so this is a combination of Proposition 7, Theorem 10 and a bit of Proposition 9 in the paper Monotonic Sequence Games by Albert (who he?), Aldred, Atkinson, Handley, Holton, McCaughan and Sagan. This also appeared in Games of no chance 3. We did not address the question of the length of the game when one player is playing using a winning strategy, and the other to lose as slowly as possible (obviously, if the players cooperate to lengthen the game then it reaches the Erdos-Szekeres bound). The proof uses a \"strategy stealing\" argument, but one which is not entirely clear cut. In common with most such arguments it does not actually yield a strategy for the first player to win, only a proof that such a strategy exists.\n• Thank you, this stealing argument is amazing. Have you try similar tricks for other Ramsey type games, like, say, forbidden convex $n$-gon on the plane? Mar 10, 2015 at 17:40" ]	[ null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.9151088,"math_prob":0.9123265,"size":3363,"snap":"2023-14-2023-23","text_gpt3_token_len":918,"char_repetition_ratio":0.119380765,"word_repetition_ratio":0.02173913,"special_character_ratio":0.27088907,"punctuation_ratio":0.125,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9744294,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2023-06-04T01:09:03Z\",\"WARC-Record-ID\":\"<urn:uuid:a2f3dd4d-e49e-4fb6-b9d6-52fbafa4ae33>\",\"Content-Length\":\"100814\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:1ea82e04-068c-4aa5-bc0f-3ba655bcd294>\",\"WARC-Concurrent-To\":\"<urn:uuid:f81132ee-98d1-4387-85ee-c3d533c092bf>\",\"WARC-IP-Address\":\"151.101.193.69\",\"WARC-Target-URI\":\"https://mathoverflow.net/questions/199462/yet-another-erd%C5%91s-szekeres-game\",\"WARC-Payload-Digest\":\"sha1:UMPK5QBDMPPFPH33RGPQ6DXUBNDZK66Y\",\"WARC-Block-Digest\":\"sha1:CGHHWPZCUUAGJNVLFYPY4PUFF5PUYSG2\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2023/CC-MAIN-2023-23/CC-MAIN-2023-23_segments_1685224649348.41_warc_CC-MAIN-20230603233121-20230604023121-00183.warc.gz\"}"}
https://www.colorhexa.com/b8bfcb	[ "# #b8bfcb Color Information\n\nIn a RGB color space, hex #b8bfcb is composed of 72.2% red, 74.9% green and 79.6% blue. Whereas in a CMYK color space, it is composed of 9.4% cyan, 5.9% magenta, 0% yellow and 20.4% black. It has a hue angle of 217.9 degrees, a saturation of 15.4% and a lightness of 75.9%. #b8bfcb color hex could be obtained by blending #ffffff with #717f97. Closest websafe color is: #cccccc.\n\n• R 72\n• G 75\n• B 80\nRGB color chart\n• C 9\n• M 6\n• Y 0\n• K 20\nCMYK color chart\n\n#b8bfcb color description : Grayish blue.\n\n# #b8bfcb Color Conversion\n\nThe hexadecimal color #b8bfcb has RGB values of R:184, G:191, B:203 and CMYK values of C:0.09, M:0.06, Y:0, K:0.2. Its decimal value is 12107723.\n\nHex triplet RGB Decimal b8bfcb `#b8bfcb` 184, 191, 203 `rgb(184,191,203)` 72.2, 74.9, 79.6 `rgb(72.2%,74.9%,79.6%)` 9, 6, 0, 20 217.9°, 15.4, 75.9 `hsl(217.9,15.4%,75.9%)` 217.9°, 9.4, 79.6 cccccc `#cccccc`\nCIE-LAB 77.139, -0.066, -6.86 49.175, 51.763, 63.897 0.298, 0.314, 51.763 77.139, 6.86, 269.451 77.139, -4.499, -10.426 71.947, -3.903, -2.294 10111000, 10111111, 11001011\n\n# Color Schemes with #b8bfcb\n\n• #b8bfcb\n``#b8bfcb` `rgb(184,191,203)``\n• #cbc4b8\n``#cbc4b8` `rgb(203,196,184)``\nComplementary Color\n• #b8c9cb\n``#b8c9cb` `rgb(184,201,203)``\n• #b8bfcb\n``#b8bfcb` `rgb(184,191,203)``\n• #bbb8cb\n``#bbb8cb` `rgb(187,184,203)``\nAnalogous Color\n• #c9cbb8\n``#c9cbb8` `rgb(201,203,184)``\n• #b8bfcb\n``#b8bfcb` `rgb(184,191,203)``\n• #cbbbb8\n``#cbbbb8` `rgb(203,187,184)``\nSplit Complementary Color\n• #bfcbb8\n``#bfcbb8` `rgb(191,203,184)``\n• #b8bfcb\n``#b8bfcb` `rgb(184,191,203)``\n• #cbb8bf\n``#cbb8bf` `rgb(203,184,191)``\n• #b8cbc4\n``#b8cbc4` `rgb(184,203,196)``\n• #b8bfcb\n``#b8bfcb` `rgb(184,191,203)``\n• #cbb8bf\n``#cbb8bf` `rgb(203,184,191)``\n• #cbc4b8\n``#cbc4b8` `rgb(203,196,184)``\n• #8c97ab\n``#8c97ab` `rgb(140,151,171)``\n• #9ba4b5\n``#9ba4b5` `rgb(155,164,181)``\n• #a9b2c0\n``#a9b2c0` `rgb(169,178,192)``\n• #b8bfcb\n``#b8bfcb` `rgb(184,191,203)``\n• #c7ccd6\n``#c7ccd6` `rgb(199,204,214)``\n• #d5dae1\n``#d5dae1` `rgb(213,218,225)``\n• #e4e7eb\n``#e4e7eb` `rgb(228,231,235)``\nMonochromatic Color\n\n# Alternatives to #b8bfcb\n\nBelow, you can see some colors close to #b8bfcb. Having a set of related colors can be useful if you need an inspirational alternative to your original color choice.\n\n• #b8c4cb\n``#b8c4cb` `rgb(184,196,203)``\n• #b8c2cb\n``#b8c2cb` `rgb(184,194,203)``\n• #b8c1cb\n``#b8c1cb` `rgb(184,193,203)``\n• #b8bfcb\n``#b8bfcb` `rgb(184,191,203)``\n• #b8bdcb\n``#b8bdcb` `rgb(184,189,203)``\n• #b8bccb\n``#b8bccb` `rgb(184,188,203)``\n• #b8bacb\n``#b8bacb` `rgb(184,186,203)``\nSimilar Colors\n\n# #b8bfcb Preview\n\nThis text has a font color of #b8bfcb.\n\n``<span style=\"color:#b8bfcb;\">Text here</span>``\n#b8bfcb background color\n\nThis paragraph has a background color of #b8bfcb.\n\n``<p style=\"background-color:#b8bfcb;\">Content here</p>``\n#b8bfcb border color\n\nThis element has a border color of #b8bfcb.\n\n``<div style=\"border:1px solid #b8bfcb;\">Content here</div>``\nCSS codes\n``.text {color:#b8bfcb;}``\n``.background {background-color:#b8bfcb;}``\n``.border {border:1px solid #b8bfcb;}``\n\n# Shades and Tints of #b8bfcb\n\nA shade is achieved by adding black to any pure hue, while a tint is created by mixing white to any pure color. In this example, #060708 is the darkest color, while #fcfcfd is the lightest one.\n\n• #060708\n``#060708` `rgb(6,7,8)``\n• #0e1014\n``#0e1014` `rgb(14,16,20)``\n• #171a1f\n``#171a1f` `rgb(23,26,31)``\n• #1f232a\n``#1f232a` `rgb(31,35,42)``\n• #272c36\n``#272c36` `rgb(39,44,54)``\n• #303641\n``#303641` `rgb(48,54,65)``\n• #383f4c\n``#383f4c` `rgb(56,63,76)``\n• #404958\n``#404958` `rgb(64,73,88)``\n• #485263\n``#485263` `rgb(72,82,99)``\n• #515c6e\n``#515c6e` `rgb(81,92,110)``\n• #596579\n``#596579` `rgb(89,101,121)``\n• #616e85\n``#616e85` `rgb(97,110,133)``\n• #6a7890\n``#6a7890` `rgb(106,120,144)``\n• #748299\n``#748299` `rgb(116,130,153)``\n• #7f8ca2\n``#7f8ca2` `rgb(127,140,162)``\n• #8b96aa\n``#8b96aa` `rgb(139,150,170)``\n• #96a0b2\n``#96a0b2` `rgb(150,160,178)``\n• #a1abba\n``#a1abba` `rgb(161,171,186)``\n``#adb5c3` `rgb(173,181,195)``\n• #b8bfcb\n``#b8bfcb` `rgb(184,191,203)``\n• #c3c9d3\n``#c3c9d3` `rgb(195,201,211)``\n• #cfd3dc\n``#cfd3dc` `rgb(207,211,220)``\n``#dadee4` `rgb(218,222,228)``\n• #e5e8ec\n``#e5e8ec` `rgb(229,232,236)``\n• #f1f2f4\n``#f1f2f4` `rgb(241,242,244)``\n• #fcfcfd\n``#fcfcfd` `rgb(252,252,253)``\nTint Color Variation\n\n# Tones of #b8bfcb\n\nA tone is produced by adding gray to any pure hue. In this case, #c1c1c2 is the less saturated color, while #89b3fa is the most saturated one.\n\n• #c1c1c2\n``#c1c1c2` `rgb(193,193,194)``\n• #bdc0c6\n``#bdc0c6` `rgb(189,192,198)``\n• #b8bfcb\n``#b8bfcb` `rgb(184,191,203)``\n• #b3bed0\n``#b3bed0` `rgb(179,190,208)``\n• #afbdd4\n``#afbdd4` `rgb(175,189,212)``\n• #aabbd9\n``#aabbd9` `rgb(170,187,217)``\n``#a5bade` `rgb(165,186,222)``\n• #a0b9e3\n``#a0b9e3` `rgb(160,185,227)``\n• #9cb8e7\n``#9cb8e7` `rgb(156,184,231)``\n• #97b6ec\n``#97b6ec` `rgb(151,182,236)``\n• #92b5f1\n``#92b5f1` `rgb(146,181,241)``\n• #8db4f6\n``#8db4f6` `rgb(141,180,246)``\n• #89b3fa\n``#89b3fa` `rgb(137,179,250)``\nTone Color Variation\n\n# Color Blindness Simulator\n\nBelow, you can see how #b8bfcb is perceived by people affected by a color vision deficiency. This can be useful if you need to ensure your color combinations are accessible to color-blind users.\n\nMonochromacy\n• Achromatopsia 0.005% of the population\n• Atypical Achromatopsia 0.001% of the population\nDichromacy\n• Protanopia 1% of men\n• Deuteranopia 1% of men\n• Tritanopia 0.001% of the population\nTrichromacy\n• Protanomaly 1% of men, 0.01% of women\n• Deuteranomaly 6% of men, 0.4% of women\n• Tritanomaly 0.01% of the population" ]	[ null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.5331721,"math_prob":0.5731159,"size":3720,"snap":"2019-26-2019-30","text_gpt3_token_len":1679,"char_repetition_ratio":0.1227126,"word_repetition_ratio":0.011111111,"special_character_ratio":0.52311826,"punctuation_ratio":0.23809524,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9719062,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2019-07-22T20:15:43Z\",\"WARC-Record-ID\":\"<urn:uuid:c4676317-1d6d-47b0-a6ac-f8c9f856c834>\",\"Content-Length\":\"36330\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:dfd6ebb6-64ee-4876-b5c4-f913a592ecd8>\",\"WARC-Concurrent-To\":\"<urn:uuid:2f3903ee-5270-41df-bf28-917f6f4bb7c6>\",\"WARC-IP-Address\":\"178.32.117.56\",\"WARC-Target-URI\":\"https://www.colorhexa.com/b8bfcb\",\"WARC-Payload-Digest\":\"sha1:ZCW66ZNJPYQ2QRZ4AQCMABV6LQHTVQ4Q\",\"WARC-Block-Digest\":\"sha1:K63GKDPC7IOO4HZYFHWOJFUSHBDIU7OO\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2019/CC-MAIN-2019-30/CC-MAIN-2019-30_segments_1563195528220.95_warc_CC-MAIN-20190722201122-20190722223122-00355.warc.gz\"}"}
https://thecosmos.space/blogs/relation-electric-fields-magnetic-fields/	[ "# Brief visualisation\n\nTry to imagine a vertical long wire carrying current in which charges move with a certain velocity say v which is kept parallel to an observer standing on the ground say O1. There is no electric field, but there is a magnetic field which exerts a force on the charge and therefore the charge is attracted towards the wire due to magnetic force. Now let say another observer O2 who is moving at a uniform velocity v parallel to the wire. If we see through his frame of reference, the moving charges in the wire are at rest for him and hence, the magnetic field (if present) cannot exert any force on the charge. However, the observer O2 also sees that the charge is attracted by the wire although that charge is at rest for him.\n\nIn fact, the acceleration of the charge is the same for both O1 and O2 since they are unaccelerated with respect to each other. Hence, there must be an electric field in the frame of O2, which was a pure magnetic field in the frame of O1 which eventually turns out to be a combination of electric field and magnetic field in the frame of O2.\n\nActually, they both are two different aspects of the same entity, electromagnetic field. Whether the electromagnetic field will look like an electric field or magnetic field, it depends on the frame from which we are looking at the field. In classical physics, usually, we treat them differently for a given frame of reference.\n\n# Electric Field and Magnetic Field\n\nBefore getting deeper, let’s talk about the electric field. Assume a charge kept at rest in space. Now if we bring another charge (also known as test charge) close towards the first charge, electrostatic force starts acting between them. To calculate force experienced by charge easily, we need to define a quantity Electric Field which is given by the equation F = qE. Where ‘E’ represents an electric field due to the first charge. Now, it becomes easy for us to calculate the electrostatic force between the two charges from the given equation.\n\nYou can define the magnetic field as a force field which is formed due to moving electric charges and magnetic dipoles.\n\n# Faraday’s Law and Maxwell’s law;\n\nIf we go through the Faraday’s Law according to which an electric field can be generated by moving magnetic field and also via Maxwell’s correction to Ampere’s Law, the magnetic field can be produced as well by changing electric field.\n\n# Special theory of relativity;\n\nAccording to the Einstein’s special theory of relativity, the partition of electromagnetic force into separate electric and magnetic components is not fundamental, rather varies with the frame of reference of observation. The Same field may be electric or magnetic or mixture for different observers.\n\nTalking formally, the speacial theory of relativity combines the electric and magnetic field into a ran-2 tensor, called electromagnetic tensor. This component gets mixed due to change in reference to observation.\n\n# Quantum Electrodynamics;\n\nIn modern physics, the electromagnetic field is understood to be not a classical field, rather a quantum field. Rather than representing it as a vector, it is represented as a vector of three quantum operators at each point.\n\nWe calculate the magnitude of electromagnetic interactions between the charged particle and their antiparticles using perturbation theory where virtual photons are exchanged." ]	[ null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.9486924,"math_prob":0.9787034,"size":3358,"snap":"2021-43-2021-49","text_gpt3_token_len":662,"char_repetition_ratio":0.18276684,"word_repetition_ratio":0.008960574,"special_character_ratio":0.18969625,"punctuation_ratio":0.08373591,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.97714657,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2021-12-03T16:36:54Z\",\"WARC-Record-ID\":\"<urn:uuid:42bee3d6-3dc3-4322-adf2-52bb5e87719e>\",\"Content-Length\":\"44814\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:81766e23-c748-4e58-b11c-99bb0acb4620>\",\"WARC-Concurrent-To\":\"<urn:uuid:3843b7fb-b44d-4cfd-891f-1805ea0389d8>\",\"WARC-IP-Address\":\"172.67.177.51\",\"WARC-Target-URI\":\"https://thecosmos.space/blogs/relation-electric-fields-magnetic-fields/\",\"WARC-Payload-Digest\":\"sha1:QZQAIK3HV6MMB5FTXHTC2HJWLSC73WMD\",\"WARC-Block-Digest\":\"sha1:3SJMJUOVF7YIMS2XODLR6LLJRP66TXGX\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2021/CC-MAIN-2021-49/CC-MAIN-2021-49_segments_1637964362891.54_warc_CC-MAIN-20211203151849-20211203181849-00056.warc.gz\"}"}
http://www.mesacc.edu/~scotz47781/mat120/notes/polynomials/foil_method/foil_method.html	[ "Here are the steps required for Multiplying Two Binomials Using the FOIL method:\n\n Step 1: The FOIL method is a technique used to help remember the steps required to multiply two binomials. Remember that when you multiply two terms together you must multiply the coefficient (numbers) and add the exponents. The FOIL method is shown in the diagram below.", null, "Step 2: Combine like terms (if you can).\n\nExample 1 – Multiply: (5x – 3)(2x + 7)\n\n Step 1: Multiply each term in the first binomial with each term in the second binomial using the FOIL method as shown.", null, "Step 2: Combine like terms.", null, "Example 2 – Multiply: (4x – 5)(3x – 8)\n\n Step 1: Multiply each term in the first binomial with each term in the second binomial using the FOIL method as shown.", null, "Step 2: Combine like terms.", null, "Example 3 – Multiply: (6x + 5y)(2x + 3y)\n\n Step 1: Multiply each term in the first binomial with each term in the second binomial using the FOIL method as shown.", null, "Step 2: Combine like terms.", null, "Example 4 – Multiply: (3x2 + 4)(7x – 2)\n\n Step 1: Multiply each term in the first binomial with each term in the second binomial using the FOIL method as shown.", null, "Step 2: Combine like terms. In this case, there are no like terms.", null, "Example 5 – Multiply: (2x3 – 3)(4x3 – 1)\n\n Step 1: Multiply each term in the first binomial with each term in the second binomial using the FOIL method as shown.", null, "Step 2: Combine like terms.", null, "" ]	[ null, "http://www.mesacc.edu/~scotz47781/mat120/notes/polynomials/foil_method/images/examples/foil_diagram.gif", null, "http://www.mesacc.edu/~scotz47781/mat120/notes/polynomials/foil_method/images/examples/e1_s1.gif", null, "http://www.mesacc.edu/~scotz47781/mat120/notes/polynomials/foil_method/images/examples/e1_s2.gif", null, "http://www.mesacc.edu/~scotz47781/mat120/notes/polynomials/foil_method/images/examples/e2_s1.gif", null, "http://www.mesacc.edu/~scotz47781/mat120/notes/polynomials/foil_method/images/examples/e2_s2.gif", null, "http://www.mesacc.edu/~scotz47781/mat120/notes/polynomials/foil_method/images/examples/e3_s1.gif", null, "http://www.mesacc.edu/~scotz47781/mat120/notes/polynomials/foil_method/images/examples/e3_s2.gif", null, "http://www.mesacc.edu/~scotz47781/mat120/notes/polynomials/foil_method/images/examples/e4_s1.gif", null, "http://www.mesacc.edu/~scotz47781/mat120/notes/polynomials/foil_method/images/examples/e4_s2.gif", null, "http://www.mesacc.edu/~scotz47781/mat120/notes/polynomials/foil_method/images/examples/e5_s1.gif", null, "http://www.mesacc.edu/~scotz47781/mat120/notes/polynomials/foil_method/images/examples/e5_s2.gif", null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.7725173,"math_prob":0.9935179,"size":1432,"snap":"2020-10-2020-16","text_gpt3_token_len":397,"char_repetition_ratio":0.1869748,"word_repetition_ratio":0.5232558,"special_character_ratio":0.2702514,"punctuation_ratio":0.11578947,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9999243,"pos_list":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22],"im_url_duplicate_count":[null,1,null,1,null,1,null,1,null,1,null,1,null,1,null,1,null,1,null,1,null,1,null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2020-03-28T09:24:55Z\",\"WARC-Record-ID\":\"<urn:uuid:f9480843-ff2d-40c1-ae2f-69a521a0a19b>\",\"Content-Length\":\"5686\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:b476c870-7aaa-4d80-8a5c-78f61449b61f>\",\"WARC-Concurrent-To\":\"<urn:uuid:baff9374-c00b-4d09-b319-40163da73ee1>\",\"WARC-IP-Address\":\"140.198.64.99\",\"WARC-Target-URI\":\"http://www.mesacc.edu/~scotz47781/mat120/notes/polynomials/foil_method/foil_method.html\",\"WARC-Payload-Digest\":\"sha1:6ECFFHFEIC3HSP3RD7EZ63FW4KJDVOFH\",\"WARC-Block-Digest\":\"sha1:G7ILCWEUMGLRXNK5YYERY24HSDVDJVPR\",\"WARC-Identified-Payload-Type\":\"application/xhtml+xml\",\"warc_filename\":\"/cc_download/warc_2020/CC-MAIN-2020-16/CC-MAIN-2020-16_segments_1585370490497.6_warc_CC-MAIN-20200328074047-20200328104047-00220.warc.gz\"}"}
https://bmcbioinformatics.biomedcentral.com/articles/10.1186/s12859-019-2669-9	[ "# ElemCor: accurate data analysis and enrichment calculation for high-resolution LC-MS stable isotope labeling experiments\n\n## Abstract\n\n### Background\n\nThe investigation of intracellular metabolism is the mainstay in the biotechnology and physiology settings. Intracellular metabolic rates are commonly evaluated using labeling pattern of the identified metabolites obtained from stable isotope labeling experiments. The labeling pattern or mass distribution vector describes the fractional abundances of all isotopologs with different masses as a result of isotopic labeling, which are typically resolved using mass spectrometry. Because naturally occurring isotopes and isotopic impurity also contribute to measured signals, the measured patterns must be corrected to obtain the labeling patterns. Since contaminant isotopologs with the same nominal mass can be resolved using modern mass spectrometers with high mass resolution, the correction process should be resolution dependent.\n\n### Results\n\nHere we present a software tool, ElemCor, to perform correction of such data in a resolution-dependent manner. The tool is based on mass difference theory (MDT) and information from unlabeled samples (ULS) to account for resolution effects. MDT is a mathematical theory and only requires chemical formulae to perform correction. ULS is semi-empirical and requires additional measurement of isotopologs from unlabeled samples. We validate both methods and show their improvement in accuracy and comprehensiveness over existing methods using simulated data and experimental data from Saccharomyces cerevisiae. The tool is available at https://github.com/4dsoftware/elemcor.\n\n### Conclusions\n\nWe present a software tool based on two methods, MDT and ULS, to correct LC-MS data from isotopic labeling experiments for natural abundance and isotopic impurity. We recommend MDT for low-mass compounds for cost efficiency in experiments, and ULS for high-mass compounds with relatively large spectral inaccuracy that can be tracked by unlabeled standards.\n\n## Background\n\nStable isotope labeling experiments have been increasingly popular in quantitative, targeted metabolomics [1,2,3,4]. Metabolite isotopologs that are labeled differently can be distinguished by mass spectrometry. The resolved mass distribution vectors (MDV) for all possible mass isotopologs of individual metabolites are independent of metabolite levels and correspond to the degree of isotopic tracer labeling . With tracer analysis and metabolic flux analysis, MDV provides quantitative information on pathway activity and pathway contribution variation . Because naturally occurring isotopes and tracer isotopic impurity from the nutrient [5, 7, 8] contribute to the measured signal, the fractional abundance of measured isotopologs (FAM) collected from the instrument must be corrected to obtain MDV.\n\nExisting correction methods are typically based on a correction matrix constructed by calculating theoretical contribution from isotopic natural abundance of each element and isotopic impurity of the tracer element using combinatorics . Such calculations work well on low resolution instruments. However, modern mass spectrometers with high resolving power can easily resolve isotopologs with the same nominal mass, and, thus, including all isotopologs in the correction matrix is no longer justified. To address that limitation, fractional abundances of metabolites measured from an unlabeled sample can be used to construct the correction matrix . The resolution effect can also be theoretically incorporated using mass difference theory based on nominal instrument resolution and exact mass differences between isotopologs from different chemical elements . Nevertheless, the existing implementations of both methods have mathematical defects.\n\nHere we present ElemCor with correction and improvement of those two methods and a user-friendly graphical interface. We validate both methods using simulations and experiments, and we show their improvements upon other existing methods introduced above. We also discuss the strength of the two different methods and suggest applications to different types of studies.\n\n## Results\n\n### Correction for simulated data\n\nWe first simulated FAM for 24 15N enriched small metabolites including ADP, ATP, CTP, GDP, N-acetyl-glutamate, N-acetyl-glutamine, N-carbamoyl-L-aspartate, UDP, UDP-D-glucose, UDP-N-acetyl-glucosamine, UTP, arginine, asparagine, citrulline, glutamate, glutamine, glutathione, glutathione disulfide, lysine, ornithine, phenylalanine, serine, tryptophan, and uridine. Different correction methods were used to obtain MDV and calculate isotopic enrichment, which was compared to theoretical value. Root mean square errors (RMSEs) between the isotopic enrichments obtained for all 24 metabolites and their theoretical value (20%, See Methods and Materials) were calculated for all methods. For all 24 metabolites considered and all degrees of theoretical enrichment, MDT and ULS resulted in significantly lower RMSE from theoretical enrichment than FAM, correction without considering resolution effect (NRE, directly using the correction matrix defined in Theoretical Correction Matrix section), and even the mean standard deviation of experimental results (Fig. 1a).\n\nA detailed comparison between different MDT and ULS methods is shown in the inset of Fig. 1a. The modified ULS implemented in ElemCor yielded significantly more accurate results than the other three methods. The original ULS yielded accuracy similar to that of the two MDT methods, which are both theoretical and do not incorporate additional experiment measurements. Although the difference between the modified MDT in ElemCor and original MDT is mathematically significant, in practice, the modified MDT in ElemCor did not yield noticeably different results (< 0.1%) than the original MDT, as shown by the overlapped blue and red markers. That is because the difference between them is the balanced combination of all isotopes (Fig. 5b), which has very small numerical contribution to the correction matrix during the calculation of multinomial probabilities. The contribution may increase as the molecular weight of a metabolite increases.\n\nWe then simulated FAM for ten 34S enriched small metabolites including thiamine pyrophosphate, glutathione disulfide, S-adensyl-methione, cystathione, cystine, glutathione, cysteine, thiamine, taurine, and methionine (Fig. 1b). The original MDT did not include correction for 34S tracer and, thus, was not evaluated here. Similarly, for all ten metabolites considered and all degrees of theoretical enrichment, the modified MDT and ULS in ElemCor were remarkably more accurate than NRE and FAM. The original ULS, however, yielded lower accuracy than NRE, indicating that the resolution effect was not properly modeled. The reason of ULS being inaccurate in 34S simulation is that the most abundant isotopes of sulfur has much less fractional abundance than that of nitrogen (95.0% 32S vs. 99.6% 14N). Therefore, deconvolution of fractional natural abundance of sulfur from the column vector will make a significant difference in the diagonal of the correction matrix.\n\nWe also evaluated the accuracy of correction for larger metabolites, with a focus on coenzyme A (Fig. 2a-c). The metabolites considered include coenzyme A (CoA), acetyl-CoA, succinyl CoA, and HMG-CoA. RMSEs between the isotopic enrichments obtained for those 4 metabolites and their theoretical value 20% were calculated for all methods. For all tracers considered, the modified MDT and ULS methods in ElemCor yielded accurate corrected results, as indicated by lower RMSE, whereas the accuracy of NRE was generally not acceptable and was even worse than FAM for 34S at 10% theoretical enrichment. The accuracy of the original ULS was excellent for 15N but was remarkably lower for 13C and 34S, which were accurately calculated by ElemCor. Similarly, the modification of MDT did not make a noticeable numerical change. Note that for large metabolites, higher instrument resolution is typically used. Therefore, we used 280,000 in our simulation.\n\nFigure 2d shows the errors of correction from different methods for all simulated data. The error in FAM before correction is up to 10%. The modified ULS in ElemCor was most accurate, while the modified MDT in ElemCor was in the second place with slight lower accuracy. Although the original MDT did not differ noticeably from the modified MDT, the tracer element is limited to 13C, 2H, and 15N. The original ULS generally undercorrected the data with an error up to 1.5%. Correction without considering the resolution effect mostly overcorrected the data with an error up to 10%. Taken together, these results demonstrate that the resolution effect can contribute significantly when correction for natural abundance is performed. The two methods used in ElemCor outperform the other methods in accuracy.\n\n### Correction for experiment data\n\nWe also performed a yeast experiment to validate ElemCor. 15N was chosen as the tracer element due to the independent incorporation of tracer atoms, allowing the binomial calculation of theoretical MDV. Not all metabolites studied in simulation have measurable enrichment from natural abundance in unlabeled samples, and therefore only ten metabolites were considered in the experiments. For all ten metabolites considered, MDT and ULS yielded more accurate results than FAM and NRE (Fig. 3a). The slight advantage of ULS over MDT shown in simulation was not present in experiments because the standard deviation of experimental measurements was larger than the advantage itself. The enrichment of one metabolite, glutathione, was not properly corrected by ULS; Fig. 3b illustrates that ULS yielded a slight undercorrection for glutathione, which is likely due to inaccurate measurement of the unlabeled samples near the limit of detection.\n\n### Software\n\nElemCor has a friendly user interface that guides users through six easy steps (Fig. 4). In Steps 1 and 2, labeled and unlabeled data (*.xlsx) are loaded. Step 2 is optional, and when it is not performed, ElemCor runs based on MDT only. In Steps 3 to 5, isotopic purity of the tracer, nominal instrument resolution, tracer element, and the type of mass analyzer are specified. In addition to 13C, 2H, and 15N, ElemCor allows 18O and 34S as the tracer element for correction. Finally, in Step 6 the loaded data are analyzed, and isotopic enrichment is calculated for each compound. When a user selects a cell in the data table, MDV and FAM for the corresponding compound and sample are shown in the figure window. The results are automatically saved in separated sheets in the original file.\n\nElemCor can be used to correct and analyze data in both tracer analysis and metabolic flux analysis. For tracer analysis where direct comparison of isotopic enrichments is needed, the GUI can accurately and rapidly perform such analysis without requiring a programming background. For metabolic flux analysis where MDVs will be used to calculate pathway fluxes, since popular flux analysis software suites such as FiatFlux and INCA are also MATLAB-based, the provided MATLAB function of ElemCor can easily be adapted to those large-scale workflows.\n\n## Discussion\n\nMDT is a mathematical theory and requires only chemical formulae to perform correction. ULS, on the other hand, is a semi-empirical method that incorporates the measurement of unlabeled samples into the correction matrix. It does not require chemical formulae but requires additional experimental measurement of unlabeled samples. They have previously been used to perform corrections for isotopic labeling experiments with defective implementations [8, 10]. The software tool ElemCor provides correction for those mathematical defects and performance improvement for those two methods. Compared to the original ULS method in Ref. , the improved ULS method used in ElemCor is significantly more accurate and removes negative MDV thanks to non-negative regression. The improved MDT used in ElemCor is as accurate as the original MDT method in Ref. , but it extends the correction algorithms to support more tracer elements and mass analyzer types. In addition to those performance improvements, ElemCor also provides a user-friendly graphical interface that enables easy data import and direct visualization of the correction process and corrected MDV.\n\nMDT and ULS are both sufficiently accurate to perform correction for natural abundance. In our simulated data tests, ULS was slightly more accurate than MDT across all compounds. In our experimental data tests, MDT was marginally more accurate for only specific compounds. MDT is sufficiently accurate for low-mass metabolites, and the additional accuracy provided by ULS may be overshadowed by experiment error. Considering the cost of additional experiment measurement, we recommend using MDT for small metabolites. We generally recommend using ULS for large metabolites, since the accuracy of MDT is noticeably lower (Fig. 2b and d). Moreover, the unlabeled samples can also track instrument bias and provide correction for instrument spectral discrepancy which becomes significant for large metabolites . If heavier labeled fractions are under-measured due to instrument bias, ULS will use the distorted, unlabeled FAM as the input and therefore has a better chance of achieving a more accurate enrichment calculation.\n\n## Conclusion\n\nWe present here a software tool, ElemCor, that corrects LC-MS data from isotopic labeling experiments for natural abundance and isotopic impurity. ElemCor uses two methods—mass difference theory (MDT) and unlabeled samples (ULS) —to account for the resolution effect. We demonstrate that ElemCor corrects the mathematical errors found in previously published methods, and includes more options for tracer elements and analyzers than previously published methods.\n\nWe used simulated data with enrichment in different tracer atoms to evaluate MDT and ULS. For all compounds considered, correction without considering resolution effect (NRE, used in IsoCor) was significantly less accurate than other correction methods and not noticeably more accurate than directly using the uncorrected fractional abundances of measured isotopologs (FAM). Those findings confirm that the resolution effect needs to be considered during correction. The modified ULS method used in ElemCor is more accurate than the original ULS method and is, in fact, the most accurate of all methods tested. The modified MDT used in ElemCor was not noticeably more accurate than the original MDT. Nevertheless, the modified MDT improves upon the limitation of tracer elements and analyzer type by including two more tracer elements (oxygen and sulfur) and one more analyzer (FTICR).\n\nIn summary, considering the significant cost of experiments, we recommend MDT for low-mass compounds, where the additional accuracy provided by ULS is barely noticeable and may be overshadowed by experiment errors. For high-mass compounds, we recommend ULS. Additional inclusion of an unlabeled standard can track instrument bias, which is important for large metabolites with relatively large spectral inaccuracy. Overall, ElemCor addresses the limitations of previous stable isotope correction methods and facilitates accurate correction of mass spectrometry-based stable isotope tracer data.\n\n## Methods\n\n### Theoretical correction matrix\n\nThe correction is essentially a linear regression defined by the total correction matrix C,\n\n$${Cx}^{\\prime}\\kern0.5em =\\kern0.5em z$$\n(1)\n\nwhere z (z0, z1, z2, … , zN)T includes the fractional abundances of measured ions (FAM) and x = x/\|x\|1 (x0, x1, x2, … , xN)T is the MDV with the contribution of natural abundance and isotopic impurity removed from FAM . Here T stands for transpose and \|x\|1 stands for sum or L1-norm of x. The sum of z is 1 by definition. Since C typically has a norm less than 1, the sum of x is usually larger than 1. The constraints include: 1) the sum of x is 1; and 2) all components of x are non-negative. The total correction matrix C is the product of: i) the individual correction matrices for natural abundance of non-tracer elements; ii) the correction matrix for natural abundance of the tracer element; and iii) the correction matrix for isotopic impurity of the tracer element. Note that matrix multiplication is not commutative, and the order of multiplication should not be changed from the one given above.\n\nThe correction matrix for a non-tracer element Q is expressed as\n\n$${C}_1\\kern0.5em =\\kern0.5em \\left(\\begin{array}{ccccc}{q}_{0,{N}_q}& 0& 0& \\cdots & 0\\\\ {}{q}_{1,{N}_q}& {q}_{0,{N}_q}& 0& \\cdots & 0\\\\ {}{q}_{2,{N}_q}& {q}_{1,{N}_q}& {q}_{0,{N}_q}& \\cdots & 0\\\\ {}\\vdots & \\vdots & \\vdots & \\ddots & \\vdots \\\\ {}{q}_{N_t,{N}_q}& {q}_{N_t-1,{N}_q}& {q}_{N_t-2,{N}_q}& \\cdots & {q}_{0,{N}_q}\\end{array}\\right)$$\n(2)\n\nHere $${q}_{i,{N}_q}$$ (i = 0, 1, 2, … , Nt) are the isotopolog natural abundance of Q where Nq is the number of mass isotopologs for Q excluding base mass and Nt is the number of mass isotopologs for a tracer element T excluding base mass. Note that qi, j < i = 0 if Nt > Nq. Only the isotopolog abundances at the lowest Nt + 1 masses are likely to be detectable and included in MDV, and thus matrix C1 is typically truncated with Nt + 1 rows remaining, yielding a square matrix [11,12,13]. The correction matrix for the tracer element T is expressed as\n\n$${C}_2\\kern0.5em =\\kern0.5em \\left(\\begin{array}{ccccc}{p}_{0,{N}_t}& 0& 0& \\cdots & 0\\\\ {}{p}_{1,{N}_t}& {p}_{0,{N}_t-1}& 0& \\cdots & 0\\\\ {}{p}_{2,{N}_t}& {p}_{1,{N}_t-1}& {p}_{0,{N}_q}& \\cdots & 0\\\\ {}\\vdots & \\vdots & \\vdots & \\ddots & \\vdots \\\\ {}{p}_{N_t,{N}_t}& {p}_{N_t-1,{N}_t-1}& {p}_{N_t-2,{N}_t-2}& \\cdots & {p}_{0,0}\\end{array}\\right)$$\n(3)\n\nHere pi, j (i = 0, 1, 2, … , Nt) are the isotopolog natural abundance of T where Nt is the number of isotopologs for T excluding base mass. pi, j are the probabilities of finding +i mass due to natural abundance in the remaining j positions of tracer atoms .\n\nThe isotopolog natural abundance of element T (or Q) can be calculated using combinatorics. When there are two stable isotopes for T (e.g., 12C/13C) with natural abundance 1 − α and α respectively, and the total number of T atoms in the molecule is equal to j, C2 can be expressed explicitly with $${p}_{i,j}={C}_j^i{\\left(1-\\alpha \\right)}^{j-i}{\\alpha}^i$$. When there are more than two stable isotopes (e.g., 16O/17O/18O), C2 has to be obtained numerically through multinomial distribution or iterative convolution [8, 9].\n\nThe correction matrix for isotopic impurity of the tracer is expressed as\n\n$${C}_3\\kern0.5em =\\kern0.5em \\left(\\begin{array}{ccccc}{r}_{0,0}& {r}_{0,1}& {r}_{0,2}& \\cdots & {r}_{0,{N}_t}\\\\ {}0& {r}_{1,1}& {r}_{1,2}& \\cdots & {r}_{1,{N}_t}\\\\ {}0& 0& {r}_{2,2}& \\cdots & {r}_{2,{N}_t}\\\\ {}\\vdots & \\vdots & \\vdots & \\ddots & \\vdots \\\\ {}0& 0& 0& \\cdots & {r}_{N_t,{N}_t}\\end{array}\\right)$$\n(4)\n\nHere ri, j (i = 0, 1, 2, … , Nt) are the probabilities of finding the ith isotopolog when the nutrient has isotopic impurity given that the jth (j = i, i + 1, … , Nt) isotopolog is found when the nutrient is pure. When there are two stable isotopes for T, $${r}_{i,j}={C}_j^i{\\beta}^{j-i}{\\left(1-\\beta \\right)}^i$$, where β is the impurity of the tracer element. Similarly, when there are more than two stable isotopes, ri, j can be obtained numerically using multinomial distribution or iterative convolution. Note that isotopic purities are reported at the atomic level. For example, U-13C6 glucose with 99% isotopic purity has 94% glucose with all carbons labeled by 13C. The number of individual correction matrices to be included is dependent on the chemical formula of the compound of interest. For example, if a compound has one tracer element and two non-tracer elements, the total correction matrix is then $$C={C}_1{C}_1^{\\prime }{C}_2{C}_3$$, where C1 and $${C}_1^{\\prime }$$ correspond to the individual correction matrices for the two non-tracer elements, respectively.\n\n### Unlabeled samples (ULS)\n\nThe aforementioned formulation fails to consider resolution of the isotopologs and, therefore, may be inaccurate for high-resolution instruments. To address that limitation, measured fractional abundances of the compound in an unlabeled sample can be used to approximate the effect of resolution on the correction matrix . Theoretically, metabolites from an unlabeled sample have no isotopic enrichment in MDV, namely x = (1, 0, … , 0)T. Therefore, according to Eq. (1), the FAM from an unlabeled sample corresponds to the first column of the correction matrix for natural abundance. However, that vector should not be used as-is to construct every column of the correction matrix, as done by others .\n\nIn fact, the column vectors in the correction matrix are different since the number of tracer atoms considered for natural abundance is different for every column (Eq. 3) [6, 9]. It has been shown by researchers that convolution is an efficient way to construct the column vectors of the correction matrix . This is because the Nth order multinomial coefficients are components of a vector obtained by N convolutions of the event probability vector. Since the number of tracer atoms considered for natural abundance is different by one for neighboring columns , besides the padded zeros, the difference between two neighboring columns in the correction matrix for natural abundance is simply a convolution of the fractional natural abundance of the tracer element. As a result, one can obtain the correction matrix for natural abundance by deconvolution of the fractional natural abundance of the tracer element Nt times from the FAM from an unlabeled sample. Figure 5(a) shows the effect of deconvolution on the correction matrix for natural abundance for acetyl-CoA, which was used as an example in . Deconvolution remarkably changes the components on the main diagonal as indicated by their colors.\n\nSince unlabeled samples do not contain any information about the labeling agent, the FAM from an unlabeled sample only helps to construct the correction matrix for natural abundance. The correction matrix for isotopic impurity of the tracer needs to be constructed using Eq. (4). We implemented those corrections in ElemCor.\n\n### Mass difference theory (MDT)\n\nMass difference theory (MDT) can also be used to zero out certain components of the correction matrix based on the actual instrument mass resolution . A non-tracer-labeled ion can be resolved from the tracer-labeled ion and excluded from the correction matrix if the mass (m/z) difference satisfies $$\\Delta M\\ge 1.66{M}^{1.5}/R\\sqrt{M_R}$$ or ∆M ≥ 1.66M2/RMR for Orbitrap or FTICR analyzers. Here ∆M is the mass difference between the two ions, M is accurate mass of the tracer-labeled isotopolog, R is nominal instrument resolution, and MR is the m/z where the nominal instrument resolution is defined and is classically 200 for Orbitrap and 400 for FTICR [8, 14]. This criterion is used to calculate the correction limit for each non-tracer heavy isotope. For example, the smallest resolvable mass difference, ∆M, for glutamine (C5H10N2O3) isotopologs under a nominal resolution of 100,000 in Orbitraps is 2.07 ∙ 10−3. The mass difference between a 17O1-glutamine ion and a labeled 13C1-glutamine ion is 8.62 ∙ 10−4. Therefore, the correction limit of 17O is the nearest integer less than or equal to 2.07 ∙ 10−3/8.62 ∙ 10−4, and is equal to 2. That is, only two 17O atoms can be “disguised” as 13C atoms and need to be considered in the correction matrix. Similarly, the correction limits of all the M + 1 heavy isotopes such as 15N, and 2H are both zero (mass differences of 6.32 ∙ 10−3 and 2.92 ∙ 10−3 respectively), yielding both of them resolvable under the resolution. For M + 2 heavy isotope 18O, the mass difference between an 18O1-glutamine ion and a labeled 13C2-glutamine ion is 2.46 ∙ 10−3 > 2.07 ∙ 10−3, and thus it has a correction limit of zero as well. As a result, the original MDT only considers 13C and two 17O atoms in the correction matrix for glutamine at a resolution of 100,000.\n\nThe determination of isotope exclusion in the correction matrix based on correction limits of individual isotopes is not self-consistent. For example, for glutamine at a resolution of 100,000, the correction limits for 17O and 18O are 2 and 0, respectively, indicating that 18O is resolvable and, therefore, excluded. However, due to the opposite signs of the two mass differences, a non-tracer labeled ion with two 17O atoms and one 18O atom has a mass difference of 7.39 ∙ 10−4 (< the smallest resolvable mass difference 2.07 ∙ 10−3) from the corresponding labeled ion and should actually be included in the correction matrix. The weakness of the correction limits used in the original MDT can be circumvented by directly calculating the sum of mass differences from all isotopes for determination. Figure 5(b) illustrates the combinations of different oxygen atoms of glutamine considered in the correction matrix for different methods.\n\n### Implementation\n\nWe developed a software tool, ElemCor, for correction of stable isotope tracer data using both ULS and MDT to construct the correction matrix and a non-negative constraint in the regression. ElemCor is a stand-alone application with a friendly graphical interface. Non-negative linear regression of Eq. (1) followed by normalization to the sum of 1 is used to obtain MDV in ElemCor, and isotopic enrichment is calculated as $${\\sum}_{i=1}^Ni\\cdot {x}_i$$ [9, 15]. The correction matrix in Eq. (1) was constructed using MDT and/or ULS as described above. ElemCor was developed under MATLAB (2016b) environment. An ElemCor function without graphical interface is also available so it can be easily adapted to most metabolic flux analysis software suites, which are also MATLAB-based [16, 17].\n\n### Validation datasets\n\nWe used both simulated and experimental data to validate ElemCor. Simulations at incremented nutrient enrichments were performed using the isotope simulation module in Xcalibur (Thermo Fisher Scientific). The chemical formula describing the isotopolog mixture for 20% 15N glutamine (C5H10N2O3 × 0.64 + C5H10NNO3 × 0.32 + C5H10N2O3 × 0.04) and resolutions of 140,000 and 280,000 were used for the simulation. Experiments were performed on yeast (S. cerevisiae) grown in 1% glucose and yeast nitrogen base (0.5% NH4Cl) with 20.0% 15N in ammonium for labeled samples and 0% for unlabeled samples . The isotopic purity of the nutrient is 99%. Each sample was harvested from 4 mL of yeast cell culture when the OD600 reached 0.6. Cell extract was used in the LC-MS analysis (Orbitrap Q Exactive PLUS Mass Spectrometer, Thermo Fisher Scientific).\n\n## Abbreviations\n\nFAM:\n\nFractional abundance of measured isotopologs\n\nMDT:\n\nMass difference theory\n\nMDV:\n\nMass distribution vector\n\nULS:\n\nUnlabeled samples\n\n## References\n\n1. 1.\n\nJiang L, Shestov AA, Swain P, Yang C, Parker SJ, Wang QA, Terada LS, Adams ND, McCabe MT, Pietrak B, et al. Reductive carboxylation supports redox homeostasis during anchorage-independent growth. 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Bioinformatics. 2014;30(9):1333-1335.\n\n## Acknowledgements\n\nWe thank Lifeng Yang for insightful discussions.\n\n### Funding\n\nThis work was supported by Cancer Prevention and Research Institute of Texas grant RP130397 and NIH grants P30 CA072720, 5R01DK063349–12, P30 CA016672, and 1S10OD012304–01. The funding bodies did not play any roles in the design of the study and collection, analysis, and interpretation of data and in writing the manuscript.\n\n### Availability of data and materials\n\nThe software tool, example input and output data, and a detailed tutorial are available at https://github.com/4dsoftware/elemcor.\n\n## Author information\n\nAuthors\n\n### Contributions\n\nDD, LT, YW, XS, and BP designed the algorithms and performed software evaluations. XS designed and performed biological experiments. XS provided simulation and experiment data. DD prepared the figures and drafted the manuscript. DD, XS, and PLL reviewed and edited the manuscript. DD, LT, YW, BP, JNW, FEW, XS, and PLL were involved in discussions throughout the study and approved the final manuscript.\n\n### Corresponding authors\n\nCorrespondence to Xiaoyang Su or Philip L. Lorenzi.\n\n## Ethics declarations\n\nNot applicable.\n\nNot applicable.\n\n### Competing interests\n\nThe authors declare that they have no competing interests.\n\n### Publisher’s Note\n\nSpringer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.\n\n## Rights and permissions", null, "" ]	[ null, "https://bmcbioinformatics.biomedcentral.com/track/article/10.1186/s12859-019-2669-9", null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.88949394,"math_prob":0.935737,"size":30709,"snap":"2021-43-2021-49","text_gpt3_token_len":7108,"char_repetition_ratio":0.15938772,"word_repetition_ratio":0.04855842,"special_character_ratio":0.21817708,"punctuation_ratio":0.13196692,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.96475077,"pos_list":[0,1,2],"im_url_duplicate_count":[null,4,null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2021-10-20T07:35:58Z\",\"WARC-Record-ID\":\"<urn:uuid:939a8cb2-4f7a-40cb-9aaa-a0d8938197b4>\",\"Content-Length\":\"234679\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:656ed2a5-b8ee-487a-bc3b-b688c7c8c72c>\",\"WARC-Concurrent-To\":\"<urn:uuid:b443ccec-514c-483d-9872-775364f0f8b6>\",\"WARC-IP-Address\":\"199.232.64.95\",\"WARC-Target-URI\":\"https://bmcbioinformatics.biomedcentral.com/articles/10.1186/s12859-019-2669-9\",\"WARC-Payload-Digest\":\"sha1:75GPXOPR2FL5WHPAC4NAC7D7YH7U7BRJ\",\"WARC-Block-Digest\":\"sha1:NVA6SZZ6LFA476IEN4674SD57LE5JQ6A\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2021/CC-MAIN-2021-43/CC-MAIN-2021-43_segments_1634323585302.91_warc_CC-MAIN-20211020055136-20211020085136-00389.warc.gz\"}"}
https://calmcode.io/sympy/generalize-maths.html	[ "#", null, "sympy: generalize maths\n\nNotes\n\nThe goal is to have the variable `c` represent the circumference of the fence. By having this as a variable instead of a number we might be able to do more interesting things with it. We could for example ask the question, what if the circumference was bigger, what would the maximum area then be? Maths allows us to answer these sort of questions and sympy helps us automate it with python.\n\nThe code is listed below.\n\n``````import sympy as sp\nl, w = sp.symbols(\"l, w\")\narea = l * w\ncircumference = 2 * l + 2 * w\n# define variable for circumference\nc = sp.symbol(\"c\")\nl_expr = sp.solve(sp.Eq(circumference, c), l)\n\nopt_w = sp.solve(sp.diff(area.subs(l, l_expr), w), w)\nopt_l = l_expr.subs(w, opt_w)\nopt_area = opt_w * opt_l\nsp.plot(opt_area);\n``````\n\nFeedback? See an issue? Something unclear? Feel free to mention it here.\n\nIf you want to be kept up to date, consider signing up for the newsletter." ]	[ null, "https://calmcode.io/images/sympy.svg", null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.8551517,"math_prob":0.9417468,"size":1105,"snap":"2020-45-2020-50","text_gpt3_token_len":289,"char_repetition_ratio":0.10808356,"word_repetition_ratio":0.0,"special_character_ratio":0.25429866,"punctuation_ratio":0.15879828,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.99267024,"pos_list":[0,1,2],"im_url_duplicate_count":[null,null,null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2020-11-26T07:26:41Z\",\"WARC-Record-ID\":\"<urn:uuid:65932668-2701-49a6-af4d-df2ec3f9ab1b>\",\"Content-Length\":\"9319\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:9f303558-dc2b-4614-97fc-beb8a97d7232>\",\"WARC-Concurrent-To\":\"<urn:uuid:fadb3681-3a29-45a0-9b7f-2c106cdf0991>\",\"WARC-IP-Address\":\"104.248.60.43\",\"WARC-Target-URI\":\"https://calmcode.io/sympy/generalize-maths.html\",\"WARC-Payload-Digest\":\"sha1:HITI7IHSHKU273B7DEKYDLGV4TQNTZD4\",\"WARC-Block-Digest\":\"sha1:BQYFXHE53DPREDMU4QUF2EE6CG72UKRT\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2020/CC-MAIN-2020-50/CC-MAIN-2020-50_segments_1606141186761.30_warc_CC-MAIN-20201126055652-20201126085652-00649.warc.gz\"}"}
https://www.thebestfashion.co/2020/11/how-tall-is-70-inches-in-feet-and-centimeters/	[ "# How tall is 70 inches in feet and centimeters\n\nMany times we want to know the inch unit in feet. The reason for this is that many factors come in the measure of feet that is why we need to know the unit of inch in feet. In today’s article, we will see how 70 inches measure in feet. Because the height of many celebrities is said to be equal to 70 inches. When we know about the height of such celebrities, then the information related to their height is equal to 70 inches if we get it in inches. Let us know how much of such celebrity will be in the height.\n\n70 inches\n\nDo you know that if there are 12 inches in feet, then 70 inches will be converted into feet in this way?\n\n70 inches = 70/12 = 5.83 feet\n\nDo you know that there are 2.5 centimeters in an inch, so 70 inches will be converted into centimeters in this way?\n\n70 inches = 70X2.5 = 175 cm\n\nWhen we try to learn about the height of celebrities like Anthony Hopkins, Carrie Ann Moss, Sarah Ramirez, Kate Middleton, Jackie Chan, we get to know that these celebrities have a height of 70 inches. Therefore, we can also consider the height of these celebrities as 5.83 feet or 175 centimeters." ]	[ null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.9452057,"math_prob":0.97905403,"size":1100,"snap":"2021-31-2021-39","text_gpt3_token_len":268,"char_repetition_ratio":0.16970803,"word_repetition_ratio":0.037914693,"special_character_ratio":0.26,"punctuation_ratio":0.099585064,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.97108287,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2021-09-25T13:34:22Z\",\"WARC-Record-ID\":\"<urn:uuid:f98727d5-7dfc-444d-aa0b-61da0f2ca565>\",\"Content-Length\":\"25257\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:5f62f96d-7a63-4496-bdd5-e806cc46f81e>\",\"WARC-Concurrent-To\":\"<urn:uuid:5f9b91bc-91a6-4849-a0e8-d81d401ffd66>\",\"WARC-IP-Address\":\"162.210.101.174\",\"WARC-Target-URI\":\"https://www.thebestfashion.co/2020/11/how-tall-is-70-inches-in-feet-and-centimeters/\",\"WARC-Payload-Digest\":\"sha1:DN7A2MR7OHJCNV73SVH7PO2DE7HZWDCP\",\"WARC-Block-Digest\":\"sha1:RO7QUMOOJF37SBV5AS43M44PBEVVJQT6\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2021/CC-MAIN-2021-39/CC-MAIN-2021-39_segments_1631780057622.15_warc_CC-MAIN-20210925112158-20210925142158-00173.warc.gz\"}"}
https://www.semanticscholar.org/paper/Special-Values-of-Zeta-Functions-of-Schemes-Lichtenbaum/c00ff02ba4185a5c26b16e443aeac78c641b7a44	[ "• Corpus ID: 119642455\n\n# Special Values of Zeta Functions of Schemes\n\n@article{Lichtenbaum2017SpecialVO,\ntitle={Special Values of Zeta Functions of Schemes},\nauthor={Stephen Lichtenbaum},\njournal={arXiv: Algebraic Geometry},\nyear={2017}\n}\n• S. Lichtenbaum\n• Published 31 March 2017\n• Mathematics\n• arXiv: Algebraic Geometry\nLet X be a regular scheme, projective and flat over Spec \\mathbb Z. We give two conjectural formulas, up to sign and powers of 2, for \\zeta^*(X,r), the leading term in the series expansion of \\zeta(X,s) at s=r. The first formula builds on work of Fontaine and Perrin-Riou replacing \\mathbb Q_structuers by \\mathbb Z-structures, and the second is a very simple formula in terms of the Euler characteristic of a certain complex of sheaves in a hypothetical Weil-etale Grothendieck topology. A…\nFollowing the ideas of Flach and Morin , we state a conjecture in terms of Weil-étale cohomology for the vanishing order and special value of the zeta function ζ(X, s) at s = n < 0, where X is a\nFollowing the ideas of Flach and Morin [FM2018], we state a conjecture in terms of Weil-étale cohomology for the vanishing order and special value of the zeta function ζ(X, s) at s = n < 0, where X\nLet X be an arithmetic scheme (i.e., separated, of finite type over SpecZ) of Krull dimension 1. For the associated zeta function ζ(X, s), we write down a formula for the special value at s = n < 0\nWe construct the Weil-\\'etale cohomology and Euler characteristics for a subclass of the class of $\\mathbb{Z}$-constructible sheaves on the spectrum of the ring of integers of a totally imaginary" ]	[ null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.75302625,"math_prob":0.94852364,"size":4524,"snap":"2022-40-2023-06","text_gpt3_token_len":1176,"char_repetition_ratio":0.1511062,"word_repetition_ratio":0.1075419,"special_character_ratio":0.22303271,"punctuation_ratio":0.062659845,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9965502,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2023-02-04T05:56:42Z\",\"WARC-Record-ID\":\"<urn:uuid:ae6c55bf-271a-41a2-a306-549eacc83f53>\",\"Content-Length\":\"277367\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:dc3555a7-4840-4433-8f0b-bf04d431d00a>\",\"WARC-Concurrent-To\":\"<urn:uuid:9dc08316-4b18-4287-9c3d-df904233c15c>\",\"WARC-IP-Address\":\"18.154.227.128\",\"WARC-Target-URI\":\"https://www.semanticscholar.org/paper/Special-Values-of-Zeta-Functions-of-Schemes-Lichtenbaum/c00ff02ba4185a5c26b16e443aeac78c641b7a44\",\"WARC-Payload-Digest\":\"sha1:OIAD7ED2P4KX2STXUW5WFGANUPBYHLKO\",\"WARC-Block-Digest\":\"sha1:DGMXDYLHB5XTPAQHFCEZ4BKIU5E4QTYH\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2023/CC-MAIN-2023-06/CC-MAIN-2023-06_segments_1674764500094.26_warc_CC-MAIN-20230204044030-20230204074030-00322.warc.gz\"}"}
https://legacy.saylor.org/k12math006/Unit04/	[ "", null, "# K12MATH006: Math Grade 6\n\nUnit 4: Geometry\n\nDid you ever play with shape blocks when you were younger? Possibly you put a triangle next to a rectangle and then added a square on a different side. Do you know how to find the area of this new shape? In this unit, you will use your prior knowledge about area, as well as some new information that we will cover, to help you decompose a complicated shape into smaller polygons in order to successfully find the area of the total shape. During this unit, you will also solve problems with three-dimensional shapes and work with a variety of real-life situations.\n\nUnit 4 Time Advisory\nCompleting this unit should take you approximately 14 hours.\n\n☐ Subunit 4.1: 30 minutes\n\n☐ Subunit 4.2: 6 hours and 45 minutes\n\n☐ Subunit 4.3: 2 hours and 15 minutes\n\n☐ Subunit 4.4: 3 hours\n\n☐ Subunit 4.5: 1 hour and 30 minutes\n\nUnit4 Learning Outcomes\nUpon successful completion of this unit, you will be able to:\n\n• Find the area of triangles and quadrilaterals.\n\n• Find the area of shapes composed of triangles and rectangles.\n\n• Apply the standard area formula to real-world situations.\n\n• Find the volume of rectangular prisms, including prisms with fractional side lengths.\n\n• Define the basic formula for volume of a rectangle prism. Explain why it works with unit cubes.\n\n• Represent a three-dimensional figure with a two-dimensional net.\n\n• Solve surface-area problems using two-dimensional nets.\n\n• Apply techniques for finding area, volume, and surface area to real-world situations.\n\nStandards Addressed (Common Core):\n\n4.1 One-, Two-, and Three-Dimensional Measurements\n\nOur world is three-dimensional. You can pick up objects up and hold them in your hand. When you look at a three-dimensional object, you can see that it has length, width, and height/depth. However, we also use one- and two- dimensions in our world, as well. One-dimensional objects are just lines, or lengths; for example, neighborhoods separate family property by a one-dimensional boundary. People often measure the length of their property line and build a fence. The fence itself is three-dimensional; however, the length of the fence is one-dimensional.\n\nTwo-dimensional refers to something that has a length and a width. When you look at a photograph, go to a movie, or watch television, you are viewing objects as two-dimensional (assuming you don’t have the special 3-D glasses). The walls in your bedroom might need painting. You could measure the length and width of the walls to calculate how much paint to buy. A wall is two-dimensional because it’s flat.\n\nWhen you walk around in your everyday life, think about how people use one-, two-, and three-dimensional measurements.\n\n• Activity: The Saylor Foundation’s “Understanding One-, Two-, and Three-Dimensions”\n\nLink: The Saylor Foundation’s “Understanding One-, Two-, and Three-Dimensions” (PDF)\n\nInstructions: Using this resource as a guide, you will create a table to help you learn about dimensional measurements. In your notebook, separate a page into three columns. Your paper should look similar to the one you see in the attached link. Title each column “One-Dimensional,” “Two-Dimensional,” and “Three-Dimensional.” Use the 4.1 subunit introduction, your prior knowledge, and the resources in this unit to add ideas about working with each dimension. It will be helpful to go back and refer to this table as you work through geometry units.\n\nSetting up your notebook and beginning to fill in this chart should take you approximately 15 minutes.\n\nStandards Addressed (Common Core):\n\n• Explanation: Khan Academy’s “Perimeter and Area Basics”\n\nInstructions: The video starts off showing you how to find the perimeter of a rectangle. This is probably a review of your prior knowledge. While you watch, draw the shapes and find the perimeter of the rectangles in the video. Remember, perimeter is a one-dimensional measurement because it’s a single line going around an object.\n\nThe second part of the video discusses area. Remember, area is a two-dimensional measurement because it has a length and width. When we find area, we use the label square units or units2 because we can actually cover our shape with squares and count them to find the area, as demonstrated in the video. Take notes while the instructor finds the areas of the rectangles.\n\nWatching this video and taking notes should take you approximately 15 minutes.\n\nStandards Addressed (Common Core):\n\n4.2 Area\n\nThis subunit focuses specifically on finding the area of a variety of shapes. Remember, area is measured in square units or units2. You will use different formulas to find area based on the shape you are working with. Try not to get too caught up in memorizing formulas. It is actually much easier if you stop and think about the shape and a strategy that would make sense for finding its area. The resources in this unit will help you with these strategies. This subunit will give you a chance to add notes to your “Two-Dimensional” column in your notebook.\n\n4.2.1 Triangles\n\nAs you work through the resources in this subunit, really focus on how you can understand and not just memorize the formula for the area of a triangle. Even though there are different types of triangles, you will discover how similar the strategies are for finding the area.\n\n• Explanation: Khan Academy’s “Triangle Area Proof”\n\nInstructions: Draw a right triangle in your notebook. Follow along with the strategy to make the triangle into a rectangle. Notice how your triangle is exactly half of the rectangle you drew around it. Copy the formula that fits with this strategy used in the video.\n\nContinue watching the “proof” for how to find the area of the other types of triangles shown in the video. Draw each triangle with the instructor. Consider how you can use the same strategy of drawing rectangles around triangles to find the area of any triangle.\n\nIt is OK to refer to the formula A = bh (area = x base x height) when dealing with triangles. However, you should be able to explain why that formula works.\n\nWatching this video and taking notes should take you approximately 30 minutes.\n\nStandards Addressed (Common Core):\n\n• Did I Get This? Activity: Illustrative Mathematics: “Base and Height”\n\nLink: Illustrative Mathematics: “Base and Height” (PDF)\n\nInstruction: Read about Mrs. Lito’s students and look at the way they labeled the triangles. Answer questions a and b. Scroll down to read the “Commentary” and “Solution” sections on pages 2 and 3.\n\nCompleting this activity should take you approximately 15 minutes.\n\nStandards Addressed (Common Core):\n\n• Did I Get This? Activity: Khan Academy’s “Area of Triangles”\n\nLink: Khan Academy’s “Area of Triangles” (HTML)\n\nInstructions: This page provides a series of practice problems that you can answer and check online. Each question has a solution worked out step-by-step if you need hints along the way. Practice solving for the area of triangles until you feel confident that you understand how to find the area of a triangle (recommended a minimum of 10 minutes of practice).\n\nCompleting these practice problems should take you approximately 15 minutes.\n\nStandards Addressed (Common Core):\n\nA quadrilateral is a four-sided shape. You probably have some experience with quadrilaterals like rectangles and squares. This subunit will review the area of some quadrilaterals.\n\n• Explanation: Khan Academy’s “Area and Perimeter”\n\nInstructions: While watching this video, press pause when the instructor shows you a new shape. Draw the shape in your notebook and find the area and perimeter of the shape. Watch and listen as he explains the process. Don’t just skip ahead to the answer because the instructor gives you some important reminders.\n\nWatching this video and completing the sample problems should take you approximately 30 minutes.\n\nStandards Addressed (Common Core):\n\n• Did I Get This? Activity: Khan Academy’s “Area 1”\n\nInstructions: This page provides a series of practice problems that you can answer and check online. Each question has a solution worked out step-by-step if you need hints along the way. Practice solving for area until you feel confident that you understand how to find the area (recommended a minimum of 10 minutes of practice).\n\nWorking these practice problems should take you approximately 15 minutes.\n\nStandards Addressed (Common Core):\n\n• Did I Get This? Activity: Khan Academy’s “Area of Squares and Rectangles”\n\nLink: Khan Academy’s “Area of Squares and Rectangles” (HTML)\n\nInstructions: This page provides a series of practice problems that you can answer and check online. Each question has a solution worked out step-by-step if you need hints along the way. Practice solving for area until you feel confident that you understand how to find the area (recommended a minimum of 10 minutes of practice).\n\nWorking these practice problems should take you approximately 15 minutes.\n\nStandards Addressed (Common Core):\n\n• Explanation: Bettina Richmond and Tom Richmond’s “Area Formulas for Basic Shapes”\n\nLink: Bettina Richmond and Tom Richmond’s “Area Formulas for Basic Shapes” (HTML)\n\nInstructions: This page provides you with animated justification for how to find the area of shapes. Watch the animation for “Area of a Rectangle” and “Area of a Parallelogram.” In your notebook, take notes of the formula and the text given to you. Also, sketch the animation the best you can. Note: there is no sound.\n\nCompleting this task should take you approximately 15 minutes.\n\nStandards Addressed (Common Core):\n\n• Explanation: CK-12: “Area of a Parallelogram”\n\nLink: CK-12: “Area of a Parallelogram” (HTML)\n\nInstructions: The previous resource showed you an animation of how to find the area of a parallelogram. This resource will also explain through words and pictures how to find the area of a parallelogram and why it works. Read through the introduction; take notes as you read through the “Guidance” section. Do the example problems and write down the vocabulary words. Complete the “Guided Practice” and watch the video. You will do some practice problems below.\n\nTaking notes, working examples, and watching the video should take you approximately 30 minutes.\n\nStandards Addressed (Common Core):\n\n• Did I Get This? Activity: Activity: Khan Academy’s “Area of Parallelograms”\n\nLink: Khan Academy’s “Area of Parallelograms” (HTML)\n\nInstructions: This page provides a series of practice problems that you can answer and check online. Each question has a solution worked out step-by-step if you need hints along the way. Practice solving for area until you feel confident that you understand how to find the area (recommended a minimum of 10 minutes of practice).\n\nCompleting this activity should take you approximately 15 minutes.\n\nStandards Addressed (Common Core):\n\n• Did I Get This? Activity: YouTube: Mathispower4u: James Sousa’s “Example: Determine the Area of a Rectangle Using Mixed Numbers”\n\nLink: YouTube: Mathispower4u: James Sousa’s “Example: Determine the Area of a Rectangle Using Mixed Numbers” (YouTube)\n\nInstructions: As you watch the video, draw the rectangle and label the side lengths. Pause the video while you solve for the area of the rectangle. (This is a good review for dealing with fractions.) Don’t forget to simplify your answer back to a mixed number. Watch the remainder of the video. Compare your answer with the instructor’s.\n\nCompleting this problem should take you approximately 15 minutes.\n\nStandards Addressed (Common Core):\n\n4.2.3 Composite Figures\n\nPicture a square and a triangle pushed up together to share a side length. This is called a composite figure. A composite figure is a figure that is made of a variety of shapes. When you are asked to find the perimeter of a composite figure, keep in mind that perimeter is the distance around an object. It is sometimes helpful to take your pencil and trace the outside of the object; you will need to add each of those side lengths together to find the perimeter. When finding the area of a composite figure, think about how you can break the object into smaller, separate shapes. Once you’ve done that, you will find the area of the smaller shapes. Then, you will add the individual areas together. Sometimes, side lengths won’t be given, but clues within the problem or the shape can help you figure out the missing side length.\n\n• Explanation: CK-12: “Area of Composite Shapes”\n\nLink: CK-12: “Area of Composite Shapes” (HTML)\n\nInstructions: This lesson will provide you with a few videos to explain how to find the area of composite shapes. Begin watching the first video. Pause the video so you can take the time to draw the shape. Solve for the area of the composite shape with the instructor. Note: at the 1:40 mark in the video, the instructor figures that the height of the triangle is 3. Do not worry about the process used to find the height. You will learn about that later in your math career.\n\nScroll down to the second video and begin watching. Pause the video to draw the shape. Try to find the area of the composite figure before the instructor. Watch the instructor’s process. Complete the next composite shape with the same process.\n\nComplete practice problems 1 - 15 at the end of the lesson.\n\nWatching these videos and completing the practice problems should take you approximately 45 minutes.\n\nStandards Addressed (Common Core):\n\n• Explanation: CK-12: “Area of Composite Shapes Involving Triangles”\n\nLink: CK-12: “Area of Composite Shapes Involving Triangles” (HTML)\n\nInstructions: Read about home plate. Think of other everyday objects that are made of a variety of shapes - the “key” on a basketball court, a hopscotch pattern, a stained glass window, and the front image of a house. Read through the “Guidance” section, draw the composite figures, and label the side lengths. Work the example problems. For each one, draw the shape and calculate the area. Check your work with the answer provided. Continue working down the page, completing the “Guided Practice” and “Practice” problems.\n\nTaking notes and completing the practice problems should take you approximately 1 hour.\n\nStandards Addressed (Common Core):\n\n• Did I Get This? Activity: YouTube: Mathispower4u: James Sousa’s “Ex: Find the Area of an L-Shaped Polygon Involving Whole Numbers”\n\nLink: YouTube: Mathispower4u: James Sousa’s “Ex: Find the Area of an L-Shaped Polygon Involving Whole Numbers” (YouTube)\n\nInstructions: Look at the composite figure. Pause the video. Draw the shape and find the area - remember to split the composite figure into smaller rectangles. To find the length of the missing sides, use the other side lengths given to you. Do not estimate, measure, or guess side lengths! Watch the remainder of the video to compare your strategy to the instructor’s. Could you have split your composite figure into rectangles in a different way than the instructor did, while still finding the area to be the same?\n\nWatching this video and taking notes should take you approximately 15 minutes.\n\nStandards Addressed (Common Core):\n\n• Did I Get This? Activity: Khan Academy’s “Interesting Perimeter and Area Problems”\n\nInstructions: Watch the video and pause it at the 0:40 mark. Draw the star shape on your paper. Try to find the perimeter based on the information given to you. Watch the instructor as he explains how to find the star perimeter. Compare your strategy to his.\n\nPause the video at 2:40. Think about how you can find the area of this shape. Break it into two smaller shapes. Draw the composite figure and find the area. If you are struggling, watch another 1:00 to see a good first step in solving for this area. Once you have found the area, watch the instructor as he continues explaining how to find the area of the composite figure. Compare your strategy to his.\n\nPause the video at 5:20. Draw the shape. Label the given side lengths. Think about how you find the other side lengths (without guessing, estimating, or measuring). If you need a hint, watch the video for another 1:10 and see what the instructor’s strategy is with the side lengths. If you feel confident, pause the video again at 6:30 and find the perimeter. Watch the remainder of the video to see how the instructor finds the perimeter.\n\nWatching this video and working through the example problems should take you approximately 30 minutes.\n\nStandards Addressed (Common Core):\n\n• Did I Get This? Activity: YouTube: Mathispower4u: James Sousa’s “Ex: Determine the Area of an L-Shaped Polygon Using Decimals”\n\nLink: YouTube: Mathispower4u: James Sousa’s “Ex: Determine the Area of an L-Shaped Polygon Using Decimals” (YouTube)\n\nInstructions: Look at the composite figure. Pause the video. Draw the shape and find the area - remember to split the composite figure into smaller rectangles. To find the length of the missing sides, use the other side lengths given to you. Do not estimate, measure, or guess side lengths! Watch the remainder of the video to compare your strategy to the instructor’s. Could you have split your composite figure into rectangles in a different way than the instructor did, while still finding the area to be the same?\n\nCompleting this problem should take you approximately 15 minutes.\n\nStandards Addressed (Common Core):\n\n• Did I Get This? Activity: Illustrative Mathematics: “Finding Areas of Polygons, Variation 1”\n\nLink: Illustrative Mathematics: “Finding Areas of Polygons, Variation 1” (HTML)\n\nInstructions: This activity is an opportunity for you to explore different strategies for finding area. Keep in mind that when you have a composite figure, there is not a simple formula for finding area. You will need to problem solve and think outside the box as you work to find two strategies for finding the area of each polygon. Once you have answers, scroll down and read the “Commentary” and “Solution” sections on pages 2 and 3.\n\nCompleting this activity should take you approximately 30 minutes.\n\nStandards Addressed (Common Core):\n\n• Did I Get This? Activity: YouTube: Mathispower4u: James Sousa’s “Area Application - Area of an Inner Room with an Outer Footing”\n\nLink: YouTube: Mathispower4u: James Sousa’s “Area Application - Area of an Inner Room with an Outer Footing” (YouTube)\n\nInstructions: Read the problem with the instructor. Pause the video at the 0:25 mark. Draw the room and the outer footing. Solve for the inside dimensions and area of the inner room. Watch the remainder of the video to compare your strategy with the instructor’s.\n\nWatching this video and completing the activity should take you approximately 15 minutes.\n\nStandards Addressed (Common Core):\n\n4.3 Surface Area\n\nHave you ever wrapped a gift for someone? You have to make sure you cut off enough wrapping paper so that it covers the entire gift. The box that gift is in is a three-dimensional box; in math it would be called a rectangular prism (although you might just think of it as a shoebox). The wrapping paper covers the gift - each side, or face, of the box is a two-dimensional surface. To figure out how much wrapping paper is needed, mathematicians find the surface area. This subunit focuses on how to find surface area and introduces you important vocabulary you need to know when dealing with two- and three-dimensional objects. This subunit will give you a chance to add notes to your “Two-Dimensional” and “Three-Dimensional” columns in your notebook.\n\n4.3.1 Nets - Explanation: Explanation: CK-12: “Surface Area of Prisms”\n\n``````Link: CK-12: [“Surface Area of\nPrisms”](http://www.ck12.org/geometry/Surface-Area-of-Prisms/lesson/Surface-Area-of-Prisms/) (HTML)\n\nInstructions: The purpose of this lesson is to help you learn some\nvocabulary concerning two- and three-dimensional measurements. While\nyou read through the lesson, do not focus on the mathematical\ncomputations; you will work on the math a little later.\n\nRead the introduction. In the “Guidance” section, take notes on\nwords such as three-dimensional, prism, surface area, and\nnet. After the third “sticky note” with a net on it, skip down\nto examples A, B, and C. Go through each true/false question.\n\nSkip down to the “Vocabulary” section. Make sure you have each of\nthe vocabulary words clearly defined in your notebook.\n\nIf you can, cut out a net from a piece of your notebook paper (or\ngraph paper if you have it). Fold the net to form a\nthree-dimensional rectangular prism or cube. Tape your rectangular\nprism together and use it as an example as you continue through this\nunit.\n\nTaking notes and building a net should take you approximately 45\nminutes.\n\nStandards Addressed (Common Core):\n\n- [CCSS.Math.Content.6.G.A.4](http://www.corestandards.org/Math/Content/6/G/A/4)\n\nattributed to CK-12.\n``````\n\n4.3.2 Parts of a Three-Dimensional Object - Explanation: CK-12: “Faces, Edges, and Vertices of Solids”\n\n``````Link: CK-12: [“Faces, Edges, and Vertices of\nSolids”](http://www.ck12.org/geometry/Faces-Edges-and-Vertices-of-Solids/lesson/Faces-Edges-and-Vertices-of-Solids/) (HTML)\n\nInstructions: This lesson will give you some more vocabulary as you\nwork with three-dimensional objects. As you read the “Guidance”\nsection, take notes on the vocabulary words: faces, edges, and\nvertices.\n\nDraw a rectangular prism, or use an actual object that you can point\nto and label - a shoebox or even a computer are examples of a\nrectangular prism. Objects are often described by their number of\nfaces, edges, or vertices. The word face is used when finding\nsurface area (see subunit 4.3.3), so make sure you understand these\nvocabulary words.\n\nComplete the examples, read through and take notes on the\n“Vocabulary” section (you will not be expected to know about a\ncylinder, pyramid, cone, or sphere as a sixth-grade student), and\ncomplete the “Guided Practice” questions.\n\nWatch the video for another view of how to label faces, edges, and\nvertices. Complete problems 1 - 7 in the “Practice” section.\n\nTaking notes, completing the practice problems, and watching the\nvideo should take you approximately 30 minutes.\n\nStandards Addressed (Common Core):\n\n- [CCSS.Math.Content.6.G.A.4](http://www.corestandards.org/Math/Content/6/G/A/4)\n\nattributed to CK-12.\n``````\n\n4.3.3 Solving for Surface Area - Explanation: CK-12: “Surface Area of Rectangular Prism”\n\n``````Link: CK-12: [“Surface Area of Rectangular\nPrisms”](http://www.ck12.org/geometry/Surface-Area-of-Rectangular-Prisms/lesson/Surface-Area-of-Rectangular-Prisms/) (HTML)\n\nInstructions: Read through the introduction about wrapping the doll\nhouse. Take notes as you read through the “Guidance” section. Solve\nthe example problems given to you. Draw both a rectangular prism and\na net for each problem. This will help you recognize how a\ntwo-dimensional net is related to a three-dimensional rectangular\nprism. The formula for the surface area of a rectangular prism is\nlong and complex. However, write it down and think about how each\npart related to the actual shape.\n\nWrite down the vocabulary words, and then continue on to the “Guided\nPractice” section. Watch the video provided. Notice there is not a\nlid on the box. This takes away an entire face.\n\nAnswer questions 1 - 10 in the “Practice” section.\n\nTaking notes, watching the video, and completing the practice\nproblems should take you approximately 1 hour.\n\nStandards Addressed (Common Core):\n\n- [CCSS.Math.Content.6.G.A.4](http://www.corestandards.org/Math/Content/6/G/A/4)\n\nattributed to CK-12.\n``````\n\n4.4 Volume\n\nRemember the gift you were wrapping in the previous subunit? What’s inside? How do you know if you have a box that is big enough for the object you are wrapping? Whenever you are thinking about the space inside of something, you are dealing with volume. Volume is a three-dimensional measurement because it has a length, width and a height (recall the one-dimensional measurement has only a length; two-dimensional measurements have a length and a width). Volume adds a third dimension - height, or sometimes called depth. Have you ever been to a swimming pool? The amount of water in the pool is the volume. Have you ever crammed your suitcase full of clothes before you went on a trip? The amount of stuff inside your luggage is volume. This subunit focuses on how to find volume and will give you important vocabulary for dealing with volume. This subunit will give you a chance to add notes to your “Three-Dimensional” column in your notebook.\n\n4.4.1 Using Cubic Units - Explanation: CK-12: “Volume of Prisms Using Unit Cubes”\n\n``````Link: CK-12: [“Volume of Prisms Using Unit\nCubes”](http://www.ck12.org/geometry/Volume-of-Prisms-Using-Unit-Cubes/lesson/Volume-of-Prisms-Using-Unit-Cubes/) (HTML)\n\nInstruction: Read through and take notes on the “Guidance” section.\nNotice the label for volume is in cubic units or units3 because\nvolume involves counting the cubes to fill an object. Complete the\nproblems in the “Examples” section in your notebook, take notes on\nthe “Vocabulary” (as a sixth-grade student you do not need to know\ntriangular prisms), and complete the “Guided Practice” sections.\nStop there for now. You will do some practice with unit cubes in the\nnext resource.\n\nReading this lesson and taking notes should take you approximately\n30 minutes.\n\nStandards Addressed (Common Core):\n\n- [CCSS.Math.Content.6.G.A.2](http://www.corestandards.org/Math/Content/6/G/A/2)\n\nattributed to CK-12.\n``````\n• Did I Get This? Activity: Illustrative Mathematics: “Computing Volume Progression 1”\n\nLink: Illustrative Mathematics: “Computing Volume Progression 1” (HTML)\n\nInstructions: Think about filling a box in the shape of a cube (remember a cube has all equal sides). Answer questions a and b. Check your answers on the second page.\n\nCompleting this activity should take you approximately 15 minutes.\n\nStandards Addressed (Common Core):\n\n4.4.2 Understanding the Equation V = lwh and V = bh - Explanation: CK-12: “Volume of Rectangular Prism”\n\n``````Link: CK-12: [“Volume of Rectangular\nPrism”](http://www.ck12.org/geometry/Volume-of-Rectangular-Prisms/lesson/Volume-of-Rectangular-Prisms/) (HTML)\n\nInstructions: As you read through the introduction, notice that\nCandice and Trevor are “filling” their boxes with the packing\npeanuts. When you see the word “fill” you can often know that the\nproblem involves volume. Continue reading and take notes on the\n“Guidance” section. Draw some rectangular prisms (do your best with\nthree-dimensional drawing), label each side “length,” “width,” and\n“height.” As you read you will notice there are two formulas. Make\nsure you not only know how to use each formula, but that you also\nknow the difference between them and can explain how they work.\nContinue with the example problems, vocabulary, and guided practice.\nIn the “Practice” section, complete problems 1 - 10. Draw and label\na rectangular prism for at least half of the practice problems to\nget an image of how the prism looks.\n\nReading this lesson, taking notes, and completing the practice\nproblems should take you approximately 45 minutes.\n\nStandards Addressed (Common Core):\n\n- [CCSS.Math.Content.6.G.A.2](http://www.corestandards.org/Math/Content/6/G/A/2)\n\nattributed to CK-12.\n``````\n• Activity: Howard County Public School System’s “Student Homework”\n\nLink: Howard County Public School System’s “Student Homework” (HTML)\n\nInstructions: Scroll down to the third section, 6.G.2. Download the document “6.G.2 Student Homework.docx.” Answer the three questions in this activity. Notice that the cubes are inch on each side. The boxes have fractional side lengths as well. You will have to use both your volume skills and your fraction skills to solve these problems.\n\nCheck that your answers here (PDF) when you are finished.\n\nCompleting these practice problems should take you approximately 30 minutes.\n\nStandards Addressed (Common Core):\n\n• Did I Get This? Activity: Illustrative Mathematics: “Computing Volume Progression 2”\n\nLink: Illustrative Mathematics: “Computing Volume Progression 2” (HTML)\n\nInstructions: Answer questions a and b. It might be helpful to draw a model of what the water level will look like in question b. Scroll down and read the commentary and the solution on the second page.\n\nCompleting this activity and checking your work should take you approximately 30 minutes.\n\nStandards Addressed (Common Core):\n\n• Did I Get This? Activity: Illustrative Mathematics: “Computing Volume Progression 3\"\n\nLink: Illustrative Mathematics: “Computing Volume Progression 3” (HTML)\n\nInstructions: As you read this problem, keep in mind that you will need to do some converting of units before you work to find the height of the tank. A good strategy to consider is that although the problem is talking about water, volume is measured in cubed units. Think about the amount of space the first layer (height of 1 cm) takes up. Consider that you will need to see how many 50 cm by 60 cm layers should be stacked to fill the tank full. Scroll down and read the commentary and the solution on the second page.\n\nCompleting this activity and checking your work should take you approximately 30 minutes.\n\nStandards Addressed (Common Core):\n\n4.5 Real-Life Situations and Word Problems Involving Geometric Shapes\n\nPainting your bedroom walls, filling a swimming pool with water, planting grass seed in the yard, and wrapping a gift are just a few examples of real-life situations when you will use geometry. Understanding how to find perimeter, area, surface area, and volume are all very important. However, the initial task is to know what the question is asking and to understand what type of formula will be used. As you go through this subunit, stop and think about each situation. Consider key words that might trigger ideas to know if you are dealing with one-, two-, or three-dimensional measurements. This subunit will give you a chance to use the notes from your notebook. You can also add additional real-life examples to each of the columns in your notebook.\n\nClick here to see an example of how the “Understanding One-, Two-, and Three-Dimensions” might look at this point in the unit.\n\n• Did I Get This? Activity: Illustrative Mathematics: “Banana Bread”\n\nInstructions: Read the problem on the first page about making banana bread. Solve the problem and describe why you think your answer makes sense. Scroll down to second page and read the “Commentary” and “Solution” sections. Compare your strategy with the one provided.\n\nCompleting this activity and checking your work should take you approximately 15 minutes.\n\nStandards Addressed (Common Core):\n\n• Did I Get This? Activity: Illustrative Mathematics: “Painting a Barn”\n\nLink: Illustrative Mathematics: “Painting a Barn” (HTML)\n\nInstructions: Read and solve the problem on the first page. Explain your work. Refer to the notes you made throughout this unit to help you consider what math operation to use when painting walls. Scroll down to the second page and read the “Commentary” and “Solution” sections. Compare your strategy with the one provided.\n\nCompleting this activity and checking your work should take you approximately 30 minutes.\n\nStandards Addressed (Common Core):\n\n• Checkpoint: Illustrative Mathematics: “Computing Volume Progression 4”\n\nLink: Illustrative Mathematics: “Computing Volume Progression 4” (HTML)\n\nInstructions: Read the problem on the first page about taking the stone out of the rectangular tank. Solve the problem and describe why you think your answer makes sense. Scroll down to the second page and read the “Commentary” and “Solution” sections. Compare your strategy with the one provided.\n\nCompleting this task and checking your work should take you approximately 15 minutes.\n\nStandards Addressed (Common Core):\n\n• Checkpoint: Illustrative Mathematics: “Christo’s Building”\n\nLink: Illustrative Mathematics: “Christo’s Building” (HTML)\n\nInstructions: Read through the problem and answer all questions. Take your time thinking about what each question is asking and how to solve it. Draw models and show your work. Scroll down and read through the “Commentary” and “Solution” sections on pages 2 and 3.\n\nCompleting this activity and checking your work should take you approximately 30 minutes.\n\nStandards Addressed (Common Core):\n\n`````` Instructions: Test your knowledge by completing this checkpoint." ]	[ null, "https://legacy.saylor.org/images/loader.gif", null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.8916181,"math_prob":0.66907126,"size":37095,"snap":"2019-13-2019-22","text_gpt3_token_len":8295,"char_repetition_ratio":0.13434526,"word_repetition_ratio":0.33555868,"special_character_ratio":0.21159185,"punctuation_ratio":0.13043478,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9594211,"pos_list":[0,1,2],"im_url_duplicate_count":[null,null,null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2019-03-22T00:36:17Z\",\"WARC-Record-ID\":\"<urn:uuid:18010366-90e5-4b1b-a5b6-3a7707121f42>\",\"Content-Length\":\"61585\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:764ee105-2ccc-4e52-bfa3-76cbb022d809>\",\"WARC-Concurrent-To\":\"<urn:uuid:e40c82d4-4bad-43d1-8019-c27269f08be0>\",\"WARC-IP-Address\":\"34.235.49.221\",\"WARC-Target-URI\":\"https://legacy.saylor.org/k12math006/Unit04/\",\"WARC-Payload-Digest\":\"sha1:2RWP5DWTJULBAV2BPCJMDUFYXJCHEAEF\",\"WARC-Block-Digest\":\"sha1:WTTAMCHCBLRYHSJHR5XB6HS2AINZQYXC\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2019/CC-MAIN-2019-13/CC-MAIN-2019-13_segments_1552912202588.97_warc_CC-MAIN-20190321234128-20190322020128-00202.warc.gz\"}"}
https://www.clutchprep.com/physics/practice-problems/48872/an-equilateral-triangular-loop-moves-into-a-0-50-t-magnetic-field-with-a-constan	[ "Problem: An equilateral triangular loop moves into a 0.50 T magnetic field with a constant speed of v = 5 m/s as shown. The loop has sides of length L = 0.50 m, a total resistance of 0.10 Ω, and it enters the field at t = 0 s. What is the direction of the induced current in the loop? Provide a complete justification for your answer.\n\nFREE Expert Solution\n90% (422 ratings)\nProblem Details\n\nAn equilateral triangular loop moves into a 0.50 T magnetic field with a constant speed of v = 5 m/s as shown. The loop has sides of length L = 0.50 m, a total resistance of 0.10 Ω, and it enters the field at t = 0 s. What is the direction of the induced current in the loop? Provide a complete justification for your answer.", null, "" ]	[ null, "https://cdn.clutchprep.com/problem_images/48872-777b455a34227f9f.png", null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.92461026,"math_prob":0.9939074,"size":1192,"snap":"2022-05-2022-21","text_gpt3_token_len":284,"char_repetition_ratio":0.10016835,"word_repetition_ratio":0.56621003,"special_character_ratio":0.2374161,"punctuation_ratio":0.104,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.97062165,"pos_list":[0,1,2],"im_url_duplicate_count":[null,3,null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2022-01-19T22:50:36Z\",\"WARC-Record-ID\":\"<urn:uuid:fab96d3e-04d2-42a7-9f65-802e1596f80a>\",\"Content-Length\":\"82110\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:ea5c53ef-7b07-436f-acf3-abfa04611059>\",\"WARC-Concurrent-To\":\"<urn:uuid:52b1d3f2-8367-4dd0-923c-599c2e92569a>\",\"WARC-IP-Address\":\"54.91.59.199\",\"WARC-Target-URI\":\"https://www.clutchprep.com/physics/practice-problems/48872/an-equilateral-triangular-loop-moves-into-a-0-50-t-magnetic-field-with-a-constan\",\"WARC-Payload-Digest\":\"sha1:ORFP5ERSCMYDJEAAUQN6HJCIDRROROP2\",\"WARC-Block-Digest\":\"sha1:SLODEWMSZWYCS23DCURPDXRAKQDAWQMO\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2022/CC-MAIN-2022-05/CC-MAIN-2022-05_segments_1642320301592.29_warc_CC-MAIN-20220119215632-20220120005632-00060.warc.gz\"}"}
https://www.biostars.org/p/127978/	[ "Question: How to make heatmap more bright with proper color spectrum ?\n0\njack800 wrote:\n\nHi,\n\nI need to create heatmap with bright colors which different values have different colors. all the enteris of my matrix is between -0.1- 0.9.\n\nI have tried the following command, but the heatmap is not interesting. How can I make it nice with more colors such that the different values have differnetiable color ?\n\nHere is my command :\n\n```heatmap.2(Gene, col=greenred(10), key=TRUE, symkey=FALSE, symm=F,symbreaks=T,cexRow=1,cexCol=1,margins=c(6,11), scale= \"column\" ,trace=\"none\",srtCol=45)\ndev.off()```\n\nHere is the image :", null, "modified 5.1 years ago by Manvendra Singh2.1k • written 5.1 years ago by jack800\n1\n\nHave you tried rainbow() instead of greenred()?\n\n1\n\nWhat about scale = \"row\"? Assuming rows represent different genes the heatmap just highlights most abundant genes when scale = \"column\".\n\nBTW, linking to an image in your gmail inbox won't work. You need to post it somewhere and then link to it there. Well, unless you want to give us your email login details, but that's a bad idea...\n\ni thought that I have copeid inot the post. well, where can I upload image ?\n\n1\n\nImgur, tinypic, postimage, etc. Just google \"free image hosting\" for more sites than you could ever need.\n\nI updated the post.\n\n2\nDevon Ryan94k wrote:\n\nYour problem is largely due to using symmetric colors. The other thing to consider is using manual color breaks. See the answer from \"Lippy\" here for an example of how to change these.\n\nthe enteris of the matrix is the correlation value and I don't scale it. I did with colum(which was wrong and I tried without scaling , the plot more or less was the same)/\n\n2\nManvendra Singh2.1k wrote:\n\nthe problem is with your range of values thatswhy on the either ways to zero its 1;9 and then you scale coloumns, try with scaling rows and add breaks in your heatmap command as;\n\n`breaks=c(-0.1,0,0.2,0.3,0.4,0.5,0.6,0.7,0,8,0.9)`\n\nor better to plot a heatmap of z-score and use colors from RColorBrewer\n\ne.g. I would do in this way\n\n```library(gplots)\nlibrary(RColorBrewer)\nhr <- hclust(as.dist(1-cor(t(Gene), method=\"pearson\")), method=\"average\");\nhc <- hclust(as.dist(1-cor(Gene, method=\"spearman\")), method=\"average\")\n# Cuts the tree and creates color vector for clusters.\nmycl <- cutree(hr,k=4, h=max(hr\\$height)/1.5);\nmycolhc <- rainbow(length(unique(mycl)), start=0.1, end=0.9)\nmycolhc <- mycolhc[as.vector(mycl)] ; myheatcol <- bluered(75)\nheatmap.2(as.matrix(Gene), Colv=as.dendrogram(hc),\ncol=myheatcol, scale=\"row\", density.info=\"none\",\ntrace=\"none\", RowSideColors=mycolhc,cexRow=1.5,\ncexCol=1.5, keysize=1,margins=c(20,10))```\n\nhth" ]	[ null, "http://i.imgur.com/GSwFOvY.png", null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.87112457,"math_prob":0.90362054,"size":1105,"snap":"2020-10-2020-16","text_gpt3_token_len":284,"char_repetition_ratio":0.1071753,"word_repetition_ratio":0.0,"special_character_ratio":0.2479638,"punctuation_ratio":0.18410042,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.98073125,"pos_list":[0,1,2],"im_url_duplicate_count":[null,4,null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2020-02-21T03:54:22Z\",\"WARC-Record-ID\":\"<urn:uuid:1b024c92-a729-487c-a40a-8f754e7e4e79>\",\"Content-Length\":\"41374\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:cb4489ff-a7c9-4b15-b15e-9c1cae4c2d7c>\",\"WARC-Concurrent-To\":\"<urn:uuid:55ecb954-74ce-4c03-8cf0-a9aeeef0df80>\",\"WARC-IP-Address\":\"69.164.220.180\",\"WARC-Target-URI\":\"https://www.biostars.org/p/127978/\",\"WARC-Payload-Digest\":\"sha1:VAPQ5X3Y34IE73RW2U6NACUNCHHKUAPJ\",\"WARC-Block-Digest\":\"sha1:D3CNZBPBPG5WSGYWK6ACQNZS6RQBVA6P\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2020/CC-MAIN-2020-10/CC-MAIN-2020-10_segments_1581875145438.12_warc_CC-MAIN-20200221014826-20200221044826-00451.warc.gz\"}"}
https://tex.stackexchange.com/questions/401886/how-to-create-simple-animations-with-animate?noredirect=1	[ "# How to create simple animations with animate?\n\nI couldn't find code examples that were simple enough to help me start using animate package. I would like to ask you to provide some of your own examples that would be simple enough to build upon.\n\nIt would help me the most if you follow these guidelines:\n\n• two animation examples are needed\n• both examples should use standalone class and as less functionality not related to animate and tikz packages as possible\n• animation in each example should have no more than 4 frames (for simplicity's sake)\n• one animation should advance to next frame only when mouse button is clicked\n• another animation should advance through frames automatically but each frame should pause (or appear) for different number of seconds than other frames\n\nIt would also help if you breakdown the structure of a text file used by animate to store frame data and what options are possible on each frame and how to use them (package's documentation seems to lack information on this).\n\nUpdate 3: Split the answer into two parts to shorten it as requested by @ArtificialStupidity here.\n\nUpdate 2: Added images produced with the new export option added by @AlexG in his answer here\n\n# First answer: use of the \\animategraphics command\n\nEntirely realized with the animate and tikz packages.\n\nThe animate package has a large number of options that give it great power. For example you can run the animation in one direction then in the other (palindrome), step by step (step), with control buttons (controls), looping infinitely (loop) and many others that you can discover on its manual.", null, "• It shows that the area of a parallelogram is equal to that of a rectangle by cutting and resizing.\n• It has 198 images, more than the 4 maximum requested. This provides a convenient method for generating the timeline file.\n• Its speed varies: it increases then decreases, its last image remains visible longer.\n• It explains the \\animategraphics command and the very powerful timeline option.\n\n### To start\n\nI wrote a file that builds the animation with tikz in standalone class. This tex file creates the 198 images of the animation called parallelogramme.pdf.\n\n\\documentclass[tikz]{standalone}\n\\usepackage{animate}\n\\usepackage{fontawesome}\n\\begin{document}\n\n\\foreach \\y in {0,.2,...,3}{% cut out the parallelogram\n\\begin{tikzpicture}\n\\useasboundingbox (-2.5,-.5) rectangle (4,4);\n\\fill[green!40](-1,0)--(-1,3)--(3,3)--(2,0)--cycle;\n\\fill[green!40](-2,0)--(-1,3)--(-1,0)--cycle;\n\\draw[dashed](-1,0)--(-1,\\y);\n\\end{tikzpicture}\n}\n\n\\begin{tikzpicture}% circle the cut out in dotted lines\n\\useasboundingbox (-2.5,-.5) rectangle (4,4);\n\\fill[green!40](-1,0)--(-1,3)--(3,3)--(2,0)--cycle;\n\\fill[green!40](-2,0)--(-1,3)--(-1,0)--cycle;\n\\draw[densely dotted](-1,0)--(-1,3)--(-2,0)--cycle;\n\\end{tikzpicture}\n\n\\foreach \\iangle in {180,179,...,0}{% move the triangle\n\\begin{tikzpicture}\n\\useasboundingbox (-2.5,-.5) rectangle (4,4);\n\\fill[green!40](-1,0)--(-1,3)--(3,3)--(2,0)--cycle;\n\n\\fill[green,opacity=.4] (\\iangle:20mm and 8mm)--([shift={(1,3)}]\\iangle:20mm and 8mm)--([shift={(1,0)}]\\iangle:20mm and 8mm)--cycle;\n\\node at ([shift={(.6,.2)}]\\iangle:20mm and 8mm)[black]{\\faHandPointerO};\n\\draw[densely dotted,thin](-1,3)--(-2,0)--(-1,0)--cycle;\n\\ifthenelse {\\iangle=0}{\\draw[densely dotted,fill=green!40](2,0)--(3,0)--(3,3)--cycle;\n\\node at ([shift={(.6,.2)}]0:20mm and 8mm)[black]{\\faHandPointerO};}{}\n\\end{tikzpicture}\n}\n\n\\end{document}\n\n\nYou will notice that the standalone documentclass is written with the dedicated tikz option that creates a single pdf page for each tikz graphic: documentclass[tikz]{standalone} and not\n\n\\documentclass{standalone}\n\\usepackage{tikz}\n\n\nbecause this last way of doing it creates a single standalone file with all tikz graphics, it looks like this:", null, "or gives an error if there are too many tikz graphics:\n\nDimension too large.\n\n### Create an image in gif format?\n\nIf you want to create an.gif image, you can use the Imagemagick software by opening a command line in the folder containing the series of pdf images you just created.\n\nTo get the .gif image set at the beginning, I copied and pasted this command: see @nox explanation here: https://tex.stackexchange.com/a/443304/138900\n\nTo vary the scrolling speeds, I chose :\n\n convert -density 100 -loop 0 -background white -alpha remove -delay 100 parallelogramme.pdf -delay 10 parallelogramme.pdf[1-16] -delay 8 para-un-pdf.pdf[17-40] -delay 4 parallelogramme.pdf[41-196] -delay 300 parallelogramme.pdf parallelogramme.gif\n\n\n### Make the pdf animation:\n\nTo make a pdf animation with the animate package, we could use the command \\animategraphics and without any other option, we lose the speed variation (here 30 frames per second):\n\n[![animation-without-speed-variation]]\n\n\\documentclass[tikz]{standalone}\n\\usepackage{animate}\n\n\\begin{document}\n\\animategraphics{30}{para-un-pdf}{}{}\n\\end{document}\n\n\n## Change the speed with the timeline option:\n\nTo vary the speeds, either you increase the number of frames, but this makes the file and compilation heavier, or you use the timeline option.\n\n## The power of the timeline file:\n\nThe timeline file describes and composes each image of the animation, then:\n\n• Each line compose a single image. So there are as many lines as there are images.\n• Each pdf pages are considered transparencies. An image (a frame) is now a stack of different pages (transparencies) of the pdf.\n• It allows the same transparency to be reused several times at different points in the animation.\n• It can also speeds up, slows down or stops animation.\n• The first page of the pdf is transparency number 0.\n• the second is numbered 1, etc.\n\n### Each line of the timeline file is composed as follows:\n\n[]:[<frame rate>]:[<transparencies>][:<JavaScript>]\n\n• The first element [] is either * or empty. If there is * then the animation stops at that image.\n• The second element [<frame rate>] is either empty or indicates the number of frames per second.\n• The third element [<transparencies>] indicates the stacking of transparencies.\n• I refer you to the package manual for the [:<JavaScript>] option and much more explanations.\n\nFor example, if the first 5 lines of the timeline file are:\n\n::0x0,1x18\n:10:3\n::4\n::5\n::6\n\n• ::0x0,1x18 : then the transparency 0 is copied on all the following images; above it is the transparent 1 repeated 18 times (line 0 to 17);\n• :10:3 : the second image is composed of the transparent 3 (added to the existing stack of transparencies) with a speed of 10 frame per second;\n• ::4 the fourth transparency is added to the stack that will make the third image and the animation stops on this image.\n• etc.\n\nIf you want to modify the background image during animation, you will have to replace commas (,) by semicolons (;)that will create overlay layers. See manual for more details.\n\n### Generate the timeline file:\n\nTo avoid having to write manually a 198 lines timeline file, we use the LaTeX (or TeX ?) \\write command. This tex file creates the timeline file called agencement.txt:\n\n\\documentclass{article}\n\\usepackage[T1]{fontenc}\n\\usepackage{multido}\n\\usepackage{ifthen}\n\n\\newwrite\\Fichier\n\\immediate\\openout\\Fichier=agencement.txt\n\\immediate\\write\\Fichier{:2:0}\n\\immediate\\write\\Fichier{:10:1}\n\\multido{\\ix=2+1}{14}%\n{%\n\\immediate\\write\\Fichier{::\\ix}%\n}\n\\immediate\\write\\Fichier{:1:16}\n\\immediate\\write\\Fichier{:12.5:17}\n\\multido{\\ix=18+1}{23}{%\n\\immediate\\write\\Fichier{::\\ix}\n}\n\\immediate\\write\\Fichier{:25:41}\n\\multido{\\ix=42+1}{155}{%\n\\immediate\\write\\Fichier{::\\ix}\n}\n\\immediate\\write\\Fichier{:.3:197}\n\\immediate\\closeout\\Fichier% Don't forget to close the file\n\n\\begin{document}\nTimeline file created\n\\end{document}\n\n\nThis tex file generates the animation on pdf called parallelogramme-animated.pdf. It use the previously created parallelogramme.pdf and the timeline file agencement.txt:\n\n\\documentclass[tikz]{standalone}\n\\usepackage{animate}\n\n\\begin{document}\n\n\\animategraphics[loop,timeline=agencement.txt]{30}{parallelogramme}{}{}\n\n\\end{document}\n\n\n### Write a stack of transparencies?\n\nThis lightens considerably the compilation and the pdf animation. The very large number of actions allowed by this timeline file complicates its writing.\n\nI do it these time. I rewrote the parallelogramme.tex file to make it generate not images, but differents transparencies called parallelogramme-bis.tex\n\n\\documentclass[tikz]{standalone}\n\\usepackage{animate}\n\\usepackage{fontawesome}\n\\begin{document}\n\n\\begin{tikzpicture}% remaining cut - transparent 0\n\\useasboundingbox (-2.5,-.5) rectangle (4,4);\n\\fill[green!40](-1,0)--(-1,3)--(3,3)--(2,0)--cycle;\n\\end{tikzpicture}\n\n\\begin{tikzpicture}% triangle cut out green- transparent 1\n\\useasboundingbox (-2.5,-.5) rectangle (4,4);\n\\fill[green!40](-2,0)--(-1,3)--(-1,0)--cycle;\n\\end{tikzpicture}\n\n\\begin{tikzpicture}% circle the cutout in dotted lines - transparent 2\n\\useasboundingbox (-2.5,-.5) rectangle (4,4);\n\\draw[densely dotted](-1,0)--(-1,3)--(-2,0)--cycle;\n\\end{tikzpicture}\n\n\\foreach \\y in {0,.2,...,3}{% cut out the parallelogram - transparent 3-18\n\\begin{tikzpicture}\n\\useasboundingbox (-2.5,-.5) rectangle (4,4);\n\\draw[dashed](-1,0)--(-1,\\y);\n\\end{tikzpicture}\n}\n\\foreach \\iangle in {179,...,0}{% move parallelogram - transparent 19-199\n\\begin{tikzpicture}\n\\useasboundingbox (-2.5,-.5) rectangle (4,4);\n\\fill[green,opacity=.4] (\\iangle:20mm and 8mm)--([shift={(1,3)}]\\iangle:20mm and 8mm)--([shift={(1,0)}]\\iangle:20mm and 8mm)--cycle;\n\\node at ([shift={(.6,.2)}]\\iangle:20mm and 8mm)[black]{\\faHandPointerO};\n\\ifthenelse {\\iangle=0}{\\draw[densely dotted,fill=green!40](2,0)--(3,0)--(3,3)--cycle;\n\\node at ([shift={(.6,.2)}]0:20mm and 8mm)[black]{\\faHandPointerO};}{}\n\\end{tikzpicture}\n}\n\\end{document}\n\n\nWe can create the timeline file called agencement-bis.txtwith LaTeX:\n\n\\documentclass{article}\n\\usepackage[T1]{fontenc}\n\\usepackage{multido}\n\\usepackage{ifthen}\n\n\\newwrite\\Fichier\n\\immediate\\openout\\Fichier=agencement-bis.txt\n\n\\immediate\\write\\Fichier{::0x0,1x18}\n\\immediate\\write\\Fichier{:10:3}\n\\multido{\\ix=4+1}{14}%\n{%\n\\immediate\\write\\Fichier{::\\ix}%\n}\n\\immediate\\write\\Fichier{:12.5:2x0}\n\\multido{\\ix=18+1}{180}%\n{%\n\\ifthenelse {\\ix=17}{\\immediate\\write\\Fichier{:12.5:\\ix}}{\n\\ifthenelse {\\ix=40}{\\immediate\\write\\Fichier{:25:\\ix}}{\n\\immediate\\write\\Fichier{::\\ix}}\n}\n}\n\\immediate\\write\\Fichier{:.3:198}\n\\immediate\\closeout\\Fichier% always close the file\n\\begin{document}\n\ntimeline file created.\n\n\\end{document}\n\n\nThe final animation is created with the \\animategraphics from the previously created parallelogramme-bis.pdf and the timeline agencement-bis.txt\n\n\\documentclass[tikz]{standalone}\n\\usepackage{animate}\n\\begin{document}\n\\animategraphics[loop,timeline=agencement-bis.txt]{30}{parallelogramme-bis}{}{}\n\\end{document}", null, "Translated with www.DeepL.com/Translator\n\n• Thank you for this nice tutorial, in particular for the timeline part! – AlexG Aug 10 '18 at 11:26\n• @AlexG I love all your interactive packages (animate, ocgx, media9) with which I make beautiful beamer presentations that my students admire, thanks to you! – AndréC Aug 10 '18 at 11:44\n• Thank you! (It's ocgx2; ocgx is by P. Gaborit.) – AlexG Aug 10 '18 at 11:53\n\nThe following two MWEs should give you a general idea on how you could use the animate package. In order to successfully compile these examples, you will need four images called example_1 to example_4 in the same directory as your .tex file.\n\n1: Animation proceeds to the next transparency only when clicking on the mouse button: (Please note the step option)\n\n\\documentclass{standalone}\n\\usepackage{graphicx}\n\\usepackage[step]{animate}\n\n\\begin{document}\n\\animategraphics[width=\\linewidth]{12}{example_}{1}{4}%\n\\end{document}\n\n\n2: Animation automatically proceeds to the next transparency with different framerate for each step: (Note: For a more in-depth explanation of a timeline please refer to the animate manual)\n\n\\documentclass{standalone}\n\\usepackage{graphicx}\n\\usepackage{animate}\n\\usepackage{filecontents}\n\n\\begin{filecontents}{mytimeline.txt}\n:0.5:0 % 1/0.51s=2s\n:0.2:1\n:10:2\n:1:3\n\\end{filecontents}\n\n\\begin{document}\n\\animategraphics[timeline=mytimeline.txt,width=\\linewidth]{12}{example_}{1}{4}%\n\\end{document}\n\n• For the last frame to be shown for 1 s (1 frame/s), one could add the loop option. – AlexG Nov 18 '17 at 16:30\n\nUpdate: Split the previous answer into two parts to shorten it as requested by @ArtificialStupidity here.\n\n# Second answer: use of the animateinline environment", null, "• It illustrates the different stages of construction with the ruler and the compass of an ove;\n• it illustrates the notion of stacking layers of transparencies that allow the screen background to be modified without hiding the previously stacked transparencies.\n• it explains the use of the animateinline environment.\n• although the gif animation above does not show a pause, the pdf animation below generates one for each image.\n\n### Create images or transparencies?\n\nDuring a ruler and compass construction, the figure is constructed successively with small drawings that are added to each other. So creating transparencies that stack one on top of the other naturally reproduces this way of building.\n\nThe animation has 7 images made with the following 9 transparencies:", null, "", null, "The first 7 transparencies represent the stages of construction, the last 2 are the backgrounds used to illustrate the underlay of transparencies.\n\nOne background is green, the other is the pattern pattern=dots. When the latter pattern is stacked, the backgrounds below it remain visible. The green background hides all the backgrounds below him.\n\n## Create an image in gif format?\n\nIt is now possible thanks to the magnificent update published on August 22, 2018 and this without needing as before to rewrite all the code. To do this, simply add the export option either to the standalone package or to the animate package:\n\n\\documentclass{standalone} \\usepackage[export]{animate}\n\nor \\documentclass[export]{standalone} \\usepackage{animate}\n\nThis produces a pdf file consisting of a series of individual pages that can be easily converted to gif with, for example, Imagemagick as shown above.\n\nThe images produced with the export option are the following 7:", null, "", null, "### The animateinline environment\n\n• it allows to group in a single file the creation of each of the transparents as well as their animation and as long as to make the generation of the timeline file.\n• it has two commands \\newframe and \\multiframe which allow to create, either images independent of each other, or transparencies whose stacking will form images.\n• it allows to factor the start and end code of each tikzpicture environment which is repeated with each creation of frames thanks to the begin and end options.\n\n## The power of the timeline file:\n\nThe timeline file describes and composes each image of the animation, then:\n\n• Each line compose a single image. So there are as many lines as there are images.\n• Each pdf pages are considered transparencies. An image (a frame) is now a stack of different pages (transparencies) of the pdf.\n• It allows the same transparency to be reused several times at different points in the animation.\n• It can also speeds up, slows down or stops animation.\n• The first page of the pdf is transparency number 0.\n• the second is numbered 1, etc.\n\n### Each line of the timeline file is composed as follows:\n\n[]:[<frame rate>]:[<transparencies>][:<JavaScript>]\n\n• The first element [] is either or empty. If there is * then the animation stops at that image.\n• The second element [<frame rate>] is either empty or indicates the number of frames per second.\n• The third element [<transparencies>] indicates the stacking of transparencies.\n• I refer you to the package manual for the [:<JavaScript>] option and much more explanations.\n\nFor example, if the first 5 lines of the timeline file are:\n\n::0x0,1x18\n:10:3\n::4\n::5\n::6\n\n• ::0x0,1x18 : then the transparency 0 is copied on all the following images; above it is the transparent 1 repeated 18 times (line 0 to 17);\n• :10:3 : the second image is composed of the transparent 3 (added to the existing stack of transparencies) with a speed of 10 frame per second;\n• ::4 the fourth transparency is added to the stack that will make the third image and the animation stops on this image.\n• etc.\n\nIf you want to modify the background image during animation, you will have to replace commas (,) by semicolons (;)that will create overlay layers. See manual for more details.\n\n### The begin and end options:\n\nFor example, below the opening is always composed of the code\n\nbegin{tikzpicture}\n\\useasboundingbox (-2.5,-2.5) rectangle (4,2.5);\n\n\nwe created a \\Debut command that will write these lines automatically to each new frame created with a \\newframe or multiframe command:\n\nNewcommand{\\Debut}{% Systematic start of drawing\n\\begin{tikzpicture}\n\\useasboundingbox (-2.5,-2.5) rectangle (4,2.5);}\n\n\nSimilarly, for closing the tikz environment, with the following command:\n\nNewcommand{\\Fin}{\\end{tikzpicture} }\n\n\n### Animation code\n\n \\documentclass{standalone}\n\\usepackage{tikz}\n\\usetikzlibrary{patterns}\n\\usepackage{animate}\n\n% creation of the ove.txt timeline file\n\\newwrite\\Fichier\n\\immediate\\openout\\Fichier=ove.txt\n\\immediate\\write\\Fichier{:.5:7x0;0x0}% the dots background is stacked first in all images\n\\immediate\\write\\Fichier{::8;1x0}% the green background is opaque and hides the dot background\n\\immediate\\write\\Fichier{::;2x0}% nothing covers the background of the transparency 0 which is therefore visible again\n\\immediate\\write\\Fichier{::c;4x0}% the letter c deletes all transparencies that have been added in the stack\n\\immediate\\write\\Fichier{::8x2;3x0}% the green background will be visible twice, so until the next image\n\\immediate\\write\\Fichier{::7x1;5x0}% the dots background is visible only once, so writing x1 is useless\n\\immediate\\write\\Fichier{::;6} %the stack is empty and no background is added, so there is no visible background\n\\immediate\\closeout\\Fichier% always close the file\n\n\\newcommand{\\Debut}{% Systematic start of drawing\n\\begin{tikzpicture}\n\\useasboundingbox (-2.5,-2.5) rectangle (4,2.5);}\n\\newcommand{\\Fin}{\\end{tikzpicture} }% Systematic end of drawing\n\\begin{document}\n% Step through the animation one frame at a time per mouse-click. The <frame rate> argument will be ignored.\n\\begin{animateinline}[autoplay,step,begin={\\Debut},end={\\Fin},timeline=ove.txt]{.5}\n% perpendicular straight - transparent 0\n\\draw[thick] (-2.5,0)--(4,0);\n\\draw[thick] (0,-2.5)--(0,2.5);\n\\newframe% circle - transparent 1\n\\draw[thick] (0,0) circle (2cm);\n\\newframe% half-line 1 - transparent 2\n\\draw[thick] (0,-2)--(3,1);\n\\newframe% half-line 2 - transparent 3\n\\draw[thick] (0,2)--(3,-1);\n\\newframe% arc 1 - transparent 4\n\\newframe% arc 2 - transparent 5\n\\newframe% arc 3 - transparent 6\n\\newframe% dots screen background - transparent 7\n\\fill[pattern=dots] (-2.5,-2.5) rectangle (4,2.5);\n\\newframe% green screen background - transparent 8\n\\fill[green!30] (-2.5,-2.5) rectangle (4,2.5);\n\\end{animateinline}\n\n\\end{document}\n\n\n### Write a sequence of images?\n\nThe file that creates the same sequence of images called ove.pdf is:\n\n\\documentclass[tikz]{standalone}\n\\usepackage{animate}\n\\usetikzlibrary{patterns}\n\\tikzset{every path/.style=thick}\n\\begin{document}\n\n\\begin{tikzpicture}% perpendiculars - 1\n\\fill[pattern=dots] (-2.5,-2.5) rectangle (4,2.5);\n\\draw (-2.5,0)--(4,0);\n\\draw (0,-2.5)--(0,2.5);\n\\end{tikzpicture}\n\n\\begin{tikzpicture}% circle - image 2\n\\fill[green!30] (-2.5,-2.5) rectangle (4,2.5);\n\\draw (-2.5,0)--(4,0);\n\\draw (0,-2.5)--(0,2.5);\n\\draw (0,0) circle (2cm);\n\\end{tikzpicture}\n\n\\begin{tikzpicture}% half-line 1 - image 3\n\\fill[pattern=dots](-2.5,-2.5) rectangle (4,2.5);\n\\draw (-2.5,0)--(4,0);\n\\draw (0,-2.5)--(0,2.5);\n\\draw (0,0) circle (2cm);\n\\draw (0,-2)--(3,1);\n\\end{tikzpicture}\n\n\\begin{tikzpicture}% arc 1 - image 4\n\\useasboundingbox (-2.5,-2.5) rectangle (4,2.5);\n\\draw (-2.5,0)--(4,0);\n\\draw (0,-2.5)--(0,2.5);\n\\draw (0,0) circle (2cm);\n\\draw (0,-2)--(3,1);\n\\end{tikzpicture}\n\n\\begin{tikzpicture}% half line 2 - image 5\n\\fill[green!30] (-2.5,-2.5) rectangle (4,2.5);\n\\fill[pattern=dots](-2.5,-2.5) rectangle (4,2.5);\n\\draw (-2.5,0)--(4,0);\n\\draw (0,-2.5)--(0,2.5);\n\\draw (0,0) circle (2cm);\n\\draw (0,-2)--(3,1);\n\\draw (0,2)--(3,-1);\n\\end{tikzpicture}\n\n\\begin{tikzpicture}% arc 2 - image 6\n\\useasboundingbox (-2.5,-2.5) rectangle (4,2.5);\n\\draw (-2.5,0)--(4,0);\n\\draw (0,-2.5)--(0,2.5);\n\\draw (0,0) circle (2cm);\n\\draw (0,-2)--(3,1);\n\\draw (0,2)--(3,-1);\n\\end{tikzpicture}\n\n\\begin{tikzpicture}% arc 3 - image 7\n\\useasboundingbox (-2.5,-2.5) rectangle (4,2.5);\n\\draw (-2.5,0)--(4,0);\n\\draw (0,-2.5)--(0,2.5);\n\\draw (0,0) circle (2cm);\n\\draw (0,-2)--(3,1);\n\\draw (0,2)--(3,-1);\n\\end{tikzpicture}\n\\end{document}\n\n\nThe file that creates the animation from the ove.pdf file containing the images is:\n\n\\documentclass[tikz]{standalone}\n\\usepackage{animate}\n\n\\begin{document}\n\\animategraphics[autoplay,step]{1}{ove}{}{}\n\\end{document}\n\n\n### Weight of the animation pdf:\n\n• By stacking transparencies created in the standalone class with the animateinline environment, the pdf animation weighs on my computer 11172 bytes .\n• When the animation is a succession of images created with the \\animategraphics command in the standalone class, the animation weighs 18539 bytes.\n\nThe creation of transparencies is 40% lighter in this case than a succession of images.\n\nI hope I have said what is essential so that everyone can do the same. I hope too I have been clear, if not, say so, I will try to explain better.\n\nTranslated with www.DeepL.com/Translator" ]	[ null, "https://i.stack.imgur.com/nPKG3.gif", null, "https://i.stack.imgur.com/Rzphg.jpg", null, "https://i.stack.imgur.com/nPKG3.gif", null, "https://i.stack.imgur.com/5YZv3.gif", null, "https://i.stack.imgur.com/RjBLa.jpg", null, "https://i.stack.imgur.com/NGVxD.jpg", null, "https://i.stack.imgur.com/2YZ9W.png", null, "https://i.stack.imgur.com/wePb2.png", null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.6912637,"math_prob":0.9168264,"size":9820,"snap":"2019-51-2020-05","text_gpt3_token_len":3006,"char_repetition_ratio":0.15627547,"word_repetition_ratio":0.064291246,"special_character_ratio":0.30142567,"punctuation_ratio":0.19110459,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9653464,"pos_list":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16],"im_url_duplicate_count":[null,null,null,9,null,null,null,9,null,9,null,9,null,9,null,9,null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2020-01-22T23:02:09Z\",\"WARC-Record-ID\":\"<urn:uuid:5f919e84-6693-464b-8b8d-daeeecd2f2d2>\",\"Content-Length\":\"178295\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:2805da0b-26a9-4ab2-9ca2-7c4f40474dcf>\",\"WARC-Concurrent-To\":\"<urn:uuid:da137e37-44ba-41f8-a80a-e4db40223a97>\",\"WARC-IP-Address\":\"151.101.193.69\",\"WARC-Target-URI\":\"https://tex.stackexchange.com/questions/401886/how-to-create-simple-animations-with-animate?noredirect=1\",\"WARC-Payload-Digest\":\"sha1:LTSXRUQOLES4MUGBGQLFVIAKY52NLY34\",\"WARC-Block-Digest\":\"sha1:53KPZ5AHMIVQ7DQTK637NHMZHGYAWDJ4\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2020/CC-MAIN-2020-05/CC-MAIN-2020-05_segments_1579250607596.34_warc_CC-MAIN-20200122221541-20200123010541-00083.warc.gz\"}"}
https://isabelle.in.tum.de/repos/isabelle/rev/acc3b7dd0b21	[ "author eberlm Sat, 15 Jul 2017 14:32:02 +0100 changeset 66276 acc3b7dd0b21 parent 66275 2c1d223c5417 child 66277 512b0dc09061\nMore material on powers for HOL-Computational_Algebra/HOL-Number_Theory\n```--- a/src/HOL/Computational_Algebra/Computational_Algebra.thy\tWed Jul 12 18:42:32 2017 +0200\n+++ b/src/HOL/Computational_Algebra/Computational_Algebra.thy\tSat Jul 15 14:32:02 2017 +0100\n@@ -9,9 +9,11 @@\nFraction_Field\nFundamental_Theorem_Algebra\nNormalized_Fraction\n+ Nth_Powers\nPolynomial_FPS\nPolynomial\nPrimes\n+ Squarefree\nbegin\n\nend```\n```--- a/src/HOL/Computational_Algebra/Factorial_Ring.thy\tWed Jul 12 18:42:32 2017 +0200\n+++ b/src/HOL/Computational_Algebra/Factorial_Ring.thy\tSat Jul 15 14:32:02 2017 +0100\n@@ -13,6 +13,12 @@\n\nsubsection \\<open>Irreducible and prime elements\\<close>\n\n+(* TODO: Move ? )\n+lemma (in semiring_gcd) prod_coprime' [rule_format]:\n+ \"(\\<forall>i\\<in>A. gcd a (f i) = 1) \\<longrightarrow> gcd a (\\<Prod>i\\<in>A. f i) = 1\"\n+ using prod_coprime[of A f a] by (simp add: gcd.commute)\n+\n+\ncontext comm_semiring_1\nbegin\n\n@@ -217,6 +223,7 @@\nqed\nqed\nwith that show thesis by blast\n+\nqed\n\ncontext\n@@ -464,6 +471,9 @@\nlemma prime_dvd_prod_mset_iff: \"prime p \\<Longrightarrow> p dvd prod_mset A \\<longleftrightarrow> (\\<exists>x. x \\<in># A \\<and> p dvd x)\"\nby (induction A) (simp_all add: prime_elem_dvd_mult_iff prime_imp_prime_elem, blast+)\n\n+lemma prime_dvd_prod_iff: \"finite A \\<Longrightarrow> prime p \\<Longrightarrow> p dvd prod f A \\<longleftrightarrow> (\\<exists>x\\<in>A. p dvd f x)\"\n+ by (auto simp: prime_dvd_prod_mset_iff prod_unfold_prod_mset)\n+\nlemma primes_dvd_imp_eq:\nassumes \"prime p\" \"prime q\" \"p dvd q\"\nshows \"p = q\"\n@@ -1104,6 +1114,40 @@\nby (intro multiplicity_geI ) (auto intro: dvd_trans[OF multiplicity_dvd' assms(1)])\nqed (insert assms, auto simp: multiplicity_unit_left)\n\n+lemma prime_power_inj:\n+ assumes \"prime a\" \"a ^ m = a ^ n\"\n+ shows \"m = n\"\n+proof -\n+ have \"multiplicity a (a ^ m) = multiplicity a (a ^ n)\" by (simp only: assms)\n+ thus ?thesis using assms by (subst (asm) (1 2) multiplicity_prime_power) simp_all\n+qed\n+\n+lemma prime_power_inj':\n+ assumes \"prime p\" \"prime q\"\n+ assumes \"p ^ m = q ^ n\" \"m > 0\" \"n > 0\"\n+ shows \"p = q\" \"m = n\"\n+proof -\n+ from assms have \"p ^ 1 dvd p ^ m\" by (intro le_imp_power_dvd) simp\n+ also have \"p ^ m = q ^ n\" by fact\n+ finally have \"p dvd q ^ n\" by simp\n+ with assms have \"p dvd q\" using prime_dvd_power[of p q] by simp\n+ with assms show \"p = q\" by (simp add: primes_dvd_imp_eq)\n+ with assms show \"m = n\" by (simp add: prime_power_inj)\n+qed\n+\n+lemma prime_power_eq_one_iff [simp]: \"prime p \\<Longrightarrow> p ^ n = 1 \\<longleftrightarrow> n = 0\"\n+ using prime_power_inj[of p n 0] by auto\n+\n+lemma one_eq_prime_power_iff [simp]: \"prime p \\<Longrightarrow> 1 = p ^ n \\<longleftrightarrow> n = 0\"\n+ using prime_power_inj[of p 0 n] by auto\n+\n+lemma prime_power_inj'':\n+ assumes \"prime p\" \"prime q\"\n+ shows \"p ^ m = q ^ n \\<longleftrightarrow> (m = 0 \\<and> n = 0) \\<or> (p = q \\<and> m = n)\"\n+ using assms\n+ by (cases \"m = 0\"; cases \"n = 0\")\n+ (auto dest: prime_power_inj'[OF assms])\n+\nlemma prime_factorization_0 [simp]: \"prime_factorization 0 = {#}\"\nby (simp add: multiset_eq_iff count_prime_factorization)\n\n@@ -1265,6 +1309,12 @@\nfinally show ?thesis .\nqed\n\n+lemma prime_factorization_prod:\n+ assumes \"finite A\" \"\\<And>x. x \\<in> A \\<Longrightarrow> f x \\<noteq> 0\"\n+ shows \"prime_factorization (prod f A) = (\\<Sum>n\\<in>A. prime_factorization (f n))\"\n+ using assms by (induction A rule: finite_induct)\n+ (auto simp: Sup_multiset_empty prime_factorization_mult)\n+\nlemma prime_elem_multiplicity_mult_distrib:\nassumes \"prime_elem p\" \"x \\<noteq> 0\" \"y \\<noteq> 0\"\nshows \"multiplicity p (x y) = multiplicity p x + multiplicity p y\"\n@@ -1462,6 +1512,18 @@\nthus ?thesis by simp\nqed\n\n+lemma inj_on_Prod_primes:\n+ assumes \"\\<And>P p. P \\<in> A \\<Longrightarrow> p \\<in> P \\<Longrightarrow> prime p\"\n+ assumes \"\\<And>P. P \\<in> A \\<Longrightarrow> finite P\"\n+ shows \"inj_on Prod A\"\n+proof (rule inj_onI)\n+ fix P Q assume PQ: \"P \\<in> A\" \"Q \\<in> A\" \"\\<Prod>P = \\<Prod>Q\"\n+ with prime_factorization_unique'[of \"mset_set P\" \"mset_set Q\"] assms[of P] assms[of Q]\n+ have \"mset_set P = mset_set Q\" by (auto simp: prod_unfold_prod_mset)\n+ with assms[of P] assms[of Q] PQ show \"P = Q\" by simp\n+qed\n+\n+\n\nsubsection \\<open>GCD and LCM computation with unique factorizations\\<close>\n\n@@ -1729,6 +1791,16 @@\nusing assms\nby (simp add: prime_factorization_Lcm_factorial Lcm_eq_Lcm_factorial Lcm_factorial_eq_0_iff)\n\n+lemma prime_factors_gcd [simp]:\n+ \"a \\<noteq> 0 \\<Longrightarrow> b \\<noteq> 0 \\<Longrightarrow> prime_factors (gcd a b) =\n+ prime_factors a \\<inter> prime_factors b\"\n+ by (subst prime_factorization_gcd) auto\n+\n+lemma prime_factors_lcm [simp]:\n+ \"a \\<noteq> 0 \\<Longrightarrow> b \\<noteq> 0 \\<Longrightarrow> prime_factors (lcm a b) =\n+ prime_factors a \\<union> prime_factors b\"\n+ by (subst prime_factorization_lcm) auto\n+\nsubclass semiring_gcd\nby (standard, unfold gcd_eq_gcd_factorial lcm_eq_lcm_factorial)\n(rule gcd_lcm_factorial; assumption)+```\n```--- /dev/null\tThu Jan 01 00:00:00 1970 +0000\n+++ b/src/HOL/Computational_Algebra/Nth_Powers.thy\tSat Jul 15 14:32:02 2017 +0100\n@@ -0,0 +1,307 @@\n+(\n+ File: HOL/Computational_Algebra/Nth_Powers.thy\n+ Author: Manuel Eberl <[email protected]>\n+\n+ n-th powers in general and n-th roots of natural numbers\n+)\n+section \\<open>\\$n\\$-th powers and roots of naturals\\<close>\n+theory Nth_Powers\n+ imports Primes\n+begin\n+\n+subsection \\<open>The set of \\$n\\$-th powers\\<close>\n+\n+definition is_nth_power :: \"nat \\<Rightarrow> 'a :: monoid_mult \\<Rightarrow> bool\" where\n+ \"is_nth_power n x \\<longleftrightarrow> (\\<exists>y. x = y ^ n)\"\n+\n+lemma is_nth_power_nth_power [simp, intro]: \"is_nth_power n (x ^ n)\"\n+ by (auto simp add: is_nth_power_def)\n+\n+lemma is_nth_powerI [intro?]: \"x = y ^ n \\<Longrightarrow> is_nth_power n x\"\n+ by (auto simp: is_nth_power_def)\n+\n+lemma is_nth_powerE: \"is_nth_power n x \\<Longrightarrow> (\\<And>y. x = y ^ n \\<Longrightarrow> P) \\<Longrightarrow> P\"\n+ by (auto simp: is_nth_power_def)\n+\n+\n+abbreviation is_square where \"is_square \\<equiv> is_nth_power 2\"\n+\n+lemma is_zeroth_power [simp]: \"is_nth_power 0 x \\<longleftrightarrow> x = 1\"\n+ by (simp add: is_nth_power_def)\n+\n+lemma is_first_power [simp]: \"is_nth_power 1 x\"\n+ by (simp add: is_nth_power_def)\n+\n+lemma is_first_power' [simp]: \"is_nth_power (Suc 0) x\"\n+ by (simp add: is_nth_power_def)\n+\n+lemma is_nth_power_0 [simp]: \"n > 0 \\<Longrightarrow> is_nth_power n (0 :: 'a :: semiring_1)\"\n+ by (auto simp: is_nth_power_def power_0_left intro!: exI[of _ 0])\n+\n+lemma is_nth_power_0_iff [simp]: \"is_nth_power n (0 :: 'a :: semiring_1) \\<longleftrightarrow> n > 0\"\n+ by (cases n) auto\n+\n+lemma is_nth_power_1 [simp]: \"is_nth_power n 1\"\n+ by (auto simp: is_nth_power_def intro!: exI[of _ 1])\n+\n+lemma is_nth_power_Suc_0 [simp]: \"is_nth_power n (Suc 0)\"\n+ by (simp add: One_nat_def [symmetric] del: One_nat_def)\n+\n+lemma is_nth_power_conv_multiplicity:\n+ fixes x :: \"'a :: factorial_semiring\"\n+ assumes \"n > 0\"\n+ shows \"is_nth_power n (normalize x) \\<longleftrightarrow> (\\<forall>p. prime p \\<longrightarrow> n dvd multiplicity p x)\"\n+proof (cases \"x = 0\")\n+ case False\n+ show ?thesis\n+ proof (safe intro!: is_nth_powerI elim!: is_nth_powerE)\n+ fix y p :: 'a assume : \"normalize x = y ^ n\" \"prime p\"\n+ with assms and False have [simp]: \"y \\<noteq> 0\" by (auto simp: power_0_left)\n+ have \"multiplicity p x = multiplicity p (y ^ n)\"\n+ by (subst (1) [symmetric]) simp\n+ with False and * and assms show \"n dvd multiplicity p x\"\n+ by (auto simp: prime_elem_multiplicity_power_distrib)\n+ next\n+ assume : \"\\<forall>p. prime p \\<longrightarrow> n dvd multiplicity p x\"\n+ have \"multiplicity p ((\\<Prod>p\\<in>prime_factors x. p ^ (multiplicity p x div n)) ^ n) =\n+ multiplicity p x\" if \"prime p\" for p\n+ proof -\n+ from that and have \"n dvd multiplicity p x\" by blast\n+ have \"multiplicity p x = 0\" if \"p \\<notin> prime_factors x\"\n+ using that and \\<open>prime p\\<close> by (simp add: prime_factors_multiplicity)\n+ with that and * and assms show ?thesis unfolding prod_power_distrib power_mult [symmetric]\n+ by (subst multiplicity_prod_prime_powers) (auto simp: in_prime_factors_imp_prime elim: dvdE)\n+ qed\n+ with assms False\n+ have \"normalize x = normalize ((\\<Prod>p\\<in>prime_factors x. p ^ (multiplicity p x div n)) ^ n)\"\n+ by (intro multiplicity_eq_imp_eq) (auto simp: multiplicity_prod_prime_powers)\n+ thus \"normalize x = normalize (\\<Prod>p\\<in>prime_factors x. p ^ (multiplicity p x div n)) ^ n\"\n+ by (simp add: normalize_power)\n+ qed\n+qed (insert assms, auto)\n+\n+lemma is_nth_power_conv_multiplicity_nat:\n+ assumes \"n > 0\"\n+ shows \"is_nth_power n (x :: nat) \\<longleftrightarrow> (\\<forall>p. prime p \\<longrightarrow> n dvd multiplicity p x)\"\n+ using is_nth_power_conv_multiplicity[OF assms, of x] by simp\n+\n+lemma is_nth_power_mult:\n+ assumes \"is_nth_power n a\" \"is_nth_power n b\"\n+ shows \"is_nth_power n (a * b :: 'a :: comm_monoid_mult)\"\n+proof -\n+ from assms obtain a' b' where \"a = a' ^ n\" \"b = b' ^ n\" by (auto elim!: is_nth_powerE)\n+ hence \"a * b = (a' * b') ^ n\" by (simp add: power_mult_distrib)\n+ thus ?thesis by (rule is_nth_powerI)\n+qed\n+\n+lemma is_nth_power_mult_coprime_natD:\n+ fixes a b :: nat\n+ assumes \"coprime a b\" \"is_nth_power n (a * b)\" \"a > 0\" \"b > 0\"\n+ shows \"is_nth_power n a\" \"is_nth_power n b\"\n+proof -\n+ have A: \"is_nth_power n a\" if \"coprime a b\" \"is_nth_power n (a * b)\" \"a \\<noteq> 0\" \"b \\<noteq> 0\" \"n > 0\"\n+ for a b :: nat unfolding is_nth_power_conv_multiplicity_nat[OF \\<open>n > 0\\<close>]\n+ proof safe\n+ fix p :: nat assume p: \"prime p\"\n+ from \\<open>coprime a b\\<close> have \"\\<not>(p dvd a \\<and> p dvd b)\"\n+ using coprime_common_divisor_nat[of a b p] p by auto\n+ moreover from that and p\n+ have \"n dvd multiplicity p a + multiplicity p b\"\n+ by (auto simp: is_nth_power_conv_multiplicity_nat prime_elem_multiplicity_mult_distrib)\n+ ultimately show \"n dvd multiplicity p a\"\n+ by (auto simp: not_dvd_imp_multiplicity_0)\n+ qed\n+ from A[of a b] assms show \"is_nth_power n a\" by (cases \"n = 0\") simp_all\n+ from A[of b a] assms show \"is_nth_power n b\"\n+ by (cases \"n = 0\") (simp_all add: gcd.commute mult.commute)\n+qed\n+\n+lemma is_nth_power_mult_coprime_nat_iff:\n+ fixes a b :: nat\n+ assumes \"coprime a b\"\n+ shows \"is_nth_power n (a * b) \\<longleftrightarrow> is_nth_power n a \\<and>is_nth_power n b\"\n+ using assms\n+ by (cases \"a = 0\"; cases \"b = 0\")\n+ (auto intro: is_nth_power_mult dest: is_nth_power_mult_coprime_natD[of a b n]\n+ simp del: One_nat_def)\n+\n+lemma is_nth_power_prime_power_nat_iff:\n+ fixes p :: nat assumes \"prime p\"\n+ shows \"is_nth_power n (p ^ k) \\<longleftrightarrow> n dvd k\"\n+ using assms\n+ by (cases \"n > 0\")\n+ (auto simp: is_nth_power_conv_multiplicity_nat prime_elem_multiplicity_power_distrib)\n+\n+lemma is_nth_power_nth_power':\n+ assumes \"n dvd n'\"\n+ shows \"is_nth_power n (m ^ n')\"\n+proof -\n+ from assms have \"n' = n' div n * n\" by simp\n+ also have \"m ^ \\<dots> = (m ^ (n' div n)) ^ n\" by (simp add: power_mult)\n+ also have \"is_nth_power n \\<dots>\" by simp\n+ finally show ?thesis .\n+qed\n+\n+definition is_nth_power_nat :: \"nat \\<Rightarrow> nat \\<Rightarrow> bool\"\n+ where [code_abbrev]: \"is_nth_power_nat = is_nth_power\"\n+\n+lemma is_nth_power_nat_code [code]:\n+ \"is_nth_power_nat n m =\n+ (if n = 0 then m = 1\n+ else if m = 0 then n > 0\n+ else if n = 1 then True\n+ else (\\<exists>k\\<in>{1..m}. k ^ n = m))\"\n+ by (auto simp: is_nth_power_nat_def is_nth_power_def power_eq_iff_eq_base self_le_power)\n+\n+\n+(* TODO: Harmonise with Discrete.sqrt )\n+\n+subsection \\<open>The \\$n\\$-root of a natural number\\<close>\n+\n+definition nth_root_nat :: \"nat \\<Rightarrow> nat \\<Rightarrow> nat\" where\n+ \"nth_root_nat k n = (if k = 0 then 0 else Max {m. m ^ k \\<le> n})\"\n+\n+lemma zeroth_root_nat [simp]: \"nth_root_nat 0 n = 0\"\n+ by (simp add: nth_root_nat_def)\n+\n+lemma nth_root_nat_aux1:\n+ assumes \"k > 0\"\n+ shows \"{m::nat. m ^ k \\<le> n} \\<subseteq> {..n}\"\n+proof safe\n+ fix m assume \"m ^ k \\<le> n\"\n+ show \"m \\<le> n\"\n+ proof (cases \"m = 0\")\n+ case False\n+ with assms have \"m ^ 1 \\<le> m ^ k\" by (intro power_increasing) simp_all\n+ also note \\<open>m ^ k \\<le> n\\<close>\n+ finally show ?thesis by simp\n+ qed simp_all\n+qed\n+\n+lemma nth_root_nat_aux2:\n+ assumes \"k > 0\"\n+ shows \"finite {m::nat. m ^ k \\<le> n}\" \"{m::nat. m ^ k \\<le> n} \\<noteq> {}\"\n+proof -\n+ from assms have \"{m. m ^ k \\<le> n} \\<subseteq> {..n}\" by (rule nth_root_nat_aux1)\n+ moreover have \"finite {..n}\" by simp\n+ ultimately show \"finite {m::nat. m ^ k \\<le> n}\" by (rule finite_subset)\n+next\n+ from assms show \"{m::nat. m ^ k \\<le> n} \\<noteq> {}\" by (auto intro!: exI[of _ 0] simp: power_0_left)\n+qed\n+\n+lemma\n+ assumes \"k > 0\"\n+ shows nth_root_nat_power_le: \"nth_root_nat k n ^ k \\<le> n\"\n+ and nth_root_nat_ge: \"x ^ k \\<le> n \\<Longrightarrow> x \\<le> nth_root_nat k n\"\n+ using Max_in[OF nth_root_nat_aux2[OF assms], of n]\n+ Max_ge[OF nth_root_nat_aux2(1)[OF assms], of x n] assms\n+ by (auto simp: nth_root_nat_def)\n+\n+lemma nth_root_nat_less:\n+ assumes \"k > 0\" \"x ^ k > n\"\n+ shows \"nth_root_nat k n < x\"\n+proof -\n+ from \\<open>k > 0\\<close> have \"nth_root_nat k n ^ k \\<le> n\" by (rule nth_root_nat_power_le)\n+ also have \"n < x ^ k\" by fact\n+ finally show ?thesis by (rule power_less_imp_less_base) simp_all\n+qed\n+\n+lemma nth_root_nat_unique:\n+ assumes \"m ^ k \\<le> n\" \"(m + 1) ^ k > n\"\n+ shows \"nth_root_nat k n = m\"\n+proof (cases \"k > 0\")\n+ case True\n+ from nth_root_nat_less[OF \\<open>k > 0\\<close> assms(2)]\n+ have \"nth_root_nat k n \\<le> m\" by simp\n+ moreover from \\<open>k > 0\\<close> and assms(1) have \"nth_root_nat k n \\<ge> m\"\n+ by (intro nth_root_nat_ge)\n+ ultimately show ?thesis by (rule antisym)\n+qed (insert assms, auto)\n+\n+lemma nth_root_nat_0 [simp]: \"nth_root_nat k 0 = 0\" by (simp add: nth_root_nat_def)\n+lemma nth_root_nat_1 [simp]: \"k > 0 \\<Longrightarrow> nth_root_nat k 1 = 1\"\n+ by (rule nth_root_nat_unique) (auto simp del: One_nat_def)\n+lemma nth_root_nat_Suc_0 [simp]: \"k > 0 \\<Longrightarrow> nth_root_nat k (Suc 0) = Suc 0\"\n+ using nth_root_nat_1 by (simp del: nth_root_nat_1)\n+\n+lemma first_root_nat [simp]: \"nth_root_nat 1 n = n\"\n+ by (intro nth_root_nat_unique) auto\n+\n+lemma first_root_nat' [simp]: \"nth_root_nat (Suc 0) n = n\"\n+ by (intro nth_root_nat_unique) auto\n+\n+\n+lemma nth_root_nat_code_naive':\n+ \"nth_root_nat k n = (if k = 0 then 0 else Max (Set.filter (\\<lambda>m. m ^ k \\<le> n) {..n}))\"\n+proof (cases \"k > 0\")\n+ case True\n+ hence \"{m. m ^ k \\<le> n} \\<subseteq> {..n}\" by (rule nth_root_nat_aux1)\n+ hence \"Set.filter (\\<lambda>m. m ^ k \\<le> n) {..n} = {m. m ^ k \\<le> n}\"\n+ by (auto simp: Set.filter_def)\n+ with True show ?thesis by (simp add: nth_root_nat_def Set.filter_def)\n+qed simp\n+\n+function nth_root_nat_aux :: \"nat \\<Rightarrow> nat \\<Rightarrow> nat \\<Rightarrow> nat \\<Rightarrow> nat\" where\n+ \"nth_root_nat_aux m k acc n =\n+ (let acc' = (k + 1) ^ m\n+ in if k \\<ge> n \\<or> acc' > n then k else nth_root_nat_aux m (k+1) acc' n)\"\n+ by auto\n+termination by (relation \"measure (\\<lambda>(_,k,_,n). n - k)\", goal_cases) auto\n+\n+lemma nth_root_nat_aux_le:\n+ assumes \"k ^ m \\<le> n\" \"m > 0\"\n+ shows \"nth_root_nat_aux m k (k ^ m) n ^ m \\<le> n\"\n+ using assms\n+ by (induction m k \"k ^ m\" n rule: nth_root_nat_aux.induct) (auto simp: Let_def)\n+\n+lemma nth_root_nat_aux_gt:\n+ assumes \"m > 0\"\n+ shows \"(nth_root_nat_aux m k (k ^ m) n + 1) ^ m > n\"\n+ using assms\n+proof (induction m k \"k ^ m\" n rule: nth_root_nat_aux.induct)\n+ case (1 m k n)\n+ have \"n < Suc k ^ m\" if \"n \\<le> k\"\n+ proof -\n+ note that\n+ also have \"k < Suc k ^ 1\" by simp\n+ also from \\<open>m > 0\\<close> have \"\\<dots> \\<le> Suc k ^ m\" by (intro power_increasing) simp_all\n+ finally show ?thesis .\n+ qed\n+ with 1 show ?case by (auto simp: Let_def)\n+qed\n+\n+lemma nth_root_nat_aux_correct:\n+ assumes \"k ^ m \\<le> n\" \"m > 0\"\n+ shows \"nth_root_nat_aux m k (k ^ m) n = nth_root_nat m n\"\n+ by (rule sym, intro nth_root_nat_unique nth_root_nat_aux_le nth_root_nat_aux_gt assms)\n+\n+lemma nth_root_nat_naive_code [code]:\n+ \"nth_root_nat m n = (if m = 0 \\<or> n = 0 then 0 else if m = 1 \\<or> n = 1 then n else\n+ nth_root_nat_aux m 1 1 n)\"\n+ using nth_root_nat_aux_correct[of 1 m n] by (auto simp: )\n+\n+\n+lemma nth_root_nat_nth_power [simp]: \"k > 0 \\<Longrightarrow> nth_root_nat k (n ^ k) = n\"\n+ by (intro nth_root_nat_unique order.refl power_strict_mono) simp_all\n+\n+lemma nth_root_nat_nth_power':\n+ assumes \"k > 0\" \"k dvd m\"\n+ shows \"nth_root_nat k (n ^ m) = n ^ (m div k)\"\n+proof -\n+ from assms have \"m = (m div k) k\" by simp\n+ also have \"n ^ \\<dots> = (n ^ (m div k)) ^ k\" by (simp add: power_mult)\n+ also from assms have \"nth_root_nat k \\<dots> = n ^ (m div k)\" by simp\n+ finally show ?thesis .\n+qed\n+\n+lemma nth_root_nat_mono:\n+ assumes \"m \\<le> n\"\n+ shows \"nth_root_nat k m \\<le> nth_root_nat k n\"\n+proof (cases \"k = 0\")\n+ case False\n+ with assms show ?thesis unfolding nth_root_nat_def\n+ using nth_root_nat_aux2[of k m] nth_root_nat_aux2[of k n]\n+ by (auto intro!: Max_mono)\n+qed auto\n+\n+end\n\\ No newline at end of file```\n```--- /dev/null\tThu Jan 01 00:00:00 1970 +0000\n+++ b/src/HOL/Computational_Algebra/Squarefree.thy\tSat Jul 15 14:32:02 2017 +0100\n@@ -0,0 +1,360 @@\n+(\n+ File: HOL/Computational_Algebra/Squarefree.thy\n+ Author: Manuel Eberl <[email protected]>\n+\n+ Squarefreeness and decomposition of ring elements into square part and squarefree part\n+)\n+section \\<open>Squarefreeness\\<close>\n+theory Squarefree\n+imports Primes\n+begin\n+\n+(* TODO: Generalise to n-th powers )\n+\n+definition squarefree :: \"'a :: comm_monoid_mult \\<Rightarrow> bool\" where\n+ \"squarefree n \\<longleftrightarrow> (\\<forall>x. x ^ 2 dvd n \\<longrightarrow> x dvd 1)\"\n+\n+lemma squarefreeI: \"(\\<And>x. x ^ 2 dvd n \\<Longrightarrow> x dvd 1) \\<Longrightarrow> squarefree n\"\n+ by (auto simp: squarefree_def)\n+\n+lemma squarefreeD: \"squarefree n \\<Longrightarrow> x ^ 2 dvd n \\<Longrightarrow> x dvd 1\"\n+ by (auto simp: squarefree_def)\n+\n+lemma not_squarefreeI: \"x ^ 2 dvd n \\<Longrightarrow> \\<not>x dvd 1 \\<Longrightarrow> \\<not>squarefree n\"\n+ by (auto simp: squarefree_def)\n+\n+lemma not_squarefreeE [case_names square_dvd]:\n+ \"\\<not>squarefree n \\<Longrightarrow> (\\<And>x. x ^ 2 dvd n \\<Longrightarrow> \\<not>x dvd 1 \\<Longrightarrow> P) \\<Longrightarrow> P\"\n+ by (auto simp: squarefree_def)\n+\n+lemma not_squarefree_0 [simp]: \"\\<not>squarefree (0 :: 'a :: comm_semiring_1)\"\n+ by (rule not_squarefreeI[of 0]) auto\n+\n+lemma squarefree_factorial_semiring:\n+ assumes \"n \\<noteq> 0\"\n+ shows \"squarefree (n :: 'a :: factorial_semiring) \\<longleftrightarrow> (\\<forall>p. prime p \\<longrightarrow> \\<not>p ^ 2 dvd n)\"\n+ unfolding squarefree_def\n+proof safe\n+ assume : \"\\<forall>p. prime p \\<longrightarrow> \\<not>p ^ 2 dvd n\"\n+ fix x :: 'a assume x: \"x ^ 2 dvd n\"\n+ {\n+ assume \"\\<not>is_unit x\"\n+ moreover from assms and x have \"x \\<noteq> 0\" by auto\n+ ultimately obtain p where \"p dvd x\" \"prime p\"\n+ using prime_divisor_exists by blast\n+ with * have \"\\<not>p ^ 2 dvd n\" by blast\n+ moreover from \\<open>p dvd x\\<close> have \"p ^ 2 dvd x ^ 2\" by (rule dvd_power_same)\n+ ultimately have \"\\<not>x ^ 2 dvd n\" by (blast dest: dvd_trans)\n+ with x have False by contradiction\n+ }\n+ thus \"is_unit x\" by blast\n+qed auto\n+\n+lemma squarefree_factorial_semiring':\n+ assumes \"n \\<noteq> 0\"\n+ shows \"squarefree (n :: 'a :: factorial_semiring) \\<longleftrightarrow>\n+ (\\<forall>p\\<in>prime_factors n. multiplicity p n = 1)\"\n+proof (subst squarefree_factorial_semiring [OF assms], safe)\n+ fix p assume \"\\<forall>p\\<in>#prime_factorization n. multiplicity p n = 1\" \"prime p\" \"p^2 dvd n\"\n+ with assms show False\n+ by (cases \"p dvd n\")\n+ (auto simp: prime_factors_dvd power_dvd_iff_le_multiplicity not_dvd_imp_multiplicity_0)\n+qed (auto intro!: multiplicity_eqI simp: power2_eq_square [symmetric])\n+\n+lemma squarefree_factorial_semiring'':\n+ assumes \"n \\<noteq> 0\"\n+ shows \"squarefree (n :: 'a :: factorial_semiring) \\<longleftrightarrow>\n+ (\\<forall>p. prime p \\<longrightarrow> multiplicity p n \\<le> 1)\"\n+ by (subst squarefree_factorial_semiring'[OF assms]) (auto simp: prime_factors_multiplicity)\n+\n+lemma squarefree_unit [simp]: \"is_unit n \\<Longrightarrow> squarefree n\"\n+proof (rule squarefreeI)\n+ fix x assume \"x^2 dvd n\" \"n dvd 1\"\n+ hence \"is_unit (x^2)\" by (rule dvd_unit_imp_unit)\n+ thus \"is_unit x\" by (simp add: is_unit_power_iff)\n+qed\n+\n+lemma squarefree_1 [simp]: \"squarefree (1 :: 'a :: algebraic_semidom)\"\n+ by simp\n+\n+lemma squarefree_minus [simp]: \"squarefree (-n :: 'a :: comm_ring_1) \\<longleftrightarrow> squarefree n\"\n+ by (simp add: squarefree_def)\n+\n+lemma squarefree_mono: \"a dvd b \\<Longrightarrow> squarefree b \\<Longrightarrow> squarefree a\"\n+ by (auto simp: squarefree_def intro: dvd_trans)\n+\n+lemma squarefree_multD:\n+ assumes \"squarefree (a * b)\"\n+ shows \"squarefree a\" \"squarefree b\"\n+ by (rule squarefree_mono[OF _ assms], simp)+\n+\n+lemma squarefree_prime_elem:\n+ assumes \"prime_elem (p :: 'a :: factorial_semiring)\"\n+ shows \"squarefree p\"\n+proof -\n+ from assms have \"p \\<noteq> 0\" by auto\n+ show ?thesis\n+ proof (subst squarefree_factorial_semiring [OF \\<open>p \\<noteq> 0\\<close>]; safe)\n+ fix q assume : \"prime q\" \"q^2 dvd p\"\n+ with assms have \"multiplicity q p \\<ge> 2\" by (intro multiplicity_geI) auto\n+ thus False using assms \\<open>prime q\\<close> prime_multiplicity_other[of q \"normalize p\"]\n+ by (cases \"q = normalize p\") simp_all\n+ qed\n+qed\n+\n+lemma squarefree_prime:\n+ assumes \"prime (p :: 'a :: factorial_semiring)\"\n+ shows \"squarefree p\"\n+ using assms by (intro squarefree_prime_elem) auto\n+\n+lemma squarefree_mult_coprime:\n+ fixes a b :: \"'a :: factorial_semiring_gcd\"\n+ assumes \"coprime a b\" \"squarefree a\" \"squarefree b\"\n+ shows \"squarefree (a b)\"\n+proof -\n+ from assms have nz: \"a * b \\<noteq> 0\" by auto\n+ show ?thesis unfolding squarefree_factorial_semiring'[OF nz]\n+ proof\n+ fix p assume p: \"p \\<in> prime_factors (a * b)\"\n+ {\n+ assume \"p dvd a \\<and> p dvd b\"\n+ hence \"p dvd gcd a b\" by simp\n+ also have \"gcd a b = 1\" by fact\n+ finally have False using nz using p by (auto simp: prime_factors_dvd)\n+ }\n+ hence \"\\<not>(p dvd a \\<and> p dvd b)\" by blast\n+ moreover from p have \"p dvd a \\<or> p dvd b\" using nz\n+ by (auto simp: prime_factors_dvd prime_dvd_mult_iff)\n+ ultimately show \"multiplicity p (a * b) = 1\" using nz p assms(2,3)\n+ by (auto simp: prime_elem_multiplicity_mult_distrib prime_factors_multiplicity\n+ not_dvd_imp_multiplicity_0 squarefree_factorial_semiring')\n+ qed\n+qed\n+\n+lemma squarefree_prod_coprime:\n+ fixes f :: \"'a \\<Rightarrow> 'b :: factorial_semiring_gcd\"\n+ assumes \"\\<And>a b. a \\<in> A \\<Longrightarrow> b \\<in> A \\<Longrightarrow> a \\<noteq> b \\<Longrightarrow> coprime (f a) (f b)\"\n+ assumes \"\\<And>a. a \\<in> A \\<Longrightarrow> squarefree (f a)\"\n+ shows \"squarefree (prod f A)\"\n+ using assms\n+ by (induction A rule: infinite_finite_induct)\n+ (auto intro!: squarefree_mult_coprime prod_coprime')\n+\n+lemma squarefree_powerD: \"m > 0 \\<Longrightarrow> squarefree (n ^ m) \\<Longrightarrow> squarefree n\"\n+ by (cases m) (auto dest: squarefree_multD)\n+\n+lemma squarefree_power_iff:\n+ \"squarefree (n ^ m) \\<longleftrightarrow> m = 0 \\<or> is_unit n \\<or> (squarefree n \\<and> m = 1)\"\n+proof safe\n+ assume \"squarefree (n ^ m)\" \"m > 0\" \"\\<not>is_unit n\"\n+ show \"m = 1\"\n+ proof (rule ccontr)\n+ assume \"m \\<noteq> 1\"\n+ with \\<open>m > 0\\<close> have \"n ^ 2 dvd n ^ m\" by (intro le_imp_power_dvd) auto\n+ from this and \\<open>\\<not>is_unit n\\<close> have \"\\<not>squarefree (n ^ m)\" by (rule not_squarefreeI)\n+ with \\<open>squarefree (n ^ m)\\<close> show False by contradiction\n+ qed\n+qed (auto simp: is_unit_power_iff dest: squarefree_powerD)\n+\n+definition squarefree_nat :: \"nat \\<Rightarrow> bool\" where\n+ [code_abbrev]: \"squarefree_nat = squarefree\"\n+\n+lemma squarefree_nat_code_naive [code]:\n+ \"squarefree_nat n \\<longleftrightarrow> n \\<noteq> 0 \\<and> (\\<forall>k\\<in>{2..n}. \\<not>k ^ 2 dvd n)\"\n+proof safe\n+ assume : \"\\<forall>k\\<in>{2..n}. \\<not> k\\<^sup>2 dvd n\" and n: \"n > 0\"\n+ show \"squarefree_nat n\" unfolding squarefree_nat_def\n+ proof (rule squarefreeI)\n+ fix k assume k: \"k ^ 2 dvd n\"\n+ have \"k dvd n\" by (rule dvd_trans[OF _ k]) auto\n+ with n have \"k \\<le> n\" by (intro dvd_imp_le)\n+ with bspec[OF , of k] k have \"\\<not>k > 1\" by (intro notI) auto\n+ moreover from k and n have \"k \\<noteq> 0\" by (intro notI) auto\n+ ultimately have \"k = 1\" by presburger\n+ thus \"is_unit k\" by simp\n+ qed\n+qed (auto simp: squarefree_nat_def squarefree_def intro!: Nat.gr0I)\n+\n+\n+\n+definition square_part :: \"'a :: factorial_semiring \\<Rightarrow> 'a\" where\n+ \"square_part n = (if n = 0 then 0 else\n+ normalize (\\<Prod>p\\<in>prime_factors n. p ^ (multiplicity p n div 2)))\"\n+\n+lemma square_part_nonzero:\n+ \"n \\<noteq> 0 \\<Longrightarrow> square_part n = normalize (\\<Prod>p\\<in>prime_factors n. p ^ (multiplicity p n div 2))\"\n+ by (simp add: square_part_def)\n+\n+lemma square_part_0 [simp]: \"square_part 0 = 0\"\n+ by (simp add: square_part_def)\n+\n+lemma square_part_unit [simp]: \"is_unit x \\<Longrightarrow> square_part x = 1\"\n+ by (auto simp: square_part_def prime_factorization_unit)\n+\n+lemma square_part_1 [simp]: \"square_part 1 = 1\"\n+ by simp\n+\n+lemma square_part_0_iff [simp]: \"square_part n = 0 \\<longleftrightarrow> n = 0\"\n+ by (simp add: square_part_def)\n+\n+lemma normalize_uminus [simp]:\n+ \"normalize (-x :: 'a :: {normalization_semidom, comm_ring_1}) = normalize x\"\n+ by (rule associatedI) auto\n+\n+lemma multiplicity_uminus_right [simp]:\n+ \"multiplicity (x :: 'a :: {factorial_semiring, comm_ring_1}) (-y) = multiplicity x y\"\n+proof -\n+ have \"multiplicity x (-y) = multiplicity x (normalize (-y))\"\n+ by (rule multiplicity_normalize_right [symmetric])\n+ also have \"\\<dots> = multiplicity x y\" by simp\n+ finally show ?thesis .\n+qed\n+\n+lemma multiplicity_uminus_left [simp]:\n+ \"multiplicity (-x :: 'a :: {factorial_semiring, comm_ring_1}) y = multiplicity x y\"\n+proof -\n+ have \"multiplicity (-x) y = multiplicity (normalize (-x)) y\"\n+ by (rule multiplicity_normalize_left [symmetric])\n+ also have \"\\<dots> = multiplicity x y\" by simp\n+ finally show ?thesis .\n+qed\n+\n+lemma prime_factorization_uminus [simp]:\n+ \"prime_factorization (-x :: 'a :: {factorial_semiring, comm_ring_1}) = prime_factorization x\"\n+ by (rule prime_factorization_cong) simp_all\n+\n+lemma square_part_uminus [simp]:\n+ \"square_part (-x :: 'a :: {factorial_semiring, comm_ring_1}) = square_part x\"\n+ by (simp add: square_part_def)\n+\n+lemma prime_multiplicity_square_part:\n+ assumes \"prime p\"\n+ shows \"multiplicity p (square_part n) = multiplicity p n div 2\"\n+proof (cases \"n = 0\")\n+ case False\n+ thus ?thesis unfolding square_part_nonzero[OF False] multiplicity_normalize_right\n+ using finite_prime_divisors[of n] assms\n+ by (subst multiplicity_prod_prime_powers)\n+ (auto simp: not_dvd_imp_multiplicity_0 prime_factors_dvd multiplicity_prod_prime_powers)\n+qed auto\n+\n+lemma square_part_square_dvd [simp, intro]: \"square_part n ^ 2 dvd n\"\n+proof (cases \"n = 0\")\n+ case False\n+ thus ?thesis\n+ by (intro multiplicity_le_imp_dvd)\n+ (auto simp: prime_multiplicity_square_part prime_elem_multiplicity_power_distrib)\n+qed auto\n+\n+lemma prime_multiplicity_le_imp_dvd:\n+ assumes \"x \\<noteq> 0\" \"y \\<noteq> 0\"\n+ shows \"x dvd y \\<longleftrightarrow> (\\<forall>p. prime p \\<longrightarrow> multiplicity p x \\<le> multiplicity p y)\"\n+ using assms by (auto intro: multiplicity_le_imp_dvd dvd_imp_multiplicity_le)\n+\n+lemma dvd_square_part_iff: \"x dvd square_part n \\<longleftrightarrow> x ^ 2 dvd n\"\n+proof (cases \"x = 0\"; cases \"n = 0\")\n+ assume nz: \"x \\<noteq> 0\" \"n \\<noteq> 0\"\n+ thus ?thesis\n+ by (subst (1 2) prime_multiplicity_le_imp_dvd)\n+ (auto simp: prime_multiplicity_square_part prime_elem_multiplicity_power_distrib)\n+qed auto\n+\n+\n+definition squarefree_part :: \"'a :: factorial_semiring \\<Rightarrow> 'a\" where\n+ \"squarefree_part n = (if n = 0 then 1 else n div square_part n ^ 2)\"\n+\n+lemma squarefree_part_0 [simp]: \"squarefree_part 0 = 1\"\n+ by (simp add: squarefree_part_def)\n+\n+lemma squarefree_part_unit [simp]: \"is_unit n \\<Longrightarrow> squarefree_part n = n\"\n+ by (auto simp add: squarefree_part_def)\n+\n+lemma squarefree_part_1 [simp]: \"squarefree_part 1 = 1\"\n+ by simp\n+\n+lemma squarefree_decompose: \"n = squarefree_part n * square_part n ^ 2\"\n+ by (simp add: squarefree_part_def)\n+\n+lemma squarefree_part_uminus [simp]:\n+ assumes \"x \\<noteq> 0\"\n+ shows \"squarefree_part (-x :: 'a :: {factorial_semiring, comm_ring_1}) = -squarefree_part x\"\n+proof -\n+ have \"-(squarefree_part x * square_part x ^ 2) = -x\"\n+ by (subst squarefree_decompose [symmetric]) auto\n+ also have \"\\<dots> = squarefree_part (-x) * square_part (-x) ^ 2\" by (rule squarefree_decompose)\n+ finally have \"(- squarefree_part x) * square_part x ^ 2 =\n+ squarefree_part (-x) * square_part x ^ 2\" by simp\n+ thus ?thesis using assms by (subst (asm) mult_right_cancel) auto\n+qed\n+\n+lemma squarefree_part_nonzero [simp]: \"squarefree_part n \\<noteq> 0\"\n+ using squarefree_decompose[of n] by (cases \"n \\<noteq> 0\") auto\n+\n+lemma prime_multiplicity_squarefree_part:\n+ assumes \"prime p\"\n+ shows \"multiplicity p (squarefree_part n) = multiplicity p n mod 2\"\n+proof (cases \"n = 0\")\n+ case False\n+ hence n: \"n \\<noteq> 0\" by auto\n+ have \"multiplicity p n mod 2 + 2 * (multiplicity p n div 2) = multiplicity p n\" by simp\n+ also have \"\\<dots> = multiplicity p (squarefree_part n * square_part n ^ 2)\"\n+ by (subst squarefree_decompose[of n]) simp\n+ also from assms n have \"\\<dots> = multiplicity p (squarefree_part n) + 2 * (multiplicity p n div 2)\"\n+ by (subst prime_elem_multiplicity_mult_distrib)\n+ (auto simp: prime_elem_multiplicity_power_distrib prime_multiplicity_square_part)\n+ finally show ?thesis by (subst (asm) add_right_cancel) simp\n+qed auto\n+\n+lemma prime_multiplicity_squarefree_part_le_Suc_0 [intro]:\n+ assumes \"prime p\"\n+ shows \"multiplicity p (squarefree_part n) \\<le> Suc 0\"\n+ by (simp add: assms prime_multiplicity_squarefree_part)\n+\n+lemma squarefree_squarefree_part [simp, intro]: \"squarefree (squarefree_part n)\"\n+ by (subst squarefree_factorial_semiring'')\n+ (auto simp: prime_multiplicity_squarefree_part_le_Suc_0)\n+\n+lemma squarefree_decomposition_unique:\n+ assumes \"square_part m = square_part n\"\n+ assumes \"squarefree_part m = squarefree_part n\"\n+ shows \"m = n\"\n+ by (subst (1 2) squarefree_decompose) (simp_all add: assms)\n+\n+lemma normalize_square_part [simp]: \"normalize (square_part x) = square_part x\"\n+ by (simp add: square_part_def)\n+\n+lemma square_part_even_power': \"square_part (x ^ (2 * n)) = normalize (x ^ n)\"\n+proof (cases \"x = 0\")\n+ case False\n+ have \"normalize (square_part (x ^ (2 * n))) = normalize (x ^ n)\" using False\n+ by (intro multiplicity_eq_imp_eq)\n+ (auto simp: prime_multiplicity_square_part prime_elem_multiplicity_power_distrib)\n+ thus ?thesis by simp\n+qed (auto simp: power_0_left)\n+\n+lemma square_part_even_power: \"even n \\<Longrightarrow> square_part (x ^ n) = normalize (x ^ (n div 2))\"\n+ by (subst square_part_even_power' [symmetric]) auto\n+\n+lemma square_part_odd_power': \"square_part (x ^ (Suc (2 * n))) = normalize (x ^ n * square_part x)\"\n+proof (cases \"x = 0\")\n+ case False\n+ have \"normalize (square_part (x ^ (Suc (2 * n)))) = normalize (square_part x * x ^ n)\"\n+ proof (rule multiplicity_eq_imp_eq, goal_cases)\n+ case (3 p)\n+ hence \"multiplicity p (square_part (x ^ Suc (2 * n))) =\n+ (2 * (n * multiplicity p x) + multiplicity p x) div 2\"\n+ by (subst prime_multiplicity_square_part)\n+ (auto simp: False prime_elem_multiplicity_power_distrib algebra_simps simp del: power_Suc)\n+ also from 3 False have \"\\<dots> = multiplicity p (square_part x * x ^ n)\"\n+ by (subst div_mult_self4) (auto simp: prime_multiplicity_square_part\n+ prime_elem_multiplicity_mult_distrib prime_elem_multiplicity_power_distrib)\n+ finally show ?case .\n+ qed (insert False, auto)\n+ thus ?thesis by (simp add: mult_ac)\n+qed auto\n+\n+lemma square_part_odd_power:\n+ \"odd n \\<Longrightarrow> square_part (x ^ n) = normalize (x ^ (n div 2) * square_part x)\"\n+ by (subst square_part_odd_power' [symmetric]) auto\n+\n+end\n\\ No newline at end of file```\n```--- a/src/HOL/Library/Multiset.thy\tWed Jul 12 18:42:32 2017 +0200\n+++ b/src/HOL/Library/Multiset.thy\tSat Jul 15 14:32:02 2017 +0100\n@@ -1813,6 +1813,12 @@\nlemma mset_zero_iff_right[simp]: \"({#} = mset x) = (x = [])\"\nby (induct x) auto\n\n+lemma count_mset_gt_0: \"x \\<in> set xs \\<Longrightarrow> count (mset xs) x > 0\"\n+ by (induction xs) auto\n+\n+lemma count_mset_0_iff [simp]: \"count (mset xs) x = 0 \\<longleftrightarrow> x \\<notin> set xs\"\n+ by (induction xs) auto\n+\nlemma mset_single_iff[iff]: \"mset xs = {#x#} \\<longleftrightarrow> xs = [x]\"\nby (cases xs) auto\n\n@@ -1949,6 +1955,9 @@\ndefines mset_set = \"folding.F add_mset {#}\"\nby standard (simp add: fun_eq_iff)\n\n+lemma sum_multiset_singleton [simp]: \"sum (\\<lambda>n. {#n#}) A = mset_set A\"\n+ by (induction A rule: infinite_finite_induct) auto\n+\nlemma count_mset_set [simp]:\n\"finite A \\<Longrightarrow> x \\<in> A \\<Longrightarrow> count (mset_set A) x = 1\" (is \"PROP ?P\")\n\"\\<not> finite A \\<Longrightarrow> count (mset_set A) x = 0\" (is \"PROP ?Q\")\n@@ -2001,6 +2010,25 @@\nlemma mset_set_set: \"distinct xs \\<Longrightarrow> mset_set (set xs) = mset xs\"\nby (induction xs) simp_all\n\n+lemma count_mset_set': \"count (mset_set A) x = (if finite A \\<and> x \\<in> A then 1 else 0)\"\n+ by auto\n+\n+lemma subset_imp_msubset_mset_set:\n+ assumes \"A \\<subseteq> B\" \"finite B\"\n+ shows \"mset_set A \\<subseteq># mset_set B\"\n+proof (rule mset_subset_eqI)\n+ fix x :: 'a\n+ from assms have \"finite A\" by (rule finite_subset)\n+ with assms show \"count (mset_set A) x \\<le> count (mset_set B) x\"\n+ by (cases \"x \\<in> A\"; cases \"x \\<in> B\") auto\n+qed\n+\n+lemma mset_set_set_mset_msubset: \"mset_set (set_mset A) \\<subseteq># A\"\n+proof (rule mset_subset_eqI)\n+ fix x show \"count (mset_set (set_mset A)) x \\<le> count A x\"\n+ by (cases \"x \\<in># A\") simp_all\n+qed\n+\ncontext linorder\nbegin\n\n@@ -2071,6 +2099,23 @@\nfinally show ?case by simp\nqed simp_all\n\n+lemma msubset_mset_set_iff [simp]:\n+ assumes \"finite A\" \"finite B\"\n+ shows \"mset_set A \\<subseteq># mset_set B \\<longleftrightarrow> A \\<subseteq> B\"\n+ using subset_imp_msubset_mset_set[of A B]\n+ set_mset_mono[of \"mset_set A\" \"mset_set B\"] assms by auto\n+\n+lemma mset_set_eq_iff [simp]:\n+ assumes \"finite A\" \"finite B\"\n+ shows \"mset_set A = mset_set B \\<longleftrightarrow> A = B\"\n+proof -\n+ from assms have \"mset_set A = mset_set B \\<longleftrightarrow> set_mset (mset_set A) = set_mset (mset_set B)\"\n+ by (intro iffI equalityI set_mset_mono) auto\n+ also from assms have \"\\<dots> \\<longleftrightarrow> A = B\" by simp\n+ finally show ?thesis .\n+qed\n+\n+\n(* Contributed by Lukas Bulwahn )\nlemma image_mset_mset_set:\nassumes \"inj_on f A\"```\n```--- a/src/HOL/Number_Theory/Number_Theory.thy\tWed Jul 12 18:42:32 2017 +0200\n+++ b/src/HOL/Number_Theory/Number_Theory.thy\tSat Jul 15 14:32:02 2017 +0100\n@@ -2,7 +2,7 @@\nsection \\<open>Comprehensive number theory\\<close>\n\ntheory Number_Theory\n-imports Fib Residues Eratosthenes Quadratic_Reciprocity Pocklington\n+imports Fib Residues Eratosthenes Quadratic_Reciprocity Pocklington Prime_Powers\nbegin\n\nend```\n```--- /dev/null\tThu Jan 01 00:00:00 1970 +0000\n+++ b/src/HOL/Number_Theory/Prime_Powers.thy\tSat Jul 15 14:32:02 2017 +0100\n@@ -0,0 +1,315 @@\n+(\n+ File: HOL/Number_Theory/Prime_Powers.thy\n+ Author: Manuel Eberl <[email protected]>\n+\n+ Prime powers and the Mangoldt function\n+)\n+section \\<open>Prime powers\\<close>\n+theory Prime_Powers\n+ imports Complex_Main \"~~/src/HOL/Computational_Algebra/Primes\"\n+begin\n+\n+definition aprimedivisor :: \"'a :: normalization_semidom \\<Rightarrow> 'a\" where\n+ \"aprimedivisor q = (SOME p. prime p \\<and> p dvd q)\"\n+\n+definition primepow :: \"'a :: normalization_semidom \\<Rightarrow> bool\" where\n+ \"primepow n \\<longleftrightarrow> (\\<exists>p k. prime p \\<and> k > 0 \\<and> n = p ^ k)\"\n+\n+definition primepow_factors :: \"'a :: normalization_semidom \\<Rightarrow> 'a set\" where\n+ \"primepow_factors n = {x. primepow x \\<and> x dvd n}\"\n+\n+lemma primepow_gt_Suc_0: \"primepow n \\<Longrightarrow> n > Suc 0\"\n+ using one_less_power[of \"p::nat\" for p] by (auto simp: primepow_def prime_nat_iff)\n+\n+lemma\n+ assumes \"prime p\" \"p dvd n\"\n+ shows prime_aprimedivisor: \"prime (aprimedivisor n)\"\n+ and aprimedivisor_dvd: \"aprimedivisor n dvd n\"\n+proof -\n+ from assms have \"\\<exists>p. prime p \\<and> p dvd n\" by auto\n+ from someI_ex[OF this] show \"prime (aprimedivisor n)\" \"aprimedivisor n dvd n\"\n+ unfolding aprimedivisor_def by (simp_all add: conj_commute)\n+qed\n+\n+lemma\n+ assumes \"n \\<noteq> 0\" \"\\<not>is_unit (n :: 'a :: factorial_semiring)\"\n+ shows prime_aprimedivisor': \"prime (aprimedivisor n)\"\n+ and aprimedivisor_dvd': \"aprimedivisor n dvd n\"\n+proof -\n+ from someI_ex[OF prime_divisor_exists[OF assms]]\n+ show \"prime (aprimedivisor n)\" \"aprimedivisor n dvd n\"\n+ unfolding aprimedivisor_def by (simp_all add: conj_commute)\n+qed\n+\n+lemma aprimedivisor_of_prime [simp]:\n+ assumes \"prime p\"\n+ shows \"aprimedivisor p = p\"\n+proof -\n+ from assms have \"\\<exists>q. prime q \\<and> q dvd p\" by auto\n+ from someI_ex[OF this, folded aprimedivisor_def] assms show ?thesis\n+ by (auto intro: primes_dvd_imp_eq)\n+qed\n+\n+lemma aprimedivisor_pos_nat: \"(n::nat) > 1 \\<Longrightarrow> aprimedivisor n > 0\"\n+ using aprimedivisor_dvd'[of n] by (auto elim: dvdE intro!: Nat.gr0I)\n+\n+lemma aprimedivisor_primepow_power:\n+ assumes \"primepow n\" \"k > 0\"\n+ shows \"aprimedivisor (n ^ k) = aprimedivisor n\"\n+proof -\n+ from assms obtain p l where l: \"prime p\" \"l > 0\" \"n = p ^ l\"\n+ by (auto simp: primepow_def)\n+ from l assms have : \"prime (aprimedivisor (n ^ k))\" \"aprimedivisor (n ^ k) dvd n ^ k\"\n+ by (intro prime_aprimedivisor[of p] aprimedivisor_dvd[of p] dvd_power;\n+ simp add: power_mult [symmetric])+\n+ from * l have \"aprimedivisor (n ^ k) dvd p ^ (l * k)\" by (simp add: power_mult)\n+ with assms * l have \"aprimedivisor (n ^ k) dvd p\"\n+ by (subst (asm) prime_dvd_power_iff) simp_all\n+ with l assms have \"aprimedivisor (n ^ k) = p\"\n+ by (intro primes_dvd_imp_eq prime_aprimedivisor l) (auto simp: power_mult [symmetric])\n+ moreover from l have \"aprimedivisor n dvd p ^ l\"\n+ by (auto intro: aprimedivisor_dvd simp: prime_gt_0_nat)\n+ with assms l have \"aprimedivisor n dvd p\"\n+ by (subst (asm) prime_dvd_power_iff) (auto intro!: prime_aprimedivisor simp: prime_gt_0_nat)\n+ with l assms have \"aprimedivisor n = p\"\n+ by (intro primes_dvd_imp_eq prime_aprimedivisor l) auto\n+ ultimately show ?thesis by simp\n+qed\n+\n+lemma aprimedivisor_prime_power:\n+ assumes \"prime p\" \"k > 0\"\n+ shows \"aprimedivisor (p ^ k) = p\"\n+proof -\n+ from assms have : \"prime (aprimedivisor (p ^ k))\" \"aprimedivisor (p ^ k) dvd p ^ k\"\n+ by (intro prime_aprimedivisor[of p] aprimedivisor_dvd[of p]; simp add: prime_nat_iff)+\n+ from assms have \"aprimedivisor (p ^ k) dvd p\"\n+ by (subst (asm) prime_dvd_power_iff) simp_all\n+ with assms * show \"aprimedivisor (p ^ k) = p\" by (intro primes_dvd_imp_eq)\n+qed\n+\n+lemma prime_factorization_primepow:\n+ assumes \"primepow n\"\n+ shows \"prime_factorization n =\n+ replicate_mset (multiplicity (aprimedivisor n) n) (aprimedivisor n)\"\n+ using assms\n+ by (auto simp: primepow_def aprimedivisor_prime_power prime_factorization_prime_power)\n+\n+lemma primepow_decompose:\n+ assumes \"primepow n\"\n+ shows \"aprimedivisor n ^ multiplicity (aprimedivisor n) n = n\"\n+proof -\n+ from assms have \"n \\<noteq> 0\" by (intro notI) (auto simp: primepow_def)\n+ hence \"n = unit_factor n * prod_mset (prime_factorization n)\"\n+ by (subst prod_mset_prime_factorization) simp_all\n+ also from assms have \"unit_factor n = 1\" by (auto simp: primepow_def unit_factor_power)\n+ also have \"prime_factorization n =\n+ replicate_mset (multiplicity (aprimedivisor n) n) (aprimedivisor n)\"\n+ by (intro prime_factorization_primepow assms)\n+ also have \"prod_mset \\<dots> = aprimedivisor n ^ multiplicity (aprimedivisor n) n\" by simp\n+ finally show ?thesis by simp\n+qed\n+\n+lemma prime_power_not_one:\n+ assumes \"prime p\" \"k > 0\"\n+ shows \"p ^ k \\<noteq> 1\"\n+proof\n+ assume \"p ^ k = 1\"\n+ hence \"is_unit (p ^ k)\" by simp\n+ thus False using assms by (simp add: is_unit_power_iff)\n+qed\n+\n+lemma zero_not_primepow [simp]: \"\\<not>primepow 0\"\n+ by (auto simp: primepow_def)\n+\n+lemma one_not_primepow [simp]: \"\\<not>primepow 1\"\n+ by (auto simp: primepow_def prime_power_not_one)\n+\n+lemma primepow_not_unit [simp]: \"primepow p \\<Longrightarrow> \\<not>is_unit p\"\n+ by (auto simp: primepow_def is_unit_power_iff)\n+\n+lemma unit_factor_primepow: \"primepow p \\<Longrightarrow> unit_factor p = 1\"\n+ by (auto simp: primepow_def unit_factor_power)\n+\n+lemma aprimedivisor_primepow:\n+ assumes \"prime p\" \"p dvd n\" \"primepow (n :: 'a :: factorial_semiring)\"\n+ shows \"aprimedivisor (p * n) = p\" \"aprimedivisor n = p\"\n+proof -\n+ from assms have [simp]: \"n \\<noteq> 0\" by auto\n+ define q where \"q = aprimedivisor n\"\n+ with assms have q: \"prime q\" by (auto simp: q_def intro!: prime_aprimedivisor)\n+ from \\<open>primepow n\\<close> have n: \"n = q ^ multiplicity q n\"\n+ by (simp add: primepow_decompose q_def)\n+ have nz: \"multiplicity q n \\<noteq> 0\"\n+ proof\n+ assume \"multiplicity q n = 0\"\n+ with n have n': \"n = unit_factor n\" by simp\n+ have \"is_unit n\" by (subst n', rule unit_factor_is_unit) (insert assms, auto)\n+ with assms show False by auto\n+ qed\n+ with \\<open>prime p\\<close> \\<open>p dvd n\\<close> q have \"p dvd q\"\n+ by (subst (asm) n) (auto intro: prime_dvd_power)\n+ with \\<open>prime p\\<close> q have \"p = q\" by (intro primes_dvd_imp_eq)\n+ thus \"aprimedivisor n = p\" by (simp add: q_def)\n+\n+ define r where \"r = aprimedivisor (p * n)\"\n+ with assms have r: \"r dvd (p * n)\" \"prime r\" unfolding r_def\n+ by (intro aprimedivisor_dvd[of p] prime_aprimedivisor[of p]; simp)+\n+ hence \"r dvd q ^ Suc (multiplicity q n)\"\n+ by (subst (asm) n) (auto simp: \\<open>p = q\\<close> dest: dvd_unit_imp_unit)\n+ with r have \"r dvd q\"\n+ by (auto intro: prime_dvd_power_nat simp: prime_dvd_mult_iff dest: prime_dvd_power)\n+ with r q have \"r = q\" by (intro primes_dvd_imp_eq)\n+ thus \"aprimedivisor (p * n) = p\" by (simp add: r_def \\<open>p = q\\<close>)\n+qed\n+\n+lemma power_eq_prime_powerD:\n+ fixes p :: \"'a :: factorial_semiring\"\n+ assumes \"prime p\" \"n > 0\" \"x ^ n = p ^ k\"\n+ shows \"\\<exists>i. normalize x = normalize (p ^ i)\"\n+proof -\n+ have \"normalize x = normalize (p ^ multiplicity p x)\"\n+ proof (rule multiplicity_eq_imp_eq)\n+ fix q :: 'a assume \"prime q\"\n+ from assms have \"multiplicity q (x ^ n) = multiplicity q (p ^ k)\" by simp\n+ with \\<open>prime q\\<close> and assms have \"n * multiplicity q x = k * multiplicity q p\"\n+ by (subst (asm) (1 2) prime_elem_multiplicity_power_distrib) (auto simp: power_0_left)\n+ with assms and \\<open>prime q\\<close> show \"multiplicity q x = multiplicity q (p ^ multiplicity p x)\"\n+ by (cases \"p = q\") (auto simp: multiplicity_distinct_prime_power prime_multiplicity_other)\n+ qed (insert assms, auto simp: power_0_left)\n+ thus ?thesis by auto\n+qed\n+\n+\n+lemma primepow_power_iff:\n+ assumes \"unit_factor p = 1\"\n+ shows \"primepow (p ^ n) \\<longleftrightarrow> primepow (p :: 'a :: factorial_semiring) \\<and> n > 0\"\n+proof safe\n+ assume \"primepow (p ^ n)\"\n+ hence n: \"n \\<noteq> 0\" by (auto intro!: Nat.gr0I)\n+ thus \"n > 0\" by simp\n+ from assms have [simp]: \"normalize p = p\"\n+ using normalize_mult_unit_factor[of p] by (simp only: mult.right_neutral)\n+ from \\<open>primepow (p ^ n)\\<close> obtain q k where : \"k > 0\" \"prime q\" \"p ^ n = q ^ k\"\n+ by (auto simp: primepow_def)\n+ with power_eq_prime_powerD[of q n p k] n\n+ obtain i where eq: \"normalize p = normalize (q ^ i)\" by auto\n+ with primepow_not_unit[OF \\<open>primepow (p ^ n)\\<close>] have \"i \\<noteq> 0\"\n+ by (intro notI) (simp add: normalize_1_iff is_unit_power_iff del: primepow_not_unit)\n+ with \\<open>normalize p = normalize (q ^ i)\\<close> \\<open>prime q\\<close> show \"primepow p\"\n+ by (auto simp: normalize_power primepow_def intro!: exI[of _ q] exI[of _ i])\n+next\n+ assume \"primepow p\" \"n > 0\"\n+ then obtain q k where : \"k > 0\" \"prime q\" \"p = q ^ k\" by (auto simp: primepow_def)\n+ with \\<open>n > 0\\<close> show \"primepow (p ^ n)\"\n+ by (auto simp: primepow_def power_mult intro!: exI[of _ q] exI[of _ \"k * n\"])\n+qed\n+\n+lemma primepow_prime [simp]: \"prime n \\<Longrightarrow> primepow n\"\n+ by (auto simp: primepow_def intro!: exI[of _ n] exI[of _ \"1::nat\"])\n+\n+lemma primepow_prime_power [simp]:\n+ \"prime (p :: 'a :: factorial_semiring) \\<Longrightarrow> primepow (p ^ n) \\<longleftrightarrow> n > 0\"\n+ by (subst primepow_power_iff) auto\n+\n+(* TODO Generalise )\n+lemma primepow_multD:\n+ assumes \"primepow (a b :: nat)\"\n+ shows \"a = 1 \\<or> primepow a\" \"b = 1 \\<or> primepow b\"\n+proof -\n+ from assms obtain p k where k: \"k > 0\" \"a * b = p ^ k\" \"prime p\"\n+ unfolding primepow_def by auto\n+ then obtain i j where \"a = p ^ i\" \"b = p ^ j\"\n+ using prime_power_mult_nat[of p a b] by blast\n+ with \\<open>prime p\\<close> show \"a = 1 \\<or> primepow a\" \"b = 1 \\<or> primepow b\" by auto\n+qed\n+\n+lemma primepow_mult_aprimedivisorI:\n+ assumes \"primepow (n :: 'a :: factorial_semiring)\"\n+ shows \"primepow (aprimedivisor n * n)\"\n+ by (subst (2) primepow_decompose[OF assms, symmetric], subst power_Suc [symmetric],\n+ subst primepow_prime_power)\n+ (insert assms, auto intro!: prime_aprimedivisor' dest: primepow_gt_Suc_0)\n+\n+lemma aprimedivisor_vimage:\n+ assumes \"prime (p :: 'a :: factorial_semiring)\"\n+ shows \"aprimedivisor -` {p} \\<inter> primepow_factors n = {p ^ k \|k. k > 0 \\<and> p ^ k dvd n}\"\n+proof safe\n+ fix q assume q: \"q \\<in> primepow_factors n\"\n+ hence q': \"q \\<noteq> 0\" \"q \\<noteq> 1\" by (auto simp: primepow_def primepow_factors_def prime_power_not_one)\n+ let ?n = \"multiplicity (aprimedivisor q) q\"\n+ from q q' have \"q = aprimedivisor q ^ ?n \\<and> ?n > 0 \\<and> aprimedivisor q ^ ?n dvd n\"\n+ using prime_aprimedivisor'[of q] aprimedivisor_dvd'[of q]\n+ by (subst primepow_decompose [symmetric])\n+ (auto simp: primepow_factors_def multiplicity_gt_zero_iff unit_factor_primepow\n+ intro: dvd_trans[OF multiplicity_dvd])\n+ thus \"\\<exists>k. q = aprimedivisor q ^ k \\<and> k > 0 \\<and> aprimedivisor q ^ k dvd n\" ..\n+next\n+ fix k :: nat assume k: \"p ^ k dvd n\" \"k > 0\"\n+ with assms show \"p ^ k \\<in> aprimedivisor -` {p}\"\n+ by (auto simp: aprimedivisor_prime_power)\n+ with assms k show \"p ^ k \\<in> primepow_factors n\"\n+ by (auto simp: primepow_factors_def primepow_def aprimedivisor_prime_power intro: Suc_leI)\n+qed\n+\n+lemma primepow_factors_altdef:\n+ fixes x :: \"'a :: factorial_semiring\"\n+ assumes \"x \\<noteq> 0\"\n+ shows \"primepow_factors x = {p ^ k \|p k. p \\<in> prime_factors x \\<and> k \\<in> {0<.. multiplicity p x}}\"\n+proof (intro equalityI subsetI)\n+ fix q assume \"q \\<in> primepow_factors x\"\n+ then obtain p k where pk: \"prime p\" \"k > 0\" \"q = p ^ k\" \"q dvd x\"\n+ unfolding primepow_factors_def primepow_def by blast\n+ moreover have \"k \\<le> multiplicity p x\" using pk assms by (intro multiplicity_geI) auto\n+ ultimately show \"q \\<in> {p ^ k \|p k. p \\<in> prime_factors x \\<and> k \\<in> {0<.. multiplicity p x}}\"\n+ by (auto simp: prime_factors_multiplicity intro!: exI[of _ p] exI[of _ k])\n+qed (auto simp: primepow_factors_def prime_factors_multiplicity multiplicity_dvd')\n+\n+lemma finite_primepow_factors:\n+ assumes \"x \\<noteq> (0 :: 'a :: factorial_semiring)\"\n+ shows \"finite (primepow_factors x)\"\n+proof -\n+ have \"finite (SIGMA p:prime_factors x. {0<..multiplicity p x})\"\n+ by (intro finite_SigmaI) simp_all\n+ hence \"finite ((\\<lambda>(p,k). p ^ k) ` \\<dots>)\" (is \"finite ?A\") by (rule finite_imageI)\n+ also have \"?A = primepow_factors x\"\n+ using assms by (subst primepow_factors_altdef) fast+\n+ finally show ?thesis .\n+qed\n+\n+\n+definition mangoldt :: \"nat \\<Rightarrow> 'a :: real_algebra_1\" where\n+ \"mangoldt n = (if primepow n then of_real (ln (real (aprimedivisor n))) else 0)\"\n+\n+lemma of_real_mangoldt [simp]: \"of_real (mangoldt n) = mangoldt n\"\n+ by (simp add: mangoldt_def)\n+\n+lemma mangoldt_sum:\n+ assumes \"n \\<noteq> 0\"\n+ shows \"(\\<Sum>d \| d dvd n. mangoldt d :: 'a :: real_algebra_1) = of_real (ln (real n))\"\n+proof -\n+ have \"(\\<Sum>d \| d dvd n. mangoldt d :: 'a) = of_real (\\<Sum>d \| d dvd n. mangoldt d)\" by simp\n+ also have \"(\\<Sum>d \| d dvd n. mangoldt d) = (\\<Sum>d \\<in> primepow_factors n. ln (real (aprimedivisor d)))\"\n+ using assms by (intro sum.mono_neutral_cong_right) (auto simp: primepow_factors_def mangoldt_def)\n+ also have \"\\<dots> = ln (real (\\<Prod>d \\<in> primepow_factors n. aprimedivisor d))\"\n+ using assms finite_primepow_factors[of n]\n+ by (subst ln_prod [symmetric])\n+ (auto simp: primepow_factors_def intro!: aprimedivisor_pos_nat\n+ intro: Nat.gr0I primepow_gt_Suc_0)\n+ also have \"primepow_factors n =\n+ (\\<lambda>(p,k). p ^ k) ` (SIGMA p:prime_factors n. {0<..multiplicity p n})\"\n+ (is \"_ = _ ` ?A\") by (subst primepow_factors_altdef[OF assms]) fast+\n+ also have \"prod aprimedivisor \\<dots> = (\\<Prod>(p,k)\\<in>?A. aprimedivisor (p ^ k))\"\n+ by (subst prod.reindex)\n+ (auto simp: inj_on_def prime_power_inj'' prime_factors_multiplicity\n+ prod.Sigma [symmetric] case_prod_unfold)\n+ also have \"\\<dots> = (\\<Prod>(p,k)\\<in>?A. p)\"\n+ by (intro prod.cong refl) (auto simp: aprimedivisor_prime_power prime_factors_multiplicity)\n+ also have \"\\<dots> = (\\<Prod>x\\<in>prime_factors n. \\<Prod>k\\<in>{0<..multiplicity x n}. x)\"\n+ by (rule prod.Sigma [symmetric]) auto\n+ also have \"\\<dots> = (\\<Prod>x\\<in>prime_factors n. x ^ multiplicity x n)\"\n+ by (intro prod.cong refl) (simp add: prod_constant)\n+ also have \"\\<dots> = n\" using assms by (intro prime_factorization_nat [symmetric]) simp\n+ finally show ?thesis .\n+qed\n+\n+end\n\\ No newline at end of file```" ]	[ null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.5588176,"math_prob":0.99323094,"size":48986,"snap":"2021-04-2021-17","text_gpt3_token_len":16006,"char_repetition_ratio":0.22218366,"word_repetition_ratio":0.19039032,"special_character_ratio":0.3510799,"punctuation_ratio":0.11968651,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.99989295,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2021-04-11T19:11:05Z\",\"WARC-Record-ID\":\"<urn:uuid:179f235e-db90-44ef-b257-eef3d928c0fb>\",\"Content-Length\":\"148592\",\"Content-Type\":\"application/http; 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https://virtual.aistats.org/virtual/2021/poster/1834	[ "## Approximation Algorithms for Orthogonal Non-negative Matrix Factorization\n\n### Moses Charikar · Lunjia Hu\n\nKeywords: [ Models and Methods ] [ Matrix and Tensor Factorization ]\n\nAbstract: In the non-negative matrix factorization (NMF) problem, the input is an $m\\times n$ matrix $M$ with non-negative entries and the goal is to factorize it as $M\\approx AW$. The $m\\times k$ matrix $A$ and the $k\\times n$ matrix $W$ are both constrained to have non-negative entries. This is in contrast to singular value decomposition, where the matrices $A$ and $W$ can have negative entries but must satisfy the orthogonality constraint: the columns of $A$ are orthogonal and the rows of $W$ are also orthogonal. The orthogonal non-negative matrix factorization (ONMF) problem imposes both the non-negativity and the orthogonality constraints, and previous work showed that it leads to better performances than NMF on many clustering tasks. We give the first constant-factor approximation algorithm for ONMF when one or both of $A$ and $W$ are subject to the orthogonality constraint. We also show an interesting connection to the correlation clustering problem on bipartite graphs. Our experiments on synthetic and real-world data show that our algorithm achieves similar or smaller errors compared to previous ONMF algorithms while ensuring perfect orthogonality (many previous algorithms do not satisfy the hard orthogonality constraint)." ]	[ null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.8980616,"math_prob":0.9891901,"size":1250,"snap":"2022-05-2022-21","text_gpt3_token_len":272,"char_repetition_ratio":0.14847513,"word_repetition_ratio":0.0,"special_character_ratio":0.1936,"punctuation_ratio":0.056872036,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.99743044,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2022-05-29T11:29:59Z\",\"WARC-Record-ID\":\"<urn:uuid:92aa0652-bc7e-4b6e-b6c0-4f4dfc0d890f>\",\"Content-Length\":\"11616\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:61752e01-773a-488c-b10b-2bb0e3b96a0b>\",\"WARC-Concurrent-To\":\"<urn:uuid:d4f4d79d-8a89-41e8-bf38-16b43e21889a>\",\"WARC-IP-Address\":\"198.202.70.65\",\"WARC-Target-URI\":\"https://virtual.aistats.org/virtual/2021/poster/1834\",\"WARC-Payload-Digest\":\"sha1:CJGMUF66PJO5PP6EKTGVXHSRFBSWY27D\",\"WARC-Block-Digest\":\"sha1:4APPXEOV7KCHMFYWKYUUTW6USYXV74XV\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2022/CC-MAIN-2022-21/CC-MAIN-2022-21_segments_1652662644142.66_warc_CC-MAIN-20220529103854-20220529133854-00017.warc.gz\"}"}
https://answers.everydaycalculation.com/lcm/30-12	[ "Solutions by everydaycalculation.com\n\n## What is the LCM of 30 and 12?\n\nThe lcm of 30 and 12 is 60.\n\n#### Steps to find LCM\n\n1. Find the prime factorization of 30\n30 = 2 × 3 × 5\n2. Find the prime factorization of 12\n12 = 2 × 2 × 3\n3. Multiply each factor the greater number of times it occurs in steps i) or ii) above to find the lcm:\n\nLCM = 2 × 2 × 3 × 5\n4. LCM = 60\n\nMathStep (Works offline)", null, "Download our mobile app and learn how to find LCM of upto four numbers in your own time:" ]	[ null, "https://answers.everydaycalculation.com/mathstep-app-icon.png", null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.75913537,"math_prob":0.9990645,"size":467,"snap":"2020-24-2020-29","text_gpt3_token_len":157,"char_repetition_ratio":0.12958963,"word_repetition_ratio":0.0,"special_character_ratio":0.41327623,"punctuation_ratio":0.082474224,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.99603856,"pos_list":[0,1,2],"im_url_duplicate_count":[null,null,null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2020-07-14T04:19:35Z\",\"WARC-Record-ID\":\"<urn:uuid:c3044f23-281e-43ed-879c-de13ebd26efa>\",\"Content-Length\":\"5792\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:5e903b22-929a-4952-810f-16a0fe48c18d>\",\"WARC-Concurrent-To\":\"<urn:uuid:88baf7d0-5d8c-42ce-8a01-1ca81f5b514b>\",\"WARC-IP-Address\":\"96.126.107.130\",\"WARC-Target-URI\":\"https://answers.everydaycalculation.com/lcm/30-12\",\"WARC-Payload-Digest\":\"sha1:V6SOF5ZHLFMNYIO4P3X74SCHCYAV4RIS\",\"WARC-Block-Digest\":\"sha1:374YMKGMWPTMO5J7SZNRQYSDFEIJVTAK\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2020/CC-MAIN-2020-29/CC-MAIN-2020-29_segments_1593657147917.99_warc_CC-MAIN-20200714020904-20200714050904-00059.warc.gz\"}"}
https://www.schoolofhaskell.com/user/Geraldus/algebraic-data-types-adts-with-aeson	[ "# Algebraic Data Types (ADTs) with Aeson\n\nAs of March 2020, School of Haskell has been switched to read-only mode.\n\n### Sections\n\nThis is tiny cheat sheet, which will help with decoding JSON data to Haskell's ADT.\n\n``````data YesNo = Yes\n\| No\nderiving Show``````\n\nFirst of all let's think how we could express our Haskell values in JSON format? According to Aeson's documentation:\n\nSo stick to objects (e.g. maps in Haskell) or arrays (lists or vectors in Haskell):\n\nTo avoid possible decoding pitfalls we should stick with objects and arrays. Here are possible (or valid) respresentations:\n\n``````For Yes value: { \"value\" : \"yes\" } { \"value\" : 1 } { \"value\" : 1.0 }\nNo value: { \"value\" : \"no\" } { \"value\" : 0 } { \"value\" : 0.0 }``````\n\nLet's declare `FromJSON` instance for our `YesNo` data type:\n\n``````instance FromJSON YesNo where\n\n-- parseJSON takes a Value, it could be one of follwing data constructors:\n-- Object, Array, String, Number, Bool or Null.\n-- First of all we expect an Object, it is defined as Object !Object,\n-- where second Object is just a type synonym for HashMap Text Value. In\n-- our case we should choose somehow our Haskell value constructor\n-- according to recieved value.\n-- So, `o` is actually a HashMap, and all we need is to lookup key \"type\"\n-- We should use strict Text for key:\nparseJSON (Object o) = case HML.lookup (pack \"type\") o of\n-- value of entity has type Value\nJust (String t) -> fromString (TL.unpack (TL.fromStrict t))\nJust (Number n) -> fromNum n\n-- Other cases are invalid\n_ -> empty\nwhere fromString :: String -> Parser YesNo\nfromString \"yes\" = pure Yes\nfromString \"no\" = pure No\nfromString _ = empty\nfromNum n\n\| n == 1 \|\| n == 1.0 = pure Yes\n\| n == 0 \|\| n == 0.0 = pure No\n\| otherwise = empty``````\n\nLet's also declare `ToJSON` instance for our Haskell data type:\n\n``````instance ToJSON YesNo where\ntoJSON Yes = object [ \"value\" .= String \"yes\" ]\ntoJSON No = object [ \"value\" .= String \"no\" ]``````\n\nNow we can play with it and test it:\n\n``````{-# LANGUAGE OverloadedStrings #-}\n\nimport Data.Aeson\nimport Data.Aeson.Types ( Parser )\n\nimport Data.Text ( Text, pack )\nimport qualified Data.Text.Lazy as TL ( fromStrict, unpack )\nimport qualified Data.ByteString.Lazy as BSL ( toChunks )\nimport qualified Data.ByteString.Lazy.Char8 as C ( fromChunks, unpack )\n\nimport qualified Data.HashMap.Lazy as HML ( lookup )\n\nimport Control.Applicative ( empty, pure )\n\ndata YesNo = Yes\n\| No\nderiving Show\n\ninstance FromJSON YesNo where\nparseJSON (Object o) = case HML.lookup \"value\" o of\nJust (String t) -> fromString (TL.unpack (TL.fromStrict t))\nJust (Number n) -> fromNum n\n_ -> empty\nwhere fromString :: String -> Parser YesNo\nfromString \"yes\" = pure Yes\nfromString \"no\" = pure No\nfromString _ = empty\nfromNum n\n\| n == 1 \|\| n == 1.0 = pure Yes\n\| n == 0 \|\| n == 0.0 = pure No\n\| otherwise = empty\n\ninstance ToJSON YesNo where\ntoJSON Yes = object [ \"value\" .= String \"yes\" ]\ntoJSON No = object [ \"value\" .= String \"no\" ]\n\n-- show\nmain = do\nlet yesJ = encode Yes\nyesJS = C.unpack \\$ C.fromChunks \\$ BSL.toChunks yesJ\nputStrLn \"Encoded:\"\nprint yesJS\nputStrLn \"Decoding:\"\nprint (decode yesJ :: Maybe Value)\nprint (decode yesJ :: Maybe YesNo)``````\n\nNow, let's imagine we are designing some kind of API. Assume that we store in database pairs of person's name and his/her cash amount.\n\n``````data Command = NotCommand\n\| WrongArg String\n\| CommandCreate { name :: Text, value :: Double }\n\| CommandUpdate { id :: Int, value :: Double }\n\| CommandDelete { id :: Int }\nderiving Show``````\n\nHere are valid representations of out `Command` data type:\n\n`Create` command example\n\n``````{ \"type\" : \"command\",\n\"name\" : \"create\",\n\"data\" : {\n\"name\" : \"Arthur\",\n\"value\" : 100.0\n}\n}``````\n\n`Update` command example\n\n``````{ \"type\" : \"command\",\n\"name\" : \"update\",\n\"data\" : {\n\"id\" : 1,\n\"value\" : 90.0\n}\n}``````\n\n`Delete` command example\n\n``````{ \"type\" : \"command\",\n\"name\" : \"delete\",\n\"data\" : 1\n}``````\n\nSo, we expect key `type`, which always should be `\"command\"`, to distinguish commands we use key `name`, and each command have third mandatory key `data`, which differs for each command.\n\nWe could declare following `FromJSON` instance for `Command`:\n\n``````instance FromJSON Command where\n-- First of all we lookup for mandatory key `type`\nparseJSON (Object o) = case HML.lookup \"type\" o of\nJust (String \"command\") -> let dt = HML.lookup \"data\" o\nin case HML.lookup \"name\" o of\n-- Then we lookup for key `name`, to distinguish commands\nJust (String \"create\") -> createCmd dt\nJust (String \"update\") -> updateCmd dt\nJust (String \"delete\") -> CommandDelete <\\$> o .: \"data\"\n_ -> unrecognizedCommand\n_ -> pure NotCommand\nwhere createCmd Nothing = missingData\ncreateCmd (Just (Object d)) = CommandCreate <\\$> d .: \"name\" <> d .: \"value\"\ncreateCmd _ = incorrectData\nupdateCmd Nothing = missingData\nupdateCmd (Just (Object d)) = CommandUpdate <\\$> d .: \"id\" <> d .: \"value\"\nupdateCmd _ = incorrectData\n\nmissingData = pure \\$ WrongArg \"Missing mandatory `data` key.\"\nincorrectData = pure \\$ WrongArg \"Incorrect data received.\"\nunrecognizedCommand = pure \\$ WrongArg \"Unrecognized command name.\"\nparseJSON _ = pure NotCommand``````\n\nThere is nothing special about `ToJSON` instance, so let's just omit its declaration and test our code!\n\n``````{-# LANGUAGE OverloadedStrings #-}\n\nimport Data.Aeson\n\nimport Data.Text ( Text, pack )\nimport qualified Data.Text.Lazy as TL ( fromStrict, unpack )\n\nimport qualified Data.HashMap.Lazy as HML ( lookup )\n\nimport Control.Applicative ( empty, pure, (<\\$>), (<>) )\nimport qualified Data.ByteString.Lazy.Char8 as BSCL\n\ndata Command = NotCommand\n\| WrongArg String\n\| CommandCreate { name :: Text, value :: Double }\n\| CommandUpdate { id :: Int, value :: Double }\n\| CommandDelete { id :: Int }\nderiving Show\n\ninstance FromJSON Command where\nparseJSON (Object o) = case HML.lookup \"type\" o of\nJust (String \"command\") -> let dt = HML.lookup \"data\" o\nin case HML.lookup \"name\" o of\nJust (String \"create\") -> createCmd dt\nJust (String \"update\") -> updateCmd dt\nJust (String \"delete\") -> CommandDelete <\\$> o .: \"data\"\n_ -> unrecognizedCommand\n_ -> pure NotCommand\nwhere createCmd Nothing = missingData\ncreateCmd (Just (Object d)) = CommandCreate <\\$> d .: \"name\" <> d .: \"value\"\ncreateCmd _ = incorrectData\nupdateCmd Nothing = missingData\nupdateCmd (Just (Object d)) = CommandUpdate <\\$> d .: \"id\" <*> d .: \"value\"\nupdateCmd _ = incorrectData\n\nmissingData = pure \\$ WrongArg \"Missing mandatory `data` key.\"\nincorrectData = pure \\$ WrongArg \"Incorrect data received.\"\nunrecognizedCommand = pure \\$ WrongArg \"Unrecognized command name.\"\nparseJSON _ = pure NotCommand\n\ninstance ToJSON Command where\ntoJSON NotCommand = String \"Not a command\"\ntoJSON (WrongArg t) = String (pack t)\ntoJSON (CommandCreate n v) = object [ \"type\" .= String \"command\"\n, \"name\" .= String \"create\"\n, \"data\" .= object [ \"name\" .= String n\n, \"value\" .= toJSON v\n]\n]\ntoJSON (CommandUpdate i v) = object [ \"type\" .= String \"command\"\n, \"name\" .= String \"update\"\n, \"data\" .= object [ \"id\" .= toJSON i\n, \"value\" .= toJSON v\n]\n]\ntoJSON (CommandDelete i) = object [ \"type\" .= String \"command\"\n, \"name\" .= String \"delete\"\n, \"data\" .= toJSON i\n]\n-- show\nmain = do\nlet c = encode \\$ CommandCreate \"Svetlana\" 100.0\nprint (decode c :: Maybe Command)\nprint (decode \"{\\\"type\\\":\\\"command\\\",\\\"name\\\":\\\"create\\\"}\" :: Maybe Command)\nprint (decode \"{\\\"type\\\":\\\"command\\\",\\\"name\\\":\\\" reate\\\",\\\"data\\\":{\\\"name\\\":\\\"Svetlana\\\",\\\"value\\\":100.0}}\" :: Maybe Command)\nprint (decode \"{\\\"type\\\":\\\"command\\\",\\\"name\\\":\\\"create\\\",\\\"data\\\":{\\\" ame\\\":\\\"Svetlana\\\",\\\"value\\\":100.0}}\" :: Maybe Command)``````" ]	[ null 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http://www.piping-designer.com/index.php/properties/classical-mechanics/252-characteristic-length	[ "# Characteristic Length\n\nWritten by Jerry Ratzlaff on . Posted in Classical Mechanics\n\nCharacteristic length, abbreviated as $$l_c$$, is the scale of a physical system. The length is used in 2D and 3D systems for Fluid Dynamics and Thermodynamics defining the parameter of the system.\n\nExamples of:\n\n## Formulas that use Characteristic Length\n\n $$\\large{ l_c = \\frac{ V } { A } }$$ $$\\large{ l_c = \\sqrt { \\frac{ \\alpha \\; t }{ Fo } } }$$ (Fourier number) $$\\large{ l_c =\\frac{ Nu \\; k }{ h } }$$ (Nusselt number) $$\\large{ l_c = \\frac {Pe \\; k}{ v \\; \\rho \\; C } }$$ (Peclet number) $$\\large{ l_c = \\frac{ Re \\; \\mu }{\\rho \\; v } }$$ (Reynolds number) $$\\large{ l_c = \\frac { Sh \\; D} {K} }$$ (Sherwood number)\n\n### Where:\n\n$$\\large{ l_c }$$ = characteristic length\n\n$$\\large{ A }$$ = area of object surface\n\n$$\\large{ \\rho }$$ (Greek symbol rho) = density\n\n$$\\large{ D }$$ = diffusion coefficient\n\n$$\\large{ \\mu }$$ (Greek symbol mu) = dynamic viscosity\n\n$$\\large{ Fo }$$ = Fourier number\n\n$$\\large{ C }$$ = heat capacity\n\n$$\\large{ h }$$ = heat transfer coefficient\n\n$$\\large{ K }$$ = mass transfer coefficient\n\n$$\\large{ Nu }$$ = Nusselt number\n\n$$\\large{ Pe }$$ = Peclet number\n\n$$\\large{ Re }$$ = Reynolds number\n\n$$\\large{ Sh }$$ = Sherwood number\n\n$$\\large{ k }$$ = thermal conductivity\n\n$$\\large{ \\alpha }$$ (Greel symbol alpha) = thermal diffusivity\n\n$$\\large{ t }$$ = time\n\n$$\\large{ v }$$ = velocity\n\n$$\\large{ V }$$ =" ]	[ null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.6248852,"math_prob":1.000009,"size":1353,"snap":"2020-24-2020-29","text_gpt3_token_len":453,"char_repetition_ratio":0.24610823,"word_repetition_ratio":0.008510638,"special_character_ratio":0.3939394,"punctuation_ratio":0.07526882,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":1.0000094,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2020-07-07T12:40:54Z\",\"WARC-Record-ID\":\"<urn:uuid:644cd97e-443b-43dc-9da8-f4782fb77c10>\",\"Content-Length\":\"29586\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:6af28ebc-087b-49b0-bff0-019fd0902883>\",\"WARC-Concurrent-To\":\"<urn:uuid:689a49f6-165e-4e0b-b486-0b8b2eebcf17>\",\"WARC-IP-Address\":\"23.235.195.68\",\"WARC-Target-URI\":\"http://www.piping-designer.com/index.php/properties/classical-mechanics/252-characteristic-length\",\"WARC-Payload-Digest\":\"sha1:PEGSHI7J76BAYXT3ELNJAFQ2DFNGJ6OX\",\"WARC-Block-Digest\":\"sha1:UDO3ATKFJ5G6AZ4MMSY7AJMAJNJMO6SP\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2020/CC-MAIN-2020-29/CC-MAIN-2020-29_segments_1593655892516.24_warc_CC-MAIN-20200707111607-20200707141607-00336.warc.gz\"}"}
https://iq.opengenus.org/tim-sort/	[ "×\n\nSearch anything:\n\n# Tim Sort\n\n#### Algorithms Sorting Algorithms", null, "Get this book -> Problems on Array: For Interviews and Competitive Programming\n\nReading time: 30 minutes \| Coding time: 15 minutes\n\nTim Sort is a hybrid stable sorting algorithm that takes advantage of common patterns in data, and utilizes a combination of an improved Merge sort and Binary Insertion sort along with some internal logic to optimize the manipulation of large scale real-world data. Tim Sort was first implemented in 2002 by Tim Peters for use in Python.\n\n### Operation\n\nTim sort uses Binary insertion sort and improved merge sort by using galloping in a combination. Binary insertion sort is the best method to sort when data is already or partially sorted or the length of run is smaller than MIN_RUN and merge sort is best when the input is large.\n\nTim Sort is complex, even by algorithmic standards. The operation is best broken down into parts:\n\n## Run (dividing slot)\n\nAn input array is divided into different sub-arrays, count of elements inside a sub-array is defined as a RUN, the minimum value of such runs is a MIN_RUN which is a power of 2 not more than 32 (or 64). These sub-arrays are usually partially ordered (strictly ascending or stricly descending).When the array is decomposed into several runs, the runs of ascending order remain unchanged, and the runs of strictly descending order are reversed. Finally, several runs of ascending order are obtained.\n\nExample:\n\nConsider an array of goals:\n\n1 5 9 8 6 4 5 6 7\n\nWe can see that [1, 5, 9] conforms to ascending order, [8, 6, 4] conforms to strict descending order, [4, 5, 6, 7] conforms to ascending order.\n\nFlip the strictly descending run and then add together a new array:\n\n1 5 9 6 8 4 5 6 7\n\n## Merge run\n\nLet's assume that the new array is:\n\n.... 6 7 8 9 10 1 2 3 4 ....\n\nIf we merge two runs using Mergesort alone\nrun1: [6, 7, 8, 9, 10]\nrun2: [1, 2, 3, 4, 5]\n\nMergesort compares the first element of the two runs\n1 < 6, get 1\n2 < 6, get 2\n3 < 6, get 3\n4 < 6, get 4\n5 < 6, get 5\n\nWe found that traversing the entire run directly would eventually consume more number of merge operations if the run was longer. Because every run is in ascending order, there's no need to compare them one by one and the concept of Galloping becomes handy.\n\n## Galloping\n\nFor example, to merge the following two runs:\nrun1: [101, 102, 103, ... 200]\nrun2: [1, 2, 3, ..., 100]\nInstead of comparing the elements one by one, we compare them by increasing powers of 2^n where n >= 0.\n\nrun1 > run2\nrun1 > run2\nrun1 > run2\nrun1 > run2\nrun1 > run2\n...\nrun1 > run2[2^n - 1]\nrun1 <= run2[2^(n+1) - 1]\n\nWe get a result run2 [2 ^ n - 1] < run1 <= run2 [2 ^ (n + 1) - 1], Since run2 is sorted, we use Binary Search to locate run1 in run2 very efficiently by defining left = run2[2^n - 1], right = run[2^(n+1) - 1]. Hence. the number of times we merge two runs is reduced from O(N) to O(logN).\n\n## Why not skip Galloping and just do Binary Search?\n\nLet's give an example:\nrun1: [1, 3, 5, 7, 9 ... 2n+1]\nrun2: [0, 2, 4, 6, 8 ... 2n]\nIn this way, run1 is only larger than run2. Each time we do Binary Search, we can only get a number of results, plus n times of Binary Search, so the time complexity changes from O(N) to O(NlogN).\n\n## Stack determines the order\n\nWhen the original array becomes a set of various ascending runs, we need to merge two runs, but if we merge a long run with a shorter run, it will take a longer time to compare a comparitively shorter run. So Timsort maintains a stack with all the run lengths on the stack, while satisfying that the length of run on the back stack is longer than the sum of the length of run on the first two stacks, so that the length of run always decreases.\n\nStack : [... runA, runB, runC]\nrunA > runB + runC\nrunB > runC\nThis avoids run merges with too much difference in length.\n\nFor those who prefer bullets:\n\n• Establish a minrun size that is a power of 2 (usually 32, never more than 64 or your Binary Insertion Sort will lose efficiency)\n• Find a run in the first minrun of data.\n• If the run is not at least minrun in length, use Insertion Sort to grab subsequent or prior items and insert them into the run until it is the correct minimum size.\n• Repeat until the entire array is divided into sorted subsections.\n• Use the latter half of Merge Sort to join the ordered arrays.\n\n## Complexity Analysis\n\nBy design TimSort is well suited for partially sorted data with the best case being totally sorted data. It falls into the adaptive sort family. Taking the number of runs ρ as a (natural) parameter for a refined analysis we obtained:\n\nTimSort runs in O(N + Nlogρ) time.\n\n• Worst case time complexity: `Θ(NlogN)`\n• Average case time complexity: `Θ(NlogN)`\n• Best case time complexity: `Θ(N)`\n• Space complexity: `Θ(N)`\n\n## Implementation\n\nImplementation of Tim Sort algorithm:\n\n• C\n• C++\n• Java\n• Python\n\n### C\n\n``````\n#include<stdio.h>\n#define min(a,b) (a)<(b)?(a):(b)\nconst int RUN = 32;\n// this function sorts array from left index to\n// right index which is of size atmost RUN\nvoid insertionSort(int arr[], int left, int right)\n{\nfor (int i = left + 1; i <= right; i++)\n{\nint temp = arr[i];\nint j = i - 1;\nwhile (j >= left && arr[j] > temp)\n{\narr[j+1] = arr[j];\nj--;\n}\narr[j+1] = temp;\n}\n}\n// merge function merges the sorted runs\nvoid merge(int arr[], int l, int m, int r)\n{\n// original array is broken in two parts\n// left and right array\nint len1 = m - l + 1, len2 = r - m;\nint left[len1], right[len2];\nfor (int i = 0; i < len1; i++)\nleft[i] = arr[l + i];\nfor (int i = 0; i < len2; i++)\nright[i] = arr[m + 1 + i];\nint i = 0;\nint j = 0;\nint k = l;\n// after comparing, we merge those two array\n// in larger sub array\nwhile (i < len1 && j < len2)\n{\nif (left[i] <= right[j])\n{\narr[k] = left[i];\ni++;\n}\nelse\n{\narr[k] = right[j];\nj++;\n}\nk++;\n}\n// copy remaining elements of left, if any\nwhile (i < len1)\n{\narr[k] = left[i];\nk++;\ni++;\n}\n// copy remaining element of right, if any\nwhile (j < len2)\n{\narr[k] = right[j];\nk++;\nj++;\n}\n}\nvoid timSort(int arr[], int n)\n{\n// Sort individual subarrays of size RUN\nfor (int i = 0; i < n; i+=RUN)\n{\ninsertionSort(arr, i, min((i+31), (n-1)));\n}\n// start merging from size RUN (or 32). It will merge\n// to form size 64, then 128, 256 and so on ....\nfor (int size = RUN; size < n; size = 2size)\n{\n// pick starting point of left sub array. We\n// are going to merge arr[left..left+size-1]\n// and arr[left+size, left+2size-1]\n// After every merge, we increase left by 2size\nfor (int left = 0; left < n; left += 2size)\n{\n// find ending point of left sub array\n// mid+1 is starting point of right sub array\nint mid = left + size - 1;\nint right = min((left + 2size - 1), (n-1));\n// merge sub array arr[left.....mid] &\n// arr[mid+1....right]\nmerge(arr, left, mid, right);\n}\n}\n}\nint main()\n{\nint a[] = {12,1,20,2,3,123,54,332},i;\nint n = sizeof(a)/sizeof(a);\nprintf(\"Printing Array elements \\n\");\nfor (i = 0; i < n; i++)\nprintf(\"%d \", a[i]);\nprintf(\"\\n\");\ntimSort(a, n);\nprintf(\"Printing sorted array elements \\n\");\nfor (i = 0; i < n; i++)\nprintf(\"%d \", a[i]);\nprintf(\"\\n\");\nreturn 0;\n}\n``````\n\n### C++\n\n``````\n#include<bits/stdc++.h>\nusing namespace std;\nconst int RUN = 32;\n// this function sorts array from left index to\n// right index which is of size atmost RUN\nvoid insertionSort(int arr[], int left, int right)\n{\nfor (int i = left + 1; i <= right; i++)\n{\nint temp = arr[i];\nint j = i - 1;\nwhile (j >= left && arr[j] > temp)\n{\narr[j+1] = arr[j];\nj--;\n}\narr[j+1] = temp;\n}\n}\n// merge function merges the sorted runs\nvoid merge(int arr[], int l, int m, int r)\n{\n// original array is broken in two parts\n// left and right array\nint len1 = m - l + 1, len2 = r - m;\nint left[len1], right[len2];\nfor (int i = 0; i < len1; i++)\nleft[i] = arr[l + i];\nfor (int i = 0; i < len2; i++)\nright[i] = arr[m + 1 + i];\nint i = 0;\nint j = 0;\nint k = l;\n// after comparing, we merge those two array\n// in larger sub array\nwhile (i < len1 && j < len2)\n{\nif (left[i] <= right[j])\n{\narr[k] = left[i];\ni++;\n}\nelse\n{\narr[k] = right[j];\nj++;\n}\nk++;\n}\n// copy remaining elements of left, if any\nwhile (i < len1)\n{\narr[k] = left[i];\nk++;\ni++;\n}\n// copy remaining element of right, if any\nwhile (j < len2)\n{\narr[k] = right[j];\nk++;\nj++;\n}\n}\nvoid timSort(int arr[], int n)\n{\n// Sort individual subarrays of size RUN\nfor (int i = 0; i < n; i+=RUN)\n{\ninsertionSort(arr, i, min((i+31), (n-1)));\n}\n// start merging from size RUN (or 32). It will merge\n// to form size 64, then 128, 256 and so on ....\nfor (int size = RUN; size < n; size = 2size)\n{\n// pick starting point of left sub array. We\n// are going to merge arr[left..left+size-1]\n// and arr[left+size, left+2size-1]\n// After every merge, we increase left by 2size\nfor (int left = 0; left < n; left += 2size)\n{\n// find ending point of left sub array\n// mid+1 is starting point of right sub array\nint mid = left + size - 1;\nint right = min((left + 2size - 1), (n-1));\n// merge sub array arr[left.....mid] &\n// arr[mid+1....right]\nmerge(arr, left, mid, right);\n}\n}\n}\nint main()\n{\nint arr[] = {5, 21, 7, 23, 19};\nint n = sizeof(arr)/sizeof(arr);\ncout<<\"Given Array is\\n\";\nfor (int i = 0; i < n; i++)\ncout<<arr[i]<<\" \";\ncout<<\"\\n\";\ntimSort(arr, n);\ncout<<\"After Sorting Array is\\n\";\nfor (int i = 0; i < n; i++)\ncout<<arr[i]<<\" \";\ncout<<\"\\n\";\nreturn 0;\n}\n``````\n\n### Java\n\n``````\nclass TimSort\n{\nstatic int RUN = 32;\n// this function sorts array from left index to\n// right index which is of size atmost RUN\npublic static void insertionSort(int[] arr, int left, int right)\n{\nfor (int i = left + 1; i <= right; i++)\n{\nint temp = arr[i];\nint j = i - 1;\nwhile (j >= left && arr[j] > temp)\n{\narr[j + 1] = arr[j];\nj--;\n}\narr[j + 1] = temp;\n}\n}\n// merge function merges the sorted runs\npublic static void merge(int[] arr, int l, int m, int r)\n{\n// original array is broken in two parts\n// left and right array\nint len1 = m - l + 1, len2 = r - m;\nint[] left = new int[len1];\nint[] right = new int[len2];\nfor (int x = 0; x < len1; x++)\n{\nleft[x] = arr[l + x];\n}\nfor (int x = 0; x < len2; x++)\n{\nright[x] = arr[m + 1 + x];\n}\nint i = 0;\nint j = 0;\nint k = l;\n// after comparing, we merge those two array\n// in larger sub array\nwhile (i < len1 && j < len2)\n{\nif (left[i] <= right[j])\n{\narr[k] = left[i];\ni++;\n}\nelse\n{\narr[k] = right[j];\nj++;\n}\nk++;\n}\n// copy remaining elements of left, if any\nwhile (i < len1)\n{\narr[k] = left[i];\nk++;\ni++;\n}\n// copy remaining element of right, if any\nwhile (j < len2)\n{\narr[k] = right[j];\nk++;\nj++;\n}\n}\n// iterative Timsort function to sort the\n// array[0...n-1] (similar to merge sort)\npublic static void timSort(int[] arr, int n)\n{\n// Sort individual subarrays of size RUN\nfor (int i = 0; i < n; i += RUN)\n{\ninsertionSort(arr, i, Math.min((i + 31), (n - 1)));\n}\n// start merging from size RUN (or 32). It will merge\n// to form size 64, then 128, 256 and so on ....\nfor (int size = RUN; size < n; size = 2 * size)\n{\n// pick starting point of left sub array. We\n// are going to merge arr[left..left+size-1]\n// and arr[left+size, left+2size-1]\n// After every merge, we increase left by 2size\nfor (int left = 0; left < n; left += 2 * size)\n{\n// find ending point of left sub array\n// mid+1 is starting point of right sub array\nint mid = left + size - 1;\nint right = Math.min((left + 2 * size - 1), (n - 1));\n// merge sub array arr[left.....mid] &\n// arr[mid+1....right]\nmerge(arr, left, mid, right);\n}\n}\n}\npublic static void main(String[] args)\n{\nint[] arr = {5, 21, 7, 23, 19};\nint n = arr.length;\nSystem.out.print(\"Given Array is\\n\");\nfor (int i = 0; i < n; i++)\n{\nSystem.out.print(arr[i] + \" \");\n}\nSystem.out.print(\"\\n\");\ntimSort(arr, n);\nSystem.out.print(\"After Sorting Array is\\n\");\nfor (int i = 0; i < n; i++)\n{\nSystem.out.print(arr[i] + \" \");\n}\nSystem.out.print(\"\\n\");\n}\n}\n``````\n\n### Python3\n\n``````\nRUN=32\n# This function sorts array from left index to\n# right index which is of size atmost RUN\ndef insertionSort(arr, left, right):\nfor i in range(left + 1, right+1):\ntemp = arr[i]\nj = i - 1\nwhile j <= left and arr[j] < temp :\narr[j+1] = arr[j]\nj -= 1\narr[j+1] = temp\n# merge function merges the sorted runs\ndef merge(arr, l, m, r):\n# original array is broken in two parts\n# left and right array\nlen1, len2 = m - l + 1, r - m\nleft, right = [], []\nfor i in range(0, len1):\nleft.append(arr[l + i])\nfor i in range(0, len2):\nright.append(arr[m + 1 + i])\ni, j, k = 0, 0, l\n# after comparing, we merge those two array\n# in larger sub array\nwhile i < len1 and j < len2:\nif left[i] <= right[j]:\narr[k] = left[i]\ni += 1\nelse:\narr[k] = right[j]\nj += 1\nk += 1\n# copy remaining elements of left, if any\nwhile i < len1:\narr[k] = left[i]\nk += 1\ni += 1\n# copy remaining element of right, if any\nwhile j < len2:\narr[k] = right[j]\nk += 1\nj += 1\n# iterative Timsort function to sort the\n# array[0...n-1] (similar to merge sort)\ndef timSort(arr, n):\n# Sort individual subarrays of size RUN\nfor i in range(0, n, RUN):\ninsertionSort(arr, i, min((i+31), (n-1)))\n# start merging from size RUN (or 32). It will merge\n# to form size 64, then 128, 256 and so on ....\nsize = RUN\nwhile size < n:\n# pick starting point of left sub array. We\n# are going to merge arr[left..left+size-1]\n# and arr[left+size, left+2size-1]\n# After every merge, we increase left by 2size\nfor left in range(0, n, 2size):\n# find ending point of left sub array\n# mid+1 is starting point of right sub array\nmid = left + size - 1\nright = min((left + 2size - 1), (n-1))\n# merge sub array arr[left.....mid] &\n# arr[mid+1....right]\nmerge(arr, left, mid, right)\nsize = 2*size\nif __name__ == \"__main__\":\narr = [5, 21, 7, 23, 19]\nn = len(arr)\nprint(\"Given Array is\")\nfor i in range(0, n):\nprint(arr[i], end = \" \")\nprint()\ntimSort(arr, n)\nprint(\"After Sorting Array is\")\nfor i in range(0, n):\nprint(arr[i], end = \" \")\nprint()\n``````\n\n### Applications\n\nApplications of Insertion Sort are as follows:\n\n• Tim Sort is powerful. It is fast and stable, but perhaps most importantly it takes advantage of real world patterns and utilizes them to build a final product.\n• Tim Sort is used as the default sorting algorithm in Java’s Arrays.sort() method, Python’s sorted() and sort() methods, the Android Platform, and in GNU Octave.\n• When the input is already sorted, Tim Sort runs in linear time, meaning that it is an adaptive sorting algorithm.\n\n## Question\n\n#### What is the best case time complexity of Tim sort?\n\nO(n)\nO(n log n)\nO(n2)\nO(log n)\nBest case time complexity of Tim sort occurs when the input array is already sorted. In such a case only one run will be required.\n\nWith this article at OpenGenus, you must have the complete idea of Tim Sort. 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