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http://sangho.ai/paper/review/FB/	[ "# Practical Lessons from Predicting Clicks on Ads at Facebook\n\nPublished:\n\nWant to share what I learned, feeled after I read and study the paper. Thesedays, usually study with machine learning, so will start with that category first.\n\nThe first paper to study is ‘Practical Lessons from Predicting Clicks on Ads at Facebook’ which has issued from Facebook team for predicting clicks on Ads at Facebook.\n\nLet’s start the journey.\n\n### 0. Introduction\n\nAccording to the paper, over 750 million daily active users and over 1 million active advertisers are being predicted to click on Facebook ads. Authors has improved the click prediction rate by combining decision trees with logistic regression and they described this outperformed either of these mathods on its own by over 3%.\n\n### 1-1) Data\n\nUsed offline data selected from an arbitrary week of the 2013, 4Q Then partitioned into training and test set\n\n### 1-2) Evaluation metrics\n\nUses accuracy of prediction instead of profit-revenue related metrics. In this work, uses NE and calibration as major evaluation metric.\n\n### NE (Normalized Entropy, Normalized Cross Entropy)\n\n$NE=\\frac{-\\frac{1}{N}\\sum_{i=1}^{n}\\left ( \\frac{1+y_{i}}{2} \\log \\left ( p_{i} \\right ) + \\frac{1-y_{i}}{2} \\log \\left ( 1-p_{i} \\right )\\right )}{-(p\\log(p)+(1-p)log(1-p))}$\n\nNE is the predictive log loss normalized by the entropy of the background CTR then it can be set as above, when training set has N examples with labels $y_{i}\\in\\left \\{-1,1 \\right \\}$ and estimated probability of click $p_{i}$. ($i=1,2,3,...N$) So as prediction accuracy is higer (as number of $y_{i}$ is larger), NE will be lower.\n\n### Calibration (Normalized Entropy, Normalized Cross Entropy)\n\nCalibration is the ratio of the average estimated CTR and empirical CTR. In other words, it is the ratio of the number of expected clicks to the number of actually observed clicks as below. $\\frac{number \\thinspace of \\thinspace actually \\thinspace observed \\thinspace clicks}{number \\thinspace of \\thinspace expected \\thinspace clicks}$\n\n### 2-1) Model structure", null, "Suggests the concatenation of boosted decision trees and of a probabilistic sparse linear classifier. For training, used SGD based linear regression and *BOPR (Bayesian online learning scheme for probit regression) To imporve the accuracy of linear classifier, there are 2 ways. 1 : Treat continuous features as categorical feature 2 : Build tuple input features like crating a new categorical feature that taking all possible values with Cartesian product. (Of course useless combinations can be pruned out)\n\nAnd boosted decision trees are a powerful and very convenient way to implement non-linear and tuple transformations described above. When you see the tree above, and assume each node has binary output like 0,1,0,1,0. Then linear classifier has binary vector input as [0,1,0,1,0]. The NE comparison table shows combination of decision tree and LR decreases NE by more than 3.4% relative to the model with no tree transforms.", null, "### 2-2) Data freshness\n\nExperiment shows data freshness effects the prediction accuracy. To maximize the data freshness, the linear online classifier is one option to train directly as the labelled Ads impressions arrive. From experiments, it shows “Per-coordinate learning rate” $\\eta_{t,i}=\\frac{\\alpha }{\\beta+\\sqrt{\\sum_{j=1}^{t}\\bigtriangledown_{j,i}^{2}}}$ has the best prediction rate for SGD-based online learning of logistic regression, compared to “per-weight square root”,”per-weight”,”global” or “constant” rate.\n\nTrained the data set and compared the prediction accuracy between LR and BOPR. Relative to LR, BOPR has a liitle bit lower NE (99.82% / 100%). One advantage of LR is smaller model size because there is only a weight associated to each feature. One advantage of BOPR is that being a Bayesian formulation, it provides a full predictive distribution over the probability of click. This can be used to compute percentiles of the predictive distribution for explore/exploit learning schemes.\n\n### 3-1) Number of Boosting tree\n\nProceed experiments with tree number 1 ~ 2000. As the number of trees increases NE decreases, but the gain from adding trees yields diminishing return as the graph of -log(x).\n\n### 3-2) Boosting features\n\nUsually, small number of features contributes the majority of explanatory power while the remaining features have only a marginal contribution. And historical features provide considerably more explanatory power than contextual features. After ordering features by importance, top 10 features are all historical features and there are only 2 features among the top 20 features.\n\nThe value of contextual features depends exclusively on current information regarding the context in which an ad is to be shown, such as the device used by the users or the current page that the user is on.\nOn the contrary, the historical features depend on previous interaction for the ad or user, for example the click through rate of the ad in last week, or the average click through rate of the user\n\n\n### 3-3) Uniform subsampling, Negative downsampling, Model Re-calibration\n\nTo reduce the cost of training, uniform subsampling is considered. To improve class imbalance between +1 and -1, negative downsampling can improve the performance After downsampling with negative label, it needs to re-calibrate the model because negative downsampling effects the performance as well. (For example, if the average CTR before samling is 0.1% and we do a 0.01 negative downsampling, the empirical CTR will become roughly 10%)\n\nCategories:" ]	[ null, "https://github.com/puhuk/puhuk.github.io/blob/master/img/FB-tree.PNG", null, "https://github.com/puhuk/puhuk.github.io/blob/master/img/FB-comp_table.PNG", null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.8894722,"math_prob":0.9567523,"size":5589,"snap":"2022-40-2023-06","text_gpt3_token_len":1281,"char_repetition_ratio":0.10778872,"word_repetition_ratio":0.014285714,"special_character_ratio":0.22508499,"punctuation_ratio":0.09645669,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9908923,"pos_list":[0,1,2,3,4],"im_url_duplicate_count":[null,2,null,2,null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2022-10-05T05:52:46Z\",\"WARC-Record-ID\":\"<urn:uuid:7bed4a0d-82ad-4564-a9f6-b6018d33a153>\",\"Content-Length\":\"20382\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:cb0d3d4a-1841-478d-bdfa-4a8db090f91a>\",\"WARC-Concurrent-To\":\"<urn:uuid:b5bb5892-fb7d-47c8-81e7-dc5f63c8432b>\",\"WARC-IP-Address\":\"185.199.109.153\",\"WARC-Target-URI\":\"http://sangho.ai/paper/review/FB/\",\"WARC-Payload-Digest\":\"sha1:4AMFTZ3SX5SIEMTIJ4GKFOAHGZGY6LKZ\",\"WARC-Block-Digest\":\"sha1:GZFPH5GSJE43CMHCVEPZXFULS2HIU2SJ\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2022/CC-MAIN-2022-40/CC-MAIN-2022-40_segments_1664030337537.25_warc_CC-MAIN-20221005042446-20221005072446-00407.warc.gz\"}"}
https://docs.telerik.com/devtools/android/controls/chart/axes/chart-axes-categorical	[ "When RadCartesianChartView visualizes CategoricalSeries, it needs an axis that can represent the different categories. The CategoricalAxis extends the base CartesianAxis class and is used to displays a range of categories. Categories are built depending on the Category value of each CategoricalDataPoint present in the owning CategoricalSeries chart series. The axis is divided into discrete slots and each data point is visualized in the slot corresponding to its categorical value.\n\nThe CategoricalAxis extends the base CartesianAxis class and is used to displays a range of categories. Categories are built depending on the `Category` value of each `CategoricalDataPoint` present in the owning CategoricalSeries chart series. The axis is divided into discrete slots and each data point is visualized in the slot corresponding to its categorical value.\n\n## Example\n\nYou can read from the Getting Started page how to define the `MonthResult` type and declare the initData() method.\n\nAfter you create the method for initialization of sample data, you can create a RadCartesianChartView with LineSeries by adding the following code to the onCreate() method of your Activity.\n\n`````` initData();\n\nLineSeries lineSeries = new LineSeries();\nlineSeries.setCategoryBinding(new PropertyNameDataPointBinding(\"Month\"));\nlineSeries.setValueBinding(new PropertyNameDataPointBinding(\"Result\"));\nlineSeries.setData(this.monthResults);\n\nCategoricalAxis horizontalAxis = new CategoricalAxis();\nchartView.setHorizontalAxis(horizontalAxis);\n\nLinearAxis verticalAxis = new LinearAxis();\nchartView.setVerticalAxis(verticalAxis);\n\nViewGroup rootView = (ViewGroup)findViewById(R.id.container);\n``````\n`````` InitData();\n\nLineSeries lineSeries = new LineSeries();\nlineSeries.CategoryBinding = new MonthResultDataBinding (\"Month\");\nlineSeries.ValueBinding = new MonthResultDataBinding (\"Result\");\nlineSeries.Data = (Java.Lang.IIterable)this.monthResults;\n\nCategoricalAxis horizontalAxis = new CategoricalAxis();\nchartView.HorizontalAxis = horizontalAxis;\n\nLinearAxis verticalAxis = new LinearAxis();\nchartView.VerticalAxis = verticalAxis;\n\nViewGroup rootView = (ViewGroup)FindViewById(Resource.Id.container);\n``````\n\nThis example assumes that you root container has id `container`\n\nHere's the result:", null, "## Features\n\n### Plot Mode\n\nThe CategoricalAxis allows you to define how exactly the axis will be plotted on the viewport of the chart. The possible values are:\n\n• AxisPlotMode.BETWEEN_TICKS: Points are plotted in the middle of the range, defined between each two ticks.\n• AxisPlotMode.ON_TICKS: Points are plotted over each tick.\n• AxisPlotMode.ON_TICKS_PADDED: Points are plotted over each tick with half a step padding applied on both ends of the axis.\n\nYou can get the current value with the getPlotMode() method and change the value with the setPlotMode(AxisPlotMode) method.\n\n### Gap Length\n\nDefines the distance (in logical units) between two adjacent categories. Default value is `0.3`. For example if you have BarSeries, you can decrease the space between the bars from the different categories by setting the Gap Length to a value lower than `0.3`. You can get the current value with getGapLength() and set a new value with setGapLength(double). The possible values are from the `(0, 1)` interval.\n\n### Major Tick Interval\n\nDefines the step at which major ticks are generated. The default and also minimum value is 1. This property also affects axis labels as they are generated on a per major tick basis. You can get the current value with the getMajorTicksInterval() method and set a new value with setMajorTickInterval(int). For example, if you don't want to display all ticks, but instead only half of them (display the first, third, fifth, etc. ticks), you should set the major tick interval to `2`:\n\n`````` horizontalAxis.setMajorTickInterval(2);\n``````\n`````` horizontalAxis.MajorTickInterval = 2;\n``````" ]	[ null, "https://docs.telerik.com/devtools/android/controls/chart/axes/images/chart-axes-categorical-1.png", null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.71218413,"math_prob":0.90350145,"size":3702,"snap":"2019-13-2019-22","text_gpt3_token_len":766,"char_repetition_ratio":0.16657653,"word_repetition_ratio":0.29039302,"special_character_ratio":0.18071313,"punctuation_ratio":0.15437393,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9748547,"pos_list":[0,1,2],"im_url_duplicate_count":[null,4,null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2019-03-23T16:24:18Z\",\"WARC-Record-ID\":\"<urn:uuid:99d6b947-3d53-4319-9f18-30d3d16f4b82>\",\"Content-Length\":\"11540\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:115201b7-58cf-4c60-b14f-8124f123e6ae>\",\"WARC-Concurrent-To\":\"<urn:uuid:c5b539b7-e9b1-449c-a209-948b3cd63ba5>\",\"WARC-IP-Address\":\"50.56.17.213\",\"WARC-Target-URI\":\"https://docs.telerik.com/devtools/android/controls/chart/axes/chart-axes-categorical\",\"WARC-Payload-Digest\":\"sha1:PZ444MTSM5ESI5BCHEYKVDED6K2W45IN\",\"WARC-Block-Digest\":\"sha1:NFSV32X4QVZ53ECAN5PYMYSGZ7RG7CNE\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2019/CC-MAIN-2019-13/CC-MAIN-2019-13_segments_1552912202889.30_warc_CC-MAIN-20190323161556-20190323183556-00507.warc.gz\"}"}
https://5doc.co/document/zx5n0llv-regular-orbits.html	[ "# Regular orbits\n\nN/A\nN/A\nProtected\n\nShare \"Regular orbits\"\n\nCopied!\n82\n0\n0\n\nFull text\n\n(1)\n\n### Orbits of linear groups\n\nMartin Liebeck\n\nImperial College London\n\n(2)\n\nLetG ≤GLn(F) =GL(V), Fa field, G finite.\n\nWill discuss results on orbits ofG onV:\n\n1 Regular orbits\n\n2 Number of orbits\n\n3 Arithmetic conditions on orbit sizes, applications\n\nCorresponding aﬃne permutation groupH :=VG ≤AGL(V), V = translation subgroup,G =H0 stabilizer of zero vector. Orbits ofG aresuborbits ofH, andHis primitive iﬀ G is irreducible onV.\n\n(3)\n\nLetG ≤GLn(F) =GL(V), Fa field, G finite.\n\nWill discuss results on orbits ofG onV:\n\n1 Regular orbits\n\n2 Number of orbits\n\n3 Arithmetic conditions on orbit sizes, applications\n\nCorresponding aﬃne permutation groupH :=VG ≤AGL(V), V = translation subgroup,G =H0 stabilizer of zero vector. Orbits ofG aresuborbits ofH, andHis primitive iﬀ G is irreducible onV.\n\n(4)\n\nLetG ≤GLn(F) =GL(V), Fa field, G finite.\n\nWill discuss results on orbits ofG onV:\n\n1 Regular orbits\n\n2 Number of orbits\n\n3 Arithmetic conditions on orbit sizes, applications\n\nCorresponding aﬃne permutation groupH :=VG ≤AGL(V), V = translation subgroup,G =H0 stabilizer of zero vector. Orbits ofG aresuborbits ofH, andHis primitive iﬀ G is irreducible onV.\n\n(5)\n\nLetG ≤GLn(F) =GL(V), Fa field, G finite.\n\nWill discuss results on orbits ofG onV:\n\n1 Regular orbits\n\n2 Number of orbits\n\n3 Arithmetic conditions on orbit sizes, applications\n\nCorresponding aﬃne permutation groupH :=VG ≤AGL(V), V = translation subgroup,G =H0 stabilizer of zero vector. Orbits ofG aresuborbits ofH, andHis primitive iﬀ G is irreducible onV.\n\n(6)\n\nLetG ≤GLn(F) =GL(V), Fa field, G finite.\n\nWill discuss results on orbits ofG onV:\n\n1 Regular orbits\n\n2 Number of orbits\n\n3 Arithmetic conditions on orbit sizes, applications\n\nCorresponding aﬃne permutation groupH:=VG ≤AGL(V), V = translation subgroup,G =H0 stabilizer of zero vector. Orbits ofG aresuborbits ofH, andHis primitive iﬀ G is irreducible onV.\n\n(7)\n\n### Regular orbits\n\nLetG ≤GLn(F) =GL(V). ThenG has aregular orbit onV if∃ v∈V\\0 such thatGv = 1. Regular orbit is vG ={vg :g ∈G}, size\|G\|.\n\nIfH=VG ≤AGL(V) is corresponding aﬃne permutation group onV, this saysH0v = 1, i.e. H has abase of size 2.\n\nDo regular orbits exist?\n\nSometimes no: eg. ifG =GLn(q) (where F=Fq finite). In general, no regular orbit if\|G\| ≥ \|V\|.\n\nSometimes yes: eg. G =�s�,s a Singer cycle inGLn(q) of order qn−1.\n\n(8)\n\n### Regular orbits\n\nLetG ≤GLn(F) =GL(V). ThenG has a regular orbit onV if∃ v∈V\\0 such thatGv = 1. Regular orbit is vG ={vg :g ∈G}, size\|G\|.\n\nIfH=VG ≤AGL(V) is corresponding aﬃne permutation group onV, this saysH0v = 1, i.e. H has abase of size 2.\n\nDo regular orbits exist?\n\nSometimes no: eg. ifG =GLn(q) (where F=Fq finite). In general, no regular orbit if\|G\| ≥ \|V\|.\n\nSometimes yes: eg. G =�s�,s a Singer cycle inGLn(q) of order qn−1.\n\n(9)\n\n### Regular orbits\n\nLetG ≤GLn(F) =GL(V). ThenG has a regular orbit onV if∃ v∈V\\0 such thatGv = 1. Regular orbit is vG ={vg :g ∈G}, size\|G\|.\n\nIfH=VG ≤AGL(V) is corresponding aﬃne permutation group onV, this saysH0v = 1, i.e. H has abase of size 2.\n\nDo regular orbits exist?\n\nSometimes no: eg. ifG =GLn(q) (where F=Fq finite). In general, no regular orbit if\|G\| ≥ \|V\|.\n\nSometimes yes: eg. G =�s�,s a Singer cycle inGLn(q) of order qn−1.\n\n(10)\n\n### Regular orbits\n\nLetG ≤GLn(F) =GL(V). ThenG has a regular orbit onV if∃ v∈V\\0 such thatGv = 1. Regular orbit is vG ={vg :g ∈G}, size\|G\|.\n\nIfH=VG ≤AGL(V) is corresponding aﬃne permutation group onV, this saysH0v = 1, i.e. H has abase of size 2.\n\nDo regular orbits exist?\n\nSometimes no: eg. ifG =GLn(q) (where F=Fq finite). In general, no regular orbit if\|G\| ≥ \|V\|.\n\nSometimes yes: eg. G =�s�,s a Singer cycle inGLn(q) of order qn−1.\n\n(11)\n\n### Regular orbits\n\nLetG ≤GLn(F) =GL(V). ThenG has a regular orbit onV if∃ v∈V\\0 such thatGv = 1. Regular orbit is vG ={vg :g ∈G}, size\|G\|.\n\nIfH=VG ≤AGL(V) is corresponding aﬃne permutation group onV, this saysH0v = 1, i.e. H has abase of size 2.\n\nDo regular orbits exist?\n\nSometimes no: eg. ifG =GLn(q) (where F=Fq finite).\n\nIn general, no regular orbit if\|G\| ≥ \|V\|.\n\nSometimes yes: eg. G =�s�,s a Singer cycle inGLn(q) of order qn−1.\n\n(12)\n\n### Regular orbits\n\nLetG ≤GLn(F) =GL(V). ThenG has a regular orbit onV if∃ v∈V\\0 such thatGv = 1. Regular orbit is vG ={vg :g ∈G}, size\|G\|.\n\nIfH=VG ≤AGL(V) is corresponding aﬃne permutation group onV, this saysH0v = 1, i.e. H has abase of size 2.\n\nDo regular orbits exist?\n\nSometimes no: eg. ifG =GLn(q) (where F=Fq finite). In general, no regular orbit if\|G\| ≥ \|V\|.\n\nSometimes yes: eg. G =�s�,s a Singer cycle inGLn(q) of order qn−1.\n\n(13)\n\n### Regular orbits\n\nLetG ≤GLn(F) =GL(V). ThenG has a regular orbit onV if∃ v∈V\\0 such thatGv = 1. Regular orbit is vG ={vg :g ∈G}, size\|G\|.\n\nIfH=VG ≤AGL(V) is corresponding aﬃne permutation group onV, this saysH0v = 1, i.e. H has abase of size 2.\n\nDo regular orbits exist?\n\nSometimes no: eg. ifG =GLn(q) (where F=Fq finite). In general, no regular orbit if\|G\| ≥ \|V\|.\n\nSometimes yes: eg. G =�s�,s a Singer cycle inGLn(q) of order qn−1.\n\n(14)\n\n### Regular orbits\n\nWhenFis infinite:\n\nLemma Let G ≤GLn(F) =GL(V), G finite, Finfinite. Then G has a regular orbit on V .\n\nProof Suppose false. ThenGv �= 1 ∀v ∈V. So everyv ∈V lies inCV(g) :={v ∈V :vg =v} for someg ∈G\\1, ie.\n\nV = �\n\ng∈G\\1\n\nCV(g).\n\nBut asFis infinite, V is not a union of finitely many proper subspaces. Contradiction.\n\nForF finite this argument shows ∃regular orbit ifF finite and\n\n\|F\|>\|G\|.\n\n(15)\n\n### Regular orbits\n\nWhenFis infinite:\n\nLemma Let G ≤GLn(F) =GL(V), G finite,F infinite. Then G has a regular orbit on V .\n\nProof Suppose false. ThenGv �= 1 ∀v ∈V. So everyv ∈V lies inCV(g) :={v ∈V :vg =v} for someg ∈G\\1, ie.\n\nV = �\n\ng∈G\\1\n\nCV(g).\n\nBut asFis infinite, V is not a union of finitely many proper subspaces. Contradiction.\n\nForF finite this argument shows ∃regular orbit ifF finite and\n\n\|F\|>\|G\|.\n\n(16)\n\n### Regular orbits\n\nWhenFis infinite:\n\nLemma Let G ≤GLn(F) =GL(V), G finite,F infinite. Then G has a regular orbit on V .\n\nProof Suppose false. ThenGv �= 1 ∀v ∈V.\n\nSo every v ∈V lies inCV(g) :={v ∈V :vg =v} for someg ∈G\\1, ie.\n\nV = �\n\ng∈G\\1\n\nCV(g).\n\nBut asFis infinite, V is not a union of finitely many proper subspaces. Contradiction.\n\nForF finite this argument shows ∃regular orbit ifF finite and\n\n\|F\|>\|G\|.\n\n(17)\n\n### Regular orbits\n\nWhenFis infinite:\n\nLemma Let G ≤GLn(F) =GL(V), G finite,F infinite. Then G has a regular orbit on V .\n\nProof Suppose false. ThenGv �= 1 ∀v ∈V. So everyv ∈V lies inCV(g) :={v ∈V :vg =v} for someg ∈G\\1, ie.\n\nV = �\n\ng∈G\\1\n\nCV(g).\n\nBut asFis infinite, V is not a union of finitely many proper subspaces. Contradiction.\n\nForF finite this argument shows ∃regular orbit ifF finite and\n\n\|F\|>\|G\|.\n\n(18)\n\n### Regular orbits\n\nWhenFis infinite:\n\nLemma Let G ≤GLn(F) =GL(V), G finite,F infinite. Then G has a regular orbit on V .\n\nProof Suppose false. ThenGv �= 1 ∀v ∈V. So everyv ∈V lies inCV(g) :={v ∈V :vg =v} for someg ∈G\\1, ie.\n\nV = �\n\ngG\\1\n\nCV(g).\n\nBut asFis infinite, V is not a union of finitely many proper subspaces. Contradiction.\n\nForF finite this argument shows ∃regular orbit ifF finite and\n\n\|F\|>\|G\|.\n\n(19)\n\n### Regular orbits\n\nWhenFis infinite:\n\nLemma Let G ≤GLn(F) =GL(V), G finite,F infinite. Then G has a regular orbit on V .\n\nProof Suppose false. ThenGv �= 1 ∀v ∈V. So everyv ∈V lies inCV(g) :={v ∈V :vg =v} for someg ∈G\\1, ie.\n\nV = �\n\ngG\\1\n\nCV(g).\n\nBut asFis infinite, V is not a union of finitely many proper subspaces. Contradiction.\n\nForF finite this argument shows ∃regular orbit ifF finite and\n\n\|F\|>\|G\|.\n\n(20)\n\n### Regular orbits\n\nWhenFis infinite:\n\nLemma Let G ≤GLn(F) =GL(V), G finite,F infinite. Then G has a regular orbit on V .\n\nProof Suppose false. ThenGv �= 1 ∀v ∈V. So everyv ∈V lies inCV(g) :={v ∈V :vg =v} for someg ∈G\\1, ie.\n\nV = �\n\ngG\\1\n\nCV(g).\n\nBut asFis infinite, V is not a union of finitely many proper subspaces. Contradiction.\n\nForF finite this argument shows∃ regular orbit ifF finite and\n\n\|F\|>\|G\|.\n\n(21)\n\n### Regular orbits\n\nWhenF=Fq finite,G ≤GLn(q) =GL(V):\n\nAim (i) If\|G\| ≥ \|V\|,G has no regular orbit onV\n\n(ii) If\|G\|<\|V\|, proveG has a regular orbit, with the following exceptions....????\n\nFar oﬀ. Delicate:\n\nExample 1 Let G =Sc <GLc−1(p) =GL(V), where p >c and V ={(a1, . . . ,ac) :ai ∈Fp,�\n\nai = 0}. Then G has regular orbits onV. Number of regular orbits is\n\n1\n\nc!(p−1)(p−2)· · ·(p−c+ 1).\n\nExample 2 Let G =Sc×C2 <GLc1(p) =GL(V), where p=c+ 1, V as above, C2 =�−1V�. ThenG has no regular orbits onV.\n\n(22)\n\n### Regular orbits\n\nWhenF=Fq finite,G ≤GLn(q) =GL(V):\n\nAim (i) If\|G\| ≥ \|V\|,G has no regular orbit onV\n\n(ii) If\|G\|<\|V\|, proveG has a regular orbit, with the following exceptions....????\n\nFar oﬀ. Delicate:\n\nExample 1 Let G =Sc <GLc−1(p) =GL(V), where p >c and V ={(a1, . . . ,ac) :ai ∈Fp,�\n\nai = 0}. Then G has regular orbits onV. Number of regular orbits is\n\n1\n\nc!(p−1)(p−2)· · ·(p−c+ 1).\n\nExample 2 Let G =Sc×C2 <GLc1(p) =GL(V), where p=c+ 1, V as above, C2 =�−1V�. ThenG has no regular orbits onV.\n\n(23)\n\n### Regular orbits\n\nWhenF=Fq finite,G ≤GLn(q) =GL(V):\n\nAim (i) If\|G\| ≥ \|V\|,G has no regular orbit onV\n\n(ii) If\|G\|<\|V\|, proveG has a regular orbit, with the following exceptions....????\n\nFar oﬀ. Delicate:\n\nExample 1 Let G =Sc <GLc−1(p) =GL(V), where p >c and V ={(a1, . . . ,ac) :ai ∈Fp,�\n\nai = 0}. ThenG has regular orbits onV. Number of regular orbits is\n\n1\n\nc!(p−1)(p−2)· · ·(p−c+ 1).\n\nExample 2 Let G =Sc×C2 <GLc1(p) =GL(V), where p=c+ 1, V as above, C2 =�−1V�. ThenG has no regular orbits onV.\n\n(24)\n\n### Regular orbits\n\nWhenF=Fq finite,G ≤GLn(q) =GL(V):\n\nAim (i) If\|G\| ≥ \|V\|,G has no regular orbit onV\n\n(ii) If\|G\|<\|V\|, proveG has a regular orbit, with the following exceptions....????\n\nFar oﬀ. Delicate:\n\nExample 1 Let G =Sc <GLc−1(p) =GL(V), where p >c and V ={(a1, . . . ,ac) :ai ∈Fp,�\n\nai = 0}. ThenG has regular orbits onV. Number of regular orbits is\n\n1\n\nc!(p−1)(p−2)· · ·(p−c+ 1).\n\nExample 2 Let G =Sc×C2 <GLc1(p) =GL(V), where p=c+ 1, V as above, C2 =�−1V�. ThenG has no regular orbits onV.\n\n(25)\n\n### More on egular orbits\n\nIn general letG <GLn(q) with q varying, but fixed Brauer\n\ncharacter ofG. There is a poly f(x) of degree n such that number of regular orbits ofG is f(q). So ∃regular orbits for all but at mostn values ofq.\n\nEg. G =Sc <GLc−1(q), char(Fq)>n: here\n\nf(q) = c!1(q−1)(q−2)· · ·(q−c+ 1). This is a poly in q with roots equal to the exponents of the Weyl groupW(Ac1)∼=Sc. Same holds for all finite reflection groups in their natural\n\nrepresentations (Orlik-Solomon). Eg. forG =W(F4)<GL4(q), number of regular orbits\n\n1\n\n\|W(F4)\|(q−1)(q−5)(q−7)(q−11).\n\nNot always so nice, eg. forG =PSL2(7)<GL3(q) (of index 2 in a unitary reflection group),f(q) = 1681 (q−1)(q2+q−48).\n\n(26)\n\n### More on egular orbits\n\nIn general letG <GLn(q) with q varying, but fixed Brauer\n\ncharacter ofG. There is a poly f(x) of degree n such that number of regular orbits ofG is f(q).\n\nSo ∃regular orbits for all but at mostn values ofq.\n\nEg. G =Sc <GLc−1(q), char(Fq)>n: here\n\nf(q) = c!1(q−1)(q−2)· · ·(q−c+ 1). This is a poly in q with roots equal to the exponents of the Weyl groupW(Ac1)∼=Sc. Same holds for all finite reflection groups in their natural\n\nrepresentations (Orlik-Solomon). Eg. forG =W(F4)<GL4(q), number of regular orbits\n\n1\n\n\|W(F4)\|(q−1)(q−5)(q−7)(q−11).\n\nNot always so nice, eg. forG =PSL2(7)<GL3(q) (of index 2 in a unitary reflection group),f(q) = 1681 (q−1)(q2+q−48).\n\n(27)\n\n### More on egular orbits\n\nIn general letG <GLn(q) with q varying, but fixed Brauer\n\ncharacter ofG. There is a poly f(x) of degree n such that number of regular orbits ofG is f(q). So ∃regular orbits for all but at mostn values ofq.\n\nEg. G =Sc <GLc−1(q), char(Fq)>n: here\n\nf(q) = c!1(q−1)(q−2)· · ·(q−c+ 1). This is a poly in q with roots equal to the exponents of the Weyl groupW(Ac1)∼=Sc. Same holds for all finite reflection groups in their natural\n\nrepresentations (Orlik-Solomon). Eg. forG =W(F4)<GL4(q), number of regular orbits\n\n1\n\n\|W(F4)\|(q−1)(q−5)(q−7)(q−11).\n\nNot always so nice, eg. forG =PSL2(7)<GL3(q) (of index 2 in a unitary reflection group),f(q) = 1681 (q−1)(q2+q−48).\n\n(28)\n\n### More on egular orbits\n\nIn general letG <GLn(q) with q varying, but fixed Brauer\n\ncharacter ofG. There is a poly f(x) of degree n such that number of regular orbits ofG is f(q). So ∃regular orbits for all but at mostn values ofq.\n\nEg. G =Sc <GLc1(q), char(Fq)>n: here\n\nf(q) = c!1(q−1)(q−2)· · ·(q−c+ 1). This is a poly in q with roots equal to the exponents of the Weyl groupW(Ac1)∼=Sc.\n\nSame holds for all finite reflection groups in their natural\n\nrepresentations (Orlik-Solomon). Eg. forG =W(F4)<GL4(q), number of regular orbits\n\n1\n\n\|W(F4)\|(q−1)(q−5)(q−7)(q−11).\n\nNot always so nice, eg. forG =PSL2(7)<GL3(q) (of index 2 in a unitary reflection group),f(q) = 1681 (q−1)(q2+q−48).\n\n(29)\n\n### More on egular orbits\n\nIn general letG <GLn(q) with q varying, but fixed Brauer\n\ncharacter ofG. There is a poly f(x) of degree n such that number of regular orbits ofG is f(q). So ∃regular orbits for all but at mostn values ofq.\n\nEg. G =Sc <GLc1(q), char(Fq)>n: here\n\nf(q) = c!1(q−1)(q−2)· · ·(q−c+ 1). This is a poly in q with roots equal to the exponents of the Weyl groupW(Ac1)∼=Sc. Same holds for all finite reflection groups in their natural\n\nrepresentations (Orlik-Solomon). Eg. forG =W(F4)<GL4(q), number of regular orbits\n\n1\n\n\|W(F4)\|(q−1)(q−5)(q−7)(q−11).\n\nNot always so nice, eg. forG =PSL2(7)<GL3(q) (of index 2 in a unitary reflection group),f(q) = 1681 (q−1)(q2+q−48).\n\n(30)\n\n### More on egular orbits\n\nIn general letG <GLn(q) with q varying, but fixed Brauer\n\ncharacter ofG. There is a poly f(x) of degree n such that number of regular orbits ofG is f(q). So ∃regular orbits for all but at mostn values ofq.\n\nEg. G =Sc <GLc1(q), char(Fq)>n: here\n\nf(q) = c!1(q−1)(q−2)· · ·(q−c+ 1). This is a poly in q with roots equal to the exponents of the Weyl groupW(Ac1)∼=Sc. Same holds for all finite reflection groups in their natural\n\nrepresentations (Orlik-Solomon). Eg. forG =W(F4)<GL4(q), number of regular orbits\n\n1\n\n\|W(F4)\|(q−1)(q−5)(q−7)(q−11).\n\nNot always so nice, eg. forG =PSL2(7)<GL3(q) (of index 2 in a unitary reflection group),f(q) = 1681 (q−1)(q2+q−48).\n\n(31)\n\n### Still more on regular orbits\n\nGeneral theory (Pahlings-Plesken): G <GLn(q) with q varying, fixed Brauer character. For each subgroupH <G there is a poly fH(q) of degree dimCV(H) such that number of orbits ofG with stabilizer conjugate toH isfH(q). ∃methods for computing these polys using “table of marks” ofG.\n\nEg. here’s the table of marks forA5: A5/C1 60\n\nA5/C2 30 2 A5/C3 20 2 A5/V4 15 3 3\n\nA5/C5 12 2\n\nA5/S3 10 2 1 1\n\nA5/D10 6 2 1 1\n\nA5/A4 5 1 2 1 1\n\nA5/A5 1 1 1 1 1 1 1 1 1\n\nC1 C2 C3 V4 C5 S3 D10 A4 A5\n\n(32)\n\n### Still more on regular orbits\n\nGeneral theory (Pahlings-Plesken): G <GLn(q) with q varying, fixed Brauer character. For each subgroupH <G there is a poly fH(q) of degree dimCV(H) such that number of orbits ofG with stabilizer conjugate toH isfH(q). ∃ methods for computing these polys using “table of marks” ofG.\n\nEg. here’s the table of marks forA5: A5/C1 60\n\nA5/C2 30 2 A5/C3 20 2 A5/V4 15 3 3\n\nA5/C5 12 2\n\nA5/S3 10 2 1 1\n\nA5/D10 6 2 1 1\n\nA5/A4 5 1 2 1 1\n\nA5/A5 1 1 1 1 1 1 1 1 1\n\nC1 C2 C3 V4 C5 S3 D10 A4 A5\n\n(33)\n\n### Still more on regular orbits\n\nGeneral theory (Pahlings-Plesken): G <GLn(q) with q varying, fixed Brauer character. For each subgroupH <G there is a poly fH(q) of degree dimCV(H) such that number of orbits ofG with stabilizer conjugate toH isfH(q). ∃ methods for computing these polys using “table of marks” ofG.\n\nEg. here’s the table of marks forA5: A5/C1 60\n\nA5/C2 30 2 A5/C3 20 2 A5/V4 15 3 3\n\nA5/C5 12 2\n\nA5/S3 10 2 1 1\n\nA5/D10 6 2 1 1\n\nA5/A4 5 1 2 1 1\n\nA5/A5 1 1 1 1 1 1 1 1 1\n\nC1 C2 C3 V4 C5 S3 D10 A4 A5\n\n(34)\n\n### Yet more on regular orbits\n\nRegular orbits ofG <GLn(q) crop up in several areas. A couple of examples:\n\n1. Supposeallorbits regular, ie. Gv = 1∀v ∈V\\0. Then aﬃne groupH=VG ≤AGL(V) is a Frobenius group(Hvw = 1∀v,w) andG aFrobenius complement. Classified by Zassenhaus. Eg. SL2(5)<GL2(q) (char >5)\n\nOrSL2(5)⊗(Cr.Cs)<GL2(q)⊗GLr(q)<GL2r(q) (r\|s−1, r,s >5)\n\n(35)\n\n### Yet more on regular orbits\n\nRegular orbits ofG <GLn(q) crop up in several areas. A couple of examples:\n\n1. Supposeallorbits regular, ie. Gv = 1∀v ∈V\\0. Then aﬃne groupH=VG ≤AGL(V) is a Frobenius group(Hvw = 1∀v,w) andG aFrobenius complement. Classified by Zassenhaus. Eg. SL2(5)<GL2(q) (char >5)\n\nOrSL2(5)⊗(Cr.Cs)<GL2(q)⊗GLr(q)<GL2r(q) (r\|s−1, r,s >5)\n\n(36)\n\n### Yet more on regular orbits\n\nRegular orbits ofG <GLn(q) crop up in several areas. A couple of examples:\n\n1. Supposeallorbits regular, ie. Gv = 1∀v ∈V\\0. Then aﬃne groupH=VG ≤AGL(V) is a Frobenius group(Hvw = 1∀v,w) andG aFrobenius complement. Classified by Zassenhaus.\n\nEg. SL2(5)<GL2(q) (char >5)\n\nOrSL2(5)⊗(Cr.Cs)<GL2(q)⊗GLr(q)<GL2r(q) (r\|s−1, r,s >5)\n\n(37)\n\n### Yet more on regular orbits\n\nRegular orbits ofG <GLn(q) crop up in several areas. A couple of examples:\n\n1. Supposeallorbits regular, ie. Gv = 1∀v ∈V\\0. Then aﬃne groupH=VG ≤AGL(V) is a Frobenius group(Hvw = 1∀v,w) andG aFrobenius complement. Classified by Zassenhaus.\n\nEg. SL2(5)<GL2(q) (char >5)\n\nOrSL2(5)⊗(Cr.Cs)<GL2(q)⊗GLr(q)<GL2r(q) (r\|s−1, r,s >5)\n\n(38)\n\n### Yet more on regular orbits\n\nRegular orbits ofG <GLn(q) crop up in several areas. A couple of examples:\n\n1. Supposeallorbits regular, ie. Gv = 1∀v ∈V\\0. Then aﬃne groupH=VG ≤AGL(V) is a Frobenius group(Hvw = 1∀v,w) andG aFrobenius complement. Classified by Zassenhaus.\n\nEg. SL2(5)<GL2(q) (char >5)\n\nOrSL2(5)⊗(Cr.Cs)<GL2(q)⊗GLr(q)<GL2r(q) (r\|s−1, r,s >5)\n\n(39)\n\n### Yet more on regular orbits\n\n2. The k(GV)-problem This is\n\nConjecture Let G <GLn(p) =GL(V), G a p-group. The number of conjugacy classes k(VG) in the semidirect product VG satisfies\n\nk(VG)≤ \|V\|.\n\nEquivalent top-soluble case of Brauer’sk(B)-problem. Equality can hold, eg. G =�Singer−cycle� ∼=Cpn1\n\nRobinson-Thompson reduction: conjecture proved if we can show that forG of “simple type” or ”extraspecial type”, there exists a regular orbit ofG on V.\n\n“Simple type”: G has an irreducible normal subgroup H such that H/Z(H) is non-abelian simple.\n\n(40)\n\n### Yet more on regular orbits\n\n2. The k(GV)-problem This is\n\nConjecture Let G <GLn(p) =GL(V), G a p-group. The number of conjugacy classes k(VG) in the semidirect product VG satisfies\n\nk(VG)≤ \|V\|.\n\nEquivalent top-soluble case of Brauer’sk(B)-problem. Equality can hold, eg. G =�Singer−cycle� ∼=Cpn1\n\nRobinson-Thompson reduction: conjecture proved if we can show that forG of “simple type” or ”extraspecial type”, there exists a regular orbit ofG on V.\n\n“Simple type”: G has an irreducible normal subgroup H such that H/Z(H) is non-abelian simple.\n\n(41)\n\n### Yet more on regular orbits\n\n2. The k(GV)-problem This is\n\nConjecture Let G <GLn(p) =GL(V), G a p-group. The number of conjugacy classes k(VG) in the semidirect product VG satisfies\n\nk(VG)≤ \|V\|.\n\nEquivalent top-soluble case of Brauer’sk(B)-problem.\n\nEquality can hold, eg. G =�Singer−cycle� ∼=Cpn1\n\nRobinson-Thompson reduction: conjecture proved if we can show that forG of “simple type” or ”extraspecial type”, there exists a regular orbit ofG on V.\n\n“Simple type”: G has an irreducible normal subgroup H such that H/Z(H) is non-abelian simple.\n\n(42)\n\n### Yet more on regular orbits\n\n2. The k(GV)-problem This is\n\nConjecture Let G <GLn(p) =GL(V), G a p-group. The number of conjugacy classes k(VG) in the semidirect product VG satisfies\n\nk(VG)≤ \|V\|.\n\nEquivalent top-soluble case of Brauer’sk(B)-problem.\n\nEquality can hold, eg. G =�Singer−cycle� ∼=Cpn1\n\nRobinson-Thompson reduction: conjecture proved if we can show that forG of “simple type” or ”extraspecial type”, there exists a regular orbit ofG on V.\n\n“Simple type”: G has an irreducible normal subgroup H such that H/Z(H) is non-abelian simple.\n\n(43)\n\n### Yet more on regular orbits\n\n2. The k(GV)-problem This is\n\nConjecture Let G <GLn(p) =GL(V), G a p-group. The number of conjugacy classes k(VG) in the semidirect product VG satisfies\n\nk(VG)≤ \|V\|.\n\nEquivalent top-soluble case of Brauer’sk(B)-problem.\n\nEquality can hold, eg. G =�Singer−cycle� ∼=Cpn1\n\nRobinson-Thompson reduction: conjecture proved if we can show that forG of “simple type” or ”extraspecial type”, there exists a regular orbit ofG onV.\n\n“Simple type”: G has an irreducible normal subgroup H such that H/Z(H) is non-abelian simple.\n\n(44)\n\n### Yet more on regular orbits\n\n2. The k(GV)-problem This is\n\nConjecture Let G <GLn(p) =GL(V), G a p-group. The number of conjugacy classes k(VG) in the semidirect product VG satisfies\n\nk(VG)≤ \|V\|.\n\nEquivalent top-soluble case of Brauer’sk(B)-problem.\n\nEquality can hold, eg. G =�Singer−cycle� ∼=Cpn1\n\nRobinson-Thompson reduction: conjecture proved if we can show that forG of “simple type” or ”extraspecial type”, there exists a regular orbit ofG onV.\n\n“Simple type”: G has an irreducible normal subgroup H such that H/Z(H) is non-abelian simple.\n\n(45)\n\n### Last one on regular orbits\n\nTheorem (Hall-L-Seitz, Goodwin, Kohler-Pahlings, Riese) If G <GLn(p) is a p-group of simple type, then G has a regular orbit unless one of:\n\n(i) Ac�G <GLc1(p), p >c\n\n(ii) 23 exceptional cases, all with n≤10,p ≤61.\n\nEventuallyk(GV)-conjecture proved (Gluck, Magaard, Riese, Schmidt 2004)\n\nGeneral classification of linear groups with/without regular orbits is out of reach at the moment. Need substitutes...\n\n(46)\n\n### Last one on regular orbits\n\nTheorem (Hall-L-Seitz, Goodwin, Kohler-Pahlings, Riese) If G <GLn(p) is a p-group of simple type, then G has a regular orbit unless one of:\n\n(i) Ac�G <GLc1(p), p >c\n\n(ii) 23 exceptional cases, all with n≤10,p ≤61.\n\nEventuallyk(GV)-conjecture proved (Gluck, Magaard, Riese, Schmidt 2004)\n\nGeneral classification of linear groups with/without regular orbits is out of reach at the moment. Need substitutes...\n\n(47)\n\n### Last one on regular orbits\n\nTheorem (Hall-L-Seitz, Goodwin, Kohler-Pahlings, Riese) If G <GLn(p) is a p-group of simple type, then G has a regular orbit unless one of:\n\n(i) Ac�G <GLc1(p), p >c\n\n(ii) 23 exceptional cases, all with n≤10,p ≤61.\n\nEventuallyk(GV)-conjecture proved (Gluck, Magaard, Riese, Schmidt 2004)\n\nGeneral classification of linear groups with/without regular orbits is out of reach at the moment. Need substitutes...\n\n(48)\n\n### Last one on regular orbits\n\nTheorem (Hall-L-Seitz, Goodwin, Kohler-Pahlings, Riese) If G <GLn(p) is a p-group of simple type, then G has a regular orbit unless one of:\n\n(i) Ac�G <GLc1(p), p >c\n\n(ii) 23 exceptional cases, all with n≤10,p ≤61.\n\nEventuallyk(GV)-conjecture proved (Gluck, Magaard, Riese, Schmidt 2004)\n\nGeneral classification of linear groups with/without regular orbits is out of reach at the moment. Need substitutes...\n\n(49)\n\n### Few orbits\n\nLetG ≤GLn(q) =GL(V). There are results classifying groups G with few orbits onV\\0.\n\nOne orbit G transitive on V\\0. Equivalently, aﬃne group VG ≤AGL(V) is 2-transitive.\n\nHering’s theorem Classification of transitive linear groups G ≤GLn(q):\n\n(i)G ≥SLn(q),Spn(q)\n\n(ii)G ≥G2(q) (n= 6, q even) (iii)G ≤ΓL1(qn)\n\n(iv)∼10 exceptions, all with \|V\| ≤592 (eg. F59◦SL2(5)<GL2(59)).\n\nTwo orbits ∃ similar classification (L) – hence the rank 3 aﬃne permutation groups.\n\nThree, four,... orbits: can be done if desperate\n\n(50)\n\n### Few orbits\n\nLetG ≤GLn(q) =GL(V). There are results classifying groups G with few orbits onV\\0.\n\nOne orbit G transitive on V\\0. Equivalently, aﬃne group VG ≤AGL(V) is 2-transitive.\n\nHering’s theorem Classification of transitive linear groups G ≤GLn(q):\n\n(i)G ≥SLn(q),Spn(q)\n\n(ii)G ≥G2(q) (n= 6, q even) (iii)G ≤ΓL1(qn)\n\n(iv)∼10 exceptions, all with \|V\| ≤592 (eg. F59◦SL2(5)<GL2(59)).\n\nTwo orbits ∃ similar classification (L) – hence the rank 3 aﬃne permutation groups.\n\nThree, four,... orbits: can be done if desperate\n\n(51)\n\n### Few orbits\n\nLetG ≤GLn(q) =GL(V). There are results classifying groups G with few orbits onV\\0.\n\nOne orbit G transitive on V\\0. Equivalently, aﬃne group VG ≤AGL(V) is 2-transitive.\n\nHering’s theorem Classification of transitive linear groups G ≤GLn(q):\n\n(i)G ≥SLn(q),Spn(q)\n\n(ii)G ≥G2(q) (n= 6, q even) (iii)G ≤ΓL1(qn)\n\n(iv)∼10 exceptions, all with \|V\| ≤592 (eg. F59◦SL2(5)<GL2(59)).\n\nTwo orbits ∃ similar classification (L) – hence the rank 3 aﬃne permutation groups.\n\nThree, four,... orbits: can be done if desperate\n\n(52)\n\n### Few orbits\n\nLetG ≤GLn(q) =GL(V). There are results classifying groups G with few orbits onV\\0.\n\nOne orbit G transitive on V\\0. Equivalently, aﬃne group VG ≤AGL(V) is 2-transitive.\n\nHering’s theorem Classification of transitive linear groups G ≤GLn(q):\n\n(i)G ≥SLn(q),Spn(q)\n\n(ii)G ≥G2(q) (n= 6, q even) (iii)G ≤ΓL1(qn)\n\n(iv)∼10 exceptions, all with \|V\| ≤592 (eg.\n\nF59◦SL2(5)<GL2(59)).\n\nTwo orbits ∃ similar classification (L) – hence the rank 3 aﬃne permutation groups.\n\nThree, four,... orbits: can be done if desperate\n\n(53)\n\n### Few orbits\n\nLetG ≤GLn(q) =GL(V). There are results classifying groups G with few orbits onV\\0.\n\nOne orbit G transitive on V\\0. Equivalently, aﬃne group VG ≤AGL(V) is 2-transitive.\n\nHering’s theorem Classification of transitive linear groups G ≤GLn(q):\n\n(i)G ≥SLn(q),Spn(q)\n\n(ii)G ≥G2(q) (n= 6, q even) (iii)G ≤ΓL1(qn)\n\n(iv)∼10 exceptions, all with \|V\| ≤592 (eg.\n\nF59◦SL2(5)<GL2(59)).\n\nTwo orbits ∃ similar classification (L) – hence the rank 3 aﬃne permutation groups.\n\nThree, four,... orbits: can be done if desperate\n\n(54)\n\n### Few orbits\n\nLetG ≤GLn(q) =GL(V). There are results classifying groups G with few orbits onV\\0.\n\nOne orbit G transitive on V\\0. Equivalently, aﬃne group VG ≤AGL(V) is 2-transitive.\n\nHering’s theorem Classification of transitive linear groups G ≤GLn(q):\n\n(i)G ≥SLn(q),Spn(q)\n\n(ii)G ≥G2(q) (n= 6, q even) (iii)G ≤ΓL1(qn)\n\n(iv)∼10 exceptions, all with \|V\| ≤592 (eg.\n\nF59◦SL2(5)<GL2(59)).\n\nTwo orbits ∃ similar classification (L) – hence the rank 3 aﬃne permutation groups.\n\nThree, four,... orbits: can be done if desperate\n\n(55)\n\n### Arithmetic conditions on orbit sizes\n\nHalf-transitivity G ≤GLn(q) =GL(V) is 12-transitive if all orbits ofG onV\\0 have equal size. (Aﬃne groupVG ≤AGL(V) is then\n\n3\n\n2-transitive.) Many examples, eg.\n\n(a)G transitive\n\n(b)G ≤ �s�,s Singer cycle of order qn−1\n\n(c)G a Frobenius complement, eg. SL2(5)<GL2(q) (d)G =S(q)<GL2(q) (q odd), where\n\nS(q) ={\n\n� a 0 0 ±a1\n\n� ,\n\n� 0 a\n\n±a1 0\n\n:a∈Fq} Passman 1969The soluble 12-transitive linear groups are: Frobenius complements, subgroups of ΓL1(qn), S(q), and 6 exceptions withqn≤172.\n\nGeneral case....???\n\n(56)\n\n### Arithmetic conditions on orbit sizes\n\nHalf-transitivity G ≤GLn(q) =GL(V) is 12-transitive if all orbits ofG on V\\0 have equal size. (Aﬃne groupVG ≤AGL(V) is then\n\n3\n\n2-transitive.)\n\nMany examples, eg. (a)G transitive\n\n(b)G ≤ �s�,s Singer cycle of order qn−1\n\n(c)G a Frobenius complement, eg. SL2(5)<GL2(q) (d)G =S(q)<GL2(q) (q odd), where\n\nS(q) ={\n\n� a 0 0 ±a1\n\n� ,\n\n� 0 a\n\n±a1 0\n\n:a∈Fq} Passman 1969The soluble 12-transitive linear groups are: Frobenius complements, subgroups of ΓL1(qn), S(q), and 6 exceptions withqn≤172.\n\nGeneral case....???\n\n(57)\n\n### Arithmetic conditions on orbit sizes\n\nHalf-transitivity G ≤GLn(q) =GL(V) is 12-transitive if all orbits ofG on V\\0 have equal size. (Aﬃne groupVG ≤AGL(V) is then\n\n3\n\n2-transitive.) Many examples, eg.\n\n(a)G transitive\n\n(b)G ≤ �s�,s Singer cycle of order qn−1\n\n(c)G a Frobenius complement, eg. SL2(5)<GL2(q) (d)G =S(q)<GL2(q) (q odd), where\n\nS(q) ={\n\n� a 0 0 ±a1\n\n� ,\n\n� 0 a\n\n±a1 0\n\n:a∈Fq} Passman 1969The soluble 12-transitive linear groups are: Frobenius complements, subgroups of ΓL1(qn), S(q), and 6 exceptions withqn≤172.\n\nGeneral case....???\n\n(58)\n\n### Arithmetic conditions on orbit sizes\n\nHalf-transitivity G ≤GLn(q) =GL(V) is 12-transitive if all orbits ofG on V\\0 have equal size. (Aﬃne groupVG ≤AGL(V) is then\n\n3\n\n2-transitive.) Many examples, eg.\n\n(a) G transitive\n\n(b)G ≤ �s�,s Singer cycle of order qn−1\n\n(c)G a Frobenius complement, eg. SL2(5)<GL2(q) (d)G =S(q)<GL2(q) (q odd), where\n\nS(q) ={\n\n� a 0 0 ±a1\n\n� ,\n\n� 0 a\n\n±a1 0\n\n:a∈Fq} Passman 1969The soluble 12-transitive linear groups are: Frobenius complements, subgroups of ΓL1(qn), S(q), and 6 exceptions withqn≤172.\n\nGeneral case....???\n\n(59)\n\n### Arithmetic conditions on orbit sizes\n\nHalf-transitivity G ≤GLn(q) =GL(V) is 12-transitive if all orbits ofG on V\\0 have equal size. (Aﬃne groupVG ≤AGL(V) is then\n\n3\n\n2-transitive.) Many examples, eg.\n\n(a) G transitive\n\n(b) G ≤ �s�,s Singer cycle of orderqn−1\n\n(c)G a Frobenius complement, eg. SL2(5)<GL2(q) (d)G =S(q)<GL2(q) (q odd), where\n\nS(q) ={\n\n� a 0 0 ±a1\n\n� ,\n\n� 0 a\n\n±a1 0\n\n:a∈Fq} Passman 1969The soluble 12-transitive linear groups are: Frobenius complements, subgroups of ΓL1(qn), S(q), and 6 exceptions withqn≤172.\n\nGeneral case....???\n\n(60)\n\n### Arithmetic conditions on orbit sizes\n\nHalf-transitivity G ≤GLn(q) =GL(V) is 12-transitive if all orbits ofG on V\\0 have equal size. (Aﬃne groupVG ≤AGL(V) is then\n\n3\n\n2-transitive.) Many examples, eg.\n\n(a) G transitive\n\n(b) G ≤ �s�,s Singer cycle of orderqn−1\n\n(c) G a Frobenius complement, eg. SL2(5)<GL2(q)\n\n(d)G =S(q)<GL2(q) (q odd), where S(q) ={\n\n� a 0 0 ±a1\n\n� ,\n\n� 0 a\n\n±a1 0\n\n:a∈Fq} Passman 1969The soluble 12-transitive linear groups are: Frobenius complements, subgroups of ΓL1(qn), S(q), and 6 exceptions withqn≤172.\n\nGeneral case....???\n\n(61)\n\n### Arithmetic conditions on orbit sizes\n\nHalf-transitivity G ≤GLn(q) =GL(V) is 12-transitive if all orbits ofG on V\\0 have equal size. (Aﬃne groupVG ≤AGL(V) is then\n\n3\n\n2-transitive.) Many examples, eg.\n\n(a) G transitive\n\n(b) G ≤ �s�,s Singer cycle of orderqn−1\n\n(c) G a Frobenius complement, eg. SL2(5)<GL2(q) (d) G =S(q)<GL2(q) (q odd), where\n\nS(q) ={\n\n� a 0 0 ±a1\n\n� ,\n\n� 0 a\n\n±a1 0\n\n:a∈Fq}\n\nPassman 1969The soluble 12-transitive linear groups are: Frobenius complements, subgroups of ΓL1(qn), S(q), and 6 exceptions withqn≤172.\n\nGeneral case....???\n\n(62)\n\n### Arithmetic conditions on orbit sizes\n\nHalf-transitivity G ≤GLn(q) =GL(V) is 12-transitive if all orbits ofG on V\\0 have equal size. (Aﬃne groupVG ≤AGL(V) is then\n\n3\n\n2-transitive.) Many examples, eg.\n\n(a) G transitive\n\n(b) G ≤ �s�,s Singer cycle of orderqn−1\n\n(c) G a Frobenius complement, eg. SL2(5)<GL2(q) (d) G =S(q)<GL2(q) (q odd), where\n\nS(q) ={\n\n� a 0 0 ±a1\n\n� ,\n\n� 0 a\n\n±a1 0\n\n:a∈Fq} Passman 1969The soluble 12-transitive linear groups are:\n\nFrobenius complements, subgroups of ΓL1(qn),S(q), and 6 exceptions withqn≤172.\n\nGeneral case....???\n\n(63)\n\n### Arithmetic conditions on orbit sizes\n\nHalf-transitivity G ≤GLn(q) =GL(V) is 12-transitive if all orbits ofG on V\\0 have equal size. (Aﬃne groupVG ≤AGL(V) is then\n\n3\n\n2-transitive.) Many examples, eg.\n\n(a) G transitive\n\n(b) G ≤ �s�,s Singer cycle of orderqn−1\n\n(c) G a Frobenius complement, eg. SL2(5)<GL2(q) (d) G =S(q)<GL2(q) (q odd), where\n\nS(q) ={\n\n� a 0 0 ±a1\n\n� ,\n\n� 0 a\n\n±a1 0\n\n:a∈Fq} Passman 1969The soluble 12-transitive linear groups are:\n\nFrobenius complements, subgroups of ΓL1(qn),S(q), and 6 exceptions withqn≤172.\n\nGeneral case....???\n\n(64)\n\n### Arithmetic conditions\n\np-exceptional groups Say G ≤GLn(p) =GL(V) is p-exceptional ifp divides G and all orbits ofG onV have size coprime to p. Ties in with previous notions:\n\n(a)G p-exceptional ⇒ G hasno regular orbit onV (b)G transitive,p\| \|G\| ⇒G p-exceptional\n\n(c)G 12-transitive,p\| \|G\| ⇒ G p-exceptional\n\n∃many examples, eg. V =Wk,G =H wrK where H≤GL(W) is transitive onW\\0 and K ≤Sk has all orbits on the power set of {1, . . . ,k}of p-size. (Eg. K ap-group)\n\nCanp-exceptional linear groups be classified?\n\nYes, at least the irreducible ones ((Giudici, L, Praeger, Saxl, Tiep)...\n\n(65)\n\n### Arithmetic conditions\n\np-exceptional groups Say G ≤GLn(p) =GL(V) is p-exceptional ifp divides G and all orbits ofG onV have size coprime to p.\n\nTies in with previous notions:\n\n(a)G p-exceptional ⇒ G hasno regular orbit onV (b)G transitive,p\| \|G\| ⇒G p-exceptional\n\n(c)G 12-transitive,p\| \|G\| ⇒ G p-exceptional\n\n∃many examples, eg. V =Wk,G =H wrK where H≤GL(W) is transitive onW\\0 and K ≤Sk has all orbits on the power set of {1, . . . ,k}of p-size. (Eg. K ap-group)\n\nCanp-exceptional linear groups be classified?\n\nYes, at least the irreducible ones ((Giudici, L, Praeger, Saxl, Tiep)...\n\n(66)\n\n### Arithmetic conditions\n\np-exceptional groups Say G ≤GLn(p) =GL(V) is p-exceptional ifp divides G and all orbits ofG onV have size coprime to p.\n\nTies in with previous notions:\n\n(a)G p-exceptional ⇒ G hasno regular orbit onV (b)G transitive,p\| \|G\| ⇒G p-exceptional\n\n(c)G 12-transitive,p\| \|G\| ⇒ G p-exceptional\n\n∃many examples, eg. V =Wk,G =H wrK where H≤GL(W) is transitive onW\\0 and K ≤Sk has all orbits on the power set of {1, . . . ,k}of p-size. (Eg. K ap-group)\n\nCanp-exceptional linear groups be classified?\n\nYes, at least the irreducible ones ((Giudici, L, Praeger, Saxl, Tiep)...\n\n(67)\n\n### Arithmetic conditions\n\np-exceptional groups Say G ≤GLn(p) =GL(V) is p-exceptional ifp divides G and all orbits ofG onV have size coprime to p.\n\nTies in with previous notions:\n\n(a)G p-exceptional ⇒ G hasno regular orbit onV\n\n(b)G transitive,p\| \|G\| ⇒G p-exceptional (c)G 12-transitive,p\| \|G\| ⇒ G p-exceptional\n\n∃many examples, eg. V =Wk,G =H wrK where H≤GL(W) is transitive onW\\0 and K ≤Sk has all orbits on the power set of {1, . . . ,k}of p-size. (Eg. K ap-group)\n\nCanp-exceptional linear groups be classified?\n\nYes, at least the irreducible ones ((Giudici, L, Praeger, Saxl, Tiep)...\n\n(68)\n\n### Arithmetic conditions\n\np-exceptional groups Say G ≤GLn(p) =GL(V) is p-exceptional ifp divides G and all orbits ofG onV have size coprime to p.\n\nTies in with previous notions:\n\n(a)G p-exceptional ⇒ G hasno regular orbit onV (b)G transitive, p\| \|G\| ⇒G p-exceptional\n\n(c)G 12-transitive,p\| \|G\| ⇒ G p-exceptional\n\n∃many examples, eg. V =Wk,G =H wrK where H≤GL(W) is transitive onW\\0 and K ≤Sk has all orbits on the power set of {1, . . . ,k}of p-size. (Eg. K ap-group)\n\nCanp-exceptional linear groups be classified?\n\nYes, at least the irreducible ones ((Giudici, L, Praeger, Saxl, Tiep)...\n\n(69)\n\n### Arithmetic conditions\n\np-exceptional groups Say G ≤GLn(p) =GL(V) is p-exceptional ifp divides G and all orbits ofG onV have size coprime to p.\n\nTies in with previous notions:\n\n(a)G p-exceptional ⇒ G hasno regular orbit onV (b)G transitive, p\| \|G\| ⇒G p-exceptional\n\n(c)G 12-transitive,p\| \|G\| ⇒ G p-exceptional\n\n∃many examples, eg. V =Wk,G =H wrK where H≤GL(W) is transitive onW\\0 and K ≤Sk has all orbits on the power set of {1, . . . ,k}of p-size. (Eg. K ap-group)\n\nCanp-exceptional linear groups be classified?\n\nYes, at least the irreducible ones ((Giudici, L, Praeger, Saxl, Tiep)...\n\n(70)\n\n### Arithmetic conditions\n\np-exceptional groups Say G ≤GLn(p) =GL(V) is p-exceptional ifp divides G and all orbits ofG onV have size coprime to p.\n\nTies in with previous notions:\n\n(a)G p-exceptional ⇒ G hasno regular orbit onV (b)G transitive, p\| \|G\| ⇒G p-exceptional\n\n(c)G 12-transitive,p\| \|G\| ⇒ G p-exceptional\n\n∃many examples, eg. V =Wk,G =H wrK where H≤GL(W) is transitive onW\\0 andK ≤Sk has all orbits on the power set of {1, . . . ,k}of p-size. (Eg. K ap-group)\n\nCanp-exceptional linear groups be classified?\n\nYes, at least the irreducible ones ((Giudici, L, Praeger, Saxl, Tiep)...\n\n(71)\n\n### Arithmetic conditions\n\np-exceptional groups Say G ≤GLn(p) =GL(V) is p-exceptional ifp divides G and all orbits ofG onV have size coprime to p.\n\nTies in with previous notions:\n\n(a)G p-exceptional ⇒ G hasno regular orbit onV (b)G transitive, p\| \|G\| ⇒G p-exceptional\n\n(c)G 12-transitive,p\| \|G\| ⇒ G p-exceptional\n\n∃many examples, eg. V =Wk,G =H wrK where H≤GL(W) is transitive onW\\0 andK ≤Sk has all orbits on the power set of {1, . . . ,k}of p-size. (Eg. K ap-group)\n\nCanp-exceptional linear groups be classified?\n\nYes, at least the irreducible ones ((Giudici, L, Praeger, Saxl, Tiep)...\n\n(72)\n\n### Arithmetic conditions\n\np-exceptional groups Say G ≤GLn(p) =GL(V) is p-exceptional ifp divides G and all orbits ofG onV have size coprime to p.\n\nTies in with previous notions:\n\n(a)G p-exceptional ⇒ G hasno regular orbit onV (b)G transitive, p\| \|G\| ⇒G p-exceptional\n\n(c)G 12-transitive,p\| \|G\| ⇒ G p-exceptional\n\n∃many examples, eg. V =Wk,G =H wrK where H≤GL(W) is transitive onW\\0 andK ≤Sk has all orbits on the power set of {1, . . . ,k}of p-size. (Eg. K ap-group)\n\nCanp-exceptional linear groups be classified?\n\nYes, at least the irreducible ones ((Giudici, L, Praeger, Saxl, Tiep)...\n\n(73)\n\n### Arithmetic conditions\n\n(G ≤GLn(p) =GL(V) isp-exceptionalifp dividesG and all orbits ofG onV have size coprime to p.)\n\nTheorem Let G ≤GLd(p) =GL(V) be p-exceptional. Suppose G acts irreducibly and primitively on V . Then one of:\n\n(i) G transitive on V\\0 (ii) G ≤ΓL1(pn)\n\n(iii) G =Ac,Sc <GLc(2), c = 2r −1 or 2r −2,�= 1 or 2 (iv) G =SL2(5)<GL4(3), orbits1,40,40\n\nPSL2(11)<GL5(3), orbits1,22,110,110 M11<GL5(3), orbits1,22,220\n\nM23<GL11(2), orbits 1,23,253,1771\n\nAlso have a classification of the imprimitivep-exceptional groups: V =Wk,G ≤HwrK whereH≤GL(W) is transitive onW\\0 andK ≤Sk has all orbits on the power set of {1, . . . ,k}of p-size.\n\n(74)\n\n### Arithmetic conditions\n\n(G ≤GLn(p) =GL(V) isp-exceptionalifp dividesG and all orbits ofG onV have size coprime to p.)\n\nTheorem Let G ≤GLd(p) =GL(V) be p-exceptional. Suppose G acts irreducibly and primitively on V . Then one of:\n\n(i) G transitive on V\\0 (ii) G ≤ΓL1(pn)\n\n(iii) G =Ac,Sc <GLc(2), c = 2r −1 or2r −2,�= 1 or 2 (iv) G =SL2(5)<GL4(3), orbits1,40,40\n\nPSL2(11)<GL5(3), orbits1,22,110,110 M11<GL5(3), orbits1,22,220\n\nM23<GL11(2), orbits 1,23,253,1771\n\nAlso have a classification of the imprimitivep-exceptional groups: V =Wk,G ≤HwrK whereH≤GL(W) is transitive onW\\0 andK ≤Sk has all orbits on the power set of {1, . . . ,k}of p-size.\n\n(75)\n\n### Arithmetic conditions\n\n(G ≤GLn(p) =GL(V) isp-exceptionalifp dividesG and all orbits ofG onV have size coprime to p.)\n\nTheorem Let G ≤GLd(p) =GL(V) be p-exceptional. Suppose G acts irreducibly and primitively on V . Then one of:\n\n(i) G transitive on V\\0 (ii) G ≤ΓL1(pn)\n\n(iii) G =Ac,Sc <GLc(2), c = 2r −1 or2r −2,�= 1 or 2 (iv) G =SL2(5)<GL4(3), orbits1,40,40\n\nPSL2(11)<GL5(3), orbits1,22,110,110 M11<GL5(3), orbits1,22,220\n\nM23<GL11(2), orbits 1,23,253,1771\n\nAlso have a classification of the imprimitivep-exceptional groups:\n\nV =Wk,G ≤HwrK whereH≤GL(W) is transitive onW\\0 andK ≤Sk has all orbits on the power set of {1, . . . ,k}of p-size.\n\n(76)\n\n### Consequences\n\nRecallG 12-transitive ⇒all orbits on V\\0 have same size⇒ G is p-exceptional. Hence\n\nTheorem If G ≤GLd(p) is 12-transitive and p divides \|G\|, then one of:\n\n(i) G is transitive on V\\0 (ii) G ≤ΓL1(pd)\n\n(iii) G =SL2(5)<GL4(3), orbits1,40,40.\n\n(77)\n\n### Consequences\n\nRecallG 12-transitive ⇒all orbits on V\\0 have same size⇒ G is p-exceptional. Hence\n\nTheorem If G ≤GLd(p) is 12-transitive and p divides \|G\|, then one of:\n\n(i) G is transitive on V\\0 (ii) G ≤ΓL1(pd)\n\n(iii) G =SL2(5)<GL4(3), orbits1,40,40.\n\n(78)\n\n### Consequences\n\nRecallG 12-transitive ⇒all orbits on V\\0 have same size⇒ G is p-exceptional. Hence\n\nTheorem If G ≤GLd(p) is 12-transitive and p divides \|G\|, then one of:\n\n(i) G is transitive on V\\0 (ii) G ≤ΓL1(pd)\n\n(iii) G =SL2(5)<GL4(3), orbits1,40,40.\n\n(79)\n\n### Consequences\n\nGluck-Wolf theorem 1984 Let p be a prime and G a finite p-soluble group. Suppose N�G and N has an irreducible character φsuch thatχ(1)/φ(1) is coprime to p for allχ⊆φG. Then G/N has abelian Sylow p-subgroups.\n\nThis implies Brauer’sheight zeroconjecture forp-soluble groups. Using our classification ofp-exceptional groups, Tiep and Navarro have proved the Gluck-Wolf theorem for arbitrary finite groupsG. May lead to the complete solution of the height zero conjecture.\n\n(80)\n\n### Consequences\n\nGluck-Wolf theorem 1984 Let p be a prime and G a finite p-soluble group. Suppose N�G and N has an irreducible character φsuch thatχ(1)/φ(1) is coprime to p for allχ⊆φG. Then G/N has abelian Sylow p-subgroups.\n\nThis implies Brauer’sheight zeroconjecture forp-soluble groups. Using our classification ofp-exceptional groups, Tiep and Navarro have proved the Gluck-Wolf theorem for arbitrary finite groupsG. May lead to the complete solution of the height zero conjecture.\n\n(81)\n\n### Consequences\n\nGluck-Wolf theorem 1984 Let p be a prime and G a finite p-soluble group. Suppose N�G and N has an irreducible character φsuch thatχ(1)/φ(1) is coprime to p for allχ⊆φG. Then G/N has abelian Sylow p-subgroups.\n\nThis implies Brauer’sheight zeroconjecture forp-soluble groups.\n\nUsing our classification ofp-exceptional groups, Tiep and Navarro have proved the Gluck-Wolf theorem for arbitrary finite groupsG. May lead to the complete solution of the height zero conjecture.\n\n(82)\n\n### Consequences\n\nGluck-Wolf theorem 1984 Let p be a prime and G a finite p-soluble group. Suppose N�G and N has an irreducible character φsuch thatχ(1)/φ(1) is coprime to p for allχ⊆φG. Then G/N has abelian Sylow p-subgroups.\n\nThis implies Brauer’sheight zeroconjecture forp-soluble groups.\n\nUsing our classification ofp-exceptional groups, Tiep and Navarro have proved the Gluck-Wolf theorem for arbitrary finite groupsG. 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A strong research and development effort, particularly into the integration of control methods, is essential to the\n\nWe expect that it is possible to reduce the classification to a finite calculation, and that the p-groups of a given coclass can be partitioned into finitely many families, where\n\n• Additional High Conservation Value Vegetation (AHCVV) means areas of vegetation which were found during ground-truthing which would otherwise meet the definition of Existing\n\nTheorem 1.18 If we have a sequence of groups and homomorphisms linking them which contains the following part which is exact at G:.. 1 → G → θ H then θ is\n\nWe show that it is unique up to isomorphism among those having a point a whose stabilizer in the automorphism group both fixes setwise every line on a and contains a subgroup that\n\nIntermediate growth is inherited by subgroups of finite index, so the same applies to the groups N(T ) ≤ Γ and the associated maps.... Example Nilpotent-by-finite groups have\n\nFor all but E 8 (q) in even characteristic, our algorithms to construct the SL 2 subgroups and to label the root and toral elements are black-box provided that the algorithms\n\nMore generally, G with large abelian factor group may have Cayley graphs with diameter proportional to \|G\|... The diameter\n\nGow proved that the conjecture holds for the symplectic groups P Sp 2n (q) if q ≡ 1 mod 4, and in proved that every semisimple element of a finite simple group of Lie type\n\n5.15 At the time of Mr C’s requests for access to the NDIS, the NDIA did not have any policy or guideline dealing specifically with incarcerated individuals and access to the NDIS.\n\nClassification of reflexible regular Cayley maps on abelian groups. Classification of t-balanced regular Cayley map on\n\nRich theory of cartesian decompositions preserved by groups with a transitive minimal normal subgroup.\n\nexistence. In making such an estimate, the Government Printer was requested to take as understood that each author body would have its report printed by the" ]	[ null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.7140042,"math_prob":0.99096894,"size":39502,"snap":"2023-40-2023-50","text_gpt3_token_len":14502,"char_repetition_ratio":0.15362804,"word_repetition_ratio":0.98404336,"special_character_ratio":0.3195028,"punctuation_ratio":0.15379596,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.99729943,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2023-11-28T20:10:42Z\",\"WARC-Record-ID\":\"<urn:uuid:b544356b-3b24-4523-8177-731ff5e41277>\",\"Content-Length\":\"216499\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:070dc26e-8a2d-4f9c-b81f-883ae76e3b55>\",\"WARC-Concurrent-To\":\"<urn:uuid:a3923ac6-9e88-4269-8ad0-84073271b2d5>\",\"WARC-IP-Address\":\"104.21.80.99\",\"WARC-Target-URI\":\"https://5doc.co/document/zx5n0llv-regular-orbits.html\",\"WARC-Payload-Digest\":\"sha1:4FHLQMRJP44RL5CMQTJR63VU3NQ6S5QP\",\"WARC-Block-Digest\":\"sha1:Q6AVFB43IE5PKNC445KPVLB4DRAJ3OZ7\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2023/CC-MAIN-2023-50/CC-MAIN-2023-50_segments_1700679099942.90_warc_CC-MAIN-20231128183116-20231128213116-00031.warc.gz\"}"}
https://testbook.com/question-answer/iff-left-x-dfrac-1-x-right--611225b9a66c9e5b9eec2339	[ "# If $$f \\left( x + \\dfrac{ 1 } { x } \\right) = x^3 + \\dfrac{1}{x^3}$$, then f(√3) is equal to\n\nThis question was previously asked in\nUP TGT Mathematics 2021 Official Paper\nView all UP TGT Papers >\n1. 0\n2. 1\n3. √3\n4. 3√3\n\nOption 1 : 0\nFree\nCT 1: Indian History\n44.9 K Users\n10 Questions 10 Marks 6 Mins\n\n## Detailed Solution\n\nConcept:\n\nAccording to the rule of a linear function, we know that, if, f(x) = x\n\n$$⇒ f(ax)=ax$$\n\nwhere a is the coefficient of x.\n\nFormula used:\n\n• a3 + b3 = (a + b) (a2 + b2 - ab) ----(1)\n\n• (a + b)2 = a2 + b2 + 2ab ----(2)\n\n• a3 + b3 = (a + b) {(a + b)2 - 3ab} [using (1) & (2)] ----(3)\n\nCalculation:\n\nWe have, $$f \\left( x + \\dfrac{ 1 } { x } \\right) = x^3 + \\dfrac{1}{x^3}$$\n\nUsing equation (3), we get\n\n$$⇒ f \\left( x + \\dfrac{ 1 } { x } \\right) = \\left( x + \\dfrac{ 1 } { x } \\right)\\left \\{ \\left( x + \\dfrac{ 1 } { x } \\right)^{2} \\ - \\ 3 \\right \\}$$\n\nLet $$x \\ + \\ \\frac{1}{x} = t$$\n\n⇒ f(t) = t (t2 - 3)\n\nOn putting t = √3, we get\n\n⇒ f(√3) = √3 {(√3)2 - 3}\n\n⇒ f(√3) = √3 (3 - 3)\n\n⇒ f(√3) = √3 (0)\n\n⇒ f(√3) = 0" ]	[ null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.78725666,"math_prob":1.0000081,"size":1582,"snap":"2023-14-2023-23","text_gpt3_token_len":598,"char_repetition_ratio":0.121673,"word_repetition_ratio":0.14018692,"special_character_ratio":0.408976,"punctuation_ratio":0.069767445,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.99999285,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2023-05-29T06:10:13Z\",\"WARC-Record-ID\":\"<urn:uuid:88c58ca9-582e-43df-b215-c4b64cd196ba>\",\"Content-Length\":\"162679\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:ea477da8-c047-45b1-af11-f17d8314e098>\",\"WARC-Concurrent-To\":\"<urn:uuid:613ba3e1-6539-4cd7-9402-9851b0e087ef>\",\"WARC-IP-Address\":\"104.22.44.238\",\"WARC-Target-URI\":\"https://testbook.com/question-answer/iff-left-x-dfrac-1-x-right--611225b9a66c9e5b9eec2339\",\"WARC-Payload-Digest\":\"sha1:RJWINGDE7QZTDCN44JOWE3QLK5UUL6SD\",\"WARC-Block-Digest\":\"sha1:3WN2ZA2VSZ5ZA6G3EV7YU5WGLIEAZDF5\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2023/CC-MAIN-2023-23/CC-MAIN-2023-23_segments_1685224644683.18_warc_CC-MAIN-20230529042138-20230529072138-00118.warc.gz\"}"}
https://tc.copernicus.org/articles/16/4319/2022/tc-16-4319-2022.html	[ "https://doi.org/10.5194/tc-16-4319-2022\nhttps://doi.org/10.5194/tc-16-4319-2022", null, "# Understanding wind-driven melt of patchy snow cover\n\nLuuk D. van der Valk, Adriaan J. Teuling, Luc Girod, Norbert Pirk, Robin Stoffer, and Chiel C. van Heerwaarden\nAbstract\n\nThe representation of snow processes in most large-scale hydrological and climate models is known to introduce considerable uncertainty into the predictions and projections of water availability. During the critical snowmelt period, the main challenge in snow modeling is that net radiation is spatially highly variable for a patchy snow cover, resulting in large horizontal differences in temperatures and heat fluxes. When a wind blows over such a system, these differences can drive advection of sensible and latent heat from the snow-free areas to the snow patches, potentially enhancing the melt rates at the leading edge and increasing the variability of subgrid melt rates. To get more insight into these processes, we examine the melt along the upwind and downwind edges of a 50 m long snow patch in the Finseelvi catchment, Norway, and try to explain the observed behavior with idealized simulations of heat fluxes and air movement over patchy snow. The melt of the snow patch was monitored from 11 June until 15 June 2019 by making use of height maps obtained through the photogrammetric structure-from-motion principle. A vertical melt of 23 ± 2.0 cm was observed at the upwind edge over the course of the field campaign, whereas the downwind edge melted only 3 ± 0.4 cm. When comparing this with meteorological measurements, we estimate the turbulent heat fluxes to be responsible for 60 % to 80 % of the upwind melt, of which a significant part is caused by the latent heat flux. The melt at the downwind edge approximately matches the melt occurring due to net radiation. To better understand the dominant processes, we represented this behavior in idealized direct numerical simulations, which are based on the measurements on a single snow patch by and resemble a flat, patchy snow cover with typical snow patch sizes of 15, 30, and 60 m. Using these simulations, we found that the reduction of the vertical temperature gradient over the snow patch was the main cause of the reductions in vertical sensible heat flux over distance from the leading edge, independent of the typical snow patch size. Moreover, we observed that the sensible heat fluxes at the leading edge and the decay over distance were independent of snow patch size as well, which resulted in a 15 % and 25 % reduction in average snowmelt for, respectively, a doubling and quadrupling of the typical snow patch size. These findings lay out pathways to include the effect of highly variable turbulent heat fluxes based on the typical snow patch size in large-scale hydrological and climate models to improve snowmelt modeling.\n\nShare\nDates\n1 Introduction\n\nSnow plays a crucial role for much of the world's population through the important water and climate services that it provides. Snowmelt is crucial for more than one-sixth of the world population for agricultural purposes or human consumption . Also, snow is a natural way to store water during the cold months; it is then released during spring and summer, when the water demands are higher (e.g., ). Changes in this snow cover, due to shifts in precipitation and temperature, have an impact on society and ecosystems , amongst other things. Strong changes in snow cover have been observed over the past decades in many regions, including Europe and the United States . To be able to assess the effects of changes in the snow cover, a more thorough understanding of the influence of snow processes on discharge is needed , and uncertainties in hydrological models due to snow storage and melt need to be overcome . Additional uncertainties are introduced by the transformation of a continuous snow cover into a patchy snow cover, as it is challenging to correctly capture the snow patches and associated processes even with relatively advanced hydrological and climate models (e.g., ). Uncertainty related to the modeling of snow processes might even lead to disagreement in the sign of projected mean streamflow changes . To reduce the modeling uncertainty of patchy snow covers, a deeper understanding of the snowmelt processes that occur under spatial variability is needed.\n\nThe main processes that cause snow to melt are different for a patchy snow cover than a continuous snow cover. For a continuous snow cover, the surface energy balance, which is responsible for snowmelt, is dominated by radiation (e.g., ). For this type of cover, the turbulent heat fluxes are mainly driven by large-scale air mass movement affecting the ambient air temperature or moisture and wind speed, which could cause these fluxes to be significant during brief periods . However, over longer periods, such as weeks, air temperature and moisture gradients near the surface are generally too low to generate significant turbulent heat fluxes (Hock2005). When the snow cover turns patchy, the net radiation becomes spatially highly variable due to variations in surface albedo and emissivity. This spatial variability can act at scales on the order of meters, such that a highly heterogeneous surface arises with a relatively warm (and possibly wet) snow-free area adjacent to a relatively cold (and drier) snow patch. When a wind blows over this horizontally heterogeneous surface, internal boundary layers form downwind of the transitions, due to changes in the surface conditions (e.g., ). The heterogeneity of these internal boundary layers induces a system in which the turbulent heat fluxes can highly vary spatially, partly due to the advection of sensible and latent heat . These systems are often described as separate growing boundary layers that follow a power law as a function of the fetch (e.g., ). For the stable internal boundary layers, i.e., those over snow patches, the air close to the snow surface can decouple from the warmer air above, due to either large temperature differences between both or through cold-air pooling, eventually limiting the transfer of sensible and latent heat from the atmosphere to the snow . Moreover, the influence of the turbulent heat fluxes on the total amount of melt increases with decreasing snow cover fraction . It has been suggested that this process can be responsible for up to 50 % of the total snowmelt (e.g., ). Additionally, similar processes have been found to potentially significantly contribute to snowmelt on ice fields (e.g., ) and glaciers . This stresses the potentially significant contribution of lateral transport of sensible and latent heat to a melting patchy snow cover, though it opens the question of what the hydrological relevance on larger scales is.\n\nIn spite of its potential importance, the lateral transport of sensible and latent heat is a rather underrepresented process in hydrological or even more complex atmospheric snow surface models, as these focus mainly on vertical melt processes. Traditionally, simple models of snowmelt in hydrological models use the so-called temperature index, which performs generally well with few computational costs. However, the performance of these models decreases significantly when increasing temporal resolution, adding a spatial component to the model, applying it beyond the period or domain of calibration (e.g., climate change), or during rain-on-snow events (e.g., ). The spatial component is especially important in mountainous regions due to topographical effects such as cold-air pooling and shading, which are not taken into account with the temperature index. Furthermore, the exclusion of the wind speed in temperature-index models can cause a bias for the amount of modeled snowmelt, especially when turbulence-driven snowmelt becomes increasingly important, for example for patchy snow covers (e.g., ). Even when using complex atmospheric snow surface models such as Alpine3D with ARPS , local-scale advection associated with subgrid variation of snow and bare ground is excluded. A few models, such as Alpine3D and CHM , do include wind-driven processes such as snow redistribution and turbulent heat fluxes, but models most often parameterize the subgrid turbulent fluxes with the average temperature or moisture at the surface and the lowest atmospheric layer per grid cell, and they do not account for melt fluxes with subgrid spatial variations. As potential solutions, propose a simple model to include local-scale advection of the sensible and latent heat to snowmelt using scaling laws, while develops a more complex approach based on the integration of the energy equation as suggested by using mixing length theory (Weismann1977). Considering the subgrid heterogeneity of the melting fluxes would significantly improve snow cover and discharge predictions . Still, implementing the subgrid turbulent fluxes using a parametrization is difficult, mainly due to the small spatial scales of the local-scale advection and the interplay between snow cover area and melt rates , as well as the process is less well understood on a catchment scale . As a consequence, more field observations should be performed to study its importance in various environmental settings. Additionally, these should be combined with other modeling approaches that can serve as a tool to improve our understanding of the process on small and larger scales. Such progress will eventually enable the implementation of lateral transport of the sensible and latent heat.\n\nDedicated observations are needed to quantify the importance of the sensible and latent heat advection in snowmelt. Several field measurements have focused on the meteorological surface characteristics, such as the development of an internal boundary layer (IBL) as a consequence of the heterogeneous snow cover or the influence of a topographical depression on cold-air pooling and subsequent snowmelt . Other field measurement campaigns have estimated the turbulent heat fluxes and the accompanying mechanisms for a single isolated snow patch (e.g., ). However, all of these experiments focused on relatively brief periods of time with a maximum time span of a day, during which the conditions for local-scale advection of sensible and latent heat were often ideal. For small areas in complex terrain and longer periods, related the measured snowmelt to turbulence, while estimated the role of advection of the sensible heat by comparing estimated sensible heat fluxes with and without advection. Yet, estimates of a multiple-day contribution of the turbulent heat fluxes to the melt of a single snow patch are lacking, which could provide additional insights into the role of these fluxes over longer timescales.\n\nWhen observing the melt of a snow patch over the course of multiple days, high spatial variability of melt rates complicates the observations. Single-point measurements might not represent the region of interest, especially for seasonal snow covers . This advocates the use of spatial field observations at high temporal and spatial resolution for patchy snow covers. Ground-based methods fulfilling these requirements include, amongst others, terrestrial laser scanning or georectification of oblique time-lapse photography . Additionally, aerial platforms such as aerial laser scanning, either manned (e.g., ) or unmanned (e.g., ), can be used. The application of structure-from-motion (SfM) photogrammetry is possible from both positions as a relatively cheap method to monitor snowmelt. Its usage was explored back in the 1960s (e.g., ), but it was sidelined due to technical constraints. Recently, due to technical development that has increased its accuracy with lower computational costs, SfM photogrammetry has been used to successfully study seasonal snow covers with low-cost imaging equipment and software for analysis (e.g., ). Therefore, this method offers a promising way to monitor snowmelt with low costs and reasonable accuracy.\n\nTo eliminate parametrization uncertainties, a relatively new type of simulation, direct numerical simulation (DNS), can be applied to model local-scale advection, as it resolves all the relevant spatial and temporal scales of turbulence. This simulation type has already proven its value in the field of fluid dynamics and allows very detailed information to be extracted from the turbulent flow, enhancing our process-based understanding of local-scale advection. As a consequence of the high resolution, these simulations do not need any turbulence parametrizations based on stability corrections and the Monin–Obukhov assumptions, in contrast to numerical atmospheric boundary layer models or large-eddy simulations . These methods assume horizontal homogeneity and constant turbulent fluxes throughout the surface layer, but these assumptions are violated for a patchy snow cover, thus introducing a large uncertainty (e.g., ). successfully showed the potential of DNS to simulate snow and ice melt in complex terrain. Through several sensitivity tests, they assessed the influence of surface properties on the micrometeorology and the subsequent effect on the turbulent heat fluxes. Moreover, the simulations are used to further enhance our insight into the fundamental processes of melting ice cliffs. Thus, DNSs show high potential to be used as a tool to further enhance our understanding of local-scale advection and the eventual implementation of the process in larger models through the derivation of new parametrizations accounting for varying surface and meteorological characteristics.\n\nWhereas previous studies pioneered examinations of the lateral advection of sensible and latent heat, most often for a single snow patch, the question of how this process is affected by the typical spatial and temporal scales within a catchment remains. Moreover, new modeling attempts need to be undertaken to increase our understanding of the wind-driven processes occurring near the surfaces of melting patchy snow covers. Therefore, in this study, we aim to assess the role of horizontal advection of the sensible and latent heat in snowmelt for a snow patch in a real-world case and an idealized environment. In the real-world case, we will identify the role of the locally advected sensible and latent heat in the melting of a snow patch in the Finseelvi catchment through studying the vertical turbulent heat fluxes with SfM photogrammetry observations over the course of multiple days. The resulting snowmelt is compared to local meteorological measurements to put the snowmelt into perspective and extract the influence of the turbulent fluxes on this melt. Subsequently, we try to uncover the behavior of the vertical sensible heat fluxes during snowmelt – including the local-scale advection of sensible heat – in an idealized environment with DNS, allowing us to extract detailed information on a wind blowing over a small, flat domain with a patchy snow cover. This allows us to illustrate the performance of DNS as a tool to understand the real-world behavior and to try to explain this with idealized simulations. To do so, we perform these simulations with the computational fluid dynamics code MicroHH. We use the measurements of on a single snow patch as a basis for designing our numerical experiments, and choose to focus on the sensible heat flux. These measurements are done with close-to-idealized settings on a flat surface. Subsequently, we investigate the influence of enlarging snow patches on the vertical sensible heat fluxes into the snow, and the implications for snowmelt modeling.\n\n2 Snowmelt observations\n\nThe snowmelt observations were done in the Finseelvi catchment near Finse, Norway (17.6 km2; 6036 N, 730 E). This catchment is a snowmelt-dominated headwater basin with the discharge outlet located at an altitude of approximately 1340 m a.s.l. (Fig. 1). The catchment has a relatively smooth topography, which decreases the spatial complexity of the turbulent heat fluxes over snow patches (e.g., ). This, combined with a high contribution of turbulent heat fluxes to snowmelt, which is generally the case for maritime climates , makes the Finseelvi catchment a suitable location for assessing the influence of vertical turbulent heat fluxes on melting patchy snow covers. In the same region, hydrological studies such as have applied SfM timelapse photogrammetry for a patchy snow cover, and estimated the exchanges of mass and energy for a melting continuous snow cover. Moreover, at a distance of approximately 2.5 km to the southeast from the outlet of the catchment, meteorological measurements were taken (Fig. 1). The meteorological data were retrieved from a meteorological flux tower at a temporal resolution of 30 min and includes, amongst others, temperature, precipitation, wind speed and direction, incoming radiation, and relative humidity.", null, "Figure 1Overview of the Finseelvi research area, with the location of the snow patch and the daily averaged discharge and snow cover fraction data for the catchment shown. The map of Norway was obtained from https://norgeibilder.no/ (last access: 23 March 2020); the catchment area and the streams were respectively obtained through the University of Oslo and the Norwegian Water Resources and Energy Directorate (http://nedlasting.nve.no/gis/, last access: 7 January 2020). The zoomed image was made by the Sentinel-2 satellite on 13 June 2018, with the snow patch (white areas are snow covered) and local wind direction indicated as experienced during the field campaign (the local wind direction resembled the measured direction at the meteorological tower).\n\nWe assess the influence of the vertical turbulent heat fluxes on a melting single snow patch within the Finseelvi catchment. From this snow patch (Fig. 1), daily height maps are obtained through photogrammetry performed over the course of 11 June until 15 June 2019 for the upwind and downwind edges of the snow patch. As the dominant wind direction experienced at the snow patch resembled the measured wind direction at the meteorological tower, which was constant throughout the field campaign (Table 4 in Sect. 4), we assume that the upwind and downwind locations of the snow patch were approximately constant. The length of the snow patch was approximately 50 m, with the maximum snow depth in the regions of interest estimated to be on the order of 0.5 m. Selection of the location was based on the following criteria: a relatively flat surface and the absence of complex topographical features nearby, which could complicate the incoming radiation by causing, for instance, partial shading. The height maps enable us to derive the amount of snowmelt during these 5 d for this single snow patch and assess the role of the vertical turbulent heat fluxes. To do so, the meteorological data are averaged over the period between the photogrammetry observations and compared with the daily melt observations. Also, three snow samples were taken to determine the snow density, such that the measured height changes could be converted to a volume. The samples were taken by digging a small snow pit and collecting 100 mL samples at 5, 25, and 45 cm below the snow surface on 14 June. The snowpit was dug in snow adjacent to the snow patch so that the measurements would be similar to those for the snow patch but they would not affect the photogrammetry observations. We are aware that samples taken on a single day and location do not reflect the potentially complex temporal and spatial dynamics of the snow density. However, we assume the variations occurring at these scales to be relatively small compared to other uncertainties introduced by our analysis when computing estimates of the contribution of the vertical turbulent heat fluxes to the snowmelt.\n\nThe height maps are obtained by applying the photogrammetric principle of structure from motion (SfM) using MicMac . A total of 1087 pictures (610 for the upwind edge, 477 for the downwind edge) were taken from various positions at both edges and at time points spread out over the 5 d using a Xiaomi A2 smartphone camera. By using a ground-based camera, we were not able to obtain height maps covering the entire snow patch, which does prohibit a detailed analysis of the snowmelt. Yet, by using this method, we illustrate that it is still possible to come up with relatively decent snowmelt estimates using a simple and cheap method. The method is based on that of , who studied the melt of a relatively large snowfield with three time-lapse cameras over the course of an ablation season. In this study, MicMac is used to initially determine the camera positions and orientations. Subsequently, tie points appearing on multiple images for all days are obtained, such that the orientations of the camera positions can be determined. Considering images from all days during the tie point retrieval allows the eventual height models per day to be coregistered from the start. Eventually, each time step is processed using the MicMac implementation of multiview stereopsis to create point clouds for each day, which are interpolated into orthoimages and digital elevation models (DEM). The orthoimages are in RGB colors, allowing snow to be distinguished from bare ground and the amount of snowmelt to be determined for snow-covered pixels (0.04 × 0.04 m) in the DEM.\n\nBoth types of grids are available for each of the 5 d and for both locations (the upwind and downwind edges), such that we have 20 grids in total. For both locations, the following post-processing procedure is performed after obtaining the grids:\n\n1. For all grids, remove isolated groups of cells which are smaller than 0.05 m2.\n\n2. For all grids, apply a median filter of 5 × 5 pixels to diminish the influence of noise located within the areas of interest but maintain the sharp transitions between snow and snow-free surfaces.\n\n3. Compute the median height of the bare ground cells per day. The cells selected for this computation should be covered by all grids and already be bare ground on the first day.\n\n4. Compute correction heights by comparing the daily median heights of the bare ground with the median height on the first day (step 3).\n\n5. Remove bare ground cells from the DEM based on orthoimages of the same day, such that only snow-covered cells remain for each day.\n\n6. Apply the correction height (step 4) to the snow-covered DEM (step 5) for each day.\n\n7. For snow-covered grid cells that are present on each day (step 5), we calculate height differences between the DEM for the first day and the DEMs for the other days (step 6). We remove absolute height differences of larger than 50 cm, as larger values are highly unlikely to occur.\n\nThe resulting height differences over time correspond to 6.7 and 30.7 m2, respectively, for the upwind and downwind edges. We chose to solely use grid cells that are continuously covered by snow and have a recorded height change on each day to reduce the chance of cells being random scatter. Additionally, this method does not include cells with relatively shallow snow depths, for which the recorded melt could be affected by the presence of the bare ground below the snow.\n\nWe are aware that these filters have an effect on the number of analyzed grid cells and could cause an underestimation of the amount of snowmelt, especially on the upwind edge, due to the varying locations of snow-covered grid cells and the retreating snow line (Fig. A1). For the downwind edge, the approximately constant locations of the snow-covered grid cells combined with only minor retreat at this edge causes this area to be significantly larger. Due to the issue with the sizes of the covered areas, we decided to treat both the edges as a “point” and not consider the spatial distribution of the recorded melt.\n\nTo determine the snowmelt based on the net radiation, the measured incoming radiation is combined with estimates of the outgoing radiation based on common snow characteristics. The outgoing shortwave radiation is calculated by assuming a snow albedo of between 0.6 and 0.8. These values are based on , who reports albedos of approximately 0.8 for the same region in May. We use both of these values in the following calculations to account for the uncertainty in the albedo due to spatial and temporal variations and other potential uncertainties in the shortwave component. We also assume an appropriate estimate of the longwave radiation component for the larger region. Furthermore, we assume that the snowpack is continuously ripe throughout research period, given that the air temperature was continuously above freezing point and the largest discharge peak had already taken place 1.5 months before the observation period. This allows us to compute the outgoing longwave radiation according to the law of Stefan–Boltzmann (i.e., ${\\mathrm{LW}}^{↑}=\\mathit{ϵ}\\mathit{\\sigma }{T}_{\\mathrm{sn}}^{\\mathrm{4}}$). It is assumed that the emissivity ϵ is close to unity and the snow surface temperature is 273.15 K, such that the outgoing longwave radiation is approximately 315 W m−2. This results in the following equation for net radiation:\n\n$\\begin{array}{}\\text{(1)}& {R}_{\\mathrm{net}}=\\left(\\mathrm{1}-\\mathit{\\alpha }\\right){\\mathrm{SW}}^{↓}+{\\mathrm{LW}}^{↓}-\\mathit{\\sigma }{T}_{\\mathrm{sn}}^{\\mathrm{4}},\\end{array}$\n\nin which Rnet (W m−2) is the net radiation, α () is the snow albedo, and σ (W m−2 K−4) is the constant of Stefan–Boltzmann. Subsequently, we calculate the amount of snowmelt caused by the net radiation by combining the net radiation with the snow density and the constant for latent heat of fusion, which is 334 KJ kg−1. This can be recalculated such that the radiation-driven melt is eventually expressed as a height change:\n\n$\\begin{array}{}\\text{(2)}& {M}_{\\mathrm{R}}=\\frac{{R}_{\\mathrm{net}}×\\mathrm{\\Delta }t}{{\\mathit{\\rho }}_{\\mathrm{sn}}×{L}_{\\mathrm{f}}}\\phantom{\\rule{0.125em}{0ex}},\\end{array}$\n\nin which MR (m) is the snowmelt due to radiation expressed as a height change, Δt (s) is the time between photogrammetry observations, ρsn (kg m−3) is the snow density, and Lf (J kg−1) the constant of the latent heat of fusion. Subsequently, we assume that the total melt is caused by radiation and the vertical turbulent heat fluxes (e.g., ), such that the difference between the total observed melt and the computed radiation-driven melt can be attributed to these turbulent heat fluxes.\n\nTo obtain an indication of the influence of the latent heat on the melt, the relative humidity is used to calculate the vapor pressure difference between the air and snow surface (eesn) and, subsequently, the specific humidity difference (qqsn). To calculate eesn, we assume the vapor pressure of the snow to be the saturated vapor pressure of air at 0 C, which is 0.613 kPa. We are aware that the relative humidity used (measured at the meteorological tower) probably does not reflect the exact conditions at the snow patch. Therefore, we only use these computed values as an indication of the contribution of the latent heat to the melt.\n\nWhen comparing the measured meteorological conditions with the snowmelt, especially the net radiation (Eqs. 1 and 2), the spatial variability of these conditions should be considered. For this comparison, we apply the measured values without performing any additional computations other than time averaging, introducing additional uncertainty. For example, the snow patch was located at the bottom of a north–south-oriented side valley, which caused shading at sunrise and sunset. The most prominent mountains to the east and west of the snow patch were approximately 150 to 200 m higher than and 1 to 1.5 km distant from the snow patch. The meteorological measurements were done in an east–west-oriented main valley, such that less shading occurs at sunrise and sunset.", null, "Figure 2Conceptual situation sketches. Sketches of one snow patch and the adjacent bare ground (i.e., an element) (a) and an exemplary horizontal domain with indicated element and snow patches (b). The parameters represent the viscosity ν, thermal diffusivity κ, wind speed u, height of the atmospheric surface layer δ, average size of one snow patch element λelem (consisting of the length of the snow patch λsn and the adjacent bare ground: ${\\mathit{\\lambda }}_{\\mathrm{elem}}\\equiv {\\mathit{\\lambda }}_{\\mathrm{bg}}+{\\mathit{\\lambda }}_{\\mathrm{sn}}$), and the buoyancies of the snow bsn and the bare ground bbg.\n\n3 Idealized system\n\n## 3.1 System description\n\nWe used an idealized system (Fig. 2) to study the turbulent heat fluxes in detail and understand the behavior observed in the field. As this is one of the first studies to use DNS to investigate the role of the vertical turbulent heat fluxes and local heat advection in these systems, we focus on the sensible heat flux, even though MicroHH allows us to include the latent heat flux (e.g., ). Instead of using our own measurements, the idealized system is based on the measurements of a single snow patch done by , due to the availability of relatively high-resolution measurements and the similarity to an idealized system in which the contribution of the local-scale advection of the sensible and latent heat to the total melt is relatively large. We are aware that this complicates the comparison between our observations and simulations. The prescribed conditions within simulations of the idealized system are based on the observations of and obtained through a dimensional analysis (elaborated on in Sect. 3.2). We assume that our idealized system consists of, on average, a near-neutral atmosphere above a patchy surface with heterogeneous properties and can be described by the following variables:\n\n$\\begin{array}{}\\text{(3)}& \\left(\\mathit{\\nu },\\mathit{\\kappa },u,\\mathit{\\delta },{\\mathit{\\lambda }}_{\\mathrm{elem}},{\\mathit{\\lambda }}_{\\mathrm{sn}},{b}_{\\mathrm{sn}},{b}_{\\mathrm{bg}}\\right).\\end{array}$\n\nThe viscosity ν (m2 s−1), thermal diffusivity κ (m2 s−1), and wind speed u (m s−1) describe the properties of the neutral atmosphere. δ (m) is the height of the atmospheric surface layer, λelem (m) represents the average size of one snow patch element, consisting of the snow patch itself (λsn) and the adjacent bare ground (${\\mathit{\\lambda }}_{\\mathrm{elem}}\\equiv {\\mathit{\\lambda }}_{\\mathrm{bg}}+{\\mathit{\\lambda }}_{\\mathrm{sn}}$). The buoyancies of the surface layer over the snow (bsn) and the bare ground (bbg) are defined as\n\n$\\begin{array}{}\\text{(4)}& b\\equiv \\frac{g}{{\\mathit{\\theta }}_{\\mathrm{atm}}}\\left(\\mathit{\\theta }-{\\mathit{\\theta }}_{\\mathrm{atm}}\\right),\\end{array}$\n\nin which g (m s−2) is the gravitational acceleration, θatm (K) is the temperature of the atmosphere, and θ is, in our case, the temperature of snow, bare ground or atmosphere to be recalculated to the buoyancy b in this equation. This definition causes the temperature dimension to cancel out and m s−2 to remain as the unit. In the simulations, we assumed for each simulation that the simulated atmosphere is initially well mixed, such that the temperature of the atmosphere is constant with height (similar to ). Furthermore, we assumed that the horizontal extent is orders of magnitude larger than the elements so that this does not have an influence on the physics in the model.\n\nTo assess the impact of the snow patch size, we varied λsn in our simulations. Specifically, we doubled and quadrupled λsn compared to a reference simulation with 15 m long snow patches, similar to . Therefore, in the remainder of this paper, we will denote the reference simulation by P15m and the simulations with a doubled and a quadrupled snow patch size as P30m and P60m, respectively. The patches in P60m simulation (and some larger patches in the P30m simulation) can be compared with our own observations to study the dominant processes. Although the meteorology in the simulations is not based on the field observations, it allows for a qualitative analysis of the processes.\n\nFurthermore, an additional simulation is performed in which the system is the same as in the P15m simulation except that stability effects are excluded. This allows us to identify the influences of stability on the vertical sensible heat fluxes, as well as the dominant turbulent character. In this simulation, temperature does not influence the buoyancy of the air. To refer to this simulation in figures, we append “NB” (i.e., the abbreviation for “No Buoyancy”) to the name, i.e., P15m-NB.\n\n## 3.2 Dimensional analysis\n\n### 3.2.1 Parameter derivation\n\nTo create a system physically similar to the measurements done by within our modeling domain, a dimensional analysis is performed according to Buckingham's Pi theorem . The modeling domain in which we fit these measurements is based on the atmospheric turbulent channel flow from .\n\nAll of the descriptive variables (Sect. 3.1) are nondimensionalized by using δ and u as repeating variables, since these cover both the primary dimensions: [L] and [LT−1], respectively, and are constant throughout all the numerical experiments (Sect. 3.4). This results in six dimensionless groups, which can be combined into\n\n$\\begin{array}{}\\text{(5)}& \\left(\\frac{\\mathit{\\nu }}{\\mathit{\\kappa }},\\frac{u\\mathit{\\delta }}{\\mathit{\\nu }},\\frac{{\\mathit{\\lambda }}_{\\mathrm{sn}}^{\\mathrm{2}}}{{\\mathit{\\lambda }}_{\\mathrm{elem}}^{\\mathrm{2}}},\\frac{\\mathit{\\delta }}{{\\mathit{\\lambda }}_{\\mathrm{elem}}},-\\frac{{b}_{\\mathrm{sn}}\\mathit{\\delta }}{{u}^{\\mathrm{2}}},-\\frac{{b}_{\\mathrm{bg}}\\mathit{\\delta }}{{u}^{\\mathrm{2}}}\\right).\\end{array}$\n\nThe nondimensional parameters can be interpreted as follows:\n\n• $\\frac{\\mathit{\\nu }}{\\mathit{\\kappa }}$ is the Prandtl number (Pr). This is assumed to be constant throughout this study, such that the flow over the patchy surface always has the same characteristics and does not influence the outcome of the simulations. In this study, the number is set to 1 instead of 0.71, which is the atmospheric value. This deviation has negligible impacts on the flow and allows for simpler scaling when analyzing the simulations .\n\n• $\\frac{u\\mathit{\\delta }}{\\mathit{\\nu }}$ is the Reynolds number (Re). For the same reason as the Prandtl number, this parameter is taken to be constant throughout the study. During this study, the same Reynolds number as in is taken: 1.10×104. Therefore, we impose a wind speed of 0.11 m s−1, a 1 m vertical extent, and a viscosity of $\\mathrm{1.0}×{\\mathrm{10}}^{-\\mathrm{5}}$ m2 s−1.\n\n• $\\frac{{\\mathit{\\lambda }}_{\\mathrm{sn}}^{\\mathrm{2}}}{{\\mathit{\\lambda }}_{\\mathrm{elem}}^{\\mathrm{2}}}$ is the snow cover fraction (SCF) and varies between zero and 1. If this parameter is 1, the field is completely snow covered, whereas values close to zero represent low snow-cover fractions. For all simulations, this dimensionless group is set to a value of 0.25, similar to , as does not provide any information about this.\n\n• $\\frac{\\mathit{\\delta }}{{\\mathit{\\lambda }}_{\\mathrm{elem}}}$ is a measure of the size of a surface element compared to the height of the system. This nondimensional number will decrease if the typical element size increases. If the snow cover fraction is kept constant, changes in this variable will affect the typical snow patch size, meaning that this variable can be interpreted as a measure of the relative snow patch size.\n\n• $-\\frac{{b}_{\\mathrm{sn}}\\mathit{\\delta }}{{u}^{\\mathrm{2}}}$ is the bulk Richardson number above the snow patches (Risn). Positive and negative values indicate, respectively, stability and instability, whereas the absolute values specify the magnitude of the (in)stability.\n\n• $-\\frac{{b}_{\\mathrm{bg}}\\mathit{\\delta }}{{u}^{\\mathrm{2}}}$ is the bulk Richardson number above the bare ground (Ribg). For Ribg, the same principles hold as for Risn.\n\nWe ensure that the increases in snow patch size λsn that are necessary for our analysis only affect the relative element size $\\frac{\\mathit{\\delta }}{{\\mathit{\\lambda }}_{\\mathrm{elem}}}$. We do this by changing λsn in tandem with λelem such that the snow cover fraction $\\frac{{\\mathit{\\lambda }}_{\\mathrm{sn}}^{\\mathrm{2}}}{{\\mathit{\\lambda }}_{\\mathrm{elem}}^{\\mathrm{2}}}$ remains constant. This allows us to study the impact of increasing the snow patch size separately from the snow cover fraction.\n\n### 3.2.2 Parameter estimation\n\nThe measurements of are first used to calculate the values of the defined dimensionless parameters (Table 1). Note that not all values of the variables used in the dimensionless parameters are known and are therefore estimated based on other literature, such as (Table 2). Subsequently, the values of the variables from are filled into the nondimensional parameters so that the remaining dimensionless parameters can be calculated for all simulations.\n\nIt should be noted that, for the P60m simulation, the nondimensional number $\\frac{\\mathit{\\delta }}{{\\mathit{\\lambda }}_{\\mathrm{elem}}}$ is rather low and could potentially affect the outcomes. During this simulation, the number is 0.83, whereas the number was originally designed to be approximately 1 or higher so that the relative snow patch size would not be too large. The implications of this could be that the largest turbulent structures arising due to the snow patches do not have enough space to develop and are therefore affected by the horizontal domain of the simulation. However, the results show no clear deviations from the expectation, and thus we assume that the influence of this value of the nondimensional number is relatively minor to nonexistent.\n\nTable 1Dimensionless parameter values. Overview of the values of the dimensionless parameters applied during the simulations.", null, "* These are not, in fact, real Richardson numbers, as buoyancy does not affect the flow in the P15m-NB simulation.\n\nTable 2Variable values applied in the simulations. Overview of the values used for obtaining the dimensionless parameters in the simulations.", null, "* Values were estimated based on .\n\n## 3.3 Rescaling the model results to reality\n\nAfter nondimensionalizing the system for the simulations, the outcomes of the simulations are rescaled back to realistic scales to compare with the observations. In order to do so, the nondimensional numbers are used, as these numbers characterize the dominant processes in the considered idealized system.\n\nWhen rescaling the surface heat fluxes back to reality, Risn influences the rescaling factor that is used. This factor indicates whether shear-driven turbulence or buoyancy-driven turbulence or a balance between the two is more dominant in the system, and thus also which process predominantly affects the surface buoyancy flux scaling. We assume shear-driven turbulence to be dominant near the surface, such that the rescaling equation becomes\n\n$\\begin{array}{}\\text{(6)}& {B}_{\\mathrm{0}}=u{b}_{\\mathrm{sn}}\\phantom{\\rule{0.125em}{0ex}},\\end{array}$\n\nin which B0 is the typical buoyancy surface flux (m2 s−3), bsn is the surface buoyancy of snow (m s−2), and u is the wind speed (m s−1). This equation is derived from the original equation for surface buoyancy flux implemented in the model\n\nin which δv is the depth of the viscous sublayer (m), which is assumed to be of the same order of magnitude as δz, whereas bsn is assumed to be a measure of δb. For δv, we use the viscous sublayer instead of the boundary layer, as the latter is approximately neutral, such that buoyancy is constantly zero throughout the layer except near the surface.\n\nEquation (8) follows from a definition of the height of the viscous sublayer δv for conditions in which shear dominates, such that δv is determined by the wind and viscosity. This allows us to set up a relation between these three variables:\n\n$\\begin{array}{}\\text{(8)}& \\begin{array}{rl}{\\mathit{\\delta }}_{\\mathrm{v}}& =f\\left(u,\\mathit{\\nu }\\right),\\\\ & \\sim \\frac{\\mathit{\\nu }}{u}.\\end{array}\\end{array}$\n\nImplementing this in Eq. (7) gives Eq. (6):\n\n$\\begin{array}{}\\text{(9)}& {B}_{\\mathrm{0}}\\sim \\mathit{\\kappa }\\frac{{b}_{\\mathrm{sn}}}{\\left(\\frac{\\mathit{\\nu }}{u}\\right)}=\\mathit{\\kappa }u\\frac{{b}_{\\mathrm{sn}}}{\\mathit{\\nu }}=u{b}_{\\mathrm{sn}},\\end{array}$\n\nas κ and ν cancel out since Pr (i.e., $\\frac{\\mathit{\\nu }}{\\mathit{\\kappa }}$) is unity in this study. Subsequently, this equation can be used to rescale the simulated surface buoyancy fluxes back to realistic values according to\n\n$\\begin{array}{}\\text{(10)}& \\frac{{B}_{\\mathrm{0},\\mathrm{sim}}}{{B}_{\\mathrm{0},\\mathrm{real}}}=\\frac{{u}_{\\mathrm{sim}}{b}_{\\mathrm{sn},\\mathrm{sim}}}{{u}_{\\mathrm{real}}{b}_{\\mathrm{sn},\\mathrm{real}}}.\\end{array}$\n\nFrom Table 2, it follows that ${b}_{\\mathrm{sn},\\mathrm{sim}}=-\\mathrm{0.008}$ m s−2, ${b}_{\\mathrm{sn},\\mathrm{real}}=-\\mathrm{0.28}$ m s−2, usim=0.11 m s−1, and ureal=6.4 m s−2, so the equation reduces to\n\n$\\begin{array}{}\\text{(11)}& \\frac{{B}_{\\mathrm{0},\\mathrm{sim}}}{{B}_{\\mathrm{0},\\mathrm{real}}}=\\frac{\\mathrm{0.11}×-\\mathrm{0.008}}{\\mathrm{6.4}×-\\mathrm{0.28}}=\\mathrm{4.9}×{\\mathrm{10}}^{-\\mathrm{4}}.\\end{array}$\n\nThus, the simulated values for the surface buoyancy fluxes are divided by $\\mathrm{4.9}×{\\mathrm{10}}^{-\\mathrm{4}}$ to compute the realistic values.\n\nSubsequently, the outcome for the realistic surface heat fluxes can be transformed into W m−2 according to\n\n$\\begin{array}{}\\text{(12)}& {H}_{\\mathrm{real}}=\\mathit{\\rho }{c}_{\\mathrm{p}}\\frac{{\\mathit{\\theta }}_{\\mathrm{0}}}{g}{B}_{\\mathrm{real}},\\end{array}$\n\nin which H is the sensible heat flux into the surface (W m−2), ρ is the density of the air (kg m−3), cp is the specific heat capacity (J kg−1 K−1), θ0 is the reference temperature (273 K in this case), and g the gravitational acceleration (m s−2).\n\nTo rescale the simulated time back to reality, the bulk Richardson number above snow (${\\mathit{Ri}}_{\\mathrm{sn}}=-\\frac{{b}_{\\mathrm{sn}}\\mathit{\\delta }}{{u}^{\\mathrm{2}}}$) is used for obtaining the timescale. From this number, the following timescale is derived:\n\n$\\begin{array}{}\\text{(13)}& \\frac{\\mathit{\\delta }}{u}={t}_{\\mathrm{adv}}\\phantom{\\rule{0.125em}{0ex}},\\end{array}$\n\nWhen calculating the ratios between the simulated and measured timescales by making use of the values in Table 2, the following ratio between the simulated and realistic timescales arises:\n\n$\\begin{array}{}\\text{(14)}& \\frac{{t}_{\\mathrm{adv},\\mathrm{sim}}}{{t}_{\\mathrm{adv},\\mathrm{meas}}}=\\frac{\\mathrm{9.09}}{\\mathrm{15.63}}=\\mathrm{0.58}.\\end{array}$\n\nThus, the simulated time is divided by 0.58 to compute the realistic value.\n\n## 3.4 Model description and setup\n\nIn this study, the model simulations are performed using the MicroHH 2.0 code, which was primarily made for the DNS of atmospheric flows over complex surfaces by . When solving the conservation equations for mass, momentum, and energy, MicroHH makes use of the Boussinesq approximation, such that the evolution of the system for a velocity vector with element ui, buoyancy b, and volume is described by\n\n$\\begin{array}{}\\text{(15)}& \\begin{array}{rl}\\frac{\\partial {u}_{i}}{\\partial t}+\\frac{\\partial {u}_{j}{u}_{i}}{\\partial {x}_{j}}& =-\\frac{\\partial \\mathit{\\pi }}{\\partial {x}_{i}}+{\\mathit{\\delta }}_{i\\mathrm{3}}b+\\mathit{\\nu }\\frac{{\\partial }^{\\mathrm{2}}{u}_{i}}{\\partial {x}_{j}^{\\mathrm{2}}},\\\\ \\frac{\\partial b}{\\partial t}+\\frac{\\partial {u}_{j}b}{\\partial {x}_{j}}& =\\mathit{\\kappa }\\frac{{\\partial }^{\\mathrm{2}}b}{\\partial {x}_{j}^{\\mathrm{2}}},\\\\ \\frac{\\partial {u}_{j}}{\\partial {x}_{j}}& =\\mathrm{0},\\end{array}\\end{array}$\n\nin which π is a modified pressure . Moreover, MicroHH uses periodic boundary conditions in the horizontal directions, which implies that we simulate a wind blowing over an infinite snow field. In the following parts of the article, when we report vertical sensible heat fluxes into the surface, these were recalculated from the surface buoyancy flux B (m2 s−3) computed in the model equation according to\n\nwhich can be recalculated to give realistic sensible heat fluxes following the steps in Sect. 3.2. Similarly to (Eq. 2), the horizontally advected sensible heat is computed via\n\n$\\begin{array}{}\\text{(17)}& {H}_{\\mathrm{adv}}=\\underset{z=\\mathrm{0}\\phantom{\\rule{0.125em}{0ex}}\\mathrm{m}}{\\overset{z=\\mathrm{2}\\phantom{\\rule{0.125em}{0ex}}\\mathrm{m}}{\\int }}\\mathit{\\rho }{c}_{\\mathrm{p}}\\stackrel{\\mathrm{‾}}{u}\\frac{\\partial \\stackrel{\\mathrm{‾}}{\\mathit{\\theta }}}{\\partial x}\\mathrm{d}z,\\end{array}$\n\nin which z (m) is the elevation above the surface, ρ (kg m−3) is the air density, cp (1005 J kg−1 K−1) is the specific heat capacity of air, and x (m) is the distance from the leading edge of the snow patch. As well as integrating over a 2 m profile height, we also integrate over a 4 m height.\n\nThe starting point for designing the numerical experiments is the atmospheric turbulent channel flow with a reduced Reynolds number (Re) designed by . This simulation is also used as a spin-up, during which we have no buoyancy effects included in the flow. The shear Reynolds number Reτ obtained from the measurements performed by is relatively high compared to that of : $\\sim \\mathrm{6}×{\\mathrm{10}}^{\\mathrm{6}}$ vs. 590, but the results of suggest that the bulk statistics, i.e., the means and variances, for at least the initial neutral channel flow are hardly affected Reτ. This channel flow is simulated in MicroHH at a resolution of 384 × 192 × 128 grid points for a domain size of 2π m ×π m × 2 m. The flow is forced in the x direction by imposing an average wind speed, which is, in this case, 0.11 m s−1. At the bottom and top boundaries, no slip and no penetration conditions are applied to the velocities (i.e., the flow velocity at the boundary is zero). Overall, showed that MicroHH is able to reproduce this turbulent channel flow.\n\nInitially, the turbulent channel flow from used as a spin-up is simulated until 1800 s so that the turbulent channel flow is well developed. Subsequently, for each simulation, which takes 900 s, this turbulent channel flow is adapted such that an atmospheric flow over a patchy snow surface is obtained. At the bottom boundary, a pattern of surface buoyancies that depends on the simulation is prescribed, such that the surface characteristics determined during the dimensional analysis are fulfilled. For snow and bare ground, the surface buoyancy is respectively $-\\mathrm{8}×{\\mathrm{10}}^{-\\mathrm{3}}$ and 0.0 m s−2 (i.e., 273 and 280.9 K in reality), which is elaborated on in Sect. 3.2. The implemented surface for P15m and P15m-NB (P30m, P60m) contains snow patch lengths of 0.15 m (0.30 m, 0.60 m) on average, and average element lengths of 0.30 m (0.60 m, 1.20 m) (Table 2; multiplied with 100 in Fig. 3). These surfaces enable the model to solve the periodic boundary conditions, because we ensure that the patches at opposing walls fit together and that flow that leaves the system on one side continues over the same snow patch when it reenters the system on the opposite wall. These patterns are created by generating noise in the Fourier space around specific wavelengths, prescribed in the form of 2D power spectra. When transforming these back to physical space using the inverse fast Fourier transform, a 2D field with dominant patterns is obtained. For a more elaborate explanation of generation, see Appendix B.", null, "Figure 3Generated realistic surface temperatures for the P15m and P15m-NB (a), P30m (b), and P60m (c) simulations.\n\n4 Field observations\n\nAt the upwind edge of the snow patch, the snow surface decreased by approximately 23 cm, whereas for the downwind edge, this decrease was approximately 3 cm (Fig. 4). For the upwind edge, the height change is computed relative to 11 June 17:00 LT. For the downwind edge, the height change is computed relative to 12 June 16:00 LT, due to a low coverage of bare ground pixels on 11 June (Fig. A1), which are used for computing the vertical correction employed to align all height maps (Table 3). For the upwind edge, the melt relative to 12 June 16:00 LT is estimated to be 21 cm.", null, "Figure 4Boxplot of height changes for the upwind and downwind edges of a snow patch recorded with SfM photogrammetry from 11 June 2019 until 15 June 2019. For the upwind edge, 11 June is taken as the reference date. For the downwind edge, the first measurement on 11 June is not used as the reference, due to a low coverage of grid cells. The melt is estimated with an error of ±2.0 and ±0.4 cm, respectively, for the upwind and downwind edges.\n\nThe average standard error after applying the vertical correction for the bare ground grid cells that are present throughout the all height maps is 1.38 cm for the upwind edge and 0.29 cm for the downwind edge. The propagation of these errors ($\\sqrt{\\mathrm{2}{\\mathit{\\sigma }}_{\\mathrm{\\Delta }\\stackrel{\\mathrm{‾}}{h}}^{\\mathrm{2}}}$), which is caused by relating the height changes in two height models, allows us to estimate the melt with an error of ±2.0 and ±0.4 cm, respectively, for the upwind and downwind edges. Usually, these standard errors are calculated over other grid cells rather than those used for the vertical correction, such that these points are independent. In this study, however, the standard error and vertical correction are computed for all bare-ground grid cells present throughout the research period, since the relatively small spatial scale causes all grid cells to be related. Considering the errors, for the upwind edge, significant melt was recorded on 13 June and onwards as compared to 11 June, used as reference. For the downwind edge, significant melt was recorded on 13 and 15 June, taking 12 June as reference. For the third period from 13 to 14 June, an increase in snow height relative to the previous day was recorded, but this increase was not significant due to overlapping error estimates.\n\nTable 3Vertical shift correction (cm) applied in the individual height models per day for the upwind and downwind locations.", null, "* Only computed with the bare ground grid cells present in this height map.\n\n## 4.1 Relation between snowmelt and meteorology\n\nRelating the meteorological circumstances (Table 4) to the measured snowmelt allows us to estimate the contribution of the vertical turbulent heat fluxes to the total amount of snowmelt. To do so, the snow density measurements are used. The snow densities at 5, 25, and 45 cm below the snow surface are, respectively, 556, 551, and 610 kg m−3. It follows from these densities that the snow density is approximately constant near the surface, whereas the snow is more compressed or stores water further down. Even though these densities are relatively high, we consider the values to be realistic considering that the largest discharge peak took place 1.5 months before the observation period (Fig. 1) and given the observation that the snowpack was relatively wet in the field.\n\nTable 4Average meteorological measurements and calculated variables in between the photogrammetry observations. T2 m is the air temperature measured at 2 m; u10 m and udir are, respectively, the wind speed and wind direction in degrees from the north and measured at 10 m; Pr is the summed precipitation during the period; SW and LW are, respectively, the incoming shortwave and longwave radiation; RH2 m is the measured relative humidity at 2 m; and Pa is the air pressure. The net radiation (Rnet), subsequent melt (MR), and specific humidity difference (qqsn) were computed based on a combination of the measured variables. The ranges for Rnet and MR are caused by applying two values for the albedo, i.e., 0.6 and 0.8, to account for uncertainties in the shortwave radiation component (see Sect. 2).", null, "The minimum estimated melt due to net radiation from 12 June until 15 June was 3.9 cm (Table 4), which is 0.5 cm larger than the melt estimate including the error for the downwind edge. As the snow patch was approximately 50 m in length, the turbulent heat fluxes into the snow likely reduced to negligible values at the downwind edge (e.g., when extrapolating the measurements of ), so this mismatch is most likely caused by uncertainties in the net radiation estimates. When assuming this radiation to be an appropriate estimate and that it is homogeneously spread over the patch, the estimated contribution of the vertical turbulent heat fluxes at the upwind edge is 13.0 to 18.2 ± 2.0 cm. For this, we also assume that the residual of the difference between the observed snowmelt and radiation-driven snowmelt is caused by these turbulent heat fluxes (e.g., ). When using Eq. (2) to compute the average vertical turbulent heat fluxes during the period, we estimate these fluxes at the upwind edge to be between 73 and 102 ± 11 W m−2. These are of the same order as those found in other studies in somewhat similar conditions, such as , , and . reported a sensible heat flux into the snow of up to 50 W m−2 for typical weather situations in alpine areas, whereas the latent heat was negligible or even contributed to the cooling of the snow. reported downward turbulent heat fluxes of over 100 W m−2 for a large single snow patch. estimated the full energy balance for 15 d in May of a relatively homogeneous snow cover near Finse and reported a sensible heat flux of approximately 20 W m−2 and a latent heat flux that was negligible on average. For our observations, however, it is expected the latent heat flux also had a significant contribution from the high relative humidity, with a subsequent difference in specific humidity between the air and snow of at least 0.9 g kg−1 (Table 4) during the field campaign, which is common for Finseelvi as it is located in a maritime climate. Our results show similar moisture gradients to , who found a significant influence of the latent heat on the snowmelt, such that we expect this to be a significant part of our estimated turbulence-driven melt. Overall, by comparing the overall melt with the melt driven by the radiation and taking the residual as the turbulence-driven melt, we estimate the contribution of the turbulent heat fluxes to the snowmelt to be roughly 60 to 80 % for the upwind edge of the snow patch. Extrapolating this to the entire catchment and applying the relations reported by for two test sites in the Alps, we estimate the contribution of the fluxes to the total melt to be maximally on the order of 10 % under the assumption that the entire catchment behaves similarly.\n\nIt is likely that the incoming radiation for the entire snow patch is overestimated, due to topography. The incoming solar radiation at the snow patch is blocked by mountains at low solar angles in the east and west, which does not apply at the location of the meteorological observations. Moreover, the incoming longwave radiation reduces with height, as was found in the Alps by , such that this snow patch – located 150 m higher – received less incoming longwave radiation. Lastly, other characteristics influencing the net radiation, such as a varying snow albedo and slightly different slopes for the upwind and downwind edges, are also identified as potential influences on the results.\n\n5 Model simulations\n\n## 5.1 System characteristics\n\nIn the idealized simulations, the time-averaged sensible turbulent heat fluxes resemble the implemented surface pattern of snow patches (Fig. 5). There is a negative surface flux on the snow patches, whereas the surface flux is positive at the bare ground. For each snow patch, there is a clear pattern that is somewhat similar to the observations. The leading edge of the snow patch shows the highest fluxes towards the surface, $\\sim -\\mathrm{500}$ W m−2, which decreases downwind of the leading edge until the end of the patch. Subsequently, at the trailing edge of the snow patch and the leading edge of the bare ground, the sensible heat flux changes sign, as the bare ground is relatively warm compared to the colder air coming from above the snow patch, resulting in ∼300 W m−2. The air warms when flowing over the bare ground, such that the air arriving at the next snow patch is relatively warm compared to that at the cold snow patch, causing a high downward flux. These fluxes at the leading edges of the snow patches are relatively high, mostly due to the presence of ideal circumstances for wind-driven melt, including local-scale advection of sensible heat (i.e., high wind speeds and relatively large temperature differences). Compared to our own observations, the sensible heat fluxes at the leading edge are much larger than the observed combined turbulent heat fluxes. This is in line with our expectations because of the inclusion of nonideal circumstances for local-scale advection of the sensible and latent heat during the field campaign.", null, "Figure 5Time-averaged surface sensible turbulent heat fluxes. Sensible turbulent heat fluxes in the P15m simulation averaged from 2000 seconds until 2700 s. Negative values indicate a downward flux, i.e., over snow patches, whereas positive values resemble an upward flux, i.e., over bare ground.\n\n## 5.2 Total snowmelt\n\nWhen comparing the average sensible heat fluxes for all the snow patches in the domain, clear differences arise between all simulations (Fig. 6). The highest sensible heat fluxes towards the snow (i.e., the most negative fluxes) are found in the simulation without buoyancy effects. This also causes the total sensible heat flux in this simulation to be significantly lower than in the other simulations. Furthermore, increasing the snow patch size reduces the heat fluxes into the snow patches. The heat fluxes of the simulation with a doubled snow patch size (P30m) are decreased by approximately 15 % relative to the P15m simulation. For the simulation with a quadrupled snow patch size (P60m), the heat fluxes are reduced by approximately 25 %. This is in contrast to the results of , who report only a minor influence of snow patch size on the amount of melt. Our findings are more in line with the results of , who based their work on a 2D boundary layer model with a regular tiled surface pattern. Potentially, the differences from are caused by the inability of ARPS to fully resolve the leading edge effect, as the resolution is too coarse to resolve the thin internal stable boundary layer formed over snow patches and the Monin–Obukhov assumptions are violated. Neither of these limitations apply for DNS.", null, "Figure 6Domain-averaged sensible heat fluxes for different surfaces. The figure shows time series of the averaged surface sensible heat fluxes for the bare ground, snow, and total surface after the introduction of the temperature differences at the surface.\n\nThe total heat fluxes for the simulations with 15, 30, and 60 m snow patches coincide approximately, as the differences arising at the snow surface are compensated for at the bare ground surface. So, although the total fluxes are equal, the snowmelt does vary with snow patch size. The simulation without buoyancy effects (P15m-NB) has a significantly reduced total heat flux compared to the original simulation (P15m). This is caused by similar averaged heat fluxes for the bare ground for both simulations, whereas this is not the case above the snow patches. This suggests that stability has little effect on the fluxes above the bare ground, whereas the surface heat fluxes above the snow are affected by stability.\n\nMoreover, the largest adjustment of the sensible heat fluxes after the initiation of the simulations is done after less than 200 s. However, a minor long-term trend is still present for each simulation. For this study, we assume that after the largest adjustment, the dominant processes are well developed and suffice to understand the system. We expect that, eventually, the total summed surface sensible heat fluxes will go to zero, due to the infinite blowing of the air over the snow cover without any other heat fluxes than those originating from the snow patches and bare ground. However, as the volume of the channel is relatively large compared to the heat fluxes, it takes a relatively long time before the whole system has cooled and reached an equilibrium.\n\n## 5.3 Surface fluxes for individual patches\n\nThe surface fluxes for all individual snow patches in each simulation show a similar behavior over distance from the leading edge, but differences occur above snow patches due to varying fluxes at the leading edge (Fig. 7). The linear behavior of the surface fluxes as a function of the distance from the upwind edge on logarithmic axes implies that the fluxes decay over distance from the leading edge according to a power law. The power laws take the following approximate forms:\n\n$\\begin{array}{}\\text{(18)}& {H}_{\\mathrm{sn}}\\left(x\\right)\\equiv {C}_{\\mathrm{sn}}{x}^{-\\mathrm{0.35}},\\end{array}$\n\nin which Hsn (W m−2) is the sensible heat flux, x (m) is the distance from the leading edge, and C (W m−1.65) is a constant representing the initial conditions at the leading edge of each patch.", null, "Figure 7Time-averaged surface sensible heat fluxes over single patches of snow. The surface sensible heat fluxes for individual patches with snow as a function of distance from the leading edge on a log-log scale at y=1.1 m are shown. The dashed black line is a trendline implemented to show the approximate power law decay of the fluxes over distance. The fluxes of the snow are multiplied with −1 so that these values can also be plotted on logarithmic axes.\n\nOur simulated vertical sensible heat fluxes into the snow are approximately 500 W m−2 at the upwind edge and 200–300 W m−2 at the downwind edge. In comparison to our field observations, these sensible heat fluxes at both edges are relatively high. At the upwind edge, the simulated sensible heat fluxes are approximately 5 times larger than the derived contribution of the combined turbulent heat fluxes to the measured snowmelt. We reckon that the simulated values are large, though it should be noted that the simulations are based on highly ideal conditions for turbulence-driven melt and local-scale advection of sensible and latent heat, whereas the conditions during the measurements were not ideal (e.g., nighttime melt was included). At the downwind edge, the measurements suggest an approximately negligible contribution of the vertical turbulent heat fluxes to the snowmelt, whereas, at comparable snow patches, the simulations show a significant contribution of the sensible heat flux (∼200–300 W m−2). Thus, the simulated decay of the sensible heat flux seems to be an underestimation in comparison to field observations. We expect that the comparable behavior of the sensible heat fluxes between patches within the idealized system is also occurring within the Finseelvi catchment for patches within similar local conditions.\n\nFigure 7 shows that the length of snow patches is the main cause of less snowmelt for larger patches, which was also found by . The power laws are approximately the same for each patch and simulation, whereas Csn is the same for each simulation except in the simulation without buoyancy effects. This behavior explains why larger snow patches reduce the average surface fluxes into the snow patches.\n\nThe behavior of the simulation without buoyancy effects is striking, as the decay of the fluxes for this simulation is similar to the decay of the fluxes for the other simulations, suggesting that stability has little influence on the decay of the surface heat fluxes, i.e., shear turbulence dominates. However, this simulation has a relatively low initial surface flux above the snow (a higher absolute flux in Fig. 7), indicating an effect of the stability on the leading edge conditions. Thus, the differences in total snowmelt (Fig. 6) between the P15m and P15m-NB simulations solely occur due to these differences at the leading edge.\n\nThe decay of the sensible heat fluxes is a consequence of the decreasing temperature gradients in the IBL (Fig. 8a). Wind speed, another important component affecting the sensible heat flux, remains constant over a snow patch (Fig. 8b). Moreover, surface roughness is the same for the entire domain. Strikingly, the average temperature within the IBL remains constant and does not depend on the height of this IBL, as the shape of the vertical temperature profile remains constant while the flow proceeds over the snow patch (Fig. 8a inset). Yet, it should be noted that the reported values for the height of the IBL are relatively high compared to .", null, "Figure 8Time-averaged vertical temperature (a) and wind (b) profiles and horizontally advected sensible energy Hadv at 2 and 4 m integration height and vertical sensible heat flux at the surface Hsn (c) along the snow patch at y=83 m and x=150 m in the P30m simulation. The markers in the temperature profile resemble the IBL height, which is computed as the first value (seen from the bottom) of the gradient of the temperature with height below 0.1 K m−1. The first two upwind grid cells and two downwind grid cells have been removed, as these are located in the transition region from bare ground to snow and vice versa. The labels with “x=” show the locations of vertical profile related to the downwind distance of the vertical profile from the leading edge, whereas the line color indicates the distance from the leading edge and goes from purple to red. In the inset graph, the vertical temperature profile has been normalized to the height of the IBL and the prescribed temperatures of the surface and atmosphere. The advected energy as a function of distance from the leading edge is based on Eq. (17).\n\n6 Discussion\n\nThis study aimed to assess the role of local-scale advection of sensible and latent heat fluxes on snowmelt. In order to do so, complementary field observations and simulations were performed. On small spatial scales, the largest melt differences due to the combined turbulent heat fluxes occur at the opposing edges of snow patches. Our results show that the upwind edge of a single snow patch in the Finseelvi catchment, which is approximately 50 m in length, melted 23 ± 2.0 cm over the course of 5 d, whereas the downwind edge melted just 3 ± 0.4 cm in 4 d. As the snow patch was approximately 50 m long, the vertical turbulent heat fluxes likely reduced to negligible values at the downwind edge, due to the leading edge effect, such that the main cause of melt at this edge was the net radiation. The simulations allowed us to extract detailed information on the atmospheric flow and were used as a tool to provide insight into the evolution of the fluxes and temperature over the patchy snow cover. The sensible heat fluxes reduce over distance from the upwind edge, following a constant power law which likely depends on the meteorological circumstances. In the simulations, this resulted in a reduction of 15 % and 25 % for, respectively, a doubling and a quadrupling in snow patch size. The simulations reveal that the reducing sensible heat fluxes over distance from the leading edge are caused by the reducing temperature gradients, pointing out the major role of the horizontally advected sensible heat, which we expect behaved similarly in our field observations. Other important factors for the turbulent heat fluxes, i.e., wind speed and surface roughness, are constant over distance from the leading edge in the simulations.\n\nHowever, the simulations lacked surface roughness differences for the snow and bare ground and topographical variations. Including the transition from a rough (bare ground) to a smooth (snow) surface would likely diminish the IBL growth due to increased turbulence levels and enhance the vertical sensible heat fluxes as a consequence of the larger temperature gradients in the IBL (Garratt1990). It should be noted that this mostly holds for shear-dominated turbulence, whereas at lower wind speeds, the influence of thermal turbulence on the IBL should also be considered. Moreover, the common formation of snow patches in topographical depressions causes atmospheric decoupling and reduced vertical turbulent heat fluxes at low and moderate wind speeds, especially downstream of the upwind edge . In the Finseelvi catchment, snow patches have formed to some extent in these depressions, while this does not hold for , Also, the choice to base the numerical experiments on instead of our own observations complicates an exact comparison. Lastly, the exclusion of external forcings, such as radiation or large-scale advection, caused the simulations to approach equilibrium, due to the compensation between the sensible heat fluxes at the snow patches and the bare ground. This is advantageous for investigating the behavior of the system in relation to snow patch size, but makes it more difficult to compare other characteristics (e.g., temperature) with, for example, , who did include external forcings, such as incoming radiation. Overall, these mechanisms greatly increase the uncertainty and made us decide not to directly compare between the simulations and observations.\n\nOn a catchment scale, these simulation results imply that differences in snowmelt within a highly idealized catchment occur solely due to snow patch length. The sensible heat fluxes into the snow at the upwind edges of the patches are independent of snow patch size and show the same decay over distance from the leading edge, such that systems with typically larger patches have, on average, reduced sensible heat fluxes into the snow. The major cause of these fluxes seems to be the horizontal advection of sensible heat. It should be noted that the latent heat flux, which is not considered in these simulations, can also play a significant role in the amount of snowmelt . For this flux, we expect similar mechanisms in our simulations based on the observations of . Variations in surface roughness and topography mostly create micrometeorological circumstances which differ substantially from the average circumstances within a catchment, for example through shading or a slope-induced drainage flow, and thus also affect snowmelt. We expect that the important role of the horizontally advected sensible heat and the identical behavior of the vertical sensible heat flux between patches, both of which were found in the simulations, are also applicable to our field observations, given the probable larger-scale approximate equilibrium of the atmosphere. However, anomalies in this behavior can be found due to varying micrometeorological conditions. Overall, our results imply that the performance of snowmelt prediction would improve if the snow patch size distribution was also considered. Information on and the usage of these distributions can be obtained with various methods, ranging from relatively simple methods, for example scaling laws (e.g., ), to more complex methods, for example assimilating various satellite retrievals (e.g., ).\n\nThe melt estimates obtained with SfM photogrammetry are in line with our expectations based on rough visual estimates made during the field campaign, whereas the estimated errors are relatively small. The errors are of the same order of magnitude as those found for high-accuracy snow depth estimates obtained from time-lapse photography (e.g., ), though it should be noted that those studies focus on somewhat larger areas. Overall, this illustrates that the influence of the vertical turbulent heat fluxes on melting snow patches is widespread and can even be observed with relatively simple and cheap methods. One of the potential limitations of this study is the choice to solely use grid cells that are continuously covered, causing the amount of available grid cells to reduce drastically and the snowmelt to be underestimated, especially at the upwind edge due to the retreating snow line. Advantageously, this shrinks the chance of grid cells being randomly scattered, which could result in unrealistic height changes. Moreover, the weather conditions on multiple days during the field campaign complicated the identification of tie points, due to the limited amount of light . Lastly, a small angle between the camera position and the horizontal snow surface was uncommon, as the method was often applied with camera positions at higher angles of incidence (e.g., with drone imagery). Overall, this meant that not all of the objects were captured from multiple perspectives, again complicating tie point identification. A possible solution to these limitations could be to add passive control points in the snow to create more tie points for the software to connect to. For more precise radiation and thus melt estimates, this study could have benefited from radiation modeling using high-resolution terrain information (cf. e.g., ). Future studies could also make use of lidar scanners, which have have recently gone into mass production; hence, we are seeing a corresponding drop in cost.\n\nA potential weakness of our simulations is the application of a low Reτ compared to the observations of (i.e., 590 vs. $\\sim \\mathrm{6}×{\\mathrm{10}}^{\\mathrm{6}}$). This saves computational costs, which are relatively high for DNS, and was done based on the results of , who showed large differences between Reτ=180 and Reτ=395, whereas Reτ=590 had similar bulk quantities and variances to the latter simulation. Adjusting Reτ possibly affected the surface momentum fluxes, of which thus also the heat fluxes. Furthermore, the low Re likely caused the fluxes in the IBL to be predominantly diffusive, whereas, in reality, turbulent fluxes are more likely to dominate. As diffusive timescales are typically larger than turbulent timescales, the typical time- and length scales of the processes in the IBL are relatively large compared to reality, such that one of the processes affected by this could be the decay of the vertical sensible heat fluxes over distance from the leading edge of a snow patch. To uncover whether the sensible heat fluxes and the decay are whether the sensible heat fluxes and the decay are dependent on Reτ, we recommend performing simulations with higher Reτ. Increasing Re will reduce the scale of the smallest eddies, i.e., the Kolmogorov length scale (Pope2000), such that an enhancement of the resolution will possibly also be required. Taken together with the influence of using an idealized system, this means that our formulated relationship between the sensible heat flux at the surfaces of the snow patches and the distance from the upwind edge (i.e., ${H}_{\\mathrm{sn}}\\sim {x}^{-\\mathrm{0.35}}$) should be approached with caution. However, the method does illustrate the use of DNS in coming up with potentially useful relationships. Future studies would need to look further into the abovedescribed behavior, especially when more comparable high-resolution data are available.\n\nMoreover, we can identify some inaccuracies during the nondimensional scaling of the wind speed and the temperature difference between the snow and atmosphere. As reported a wind speed of 6.4 m s−1, this value is also considered in our dimensional analysis and related to 0.11 m s−1, which was the average wind speed over the whole channel in the case of . However, the reported wind speed was measured at 1.8 m above the ground, thus implying that the average wind speed for the whole air column under consideration would be higher. Consequently, this affects the leading edge effect due to increased wind shear, and thus also the fluxes towards the surface. Also, the temperature difference between the atmosphere and snow has possibly been overestimated, causing an increased sensible heat flux. The graphs presented by show the temperatures of the bare ground and the atmosphere to be constant near the surface, 6.4 C. However, for the dimensional analysis, the atmospheric temperature mentioned by , i.e., 7.9 C, was used. Overall, these differences in assumptions between the simulations and the field observations make a one-to-one comparison difficult. Yet, the general behavior found in the simulations is similar to that reported in previous literature, i.e., temperature profiles and melting patterns (e.g., ), and shows the potential of DNS as a modeling tool to understand the melting of a patchy snow cover, especially considering that DNS does not violate the assumptions for Monin–Obukhov bulk formulations and is able to resolve the leading edge effect, in contrast to modeling studies with coarser spatial resolutions, which could lead to major errors . As such, this type of simulation is expected to provide more realistic behavior of the leading edge effect on the vertical turbulent heat fluxes, especially when combining it with case-specific boundary conditions.\n\nIn the studied system, the influence of stability on the relative decay of the surface fluxes over distance from the leading edge seems to be negligible, since the sensible heat fluxes in the simulations with and without buoyancy effects show the same decay. However, the snowmelt in the simulation without buoyancy effects is still higher, as the absolute sensible heat fluxes at the leading edge are highest for this simulation. We expect that the decay is similar due to the relatively high wind speeds compared to the temperature difference between the snow and bare ground: 6.4 m s−1 and 8 K, respectively. Overall, this causes the shear-induced turbulence to dominate over the buoyancy-induced turbulence. As multiple studies have suggested a role of stability in snowmelt (e.g., ), stable regions (i.e., snow patches) could have a much larger impact on the amount of wind-driven snowmelt, especially when reducing the turbulence to the edge of collapse. Therefore, it would be interesting to reduce the wind speed and increase the temperature difference between the snow and bare ground to identify the Risn that is needed for stability to become a more important influence on the sensible heat fluxes.\n\n7 Conclusions\n\nIn this study, we examined the melt of a 50 m long snow patch in the Finseelvi catchment, Norway, and investigated the observed melt with highly idealized simulations. The melt estimates, obtained with relatively simple and cheap structure-from-motion photogrammetry, for the upwind and downwind edges of the snow patch are feasible, and the estimated errors are in line with previous studies. The combined influence of the sensible and latent heat fluxes on the snowmelt at the upwind edge is estimated to be between 60 % and 80 %, while this contribution for the entire catchment would be maximally on the order of 10 %, based on previous studies. This estimate is based on the difference between the recorded melt and the net radiation of the snow patch determined from measurements at a meteorological tower near the catchment. This shows that, under specific circumstances, the local advection of the sensible and latent heat can be of major importance in the snowmelt of a patchy snow cover, expressing the necessity of a sound implementation of this process when modeling snowmelt.\n\nIn the idealized simulations based on measurements done by on a single 15 m snow patch on a flat surface, the sensible heat fluxes reduce over distance from the leading edge, following a constant power law. These reductions are caused by the cooling of the air above the snow patch while the wind speed and surface roughness are constant over the snow patch. Other simulations, in which the typical snow patch length is doubled and quadrupled, show exactly the same behavior over snow patches, such that larger snow patches receive less sensible heat on average. Domain-averaged sensible heat fluxes even reduced by 15 % and 25 %, respectively, with a doubling and a quadrupling of the typical snow patch size. Overall, this implies that the sensible (and likely also the untested latent) heat fluxes have a lower influence on the snowmelt in catchments with typically larger snow patches.\n\nWhen comparing the simulated behavior to the observed melt in the field, the observed vertical turbulent heat fluxes at the upwind edge are of the same order of magnitude as the simulations, especially when considering the inclusion of the diurnal cycle in these estimates. Moreover, based on the simulations, it is expected that the behavior found in the simulations also explains the reductions found in the field. However, it should be noted that the decay of sensible heat fluxes over distance from the leading edge measured by was higher than the simulated decay, for which some potential causes can be identified, such as a Reynolds number that is too low and inaccuracies in the nondimensionalization.\n\nYet, the idealized simulations have shown the potential of direct numerical simulations to simulate a patchy snow cover, especially when compared to the errors found for other simulation types. All performed simulations show the ability to simulate the leading edge effect, and clear IBLs form over snow patches. For comparison, studies that make use of large-eddy simulations, such as , report large errors compared with measurements. For our study, the flow characteristics are similar to controlled and field measurements. Aside from the measurements done by in the field, the measurements done by in a wind tunnel also show similar shapes for the temperature profiles above snow patches. Some characteristics vary compared to observations, such as the height of our IBLs compared to , but the general outcome seems promising for future research. Overall, the simulations allow us to extract very detailed information on the atmospheric behavior above a snow patch and can be used as a tool for achieving a better understanding of melting patchy snow covers.\n\nAppendix A: Orthoimages", null, "Figure A1The resulting height change maps used for Fig. 4 plotted over the real-color orthoimages, which are used for distinguishing snow from bare ground, after removing isolated groups of cells and median filtering. The insets show zoomed areas of the upwind edge for more detail.\n\nAppendix B: Surface generation\n\nTo create the surfaces for the simulations, noise is generated in Fourier space such that seemingly random patterns with a specified wavelength arise. These wavelengths are prescribed in the form of 2D power spectra. This method is applied so that patches at opposing walls fit together and flow that leaves the system on one side continues over the same snow patch when it reenters the system on the opposite wall. This enables the model to solve the periodic boundary conditions.\n\nInitially, a field with random phases between 0 and 2π is generated. These phases are applied in Euler's law (i.e., ${e}^{i\\mathit{\\phi }}=\\mathrm{cos}\\mathit{\\phi }+i\\mathrm{sin}\\mathit{\\phi }$) such that the phases are described in exponential form. The phases are multiplied with the desired magnitude per phase such that the Fourier space is generated ($z=\|z\|{e}^{i\\mathit{\\phi }}$ and Figs. B1B3). Eventually, a 2D field with dominant patterns is obtained by returning to physical space by using the inverse fast Fourier transform in Fourier space. Also, to avoid numerical instabilities in MicroHH, a Gaussian filter is applied on the surface with a standard deviation of 1 grid cell. When this filter overlaps the edges of the domain, the values at the opposing edge are applied.\n\nThe specified spectra for all generated surfaces consist of two broad peaks (Figs. B1B3). The peak with the lowest wavenumber has the higher factor and, thus, gives the dominant structures to the snow patch distribution. The peak with the higher wavenumber (i.e., 3 times the average wavenumber of the main peak) has a lower factor, such that some smaller fluctuations occur within the larger structures, giving the patches a more realistic appearance. For the surfaces in the P30m and P60m simulations, the wavenumber of the main peak is reduced by a factor of 2 and 4, respectively, compared to the surfaces in the P15m and P15m-NB simulations. This implies an average snow patch length in reality of 15, 30, and 60 m for the P15m (and P15m-NB), P30m, and P60m simulations, respectively.", null, "Figure B1Generated surface temperature and the applied Fourier space in absolute form for the P15m and P15m-NB simulations. The generated surface temperature (a) was obtained by applying the inverse fast Fourier transform in Fourier space (b).", null, "Figure B2Generated surface temperature and the applied Fourier space in absolute form for the P30m simulation. The generated surface temperature (a) was obtained by applying the inverse fast Fourier transform in Fourier space (b).", null, "Figure B3Generated surface temperature and the applied Fourier space in absolute form for the P60m simulation. The generated surface temperature (a) was obtained by applying the inverse fast Fourier transform in Fourier space (b).\n\nCode and data availability\n\nFor the snowmelt observations, the images used and a brief description of the photogrammetry workflow are available at https://doi.org/10.5281/zenodo.4704873 . For the numerical experiments, exemplary input files and model output can be found at https://doi.org/10.5281/zenodo.4705288 .\n\nAuthor contributions\n\nLDvdV carried out the research under the supervision of AJT, NP, RS, and CCvH. LDvdV designed the numerical experiment together with CCvH and RS, and LDvdV designed the field experiment together with AJT and NP. LG helped perform the photogrammetry analysis. LDvdV prepared the manuscript, with contributions from all co-authors.\n\nCompeting interests\n\nThe contact author has declared that none of the authors have any competing interests.\n\nDisclaimer\n\nPublisher’s note: Copernicus Publications remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.\n\nAcknowledgements\n\nWe would like to thank the anonymous reviewer and Rebecca Mott for their valuable feedback that helped to improve this manuscript. The simulations were carried out on the Dutch national e-infrastructure with the support of SURF Cooperative.\n\nReview statement\n\nThis paper was edited by Ketil Isaksen and reviewed by Rebecca Mott and one anonymous referee.\n\nReferences\n\nAalstad, K., Westermann, S., Schuler, T. V., Boike, J., and Bertino, L.: Ensemble-based assimilation of fractional snow-covered area satellite retrievals to estimate the snow distribution at Arctic sites, The Cryosphere, 12, 247–270, https://doi.org/10.5194/tc-12-247-2018, 2018. a\n\nAnderson, B., Mackintosh, A., Stumm, D., George, L., Kerr, T., Winter-Billington, A., and Fitzsimons, S.: Climate sensitivity of a high-precipitation glacier in New Zealand, J. Glaciology, 56, 114–128, https://doi.org/10.3189/002214310791190929, 2010. a\n\nBarnett, T. P., Adam, J. C., and Lettenmaier, D. P.: Potential impacts of a warming climate on water availability in snow-dominated regions, Nature, 438, 303–309, https://doi.org/10.1038/nature04141, 2005. a\n\nBerghuijs, W. R., Woods, R. A., and Hrachowitz, M.: A precipitation shift from snow towards rain leads to a decrease in streamflow, Nat. Clim. Change, 4, 583–586, https://doi.org/10.1038/NCLIMATE2246, 2014. a, b\n\nBonekamp, P. N. J., van Heerwaarden, C. C., Steiner, J. F., and Immerzeel, W. W.: Using 3D turbulence-resolving simulations to understand the impact of surface properties on the energy balance of a debris-covered glacier, The Cryosphere, 14, 1611–1632, https://doi.org/10.5194/tc-14-1611-2020, 2020. a, b, c\n\nBrandenberger, A. J.: Map of the McCall Glacier, Brooks Range, Alaska, American Geographical Society, New York, AGS Report, 11, https://collections.lib.uwm.edu/digital/collection/agdm/id/6815/(last access: 23 September 2022), 1959. a\n\nBuckingham, E.: On physically similar systems; illustrations of the use of dimensional equations, Phys. Rev., 4, 345–376, https://doi.org/10.1103/PhysRev.4.345, 1914. a\n\nBühler, Y., Adams, M. S., Bösch, R., and Stoffel, A.: Mapping snow depth in alpine terrain with unmanned aerial systems (UASs): potential and limitations, The Cryosphere, 10, 1075–1088, https://doi.org/10.5194/tc-10-1075-2016, 2016. a\n\nCimoli, E., Marcer, M., Vandecrux, B., Bøggild, C. E., Williams, G., and Simonsen, S. B.: Application of low-cost UASs and digital photogrammetry for high-resolution snow depth mapping in the Arctic, Remote Sensing, 9, 1144, https://doi.org/10.3390/rs9111144, 2017. a\n\nConway, J. and Cullen, N.: Constraining turbulent heat flux parameterization over a temperate maritime glacier in New Zealand, Ann. Glaciol., 54, 41–51, https://doi.org/10.3189/2013AoG63A604, 2013. a\n\nDadic, R., Mott, R., Lehning, M., Carenzo, M., Anderson, B., and Mackintosh, A.: Sensitivity of turbulent fluxes to wind speed over snow surfaces in different climatic settings, Adv. Water Resour., 55, 178–189, https://doi.org/10.1016/j.advwatres.2012.06.010, 2013. a\n\nDeBeer, C. M. and Pomeroy, J. W.: Influence of snowpack and melt energy heterogeneity on snow cover depletion and snowmelt runoff simulation in a cold mountain environment, J. Hydrol., 553, 199–213, https://doi.org/10.1016/j.jhydrol.2017.07.051, 2017. a\n\nDeems, J. S., Painter, T. H., and Finnegan, D. C.: Lidar measurement of snow depth: a review, J. Glaciol., 59, 467–479, https://doi.org/10.3189/2013JoG12J154, 2013. a\n\nDong, C. and Menzel, L.: Snow process monitoring in montane forests with time-lapse photography, Hydrol. Process., 31, 2872–2886, https://doi.org/10.1002/hyp.11229, 2017. a\n\nDozier, J. and Warren, S. G.: Effect of viewing angle on the infrared brightness temperature of snow, Water Resour. Res., 18, 1424–1434, https://doi.org/10.1029/WR018i005p01424, 1982. a\n\nEgli, L., Jonas, T., Grünewald, T., Schirmer, M., and Burlando, P.: Dynamics of snow ablation in a small Alpine catchment observed by repeated terrestrial laser scans, Hydrol. Process., 26, 1574–1585, https://doi.org/10.1002/hyp.8244, 2012. a\n\nEssery, R., Granger, R., and Pomeroy, J.: Boundary-layer growth and advection of heat over snow and soil patches: Modellling and parameterization, Hydrol. Process., 20, 953–967, https://doi.org/10.1002/hyp.6122, 2006. a, b, c\n\nFilhol, S., Perret, A., Girod, L., Sutter, G., Schuler, T., and Burkhart, J.: Time-Lapse Photogrammetry of Distributed Snow Depth During Snowmelt, Water Resour. Res., 55, 7916–7926, https://doi.org/10.1029/2018WR024530, 2019. a, b, c\n\nFontrodona Bach, A., Van der Schrier, G., Melsen, L., Klein Tank, A., and Teuling, A.: Widespread and accelerated decrease of observed mean and extreme snow depth over Europe, Geophys. Res. Lett., 45, 12–312, https://doi.org/10.1029/2018GL079799, 2018. a\n\nFujita, K., Hiyama, K., Iida, H., and Ageta, Y.: Self-regulated fluctuations in the ablation of a snow patch over four decades, Water Resour. Res., 46, W11541, https://doi.org/10.1029/2009WR008383, 2010. a, b\n\nGarratt, J. R.: The internal boundary layer – A review, Bound.-Lay. Meteorol., 50, 171–203, https://doi.org/10.1161/01.RES.80.6.877, 1990. a, b\n\nGarvelmann, J., Pohl, S., and Weiler, M.: From observation to the quantification of snow processes with a time-lapse camera network, Hydrol. Earth Syst. Sci., 17, 1415–1429, https://doi.org/10.5194/hess-17-1415-2013, 2013. a\n\nGirod, L., Nuth, C., Kääb, A., Etzelmüller, B., and Kohler, J.: Terrain changes from images acquired on opportunistic flights by SfM photogrammetry, The Cryosphere, 11, 827–840, https://doi.org/10.5194/tc-11-827-2017, 2017. a\n\nGolombek, R., Kittelsen, S. A., and Haddeland, I.: Climate change: impacts on electricity markets in Western Europe, Climatic Change, 113, 357–370, https://doi.org/10.1007/s10584-011-0348-6, 2012. a\n\nGranger, R. J., Pomeroy, J. W., and Parviainen, J.: Boundary-layer integration approach to advection of sensible heat to a patchy snow cover, Hydrol. Process., 16, 3559–3569, https://doi.org/10.1002/hyp.1227, 2002. a, b, c, d\n\nGranger, R. J., Essery, R., and Pomeroy, J. W.: Boundary-layer growth over snow and soil patches: Field observations, Hydrol. Process., 20, 943–951, https://doi.org/10.1002/hyp.6123, 2006. a, b\n\nGroffman, P. M., Driscoll, C. T., Fahey, T. J., Hardy, J. P., Fitzhugh, R. D., and Tierney, G. L.: Colder soils in a warmer world: a snow manipulation study in a northern hardwood forest ecosystem, Biogeochemistry, 56, 135–150, https://doi.org/10.1023/A:1013039830323, 2001. a\n\nGrünewald, T., Schirmer, M., Mott, R., and Lehning, M.: Spatial and temporal variability of snow depth and ablation rates in a small mountain catchment, The Cryosphere, 4, 215–225, https://doi.org/10.5194/tc-4-215-2010, 2010. a\n\nGrünewald, T., Wolfsperger, F., and Lehning, M.: Snow farming: conserving snow over the summer season, The Cryosphere, 12, 385–400, https://doi.org/10.5194/tc-12-385-2018, 2018. a\n\nHamilton, T. D.: Comparative glacier photographs from northern Alaska, J. Glaciol., 5, 479–487, https://doi.org/10.3189/S0022143000018451, 1965. a\n\nHarder, P., Pomeroy, J. W., and Helgason, W.: Local-Scale Advection of Sensible and Latent Heat During Snowmelt, Geophys. Res. Lett., 44, 9769–9777, https://doi.org/10.1002/2017GL074394, 2017. a, b, c, d, e, f, g, h, i, j, k, l, m, n, o, p, q, r, s, t, u, v, w, x, y, z, aa, ab, ac, ad, ae, af, ag, ah\n\nHarder, P., Pomeroy, J. W., and Helgason, W. D.: A simple model for local-scale sensible and latent heat advection contributions to snowmelt, Hydrol. Earth Syst. Sci., 23, 1–17, https://doi.org/10.5194/hess-23-1-2019, 2019. a, b, c, d, e\n\nHarder, P., Pomeroy, J. W., and Helgason, W. D.: Improving sub-canopy snow depth mapping with unmanned aerial vehicles: lidar versus structure-from-motion techniques, The Cryosphere, 14, 1919–1935, https://doi.org/10.5194/tc-14-1919-2020, 2020. a\n\nHarding, R.: Exchanges of energy and mass associated with a melting snowpack, in: Modelling Snowmelt-Induced Processes, Budapest, July 1986, edited by: Morris, E. M., IAHS Publication, 3–15, ISBN 9780947571603, 1986. a, b, c, d\n\nHärer, S., Bernhardt, M., Corripio, J. G., and Schulz, K.: PRACTISE – Photo Rectification And ClassificaTIon SoftwarE (V.1.0), Geosci. Model Dev., 6, 837–848, https://doi.org/10.5194/gmd-6-837-2013, 2013. a\n\nHock, R.: Temperature index melt modelling in mountain areas, J. Hydrol., 282, 104–115, https://doi.org/10.1016/S0022-1694(03)00257-9, 2003. a\n\nHock, R.: Glacier melt: a review of processes and their modelling, Prog. Phys. Geog., 29, 362–391, https://doi.org/10.1191/0309133305pp453ra, 2005. a\n\nHojatimalekshah, A., Uhlmann, Z., Glenn, N. F., Hiemstra, C. A., Tennant, C. J., Graham, J. D., Spaete, L., Gelvin, A., Marshall, H.-P., McNamara, J. P., and Enterkine, J.: Tree canopy and snow depth relationships at fine scales with terrestrial laser scanning, The Cryosphere, 15, 2187–2209, https://doi.org/10.5194/tc-15-2187-2021, 2021. a\n\nJacobs, J. M., Hunsaker, A. G., Sullivan, F. B., Palace, M., Burakowski, E. A., Herrick, C., and Cho, E.: Snow depth mapping with unpiloted aerial system lidar observations: a case study in Durham, New Hampshire, United States, The Cryosphere, 15, 1485–1500, https://doi.org/10.5194/tc-15-1485-2021, 2021. a\n\nKawamura, H., Ohsaka, K., Abe, H., and Yamamoto, K.: DNS of turbulent heat transfer in channel flow with low to medium-high Prandtl number fluid, Int. J. Heat Fluid Fl., 19, 482–491, https://doi.org/10.1016/S0142-727X(98)10026-7, 1998. a\n\nKumar, M., Marks, D., Dozier, J., Reba, M., and Winstral, A.: Evaluation of distributed hydrologic impacts of temperature-index and energy-based snow models, Adv. Water Resour., 56, 77–89, https://doi.org/10.1016/j.advwatres.2013.03.006, 2013. a\n\nLehning, M., Völksch, I., Gustafsson, D., Nguyen, T. A., Stähli, M., and Zappa, M.: ALPINE3D: a detailed model of mountain surface processes and its application to snow hydrology, Hydrol. Process., 20, 2111–2128, https://doi.org/10.1002/hyp.6204, 2006. a\n\nLejeune, Y., Bouilloud, L., Etchevers, P., Wagnon, P., Chevallier, P., Sicart, J.-E., Martin, E., and Habets, F.: Melting of snow cover in a tropical mountain environment in Bolivia: Processes and modeling, J. Hydrometeorol., 8, 922–937, https://doi.org/10.1175/JHM590.1, 2007. a\n\nListon, G. E.: Local Advection of Momentum, Heat and Moisture during the Melt of Patch Snow Covers, J. Appl. Meteorol., 34, 1705–1716, https://journals.ametsoc.org/view/journals/apme/34/7/1520-0450-34_7_1705.xml?tab_body=abstract-display (last access: 4 October 2022), 1995. a\n\nListon, G. E.: Representing subgrid snow cover heterogeneities in regional and global models, J. Climate, 17, 1381–1397, https://doi.org/10.1175/1520-0442(2004)017<1381:RSSCHI>2.0.CO;2, 2004. a\n\nLoth, B. and Graf, H.-F.: Modeling the snow cover in climate studies: 2. The sensitivity to internal snow parameters and interface processes, J. Geophys. Res.-Atmos., 103, 11329–11340, https://doi.org/10.1029/97JD01412, 1998. a\n\nMale, D. H. and Granger, R. J.: Snow surface energy exchange, Water Resour. Res., 17, 609–627, https://doi.org/10.1029/WR017i003p00609, 1981. a\n\nMarsh, P., Pomeroy, J., and Neumann, N.: Sensible heat flux and local advection over a heterogeneous landscape at an Arctic tundra site during snowmelt, Ann. Glaciol., 25, 132–136, https://doi.org/10.3189/S0260305500013926, 1997. a\n\nMarsh, P., Neumann, N., Essery, R., and Pomeroy, J.: Model estimates of local advection of sensible heat over a patchy snow cover, in: Interactions between the Cryosphere, Climate and Greenhouse Gases, 103–110, ISBN 9781901502909, 1999. a, b, c, d\n\nMarty, C., Philipona, R., Fröhlich, C., and Ohmura, A.: Altitude dependence of surface radiation fluxes and cloud forcing in the alps: results from the alpine surface radiation budget network, Theor. Appl. Climatol., 72, 137–155, https://doi.org/10.1007/s007040200019, 2002. a\n\nMelsen, L. A., Addor, N., Mizukami, N., Newman, A. J., Torfs, P. J. J. F., Clark, M. P., Uijlenhoet, R., and Teuling, A. J.: Mapping (dis)agreement in hydrologic projections, Hydrol. Earth Syst. Sci., 22, 1775–1791, https://doi.org/10.5194/hess-22-1775-2018, 2018. a, b\n\nMoin, P. and Mahesh, K.: Direct Numerical Simulation: A Tool in Turbulence Research, Annu. Rev. Fluid Mech., 30, 539–578, https://doi.org/10.1146/annurev.fluid.30.1.539, 1998. a\n\nMoser, R. D., Kim, J., and Mansour, N. N.: Direct numerical simulation of turbulent channel flow up to Reτ=590, Phys. Fluids, 11, 943–945, https://doi.org/10.1063/1.869966, 1999. a, b, c, d, e, f, g, h, i, j\n\nMote, P. W., Li, S., Lettenmaier, D. P., Xiao, M., and Engel, R.: Dramatic declines in snowpack in the western US, npj Clim. Atmos. Sci., 1, 2, https://doi.org/10.1038/s41612-018-0012-1, 2018. a\n\nMott, R., Egli, L., Grünewald, T., Dawes, N., Manes, C., Bavay, M., and Lehning, M.: Micrometeorological processes driving snow ablation in an Alpine catchment, The Cryosphere, 5, 1083–1098, https://doi.org/10.5194/tc-5-1083-2011, 2011. a, b, c\n\nMott, R., Gromke, C., Grünewald, T., and Lehning, M.: Relative importance of advective heat transport and boundary layer decoupling in the melt dynamics of a patchy snow cover, Adv. Water Resour., 55, 88–97, https://doi.org/10.1016/j.advwatres.2012.03.001, 2013. a\n\nMott, R., Daniels, M., and Lehning, M.: Atmospheric Flow Development and Associated Changes in Turbulent Sensible Heat Flux over a Patchy Mountain Snow Cover, J. Hydrometeorol., 16, 1315–1340, https://doi.org/10.1175/JHM-D-14-0036.1, 2015. a, b\n\nMott, R., Paterna, E., Horender, S., Crivelli, P., and Lehning, M.: Wind tunnel experiments: cold-air pooling and atmospheric decoupling above a melting snow patch, The Cryosphere, 10, 445–458, https://doi.org/10.5194/tc-10-445-2016, 2016. a, b, c, d\n\nMott, R., Schlögl, S., Dirks, L., and Lehning, M.: Impact of Extreme Land Surface Heterogeneity on Micrometeorology over Spring Snow Cover, J. Hydrometeorol., 18, 2705–2722, https://doi.org/10.1175/JHM-D-17-0074.1, 2017. a, b\n\nMott, R., Vionnet, V., and Grünewald, T.: The seasonal snow cover dynamics: review on wind-driven coupling processes, Front. Earth Sci., 6, 197, https://doi.org/10.3389/feart.2018.00197, 2018. a, b, c, d\n\nMott, R., Wolf, A., Kehl, M., Kunstmann, H., Warscher, M., and Grünewald, T.: Avalanches and micrometeorology driving mass and energy balance of the lowest perennial ice field of the Alps: a case study, The Cryosphere, 13, 1247–1265, https://doi.org/10.5194/tc-13-1247-2019, 2019. a\n\nMott, R., Stiperski, I., and Nicholson, L.: Spatio-temporal flow variations driving heat exchange processes at a mountain glacier, The Cryosphere, 14, 4699–4718, https://doi.org/10.5194/tc-14-4699-2020, 2020. a\n\nNolan, M., Larsen, C., and Sturm, M.: Mapping snow depth from manned aircraft on landscape scales at centimeter resolution using structure-from-motion photogrammetry, The Cryosphere, 9, 1445–1463, https://doi.org/10.5194/tc-9-1445-2015, 2015. a, b\n\nOlyphant, G. A. and Isard, S. A.: The role of advection in the energy balance of late-lying snowfields: Niwot Ridge, Front Range, Colorado, Water Resour. Res., 24, 1962–1968, https://doi.org/10.1029/WR024i011p01962, 1988. a, b\n\nPainter, T. H., Berisford, D. F., Boardman, J. W., Bormann, K. J., Deems, J. S., Gehrke, F., Hedrick, A., Joyce, M., Laidlaw, R., Marks, D., Mattmann, C., McGurk, B., Ramirez, P., Richardson, M., Skiles, S. M., Seidel, F. C., and Winstral, A.: The Airborne Snow Observatory: Fusion of scanning lidar, imaging spectrometer, and physically-based modeling for mapping snow water equivalent and snow albedo, Remote Sens. Environ., 184, 139–152, https://doi.org/10.1016/j.rse.2016.06.018, 2016. a\n\nPlüss, C. and Mazzoni, R.: The Role of Turbulent Heat Fluxes in the Energy Balance of High Alpine Snow Cover, Hydrol. Res., 25, 25–38, https://doi.org/10.2166/nh.1994.0017, 1994. a, b\n\nPohl, S. and Marsh, P.: Modelling the spatial–temporal variability of spring snowmelt in an arctic catchment, Hydrol. Process., 20, 1773–1792, https://doi.org/10.1002/hyp.5955, 2006. a, b\n\nPope, S.: Turbulent Flows, Cambridge University Press, ISBN 9780511840531, 2000. a\n\nRupnik, E., Daakir, M., and Deseilligny, M. P.: MicMac – a free, open-source solution for photogrammetry, Open Geospatial Data, Software and Standards, 2, 14, https://doi.org/10.1186/s40965-017-0027-2, 2017. a\n\nSauter, T. and Galos, S. P.: Effects of local advection on the spatial sensible heat flux variation on a mountain glacier, The Cryosphere, 10, 2887–2905, https://doi.org/10.5194/tc-10-2887-2016, 2016. a, b\n\nSchlögl, S., Lehning, M., Nishimura, K., Huwald, H., Cullen, N. J., and Mott, R.: How do Stability Corrections Perform in the Stable Boundary Layer Over Snow?, Bound.-Lay. Meteorol., 165, 161–180, https://doi.org/10.1007/s10546-017-0262-1, 2017. a\n\nSchlögl, S., Lehning, M., and Mott, R.: How Are Turbulent Sensible Heat Fluxes and Snow Melt Rates Affected by a Changing Snow Cover Fraction?, Front. Earth Sci., 6, 154, https://doi.org/10.3389/feart.2018.00154, 2018a. a, b, c, d, e, f, g, h, i\n\nSchlögl, S., Lehning, M., Fierz, C., and Mott, R.: Representation of Horizontal Transport Processes in Snowmelt Modeling by Applying a Footprint Approach, Front. Earth Sci., 6, 120, https://doi.org/10.3389/feart.2018.00120, 2018b. a, b, c\n\nSicart, J. E., Hock, R., and Six, D.: Glacier melt, air temperature, and energy balance in different climates: The Bolivian Tropics, the French Alps, and northern Sweden, J. Geophys. Res.-Atmos., 113, D24113, https://doi.org/10.1029/2008JD010406, 2008. a\n\nSilantyeva, O., Burkhart, J. F., Bhattarai, B. C., Skavhaug, O., and Helset, S.: Operational hydrology in highly steep areas: evaluation of tin-based toolchain, EGU General Assembly 2020, Online, 4–8 May 2020, EGU2020-8172, https://doi.org/10.5194/egusphere-egu2020-8172, 2020. a\n\nSturm, M. and Benson, C.: Scales of spatial heterogeneity for perennial and seasonal snow layers, Ann. Glaciol., 38, 253–260, https://doi.org/10.3189/172756404781815112, 2004. a\n\nSturm, M., Goldstein, M. A., and Parr, C.: Water and life from snow: A trillion dollar science question, Water Resour. Res., 53, 3534–3544, https://doi.org/10.1002/2017WR020840, 2017. a\n\nvan der Valk, L. D., Teuling, A. J., Girod, L., Pirk, N., Stoffer, R., and van Heerwaarden, C. C.: Snowmelt observations: Understanding wind-driven melt of patchy snow cover, Zenodo [data set], https://doi.org/10.5281/zenodo.4704873, 2021a. a\n\nvan der Valk, L. D., Teuling, A. J., Girod, L., Pirk, N., Stoffer, R., and van Heerwaarden, C. C.: Simulation output: Understanding wind-driven melt of patchy snow cover, Zenodo [data set], https://doi.org/10.5281/zenodo.4705288, 2021b. a\n\nvan Heerwaarden, C. C. and Mellado, J. P.: Growth and Decay of a Convective Boundary Layer over a Surface with a Constant Temperature, J. Atmos. Sci., 73, 2165–2177, https://doi.org/10.1175/JAS-D-15-0315.1, 2016. a\n\nvan Heerwaarden, C. C., van Stratum, B. J. H., Heus, T., Gibbs, J. A., Fedorovich, E., and Mellado, J. P.: MicroHH 1.0: a computational fluid dynamics code for direct numerical simulation and large-eddy simulation of atmospheric boundary layer flows, Geosci. Model Dev., 10, 3145–3165, https://doi.org/10.5194/gmd-10-3145-2017, 2017. a, b\n\nVionnet, V., Marsh, C. B., Menounos, B., Gascoin, S., Wayand, N. E., Shea, J., Mukherjee, K., and Pomeroy, J. W.: Multi-scale snowdrift-permitting modelling of mountain snowpack, The Cryosphere, 15, 743–769, https://doi.org/10.5194/tc-15-743-2021, 2021. a\n\nViviroli, D., Dürr, H. H., Messerli, B., Meybeck, M., and Weingartner, R.: Mountains of the world, water towers for humanity: Typology, mapping, and global significance, Water Resour. Res., 43, W07447, https://doi.org/10.1029/2006WR005653, 2007. a\n\nWeismann, R. N.: Snowmelt: A Two-Dimensional Turbulent Diffusion Model, Water Resour. Res., 13, 337–342, https://doi.org/10.1029/WR013i002p00337, 1977. a, b\n\nWheeler, J. A., Cortés, A. J., Sedlacek, J., Karrenberg, S., van Kleunen, M., Wipf, S., Hoch, G., Bossdorf, O., and Rixen, C.: The snow and the willows: earlier spring snowmelt reduces performance in the low-lying alpine shrub Salix herbacea, J. Ecol., 104, 1041–1050, https://doi.org/10.1111/1365-2745.12579, 2016. a\n\nXue, M., Droegemeier, K. K., Wong, V., Shapiro, A., Brewster, K., Carr, F., Weber, D., Liu, Y., and Wang, D.: The Advanced Regional Prediction System (ARPS) – A multi-scale nonhydrostatic atmospheric simulation and prediction tool. Part II: Model physics and applications, Meteorol. Atmos. Phys., 76, 143–165, https://doi.org/10.1007/s007030170027, 2001. a" ]	[ null, "https://tc.copernicus.org/articles/16/4319/2022/tc-16-4319-2022-avatar-thumb150.png", null, "https://tc.copernicus.org/articles/16/4319/2022/tc-16-4319-2022-f01-thumb.png", null, "https://tc.copernicus.org/articles/16/4319/2022/tc-16-4319-2022-f02-thumb.png", null, "https://tc.copernicus.org/articles/16/4319/2022/tc-16-4319-2022-t01-thumb.png", null, "https://tc.copernicus.org/articles/16/4319/2022/tc-16-4319-2022-t02-thumb.png", null, "https://tc.copernicus.org/articles/16/4319/2022/tc-16-4319-2022-f03-thumb.png", null, "https://tc.copernicus.org/articles/16/4319/2022/tc-16-4319-2022-f04-thumb.png", null, "https://tc.copernicus.org/articles/16/4319/2022/tc-16-4319-2022-t03-thumb.png", null, "https://tc.copernicus.org/articles/16/4319/2022/tc-16-4319-2022-t04-thumb.png", null, "https://tc.copernicus.org/articles/16/4319/2022/tc-16-4319-2022-f05-thumb.png", null, "https://tc.copernicus.org/articles/16/4319/2022/tc-16-4319-2022-f06-thumb.png", null, "https://tc.copernicus.org/articles/16/4319/2022/tc-16-4319-2022-f07-thumb.png", null, "https://tc.copernicus.org/articles/16/4319/2022/tc-16-4319-2022-f08-thumb.png", null, "https://tc.copernicus.org/articles/16/4319/2022/tc-16-4319-2022-f09-thumb.png", null, "https://tc.copernicus.org/articles/16/4319/2022/tc-16-4319-2022-f10-thumb.png", null, "https://tc.copernicus.org/articles/16/4319/2022/tc-16-4319-2022-f11-thumb.png", null, "https://tc.copernicus.org/articles/16/4319/2022/tc-16-4319-2022-f12-thumb.png", null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.8950891,"math_prob":0.9290374,"size":99771,"snap":"2023-14-2023-23","text_gpt3_token_len":24281,"char_repetition_ratio":0.1793579,"word_repetition_ratio":0.05417305,"special_character_ratio":0.24834871,"punctuation_ratio":0.1905919,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.951795,"pos_list":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34],"im_url_duplicate_count":[null,null,null,2,null,2,null,2,null,2,null,2,null,2,null,2,null,2,null,2,null,2,null,2,null,2,null,2,null,2,null,2,null,2,null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2023-05-30T13:30:19Z\",\"WARC-Record-ID\":\"<urn:uuid:153fa373-7a19-4ac1-9862-86be8f409b99>\",\"Content-Length\":\"456318\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:bd9f34f4-c060-4b0a-bb20-27d67644e7f1>\",\"WARC-Concurrent-To\":\"<urn:uuid:635a7fa3-4e87-4f71-8e4b-9c76b2a962e1>\",\"WARC-IP-Address\":\"81.3.21.103\",\"WARC-Target-URI\":\"https://tc.copernicus.org/articles/16/4319/2022/tc-16-4319-2022.html\",\"WARC-Payload-Digest\":\"sha1:V7FRPKDGPHBAH4JNKL2TXJQEIZUQ6ZNO\",\"WARC-Block-Digest\":\"sha1:NVMFWRIGTWZBTEYE56LCVZAVD7LKLCIZ\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2023/CC-MAIN-2023-23/CC-MAIN-2023-23_segments_1685224645810.57_warc_CC-MAIN-20230530131531-20230530161531-00149.warc.gz\"}"}
http://activepatience.com/photosynthesis-diagrams-worksheet/photosynthesis-diagrams-worksheet-blank-leaf-diagram-photosynthesis-house-wiring-diagram-symbols-today-photosynthesis-activity-photosynthesis-worksheet-photosynthesis-diagrams-worksheet-part-2/	[ "# Photosynthesis Diagrams Worksheet Blank Leaf Diagram Photosynthesis House Wiring Diagram Symbols Today Photosynthesis Activity Photosynthesis Worksheet Photosynthesis Diagrams Worksheet Part 2", null, "photosynthesis diagrams worksheet blank leaf diagram photosynthesis house wiring diagram symbols today photosynthesis activity photosynthesis worksheet photosynthesis diagrams worksheet part 2.\n\nstructures photosynthesis diagram worksheet wiring diagrams answers key part 1,photosynthesis diagram worksheet high school diagrams answers key impressive flow chart structures of,photosynthesis diagram worksheet pdf and cellular respiration diagrams structures of answers answer key part 1,photosynthesis diagrams worksheet part 2 diagram high school plant respiration to label custom wiring o pdf,photosynthesis diagrams worksheet answers key biology junction answer structures of,photosynthesis diagram worksheet high school diagrams biology junction answer key answers carbon fixation in definition reactions video botany,photosynthesis diagrams worksheet answers part 1 and cellular respiration diagram data wiring key,photosynthesis diagrams worksheet help electrical wiring diagram pdf high school biology junction answers,cellular respiration diagram worksheet photosynthesis and diagrams answers key biology junction answer,structures of photosynthesis diagram answers download wiring diagrams worksheet biology junction high school.\n\nPosted on" ]	[ null, "http://activepatience.com/wp-content/uploads/2018/10/photosynthesis-diagrams-worksheet-blank-leaf-diagram-photosynthesis-house-wiring-diagram-symbols-today-photosynthesis-activity-photosynthesis-worksheet-photosynthesis-diagrams-worksheet-part-2.jpg", null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.71830857,"math_prob":0.6166137,"size":1430,"snap":"2019-13-2019-22","text_gpt3_token_len":221,"char_repetition_ratio":0.29663393,"word_repetition_ratio":0.012345679,"special_character_ratio":0.12727273,"punctuation_ratio":0.059139784,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.96181047,"pos_list":[0,1,2],"im_url_duplicate_count":[null,1,null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2019-05-20T09:23:49Z\",\"WARC-Record-ID\":\"<urn:uuid:4c265671-4605-4645-8d3e-a4f2621baaf1>\",\"Content-Length\":\"46646\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:0b3f1c6b-60a7-4e48-8f40-baad0ceb411f>\",\"WARC-Concurrent-To\":\"<urn:uuid:6675e40a-e9a3-4503-9830-7921f747ff16>\",\"WARC-IP-Address\":\"104.28.29.198\",\"WARC-Target-URI\":\"http://activepatience.com/photosynthesis-diagrams-worksheet/photosynthesis-diagrams-worksheet-blank-leaf-diagram-photosynthesis-house-wiring-diagram-symbols-today-photosynthesis-activity-photosynthesis-worksheet-photosynthesis-diagrams-worksheet-part-2/\",\"WARC-Payload-Digest\":\"sha1:5O7MLOSECLNV6XXOXBB64HABCIUSKMWS\",\"WARC-Block-Digest\":\"sha1:KLCUDB357OUNSKS6YWKEPRCGEZT7PD57\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2019/CC-MAIN-2019-22/CC-MAIN-2019-22_segments_1558232255837.21_warc_CC-MAIN-20190520081942-20190520103942-00002.warc.gz\"}"}
http://ma3110fall2009.wikidot.com/exercise-7-3-9	[ "Exercise 7.3.9\n##### Claim\n\nThe union of two countable sets is countable.\n\n##### Proof\n\nWe will prove this in three cases.\n\n1. Let both $A$ and $B$ be finite. Then $A \\cup B$ is finite, and therefore is also count-\nable.\n\n2. Assume one of $A$ and $B$ is finite and the other is countably infinite. Assume that $A$ is finite. Since $B$ is countably infinite, there exists a function $f : B \\to \\mathbb{N}$ which is a one-to-one correspondence. For $b_i \\in B$ and $i \\in \\mathbb{N}$, let $f(b_i) = i$. Let $C = A-B$. Therefore $A \\cup B = B \\cup C$. If\n$C$ is an empty set , then $A \\cup B = B \\cup C$ and is countably infinite. If $C$ is a non-empty set, let $C = \\{{c_1, c_2, \\dots c_n\\}$. Define a new function $g : B \\cup C \\to \\mathbb{N}$ via $g(c_j) = j$ and $g(b_i) = n + i$. Therefore $g$ is a one to one correspondence. Therefore $B \\cup C$ is countable and $B \\cup C = A \\cup B$, so $A \\cup B$ is also countable.\n\n3. Let $A$ and $B$ be infinite. Let $C = A-B$. Then $A \\cup B = A \\cup C$. If $C$ is countably infinite then $A \\cup B = A \\cup C$ is countable by part 2. If $C$ is finite then $A \\cup B = A \\cup C$ is also finite and therefore countable.\n\n$\\blacksquare$" ]	[ null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.8239707,"math_prob":1.0000036,"size":1143,"snap":"2020-24-2020-29","text_gpt3_token_len":400,"char_repetition_ratio":0.19578578,"word_repetition_ratio":0.081196584,"special_character_ratio":0.35870516,"punctuation_ratio":0.124060154,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":1.00001,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2020-07-06T03:28:30Z\",\"WARC-Record-ID\":\"<urn:uuid:62977358-2214-4b78-8ed1-d2067bd08592>\",\"Content-Length\":\"21905\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:521231ce-140a-4612-8470-ae88003a447a>\",\"WARC-Concurrent-To\":\"<urn:uuid:bfbe1855-d1dd-45c9-8c7a-ce4e3947fd07>\",\"WARC-IP-Address\":\"107.20.139.176\",\"WARC-Target-URI\":\"http://ma3110fall2009.wikidot.com/exercise-7-3-9\",\"WARC-Payload-Digest\":\"sha1:DAAA6CQO2DXZNZMQEAP6QY5LMKNLGWOR\",\"WARC-Block-Digest\":\"sha1:FWIBTH2NJ36REM4FIYMD7I7MQT3J65XS\",\"WARC-Identified-Payload-Type\":\"application/xhtml+xml\",\"warc_filename\":\"/cc_download/warc_2020/CC-MAIN-2020-29/CC-MAIN-2020-29_segments_1593655890092.28_warc_CC-MAIN-20200706011013-20200706041013-00209.warc.gz\"}"}
http://www.stumblingrobot.com/2016/03/10/determine-convergence-series-1n-2n100-3n1n/	[ "Home » Blog » Determine the convergence of the series (-1)n ((2n+100) / (3n+1))n\n\nDetermine the convergence of the series (-1)n ((2n+100) / (3n+1))n\n\nConsider the series", null, "Determine whether the series converges or diverges. If it converges, determine whether it converges conditionally or absolutely.\n\nThe given series is absolutely convergent.\n\nProof. To see this is absolutely convergent, first we have", null, "Then, looking to apply the root test, we have", null, "And so,", null, "Hence, by the root test, the series converges. Therefore, the given series is absolutely convergent", null, "" ]	[ null, "http://www.stumblingrobot.com/wp-content/ql-cache/quicklatex.com-da65c3b0fa955c325b9e03c3a13448b0_l3.png", null, "http://www.stumblingrobot.com/wp-content/ql-cache/quicklatex.com-f4c9eddb66573f688a741c5b4649b619_l3.png", null, "http://www.stumblingrobot.com/wp-content/ql-cache/quicklatex.com-12673f78abf90ff8211e09f3f2b30648_l3.png", null, "http://www.stumblingrobot.com/wp-content/ql-cache/quicklatex.com-3453b7809248e5e8c097916487031b92_l3.png", null, "http://www.stumblingrobot.com/wp-content/ql-cache/quicklatex.com-d108977a21a0e2720762a2b36c18838e_l3.png", null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.8160562,"math_prob":0.98402053,"size":404,"snap":"2019-43-2019-47","text_gpt3_token_len":89,"char_repetition_ratio":0.205,"word_repetition_ratio":0.0,"special_character_ratio":0.18316832,"punctuation_ratio":0.17333333,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9886885,"pos_list":[0,1,2,3,4,5,6,7,8,9,10],"im_url_duplicate_count":[null,2,null,2,null,2,null,2,null,null,null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2019-10-15T18:38:57Z\",\"WARC-Record-ID\":\"<urn:uuid:569a5b2e-2b19-4d67-a9b9-32f76dadce75>\",\"Content-Length\":\"56348\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:5e9d742f-14d8-403b-b557-d5255097fd88>\",\"WARC-Concurrent-To\":\"<urn:uuid:92be447c-71ad-4740-bcf2-a294aca495e8>\",\"WARC-IP-Address\":\"194.1.147.70\",\"WARC-Target-URI\":\"http://www.stumblingrobot.com/2016/03/10/determine-convergence-series-1n-2n100-3n1n/\",\"WARC-Payload-Digest\":\"sha1:5LFR4OUSBMSIE4MZ2SURJJ5V7MTIC7LN\",\"WARC-Block-Digest\":\"sha1:QQ6EGH3QCB56WV7POIDBO3GWNXAM2N7Y\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2019/CC-MAIN-2019-43/CC-MAIN-2019-43_segments_1570986660231.30_warc_CC-MAIN-20191015182235-20191015205735-00024.warc.gz\"}"}
https://community.cloudera.com/t5/Support-Questions/Pandas-udf-with-a-tuple-pyspark/m-p/190142	[ "Support Questions\n\n# Pandas_udf with a tuple? (pyspark)", null, "alexander_witte\nExplorer\n\nHi!\n\nI have a UDF that returns a tuple object:\n\n```stringSchema = StructType([\nStructField(\"fixedRoute\", StringType(), False),\nStructField(\"accuracy\", IntegerType(), False)])\n\ndef stringClassifier(x,y,z):\n... do some code\nreturn (value1,value2)\nstringClassifier_udf = udf(stringClassifier, stringSchema)\n\n```\n\nI use it in a dataframe like this:\n\n```df = df.select(['route', 'routestring', stringClassifier_udf(x,y,z).alias('newcol')])\n```\n\nThis works fine. I later split that tuple into two distinct columns. The UDF however does some string matching and is somewhat slow as it collects to the driver and then filters through a 10k item list to match a string. (it does this for every row). I've been reading about pandas_udf and Apache Arrow and was curious if running this same function would be possible with pandas_udf... or if this would be help improve the performance..? I think my hangup is that the return value of the UDF is a tuple item... here is my attempt:\n\n```from pyspark.sql.functions import pandas_udf, PandasUDFType\n\nstringSchema = StructType([\nStructField(\"fixedRoute\", StringType(), False),\nStructField(\"accuracy\", IntegerType(), False)])\n\n@pandas_udf(stringSchema)\ndef stringClassifier(x,y,z):\n... do some code\nreturn (value1,value2)\n\n```\n\nOf course this is gives me errors and I've tried decorating the function with: @pandas_udf('list', PandasUDFType.SCALAR)\n\nMy errors looks like this:\n\n`NotImplementedError: Invalid returnType with scalar Pandas UDFs: StructType(List(StructField(fixedRoute,StringType,false),StructField(accuracy,IntegerType,false))) is not supported`\n\nAny idea if there is a way to make this work?\n\nThanks!\n\n2 REPLIES 2", null, "o912451\nNew Contributor\n\nIt looks like you are using a scalar pandas_udf type, which doesn't support returning structs currently. I believe the return type you want is an array of strings, which is supported, so this should work. Try this:\n\n```@pandas_udf(\"array<string>\")\ndef stringClassifier(x,y,z):\n# return a pandas series of a list of strings, that is same length as input - for example\ns = pd.Series([[u\"a\", u\"b\"]] * len(x))\nreturn s```\n\nIf you are using Python 2, make sure your strings are in unicode otherwise they might get interpreted as bytes. Hope that helps!", null, "alexander_witte\nExplorer\n\nHey Bryan thanks so much for taking the time! I think I'm almost there! The hint about the unicode issue helping me get past the first slew of errors. I seem to be running into a length one now however:\n\n```@pandas_udf(\"array<string>\")\ndef stringClassifier(lookupstring, first, last):\n\nlookupstring = lookupstring.to_string().encode(\"utf-8\")\nfirst = first.to_string().encode(\"utf-8\")\nlast = last.to_string().encode(\"utf-8\")\n\n#this part takes the 3 strings above and reaches out to another library to do a string match\nresult = process.extract(lookupstring, lookup_list, limit=4000)\nmatch_list = [item for item in result if item.startswith(first) and item.endswith(last)]\nresult2 = process.extractOne(lookupstring, match_list)\n\nif result2 is not None and result2 > 75:\nelse:\nfail = [\"N/A\",\"0\"]\nreturn pd.Series(fail)\n```\n\nRuntimeError: Result vector from pandas_udf was not the required length: expected 1, got 2\n\nI'm initially passing three strings as variables to the function which then get passed to another library. The result is a tuple which I covert to a list then to a pandas Series object. I'm curious how I can make a 2 item array object a length of 1 ..? I'm obviously missing some basics here.\n\n@Bryan C", null, "Take a Tour of the Community\nDon't have an account?" ]	[ null, "https://community.cloudera.com/html/rank_icons/Rank-4-Explorer.svg", null, "https://community.cloudera.com/html/rank_icons/Rank-3-New-Contributor.svg", null, "https://community.cloudera.com/html/rank_icons/Rank-4-Explorer.svg", null, "https://community.cloudera.com/skins/images/1DE9D8BE8753F1277D0F9516815AB0CE/responsive_peak/images/icon_anonymous_message.png", null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.78449047,"math_prob":0.75509596,"size":3500,"snap":"2022-27-2022-33","text_gpt3_token_len":866,"char_repetition_ratio":0.1201373,"word_repetition_ratio":0.036734693,"special_character_ratio":0.25257143,"punctuation_ratio":0.17083947,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9732854,"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\":\"2022-07-03T00:26:30Z\",\"WARC-Record-ID\":\"<urn:uuid:fac1e790-db76-450d-b0ab-cb9d2d8b27fb>\",\"Content-Length\":\"166728\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:12e7479f-e760-4bdf-b3ff-526d4940ba7d>\",\"WARC-Concurrent-To\":\"<urn:uuid:d8ee9171-54ce-4265-82f3-34526eaddd68>\",\"WARC-IP-Address\":\"13.32.208.36\",\"WARC-Target-URI\":\"https://community.cloudera.com/t5/Support-Questions/Pandas-udf-with-a-tuple-pyspark/m-p/190142\",\"WARC-Payload-Digest\":\"sha1:PVPWSB3BLKHUPY4OTZL5NBSJW6LCCYSW\",\"WARC-Block-Digest\":\"sha1:47L3IVBVJH5D5ENHKKZ65XAVGAWEX3I3\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2022/CC-MAIN-2022-27/CC-MAIN-2022-27_segments_1656104205534.63_warc_CC-MAIN-20220702222819-20220703012819-00405.warc.gz\"}"}
https://nrich.maths.org/1324	[ "# Whole Number Dynamics V\n\n##### Age 14 to 18\n\nPublished 1998 Revised 2011\n\nThe last article Whole Number Dynamics IV was about the rule which takes $N$ to $N^{\\prime}$ where $N$ was written in the form $N=10M+R$ and the remainder $R$ lies between $0$ and $9$ inclusive, and $N^{\\prime}=10R-M$.\n\nWe suggested there that if we start with any whole number and then apply the rule $N \\to N^{\\prime}$ repeatedly, we will eventually reach $0$, or end in a cycle of four numbers. If you check many cases you will find this is so, but of course, this does not prove that it is so.\n\nWe also saw that for any whole number $K$ there are exactly ten numbers which go to $K$ in one step, and these are the numbers: $$-10K, 101-10K, 202-10K, 303-10K, ......, 808-10K, 909-10K$$ As a result of this, we then showed that if $N$ eventually reaches $0$ (after applying the rule sufficiently many times) then $N$ must be a multiple of $101$.\n\nNow it is important to understand that the statement\n\n(1) if $N$ is a multiple of $101$, then $N$ eventually reaches $0$,\n\nis not the same as the statement:\n\n(2) if $N$ eventually reaches $0$, then $N$ is a multiple of $101$.\n\nA good way to see that these are different statements is to consider the following two (very similar) statements:\n\n(3) if a whole number is a squared number, then it is positive or zero,\n\nand\n\n(4) if a whole number is positive or zero, then it is a squared number.\n\nObviously, (3) is true and (4) is false so that these two forms of a statement must be different. Returning to consider the statements (1) and (2), we recall that we have proved (2) in the last article; we shall now prove (1). Note that when combined together, (1) and (2) describe exactly those numbers which will eventually arrive at 0.\n\nTo show that (1) is true, let us consider any whole number $N$ that is a multiple of $101$; we want to show that this eventually reaches $0$. We can write $N= 101 P$, say, where $P$ is a whole number, and we can always write $P$ in the form $P =10 M+ R$, where $R$ is the remainder of $P$ when we divide $P$ by $10$. This means that: $$N = 101P = 101(10M + R) = 1010M + 100R + R = 10 X (101M + 10R) +R$$ so that $R$ is also the remainder of $N$.\n\nApplying the rule to $N$, we see that $N$ goes to $N^{\\prime}$, where $$N^{\\prime} = 10 R - (101 M+ 10 R) = -101 M .$$ This shows, for example, that multiples of $101$ always go to multiples of $101$ (and we also know from last time that they can only come from multiples of $101$), so clearly the number $101$ plays an important role here!\n\nLet us look at what we have just proved a little more closely.\n\nIn fact, we have just seen that if $N = 101(10 M+R)$ then $N^{\\prime}= 101 \\times(-M)$ , and roughly speaking,this says that when we apply the rule we just `drop' the units figure and change the sign). This shows, for example, that:\n\n\\begin{eqnarray} 101 \\times 38792 & \\to & 101 \\times (-3879) \\\\ & \\to & 101 \\times 387 \\\\ & \\to & 101 \\times (-38) \\\\ & \\to & 101 \\times 3 = 303 \\\\ & \\to & 0 \\end{eqnarray}\n\nA similar argument works with $38792$ replaced by any integer and this shows that if a whole number $N$ is a multiple of $101$, then it eventually reaches $0$.\n\nLet us look again at the problem posed at the end of the last article. We asked there whether or not the number $123456$ ends up at $0$ or in a cycle of four numbers? Using a calculator, we see that $123456$ is not a multiple of $101$ so that (by what we have just shown) it cannot end up at $0$.\n\nWhat about the number $12345678987654321$? This number is too big to put on a calculator so we need to find another approach for it would clearly be a very long task indeed to keep on applying the rule to this number! How do we handle this number?\n\nWe want to decide whether or not the number $12345678987654321$ is a multiple of $101$. Of course if a whole number $X$ is a multiple of $101$ then the number $12345678987654321 - X$ is also a multiple of $101$ and conversely, and using this we see that it is enough to subtract multiples of $101$ from $12345678987654321$ and then check whether or not the answer is a multiple of $101$. Better still, if P is any whole number then:\n\n$$10000P = 9999P + P = (99 \\times101)P + P$$ so that $10000 P$ is a multiple of $101$ plus $P$.\n\nThus, $$12345678987654321 = (1234567898765 x 10000) + 4321 = 1234567898765 + 101Q + 4321$$ for some whole number $Q$. Applying this again, we get $$1234567898765 = (123456789 x 10000) + 8765 = 123456789 + 101S + 8765$$ for some whole number $S$, and again, $$123456789 = (12345 x 10000) + 6789 = 12345 + 101T + 6789$$ for some whole number $T$.\n\nPutting all these together, we find that the two numbers $12345678987654321$ and $( 4321 + 8765 + 6789 + 12345)$ differ by a multiple of $101$. It is enough, therefore, to check whether $(4321+8765+6789+12345)$ is, or is not, a multiple of $101$, and we have now reduced the problem to one that we can do on a calculator.\n\nYou can now answer the question : does $12345678987654321$ eventually reach $0$ or not ?\n\nA final question : does $8765432123456789$ eventually reach $0$ or not?" ]	[ null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.8953377,"math_prob":0.99993443,"size":4989,"snap":"2021-43-2021-49","text_gpt3_token_len":1490,"char_repetition_ratio":0.15827483,"word_repetition_ratio":0.026399156,"special_character_ratio":0.38925636,"punctuation_ratio":0.097345136,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.99997973,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2021-12-08T00:08:28Z\",\"WARC-Record-ID\":\"<urn:uuid:daf5e61a-0d9b-47e3-88e2-3c1f15306b82>\",\"Content-Length\":\"15853\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:31b22c03-073a-4e95-b135-cd2132ea6a58>\",\"WARC-Concurrent-To\":\"<urn:uuid:cff0b118-c95a-4ffe-8dd6-f58f63c986a6>\",\"WARC-IP-Address\":\"131.111.18.195\",\"WARC-Target-URI\":\"https://nrich.maths.org/1324\",\"WARC-Payload-Digest\":\"sha1:WO2GDIYSF4XFLNEUXUVLFAGVNBXZUVZN\",\"WARC-Block-Digest\":\"sha1:I3H77XEX4KO6ZFAQUES2RDZTXS3FKHRW\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2021/CC-MAIN-2021-49/CC-MAIN-2021-49_segments_1637964363420.81_warc_CC-MAIN-20211207232140-20211208022140-00326.warc.gz\"}"}
https://www.emathhelp.net/calculators/linear-algebra/unit-vector-calculator/?u=1%2C+2%2C+1	[ "# Unit Vector Calculator\n\n## Calculate unit vectors step by step\n\nThe calculator will find the unit vector in the direction of the given vector, with steps shown.\n\n$\\langle$ $\\rangle$\nComma-separated.\n\nIf the calculator did not compute something or you have identified an error, or you have a suggestion/feedback, please write it in the comments below.\n\nFind the unit vector in the direction of $\\mathbf{\\vec{u}} = \\left\\langle 1, 2, 1\\right\\rangle$." ]	[ null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.7149046,"math_prob":0.9867753,"size":267,"snap":"2022-40-2023-06","text_gpt3_token_len":69,"char_repetition_ratio":0.18250951,"word_repetition_ratio":0.15789473,"special_character_ratio":0.24719101,"punctuation_ratio":0.09803922,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.995137,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2022-10-05T03:25:31Z\",\"WARC-Record-ID\":\"<urn:uuid:98ac2e09-4de2-448f-8def-36dd20e6f090>\",\"Content-Length\":\"25031\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:d7714e1d-138d-4b16-9e70-eca2a70d915a>\",\"WARC-Concurrent-To\":\"<urn:uuid:44e3332e-f158-4169-bb7e-f2aed08b81f1>\",\"WARC-IP-Address\":\"69.55.60.125\",\"WARC-Target-URI\":\"https://www.emathhelp.net/calculators/linear-algebra/unit-vector-calculator/?u=1%2C+2%2C+1\",\"WARC-Payload-Digest\":\"sha1:6KI2YERKEGYJHJX3Y4PARTIPG7GWDGWA\",\"WARC-Block-Digest\":\"sha1:6RBVPGBYSOUHNKY6QNI5ARQDYNITM765\",\"WARC-Identified-Payload-Type\":\"application/xhtml+xml\",\"warc_filename\":\"/cc_download/warc_2022/CC-MAIN-2022-40/CC-MAIN-2022-40_segments_1664030337531.3_warc_CC-MAIN-20221005011205-20221005041205-00733.warc.gz\"}"}
https://apps.dtic.mil/sti/citations/AD0613915	[ "# Abstract:\n\nThe flow near the stagnation streamline of a blunt body is often analyzed by using the approximation of local similarity, which reduces the equations of motion to a system of ordinary differential equations. This scheme is equivalent to truncating at one term a powerseries expansion of the flow variables from the stagnation point, neglecting backward influence. The accuracy of such a truncation is examined. The principal assumption is that the Navier-Stokes equations are valid. In addition, it is assumed that the validity of the first truncation can be evaluated by comparing it with the second. The conclusion is that the usual assumption of local similarity is remarkably accurate for predicting flow quantities near the stagnation streamline. Author" ]	[ null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.8931505,"math_prob":0.98560977,"size":1050,"snap":"2021-21-2021-25","text_gpt3_token_len":227,"char_repetition_ratio":0.107074566,"word_repetition_ratio":0.0,"special_character_ratio":0.19238095,"punctuation_ratio":0.123655915,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9795135,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2021-05-13T20:09:03Z\",\"WARC-Record-ID\":\"<urn:uuid:1d4c3321-a437-4c74-9fab-c9c874e66654>\",\"Content-Length\":\"17705\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:ae5a0c7e-2f91-4825-992b-e1d1698f4279>\",\"WARC-Concurrent-To\":\"<urn:uuid:83054d41-1d71-4d8d-b91b-0f70eefbf3cd>\",\"WARC-IP-Address\":\"131.84.180.30\",\"WARC-Target-URI\":\"https://apps.dtic.mil/sti/citations/AD0613915\",\"WARC-Payload-Digest\":\"sha1:UKO2BVIIXM3VGPIWWRYOK37T3UA33O46\",\"WARC-Block-Digest\":\"sha1:P2SC5DXTRSGIMJEG6HA7UFAQ4CNT4YDR\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2021/CC-MAIN-2021-21/CC-MAIN-2021-21_segments_1620243991943.36_warc_CC-MAIN-20210513173321-20210513203321-00465.warc.gz\"}"}
http://www.quantopia.net/monte-carlo-in-quant-finance/	[ "# Monte Carlo in Quantitative Finance\n\nIn this post I’m going to give a little introduction to Monte Carlo as a method for integration, and try to get server-side scripting working via WordPress!\n\nMonte Carlo is fundamentally a technique for doing numerical integration. If we’re confronted with an integral that we can’t solve analytically, we have an array of possible techniques available to us. For 1d integrals, probably the easiest thing for well-behaved functions is to use Simpson’s rule and integrate across the part we’re interested in. But to do this in many dimensions can be tricky – for each dimension we probably want the same number of partitions, so the overall number of function evaluations scales exponentially with dimension – very bad news! By contrast, Monte Carlo essentially samples the function at random points and takes an average. The higher value points contribute more to the average than lower value points, and the overall error in the average scales as , giving a good approximation relatively quickly for high dimensional problems.\n\nThe quintessential example of a Monte Carlo experiment is a simple process to approximate pi. Consider drawing a circle inscribed into a square, just as shown in the following image:", null, "Now, imagine scattering dried cous-cous over the whole area. They will land randomly all over the surface, and first of all we will sweep away any that fall outside of the square.\n\nWhat next? Well, if we count the number of grains that fell inside the circle, and compare that to the total amount that fell inside the square, what do we expect the ratio to be? As long as they’re not too clustered, the expectation is that it will be , which is of course just the ratio of the areas.\n\nRather than actually counting grains, we can simulate this process on the computer much more quickly. To represent each grain, we simulate two uniform variables, each on the interval [-0.5,0.5]. We treat these as the grain’s x-coordinate and y-coordinate, and calculate the distance from the origin (0,0) by taking the sum of the squares of these. If the sum is less than the square of the radius of the circle (ie. 0.25) then the grain is ‘inside’ the circle, if it is greater then the grain is ‘outside’.\n\nHere is the simulation run for 1000 grains of cous-cous (refresh for a repeat attempt):\n\nGrains Simulated: 1000\nGrains Inside Circle: 785\nOur estimate of pi is 3.14\n\nThe estimate here is probably fairly close – you may have been lucky (or unlucky!), but we claimed that the estimate will converge to the real value of pi with a progressively larger number of grains [exercise: can you demonstrate this using the central limit theorem?]. Well, below you’ll find another simulation, this time going up to a little over a million grains but making estimates each time the number of grains is doubled, and the error in the estimate is compared to the number of grains so far at each step.\n\n number of steps estimate error 2 2 -1.141593 4 3 -0.141593 8 2.5 -0.641593 16 3.25 0.108407 32 3.375 0.233407 64 3.25 0.108407 128 3.1875 0.045907 256 3.03125 -0.110343 512 3.023438 -0.118155 1024 3.054688 -0.086905 2048 3.099609 -0.041983 4096 3.12793 -0.013663 8192 3.133789 -0.007804 16384 3.125488 -0.016104 32768 3.127808 -0.013785 65536 3.138855 -0.002738 131072 3.144653 0.003061 262144 3.143097 0.001504 524288 3.142647 0.001054 1048576 3.142952 0.001359\n\nIt should b fairly straight-forward to copy-paste these figures into excel – does the error fall like as claimed? In fact, the central limit theorem tells us that the mean of the ratio of grains inside should go to a normal distribution with standard deviation proportional to this quantity, so we should expect it to be outside the bound about 30% of the time (if you have calculated the right constant!).\n\nOf course, in the 2 dimensional circle case, a better idea might be to try and put points evenly over the square and count how many of these fall inside/outside. As mentioned before, the benefits of Monte Carlo are most pronounced when there are many dimensions involved. But, this is something like the procedure involved in quasi-Monte Carlo, a procedure that I’ll talk about some other time that doesn’t use random numbers at all…\n\n-QuantoDrifter" ]	[ null, "http://www.quantopia.net/wp-content/uploads/2012/12/squareCircle.png", null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.8787424,"math_prob":0.98437804,"size":4264,"snap":"2019-13-2019-22","text_gpt3_token_len":1045,"char_repetition_ratio":0.106807515,"word_repetition_ratio":0.0,"special_character_ratio":0.3008912,"punctuation_ratio":0.12270642,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9829899,"pos_list":[0,1,2],"im_url_duplicate_count":[null,10,null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2019-03-19T23:43:42Z\",\"WARC-Record-ID\":\"<urn:uuid:8b025697-c6cb-46fd-8221-2ecbdc9ef93b>\",\"Content-Length\":\"41030\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:c143cd54-0996-4365-9646-0b5218e4b327>\",\"WARC-Concurrent-To\":\"<urn:uuid:c4e37910-8fd7-463d-968a-91d3d1fe85a4>\",\"WARC-IP-Address\":\"5.77.50.192\",\"WARC-Target-URI\":\"http://www.quantopia.net/monte-carlo-in-quant-finance/\",\"WARC-Payload-Digest\":\"sha1:77SNVID6D6UFFZXC6AKAZDRYHWQLB64J\",\"WARC-Block-Digest\":\"sha1:7A7MXNC7SAQTMCDQG62K3CDDHSBVCPQR\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2019/CC-MAIN-2019-13/CC-MAIN-2019-13_segments_1552912202161.73_warc_CC-MAIN-20190319224005-20190320010005-00341.warc.gz\"}"}
https://www.exceldemy.com/countifs-with-multiple-criteria-and-or-logic/	[ "# Excel COUNTIFS with Multiple Criteria and OR Logic (3 Examples)\n\nGet FREE Advanced Excel Exercises with Solutions!\n\nIn this article, we will learn to use the COUNTIFS function with multiple criteria and OR logic in Excel. The COUNTIFS function counts the number of cells that satisfies a set of given conditions or criteria. Today, we will demonstrate 3 ideal examples. Using these examples, you can use the COUNTIFS function with multiple criteria and OR logic. Also, we will show an alternative way to count with multiple criteria. So, without further delay, let’s start the discussion.\n\n## 3 Ideal Examples of Excel COUNTIFS with Multiple Criteria and OR Logic\n\nTo explain the examples, we will use a dataset that contains information about the Payment Status and Quantity of some Products. We will use the COUNTIFS function to count the number of cells based on multiple criteria and OR logic. For example, we want to count the number of laptops and keyboards with pending or paid status. In that case, we can use the following examples. Also, you can apply them to solve your problems easily.", null, "### 1. Use Excel SUM and COUNTIFS Functions with Multiple Criteria and OR Logic\n\nIn the first example, we will use the SUM and COUNTIFS functions with multiple criteria and OR logic. Here, we will count the number of cells that contain a Laptop or Keyboard with the Paid or Pending payment status. In the dataset below, you can see 4 sets of cells that satisfy the conditions.", null, "Let’s follow the steps below to see how we can create a formula to count the number of cells with multiple criteria.\n\nSTEPS:\n\n• First of all, select Cell F13 and type the formula below:\n`=SUM(COUNTIFS(B5:B13,{\"Laptop\",\"Keyboard\"},D5:D13,{\"Paid\";\"Pending\"}))`\n• Secondly, press Enter to see the result.", null, "Here, we have used the SUM and COUNTIFS functions together to get the result. Inside the COUNTIFS function, we have entered the criteria range and criteria.\n\nThe first criteria look for a Laptop or Keyboard in the range B5:B13 and the second criteria looks for Paid or Pending payment status in the range D5:D13.\n\n🔎 How Does the Formula Work?\n\n• COUNTIFS(B5:B13,{“Laptop”,”Keyboard”},D5:D13,{“Paid”;”Pending”}): The output of this part is an array. And that array is {2,0,1,1}. The array can be displayed like the picture below:", null, "It shows that the number of Laptops with the Paid status is 2 and the Pending status is 1. Similarly, the number of Keyboards with Paid status is 0 and the Pending status is 1. So, the total number is 4.\n\n• SUM(COUNTIFS(B5:B13,{“Laptop”,”Keyboard”},D5:D13,{“Paid”;”Pending”})): The SUM function adds up the array {2,0,1,1} and displays the result which is 4.\n\n### 2. Excel COUNTIFS Function with Plus Operator to Count Cells with Multiple Criteria\n\nWe can also use the COUNTIFS function with the Plus (+) operator to count cells with multiple criteria and OR logic. Here, our goal is the same as in the previous example. We will try to count the number of cells that contain the Laptop and Keyboard with Paid or Pending status. For that purpose, we will use a slightly different dataset from the previous one. Here, we have typed the Status in Cell F9 and Cell F10.", null, "Let’s follow the steps below to see how we can implement the formula with the Plus (+) operator.\n\nSTEPS:\n\n• Firstly, select Cell F13 and type the formula below:\n`=COUNTIFS(B5:B13,\\$F\\$6,D5:D13,\\$F\\$9)+COUNTIFS(B5:B13,\\$F\\$6,D5:D13,\\$F\\$10)+COUNTIFS(B5:B13,\\$F\\$7,D5:D13,\\$F\\$9)+COUNTIFS(B5:B13,\\$F\\$7,D5:D13,\\$F\\$10)`\n• After that, press Enter to see the result.", null, "Here, we have used absolute cell reference instead of typing the text inside the formula.\n\n🔎 How Does the Formula Work?\n\nWe can break the formula into four parts and each part counts the cell number specified by the condition.\n\n• COUNTIFS(B5:B13,\\$F\\$6,D5:D13,\\$F\\$9): This is the first part of the formula. It counts the number of cells that contain Keyboard in the range B5:B13 and Pending in the range D5:D13.\n• COUNTIFS(B5:B13,\\$F\\$6,D5:D13,\\$F\\$10): The second part counts the number of cells that contain Keyboard in the range B5:B13 and Paid in the range D5:D13.\n• COUNTIFS(B5:B13,\\$F\\$7,D5:D13,\\$F\\$9): Similarly, the third part counts the number of cells that contain Laptop in the range B5:B13 and Pending in the range D5:D13.\n• COUNTIFS(B5:B13,\\$F\\$7,D5:D13,\\$F\\$10): The last part counts the cell numbers that store Laptop in the range B5:B13 and Paid in the range D5:D13.\n• The Plus (+) operator sums the value and shows the summation.\n\n### 3. Count Cells Using Excel COUNTIFS Formula with OR as well as AND Logic\n\nIn the third example, we will count cells using the COUNTIFS function with OR as well as AND logic. Here, we will count the number of cells that contain a Laptop or Keyboard in the range B5:B13 and payment status Paid in the range D5:D13. For that purpose, we will use the dataset below.", null, "Let’s observe the steps below to see how we can count cell numbers with OR as well as AND logic.\n\nSTEPS:\n\n• In the first place, select Cell F12 and type the formula below:\n`=COUNTIFS(B5:B13,F6:F7,D5:D13,F10)`\n• Now, press Ctrl + Shift + Enter to see the result.", null, "This formula is an array formula. That is why we got an array {0,2} in the result.\n\n• The first condition looks for the text Keyboard or Laptop in the range B5:B13.\n• Similarly, the second condition looks for the word Paid in the range D5:D13.\n• To get the summation of the array {0,2}, you need to use the formula below:\n`=SUM(COUNTIFS(B5:B13,F6:F7,D5:D13,F10))`\n\nIf you insert this formula in Cell F12, then it will show 2 in that cell.\n\n## Alternative to Excel COUNTIFS Function to Count Cells with Multiple Sets of OR Criteria\n\nThe alternative to the COUNTIFS function is to use the SUMPRODUCT function. Inside the SUMPRODUCT function, we will have to use the ISNUMBER function and the MATCH function. To understand the use of the SUMPRODUCT function, we will use the dataset of Example 1. Here, we want to find the count of cells that contain the word Laptop or Keyboard with the payment status Paid or Pending.\n\nTo do so, you need to follow the steps below:\n\nSTEPS:\n\n• In the beginning, select Cell F13 and type the formula below:\n`=SUMPRODUCT(ISNUMBER(MATCH(B5:B13,{\"Laptop\",\"Keyboard\"},0))ISNUMBER(MATCH(D5:D13,{\"Paid\",\"Pending\"},0)))`\n• After that, press Enter to get the result.", null, "Here, the MATCH function looks for the exact match of the texts in the desired range. After that, the ISNUMBER function checks if the returned value is a number or not. If it is a number, then it returns TRUE, otherwise FALSE. Let’s go through the formula breakdown to learn more.\n\n🔎 How Does the Formula Work?\n\n• MATCH(B5:B13,{“Laptop”,”Keyboard”},0): This part returns the array:\n\n`{#N/A, 2, #N/A, 1, 2, #N/A, 2, 1, 1}`\n\n• ISNUMBER(MATCH(B5:B13,{“Laptop”,”Keyboard”},0)): It checks if the returned array contains a number. The output of this part is:\n\n`{FALSE, TRUE, FALSE, TRUE, TRUE, FALSE, TRUE, TRUE, TRUE}`\n\n• ISNUMBER(MATCH(D5:D13,{“Paid”,”Pending”},0)): Similarly, the output of this part is:\n\n`{TRUE, FALSE, TRUE, TRUE, TRUE, TRUE, FALSE, TRUE, TRUE}`\n\n• ISNUMBER(MATCH(B5:B13,{“Laptop”,”Keyboard”},0))ISNUMBER(MATCH(D5:D13,{“Paid”,”Pending”},0)): This part multiplies the previous two arrays. So, the result becomes:\n\n`{0, 0, 0, 1, 1, 0, 0, 1, 1}`\n\n• Finally, the SUMPRODUCT function returns the sum of the products of that array which is 4.\n\n## How to Apply COUNTIFS Function with Dynamic OR Logic in Excel\n\nIn the previous sections, we showed the COUNTIFS function with a formula that contained hardcore values. In that case, if you need change, then you have to make change changes inside the formula.\n\nBut we can also use dynamic OR logic inside a formula. For that purpose, we need to define names before applying the formula. Here, we will count the number of cells that contain the text Laptop or Keyboard in the range B5:B13. Let’s follow the steps below to see how to define names in Excel and use it inside the formula.\n\nSTEPS:\n\n• Firstly, select the range F5:F6.", null, "• Secondly, go to the Formulas tab and select Define Name.", null, "• A box will appear.\n• Type a name in the Name field and click OK to proceed.\n• Here, we have named the range F5:F6 as the Product.", null, "• After that, select Cell F13 and type the formula below:\n`=SUM(COUNTIFS(B5:B13,Product))`\n• Press Enter to see the result.", null, "• Finally, if you remove the word Laptop from Cell F6, then the result of Cell F13 will automatically update.", null, "## How to Use COUNTIFS Function with Wildcard Characters in Excel\n\nIn Excel, we can use the Question Mark (?) and Asterisk () wildcard characters in the COUNTIFS function. The Question Mark (?) matches a single character and the Asterisk () symbol matches a sequence of characters. To demonstrate the wildcard characters, we will use the dataset below. From the dataset, we will count the number of cells with payment status.", null, "Let’s follow the steps below to see how we can use the wildcard characters.\n\nSTEPS:\n\n• First of all, select Cell F11 and type the formula below:\n`=COUNTIFS(B5:B13,\"*\",D5:D13,\"<>\"&\"\")`\n• After that, press Enter to see the result.", null, "This formula matches the sequence of the range B5:B13 & D5:D13 and finds the non-empty cells.\n\n## Related Articles", null, "Mursalin Ibne Salehin\n\nHi there! This is Mursalin. I am currently working as a Team Leader at ExcelDemy. I am always motivated to gather knowledge from different sources and find solutions to problems in easier ways. I manage and help the writers to develop quality content in Excel and VBA-related topics.\n\nWe will be happy to hear your thoughts", null, "Advanced Excel Exercises with Solutions PDF", null, "", null, "" ]	[ null, "data:image/svg+xml,%3Csvg%20xmlns='http://www.w3.org/2000/svg'%20viewBox='0%200%20532%20392'%3E%3C/svg%3E", null, "data:image/svg+xml,%3Csvg%20xmlns='http://www.w3.org/2000/svg'%20viewBox='0%200%20549%20396'%3E%3C/svg%3E", null, "data:image/svg+xml,%3Csvg%20xmlns='http://www.w3.org/2000/svg'%20viewBox='0%200%20580%20455'%3E%3C/svg%3E", null, "data:image/svg+xml,%3Csvg%20xmlns='http://www.w3.org/2000/svg'%20viewBox='0%200%20240%20119'%3E%3C/svg%3E", null, "data:image/svg+xml,%3Csvg%20xmlns='http://www.w3.org/2000/svg'%20viewBox='0%200%20543%20394'%3E%3C/svg%3E", null, "data:image/svg+xml,%3Csvg%20xmlns='http://www.w3.org/2000/svg'%20viewBox='0%200%20576%20490'%3E%3C/svg%3E", null, "data:image/svg+xml,%3Csvg%20xmlns='http://www.w3.org/2000/svg'%20viewBox='0%200%20538%20396'%3E%3C/svg%3E", null, "data:image/svg+xml,%3Csvg%20xmlns='http://www.w3.org/2000/svg'%20viewBox='0%200%20552%20425'%3E%3C/svg%3E", null, "data:image/svg+xml,%3Csvg%20xmlns='http://www.w3.org/2000/svg'%20viewBox='0%200%20701%20453'%3E%3C/svg%3E", null, "data:image/svg+xml,%3Csvg%20xmlns='http://www.w3.org/2000/svg'%20viewBox='0%200%20538%20396'%3E%3C/svg%3E", null, "data:image/svg+xml,%3Csvg%20xmlns='http://www.w3.org/2000/svg'%20viewBox='0%200%20497%20130'%3E%3C/svg%3E", null, "data:image/svg+xml,%3Csvg%20xmlns='http://www.w3.org/2000/svg'%20viewBox='0%200%20320%20239'%3E%3C/svg%3E", null, "data:image/svg+xml,%3Csvg%20xmlns='http://www.w3.org/2000/svg'%20viewBox='0%200%20550%20436'%3E%3C/svg%3E", null, "data:image/svg+xml,%3Csvg%20xmlns='http://www.w3.org/2000/svg'%20viewBox='0%200%20533%20398'%3E%3C/svg%3E", null, "data:image/svg+xml,%3Csvg%20xmlns='http://www.w3.org/2000/svg'%20viewBox='0%200%20533%20398'%3E%3C/svg%3E", null, "data:image/svg+xml,%3Csvg%20xmlns='http://www.w3.org/2000/svg'%20viewBox='0%200%20567%20436'%3E%3C/svg%3E", null, "data:image/svg+xml,%3Csvg%20xmlns='http://www.w3.org/2000/svg'%20viewBox='0%200%2069%2069'%3E%3C/svg%3E", null, "https://www.exceldemy.com/countifs-with-multiple-criteria-and-or-logic/", null, "https://www.exceldemy.com/countifs-with-multiple-criteria-and-or-logic/", null, "data:image/svg+xml,%3Csvg%20xmlns='http://www.w3.org/2000/svg'%20viewBox='0%200%20160%2050'%3E%3C/svg%3E", null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.82934266,"math_prob":0.9547804,"size":9816,"snap":"2023-40-2023-50","text_gpt3_token_len":2464,"char_repetition_ratio":0.16123115,"word_repetition_ratio":0.1579272,"special_character_ratio":0.25855747,"punctuation_ratio":0.17117538,"nsfw_num_words":1,"has_unicode_error":false,"math_prob_llama3":0.99169713,"pos_list":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40],"im_url_duplicate_count":[null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,2,null,2,null,null,null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2023-09-21T10:23:08Z\",\"WARC-Record-ID\":\"<urn:uuid:6468481b-0980-4e9f-97a1-3febbc98642e>\",\"Content-Length\":\"239414\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:06d97fd0-6bb8-4133-942d-f2d5e5f8995d>\",\"WARC-Concurrent-To\":\"<urn:uuid:056dca10-3b3c-4c41-95cf-0dddfdd6b1d5>\",\"WARC-IP-Address\":\"104.21.6.27\",\"WARC-Target-URI\":\"https://www.exceldemy.com/countifs-with-multiple-criteria-and-or-logic/\",\"WARC-Payload-Digest\":\"sha1:LLW2ZPBYKJGATO25GXH5AFSB7HBBAXZE\",\"WARC-Block-Digest\":\"sha1:YX3RQAIBS3IX3DPEA5MMI4DYO5JGAC57\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2023/CC-MAIN-2023-40/CC-MAIN-2023-40_segments_1695233505362.29_warc_CC-MAIN-20230921073711-20230921103711-00880.warc.gz\"}"}
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https://www.studystack.com/flashcard-2151117	[ "Save", null, "or", null, "or", null, "taken", null, "Make sure to remember your password. If you forget it there is no way for StudyStack to send you a reset link. You would need to create a new account.\n\nfocusNode\nDidn't know it?\nclick below\n\nKnew it?\nclick below\nDon't Know\nRemaining cards (0)\nKnow\n0:00\nEmbed Code - If you would like this activity on your web page, copy the script below and paste it into your web page.\n\nNormal Size Small Size show me how\n\n# Module 5 - Linear Eq\n\n### Math 113 Online: Module 5 - Linear Equations and Substitution Method\n\nDetermine whether each ordered pair is a solution of the system of linear equations. {x+y=8 and 3x+4y=29 a) (2,6) b) (3,5) Plug in each ordered pair into both equations to determine if said ordered pairs are solutions to the linear equations. 2+6=9; 3(2)+4(6)=29/6+24=30 and 3+5=8; 3(3)+4(5)=29/9+20=29. Ordered pair a isn't a solution, but pair b is.\nSolve the System of linear equations by graphing. {x+y=9 and x-y=5 Once you've graphed both equations, find the point at which both lines intersect. The ordered pair should satisfy the equations. The solution of the system is (7,2) because it completes both linear equations. 7+2=9;7-2=5\nWithout graphing, decide. a) Are the graphs of the equations identical lines, parallel lines, or lines intersecting at a single point? and b) How many solutions does the system have? {9x+7=27 and x+8y=8 Rewrite both equations in slope intercept form y=-9+27 and y=-1/8x+1. Identify the slopes and y-intercepts for both lines. The slopes for the lines are -9 and -1/8. The slopes are different, and must intersect at one point; meaning one solution.\nSolve the system of equations using the substitution method. {x+y=10 and x=4y The first step is to solve one of the equations for one variable. The second equation; x=4y has already been solved for x, so we'll use it. 4y+y=10 (Combine like terms) 5y=10; y=2 x=4(2); x=8. So the solution is (8,2)\nSolve the system of equations by the substitution method. {y=5x+7 and y=7x+8 Substitute 5x+7 for y in the 2 equation. 5x+7=7x+8 isolate the x terms on the left side by subtracting 7x and adding 7. -2x=1; x=-1/2 Plug in x for the 1st equation. y=5(-1/2)+7; y=-5/2+7. Simplify. y=9/2 So the ordered pair is (-1/2,9/2)\nSolve the system of equations by the substitution method. {12x+3y=8 and -4x=y+8 Neither equation is solved for x or y, so we'll isolate y on the right side of equation 2. -4x-8=y Now fill in the 1st equation 12x+3(-4x-8)=8 Use the distributive property on the left side. 12x-12x-24=8 -24=8 is a false statement/no solution.\nSolve the system of equations by the substitution method. {4x-y=3 and 5x-2y=12 Neither is solved, so we'll isolate y in the 1st equation. -y=3-4x Solve for y y=4x-3 Now substitute y in the 2nd equation. 5x-2(4x-3)=12 Distributive property. 5x-8x+6=12; -3x+6=12, isolate x. -3x=6, solve for x, x=-2. Replace x in the first equation.\nSolve the system of equations by the substitution method. {4x-y=3 and 5x-2y=12 (Continued) y=4(-2)-3, y=-8-3, y=-11. So the ordered pair is (-2,-11)\nSolve the system of equations by the substitution method. {3x+12y=15 and 4x+18y=22 Neither is solved, so we'll isolate x in the 1st equation. 3x=15-12y, solve for x. x=5-4y, now substitute x in the 2nd equation. 4(5-4y)+18y=22 Distribute. 20-16y+18y=22, combine like terms. 20+2y=22 isolate y on the left side. 2y=2; y=1\nSolve the system of equations by the substitution method. {3x+12y=15 and 4x+18y=22 (Continued) Replace y in the first equation. x=5-4(1) Solve. x=1 So the ordered pair is (1,1)\nCreated by: 1118006241566176\n\nVoices\n\nUse these flashcards to help memorize information. Look at the large card and try to recall what is on the other side. Then click the card to flip it. If you knew the answer, click the green Know box. 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https://www.quantopian.com/posts/help-for-a-newby-mean-reversion	[ "Help for a newby (mean reversion)\n\nI very new to this, and I've written this (my first) mean reversion algo which calculates a hedge ratio using linear regression between USO and GLD and then buys/sells depending on the deviation of the portfolio price from the mean. It hasn't turned out very well, unsurprisingly, and the results vary wildly depending on what I set as the lookback period. I was wondering if anyone could point out if I have fundamentally misunderstood how this strategy works. Pointers on my terrible, terrible code would also be appreciated.\n\n4\nTotal Returns\n--\nAlpha\n--\nBeta\n--\nSharpe\n--\nSortino\n--\nMax Drawdown\n--\nBenchmark Returns\n--\nVolatility\n--\n Returns 1 Month 3 Month 6 Month 12 Month\n Alpha 1 Month 3 Month 6 Month 12 Month\n Beta 1 Month 3 Month 6 Month 12 Month\n Sharpe 1 Month 3 Month 6 Month 12 Month\n Sortino 1 Month 3 Month 6 Month 12 Month\n Volatility 1 Month 3 Month 6 Month 12 Month\n Max Drawdown 1 Month 3 Month 6 Month 12 Month\nimport pandas as pd\nimport numpy as np\nfrom pandas.stats.api import ols\n\ndef initialize(context):\n\n#trading only GLD and USO ETfs\ncontext.securities = symbols('GLD', 'USO')\n\n#pretty arbitrary lookback for moving mean and std of portfolio price\ncontext.lookback = 10\n\n#to track portfolio price\ncontext.yPort = []\n\n# Rebalance every day, 1 hour after market open.\nschedule_function(my_rebalance, date_rules.every_day(), time_rules.market_close(minutes = 10))\n\n# Record tracking variables at the end of each day.\nschedule_function(my_record_vars, date_rules.every_day(), time_rules.market_close())\n\nlb = context.lookback\n\n#ETF recent prices\nx = data.history(symbol('GLD'), 'close', lb, '1d')\ny = data.history(symbol('USO'), 'close', lb, '1d')\n\n#linear regression to get hedge ratio (coffeccient of x)\ndf = pd.DataFrame({'GLD':x, 'USO':y})\n\nres = ols(y=df['USO'], x=df['GLD'])\n\ncontext.hr = res.beta['x']\n\ndef my_rebalance(context,data):\nlb = context.lookback\n\n#get current ETF prices at near end of day\nx = data.current(symbol('GLD'), 'price')\ny = data.current(symbol('USO'), 'price')\n\n#price of unit portfolio\nport = y - context.hrx\n\n#keep track of portfolio prices\ncontext.yPort.append(port)\n\n#only need previous lookback number of prices\nif len(context.yPort) > lb + 1:\ncontext.yPort = context.yPort[1::]\n\n#after enough portfolio prices have been determined\nif len(context.yPort) == lb + 1:\n#moving average and std of portfolio prices\nma = np.mean(context.yPort[:lb])\nms = np.std(context.yPort[:lb])\n\n#Z-score of current portfolio value\nz = (port - ma)/ms\n\n#order ETFs depending on current hedge ratio and multiplied by Z-score, pumped up by an arbitrary factor\norder_target_value(symbol('GLD'), 10000zcontext.hrx)\norder_target_value(symbol('USO'), -10000zy)\n\ndef my_record_vars(context, data):\nrecord(leverage = context.account.leverage)\nrecord(hedgeRatio = context.hr)\n\n\nThere was a runtime error.\n7 responses\n\nSome changes I've made, which seem to help reduce the volatility and drawdown:\n- order on market open, using yesterday's close data (I guessed more liquidity is better than trying to predict the close)\n- compare current day's closing price to moving average of last 10 days including itself (this should improve accuracy)\n- grow position size as portfolio grows (instead of fixed position sizing)\n\n6\nTotal Returns\n--\nAlpha\n--\nBeta\n--\nSharpe\n--\nSortino\n--\nMax Drawdown\n--\nBenchmark Returns\n--\nVolatility\n--\n Returns 1 Month 3 Month 6 Month 12 Month\n Alpha 1 Month 3 Month 6 Month 12 Month\n Beta 1 Month 3 Month 6 Month 12 Month\n Sharpe 1 Month 3 Month 6 Month 12 Month\n Sortino 1 Month 3 Month 6 Month 12 Month\n Volatility 1 Month 3 Month 6 Month 12 Month\n Max Drawdown 1 Month 3 Month 6 Month 12 Month\nimport pandas as pd\nimport numpy as np\nfrom pandas.stats.api import ols\n\ndef initialize(context):\n\n#trading only GLD and USO ETfs\ncontext.securities = symbols('GLD', 'USO')\n\n#pretty arbitrary lookback for moving mean and std of portfolio price\ncontext.lookback = 10\n\n#to track portfolio price\ncontext.yPort = []\n\n# Rebalance every day, 1 hour after market open.\nschedule_function(my_rebalance, date_rules.every_day(), time_rules.market_open())\n\n# Record tracking variables at the end of each day.\nschedule_function(my_record_vars, date_rules.every_day(), time_rules.market_close())\n\nlb = context.lookback\n\n#ETF recent prices\nx = data.history(symbol('GLD'), 'close', lb, '1d')\ny = data.history(symbol('USO'), 'close', lb, '1d')\n\n#linear regression to get hedge ratio (coffeccient of x)\ndf = pd.DataFrame({'GLD':x, 'USO':y})\n\nres = ols(y=df['USO'], x=df['GLD'])\n\ncontext.hr = res.beta['x']\n\n#price of unit portfolio\nport = y[-1] - context.hrx[-1]\ncontext.port = port\ncontext.x = x[-1]\ncontext.y = y[-1]\n\n#keep track of portfolio prices\ncontext.yPort.append(port)\n\ncontext.z = 0\n\n#only need previous lookback number of prices\nif len(context.yPort) >= lb:\ncontext.yPort = context.yPort[-lb:]\n\nma = np.mean(context.yPort)\nms = np.std(context.yPort)\n\n#Z-score of current portfolio value\ncontext.z = (port - ma)/ms\n\ndef my_rebalance(context,data):\n\nz = context.z\ny = context.y\nx = context.x\nacc = context.portfolio.portfolio_value\n#order ETFs depending on current hedge ratio and multiplied by Z-score, pumped up by an arbitrary factor\norder_target_value(symbol('GLD'), accz/100context.hrx)\norder_target_value(symbol('USO'), -accz/100y)\n#order_target_value(symbol('GLD'), 10000zcontext.hrx)\n#order_target_value(symbol('USO'), -10000zy)\n\ndef my_record_vars(context, data):\n#record(leverage = context.account.leverage)\n#record(hedgeRatio = context.hr)\nrecord(port=context.port)\n\n\nThere was a runtime error.\n\nAlso I was wondering if you actually need two lookback periods. One is for establishing the hedge ratio. If it's relatively stable this could be longer than the current 10 days. The other is for establishing direction of mean reversion. This will be related to the half life of mean reversion, and may vary. I believe it's an output of OLS?\n\nUpdate: the half life of mean reversion can be worked out from the auto-correlation slope. In other words the beta of the series returns regressed on lagged version of itself. Ernie Chan gives the matlab code in his book.\n\nFound this:\n\nhttp://epchan.blogspot.co.uk/2011/06/when-cointegration-of-pair-breaks-down.html\n\nI noticed the algo was really unstable with the lookback length. Looking at the chart of the hedge ratio, I can see its very unstable, and yet I would expect the hedge ratio to only change relatively slowly. So, I use an exponential smoothing on the hedge ratio. This seems to help stabilise the performance for a wider range of lookback parameters.\n\n9\nTotal Returns\n--\nAlpha\n--\nBeta\n--\nSharpe\n--\nSortino\n--\nMax Drawdown\n--\nBenchmark Returns\n--\nVolatility\n--\n Returns 1 Month 3 Month 6 Month 12 Month\n Alpha 1 Month 3 Month 6 Month 12 Month\n Beta 1 Month 3 Month 6 Month 12 Month\n Sharpe 1 Month 3 Month 6 Month 12 Month\n Sortino 1 Month 3 Month 6 Month 12 Month\n Volatility 1 Month 3 Month 6 Month 12 Month\n Max Drawdown 1 Month 3 Month 6 Month 12 Month\nimport pandas as pd\nimport numpy as np\nfrom pandas.stats.api import ols\n\ndef initialize(context):\n\n#trading only GLD and USO ETfs\ncontext.securities = symbols('GLD', 'USO')\n\n#pretty arbitrary lookback for moving mean and std of portfolio price\ncontext.lookback = 10\ncontext.hr = 0.10\n\n#to track portfolio price\ncontext.yPort = []\n\n# Rebalance every day, 1 hour after market open.\nschedule_function(my_rebalance, date_rules.every_day(), time_rules.market_open())\n\n# Record tracking variables at the end of each day.\nschedule_function(my_record_vars, date_rules.every_day(), time_rules.market_close())\n\nlb = context.lookback\n\n#ETF recent prices\nx = data.history(symbol('GLD'), 'close', lb, '1d')\ny = data.history(symbol('USO'), 'close', lb, '1d')\n\n#linear regression to get hedge ratio (coffeccient of x)\ndf = pd.DataFrame({'GLD':x, 'USO':y})\n\nres = ols(y=df['USO'], x=df['GLD'])\n\n#smooth the HR so it doesn't change so rapidly\ncontext.hr = context.hr(1.0-1.0/lb)+res.beta['x'](1.0/lb)\n\n#price of unit portfolio\nport = y[-1] - context.hrx[-1]\ncontext.port = port\ncontext.x = x[-1]\ncontext.y = y[-1]\n\n#keep track of portfolio prices\ncontext.yPort.append(port)\n\ncontext.z = 0\n\n#only need previous lookback number of prices\nif len(context.yPort) >= lb:\ncontext.yPort = context.yPort[-lb:]\n\nma = np.mean(context.yPort)\nms = np.std(context.yPort)\n\n#Z-score of current portfolio value\ncontext.z = (port - ma)/ms\n\ndef my_rebalance(context,data):\n\nz = context.z\ny = context.y\nx = context.x\nacc = context.portfolio.portfolio_value\n#order ETFs depending on current hedge ratio and multiplied by Z-score, pumped up by an arbitrary factor\norder_target_value(symbol('GLD'), accz/100context.hrx)\norder_target_value(symbol('USO'), -accz/100y)\n#order_target_value(symbol('GLD'), 10000zcontext.hrx)\n#order_target_value(symbol('USO'), -10000zy)\n\ndef my_record_vars(context, data):\n#record(leverage = context.account.leverage)\nrecord(hedgeRatio = context.hr)\n#record(port=context.port)\n\n\nThere was a runtime error.\n\nThats very helpful, thanks- yeah, I thought it was strange how much the hedge ratio was fluctuating, I wonder if it is due to having a short lookback period for the linear regression.\n\nThis paper looks good. Updates the hedge ratio using a kalman filter." ]	[ null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.56848484,"math_prob":0.81022173,"size":2968,"snap":"2020-34-2020-40","text_gpt3_token_len":752,"char_repetition_ratio":0.1565452,"word_repetition_ratio":0.06763285,"special_character_ratio":0.27156335,"punctuation_ratio":0.14548802,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9843586,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2020-08-03T17:36:20Z\",\"WARC-Record-ID\":\"<urn:uuid:a86c356a-d3fc-4d5b-ae59-c20411028aee>\",\"Content-Length\":\"79127\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:f16016e9-74fe-48eb-a0f4-57ad3716e760>\",\"WARC-Concurrent-To\":\"<urn:uuid:d6a56301-5b1e-4518-9e74-d90cfa674ff3>\",\"WARC-IP-Address\":\"172.67.8.162\",\"WARC-Target-URI\":\"https://www.quantopian.com/posts/help-for-a-newby-mean-reversion\",\"WARC-Payload-Digest\":\"sha1:FC5DLJQSBLIIOPOCRMSUFGCCENVMSWCZ\",\"WARC-Block-Digest\":\"sha1:PUGMNVYVNBRGSNN4BMHOJQ6EJGDNQQ25\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2020/CC-MAIN-2020-34/CC-MAIN-2020-34_segments_1596439735823.29_warc_CC-MAIN-20200803170210-20200803200210-00416.warc.gz\"}"}
https://www.pushvps.com/44199.html	[ "###什么是C++11\n\nC++11是曾经被叫做C++0x，是对目前C++语言的扩展和修正，C++11不仅包含核心语言的新机能，而且扩展了C++的标准程序库（STL），并入了大部分的C++ Technical Report 1（TR1）程序库(数学的特殊函数除外)。\n\nC++11包括大量的新特性：包括lambda表达式，类型推导关键字auto、decltype，和模板的大量改进。\n\n###新的关键字\n\n####auto\n\nC++11中引入auto第一种作用是为了自动类型推导\n\nauto的自动类型推导，用于从初始化表达式中推断出变量的数据类型。通过auto的自动类型推导，可以大大简化我们的编程工作\n\nauto实际上实在编译时对变量进行了类型推导，所以不会对程序的运行效率造成不良影响\n\n``````<!-- lang: cpp -->\nauto a; // 错误，auto是通过初始化表达式进行类型推导，如果没有初始化表达式，就无法确定a的类型\nauto i = 1;\nauto d = 1.0;\nauto str = \"Hello World\";\nauto ch = 'A';\nauto func = less<int>();\nvector<int> iv;\nauto ite = iv.begin();\nauto p = new foo() // 对自定义类型进行类型推导\n``````\n\nauto不光有以上的应用，它在模板中也是大显身手，比如下例这个加工产品的例子中，如果不使用auto就必须声明Product这一模板参数：\n\n``````template <typename Product, typename Creator>\nvoid processProduct(const Creator& creator) {\nProduct* val = creator.makeObject();\n// do somthing with val\n}\n.\n``````\n\n``````template <typename Creator>\nvoid processProduct(const Creator& creator) {\nauto val = creator.makeObject();\n// do somthing with val\n}\n``````\n\n####decltype\n\ndecltype实际上有点像auto的反函数，auto可以让你声明一个变量，而decltype则可以从一个变量或表达式中得到类型，有实例如下：\n\n``````int x = 3;\ndecltype(x) y = x;\n``````\n\n``````template <typename Creator>\nauto processProduct(const Creator& creator) -> decltype(creator.makeObject()) {\nauto val = creator.makeObject();\n// do somthing with val\n}\n``````\n\n####nullptr\n\nnullptr是为了解决原来C++中NULL的二义性问题而引进的一种新的类型，因为NULL实际上代表的是0，\n\n``````void F(int a){\ncout<<a<<endl;\n}\n\nvoid F(int p){\nassert(p != NULL);\n\ncout<< p <<endl;\n}\n\nint main(){\n\nint p = nullptr;\nint q = NULL;\nbool equal = ( p == q ); // equal的值为true，说明p和q都是空指针\nint a = nullptr; // 编译失败，nullptr不能转型为int\nF(0); // 在C++98中编译失败，有二义性；在C++11中调用F（int）\nF(nullptr);\n\nreturn 0;\n}\n``````\n\n###序列for循环\n\n``````map<string, int> m{{\"a\", 1}, {\"b\", 2}, {\"c\", 3}};\nfor (auto p : m){\ncout<<p.first<<\" : \"<<p.second<<endl;\n}\n``````\n\n###Lambda表达式\n\nlambda表达式类似Javascript中的闭包，它可以用于创建并定义匿名的函数对象，以简化编程工作。Lambda的语法如下：\n\n[函数对象参数]（操作符重载函数参数）->返回值类型{函数体}\n\n``````vector<int> iv{5, 4, 3, 2, 1};\nint a = 2, b = 1;\n\nfor_each(iv.begin(), iv.end(), [b](int &x){cout<<(x + b)<<endl;}); // (1)\n\nfor_each(iv.begin(), iv.end(), [=](int &x){x = (a + b);});\t\t// (2)\n\nfor_each(iv.begin(), iv.end(), [=](int &x)->int{return x * (a + b);});// (3)\n``````\n• []内的参数指的是Lambda表达式可以取得的全局变量。(1)函数中的b就是指函数可以得到在Lambda表达式外的全局变量，如果在[]中传入=的话，即是可以取得所有的外部变量，如（2）和（3）Lambda表达式\n• ()内的参数是每次调用函数时传入的参数\n• ->后加上的是Lambda表达式返回值的类型，如（3）中返回了一个int类型的变量\n\n###变长参数的模板\n\n``````auto p = make_pair(1, \"C++ 11\");\n``````\n\n``````auto t1 = make_tuple(1, 2.0, \"C++ 11\");\nauto t2 = make_tuple(1, 2.0, \"C++ 11\", {1, 0, 2});\n``````\n\n``````template<typename head, typename... tail>\nPrint(tail...);\n}\n``````\n\nPrint中可以传入多个不同种类的参数，如下：\n\n``````Print(1, 1.0, \"C++11\");\n``````\n\n###更加优雅的初始化方法\n\n``````int arr = {1, 2, 3}\nvector<int> v(arr, arr + 3);\n``````\n\n``````int arr{1, 2, 3};\nvector<int> iv{1, 2, 3};\nmap<int, string>{{1, \"a\"}, {2, \"b\"}};\nstring str{\"Hello World\"};\n``````\n\n###然后呢...\n\nToWrting的C++11系列博文\n\nC++11的编译器支持列表", null, "", null, "" ]	[ null, "https://ae01.alicdn.com/kf/H67bb680a41e24b45a15de1e6cafc29a2y.jpg", null, "https://ae01.alicdn.com/kf/Ha77c54e7b28c488e9a4aea6159695b1av.jpg", null ]	{"ft_lang_label":"__label__zh","ft_lang_prob":0.7355156,"math_prob":0.9948246,"size":3896,"snap":"2020-34-2020-40","text_gpt3_token_len":2339,"char_repetition_ratio":0.09429599,"word_repetition_ratio":0.09749304,"special_character_ratio":0.31057495,"punctuation_ratio":0.21580547,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.97022367,"pos_list":[0,1,2,3,4],"im_url_duplicate_count":[null,null,null,null,null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2020-08-07T01:15:01Z\",\"WARC-Record-ID\":\"<urn:uuid:622daf82-6e36-4536-9861-857e796fd62e>\",\"Content-Length\":\"22391\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:5536c717-6f45-4094-b7c2-6112ca34ec4d>\",\"WARC-Concurrent-To\":\"<urn:uuid:ad917937-3c56-4789-84c9-5e839c733907>\",\"WARC-IP-Address\":\"104.26.2.99\",\"WARC-Target-URI\":\"https://www.pushvps.com/44199.html\",\"WARC-Payload-Digest\":\"sha1:GQNWZ4MUBT5GQVB3QFD33GE5UT3PGB4B\",\"WARC-Block-Digest\":\"sha1:3RBL4IAFXQHONGVRC6DXSY74XVZEPMAB\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2020/CC-MAIN-2020-34/CC-MAIN-2020-34_segments_1596439737050.56_warc_CC-MAIN-20200807000315-20200807030315-00492.warc.gz\"}"}
http://library.kaist.ac.kr/search/detail/view.do?bibCtrlNo=467748&flag=dissertation	[ "#", null, "", null, "서지주요정보\n70Na${}_{1/2}$Bi${}_{1/2}$TiO${}_{3}$-30BaTiO${}_{3}$ 계에서 Na${}_{2}$O 첨가량에 따른 입자성장거동 = Grain Growth Behavior in 70Na${}_{1/2}$Bi${}_{1/2}$TiO${}_{3}$-30BaTiO${}_{3}$ System with Change of Na${}_{2}$O Addition\n서명 / 저자 70Na${}_{1/2}$Bi${}_{1/2}$TiO${}_{3}$-30BaTiO${}_{3}$ 계에서 Na${}_{2}$O 첨가량에 따른 입자성장거동 = Grain Growth Behavior in 70Na${}_{1/2}$Bi${}_{1/2}$TiO${}_{3}$-30BaTiO${}_{3}$ System with Change of Na${}_{2}$O Addition / 박지훈.\n저자명 박지훈 ; Park, Ji-Hoon\n발행사항 [대전 : 한국과학기술원, 2011].\nOnline Access 원문보기 원문인쇄\n\n등록번호\n\n8022578\n\n소장위치/청구기호\n\n학술문화관(문화관) 보존서고\n\nMAME 11014\n\n도서상태\n\n이용가능\n\n반납예정일\n\n#### 초록정보\n\nRelaxor-based piezoelectric single crystals such as Pb($Zn_{1/3}Nb_{2/3}$)$TiO_3$-$PbTiO_3$ (PZN-PT) and Pb($Mg_{1/3}Nb_{2/3})$TiO_3$-$PbTiO_3$(PMN-PT) are considered as promising for potential applications in medical ultrasonic, sonar transducers and solid state actuators owing to their ultrahigh coupling coefficient as high as 94\\% and piezoelectric coefficient ($d_{33}$) of 2500 pC/N in contrast to conventional Pb(Zr,Ti)$O_3$(PZT) ceramics with$k_{33}$~ 75\\% and$d_{33 }$~ 500 pC/N. However, the toxicity of PbO and its high vapor pressure during sintering has led to a demand for alternative eco-friendly lead-free materials as several countries have already restricted the utility of hazardous substances in electrical and electronic equipments. Therefore, attention has recently been focused on the fabrication of lead-free piezoelectric ceramics and a parallel search for high performance relaxor-based lead-free piezoelectric single crystals. The solid solution 100-$\\textit{x}(Na_{0.5}Bi_{0.5}$)$TiO_{3}$-$\\textit{x}BaTiO_{3}$(NBT-$\\textit{x}$BT) emerged as one the lead-free systems that can replace lead-based pizoelectrics. Over last decade, tremendous efforts have been made to fabricate high quality ceramics and single crystals of NBT-$\\textit{x}$BT, particularly the compositions close to morphotropic phase boundary. In producing NBT-\\textit{x}BT single crystals by melt or solution related techniques such as Bridgeman, Czochralski, top-seeded-solution growth (TSSG), and flux techniques, however, the inhomogeneity problem could not be overcome due to incongruent solidification of the compound and volatilization of low melting-point elements, bismuth and sodium, during crystal growth. To overcome this inherent and critical problem, the solid-state crystal growth (SSCG) technique, which utilizes the frequently observed phenomenon of abnormal grain growth in polycrystals, has been developed. In addition, this technique is much simpler and more cost effective than the conventional techniques. To be successful with the SSCG technique, however, coarsening of fine matrix grains needs to be suppressed during the growth of a seed crystal embedded into the powder compact. Therefore, it is very essential to study the interface morphology and grain growth behaviour of any system before growing the crystal via SSCG. For this present study, a composition ($Na_{0.5}Bi_{0.5}$)$TiO_{3}$-$30BaTiO_{3}$(NBT-30BT) that exhibited relatively strong relaxor behaviour with adequate dielectric permittivity and piezoelectric properties has been considered for the single crystal growth by SSCG technique. Prior to crystal growth, the grain coarsening behaviour of NBT-30BT system has been investigated with varying mol\\% of excess$Na_{2}CO_{3}$. The powders of NBT-30BT, NBT-30BT with 0.2 and 0.5 mol\\%$Na_{2}CO_{3}$(0.2N 30BT, 0.5N 30BT) were prepared by conventional solid state reaction and sintered at different temperatures (1100-1200℃) and durations (10 min-10 h). When the mol\\% of$Na_{2}CO_{3}$increased, the 3-dimensional grain structure changed from rounded to cube shape with more faceted interface and the grain growth behaviour changes toward the abnormal grain growth behaviour. Grain growth behaviour with sintering temperature is explained in terms of the effect of excess$Na_{2}CO_{3}$on interface-reaction controlled grain growth and the critical driving force. The optimal conditions have been figured out and single crystals of 0.2N and 0.5N 30BT with dimension 4 mm x 4 mm x 0.5 mm have been grown using (110)$SrTiO_{3}$seed crystal by SSCG. Pb($Zn_{1/3}Nb_{2/3}$)$TiO_{3}$-$PbTiO_{3}$(PZN-PT), Pb($Mg_{1/3}Nb_{2/3}$)$TiO_{3}$-$PbTiO_{3}$(PMN-PT) 압전단결정은 94\\%의 매우 높은 coupling coefficient와 2500 pC/N에 달하는 압전상수를 갖기 때문에 medical ultrasonic, sonar transducers 그리고 solid state actuators와 같은 소자에 응용이 된다. 하지만, PbO의 독성과 소결시 이 물질의 높은 증기압 때문에 전세계적으로 전자, 전기부품소자를 친환경적인 물질로 대체하고자 하는 추세이다. 그러므로, 최근의 연구는 relaxor-based 비연계압전세라믹스의 제조 및 비연계압전단결정에 집중되고 있다. 고용체인 100-$\\textit{x}(Na_{0.5}Bi_{0.5}$)$TiO_{3}$-$\\textit{x}BaTiO_{3}$(NBT-$\\textit{x}$BT) 연계압전세라믹스를 대체할 물질 중 하나로 떠오르고 있다. 최근 10년동안, MPB조성근처 NBT-BT계의 다결정체 및 단결정에 대한 연구가 이루어졌다. 하지만 Bridgeman, Czochralski, top-seeded-solution growth (TSSG), 그리고 flux techniques을 이용한 NBT-BT계의 단결정성장실험에서의 조성불균일문제가 아직도 해결되지 못했다. 이러한 문제를 해결하기 위해서 다결정체에서의 비정상입자성장을 이용한 고상단결정성장법(SSCG)을 통한 단결정의 성장에 대한 연구가 이루어졌다. SSCG법을 통한 단결정성장을 위해서는 종자단결정이 성장하는 동안 matrix 입자에서의 입자성장이 억제되어야 한다. 따라서, SSCG법을 통해 단결정을 성장시키기에 앞서 다결정체 내에서 입계의 구조 및 입자성장거동을 관찰하는 과정이 필요하다. 본 연구에서는, 상대적으로 강한 relaxor거동을 보이는 ($Na_{0.5}Bi_{0.5}$)$TiO_{3}$-$30BaTiO_{3}$(NBT-30BT)계의 입자성장거동 및 SSCG법을 통한 단결정성장가능성에 대해 연구하였다. 고상단결정성장에 유리한 조건을 찾기 위하여 NBT-30BT 에$Na_{2}CO_{3}$의 양을 0.2~0.5 mol\\%로 변화시켜가며 1100-1200℃ 에서 다양한 시간 동안 입자성장거동을 관찰하였다.$Na_{2}CO_{3}$의 mol\\% 가 증가함에 따라, 3차원 입자모양은 round shape에서 faceted shape으로 변화하였으며, 점점 비정상입자성장거동을 보였다. 이러한 입자성장거동은 첨가된$Na_{2}CO_{3}\\$의 양의 변화에 따른 입자성장의 임계구동력과 최대입자성장구동력과의 관계로 설명할 수 있다.\n\n#### 서지기타정보\n\n청구기호 {MAME 11014 iii, 43 p. : 삽도 ; 26 cm 한국어 저자명의 영문표기 : Ji-Hoon Park 지도교수의 한글표기 : 강석중 지도교수의 영문표기 : Suk-Joong L Kang 학위논문(석사) - 한국과학기술원 : 신소재공학과, 참고문헌 : p. 41-43 비연계 압전특성 입자성장 미세조직 고상단결정성장 Lead-free Piezoelectrics Grain growth Microstructure SSCG\nQR CODE", null, "" ]	[ null, "http://library.kaist.ac.kr/common/images/logo_top.png", null, "http://library.kaist.ac.kr/common/images/bookcover0.jpg", null, "http://chart.googleapis.com/chart", null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.56618553,"math_prob":0.98051494,"size":4942,"snap":"2019-51-2020-05","text_gpt3_token_len":1930,"char_repetition_ratio":0.11077359,"word_repetition_ratio":0.00295858,"special_character_ratio":0.26325375,"punctuation_ratio":0.08955224,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9808088,"pos_list":[0,1,2,3,4,5,6],"im_url_duplicate_count":[null,null,null,null,null,null,null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2020-01-24T22:45:44Z\",\"WARC-Record-ID\":\"<urn:uuid:23875f3b-bd6b-4660-8d6e-ade5c5b42ded>\",\"Content-Length\":\"102044\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:551126ce-6104-40eb-bf21-884ec25bfe67>\",\"WARC-Concurrent-To\":\"<urn:uuid:2ac89412-d49f-4d41-9484-0729507c1fa0>\",\"WARC-IP-Address\":\"143.248.118.12\",\"WARC-Target-URI\":\"http://library.kaist.ac.kr/search/detail/view.do?bibCtrlNo=467748&flag=dissertation\",\"WARC-Payload-Digest\":\"sha1:VZ2MINPVTKPGWM3ZVEFDVXX33RBPKKM6\",\"WARC-Block-Digest\":\"sha1:6ZYJA5VIGIFKLR64NNVB7K6MEJMLRU2O\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2020/CC-MAIN-2020-05/CC-MAIN-2020-05_segments_1579250626449.79_warc_CC-MAIN-20200124221147-20200125010147-00038.warc.gz\"}"}
https://courses.cs.washington.edu/courses/cse341/98au/java/jdk1.2beta4/docs/api/java/lang/Math.html	[ "Java Platform 1.2\nBeta 4\n\n## Class java.lang.Math\n\n```java.lang.Object\n\|\n+--java.lang.Math\n```\n\npublic final class Math\nextends Object\nThe class `Math` contains methods for performing basic numeric operations such as the elementary exponential, logarithm, square root, and trigonometric functions.\n\nTo help ensure portability of Java programs, the definitions of many of the numeric functions in this package require that they produce the same results as certain published algorithms. These algorithms are available from the well-known network library `netlib` as the package \"Freely Distributable Math Library\" (`fdlibm`). These algorithms, which are written in the C programming language, are then to be understood as executed with all floating-point operations following the rules of Java floating-point arithmetic.\n\nThe network library may be found on the World Wide Web at:\n\n``` http://netlib.att.com/\n```\n\nthen perform a keyword search for \"`fdlibm`\".\n\nThe Java math library is defined with respect to the version of `fdlibm` dated January 4, 1995. Where `fdlibm` provides more than one definition for a function (such as `acos`), use the \"IEEE 754 core function\" version (residing in a file whose name begins with the letter `e`).\n\nSince:\nJDK1.0\n\n Field Summary static double E The `double` value that is closer than any other to `e`, the base of the natural logarithms. static double PI The `double` value that is closer than any other to pi, the ratio of the circumference of a circle to its diameter.\n\n Method Summary static double abs(double a) Returns the absolute value of a `double` value. static float abs(float a) Returns the absolute value of a `float` value. static int abs(int a) Returns the absolute value of an `int` value. static long abs(long a) Returns the absolute value of a `long` value. static double acos(double a) Returns the arc cosine of an angle, in the range of 0.0 through pi. static double asin(double a) Returns the arc sine of an angle, in the range of -pi/2 through pi/2. static double atan(double a) Returns the arc tangent of an angle, in the range of -pi/2 through pi/2. static double atan2(double a, double b) Converts rectangular coordinates (`b`, `a`) to polar (r, theta). static double ceil(double a) Returns the smallest (closest to negative infinity) `double` value that is not less than the argument and is equal to a mathematical integer. static double cos(double a) Returns the trigonometric cosine of an angle. static double exp(double a) Returns the exponential number e (i.e. static double floor(double a) Returns the largest (closest to positive infinity) `double` value that is not greater than the argument and is equal to a mathematical integer. static double IEEEremainder(double f1, double f2) Computes the remainder operation on two arguments as prescribed by the IEEE 754 standard. static double log(double a) Returns the natural logarithm (base e) of a `double` value. static double max(double a, double b) Returns the greater of two `double` values. static float max(float a, float b) Returns the greater of two `float` values. static int max(int a, int b) Returns the greater of two `int` values. static long max(long a, long b) Returns the greater of two `long` values. static double min(double a, double b) Returns the smaller of two `double` values. static float min(float a, float b) Returns the smaller of two `float` values. static int min(int a, int b) Returns the smaller of two `int` values. static long min(long a, long b) Returns the smaller of two `long` values. static double pow(double a, double b) Returns of value of the first argument raised to the power of the second argument. static double random() Returns a random number between `0.0` and `1.0`. static double rint(double a) returns the closest integer to the argument. static long round(double a) Returns the closest `long` to the argument. static int round(float a) Returns the closest `int` to the argument. static double sin(double a) Returns the trigonometric sine of an angle. static double sqrt(double a) Returns the square root of a `double` value. static double tan(double a) Returns the trigonometric tangent of an angle. static double toDegrees(double angrad) Converts an angle measured in radians to the equivalent angle measured in degrees. static double toRadians(double angdeg) Converts an angle measured in degrees to the equivalent angle measured in radians.\n\n Methods inherited from class java.lang.Object clone , equals , finalize , getClass , hashCode , notify , notifyAll , toString , wait , wait , wait\n\n Field Detail\n\n### E\n\n`public static final double E`\nThe `double` value that is closer than any other to `e`, the base of the natural logarithms.\n\n### PI\n\n`public static final double PI`\nThe `double` value that is closer than any other to pi, the ratio of the circumference of a circle to its diameter.\n Method Detail\n\n### sin\n\n`public static double sin(double a)`\nReturns the trigonometric sine of an angle.\nParameters:\n``` a``` - an angle, in radians.\nReturns:\nthe sine of the argument.\n\n### cos\n\n`public static double cos(double a)`\nReturns the trigonometric cosine of an angle.\nParameters:\n``` a``` - an angle, in radians.\nReturns:\nthe cosine of the argument.\n\n### tan\n\n`public static double tan(double a)`\nReturns the trigonometric tangent of an angle.\nParameters:\n``` a``` - an angle, in radians.\nReturns:\nthe tangent of the argument.\n\n### asin\n\n`public static double asin(double a)`\nReturns the arc sine of an angle, in the range of -pi/2 through pi/2.\nParameters:\n``` a``` - an angle, in radians.\nReturns:\nthe arc sine of the argument.\n\n### acos\n\n`public static double acos(double a)`\nReturns the arc cosine of an angle, in the range of 0.0 through pi.\nParameters:\n``` a``` - an angle, in radians.\nReturns:\nthe arc cosine of the argument.\n\n### atan\n\n`public static double atan(double a)`\nReturns the arc tangent of an angle, in the range of -pi/2 through pi/2.\nParameters:\n``` a``` - an angle, in radians.\nReturns:\nthe arc tangent of the argument.\n\n`public static double toRadians(double angdeg)`\nConverts an angle measured in degrees to the equivalent angle measured in radians.\nParameters:\n``` angdeg``` - an angle, in degrees\nReturns:\nthe measurement of the angle `angdeg` in radians.\nSince:\nJDK1.2\n\n### toDegrees\n\n`public static double toDegrees(double angrad)`\nConverts an angle measured in radians to the equivalent angle measured in degrees.\nParameters:\n``` angrad``` - an angle, in radians\nReturns:\nthe measurement of the angle `angrad` in degrees.\nSince:\nJDK1.2\n\n### exp\n\n`public static double exp(double a)`\nReturns the exponential number e (i.e., 2.718...) raised to the power of a `double` value.\nParameters:\n``` a``` - a `double` value.\nReturns:\nthe value ea, where e is the base of the natural logarithms.\n\n### log\n\n`public static double log(double a)`\nReturns the natural logarithm (base e) of a `double` value.\nParameters:\n``` a``` - a number greater than `0.0`.\nReturns:\nthe value ln `a`, the natural logarithm of `a`.\n\n### sqrt\n\n`public static double sqrt(double a)`\nReturns the square root of a `double` value.\nParameters:\n``` a``` - a `double` value.\nReturns:\nthe square root of `a`. If the argument is NaN or less than zero, the result is NaN.\n\n### IEEEremainder\n\n```public static double IEEEremainder(double f1,\ndouble f2)```\nComputes the remainder operation on two arguments as prescribed by the IEEE 754 standard. The remainder value is mathematically equal to `f1 - f2` × n, where n is the mathematical integer closest to the exact mathematical value of the quotient `f1/f2`, and if two mathematical integers are equally close to `f1/f2`, then n is the integer that is even. If the remainder is zero, its sign is the same as the sign of the first argument.\nParameters:\n``` f1``` - the dividend.\n``` f2``` - the divisor.\nReturns:\nthe remainder when `f1` is divided by `f2`.\n\n### ceil\n\n`public static double ceil(double a)`\nReturns the smallest (closest to negative infinity) `double` value that is not less than the argument and is equal to a mathematical integer.\nParameters:\n``` a``` - a `double` value.\nReturns:\nthe smallest (closest to negative infinity) `double` value that is not less than the argument and is equal to a mathematical integer.\n\n### floor\n\n`public static double floor(double a)`\nReturns the largest (closest to positive infinity) `double` value that is not greater than the argument and is equal to a mathematical integer.\nParameters:\n``` a``` - a `double` value.\n``` a``` - an assigned value.\nReturns:\nthe largest (closest to positive infinity) `double` value that is not greater than the argument and is equal to a mathematical integer.\n\n### rint\n\n`public static double rint(double a)`\nreturns the closest integer to the argument.\nParameters:\n``` a``` - a `double` value.\nReturns:\nthe closest `double` value to `a` that is equal to a mathematical integer. If two `double` values that are mathematical integers are equally close to the value of the argument, the result is the integer value that is even.\n\n### atan2\n\n```public static double atan2(double a,\ndouble b)```\nConverts rectangular coordinates (`b``a`) to polar (r, theta). This method computes the phase theta by computing an arc tangent of `a/b` in the range of -pi to pi.\nParameters:\n``` a``` - a `double` value.\n``` b``` - a `double` value.\nReturns:\nthe theta component of the point (rtheta) in polar coordinates that corresponds to the point (ba) in Cartesian coordinates.\n\n### pow\n\n```public static double pow(double a,\ndouble b)```\nReturns of value of the first argument raised to the power of the second argument.\n\nIf (`a == 0.0`), then `b` must be greater than `0.0`; otherwise an exception is thrown. An exception also will occur if (`a <= 0.0`) and `b` is not equal to a whole number.\n\nParameters:\n``` a``` - a `double` value.\n``` b``` - a `double` value.\nReturns:\nthe value `ab`.\nThrows:\nArithmeticException - if (`a == 0.0`) and (`b <= 0.0`), or if (`a <= 0.0`) and `b` is not equal to a whole number.\n\n### round\n\n`public static int round(float a)`\nReturns the closest `int` to the argument.\n\nIf the argument is negative infinity or any value less than or equal to the value of `Integer.MIN_VALUE`, the result is equal to the value of `Integer.MIN_VALUE`.\n\nIf the argument is positive infinity or any value greater than or equal to the value of `Integer.MAX_VALUE`, the result is equal to the value of `Integer.MAX_VALUE`.\n\nParameters:\n``` a``` - a `float` value.\nReturns:\nthe value of the argument rounded to the nearest `int` value.\n`Integer.MAX_VALUE`, `Integer.MIN_VALUE`\n\n### round\n\n`public static long round(double a)`\nReturns the closest `long` to the argument.\n\nIf the argument is negative infinity or any value less than or equal to the value of `Long.MIN_VALUE`, the result is equal to the value of `Long.MIN_VALUE`.\n\nIf the argument is positive infinity or any value greater than or equal to the value of `Long.MAX_VALUE`, the result is equal to the value of `Long.MAX_VALUE`.\n\nParameters:\n``` a``` - a `double` value.\nReturns:\nthe value of the argument rounded to the nearest `long` value.\n`Long.MAX_VALUE`, `Long.MIN_VALUE`\n\n### random\n\n`public static double random()`\nReturns a random number between `0.0` and `1.0`. Random number generators are often referred to as pseudorandom number generators because the numbers produced tend to repeat themselves after a period of time.\nReturns:\na pseudorandom `double` between `0.0` and `1.0`.\n`Random.nextDouble()`\n\n### abs\n\n`public static int abs(int a)`\nReturns the absolute value of an `int` value. If the argument is not negative, the argument is returned. If the argument is negative, the negation of the argument is returned.\n\nNote that if the argument is equal to the value of `Integer.MIN_VALUE`, the most negative representable `int` value, the result is that same value, which is negative.\n\nParameters:\n``` a``` - an `int` value.\nReturns:\nthe absolute value of the argument.\n`Integer.MIN_VALUE`\n\n### abs\n\n`public static long abs(long a)`\nReturns the absolute value of a `long` value. If the argument is not negative, the argument is returned. If the argument is negative, the negation of the argument is returned.\n\nNote that if the argument is equal to the value of `Long.MIN_VALUE`, the most negative representable `long` value, the result is that same value, which is negative.\n\nParameters:\n``` a``` - a `long` value.\nReturns:\nthe absolute value of the argument.\n`Long.MIN_VALUE`\n\n### abs\n\n`public static float abs(float a)`\nReturns the absolute value of a `float` value. If the argument is not negative, the argument is returned. If the argument is negative, the negation of the argument is returned.\nParameters:\n``` a``` - a `float` value.\nReturns:\nthe absolute value of the argument.\n\n### abs\n\n`public static double abs(double a)`\nReturns the absolute value of a `double` value. If the argument is not negative, the argument is returned. If the argument is negative, the negation of the argument is returned.\nParameters:\n``` a``` - a `double` value.\nReturns:\nthe absolute value of the argument.\n\n### max\n\n```public static int max(int a,\nint b)```\nReturns the greater of two `int` values.\nParameters:\n``` a``` - an `int` value.\n``` b``` - an `int` value.\nReturns:\nthe larger of `a` and `b`.\n\n### max\n\n```public static long max(long a,\nlong b)```\nReturns the greater of two `long` values.\nParameters:\n``` a``` - a `long` value.\n``` b``` - a `long` value.\nReturns:\nthe larger of `a` and `b`.\n\n### max\n\n```public static float max(float a,\nfloat b)```\nReturns the greater of two `float` values. If either value is `NaN`, then the result is `NaN`. Unlike the the numerical comparison operators, this method considers negative zero to be strictly smaller than positive zero.\nParameters:\n``` a``` - a `float` value.\n``` b``` - a `float` value.\nReturns:\nthe larger of `a` and `b`.\n\n### max\n\n```public static double max(double a,\ndouble b)```\nReturns the greater of two `double` values. If either value is `NaN`, then the result is `NaN`. Unlike the the numerical comparison operators, this method considers negative zero to be strictly smaller than positive zero.\nParameters:\n``` a``` - a `double` value.\n``` b``` - a `double` value.\nReturns:\nthe larger of `a` and `b`.\n\n### min\n\n```public static int min(int a,\nint b)```\nReturns the smaller of two `int` values.\nParameters:\n``` a``` - an `int` value.\n``` b``` - an `int` value.\nReturns:\nthe smaller of `a` and `b`.\n\n### min\n\n```public static long min(long a,\nlong b)```\nReturns the smaller of two `long` values.\nParameters:\n``` a``` - a `long` value.\n``` b``` - a `long` value.\nReturns:\nthe smaller of `a` and `b`.\n\n### min\n\n```public static float min(float a,\nfloat b)```\nReturns the smaller of two `float` values. If either value is `NaN`, then the result is `NaN`. Unlike the the numerical comparison operators, this method considers negative zero to be strictly smaller than positive zero.\nParameters:\n``` a``` - a `float` value.\n``` b``` - a `float` value.\nReturns:\nthe smaller of `a` and `b.`\n\n### min\n\n```public static double min(double a,\ndouble b)```\nReturns the smaller of two `double` values. If either value is `NaN`, then the result is `NaN`. Unlike the the numerical comparison operators, this method considers negative zero to be strictly smaller than positive zero.\nParameters:\n``` a``` - a `double` value.\n``` b``` - a `double` value.\nReturns:\nthe smaller of `a` and `b`.\n\nJava Platform 1.2\nBeta 4\n\nSubmit a bug or feature" ]	[ null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.6590068,"math_prob":0.9641219,"size":11337,"snap":"2022-27-2022-33","text_gpt3_token_len":2895,"char_repetition_ratio":0.23444808,"word_repetition_ratio":0.4334698,"special_character_ratio":0.23595308,"punctuation_ratio":0.13132635,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9995446,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2022-07-07T17:36:12Z\",\"WARC-Record-ID\":\"<urn:uuid:f20afc5c-6c77-4d9e-ae32-963299e003cd>\",\"Content-Length\":\"40567\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:81a43fd0-4956-49e0-86fb-b64aa5decfa5>\",\"WARC-Concurrent-To\":\"<urn:uuid:63e2db14-a536-4727-b9bb-65f91aef7d05>\",\"WARC-IP-Address\":\"128.208.1.193\",\"WARC-Target-URI\":\"https://courses.cs.washington.edu/courses/cse341/98au/java/jdk1.2beta4/docs/api/java/lang/Math.html\",\"WARC-Payload-Digest\":\"sha1:BF55T4ST6Z45FITXAWUMTFDRFQGVQYD4\",\"WARC-Block-Digest\":\"sha1:3PVVBHCUASWXQQLIPAIITAOCZPE7TKJV\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2022/CC-MAIN-2022-27/CC-MAIN-2022-27_segments_1656104495692.77_warc_CC-MAIN-20220707154329-20220707184329-00242.warc.gz\"}"}
https://mathematica.stackexchange.com/questions/196921/ndsolve-gives-unexpected-results-when-using-the-method-of-lines	[ "# NDSolve gives unexpected results when using the method of lines\n\nI've been trying to solve the following PDE using the NDSolve function but it seems something is not working properly. The PDE is a the heat equation on polar coordinates and assuming angular simmetry:\n\n$$u_t (t, r) = \\alpha(r) \\frac{1}{r} \\partial_r (r u_r (t, r))$$\n\nwith boundary conditions $$u_t(t, r_{in}) = p$$ and $$u(t, r_{max}) = 0$$, and initial condition $$u(0,r)=0$$ for $$r\\in(r_{in}, r_{max})$$. The function $$\\alpha$$ represents the diffusion coefficient of the temperature in the media.\n\nThe idea is that $$r_{max}$$ is large enough so that $$u(t,r)$$ is alsmost flat and equal to zero close to $$r_{max}$$ through all the time window integrated.\n\nI have tried different $$\\alpha(r)$$ functions, which are\n\nalpStep[r_] :=If[r < rout, aA, aB];\nalpLin[r_] :=If[r < rout, aA + (aB - aA)(r-rin)/(rout - rin), aB];\nalpA[r_] := aA;\nalpB[r_] := aB;\n\n\n\nwhere $$r_{out}\\in(r_{ín},r_{max})$$ is a critical radius beyond which the diffusion remains constant.\n\nThe unexpected result is the following. I set aA > aB, and I integrate the different PDEs (with the different alp functions) by means of NDSolve. It turns out that the solution associated to the alpStep function takes values much more higher than the other functions, and I would expect it to be somewhere in between the solutions associated to the functions alpA and alpB. May be the problem is due to the discontinuous diffusion, but I can't see why.\n\nThe code is the following:\n\nrin = 0.05;\nrout = 0.15;\nrmax = 5;\np = 0.01;\ntend = 2410;\n\naA = 0.01;\naB = 0.001;\nalpStep[r_] := If[r < rout, aA, aB];\nalpLin[r_] := If[r < rout, aA + (aB - aA) (r - rin)/(rout - rin), aB];\nalpA[r_] := aA;\nalpB[r_] := aB;\n\nopts = Method -> {\"MethodOfLines\",\n\"SpatialDiscretization\" -> {\"FiniteElement\",\n\"MeshOptions\" -> {\"MaxCellMeasure\" -> 0.001}}};\n\nWith[{u = u[t, r]}, eqn = alpStep[r] ((1/r) D[r D[u, r], r]) - D[u, t];\nrobinbc = NeumannValue[palpStep[rin], r == rin];\nbc = DirichletCondition[u == 0, r == rmax];\nic = u == 0 /. {t -> 0};]\nsolStepAB =\nNDSolveValue[{eqn == robinbc, bc, ic},\nu, {t, 0, tend}, {r, rin, rmax}, opts];\n\nWith[{u = u[t, r]}, eqn = alpLin[r] ((1/r) D[r D[u, r], r]) - D[u, t];\nrobinbc = NeumannValue[palpLin[rin], r == rin];\nbc = DirichletCondition[u == 0, r == rmax];\nic = u == 0 /. {t -> 0};]\nsolLinAB =\nNDSolveValue[{eqn == robinbc, bc, ic},\nu, {t, 0, tend}, {r, rin, rmax}, opts];\n\nWith[{u = u[t, r]}, eqn = alpA[r] ((1/r) D[r D[u, r], r]) - D[u, t];\nrobinbc = NeumannValue[palpA[rin], r == rin];\nbc = DirichletCondition[u == 0, r == rmax];\nic = u == 0 /. {t -> 0};]\nsolConA =\nNDSolveValue[{eqn == robinbc, bc, ic},\nu, {t, 0, tend}, {r, rin, rmax}, opts];\n\nWith[{u = u[t, r]}, eqn = alpB[r] ((1/r) D[r D[u, r], r]) - D[u, t];\nrobinbc = NeumannValue[p*alpB[rin], r == rin];\nbc = DirichletCondition[u == 0, r == rmax];\nic = u == 0 /. {t -> 0};]\nsolConB =\nNDSolveValue[{eqn == robinbc, bc, ic},\nu, {t, 0, tend}, {r, rin, rmax}, opts];\n\nGrid[{{\nPlot[{solStepAB[t, rin], solLinAB[t, rin], solConA[t, rin],\nsolConB[t, rin]}, {t, tend/1000, tend}],\nPlot[{solStepAB[t, rout], solLinAB[t, rout], solConA[t, rout],\nsolConB[t, rout]}, {t, tend/1000, tend}],\nPlot[{solStepAB[t, rout + (rout - rin)],\nsolLinAB[t, rout + (rout - rin)],\nsolConA[t, rout + (rout - rin)],\nsolConB[t, rout + (rout - rin)]}, {t, tend/1000, tend}]\n}}]\n\n\n\nThe picture with the temperature profiles for the different solutions alpStep (blue), alpLin (Orange), alpA (Green) and alpB (Red) at $$r = r_{in}, r_{out}, 2 r_{out} - r_{in}$$ from left to right is the following, where it can be seen that the profile associated to alpStep is much higher that the others:", null, "• I am not a 100% I understand your question but have a look at FEMDocumentation/tutorial/FiniteElementBestPractice#588198981 in the help system or online here Apr 25, 2019 at 5:52\n• Maybe similar to this Apr 25, 2019 at 6:23\n• Thanks for the reply, I'll take a look. Apr 25, 2019 at 13:44\n• Were you able to figure this out? Apr 29, 2019 at 5:16\n\nYes! The problem was that some information was missing in the problem I stated above. Specifically, due to the discontinuity of the function $$\\alpha$$ at $$r_{out}$$, the solution fails to be differentiable at that point, and multiple solutions exist. NDSolve gives one of these following some criterium related to fluxes (the heat flux at one side of the discontinuity has to be equal to the heat flux at the other side). Notice that the flux depends on the thermal conductivity $$k$$, which is not explicitly stated in the problem above. In general the thermal diffusivity $$\\alpha$$ is equal to the conductivity $$k$$ over the heat capacity $$c$$ (i.e. $$\\alpha = k / c$$). In order to use properly the NDSolve function in this case (in the sense that the continuity of the flux is preserved at $$r_{out}$$) one should avoid working with diffusion coefficient and use $$k$$ and $$c$$ instead, that is one should write\nWith[{u = u[t, r]}, eqn = k[r] ((1/r) D[r D[u, r], r]) - c[r] D[u, t];" ]	[ null, "https://i.stack.imgur.com/Uz67U.jpg", null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.7413816,"math_prob":0.9995548,"size":3671,"snap":"2022-27-2022-33","text_gpt3_token_len":1317,"char_repetition_ratio":0.102536134,"word_repetition_ratio":0.27154472,"special_character_ratio":0.38082266,"punctuation_ratio":0.2326139,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9999639,"pos_list":[0,1,2],"im_url_duplicate_count":[null,5,null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2022-08-18T04:28:53Z\",\"WARC-Record-ID\":\"<urn:uuid:bbd52c5e-8103-48ec-9f9c-ba4d0dbb0b1a>\",\"Content-Length\":\"231899\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:f29117a7-badb-4114-aaca-8da2b297a54a>\",\"WARC-Concurrent-To\":\"<urn:uuid:15a8fa2d-f045-4f49-877a-3535a03d91a1>\",\"WARC-IP-Address\":\"151.101.65.69\",\"WARC-Target-URI\":\"https://mathematica.stackexchange.com/questions/196921/ndsolve-gives-unexpected-results-when-using-the-method-of-lines\",\"WARC-Payload-Digest\":\"sha1:6X32I4N5SW3FJP6KINQM2TV3YCBWKQE2\",\"WARC-Block-Digest\":\"sha1:66X5LQWKJWDOQDAQG7XJUF7ND36SSBG5\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2022/CC-MAIN-2022-33/CC-MAIN-2022-33_segments_1659882573163.7_warc_CC-MAIN-20220818033705-20220818063705-00265.warc.gz\"}"}
https://www.sanfoundry.com/electronic-devices-circuits-questions-answers-hall-effect/	[ "# Electronic Devices and Circuits Questions and Answers – The Hall Effect\n\nThis set of Electronic Devices and Circuits Multiple Choice Questions & Answers (MCQs) focuses on “The Hall Effect”.\n\n1. In the Hall Effect, the directions of electric field and magnetic field are parallel to each other.\nThe above statement is\na) True\nb) False\n\nExplanation: To make Lorentz force into the effect, the electric field and magnetic field should be perpendicular to each other.\n\n2. Which of the following parameters can’t be found with Hall Effect?\na) Polarity\nb) Conductivity\nc) Carrier concentration\nd) Area of the device\n\nExplanation: The Hall Effect is used for finding the whether the semiconductor is of n-type or p-type, mobility, conductivity and the carrier concentration.\n\n3. In the Hall Effect, the electric field is in x direction and the velocity is in y direction. What is the direction of the magnetic field?\na) X\nb) Y\nc) Z\nd) XY plane\n\nExplanation: The Hall Effect satisfies the Lorentz’s Force which is\nE=vxB\nSo, the direction of the velocity, electric field and magnetic field should be perpendicular to each other.\n\n4. What is the velocity when the electric field is 5V/m and the magnetic field is 5A/m?\na) 1m/s\nb) 25m/s\nc) 0.2m/s\nd) 0.125m/s\n\nExplanation: E=vxB\nv=E/B\n=5/5\n=1m/s.\n\n5. Calculate the hall voltage when the Electric Field is 5V/m and height of the semiconductor is 2cm.\na) 10V\nb) 1V\nc) 0.1V\nd) 0.01V\n\nExplanation: Vh=Ed\n=52/100\n=0.1V.\n\n6. Which of the following formulae doesn’t account for correct expression for J?\na) ρv\nb) I/wd\nc) σE\nd) µH\n\nExplanation: B=µH\nSo, option µH is correct option.\n\n7. Calculate the Hall voltage when B=5A/m, I=2A, w=5cm and n=1020.\na) 3.125V\nb) 0.3125V\nc) 0.02V\nd) 0.002V\n\nExplanation: Vh=BI/wρ\n=5=2/ (510-21051.610-19)\n=0.002V.\n\n8. Calculate the Hall Effect coefficient when number of electrons in a semiconductor is 1020.\na) 0.625\nb) 0.0625\nc) 6.25\nd) 62.5\n\nExplanation: R=1/ρ\n=1/(1.610-191020)\n=0.0625.\n\n9. What is the conductivity when the Hall Effect coefficient is 5 and mobility is 5cm2 /s.\na) 100 S/m\nb) 10 S/m\nc) 0.0001S/m\nd) 0.01 S/m\n\nExplanation: µ=σR\nσ =µ/R\n=5*10-4/5\n=0.0001 S/m.\n\n10. In Hall Effect, the electric field applied is perpendicular to both current and magnetic field?\na) True\nb) False\n\nExplanation: In Hall Effect, the electric field is perpendicular to both current and magnetic field so that the force due to magnetic field can be balanced by the electric field or vice versa.\n\nSanfoundry Global Education & Learning Series – Electronic Devices and Circuits.\n\nTo practice all areas of Electronic Devices and Circuits, here is complete set of 1000+ Multiple Choice Questions and Answers.", null, "" ]	[ null, "data:image/svg+xml,%3Csvg%20xmlns=%22http://www.w3.org/2000/svg%22%20viewBox=%220%200%20150%20150%22%3E%3C/svg%3E", null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.6991632,"math_prob":0.9766574,"size":2697,"snap":"2023-40-2023-50","text_gpt3_token_len":837,"char_repetition_ratio":0.16747122,"word_repetition_ratio":0.08888889,"special_character_ratio":0.29477197,"punctuation_ratio":0.12790698,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9960417,"pos_list":[0,1,2],"im_url_duplicate_count":[null,null,null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2023-12-10T05:15:52Z\",\"WARC-Record-ID\":\"<urn:uuid:3ac8bbb3-d59e-445d-b5fe-add20dee813d>\",\"Content-Length\":\"176865\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:f15968c6-514c-4d9b-b739-fc64cf83e69d>\",\"WARC-Concurrent-To\":\"<urn:uuid:bf8ac72b-b3ba-41b9-a38a-5cb5da33d5b8>\",\"WARC-IP-Address\":\"172.67.82.182\",\"WARC-Target-URI\":\"https://www.sanfoundry.com/electronic-devices-circuits-questions-answers-hall-effect/\",\"WARC-Payload-Digest\":\"sha1:IMPHTC6OBOIHNT367D3WBRO6C7BNELBC\",\"WARC-Block-Digest\":\"sha1:RX234L3CV2NYRPTE2DAAZDQ4KQ66D47A\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2023/CC-MAIN-2023-50/CC-MAIN-2023-50_segments_1700679101195.85_warc_CC-MAIN-20231210025335-20231210055335-00316.warc.gz\"}"}
https://community.tableau.com/thread/222974	[ "10 Replies Latest reply on Dec 22, 2016 10:36 AM by Shinichiro Murakami\n\nVariances between standard and current month in a dynamic report\n\nHi All,\n\nPlease help!! I am sure it supposed to be simple...but it is taking me ages to try and figure this out!\n\nI am trying to create a report whereby I show the cost per piece over the last 12 months - my report is dynamic so the user can change the current month and thus the preceding 11 months before.\n\nThe problem I am encountering is that I am trying to show a variance column based on the \"current month\" and the Dec of the previous year.\n\nI have taken the following steps:\n\n1) I have created the formula where [Date Control] is my parameter to select the month.\n\nCurrent Month Cost Per Unit: IF ([Date]) = [Date Control] THEN [Cost Per Unit] ELSE 0 END\n\nLast Month Last Year (LM LY) Cost Per Unit: IF ([Date]) = [LM LY] THEN [Cost Per Unit] ELSE 0 END\n\n* Variance Calc*: [LM LY Cost Per Unit] - [Current Month Cost Per Unit]\n\n2) I have added the variance calc to my rows as a discrete value - The problem here is that rather than each part sitting on one line, when I add this calculation, it sits on three lines - one line has the \"current month\" cost, one the Dec'15 cost and the other has the remaining months. I then get a variance for both the Dec'15 row and the current month row - summed up, the variance would be correct but the current way it is displayed is not helpful:", null, "(This is what it looked like before I added the variance calc):", null, "3) I thought I had a solution - to duplicate the report and create a formula firstly to only show the previous dec and the current month. This would then enable me to create the below formula:\n\nSUM([Cost Per Unit]) - LOOKUP(ZN(SUM([Cost Per Unit])), -1)\n\nHowever, although the calculation is then correct (i.e. it actually subtracts the values between Dec15 and Nov16, it still shows on two lines and shows a \"Null\":", null, "The report I am trying to produce is very long, with many part costs, and so the idea of a second worksheet may prove difficult when lining it up in the Dashboard...I have already noticed that the order doesn't always stay the same etc, and I also wouldn't know how to hide the Dec month in the second worksheet (as it will already be one of the months showing in the first worksheet).\n\nSo my queries are as follows:\n\n1) How would you go about trying to present this long report with variance column for current month versus previous Dec (I will need to present the months in chronological order, so the previous Dec will never be in the same position compared to the current month)?\n\n2) How can I display just one row for each part so that comparison is easy and the variance calculates properly?\n\n2) How can I get the variance column to sit as the last column in the report?\n\n3) How can I apply a different colour only to the variance column?\n\nI'd be so grateful for any help you can offer, this is starting to give me a headache!\n\nNicola\n\n• 1. Re: Variances between standard and current month in a dynamic report\n\nHi Nicola,\n\nThank you for describing the issue very in detail, but to be honest, I hesitate to read thru your question without seeing your data.\n\nThat's really in vain, in case of without data.\n\nPackaged workbooks and flows: when, why, how\n\nThanks,\n\nShin\n\n• 2. Re: Variances between standard and current month in a dynamic report\n\nHi Shin,\n\nI have tried to replicate what I am doing with the dummy Superstore data...but unfortunately there seems to be something odd with my variance formula...it is only working for some of the Product IDs - the rest it just returns zero. Perhaps you could show me how you might go about creating a variance column between the previous Dec and the current month and I will see if I can replicate it in my own data - as you say, this might be an easier way than answering my query above!\n\nThanks a lot,\n\nNicola\n\n• 3. Re: Variances between standard and current month in a dynamic report\n\nNicala,\n\nTo be honest, to meet all of you request is quite troublesome, in other words, very difficult for beginners.\n\nBut anyways, I put some solutions here.\n\nHere are two versions, with variance different(Version 2) color and Not(version 1)\n\nversion 1 is easier and less steps, but still you need time to digest it.", null, "[Month Filter]\n\nif attr(datetrunc('month',[Order Date]))=attr(datetrunc('month',[Control Month]))\n\nor\n\nthen \"show\" else \"hide\" end\n\n[Variance Current - Dec] <== as Discrete\n\nint(window_sum(ZN(sum(if datetrunc('month',[Order Date])=datetrunc('month',[Control Month])\n\nthen [Sales] end)))\n\n-\n\nthen [Sales] end))))", null, "", null, "version 2 : Detail concept is here, it's requires around 100 steps by itself\n\nSuch a simple view, it required 100 steps!! - Having Multiple KPIs - - Still Struggling with Excel ?? <Tableau's Room>", null, "[Last December Sales]\n\nthen [Sales] end)))\n\n[This month Sales]\n\nint(window_sum(ZN(sum(if datetrunc('month',[Order Date])=datetrunc('month',[Control Month])\n\nthen [Sales] end))))", null, "Thanks,\n\nShin\n\n• 4. Re: Variances between standard and current month in a dynamic report\n\nHi Shin,\n\nThanks a lot for your reply. I will work through it today and see if I can make it work...may I ask though - if I want to display all the other months (previous 11 months + current month)...will it still work?\n\nThanks\n\nNicola\n\n• 5. Re: Variances between standard and current month in a dynamic report\n\nIn version two, you need add column X 2 fields. I personally persuade better give up coloring variance.\n\nYou might use dashboard instead. Though, using dashboard has different constraints and need to see detail.\n\nThe final goal is to balance between \"workload and durability\" and \"interactive function and visibility\".\n\nThanks,\n\nShin\n\n• 6. Re: Variances between standard and current month in a dynamic report\n\nHi Shin,\n\nThanks for taking the time to explain all this.\n\nOne of the issues with the standard version is that the Variance column still appears first - ideally I should like to see this last - after the current month. I think I can get the order to change for the colour one - so I will try to replicate this, I have worked with something previously you suggested using this approach so hopefully I can get my head around it.\n\nCan you help me understand though what the \"INT\" and \"Window_Sum\" are doing in your formula? The rest of the formula makes sense but I can't quite understand this part.\n\nint(window_sum(ZN(sum(if datetrunc('month',[Order Date])=datetrunc('month',[Control Month])\n\nthen [Sales] end)))\n\n-\n\nthen [Sales] end))))\n\nI actually was playing around yesterday and came up with the attached (with only 3 months as an example), however, the problem here is that because I have to create a calculated field for all the 12 months and because it is dynamic...I can't name the columns with the month name - is there anyway I can change this so it picks up the correct month name? Without the month names, I can't share this with my end users.\n\nIn fact...I think I will encounter the same issue with regards to the months names in the colour version?\n\nThanks\n\nNicola\n\n• 7. Re: Variances between standard and current month in a dynamic report\n\nI'm not sure I remember correctly, but maybe ,window_sum is used to calculate total regardless of dimensions. Standard sum's case, that calculation becomes null because of dimensions settings.\n\nInt is changing decimal to whole number. To use numbers as dimensions, decimal brings trouble.\n\nAnd you are right, bringing Month Name is another challenge which I don't have immediate answer right now.\n\nThanks,\n\nShin\n\n• 8. Re: Variances between standard and current month in a dynamic report\n\nThere are \"1\" value difference , maybe 'round' is better than 'Int', BTW.\n\nThanks,\n\nShin\n\n• 9. Re: Variances between standard and current month in a dynamic report\n\nGive me one more try.\n\nKind of crazy approach but might be providing some hints for other case.", null, "Using standard method\n\n[Variance Current - Dec (continuous)2] <== Only shows variance on \"Control Month\"\n\nif attr(datetrunc('month',[Order Date]))= datetrunc('month',[Control Month]) then\n\nint(window_sum(ZN(sum(if datetrunc('month',[Order Date])=datetrunc('month',[Control Month])\n\nthen [Sales] end)))\n\n-\n\nthen [Sales] end))))\n\nend\n\n[Variance Current - Dec (continuous)2 Negative] <== Only shows values if negative with \"\|\" text separator\n\nif [Variance Current - Dec (continuous)2]<0 then \"\| \"+str([Variance Current - Dec (continuous)2]) end\n\n[Variance Current - Dec (continuous)2 positive] <== Only shows values if positive with \"\|\" text separator\n\nif [Variance Current - Dec (continuous)2]>=0 then \"\| \"+str([Variance Current - Dec (continuous)2]) end", null, "That's it.\n\nThanks,\n\nShin\n\n• 10. Re: Variances between standard and current month in a dynamic report\n\nAnd one more option to show the variance to the right end.\n\nOnly available when the latest month > parameter control month", null, "if DATEDIFF('month', [Order Date], [Control Month]) = -1\n\nthen \"Variance\" else datename('month', [Order Date]) end\n\n[Relative Date +1]\n\nIF\n\nDATEDIFF('month', [Order Date], #2014-12-01#) = 0 OR\n\nDATEDIFF('month', [Order Date], [Control Month]) <11 and\n\nDATEDIFF('month', [Order Date], [Control Month]) >= -1\n\nTHEN \"Show\" ELSE \"Hide\" END\n\n[Variance Current - Dec (continuous)2]\n\nint(window_sum(ZN(sum(if datetrunc('month',[Order Date])=datetrunc('month',[Control Month])\n\nthen [Sales] end)))\n\n-\n\nthen [Sales] end))))\n\nend\n\n[Sales SM]\n\nif attr([Header Date])<>\"Variance\" then sum([Sales]) end\n\n[Variance Current - Dec (continuous)2 Negative]\n\nif attr([Header Date]) =\"Variance\" and [Variance Current - Dec (continuous)2]<0 then [Variance Current - Dec (continuous)2] end\n\n[Variance Current - Dec (continuous)2 positive]\n\nif attr([Header Date])=\"Variance\" and [Variance Current - Dec (continuous)2]>=0 then [Variance Current - Dec (continuous)2] end\n\nThanks,\n\nShin" ]	[ null, "https://community.tableau.com/servlet/JiveServlet/downloadImage/2-561585-212238/pastedImage_3.png", null, "https://community.tableau.com/servlet/JiveServlet/downloadImage/2-561585-212239/pastedImage_7.png", null, "https://community.tableau.com/servlet/JiveServlet/downloadImage/2-561585-212207/pastedImage_0.png", null, "https://community.tableau.com/servlet/JiveServlet/downloadImage/2-562406-212849/pastedImage_0.png", null, "https://community.tableau.com/servlet/JiveServlet/downloadImage/2-562406-212850/435-450/pastedImage_3.png", null, "https://community.tableau.com/servlet/JiveServlet/downloadImage/2-562406-212851/499-283/pastedImage_4.png", null, "https://community.tableau.com/servlet/JiveServlet/downloadImage/2-562406-212852/495-289/pastedImage_8.png", null, "https://community.tableau.com/servlet/JiveServlet/downloadImage/2-562406-212854/522-270/pastedImage_11.png", null, "https://community.tableau.com/servlet/JiveServlet/downloadImage/2-563216-213293/581-261/pastedImage_0.png", null, "https://community.tableau.com/servlet/JiveServlet/downloadImage/2-563216-213294/490-195/pastedImage_5.png", null, "https://community.tableau.com/servlet/JiveServlet/downloadImage/2-563277-213738/527-225/pastedImage_0.png", null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.93138546,"math_prob":0.8738502,"size":2897,"snap":"2019-26-2019-30","text_gpt3_token_len":668,"char_repetition_ratio":0.13930176,"word_repetition_ratio":0.018348623,"special_character_ratio":0.24266483,"punctuation_ratio":0.07757167,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9835484,"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\":\"2019-06-25T07:02:08Z\",\"WARC-Record-ID\":\"<urn:uuid:8ee4fac8-063c-4679-a0a3-c26207ae0824>\",\"Content-Length\":\"154317\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:fd0a563e-8121-4794-b31e-f3577c0af319>\",\"WARC-Concurrent-To\":\"<urn:uuid:bac6040a-665b-4e57-867f-37b1ded9dcec>\",\"WARC-IP-Address\":\"204.93.79.205\",\"WARC-Target-URI\":\"https://community.tableau.com/thread/222974\",\"WARC-Payload-Digest\":\"sha1:HTKFRJI3OTR4AYBJRIBLYXYLCIN2YI5C\",\"WARC-Block-Digest\":\"sha1:27FLG4X3AGDB6OXHXGVX4A7K7XIIDEK3\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2019/CC-MAIN-2019-26/CC-MAIN-2019-26_segments_1560627999800.5_warc_CC-MAIN-20190625051950-20190625073950-00458.warc.gz\"}"}
https://www.datasciencemadesimple.com/len-function-python-string-length-python/	[ "# len() function in python – Get string length in python\n\nlen() function in python returns the length of the string. lets see an example on how to get length of a string in python with the help of length function – len().\n\nSyntax for len() function in python – string length function :\n\nlen( str )\n\n#### Example of len() function in python for strings :\n\nLength function in python –len() returns the length of the string in python, lets see with an example\n\n```str = \"Beauty of democracy!!\";\nprint \"Length of the string is : \", len(str)\n```\n\nThe output will be\n\nLength of the string is: 21\n\n#### Example of len() Function for list in python – number of elements in the list:\n\nLength function – len() also returns the number of elements in the list. Lets see with an example\n\n```List1=[456,\"Example\",\"decode\",\"435hyud\"]\nList2=[\"Democracy\",\"Development\"]\n\nprint \"Length of the first List : \", len(List1)\nprint \"Length of the second List : \", len(List2)\n```\n\nThe output will be\n\nLength of the first List : 4\nLength of the second List : 2\n\nlength of the string in columns of dataframe in python can be referred here" ]	[ null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.68612134,"math_prob":0.9499348,"size":1045,"snap":"2020-45-2020-50","text_gpt3_token_len":248,"char_repetition_ratio":0.20557156,"word_repetition_ratio":0.113513514,"special_character_ratio":0.26698565,"punctuation_ratio":0.11707317,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.98734266,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2020-11-27T10:57:02Z\",\"WARC-Record-ID\":\"<urn:uuid:bbc44081-c1c6-4c05-bf4b-02d9676760d6>\",\"Content-Length\":\"52348\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:1e64ef47-75bc-4184-bf81-c5b764bc5674>\",\"WARC-Concurrent-To\":\"<urn:uuid:621d03ac-5e83-4edf-99fb-8d45e02f4561>\",\"WARC-IP-Address\":\"104.27.148.144\",\"WARC-Target-URI\":\"https://www.datasciencemadesimple.com/len-function-python-string-length-python/\",\"WARC-Payload-Digest\":\"sha1:MX6T3T2YCFKFXXX7O7IFGFHFHO2A2GHO\",\"WARC-Block-Digest\":\"sha1:OTUIRE6C4TLN66ROD55CNRSSXVEMDYHI\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2020/CC-MAIN-2020-50/CC-MAIN-2020-50_segments_1606141191692.20_warc_CC-MAIN-20201127103102-20201127133102-00210.warc.gz\"}"}
https://www.w3resource.com/python-exercises/dictionary/python-data-type-dictionary-exercise-37.php	[ " Python: Replace dictionary values with their average - w3resource\n\n# Python Exercise: Replace dictionary values with their average\n\n## Python dictionary: Exercise-37 with Solution\n\nWrite a Python program to replace dictionary values with their average.\n\nSample Solution:-\n\nPython Code:\n\n``````def sum_math_v_vi_average(list_of_dicts):\nfor d in list_of_dicts:\nn1 = d.pop('V')\nn2 = d.pop('VI')\nd['V+VI'] = (n1 + n2)/2\nreturn list_of_dicts\nstudent_details= [\n{'id' : 1, 'subject' : 'math', 'V' : 70, 'VI' : 82},\n{'id' : 2, 'subject' : 'math', 'V' : 73, 'VI' : 74},\n{'id' : 3, 'subject' : 'math', 'V' : 75, 'VI' : 86}\n]\nprint(sum_math_v_vi_average(student_details))\n```\n```\n\nSample Output:\n\n```[{'subject': 'math', 'id': 1, 'V+VI': 76.0}, {'subject': 'math', 'id': 2, 'V+VI': 73.5}, {'subject': 'math', '\nid': 3, 'V+VI': 80.5}]\n```\n\n## Visualize Python code execution:\n\nThe following tool visualize what the computer is doing step-by-step as it executes the said program:\n\nPython Code Editor:\n\nHave another way to solve this solution? Contribute your code (and comments) through Disqus.\n\nWhat is the difficulty level of this exercise?\n\n\n\nInviting useful, relevant, well-written and unique guest posts" ]	[ null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.56576353,"math_prob":0.4776399,"size":1668,"snap":"2019-51-2020-05","text_gpt3_token_len":450,"char_repetition_ratio":0.1298077,"word_repetition_ratio":0.023715414,"special_character_ratio":0.30035973,"punctuation_ratio":0.21782178,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.99602705,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2019-12-16T10:53:55Z\",\"WARC-Record-ID\":\"<urn:uuid:46936dea-38c7-4cfc-99d1-379ca630f495>\",\"Content-Length\":\"111501\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:924e6f1a-ada6-4768-a35c-aef749be1d05>\",\"WARC-Concurrent-To\":\"<urn:uuid:30887a90-73b4-4ee3-9c97-b94fcdf57c56>\",\"WARC-IP-Address\":\"104.26.14.93\",\"WARC-Target-URI\":\"https://www.w3resource.com/python-exercises/dictionary/python-data-type-dictionary-exercise-37.php\",\"WARC-Payload-Digest\":\"sha1:ZEXDAZJGYIP72DO7A7AG2NWZCWU7AHPK\",\"WARC-Block-Digest\":\"sha1:4S3QVGPZJO32YYXWXQJZT7X227LUE5JT\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2019/CC-MAIN-2019-51/CC-MAIN-2019-51_segments_1575541319511.97_warc_CC-MAIN-20191216093448-20191216121448-00190.warc.gz\"}"}
https://forums.swift.org/t/checking-if-dependentmembertype-conformance-requirement-is-satisfied-by-substitutionmap/25053	[ "# Checking if `DependentMemberType` conformance requirement is satisfied by SubstitutionMap\n\nHi compiler experts,\n\nI'm wondering how to check (in a SIL transform) whether a conformance requirement with a `DependentMemberType` lhs type is satisfied either in a given `SubstitutionMap` or in a given module.\n\nContext is the following program. 1 shows the array of `Requirement`s and 2 shows the substitution map. The \"given module\" is the current module.\n\n``````// In stdlib:\n// protocol Differentiable {\n// associatedtype TangentVector: Differentiable & AdditiveArithmetic\n// where ...\n// }\n\nprotocol MyProtocol {\nassociatedtype Scalar\n}\nextension MyProtocol where Scalar : FloatingPoint {\n// 1. `@differentiable` attribute declares an array of `Requirement`s in\n// a trailing where clause.\n//\n// Relevant to the question: the `Self.TangentVector : MyProtocol`\n// requirement is a conformance requirement, where the lhs\n// `Self.TangentVector` is a `DependentMemberType` type:\n//\n// (dependent_member_type assoc_type=Swift.(file).Differentiable.TangentVector\n// (base=generic_type_param_type depth=0 index=0 decl=main.(file).MyProtocol extension.Self))\n@differentiable(\nwhere Self : Differentiable, Scalar : Differentiable,\nSelf.TangentVector : MyProtocol\n)\nstatic func +(lhs: Self, rhs: Self) -> Self { ... }\n}\n\nstruct Dummy<Scalar> : MyProtocol {}\nextension Dummy : Differentiable where Scalar : Differentiable {\ntypealias TangentVector = Dummy\n}\n\nlet fn = { (x: Dummy<Float>) -> Dummy<Float> in\n// 2. Here, I need to check that the requirements declared in the\n// `@differentiable` attribute above are met.\n//\n// I have access to the substitution map of the `apply` instruction\n// corresponding to the `+` application:\n//\n// (substitution_map generic_signature=<τ_0_0 where τ_0_0 : MyProtocol, τ_0_0.Scalar : FloatingPoint>\n// (substitution τ_0_0 -> Dummy<Float>)\n// (conformance type=τ_0_0\n// (specialized_conformance type=Dummy<Float> protocol=MyProtocol\n// (substitution_map generic_signature=<τ_0_0>\n// (substitution τ_0_0 -> Float))\n// (conditional requirements unable to be computed)\n// (normal_conformance type=Dummy<Scalar> protocol=MyProtocol\n// (assoc_type req=Scalar type=Scalar))))\n// (conformance type=τ_0_0.Scalar\n// (normal_conformance type=Float protocol=FloatingPoint lazy)))\nreturn x + x\n}\n``````\n\nReposting the key information: how can I check whether the `Self.TangentVector : MyProtocol` conformance requirement, whose LHS is a `DependentMemberType`:\n\n``````(dependent_member_type assoc_type=Swift.(file).Differentiable.TangentVector\n(base=generic_type_param_type depth=0 index=0 decl=main.(file).MyProtocol extension.Self))\n``````\n\nIs satisfied given the current module and the following substitution map?\n\n``````(substitution_map generic_signature=<τ_0_0 where τ_0_0 : MyProtocol, τ_0_0.Scalar : FloatingPoint>\n(substitution τ_0_0 -> Dummy<Float>)\n(conformance type=τ_0_0\n(specialized_conformance type=Dummy<Float> protocol=MyProtocol\n(substitution_map generic_signature=<τ_0_0>\n(substitution τ_0_0 -> Float))\n(conditional requirements unable to be computed)\n(normal_conformance type=Dummy<Scalar> protocol=MyProtocol\n(assoc_type req=Scalar type=Scalar))))\n(conformance type=τ_0_0.Scalar\n(normal_conformance type=Float protocol=FloatingPoint lazy)))\n``````\n\nIntuitively, the requirement is satisfied: `Dummy<Float>` conforms to `Differentiable` and `Dummy<Float>.TangentVector` is equal to `Dummy<Float>` and also conforms to `Differentiable`.\n\nHowever, the conformance of `Dummy<Float>` to `Differentiable` is missing from the substitution map, and it's not clear how to remap `τ_0_0` from the substitution map to `Self` in the conformance requirement in a robust way.\n\nHere's the current hacky logic for checking this requirement: it calls `DependentMemberType:: substBaseType` on the first replacement type in the substitution map, if the substitution map has only one replacement type (acting as `Self`). It's not robust, and the entire `checkRequirementsSatisfied` function can likely be improved.\n\nAny advice would be greatly appreciated!\n\ncc team: @rxwei @bartchr808\ncc @Slava_Pestov\n\nWhy can't you just call SubstitutionMap::lookupConformance()?\n\nActually, I think what you want here is to do `module->lookupConformance(subMap.subst(myType), myProto)`. This should work with any Type, not just DependentMemberType.\n\nI don't believe the conformance of `τ_0_0` to `Differentiable` actually exists in the substitution map.\n\nThanks! Let me give this a try.\n\nEdit: actually, I believe `module->lookupConformance(subMap.subst(myType), myProto)` is already attempted, but it doesn't work for some reason:\n\n`````` if (auto origFirstType = firstType.subst(substMap)) {\nif (!origFirstType->hasError() &&\nswiftModule->lookupConformance(origFirstType, protocol)) {\ncontinue; // continue loop before diagnosing unmet requirement\n}\n}\n``````\n\nLet me see why it doesn't work.\n\nOh I see. That's weird and it suggests that the substitution map was built from the wrong generic signature. If the original generic signature had a requirement τ_0_0 : Differentiable, it would work.\n\nHowever, if that is not an option I think then you need something like:\n\n``````auto substType = type.subst(QuerySubstitutionMap(subMap), LookUpConformanceInModule(swiftModule))\n``````\n\nThen we will find the conformance in the module in order to resolve the correct type witness for τ_0_0.TangentVector (since it can't be found in the substitution map).\n\n1 Like\n\nFWIW, subst() will return the null type Type() on failure, not a type containing error types, so your code above will segfault in this case. If you want to return the original type with failed substitutions replaced by ErrorTypes, call `subst()` with SubstitutionFlags::UseErrorTypes.\n\n1 Like\n\nThis seems promising, thanks! Let me try it and report back." ]	[ null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.57185876,"math_prob":0.4092393,"size":3889,"snap":"2023-40-2023-50","text_gpt3_token_len":951,"char_repetition_ratio":0.17142858,"word_repetition_ratio":0.034042552,"special_character_ratio":0.23553613,"punctuation_ratio":0.14801444,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9594277,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2023-11-29T22:54:48Z\",\"WARC-Record-ID\":\"<urn:uuid:aa15c532-389b-4519-bc1a-fcdac423bd5c>\",\"Content-Length\":\"41784\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:bebf6b7b-1881-47aa-9007-8bb8251ee6ab>\",\"WARC-Concurrent-To\":\"<urn:uuid:2ee714a8-27dc-49f3-8aa0-4dfd0cd893f0>\",\"WARC-IP-Address\":\"184.105.99.75\",\"WARC-Target-URI\":\"https://forums.swift.org/t/checking-if-dependentmembertype-conformance-requirement-is-satisfied-by-substitutionmap/25053\",\"WARC-Payload-Digest\":\"sha1:XCQZQRM3RKVSI37E5YQNXRRSTCG6WGLT\",\"WARC-Block-Digest\":\"sha1:GETVNIYLMQJY2FA26HQEJAG7Y4SWRGYX\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2023/CC-MAIN-2023-50/CC-MAIN-2023-50_segments_1700679100146.5_warc_CC-MAIN-20231129204528-20231129234528-00177.warc.gz\"}"}
https://unacademy.com/lesson/linear-equations-with-two-variables/HZWDWLEC	[ "to enroll in courses, follow best educators, interact with the community and track your progress.\nEnroll\n204\nLinear Equations with Two Variables\n799 plays\n\nMore\nIn this lesson we will discuss, What are linear equations with two variables, a couple of techniques to solve equations with two variables and some sample problems.\n\nU\nGP\nalso one suggesation if possible please add extra questions of your choise after explaining a perticular chapter may be some previous year question and explain it. it whould surely clear understanding basic of a perticular chapter.\nGP\nvery simple to understand. thank you so much. i appriciate your effort. please continue. please add more leson. if possible please cover all topics of bank and csat-2 paper qa in one series lesson if possible. explanation is outstanding.\nGP\n1. Linear Equations with 2 variables\n\n2. About Me Shubham Agrawal IIT Roorkee 2013 graduate Left job to pursue my passion- Education\n\n3. Linear Equations with 2 variables -> 0 Usually of the form ax + by + c Represents a line in x y plane . .\n\n4. Linear Equations with 2 variables -> . Usually of the form ax + by + c 0 Represents a line in x y plane .For every value of x, there's a corresponding value of y\n\n5. Linear Equations with 2 variables -> If there are 2 equations of the form . 1st Method 2. 2 abb 2 2.\n\n6. Linear Equations with 2 variables -> Q.\n\n7. Linear Equations with 2 variables -> Q. f- 2\n\n8. Linear Equations with 2 variables -> Q. f- TL YL\n\n9. Linear Equations with 2 variables -> If there are 2 equations of the form . a2x + by+20 a2 x 2nd Method Subtract or add one equation with the other to eliminate either x or y\n\n10. Linear Equations with 2 variables -> Q. 41+24-620 tD\n\n11. Linear Equations with 2 variables -> Represents intersection of the two lines in x y plane .\n\n12. Applications -> Q. The sum of the digits of a two digit number is 16. If the number formed by reversing the digits is less than the original number by 18. Find the original number.\n\n13. Applications -> Q. The ratio of the ages of the father and the son at present is 3: 1. Four years earliar, the ratio was 4: 1. What are the present ages of the son and the father?\n\n14. Applications -> Q. The ratio of the ages of the father and the son at present is 3: 1. Four years earliar, the ratio was 4: 1. What are the present ages of the son and the father?\n\n15. Applications -> Q. The ratio of the ages of the father and the son at present is 3: 1. Four years earliar, the ratio was 4: 1. What are the present ages of the son and the father? F- n e)\n\n16. Applications -> Q. The ratio of the ages of the father and the son at present is 3: 1. Four years earliar, the ratio was 4: 1. What are the present ages of the son and the father? YL F- n e)\n\n17. 6 12L2 12" ]	[ null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.8893104,"math_prob":0.9954032,"size":2763,"snap":"2019-51-2020-05","text_gpt3_token_len":772,"char_repetition_ratio":0.16853933,"word_repetition_ratio":0.41541353,"special_character_ratio":0.26275787,"punctuation_ratio":0.10745234,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9974576,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2019-12-11T22:24:00Z\",\"WARC-Record-ID\":\"<urn:uuid:1a841899-eb75-4107-907c-3390b8f345be>\",\"Content-Length\":\"823976\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:50201bf4-8ae9-4591-857f-f55f9fa66ab2>\",\"WARC-Concurrent-To\":\"<urn:uuid:766471e9-1a6c-4e09-9fcd-d310c69d833b>\",\"WARC-IP-Address\":\"52.74.39.149\",\"WARC-Target-URI\":\"https://unacademy.com/lesson/linear-equations-with-two-variables/HZWDWLEC\",\"WARC-Payload-Digest\":\"sha1:7YY2JDQOAO2RVDI3QSHPHEHB4ZYUQZAL\",\"WARC-Block-Digest\":\"sha1:T3PTBNJDZYPMZVGKZ56QPRL4B23RALLA\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2019/CC-MAIN-2019-51/CC-MAIN-2019-51_segments_1575540533401.22_warc_CC-MAIN-20191211212657-20191212000657-00082.warc.gz\"}"}
http://randomjohn.info/math-vocabulary-worksheets/collection-of-free-grade-math-vocabulary-worksheets-ready-to-download-or-print-please-do-multiplication-7-worksheet-library-and-fifth-words-num/	[ "# Collection Of Free Grade Math Vocabulary Worksheets Ready To Download Or Print Please Do Multiplication 7 Worksheet Library And Fifth Words Num", null, "collection of free grade math vocabulary worksheets ready to download or print please do multiplication 7 worksheet library and fifth words num.\n\nmath vocabulary worksheets lesson plans for grade related files 1st words to numbers word problems kindergarten,math key words worksheets algebra worksheet grade best solutions vocabulary definition esl,math vocabulary list with definitions kindergarten worksheets words maths for free collection of third grade 1,basic mathematics vocabulary poetry match worksheet free worksheets grade math second key words kindergarten,4th grade math vocabulary worksheets pdf maths positional words fourth,math positional words worksheets kindergarten vocabulary 6th grade 1st 5,maths vocabulary revision sorting worksheet activity sheet back to esl math worksheets basic for students,6th grade math vocabulary worksheets pdf flashcards for free printable first word problems preschoolers 2nd,math vocabulary worksheets pdf esl grade word searches social studies search kindergarten words,math word problems worksheets vocabulary pdf definition worksheet printable for students free activity sheets." ]	[ null, "http://randomjohn.info/wp-content/uploads/2019/05/collection-of-free-grade-math-vocabulary-worksheets-ready-to-download-or-print-please-do-multiplication-7-worksheet-library-and-fifth-words-num.jpg", null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.7342276,"math_prob":0.40612668,"size":1132,"snap":"2019-13-2019-22","text_gpt3_token_len":184,"char_repetition_ratio":0.25,"word_repetition_ratio":0.013888889,"special_character_ratio":0.1475265,"punctuation_ratio":0.065476194,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.95492256,"pos_list":[0,1,2],"im_url_duplicate_count":[null,1,null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2019-05-21T19:19:51Z\",\"WARC-Record-ID\":\"<urn:uuid:6c5ec287-06dc-4a20-8bfd-ea1de778fb30>\",\"Content-Length\":\"55539\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:42a0813b-e89a-4521-b9f2-ba35ff03ea60>\",\"WARC-Concurrent-To\":\"<urn:uuid:564c89b0-4d36-4834-9c79-a095308d33db>\",\"WARC-IP-Address\":\"104.27.139.32\",\"WARC-Target-URI\":\"http://randomjohn.info/math-vocabulary-worksheets/collection-of-free-grade-math-vocabulary-worksheets-ready-to-download-or-print-please-do-multiplication-7-worksheet-library-and-fifth-words-num/\",\"WARC-Payload-Digest\":\"sha1:YGZ7WHC7L7LBYFG2OB6NRI3SNYPGHRIK\",\"WARC-Block-Digest\":\"sha1:PFSJUH3VAXC7S4GTXFJXVU7FT4QVCYDF\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2019/CC-MAIN-2019-22/CC-MAIN-2019-22_segments_1558232256546.11_warc_CC-MAIN-20190521182616-20190521204616-00551.warc.gz\"}"}
https://handbook.fca.org.uk/handbook?related-provisions-for-provision=BIPRU%207.2.24&date=2016-10-03	[ "# Related provisions for BIPRU 7.2.24\n\n1 - 2 of 2 items.\n\n#### Similar To\n\nTo access the FCA Handbook Archive choose a date between 1 January 2001 and 31 December 2004 (From field only).\n\nBIPRU 7.2.21RRP\nInterest rate swaps or foreign currency swaps without deferred starts must be treated as the two notional positions (one long, one short) shown in the table in BIPRU 7.2.22R.\nBIPRU 7.2.22RRP\n\nTable: Interest rate and foreign currency swaps\n\nThis table belongs to BIPRU 7.2.21R\n\n Paying leg (which must be treated as a short position in a zero-specific-risk security) Receiving leg (which must be treated as a long position in a zero-specific-risk security) Receiving fixed and paying floating Coupon equals the floating rate and maturity equals the reset date Coupon equals the fixed rate of the swap and maturity equals the maturity of the swap Paying fixed and receiving floating Coupon equals the fixed rate of the swap and maturity equals the maturity of the swap Coupon equals the floating rate and maturity equals the reset date Paying floating and receiving floating Coupon equals the floating rate and maturity equals the reset date Coupon equals the floating rate and maturity equals the reset date\nBIPRU 7.2.25RRP\n\nTable: Deferred start interest rate and foreign currency swaps\n\nThis table belongs to BIPRU 7.2.24R\n\n Paying leg (which must be treated as a short position in a zero-specific-risk security with a coupon equal to the fixed rate of the swap) Receiving leg (which must be treated as a long position in a zero-specific-risk security with a coupon equal to the fixed rate of the swap) Receiving fixed and paying floating maturity equals the start date of the swap maturity equals the maturity of the swap Paying fixed and receiving floating maturity equals the maturity of the swap maturity equals the start date of the swap\nBIPRU 7.2.46AGRP\n3BIPRU 7.2.43 R includes both actual and notional positions. However, notional positions in a zero-specific-risk security do not attract specific risk. For example:(1) interest-rate swaps, foreign-currency swaps, FRAs, interest-rate futures, foreign-currencyforwards, foreign-currencyfutures, and the cash leg of repurchase agreements and reverse repurchase agreements create notional positions which will not attract specific risk; while(2) futures, forwards and swaps which are based\nIFPRU 6.2.10GRP\nInterest-rate swaps or foreign currency swaps with a deferred start should be treated as the two notional positions (one long, one short). The paying leg should be treated as a short position in a zero specific risk security with a coupon equal to the fixed rate of the swap. The receiving leg should be treated as a long position in a zero specific risk security, which also has a coupon equal to the fixed rate of the swap." ]	[ null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.8459303,"math_prob":0.9806719,"size":1503,"snap":"2020-24-2020-29","text_gpt3_token_len":335,"char_repetition_ratio":0.19146097,"word_repetition_ratio":0.81322956,"special_character_ratio":0.20958084,"punctuation_ratio":0.030303031,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9597255,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2020-06-01T09:12:54Z\",\"WARC-Record-ID\":\"<urn:uuid:92f8c2e2-2547-4893-a13f-a69656ee3ecd>\",\"Content-Length\":\"38861\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:0f7ef97f-ae10-4fef-ae75-6e465a48fc7b>\",\"WARC-Concurrent-To\":\"<urn:uuid:240a6f6e-cdce-4656-95ce-9e0469c4c9f0>\",\"WARC-IP-Address\":\"35.178.221.231\",\"WARC-Target-URI\":\"https://handbook.fca.org.uk/handbook?related-provisions-for-provision=BIPRU%207.2.24&date=2016-10-03\",\"WARC-Payload-Digest\":\"sha1:IT3IRDCJT6MLKH63PBBK3KLC2ENCAAQ2\",\"WARC-Block-Digest\":\"sha1:O7RR5QGAWGOE3IC2L2AXWSXE67HVNVSX\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2020/CC-MAIN-2020-24/CC-MAIN-2020-24_segments_1590347415315.43_warc_CC-MAIN-20200601071242-20200601101242-00565.warc.gz\"}"}
https://ronaldgrey.com/2011/12/09/fisca-stimulus-to-guarantee-results/	[ "# Fiscal Stimulus to Guarantee Results", null, "Figure 1. Pareto plot shows Obama’s 2009 stimulus spending was wasteful.\n\nMuch of the wasteful spending by Barack Obama – such as his failed stimulus of 2009 – could be prevented through the use of a simple statistical tool\n\n### The Economy’s Performance Gap\n\nBarack Obama’s handling of the economy — America’s most important problem — has been disappointing.\n\n### Obama Defined by Maldistribution Curve\n\nTo determine the degree to which the maldistribution of Obama’s stimulus spending is associated with the Pareto distribution with probability density function (PDF)", null, "$P(x) = \\displaystyle\\frac{ab^a}{x^{a+1}}$ for", null, "$x \\geq b$\n\nnormalized values for the Pareto PDF were plotted as a function of normalized values for spending. Linear regression by method of least squares fitting revealed a near-perfect correlation between the values, with the fitted line’s slope (R) equal to 0.9962 (Figure 2).", null, "" ]	[ null, "https://ronaldgrey.files.wordpress.com/2011/12/pareto.png", null, "https://s0.wp.com/latex.php", null, "https://s0.wp.com/latex.php", null, "https://ronaldgrey.files.wordpress.com/2011/12/correlation4.png", null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.95009804,"math_prob":0.96052116,"size":3475,"snap":"2023-14-2023-23","text_gpt3_token_len":700,"char_repetition_ratio":0.12762892,"word_repetition_ratio":0.02247191,"special_character_ratio":0.20115107,"punctuation_ratio":0.08169935,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9609814,"pos_list":[0,1,2,3,4,5,6,7,8],"im_url_duplicate_count":[null,5,null,null,null,null,null,5,null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2023-03-25T11:40:23Z\",\"WARC-Record-ID\":\"<urn:uuid:5a8114fa-ff7c-4755-a730-42a10d32c845>\",\"Content-Length\":\"92776\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:1df66081-97df-492c-8e8b-9b034990cafc>\",\"WARC-Concurrent-To\":\"<urn:uuid:e681fa56-650a-46c0-b545-6d026c4fe26e>\",\"WARC-IP-Address\":\"192.0.78.25\",\"WARC-Target-URI\":\"https://ronaldgrey.com/2011/12/09/fisca-stimulus-to-guarantee-results/\",\"WARC-Payload-Digest\":\"sha1:APFXVARJZKH2M3QUUEYDUF6H7LWZRRVK\",\"WARC-Block-Digest\":\"sha1:ULU6X6ZHIOSDZE5GATCVE2CK7HTLWKTW\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2023/CC-MAIN-2023-14/CC-MAIN-2023-14_segments_1679296945323.37_warc_CC-MAIN-20230325095252-20230325125252-00064.warc.gz\"}"}
https://plus.maths.org/content/comment/reply/2308/8116	[ "### Engineer\n\nThink of a cube. This has 6 faces, 12 edges and 8 vertices, so E-V=4\nNow take one of the edges, and add a midpoint vertex. This divides the edge into two, so you also end up adding an edge ( and two of the faces now have 5 edges instead of 4, but that is irrelevant) So now you have E=13,V=9 and still E-V=4\n\nIt is clear that this can be repeated as many times as you want. So yes, there are really an unlimited number of possibilities!" ]	[ null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.95641077,"math_prob":0.84917116,"size":880,"snap":"2020-45-2020-50","text_gpt3_token_len":239,"char_repetition_ratio":0.10958904,"word_repetition_ratio":0.9714286,"special_character_ratio":0.27045456,"punctuation_ratio":0.103286386,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9863828,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2020-11-26T10:26:52Z\",\"WARC-Record-ID\":\"<urn:uuid:59d979e9-4604-4388-8c77-1c207cfddcb5>\",\"Content-Length\":\"24622\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:7ebbbcd8-4424-4374-9efd-4a723af051b4>\",\"WARC-Concurrent-To\":\"<urn:uuid:ea470bca-0ffd-401a-b4bf-1117ab683571>\",\"WARC-IP-Address\":\"131.111.24.106\",\"WARC-Target-URI\":\"https://plus.maths.org/content/comment/reply/2308/8116\",\"WARC-Payload-Digest\":\"sha1:XVVSSJXBVCKYQSBQFA2JBC25UXCRPUEW\",\"WARC-Block-Digest\":\"sha1:UAN7QOKFCEHLUM7LCXEMNWXZ23SJZE5O\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2020/CC-MAIN-2020-50/CC-MAIN-2020-50_segments_1606141187753.32_warc_CC-MAIN-20201126084625-20201126114625-00247.warc.gz\"}"}
https://www.iktsoft.net/keiji-en/articles/4694/	[ "« Top\n\n## 3D Color Analysis\n\nWednesday, July 04, 2001\nA visualization of the RGB color space provides interesting insight about the color distribution of the picture.\nObjective\nThe purposes of this program is to grasp the pattern of the colors used in the image or to see the color variation which define the image.\nIn order to do so, this program convert each pixel into 3D space according to its RGB value.", null, "To achieve it, this program has such functions as follows:\n\n• Load .jpg, .png, .bmp file.\n• Show total image in left screen.\n• Plot each pixel in right screen.\nThose pixels are plotted in 3D space using various projections.\n\nScreen snapshot of Color Spatioplotter", null, "iOS version is now available.\n\nThe Method of projection of color in 3D space\nThere are various method to plot color value into 3D space.\nThe projection methods are as follows:\n\nRGB(x=R, y=G, z=B)", null, "RGB (rotated)", null, "", null, "a. Line (0,0,0)-(1,1,1) of Fig.1 is converted as Z-axis.\nb. Emphasized to XY plane.\n\nHSV,polar coordinate (x=Scos(H), y=S*sin(H), z=V)", null, "HSV(x=H, y=S, z=V)", null, "CMY(K)(x=C, y=M, z=Y)\nThe value 'K' is not shown in the space.", null, "CMYK (rotated)", null, "", null, "a. Line (0,0,0)-(1,1,1) of Fig.5 is converted as Z-axis.\nb. Emphasized to XY space.\nc. Multiplying z value by 'K' value." ]	[ null, "https://www.iktsoft.net/ws/f/162/454.png", null, "https://www.iktsoft.net/ws/f/162/455.jpg", null, "https://www.iktsoft.net/ws/f/162/456.jpg", null, "https://www.iktsoft.net/ws/f/162/457.jpg", null, "https://www.iktsoft.net/ws/f/162/458.jpg", null, "https://www.iktsoft.net/ws/f/162/459.jpg", null, "https://www.iktsoft.net/ws/f/162/460.jpg", null, "https://www.iktsoft.net/ws/f/162/461.jpg", null, "https://www.iktsoft.net/ws/f/162/462.jpg", null, "https://www.iktsoft.net/ws/f/162/463.jpg", null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.83926743,"math_prob":0.975527,"size":1197,"snap":"2021-31-2021-39","text_gpt3_token_len":335,"char_repetition_ratio":0.10813076,"word_repetition_ratio":0.050251257,"special_character_ratio":0.26900584,"punctuation_ratio":0.1696113,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9925128,"pos_list":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20],"im_url_duplicate_count":[null,4,null,4,null,4,null,4,null,4,null,4,null,4,null,4,null,4,null,4,null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2021-07-28T16:40:50Z\",\"WARC-Record-ID\":\"<urn:uuid:320ec747-dc51-43ed-bfd5-6de1841a89bc>\",\"Content-Length\":\"20854\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:81ee3031-1e7c-4859-9754-0317322f2370>\",\"WARC-Concurrent-To\":\"<urn:uuid:a5969f13-f430-4555-aac1-b2d1195a1be2>\",\"WARC-IP-Address\":\"96.31.34.118\",\"WARC-Target-URI\":\"https://www.iktsoft.net/keiji-en/articles/4694/\",\"WARC-Payload-Digest\":\"sha1:NBDU74BY47X4YI7C6YI5TXCO52LRJDIB\",\"WARC-Block-Digest\":\"sha1:F7VYGZ4FYK3HPBHT5IKGZSJB62SKDCXX\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2021/CC-MAIN-2021-31/CC-MAIN-2021-31_segments_1627046153739.28_warc_CC-MAIN-20210728154442-20210728184442-00108.warc.gz\"}"}
https://computergraphics.stackexchange.com/questions/4703/algorithm-to-map-a-triangular-mesh-onto-plane-preserving-angles-and-distances-fr	[ "# Algorithm to map a triangular mesh onto plane preserving angles and distances from one vertex\n\nI have a 3D triangular mesh representing cortical surface of a brain. I also have one vertex of interest. I would like to unfold a neighborhood of this vertex in such a way that both the angles (their ratios) and distances of paths from this vertex are preserved as much as possible.\n\nI read some reviews on mesh parameterization techniques (eg. this one by Scheffer et al.). My understanding is that all the algorithms try to preserve a certain property (angle, distance, area) or a combination of those properties for all of the vertices or faces. In my case I need to preserve angles and distance of paths going from only one point. My intuition tells me that what I need should be quite simple because I demand much less from the output than all those parameterization algorithms. Yet I cannot come up with the solution. Any help is greatly appreciated.\n\n• Are you looking to preserve the ratio between angles, to keep things in proportion, or to preserve the exact angle values? In general the angles cannot be preserved as angles around a point do not necessarily add up to 360 degrees in a non-planar surface. Feb 13 '17 at 16:42\n• The ratio. I will edit the post. Thank you for pointing this out. Feb 13 '17 at 17:13\n• @trichoplax Angles around a point do add up to 360° as long as the surface is smooth (a small patch around the point looks like a plane). You might be thinking of the angles inside a polygon? For instance, the interior angles of a triangle don't add up to 180° on a curved surface in general. Feb 13 '17 at 22:20\n• Please see Nathan's comment which explains my misunderstanding (thanks Nathan - that is indeed what I was confusing it with) Feb 14 '17 at 14:45\n• @NathanReed I'm doubting myself now, but I think my error was in terminology (talking about surface instead of mesh), and that the intended point stands. Although the smooth surface that is being approximated by a triangle mesh is locally planar, the mesh itself, as a polyhedron, does not have angles around each vertex that sum to 360°. So an angle preserving projection can be made from a surface to a plane, but the angles are dependent on which surface was chosen to fit to the vertices of the mesh prior to projection (unless the original surface is known). Is this relevant here? Feb 14 '17 at 16:09" ]	[ null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.95283973,"math_prob":0.70015484,"size":856,"snap":"2021-43-2021-49","text_gpt3_token_len":173,"char_repetition_ratio":0.09741784,"word_repetition_ratio":0.0,"special_character_ratio":0.19859813,"punctuation_ratio":0.07926829,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.98075795,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2021-10-27T04:12:37Z\",\"WARC-Record-ID\":\"<urn:uuid:c98a876d-da4a-4e89-90e0-8c0361cc2fef>\",\"Content-Length\":\"170076\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:89886c6e-9b09-49dc-a751-3f9939e85e24>\",\"WARC-Concurrent-To\":\"<urn:uuid:30f895fc-f7bd-4706-842e-d560375687fa>\",\"WARC-IP-Address\":\"151.101.1.69\",\"WARC-Target-URI\":\"https://computergraphics.stackexchange.com/questions/4703/algorithm-to-map-a-triangular-mesh-onto-plane-preserving-angles-and-distances-fr\",\"WARC-Payload-Digest\":\"sha1:PVZ3LI57WERLQ6TB6GDOL4XSOQ3BJ37J\",\"WARC-Block-Digest\":\"sha1:4U47SXCDIV5MAUILUHQZW6CX7QLCG4XF\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2021/CC-MAIN-2021-43/CC-MAIN-2021-43_segments_1634323588053.38_warc_CC-MAIN-20211027022823-20211027052823-00193.warc.gz\"}"}
https://www.esaral.com/q/an-aeroplane-with-its-wings-spread-10-m-55019	[ "# An aeroplane, with its wings spread 10 m,\n\nQuestion:\n\nAn aeroplane, with its wings spread $10 \\mathrm{~m}$, is flying at a speed of $180 \\mathrm{~km} / \\mathrm{h}$ in a horizontal direction. The total intensity of earth's field at that part is $2.5 \\times 10^{-4} \\mathrm{~Wb} / \\mathrm{m}^{2}$ and the angle of dip is $60^{\\circ}$. The emf induced between the tips of the plane wings will be\n\n1. (1) $88.37 \\mathrm{mV}$\n\n2. (2) $62.50 \\mathrm{mV}$\n\n3. (3) $54.125 \\mathrm{mV}$\n\n4. (4) $108.25 \\mathrm{mV}$\n\nCorrect Option: , 4\n\nSolution:", null, "$\\sum=B \\perp v \\xi$\n\n$\\sin 60^{\\circ}-\\frac{B_{n}}{u}$", null, "$\\frac{\\sqrt{3}}{2}=\\frac{B_{3}}{B}$\n\n$B V=\\frac{\\sqrt{3}}{2} B$\n\n$E=\\frac{\\sqrt{3}}{2} B \\ell v$\n\n$=\\frac{\\sqrt{3}}{2} \\times 2.5 \\times 10^{-4} \\times 10 \\times 180 \\times \\frac{5}{18}$\n\n$=\\frac{\\sqrt{3}}{2} \\times 2.5 \\times 5 \\times 10^{-2}=10.825 \\times 10^{-2}=108.25 \\mathrm{mV}$" ]	[ null, "https://www.esaral.com/media/uploads/2022/01/12/image91040.png", null, "https://www.esaral.com/media/uploads/2022/01/12/image14311.png", null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.5334352,"math_prob":1.0000069,"size":716,"snap":"2023-14-2023-23","text_gpt3_token_len":290,"char_repetition_ratio":0.2008427,"word_repetition_ratio":0.0,"special_character_ratio":0.4567039,"punctuation_ratio":0.083333336,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9999931,"pos_list":[0,1,2,3,4],"im_url_duplicate_count":[null,1,null,1,null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2023-04-01T13:09:44Z\",\"WARC-Record-ID\":\"<urn:uuid:84ff0449-dc2e-4032-b5c6-12317f7711be>\",\"Content-Length\":\"24678\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:f4f8da50-656a-4bcd-b284-acd169edf212>\",\"WARC-Concurrent-To\":\"<urn:uuid:4ba08a57-21fe-44ae-a68b-f135007f1118>\",\"WARC-IP-Address\":\"104.21.61.187\",\"WARC-Target-URI\":\"https://www.esaral.com/q/an-aeroplane-with-its-wings-spread-10-m-55019\",\"WARC-Payload-Digest\":\"sha1:Q6GYGSKR3PSQ7O7PDJSYBEL2XYFWYOZJ\",\"WARC-Block-Digest\":\"sha1:YN4TJIRV55KFQGCV36V72QLEB5FY26GV\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2023/CC-MAIN-2023-14/CC-MAIN-2023-14_segments_1679296950030.57_warc_CC-MAIN-20230401125552-20230401155552-00540.warc.gz\"}"}
https://chem.libretexts.org/Bookshelves/Organic_Chemistry/Organic_Chemistry_(McMurry)/02%3A_Polar_Covalent_Bonds_Acids_and_Bases/2.04%3A_Resonance	[ "# 2.4: Resonance\n\n$$\\newcommand{\\vecs}{\\overset { \\rightharpoonup} {\\mathbf{#1}} }$$ $$\\newcommand{\\vecd}{\\overset{-\\!-\\!\\rightharpoonup}{\\vphantom{a}\\smash {#1}}}$$$$\\newcommand{\\id}{\\mathrm{id}}$$ $$\\newcommand{\\Span}{\\mathrm{span}}$$ $$\\newcommand{\\kernel}{\\mathrm{null}\\,}$$ $$\\newcommand{\\range}{\\mathrm{range}\\,}$$ $$\\newcommand{\\RealPart}{\\mathrm{Re}}$$ $$\\newcommand{\\ImaginaryPart}{\\mathrm{Im}}$$ $$\\newcommand{\\Argument}{\\mathrm{Arg}}$$ $$\\newcommand{\\norm}{\\\| #1 \\\|}$$ $$\\newcommand{\\inner}{\\langle #1, #2 \\rangle}$$ $$\\newcommand{\\Span}{\\mathrm{span}}$$ $$\\newcommand{\\id}{\\mathrm{id}}$$ $$\\newcommand{\\Span}{\\mathrm{span}}$$ $$\\newcommand{\\kernel}{\\mathrm{null}\\,}$$ $$\\newcommand{\\range}{\\mathrm{range}\\,}$$ $$\\newcommand{\\RealPart}{\\mathrm{Re}}$$ $$\\newcommand{\\ImaginaryPart}{\\mathrm{Im}}$$ $$\\newcommand{\\Argument}{\\mathrm{Arg}}$$ $$\\newcommand{\\norm}{\\\| #1 \\\|}$$ $$\\newcommand{\\inner}{\\langle #1, #2 \\rangle}$$ $$\\newcommand{\\Span}{\\mathrm{span}}$$\n\n##### Objective\n\nAfter completing this section, you should be able to\n\n• draw resonance forms for molecules and ions.\n##### Key Terms\n\nMake certain that you can define, and use in context, the key term below.\n\n• resonance form\n• delocalization\n\n## Resonance Delocalization\n\nSometimes, even when formal charges are considered, the bonding in some molecules or ions cannot be described by a single Lewis structure. Resonance is a way of describing delocalized electrons within certain molecules or polyatomic ions where the bonding cannot be expressed by a single Lewis formula. A molecule or ion with such delocalized electrons is represented by several contributing structures (also called resonance contributors or canonical forms). Resonance contributors involve the ‘imaginary movement’ of pi-bonded electrons or of lone-pair electrons that are adjacent to pi bonds. Note, sigma bonds cannot be broken during resonance – if you show a sigma bond forming or breaking, you are showing a chemical reaction taking place. Likewise, the positions of atoms in the molecule cannot change between resonance contributors.\n\nWhen looking at the structure of the molecule, formate, we see that there are two equivalent structures possible. Which one is correct? There are two simple answers to this question: 'both' and 'neither one'. Both ways of drawing the molecule are equally acceptable approximations of the bonding picture for the molecule, but neither one, by itself, is an accurate picture of the delocalized pi bonds. The two alternative drawings, however, when considered together, give a much more accurate picture than either one on its own. This is because they imply, together, that the carbon-carbon bonds are not double bonds, not single bonds, but about halfway in between.", null, "Formate Ion Structures are Equivalent in Energy\n\nWhen it is possible to draw more than one valid structure for a compound or ion, we have identified resonance contributors: two or more different Lewis structures depicting the same molecule or ion that, when considered together, do a better job of approximating delocalized pi-bonding than any single structure. By convention, resonance contributors are linked by a double-headed arrow, and are sometimes enclosed by brackets:", null, "The depiction of formate using the two resonance contributors A and B in the figure above does not imply that the molecule at one moment looks like structure A, then at the next moment shifts to look like structure B. Rather, at all moments, the molecule is a combination, or resonance hybrid of both A and B. Each individual resonance contributor of the formate ion is drawn with one carbon-oxygen double bond (120 pm) and one carbon-oxygen single bond (135 pm), with a negative formal charge located on the single-bonded oxygen. However, the two carbon-oxygen bonds in formate are actually the same length (127 pm) which implies that neither resonance contributor is correct. Although there is an overall negative formal charge on the formate ion, it is shared equally between the two oxygens. Therefore, the formate ion can be more accurately depicted by a pair of resonance contributors. Alternatively, a single structure can be used, with a dashed line depicting the resonance-delocalized pi bond and the negative charge located in between the two oxygens.", null, "The electrostatic potential map of formate shows that there is an equal amount of electron density (shown in red) around each oxygen.", null, "Valence bond theory can be used to develop a picture of the bonding in a carboxylate group. We know that the carbon must be sp2-hybridized, (the bond angles are close to 120˚, and the molecule is planar), and we will treat both oxygens as being sp2-hybridized as well. Both carbon-oxygen sigma bonds, then, are formed from the overlap of carbon sp2 orbitals and oxygen sp2 orbitals.", null, "In addition, the carbon and both oxygens each have an un-hybridized 2pz orbital situated perpendicular to the plane of the sigma bonds. These three 2pz orbitals are parallel to each other, and can overlap in a side-by-side fashion to form a delocalized pi bond.", null, "", null, "Overall, the situation is one of three parallel, overlapping 2pz orbitals sharing four delocalized pi electrons. Because there is one more electron than there are 2pz orbitals, the system has an overall charge of 1–. Resonance contributors are used to approximate overlapping 2pz orbitals and delocalized pi electrons. Molecules with resonance are usually drawn showing only one resonance contributor for the sake of simplicity. However, identifying molecules with resonance is an important skill in organic chemistry.\n\nThis example shows an important exception to the general rules for determining the hybridization of an atom. The oxygen with the negative charge appears to be sp3 hybridized because it is surrounded by four electron groups. However, this representation of the oxygen atom is not correct because it is actually part of a resonance hybrid. A pair of lone pair of electrons on the negatively charged oxygen are not localized in an sp3 orbital, rather, they are delocalized as part of a conjugated pi system. The stability gained though resonance is enough to cause the expected sp3 to become sp2. The sp2 hybridization gives the oxygen a p orbital allowing it to participate in conjugation. As a general rule sp3 hybridized atoms with lone pair electrons tend to become sp2 hybridized when adjacent to a conjugated system.", null, "### Example: Carbonate (CO32−)\n\nLike formate, the electronic structure of the carbonate ion cannot be described by a single Lewis electron structure. Unlike O3, though, the Lewis structures describing CO32 has three equivalent representations.", null, "1. Because carbon is the least electronegative element, we place it in the central position:", null, "2. Carbon has 4 valence electrons, each oxygen has 6 valence electrons, and there are 2 more for the 2− charge. This gives 4 + (3 × 6) + 2 = 24 valence electrons.\n\n3. Six electrons are used to form three bonding pairs between the oxygen atoms and the carbon:", null, "4. We divide the remaining 18 electrons equally among the three oxygen atoms by placing three lone pairs on each and indicating the 2− charge:", null, "5. There are no electrons left for the central atom.", null, "6. At this point, the carbon atom has only 6 valence electrons, so we must take one lone pair from an oxygen and use it to form a carbon–oxygen double bond. In this case, however, there are three possible choices:", null, "As with formate, none of these structures describes the bonding exactly. Each predicts one carbon–oxygen double bond and two carbon–oxygen single bonds, but experimentally all C–O bond lengths are identical. We can write resonance structures (in this case, three of them) for the carbonate ion:", null, "As the case for formate, the actual structure involves the formation of a molecular orbital from pz orbitals centered on each atom and sitting above and below the plane of the CO32 ion.", null, "##### Example 2.4.1\n\nBenzene is a common organic solvent that was previously used in gasoline; it is no longer used for this purpose, however, because it is now known to be a carcinogen. The benzene molecule (C6H6) consists of a regular hexagon of carbon atoms, each of which is also bonded to a hydrogen atom. Use resonance structures to describe the bonding in benzene.\n\nGiven: molecular formula and molecular geometry\n\nStrategy:\n\nA Draw a structure for benzene illustrating the bonded atoms. Then calculate the number of valence electrons used in this drawing.\n\nB Subtract this number from the total number of valence electrons in benzene and then locate the remaining electrons such that each atom in the structure reaches an octet.\n\nC Draw the resonance structures for benzene.\n\nSolution:\n\nA Each hydrogen atom contributes 1 valence electron, and each carbon atom contributes 4 valence electrons, for a total of (6 × 1) + (6 × 4) = 30 valence electrons. If we place a single bonding electron pair between each pair of carbon atoms and between each carbon and a hydrogen atom, we obtain the following:", null, "Each carbon atom in this structure has only 6 electrons and has a formal charge of 1+, but we have used only 24 of the 30 valence electrons.\n\nB If the 6 remaining electrons are uniformly distributed pair-wise on alternate carbon atoms, we obtain the following:", null, "Three carbon atoms now have an octet configuration and a formal charge of 1−, while three carbon atoms have only 6 electrons and a formal charge of 1+. We can convert each lone pair to a bonding electron pair, which gives each atom an octet of electrons and a formal charge of 0, by making three C=C double bonds.\n\nC There are, however, two ways to do this:", null, "Each structure has alternating double and single bonds, but experimentation shows that each carbon–carbon bond in benzene is identical, with bond lengths (139.9 pm) intermediate between those typically found for a C–C single bond (154 pm) and a C=C double bond (134 pm). We can describe the bonding in benzene using the two resonance structures, but the actual electronic structure is an average of the two. The existence of multiple resonance structures for aromatic hydrocarbons like benzene is often indicated by drawing either a circle or dashed lines inside the hexagon:\n\nThis combination of p orbitals for benzene can be visualized as a ring with a node in the plane of the carbon atoms. As can be seen in an electrostatic potential map of benzene, the electrons are distributed symmetrically around the ring.", null, "##### Exercise $$\\PageIndex{1}$$\n\nThe sodium salt of nitrite is used to relieve muscle spasms. Draw two resonance structures for the nitrite ion (NO2).", null, "## Exercises\n\n#### Questions\n\nQ2.4.1\n\nDraw the resonance structures for the following molecule:", null, "#### Solutions\n\nS2.4.1", null, "### Extra example - O3\n\nA molecule or ion with such delocalized electrons is represented by several contributing structures (also called resonance structures or canonical forms). Such is the case for ozone (O3), an allotrope of oxygen with a V-shaped structure and an O–O–O angle of 117.5°.\n\n1. We know that ozone has a V-shaped structure, so one O atom is central:", null, "", null, "2. Each O atom has 6 valence electrons, for a total of 18 valence electrons.", null, "3. Assigning one bonding pair of electrons to each oxygen–oxygen bond gives", null, "with 14 electrons left over.", null, "4. If we place three lone pairs of electrons on each terminal oxygen, we obtain", null, "and have 2 electrons left over.", null, "5. At this point, both terminal oxygen atoms have octets of electrons. We therefore place the last 2 electrons on the central atom:", null, "6. The central oxygen has only 6 electrons. We must convert one lone pair on a terminal oxygen atom to a bonding pair of electrons—but which one? Depending on which one we choose, we obtain either", null, "Which is correct? In fact, neither is correct. Both predict one O–O single bond and one O=O double bond. If the bonds were of different types (one single and one double, for example), they would have different lengths. It turns out, however, that both O–O bond distances are identical, 127.2 pm, which is shorter than a typical O–O single bond (148 pm) and longer than the O=O double bond in O2 (120.7 pm).", null, "Equivalent Lewis dot structures, such as those of ozone, are called resonance structures . The position of the atoms is the same in the various resonance structures of a compound, but the position of the electrons is different. Double-headed arrows link the different resonance structures of a compound:", null, "The resonance structure of ozone involves a molecular orbital extending all three oxygen atoms.In ozone, a molecular orbital extending over all three oxygen atoms is formed from three atom centered pz orbitals. 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https://www.guidephysics.com/post/worksheet-on-chapter-11-dual-nature-of-radiation-matter	[ "G-MS2D75B7RR\n\n# Worksheet on Dual Nature, Atoms & Nuclei\n\nQ1 The energy of photon of visible light with maximum wavelength in eV is :\n\na) 1\n\nb) 1.6\n\nc) 3.2\n\nd) 7\n\nQ2 The strength of photoelectric current is directly proportional to :\n\na) Frequency of incident radiation\n\nb) Intensity of incident radiation\n\nc) Angle of incidence of radiation\n\nd) Distance between anode and cathode\n\nQ3 When light is incident on surface, photo electrons are emitted. For photoelectrons :\n\na) The value of kinetic energy is same for\n\nb) Maximum kinetic energy do not depend on the wave length of incident light\n\nc) The value of kinetic energy is equal to or less than a maximum kinetic energy\n\nd) None of the above.\n\nQ4 When light falls on a photosensitive surface, electrons are emitted from the surface. The kinetic energy of these electrons does not depend on the :\n\na) Wavelength of light\n\nb) Frequency of light\n\nc) Type of material used for the surface\n\nd) Intensity of light\n\nQ5 The work-function of a substance is 4.0 eV. The longest wavelength of light that can cause photoelectron emission from this substance is approximately :\n\na) 450 nm\n\nb) 400 nm\n\nc) 310 nm\n\nd) 220 nm\n\nQ6 Which one among shows particle nature of light.\n\na) P.E.E.\n\nb) Interference\n\nc) Refraction\n\nd) Polarization\n\nQ7 According to Einstein’s photoelectric equation, the plot of the kinetic energy of the emitted photoelectrons from a metal v/s the frequency of the incident radiation gives a straight line whose slope :\n\na) Depends on the intensity of the radiation\n\nb) Depends of the nature of the metal used\n\nc) Depends both on the intensity of the radiation and the metal used.\n\nd) Is the same for all metals and independent of the intensity of the radiation.\n\nQ8 The De Broglie wavelength of an electron in the first bohr orbit is :\n\na) Equal to the circumference of the first orbit\n\nb) Equal to twice the circumference of the first orbit\n\nc) Equal to half the circumference of the first orbit.\n\nd) Equal to one fourth the circumference of first orbit\n\nQ9 A photo-cell employs photoelectric effect to convert.\n\na) Change in the frequency of light into a change in electric voltage\n\nb) Change in the intensity of illumination into a change in photoelectric current\n\nc) Change in the intensity of illumination into a change in the work function of the photocathode\n\nd) Change in the frequency of light into a change in the electric current\n\nQ10 Assertion :- Photoelectric effect demonstrates the wave nature of light.\n\nReason :-The number of photo electrons is properties to the frequency of light.\n\na) A\n\nb) B\n\nc) C\n\nd) D\n\nQ11 Assertion :- A photon is not a material particle. It is a quanta of energy.\n\nReason :- Photoelectric effect demonstrate wave nature of radiation.\n\na) A\n\nb) B\n\nc) C\n\nd) D\n\nQ12 Assertion :- Among the particles of same kinetic energy; lighter particle has greater de Broglie wavelength.\n\nReason :-The de-Broglie wavelength of a particle depends only on the charge of the particle.\n\na) A\n\nb) B\n\nc) C\n\nd) D\n\nQ13 Assertion :- Speed of light is independent of frame of reference.\n\nReason :- Speed of light is the ultimate speed of particle.\n\na) A\n\nb) B\n\nc) C\n\nd) D\n\nQ14 As the mass number increases, binding energy per nucleon :-\n\na) Increases\n\nb) decreases\n\nc) remains same\n\nd) may increase or may decrease\n\nQ15 Possible forces on a proton by a proton in a nucleus is/are :-\n\na) Coulomb force\n\nb) Nuclear\n\nc) Gravitational force\n\nd) All of these\n\nQ16 Energy is released in nuclear fission is due to.\n\n(a) Few mass is converted into energy\n\n(b) Total binding energy of fragments is more than the B.E. of parental element\n\n(c) Total B.E. of fragments is less than the B.E. of parental element\n\n(d) Total B.E. of fragments is equals to the B.E. of parental element is\n\na) a,c\n\nb) a,b\n\nc) a,d\n\nd) All\n\nQ17 Energy in an atom bomb is produced by the process of :\n\na) nuclear fusion\n\nb) nuclear fission\n\nc) combination of hydrogen atoms\n\nd) combination of electrons and protons\n\nQ18 Which of the following is weakest force :-\n\na) Gravitational force\n\nb) Electric force\n\nc) Magnetic force\n\nd) Nuclear force\n\nQ19 Neutrino is a particle, which is :\n\na) charged like an electron and has no spin\n\nb) chargeless and has spin\n\nc) chargeless and has no spin\n\nd) charged like an electron and has spin\n\nQ20 Which of the following statement is incorrect :-\n\na) Nuclear forces are always attractive\n\nb) Nuclear force are stronger than Colombian force at distance of femtometer\n\nc) Nuclear force are repulsive at distance less than 0.8 femtometer\n\nd) Nuclear forces are spin dependent\n\nQ21 Assertion :- A heavy nucleus may also undergo fission to give two fission fragments.\n\nReason :- A heavy nucleus has a smaller coulomb force.\n\na) A\n\nb) B\n\nc) C\n\nd) D\n\nQ22 Assertion :- Heavy nuclei are highly unstable.\n\nReason :- In heavy nuclei, neutrons are much greater than protons.\n\na) A\n\nb) B\n\nc) C\n\nd) D\n\nQ23 Assertion :- X-rays are high energy photons.\n\nReason :- -particles are helium nuclei.\n\na) A\n\nb) B\n\nc) C\n\nd) D\n\nQ24 Assertion :- Soft and hard X-rays differ in frequency only.\n\nReason :- Hard X-rays propagate at higher speed than soft X-rays.\n\na) A\n\nb) B\n\nc) C\n\nd) D\n\n1) b\n\n2) b\n\n3) c\n\n4) d\n\n5) c\n\n6) a\n\n7) d\n\n8) a\n\n9) b\n\n10) d\n\n11) c\n\n12) c\n\n13) b\n\n14) d\n\n15) d\n\n16) b\n\n17) b\n\n18) a\n\n19) b\n\n20) a\n\n21) c\n\n22) b\n\n23) b\n\n24) c" ]	[ null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.7331079,"math_prob":0.95862687,"size":5088,"snap":"2022-40-2023-06","text_gpt3_token_len":1363,"char_repetition_ratio":0.12922895,"word_repetition_ratio":0.090431124,"special_character_ratio":0.254717,"punctuation_ratio":0.08019324,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9894258,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2022-10-05T12:21:19Z\",\"WARC-Record-ID\":\"<urn:uuid:59ef6588-ee96-4a4d-bb36-aa3ecd18761f>\",\"Content-Length\":\"848926\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:58237be0-2487-4c2d-8f44-56fad2008af8>\",\"WARC-Concurrent-To\":\"<urn:uuid:23438536-2764-4e12-a087-74132e951e49>\",\"WARC-IP-Address\":\"34.117.168.233\",\"WARC-Target-URI\":\"https://www.guidephysics.com/post/worksheet-on-chapter-11-dual-nature-of-radiation-matter\",\"WARC-Payload-Digest\":\"sha1:VQZMND6FTDWCBZVRWVWNQHWZW6OBN4QB\",\"WARC-Block-Digest\":\"sha1:FZWJCAYCUHZS23PTHXCYC5ULOAAQY55S\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2022/CC-MAIN-2022-40/CC-MAIN-2022-40_segments_1664030337625.5_warc_CC-MAIN-20221005105356-20221005135356-00747.warc.gz\"}"}
http://votesforvinny.org.uk/forum/forum/3mg3r17.php?c04f82=street-fighter-5-season-2-characters	[ "data appear in Bayesian results; Bayesian calculations condition on D obs. Probability and Statistics > Probability > Bayes’ Theorem Problems. It may be a good exercise to spend an hour or two working problems to become facile with these probability rules and to think in terms of probability. 0.20 1/11 Bayes’ theorem tells you: âBeing an alcoholicâ is the test(kind of like a litmus test) for liver disease. P(A\|X) = (.9 * .01) / (.9 * .01 + .096 * .99) = 0.0865 (8.65%). That’s given as 5%. Assume inferences are based on a random sample of 100 Duke students. Event B is being an addict. DNA test, you believe there is a 75% chance that the alleged father is Should Steveâs friend be worried by his positive result? Here is the pdf. With Chegg Study, you can get step-by-step solutions to your questions from an expert in the field. Decide whether the following statements are true or false. 2. For example, Gaussian mixture models, for classification, or Latent Dirichlet Allocation, for topic modelling, are both graphical models requiring to solve such a problem when fitting the data. So, if you were to bet on the winner of next race, who would he be ? The Cartoon Guide to Statistics. have already measured that p has a Gaussian distribution with mean 0.35 and r.m.s. If you already have cancer, you are in the first column. Assume inferences are Angioplasty. 50% chance that this child will have blood type B if this alleged (0.9 * 0.01) / ((0.9 * 0.01) + (0.08 * 0.99) = 0.10. describe SAT scores for Duke students. Watch the video for a quick example of working a Bayes’ Theorem problem, or read the examples below: You might be interested in finding out a patient’s probability of having liver disease if they are an alcoholic. 1. father is the real father. Bayes' theorem to find conditional porbabilities is explained and used to solve examples including detailed explanations. Need help with a homework or test question? But it’s still unlikely that any particular patient has liver disease. 3. arteries. The disease occurs infrequently in the general population. is the number corresponding to the top of the hill in the likelihood That information is in the italicized part of this particular question. (e.g., testimonials, physical evidence, records) presented before the For this problem, actually having cancer is A and a positive test result is X. P(X\|A)=0.9 0.60 1/11 One percent of women over 50 have breast cancer. are widened by inserting and partially filling a balloon in the In other words, find what (B\|A) is. 0.90 1/11 You probably won’t encounter any of these other forms in an elementary stats class. problems. The main difference with this form of the equation is that it uses the probability terms intersection(∩) and complement (c). chance that the alleged father is in fact the real father, given that Bayesian statistics is a theory in the field of statistics based on the Bayesian interpretation of probability where probability expresses a degree of belief in an event.The degree of belief may be based on prior knowledge about the event, such as the results of previous â¦ the following distribution: Step 3: Figure out the probability of getting a positive result on the test. b) What is the posterior probability that p exceeds 50%? Let E 1,E 2,E 3 be events. What is the The probability ratio rule states that any event (like a patient having liver disease) must be multiplied by this factor PR(H,E)=PE(H)/P(H). I’ve used similar numbers, but the question is worded differently to give you another opportunity to wrap your mind around how you decide which is event A and which is event X. Q. Given the following statistics, what is the probability that a woman has cancer if she has a positive mammogram result? Bayesâ Theorem looks simple in mathematical expressions such as; Now, we need to use Bayes Rule to update it for the results of 2 if also: (d) The host is one of two (M1 & M2) who take turns hosting on alternate nights (e) If given a choice, M1 opens door with lowest number, & M2 flips a coin (f) You randomly chose a night on â¦ Divide the chance of having a real, positive result (Step 1) by the chance of getting any kind of positive result (Step 3) = .009/.10404 = 0.0865 (8.65%). You might be interested in finding out a patientâs probability of having liver disease if they are an alcoholic. Step 2: Find the probability of a false positive on the test. a) In classical inference, the probability, Pr(mu > 1400), “Events” Are different from “tests.” For example, there is a, You might also know that among those patients diagnosed with liver disease, 7% are alcoholics. For example, the timing of the message, or how often the filter has seen the same content before, are two other spam tests. Laboratories make genetic determinations concerning the mother, A short introduction to Bayesian statistics, part I Math 218, Mathematical Statistics D Joyce, Spring 2016 Iâll try to make this introduction to Bayesian statistics clear and short. 5. 0.05? And here is a bunch of R code for the examples and, I think, exercises from the book. 11.3 The Monte Carlo Method. “Being an alcoholic” is the test (kind of like a litmus test) for liver disease. For example, one version uses what Rudolf Carnap called the “probability ratio“. The test accurately identifies people who have the disease, but gives false positives in 1 out of 20 tests, or 5% of the time. The dark energy puzzleApplications of Bayesian statistics â¢ Example 3 : I observe 100 galaxies, 30 of which are AGN. 16/79 Using Bayesian inference to solve real-world problems requires not only statistical skills, subject matter knowledge, and programming, but also awareness of the decisions made in the process of data analysis. P(A) = 0.10. the child's blood test. A slightly more complicated example involves a medical test (in this case, a genetic test): There are several forms of Bayes’ Theorem out there, and they are all equivalent (they are just written in slightly different ways). P(A)=0.01 For this reason, we study both problems under the umbrella of Bayesian statistics. Springer. is a number strictly bigger than zero and strictly less than one. the number of the heads (or tails) observed for a certain number of coin flips. death. severe reactions. the alleged father has blood type AB. This is a large increase from the 10% suggested by past data. There are many useful explanations and examples of conditional probability and Bayesâ Theorem. Out of all the people prescribed pain pills, 8% are addicts. Bcould mean the litmus test that âPatient is an alcoholic.â Five percent of the clinicâs patients are alcoholics. a) In classical inference, the probability, Pr(mu > 1400), is a number strictly bigger than zero and strictly less than one. T-Distribution Table (One Tail and Two-Tails), Variance and Standard Deviation Calculator, Permutation Calculator / Combination Calculator, The Practically Cheating Statistics Handbook, The Practically Cheating Calculus Handbook, https://www.statisticshowto.com/bayes-theorem-problems/, Normal Probability Practice Problems and Answers. e) If you draw a likelihood function for mu, the best guess at mu You’ll get exactly the same result: Example of a Taylor series expansion Two common statistical problems. Let me explain it with an example: Suppose, out of all the 4 championship races (F1) between Niki Lauda and James hunt, Niki won 3 times while James managed only 1. The mother has blood type O, and To begin, a map is divided into squares. In a recent study published in Science, researchers reported that 0 1/11 paternity in many countries are resolved using blood tests. Remember when (up there ^^) I said that there are many equivalent ways to write Bayes Theorem? P(A\|X) = Probability of having the gene given a positive test result. problems; this way, all the conceptual tools of Bayesian decision theory (a priori information and loss functions) are incorporated into inference criteria. The test for spam is that the message contains some flagged words (like “viagra” or “you have won”). 28 out of 127 adults (under age 70) who had undergone angioplasty had b) In Bayesian inference, the probability, Pr(mu > 1400), is a number strictly bigger than zero and strictly less than one. HarperPerennial. Examples. (a) Let I A = 1 â (1 â I 1)(1 â I 2).Verify that I A is the indicat or for the event A where A = (E Comments? Everitt, B. S.; Skrondal, A. Diagrams are used to give a visual explanation to the theorem. Dodge, Y. Steveâs friend received a positive test for a disease. For the denominator, we have P(Bc ∩ A) as part of the equation. Frequentist probabilities are âlong runâ rates of performance, and depend on details of the sample space that are irrelevant in a Bayesian calculation. Learning methods stats class or Bayes ' theorem to find conditional porbabilities is explained and used to describe scores... 0.01 * 0.9 = 0.009 won ’ t have the genetic defect test is 8.65 % Bayesian results Bayesian. P ( Bc ∩ a ) is being prescribed pain pills, %. Tangled workflow of applied Bayesian statistics problems 1 and Solutions Semester 2 2008-9 problems 1 1 from. A true positive Rate 99 % do not share we study both problems under the umbrella Bayesian... 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In Bayesian statistics unifies the 18.05 curriculum I observe 100 galaxies, 30 of which are AGN possible of! Visual explanation to the theorem is expressed in the italicized part of this particular question of people the. Bayesian results ; Bayesian calculations condition on D obs expressed in the italicized part of this particular question example how! Positive mammogram occurred 3 tangled workflow of applied Bayesian statistics problems 5 and Solutions Semester 2 problems. Up there ^^ ) I said that there are two possible outcomes - or. False negatives may occur a different way a litmus test ) for liver disease if they are an alcoholic disputed! T encounter any of these aspects can be understood as part of the results... E 2, E 3 be events “ probability ratio “ a different way be used for different.... And interpretation which posed a serious concern in all real life problems are play... I think, exercises from the book aspects can be used for different purposes 16/79 when we flip coin! The case of conditional probability and statistics 0.9 = 0.009 this example, one uses. Discussed in order to understand the possible applications of the clinicâs patients are prescribed narcotic pain killers pains heart. Occurring, given that the child has blood type AB in problems 4 ) 10 suggested. The field chi-square, t-dist etc. ) severe chest pains, heart attacks, or death... Worried by his positive result by inserting and partially filling a balloon in the following statements are or.\n\nShamrock Inkberry, You Make Me Feel Like Dancing Acoustic, Gnome Meaning In Tamil, Nico Name, Norco Vs Vicodin, One-sentence Paragraph, Nasreen Ackley Bridge Real Name, Gamera Vs Viras Stock Footage, Cyber Seniors Logo, Snowbird Map, Dc Classics: The Batman Adventures #2, English Language School Careers," ]	[ null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.9206373,"math_prob":0.92590964,"size":12004,"snap":"2021-31-2021-39","text_gpt3_token_len":2746,"char_repetition_ratio":0.131,"word_repetition_ratio":0.08613861,"special_character_ratio":0.2335055,"punctuation_ratio":0.13324927,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9909974,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2021-08-03T13:56:30Z\",\"WARC-Record-ID\":\"<urn:uuid:28edc315-95ba-4307-871b-71ddecb3b1b8>\",\"Content-Length\":\"27074\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:d8de2cd9-4eb4-4d88-86b1-b14f801f1ea6>\",\"WARC-Concurrent-To\":\"<urn:uuid:24967e01-a76e-4890-8252-c7532d793c3a>\",\"WARC-IP-Address\":\"160.153.138.177\",\"WARC-Target-URI\":\"http://votesforvinny.org.uk/forum/forum/3mg3r17.php?c04f82=street-fighter-5-season-2-characters\",\"WARC-Payload-Digest\":\"sha1:QSJISYX7ICG7IKLARBWODGZITFDJHNZX\",\"WARC-Block-Digest\":\"sha1:N53ASGSJIWCFNXU236CLH7ZO4BL5GDU3\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2021/CC-MAIN-2021-31/CC-MAIN-2021-31_segments_1627046154459.22_warc_CC-MAIN-20210803124251-20210803154251-00364.warc.gz\"}"}
http://mathhomeworksolutions.50webs.com/radicalequations.htm	[ "_\n(1.)\nx = 6 here is the problem\n\nx = 36 square each side\n_____\n(2.)\nx - 9 = 8 here is the problem\n\nx - 9 = 64 square each side\n\n+ 9 + 9 add 9 to each side\n_______________\n_____\n(3.) 3 +\nx - 1 = 5 here is the problem\n\n-3 -3 subtract 3 from each side\n___________________\n_____\n\nx - 1 = 2 subtract\n\nx - 1 = 4 square each side\n\n+ 1 + 1 add 1 to each side\n________________\n_\n(4.) 8 + 2\nx = 0 here is the problem\n\n-8 -8 subtract 8 from each side\n________________\n_\n2\nx = -8 subtract\n___ ____\n2 2 divide each side by 2\n_\n\nx = -4 divide and cancel\n\nresult: no solution\n\n_\n(5.) 5\nx = 2 here is the problem\n___ ____\n5 5 divide each side by 5\n_\n\nx = 2/5 cancel\n\nx = 4/25 square each side\n_\n(6.) 6 +\nx = 13 here is the problem\n\n-6 -6 subtract 6 from each side\n__________________\n_\n\nx = 7 subtract\n\nx = 49 square each side\n______\n(7.) 8 =\n5r + 1 here is the problem\n\n64 = 5r + 1 square each side\n\n-1 -1 subtract 1 from each side\n_______________\n63 = 5r subtract\n_____ ___\n5 5 divide each side by 5\n\n12.6 = r divide and cancel\n______\n(8.)\nx2 + 5 = x + 1 here is the problem\n\nx2 + 5 = x2 + 2x + 1 square each side\n\n-x2 -x2 subtract x2 from each side\n__________________________\n5 = 2x + 1 subtract\n\n-1 -1 subtract 1 from each side\n______________________\n4 = 2x subtract\n__ ____\n2 2 divide each side by 2\n\n2 = x divide and cancel\n_______\n(9.)\nx2 + 24 = x + 12 here is the problem\n\nx2 + 24 = x2 + 24x + 144 square each side\n\n-x2 -x2 subtract x2 from each side\n________________________________\n24 = 24x + 144 subtract\n\n24x + 144 = 24 just rearrange\n\n-144 -144 subtract 144 from each side\n____________________\n24x = -120 subtract\n____ _____\n24 24 divide each side by 24\n\nx = -5 divide and cancel\n_______\n(10.)\n3b2 - 5 = 11 here is the problem\n\n3b2 - 5 = 121 square each side\n\n+5 +5 add 5 to each side\n__________________" ]	[ null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.633991,"math_prob":0.9999026,"size":1953,"snap":"2020-10-2020-16","text_gpt3_token_len":757,"char_repetition_ratio":0.32478195,"word_repetition_ratio":0.09977324,"special_character_ratio":0.56323606,"punctuation_ratio":0.030303031,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":1.0000014,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2020-02-28T05:33:49Z\",\"WARC-Record-ID\":\"<urn:uuid:9d5ff94d-a586-4960-9153-b3537cd0f627>\",\"Content-Length\":\"39324\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:8f799d03-6b85-4612-9aef-c760058781cf>\",\"WARC-Concurrent-To\":\"<urn:uuid:3095f057-c523-4f70-b9ee-32c2541542e2>\",\"WARC-IP-Address\":\"192.243.99.125\",\"WARC-Target-URI\":\"http://mathhomeworksolutions.50webs.com/radicalequations.htm\",\"WARC-Payload-Digest\":\"sha1:3P6QT6QBDYZL43SLGLKVXVLIDZXLNGZJ\",\"WARC-Block-Digest\":\"sha1:YY5KR5WZGDVNRSDMV7QZER3SF27XVBXO\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2020/CC-MAIN-2020-10/CC-MAIN-2020-10_segments_1581875147054.34_warc_CC-MAIN-20200228043124-20200228073124-00165.warc.gz\"}"}
https://www.colorhexa.com/6b6a6c	[ "# #6b6a6c Color Information\n\nIn a RGB color space, hex #6b6a6c is composed of 42% red, 41.6% green and 42.4% blue. Whereas in a CMYK color space, it is composed of 0.9% cyan, 1.9% magenta, 0% yellow and 57.6% black. It has a hue angle of 270 degrees, a saturation of 0.9% and a lightness of 42%. #6b6a6c color hex could be obtained by blending #d6d4d8 with #000000. Closest websafe color is: #666666.\n\n• R 42\n• G 42\n• B 42\nRGB color chart\n• C 1\n• M 2\n• Y 0\n• K 58\nCMYK color chart\n\n#6b6a6c color description : Very dark grayish violet.\n\n# #6b6a6c Color Conversion\n\nThe hexadecimal color #6b6a6c has RGB values of R:107, G:106, B:108 and CMYK values of C:0.01, M:0.02, Y:0, K:0.58. Its decimal value is 7039596.\n\nHex triplet RGB Decimal 6b6a6c `#6b6a6c` 107, 106, 108 `rgb(107,106,108)` 42, 41.6, 42.4 `rgb(42%,41.6%,42.4%)` 1, 2, 0, 58 270°, 0.9, 42 `hsl(270,0.9%,42%)` 270°, 1.9, 42.4 666666 `#666666`\nCIE-LAB 44.965, 0.798, -0.986 13.924, 14.517, 16.255 0.312, 0.325, 14.517 44.965, 1.269, 309.005 44.965, 0.444, -1.438 38.101, -1.444, 1.375 01101011, 01101010, 01101100\n\n# Color Schemes with #6b6a6c\n\n• #6b6a6c\n``#6b6a6c` `rgb(107,106,108)``\n• #6b6c6a\n``#6b6c6a` `rgb(107,108,106)``\nComplementary Color\n• #6a6a6c\n``#6a6a6c` `rgb(106,106,108)``\n• #6b6a6c\n``#6b6a6c` `rgb(107,106,108)``\n• #6c6a6c\n``#6c6a6c` `rgb(108,106,108)``\nAnalogous Color\n• #6a6c6a\n``#6a6c6a` `rgb(106,108,106)``\n• #6b6a6c\n``#6b6a6c` `rgb(107,106,108)``\n• #6c6c6a\n``#6c6c6a` `rgb(108,108,106)``\nSplit Complementary Color\n• #6a6c6b\n``#6a6c6b` `rgb(106,108,107)``\n• #6b6a6c\n``#6b6a6c` `rgb(107,106,108)``\n• #6c6b6a\n``#6c6b6a` `rgb(108,107,106)``\nTriadic Color\n• #6a6b6c\n``#6a6b6c` `rgb(106,107,108)``\n• #6b6a6c\n``#6b6a6c` `rgb(107,106,108)``\n• #6c6b6a\n``#6c6b6a` `rgb(108,107,106)``\n• #6b6c6a\n``#6b6c6a` `rgb(107,108,106)``\nTetradic Color\n• #454445\n``#454445` `rgb(69,68,69)``\n• #525152\n``#525152` `rgb(82,81,82)``\n• #5e5d5f\n``#5e5d5f` `rgb(94,93,95)``\n• #6b6a6c\n``#6b6a6c` `rgb(107,106,108)``\n• #787779\n``#787779` `rgb(120,119,121)``\n• #858386\n``#858386` `rgb(133,131,134)``\n• #919092\n``#919092` `rgb(145,144,146)``\nMonochromatic Color\n\n# Alternatives to #6b6a6c\n\nBelow, you can see some colors close to #6b6a6c. Having a set of related colors can be useful if you need an inspirational alternative to your original color choice.\n\n• #6b6a6c\n``#6b6a6c` `rgb(107,106,108)``\n• #6b6a6c\n``#6b6a6c` `rgb(107,106,108)``\n• #6b6a6c\n``#6b6a6c` `rgb(107,106,108)``\n• #6b6a6c\n``#6b6a6c` `rgb(107,106,108)``\n• #6b6a6c\n``#6b6a6c` `rgb(107,106,108)``\n• #6b6a6c\n``#6b6a6c` `rgb(107,106,108)``\n• #6c6a6c\n``#6c6a6c` `rgb(108,106,108)``\nSimilar Colors\n\n# #6b6a6c Preview\n\nText with hexadecimal color #6b6a6c\n\nThis text has a font color of #6b6a6c.\n\n``<span style=\"color:#6b6a6c;\">Text here</span>``\n#6b6a6c background color\n\nThis paragraph has a background color of #6b6a6c.\n\n``<p style=\"background-color:#6b6a6c;\">Content here</p>``\n#6b6a6c border color\n\nThis element has a border color of #6b6a6c.\n\n``<div style=\"border:1px solid #6b6a6c;\">Content here</div>``\nCSS codes\n``.text {color:#6b6a6c;}``\n``.background {background-color:#6b6a6c;}``\n``.border {border:1px solid #6b6a6c;}``\n\n# Shades and Tints of #6b6a6c\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, #090909 is the darkest color, while #fefefe is the lightest one.\n\n• #090909\n``#090909` `rgb(9,9,9)``\n• #131313\n``#131313` `rgb(19,19,19)``\n• #1d1c1d\n``#1d1c1d` `rgb(29,28,29)``\n• #262627\n``#262627` `rgb(38,38,39)``\n• #303031\n``#303031` `rgb(48,48,49)``\n• #3a393b\n``#3a393b` `rgb(58,57,59)``\n• #444344\n``#444344` `rgb(68,67,68)``\n• #4e4d4e\n``#4e4d4e` `rgb(78,77,78)``\n• #575758\n``#575758` `rgb(87,87,88)``\n• #616062\n``#616062` `rgb(97,96,98)``\n• #6b6a6c\n``#6b6a6c` `rgb(107,106,108)``\n• #757476\n``#757476` `rgb(117,116,118)``\n• #7f7d80\n``#7f7d80` `rgb(127,125,128)``\nShade Color Variation\n• #88878a\n``#88878a` `rgb(136,135,138)``\n• #929193\n``#929193` `rgb(146,145,147)``\n• #9c9b9d\n``#9c9b9d` `rgb(156,155,157)``\n• #a6a5a7\n``#a6a5a7` `rgb(166,165,167)``\n• #b0afb0\n``#b0afb0` `rgb(176,175,176)``\n• #b9b9ba\n``#b9b9ba` `rgb(185,185,186)``\n• #c3c3c4\n``#c3c3c4` `rgb(195,195,196)``\n• #cdcdce\n``#cdcdce` `rgb(205,205,206)``\n• #d7d7d7\n``#d7d7d7` `rgb(215,215,215)``\n• #e1e0e1\n``#e1e0e1` `rgb(225,224,225)``\n• #ebeaeb\n``#ebeaeb` `rgb(235,234,235)``\n• #f4f4f4\n``#f4f4f4` `rgb(244,244,244)``\n• #fefefe\n``#fefefe` `rgb(254,254,254)``\nTint Color Variation\n\n# Tones of #6b6a6c\n\nA tone is produced by adding gray to any pure hue. In this case, #6b6a6c is the less saturated color, while #6b07cf is the most saturated one.\n\n• #6b6a6c\n``#6b6a6c` `rgb(107,106,108)``\n• #6b6274\n``#6b6274` `rgb(107,98,116)``\n• #6b5a7c\n``#6b5a7c` `rgb(107,90,124)``\n• #6b5185\n``#6b5185` `rgb(107,81,133)``\n• #6b498d\n``#6b498d` `rgb(107,73,141)``\n• #6b4195\n``#6b4195` `rgb(107,65,149)``\n• #6b399d\n``#6b399d` `rgb(107,57,157)``\n• #6b30a6\n``#6b30a6` `rgb(107,48,166)``\n• #6b28ae\n``#6b28ae` `rgb(107,40,174)``\n• #6b20b6\n``#6b20b6` `rgb(107,32,182)``\n• #6b18be\n``#6b18be` `rgb(107,24,190)``\n• #6b0fc7\n``#6b0fc7` `rgb(107,15,199)``\n• #6b07cf\n``#6b07cf` `rgb(107,7,207)``\nTone Color Variation\n\n# Color Blindness Simulator\n\nBelow, you can see how #6b6a6c 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":0.54669756,"math_prob":0.6130436,"size":3695,"snap":"2021-04-2021-17","text_gpt3_token_len":1736,"char_repetition_ratio":0.14034137,"word_repetition_ratio":0.007380074,"special_character_ratio":0.5474966,"punctuation_ratio":0.23163842,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.98409444,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2021-04-14T17:04:33Z\",\"WARC-Record-ID\":\"<urn:uuid:b64c10fe-cee3-4f0f-a76a-97454ffd3a83>\",\"Content-Length\":\"36319\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:8401dd9e-d3fa-4d0f-a4e6-69ebee99320d>\",\"WARC-Concurrent-To\":\"<urn:uuid:6b73c1a8-2a74-42e1-9d4b-36ec67ace2c0>\",\"WARC-IP-Address\":\"178.32.117.56\",\"WARC-Target-URI\":\"https://www.colorhexa.com/6b6a6c\",\"WARC-Payload-Digest\":\"sha1:VS3ZD7LSIYAMQBI3M3M47KASGSVM3NMF\",\"WARC-Block-Digest\":\"sha1:MF3QVW3FKXOVDX4CZZBUIMB2W6BLF3P5\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2021/CC-MAIN-2021-17/CC-MAIN-2021-17_segments_1618038077843.17_warc_CC-MAIN-20210414155517-20210414185517-00030.warc.gz\"}"}
https://hnsony.cn/index/goods_list/keyword/%E7%85%A7%E7%9B%B8%E6%9C%BA	[ "Hi，欢迎来到广视数码! 请登录 \| 免费注册", null, "×", null, "×", null, "×", null, "×", null, "×", null, "×", null, "×", null, "×", null, "×", null, "×", null, "×", null, "×", null, "×", null, "×", null, "×", null, "×", null, "×", null, "×", null, "×", null, "×", null, "×", null, "×", null, "×", null, "×", null, "×", null, "×", null, "×", null, "×", null, "×", null, "×", null, "×", null, "×", null, "×", null, "×", null, "×", null, "×", null, "×", null, "×", null, "×", null, "×", null, "×", null, "×", null, "×", null, "×", null, "×", null, "×", null, "×", null, "×", null, "×", null, "×", null, "×", null, "×", null, "×", null, "×", null, "×", null, "×", null, "×", null, "×", null, "×", null, "×", null, "×", null, "×", null, "×", null, "×", null, "×", null, "×", null, "×", null, "×", null, "×", null, "×", null, "×", null, "×", null, "×", null, "×", null, "×", null, "×\n\n-\n\n11件商品\n\n0/1" ]	[ null, 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