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https://support.minitab.com/en-us/minitab/19/help-and-how-to/statistical-modeling/regression/supporting-topics/logistic-regression/what-is-a-generalized-linear-model/	[ "# What is a generalized linear model?\n\nBoth generalized linear models and least squares regression investigate the relationship between a response variable and one or more predictors. A practical difference between them is that generalized linear model techniques are usually used with categorical response variables. Least squares regression is usually used with continuous response variables. For a thorough description of generalized linear models, see 1\n\nBoth generalized linear model techniques and least squares regression techniques estimate parameters in the model so that the fit of the model is optimized. Least squares minimizes the sum of squared errors to obtain maximum likelihood estimates of the parameters. Generalized linear models obtain maximum likelihood estimates of the parameters using an iterative-reweighted least squares algorithm.\n\nFor example, you could use a generalized linear model to study the relationship between machinists' years of experience (a nonnegative continuous variable), and their participation in an optional training program (a binary variable: either yes or no), to predict whether their products meet specifications (a binary variable: either yes or no). The first two variables are the predictors; the third is the categorical response.\n\n## Types of logistic regression\n\nMinitab Statistical Software provides four generalized linear model techniques that you can use to assess the relationship between one or more predictor variables and a response variable of the following types. The previous example uses binary logistic regression because the response variable has two levels.\n\nVariable type Number of categories Characteristics Examples\n\nBinary\n\n2\n\nTwo levels\n\nPass/Fail\n\nYes/No\n\nHigh/Low\n\nOrdinal\n\n3 or more\n\nNatural ordering of the levels\n\nTaste (Mild, Medium, Hot)\n\nMedical condition (Critical, Serious, Stable, Good)\n\nSurvey results (Disagree, Neutral, Agree)\n\nNominal\n\n3 or more\n\nNo natural ordering of the levels\n\nTaste (Bitter, Sweet, Sour)\n\nColor (Red, Blue, Black)\n\nSchool subject (Math, Science, Art)\n\nPoisson\n\n3 or more\n\nThe response variable describes the number of times an event occurs in a finite observation space.\n\n0, 1, 2, ...\n###### Note\n\nFor a model that has one continuous predictor and a binary response variable, Minitab provides a fifth technique. A Binary Fitted Line Plot quickly describes the relationship between the predictor and the response.\n\n1 P. McCullagh and J. A. Nelder (1992). Generalized Linear Models. Chapman & Hall.\nBy using this site you agree to the use of cookies for analytics and personalized content. Read our policy" ]	[ null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.8429636,"math_prob":0.9412047,"size":2352,"snap":"2020-24-2020-29","text_gpt3_token_len":459,"char_repetition_ratio":0.13458262,"word_repetition_ratio":0.042613637,"special_character_ratio":0.18962584,"punctuation_ratio":0.104166664,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9819999,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2020-07-06T09:21:46Z\",\"WARC-Record-ID\":\"<urn:uuid:7f6edb49-ba32-461e-a5c8-c161d35868e7>\",\"Content-Length\":\"12018\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:ae747ac8-49e3-4233-82ed-ea6e869bdcbd>\",\"WARC-Concurrent-To\":\"<urn:uuid:b4a903c5-1ebc-4d4f-b73c-26bba70402c0>\",\"WARC-IP-Address\":\"23.96.207.177\",\"WARC-Target-URI\":\"https://support.minitab.com/en-us/minitab/19/help-and-how-to/statistical-modeling/regression/supporting-topics/logistic-regression/what-is-a-generalized-linear-model/\",\"WARC-Payload-Digest\":\"sha1:L2KPBFTWHH23VF5CP23LKHANWFIAMKHM\",\"WARC-Block-Digest\":\"sha1:2IUYA327LMSXLYM4VVPMMZ4AWOL6ORNB\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2020/CC-MAIN-2020-29/CC-MAIN-2020-29_segments_1593655890157.10_warc_CC-MAIN-20200706073443-20200706103443-00254.warc.gz\"}"}
https://notebook.community/Akira794/ProbabilisticRobot/MCL/mcl_py2	[ "# mcl_py2\n\n``````\n\nIn :\n\nimport numpy as np\nimport copy\nimport math, random\nimport matplotlib.pyplot as plt\nfrom matplotlib.patches import Ellipse\nfrom scipy.stats import norm\n\ndef draw_landmarks(landmarks):\nxs = [ e for e in landmarks ]\nys = [ e for e in landmarks ]\nplt.scatter(xs,ys, s=300, marker=\"\", label =\"landmarks\", color=\"orange\")\n\ndef draw_robot(pose):\nplt.quiver([pose],[pose],[math.cos(pose)],[math.sin(pose)],color=\"red\",label=\"actual robot motion\")\n\ndef relative_landmark_pos(pose,landmark):\nx,y,th =pose\nlx,ly = landmark\ndistance = math.sqrt((x-lx)2 + (y-ly)2)\ndirection = math.atan2(ly-y, lx-x) - th\n\nreturn (distance, direction, lx, ly)\n\ndef draw_observation(pose, measurment):\nx,y,th =pose\ndistance, direction,lx,ly = measurment\nlx = distance math.cos(th + direction) + x\nly = distance * math.sin(th + direction) + y\nplt.plot([x, lx],[y, ly],color=\"pink\")\n\ndef draw_observations(pose, measurements):\nfor m in measurements:\ndraw_observation(pose, m)\n\ndef observation(pose, landmark):\nactual_distance, actual_direction,lx,ly = relative_landmark_pos(pose,landmark)\n\nif(math.cos(actual_direction) < 0.0 ):\nreturn None\n\nmeasured_distance = random.gauss(actual_distance, actual_distance0.1) #10% error\nmeasured_direction = random.gauss(actual_direction,5.0/180.0math.pi) #5deg error\nreturn (measured_distance, measured_direction,lx,ly)\n\ndef observations(pose,landmarks):\nreturn filter(lambda x: x != None, [ observation(pose,e) for e in landmarks])\n\ndef likelihood(pose, measurement):\nx,y,th = pose\ndistance, direction,lx,ly = measurement\n\nrel_distance, rel_direction, tmp_x, tmp_y = relative_landmark_pos(pose,(lx,ly))\n\nreturn norm.pdf(x = distance - rel_distance, loc = 0.0, scale = rel_distance / 10.0) \\\n* norm.pdf(x = direction - rel_direction, loc = 0.0, scale = 5.0/180.0 * math.pi)\n\ndef change_weights(particles, measurement):\nfor p in particles:\np.weight = likelihood(p.pose, measurement)\n\nws = [ p.weight for p in particles ]\ns = sum(ws)\nfor p in particles:\np.weight = p.weight / s\n\nclass Particles:\ndef __init__(self,pose, w ):\nself.pose = np.array(pose)\nself.weight = w\n\ndef __repr__(self):\nreturn \"pose: \" + str(self.pose) + \"weight: \" + str(self.weight)\n\ndef motion(pose,u):\np_x, p_y, p_th = pose\nfw, rot = u\n\nactual_fw = random.gauss(fw, fw/10)\ndir_error = random.gauss(0.0, math.pi/ 180.0 3.0)\nactual_rot = random.gauss(rot,rot/10)\n\np_x += actual_fw * math.cos(p_th + dir_error)\np_y += actual_fw * math.sin(p_th + dir_error)\np_th += actual_rot + dir_error\n\nreturn np.array([p_x, p_y, p_th])\n\ndef draw(pose,particles):\nfig = plt.figure(i, figsize= (8, 8))\nsp.set_xlim(-1.0,1.0)\nsp.set_ylim(-0.5,1.5)\n\nxs = [e.pose for e in particles]\nys = [e.pose for e in particles]\nvxs = [math.cos(e.pose)e.weight for e in particles]\nvys = [math.sin(e.pose)e.weight for e in particles]\nplt.quiver(xs,ys,vxs,vys,color=\"blue\",label=\"particles\")\n\nplt.quiver([pose],[pose],[math.cos(pose)],[math.sin(pose)],color=\"red\",label=\"actual robot motion\")\n\nif __name__ == '__main__':\nactual_x = np.array([0.0,0.0,0.0])\nu = np.array([0.2,math.pi / 180.0 * 20])\nactual_landmarks =[np.array([-0.5,0.0]),np.array([0.5,0.0]),np.array([0.0,0.5])]\nparticles = [Particles([0.0,0.0,0.0],1.0/100) for i in range(100)] ## 100 ##\n\npath = [actual_x]\nparticle_path = [copy.deepcopy(particles)]\nmeasurements = [observations(actual_x, actual_landmarks)]\n\nfor i in range(15):\nactual_x = motion(actual_x,u)\npath.append(actual_x)\n\nms = observations(actual_x, actual_landmarks)\nmeasurements.append(ms)\n\nfor p in particles:\np.pose = motion(p.pose,u)\n\nfor m in ms:\nchange_weights(particles, m)\n\npointer = random.uniform(0.0,1.0/len(particles))\nnew_particles = []\nparticles_num = len(particles)\n\naccum =[]\nsm = 0.0\nfor p in particles:\naccum.append(p.weight + sm)\nsm += p.weight\n\nwhile pointer < 1.0:\nif accum >= pointer:\nnew_particles.append(\nParticles(copy.deepcopy(particles.pose),1.0/particles_num)\n)\npointer += 1.0/particles_num\nelse:\naccum.pop(0)\nparticles.pop(0)\n\nparticles = new_particles\n\nparticle_path.append(copy.deepcopy(particles))\n\nfor i,p in enumerate(path):\ndraw(path[i],particle_path[i])\ndraw_landmarks(actual_landmarks)\ndraw_observations(path[i], measurements[i])\nplt.show()\n\n``````\n``````\n\nIn [ ]:\n\n``````" ]	[ null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.59075433,"math_prob":0.9982792,"size":4220,"snap":"2021-31-2021-39","text_gpt3_token_len":1301,"char_repetition_ratio":0.16911764,"word_repetition_ratio":0.012931035,"special_character_ratio":0.33341232,"punctuation_ratio":0.27380952,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.99985147,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2021-07-30T23:38:20Z\",\"WARC-Record-ID\":\"<urn:uuid:d86ffb88-62ef-464a-b457-183e8ac7605b>\",\"Content-Length\":\"55778\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:0881df0a-adcb-4160-afff-8a269b52a5df>\",\"WARC-Concurrent-To\":\"<urn:uuid:ed5ae6fe-1e01-44f2-9c42-38cd23f86ed7>\",\"WARC-IP-Address\":\"216.239.32.21\",\"WARC-Target-URI\":\"https://notebook.community/Akira794/ProbabilisticRobot/MCL/mcl_py2\",\"WARC-Payload-Digest\":\"sha1:PC66H36Y4FRVVZLFZTCUFSL7K6CACPQ7\",\"WARC-Block-Digest\":\"sha1:E5RKHIUTH2W5IV6E5XEF7I56KWWPGN75\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2021/CC-MAIN-2021-31/CC-MAIN-2021-31_segments_1627046154032.75_warc_CC-MAIN-20210730220317-20210731010317-00169.warc.gz\"}"}
https://help.imsl.com/java/7.2/manual/api/com/imsl/datamining/KohonenSOM.html	[ "JMSLTM Numerical Library 7.2.0\ncom.imsl.datamining\n\n## Class KohonenSOM\n\n• All Implemented Interfaces:\nSerializable, Cloneable\n\n```public class KohonenSOM\nextends Object\nimplements Serializable, Cloneable```\nA Kohonen self organizing map.\n\nA self-organizing map (SOM), also known as a Kohonen map or Kohonen SOM, is a technique for gathering high-dimensional data into clusters that are constrained to lie in low dimensional space, usually two dimensions. A Kohonen map is a widely used technique for the purpose of feature extraction and visualization for very high dimensional data in situations where classifications are not known beforehand. The Kohonen SOM is equivalent to an artificial neural network having inputs linked to every node in the network. Self-organizing maps use a neighborhood function to preserve the topological properties of the input space.\n\nIn a Kohonen map, nodes are arranged in a rectangular or hexagonal grid or lattice. The input is connected to each node, and the output of the Kohonen map is the zero-based (i, j) index of the node that is closest to the input. A Kohonen map involves two steps: training and forecasting. Training builds the map using input examples (vectors), and forecasting classifies a new input.\n\nDuring training, an input vector is fed to the network. The input's Euclidean distance from all the nodes is calculated. The node with the shortest distance is identified and is called the Best Matching Unit, or BMU. After identifying the BMU, the weights of the BMU and the nodes closest to it in the SOM lattice are updated towards the input vector. The magnitude of the update decreases with time and with distance (within the lattice) from the BMU. The weights of the nodes surrounding the BMU are updated according to:", null, "where", null, "represents the node weights,", null, "is the monotonically decreasing learning coefficient function,", null, "is the neighborhood function, d is the lattice distance between the node and the BMU, and", null, "is the input vector.\n\nThe monotonically decreasing learning coefficient function", null, "is a scalar factor that defines the size of the update correction. The value of", null, "decreases with the step index t.\n\nThe neighborhood function", null, "depends on the lattice distance d between the node and the BMU, and represents the strength of the coupling between the node and BMU. In the simplest form, the value of", null, "is 1 for all nodes closest to the BMU and 0 for others, but a Gaussian function is also commonly used. Regardless of the functional form, the neighborhood function shrinks with time (Hollm�n, 15.2.1996). Early on, when the neighborhood is broad, the self-organizing takes place on the global scale. When the neighborhood has shrunk to just a couple of nodes, the weights converge to local estimates.\n\nNote that in a rectangular grid, the BMU has four closest nodes for the Von Neumann neighborhood type, or eight closest nodes for the Moore neighborhood type. In a hexagonal grid, the BMU has six closest nodes.\n\nDuring training, this process is repeated for a number of iterations on all input vectors.\n\nDuring forecasting, the node with the shortest Euclidean distance is the winning node, and its (i, j) index is the output.\n\nExample, Serialized Form\n• ### Field Summary\n\nFields\nModifier and Type Field and Description\n`static int` `GRID_HEXAGONAL`\nIndicates a hexagonal grid.\n`static int` `GRID_RECTANGULAR`\nIndicates a rectangular grid.\n`static int` `TYPE_MOORE`\nIndicates a Moore neighborhood type.\n`static int` `TYPE_VON_NEUMANN`\nIndicates a Von Neumann neighborhood type.\n• ### Constructor Summary\n\nConstructors\nConstructor and Description\n```KohonenSOM(int dim, int nrow, int ncol)```\nConstructor for a `KohonenSOM` object.\n• ### Method Summary\n\nMethods\nModifier and Type Method and Description\n`int[]` `forecast(double[] input)`\nReturns a forecast computed using the `KohonenSOM` object.\n`int[][]` `forecast(double[][] input)`\nReturns forecasts computed using the `KohonenSOM` object.\n`int` `getDimension()`\nReturns the number of weights for each node.\n`int` `getGridType()`\nReturns the grid type.\n`int` `getNeighborhoodType()`\nReturns the neighborhood type for the rectangular grid.\n`int` `getNumberOfColumns()`\nReturns the number of columns of the node grid.\n`int` `getNumberOfRows()`\nReturns the number of rows of the node grid.\n`double[][][]` `getWeights()`\nReturns the weights of the nodes.\n`double[]` ```getWeights(int i, int j)```\nReturns the weights of the node at (i, j) in the node grid.\n`boolean` `isWrapAround()`\nReturns whether the opposite edges are connected or not.\n`void` `setGridType(int type)`\nSets the grid type.\n`void` `setNeighborhoodType(int type)`\nSets the neighborhood type.\n`void` `setWeights()`\nSets the weights of the nodes using random numbers.\n`void` `setWeights(double[][][] weights)`\nSets the weights of the nodes.\n`void` ```setWeights(int i, int j, double[] weights)```\nSets the weights of the node at (i, j) in the node grid.\n`void` `setWeights(Random random)`\nSets the weights of the nodes using a `Random` object.\n`void` `wrapAround()`\nSets a flag to indicate the map should wrap around or connect opposite edges.\n• ### Methods inherited from class java.lang.Object\n\n`clone, equals, finalize, getClass, hashCode, notify, notifyAll, toString, wait, wait, wait`\n• ### Field Detail\n\n• #### GRID_HEXAGONAL\n\n`public static final int GRID_HEXAGONAL`\nIndicates a hexagonal grid.\nConstant Field Values\n• #### GRID_RECTANGULAR\n\n`public static final int GRID_RECTANGULAR`\nIndicates a rectangular grid.\nConstant Field Values\n• #### TYPE_MOORE\n\n`public static final int TYPE_MOORE`\nIndicates a Moore neighborhood type.\nConstant Field Values\n• #### TYPE_VON_NEUMANN\n\n`public static final int TYPE_VON_NEUMANN`\nIndicates a Von Neumann neighborhood type.\nConstant Field Values\n• ### Constructor Detail\n\n• #### KohonenSOM\n\n```public KohonenSOM(int dim,\nint nrow,\nint ncol)```\nConstructor for a `KohonenSOM` object.\nParameters:\n`dim` - An `int` scalar containing the number of weights for each node in the node grid. `dim` must be greater than zero.\n`nrow` - An `int` scalar containing the number of rows in the node grid. `nrow` must be greater than zero.\n`ncol` - An `int` scalar containing the number of columns in the node grid. `ncol` must be greater than zero.\n• ### Method Detail\n\n• #### forecast\n\n`public int[] forecast(double[] input)`\nReturns a forecast computed using the `KohonenSOM` object.\nParameters:\n`input` - A `double` array containing the input data. `input.length` must be equal to `dim`.\nReturns:\nAn `int` array of length 2 containing the (i, j) index of the output node.\n• #### forecast\n\n`public int[][] forecast(double[][] input)`\nReturns forecasts computed using the `KohonenSOM` object.\nParameters:\n`input` - A `double` matrix containing `input.length` observations of data. `input[i].length` must be equal to `dim`.\nReturns:\nAn `int` matrix containing the output indices of the nodes. The i-th row contains the (i, j) index of the output node for `input[i]`.\n• #### getDimension\n\n`public int getDimension()`\nReturns the number of weights for each node.\nReturns:\nAn `int` scalar containing the number of weights for each node.\n• #### getGridType\n\n`public int getGridType()`\nReturns the grid type.\nReturns:\nAn `int` scalar containing the grid type. The return value is either `KohonenSOM.GRID_RECTANGULAR` or `KohonenSOM.GRID_HEXAGONAL`\n• #### getNeighborhoodType\n\n`public int getNeighborhoodType()`\nReturns the neighborhood type for the rectangular grid.\nReturns:\nAn `int` scalar containing the neighborhood type. The return value is either `KohonenSOM.TYPE_VON_NEUMANN` or `KohonenSOM.TYPE_MOORE`\n• #### getNumberOfColumns\n\n`public int getNumberOfColumns()`\nReturns the number of columns of the node grid.\nReturns:\nAn `int` scalar containing the number of columns of the node grid.\n• #### getNumberOfRows\n\n`public int getNumberOfRows()`\nReturns the number of rows of the node grid.\nReturns:\nAn `int` scalar containing the number of rows of the node grid.\n• #### getWeights\n\n`public double[][][] getWeights()`\nReturns the weights of the nodes.\nReturns:\nAn `nrow` by `ncol` matrix of double arrays containing the weights of the nodes.\n• #### getWeights\n\n```public double[] getWeights(int i,\nint j)```\nReturns the weights of the node at (i, j) in the node grid.\nParameters:\n`i` - An `int` scalar containing the row index of the node in the node grid, where", null, ".\n`j` - An `int` scalar containing the column index of the node in the node grid, where", null, ".\nReturns:\nA `double` array containing the weights of the node at (i, j) in the node grid.\n• #### isWrapAround\n\n`public boolean isWrapAround()`\nReturns whether the opposite edges are connected or not.\nReturns:\nA `boolean` indicating whether or not the opposite edges are connected. It is true if the opposite edges are connected. Otherwise, it is false.\n• #### setGridType\n\n`public void setGridType(int type)`\nSets the grid type.\nParameters:\n`type` - An `int` scalar containing the grid type, rectangular (`KohonenSOM.GRID_RECTANGULAR`) or hexagonal (`KohonenSOM.GRID_HEXAGONAL`).\n\nDefault: `type` = `GRID_RECTANGULAR`.\n\n `type` Description `GRID_RECTANGULAR` Use a rectangular grid (`type` = 0). `GRID_HEXAGONAL` Use a hexagonal grid (`type` = 1).\n• #### setNeighborhoodType\n\n`public void setNeighborhoodType(int type)`\nSets the neighborhood type.\nParameters:\n`type` - An `int` scalar containing the neighborhood type, Von Neumann (`KohonenSOM.TYPE_VON_NEUMANN`) or Moore (`KohonenSOM.TYPE_MOORE`). This method is ignored for a hexagonal grid.\n\nDefault: `type` = `TYPE_VON_NEUMANN`.\n\n `type` Description `TYPE_VON_NEUMANN` Use the Von Neumann (`type` = 0) neighborhood type. `TYPE_MOORE` Use the Moore (`type` = 1) neighborhood type.\n• #### setWeights\n\n`public void setWeights()`\nSets the weights of the nodes using random numbers. The weights are in [0.0, 1.0].\n• #### setWeights\n\n`public void setWeights(double[][][] weights)`\nSets the weights of the nodes.\nParameters:\n`weights` - An `nrow` by `ncol` matrix of double arrays containing the weights of the nodes. `weights[i][j].length` must be equal to `dim`.\n• #### setWeights\n\n```public void setWeights(int i,\nint j,\ndouble[] weights)```\nSets the weights of the node at (i, j) in the node grid.\nParameters:\n`i` - An `int` scalar containing the row index of the node in the node grid, where", null, ".\n`j` - An `int` scalar containing the column index of the node in the node grid, where", null, ".\n`weights` - A `double` array containing the weights. `weights.length` must be equal to `dim`.\n• #### setWeights\n\n`public void setWeights(Random random)`\nSets the weights of the nodes using a `Random` object. The weights are generated using the `Random.nextDouble` method.\nParameters:\n`random` - A `Random` object used to generate random numbers for the nodes.\n• #### wrapAround\n\n`public void wrapAround()`\nSets a flag to indicate the map should wrap around or connect opposite edges. A hexagonal grid must have an even number of rows to wrap around. By default, opposite edges are not connected.\nJMSLTM Numerical Library 7.2.0" ]	[ null, "https://help.imsl.com/java/7.2/manual/api/com/imsl/datamining/eqn_0146.png", null, "https://help.imsl.com/java/7.2/manual/api/com/imsl/datamining/eqn_0147.png", null, "https://help.imsl.com/java/7.2/manual/api/com/imsl/datamining/eqn_0148.png", null, "https://help.imsl.com/java/7.2/manual/api/com/imsl/datamining/eqn_0149.png", null, "https://help.imsl.com/java/7.2/manual/api/com/imsl/datamining/eqn_0150.png", null, "https://help.imsl.com/java/7.2/manual/api/com/imsl/datamining/eqn_0151.png", null, "https://help.imsl.com/java/7.2/manual/api/com/imsl/datamining/eqn_0152.png", null, "https://help.imsl.com/java/7.2/manual/api/com/imsl/datamining/eqn_0153.png", null, "https://help.imsl.com/java/7.2/manual/api/com/imsl/datamining/eqn_0154.png", null, "https://help.imsl.com/java/7.2/manual/api/com/imsl/datamining/eqn_0157.png", null, "https://help.imsl.com/java/7.2/manual/api/com/imsl/datamining/eqn_0158.png", null, "https://help.imsl.com/java/7.2/manual/api/com/imsl/datamining/eqn_0155.png", null, "https://help.imsl.com/java/7.2/manual/api/com/imsl/datamining/eqn_0156.png", null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.72477967,"math_prob":0.90932024,"size":8499,"snap":"2022-27-2022-33","text_gpt3_token_len":2027,"char_repetition_ratio":0.17245439,"word_repetition_ratio":0.18027961,"special_character_ratio":0.21731968,"punctuation_ratio":0.11126006,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9900784,"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],"im_url_duplicate_count":[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\":\"2022-08-15T04:34:36Z\",\"WARC-Record-ID\":\"<urn:uuid:cb5a29d8-b381-4490-823f-cb074f48b349>\",\"Content-Length\":\"35900\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:41cbcc3b-5e09-4457-b95f-66e0becab23f>\",\"WARC-Concurrent-To\":\"<urn:uuid:30cd044f-5c27-474c-8ba4-17efbd9ceebe>\",\"WARC-IP-Address\":\"13.91.82.80\",\"WARC-Target-URI\":\"https://help.imsl.com/java/7.2/manual/api/com/imsl/datamining/KohonenSOM.html\",\"WARC-Payload-Digest\":\"sha1:7J64G2JOBV3ND44PV27MW4P2VYRCMWEL\",\"WARC-Block-Digest\":\"sha1:CBVOEPBNSHBDWV37TIOPX7O3AXQHXG25\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2022/CC-MAIN-2022-33/CC-MAIN-2022-33_segments_1659882572127.33_warc_CC-MAIN-20220815024523-20220815054523-00228.warc.gz\"}"}
https://telliott99.blogspot.com/2009/08/	[ "## Monday, August 31, 2009\n\n### Connecting ln(x) and 1/x\n\nI have more from Strang. We're going to look at the the limit of Δy/Δx in the context of the exponential. Following the book, we use the symbol h for Δx, the small change in x. So we have:\n\n `dy/dx = lim(h->0) [y(x+h) - y(x)] / h]`\n\nIn this case, y(x) = bx so we want\n\n `dy/dx = lim(h->0) [ bx+h - bx ] / h ]`\n\nAs he says, the key idea is that we can split the first part\n\n `bx+h = bx bh`\n\nand then factor out bx and move it outside the limit:\n\n `dy/dx = bx lim(h->0) [bh - 1] / h ]`\n\nThe term in the limit is something. It depends on what the limit converges too. However, the important thing is that it is not a variable but a constant. Thus,\n\n `dy/dx = c bx`\n\nThis is the equation that I used in the last post. Starting from here there is a quick demonstration of something that we proved in a more roundabout fashion before. Start with the equation above and invert it:\n\n `dx/dy = 1 / c bx`\n\nbut\n\n `y = bxx = logb ydx/dy = 1/cyd/dy logb(y) = 1/cy`\n\nBeautiful! Strang says \"that proof was...powerful too quick.\" So he does the following.\n\n `y = bxf(y) = xf(bx) = xf'(bx) (c bx) = 1 # this is the chain rulef'(bx) = 1 / (c bx) # identify bx as yf'(y) = 1/cy`\n\nf(y) converts y to x, it takes the logarithm to base b of x.\n\n### More about e", null, "Previously I posted about a few of the remarkable properties of e, the base of natural logarithms. I said that ex is the function that is the derivative of itself.\n\nActually, that's not the whole truth. Every function bx is the derivative of itself, you just have to multiply by a constant. The thing is that the constant is 1 when b = e. As you can see in the code below, we obtain the slope of the curve bx at x as loge b * bx. The constant to multiply by is loge b, which equals 1 when b = e.\n\nFor more details, see Chapter 6 in Strang.\n\n `color.list=c( 'red','magenta', 'cyan','thistle4', 'salmon')plot(1:15,xlim=c(0.5,1.8), ylim=c(0,15),type='n')L = 2:6L2 = c(1,1.5)for (i in 1:length(L)) { b = L[i] col=color.list[i] curve(bx,lwd=10, add=T,col=col) for (j in 1:length(L2)) { x = L2[j] dx = 0.35 x0 = x - dx x1 = x + dx y = bx points(x,y,pch=16, col='blue',cex=2) m = log(b)y # slope is const y y0 = y - mdx y1 = y + mdx lines(c(x0,x1), c(y0,y1),lwd=3, col='blue') } }`\n\n## Sunday, August 30, 2009\n\n### Slope of the sine curve\n\nIn another post about representing the sine and cosine as infinite series, I tried to show how the series form easily demonstrates the correctness of the derivatives of these functions, that if\n\n `y = sin(x)dy/dx = cos(x)y = cos(x)dy/dx = -sin(x)`\n\nBut unfortunately this argument is circular (I think), since the series comes from a Taylor series, which is generated using the first derivative (and the second, third and more derivatives as well).\n\nIn this post, I want to do two things. Strang has a nice picture which makes clear the relationship between position on the unit circle and speed. This is it:", null, "The triangles for velocity and position are similar, just rotated by π / 2.\n\nIt is clear from the diagram that at any point, the y-component of the velocity is cos(t), while the y-position is sin(t). Thus, the rate of change of sin(t) is cos(t). This is the result we've been seeking. Similarly, the rate of change of the x-position, cos(t), is -sin(t).\n\nStrang also derives this result more rigorously starting on p. 64. That derivation is a bit complicated, although not too bad, and I won't follow the whole thing here. It uses his standard approach as follows:\n\n `dy / dx = lim(h -> 0) [sin(x + h) - sin(x)] / h`\n\nApplying a result found using just the Pythagorean theorem earlier (p. 31) for sin (s + t):\n\n `sin(s + t) = sin(s) cos(t) + cos(s) sin(t)cos(s + t) = cos(s) cos(t) - sin(s) sin(t)`\n\nHe comes up with this expression for:\n\n `Δy / Δx = sin x [cos(h) - 1 / h] + cos x (sin h / h)`\n\nThe problem is then to determine what happens to these two expressions in the limit as h -> 0. The first one is more interesting. As h gets small, \| cos(h)-1 \| gets smaller like h2, so the ratio goes to 0.\n\nR can help us see better.\n\n `h = 1for (i in 1:6) { h = h/10 print (h) print (cos(h)-1) }`\n\n ` 0.1 -0.004995835 0.01 -4.999958e-05 0.001 -5e-07 1e-04 -5e-09 1e-05 -5e-11 1e-06 -5.000445e-13`\n\nHere is the plot for the second one, which converges to 1, leaving us with simply cos(x):", null, "`f<-function(x) { x }curve(sin(x),from=0,to=0.5, ylim=c(0,0.6), lwd=3,col='red')curve(f,from=0,to=0.5, lwd=3,col='blue',add=T)`\n\n## Friday, August 28, 2009\n\n### Feynman and Kepler\n\nWhen I was in college I bought a book containing lectures that Richard Feynman gave at Cornell in 1964 (the Messenger series). The book is called The Character of Physical Law. Quite simply, it was and is wonderful.\n\nNow, years later it turns out that the lectures were taped, and that after some wrangling, a famous guy (let's call him Bill) bought the rights to this material. Recently, Bill made the video available online. There's just one small problem. I promised myself I would never again install any software made by Bill and his friends on my computer. But, in order to view the lectures, he requires me to install the Silverlight plug-in for my browser (it's undoubtedly a DRM thing). In typical Bill fashion, if you go to the link for the videos, the download of the plug-in starts automatically. Long story short---I held my nose and did it. So, thanks for the lectures (link).\n\nIn Feynman's second lecture (the Relation of Mathematics and Physics) he talks about Kepler's Second Law: \"A line joining a planet and the sun sweeps out equal areas during equal intervals of time.\" Feynman uses an argument based on vectors to show this. I didn't understand what Feynman said (and the book is just a transcription of the talk), but I googled around and found a post here. Unfortunately, the post is truncated at the critical point. The author was kind enough to write me with a detailed explanation. In the spirit of this blog as the place I post my homework for a self-taught course in anything I'm interested in, here is my version of his explanation of Feynman's explanation of Kepler's law.", null, "Feynman uses the cross-product of two vectors.\n\nIn the diagram, the vectors A and B originate at the same point and the angle between them is θ. The cross-product A x B is a vector with magnitude = \|A\| \|B\| sin θ and its direction is perpendicular to A in the plane formed by A and B. The area of the triangle formed by A and B is one-half the magnitude of the cross-product. (I wish I knew how to produce the arrows over the labels for the vectors using html but I don't, sorry).\n\nIn the context of our problem, A is the vector from the sun to our planet at time-zero, and B is the vector at time dt. The magnitude of B is not equal to that of A, in general, because the orbit is an ellipse. So we can replace the labels by A = r and B = r + dr. The area being swept out (or its tiny component dA in a very short time dt) is proportional to the cross-product (neglect the factor of one-half):\n\n `dA = r x (r + dr)`", null, "or, as Feynman wrote using the dot notation:\n\n `. .A = r x r`\n\nNow, what we are interested in is the proposition that the rate-of-change of the area is constant, that the rate-of-change of the rate-of-change of the area is zero.\n\n `..A = d2A/dt2 = 0`\n\nSo, we need to differentiate again with respect to time, and, apparently, we can use the product rule from ordinary differentiation on this cross-product:\n\n `.. .A = d/dt (r x r) .. . . = r x r + r x r`\n\n(As Feynman says: it's just playing with dots). The second term has the cross-product of a vector with itself, θ is zero and sin θ is zero so the whole thing equals zero.\n\nThe first term has the cross-product of r with the acceleration, the rate-of-change of the rate-of-change of r with time. That is equal to F/m.\n\nBut the thing is that the force acts along the radius, so now we have the cross-product of a vector with another vector that is turned by 180° from it. This product is also zero, and therefore Ä = 0.\n\n### Area of a circle: simple calculus", null, "I posted some time ago about a formula to find the area of the circle (knowing its circumference). I think it's credited to Euclid, although Google isn't being my friend at the moment, so I'm not sure.\n\nTo brush up on calculus, let's explore methods to find the area (as described in Strang's Calculus. He says:\n\"The goal here is to take a first step away from rectangles...that is our first step toward freedom, away from rectangles to rings.\"\n\nIn the example on the left panel of the figure, imagine slicing up the circle in the way of any standard integration problem to find the area under a curve. Simplify by considering only the area above the dotted line (y = 0). If the radius is R, then we can write y as a function of x:\n\n `y = (R2 + x2)1/2`\n\nImmediately, we run into a problem. I don't know a function which gives this as its derivative, so I don't know how to integrate it. Let's leave it until the end.\n\nIn the second approach, we picture the circle as being built up out of rings. As the radius r varies from 0 to R, the circumference of the ring is 2πr and we need to integrate:\n\n `A = ∫ 2πr dr = πr2 + C`\n\nevaluated between r = 0 and r = R:\n\n `A = πR2`\n\nAs Strang emphasizes, now we can see the geometrical reason why the the derivative of the area is the circumference! How fast does the area grow? It grows in proportion to the circumference.", null, "Exactly the same argument explains why the surface area of a sphere (4πr2) is the derivative of the volume (4/3πr3).\n\nThe third approach is to visualize the circle being sliced like a pie. This is really the same method we used in the previous post, which employed calculus on the sly. Now we have a series of triangles. The angle at the vertex of each thin triangle is dθ. The height is just R, and the base of each triangle is proportional to R, it is R dθ. So the area of the thin triangle is:\n\n `dA = 1/2 R2 dθ`\n\nWe integrate (R is a constant) and evaluate between θ = 2π and θ = 0:\n\n `A = 1/2 R2 θA = 1/2 R2 2πA = π R2`\n\nBack to the first method. We will try numerical integration. We will compute the area of the quarter-circle between x = 0 and x = R, and then simplify even further by considering just the unit circle with R = 1. We do this in a naive way, dividing the figure into a large number of thin segments, and calculate:\n\n `y = (1 - x2)1/2`\n\nWe get a better approximation if we divide the very first interval in half. In R:\n\n `y <- function(x) { sqrt(1 - x*2) }d = 0.00001x = seq(0,1,by=d)f = y(x)S = sum(f[-1]) + 0.5fS = Sd`\n\n `> S 0.7853982> S4 3.141593`\n\n## Thursday, August 27, 2009\n\n### Series for sine and cosine", null, "You probably know that the sine and cosine functions can also be given as a series:\n\n `sin(x) = x - x3/3! + x5/5! - x7/7! ...cos(x) = 1 - x2/2! + x4/4! - x6/6! ...`\n\nAnd furthermore, that the exponential function can also be given as a series:\n\n `ex = 1 + x/1! + x2/2! + x3/3! + x4/4! ...`\n\nThese series are really interesting. One reason is that they make clear that the formulas for the derivatives of these functions are obviously correct.\n\nTake e first:\n `ex = 1 + x/1! + x2/2! + x3/3! + x4/4! ...d/dx ex = 0 + 1/1 + 2x/2 + 3x2/(32!) + 4x3/(43!) ... = 1 + x/1! + x2/2! + x3/3! ... = ex`\n\nNow, the sine:\n\n `sin(x) = x - x3/3! + x5/5! - x7/7! ...d/dx sin(x) = 1 - 3x2/32! + 5x4/54! - 7x6/76! ... = 1 - x2/2! + x4/4! - x6/6! ... = cos(x)`\n\nThat's really spectacular. So the question I had was: where do these series come from. And I'm not sure this is the only way, but I think they come from a Taylor series. The Taylor series approximates a function f(x) at a particular point x = a by the series:\n\n `f(x) ≈ Σ (n=0 to n=∞) f(n)(a)/n! (x-a)n`\n\nwhere f(n)(a) is the nth derivative of f(x). For ex, all of these derivative terms are just ex. Evaluating the series for a = 0 these derivatives are 1, and the series becomes (as we had above):\n\n `f(x) ≈ Σ (n=0 to n=∞) xn / n!`\n\nNow, I still have a couple of problems. For example, in order to get the terms of the Taylor series for sine and cosine we will have to know the derivatives, which is also what we seek, so that is a circular argument. Also, I'm not very clear on what the a is about. The approximation to f(x) works for the \"neighborhood of a.\"\n\nR code:\n `plot(1:2pi,ylim=c(-1.2,1.2),type='n')curve(sin,from=0, to=2pi, col='blue',lwd=5)curve(cos,from=0, to=2pi, col='red',lwd=5,add=T)lines(c(0,2pi),c(0,0),lty=2,lwd=3)`\n\n## Wednesday, August 26, 2009\n\n### Duly Quoted\n\nHere's a funny (and telling) story about a job interview someone had with Johnny von Neumann (and my post here):\n\n\"Von Neumann lived in this elegant lodge house on Westcott Road in Princeton... As I parked my car and walked in, there was this very large Great Dane dog bouncing around on the front lawn. I knocked on the door and von Neumann, who was a small, quiet, modest kind of a man, came to the door and bowed to me and said, 'Bigelow, won't you come in,' and so forth, and this dog brushed between our legs and went into the living room. He proceeded to lie down on the rug in front of everybody, and we had the entire interview---whether I would come, what I knew, what the job was going to be like---and this lasted maybe forty minutes, with the dog wandering all around the house. Towards the end of it, von Neumann asked me if I always traveled with the dog. But of course it wasn't my dog, and it wasn't his either, but von Neumann---being a diplomatic, middle-European type person---he kindly avoided mentioning it until the end.\"\n\n--Julian Bigelow\nquoted in:\nWho got Einstein's Office\nEd Regis\np 110\n\n### Towers of Hanoi", null, "This famous mathematical game is described in wikipedia. Briefly, we have three pegs and N disks. We wish to move the whole stack of disks to a different peg, let's say # 3. The rules are:\n\n• Only one disk may be moved at a time.\n• Each move consists of taking the upper disk from one of the rods and sliding it onto another rod, on top of the other disks that may already be present on that rod.\n• No disk may be placed on top of a smaller disk.\n\nFor example, here is an intermediate stage in the game with 4 disks. Can you see the three moves that brought us to this point?", null, "A couple of interesting things: one is the recursive nature of Towers of Hanoi. If you already know how to solve the game with N disks, then how do you solve the game with N+1?\n\nEasy, move the first N disks to peg # 2, then move disk N+1 to peg # 3, then move all the other N disks on top. This stereotyped pattern leads to the following visual aid. (I've forgotten where I saw it). It looks like a binary ruler.", null, "This version of the ruler describes the series of moves (though not the target pegs) for the N=4 game. To extend it, add a new bar of the correct height for N = 5, then duplicate all the bars we already have.\n\nThe number of moves grows rapidly:\n\n `N moves1 12 33 74 15N 2N-1`\n\nAccording to wikipedia:\n\nThe puzzle was invented by the French mathematician Édouard Lucas in 1883. There is a legend about a Vietnamese temple which contains a large room with three time-worn posts in it surrounded by 64 golden disks. The monks of Hanoi, acting out the command of an ancient prophecy, have been moving these disks, in accordance with the rules of the puzzle, since that time. The puzzle is therefore also known as the Tower of Brahma puzzle. According to the legend, when the last move of the puzzle is completed, the world will end.\n\nWe can calculate, at one move per second, this game will take roughly 292 billion years, about 20 times the current age of the universe.\n\n `>>> x = 360024365>>> x31536000>>> 2*63/x292471208677L`\n\n### Volume of a sphere: calculus", null, "This derivation is straight from the book Strang's Calculus. We want to calculate the volume of a hemi-sphere as shown in the figure. The cross-sections of this solid are semi-circles. The area of each semicircle is:\n\n `A = 1/2 πr2 = 1/2 π(R2-x2)`\n\nTo find the volume, we add up (integrate) the areas of each little slice:\n\n ```V = ∫ A(x) dx = ∫ 1/2 π(R2-x2) dx = 1/2 π [ R2x - 1/3 x3 ]```\n\nEvaluate the expression in brackets between x = -R and +R:\n\n ```= [R3 - 1/3 R3] - [-R3 + 1/3 R3] = 2/3 R3 - - 2/3 R3 = 4/3 R3 V = 1/2 π 4/3 R3 = 2/3 π R3```\n\nThe total volume is twice this, or\n\n `V = 4/3 π R3`\n\nThe volume can be found in either cylindrical or spherical coordinates. For spherical coordinates we have:\n\n ```x = r cosθ sinφ y = r sinθ sinφ z = r cosφ```", null, "That is, φ is the angle of the vector r with the z-axis, so the z-component is r cosφ. The component orthogonal to that is the projection of r on the x,y-plane, and that is r sinφ. Then, θ is the angle of that projection (and r) with respect to the x-axis. The x- component is r sinφ cosθ (the order is not important), while the y-component is r sinφ sinθ.\n\nIn this case, the integral is a triple integral:\n\n ```2π π R V = ∫ ∫ ∫ r2 sinφ dr dφ dθ 0 0 0```", null, "From reading Chapter 14 on multiple integrals in Strang, it seems that what we do here is to first integrate with respect to r, holding the angles constant:\n\n ```2π π V = 1/3 R3 ∫ ∫ sinφ dφ dθ 0 0```\n\nUPDATE: A sharp-eyed reader caught a silly error in the part that follows, so it's been edited. I had set up the problem in the first half (see the figure at the beginning of the post) to use the hemisphere. But in the second part we have two angles, θ and φ, with different limits of integration.\n\nContinuing with the standard approach in two dimensions, we let θ go from 0 to 2π. Now φ will go from 0 to π. Imagine rotating the great circle in the xy-plane around the x-axis: we need only to go from 0 to π. We could do the hemisphere by having φ go from 0 to π/2, but it would be a weird volume, with two wedges joined at the x-axis.\n\nSo, we have the integral of sinφ dφ, which is - cosφ, evaluated between φ = 0 and φ = π =>\n ```= -cos(pi) - -(cos(0)) = -(-1) + 1 = 2 2π V = 2/3 R3 ∫ dθ 0```\n\nBut this is just θ evaluated between θ = 0 and θ = 2π => 2π, so finally we have:\n\n `V = 4/3 R3 π`\n\n(Figures are from the book)\n\nUPDATE 2: Coming back a couple of years later, I notice Google has made changes to blogger (editing the html) that retroactively messed up the formatting on this post. The limits of integration aren't shown at the correct column offset any more. My apologies, but I don't see a simple fix at the moment.\n\n### Archimedes: volume of a sphere\n\nArchimedes was one of those rare people in the history of the world who overwhelm you with their genius. Like few others, he discovered not just one but a large number of really important things. The discovery of which he was probably most proud was the method to find the volume (and surface area) of a sphere. He found that the volume of a sphere is 2/3 the volume of the cylinder that just contains it.\n\nThis was symbolized by the sphere and cylinder on his tombstone, as witnessed (years later) by Cicero. We have no idea what Archimedes looked like, but that doesn't keep people from drawing his portrait!\n\nWhat strikes me most vividly about the discovery is that Archimedes found the correct relationship by experiment---he weighed the solids, and not only that, he used the law of the lever. According to this page, the balancing was done in a fairly complex manner. We have a cylinder that can just contain the sphere and a cone whose radius and height are equal and twice the radius of the sphere. Moreover, the density of the cylinder is four times that of the other two objects.", null, "Then, by the law of the lever, the weight of the cylinder is twice the combined weights of the sphere and the cone together (an equal force from gravity when suspended at half the distance from the fulcrum). Because the density of the cylinder is 4 times greater, its volume must be also one-half the combined volumes of the sphere and the cone.\n\nWe can check that using the (now) known formulas (see my post about the cone):\n\n `Vcylinder = 2rπr2 = 2 πr3Vsphere = 4/3 πr3Vcone = 1/3 π(2r)(2r)2 = 8/3 πr3Vsphere + Vcone = 4 πr3`\n\nAccording to Archimedes in the Method (translation by Heath):\n\"For certain things which first became clear to me by a mechanical method had afterward to be demonstrated by geometry...it is of course easier, when we have previously acquired by the method some knowledge of questions, to supply the proof than it is to find the proof without any previous knowledge. This is a reason why, in the case of the theorems the proof of which Eudoxus was the first to discover, namely, that the cone is a third part of the cylinder, and the pyramid a third part of the prism, having the same base and equal height, we should give no small share of the credit to Democritus, who was the first to assert this truth...though he did not prove it.\"\n\nOnce he knew the correct answer, he was able to find his way to a rigorous derivation. Very smart.\n\nThere two things that make me wonder about the story. One is: why not just weigh objects of equal density like this:", null, "We get 4/3 for the sphere, 2/3 1/3 for the cone, and 2 for the cylinder. It should work. (Oops, see below). My guess is that Archimedes is just showing off. Note that he would have used his principle of buoyancy to determine the correct densities (and for that matter, could use the law of the lever to correct if the cylinder's density was not precisely 4x the others).\n\nThe second question is: what materials would he use? What has a density 4 times something else? From wikianswers we have:\n\n `Sand 2.80Copper 8.63Silver 10.40Gold 19.30Marble 2.56`\n\nHow about marble and silver?\n\n[UPDATE: Almost two years later, I find a silly error in this post. You'd need two cones, or put the one out at 2x the distance on the lever. Why didn't someone tell me?]\n\n### Volume of a cone: simple calculus\n\nI've been brushing up on my calculus skills, though it can be challenging for an old fart like me. (In one ear and out the other). I saw a book online (Strang's Calculus), which I like because it is succinct, and also the author has a very interesting, idiosyncratic style. I liked it so much I got a hard copy. Now that I've studied a bit, I realize that the book is more than succinct, it's dense, filled with math, and contains a lot more than just calculus. It's a great book.\n\nIn Chapter 8, Applications of the Integral, we encounter Example 11, to find the volume of a cone. I've redrawn the diagram from the book, below. The idea is to move vertically from the top to the bottom, letting x be equal to the radius at each point. At each value for x, we draw a shell (the outside surface of a cylinder) as shown in red. As we move from top to bottom, we accumulate a collection of these cylinders whose volumes are summed to get the volume of the cone.\n\nThe key is to recognize that the distance from the top of the cone to the top of each cylinder has the same relationship to x as b does to r (e.g. when x = r, then this distance = b). The smallest and largest rectangles in the figure are similar.\n\nThe height of the cylinder plus this distance equals b.", null, "So the height of each cylinder is\n\n `h = b-bx/r`\n\nFrom this point it's easy. The area of the cross-section of each shell is its diameter (2x), times &pi times the small width dx; the volume is this area times the height. I'm going to leave out the multiplication symbol I usually use () for clarity:\n\n ` = 2πxhdx = 2πx(b-bx/r)dx`\n\nWe sum (integrate) all these cylinders:\n\n ` = ∫2πx(b-bx/r) dx = ∫2πxb dx - ∫2πxb(x/r) dx = πx2b - (2/3)πx3b/r`\n\nevaluated between x = 0 and x = r:\n\n ` = πr2b - (2/3)πr2b = (1/3)πr2b`\n\n### Volume of a cone: geometric method\n\nWe're on the trail of Archimedes. Last time we found a formula for the sum of the squares of the integers from 1 to n:\n\nS = n(n+1)(2n+1)/6\n\nWe're going to use that in a derivation of the formula for the volume of a cone. To start with, we slice the cone horizontally. In the limit as the number of slices gets very large and each individual slice gets very thin, the triangular shaped wedge pieces and the little bit at the top won't contribute significantly to the volume.\n\nHere is the diagram from the post:\n\n ` + /\|\\ / \|h\\ +--+--+ /\| \|h \|\\ / \| \|r1\| \\ +--+--+--+--+ /\| \|h \|\\ / \| \| r2 \| \\ +--+-----+-----+--+ /\| \|h \|\\ / \| \| r3 \| \\ +--+--------+--------+--+ /\| \|h \|\\ / \| \| r4 \| \\ +--+-----------+-----------+--+ R`\n\nAs Dr. Peterson explains\n\nIf the cone has base radius R and height H, and we've cut it into N slices (including that empty slice at the top, with radius r0 = 0), then each cylinder will have height h = H/N, and radius r[k] = kR/N, where k is the number of the cylinder, starting with 0 at the top and ending with N-1 for the bottom cylinder.\n\nThe only thing that's tricky about this is the numbering. The triangle at the top will be ignored. This includes the cylinder numbered k = 0 with radius = kR/N = 0. The next section below it contains a cylinder with h for its height and r1 for its radius. Since we started counting at 0, the last of the N segments has k = N-1.\n\nThe volume of each individual cylinder will be:\n\n ` = π * h * rk2 = π * H/N * (kR/N)2 = π * R2 * H * k2 / N3`\n\nThe total volume will be the sum of these, for all k from 0 to N-1; since only k is different from one cylinder to the next, we can factor everything else out from the sum and get:\n\n `V = [π * R2 * H / N3 ] * Sum(k2)Sum(k2) = 0 + 1 + 4 + ... + (N-1)2`\n\nUse our formula from last time:\n\n `Sum(k2) = (N-1)(N)(2N-1) / 6`\n\nDivide by N3:\n\n `Sum(k2) = (1-1/N)(1)(2-1/N) / 6`\n\nAs N gets very large this converges to 2/6 = 1/3, so we have finally:\n\n `V = πR2H / 3`\n\nVery nice.\n\n## Tuesday, August 25, 2009\n\n### Sum of squares\n\nI got interested in learning more about Archimedes. In particular, I was amazed by his derivation of the formula for the volume of a sphere. Actually he was pretty proud of it himself---according to legend and confirmed by Cicero's testimony his gravestone \"was surmounted by a sphere and a cylinder.\"\n\nSo that's where I'm headed, but in order to get there, we need to start with a simpler problem. What is the sum of all the values n2 for n = 1 to n = k? The squares and the sum go like this:\n\n `1 14 59 1416 3025 5536 9149 14064 204`\n\nIt is hard to see the pattern, so let's look up the answer, the formula is:\n\nsum of squares = k * (k+1) * (2k+1) / 6\n\nThere is a neat proof by induction in the post. We assume that the formula is correct for n = k and then prove that if so, it is also true for n = k+1. We must also prove that it works for the \"base case\", n = 1, but that is self-evident.\n\nWe have:\n\n `n = k Σ n2 = k * (k+1) * (2k+1) / 6n = 1`\n\nTo find the sum of squares for k+1, we add (k+1)2 to both sides. Here is the right-hand expression:\n\n `k * (k+1) * (2k+1) / 6 + (k+1)(k+1)`\n\nFactor out (k+1)\n\n `(k+1) * [ k * (2k+1) / 6 + (k+1) ](k+1) * [ k * (2k+1) + 6k + 6 ] / 6(k+1) * [ 2k2 + 7k + 6 ] / 6`\n\nThe term in the brackets can be factored to give:\n\n `(k+2) * (2k+3)`\n\nwhich can be rearranged to\n\n `[(k+1) + 1] * [2(k+1) + 1]`\n\nSo we have, finally:\n\n `(k+1) [(k+1) + 1] * [2(k+1) + 1] / 6`\n\nwhich is what we wanted to prove.\n\nThere is an even better proof in the link which uses a \"telescoping sum.\"\n\n## Monday, August 24, 2009\n\n### Duly quoted\n\nDo you have ideas in your head that have been there forever, and then when you really need to know, where did it come from, you can't find the source? That's the way I feel about the subject of this post. I searched for the source of one such idea, but found only this:\n\n\"Life is an endless series of experiments.\"\n\n---Mohandas K Gandhi (link)\n\nWhile this sounds nice, of course it is fundamentally wrong. As one of my early mentors, Bart Sefton, often told me:\n\n\"Every good experiment comes with a control.\"\n\nAnd the quote I was searching for, which I have always attributed to Milan Kundera, goes like this:\n\n\"That's the trouble with life, you never do the control.\"\n\nI thought it was in The Unbearable Lightness of Being (link), but I can't find it.\n\n## Saturday, August 22, 2009\n\n### The fly and the train, contd.\n\nThinking about the famous story involving von Neumann and the infinite series, I guessed that if we have a series where:\n\neach succeeding term xn+1 is produced from the preceding term xn by multiplying by the factor 1/r, then the sum of all the terms xn+1 + xn+2... is equal to xn 1/(r-1).\n\nSo, I ran a Python simulation and it turns out to be correct!\n\n `def test(r): n = 100 # works for other n as well rL = [n] f = 1.0/r for i in range(10): rL.append(rL[-1]f) print str(r).ljust(3), print round(n1.0/sum(rL[1:]),2) for r in range(2,10): test(r)`\n\n `2 1.03 2.04 3.05 4.06 5.07 6.08 7.09 8.0`\n\nIf I'm reading the article correctly, this turns out to be a simple result from the theory of geometric series, and it looks like it works even for fractional r. Modifying the code above to test fractional r shows this is true. Department of \"learn something new every day\"....\n\nMy guess is that von Neumann saw all three pieces instantly:\n• the first term and step size of the series\n• the formula for the terms after the first one\n• the easy way\n\n### von Neumann\n\nI'm reading a biography of Johnny von Neumann by Norman Macrae. Or maybe I should say the biography, it's not like there are many choices for biographies of this great mathematician and early computer scientist. The book is not horrible and it's the only game in town, so I am reading it. It's based on material collected by Stephen White---who was somehow involved with Stan Ulam in an earlier version of the project.\n\nThe basic problem with the book is that Macrae is a whack job. To quote Mark Yasuda's review at Amazon:\nBy far the biggest problem, however, comes from MacRae's approach to the book - he insists upon inserting so much of his own world views and dogma into the body of the book, that we no longer have a biography on Von Neumann - we have Von Neumann's life used as a vehicle for MacRae's own personal views on education, politics, the Japanese economy of the 1960's through the 1980's (I never expected to see this in a Von Neumann biography), and cold war history. He takes time out to provide slanted views of Bertrand Russell and Norbert Wiener, for no reason (they barely figure in the book beyond his distorted descriptions of them) other than to insinuate that their liberal viewpoints are due to poor parenting. In sum, the book's most fatal flaw is that there's entirely too much of MacRae, and not enough Von Neumann.\n\nLeaving all that aside, the book contains yet another retelling of the famous story about von Neumann and his approach to this problem:\nTwo trains (or bicycles) are 20 miles apart and headed directly toward each other at a speed of 10 mph. A fly starts from the front of the train on the left, flies at a speed of 15 mph to the tip of the train on the right, then turns around immediately and flies back to the first. This cycle continues until the trains meet, ending everything. How far does the fly fly?\n\nThe story is that when posed this problem, von Neumann thought for a second and gave the answer (below). When asked how he did it, he answered \"I summed the infinite series, of course.\"\n\nThe easy way to do the calculation is to focus on the trains:\n\n1. The trains will meet in one hour.\n2. The fly will fly 15 miles in that hour.\n\nWhat was interesting to me is that, when you think about the series method, it is not that hard, and converges rapidly.\n\nIn the first cycle the fly and its target train start with a separation of 20 miles. The fly's trajectory covers the initial distance times the ratio of its velocity to the total velocity of approach (15/(10+15)):\n\nd1 = 20 miles * 0.6 = 12\n\nHowever, each of the trains also moves 12 miles * 10/15 = 8 miles during the same period, so the new distance for the second cycle is 20 - 28 = 4, or one-fifth of the original. If you follow this out you can see that the series we need to sum has its first term equal to 12 and each succeeding term is 1/5 of the preceding one. The first four terms sum to:\n\n12 + 2.4 + 0.48 + 0.096 = 14.976\n\nI think Johnny guessed at this point.\n\nCome to think of it, there is probably a theorem about infinite series that says if we add terms like this, the sum of all the terms obtained from 12 by multiplying by 1/5, that will be equal to 12 1/4. Anybody know?\n\n[UPDATE: Not sure if this was laziness (I was in a hurry) or being incredibly dumb. See wikipedia. This is the geometric series with r = 1/5 and a = 12. The sum is a times 1/(1-r) = 125/4 = 15. ]\n\n## Thursday, August 20, 2009\n\n### Duly Quoted\n\n\"Perhaps the worst plight of a vessel is to be caught in a gale on a lee shore. In this connection the following...rules should be observed:\n1. Never allow your vessel to be found in such a predicament...\"\n\nCallingham, Seamanship: Jottings for the Young Sailor\nas quoted in:\nLen Deighton, Horse Under Water\n\n## Tuesday, August 18, 2009\n\n### Duly Quoted\n\nFrom Ezra Klein:\n\nLove is what we call it when our neuroses are compatible.\n\n(link)\n\n### Gamma distribution", null, "I posted previously about the beta distribution here, here and here. We ran into it because it is the conjugate prior for Bayesian analysis of problems in binomial proportion. That's because the likelihood function is in the same form.\n\nThe beta distribution is a continuous probability distribution with two shape parameters α and β (or a and b). If we consider p as the random variable (and q = 1-p), then the unnormalized distribution is just:\n\n `f(x) = pa-1 qb-1`\n\nThe function has symmetry: if we switch a and b, and also p and q, everything would look the same. Varying the values for a and b can yield a wide variety of shapes. For large a and b, the plot approaches a normal distribution with mean = a / a+b.\n\nThe need to normalize the beta distribution brings us to the gamma distribution. This one can be viewed in a simple way but also can be fairly complex. The simple version is that, for positive integers, Γ(x) is just (x-1)!.\n\nThe normalizing constant for the beta distribution with parameters a and b is:\n\n `Γ(a+b) / (Γ(a) Γ(b))`\n\nThe gamma distribution is frequently used in phylogenetic models, for example, to model the distribution of variability during evolutionary time among different positions in a protein. The gamma distribution also has two parameters but these apparently have quite different roles.\n\nThey are called the shape parameter (k) and the scale parameter (θ). More generally, the unnormalized gamma distribution is:\n\n `f(x) = xk-1 exp { -x / θ }`\n\nWe can get an idea of the roles of the two variables by considering this:\n\n `mean = kθvariance = kθ2`\n\nLet's look at some plots obtained using different values for the two parameters. In the series below, k goes from 1 to 4, and then for each plot theta varies from 0.25, 0.5, 1, 2, 3 as the color goes from blue to green to magenta to red to maroon.", null, "", null, "", null, "", null, "`N = 6thetaL = c(0.25,0.5,1,2,3)color.list = c('blue','darkgreen', 'magenta','red','darkred')#k = 1,2,3,4plot(1:N,ylim=c(0,0.8),type='n')for (i in 1:5) { curve(dgamma(x, shape=k,scale=thetaL[i]), from=0,to=N,lwd=3, col=color.list[i],add=T) }`\n\n## Sunday, August 16, 2009\n\n### Geometric distribution", null, "According to wikipedia, the geometric distribution is the probability distribution which describes the number of Bernoulli trials required to obtain a single success.\n\nSo, in the simple case of a fair coin (p = 1/2):\n\n `P(X=1) = 1/2P(X=2) = 1/4 (1/2 times 1/2)P(X=3) = 1/8 (1/2 times 1/4)`\n\nAccording to mathworld, the geometric distribution is the only discrete memoryless random distribution, and is a discrete analog of the exponential distribution.\n\nThe memoryless property can be seen easily if we take the geometric series:\n\n `1/2 + 1/4 + 1/8 ...`\n\nIf we have already obtained a failure on the first trial, then we remove the first term (corresponding to success on the first trial) and then normalize by dividing by the sum of all remaining terms (1 - 1/2):\n\n ` = 2/4 + 2/8 + 2/16 ... = 1/2 + 1/4 + 1/8 ...`\n\nThe normalization is needed because we require the sum of all the terms to add up to 1 for a proper probability distribution. In general, for process with probability of success p and failure q = 1 - p:\n\n `P(X=k) = qk-1 * p`\n\nAccording to wikipedia, the mean is 1/p and the variance is q/p2. We can compare that to the exponential distribution with a pdf of:\n\n `λe-λx`\n\nand mean = 1/λ and variance = 1/λ2.\n\nIt is not clear to me at present why the expressions for the variance don't match up.\n\nR code:\n\n `x=numeric(7)x = 1for (i in 2:length(x)) { x[i] = x[i-1]/2 }plot(x,type='s', ylim=c(0,1.0), col='red',lwd=3)`\n\n## Saturday, August 15, 2009\n\n### Normally squared\n\nAccording to wikipedia, the chi-square distribution (with df = k), arises as the sum of the squares of k independent, normally distributed random variables with mean 0 and variance 1. For example, in a bivariate distribution where each variable is normally distributed, the square of the distance from the origin should be distributed in this way with df = 2.\n\n `u = rnorm(10000)v = rnorm(10000)w = u2 + v2B = seq(0,max(w)+1,by=1)hist(w,freq=F,ylim=c(0,0.5), breaks=B,col='cyan')curve(dchisq(x,df=2), lwd=5,col='red', from=0,to=10,add=T)`", null, "I found that I could also get the chi-squared distribution by multiplying two random variables (with mean not equal to 0) together, and it seems that in this case the degrees of freedom is equal to the product of the means.\n\n `u = rnorm(10000,3)v = rnorm(10000,3)w = uvB = seq(-5,max(w)+1,by=1)hist(w,freq=F,ylim=c(0,0.18), breaks=B,col='cyan')curve(dchisq(x,df=9), lwd=5,col='red', from=0,to=30,add=T)`", null, "Let's explore how to use R to solve the problem of the grade distribution from the last post. We have males and females with grades from A,B,D,F in order, stored in a matrix M:\n\n `m = c(56,60,43,8)f = c(37,63,47,5)M = t(rbind(m,f))`\n\n `> M m f[1,] 56 37[2,] 60 63[3,] 43 47[4,] 8 5`\n\n `r = chisq.test(M)`\n\n `> r Pearson's Chi-squared testdata: M X-squared = 4.1288, df = 3, p-value =0.2479`\n\nThis matches the value given in Grinstead & Snell. We can explore contributions of the individual categories to the statistic as follows.\n\n `E = r\\$expectedO = r\\$observed(O-E)2/E`\n\nWe see that the disparity in A's was certainly higher than for the other categories, but the p-value (above) is not significant.\n\n `> (O-E)2/E m f[1,] 1.0985994 1.2070139[2,] 0.2995463 0.3291068[3,] 0.3595670 0.3950506[4,] 0.2096039 0.2302885`\n\nWe see that the 95th percentile of the chi-squared distribution for df=3 is just a bit larger than 7.8:\n\n `> S = seq(7,8,by=0.1)> pchisq(S,df=3) 0.9281022 0.9312222 0.9342109 0.9370738 0.9398157 0.9424415 0.9449561 0.9473637 0.9496689 0.9518757 0.9539883`\n\nWe can do a Monte Carlo simulation (I actually do not know yet how the preceding function works, but the simulation should be just like what we did yesterday:\n\n `r = chisq.test(M,simulate.p.value=T)`\n\n `> r Pearson's Chi-squared test with simulated p-value (based on 2000 replicates)data: M X-squared = 4.1288, df = NA, p-value =0.2414`\n\nAnd Fisher's exact test (also on my study list for the future):\n\n `r = fisher.test(M)`\n\n `> r Fisher's Exact Test for Count Datadata: M p-value = 0.2479alternative hypothesis: two.sided`\n\n## Friday, August 14, 2009\n\n### Grade disparity - chi-squared analysis", null, "I recently posted about two common statistical problems: the first involves making an estimate of the population mean when the sample size is small, involving the t statistic of \"Student\", as well as the related problem of deciding whether the means for two small samples are different, including the example of paired samples. These arise commonly in biology. The second problem is deciding whether two sets of paired values are independent, or instead whether given the value for x, we can predict y. This problem involves covariance and the simplest approach is linear regression.\n\nThe third basic problem in statistics for which I want to improve my understanding is that of the independence of two discrete distributions. This will lead us to the chi-squared density as a special case of the gamma distribution.\n\nThere is a very nice discussion of this in Grinstead & Snell (pdf). Here is their table which gives a grade distribution, comparing the results for females and males in a particular class.", null, "If the distributions are independent (e.g. P(grade = A) = P(grade = A \| sex = female), then we can predict the most likely distribution, but also expect that there will be some spread in the observed breakdown by grade and sex due to random sampling. How much deviation should we expect?", null, "We can model the problem using colored balls. For example, we put 152 pink balls and 167 baby blue balls in to an urn, mix well, and then draw out in succession 93 'A's, 123 'B's, 90 'C's and 13 'D's.\n\nWe calculate for each category:\n\n(O - E)2/ E\n\nwhere O is the observed value and E the expected value based on the assumption of independence. We sum the value over all categories to obtain the statistic. According to theory, this statistic has a chi-squared (Χ2) distribution with degrees of freedom df = 3. (For this problem df is the number of grades - 1 the number of sexes - 1).\n\nIf the calculated value for the statistic exceeds 95 % of the values in the distribution for this df, we reject the null hypothesis that the two probability distributions are independent. In other words, we suspect that males and females have different grade distributions for this course.\n\nFor this data, the value of Χ2 that is exceeded 5 % of the time is 7.8, so the calculated value of 4.13 does not allow rejection.\n\nHere is code for a Python simulation (R code to plot is at the end). I redirect the output to a file like so:\n\n `python grades.py > results.txt`\n\n `import randomrL = list() # for statistic valuesF = 152; M = 167N = M + Ff = F1.0/N; m = M1.0/Nv = Falsegrades = [93,123,90,13]# expected values:EF = [fg for g in grades]EM = [mg for g in grades]def test(): print 'EF', for e in EF: print round(e,2), print print 'EM', for e in EM: print round(e,2), print R = 10000 # number of trialsfor i in range(10000): if v: test() chisq = 0 pL = ['F']F + ['M']M random.shuffle(pL) mL = list() fL = list() for i in range(4): # grades A-D n = grades[i] # how many 'A's... fL.append(pL[:n].count('F')) mL.append(pL[:n].count('M')) pL = pL[n:] if v: print 'OF',' '.join([str(e).rjust(2) for e in fL]) if v: print 'OM',' '.join([str(e).rjust(2) for e in mL]) for i in range(4): chisq += (fL[i]-EF[i])2/EF[i] chisq += (mL[i]-EM[i])2/EM[i] print round(chisq,2) if v: print`\n\n `setwd('Desktop')v = read.table( 'results.txt',head=F)B = seq(0,20,by=0.25)hist(v[,1],freq=F,breaks=B)curve(dchisq(x,3), from=0,to=20,add=T,col='blue',lwd=3)`\n\n### Regression corrected\n\nIf you read my post about regression from yesterday, you may have noticed that it has a serious problem with the way the errors were generated. What I did was this:\n\n `set.seed(157)x = runif(10)e = xfor (i in 1:length(x)) { e[i] = rnorm(1,mean=x[i],sd=0.3) }y = 2.4x + eplot(y~x,pch=16,col='blue',cex=1.8)`", null, "If we look at the errors, we see that they are dependent on the value of x! Naturally (b/c of mean=x[i]).\n\n `plot(x,e,pch=16,col='blue',cex=2)`", null, "What I should have done is something like this:\n\n `set.seed(1357)e = rnorm(10)/3y = 2.4x + eplot(x,e,pch=16,col='magenta',cex=2)`", null, "`plot(x,y,pch=16,col='darkred',cex=2)`", null, "Nevertheless, I hope the essential points are clear:\n\n• covariance is related to variance: cov(x,x) = var(x)\n• correlation is the covariance of z-scores\n• the slope of the regression line is: cov(x,y) / var(x)\n• the regression line goes through x, y\n• r ≈ cov(x,y) / sqrt(var(x)var(y))\n• the call to plot the line is abline(lm(y~x))\n\nThe proportionality for r is b/c we are not matching R's output for this. I think it is because we are missing a correction factor of n-1/n. I will have to look into that when I get back to my library at home.\n\nWith the change, we do a little better on guessing the slope:\n\n `> lm(y~x)Call:lm(formula = y ~ x)Coefficients:(Intercept) x 0.2625 2.0039 `\n\n## Thursday, August 13, 2009\n\n### Limited power\n\nI first came across Andrew Gelman's name because he co-wrote a book called 'Red State, Blue State'. I'm interested in politics and follow a few bloggers I think are smart---funny that they all turn out to be \"progressives.\" What are the chances?\n\nI haven't read that book, though I do have a book of his with ideas for teaching Statistics, and I aspire to read his book on Bayesian Analysis. He also has a blog.\n\nAnyway, somewhere I came across an article of his on the web (pdf). It's an analysis prompted by the publication of a set of articles in the Journal of Theoretical Biology with titles such as \"Beautiful Parents Have More Daughters,\" that are (as you might guess by this point) fatally flawed. It's a great read.\n\n### Basic regression", null, "Continuing with my education in statistics, I'm reading Dalgaard's book, Introductory Statistics with R. The topic of this post is linear regression by least squares.\n\nWe're going to model errors in the data as follows:\n\n `set.seed(157)x = runif(10)e = xfor (i in 1:length(x)) { e[i] = rnorm(1,mean=x[i],sd=0.3) }y = 2.4x + eplot(y~x,pch=16,col='blue',cex=1.8)`\n\n[UPDATE: this is not the correct way to model the errors. See here.]", null, "`dx = x-mean(x)dy = y-mean(y)n = length(x)sum(dx2)/(n-1)var(x)`\n\nThe variance of x is computed in the usual way:\n\n `> sum(dx2)/(n-1) 0.1182592> var(x) 0.1182592`\n\nThe covariance of x and y is defined in such a way that the variance of x is equal to the covariance of x with itself. That makes it easy to remember!\n\n `sum(dxdy)/(n-1)cov(x,y)`\n\n `> sum(dxdy)/(n-1) 0.4309724> cov(x,y) 0.4309724`\n\nCorrelation is just like covariance, only it is computed using z-scores (normalized data).\n\n `zx = (x-mean(x))/sd(x)zy = (y-mean(y))/sd(y)cov(zx,zy)cor(x,y)`\n\n `> cov(zx,zy) 0.9515839> cor(x,y) 0.9515839`\n\nAs discussed in Dalgard (Ch. 5), the estimated slope is:\n\n `β = cov(x,y) / var(x)`\n\n `> cov(x,y) / var(x) 3.644302`\n\nwhile the intercept is:\n\n `α = y - βx`\n\nThe linear regression line goes through x, y.\n\nLet R do the work:\n\n `> lm(y~x)Call:lm(formula = y ~ x)Coefficients:(Intercept) x -0.04804 3.64430`\n\nThe estimated slope is 3.64, while the true value is 2.4. I thought the problem was the last point, but it goes deeper than that.\n\n `># doing this by hand> i = 7 # index of max y value> lm(y[-i]~x[-i])Call:lm(formula = y[-i] ~ x[-i])Coefficients:(Intercept) x[-i] -0.1053 3.5803`\n\nMuch more information is available using the \"extractor function\" summary:\n\n `> summary(xy.model)Call:lm(formula = y ~ x)Residuals: Min 1Q Median 3Q -0.456059 -0.369435 -0.006057 0.239960 Max 0.814548 Coefficients: Estimate Std. Error t value(Intercept) -0.04804 0.28819 -0.167x 3.64430 0.41621 8.756 Pr(>\|t\|) (Intercept) 0.872 x 2.27e-05 ---Signif. codes: 0 ‘’ 0.001 ‘’ 0.01 ‘’ 0.05 ‘.’ 0.1 ‘ ’ 1 Residual standard error: 0.4294 on 8 degrees of freedomMultiple R-squared: 0.9055, Adjusted R-squared: 0.8937 F-statistic: 76.67 on 1 and 8 DF, p-value: 2.267e-05 `\n\nThe correlation coefficient is\n\n `r = sum(dxdy) / sqrt( sum(dx2) sum(dy*2) )`\n\nThat is\n\n `r = cov(x,y) / sqrt(var(x)var(y))`\n\n `> cov(x,y) / sqrt(var(x)*var(y)) 0.9515839`\n\nI am not sure why this doesn't match the output above.\n\nTo get the plot shown at the top of the post we use another extractor function on the linear model:\n\n `xy.model = lm(y~x)w = fitted(xy.model)`\n\nw contains the predicted values of y for each x according to the model.\n\n `plot(y~x,pch=16,col='blue',cex=1.8)xy.model = lm(y~x)abline(xy.model)segments(x,w,x,y,col='blue',lty=2)`\n\nWe can do an even fancier plot, with confidence bands, as follows.\n\n `df = data.frame(x)pp = predict(xy.model,int='p',newdata=df)pc = predict(xy.model,int='c',newdata=df)plot(y~x,pch=16,col='blue',cex=1.8)matlines(x,pc,lty=2,col='black')matlines(x,pp,lty=3,col='black')`", null, "This is promising but it is obviously going to take more work.\n\n### Core statistics for bioinformatics\n\nI found a nice overview of statistics on the internet. The author's name is Woon Wei Lee. The course page is here, and here is a link to the pdf. There are folders for other lectures which likely contain interesting stuff.\n\n### Student's t test 3\n\nThe paired t test is used when two sets of values are related, for example because each of a pair of measurements was made on the same subject.\n\nIn this case, it is the mean of the difference between the two values that is distributed according to the t distribution.\n\nThis example is from Dalgard.\n\n `pre = c(5260,5470,5640, 6180,6390,6515,6805, 7515,7515,8230,8770)post = c(3910,4220,3885, 5160,5645,4680,5265, 5975,6790,6900,7335)plot(pre,post,pch=16, col='blue',cex=2)diff = post-pre`", null, "Not only are the values correlated, but the difference is always negative:\n\n `> diff -1350 -1250 -1755 -1020 -745 -1835 -1540 -1540 -725 -1330 -1435`\n\n `t.test(pre,post,paired=T)`\n\n `> t.test(pre,post,paired=T) Paired t-testdata: pre and post t = 11.9414, df = 10,p-value = 3.059e-07alternative hypothesis: true difference in means is not equal to 0 95 percent confidence interval: 1074.072 1566.838 sample estimates:mean of the differences 1320.455`\n\nWe can do the test by hand, as follows:\n\n `> mean(diff) -1320.455> sd(diff) 366.7455> x = sd(diff)/sqrt(10)> x 115.9751> abs(mean(diff))/x 11.38567`\n\nThe question now is, what fraction of the values from the t-distribution with df = 10 are greater than 11.39?\n\n `S = seq(0,1,by=0.001)w = rt(1000000,df=10)y = quantile(w,S)round(tail(y))`\n\n `> round(tail(y)) 99.5% 99.6% 99.7% 3 3 3 99.8% 99.9% 100.0% 4 4 11`\n\nThe short answer: not very many!" ]	[ null, "https://3.bp.blogspot.com/_39NGKVWYg3o/SpvnhA8hMFI/AAAAAAAAAl4/yUsO3Vjm8do/s320/Picture+3.png", null, "https://4.bp.blogspot.com/_39NGKVWYg3o/SppPeviNW1I/AAAAAAAAAlo/vKXxEZN-BPA/s320/Picture+2.png", null, "https://1.bp.blogspot.com/_39NGKVWYg3o/SppPfPccmKI/AAAAAAAAAlw/jeoETMZ_gjY/s320/Picture+4.png", null, "https://4.bp.blogspot.com/_39NGKVWYg3o/SpfsOfJVF1I/AAAAAAAAAlY/CoGXgfdzd_c/s320/Picture+1.png", null, 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https://www.thefreedictionary.com/Newton%27s+first+law+of+motion	[ "# Newton's first law of motion\n\nAlso found in: Thesaurus, Encyclopedia, Wikipedia.\nRelated to Newton's first law of motion: Newton's third law of motion\nThesaurusAntonymsRelated WordsSynonymsLegend:\n Noun 1 Newton's first law of motion - a body remains at rest or in motion with a constant velocity unless acted upon by an external forcelaw of motion, Newton's law, Newton's law of motion - one of three basic laws of classical mechanics\nBased on WordNet 3.0, Farlex clipart collection. © 2003-2012 Princeton University, Farlex Inc.\nReferences in periodicals archive ?\n'If you search the text of Newton's first law of motion you will find million hits, does that mean Newton was guilty of plagiarism,' CIIT's rector told The Express Tribune.\nAccording to Newton's First Law of Motion, an object moving in space to move in the same direction at a constant velocity in the absence of an unbalanced force.\nWhen your plane is stationary, it's a good time to consider Isaac Newton's First Law of Motion, regarding inertia.\nNewton's First Law of Motion: Every object in a state of uniform motion (or at rest) tends to remain in that state of motion (or at rest) unless an external force is applied to it.\nThe group has met weekly, covering a variety of STEM-focused topics such as the Scientific Method, Bernoulli's Principle, Pascal's Law, Newton's first law of motion and more.\nThat's because each time the sled stops, a player has to contend with Newton's First Law of Motion, which states that an object at rest has the tendency to remain at rest.\nNEWTON'S FIRST LAW OF MOTION, also known as the principle of inertia, says \"Every body perseveres in its state of being at rest or of moving uniformly straight forward, except insofar as it is compelled to change its state by forces impressed.\" (1) I will argue that inertia is an inherent principle and that inertia and Newton's First Law are in this way natural in the Aristotelian sense.\nParticipants learned that an actor's movements onstage were more effective when they followed Newton's First Law of Motion (inertia and momentum: a body will either remain at rest or in motion unless acted upon by an outside force).\nSir Isaac Newton's First Law of Motion states that an object at rest tends to stay at rest and that an object in motion tends to stay in motion with the same speed and in the same direction unless acted upon by an unbalanced force.\nASK any paratrooper how softly he lands while parachuting and he'll quickly explain Sir Isaac Newton's First Law of Motion: \"An object in motion will remain in motion until an external force is applied.\" In other words, something has to stop the movement.\nNewton's First Law of Motion states that unless acted upon by an external force, a body at rest will remain at rest and a body in motion will remain in motion.\nNewton's first law of motion states, \"Every object in a state of uniform motion tends to remain in that state of motion, unless an external force is applied to it.\" Folks, your car is not equipped with an ejection seat to get you out a nanosecond before a crash.\n\nSite: Follow: Share:\nOpen / Close" ]	[ null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.956216,"math_prob":0.611697,"size":2511,"snap":"2021-04-2021-17","text_gpt3_token_len":542,"char_repetition_ratio":0.16114879,"word_repetition_ratio":0.10633484,"special_character_ratio":0.20708881,"punctuation_ratio":0.07628866,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9726828,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2021-01-23T02:23:19Z\",\"WARC-Record-ID\":\"<urn:uuid:19c3d0ce-dfb7-471a-8706-e471faec6167>\",\"Content-Length\":\"47026\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:4e488cb3-d00a-46d3-b530-a83ea66ca2b7>\",\"WARC-Concurrent-To\":\"<urn:uuid:660ac981-5a59-42c6-9613-f7ca1bf158fa>\",\"WARC-IP-Address\":\"209.160.67.5\",\"WARC-Target-URI\":\"https://www.thefreedictionary.com/Newton%27s+first+law+of+motion\",\"WARC-Payload-Digest\":\"sha1:ZGQQJMIDN5RVAVYUEWU2TEMHPUOJ5KZQ\",\"WARC-Block-Digest\":\"sha1:36XLVWL2X3CE3GXLPRTXEA2YF4LGL7ZI\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2021/CC-MAIN-2021-04/CC-MAIN-2021-04_segments_1610703531702.36_warc_CC-MAIN-20210123001629-20210123031629-00455.warc.gz\"}"}
https://downloads.hindawi.com/journals/ijap/2013/837639.xml	[ "IJAP International Journal of Antennas and Propagation 1687-5877 1687-5869 Hindawi Publishing Corporation 837639 10.1155/2013/837639 837639 Research Article A Modified STAP Estimator for Superresolution of Multiple Signals Wang Zhongbao Xie Junhao Ma Zilong Quan Taifan Nickel Ulrich Department of Electronic Engineering Harbin Institute of Technology Harbin 150001 China hit.edu.cn 2013 27 4 2013 2013 22 01 2013 08 04 2013 2013 Copyright © 2013 Zhongbao Wang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.\n\nA modified space-time adaptive processing (STAP) estimator is described in this paper. The estimator combines the incremental multiparameter (IMP) algorithm and the existing beam-space preprocessing techniques yielding a computationally cheap algorithm for the superresolution of multiple signals. It is a potential technique for the remote sensing of the ocean currents from the broadened first-order Bragg sea echo spectrum of shipborne high-frequency surface wave radar (HFSWR). Some simulation results and real-data analysis are shown to validate the proposed algorithm.\n\n1. Introduction\n\nThe measurement of the near-surface currents is a very difficult task by using conventional methods, especially under some harsh sea conditions. Many advanced marine measurement instruments, such as drifting buoys and acoustic current meters , have been used to collect sea state information. However, it would be very expensive to collect and interpret data from these devices for the sparse spatial sampling provided. Therefore, it is virtually impossible to form the current maps over a widespread ocean surface timely and accurately with these conventional meters.\n\nIn recent years, HFSWR has already become a powerful remote-sensing tool which receives increasing attention from oceanographers and research groups for its good ability to determine the large-scale sea state under all weather conditions [3, 4]. The measurement principle of HFSWR mainly depends on the Bragg resonant scattering theory and Doppler frequency effect theory in the sea echo spectrum of HFSWR. In the absence of ocean currents, the first-order Bragg lines would appear symmetrically above and below the zero Doppler frequency, which are caused by the ocean waves with precisely one-half wavelength of the radar moving towards and away from the radar. In practical application, some displacements will happen in the first-order Bragg lines since the near-surface currents always exist on the ocean .\n\nAs an extension of shore-based HFSWR, shipborne HFSWR not only inherits all the advantages of shore-based HFSWR but also shows some outstanding features, such as flexibility and mobility. However, some new problems emerge as the radar on board is a moving ship. One of the worst problems is that the first-order Bragg lines have been broadened into two pass bands (when the speed of ship is slow) or one low pass band (when the ship is sailing at a high speed) in the first-order Bragg sea echo spectrum of shipborne HFSWR . For these cases, it will be hard to determine the ocean currents from the broadened sea echo spectrum. Furthermore, the moving ship yields different Doppler shifts to the different azimuth sea echoes. Thus, there exists a certain space-time coupling relation in the received sea echo spectrum of shipborne HFSWR.\n\nThe novel space-time IMP algorithm was proposed by Clarke and Spence which was based on one-dimensional IMP algorithm and modified to detect and estimate multiple signals from the conventional beamwidth and (or) Doppler resolution bin. Although space-time IMP estimator effectively improves the robustness of the detection and estimation of multiple signals, the computational load of two-dimensional search process is too heavy for real-time application. Lately, Chadwick used the eigendecomposition method instead of the full-search process to reduce the heavy computational burden in his proposed polarisation-sensitive IMP algorithm. However, for surface wave radar, this method becomes invalid since the returns of HFSWR are mostly vertical polarization sensitive components.\n\nMany researchers, such as Shaw and Wilkins , used beam-space preprocessing technique to reduce the computational load and improve robust performance of high-resolution DOA (direction-of-arrival) estimation algorithms. Recently, Hassanien et al. [10, 11], proposed a new concept about beam-space preprocessing algorithm which was able to suppress the out-of-sector interferences accompanied with the updated array data. This technique was proven to be more robust than the aforementioned beam-space methods.\n\nIn this paper, we combine the space-time IMP algorithm and the adaptive beam-space preprocessing technique yielding a computationally cheap space-time adaptive estimator for detection, estimation and super-resolution of multiple signals. The proposed algorithm is validated by simulation results as well as experimental examples.\n\nThis paper is organized as follows. First we introduce the signal model. In the following section, the proposed algorithm and some simulation results are presented. The measurement of the near-surface currents of the ocean by shipborne HFSWR and some real-data analysis are presented in Section 4. The final part is the study conclusion.\n\n2. Signal Model\n\nConsider a uniform linear array (ULA) with M omnidirectional antennas and the antenna spacing d. If there are N sources that have been received from a far field with different relative delays and attenuations. The received data X is then given by as follows: (1)X=AΛBH+N, where (2)X=[x1x2xM]T,xi=[xi(1)xi(2)xi(L)],xi(l),i=1,2,,M,l=1,2,,L denotes the data received from the ith sensor at lth sampling time, [·]T denotes the transpose operation, A=[a(φ1)a(φ2)a(φN)]T is the array manifold matrix, (3)a(φi)=[1e-j2πd(M-1)sinφi/λ]T is the steering vector points to the direction φi,i=1,2,,N, λ denotes wavelength of the radar, Λ is a (N×N) diagonal matrix containing the signal magnitudes, B=[b(f1)b(f2)b(fN)] is a (L×N) matrix comprising the normalized source waveforms, b(fi)=[1e-j2πfie-j2πfi(L-1)]T is the normalized source waveform, fi,i=1,2,,N is the frequency of jth source, [·]H denotes the Hermitian transpose operation, and N is the (M×L) matrix comprising the zero mean and σ2 variance Gaussian noise.\n\nThe (M×M) covariance matrix of the received data is given by (4)R=E{XXH}=ASAH+σ2I, where E{·} is the statistical expectation operator, S=E{ΛBHBΛH} is the (L×L) source covariance matrix, and I is the identity matrix.\n\n3. Modified STAP Estimator 3.1. Space-Time IMP\n\nSpace-time IMP is a two-dimensional maximum likelihood method which uses a set of space-time calibration response vectors to match with the received data. Thus, the primary objective of space-time IMP is to maximize the “signal plus noise” to “expected noise” power ratio (SNR). If the maximum output power is over the threshold, then a target is detected and the corresponding space-time calibration response vector is recorded. In order to reduce the sidelobe leakage of the detected targets and improve the detection and estimation of the potential signals in the residual data, the detected targets are removed from the original data through an orthogonal subspace projection matrix before each iterative stage.\n\nAccording to the definition of SNR in space-time IMP, we have the following : (5)F(θ,f)=WH(θ,f)Qvec(X)vec(X)HQW(θ,f)WH(θ,f)QW(θ,f), where (6)W(θ,f)=vec{(a(θ)aH(θ)a(θ))(b(f)bH(f)b(f))H} is the (ML×1) space-time calibration response matrix, vec{·} is the vectorization operation, and Q is an (ML×ML) orthogonal projection matrix, which is given by (7)Q=I-M[MHM]-1MH, where M denotes a sum of space-time calibration response vectors corresponding to the detected signals; that is, Q projects the received data into a subspace orthogonal to the detected signals.\n\nThe adaptive beam-space preprocessing technique was first mentioned in which used the updated data for adaptive suppression of out-of-sector interferences. This technique had been shown to be more robust than the aforementioned beam-space methods.\n\nThe primary objective of the data-adaptive beam-space preprocessing is to solve the optimal beam-space matrix design problem through minimizing the output power of the transformed data, which can be expressed as follows : (8)minCtr(CHRC)subjecttoCHC=IsubjecttoCHa(θb)=1b=1,2,,BsubjecttoCHa(θ-k)γθ-kΘ-,k=1,,K, where tr{·} is the trace of a matrix, C is the (M×B) beam-space matrix, B(BM) is the beam-space dimension, · is the vector 2-norm, Θ- denotes all out-of-sector directions which are divided into K angular grids, {θb}b=1B and {θ-k}k=1K are the angles corresponding to the in-of-sector and out-of-sector directions, and γ is the stopband attenuation parameter which should meet the requirement (9)minCγsubjecttoCHC=IsubjecttoCHa(θb)=1b=1,2,,BsubjecttoCHa(θ-k)γθ-kΘ-,k=1,,K.\n\nAfter the beam-space transformation, the array steering vector matrix a and manifold matrix A have already transformed into (10)a~=CHa,A~=CHA.\n\nThen, the (B×B) covariance matrix R~ in beam-space should be rewritten as (11)R~=A~SA~H+σ2I.\n\nObviously, the dimension of the matrix R~ is lower than that of R. This fact is exploited in all beam-space-based methods to reduce computational load compared with the element-space algorithms .\n\nIn this section, we show how the conventional space-time estimator combines with the adaptive beam-space preprocessing technique to present a computationally cheap space-time adaptive beam-space IMP estimator.\n\nFollowing the discussion above, the two-dimensional discriminants shown in (5) should be modified as (12)F~(θ,f)=W~H(θ,f)Q~vec(CHX)vec(CHX)HQ~W~(θ,f)W~H(θ,f)Q~W~(θ,f), where (13)W(θ,f)=vec{(CHa(θ)aH(θ)CCHa(θ))(b(f)bH(f)b(f))H}, and Q~ has been reduced to a BL×BL matrix in beam-space domain, which is given by (14)Q~=I-M~[M~HM~]-1M~H, where M~ denotes a sum of space-time calibration response vectors corresponding to the detected signals in beam-space domain.\n\n3.4. Threshold Setting\n\nHow to select an appropriate threshold to terminate the iterative process in IMP algorithm is very important. Theoretically, when all “significant peaks” have been detected and cleared out from the received data, there is only a completely flat plane in the residual scan . However, it is impossible to accurately estimate the noise statistics from the limited received data. Furthermore, the definition of “significant peak” in IMP algorithm has not been clearly reported.\n\nIn the paper, we use a double-threshold setting method to ensure that the iterative process is halted timely. First, we check two successive scans before the next iteration. If the difference between the two scans is comparable to that of the “expected noise” level, that is, no “significant peak” appears during the last scan, then the iterative process should be halted. Besides, if the difference between two successive estimations has reached the preset threshold, which suggests that the iterations are estimating the same target, then the iterative process should be halted as well.\n\n3.5. Simulation Results\n\nSeveral simulation results are shown in this section to test the performance of the modified algorithm through comparing it with several conventional algorithms.\n\nDuring the simulations, we assume that the radar works at f=6MHz, which contains an ULA with M=8 omnidirectional sensors and the elements are spaced one-half wavelength apart. The half power beamwidth is approximately 13°. The number of snapshots L=32 and the beam-space dimension B=2 are chosen for our simulations. The adaptive beam-space matrix has been solved using the cvx optimization MATLAB toolbox. Since the minimum value in (9) is γmin=0.0676, we take the parameter γ=0.07 for (8). Furthermore, the two simulation targets (0.2Hz,86°) and (0.3Hz,90°) are also selected in the simulations.\n\nTo define a successful experiment, we use the criterion mentioned in if (15)i=12\|θ^i-θi\|<\|θ1-θ2\|, where θ^i and θi (i=1,2) are, respectively, the estimated and truth values, then the two signals are successfully resolved.\n\nFigures 1 and 2 illustrate the probability of source resolution and their root mean square error (RMSE) versus SNR in Doppler domain, respectively. The conventional space-time IMP, space-time beam-space IMP which combines space-time IMP with discrete fourier transform (DFT) matrix beam-space processing technique , and 64 points and 256 points FFT results are used for the comparison in the figures. We find that the beam-space-based algorithms show better resolution and smaller RMSE than the other algorithms in resolving the two simulation targets. Thus, it is reasonable to conclude that the beam-space-based methods require less observation time but maintain high Doppler accuracy compared with the conventional algorithms. By the way, all the simulation results shown in this section have averaged over 1000 independent Monte Carlo experiments.\n\nProbability of source resolution versus SNR in Doppler domain.\n\nDoppler estimated RMSE versus SNR.\n\nFigures 3 and 4 are the probability of source resolution and their RMSE versus SNR in azimuth domain, respectively. According to these figures, the beam-space-based algorithms show better performance than the other algorithms. Comparing these methods, the space-time adaptive beam-space IMP algorithm shows the highest robust and lowest RMSE in resolving the two simulation targets. Thus, it is reasonable to conclude that the proposed algorithm requires smaller antenna array and lower SNR threshold for detection and estimation of multiple signals when compared with the conventional DOA algorithms.\n\nProbability of source resolution versus SNR in azimuth domain.\n\nAzimuth estimated RMSE versus SNR.\n\n4. Shipborne HFSWR 4.1. Space-Time Coupling Relation\n\nIn , Xie et al. had proven that the first-order Bragg spectrums were broadened along the azimuthal directions from the real-data analysis. The authors concluded that there was a space-time coupling relation existed in the first-order Bragg sea echo spectrum of shipborne HFSWR.\n\nAssuming that both the transmitting and the receiving antennas of HFSWR are mounted on a ship which is moving in the positive direction of the x-axis at a constant speed vs(m/s), as shown in Figure 5.\n\nSchematic of shipborne HFSWR and current vector.\n\nIn the absence of ocean current, the space-time coupling relation in the first-order Bragg sea echo spectrum of shipborne HFSWR can be expressed as follows : (16)fd=2vsλcosϕ+fB, where ϕ[0,π] is the azimuth direction, fB=±g/πλ are the first-order Bragg frequencies, the positive and negative signs are, respectively, the Bragg waves moving towards and away from the radar.\n\n4.2. Current Measurement\n\nIn the presence of ocean currents, the first-order Bragg lines in (16) are shifted from the theoretical positions. The displacements are proportional to the radial current velocities. Thus, (16) should be rewritten as (17)fd=2vsλcosϕ+fB+2vc(ϕ)λ.\n\nAs shown in (17), the first-order Bragg lines are related to the azimuth directions as well as the speed of ship and currents. Therefore, the first-order Bragg peaks are not only displaced, but also broadened into two pass bands (for slow ship speed case) in the first-order Bragg sea echo spectrum of shipborne HFSWR.\n\n4.3. Real-Data Analysis\n\nThe real data used in this paper was received from the shipborne HFSWR experiments conducted on the Yellow Sea of China on September 8, 1998 . Figures 6 and 7 show the space-time coupling relation in the sixth range bin of the real-data file 1128 (containing 7 channels × 256 samples × 32 range bins and the ship speed was about 4.8 m/s), which is processed through the DFT plus DBF (Digital Beamforming) cascade processing and the proposed algorithm. As shown in the figures, the first-order Bragg lines are broadened along the azimuth directions, which tally well with the theoretical lines in (16). The displacements may be caused by ocean currents or interferences.\n\nSpace-time spectrum of real data processed by DFT and DBF cascaded processing. – – denotes the theoretical value (f0=5.283MHz, Tp=0.262s, d=14m).\n\nSpace-time spectrum of real data processed by IMP.\n\nTable 1 is an example of the DOA and Doppler estimations. The modified STAP estimator is used to process the above-mentioned real data within the section between the azimuth 40° and 90°. Since there was no information about the sea state recorded during the experiment, we here assume that the detected targets near the theoretical first-order Bragg lines are considered as ocean currents and their displacements are proportional to their radial velocities. Based on this assumption, five currents have been detected and estimated. All of them are very close to the theoretical first-order Bragg frequencies and their corresponding radial velocities are calculated in the table.\n\nDOA and Doppler estimations.\n\nTarget 1 2 3 4 5\nDOA (Deg) 62.72 50.12 64.06 52.56 63.35\nDoppler (Hz) - 0.1582 - 0.1154 - 0.1596 - 0.1292 - 0.1589\nTheoretical (Hz) - 0.1541 - 0.1220 - 0.1577 - 0.1278 - 0.1558\nVelocity (m/s) 0.1166 - 0.1869 0.0519 0.0398 0.0885\n\nTable 2 is another example for better understanding the robustness of the proposed algorithm, where we add a simulation target (-0.18Hz,80°) to the real data used previously, as shown in Figure 8. The added simulation target is detected correctly and the estimations of DOA and Doppler frequency are within 2° and 0.0025Hz of the true signal position.\n\nDOA and Doppler estimations.\n\nTarget 1 2 3 4 5\nDOA (Deg) 82.97 67.33 64.19 81.97 52.56\nDoppler (Hz) - 0.1833 - 0.1627 - 0.1567 - 0.1825 - 0.1176\nTheoretical (Hz) - 0.2130 - 0.1668 - 0.1581 - 0.2100 - 0.1238\n\nSpace-time spectrum of real data adds simulation target.\n\n5. Conclusion\n\nIn this paper, we have introduced a modified space-time adaptive processing estimator that can be used for the detection, estimation, and superresolution of multiple signals. The method combines the conventional IMP method and the existing adaptive beam-space preprocessing techniques yielding a computationally cheap algorithm for estimating the near-surface currents of the ocean from the broadening of the first-order Bragg sea echo spectrum of shipborne HFSWR. The proposed algorithm is validated by simulation results as well as experimental examples.\n\nAcknowledgment\n\nThis work is supported by the State Key Program of National Natural Science of China (Grant no. 61132005)." ]	[ null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.87870914,"math_prob":0.9637755,"size":20697,"snap":"2022-05-2022-21","text_gpt3_token_len":5134,"char_repetition_ratio":0.13241193,"word_repetition_ratio":0.07009511,"special_character_ratio":0.2427888,"punctuation_ratio":0.120236866,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9860633,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2022-01-25T23:06:30Z\",\"WARC-Record-ID\":\"<urn:uuid:ca16c971-0c64-439c-87cf-f17e5510cc70>\",\"Content-Length\":\"102862\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:450d40f3-754b-44c8-8cbe-b72ab37a6ff7>\",\"WARC-Concurrent-To\":\"<urn:uuid:93dd1c9e-34bf-47c9-9256-9507d6402a7c>\",\"WARC-IP-Address\":\"13.249.38.31\",\"WARC-Target-URI\":\"https://downloads.hindawi.com/journals/ijap/2013/837639.xml\",\"WARC-Payload-Digest\":\"sha1:GGQT3ZLOCCGXAZYSGFGEJY72FSZJNPZ7\",\"WARC-Block-Digest\":\"sha1:7BC6K57SLTNBWNEPSDWNOPN3DI6VNWRR\",\"WARC-Identified-Payload-Type\":\"application/xml\",\"warc_filename\":\"/cc_download/warc_2022/CC-MAIN-2022-05/CC-MAIN-2022-05_segments_1642320304876.16_warc_CC-MAIN-20220125220353-20220126010353-00569.warc.gz\"}"}
https://mathoverflow.net/questions/240299/how-do-i-complete-the-sketch-of-proof-in-fga-explained-5-5-8	[ "How do I complete the sketch of proof in 'FGA explained', 5.5.8?\n\nIt seems to me that the one part that is difficult to transfer to general coherent sheaves is given only one sentence: \"Because we have such a common $m$, we get as before an injective morphism from the functor $\\mathfrak{Quot}^{\\Phi,L}_{E/X/S}$ into the Grassmanian functor $\\mathfrak{Grass}(\\pi_\\ast E(r), \\Phi(r))$.\" But I see at least two serious hardships with this proof: First, why is $R^1 \\pi_{T,\\ast} \\left( \\mathcal{G}_t (r) \\right) = 0$? Second, even if that is true, the values on $T$ are quotients of $\\pi_{T,\\ast} E_T$, not of $(\\pi_\\ast E)_T$, where the latter is the pullback of the pushforward. These two sheaves are the same for some $r$ by Theorem 5.4 (3.1 in the standalone paper), but that $r$ might depend on $T$.\n\nEdit: The statement of the theorem that has a more or less thorough proof in the book \"Fundamental algebraic geometry: Grothendieck's FGA explained.\", in part 5, by Nitin Nitsure (it exists as a separate paper [Nitsure, Nitin (2005), \"Construction of Hilbert and Quot schemes\", Fundamental algebraic geometry, Math. Surveys Monogr. 123, Providence, R.I.: Amer. Math. Soc., pp. 105–137, arXiv:math/0504590, MR 2223407]), is as follows: (theorem 5.2 in the paper, 5,15 in the book)\n\nLet $S$ be a Noetherian scheme, and $X$ a closed subscheme in $\\mathbb{P}(V)$ for some vector bundle $V$, let $\\pi$ denote the stuctral morphism $X \\to S$. Let $E$ be a coherent factor-sheaf of $\\pi^\\ast(W)(\\nu)$ where $W$ is an $S$-vector bundle and $\\nu$ is an integer. Then for any integer-valued polynomial $\\Phi$, the functor $\\mathfrak{Quot}_{E/X/S}^{\\Phi,\\mathcal{O}(1)}$ is representable by a closed subscheme of the Grassmanian $Gr(W \\otimes Sym^r(V),\\Phi(r))$ for large enough r.\n\nIt differs from the original Grothendieck's statement in that in general, not all projective schemes embed into projectivisations of vector bundles (if there is an ample bundle on $S$, they do) Also, the bundle $E$ is just any coherent sheaf in Grothendieck's version of the theorem.\n\nBecause I don't know what the author meant when he wrote that the part that I pointed out generalises easily, I guess I will just write down that part of the proof (it is implicit in the text, I had to guess the specifics myself) and note where I see difficulty.\n\nFirst of all, $X$ is obviously replaced with $\\mathbb{P}(V)$. Then the factor-sheaf of $\\pi^\\ast (W) (\\nu)$ is replaced by $\\pi^\\ast(W)$ - this induces a closed embedding on the $\\mathfrak{Quot}$ sheaves. Here I don't see how to generalise - to express an arbitrary coherent sheaf as a quotient of a vector bundle. So, over an $S$-scheme $T$, on $\\mathbb{P}(H) \\times_S T$ there is an exact sequence $0 \\to \\mathcal{G} \\to E_T \\to \\mathcal{F} \\to 0.$ And what seems crucial is that all these three sheaves are flat over $T$. On fibers $s$, $\\mathcal{G}_s$ is a subsheaf of a free sheaf $E_s$, and we know the Hilbert polynome of $\\mathcal{G}$ so Mumford's theorem (5.3 in the book, 2.3 in the paper) is used to show that there exists an $m$ which does not depend on $s$ or the particular $\\mathcal{F}$ such that all three sheaves in the exact sequence on the fiber are (Castelnuovo-Mumford) $m$-regular. (perhaps for general $E$ one can emulate the proof of Mumford's theorem, uniformly bounding the dimensions of global sections of $E$ on subspaces in fibers) Then after twisting by $r$ for $r \\geq m$, the three sheaves have no higher cohomology on fibers. Then by the Flatness and Base Change Theorem (strongly using that the sheaves are flat over $S$), the higher cohomology vanish. They are also relatively globally generated (maybe after a twist by n), I hae thought of the following argument: they become \"relatively 0-regular\" after thet twist, so they are relatively globally generated. Since $R^1\\pi_{T,\\ast}(\\mathcal{G(r)})=0$, we get an exact sequence $0 \\to \\pi_{T,\\ast}(\\mathcal{G}(r)) \\to \\pi_{T,\\ast}(E(r)) \\to \\pi_{T,\\ast}(\\mathcal{F}(r)) \\to 0$, and thus a morphism to the Grassmanian. The global generation (of $\\mathcal{G}(r)$) is used to show that the morphism is injective and to find its image.\n\nThe statement of Mumford's theorem is as follows: There exists a polynome with integer coefficients $F_{d,n}$ in $n+1$ variables such that for any $\\mathcal{F}$ a coherent subsheaf of $\\oplus^d \\mathcal{O}_{\\mathbb{P}^n_k}$, $\\mathcal{F}$ is $F_{n,d}(a_0, \\cdots a_n)$-regular where $a_i$ are the coordinates of the hilbert polynome of $\\mathcal{F}$ in the binomial basis.\n\n• You're far more likely to get an answer to your question if you make it self-contained. Maybe someone knows some alg geom but doesn't have FGA explained (e.g. maybe they read FGA!) and they're not going to be answering your question as it stands. – znt Jun 2 '16 at 21:52\n• As @znt says, you should make your question self-contained. As a minimum, you should include the statement of the result that you are asking about, with a proper reference to the document in which it appears, preferably with a link. – Neil Strickland Jun 3 '16 at 6:13\n• I hope the text that I wrote is helpful. It is really hard to make the question self-contained because the statements in the text are interconnected and many of them I have not met before. Maybe I should have just posted the link to the paper, which contains the text available for anyone. – Alexei Tsybyshev Jun 3 '16 at 14:20" ]	[ null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.8858688,"math_prob":0.9973656,"size":4443,"snap":"2019-43-2019-47","text_gpt3_token_len":1336,"char_repetition_ratio":0.11308853,"word_repetition_ratio":0.005722461,"special_character_ratio":0.28561783,"punctuation_ratio":0.10964912,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9997904,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2019-10-20T20:15:04Z\",\"WARC-Record-ID\":\"<urn:uuid:8359e382-0db8-4c01-ab3b-078aedbef9c9>\",\"Content-Length\":\"114403\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:6baf46db-b375-4640-b511-d5a9fe817c92>\",\"WARC-Concurrent-To\":\"<urn:uuid:b58e52c0-bb05-469b-8225-b29b147bbd59>\",\"WARC-IP-Address\":\"151.101.193.69\",\"WARC-Target-URI\":\"https://mathoverflow.net/questions/240299/how-do-i-complete-the-sketch-of-proof-in-fga-explained-5-5-8\",\"WARC-Payload-Digest\":\"sha1:YAQU3TRNLNO6ZNULUBDWMEMXC6KTTJ3K\",\"WARC-Block-Digest\":\"sha1:6IIO4DHFUZPDBB5WA7H3FZKU7J7CVUVA\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2019/CC-MAIN-2019-43/CC-MAIN-2019-43_segments_1570986718918.77_warc_CC-MAIN-20191020183709-20191020211209-00388.warc.gz\"}"}
https://support.zabbix.com/browse/ZBX-14697	[ "", null, "# Memory leak in alert manager when zabbix database is not available\n\nXMLWordPrintable\n\n#### Details\n\n• Type:", null, "Problem report\n• Status: Closed\n• Priority:", null, "Trivial\n• Resolution: Fixed\n• Affects Version/s: 3.4.12, 4.0.0alpha9\n• Fix Version/s:\n• Component/s:\n• Labels:\n• Team:\nTeam A\n• Sprint:\nSprint 40\n• Story Points:\n0.125\n\n#### Description\n\nAlert manager does not close previous connection, this leads to memory leak after each successful reconnect.\n\n```==613== For counts of detected and suppressed errors, rerun with: -v\n==613== ERROR SUMMARY: 0 errors from 0 contexts (suppressed: 0 from 0)\n==637== at 0x4C30B06: calloc (vg_replace_malloc.c:711)\n==637== by 0x4E58C16: mysql_init (in /usr/lib64/libmariadb.so.3)\n==637== by 0x7B505E: zbx_db_connect (db.c:494)\n==637== by 0x6EA30C: DBconnect (db.c:73)\n==637== by 0x41E986: MAIN_ZABBIX_ENTRY (server.c:1178)\n==637== by 0x62FBAA: daemon_start (daemon.c:392)\n==637== by 0x41D14D: main (server.c:854)\n==637==\n==637== 173,765 (6,360 direct, 167,405 indirect) bytes in 5 blocks are definitely lost in loss record 140 of 140\n==637== at 0x4C30B06: calloc (vg_replace_malloc.c:711)\n==637== by 0x4E58C16: mysql_init (in /usr/lib64/libmariadb.so.3)\n==637== by 0x7B505E: zbx_db_connect (db.c:494)\n==637== by 0x6EA30C: DBconnect (db.c:73)\n==637== by 0x41E986: MAIN_ZABBIX_ENTRY (server.c:1178)\n==637== by 0x62FBAA: daemon_start (daemon.c:392)\n==637== by 0x41D14D: main (server.c:854)\n==637==\n==637== LEAK SUMMARY:\n==637== definitely lost: 7,632 bytes in 6 blocks\n==637== indirectly lost: 200,886 bytes in 120 blocks\n==637== possibly lost: 0 bytes in 0 blocks\n==637== still reachable: 146,440 bytes in 244 blocks\n==637== suppressed: 0 bytes in 0 blocks\n==637== Reachable blocks (those to which a pointer was found) are not shown.\n==637== To see them, rerun with: --leak-check=full --show-leak-kinds=all\n```\n\nPostgreSQL\n\n```==22638== 35,532 (896 direct, 34,636 indirect) bytes in 1 blocks are definitely lost in loss record 103 of 104\n==22638== at 0x4C2EBAB: malloc (vg_replace_malloc.c:299)\n==22638== by 0x4E4767A: ??? (in /usr/lib64/libpq.so.5.10)\n==22638== by 0x4E4DD3A: PQsetdbLogin (in /usr/lib64/libpq.so.5.10)\n==22638== by 0x783163: zbx_db_connect (db.c:542)\n==22638== by 0x6E96AF: DBconnect (db.c:73)\n==22638== by 0x43966D: MAIN_ZABBIX_ENTRY (server.c:1130)\n==22638== by 0x61A2F0: daemon_start (daemon.c:392)\n==22638== by 0x437F5D: main (server.c:832)\n==22638==\n==22638== LEAK SUMMARY:\n==22638== definitely lost: 896 bytes in 1 blocks\n==22638== indirectly lost: 34,636 bytes in 31 blocks\n==22638== possibly lost: 0 bytes in 0 blocks\n==22638== still reachable: 147,066 bytes in 251 blocks\n==22638== suppressed: 0 bytes in 0 blocks\n==22638== Reachable blocks (those to which a pointer was found) are not shown.\n==22638== To see them, rerun with: --leak-check=full --show-leak-kinds=all\n```\n\nFollowing patch fixes the issue but need proper review and testing\n\n```Index: src/zabbix_server/alerter/alert_manager.c\n===================================================================\n@@ -1960,6 +1960,7 @@\n\nif (ZBX_DB_DOWN == manager.dbstatus && time_connect + ZBX_DB_WAIT_DOWN <= now)\n{\n+\t\t\tDBclose();\nif (ZBX_DB_DOWN == (manager.dbstatus = DBconnect(ZBX_DB_CONNECT_ONCE)))\n{\n\n```\n\n#### People\n\nAssignee:", null, "Vladislavs Sokurenko\nReporter:", null, "Vladislavs Sokurenko" ]	[ null, "https://support.zabbix.com/secure/projectavatar", null, "https://support.zabbix.com/secure/viewavatar", null, "https://support.zabbix.com/images/icons/priorities/trivial.svg", null, "https://support.zabbix.com/secure/useravatar", null, "https://support.zabbix.com/secure/useravatar", null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.7544458,"math_prob":0.7653331,"size":3617,"snap":"2019-51-2020-05","text_gpt3_token_len":1250,"char_repetition_ratio":0.20619984,"word_repetition_ratio":0.2685185,"special_character_ratio":0.4547968,"punctuation_ratio":0.22886297,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.97040325,"pos_list":[0,1,2,3,4,5,6,7,8,9,10],"im_url_duplicate_count":[null,null,null,null,null,null,null,null,null,null,null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2020-01-23T12:11:03Z\",\"WARC-Record-ID\":\"<urn:uuid:1d51cadc-ae59-4b9e-925a-a5afe0582c54>\",\"Content-Length\":\"81173\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:f53c8479-2808-4b5c-aa12-f7e50a92f175>\",\"WARC-Concurrent-To\":\"<urn:uuid:950b08f7-afbf-478a-92c5-d0aead747400>\",\"WARC-IP-Address\":\"87.110.183.173\",\"WARC-Target-URI\":\"https://support.zabbix.com/browse/ZBX-14697\",\"WARC-Payload-Digest\":\"sha1:NY2KYG7M3IAO3OSURAOMRBCZW2HAIWHJ\",\"WARC-Block-Digest\":\"sha1:6TYM6MDLJ4WVR3TKQ6NEBPZFWJG2YUZK\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2020/CC-MAIN-2020-05/CC-MAIN-2020-05_segments_1579250610004.56_warc_CC-MAIN-20200123101110-20200123130110-00374.warc.gz\"}"}
https://openlab.citytech.cuny.edu/mat1175fa2011/student-learning-outcomes/	[ "# Student Learning Outcomes\n\nContent-Specific Learning Outcomes\n\nBy the end of the course, the student will :\n\n1. demonstrate the ability to manipulate algebraic expressions including polynomials, rational and radical expressions.\n2. demonstrate the ability to solve equations including linear, quadratic, rational and radical equations, as well as systems of linear equations in two variables.\n3. demonstrate the ability to apply theorems and solve problems in geometry including parallel and perpendicular lines, congruent and similar triangles, and special right triangles.\n4. identify abstract mathematical relationships between quantities.\n5. identify concrete mathematical relationships in other disciplines and outside the classroom.\n6. apply algebraic and geometric principles to analyze and solve problems in other disciplines and outside the classroom.\n\nGeneral Education Learning Outcomes\n\nBy the end of the course, the student will :\n\n1. use the scientific method (make observations, perform experiments, record and process data, and draw conclusions);\n2. make diagrams and graphically present data as a tool for problem solving;\n3. read for details, big picture concepts and themes;\n4. listen to instructor as well as peers;\n5. speak one-on-one, in meetings and formally;\n6. incorporate information from various sources;\n7. write descriptively and reflectively;\n8. effectively collaborate and work as a team.\n\nMathematics Department\nMAT 1175, Fall 2011\nProfessors E. Halleck and J. Reitz" ]	[ null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.91219753,"math_prob":0.9603095,"size":1451,"snap":"2022-27-2022-33","text_gpt3_token_len":272,"char_repetition_ratio":0.09951624,"word_repetition_ratio":0.13043478,"special_character_ratio":0.18538938,"punctuation_ratio":0.13913043,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.96783304,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2022-08-14T07:52:15Z\",\"WARC-Record-ID\":\"<urn:uuid:64195e72-7e3a-4c88-b644-90c70c21b0f6>\",\"Content-Length\":\"107150\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:c1486b12-6ef7-4c45-95d3-bb194be4b810>\",\"WARC-Concurrent-To\":\"<urn:uuid:19f15652-0efb-4627-8785-fdcbf17cb6ba>\",\"WARC-IP-Address\":\"209.68.53.53\",\"WARC-Target-URI\":\"https://openlab.citytech.cuny.edu/mat1175fa2011/student-learning-outcomes/\",\"WARC-Payload-Digest\":\"sha1:GI5ESOR5TDUY3AHDFADFQTD3REFSRLKL\",\"WARC-Block-Digest\":\"sha1:4ZYA5CZDBGCE5PO6YQDJSZJX7QGVJREF\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2022/CC-MAIN-2022-33/CC-MAIN-2022-33_segments_1659882571996.63_warc_CC-MAIN-20220814052950-20220814082950-00353.warc.gz\"}"}
https://federalprism.com/does-koh-and-cuso4-form-a-precipitate/	[ "# Does KOH and CuSO4 form a precipitate?\n\n## Does KOH and CuSO4 form a precipitate?\n\nDescription: Copper hydroxide precipitate, 1 of 3. Copper (II) hydroxide precipitate (Cu(OH)2) formed by adding 0.5 M copper sulfate (CuSO4) solution to a 0.2 M solution of potassium hydroxide (KOH). The reaction is CuSO4 + KOH -> Cu(OH)2 + K2SO4. This is an example of a double replacement reaction.\n\n## What happens when you mix HCl and KOH?\n\nPotassium hydroxide (KOH) reacts with hydrochloric acid (HCl) to produce potassium chloride (KCl), a salt and water (H2O). This is a neutralization reaction.\n\nDoes CuSO4 react with HCl?\n\nWhen concentrated hydrochloric acid is added to a very dilute solution of copper sulfate, the pale blue solution slowly turns yellow-green on the formation of a copper chloride complex.\n\nWhat happens when you mix CuSO4 and KOH?\n\n[Note: May be used as an aqueous solution.]…Search by products (Cu(OH) 2, K 2SO 4)\n\n1 KOH + CuSO4 → K2SO4 + Cu(OH)2\n2 KOH + CuSO4*5H2O → H2O + K2SO4 + Cu(OH)2\n\n### What is the reaction between Cu and H in HCl?\n\nIn the case of Cu + HCl we can see that Cu is below H on the activity series. Because of this the Cu will not be able to replace the H in HCl and there will be NO REACTION. If you are unsure if a compound is soluble when writing net ionic equations you should consult a solubility table for the compound.\n\n### How to use ionic net Equation Calculator?\n\nMethod to use the Ionic net equation calculator is as follows: 1 1: Enter the chemical equation in the “Enter the chemical equation” field. 2 2: Now click the button “Balance” to get the equalize equation. 3 3: Finally, for the specified chemical equation, a window will pop with the output. More\n\nWhat is the net ionic equation for a protium ion?\n\nKOH (aq) + H Cl(aq) → KCl(aq) +H 2O(l) …the net ionic equation is…. These days, commonly, we represent the protium ion, H +, in aqueous solution as… H 3O+ …i.e. a protonated water molecule. The following is taken from a previous answer…\n\nWhat happens when HCl (g) is bleached in water?\n\nWe may take a tank of H Cl(g), and we can bleed it in to water to give an AQUEOUS solution that we could represent as H Cl(aq) OR H 3O+ and Cl−. In each case this is a REPRESENTATION of what occurs in solution. If we bleed enuff gas in, we achieve saturation at a concentration of approx. 10.6 ⋅ mol ⋅ L−1 with respect to hydrochloric acid." ]	[ null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.85892355,"math_prob":0.96366566,"size":2357,"snap":"2023-40-2023-50","text_gpt3_token_len":681,"char_repetition_ratio":0.11644709,"word_repetition_ratio":0.018691588,"special_character_ratio":0.25922784,"punctuation_ratio":0.10612245,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.98425734,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2023-12-10T12:46:27Z\",\"WARC-Record-ID\":\"<urn:uuid:bf3c7492-9eba-43ad-b2d5-026679886122>\",\"Content-Length\":\"59218\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:7ccbf081-0cc0-463a-82eb-fe2e9a27b29c>\",\"WARC-Concurrent-To\":\"<urn:uuid:3a499c36-bf3e-47a0-9c49-83c329c4bbad>\",\"WARC-IP-Address\":\"172.67.181.88\",\"WARC-Target-URI\":\"https://federalprism.com/does-koh-and-cuso4-form-a-precipitate/\",\"WARC-Payload-Digest\":\"sha1:4LRYIWSWOH432WBA6JRQ4BFSEFYZOMCK\",\"WARC-Block-Digest\":\"sha1:ZEPV45M32DVTTSZPOZIVG4BJ74P2CHWE\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2023/CC-MAIN-2023-50/CC-MAIN-2023-50_segments_1700679102469.83_warc_CC-MAIN-20231210123756-20231210153756-00842.warc.gz\"}"}
https://www.colorhexa.com/030f4a	[ "# #030f4a Color Information\n\nIn a RGB color space, hex #030f4a is composed of 1.2% red, 5.9% green and 29% blue. Whereas in a CMYK color space, it is composed of 95.9% cyan, 79.7% magenta, 0% yellow and 71% black. It has a hue angle of 229.9 degrees, a saturation of 92.2% and a lightness of 15.1%. #030f4a color hex could be obtained by blending #061e94 with #000000. Closest websafe color is: #000033.\n\n• R 1\n• G 6\n• B 29\nRGB color chart\n• C 96\n• M 80\n• Y 0\n• K 71\nCMYK color chart\n\n#030f4a color description : Very dark blue.\n\n# #030f4a Color Conversion\n\nThe hexadecimal color #030f4a has RGB values of R:3, G:15, B:74 and CMYK values of C:0.96, M:0.8, Y:0, K:0.71. Its decimal value is 200522.\n\nHex triplet RGB Decimal 030f4a `#030f4a` 3, 15, 74 `rgb(3,15,74)` 1.2, 5.9, 29 `rgb(1.2%,5.9%,29%)` 96, 80, 0, 71 229.9°, 92.2, 15.1 `hsl(229.9,92.2%,15.1%)` 229.9°, 95.9, 29 000033 `#000033`\nCIE-LAB 7.726, 21.573, -37.527 1.444, 0.855, 6.567 0.163, 0.096, 0.855 7.726, 43.286, 299.894 7.726, -2.794, -24.282 9.248, 11.689, -35.628 00000011, 00001111, 01001010\n\n# Color Schemes with #030f4a\n\n• #030f4a\n``#030f4a` `rgb(3,15,74)``\n• #4a3e03\n``#4a3e03` `rgb(74,62,3)``\nComplementary Color\n• #03334a\n``#03334a` `rgb(3,51,74)``\n• #030f4a\n``#030f4a` `rgb(3,15,74)``\n• #1b034a\n``#1b034a` `rgb(27,3,74)``\nAnalogous Color\n• #334a03\n``#334a03` `rgb(51,74,3)``\n• #030f4a\n``#030f4a` `rgb(3,15,74)``\n• #4a1b03\n``#4a1b03` `rgb(74,27,3)``\nSplit Complementary Color\n• #0f4a03\n``#0f4a03` `rgb(15,74,3)``\n• #030f4a\n``#030f4a` `rgb(3,15,74)``\n• #4a030f\n``#4a030f` `rgb(74,3,15)``\n• #034a3e\n``#034a3e` `rgb(3,74,62)``\n• #030f4a\n``#030f4a` `rgb(3,15,74)``\n• #4a030f\n``#4a030f` `rgb(74,3,15)``\n• #4a3e03\n``#4a3e03` `rgb(74,62,3)``\n• #000000\n``#000000` `rgb(0,0,0)``\n• #010519\n``#010519` `rgb(1,5,25)``\n• #020a31\n``#020a31` `rgb(2,10,49)``\n• #030f4a\n``#030f4a` `rgb(3,15,74)``\n• #041463\n``#041463` `rgb(4,20,99)``\n• #05197b\n``#05197b` `rgb(5,25,123)``\n• #061e94\n``#061e94` `rgb(6,30,148)``\nMonochromatic Color\n\n# Alternatives to #030f4a\n\nBelow, you can see some colors close to #030f4a. Having a set of related colors can be useful if you need an inspirational alternative to your original color choice.\n\n• #03214a\n``#03214a` `rgb(3,33,74)``\n• #031b4a\n``#031b4a` `rgb(3,27,74)``\n• #03154a\n``#03154a` `rgb(3,21,74)``\n• #030f4a\n``#030f4a` `rgb(3,15,74)``\n• #03094a\n``#03094a` `rgb(3,9,74)``\n• #03034a\n``#03034a` `rgb(3,3,74)``\n• #09034a\n``#09034a` `rgb(9,3,74)``\nSimilar Colors\n\n# #030f4a Preview\n\nText with hexadecimal color #030f4a\n\nThis text has a font color of #030f4a.\n\n``<span style=\"color:#030f4a;\">Text here</span>``\n#030f4a background color\n\nThis paragraph has a background color of #030f4a.\n\n``<p style=\"background-color:#030f4a;\">Content here</p>``\n#030f4a border color\n\nThis element has a border color of #030f4a.\n\n``<div style=\"border:1px solid #030f4a;\">Content here</div>``\nCSS codes\n``.text {color:#030f4a;}``\n``.background {background-color:#030f4a;}``\n``.border {border:1px solid #030f4a;}``\n\n# Shades and Tints of #030f4a\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, #010411 is the darkest color, while #fefeff is the lightest one.\n\n• #010411\n``#010411` `rgb(1,4,17)``\n• #010724\n``#010724` `rgb(1,7,36)``\n• #020b37\n``#020b37` `rgb(2,11,55)``\n• #030f4a\n``#030f4a` `rgb(3,15,74)``\n• #04135d\n``#04135d` `rgb(4,19,93)``\n• #051770\n``#051770` `rgb(5,23,112)``\n• #051a83\n``#051a83` `rgb(5,26,131)``\n• #061e95\n``#061e95` `rgb(6,30,149)``\n• #0722a8\n``#0722a8` `rgb(7,34,168)``\n• #0826bb\n``#0826bb` `rgb(8,38,187)``\n• #082ace\n``#082ace` `rgb(8,42,206)``\n• #092ee1\n``#092ee1` `rgb(9,46,225)``\n• #0a31f4\n``#0a31f4` `rgb(10,49,244)``\n• #1b40f6\n``#1b40f6` `rgb(27,64,246)``\n• #2e50f7\n``#2e50f7` `rgb(46,80,247)``\n• #4160f7\n``#4160f7` `rgb(65,96,247)``\n• #5470f8\n``#5470f8` `rgb(84,112,248)``\n• #677ff9\n``#677ff9` `rgb(103,127,249)``\n• #7a8ffa\n``#7a8ffa` `rgb(122,143,250)``\n• #8c9ffa\n``#8c9ffa` `rgb(140,159,250)``\n• #9faffb\n``#9faffb` `rgb(159,175,251)``\n• #b2bffc\n``#b2bffc` `rgb(178,191,252)``\n• #c5cefd\n``#c5cefd` `rgb(197,206,253)``\n• #d8defd\n``#d8defd` `rgb(216,222,253)``\n• #ebeefe\n``#ebeefe` `rgb(235,238,254)``\n• #fefeff\n``#fefeff` `rgb(254,254,255)``\nTint Color Variation\n\n# Tones of #030f4a\n\nA tone is produced by adding gray to any pure hue. In this case, #242529 is the less saturated color, while #000d4d is the most saturated one.\n\n• #242529\n``#242529` `rgb(36,37,41)``\n• #21232c\n``#21232c` `rgb(33,35,44)``\n• #1e212f\n``#1e212f` `rgb(30,33,47)``\n• #1b1f32\n``#1b1f32` `rgb(27,31,50)``\n• #181d35\n``#181d35` `rgb(24,29,53)``\n• #151b38\n``#151b38` `rgb(21,27,56)``\n• #12193b\n``#12193b` `rgb(18,25,59)``\n• #0f173e\n``#0f173e` `rgb(15,23,62)``\n• #0c1541\n``#0c1541` `rgb(12,21,65)``\n• #091344\n``#091344` `rgb(9,19,68)``\n• #061147\n``#061147` `rgb(6,17,71)``\n• #030f4a\n``#030f4a` `rgb(3,15,74)``\n• #000d4d\n``#000d4d` `rgb(0,13,77)``\nTone Color Variation\n\n# Color Blindness Simulator\n\nBelow, you can see how #030f4a 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":0.5321583,"math_prob":0.7304466,"size":3633,"snap":"2019-13-2019-22","text_gpt3_token_len":1632,"char_repetition_ratio":0.12510334,"word_repetition_ratio":0.011090573,"special_character_ratio":0.5549133,"punctuation_ratio":0.23489933,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9893631,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2019-03-25T20:32:27Z\",\"WARC-Record-ID\":\"<urn:uuid:71a2c123-e49e-4acc-8ccc-d2ec10ac41b0>\",\"Content-Length\":\"36281\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:343775d3-fff3-4a22-809b-b00b41fec6c9>\",\"WARC-Concurrent-To\":\"<urn:uuid:33cbdcda-81f6-4dd3-b8ad-c7608aa0f9ac>\",\"WARC-IP-Address\":\"178.32.117.56\",\"WARC-Target-URI\":\"https://www.colorhexa.com/030f4a\",\"WARC-Payload-Digest\":\"sha1:OX4YGJL3IKV6SN4O42MKMKSGR6DG6QTK\",\"WARC-Block-Digest\":\"sha1:BMZWEBUE5FOZPQ625ZU4XARZYVNNKMYN\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2019/CC-MAIN-2019-13/CC-MAIN-2019-13_segments_1552912204300.90_warc_CC-MAIN-20190325194225-20190325220225-00112.warc.gz\"}"}
https://www.mathworks.com/matlabcentral/cody/problems/49-sums-with-excluded-digits/solutions/1603736	[ "Cody\n\n# Problem 49. Sums with Excluded Digits\n\nSolution 1603736\n\nSubmitted on 8 Aug 2018 by Shawn Neal\nThis solution is locked. To view this solution, you need to provide a solution of the same size or smaller.\n\n### Test Suite\n\nTest Status Code Input and Output\n1 Pass\nn = 20; m = 5; total = 190; assert(isequal(no_digit_sum(n,m),total))\n\ns = 2\n\n2 Pass\nn = 10; m = 5; total = 50; assert(isequal(no_digit_sum(n,m),total))\n\ns = 1\n\n3 Pass\nn = 33; m = 3; total = 396; assert(isequal(no_digit_sum(n,m),total))\n\ns = 3" ]	[ null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.62429255,"math_prob":0.9993197,"size":510,"snap":"2020-10-2020-16","text_gpt3_token_len":172,"char_repetition_ratio":0.13043478,"word_repetition_ratio":0.02247191,"special_character_ratio":0.3882353,"punctuation_ratio":0.14678898,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9990136,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2020-03-29T21:16:44Z\",\"WARC-Record-ID\":\"<urn:uuid:c355a4c9-3b6c-412c-bf25-d3efe28964bf>\",\"Content-Length\":\"73383\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:333c4011-9138-46fc-bda1-9280fffa5476>\",\"WARC-Concurrent-To\":\"<urn:uuid:69c257ea-ff8f-4310-9bb9-16cd18be4021>\",\"WARC-IP-Address\":\"104.110.193.39\",\"WARC-Target-URI\":\"https://www.mathworks.com/matlabcentral/cody/problems/49-sums-with-excluded-digits/solutions/1603736\",\"WARC-Payload-Digest\":\"sha1:3OQP3PP77MM4ZAMD6PCU2BHLQTNLPH2N\",\"WARC-Block-Digest\":\"sha1:ZP6YBDI3KEJ5R6JTM7N7QKOTJC5N26KW\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2020/CC-MAIN-2020-16/CC-MAIN-2020-16_segments_1585370496227.25_warc_CC-MAIN-20200329201741-20200329231741-00522.warc.gz\"}"}
http://cogent.psyc.bbk.ac.uk/help/version2.4/classes/buffer_graph.html	[ "COGENT Online\n CONTENTS\n COGENT Version 2.4 Help", null, "# Buffer/Graph\n\n## Introduction\n\nGraph buffers provide the functionality of a buffer with an option to view and print the buffer contents as a graph or chart. Elements of a graph buffer may take one of two forms: type(DataSet, Type, Properties) or data(DataSet, X, Y), where DataSet identifies a particular data set, Type specifies the type of graph required for that data set (see below), Properties is a list of secondary properties specifying aspects of the appearance of the particular data set (see below), and X and Y are numeric values for the data set. Graph buffer elements may be matched, added or deleted, in a way analogous to elements from other sub-types of buffer.\n\nThere is no limit to the number of data sets that may be stored in one graph buffer. Normally there will be one type/3 element and several data/3 elements for each data set. Multiple data sets within a single graph buffer are treated independently, with the limitation that the graph axes and labels are determined by graph properties (see below), and are hence shared by all data sets within a graph.\n\n## Element Types\n\nA type/3 term specifies the style or type of graph to be drawn for a given data set. The second argument must be one of scatter, line, or bar. The third argument specifies graph properties such as colour and marker style, in the form of a list of secondary properties. Thus, the following type/3 element:\n\ntype(frequency, bar, [colour(blue), filled(true)])\n\nspecifies that the graph associated with frequency should be a bar-chart drawn with blue filled bars. This could be changed to a line graph using red filled square markers (and a red line) by replacing the above element with:\n\ntype(frequency, line, [colour(red), marker(square), filled(true)])\n\nThree kinds of secondary property are recognised: colour, filled, and marker. There are four marker types: square, circle, cross and plus.\n\n## Properties\n\nGraph buffers inherit all of the properties associated with the parent buffer class, plus additional properties for controlling the appearance of the graph. The full set of properties of a graph buffer are:\n\nInitialise (possible values: Each Trial/Each Block/Each Subject/Each Experiment/Each Session; default: Each Trial)\nThe timing of buffer initialisation is determined by this property. When the value is Each Trial, the buffer will automatically initialise itself at the beginning of each trial. When the value is Each Block, the buffer will initialise itself at the beginning of each block of trials (i.e., contents will be preserved across trials within a block). Similarly, when the value is Each Subject, contents will be preserved across simulated blocks, when the value is Each Experiment, contents will be preserved across simulated subjects, and when the value is Each Session, the contents will be preserved across experiments.\n\nDecay (possible values: None/Half-Life/Linear/Fixed; default: None)\nThis property specifies whether the elements of a buffer decay with time, and if so what pattern of decay is observed.\nWhen the value is None, elements will remain in a buffer until they are either explicitly deleted or are forced out by the buffer overflowing.\nWhen the value is Half-Life, elements decay in a random fashion, but with the probability of decay begin constant on each cycle. This probability is specified in terms of a half life (specified by the Decay Constant property). The half life is the number of cycles it takes on average for half of the elements to decay.\nWhen the value is Linear, elements decay in a probabilistic fashion, but with the probability of an element decay increasing linearly with each cycle it remains in the buffer. The maximum number of cycles an element can spend in the buffer is given by the value of the Decay Constant property. (So if decay is Linear and the decay constant is 10, the probability of an element decaying on the first cycle will be 0.1, the probability of it decaying within two cycles will be 0.2, and so on, with the probability of it decaying in 10 cycles being 1.0.)\nWhen the value is Fixed, elements remain in the buffer for a fixed number of cycles (specified by the Decay Constant property).\n\nDecay Constant (possible values: 1 -- 9999; default: 20)\nThis property determines the decay rate if decay is specified for the buffer. In the case of Half-Life decay, the constant specifies the half-life (in cycles) of buffer elements. A larger number will result in a longer half-life and so a slower decay. In the case of Linear decay, the constant specifies the maximum number of cycles an element may remain in the buffer. In the case of Fixed decay, the constant specifies the number of cycles an element will remain present in the buffer after the element is added to the buffer (provided it is not ``refreshed'' via Duplicates: No, as discussed above). Again, a larger number will lead to slower decay.\n\nLimited Capacity (possible values: on/off; default: off)\nThis property determines whether the buffer has limited or unlimited capacity. If the value is Yes then the buffer's capacity is limited to the value specified by the Capacity property, and its behaviour when that capacity is exceeded is governed by the On Excess property; if the value is No, these two properties are ignored.\n\nCapacity (possible values: 1 -- 9999; default: 7)\nThis property specifies the capacity of a buffer in terms of the number of items it may hold. If Limited Capacity is not selected, this property has no effect.\n\nOn Excess (possible values: Random/Youngest/Oldest/Ignore; default: Random)\nThe value of this property determines how the buffer behaves when its capacity is exceeded. If the value is Random, then a random element will be deleted from the buffer to make way for the new element. If the value is Youngest, then the most recently added element will be deleted to make way for the new element. If the value is Oldest, then the least recently added element will be deleted to make way for the new element. If the value is Ignore, then the new element will be discarded and the buffer contents will not be altered. If Limited Capacity is not selected, this property has no effect.\n\nGrounded (Boolean; default: TRUE)\nIf this property is set, attempts to add ungrounded terms (i.e., terms containing variables) to the buffer will result in an error message. The rationale for this property is that in most applications adding ungrounded terms to a buffer is probably results from a bug in the model, so flagging such occurrences is useful. In some applications, however, it may be reasonable to have ungrounded terms in buffers (e.g., where the terms represent production-like rules). In these cases, the property should not be set.\n\nAccess (values: Random/FIFO/LIFO; default: Random):\nThe order in which the buffer's elements are accessed by match operations. Interpretation of the property's values follows that of propositional buffers.\n\nTitle (values: an arbitrary character string; default: ``Title''):\nThe graph's title, which is centred above the graph when the graph is viewed in the Current Graph page or when the graph is printed.\n\nX Label (values: an arbitrary character string; default: ``X''):\nThe label drawn beside the graph's horizontal axis.\n\nAutoscale X (Boolean; default: FALSE):\nIf this property is set, COGENT will automatically select suitable values for the minimum and maximum coordinates on the X axis (based on the data present on the graph). If the property is not set, the values of the X Min and X Max properties will be used instead.\n\nX Min (values: a real number; default: 0.0):\nThe minimum value of the horizontal coordinate of the graph.\n\nX Max (values: a real number; default: 10.0):\nThe maximum value of the horizontal coordinate of the graph.\n\nX Units (values: a positive integer: default: 5):\nThe number of units into which the horizontal axis of the graph is divided.\n\nY Label (values: an arbitrary character string; default: ``Y''):\nThe label drawn beside the graph's vertical axis.\n\nAutoscale Y (Boolean; default: FALSE):\nIf this property is set, COGENT will automatically select suitable values for the minimum and maximum coordinates on the Y axis (based on the data present on the graph). If the property is not set, the values of the Y Min and Y Max properties will be used instead.\n\nY Min (values: a real number; default: 0.0):\nThe minimum value of the vertical coordinate of the graph.\n\nY Max (values: a real number; default: 100.0):\nThe maximum value of the vertical coordinate of the graph.\n\nY Units (values: a positive integer: default: 5):\nThe number of units into which the vertical axis of the graph is divided.\n\n## The Graph View\n\nGraphical buffers may be viewed as graphs by selecting the Current Graph page of the box's notebook. Each data/3 element is used to construct a data point on the graph for a particular data set, with the second and third arguments specifying the coordinates (and the first argument specifying the data set). The standard printing procedures may be used to print these graphs.\n\n## The Buffer Element Editor\n\nRecall that all information must be represented in COGENT via Prolog terms. Buffer elements are no exception, but they are perhaps the simplest sorts of box elements in COGENT. This is reflected in the simplicity of the buffer element editor. Apart from the comment line, it contains a single text field into which the buffer element should be typed. The contents of this field should be a valid Prolog term. If not, however, COGENT does automatic syntax checking (and attempted correction) of editor elements, and so any error will be noted and (possibly) corrected.\n\n COGENT Online\n CONTENTS\n COGENT Version 2.4 Help" ]	[ null, "http://cogent.psyc.bbk.ac.uk/help/version2.4/images/buffer_graph.gif", null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.82863444,"math_prob":0.8852563,"size":9482,"snap":"2019-13-2019-22","text_gpt3_token_len":2081,"char_repetition_ratio":0.16807343,"word_repetition_ratio":0.15613148,"special_character_ratio":0.21050411,"punctuation_ratio":0.124528304,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.96821046,"pos_list":[0,1,2],"im_url_duplicate_count":[null,1,null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2019-03-27T01:53:18Z\",\"WARC-Record-ID\":\"<urn:uuid:f23ff5fe-9853-4ba0-ae93-0cb535d65d74>\",\"Content-Length\":\"12282\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:a6338a0e-d722-4fdd-a973-d477f523b2bb>\",\"WARC-Concurrent-To\":\"<urn:uuid:63011515-45d0-48ed-af5d-e1a37760c2ba>\",\"WARC-IP-Address\":\"193.61.4.225\",\"WARC-Target-URI\":\"http://cogent.psyc.bbk.ac.uk/help/version2.4/classes/buffer_graph.html\",\"WARC-Payload-Digest\":\"sha1:FUMQ2LIWBOYG4F3MBXFZCOLSIT2X6WOA\",\"WARC-Block-Digest\":\"sha1:FQCXBYS3AHCLEYLEVU5LR7E4I6FA2XWW\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2019/CC-MAIN-2019-13/CC-MAIN-2019-13_segments_1552912207146.96_warc_CC-MAIN-20190327000624-20190327022624-00428.warc.gz\"}"}
https://au.mathworks.com/help/optim/ug/generate-and-plot-a-pareto-front.html	[ "Generate and Plot Pareto Front\n\nThis example shows how to generate and plot a Pareto front for a 2-D multiobjective function using fgoalattain.\n\nThe two objective functions in this example are shifted and scaled versions of the convex function $\\sqrt{1+{x}^{2}}$. The code for the objective functions appears in the simple_mult helper function at the end of this example.\n\nBoth objective functions decrease in the region $x\\le 0$ and increase in the region $x\\ge 1$. In between 0 and 1, ${f}_{1}\\left(x\\right)$ increases and ${f}_{2}\\left(x\\right)$ decreases, so a tradeoff region exists. Plot the two objective functions for $x$ ranging from $-1/2$ to $3/2$.\n\nt = linspace(-1/2,3/2);\nF = simple_mult(t);\nplot(t,F,'LineWidth',2)\nhold on\nplot([0,0],[0,8],'g--');\nplot([1,1],[0,8],'g--');\nplot([0,1],[1,6],'k.','MarkerSize',15);\ntext(-0.25,1.5,'Minimum(f_1(x))')\ntext(.75,5.5,'Minimum(f_2(x))')\nhold off\nlegend('f_1(x)','f_2(x)')\nxlabel({'x';'Tradeoff region between the green lines'})", null, "To find the Pareto front, first find the unconstrained minima of the two objective functions. In this case, you can see in the plot that the minimum of ${f}_{1}\\left(x\\right)$ is 1, and the minimum of ${f}_{2}\\left(x\\right)$ is 6, but in general you might need to use an optimization routine to find the minima.\n\nIn general, write a function that returns a particular component of the multiobjective function. (The pickindex helper function at the end of this example returns the $k$th objective function value.) Then find the minimum of each component using an optimization solver. You can use fminbnd in this case, or fminunc for higher-dimensional problems.\n\nk = 1;\n[min1,minfn1] = fminbnd(@(x)pickindex(x,k),-1,2);\nk = 2;\n[min2,minfn2] = fminbnd(@(x)pickindex(x,k),-1,2);\n\nSet goals that are the unconstrained optima for each objective function. You can simultaneously achieve these goals only if the objective functions do not interfere with each other, meaning there is no tradeoff.\n\ngoal = [minfn1,minfn2];\n\nTo calculate the Pareto front, take weight vectors $\\left[a,1–a\\right]$ for $a$ from 0 through 1. Solve the goal attainment problem, setting the weights to the various values.\n\nnf = 2; % Number of objective functions\nN = 50; % Number of points for plotting\nonen = 1/N;\nx = zeros(N+1,1);\nf = zeros(N+1,nf);\nfun = @simple_mult;\nx0 = 0.5;\noptions = optimoptions('fgoalattain','Display','off');\nfor r = 0:N\nt = onenr; % 0 through 1\nweight = [t,1-t];\n[x(r+1,:),f(r+1,:)] = fgoalattain(fun,x0,goal,weight,...\n[],[],[],[],[],[],[],options);\nend\nfigure\nplot(f(:,1),f(:,2),'ko');\nxlabel('f_1')\nylabel('f_2')", null, "You can see the tradeoff between the two objective functions.\n\nHelper Functions\n\nThe following code creates the simple_multi function.\n\nfunction f = simple_mult(x)\nf(:,1) = sqrt(1+x.^2);\nf(:,2) = 4 + 2sqrt(1+(x-1).^2);\nend\n\nThe following code creates the pickindex function.\n\nfunction z = pickindex(x,k)\nz = simple_mult(x); % evaluate both objectives\nz = z(k); % return objective k\nend" ]	[ null, "https://au.mathworks.com/help/examples/optim/win64/GenerateAndPlotParetoFrontExample_01.png", null, "https://au.mathworks.com/help/examples/optim/win64/GenerateAndPlotParetoFrontExample_02.png", null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.73659754,"math_prob":0.9998586,"size":2796,"snap":"2022-05-2022-21","text_gpt3_token_len":784,"char_repetition_ratio":0.15114613,"word_repetition_ratio":0.01965602,"special_character_ratio":0.3050787,"punctuation_ratio":0.21314102,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9997869,"pos_list":[0,1,2,3,4],"im_url_duplicate_count":[null,2,null,2,null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2022-01-29T03:49:45Z\",\"WARC-Record-ID\":\"<urn:uuid:fc39895f-1a96-4522-b71c-731a86867445>\",\"Content-Length\":\"82275\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:8b4feedc-0f70-48bb-8adb-8b227d7e0e11>\",\"WARC-Concurrent-To\":\"<urn:uuid:1231907f-caf5-4381-b90a-39b8e9184341>\",\"WARC-IP-Address\":\"23.47.146.88\",\"WARC-Target-URI\":\"https://au.mathworks.com/help/optim/ug/generate-and-plot-a-pareto-front.html\",\"WARC-Payload-Digest\":\"sha1:SNZQCEJ3PMQ6JIKQCLPPJA635XO4QUIM\",\"WARC-Block-Digest\":\"sha1:CCH7S43HPAMSK3RWSOFWV7E32CYFDRZE\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2022/CC-MAIN-2022-05/CC-MAIN-2022-05_segments_1642320299927.25_warc_CC-MAIN-20220129032406-20220129062406-00321.warc.gz\"}"}
https://nontrivialproblems.wordpress.com/tag/scientific-units/	[ "# When Less is More, part 2: Flipping the Sign\n\nIn part 1, we looked at absolute temperature scales. Before an aside about not setting your kitchen on fire, we defined absolute zero as the point where atoms have no more energy from their motion (or kinetic energy). We’ve now also nicely set ourselves up for the “big deal”. If absolute zero is where there’s no kinetic energy, how do you get below that? Can atoms somehow have negative kinetic energy? The simple (and relevant to this discussion) answer is no.\n\nInstead, we need to add something we’ve been neglecting from our discussion of temperature. So far we’ve only talked about heat. But another concept is used in the “relevant” definition (I’ll have more to say on this below) of temperature for this experiment: entropy, which is basically the amount of disorder in a system. If you took an advanced physics or chemistry class in high school (or maybe a first-year class in either of those fields in college), you might have learned the second law of thermodynamics, which states that entropy tends to increase, and must increase in closed systems. It also defines how entropy changes in physical processes: at a given (absolute) temperature T, if an amount of heat dQ flows into a system (we can also use a negative sign for heat flowing out), then the entropy S changes by an amount dS by", null, "$\\Large dS=\\frac{\\ dQ}{\\ T}$\n\nOf course, definitions can sometimes work both ways, and that’s what physicists decided to do. Based on this, they solve for T, and we “officially” define temperature as", null, "$\\Large T=\\frac{\\ dQ}{\\ dS}$\n\nAnd so with this definition, we can see where a negative temperature comes from: anytime the heat flow and entropy change have different signs, T should be negative.\n\nWe can also be more general in that last equation, and look at temperature as a function of how entropy changes with respect to energy (if you’ve taken calculus, I mean the derivative of entropy with respect to energy), and temperature is", null, "$\\Large \\frac{\\ 1}{\\ T}=\\frac{\\ d}{\\ dE} S(E)$\n\nOf course, “anytime” would still be rare in our everyday experience. At the low energies and macroscopic scales of normal human life, heat (or energy) and entropy increase together. But scientists can set up quantum systems (read: basically any system on an atomic scale) where we break that trend. This is where a more basic understanding of entropy comes in. If you have a system where atoms can have two different energies, the greatest entropy is when half of the atoms are in each energy state. This is because that allows for the greatest number of combinations of atoms. Look at this plot of the number of total possible combinations for making a combination of k atoms from a set of n total atoms based on this formula", null, ", in this case where n is 50.\n\nAs you can see, the number of combination explodes close to the halfway point. Also note the symmetry of the combination curve in this case. So let’s say with our 50 atoms, k equals the number of atoms in the second, higher energy state. (For those who know about electron orbitals, realize here that we’re not talking about the energy levels of the electron; we’re actually talking about the energy state the whole atom. This can come up in systems where atoms are placed in magnetic fields.) In most everyday systems, you’d have a lot of atoms in the lower state, and so we have a small k. Adding energy kicks atoms up to the higher state, so k increases, and you see the number of possible configurations (and therefore the entropy), also increases. But we have the opposite happen if we were to start with a high energy system, where k is already close to 50. In that case, adding energy would decrease the entropy, and so by the third equation, we have a negative temperature. Ta da!\n\nThe idea of a “simple” energy state system having a negative temperature is actually kind of old hat; i.e., we did it already. (In fact, if you know how lasers work, this basically describes the population inversion of electrons; we just don’t typically ascribe a “temperature” to electrons confined in solids.) So what’s the big deal about this new research? While the atoms were in a gas state, the team was able to get them in a lattice arrangement in space and actually prevented motion of atoms. By rapidly adjusting the magnetic field and lasers to trap most of the atoms in a high energy state, they made what should have been an energetically unstable arrangement (like a pencil balanced on its tip) stable.\n\nNow that we have a spatially spread out system in a negative temperature, we can also test lots of interesting physics. For example, the combination of stability on millisecond time scales (which is LONG for particle physics) and high energies can allow for unique probing of the Standard Model, such as the formation of structures defined by the strong force. The researchers also point out that in this negative temperature regime, they also experience negative pressure, which is similar to the force believed to drive the expansion of the universe.\n\nSo that paragraph ends the NEW science, but I promised I would go more into the definition of temperature. Or more accurately, why many people (look at the comments on the review) seem to complain about how this really isn’t a negative absolute temperature. Especially as those slightly more in the know freak out when they learn that systems with negative absolute temperature behave as if they were hotter than a temperature of positive infinity. I’ll explain that confusing fact briefly. If you put a negative temperature object next to a positive temperature object and there’s no energy source, heat will flow from the negative temperature object. As I mentioned, the 2nd Law of Thermodynamics says entropy increases in closed systems. Adding energy to the negative temperature system would decrease the system entropy (positive temperature systems lose entropy when they lose energy, and the negative system loses entropy when it gains energy). To maximize the entropy, heat would need to flow out of the negative temperature object to the positive temperature object, bringing both objects closer to that giant entropy peak in the middle.\n\nThose who complain that the temperature is only negative because of a weird definition resort to arguments about kinetic energy. Unless you take a course on statistical mechanics, you almost never see the definition of temperature we just derived here. Instead, you typically talk about how temperature is a measure of the average kinetic energy of the atoms in a system, and absolute zero is then described as the point where all atomic motion stops. This then leads people to an obvious question, “How can atoms move less than being stopped?”\n\nIt’s a good question. The problem is that the basic definition people are using isn’t the right one. As one of the wonderful SciBloggers explains, the kinetic energy is important, but not in the way that simplification leads most people to believe. It’s not the mere average of atomic energies that matters, it’s the nature of the probability distribution of energy around the average. In statistical physics, this is described by the (Maxwell-)Boltzmann distribution.", null, "Plot of number of atoms at each velocity for a variety of temperatures based on the Boltzmann distribution (given in Celsius, not Kelvin). A plot for energy would be similar. (From Wikipedia)\n\nIn the figure, you’ll see that the average velocity does increase with temperature. Another trend is that the distribution widens with increasing temperature. And if you analyzed the graphs, you’d see that most atoms have less energy than the average energy. For a negative temperature system, these features are slightly tweaked. The average energy increases with temperature (so -1 K has a higher average energy than -100 K). The distribution widens at lower temperatures. And the big deal is that most particles in a negative temperature system have a higher kinetic energy than the average. This would look like the graph of the Boltzmann distribution if we say the temperature is negative, and so that’s why we go with that. So it’s not that physicists have lied, it’s just kind of bizarre compared to our normal experience.\n\nThere’s also one slightly more intuitive explanation that I’ve been using to explain the concept of negative temperature. First, I need to dispel another aspect of the definition that people are confused on. Atoms at absolute zero don’t have zeroes of everything else. Quantum mechanics says there is a zero-point, or minimum, energy that all objects have in a system. And I’m pretty sure the uncertainty principle requires anything with a non-zero energy to have a momentum, which means it must move. Instead, physicists define absolute zero as a minimum entropy point. In order to minimize the entropy of a negative temperature system, we have to add energy to it. In other words, we have to ADD heat to a negative temperature system to bring it to absolute zero.\n\n# When Less Is More, part 1: What Is Temperature?\n\nSo the science regions of the blogosphere recently exploded with a new paper claiming to have made a material with a negative absolute temperature. Those last three words together may sound weird to the uninitiated (and residents of the Plains and Northeast may wonder why physicists need all that newfangled equipment to go below zero degrees), but they need to be combined in that phrase in physics. If you’ve ever been prone to intense existential thought over your weather forecast, you might realize the way we think of 0 degrees in everyday life seems a bit arbitrary compared to what 0 means for other units. Most of the time when we say something is 0, we mean there’s none of that quality. A length or mass of 0 means there is nothing there. A speed of 0 means something isn’t moving. Zero current means there’s no electricity. A bank account with \\$0 means you have no money. And in units that allow negative amounts, these counteract positive values. A negative velocity means you’re going in the direction opposite your reference. A negative financial statement means you’ve lost money or owe money (and so take away from positive values of money you had).\n\nSo what do these mean for temperature? First, it’s important that while we deal with temperature everyday, it’s actually kind of an ephemeral unit. It’s not the same thing as energy, but it is related to that. And in scientific contexts, temperature is not the same as heat. Heat is defined as the transfer of energy between bodies by some thermal process, like radiation (basically how old light bulbs work), conduction (touching), or convection (heat transfer by a fluid moving, like the way you might see soup churn on a stove). So as a kind of approximate definition, we can think of temperature as a measure of how much energy something could give away as heat* (Edit: A friend of mine has come up with a good analogy: Temperature is like pressure, but for heat. Mass moves from high to low pressure areas, and heat flows from high to low temperature objects). And so now, we realize there’s nothing particularly special about 0 degrees in our everyday scales, in both Fahrenheit and Celsius. I mean, sure, there is a marker that these definitions came from. Fahrenheit’s 0 point is the temperature which a mixture of water ice and ammonium chloride (which Scandinavians might recognize as the flavoring of salty licorice) always stabilizes to. And as nearly all middle schoolers know, Celsius is 0 at the temperature at which water freezes.\n\nWhile these are cold temperatures to our human flesh, you know things at 0 degrees in either scale can still manage to give up heat (just not to you and your warm 98.6 degree F body). This is where absolute temperature comes into play. For the field of thermodynamics, where temperatures and heat transfer are important in studying energy, scientists quickly realized that their old zero points were meaningless when trying to talk intelligently about energy. So they made new scales with “absolute zero”, a point where there is no more energy (at least from the motion of atoms). This point is about -273 °C and -460 °F. This is also partially why anyone attempting to cook a recipe twice as fast by “doubling” the temperature on their oven/stove/whatever is doomed to failure, because going from say 250 °F to 500 °F on your oven hasn’t doubled the temperature. Instead, you would need to go from 710 absolute degrees Rankine (the absolute temperature scale based on the Fahrenheit intervals; ask a chemical engineering friend if they’ve worked in Rankine) to 1420 °R, which would be like 960 °F and would lead to a really amusing story with your oven manufacturer and/or the fire department.\n\nYou’ll notice a first on the blog here, and that’s the presence of a “part 1”. I haven’t actually gotten to the new science of the article yet, but I wanted to set up the background here. Scientific units actually have fascinating histories and I got incredibly caught up on researching temperature for this part, so I ended up with way more information than I originally planned on.\n\n*This is a really loose definition of temperature. Don’t ever use this on homework, ever. 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https://www.arxiv-vanity.com/papers/hep-th/0301111/	[ "# Charge density and electric charge in quantum electrodynamics\n\nG. Morchio,\nDip. di Fisica, Universita’ di Pisa and INFN, Pisa, Italy\nF. Strocchi,\nScuola Normale Superiore and INFN, Pisa, Italy\n###### Abstract\n\nThe convergence of integrals over charge densities is discussed in relation with the problem of electric charge and (non local) charged states in Quantum Electrodynamics (QED). Delicate, but physically relevant, mathematical points like the domain dependence of local charges as quadratic forms and the time smearing needed for strong convergence of integrals of charge densities are analyzed. The results are applied to QED and the choice of time smearing is shown to be crucial for the removal of the vacuum polarization effects responsible for the time dependence of the charge (Swieca phenomenon). The possibility of constructing physical charged states in the Feynman-Gupta-Bleuler gauge as limits of local state vectors is discussed, compatibly with the vanishing of the Gauss charge on local states. A modification by a gauge term of the Dirac exponential factor which yields the physical Coulomb fields from the Feynman-Gupta-Bleuler fields is shown to remove the infrared divergence of scalar products of local and physical charged states, allowing for a construction of physical charged fields with well defined correlation functions with local fields.\n\n## 1 Introduction\n\nThe simple relation between charge density and electric charge in classical electrodynamics does not extend trivially to the quantum case, because of problems due to vacuum polarization and infinite volume integration.\n\nQuite generally, the relation between local charges and global conserved charges has been extensively discussed in the seventies, in relation with the proof of the Goldstone theorem [1, 2, 3, 4] and it has become standard wisdom in the quantum field theory (QFT) framework in which all the relevant information are carried by the local states.\n\nThe problem changes substantially if the relevant charged states are non local, as it is the case of Quantum Electrodynamics(QED) . As a consequence, one cannot rely on the standard strategy of controlling the convergence of local charges on the domain of local states, and in fact the limit of local charges, as quadratic forms, crucially depends on the domain which is considered. Moreover, as discussed by Swieca , on the charged states obtained by applying Coulomb fields to the vacuum, the local charge given by the integral of the density with the standard smearing in space and time does not converge to the electric charge and its limit is even time dependent.\n\nThis difficulty requires an analysis of the convergence of suitably time-smeared integrals of the charge density; as we shall see, not only the standard time smearing does not work, but also Requardt’s space-time smearing prescription requires a modification in order to obtain the correct result for the renormalized charge. Actually, the basic point is the control of the construction of charged states, which is related to the infrared problem and is a deep non perturbative problem, both in the general algebraic approach and in the approach which uses fields operators .\n\nEven in perturbation theory a rigorous control on the construction of charged states is far from trivial. In the (positive) Coulomb gauge the (non local) charged fields are difficult to handle and the standard strategy is to use a local formulation at the expense of positivity, as in the Feynman-Gupta-Bleuler gauge. In this case, the charged states should be the obtained by an appropriate construction in terms of local (unphysical) states. Such a possibility has been advocated by Dirac and Symanzik [9, 8] who proposed explicit formulas for non local charged (Coulomb) fields in terms of the local Feynman-Gupta-Bleuler fields. Such a construction, which involves non trivial ultraviolet and infrared problems has recently been refined by Steinmann [10, 13] on the basis of a perturbative expansion.\n\nAn important issue is whether the above states can be constructed only in terms of expectations of the observables or they exist as vectors in a space in which local states are dense. In the latter case, the control of limits of local states requires a topology and the topology defined by the Wightman functions of the local fields is too weak to give a unique space; thus, the possibility of reaching the physical charged states, characterized by a Coulomb delocalization, depends on the choice of a topology. For example, the implicit use of the standard (Krein) metric on the asymptotic fields excludes the presence of charged states in the corresponding physical space, as pointed out by Zwanziger in his investigations on the infrared problem in QED . A possible non perturbative construction of physical charged states as limits of local states was discussed in Ref. , with the use of a Hilbert-Krein topology which takes into account the effects of the infrared problem. In our opinion, the non uniqueness of such Hilbert-Krein majorant topologies, which are associated to the Wightman functions of the local fields in order to obtain weakly complete inner product spaces of states, should not be regarded as a mathematical oddness, being related to the allowed large distance behaviour or ”boundary conditions” at infinity.\n\nThe possibility of constructing physical charged states as limits of local state vectors in a weak topology has been recently denied on the basis of an argument by which the local Gauss charge, corresponding to the integral of , vanishes on the local states and therefore on any weak closure of them; thus no weak closure of the local states could contain physical charged states. A main conclusion of our analysis is that the assumptions involved in the argument underestimate the delicate rôle of such topologies for the convergence of local charges in QED.\n\nIn view of the problems which arise in QED, in Section 2 we discuss in general charges defined as limits of quadratic forms, their crucial dependence on the domain and their relation to global charge operators; in particular, attention is paid to the case in which the relevant domains arise by applying non local field operators to the vacuum.\n\nIn Section 3 we consider the problem of weak convergence of local charges, which is shown to be very relevant for the Steinmann argument. Strong convergence on the vacuum is shown to be a general consequence of a stronger version of Requardt’s theorem, which also allows for an improved time smearing procedure, necessary for obtaining the correct value of the charge on Coulomb charged states. Such a time smearing procedure avoids the time dependence effects due to vacuum polarization while preserving the correct value of the charge.\n\nIn Section 4, we discuss convergence of Gauss local charges on physical charged states, on the basis of the standard local formulations of QED, the Feynman-Gupta-Bleuler gauge. Quite generally, independently of the use of a Hilbert-Krein topology, it is shown that the construction of physical charged state vectors as limits of local states in a weak topology is incompatible with convergence, in the same weak topology, of the Gauss local charges, even with a time smearing a la Requardt, on local states. A simple model is discussed which mimics the relation between charge density and charge in QED and displays the compatibility between the vanishing of the Gauss charge on a dense domain of local states and its strong convergence to a non zero electric charge on the physical space. In Section 5 we compare the construction of physical charged states of Ref. with the DSS construction analyzed by Steinmann [10, 13]. We show that the infrared divergence in the matrix elements of physical charged states with local states is avoided by the modified DSS exponential used in Ref. , which only differs from the standard factor by a gauge term. In this way one removes the obstruction pointed out by Steinmann as an argument for the impossibility of constructing physical charged fields with well defined correlation functions with local fields.\n\n## 2 Charges as limits of quadratic forms\n\nThe analysis of the charge operator in QED presents subtle features arising from the Coulomb delocalization of charged states [5, 12]. It is therefore convenient to start by an analysis of charges as integrals over a local density on general (not necessarily local) domains.\n\nIn this section we shall show that i) charges defined as limits of quadratic forms on dense domains in general (including the quantum field theory case) crucially depend on the domain; e.g. may converge to zero on and have a non zero limit on if is not dense, ii) such a phenomenon cannot occur if converges weakly on and on .\n\nQuite generally, in quantum field theory the problem of associating an (unbroken) charge to the integral over a local density\n\n QR=∫\|x\|≤Rdxj0(x,0),∂μjμ=0,\n\nis delicate and deserves special attention. Intuitively, one thinks of defining a state of charge , as satisfying\n\n QΨ=limR→∞QRΨ=qΨ,\n\nbut as emphasized by Schoer and Stichel , the limit does not exist as a weak limit, even if some smearing in time is made with and even if is a local state, briefly . In the latter case, the limit exists as a sesquilinear form on\n\n limR→∞(Φ,QRΨ)=Q(Φ,Ψ),Φ,Ψ∈D0.\n\nFurthermore, if defines an unbroken symmetry on the local fields the limit sesquilinear form defines an (hermitean) operator on .\ni) Domains and limits of quadratic forms\nIn general, the limit of hermitean operators as forms on domains , crucially depends on the domain , in particular, the limit on does not constrain the limit on , .\n\nSuch a domain dependence in general persists, as shown by the example below, even if converges to an hermitean sesquilinear form on satisfying the boundedness condition\n\n \|Q(Φ,Ψ)\|≤CΨ\|\|Φ\|\|,∀Φ,Ψ∈D, (2.1)\n\nand therefore identifies an (hermitean) operator with domain . Furthermore, even if eq.(2,1) holds, it is not at all guaranteed that, ,\n\n (χ,QΨ)=limR→∞(χ,QRΨ),∀Ψ∈D. (2.2)\n\nIn fact, such an equation means that converge weakly. By the convergence of on , weak convergence of is equivalent to the boundedness of the norms , for each fixed .\n\nIn particular, as shown by the example below, even if converges to zero , one cannot conclude that , . (This also shows that the failure of eq.(2.2) does not depend on converging to an unbounded or a bounded operator.)\n\nThe general phenomenon is that, if converge to an operator on and to an operator on , the two operators and are in general not related, in the sense of the following\n\n###### Definition 2.1\n\nTwo densely defined hermitean operators are said to be related if there is an hermitean operator of which and are restrictions. They will be said to be weakly related if there is a densely defined hermitean operator to which both and are related.\n\nThe above relations are symmetric and the second notion is strictly weaker, since e.g. different self adjoint extensions of an hermitean operator are not restrictions of the same hermitean operator. An example of limits of quadratic forms which define not weakly related operators is given below.\n\nExample. Let us consider the space of functions vanishing at the origin, the linear span of and\n\nClearly, both and are dense domains; in fact, if is orthogonal to one has\n\n cn≡(f,sinnx)=αn(f,sinx)=αnc1,n≥2.\n\nFurthermore\n\n 0=(π/2)∫π0dxf(x)=∑n≥1cn∫π0dxsinnx=2(∑n≥2α2nc1+c1)\n\nimplies , i.e. . Now, let be the multiplication operator by a regular function converging to as a distribution; then\n\n (D0,QRD0)→0,(D1,QRD1)→(D1,P1D1)≠0,\n\nwith the projection on . Thus, the limits of the hermitean operators define two bounded operators which are not even weakly related.\n\nConvergence of on to an operator constrains convergence to an operator on any domain , such that is dense.\n\n###### Proposition 2.1\n\nLet the hermitean operators converge to an operator on and to an operator on ;\ni) if is dense, then and are weakly related\nii) if , then and are related\niii) in both cases, if is essentially selfadjoint on , then is contained in the closure of\niv) if and are not related, then does not converge to an operator on .\n\nProof The hermiticity of implies that both are densely defined hermitean operators and so is their restriction to . In case ii) extends , in case i) and extend . If converge to on , both and are restrictions of , so that and are related.\n\n###### Proposition 2.2\n\nIf both and converge weakly, then the two limits define hermitean operators and which are related\n\nProof Hermiticity of the limit forms follows from that of and the existence of weak limits implies that the limit forms define operators on and on . The weak limit of exists also on and by the same argument defines an hermitean operator which extends and .\n\nAs a result, if is essentially selfadjoint, is contained in its closure and in particular if , also , in other terms if converges to zero weakly and converges weakly, then .\nii) Convergence of local charges in quantum field theory\nA general situation which occurs in quantum field theory is described in terms of translational invariant (field) algebras , a (unique translationally invariant) cyclic vector , domains\n\n D0=A0Ψ0,D1=A1Ψ0,\n\nand local hermitean charges , with domains containing and and with . In general, is the integral of the zero component of a local conserved (operator valued tempered distribution) current with suitable smearing:\n\n QR=∫d4xj0(x,x0)fR(x)α(x0)=j0(fRα), (2.3)\n fR(x)=f(\|x\|/R)∈D(R3),f(x)=1,if\|x\|≤1,f(x)=0,if\|x\|≥2,\n α∈D(R),suppα⊂[−a,a],a<1,∫dtα(t)=1.\n\nIf is a local (field) algebra and converges as , the limit defines an operator iff\n\n limR→∞(Ψ0,[QR,A0]Ψ0)=0, (2.4)\n\nequivalently iff\n\n limR→∞(D0,QRΨ0)=0. (2.5)\n\nNon local algebras may be relevant in the discussion of non local states, e.g. asymptotic states, or charged states in the Coulomb gauge; a local and a non local field algebra, and , occur in the construction of charged states in QED.\n\n###### Proposition 2.3\n\nLet an algebra invariant under translations; if on , converge to an operator , then\n\n limR→∞(D,QRΨ0)=0. (2.6)\n\nProof The spectral representations of the space translations gives\n\n ((U(a)−1)4AΨ0,QRΨ0)=∫dJA(k)(eik⋅a−1)4R3~f(Rk),∀A∈A (2.7)\n\nwhere\n\n \|(eik⋅a−1)4R3~f(Rk)P(k)\|≤\|Rk⋅a\|4R\|~f(Rk)P(Rk)\|\|P(k)P(Rk)\|≤CR→0,\n\nin the limit , the r.h.s. of eq.(2.6) converges to zero and therefore, by the density of , one has\n\n (U(a)−1)4QΨ0=0,∀a.\n\nThen, since is a normal operator, it follows that and by the uniqueness of the translationally invariant state ; actually , because .\n\nThus, under the same assumptions, one has that the charge defined in terms of the limit of the commutator , coincides with , i.e.\n\n (D,Q′AΨ0)≡limR→∞(D,[QR,A]Ψ0)=limR→∞(D,QRAΨ0). (2.8)\n\nThe domain dependence of charge operators obtained as limits of quadratic forms appears also in the above quantum field theory framework. In particular, as a result of Proposition 2.1, if converges to zero on , the convergence to a non zero operator on is excluded if is dense, but may be allowed if is not dense, even if .\n\nSuch features are illustrated and displayed by the following Example.\nExample. Let be a massless scalar field, a free Dirac field, the algebra generated by and by and the algebra generated by and by\n\n ψd(x)=ψ(x)U(x),U(x)=eiϕ(fx)\n ϕ(fx)=∫dyϕ(y)f(y−x),f∈D(R4),∫dxf(x)=1.\n\nThen we consider the local charges\n\n QRϕ≡∂0ϕ(fRα,),QRψ≡j0(fRα),jμ(x)=:¯ψγμψ:,\n QR=QRψ+QRϕ\n\nand the Fock representation of , with Fock vacuum . Since by locality\n\n limR→∞[QRϕ,A0]=0,limR→∞(D0,QRΨ0)=0,\n\nwe have\n\n limR→∞(D0,QRD0)=(D0,QψD0),\n\nwhere is the unbroken fermionic charge. On the other hand, since we have\n\n limR→∞(D,QRD)=0.\n\nIn conclusion converge to the unbroken fermionic charge on and to the zero charge on .\n\nIt is worthwhile to note that the limit of the operators does not define an operator on , where , (since the corresponding bilinear form is discontinuous on the left). Moreover, one has a symmetry breaking condition on the algebra generated by and : converges weakly (actually strongly) and\n\n limR→∞(Ψ0,[QR,ψ†ψd]Ψ0)≠0.\n\nThis fact is actually a consequence of and being not related. In general if converges on to operators which are not related, then, for the algebra generated by and , one cannot have both weak convergence of and\n\n limR→∞(Ψ0,[QR,A]Ψ0)=0. (2.9)\n\nIn fact, by eq.(2.6),\n\n (Di,QiAΨ0)≡limR→∞(Di,[QR,A]Ψ0),∀A∈Ai.\n\nNow, if eq.(2.9) holds, by a standard argument one gets an hermitean operator on , which extends and , in contrast with their being not related.\n\n## 3 Convergence of time smeared integral of charge density. The vacuum sector of QED\n\nIn this section we discuss weak and strong convergence of local charges, in particular in the vacuum sector of QED.\n\nAs found by Requardt , the weak limit of on local states can be obtained under general conditions by a suitable time smearing of the charge density, namely by considering, with as in eq.(2.3),\n\n QR≡j0(fRαR),αR(x0)≡α(\|x0\|/R)/R. (3.1)\n\nActually, one can strengthen Requardt’s theorem and obtain strong convergence (Proposition 3.1), also with a more general time smearing , which will prove necessary in the charged sectors of QED.\n\nWe recall that if is a Lorentz covariant conserved tempered current, the two point function of the charge density is of the form\n\n =−ΔJ(x−y),\n\nwith a Lorentz invariant tempered distribution of positive type; we denote by the spectral measure defined by .\n\n###### Proposition 3.1\n\nIf the spectral measure satisfies the (infrared) regularity condition\n\n dν(k2)=k2dσ(k2),dσameasure,\n\nthen, putting one has\ni)\nii) for all functions with , satisfying and ,\niii) if, for , , the above strong convergence to zero is obtained by choosing .\n\nProof In fact, one has\n\n \|\|QR,TΨ0\|\|2=∫dν(k2)d3q\|q~f(q)\|22√(\|q\|/R)2+k2)R\|~α(T√(\|q\|/R)2+k2)\|2.\n\nSince is of fast decrease,\n\n \|~α(T√(\|q\|/R)2+k2)\|2≤CN1+((T\|q\|/R)2+T2k2)N≤CN1+(T2k2)N,\n\nand since is tempered there is an such that is a finite measure. Then, by taking , one has\n\n \|\|QR,TΨ0\|\|2≤C′RT∫dσ′(s2)Ts(1+T2s2)2≡RTG(T).\n\nThe integrand function is bounded and converges to zero pointwise, when , so that by the dominated convergence theorem . Thus i) is proved; moreover strong convergence to zero holds if one chooses and ii) follows since ,\n\n ∫∞εdσ′(k2)T√k2/(1+T2k2)2=O(1/T3).\n\nIf the hypothesis of iii) holds one can bound the integral from to by\n\n C∫ε0ds2Ts(1+T2s2)2≤CT2∫∞0du2u(1+u2)2=O(1/T2).\n\nThus, the strong convergence to zero is obtained if .\n\nIn the physical vacuum sector of QED the assumptions of Proposition 3.1 for the spectral measure of the electric current are satisfied since\n\n <∂F0(x)∂F0(y)>=∫k2dρ(k2)d3k\|2√k2+k2\|−1k2eik(x−y),∂Fμ≡∂νFμν,\n\nwith the spectral measure of the two point function of . Hence, for as in ii) of Proposition 3.1,\n\n limR→∞\|\|∂F0(fR,αT(R))Ψ0\|\|2=0 (3.2)\n\nand therefore converges strongly to zero on the dense domain obtained by applying local bounded observable operators to the vacuum. Eq.(3.2) with was also obtained by D’Emilio .\n\nThe situation is completely different if one adopts the standard smearing [1, 2], with a fixed ,\n\n ~QR=j0(fRα).\n###### Proposition 3.2\n\nThe operators have the following properties\ni) they converge to zero on\nii) does not converge weakly in , nor does , a bounded local operator\niii) there are vectors such that\n\n limR→∞<Ψ,~QRΨ0>\n\ndepends on the time smearing test function (time dependence of the charge)\niv) there are operators such that,\n\n limR→∞<Ψ0,[~QR,F]Ψ0>≠0\n\n(Swieca phenomenon )\n\nProof Since in the physical vacuum sector , i) follows by locality and Maison theorem .\n\nFor ii), the same calculation done above for now gives\n\n \|\|~QRΨ0\|\|2=R∫k2dρ(k2)d3q\|q~f(q)\|22√(\|q\|/R)2+k2)R\|~α(√(\|q\|/R)2+k2)\|2,\n\nso that cannot converge weakly. Furthermore,\n\n ~QRUΨ0=[~QR,U]Ψ0+U~QRΨ0\n\nand the first term on the r.h.s converges by locality; since the second term does not converge weakly, neither does the l.h.s.\n\nIn order to construct the vector of iii) we consider\n\n ΨR≡F0i((∂iΔ−1g)fRh)Ψ0,g∈D(R3),h∈D(R).\n\nSuch vectors converge strongly to a vector , for , since the Fourier transform of is square integrable with respect to the measure defined by the Fourier transform of . Then, we have\n\n limR→∞<Ψ,~QRΨ0>=limR→∞∫dρ(k2)d3k\|2k0\|−1k2~fR(k)~α(k0)¯~g(k)¯~h(k0)→\n ¯~g(0)∫d ρ(m2)m~α(m)¯~h(m),\n\nwhich displays the dependence on .\n\nThe operators converge strongly to an operator on the dense domain the algebra of strictly localized (bounded) observables, since they converge strongly on and becomes independent of , for sufficiently large by locality. Then, we have\n\n limR→∞<Ψ0,[~QR,F]Ψ0>=¯~g(0)∫dρ(m2)m(~α(m)¯~h(m)−~α(−m)¯~h(−m))\n\nwhich does not vanish in general.\n\nThe vector reflects the infrared behaviour of ”dipole states” of the form , where is the electron field in the Coulomb gauge, constructed, e.g., according to the Dirac-Symanzik-Steinmann [8, 10] prescription. Thus, in QED, even in the vacuum sector, the naive idea of the charge as the integral of the charge density gives rise to substantial problems because of vacuum polarization effects which disappear only with a suitable time smearing. The same problems arise in the charged sectors of the Coulomb gauge, as stressed by Swieca ; they are a general consequence of the non locality of the charged Coulomb fields.\n\nIn general, the standard procedure, eq.(2.3), corresponds to taking, in the corresponding correlation functions in momentum space, the limit and gives a function in only in expectations on local states. On the other hand, Requardt time smearing corresponds to taking a limit on the light cone; in expectations on local states, it coincides with that of the standard smearing and it is independent. As discussed in the Appendix, independence does not hold on the (non local) charged states of QED and therefore a modification of Requardt’s prescription is required for QED.\n\n## 4 Charge density and charge in local formulations of QED\n\nThe relation between charge density and charge presents further subtle aspects in the charged sectors. As a consequence of the local Gauss’ law, charged states cannot be local. In this section we discuss the limit of the Gauss charges\n\n QGR=(∂F)0(fRαR)\n\nas quadratic forms on local and on physical charged states in the Feynman-Gupta-Bleuler formulation of QED and the implications on the possibility of constructing physical state vectors as weak limits of local states.\n\nIn the Coulomb gauge, since the charged fields are not local, one has to discuss the limit of local charges on domains obtained from the vacuum by a non local algebra, giving rise to the problems discussed in Sect.2.\n\nEven in perturbation theory the control of the Coulomb gauge is difficult and the standard strategy is to use a local formulation at the expense of positivity; this is the case of the Feynman or Gupta-Bleuler gauge. In this case, the charged fields and the vector potential are local but their vacuum expectation values cannot satisfy positivity; the corresponding Wightman functions define an indefinite inner product space (with the local field algebra), with inner product denoted by , which does not contains physical charged states [5, 12].\n\nAs suggested by perturbation theory, non local physical charged states may be obtained as suitable limits of local unphysical charged state vectors. A possible non perturbative construction of physical charged state vectors along these lines was discussed in .\n\nQuite generally, a crucial issue is that the definition and the control of the limit of local charged state vectors requires a topology; even in the positive case the weak topology on defined by the seminorms , i.e. by the Wightman functions is too weak; on the other hand, the inner product space does not identify a unique Hilbert-Krein majorant topology and one has different closures . For the physical interpretation, the relevant space is the physical subspace , identified by a subsidiary condition (which in QED selects gauge invariant states) and different topologies may give rise to isomorphic physical spaces.\n\nIn general, the dependence of the space on the topology should not be regarded as a mathematical oddness, since different closures of reflect different ”boundary conditions” at infinity. Even in the standard theory of unbounded hermitean operators the local domain of functions of compact support may allow different self adjoint extensions, corresponding to different boundary conditions; in the physical applications the choice of one instead of the other is dictated by physical considerations [12, 17]. In the QED case the lack of non uniqueness reflects the physical fact that different Hilbert-Krein topologies, defined by majorant inner products , correspond to different large distance behaviours of the limit states, classified in particular by the velocity parameter of their Lienard-Wiechert electromagnetic fields al large distances . Thus, the choice of the Hilbert-Krein topology is governed by physical considerations since it determines the class of vector states which one can constructively associate to the Wightman functions, i.e. the corresponding closure of the vector space . For these reasons it should not be a surprise that may allow different extensions. Even in the algebraic approach the construction of the charged states, which correspond to non local morphisms of the algebra of observables, is not under sharp control and in any case does not resolve the multiplicity associated to the large distance behaviour .\n\nThe choice of the Hilbert-Krein topology in local formulations of QED was discussed at length in also in connection with Zwanziger unsuccessful attempt to construct physical charged states, as a result of a too restrictive Hilbert-Krein topology.\n\nIt has been argued that the Gauss charge converges weakly to zero on the local states as a consequence of the vanishing of the Gauss charge commutators with local fields, and that this prevents the construction of physical state with non zero Gauss charge as limits of local states. We shall examine the weak points of this argument in order.\n\nFirst, the vanishing of the Gauss charge commutators with local fields implies the vanishing of the Gauss charge as a quadratic form on , (see eq.(2.8) and the Appendix). The vanishing of the Gauss charge on a closure of would follow (see Proposition 4.2 below) if one had weak convergence of in the topology which defines such a closure of .\n\nAs we shall see the validity of such a property is not constrained by the correlation functions of the local fields and does not hold in general. Actually, (see the Example below and the following Section) one may find a Hilbert-Krein topology which avoids the weak convergence of and allows for the construction of physical charged state vectors.\n\nThe failure of the -weak convergence of should not appear strange, since it involves a topology whose rôle is merely that of linking the physical non local charged states to the unphysical local states. It should be stressed that the Gauss charge may well converge weakly or even strongly on a dense domain of physical states, with respect to the intrinsic Hilbert topology of the physical space. This means that , (equivalently where denotes the distinguished subspace of satisfying the subsidiary condition and a dense subspace of ), one has that\n\n limR→∞<Φ,QRΨ>=limR→∞<Φ,QRGΨ>,QR≡j0(fRαR)\n\nexists, equivalently\n\n =\|\|QRGΨ\|\|2 (4.1)\n\nare bounded. This, however, does not mean that or converge weakly with respect to the Hilbert-Krein closure , since weak convergence in amounts to the boundedness of\n\n \|\|QRGΨ\|\|2HK≡(QRGΨ,QRGΨ),\n\nwhere is the majorant inner product which defines the Hilbert-Krein topology and the corresponding closure of the local states .\n\nActually, independently of any Hilbert-Krein majorant, there is a conflict between the construction of the physical charged states in terms of the Wightman functions of the local field algebra and the weak convergence of in the corresponding extension of . This difficulty is an intrinsic one, since it only involves the Wightman functions of and the existence of the physical charged states in an extension of compatible with the inner product defined by the Wightman functions, namely such that the sequences of elements of which define the extension, have convergent inner products . No reference is needed to a Hilbert-Krein majorant topology, even if, clearly, any Hilbert-Krein majorant defines a weak extension. To clarify this point we introduce the following\n\n###### Definition 4.1\n\nGiven two vector spaces and , with inner products and , we say that can be realized in a weak extension of if there exists an inner product vector space containing a weakly dense inner product subspaces isomorphic to and a subspace isomorphic to .\n\nIf and are defined by the vacuum correlation functions of two field algebras , the property of being realized in a extension of is implied by the existence of joint vacuum correlation functions of and . In the case of local formulations of QED, if the correlation functions of the physical field algebra , e.g. of the field algebra of the Coulomb gauge, can be constructed in terms of the correlation functions of the local field algebra , one has an extended field algebra generated by and , and is realized in an extension of .\n\n###### Proposition 4.1\n\nLet be a non degenerate vector space with inner product , a weakly dense subspace and ; let be hermitean charges and\n\n limR→∞=0, (4.2)\n limR→∞≠0. (4.3)\n\nThen, cannot converge in the weak topology defined by .\n\nIn concrete, if physical charged states may be obtained as limits of the local states of in a Hilbert-Krein topology , i.e. they belong to a (Hilbert-Krein) extension of and\n\n limR→∞=≠0,limR→∞=0, (4.4)\n\nthen cannot converge weakly with respect to .\n\nProof Since is dense and is non degenerate, eq.(4.2) and weak convergence imply that converges weakly to zero. Thus\n\n =limR→∞=0\n\nand again by the density of , converges weakly to zero, which is incompatible with eq.(4.3).\n\nBy eqs.(4.4) and locality and therefore, by the density of , weak convergence implies and\n\n =limR→∞=limR→∞=0.\n\nThus, the construction of physical charged states in a Hilbert-Krein extension of is incompatible with weak convergence of the Gauss charge on .\n\nThe failure of weak convergence of gives rise to the same problems and features discussed in Sect. 2; in particular the domain dependence of the limits of allows the vanishing of such a limit on compatibly with its being non zero on a domain containing non local states (as are the physical charged states).\n\nA Hilbert-Krein topology which allows the construction of physical charged states, avoiding the weak convergence of , was discussed in in terms of the properties of the asymptotic fields . The mechanism is clearly displayed by the following\nExample. Let be a (canonical) free massive Dirac field and two massless scalar fields satisfying the following (equal times) commutation relations\n\n [ϕ1,ϕ2]=0,[π1,π2]=0,[ϕi,πi]=0,πi≡∂0ϕi,i=1,2,\n [π1(x),ϕ2(y)]=[π2(x),ϕ1(y)]=−iδ(x−y).\n\nThen, the fields\n\n ϕ±≡(ϕ1±ϕ2)/√2,π±≡(π1±π2)/√2,\n ψ(x)≡U(x)ψ0(x),U(x)≡:eiϕ2:(x) (4.5)\n\nsatisfy the following commutators and anti commutators\n\n [ϕ±(x),ϕ±(y)]=±iD(x−y),[ϕ±(x),ϕ∓(y)]=0,\n [ϕ±(x),ψ(y)]=±iD(x−y)ψ(y),{ψ(x),¯ψ(y)}=iS(x−y), (4.6)\n\nwhere are the standard commutator functions for massless scalar and Dirac fields. Thus, and are local fields.\n\nOur field theory model is defined by the vacuum correlation functions of the field algebra generated by and and their Wick products; such correlation functions do not satisfy positivity.\n\nNow, we consider the following local charges\n\n QRϕ≡∂0ϕ1(fRαR),QR≡j0(fRαR),QRG≡QR−QRϕ, (4.7)\n\nwhere\n\n jμ(x)=:¯ψγμψ:(x)=:¯ψ0γμψ0:(x).\n\nThe factorization of the correlation functions of and implies that converges to an unbroken (non zero) ”electron” charge in sense of quadratic forms on and in fact the correlation functions with unequal numbers of and vanish. Actually, converges strongly with respect to any Hilbert-Krein topology chosen to turn into a pre-Hilbert space, provided it is a product over fermion and boson Fock spaces since, by positivity of the correlation functions of ,\n\n \|\|QRΨ0\|\|2HK=→0. (4.8)\n\nThe charge requires a quite different discussion. The field algebra is neutral under\n\n limR→∞[QRG,F]=0. (4.9)\n\nTherefore, putting , by the argument at the beginning of Sect.2, ii), one has\n\n limR→∞=limR→∞=0.\n\nIn the analogy with the local formulation of QED, the local charge plays the rôle of the Gauss charge, plays the rôle of the electron charge and" ]	[ null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.90982205,"math_prob":0.98672885,"size":30015,"snap":"2023-14-2023-23","text_gpt3_token_len":6352,"char_repetition_ratio":0.18050048,"word_repetition_ratio":0.050462406,"special_character_ratio":0.20063302,"punctuation_ratio":0.09980112,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.98921597,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2023-03-21T17:32:05Z\",\"WARC-Record-ID\":\"<urn:uuid:c79c6d8a-6945-436a-8ec0-8278d3d7ee5e>\",\"Content-Length\":\"1049404\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:77086db0-c3a0-4292-b3ca-bef4a121517b>\",\"WARC-Concurrent-To\":\"<urn:uuid:3a59e223-8211-483c-8ace-79f6dffa6a0d>\",\"WARC-IP-Address\":\"104.21.14.110\",\"WARC-Target-URI\":\"https://www.arxiv-vanity.com/papers/hep-th/0301111/\",\"WARC-Payload-Digest\":\"sha1:F633MDAMFC6WSIYJZUE7M6TNLRAATGFG\",\"WARC-Block-Digest\":\"sha1:NLMH7ZQEBLMAIPGE3HVOEIROVK7CRHAG\",\"WARC-Truncated\":\"length\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2023/CC-MAIN-2023-14/CC-MAIN-2023-14_segments_1679296943704.21_warc_CC-MAIN-20230321162614-20230321192614-00140.warc.gz\"}"}
https://mne.tools/dev/generated/mne.simulation.metrics.precision_score.html	[ "# mne.simulation.metrics.precision_score#\n\nmne.simulation.metrics.precision_score(stc_true, stc_est, threshold='90%', per_sample=True)[source]#\n\nCompute the precision.\n\nThe precision is the ratio `tp / (tp + fp)` where `tp` is the number of true positives and `fp` the number of false positives. The precision is intuitively the ability of the classifier not to label as positive a sample that is negative.\n\nThe best value is 1 and the worst value is 0.\n\nThreshold is used first for data binarization.\n\nParameters:\nstc_trueinstance of (Vol\|Mixed)SourceEstimate\n\nThe source estimates containing correct values.\n\nstc_estinstance of (Vol\|Mixed)SourceEstimate\n\nThe source estimates containing estimated values e.g. obtained with a source imaging method.\n\nthreshold\n\nThe threshold to apply to source estimates before computing the precision. If a string the threshold is a percentage and it should end with the percent character.\n\nper_sample`bool`\n\nIf True the metric is computed for each sample separately. If False, the metric is spatio-temporal.\n\nReturns:\nmetric`float` \| `array`, shape (n_times,)\n\nThe metric. float if per_sample is False, else array with the values computed for each time point.\n\nNotes\n\nNew in version 1.2.\n\n## Examples using `mne.simulation.metrics.precision_score`#", null, "Compare simulated and estimated source activity\n\nCompare simulated and estimated source activity" ]	[ null, "https://mne.tools/dev/_images/sphx_glr_plot_stc_metrics_thumb.png", null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.65379363,"math_prob":0.98319083,"size":1181,"snap":"2023-14-2023-23","text_gpt3_token_len":265,"char_repetition_ratio":0.13678844,"word_repetition_ratio":0.025157232,"special_character_ratio":0.20152414,"punctuation_ratio":0.14563107,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.994733,"pos_list":[0,1,2],"im_url_duplicate_count":[null,null,null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2023-06-09T09:29:44Z\",\"WARC-Record-ID\":\"<urn:uuid:c36a17bd-ccaa-4a86-b58e-8081e70d91c9>\",\"Content-Length\":\"113329\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:e23b1ec0-57ca-4768-bcf3-e118a3ebf386>\",\"WARC-Concurrent-To\":\"<urn:uuid:0bf0b088-8c60-4a5e-8364-99a729ce93aa>\",\"WARC-IP-Address\":\"185.199.108.153\",\"WARC-Target-URI\":\"https://mne.tools/dev/generated/mne.simulation.metrics.precision_score.html\",\"WARC-Payload-Digest\":\"sha1:2E6XWFHBJPWBR5VNU7S2X53F2V7QJ3G2\",\"WARC-Block-Digest\":\"sha1:M5H7MKJHLCO77BICKBRHHUR4F2XXXGXF\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2023/CC-MAIN-2023-23/CC-MAIN-2023-23_segments_1685224655446.86_warc_CC-MAIN-20230609064417-20230609094417-00136.warc.gz\"}"}
https://marvelousessays.co.uk/essays/analysis/behavior-of-compression-members.html	[ "", null, "## Free «Behavior of Compression Members» UK Essay Paper\n\nThe elaciticity of an obvject determines the extent to which the object can withold the tensile stress subjected in the form of an external force. Elaciticity depends with the material upon which the mamber is fabricated, where some elements have lower values of transition from the plastic to the elastic nature while others have hagher values. The ranges of these values exhibited by members once subjected to an external force forms the definition of the eccentricity. A bar placed in an axial position could be used to determine its eccentricity since the position allows for uniform subjection of exaternal forces, where a slender bar could bend as a result of the added force. This form of bending is known as buckling, while the nature of the bar reveals an impafection in the member, which is termed as the elasticity. For instance, the figure gives the curviture of a bar in axial positiin subjected to an external force, where the variations in the force applied gives the ideal buckling (Wiliams, 2000).\n\nThe bar bends depending on the amount of force applied, while verying the accentricities could be viable for the culculation of the Euler buckling load. This could be done experimentally, where the experimental results for the zero accentricity experiemnt and the maximum accentricity experiment are valuable in revealing the Euler Buckling load. By consideration, the Euler Buckling load is proportional to the product of the square of pie with both the modulus of elaciticity and the 2nd moment of area about the buckling axis, while it is also inveersely proportional to the square of the effective length of the strut. Additionally, the effective length of the strut depends on the flexibility of the rotation. The experiemental investigation of the accentricity and the culculation of the Euler Buckling load forms the essence of this paper (Wiliams, 2000).\n\nBehavior of the Compression Members\n\n## Buy Behavior of Compression Members essay paper online\n\nBecome our VIP client\n\nTotal price:\n\n* Final order price might be slightly different depending on\nthe current exchange rate of chosen payment system.\n\nThe experimental analysis for determination of the behavior of compression members involves the structuring of a strut with known modulus of elaciticity. The strut is places in an axial position against pivots of eqal force of attraction, where the lower pivot and the higher pivots holding the strut in the axial position are adjusted in a way that there is little influence of their individaul weights on the strut. This elaves the bending motions to the force exerted by the wights on the spring, where in this case, weights of 10n were used in an increamenting trend of 5n. this gave the resultant buckling strengnths, which were proportional to the modulus of the strut used and also varying with the amoungt of effert loaded. For instance, the ability of ataining elasticity to gain the transition of the strut from the plastic stage to the elastic state was influenced by an increase in the amount of applied force resulting from an increase in the weights of the loads. The consideration of Zero exentricity involved the application of the knive edge brackets that were housed in the groove. On the convese, the maximum accentricity was obtained through removing the housing in the grooves of the edge brackets for the knives (Megson, 2005).\n\nDifferent compression members habour different mudulus of elaciticities. This implies that the Euler bucking load for these members differs at varied eccentricities. For instance, varying the weights as exteranl forces increases the extent of deflection, while a plot of the graph of the deflection verses the quotient of the deflection and the Euler bucling load gives a straight line. This curve is also essential in culculatng the initial curviture, which is in this case termed as the accentricity. The initial shape of the strut is defined by the product of the maximum cenrtala mplitude and the sine of the quotient of the product of both pie and the distance of dispklacement along the x axis with the length of the strut. Varaitions of the strut due to increase in the external forces of influence acrued from an increase in the load of the weights gives the initial final position of the strut defined by the central amplitude. This central amplitude is vital in the culculation of the euler buckling load by use of the gradient of the graph of the central amplitude verses the quotient of the curviture and the displacement along the x axis (Megson, 2005).\n\nOn the other hand, the accentricity of the strut is culculated from the additional central deflection obtained by increasing the load of the sources of force or stress to the strut. The theoritical value of the Euler Buckling strength is culculated by the fomular: F = (pi^2)(E)(I)/ (L^2), where F is the Euler’s buckling load, I is the cross-sectional moment of inertia while E is the Young’s modulus of elasticity for the strut. By consideration, the experiment involved the use of a strut of initial curvature position represented by Y=yo/ (1-p/pE) where Y is the deflecting shape of the strut, yo is the distance of curvature in the initial position, while p is the vertical displacement accrued from the boundary conditions. This formula is a viable tool for the acquaintance of the initial eccentricity or the zero eccentricity in this case. Consequently, the maximum curvature is represented by the equation Y= ao/(1-p/pE), where Y is the maximum deflection reached by the highest weights accrued from increased loads. This formula is vital in synthesizing the maximum eccentricity.\n\nThe essence of increasing the weights in terms of loads results in the realization of the effect of buckling, which is the critical amount of force that the strut can withhold to acquire a transition from the plastic to the elastic mode of formation. A plot for the graph of the eigenvalues verses the weights gives a straight line, where the gradient of the line is suitable in the acquisition of the Euler’s buckling load. This is also important in the determination of the initial and final position of curvature since the y intercept for the graphs gives both the initial and final positions of the curvature (Ghali, 2003).\n\nZero Eccentricity\n\nBy consideration, from the formula: F = (pi^2)(E)(I)/ (L^2) using the initial curvature and replacing the values gives the Euler’s buckling load of 136.903. This value is obtainable through using 600mm as the length of the struts with a cross sectional area of 25.4 by 1.6mm. Having the values for E as 205kN/mm2, shows that the modulus of the elasticity for the strut is constant. This implies that in the theoretical value calculation of the Euler’s buckling load for the strut, the modulus of elasticity is constant. This implies that Po= (3.142205kN/mm225.4mm1.6mm)/600mm), which is equal to 136.9038kips. The value shows a slight deviation from the conventional graphical analysis, which gives 135.089kips as the gradient of the slope correlating to the graphical representation of the Euler’s buckling load. The differences emanate from the fact that the measurements for graphical analysis vary with the net weights of he load, which might not be linear (Ghali, 2003).\n\nLimited Time offer!\n\nGet 19% OFF\n\n0\n0\ndays\n:\n0\n0\nhours\n:\n0\n0\nminutes\n:\n0\n0\nseconds\n\nMoreover, the differences might have been as a result of lack of uniformity in the application of the forces emanating from the bending curvature. This results in unequal subjection of forces that are geared towards producing the buckling effects. On the other hand, the plot gives the initial curvature as being 1.067, which is acquired from the Y intercept of the graph. This implies that the members of he same group portray similar characteristics in such a way that there is a prior acquaintance to the tensile stress amid constrains of the curvature effects from the load. This implies that the members of he same elements always contain some form of curvature even without the application of an external force of curvature. This is also the value for the eccentricity of the strut at the zero eccentricity, where the Y intercept represents the initial curvature (Milliams, 2005).\n\nThe above graph shows that there is a form of force that the structural members made up of different elements exhibit some tensional force that results in the plastic characteristics. These characteristics transit upon exertion of an external force to give rise to the transition of the elements into the elastic nature. This elastic nature is what defines the curvature, which is obtainable through calculation of the Euler’s Buckling strength. This also shows that materials of he same cross-sectional area and same modulus of elasticity harbor the same values of the Euler’s buckling load at zero eccentricity. Moreover, the values for the external force need to be uniform, where it is necessary to use the axial position for the struts to depict the linearity in terms of the external force applied (Ghali, 2003).\n\nMaximum Eccentricity\n\nThis type of eccentricity involves the use of the final position of curvature of the strut, which is elucidated by the bending motion due to the external force. Once the load is placed on to the spring, there is exertion of the external force that culminates in the change in nature of the strut from the plastic nature to the elastic nature. This gives rise to the bending motion, where the plot of Y verses Y/pE gives a straight line. The slope of the line gives the Euler’s buckling load, while the Y intercept of the graph gives the final curvature position. This is the amount of force required for the strut to gain the transition from the plastic nature to the elastic nature, which varies with the material making up the strut. For instance, the use of the strut with a modulus of elasticity of 205kN/mm2 for a rod of cross-sectional area of 25.4 by 1.6mm gives a lead into the calculation of the Euler’s buckling load.>\n\nThis is from the fact that from the formula F = (pi^2)(E)(I)/ (L^2), the resultant Euler’s buckling load is equal to 135.88 kips at the maximum eccentricity. By consideration, the value is obtainable from the synthesis that leads to the calculation: (3.1422250kN/mm225.4mm1.6mm)/600mm. this gives 136.903 kips as the theoretical value for the Euler buckling load. This is the value Euler’s buckling load for the strut at the maximum eccentricity, which corresponds with the value obtained from the calculation from zero eccentricity. The differences in the values obtainable lies in the graphical analysis, where the value obtained in this case as synthesized from the slope of the straight line obtained from the plot of the final curvature with the quotient of the final curvature and the eccentric force (Wiliams, 2000).\n\nFrom the graphical analysis, it is evident that the Y intercept, which represents the final curviture lies at 0.60. this value is higher than the initial value obtained from the experiemntation with zero eccentricity. It is evident that the values differ in terms of the accentricity, which also provides a aplatform for varied Euler’s buckling load. The nature of he graphs differ in terms of the gradient, where the graphical representation for the zero accentricity is more pronounced than that for the maximum accentricity. This is due to the increase in the load of force, where the result is an increase in the force of applkciation. As the force of stretch increases, it leads to the increase in the extent of stretch. Morever, this results into the reduction in the distance of curviture in the resulting PE values (Megson, 2005).\n\nIt is also evident that there is a slight diviation in the values of the Euler buckling constant obtained from the synthesis of the graphical analysis for both the graphical analysis of the Zero eccentricity and the maximum eccentricity with that of the tehoritical value. The value for the tehoritical analysis is slightly grater than the graphical analysis in both cases. This is due to the clarity in the theoritical value, which is determined directly from the values for the buckling constant. The devaitions are as a result of the measurements for the weights, which does not give a linear observation in relation to the extent of stretch caused by the force. In addition, the element of the force of gravity, which in this case is not taken into consideration results in the final influence of the resultant datum (Megson, 2005).\n\nThe datum collected from the analysis is a vaible tool for the reflection of both the accentricity of the strut and the Euler’s buckling load, where the varaitions in the laods is in a sound correlation to the influence of stretch. For instance, the initiatl curviture for the strut is due to the normal situational mandate of structural elements belonging to the same group. The natural behaviour of these members shows that they exhibit a form of stretch at an intitial position. This force could be accelerated through increasing the external force. In this case, the exteranl force was increased through an increase in the weights of the laods. This resulted into the increse in the amount of stretch culminating in an increase in the extent of curviture. This also confirms the linearity in the resultant datum with the theoritical basis for the experiement (Milliams, 2005).\n\nLet’s earn with us!\n\nGet 10% from your friends orders!\n\nThe essence of use of standardised loads of known weights takes care of he influence of the gravitational pull, where the mass is defined in terms of the force of gravity acting on the load. On the other hand, stadnardisation of the results was achieved trhough ensuring that the supporting beams do not exert an external force on the spring balance. This was through ensuring that the parts of the supporting beams are not part of the experiemntal analysis (Megson, 2005).\n\nConclusion\n\nThe obtained value for accentricity, which is the Y intercept of the graph of curviture vuses the quotient of the curviture with the displacement as a result of the stretch is 0.17 for the zero eccentricity experiemnt and 0.6 for the maximum eccentricity. On the other hand, the value for the Euler’s accentricity forgraphical analysis is obtainable as 135.88 kips, which is slightly lower than the theoritical value for the Euler’s ccentricity that is 136.9038. these values differ because of the influence of measure of the weights in relation to the gravitational pull. The results obtained from experimental analysis are linear with the fact that the extent of curviture is proportional with the force of stretch, while the force increses with the increse in the mass of he load. This leads to a straight line graph that depicts the initial curviture vurses the quotient of the initial curviture and the extent of stretch, whose gradient gives the Euler’s buckling load while the Y intercept forms the eccentricity.", null, "" ]	[ null, "https://marvelousessays.co.uk/essays/behavior-of-compression-members.png", null, "https://marvelousessays.co.uk/files/images/holidays/close-white.png", null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.9201031,"math_prob":0.96677196,"size":24493,"snap":"2020-45-2020-50","text_gpt3_token_len":5255,"char_repetition_ratio":0.18763526,"word_repetition_ratio":0.7749815,"special_character_ratio":0.20222104,"punctuation_ratio":0.077380955,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9901219,"pos_list":[0,1,2,3,4],"im_url_duplicate_count":[null,4,null,4,null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2020-11-24T17:18:23Z\",\"WARC-Record-ID\":\"<urn:uuid:d005af33-f233-49db-89d3-2f1bba6e817b>\",\"Content-Length\":\"340864\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:454c8cbe-b5a6-4226-8740-740493939fff>\",\"WARC-Concurrent-To\":\"<urn:uuid:acc683fa-1dcb-40b0-910e-a2bb3862e981>\",\"WARC-IP-Address\":\"66.165.237.122\",\"WARC-Target-URI\":\"https://marvelousessays.co.uk/essays/analysis/behavior-of-compression-members.html\",\"WARC-Payload-Digest\":\"sha1:WQXXPLIKSLMBWBULJUN4NYDH6UKGMHTK\",\"WARC-Block-Digest\":\"sha1:OLCRSIBYNZDV3U5WKREYBZSWUWOAHCS6\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2020/CC-MAIN-2020-50/CC-MAIN-2020-50_segments_1606141176922.14_warc_CC-MAIN-20201124170142-20201124200142-00285.warc.gz\"}"}
https://www.allkidsnetwork.com/search?ty=4%2C10%2C24%2C27&mltid=2be93742-234a-4a16-8922-0805ce172eb8	[ "# Search\n\nAbout 3,743 Search Results Matching Types of Worksheet, Worksheet Section, Generator, Generator Section, Similar to Triangle Worksheet", null, "## Triangle Worksheet\n\nTrace the triangles, draw some on your own, the...", null, "## Triangle Worksheet\n\nLearn about the triangles, then trace the word ...", null, "## Triangle Worksheet\n\nTrace triangles, then draw some on your own. T...", null, "## Traceable Triangles Worksheet\n\nTrace the different size triangles and then dra...", null, "## Angles in a Triangle Worksheet\n\nKids learn that the angles of a triangle always...", null, "## Shape Names Recognition Worksheet\n\nDraw a line to match each shape with its name.", null, "## Preschool Shapes Worksheets\n\nThis collection of free worksheets will help yo...", null, "## Draw and Find Shapes Worksheets\n\nThis is our best set of shapes worksheet yet! I...", null, "## Subtraction Problems Worksheet - Shapes Theme\n\nUse the pictures to count and subtract and get ..." ]	[ null, "data:image/svg+xml,%3Csvg xmlns='http://www.w3.org/2000/svg' viewBox='0 0 170 140'%3E%3C/svg%3E", null, "data:image/svg+xml,%3Csvg xmlns='http://www.w3.org/2000/svg' viewBox='0 0 170 140'%3E%3C/svg%3E", null, "data:image/svg+xml,%3Csvg xmlns='http://www.w3.org/2000/svg' viewBox='0 0 170 140'%3E%3C/svg%3E", null, "data:image/svg+xml,%3Csvg xmlns='http://www.w3.org/2000/svg' viewBox='0 0 170 140'%3E%3C/svg%3E", null, "data:image/svg+xml,%3Csvg xmlns='http://www.w3.org/2000/svg' viewBox='0 0 170 140'%3E%3C/svg%3E", null, "data:image/svg+xml,%3Csvg xmlns='http://www.w3.org/2000/svg' viewBox='0 0 170 140'%3E%3C/svg%3E", null, "data:image/svg+xml,%3Csvg xmlns='http://www.w3.org/2000/svg' viewBox='0 0 170 140'%3E%3C/svg%3E", null, "data:image/svg+xml,%3Csvg xmlns='http://www.w3.org/2000/svg' viewBox='0 0 170 140'%3E%3C/svg%3E", null, "data:image/svg+xml,%3Csvg xmlns='http://www.w3.org/2000/svg' viewBox='0 0 170 140'%3E%3C/svg%3E", null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.87564534,"math_prob":0.659397,"size":519,"snap":"2019-51-2020-05","text_gpt3_token_len":110,"char_repetition_ratio":0.16116504,"word_repetition_ratio":0.0,"special_character_ratio":0.24277456,"punctuation_ratio":0.2601626,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.95930785,"pos_list":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18],"im_url_duplicate_count":[null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2020-01-21T01:14:36Z\",\"WARC-Record-ID\":\"<urn:uuid:2afb86e4-fc3e-46c7-a88a-857676bc839f>\",\"Content-Length\":\"84464\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:945f3dfb-6e5c-4ac5-aa75-b9658a85c9c7>\",\"WARC-Concurrent-To\":\"<urn:uuid:137e1ce3-ab90-4376-bb85-42391a4158de>\",\"WARC-IP-Address\":\"52.45.69.225\",\"WARC-Target-URI\":\"https://www.allkidsnetwork.com/search?ty=4%2C10%2C24%2C27&mltid=2be93742-234a-4a16-8922-0805ce172eb8\",\"WARC-Payload-Digest\":\"sha1:QR34GAMN5ZWPIGFYHNQY6HAYYWQ5CEWX\",\"WARC-Block-Digest\":\"sha1:IMWCVGPWDD3W47IH4LY3IRKAF2I7BCK7\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2020/CC-MAIN-2020-05/CC-MAIN-2020-05_segments_1579250601040.47_warc_CC-MAIN-20200120224950-20200121013950-00224.warc.gz\"}"}
https://socratic.org/questions/how-do-you-factor-8y-3-125	[ "# How do you factor 8y^3-125?\n\nFeb 27, 2017\n\n$8 {y}^{3} - 125 = \\left(2 y - 5\\right) \\left(4 {y}^{2} + 10 y + 25\\right)$\n\n#### Explanation:\n\nThe difference of cubes identity can be written:\n\n${a}^{3} - {b}^{3} = \\left(a - b\\right) \\left({a}^{2} + a b + {b}^{2}\\right)$\n\nUse this with $a = 2 y$ and $b = 5$ as follows:\n\n$8 {y}^{3} - 125 = {\\left(2 y\\right)}^{3} - {5}^{3}$\n\n$\\textcolor{w h i t e}{8 {y}^{3} - 125} = \\left(2 y - 5\\right) \\left({\\left(2 y\\right)}^{2} + \\left(2 y\\right) \\left(5\\right) + {5}^{2}\\right)$\n\n$\\textcolor{w h i t e}{8 {y}^{3} - 125} = \\left(2 y - 5\\right) \\left(4 {y}^{2} + 10 y + 25\\right)$" ]	[ null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.515467,"math_prob":1.0000094,"size":415,"snap":"2019-51-2020-05","text_gpt3_token_len":170,"char_repetition_ratio":0.11192214,"word_repetition_ratio":0.0,"special_character_ratio":0.4506024,"punctuation_ratio":0.044444446,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.99971753,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2019-12-14T18:09:22Z\",\"WARC-Record-ID\":\"<urn:uuid:31df2d30-6343-4ae1-b0c2-6b8d016ad8eb>\",\"Content-Length\":\"32952\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:4e6613df-4b29-4d5f-b983-094db6fa3b0a>\",\"WARC-Concurrent-To\":\"<urn:uuid:553bcda5-74af-44d9-9629-c547b5d01592>\",\"WARC-IP-Address\":\"54.221.217.175\",\"WARC-Target-URI\":\"https://socratic.org/questions/how-do-you-factor-8y-3-125\",\"WARC-Payload-Digest\":\"sha1:O3ZXD5HQRN7AHUZL3OFCRZVCGECZF5JQ\",\"WARC-Block-Digest\":\"sha1:4PKTY57BTMOWWXOBTCBAGQX4BGOZYJRL\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2019/CC-MAIN-2019-51/CC-MAIN-2019-51_segments_1575541288287.53_warc_CC-MAIN-20191214174719-20191214202719-00044.warc.gz\"}"}
https://math.libretexts.org/Courses/University_of_California_Davis/UCD_Mat_21A%3A_Differential_Calculus/1%3A_Functions/1.2%3A_Combining_Functions%3B_Shifting_and_Scaling_Graphs	[ "$$\\newcommand{\\id}{\\mathrm{id}}$$ $$\\newcommand{\\Span}{\\mathrm{span}}$$ $$\\newcommand{\\kernel}{\\mathrm{null}\\,}$$ $$\\newcommand{\\range}{\\mathrm{range}\\,}$$ $$\\newcommand{\\RealPart}{\\mathrm{Re}}$$ $$\\newcommand{\\ImaginaryPart}{\\mathrm{Im}}$$ $$\\newcommand{\\Argument}{\\mathrm{Arg}}$$ $$\\newcommand{\\norm}{\\\| #1 \\\|}$$ $$\\newcommand{\\inner}{\\langle #1, #2 \\rangle}$$ $$\\newcommand{\\Span}{\\mathrm{span}}$$\n\n# 1.2: Combining Functions; Shifting and Scaling Graphs\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\nMany functions in applications are built up from simple functions by inserting constants in various places. It is important to understand the effect such constants have on the appearance of the graph.\n\n## Horizontal shifts\n\nIf we replace $$x$$ by $$x-C$$ everywhere it occurs in the formula for $$f(x)$$, then the graph shifts over $$C$$ to the right. (If $$C$$ is negative, then this means that the graph shifts over $$\|C\|$$ to the left.) For example, the graph of $$y=(x-2)^2$$ is the $$x^2$$-parabola shifted over to have its vertex at the point 2 on the $$x$$-axis. The graph of $$y=(x+1)^2$$ is the same parabola shifted over to the left so as to have its vertex at $$-1$$ on the $$x$$-axis. Note well: when replacing $$x$$ by $$x-C$$ we must pay attention to meaning, not merely appearance. Starting with $$y=x^2$$ and literally replacing $$x$$ by $$x-2$$ gives $$y=x-2^2$$. This is $$y=x-4$$, a line with slope 1, not a shifted parabola.\n\n## Vertical shifts\n\nIf we replace $$y$$ by $$y-D$$, then the graph moves up $$D$$ units. (If $$D$$ is negative, then this means that the graph moves down $$\|D\|$$ units.) If the formula is written in the form $$y=f(x)$$ and if $$y$$ is replaced by $$y-D$$ to get $$y-D=f(x)$$, we can equivalently move D to the other side of the equation and write $$y=f(x)+D$$. Thus, this principle can be stated: to get the graph of $$y=f(x)+D$$, take the graph of $$y=f(x)$$ and move it D units up.For example, the function $$y=x^2-4x=(x-2)^2-4$$ can be obtained from $$y=(x-2)^2$$ (see the last paragraph) by moving the graph 4 units down. The result is the $$x^2$$-parabola shifted 2 units to the right and 4 units down so as to have its vertex at the point $$(2,-4)$$.\n\nWarning. Do not confuse $$f(x)+D$$ and $$f(x+D)$$. For example, if $$f(x)$$ is the function $$x^2$$, then $$f(x)+2$$ is the function $$x^2+2$$, while $$f(x+2)$$ is the function $$(x+2)^2=x^2+4x+4$$.\n\nAn important example of the above two principles starts with the circle $$x^2+y^2=r^2$$. This is the circle of radius $$r$$ centered at the origin. (As we saw, this is not a single function $$y=f(x)$$, but rather two functions $$y=\\pm\\sqrt{r^2-x^2}$$ put together; in any case, the two shifting principles apply to equations like this one that are not in the form $$y=f(x)$$.) If we replace $$x$$ by $$x-C$$ and replace $$y$$ by $$y-D$$---getting the equation $$(x-C)^2+(y-D)^2=r^2$$---the effect on the circle is to move it $$C$$ to the right and $$D$$ up, thereby obtaining the circle of radius $$r$$ centered at the point $$(C,D)$$. This tells us how to write the equation of any circle, not necessarily centered at the origin.\n\nWe will later want to use two more principles concerning the effects of constants on the appearance of the graph of a function.\n\n## Horizontal dilation\n\nIf $$x$$ is replaced by $$x/A$$ in a formula and $$A>1$$, then the effect on the graph is to expand it by a factor of $$A$$ in the $$x$$-direction (away from the $$y$$-axis). If $$A$$ is between 0 and 1 then the effect on the graph is to contract by a factor of $$1/A$$ (towards the $$y$$-axis). We use the word \"dilate'' to mean expand or contract.\n\nFor example, replacing $$x$$ by $$x/0.5=x/(1/2)=2x$$ has the effect of contracting toward the $$y$$-axis by a factor of 2. If $$A$$ is negative, we dilate by a factor of $$\|A\|$$ and then flip about the $$y$$-axis. Thus, replacing $$x$$ by $$-x$$ has the effect of taking the mirror image of the graph with respect to the $$y$$-axis. For example, the function $$y=\\sqrt{-x}$$, which has domain $$\\{x\\in R\\mid x\\le 0\\}$$, is obtained by taking the graph of $$\\sqrt{x}$$ and flipping it around the $$y$$-axis into the second quadrant.\n\n## Vertical dilation\n\nIf $$y$$ is replaced by $$y/B$$ in a formula and $$B>0$$, then the effect on the graph is to dilate it by a factor of $$B$$ in the vertical direction. As before, this is an expansion or contraction depending on whether $$B$$ is larger or smaller than one. Note that if we have a function $$y=f(x)$$, replacing $$y$$ by $$y/B$$ is equivalent to multiplying the function on the right by $$B$$: $$y=Bf(x)$$. The effect on the graph is to expand the picture away from the $$x$$-axis by a factor of $$B$$ if $$B>1$$, to contract it toward the $$x$$-axis by a factor of $$1/B$$ if $$0 < B < 1$$, and to dilate by $$\|B\|$$ and then flip about the $$x$$-axis if $$B$$ is negative.\n\n$\\left({x\\over a}\\right)^2+\\left({y\\over b}\\right)^2=1 \\qquad\\hbox{or}\\qquad {x^2\\over a^2}+{y^2\\over b^2}=1.$\n\nFinally, if we want to analyze a function that involves both shifts and dilations, it is usually simplest to work with the dilations first, and then the shifts. For instance, if we want to dilate a function by a factor of $$A$$ in the $$x$$-direction and then shift $$C$$ to the right, we do this by replacing $$x$$ first by $$x/A$$ and then by $$(x-C)$$ in the formula. As an example, suppose that, after dilating our unit circle by $$a$$ in the $$x$$-direction and by $$b$$ in the $$y$$-direction to get the ellipse in the last paragraph, we then wanted to shift it a distance $$h$$ to the right and a distance $$k$$ upward, so as to be centered at the point $$(h,k)$$. The new ellipse would have equation $$\\left({x-h\\over a}\\right)^2+\\left({y-k\\over b}\\right)^2=1.$$ Note well that this is different than first doing shifts by $$h$$ and $$k$$ and then dilations by $$a$$ and $$b$$:\n\n$\\left({x\\over a}-h\\right)^2+\\left({y\\over b}-k\\right)^2=1.$\n\nSee figure 1.4.1.", null, "Figure 1.4.1. Ellipses: $$\\left({x-1\\over 2}\\right)^2+\\left({y-1\\over 3}\\right)^2=1$$ on the left, $$\\left({x\\over 2}-1\\right)^2+\\left({y\\over 3}-1\\right)^2=1$$ on the right.\n\n## Contributors\n\n• Integrated by Justin Marshall." ]	[ null, "https://math.libretexts.org/@api/deki/files/2578/1.4.1.png", null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.89720064,"math_prob":1.0000091,"size":5576,"snap":"2021-43-2021-49","text_gpt3_token_len":1666,"char_repetition_ratio":0.14985642,"word_repetition_ratio":0.07204301,"special_character_ratio":0.33662122,"punctuation_ratio":0.0813278,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":1.0000087,"pos_list":[0,1,2],"im_url_duplicate_count":[null,9,null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2021-10-27T03:56:15Z\",\"WARC-Record-ID\":\"<urn:uuid:5b94e07b-b4a6-4765-b61d-8331bc1785ef>\",\"Content-Length\":\"101204\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:35cebba0-65f9-45c0-b243-1da7f13f8aa3>\",\"WARC-Concurrent-To\":\"<urn:uuid:96a012d7-5ce8-4d91-a3e4-0764a25644e3>\",\"WARC-IP-Address\":\"13.249.38.5\",\"WARC-Target-URI\":\"https://math.libretexts.org/Courses/University_of_California_Davis/UCD_Mat_21A%3A_Differential_Calculus/1%3A_Functions/1.2%3A_Combining_Functions%3B_Shifting_and_Scaling_Graphs\",\"WARC-Payload-Digest\":\"sha1:WUDWDPPR6RIUNPGI5RVCW46SW4WFAF2R\",\"WARC-Block-Digest\":\"sha1:7NGSUOYI5BRCAIV6BHDYUPH4GRO62Y6C\",\"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-20211027022823-20211027052823-00200.warc.gz\"}"}
https://sciencing.com/how-8086143-calculate-fan-output.html	[ "# How to Calculate Fan Output", null, "••• Jupiterimages/Photos.com/Getty Images\n\nEngineers specify a fan's output in terms of the amount of air it displaces each minute. This measurement takes into account the speed of the wind that the fan produces and also the size of the fan's blades. The fan's output, the pressure it creates and the power it consumes all relate to one another. The manufacturer's documentation also likely directly lists the fan's power consumption, which will let you calculate its total output.\n\nDivide the fan's power consumption, measured in kilowatts, by 0.746 to convert it to horsepower. If, for instance, a fan consumes 4 kW: 4 / 0.746 = 5.36 horsepower.\n\nMultiply the result by the fan's efficiency. If the fan operates, for instance, at 80 percent efficiency: 5.36 x 0.80 = 4.29 horsepower.\n\nMultiply the result by 530, a conversion constant: 4.29 x 530 = 2,273.\n\nDivide this answer by the fan's total pressure, measured in feet of water. For instance, if the fan operates at 0.2 feet of water of pressure: 2,273 / 0.2 = 11,365. The fan's output is therefore approximately 11,500 cubic feet per minute.\n\nDont Go!\n\nWe Have More Great Sciencing Articles!" ]	[ null, "https://img-aws.ehowcdn.com/360x267p/s3-us-west-1.amazonaws.com/contentlab.studiod/getty/cache.gettyimages.com/9ee968d00b5d4dd89ba8b9d8c44dd459.jpg", null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.89102185,"math_prob":0.96905303,"size":1456,"snap":"2023-40-2023-50","text_gpt3_token_len":345,"char_repetition_ratio":0.11707989,"word_repetition_ratio":0.008474576,"special_character_ratio":0.2445055,"punctuation_ratio":0.15254237,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9749779,"pos_list":[0,1,2],"im_url_duplicate_count":[null,3,null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2023-09-26T12:22:03Z\",\"WARC-Record-ID\":\"<urn:uuid:e4d2995d-032b-4024-a65c-b6c0ac7e9470>\",\"Content-Length\":\"407470\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:7e27e1b4-ad9a-4f67-b3fe-f943e6d82c19>\",\"WARC-Concurrent-To\":\"<urn:uuid:74b987fc-fa77-448a-b625-aa1ee38a030e>\",\"WARC-IP-Address\":\"23.205.107.80\",\"WARC-Target-URI\":\"https://sciencing.com/how-8086143-calculate-fan-output.html\",\"WARC-Payload-Digest\":\"sha1:4SAON7IPCTPEBJANR7BNV4VJXAVIQJ4X\",\"WARC-Block-Digest\":\"sha1:Q33ZM6XBDKB2G6MGFMHFGQ3ZT43KDSKB\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2023/CC-MAIN-2023-40/CC-MAIN-2023-40_segments_1695233510208.72_warc_CC-MAIN-20230926111439-20230926141439-00516.warc.gz\"}"}
https://www.colorhexa.com/baf6b3	[ "# #baf6b3 Color Information\n\nIn a RGB color space, hex #baf6b3 is composed of 72.9% red, 96.5% green and 70.2% blue. Whereas in a CMYK color space, it is composed of 24.4% cyan, 0% magenta, 27.2% yellow and 3.5% black. It has a hue angle of 113.7 degrees, a saturation of 78.8% and a lightness of 83.3%. #baf6b3 color hex could be obtained by blending #ffffff with #75ed67. Closest websafe color is: #ccffcc.\n\n• R 73\n• G 96\n• B 70\nRGB color chart\n• C 24\n• M 0\n• Y 27\n• K 4\nCMYK color chart\n\n#baf6b3 color description : Very soft lime green.\n\n# #baf6b3 Color Conversion\n\nThe hexadecimal color #baf6b3 has RGB values of R:186, G:246, B:179 and CMYK values of C:0.24, M:0, Y:0.27, K:0.04. Its decimal value is 12252851.\n\nHex triplet RGB Decimal baf6b3 `#baf6b3` 186, 246, 179 `rgb(186,246,179)` 72.9, 96.5, 70.2 `rgb(72.9%,96.5%,70.2%)` 24, 0, 27, 4 113.7°, 78.8, 83.3 `hsl(113.7,78.8%,83.3%)` 113.7°, 27.2, 96.5 ccffcc `#ccffcc`\nCIE-LAB 91.507, -31.304, 26.289 61.34, 79.604, 54.779 0.313, 0.407, 79.604 91.507, 40.879, 139.976 91.507, -29.762, 43.17 89.221, -33.417, 26.053 10111010, 11110110, 10110011\n\n# Color Schemes with #baf6b3\n\n• #baf6b3\n``#baf6b3` `rgb(186,246,179)``\n• #efb3f6\n``#efb3f6` `rgb(239,179,246)``\nComplementary Color\n• #dcf6b3\n``#dcf6b3` `rgb(220,246,179)``\n• #baf6b3\n``#baf6b3` `rgb(186,246,179)``\n• #b3f6ce\n``#b3f6ce` `rgb(179,246,206)``\nAnalogous Color\n• #f6b3dc\n``#f6b3dc` `rgb(246,179,220)``\n• #baf6b3\n``#baf6b3` `rgb(186,246,179)``\n• #ceb3f6\n``#ceb3f6` `rgb(206,179,246)``\nSplit Complementary Color\n• #f6b3ba\n``#f6b3ba` `rgb(246,179,186)``\n• #baf6b3\n``#baf6b3` `rgb(186,246,179)``\n• #b3baf6\n``#b3baf6` `rgb(179,186,246)``\n• #f6efb3\n``#f6efb3` `rgb(246,239,179)``\n• #baf6b3\n``#baf6b3` `rgb(186,246,179)``\n• #b3baf6\n``#b3baf6` `rgb(179,186,246)``\n• #efb3f6\n``#efb3f6` `rgb(239,179,246)``\n• #7cee6f\n``#7cee6f` `rgb(124,238,111)``\n• #91f185\n``#91f185` `rgb(145,241,133)``\n• #a5f39c\n``#a5f39c` `rgb(165,243,156)``\n• #baf6b3\n``#baf6b3` `rgb(186,246,179)``\n• #cff9ca\n``#cff9ca` `rgb(207,249,202)``\n• #e3fbe1\n``#e3fbe1` `rgb(227,251,225)``\n• #f8fef7\n``#f8fef7` `rgb(248,254,247)``\nMonochromatic Color\n\n# Alternatives to #baf6b3\n\nBelow, you can see some colors close to #baf6b3. Having a set of related colors can be useful if you need an inspirational alternative to your original color choice.\n\n• #cbf6b3\n``#cbf6b3` `rgb(203,246,179)``\n• #c5f6b3\n``#c5f6b3` `rgb(197,246,179)``\n• #c0f6b3\n``#c0f6b3` `rgb(192,246,179)``\n• #baf6b3\n``#baf6b3` `rgb(186,246,179)``\n• #b4f6b3\n``#b4f6b3` `rgb(180,246,179)``\n• #b3f6b7\n``#b3f6b7` `rgb(179,246,183)``\n• #b3f6bd\n``#b3f6bd` `rgb(179,246,189)``\nSimilar Colors\n\n# #baf6b3 Preview\n\nThis text has a font color of #baf6b3.\n\n``<span style=\"color:#baf6b3;\">Text here</span>``\n#baf6b3 background color\n\nThis paragraph has a background color of #baf6b3.\n\n``<p style=\"background-color:#baf6b3;\">Content here</p>``\n#baf6b3 border color\n\nThis element has a border color of #baf6b3.\n\n``<div style=\"border:1px solid #baf6b3;\">Content here</div>``\nCSS codes\n``.text {color:#baf6b3;}``\n``.background {background-color:#baf6b3;}``\n``.border {border:1px solid #baf6b3;}``\n\n# Shades and Tints of #baf6b3\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, #020c01 is the darkest color, while #fafef9 is the lightest one.\n\n• #020c01\n``#020c01` `rgb(2,12,1)``\n• #061d03\n``#061d03` `rgb(6,29,3)``\n• #0a2f06\n``#0a2f06` `rgb(10,47,6)``\n• #0e4008\n``#0e4008` `rgb(14,64,8)``\n• #11520a\n``#11520a` `rgb(17,82,10)``\n• #15630c\n``#15630c` `rgb(21,99,12)``\n• #19750e\n``#19750e` `rgb(25,117,14)``\n• #1c8610\n``#1c8610` `rgb(28,134,16)``\n• #209812\n``#209812` `rgb(32,152,18)``\n• #24aa14\n``#24aa14` `rgb(36,170,20)``\n• #27bb16\n``#27bb16` `rgb(39,187,22)``\n• #2bcd18\n``#2bcd18` `rgb(43,205,24)``\n• #2fde1a\n``#2fde1a` `rgb(47,222,26)``\n• #3be527\n``#3be527` `rgb(59,229,39)``\n• #4be738\n``#4be738` `rgb(75,231,56)``\n• #5aea4a\n``#5aea4a` `rgb(90,234,74)``\n• #6aec5b\n``#6aec5b` `rgb(106,236,91)``\n• #7aee6d\n``#7aee6d` `rgb(122,238,109)``\n• #8af07e\n``#8af07e` `rgb(138,240,126)``\n• #9af290\n``#9af290` `rgb(154,242,144)``\n• #aaf4a1\n``#aaf4a1` `rgb(170,244,161)``\n• #baf6b3\n``#baf6b3` `rgb(186,246,179)``\n• #caf8c5\n``#caf8c5` `rgb(202,248,197)``\n``#dafad6` `rgb(218,250,214)``\n• #eafce8\n``#eafce8` `rgb(234,252,232)``\n• #fafef9\n``#fafef9` `rgb(250,254,249)``\nTint Color Variation\n\n# Tones of #baf6b3\n\nA tone is produced by adding gray to any pure hue. In this case, #d4d5d4 is the less saturated color, while #b5fdac is the most saturated one.\n\n• #d4d5d4\n``#d4d5d4` `rgb(212,213,212)``\n• #d1d9d0\n``#d1d9d0` `rgb(209,217,208)``\n• #cfdccd\n``#cfdccd` `rgb(207,220,205)``\n• #ccdfca\n``#ccdfca` `rgb(204,223,202)``\n• #cae2c7\n``#cae2c7` `rgb(202,226,199)``\n• #c7e6c3\n``#c7e6c3` `rgb(199,230,195)``\n• #c4e9c0\n``#c4e9c0` `rgb(196,233,192)``\n• #c2ecbd\n``#c2ecbd` `rgb(194,236,189)``\n• #bfefba\n``#bfefba` `rgb(191,239,186)``\n• #bdf3b6\n``#bdf3b6` `rgb(189,243,182)``\n• #baf6b3\n``#baf6b3` `rgb(186,246,179)``\n• #b7f9b0\n``#b7f9b0` `rgb(183,249,176)``\n• #b5fdac\n``#b5fdac` `rgb(181,253,172)``\nTone Color Variation\n\n# Color Blindness Simulator\n\nBelow, you can see how #baf6b3 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":0.53712255,"math_prob":0.6814146,"size":3725,"snap":"2020-34-2020-40","text_gpt3_token_len":1714,"char_repetition_ratio":0.12604138,"word_repetition_ratio":0.011070111,"special_character_ratio":0.5197315,"punctuation_ratio":0.23756906,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9606171,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2020-09-25T09:32:50Z\",\"WARC-Record-ID\":\"<urn:uuid:07cc60ef-5986-4a9a-9ebc-1b8286008891>\",\"Content-Length\":\"36359\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:f027ab37-6c69-4501-a068-24b1734d437c>\",\"WARC-Concurrent-To\":\"<urn:uuid:c66c2ab2-24f8-4eb1-8020-c9b8661d1c0f>\",\"WARC-IP-Address\":\"178.32.117.56\",\"WARC-Target-URI\":\"https://www.colorhexa.com/baf6b3\",\"WARC-Payload-Digest\":\"sha1:HYJMI3YWDEXKSO42BATEELOSC7BFCEIA\",\"WARC-Block-Digest\":\"sha1:DDY5E35XPF3SBN3LNVGT5RNIYTW42EWH\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2020/CC-MAIN-2020-40/CC-MAIN-2020-40_segments_1600400223922.43_warc_CC-MAIN-20200925084428-20200925114428-00634.warc.gz\"}"}
https://flylib.com/books/en/2.582.1/logical_functions.html	[ "# Logical Functions\n\nThe category of functions known as the logical functions contains a strange hodgepodge of things. Chapter 8 discussed two of them: the If and Case conditional functions. The logical functions covered here include several that are new to FileMaker Pro 8.\n\nThe Let Function\n\nThe Let function enables you to simplify complex calculations by declaring variables to represent subexpressions. These variables exist only within the scope of the formula and can't be referenced in other places. As an example, this is a formula presented in Chapter 8 for extracting the last line of a text field:\n\n`Right(myText; Length(myText) - Position(myText; \"¶\"; 1; PatternCount(myText; \"¶\")))`\n\nWith the Let function, this formula could be rewritten this way:\n\n```Let ([fieldLength = Length(myText) ;\nreturnCount = PatternCount(myText; \"¶\") ;\npositionOfLastReturn = Position (myText; \"¶\"; 1; returnCount) ;\ncharactersToGrab = fieldLength - positionOfLastReturn];\nRight (myText, charactersToGrab)\n)```\n\nThe Let function takes two parameters. The first is a list of variable declarations. If you want to declare multiple variables, you need to enclose the list within square brackets and separate the individual declarations within the list with semicolons. The second parameter is some formula you want evaluated. That formula can reference any of the variables declared in the first parameter, just as it would reference any field value.", null, "If you experience unexpected behavior of a Let function, the trouble might be your variable names. For more information, see \"Naming Variables in Let Functions\" in the \"Troubleshooting\" section at the end of this chapter.\n\nNotice in this example that the third variable declared, positionOfLastReturn, references the returnCount variable, which was the second variable declared. This capability to have subsequent variables reference previously defined ones is one of the powerful aspects of the Let function because it enables you to build up a complex formula via a series of simpler ones.\n\nIt is fair to observe that the Let function is never necessary; you could rewrite any formula that uses the Let function, without using Let, either as a complex nested formula or by explicitly defining or setting fields to contain subexpressions. The main benefits of using the Let function are simplicity, clarity, and ease of maintenance. For instance, a formula that returns a person's age expressed as a number of years, months, and days could be written as shown here:\n\n```\n```\n\n[View full width]\n\nYear (Get (CurrentDate)) - Year(birthDate) - (DayOfYear(Get(CurrentDate)) < DayOfYear", null, "(birthDate)) & \" years, \" & Mod ( Month(Get(CurrentDate)) - Month (birthDate) - (Day (Get(", null, "CurrentDate)) < Day(birthDate)); 12) & \" months, and \" & (Get(CurrentDate) - Date (Month(", null, "Get(CurrentDate)) - (Day (Get(CurrentDate)) < Day(birthDate)); Day (birthDate); Year (Get(", null, "CurrentDate)))) & \" days \"\n\nThis is a fairly complex nested formula, and many subexpressions appear multiple times. Writing and debugging this formula is difficult, even when you understand the logic on which it's based. With the Let function, the formula could be rewritten this way:\n\nLet ( [ C = Get(CurrentDate); yC = Year (C) ; mC = Month (C) ; dC = Day (C) ; doyC = DayOfYear (C) ; B = birthDate; yB = Year (B) ; mB = Month (B) ; dB= Day (B) ; doyB = DayOfYear (b) ; num_years = ( yC - yB - (doyC < doyB)) ; num_months = Mod (mC - mB - (dC\n\nBecause of the extra space we've put in the formula, it's a bit longer than the original, but it's vastly easier to comprehend. If you were a developer needing to review and understand a formula written by someone else, we're sure you'd agree that you'd prefer seeing the Let version of this rather than the first version.\n\nBesides simplicity and clarity, there are also performance benefits to using the Let function. If you have a complex subexpression that you refer to multiple times during the course of a calculation, FileMaker Pro evaluates it anew each time it's referenced. If you create the subexpression as a variable within a Let statement, the subexpression is evaluated only once, no matter how many times it is subsequently referenced. In the example just shown, for instance, FileMaker would evaluate Get(CurrentDate) eight times in the first version. In the version that uses Let, it's evaluated only once. In many cases, the performance difference may be trivial or imperceptible. But other times, optimizing the evaluation of calculation formulas may be just the answer for increasing your solution's performance.\n\nThe more you use the Let function, the more likely it is that it will become one of the core functions you use. To help you become more familiar with it, we use it frequently throughout the examples in the rest of this chapter.\n\n## Quick Calculation Testing Using Let\n\nThe Let function makes it much easier to debug calculation formulas. It used to be that if you wanted to make sure that a subexpression was evaluating correctly, you'd need to create a separate field to investigate it. Using Let, you can just comment out the second parameter of the Let function and have the function return one or more of the subexpressions directly. When you've got each subexpression working as intended, just comment out the test code and uncomment the original code.\n\nTip", null, "It's not uncommon that you may want to set the same variable several times within a Let statement. A typical example occurs when you want to perform a similar operation several times on the same variable, without excessive nesting. For example, in FileMaker 7, a fragment of a Let statement that's involved in some complex text parsing might look like this:\n``` result = _TextColor( text; RGB( 255: 0; 0 ));\nresult1 = _TextFont ( result; \"TimesNewRoman\");\nresult2 = _Textsize ( result1; 14);```\n\nHere, we want to apply several text formatting operations to the value of text. We'd like to put them on successive rows, rather than building a big nested expression. We'd prefer to just keep naming the output result, but in FileMaker 7, we'll be prevented from setting a variable with the same name twice. In FileMaker 8 this behavior is permitted, and we could rewrite the code fragment as something like this:\n\n``` result = _TextColor( text; RGB( 255: 0; 0 ));\nresult = _ TextFont ( result; \"TimesNewRoman\");\nresult = _Textsize ( result; 14);```\n\nAlthough this is a great convenience when you need to do it, be aware that calculations and custom functions that use this technique will not execute correctly if the file is accessed via FileMaker Pro 7.\n\nThe Choose Function\n\nThe If and Case functions are sufficiently robust and elegant for most conditional tests that you'll write. For several types of conditional tests, however, the Choose function is a more appropriate option. As with If and Case, the value returned by the Choose function is dependent on the result of some test. What makes the Choose function different is that the test should return an integer rather than a true/false result. The test is followed by a number of possible results. The one that's chosen depends on the numeric result of the test. If the test result is 0, the first result is used. If the test result is 1, the second result is used, and so on. The syntax for Choose is as follows:\n\n`Choose (test ; result if test=0 ; result if test=1 ; result if test=2 ....)`\n\nA classic example of when a Choose function comes in handy is when you have categorical data stored as a number and you need to represent it as text. For instance, you might import demographic data in which the ethnicity of an individual is represented by an integer from 1 to 5. The following formula might be used to represent it to users:\n\n```\n```\n\n[View full width]\n\nChoose (EthnicityCode; \"\"; \"African American\"; \"Asian\"; \"Caucasian\"; \"Hispanic\"; \" Native", null, "American\")\n\nOf course, the same result could be achieved with the following formula:\n\n```\n```\n\n[View full width]\n\nCase (EthnicityCode = 1; \"African American\"; EthnicityCode = 2; \"Asian\", EthnicityCode =", null, "3; \"Caucasian\"; EthnicityCode = 4; \"Hispanic\"; EthnicityCode= 5; \"Native American\")\n\nYou should consider the Choose function in several other situations. The first is for generating random categorical data. Say your third-grade class is doing research on famous presidents, and you want to randomly assign each student one of the six presidents you have chosen. By first generating a random number from 0 to 5, you can then use the Choose function to select a president. The formula would be this:\n\n```Let ( r = Random * 6; // Generates a random number from 0 to 5\nChoose (r, \"Washington\", \"Jefferson\", \"Lincoln\", \"Roosevelt\", \"Truman\", \"Kennedy\"))```\n\nDon't worry that r isn't an integer; the Choose function ignores everything but the integer portion of a number.\n\nSeveral FileMaker Pro functions return integer numbers from 1 to n, so these naturally work well as the test for a Choose function. Most notable are the DayofWeek function, which returns an integer from 1 to 7, and the Month function, which returns an integer from 1 to 12. As an example, you could use the Month function within a Choose to figure out within which quarter of the year a given date fell:\n\n```\n```\n\n[View full width]\n\nChoose (Month(myDate)-1; \"Q1\"; \"Q1\"; \"Q1\"; \"Q2\"; \"Q2\"; \"Q2\"; \"Q3\"; \"Q3\"; \"Q3\"; \"Q4\"; \"Q4\";", null, "\"Q4\")\n\nThe -1 shifts the range of the output from 112 to 011, which is more desirable because the Choose function is zero-based, meaning that the first result corresponds to a test value of zero. There are more compact ways of determining the calendar quarter of a date, but this version is very easy to understand and offers much flexibility.\n\nAnother example of when Choose works well is when you need to combine the results of some number of Boolean tests to produce a distinct result. As an example, imagine that you have a table that contains results on Myers-Briggs personality tests. For each test given, you have scores for four pairs of personality traits (E/I, S/N, T/F, J/P). Based on which score in each pair is higher, you want to classify each participant as one of 16 personality types. Using If or Case statements, you would need a very long, complex formula to do this. With Choose, you can treat the four tests as a binary number, and then simply do a conversion back to base-10 to decode the results. The formula might look something like this:\n\n```Choose( (8 * (E>I)) + (4 * (S>N)) + (2 * (T>F)) + (J>P);\n\"Type 1 - INFP\" ; \"Type 2 - INFJ\" ; \"Type 3 - INTP\" ; \"Type 4 - INTJ\" ;\n\"Type 5 - ISFP\" ; \"Type 6 - ISFJ\" ; \"Type 7 - ISTP\" ; \"Type 8 - ISTJ\" ;\n\"Type 9 - ENFP\" ; \"Type 10 - ENFJ\" ; \"Type 11 - ENTP\" ; \"Type 12 - ENTJ\" ;\n\"Type 13 - ESFP\" ; \"Type 14 - ESFJ\" ; \"Type 15 - ESTP\" ; \"Type 16 - ESTJ\")```\n\nEach greater-than comparison is evaluated as a 1 or 0 depending on whether it represents a true or false statement for the given record. By multiplying each result by successive powers of 2, you end up with an integer from 0 to 15 that represents each of the possible outcomes. (This is similar to how flipping a coin four times generates 16 possible outcomes.)\n\nAs a final example, the Choose function can also be used anytime you need to \"decode\" a set of abbreviations into their expanded versions. Take, for example, a situation in which survey respondents have entered SA, A, N, D, or SD as a response to indicate Strongly Agree, Agree, Neutral, Disagree, or Strongly Disagree. You could map from the abbreviation to the expanded text by using a Case function like this:\n\n```Case (ResponseAbbreviation = \"SA\"; \"Strongly Agree\";\nResponseAbbreviation = \"A\"; \"Agree\" ;\nResponseAbbreviation = \"N\"; \"Neutral\" ;\nResponseAbbreviation = \"D\"; \"Disagree\" ;\nResponseAbbreviation = \"SD\"; \"Strongly Disagree\" )```\n\nYou can accomplish the same mapping by using a Choose function if you treat the two sets of choices as ordered lists. You simply find the position of an item in the abbreviation list, and then find the corresponding item from the expanded text list. The resulting formula would look like this:\n\n```\n```\n\n[View full width]\n\nLet ( [a = \"\|SA\|\|A\|\|N\|\|D\|\|SD\|\" ; r = \"\|\" & ResponseAbbreviation & \"\|\" ; pos = Position (a; r ; 1 ; 1) ; itemNumber = PatternCount (Left (a; pos-1); \"\|\") / 2]; Choose (itemNumber, \"Strongly Agree\"; \"Agree\"; \"Neutral\"; \"Disagree\"; \"Strongly", null, "Disagree\") )\n\nIn most cases, you'll probably opt for using the Case function for simple decoding of abbreviations. Sometimes, however, the list of choices isn't something you can explicitly test against (such as with the contents of a value list), and finding one's position within the list may suffice to identify a parallel position in some other list. Having the Choose function in your toolbox may offer an elegant solution to such challenges.\n\nThe GetField Function\n\nWhen writing calculation formulas, you use field names to refer abstractly to the contents of particular fields in the current record. That is, the formula for a FullName calculation might be FirstName & \" \" & LastName. FirstName and LastName are abstractions; they represent data contained in particular fields.\n\nImagine, however, that instead of knowing in advance what fields to refer to in the FullName calculation, you wanted to let users pick any fields they wanted to. So you set up two fields, which we'll call UserChoice1 and UserChoice2. How can you rewrite the FullName calculation so that it's not hard-coded to use FirstName and LastName, but rather uses the fields that users type in the two UserChoice fields?\n\nThe answer, of course, is the GetField function. GetField enables you to add another layer of abstraction to your calculation formulas. Instead of hard-coding field names in a formula, GetField allows you to place into a field the name of the field you're interested in accessing. That sounds much more complicated than it actually is. Using GetField, we might rewrite our FullName formula as shown here:\n\n`GetField (UserChoice1) & \" \" & GetField (UserChoice2)`\n\nThe GetField function takes just one parameter. That parameter can be either a literal text string or a field name. Having it be a literal text string, although possible, is not particularly useful. The function GetField(\"FirstName\") would certainly return the contents of the FirstName field, but you can achieve the same thing simply by using FirstName by itself. It's only when the parameter of the GetField function is a field or formula that it becomes interesting. In that case, the function returns the contents of the field referred to by the parameter.\n\nThere are many potential uses of GetField in a solution. Imagine, for instance, that you have a Contact table with fields called First Name, Nickname, and Last Name (among others). Sometimes contacts prefer to have their nickname appear on badges and in correspondence, and sometimes the first name is desired. To deal with this, you could create a new text field called Preferred Name and format that field as a radio button containing First Name and Nickname as the choices. When doing data entry, a user could simply check off which name should be used for correspondence. When it comes time to make a Full Name calculation field, one of your options would be the following:\n\n```Case ( Preferred Name = \"First Name\"; First Name;\nPreferred Name = \"Nickname\"; Nickname) &\n\" \" & Last Name```\n\nAnother option, far more elegant and extensible, would be the following:\n\n`GetField (PreferredName) & \" \" & Last Name`\n\nWhen there are only two choices, the Case function certainly isn't cumbersome. But if there were dozens or hundreds of fields to choose from, GetField clearly has an advantage.\n\nBuilding a Customizable List Report\n\nOne of the common uses of GetField is for building user-customizable list reports. It's really nothing more than an extension of the technique shown in the preceding example, but it's still worth looking at in depth. The idea is to have several global text fields where a user can select from a pop-up list of field names. The global text fields can be defined in any table you want. Remember, in calculation formulas, you can refer to a globally stored field from any table, even without creating a relationship to that table. The following example uses two tables: SalesPeople and Globals. The SalesPeople table has the following data fields:\n\nSalesPersonID\n\nFirstName\n\nLastName\n\nTerritory\n\nCommissionRate\n\nPhone\n\nEmail\n\nSales_2005\n\nSales_2006\n\nThe Globals table has six global text fields named gCol1 through gCol6.\n\nWith these in place, you can now create six display fields in the SalesPeople table (named ColDisplay1 through ColDisplay6) that will contain the contents of the field referred to in one of the global fields. For instance, ColDisplay1 has the following formula:\n\n`GetField (Globals::gCol1)`\n\nColDisplay2 through 6 will have similar definitions. The next step is to create a value list that contains all the fields you want the user to be able to select. The list used in this example is shown in Figure 14.1. Keep in mind that because the selection is used as part of a GetField function, the field names must appear exactly as they have been definedand any change to the underlying field names will cause the report to malfunction.\n\nFigure 14.1. Define a value list containing a list of the fields from which you want to allow a user to select for the custom report.", null, "The final task is to create a layout where users can select and see the columns for their custom list report. You might want to set up one layout where the user selects the fields and another for displaying the results, but we think it's better to take advantage of the fact that in FileMaker 8, fields in header parts of list layouts can be edited. The column headers of your report can simply be pop-up lists. Figure 14.2 shows how you would set up your layout this way.\n\nFigure 14.2. The layout for your customizable list report can be quite simple. Here, the selection fields act also as field headers.", null, "Back in Browse mode, users can now click into a column heading and select what data they want to appear there. This one layout can thus serve a wide variety of needs. Figures 14.3 and 14.4 show two examples of the types of reports that can be made.\n\nFigure 14.3. A user can customize the contents of a report simply by selecting fields from pop-up lists in the header.", null, "Figure 14.4. Here's another example of a how a user might configure the customizable list report.", null, "Extending the Customizable List Report\n\nAfter you have the simple custom report working, there are many ways you can extend it to add even more value and flexibility for your users. For instance, you might add a subsummary part that's also based on a user-specified field. A single layout can thus be a subsummary based on any field the user wants. One way to implement this is to add another pop-up list in the header of your report and a button to sort and preview the subsummary report. Figure 14.5 shows what your layout would look like after adding the subsummary part and pop-up list. BreakField is a calculation in the SalesPeople table that's defined as shown here:\n\n`GetField (Globals::gSummarizeBy)`", null, "The Preview button performs a script that sorts by the BreakField and goes to Preview mode. Figure 14.6 shows the result of running the script when Territory has been selected as the break field.\n\nFigure 14.6. Sorting by the break field and previewing shows the results of the dynamic subsummary.", null, "Caution\n\nTo be fully dynamic, any calculations you write using the GetField function are probably going to need to be unstored. Unstored calculations will not perform well over very large data sets when searching and sorting, so use caution when creating GetField routines that might need to handle large data sets.\n\nThe Evaluate Function\n\nThe Evaluate function is one of the most intriguing functions in FileMaker Pro 8. In a nutshell, it enables you to evaluate a dynamically generated or user-generated calculation formula. With a few examples, you'll easily understand what this function does. It may, however, take a bit more time and thought to understand why you'd want to use it in a solution. We start with explaining the what, and then suggest a few potential whys.\n\nThe syntax for the Evaluate function is as follows:\n\n`Evaluate ( expression {; [field1 ; field2 ;...]} )`\n\nThe expression parameter is a text string representing some calculation formula that you want evaluated. The optional additional parameter is a list of fields whose modification triggers the reevaluation of the expression.\n\nFor example, imagine that you have a text field named myFormula and another named myTrigger. You then define a new calculation field called Result, using the following formula:\n\n`Evaluate (myFormula; myTrigger)`\n\nFigure 14.7 shows some examples of what Result will contain for various entries in myFormula.\n\nFigure 14.7. Using the Evaluate function, you can have a calculation field evaluate a formula contained in a field.", null, "There's something quite profound going on here. Instead of having to \"hard-code\" calculation formulas, you can evaluate a formula that's been entered as field data. In this way, Evaluate provides an additional level of logic abstraction similar to the GetField function. In fact, if myFormula contained the name of a field, Evaluate(myFormula) and GetField(myFormula) would return exactly the same result. It might help to think of Evaluate as the big brother of GetField. Whereas GetField can return the value of a dynamically specified field, Evaluate can return the value of a dynamically specified formula.\n\nUses for the Evaluate Function\n\nA typical use for the Evaluate function is to track modification information about a particular field or fields. A timestamp field defined to auto-enter the modification time is triggered anytime any field in the record is modified. There may be times, however, when you want to know the last time that the Comments field was modified, without respect to other changes to the record. To do this, you would define a new calculation field called CommentsModTime with the following formula:\n\n`Evaluate (\"Get(CurrentTimestamp)\" ; Comments)`\n\nThe quotes around Get(CurrentTimestamp) are important, and are apt be a source of confusion. The Evaluate function expects to be fed either a quote-enclosed text string (as shown here) or a formula that yields a text string (as in the Result field earlier). For instance, if you want to modify the CommentsModTime field so that rather than just returning a timestamp, it returns something like Record last modified at: 11/28/2005 12:23:58 PM by Fred Flintstone, you would need to modify the formula to the following:\n\n```\n```\n\n[View full width]\n\nEvaluate (\"\"Record modified at: \" & Get (CurrentTimeStamp) & \" by \" & Get", null, "(AccountName)\" ; Comments)\n\nHere, because the formula you want to evaluate contains quotation marks, you must escape them by preceding them with a slash. For a formula of any complexity, this becomes difficult both to write and to read. There is, fortunately, a function called Quote that eliminates all this complexity. The Quote function returns the parameter it is passed as a quote-wrapped text string, with all internal quotes properly escaped. Therefore, you could rewrite the preceding function more simply as this:\n\n```\n```\n\n[View full width]\n\nEvaluate (Quote (\"Record modified at: \" & Get (CurrentTimeStamp) & \" by \" & Get (", null, "AccountName)) ; Comments)\n\nIn this particular case, using the Let function further clarifies the syntax:\n\n```Let ( [\ntime = Get ( CurrentTimeStamp ) ;\naccount = Get ( AccountName );\nmyExpression = Quote ( \"Record modified at: \" & time & \" by \" & account ) ] ;\n\nEvaluate ( myExpression ; Comments )\n)```\n\nEvaluation Errors\n\nYou typically find two other functions used in conjunction with the Evaluate function: IsValidExpression and EvaluationError.\n\nIsValidExpression takes as its parameter an expression, and it returns a 1 if the expression is valid, a 0 if it isn't. An invalid expression is any expression that can't be evaluated by FileMaker Pro, whether due to syntax errors or other runtime errors. If you plan to allow users to type calculation expressions into fields, be sure to use IsValidExpression to test their input to be sure it's well formed. In fact, you probably want to include a check of some kind within your Evaluate formula itself:\n\n```Let ( valid = IsValidExpression (myFormula) ;\nIf (not valid; \"Your expression was invalid\" ; Evaluate (myFormula) )```\n\nThe EvaluationError function is likewise used to determine whether there's some problem with evaluating an expression. However, it returns the actual error code corresponding to the problem. One thing to keep in mind, however, is that rather than testing the expression, you want to test the evaluation of the expression. So, as an error trap used in conjunction with an Evaluate function, you might have the following:\n\n```Let ( [result = Evaluate (myFormula) ;\nerror = EvaluationError (result) ] ;\nIf (error ; \"Error: \" & error ; result)\n)```\n\nCustomizable List Reports Redux\n\nWe mentioned previously that Evaluate could be thought of as an extension of GetField. In an example presented in the GetField section, we showed how you could use the GetField function to create user-customizable report layouts. One of the drawbacks of that method that we didn't discuss at the time is that your field names need to be user- and display-friendly. However, there is an interesting way to get around this limitation that also happens to showcase the Evaluate function. We discuss that solution here as a final example of Evaluate.", null, "Another use of Evaluate is presented in \"Passing Multivalued Parameters,\" p. 438.\n\nTo recap the earlier example, imagine that you have six global text fields (gCol1 through gCol6) in a table called Globals. Another table, called SalesPeople, has demographic and sales-related data for your salespeople. Six calculation fields in SalesPeople, called ColDisplay1 through ColDisplay6, display the contents of the demographic or sales data fields, based on a user's selection from a pop-up list containing field names. ColDisplay1, for instance, has the following formula:\n\n`GetField (Globals::gCol1)`\n\nWe now extend this solution in several ways. First, create a new table in the solution called FieldNames with the following text fields: FieldName and DisplayName. Figure 14.8 shows the data that might be entered in this table.\n\nFigure 14.8. The data in FieldName represents fields in the SalesPerson table; the DisplayName field shows more user-friendly labels that will stand in for the actual field labels.", null, "Earlier, we suggested using a hard-coded value list for the pop-up lists attached to the column selection fields. Now you'll want to change that value list so that it contains all the items in the DisplayName column of the FieldNames table. Doing this, of course, causes all the ColDisplay fields to malfunction. There is, for instance, no field called Ph. Number, so GetField (\"Ph. Number\") will not function properly. What we want now is the GetField function not to operate on the user's entry, but rather on the FieldName that corresponds to the user's DisplayName selection. That is, when the user selects Ph. Number in gCol1, ColDisplay1 should display the contents of the Phone field.\n\nYou can accomplish this result by creating a relationship from the user's selection over to the DisplayName field. Because there are six user selection fields, there need to be six relationships. This requires that you create six occurrences of the FieldNames table. Figure 14.9 shows the Relationships Graph after you have set up the six relationships. The six new table occurrences are named Fields1 through Fields6. Notice that there's also a cross-join relationship between SalesPeople and Globals. This relationship allows you to look from SalesPeople all the way over to the FieldNames table.\n\nFigure 14.9. To create six relationships from the Globals table to the FieldNames table, you need to create six occurrences of FieldNames.", null, "The final step is to alter the calculation formulas in the ColDisplay fields. Remember, instead of \"getting\" the field specified by the user, we now want to get the field related to the field label specified by the user. At first thought, you might be tempted to redefine ColDisplay1 this way:\n\n`GetField (Fields1::FieldName)`\n\nThe problem with this is that the only way that ColDisplay1 updates is if the FieldName field changes. Changing gCol1 doesn't have any effect on it. This, finally, is where Evaluate comes in. To force ColDisplay1 to update, you can use the Evaluate function instead of GetField. The second parameter of the formula can reference gCol1, thus triggering the reevaluation of the expression every time gCol1 changes. The new formula for ColDisplay1 is therefore this:\n\n`Evaluate (Fields1::FieldName ; Globals::gCol1)`\n\nThere is, in fact, still a slight problem with this formula. Even though the calculation is unstored, the field values don't refresh onscreen. The solution is to refer not merely to the related FieldName, but rather to use a Lookup function (which is covered in depth in the next section) to explicitly grab the contents of FieldName. The final formula, therefore, is the following:\n\n`Evaluate (Lookup (Fields1::FieldName) ; Globals::gCol1)`\n\nThere's one final interesting extension we will make to this technique. At this point, the Evaluate function is used simply to grab the contents of a field. It's quite possible, however, to add a field called Formula to the FieldNames table, and have the Evaluate function return the results of some formula that you define there. The formula in ColDisplay1 would simply be changed to this:\n\n`Evaluate (Lookup (Fields1::Formula) ; Globals::gCol1)`\n\nOne reason you might want to do this is to be able to add some text formatting to particular fields. For instance, you might want the Sales_2004 field displayed with a leading dollar sign. Because all the ColDisplay fields yield text results, you can't do this with ordinary field formatting. Instead, in the Formula field on the Sales_2004 record, you could type the following formula:\n\n`\"\\$ \" & Sales_2004`\n\nThere's no reason, of course, why a formula you write can't reference multiple fields. This means that you can invent new fields for users to reference simply by adding a new record to the FieldNames table. For example, you could invent a new column called Initials, defined this way:\n\n`Left (FirstName; 1) & Left (LastName; 1)`\n\nYou could even invent a column called Percent Increase that calculates the percent sales increase from 2004 to 2005. This would be the formula for that:\n\n`Round((Sales_2005 - Sales_2004) / Sales_2004 *100, 2) & \" %\"`\n\nFigure 14.10 shows the contents of the FieldNames table. Note that for columns where you just want to retrieve the value of a field (for example, FirstName), the field name itself is the entire formula.\n\nFigure 14.10. The expression in the Formula field is dynamically evaluated when a user selects a column in the customizable report.", null, "This technique is quite powerful. You can cook up new columns for the customizable report just by adding records to the FieldNames table. Figure 14.11 shows an example of a report that a user could create based on the formulas defined in FieldNames. Keep in mind that Initials, \\$ Increase, and Percent Increase have not been defined as fields anywhere.\n\nFigure 14.11. In the finished report, users can select from any of the columns defined in the FieldNames table, even those that don't explicitly exist as defined fields.", null, "The Lookup Functions\n\nIn versions of FileMaker before version 7, lookups were exclusively an auto-entry option. FileMaker 7 added two lookup functions, Lookup and LookupNext, and both are useful additions to any developer's toolkit.\n\nThe two lookup functions operate quite similarly to their cousin, the auto-entry lookup option. In essence, a lookup is used to copy a related value into the current table. Lookups (all kinds) have three necessary components: a relationship, a trigger field, and a target field. When the trigger field is modified, the target field is set to some related field value.\n\nIt's important to understand the functional differences between the lookup functions and the auto-entry option. Although they behave similarly, they're not quite equivalent. Some of the key differences include the following:\n\n• Auto-entry of a looked-up value is an option for regular text, number, date, time, or timestamp fields, which are subsequently modifiable by the user. A calculation field that includes a lookup function is not user modifiable.\n• The lookup functions can be used anywherenot just in field definitions. For instance, they can be used in formulas in scripts, record-level security settings, and calculated field validation. Auto-entering a looked-up value is limited to field definition.\n• The lookup functions can be used in conjunction with other functions to create more complex logic rules. The auto-entry options are comparatively limited.\n\nLookup\n\nThe syntax of the Lookup function is as follows:\n\n`Lookup ( sourceField {; failExpression} )`\n\nThe sourceField is the related field whose value you want to retrieve. The optional failExpression parameter is returned if there is no related record or if the sourceField is blank for the related record. If the specified relationship matches multiple related records, the value from the first related record is returned.\n\nThere are two main differences between using the Lookup function and simply referencing a related field in a formula. The first is that calculations that simply reference related fields must be unstored, but calculations that use the Lookup function to access related fields can be stored and indexed. The other difference is that changing the sourceField in the related table does not cause the Lookup to retrigger. Just as with auto-entry of a looked-up value, the Lookup function captures the sourceField as it existed at a moment in time. The alternative, simply referencing the related field, causes all the values to remain perfectly in sync: When the related value is updated, any calculations that reference it are updated as well. (The downside is that, as with all calculations that directly reference related data, such a calculation cannot be stored.)\n\nLookupNext\n\nThe LookupNext function is designed to allow you to map continuous data elements to categorical results. It has the same effect as checking the Copy Next Lower Value or Copy Next Text Formatting Functions Higher Value options when specifying an auto-entry lookup field option. Here is its syntax:\n\n`LookupNext ( sourceField ; lower/higherFlag )`\n\nThe acceptable values for the second parameter are Lower and Higher. These are keywords and shouldn't be placed in quotes.\n\nAn example should help clarify what we mean about mapping continuous data to categorical results. Imagine that you have a table that contains information about people, and that one of the fields is the person's birth date. You want to have some calculation fields that display the person's astrological information, such as a zodiac sign and ruling planet. Birth dates mapping to zodiac signs is a good example of continuous data mapping to categorical results: A range of birth dates corresponds to each zodiac sign.\n\nIn practice, two small but instructive complications arise when you try to look up zodiac signs. The first complication is that the zodiac date ranges are expressed not as full dates, but merely as months and days (for example, Cancer starts on June 22 regardless of what year it is). This means that when you set up your zodiac table, you'll use text fields rather than date fields for the start and end dates. The second complication is that Capricorn wraps around the end of the year. The easiest way to deal with this is to have two records in the Zodiac table for Capricorn, one that spans December 22December 31, and the other that spans January 1January 20.\n\nFigure 14.12 shows the full data of the Zodiac table. The StartDate and EndDate fields, remember, are actually text fields. The leading zeros are important for proper sorting.\n\nFigure 14.12. The data from the Zodiac table is looked up and is transferred to a person record based on the person's birth date.", null, "In the Person table, you need to create a calculation formula that generates a text string containing the month and date of the person's birth date, complete with leading zeros so that it's consistent with the way dates are represented in the Zodiac table. The DateMatch field is defined this way:\n\n`Right (\"00\" & Month (Birthdate); 2) & \"/\" & Right (\"00\"& Day (Birthdate); 2)`\n\nNext, create a relationship between the Person and Zodiac tables, matching the DateMatch field in Person to the StartDate field in Zodiac. This relationship is shown in Figure 14.13.\n\nFigure 14.13. By relating the Person table to Zodiac, you can look up any information you want based on the person's birth date.", null, "Obviously, many birth dates aren't start dates for one of the zodiac signs. To match to the correct zodiac record, you want to find the next lower match when no exact match is found. For instance, with a birth date of February 13 (02/13), there is no matching record where the StartDate is 02/13, so the next lowest StartDate, which is 01/21 (Aquarius), should be used.\n\nIn the Person table, therefore, you can grab any desired zodiac information by using the LookupNext function. Figure 14.14 shows an example of how this date might be displayed on a person record. The formula for ZodiacInfo is as follows:\n\n```\"Sign: \" & LookupNext (Zodiac::ZodiacSign; Lower) & \"¶\" &\n\"Symbol: \" & LookupNext (Zodiac::ZodiacSymbol; Lower) & \"¶\" &\n\"Ruling Planet: \" & LookupNext (Zodiac::RulingPlanet; Lower)```\n\nFigure 14.14. Using the LookupNext function, you can create a calculation field in the Person table that contains information from the next lower matching record.", null, "It would have been possible in the previous examples to match to the EndDate instead of the StartDate. In that case, you would simply need to match to the next higher instead of the next lower matching record.\n\nAn entirely different but perfectly valid way of approaching the problem would have been to define a more complex relationship between Person and Zodiac, in which the DateMatch was greater than or equal to the StartDate and less than or equal to the EndDate. Doing this would allow you to use the fields from the Zodiac table as plain related fields; no lookup would have been required. There are no clear advantages or disadvantages of this method over the one discussed previously.\n\nNote\n\nOther typical scenarios for using LookupNext are for things such as shipping rates based on weight ranges, price discounts based on quantity ranges, and defining cut scores based on continuous test score ranges.\n\n### Text Formatting Functions", null, "Special Edition Using FileMaker 8\nISBN: 0789735121\nEAN: 2147483647\nYear: 2007\nPages: 296", null, "" ]	[ null, "https://flylib.com/books/2/582/1/html/2/images/troubleshooting.jpg", null, "https://flylib.com/books/2/582/1/html/2/images/ccc.gif", null, "https://flylib.com/books/2/582/1/html/2/images/ccc.gif", null, "https://flylib.com/books/2/582/1/html/2/images/ccc.gif", null, "https://flylib.com/books/2/582/1/html/2/images/ccc.gif", null, "https://flylib.com/books/2/582/1/html/2/images/new.jpg", null, "https://flylib.com/books/2/582/1/html/2/images/ccc.gif", null, "https://flylib.com/books/2/582/1/html/2/images/ccc.gif", null, "https://flylib.com/books/2/582/1/html/2/images/ccc.gif", null, "https://flylib.com/books/2/582/1/html/2/images/ccc.gif", null, "https://flylib.com/books/2/582/1/html/2/images/14fig01.jpg", null, "https://flylib.com/books/2/582/1/html/2/images/14fig02.jpg", null, "https://flylib.com/books/2/582/1/html/2/images/14fig03.jpg", null, "https://flylib.com/books/2/582/1/html/2/images/14fig04.jpg", null, "https://flylib.com/books/2/582/1/html/2/images/14fig05.jpg", null, "https://flylib.com/books/2/582/1/html/2/images/14fig06.jpg", null, "https://flylib.com/books/2/582/1/html/2/images/14fig07.jpg", null, "https://flylib.com/books/2/582/1/html/2/images/ccc.gif", null, "https://flylib.com/books/2/582/1/html/2/images/ccc.gif", null, "https://flylib.com/books/2/582/1/html/2/images/arrow.jpg", null, "https://flylib.com/books/2/582/1/html/2/images/14fig08.jpg", null, "https://flylib.com/books/2/582/1/html/2/images/14fig09.jpg", null, "https://flylib.com/books/2/582/1/html/2/images/14fig10.jpg", null, "https://flylib.com/books/2/582/1/html/2/images/14fig11.jpg", null, "https://flylib.com/books/2/582/1/html/2/images/14fig12.jpg", null, "https://flylib.com/books/2/582/1/html/2/images/14fig13.jpg", null, "https://flylib.com/books/2/582/1/html/2/images/14fig14.jpg", null, "https://flylib.com/icons/4829-small.jpg", null, "https://flylib.com/media/images/top.png", null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.88836545,"math_prob":0.86007804,"size":39083,"snap":"2023-40-2023-50","text_gpt3_token_len":8517,"char_repetition_ratio":0.14550014,"word_repetition_ratio":0.016251354,"special_character_ratio":0.21712765,"punctuation_ratio":0.117704555,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9570403,"pos_list":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58],"im_url_duplicate_count":[null,null,null,null,null,null,null,null,null,null,null,4,null,null,null,null,null,null,null,null,null,4,null,4,null,4,null,4,null,4,null,4,null,4,null,null,null,null,null,null,null,4,null,4,null,4,null,4,null,4,null,4,null,4,null,null,null,null,null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2023-12-04T06:27:04Z\",\"WARC-Record-ID\":\"<urn:uuid:ec2b0150-fc36-4fb8-bed7-a8faebf1694a>\",\"Content-Length\":\"93586\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:9e626309-389c-4a47-91a6-294cf73763d7>\",\"WARC-Concurrent-To\":\"<urn:uuid:feaaadad-f883-47a4-947e-7a264ff37e27>\",\"WARC-IP-Address\":\"179.43.157.53\",\"WARC-Target-URI\":\"https://flylib.com/books/en/2.582.1/logical_functions.html\",\"WARC-Payload-Digest\":\"sha1:523NEYQ3LAEQJZDTNLZ5NPW2R3CRFN3Q\",\"WARC-Block-Digest\":\"sha1:F47WCYAM2QED22PAHGDUIVH4MNZ5UNDU\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2023/CC-MAIN-2023-50/CC-MAIN-2023-50_segments_1700679100525.55_warc_CC-MAIN-20231204052342-20231204082342-00044.warc.gz\"}"}
http://oak.go.kr/central/journallist/journaldetail.do?article_seq=16825	[ "### Budget Deficits and Exchange-Rate Crises\n\n•", null, "• #### ABSTRACT\n\nThis paper investigates currency crises in an optimizing general equilibrium model with overlapping generations. It is shown that a rise in government budget deficits financed by future taxes generates a decumulation of external assets, leading up to a speculative attack and forcing the monetary authorities to abandon the peg.\n\n• #### KEYWORD\n\nBudget deficits , foreign exchange reserves , currency crises\n\n• ### 1. Introduction\n\nThe financial and currency turmoils of the 1990s in many European, Latin-American and Asian countries have called into question the viability of fixed exchange-rate regimes and led to the development of new models on the causes of currency collapses. We can now identify at least two main theoretical explanations of crises, one based on the view that a collapse is the inevitable outcome of inconsistent macroeconomic policies and the other based on the view that a collapse results from self-fulfilling expectations. According to the so-called ‘firstgeneration’ models of currency crises, if a government finances its fiscal deficits by printing money in excess of money demand growth, while following a fixed exchange-rate policy, a gradual loss of reserves will occur, leading to a speculative attack against the currency that forces the abandonment of the fixed-rate regime eventually (see, for example, Krugman, 1979; Flood & Garber, 1984; Obstfeld, 1986; Calvo, 1987; van Wijnbergen, 1991; Calvo & Végh, 1999).\n\nThe ‘second-generation’ models of currency crises, on the other hand, show that the government’s decision to give up a fixed exchange rate depends on the net benefits of pegging; hence, the fixed rate is likely to be maintained as long as the benefits of devaluing are smaller than the costs. However, changes in market beliefs about the currency sustainability can force the government to go out of the peg. For example, if agents expect a devaluation, a speculative attack will start, forcing the government to abandon the peg, since the costs of keeping the exchange rate fixed outweigh the benefits. On the other hand, if agents expect no change in the currency rate the fixed peg will be preserved. Private expectations are self-fulfilling and multiple equilibria can occur, for given fundamentals (see, for example, Obstfeld, 1996; Velasco, 1996; Cole & Kehoe, 1996; Jeanne, 1997; Jeanne & Masson, 2000).1\n\nMore recently, the debate about the role played by fundamentals and/or selffulfilling expectations in triggering a speculative attack has been enriched by a new set of models analyzing currency crises in the context of a change in the expectations of future policy (see, for example, Burnside et al., 2001, 2003, 2004, 2006; Daniel, 2000, 2001; Corsetti & Maćkowiack, 2006; Maćkowiack, 2007; Singh, 2009). According to this view, explanations of crises do not necessarily require a period of fundamental misalignments. All that is needed is that the path of current and future government policies becomes inconsistent with the fixed peg.\n\nThis literature typically uses static models, or models with extrinsic dynamics; that is, models where the dynamics of the system come exclusively from current or anticipated future changes in exogenous variables. Such systems are, in fact, always in steady-state equilibrium in the absence of external shocks. This is in contrast to the so-called intrinsic dynamics of the system, where the economy evolves from some initial stationary state due to, for example, the accumulation of capital stock or foreign assets.2 Models with intrinsic dynamics are useful to understand currency crises and to predict the exact time of an attack. They enable us to study the dynamics of relevant macroeconomic variables.\n\nThe purpose of this paper is to deal with this kind of dynamics using a modified version of the Yaari (1965)Blanchard (1985) model. Our approach has three main advantages. First, it allows a ‘nondegenerate’ dynamic adjustment in the basic monetary model of exchange rate determination. Second, it shows that the macroeconomic equilibrium is dependent on the timing of fiscal policies. Third, the current account is allowed to play a crucial role in transmitting fiscal disturbances to the rest of the economy.\n\nAcentral finding of the paper is that a collapse may occur even as a consequence of a temporary tax cut fully financed by future taxes. In particular, following a fiscal expansion, current account imbalances and the expected depletion of foreign assets will lead to a currency crisis, forcing the monetary authorities to adopt a floating exchange-rate regime. This result is in sharp contrast with the existing literature, where monetary or fiscal policies are inconsistent or expected to be inconsistent with the exchange-rate policy. On the other hand, our theoretical results are consistent with the evidence that Asian countries that came under attack in 1997 were those that had experienced larger current account deficits on the eve of the crisis.3\n\nThe main conclusion that emerges from our analysis is that crises can occur in a flexible-price, fully optimizing framework, even when both monetary and fiscal policies are correctly designed; that is, when the intertemporal budget constraint of the government is always respected and monetary policy obeys the rules of the game. The crisis, however, is not driven by self-fulfilling expectations, as the collapse results from well-defined dynamics in the fundamentals. From this point of view, the crisis is basically real and is triggered by the macroeconomic adjustment path associated with a consistent and flexible policy rule that, nonetheless, gives rise to the conditions for a run on the central bank’s foreign reserves. The main implication of our model is that the sustainability of a fixed exchange-rate system may require not only giving up monetary sovereignty, but also strongly restraining the conduct of fiscal policy.4\n\nThe paper is organized as follows. Section 2 presents the theoretical model. Section 3 describes the dynamics of the model and the time of the speculative attack. Section 4 concludes.\n\n1The partition between first and second generation models is consistent with the classification scheme of currency crises proposed by Flood and Marion (1999), and Jeanne (2000). From this point of view, the so-called ‘third-generation’ models, elaborated to explain the more recent financial turmoils in Asia, Latin America and Russia, are considered extensions of the existing setups that explicitly include the financial side of the economy 2This distinction may be found, for example, in Turnovsky (1977, 1997) and Obstfeld and Stockman (1985). 3For an exhaustive overviewof the economic fundamentals in Asian countries in the years preceding the financial and currency crisis, see Corsetti et al. (1999) and World Bank (1999). 4Similar implications are also found in Cook and Devereux (2006).\n\n### 2. The Model\n\nConsider a small open economy described as follows.Agents have perfect foresight and consume a single tradeable good. Domestic supply of the good is exogenous. Household’s financial wealth is divided between domestic money (which is not held abroad) and internationally traded bonds. There are no barriers to trade, so that purchasing power parity (PPP) holds at all times, that is PS = P, where S is the nominal exchange rate (defined as units of domestic currency per unit of foreign currency), P is the domestic price level and P is the foreign price level. There is perfect capital mobility and domestic and foreign non-monetary assets are perfect substitutes; thus, uncovered interest parity (UIP) is always verified,\n\nwhere i and i are the domestic and foreign (constant) nominal interest rate, respectively, and\n\nis the rate of exchange depreciation. In the absence of foreign inflation the external nominal interest rate is equal to the real rate.\n\nThe demand side of the economy is described by an extended version of the Yaari–Blanchard perpetual youth model with money in the utility function.5 There is no bequest motive, and financialwealth for newly born agents is assumed to be zero. The birth and death rates are the same, so that population is constant. Let δ denote the instantaneous constant probability of death and β the subjective discount rate. For convenience, the size of each generation at birth is normalized to δ, hence total population is equal to unity.\n\nEach individual of the generation born at time s at each time period ts faces the following maximization problem:\n\nsubject to the individual consumer’s flow budget constraint\n\nand to the transversality condition\n\nwhere\n\n6\n\nis the external inflation rate, while cs, ys, ms, ws and τs denote consumption, endowment, nominal money balances, total financial wealth and lump-sum taxes, respectively. Notice that the effective discount rate of consumers is given by β + δ, where\n\n7 Each individual is assumed to receive for every period of her life an actuarial fair premium equal to a fraction δ of her financial wealth from a life insurance company operating in a perfectly competitive market. At the time of her death the remaining individual’s net wealth goes to the insurance company. For simplicity, both the endowment and the amount of lump-sum taxes are age-independent; hence, individuals of all generations have the same human wealth. In addition, the endowment is assumed to be constant over time.\n\nThe representative consumer of generation s chooses a sequence for consumption and money balances in order to maximize equation (1) subject to equations (2) and (3) for an initial level of wealth. Solving the dynamic optimization problem and aggregating the results across cohorts yield the following expressions for the time path of aggregate consumption, the portfolio balance condition, the aggregate flow budget constraint of the households and the transversality condition, respectively:\n\nwhere, η; ≡ (1 − ξ )/ξ . The upper case letter denotes the population aggregate of the generic individual economic variable. For any generic economic variable at individual level, say x, the corresponding population aggregateX can be obtained as\n\nLetting B denote traded bonds denominated in foreign currency, total financial wealth of the private sector can be expressed as:\n\nwhere we have used the PPP condition.\n\nThe public sector is viewed as a composite entity consisting of a government and a central bank. Let D denote the net stock of government debt in terms of foreign currency given by the stock of foreign-currency denominated government bonds (DG) net of official foreign reserves (R), that is D = DGR. Under the assumption that government bonds and foreign reserves yield the same interest rate, the public sector flow budget constraint can be expressed as:\n\nwhere G is public spending, μ is the growth rate of nominal money and MMH + SR with MH being the domestic component of the money supply (domestic credit) and SR is the stock of official reserves valued in home currency. Henceforth, without loss of generality we assume that Gt = 0.\n\nSubtracting equation (9) from equation (6) yields the current account balance:\n\nwhere F = BD is the net stock of the external assets of the economy. Since D = DG − Rthe net stock of external assets can also be expressed as F = B + RDG.\n\n### 2.1 Fiscal and Monetary Regimes\n\nIn order to close the modelwe need to specify the fiscal and the monetary regimes. The government is assumed to adopt a tax rule of the form:\n\nwhere Z is a transfer and\n\nTaxes are an increasing function of net government debt adjusted for seigniorage.8 The parameter restriction on α rules out any explosive paths for the public debt.\n\nThe monetary regime is described by a fixed exchange-rate system. The central bank pegs on each date t the exchange rate at the constant level\n\nstanding ready to accommodate any change in money demand in order to keep the relative price of the currency fixed; that is, to accommodate any change in money demand by selling or buying foreign currency bonds. Thus, money supply is endogenously determined according to equation (5) for a given\n\nUnder a permanently fixed exchange rate,\n\nand the domestic price level is\n\nThe foreign price P is assumed constant and normalized to one. It follows that\n\nthat\n\n### 2.2 Macroeconomic Equilibrium\n\nBy combining equations (4) and (5) with equations (8)–(11), under the fixed exchange-rate regime, the macroeconomic equilibrium of the model is described by the following set of equations:\n\ngiven the initial conditions on net public debt, on official foreign reserves, on the stock of foreign assets and on nominal money balances: D0 = 0, R0, F0 and M0. The last term of equation (12) results from the redistribution of financial wealth across generations. If the probability of death δ were equal to zero, the economy would collapse into the standard infinitely-lived representative agent model.9\n\nThe dynamic system described by equations (12)–(14) consists of one jump variable (C) and two predetermined or sluggish variables (D and F). The system must have one positive and two negative roots in order to generate a unique stable saddle-point equilibrium path. In the Appendix it is shown that this condition is always satisfied.\n\n5The approach of entering money in the utility function to allowfor money holding behavior within a Yaari–Blanchard framework, is common to a number of papers including Spaventa (1987), Marini and van der Ploeg (1988), van der Ploeg (1991), and Kawai and Maccini (1990, 1995). Similar results could also be obtained by use of cash-in-advance models (see Feenstra, 1986). For aYaari–Blanchard model with cash-in-advance, see Petrucci (2003). 6Following Blanchard (1985), this condition ensures that consumers are relatively patient, in order to ensure that the steady state-level of aggregate financial wealth is positive. 7This assumption ensures that savings are decreasing in wealth and that a steady-state value of aggregate consumption exists. See Blanchard (1985) for details. 8Similar fiscal rules are frequently adopted in the literature. See Benhabib et al. (2001) among others. 9However, for δ = 0 model stability requires that β = i∗.\n\n### 3. Fiscal Deficits and Currency Crises\n\nIn this section, we examine the dynamic effects of fiscal policy on the macrovariables of the model to derive the links between future budget deficits and currency crises in a pegged exchange-rate economy. The policy is centered on an unanticipated lump-sum tax cut; that is, a once and for all increase in Z.10 There is a fiscal deficit at time t = 0, generated by the tax cut, followed by future surpluses as debt accumulates, so as always to satisfy the intertemporal government budget constraint without recourse to seigniorage revenues. For the sake of simplicity it is assumed that up to time zero the economy has been in steady-state.\n\nAccording to Salant–Henderson’s criterion, the peg will be abandoned when the shadow exchange rate, i.e. the exchange that would prevail in the economy in a flexible exchange rate regime, is equal to the prevailing fixed rate. The hypothesis of perfect foresight implies that traders will exchange domestic currency for foreign currency before reserves run out in order to avoid capital losses. The flexible exchange-rate regime that follows the fixed-rate regime’s collapse is assumed to be permanent. For the sake of simplicity, we assume that following the collapse the monetary authorities will adopt a monetary targeting regime characterized by a zero-growth rate of the money, that is\n\nSince the analysis is based on the assumption of perfect foresight, the transitional dynamics of the economy depend on the expectations of the long-run steady-state relationships.\n\nThe steady state equilibrium is described by the set of equations (12)–(14) when\n\nand the portfolio balance condition (15). Let\n\nand\n\ndenote the long-run levels of consumption, public debt, net foreign assets and real money balances, respectively. The Appendix shows that the long-run effects of a tax cut are described by the following set of derivatives:\n\nFrom the above relationships we can see that a tax cut implies that in the new steady-state equilibrium consumption, real money balances and foreign assets are below their original levels, while the government debt is higher.11 Also, notice that the tax cut would imply a jump in consumption (through the wealth effect) and hence an increase in real money balances at t = 0. Given the fixed nominal exchange rate\n\nthe increase in real money balances is accomplished by a rise in the nominal money supply. This occurs as households accommodate any changes in money demand with portfolio shifts from foreign bonds to money (i.e., by selling foreign bonds to the central bank for domestic money). Hence, foreigncurrency assets at the central bank rise on impact, whereas the net stock of the economy external assets goes unchanged as the adjustment in B and R net out in the aggregate. Thereafter, following the contraction in consumption and real money demand along the transitional path, a reverse portfolio shift occurs, as households force the central bank to sell off their reserves for domestic money. Therefore, the model implies that the rise in the budget deficit at t = 0 generates a continuous depletion of foreign assets (or current account deficits) along the transition to the newsteady-state equilibrium, thus making the central bank open to speculative attacks.12 As shown below, this may indeed happen if the shadow exchange rate crosses the pegged rate at some point in time along the economy’s adjustment path. At the time of the attack there will be a jump increase in net government debt and a decline in both net foreign asset and money supply.13\n\n### 3.1 Solution Procedure and the Time of the Speculative Attack\n\nIn order to analyze the adjustment of the economy to the initial tax cut, when the public anticipate the collapse of the exchange-rate regime at some point in the future,we need to proceed backward in time. In particular,we first solve the model under the floating exchange-rate regime and find the exact time of the attack, say t. We can then use the results to solve the model under the fixed exchange-rate regime, that is, for 0 < t < t. Notice that perfect foresight requires that all jump variables be continuous at t = t.\n\nConsider first the model under the floating regime, which applies for t > t. The relevant equations of the system are equations (13)–(14) together with:\n\nwhere Φ = M/S. Since under the floating regime it is only the exchange rate that responds to variation in money demand, we have that all the observed changes in the level of real money balances are due to changes in the exchange rate, that is\n\nFrom equation (5), equation (18) immediately follows. Notice that the system of equations describing the economy under the peg (12)–(14) and under the float (13), (14), (17) and (18) have the same steady-state solution, that is\n\nand\n\nAs shown in the Appendix, the system of differential equations (13), (14), (17) and (18) is saddle path stable. The solution can be easily obtained by using the initial conditions on the predetermined variables, D, F andM, under a zero level of official reserves. In this way, it is possible to compute the time path for the floating exchange rate before and after the speculative attack. According to the Salant–Henderson criterion, the currency crisis will occur at the point where the shadow exchange rate (i.e., the exchange rate that would prevail in the economy after the fiscal expansion under a flexible regime if the official reserves had fallen to zero) is equal to the fixed rate, that is\n\nThe time path of the shadow floating rate is given by (see the Appendix):\n\nwhere\n\nand\n\nUsing the above result and imposing the condition\n\none can compute the time of the speculative attack. Clearly, the time of the attack also depends on the size of δ. The larger δ, the sooner the crisis will occur. For δ = 0 instead the initial fiscal expansion will not give rise to any change in consumption, in money demand and in the current account, so that equation (19) will become an identity.\n\nAfter the attack the current exchange rate will depreciate, steadily converging toward a steady state value whose level crucially depends on the initial tax cut:14\n\nwhere Δ ≡ (αi)(β + δi)(i + δ)i > 0. Of course the attack will take place only if the initial tax cut is sufficiently large to bring about a depreciation of the shadow exchange rate such that at a certain point\n\nBy contrast, the crisis will not take place if the shadow exchange rate converges towards a level below the peg, that is if\n\nIt can be easily shown that the currency crisis will occur only if the initial tax reduction Z satisfies the following condition:\n\nThe lower the probability of death δ, the larger the initial tax cut able to trigger a crisis.15 Similarly, we can compute the critical level of the initial stock reserves. For a given Z, speculators will attack the domestic currency only if the level of the initial stock of reserves is such that:\n\nIt can be shown that the critical threshold level of initial reserves is increasing in the probability of death δ.16 Proceeding backward in time, it is now possible to describe the time path for the economy under the fixed regime immediately after the fiscal expansion by solving the system of differential equations (12)–(14), given the initial conditions on the predetermined variables, and by imposing the condition associated with the assumption of perfect foresight,which requires that consumption be continuous at the time of the attack t. The analytical solutions are described in detail in the Appendix.", null, "### 3.2 Numerical Example\n\nThe adjustment process can be better described by making use of a simple numerical example.17\n\nConsider Figures 14 and assume, for example, an unanticipated tax cut at t = 0. Figures 1 and 2 plot the current time paths for consumption and the nominal exchange rate (bold lines) and for their related ‘shadow’ levels before the attack occurring at time t. There is, on impact, an increase in consumption, while the shadow exchange rate, after the initial appreciation, starts depreciating steadily.", null, "", null, "Current generations profit from the lump-sum tax cut, since they share the burden of future increases in taxation with yet unborn individuals.\n\nThe dynamics of net foreign assets is depicted in Figure 3. Following the tax cut, the economy starts reducing its holding of foreign assets to finance the higher consumption level along the transitional path. Agents accommodate additional money needed for consumption by selling initially Bt forMt at the central bank’s window. Next, as consumption and real money holdings decline, they exchange domestic money for foreign reserves, using up foreign-currency assets at the central bank. At time t there is an abrupt decline in net foreign assets as a consequence of the speculative attack and the depletion of remaining official reserves. The peg is abandoned and the economy shifts to a flexible exchange-rate regime, where the money supply is under the control of the monetary authorities adopting a simple targeting rule, as already mentioned, foreign assets decline towards the new long-run equilibrium, and the nominal exchange rate depreciates until the current account is brought back in equilibrium.", null, "", null, "Figure 4 plots the time path for net government debt, which increases gradually and at the time of the attack jumps upward because of the sudden exhaustion of official reserves, and then continues increasing converging to a new long-run equilibrium above its starting level.\n\nUnder the baseline calibration, Figures 58 show how the time of the attack t is affected by the initial level of reserves R0, the amount of the tax cut Z, the responsiveness of taxes to public debt α and the liquidity preference parameter η;, respectively.The time of the attack depends positively on the level of official foreign reserves and on the responsiveness of taxes to public debt α, but negatively on the magnitude of the fiscal expansion and on the liquidity preference parameter.", null, "", null, "", null, "Notice that in this model a crisis may occur even when the fiscal budget is showing a surplus.18 This is because, along the transitional path, a sequence of fiscal surpluses will replace the initial sequence of deficits in order to satisfy the government intertemporal budget constraint.\n\n10The effects of government deficit in optimizing models can be found, for example, in Frenkel and Razin (1987), Obstfeld (1989), Turnovsky and Sen (1991). 11It can be easily shown that if δ = 0, the long-run effects of the fiscal expansion will simply be: That is because when the probability of death is equal to zero, the Ricardian equivalence is restored and the time profile of lump-sum taxes does not affect consumption. In such circumstances a tax cut will not give rise to any current account imbalances and to any change in demand for real money balances, that is why the currency crisis will not take place. 12Strong empirical support for a positive relationship between the current account deficit and current and expected future budget deficits is found by Piersanti (2000). See Baxter (1995) for a more general discussion on this issue. 13Clearly, since movements in fundamentals and hence the speculative attack are predictable in this model, the government could avoid the sharp loss in reserves by abandoning the peg just before the attack. However, if the government did so, then speculators would incorporate this into their strategies, so introducing strategic interactions and uncertainty into the time of collapse.We do not consider these issues here, which would terribly complicate the model and make it intractable (as there is no equilibrium in pure strategies) without adding anything newto the Pastine’s (2002) results; nor we consider the optimal exit time issue as in Rebelo and Végh (2006), which nonetheless also involves strategic interactions.We simply focus on the less intricate and narrow target of describing the macroeconomic adjustment path associated with a consistent and more flexible policy rule – in contrast to those considered in the existing literature – which gives rise to the possibility, but not the certainty, of a currency crash, i.e., on the conditions for an attack. 14Recall that under under the floating regime all the observed variations in the level of real money balances are only due to changes in the exchange rate. 15Letting Z∗ denote the critical level of the tax cut above which the peg will collapse, it can be easily shown that 16Letting R*0be the critical stock of reserves below which the crisis will occur, we have that 17Figures 1–4 illustrate a numerical example based on the following parametrization: i∗ = 0.03, η; = 0.25, δ = 0.02, α = 0.04, Y = 1,G = D0 = 0, F0 = 0.5 and R0 = 0.2. The rate of time preference, β = 0.024, and the initial stock of nominal money balances, M0 = 8.46, are implied. The fiscal expansion consists of an increase in Z of 0.05 from zero. 18This result is consistent with the evidence that in most Asian countries, during the years preceding the crisis, fiscal imbalances were either in surplus or in modest deficit. See World Bank (1999).\n\n### 4. Conclusion\n\nIn this paper we have used an optimizing general equilibrium model with overlapping generations to investigate the relation between fiscal deficits and currency crises. It is shown that a rise in current and expected future budget deficits generates a depletion of foreign reserves, leading up to a currency crisis.\n\nCrises can thus occur even when policies are ‘correctly’ designed; that is, a monetary policy fully committed to maintaining the peg and the fiscal authorities respecting the intertemporal government budget constraint. The sustainability of fixed exchange-rate systems may thus require not only giving up monetary sovereignty but also imposing a more severe degree of fiscal discipline than implied by the standard solvency conditions.\n\nThe authors are very grateful to two anonymous referees for their extremely useful suggestions. The authors also wish to thank Fabio C. Bagliano, Marianne Baxter, Leonardo Becchetti, Anna Rita Bennato, Lorenzo Bini Smaghi, Giancarlo Corsetti, Andrea Costa, Alberto Dalmazzo, Bassam Fattouh, Laurence Harris, Fabrizio Mattesini, PedroM. Oviedo, Alessandro Piergallini, Cristina M. Rossi, Pasquale Scaramozzino, Salvatore Vinci and participants at the II RIEF Conference on International Economics and Finance – University of Rome ‘Tor Vergata’ for helpful comments on previous versions of this paper. The financial support of the MIUR (Grant No. 2007P8MJ7P_003) is gratefully acknowledged. The usual disclaimer applies.\n\n• []", 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, "• [Figure 1.] Exchange rate dynamics, S.", null, "• [Figure 2.] Consumption dynamics, C.", null, "• [Figure 3.] Net foreign assets dynamics, F.", null, "• [Figure 4.] Net government debt dynamics, D.", null, "• [Figure 5.] The time of the attack and R0.", null, "• [Figure 6.] The time of the attack and Z.", null, "• [Figure 7.] The time of the attack and α.", null, "• [Figure 8.] The time of the attack and η;.", null, "" ]	[ null, "http://oak.go.kr/central/images/2015/cc_img.png", null, "http://oak.go.kr//repository/journal/16825/NRF002_2011_v25n2_285_f0001.jpg", null, "http://oak.go.kr//repository/journal/16825/NRF002_2011_v25n2_285_f0002.jpg", null, "http://oak.go.kr//repository/journal/16825/NRF002_2011_v25n2_285_f0003.jpg", null, "http://oak.go.kr//repository/journal/16825/NRF002_2011_v25n2_285_f0004.jpg", null, "http://oak.go.kr//repository/journal/16825/NRF002_2011_v25n2_285_f0005.jpg", null, "http://oak.go.kr//repository/journal/16825/NRF002_2011_v25n2_285_f0006.jpg", null, "http://oak.go.kr//repository/journal/16825/NRF002_2011_v25n2_285_f0007.jpg", null, "http://oak.go.kr//repository/journal/16825/NRF002_2011_v25n2_285_f0008.jpg", null, "http://oak.go.kr/repository/journal/16825/NRF002_2011_v25n2_285_e0001.jpg", null, 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https://www.mql5.com/en/code/20927	[ "Interesting script?\nSo post a link to it -\nlet others appraise it\n\nYou liked the script? Try it in the MetaTrader 5 terminal", null, "# WEVOMO - indicator for MetaTrader 5\n\nViews:\n1494\nRating:\nPublished:\n2018.06.18 16:25\nUpdated:\n2018.10.09 11:05\n\nThe indicator displays in the price chart two moving averages: Volume Move-Adjusted Moving Average and Weight Volume Move-Adjusted Moving Average, calculated by the formula:\n\n```VOMOMA = (MOMA + VOMA)/2\nWEVOMO = (MOMA + VOMA + WMA)/3\n```\n\nwhere:\n\n```MOMA[i] = (Close[i-Period+1]Diff[i-Period+1] + Close[i-Period+2]Diff[i-Period+2] +... + Close[i]Diff[i])/Sum(Diff)\nVOMA[i] = (Close[i-Period+1]Volume[i-Period+1] + Close[i-Period+2]Volume[i-Period+2] + ... + Close[i]Volume[i])/Sum(Volume)\nWMA[i] = (Close[i-Period+1]1 + Close[i-Period+2]2 + ... + Close[i]Period)/LSum\nLSum = (Period+1)Period/2\nDiff[i] = Abs(Close[i] - Close[i-1])\n```\n\nThe indicator has three input parameters:\n\n• Period - calculation period;\n• Show Volume Move MA - display Volume Move-Adjusted MA (VOMOMA);\n• Show Weighted Volume Move MA - display Weight Volume Move-Adjusted MA (WEVOMO).", null, "Translated from Russian by MetaQuotes Software Corp.\nOriginal code: https://www.mql5.com/ru/code/20927", null, "Wiseman_HTF\n\nIndicator Wiseman with the timeframe selection option in its input parameters.", null, "RVRResistance\n\nAn indicator of the volume / bar price range ratio with a signal line and with the option of identifying the maximum/minimum price change resistance.", null, "SSD_With_Histogram\n\nA slow stochastic with a histogram.", null, "Smart_Money_Pressure_Oscillator\n\nSmart Money Pressure Oscillator" ]	[ null, "https://c.mql5.com/i/code/indicator.png", null, "https://c.mql5.com/18/68/WEVOMO.png", null, "https://c.mql5.com/i/code/indicator.png", null, "https://c.mql5.com/i/code/indicator.png", null, "https://c.mql5.com/i/code/indicator.png", null, "https://c.mql5.com/i/code/indicator.png", null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.7128863,"math_prob":0.88854045,"size":1113,"snap":"2020-10-2020-16","text_gpt3_token_len":323,"char_repetition_ratio":0.17853923,"word_repetition_ratio":0.0,"special_character_ratio":0.27583107,"punctuation_ratio":0.10837439,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9845416,"pos_list":[0,1,2,3,4,5,6,7,8,9,10,11,12],"im_url_duplicate_count":[null,null,null,1,null,null,null,null,null,null,null,null,null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2020-04-04T10:27:38Z\",\"WARC-Record-ID\":\"<urn:uuid:aaa45580-b8fa-457d-9ff8-f84c6dbf5729>\",\"Content-Length\":\"32053\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:325269c3-cf27-4260-a710-d853b7b7f9ba>\",\"WARC-Concurrent-To\":\"<urn:uuid:829b0984-a4a1-4ca6-a3c9-4825ce8d31f4>\",\"WARC-IP-Address\":\"78.140.180.100\",\"WARC-Target-URI\":\"https://www.mql5.com/en/code/20927\",\"WARC-Payload-Digest\":\"sha1:PEDXLQS6YDG2RUXQY3P2Q5NKHEQB4OKY\",\"WARC-Block-Digest\":\"sha1:LC3NGIDA2USGU62O5NKBCWIGK55HDKUP\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2020/CC-MAIN-2020-16/CC-MAIN-2020-16_segments_1585370521574.59_warc_CC-MAIN-20200404073139-20200404103139-00252.warc.gz\"}"}
https://www.geeksforgeeks.org/tcs-nqt-coding-questions-how-coding-task-evaluated-in-tcs-nqt/	[ "# TCS NQT Coding Questions & How Coding Task Evaluated in TCS NQT\n\n• Difficulty Level : Medium\n• Last Updated : 24 Jan, 2022\n\nTo know more about the TCS NQT: TCS NQT – National Qualifier Test\n\n1. How to solve the Coding Section in TCS NQT 2022?\n\nSTEP by STEP guide to solving the coding section in TCS NQT 2022.\n\nSTEP 1: Understanding the storyYou are given 30 minutes to write a program and clear all the test cases. Of this, spend the first five to six minutes on understanding the story given and figuring out what needs to be calculated. If your English is not very strong, you need to spend more time in reading and understanding the task. Don’t jump into typing your program before you understand what is going on.\n\nThese are the points that you need to understand from the task.\n\n• There are coins of several denominations.\n• Initially, there are an Even number of coins of each type.\n• One of the coins is lost. When you remove 1 from an even number, we get an Odd number. So, finally we have many denominations occurring an Even number of times but one particular denomination occurring an Odd number of times.\n• The first input we get is the total number of coins. Let’s call this N.\n• In the second line we get only N-1 values because one of the coins is missing.\n• The output is the denomination (value) of the missing coin.\n\nSTEP 2: Reading the inputs Once you know what is needed, we need to think in terms of how you write the code. At this time we don’t know how to find the answer. But we know how to take the inputs. So, the first part of the program is to read the value of N, declare an array and read N-1 values into the array. Even though the task keeps changing, reading a set of values in a loop is required in many programs. You will need to practice several programs so that you won’t waste your time in simple tasks like this in the actual exam.\n\nSTEP 3: Cracking the core logic\n\nThe next step is to figure out how to convert this data into an answer. By the time we reach this step in the exam, we should have about 20 minutes left. Let’s race against the time. Consider the example given in the question. We see that the denomination Rs.2 appears two times, Rs.1 appears two times and Rs.5 appears three times. Why is it so? We know that originally all these are present an even number of times. But, one of the coins got lost and that is the number which appears an odd number of times. In this example, we can infer that originally, there were four coins with Rs. 5 but one of them fell down. That is why we finally have three Rs.5 coins instead of four Rs.5 coins. Here is a direct way of solving this problem. Towards the end of this article, we will see a more efficient way of solving the same problem.\n\nMethod 1:\n\n• Task 1: Read one coin at a time in the loop. Take its value. Let’s call this a[j].\n• Task 2: In an inner loop, go through each coin in the list and count how many times is V occurring. For this we first need to initialize count to zero. Whenever a[j] == a[i] is true, we need to increment the counter.\n• Task 3: Once the inner loop is completed, the value of count will tell us how many times has a[i] occurred in the array.\n• Task 4: We need to check if the count is Odd. We are told that only 1 denomination will occur an odd number of times. If we find it, we can print it and exit the program.\n\nWhen we divide an Odd number with 2, we get 1 as remainder. This can be obtained using the % operator.\n\nMethod 2: Once we realize that we need to find the number which is occurring an Odd number of times, some of us can come up with an alternative method to identify it. This method is based on the EXOR operation which is a bit-wise operation performed using the ^ symbol. Here is the Truth table for XOR operations. Operands Result:\n\n```0 ^ 0 0\n0 ^ 1 1\n1 ^ 0 1\n1 ^ 1 0```\n\nFrom the above Truth table, we can conclude that N ^ N = 0, 0 ^ N = N. Let’s say we have 3 integers A, B and C. Here are a few interesting results from EXOR.\n\n```A^A=0\nA^B^A = A^A^B = B\nA^B^C^B^A^C = 0```\n• So, if we perform the EXOR operation on a series of numbers we will notice some interesting results.\n• If a particular number (say A) is occurring an Even number of times, the EXOR of all of them together is 0. i.e. A^A^A…^A =0 when the number is occurring an Even number of times.\n• If a number is occurring an Odd number of times, the EXOR of all the occurrences is same as the number itself. So, A^A^A^….^A= A when the number is occurring an Odd number of times.\n• The order in which we apply the EXOR operation doesn’t matter A^B=B^A.\n• Using these properties together, we can notice that when we take the EXOR of all the inputs, any number that occurs an even number of times will give an EXOR of 0. If\n• A is the number that occurs an Odd number of times, the overall EXOR for all its appearances will be equal to A. The overall result for all the other numbers will be zero. So, finally we get A^0 which is equal to A.\n\nOnce we know this, we can go ahead and implement this in the code. We just need to take a temporary variable to store the result. Let’s call this E and initialize it to 0. We then need to go through a loop and perform EXOR on all the given elements. The final value of EXOR is the answer we need. 2. What is TCS NQT 2022?\n\nStep 4: Validating the code\n\nThe remaining time in the exam can be spent in verifying that the code clears all the test cases. In case it fails, you can try giving your own inputs to find out when it is failing and then try to correct the algorithm. In TCS NQT 2022, you might not have a big penalty if your code is slow. So, it makes a lot of sense for you to write working code before you worry about efficiency.\n\nTCS NQT 2022(https://learning.tcsionhub.in/hub/national-qualifier-test/) is the exam conducted by TCS for recruiting freshers who are going to graduate in the year 2022. Make sure that you understand the Eligibility criteria for the test, the syllabus and test pattern.\n\n3. What is the Coding section in NQT 2022?\n\nThe coding section of the TCS NQT 2022 has one question which is usually in the form a Case study or a Story. At the end of the Caselet, they will ask us to write a program which takes the input in a particular format and produces the output as per the required format. 4. What are the other rules for the Coding section?\n\nYou can attempt the coding task in any of the 5 languages given by TCS. These are C, C++, Java, Python and Perl. You have a total of 30 minutes to solve this question. Here are the most important points to know about before you attempt the coding section of TCS NQT 2022.\n\n5. How is the coding task evaluated?\n\nThe TCS NQT 2022 is attempted by lakhs of students. The examiners are not going to read everyone’s code. Instead, they will use a computerized evaluation which will automatically assign score based ONLY ON THE OUTPUT. The main part of the coding section is “Test Cases”. Your code will be Validated against test cases. You will be given partial marks based on how many test cases are cleared.\n\n6. Do I need to make my program so efficient?\n\nWhile it is always good to write efficient programs you need to control your greed. Before you worry about efficiency, you need to ensure that your program clears at least some of the test cases. Finally, nobody reads your code – they just look at the number of test cases. Make sure that your code clears as many test cases as possible. However, if you know an efficient method, there is no reason for you to not use it. Go ahead, give it best shot. This is your playground.\n\n7. In which language should I write the code?\n\nThe advantage of choosing C, C++ and Java over scripting languages is that the compiler is very strict. The probability of any mistakes being found by the compiler is very high which means the probability of validating against the test cases is also high. On the other hand, in scripting languages like Python or Perl the probability of finding mistakes by compiler is not fruitful. Some of your coding mistakes might percolate till the time you go into the compiler stage. As we keep telling, don’t try to learn a new programming language now. Stick with a language that you already know. Build your confidence by practicing multiple tasks using it.\n\n8. Is it necessary to Validate Against Test Cases?\n\nTCS NQT 2022 has introduced the facility of validating your solution code against the actual test cases. Make sure that your code is clearing them. Keep tweaking it until you are done. However, in the last 2 minutes you need to ensure that your code is stable. Stop making any further changes. Read your code a few times to ensure that you haven’t done anything stupid in your excitement.\n\n9. Where can I get more questions like this?\n\nYou can prepare yourself for the TCS NQT with us by following this course TCS NQT Preparation Test Series.\n\nMy Personal Notes arrow_drop_up" ]	[ null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.93398494,"math_prob":0.9481603,"size":8753,"snap":"2022-27-2022-33","text_gpt3_token_len":2105,"char_repetition_ratio":0.13144359,"word_repetition_ratio":0.029218843,"special_character_ratio":0.23820405,"punctuation_ratio":0.104210526,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9888412,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2022-07-03T09:49:42Z\",\"WARC-Record-ID\":\"<urn:uuid:4aca3994-e785-48a9-b6bb-60fa054f2eea>\",\"Content-Length\":\"129920\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:68e1a8f8-f8b2-4162-8e77-06b668e2f684>\",\"WARC-Concurrent-To\":\"<urn:uuid:9c45626d-ebd0-4156-94e3-a88bd504ccdb>\",\"WARC-IP-Address\":\"23.45.180.234\",\"WARC-Target-URI\":\"https://www.geeksforgeeks.org/tcs-nqt-coding-questions-how-coding-task-evaluated-in-tcs-nqt/\",\"WARC-Payload-Digest\":\"sha1:TBIGWTK4CEY3VKCTN66Y4TKCY5AYFTMF\",\"WARC-Block-Digest\":\"sha1:BXDQMAEUKCC6W65733JACYMKAUF47TEM\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2022/CC-MAIN-2022-27/CC-MAIN-2022-27_segments_1656104215805.66_warc_CC-MAIN-20220703073750-20220703103750-00184.warc.gz\"}"}
https://zbmath.org/authors/?q=ai%3Ajiang.yaolin	[ "## Jiang, Yaolin\n\nCompute Distance To:\n Author ID: jiang.yaolin", null, "Published as: Jiang, Yao-Lin; Jiang, Yaolin; Jiang, Yao-lin; Jiang, Yao Lin; Jiang, Y.-L. more...less Further Spellings: 蒋耀林 Homepage: http://gr.xjtu.edu.cn/web/yljiang/english External Links: MGP\n Documents Indexed: 234 Publications since 1991 Co-Authors: 92 Co-Authors with 206 Joint Publications 3,593 Co-Co-Authors\nall top 5\n\n### Co-Authors\n\n 9 single-authored 17 Xu, Kangli 15 Li, Hongli 13 Teng, Zhi-dong 13 Yang, Yunbo 12 Xiao, Zhihua 11 Ding, Xiaoli 11 Wang, Xiaolong 10 Zhang, Long 9 Song, Bo 8 Chen, Cheng 8 Chen, Fang 8 Wang, Zuolei 7 Chen, Richard M. M. 7 Miao, Zhen 7 Qi, Zhen-Zhong 7 Wing, Omar 7 Xu, Zongben 7 Yang, Ping 6 Zhang, Hui 5 Chen, Haibao 5 Gao, Jing 5 Hu, Cheng 5 Kong, Xu 4 Chen, Chunyue 4 Kang, Yanmei 4 Kong, Qiongxiang 4 Lu, Yi 4 Peng, Guojun 3 Deng, Dingwen 3 Jia, Jiteng 3 Li, Yanpeng 3 Lin, Xiaolin 3 Sun, Wei 3 Wang, Yan 3 Wang, Zhaohong 3 Yang, Jun-Man 3 Yang, Zhixia 3 You, Zhaoyong 3 Yuan, Jiawei 2 Gander, Martin Jakob 2 Huang, Zu-Lan 2 Jiang, Haijun 2 Li, Jicheng 2 Li, Zhen 2 Liang, Dong 2 Liu, Wei 2 Roach, Gary Francis 2 Shi, Xuerong 2 Wang, Chen-Ye 2 Wang, Wei 2 Wang, Weigang 2 Wang, Yuying 2 Xu, Hong-Kun 1 Antillon, Armando 1 Bao, Yangjuan 1 Chen, Haodong 1 Chen, Yonghong 1 Chen, Yuxian 1 Feng, Xiaomei 1 Gao, Ning 1 Huang, Fenfen 1 Jiang, Chen 1 Kao, Yonggui 1 Kohn, Kathlén 1 Li, Baocheng 1 Li, Changpin 1 Li, Zi-Xue 1 Liang, Dongyue 1 Lin, Xiaola 1 Liu, Gui-Rong 1 Liu, Qingquan 1 Liu, Wei 1 Liu, Xianliang 1 Liu, Yaowu 1 Lu, Hongliang 1 Mei, Kenneth K. 1 Mu, Hong-liang 1 Muhammadhaji, Ahmadjan 1 Parmananda, Punit 1 Qiu, Zhiyong 1 Song, Qiu-Yan 1 Wang, Dengshan 1 Wang, Hongyong 1 Wang, Xiaoqin 1 Wang, Xiaoyun 1 Wang, Zhen 1 Wei, Guangsheng 1 Wen, Xiaoyong 1 Xie, Jianqiang 1 Xu, Jianxue 1 Xu, Jiaojiao 1 Xu, Ling 1 Xu, Wei-Wei 1 Yang, Jingyu 1 Yong, Xie 1 You, Zaoyong 1 Yu, Bo-Hao 1 Yu, Qingjian 1 Zeng, Wei 1 Zhang, Wei ...and 2 more Co-Authors\nall top 5\n\n### Serials\n\n 15 Applied Mathematics and Computation 13 Journal of the Franklin Institute 9 Numerical Algorithms 9 International Journal of Computer Mathematics 8 Journal of Computational Mathematics 8 International Journal of Systems Science. Principles and Applications of Systems and Integration 6 Computers & Mathematics with Applications 6 Journal of Computational and Applied Mathematics 5 International Journal of Control 5 Mathematical and Computer Modelling of Dynamical Systems 4 Chaos, Solitons and Fractals 4 IEEE Transactions on Circuits and Systems. I: Fundamental Theory and Applications 4 Journal on Numerical Methods and Computer Applications 4 Communications in Nonlinear Science and Numerical Simulation 4 Journal of Applied Mathematics and Computing 4 International Journal of Biomathematics 4 East Asian Journal on Applied Mathematics 3 International Journal of Modern Physics B 3 SIAM Journal on Numerical Analysis 3 Applied Numerical Mathematics 3 IMA Journal of Mathematical Control and Information 3 Applied Mathematical Modelling 3 Nonlinear Dynamics 3 Chinese Journal of Engineering Mathematics 2 Physics Letters. A 2 IEEE Transactions on Automatic Control 2 Mathematics and Computers in Simulation 2 Numerical Mathematics 2 Mathematica Numerica Sinica 2 Systems & Control Letters 2 Applied Mathematics Letters 2 SIAM Journal on Matrix Analysis and Applications 2 Mathematica Applicata 2 Journal of Scientific Computing 2 SIAM Journal on Scientific Computing 2 Journal of Difference Equations and Applications 2 Fractional Calculus & Applied Analysis 2 Discrete Dynamics in Nature and Society 2 Nonlinear Analysis. Real World Applications 2 Nonlinear Analysis. Modelling and Control 2 Computational Methods in Applied Mathematics 2 Scientia Sinica. Mathematica 2 Asian Journal of Control 1 Bulletin of the Australian Mathematical Society 1 Discrete Mathematics 1 Indian Journal of Pure & Applied Mathematics 1 Journal of Mathematical Physics 1 Linear and Multilinear Algebra 1 Mathematical Methods in the Applied Sciences 1 Mathematics of Computation 1 Acta Mathematica Sinica 1 Journal of the Korean Mathematical Society 1 Le Matematiche 1 Nonlinear Analysis. Theory, Methods & Applications. Series A: Theory and Methods 1 Numerical Functional Analysis and Optimization 1 Proceedings of the American Mathematical Society 1 Proceedings of the Edinburgh Mathematical Society. Series II 1 Journal of Xi’an Jiaotong University 1 Acta Mathematicae Applicatae Sinica 1 Journal of Mathematical Research & Exposition 1 Applied Mathematics and Mechanics. (English Edition) 1 Chinese Annals of Mathematics. Series A 1 Journal of Engineering Mathematics (Xi’an) 1 Acta Mathematicae Applicatae Sinica. English Series 1 Journal of Biomathematics 1 Numerical Methods for Partial Differential Equations 1 Japan Journal of Industrial and Applied Mathematics 1 Applications of Mathematics 1 Neurocomputing 1 International Journal of Bifurcation and Chaos in Applied Sciences and Engineering 1 Journal of Nonlinear Science 1 Numerical Linear Algebra with Applications 1 Pure and Applied Mathematics 1 Engineering Analysis with Boundary Elements 1 Communications on Applied Nonlinear Analysis 1 Journal of Mathematical Chemistry 1 Differential Equations and Dynamical Systems 1 Taiwanese Journal of Mathematics 1 Journal of Combinatorial Optimization 1 Chaos 1 European Journal of Mechanics. A. Solids 1 International Journal of Nonlinear Sciences and Numerical Simulation 1 Journal of Nonlinear Mathematical Physics 1 Mathematical Modelling and Analysis 1 Journal of Applied Mathematics 1 Acta Mathematica Scientia. Series A. (Chinese Edition) 1 Acta Mathematica Scientia. Series B. (English Edition) 1 Waves in Random and Complex Media 1 Frontiers of Mathematics in China 1 Applied Mathematical Sciences (Ruse) 1 Journal of Physics A: Mathematical and Theoretical 1 Communications in Theoretical Physics 1 IET Control Theory & Applications 1 Advances in Applied Mathematics and Mechanics 1 Communication on Applied Mathematics and Computation 1 Analysis and Mathematical Physics 1 Numerical Algebra, Control and Optimization 1 Journal of Theoretical Biology 1 Operations Research Transactions 1 Nonlinear Analysis. Theory, Methods & Applications ...and 1 more Serials\nall top 5\n\n### Fields\n\n 104 Numerical analysis (65-XX) 67 Ordinary differential equations (34-XX) 65 Systems theory; control (93-XX) 38 Partial differential equations (35-XX) 22 Biology and other natural sciences (92-XX) 15 Dynamical systems and ergodic theory (37-XX) 13 Linear and multilinear algebra; matrix theory (15-XX) 9 Operator theory (47-XX) 8 Fluid mechanics (76-XX) 7 Special functions (33-XX) 7 Integral equations (45-XX) 6 Information and communication theory, circuits (94-XX) 5 Probability theory and stochastic processes (60-XX) 5 Computer science (68-XX) 4 Real functions (26-XX) 4 Mechanics of deformable solids (74-XX) 4 Optics, electromagnetic theory (78-XX) 3 Algebraic geometry (14-XX) 3 Statistics (62-XX) 3 Statistical mechanics, structure of matter (82-XX) 2 Combinatorics (05-XX) 2 Difference and functional equations (39-XX) 2 Approximations and expansions (41-XX) 2 Harmonic analysis on Euclidean spaces (42-XX) 2 Calculus of variations and optimal control; optimization (49-XX) 2 Global analysis, analysis on manifolds (58-XX) 2 Mechanics of particles and systems (70-XX) 2 Classical thermodynamics, heat transfer (80-XX) 2 Operations research, mathematical programming (90-XX) 2 Game theory, economics, finance, and other social and behavioral sciences (91-XX) 1 General and overarching topics; collections (00-XX) 1 Functions of a complex variable (30-XX) 1 Integral transforms, operational calculus (44-XX) 1 Differential geometry (53-XX) 1 Quantum theory (81-XX)\n\n### Citations contained in zbMATH Open\n\n148 Publications have been cited 932 times in 607 Documents Cited by Year\nDynamical analysis of a fractional-order predator-prey model incorporating a prey refuge. Zbl 1377.34062\nLi, Hong-Li; Zhang, Long; Hu, Cheng; Jiang, Yao-Lin; Teng, Zhidong\n2017\nA generalization of the inexact parameterized Uzawa methods for saddle point problems. Zbl 1159.65034\nChen, Fang; Jiang, Yao-Lin\n2008\nTime domain model order reduction of general orthogonal polynomials for linear input-output systems. Zbl 1369.93117\nJiang, Yao-Lin; Chen, Hai-Bao\n2012\nGlobal Mittag-Leffler stability of coupled system of fractional-order differential equations on network. Zbl 1410.34025\nLi, Hong-Li; Jiang, Yao-Lin; Wang, Zuolei; Zhang, Long; Teng, Zhidong\n2015\nOn time-domain simulation of lossless transmission lines with nonlinear terminations. Zbl 1116.78029\nJiang, Yao-Lin\n2004\nBifurcations of a Holling-type II predator-prey system with constant rate harvesting. Zbl 1175.34062\nPeng, Guojun; Jiang, Yaolin; Li, Changpin\n2009\nAnalytical solutions for the multi-term time-space fractional advection-diffusion equations with mixed boundary conditions. Zbl 1260.35241\nDing, Xiao-Li; Jiang, Yao-Lin\n2013\nPinning adaptive and impulsive synchronization of fractional-order complex dynamical networks. Zbl 1372.34086\nLi, Hong-Li; Hu, Cheng; Jiang, Yao-Lin; Wang, Zuolei; Teng, Zhidong\n2016\nA general approach to waveform relaxation solutions of nonlinear differential-algebraic equations: the continuous-time and discrete-time cases. Zbl 1374.94906\nJiang, Yao-Lin\n2004\nGeneralized projective synchronization of a class of fractional-order chaotic systems via a scalar transmitted signal. Zbl 1220.34060\nPeng, Guojun; Jiang, Yaolin\n2008\nWaveform relaxation methods for fractional differential equations with the Caputo derivatives. Zbl 1259.65113\nJiang, Yao-Lin; Ding, Xiao-Li\n2013\nModel reduction of bilinear systems based on Laguerre series expansion. Zbl 1273.93034\nWang, Xiaolong; Jiang, Yaolin\n2012\nAnalysis of two parareal algorithms for time-periodic problems. Zbl 1283.65077\nGander, Martin J.; Jiang, Yao-Lin; Song, Bo; Zhang, Hui\n2013\nSynchronization of fractional-order complex dynamical networks via periodically intermittent pinning control. Zbl 1375.93060\nLi, Hong-Li; Hu, Cheng; Jiang, Haijun; Teng, Zhidong; Jiang, Yao-Lin\n2017\nA note on the spectra and pseudospectra of waveform relaxation operators for linear differential-algebraic equations. Zbl 0980.65081\nJiang, Yao-Lin; Wing, Omar\n2000\nWaveform relaxation for reaction-diffusion equations. Zbl 1230.65111\nLiu, Jun; Jiang, Yao-Lin\n2011\nImproving convergence performance of relaxation-based transient analysis by matrix splitting in circuit simulation. Zbl 0999.94574\nJiang, Yao-Lin; Chen, Richard M. M.; Wing, Omar\n2001\nBifurcation of a delayed gause predator-prey model with Michaelis-Menten type harvesting. Zbl 1394.92109\nLiu, Wei; Jiang, Yaolin\n2018\nGlobal stability of an SI epidemic model with feedback controls in a patchy environment. Zbl 1426.92077\nLi, Hong-Li; Zhang, Long; Teng, Zhidong; Jiang, Yao-Lin; Muhammadhaji, Ahmadjan\n2018\nA note on convergence conditions of waveform relaxation algorithms for nonlinear differential-algebraic equations. Zbl 0984.65079\nJiang, Yao-Lin; Wing, Omar\n2001\nAnti-synchronization and intermittent anti-synchronization of two identical hyperchaotic Chua systems via impulsive control. Zbl 1345.34095\nLi, Hong-Li; Jiang, Yao-Lin; Wang, Zuo-Lei\n2015\nTrigonometric Hermite wavelet approximation for the integral equations of second kind with weakly singular kernel. Zbl 1350.65144\nGao, Jing; Jiang, Yao-Lin\n2008\nMonotone waveform relaxation for systems of nonlinear differential-algebraic equations. Zbl 1049.65077\nJiang, Yao-Lin; Wing, Omar\n2000\nComputing periodic solutions of linear differential-algebraic equations by waveform relaxation. Zbl 1115.37068\nJiang, Yao-Lin; Chen, Richard M. M.\n2005\nSimplest equation method for some time-fractional partial differential equations with conformable derivative. Zbl 1415.35275\nChen, Cheng; Jiang, Yao-Lin\n2018\nAnalysis of a new parareal algorithm based on waveform relaxation method for time-periodic problems. Zbl 1308.65117\nSong, Bo; Jiang, Yao-Lin\n2014\nA parareal waveform relaxation algorithm for semi-linear parabolic partial differential equations. Zbl 1248.65090\nLiu, Jun; Jiang, Yao-Lin\n2012\n$$H_2$$ optimal reduced models of general MIMO LTI systems via the cross Gramian on the Stiefel manifold. Zbl 1364.93114\nJiang, Yaolin; Xu, Kangli\n2017\nWaveform relaxation method for fractional differential-algebraic equations. Zbl 1305.26017\nDing, Xiao-Li; Jiang, Yao-Lin\n2014\nA superlinear convergence estimate for the parareal Schwarz waveform relaxation algorithm. Zbl 1414.65018\nGander, Martin J.; Jiang, Yao-Lin; Song, Bo\n2019\nOn HSS and AHSS iteration methods for nonsymmetric positive definite Toeplitz systems. Zbl 1191.65027\nChen, Fang; Jiang, Yao-Lin\n2010\nExotic localized vector waves in a two-component nonlinear wave system. Zbl 1440.37068\nXu, Ling; Wang, Deng-Shan; Wen, Xiao-Yong; Jiang, Yao-Lin\n2020\nNumerical analysis and computation of a type of IMEX method for the time-dependent natural convection problem. Zbl 1336.65155\nYang, Yun-Bo; Jiang, Yao-Lin\n2016\nConvergence analysis of waveform relaxation for nonlinear differential-algebraic equations of index one. Zbl 1001.94058\nJiang, Yao-Lin; Chen, Richard M. M.; Wing, Omar\n2000\nSemilinear fractional differential equations based on a new integral operator approach. Zbl 1263.35215\nDing, Xiao-Li; Jiang, Yao-Lin\n2012\nNonnegative solutions of fractional functional differential equations. Zbl 1247.34007\nJiang, Yao-Lin; Ding, Xiao-Li\n2012\nA parareal algorithm based on waveform relaxation. Zbl 1256.65071\nLiu, Jun; Jiang, Yao-Lin\n2012\nModel order reduction of MIMO bilinear systems by multi-order Arnoldi method. Zbl 1344.93025\nXiao, Zhi-Hua; Jiang, Yao-Lin\n2016\nModel order reduction methods for coupled systems in the time domain using Laguerre polynomials. Zbl 1232.65119\nWang, Xiao-Long; Jiang, Yao-Lin\n2011\nDynamic analysis of a fractional-order single-species model with diffusion. Zbl 1416.92138\nLi, Hong-Li; Zhang, Long; Hu, Cheng; Jiang, Yao-Lin; Teng, Zhidong\n2017\nTime domain model order reduction using general orthogonal polynomials for K-power bilinear systems. Zbl 1338.93095\nQi, Zhen-Zhong; Jiang, Yao-Lin; Xiao, Zhi-Hua\n2016\nA new parareal waveform relaxation algorithm for time-periodic problems. Zbl 1308.65131\nSong, Bo; Jiang, Yao-Lin\n2015\nOn the determinant evaluation of quasi penta-diagonal matrices and quasi penta-diagonal Toeplitz matrices. Zbl 1286.65057\nJiang, Yao-Lin; Jia, Ji-Teng\n2013\nPractical computation of normal forms of the Bogdanov-Takens bifurcation. Zbl 1286.34059\nPeng, Guojun; Jiang, Yaolin\n2011\nTwo-sided projection methods for model reduction of MIMO bilinear systems. Zbl 1305.93046\nWang, Xiao-Long; Jiang, Yao-Lin\n2013\nAn effective fracture analysis method based on the virtual crack closure-integral technique implemented in CS-FEM. Zbl 1459.74168\nZeng, W.; Liu, G. R.; Jiang, C.; Dong, X. W.; Chen, H. D.; Bao, Y.; Jiang, Y.\n2016\nModel reduction of discrete-time bilinear systems by a Laguerre expansion technique. Zbl 1465.93022\nWang, Xiaolong; Jiang, Yaolin\n2016\nA further necessary and sufficient condition for strong convergence of nonlinear contraction semigroups and of iterative methods for accretive operators in Banach spaces. Zbl 0820.47074\nXu, Zong-Ben; Jiang, Yao-Lin; Roach, G. F.\n1995\nPeriodic waveform relaxation solutions of nonlinear dynamic equations. Zbl 1088.34520\nJiang, Yao-Lin\n2003\nModel-order reduction of coupled DAE systems via $$\\varepsilon$$ technique and Krylov subspace method. Zbl 1255.93030\nJiang, Yao-Lin; Chen, Chun-Yue; Chen, Hai-Bao\n2012\nWaveform relaxation methods for fractional functional differential equations. Zbl 1312.34011\nDing, Xiao-Li; Jiang, Yao-Lin\n2013\nOn the uniqueness and perturbation to the best rank-one approximation of a tensor. Zbl 1317.65111\nJiang, Yao-Lin; Kong, Xu\n2015\nWindowing waveform relaxation of initial value problems. Zbl 1113.65081\nJiang, Yaolin\n2006\nArnoldi-based model order reduction for linear systems with inhomogeneous initial conditions. Zbl 1380.93071\nSong, Qiu-Yan; Jiang, Yao-Lin; Xiao, Zhi-Hua\n2017\n$$H_{2}$$ optimal model order reduction by two-sided technique on Grassmann manifold via the cross-Gramian of bilinear systems. Zbl 1359.93086\nXu, Kang-Li; Jiang, Yao-Lin; Yang, Zhi-Xia\n2017\nLaguerre functions approximation for model reduction of second order time-delay systems. Zbl 1347.93072\nWang, Xiaolong; Jiang, Yaolin; Kong, Xu\n2016\nOn model reduction of $$K$$-power bilinear systems. Zbl 1290.93031\nWang, Xiao-Long; Jiang, Yao-Lin\n2014\nLie group analysis method for two classes of fractional partial differential equations. Zbl 1440.35340\nChen, Cheng; Jiang, Yao-Lin\n2015\nSchwarz waveform relaxation methods for parabolic equations in space-frequency domain. Zbl 1142.65411\nJiang, Yao-Lin; Zhang, Hui\n2008\nSymbolic algorithm for solving cyclic penta-diagonal linear systems. Zbl 1269.65027\nJia, Ji-Teng; Jiang, Yao-Lin\n2013\nMulti-order Arnoldi-based model order reduction of second-order time-delay systems. Zbl 1347.93073\nXiao, Zhi-Hua; Jiang, Yao-Lin\n2016\nComputation of universal unfolding of the double-zero bifurcation in $$Z_2$$-symmetric systems by a homological method. Zbl 1282.34047\nPeng, Guojun; Jiang, Yao Lin\n2013\nRiemannian modified Polak-Ribière-Polyak conjugate gradient order reduced model by tensor techniques. Zbl 1441.93045\nJiang, Yao-Lin; Xu, Kang-Li\n2020\nFinite-time balanced truncation for linear systems via shifted Legendre polynomials. Zbl 1425.93067\nXiao, Zhi-Hua; Jiang, Yao-Lin; Qi, Zhen-Zhong\n2019\nWaveform relaxation methods of nonlinear integral-differential-algebraic equations. Zbl 1072.65166\nJiang, Yaolin\n2005\nRunge-Kutta methods of dynamic iteration for index-2 differential-algebraic equations. Zbl 1080.65070\nSun, Wei; Jiang, Yao-Lin\n2005\nMathematical modelling on $$RLCG$$ transmission lines. Zbl 1121.93032\nJiang, Y.-L.\n2005\nDimension reduction for second-order systems by general orthogonal polynomials. Zbl 1298.93100\nXiao, Zhi-Hua; Jiang, Yao-Lin\n2014\nA general method for solving singular perturbed impulsive differential equations with two-point boundary conditions. Zbl 1090.65094\nWang, Xiao-Yun; Jiang, Yao-Lin\n2005\nModel order reduction based on general orthogonal polynomials in the time domain for coupled systems. Zbl 1291.93062\nQi, Zhen-Zhong; Jiang, Yao-Lin; Xiao, Zhi-Hua\n2014\nA trust-region method for $$H_2$$ model reduction of bilinear systems on the Stiefel manifold. Zbl 1409.93017\nYang, Ping; Jiang, Yao-Lin; Xu, Kang-Li\n2019\nA parareal approach of semi-linear parabolic equations based on general waveform relaxation. Zbl 1431.65135\nLi, Jun; Jiang, Yao-lin; Miao, Zhen\n2019\n$$H_2$$ optimal model order reduction of the discrete system on the product manifold. Zbl 1470.93034\nJiang, Yao-Lin; Wang, Wei-Gang\n2019\nAnalysis of two decoupled time-stepping finite-element methods for incompressible fluids with microstructure. Zbl 1387.65108\nYang, Yun-Bo; Jiang, Yao-Lin\n2018\nSemi-discrete Galerkin finite element method for the diffusive Peterlin viscoelastic model. Zbl 1391.76337\nJiang, Yao-Lin; Yang, Yun-Bo\n2018\nAn explicitly uncoupled VMS stabilization finite element method for the time-dependent Darcy-Brinkman equations in double-diffusive convection. Zbl 1402.65139\nYang, Yun-Bo; Jiang, Yao-Lin\n2018\nAn $$\\varepsilon$$-embedding model-order reduction approach for differential-algebraic equation systems. Zbl 1251.93042\nChen, Chun-Yue; Jiang, Yao-Lin; Chen, Hai-Bao\n2012\nExistence of periodic solutions in predator-prey with Watt-type functional response and impulsive effects. Zbl 1203.34073\nLin, Xiaolin; Jiang, Yaolin; Wang, Xiaoqin\n2010\nStructure-preserving model order reduction by general orthogonal polynomials for integral-differential systems. Zbl 1307.93093\nYuan, Jia-Wei; Jiang, Yao-Lin; Xiao, Zhi-Hua\n2015\nArnoldi-based model reduction for fractional order linear systems. Zbl 1312.93025\nJiang, Yao-Lin; Xiao, Zhi-Hua\n2015\nAn adaptive wavelet method for nonlinear differential-algebraic equations. Zbl 1162.65046\nGao, Jing; Jiang, Yao-Lin\n2007\nStability analysis for coupled systems of fractional differential equations on networks. Zbl 1370.34014\nLi, Hong-Li; Jiang, Yao-Lin; Wang, Zuolei; Feng, Xiaomei; Teng, Zhidong\n2017\nNonlinear model order reduction based on tensor Kronecker product expansion with Arnoldi process. Zbl 1347.93070\nJiang, Yao-Lin; Chen, Hai-Bao; Qi, Zhen-Zhong\n2016\nOn structured variants of modified HSS iteration methods for complex Toeplitz linear systems. Zbl 1289.65052\nChen, Fang; Jiang, Yaolin; Liu, Qingquan\n2013\nA note on the $$H^1$$-convergence of the overlapping Schwarz waveform relaxation method for the heat equation. Zbl 1295.65091\nZhang, Hui; Jiang, Yao-Lin\n2014\nConservation laws and optimal system of extended quantum Zakharov-Kuznetsov equation. Zbl 1420.35071\nJiang, Yao-Lin; Lu, Yi; Chen, Cheng\n2016\nTime domain and frequency domain model order reduction for discrete time-delay systems. Zbl 1483.93048\nWang, Zhao-Hong; Jiang, Yao-Lin; Xu, Kang-Li\n2020\nStructure preserving balanced proper orthogonal decomposition for second-order form systems via shifted Legendre polynomials. Zbl 1432.93046\nXiao, Zhi-Hua; Jiang, Yao-Lin; Qi, Zhen-Zhong\n2019\nRobust exponential stability of fractional-order coupled quaternion-valued neural networks with parametric uncertainties and impulsive effects. Zbl 07512492\nLi, Hong-Li; Kao, Yonggui; Hu, Cheng; Jiang, Haijun; Jiang, Yao-Lin\n2021\nFrequency-limited reduced models for linear and bilinear systems on the Riemannian manifold. Zbl 1471.93182\nJiang, Yao-Lin; Xu, Kang-Li\n2021\nHigh-order finite difference methods for a second order dual-phase-lagging models of microscale heat transfer. Zbl 1411.80005\nDeng, Dingwen; Jiang, Yaolin; Liang, Dong\n2017\nAn efficient hybrid reduction method for time-delay systems using Hermite expansions. Zbl 1416.93040\nWang, Xiaolong; Jiang, Yao-Lin\n2019\nA semi-analytic method for computing the long-time order parameter dynamics in mean-field coupled overdamped oscillators with colored noises. Zbl 1227.37018\nKang, Yan-Mei; Jiang, Yao-Lin\n2008\nConvergence conditions on waveform relaxation of general differential-algebraic equations. Zbl 1208.65120\nJiang, Yao-Lin\n2010\nOn product-type generalized block AOR method for augmented linear systems. Zbl 1264.65043\nChen, Fang; Gao, Ning; Jiang, Yao-Lin\n2012\nSubtracting a best rank-1 approximation from $$p\\times p\\times 2$$ ($$p\\geq 2$$) tensors. Zbl 1274.15002\nKong, Xu; Jiang, Yao-Lin\n2012\nLie group analysis and invariant solutions for nonlinear time-fractional diffusion-convection equations. Zbl 1377.34008\nChen, Cheng; Jiang, Yao-Lin\n2017\nAn approach to $$H_{2,\\omega}$$ model reduction on finite interval for bilinear systems. Zbl 1373.93074\nXu, Kang-Li; Jiang, Yao-Lin\n2017\nAnti-synchronization and intermittent anti-synchronization of two identical delay hyperchaotic Chua systems via linear control. Zbl 1358.93091\nLi, Hong-Li; Wang, Zuolei; Jiang, Yao-Lin; Zhang, Long; Teng, Zhidong\n2017\nA note on the ranks of $$2\\times 2\\times 2$$ and $$2\\times 2\\times 2\\times 2$$ tensors. Zbl 1282.15004\nKong, Xu; Jiang, Yao-Lin\n2013\nOptimal convergence and long-time conservation of exponential integration for Schrödinger equations in a normal or highly oscillatory regime. Zbl 07488706\nWang, Bin; Jiang, Yaolin\n2022\nRobust exponential stability of fractional-order coupled quaternion-valued neural networks with parametric uncertainties and impulsive effects. Zbl 07512492\nLi, Hong-Li; Kao, Yonggui; Hu, Cheng; Jiang, Haijun; Jiang, Yao-Lin\n2021\nFrequency-limited reduced models for linear and bilinear systems on the Riemannian manifold. Zbl 1471.93182\nJiang, Yao-Lin; Xu, Kang-Li\n2021\nDimension reduction for $$k$$-power bilinear systems using orthogonal polynomials and Arnoldi algorithm. Zbl 1483.93045\nQi, Zhen-Zhong; Jiang, Yao-Lin; Xiao, Zhi-Hua\n2021\nUnconditional optimal error estimates of linearized backward Euler Galerkin FEMs for nonlinear Schrödinger-Helmholtz equations. Zbl 1475.65135\nYang, Yun-Bo; Jiang, Yao-Lin\n2021\nAnalysis of two new parareal algorithms based on the Dirichlet-Neumann/Neumann-Neumann waveform relaxation method for the heat equation. Zbl 1471.65130\nSong, Bo; Jiang, Yao-Lin; Wang, Xiaolong\n2021\nUnconditional optimal error estimates of linearized, decoupled and conservative Galerkin FEMs for the Klein-Gordon-Schrödinger equation. Zbl 1476.65259\nYang, Yun-Bo; Jiang, Yao-Lin; Yu, Bo-Hao\n2021\nUnconditional optimal error estimates of linearized second-order BDF Galerkin FEMs for the Landau-Lifshitz equation. Zbl 1459.65191\nYang, Yun-Bo; Jiang, Yao-Lin\n2021\nOn the complexity of all $$( g , f )$$-factors problem. Zbl 1453.05103\nLu, Hongliang; Wang, Wei; Jiang, Yaolin\n2021\nExotic localized vector waves in a two-component nonlinear wave system. Zbl 1440.37068\nXu, Ling; Wang, Deng-Shan; Wen, Xiao-Yong; Jiang, Yao-Lin\n2020\nRiemannian modified Polak-Ribière-Polyak conjugate gradient order reduced model by tensor techniques. Zbl 1441.93045\nJiang, Yao-Lin; Xu, Kang-Li\n2020\nTime domain and frequency domain model order reduction for discrete time-delay systems. Zbl 1483.93048\nWang, Zhao-Hong; Jiang, Yao-Lin; Xu, Kang-Li\n2020\nVibration suppression of cantilevered piezoelectric laminated composite rectangular plate subjected to aerodynamic force in hygrothermal environment. Zbl 1473.74062\nLu, S. F.; Jiang, Y.; Zhang, W.; Song, X. J.\n2020\nTwo-grid stabilized FEMs based on Newton type linearization for the steady-state natural convection problem. Zbl 1488.65666\nYang, Yunbo; Jiang, Yaolin; Kong, Qiongxiang\n2020\nTime domain model reduction of time-delay systems via orthogonal polynomial expansions. Zbl 1433.93024\nWang, Xiaolong; Jiang, Yaolin\n2020\nA superlinear convergence estimate for the parareal Schwarz waveform relaxation algorithm. Zbl 1414.65018\nGander, Martin J.; Jiang, Yao-Lin; Song, Bo\n2019\nFinite-time balanced truncation for linear systems via shifted Legendre polynomials. Zbl 1425.93067\nXiao, Zhi-Hua; Jiang, Yao-Lin; Qi, Zhen-Zhong\n2019\nA trust-region method for $$H_2$$ model reduction of bilinear systems on the Stiefel manifold. Zbl 1409.93017\nYang, Ping; Jiang, Yao-Lin; Xu, Kang-Li\n2019\nA parareal approach of semi-linear parabolic equations based on general waveform relaxation. Zbl 1431.65135\nLi, Jun; Jiang, Yao-lin; Miao, Zhen\n2019\n$$H_2$$ optimal model order reduction of the discrete system on the product manifold. Zbl 1470.93034\nJiang, Yao-Lin; Wang, Wei-Gang\n2019\nStructure preserving balanced proper orthogonal decomposition for second-order form systems via shifted Legendre polynomials. Zbl 1432.93046\nXiao, Zhi-Hua; Jiang, Yao-Lin; Qi, Zhen-Zhong\n2019\nAn efficient hybrid reduction method for time-delay systems using Hermite expansions. Zbl 1416.93040\nWang, Xiaolong; Jiang, Yao-Lin\n2019\nAn unconstrained $$H_2$$ model order reduction optimisation algorithm based on the Stiefel manifold for bilinear systems. Zbl 1416.93041\nXu, Kang-Li; Jiang, Yao-Lin\n2019\nModeling and dynamics of an ecological-economic model. Zbl 1416.92139\nLiu, Wei; Jiang, Yaolin\n2019\nLie group analysis and dynamical behavior for classical Boussinesq-Burgers system. Zbl 1409.35021\nJiang, Yao-Lin; Chen, Cheng\n2019\nLie symmetry analysis and dynamic behaviors for nonlinear generalized Zakharov system. Zbl 1418.35086\nChen, Cheng; Jiang, Yao-Lin\n2019\nModel order reduction for discrete-time linear systems with the discrete-time polynomials. Zbl 1425.93063\nJiang, Yao-Lin; Yang, Jun-Man; Xu, Kang-Li\n2019\nThe Arrow-Hurwicz iterative finite element method for the stationary magnetohydrodynamics flow. Zbl 1428.76108\nYang, Yun-Bo; Jiang, Yao-Lin; Kong, Qiong-Xiang\n2019\nBifurcation of a delayed gause predator-prey model with Michaelis-Menten type harvesting. Zbl 1394.92109\nLiu, Wei; Jiang, Yaolin\n2018\nGlobal stability of an SI epidemic model with feedback controls in a patchy environment. Zbl 1426.92077\nLi, Hong-Li; Zhang, Long; Teng, Zhidong; Jiang, Yao-Lin; Muhammadhaji, Ahmadjan\n2018\nSimplest equation method for some time-fractional partial differential equations with conformable derivative. Zbl 1415.35275\nChen, Cheng; Jiang, Yao-Lin\n2018\nAnalysis of two decoupled time-stepping finite-element methods for incompressible fluids with microstructure. Zbl 1387.65108\nYang, Yun-Bo; Jiang, Yao-Lin\n2018\nSemi-discrete Galerkin finite element method for the diffusive Peterlin viscoelastic model. Zbl 1391.76337\nJiang, Yao-Lin; Yang, Yun-Bo\n2018\nAn explicitly uncoupled VMS stabilization finite element method for the time-dependent Darcy-Brinkman equations in double-diffusive convection. Zbl 1402.65139\nYang, Yun-Bo; Jiang, Yao-Lin\n2018\nWaveform relaxation of partial differential equations. Zbl 1433.65159\nJiang, Yao-Lin; Miao, Zhen\n2018\nA parameterised model order reduction method for parametric systems based on Laguerre polynomials. Zbl 1397.93041\nYuan, Jia-Wei; Jiang, Yao-Lin\n2018\nCoupling parareal and Dirichlet-Neumann/Neumann-Neumann waveform relaxation methods for the heat equation. Zbl 1450.65111\nJiang, Yao-Lin; Song, Bo\n2018\n$$H_2$$ optimal model order reduction on the Stiefel manifold for the MIMO discrete system by the cross Gramian. Zbl 1485.93105\nWang, Wei-Gang; Jiang, Yao-Lin\n2018\nAnalysis of some projection methods for the incompressible fluids with microstructure. Zbl 1395.65129\nJiang, Yao-Lin; Yang, Yun-Bo\n2018\nInterpolatory model order reduction method for second order systems. Zbl 1391.93052\nQiu, Zhi-Yong; Jiang, Yao-Lin; Yuan, Jia-Wei\n2018\nDynamical analysis of a fractional-order predator-prey model incorporating a prey refuge. Zbl 1377.34062\nLi, Hong-Li; Zhang, Long; Hu, Cheng; Jiang, Yao-Lin; Teng, Zhidong\n2017\nSynchronization of fractional-order complex dynamical networks via periodically intermittent pinning control. Zbl 1375.93060\nLi, Hong-Li; Hu, Cheng; Jiang, Haijun; Teng, Zhidong; Jiang, Yao-Lin\n2017\n$$H_2$$ optimal reduced models of general MIMO LTI systems via the cross Gramian on the Stiefel manifold. Zbl 1364.93114\nJiang, Yaolin; Xu, Kangli\n2017\nDynamic analysis of a fractional-order single-species model with diffusion. Zbl 1416.92138\nLi, Hong-Li; Zhang, Long; Hu, Cheng; Jiang, Yao-Lin; Teng, Zhidong\n2017\nArnoldi-based model order reduction for linear systems with inhomogeneous initial conditions. Zbl 1380.93071\nSong, Qiu-Yan; Jiang, Yao-Lin; Xiao, Zhi-Hua\n2017\n$$H_{2}$$ optimal model order reduction by two-sided technique on Grassmann manifold via the cross-Gramian of bilinear systems. Zbl 1359.93086\nXu, Kang-Li; Jiang, Yao-Lin; Yang, Zhi-Xia\n2017\nStability analysis for coupled systems of fractional differential equations on networks. Zbl 1370.34014\nLi, Hong-Li; Jiang, Yao-Lin; Wang, Zuolei; Feng, Xiaomei; Teng, Zhidong\n2017\nHigh-order finite difference methods for a second order dual-phase-lagging models of microscale heat transfer. Zbl 1411.80005\nDeng, Dingwen; Jiang, Yaolin; Liang, Dong\n2017\nLie group analysis and invariant solutions for nonlinear time-fractional diffusion-convection equations. Zbl 1377.34008\nChen, Cheng; Jiang, Yao-Lin\n2017\nAn approach to $$H_{2,\\omega}$$ model reduction on finite interval for bilinear systems. Zbl 1373.93074\nXu, Kang-Li; Jiang, Yao-Lin\n2017\nAnti-synchronization and intermittent anti-synchronization of two identical delay hyperchaotic Chua systems via linear control. Zbl 1358.93091\nLi, Hong-Li; Wang, Zuolei; Jiang, Yao-Lin; Zhang, Long; Teng, Zhidong\n2017\nSymplectic waveform relaxation methods for Hamiltonian systems. Zbl 1410.65479\nLu, Yi; Jiang, Yao-Lin; Song, Bo\n2017\nA delayed predator-prey system with impulsive diffusion between two patches. Zbl 1366.92108\nLi, Hong-Li; Zhang, Long; Teng, Zhi-Dong; Jiang, Yao-Lin\n2017\nNonlinear dynamical behaviour in a predator-prey model with harvesting. Zbl 1376.92046\nLiu, Wei; Jiang, Yaolin\n2017\nModeling and analysis of a predator-prey system with time delay. Zbl 1361.92058\nLiu, Wei; Jiang, Yaolin\n2017\nA periodic single species model with intermittent unilateral diffusion in two patches. Zbl 1357.92063\nLi, Hong-Li; Zhang, Long; Teng, Zhidong; Jiang, Yao-Lin\n2017\n$$\\mathcal{H}_{2}$$ model order reduction for bilinear systems based on the cross gramian. Zbl 1397.93040\nYang, Ping; Xu, Kang-Li; Jiang, Yao-Lin\n2017\n$$H_2$$ optimal model reduction of coupled systems on the Grassmann manifold. Zbl 1488.65208\nYang, Ping; Jiang, Yao-Lin\n2017\nPinning adaptive and impulsive synchronization of fractional-order complex dynamical networks. Zbl 1372.34086\nLi, Hong-Li; Hu, Cheng; Jiang, Yao-Lin; Wang, Zuolei; Teng, Zhidong\n2016\nNumerical analysis and computation of a type of IMEX method for the time-dependent natural convection problem. Zbl 1336.65155\nYang, Yun-Bo; Jiang, Yao-Lin\n2016\nModel order reduction of MIMO bilinear systems by multi-order Arnoldi method. Zbl 1344.93025\nXiao, Zhi-Hua; Jiang, Yao-Lin\n2016\nTime domain model order reduction using general orthogonal polynomials for K-power bilinear systems. Zbl 1338.93095\nQi, Zhen-Zhong; Jiang, Yao-Lin; Xiao, Zhi-Hua\n2016\nAn effective fracture analysis method based on the virtual crack closure-integral technique implemented in CS-FEM. Zbl 1459.74168\nZeng, W.; Liu, G. R.; Jiang, C.; Dong, X. W.; Chen, H. D.; Bao, Y.; Jiang, Y.\n2016\nModel reduction of discrete-time bilinear systems by a Laguerre expansion technique. Zbl 1465.93022\nWang, Xiaolong; Jiang, Yaolin\n2016\nLaguerre functions approximation for model reduction of second order time-delay systems. Zbl 1347.93072\nWang, Xiaolong; Jiang, Yaolin; Kong, Xu\n2016\nMulti-order Arnoldi-based model order reduction of second-order time-delay systems. Zbl 1347.93073\nXiao, Zhi-Hua; Jiang, Yao-Lin\n2016\nNonlinear model order reduction based on tensor Kronecker product expansion with Arnoldi process. Zbl 1347.93070\nJiang, Yao-Lin; Chen, Hai-Bao; Qi, Zhen-Zhong\n2016\nConservation laws and optimal system of extended quantum Zakharov-Kuznetsov equation. Zbl 1420.35071\nJiang, Yao-Lin; Lu, Yi; Chen, Cheng\n2016\nDynamic properties of a delayed predator-prey system with Ivlev-type functional response. Zbl 1354.37095\nLiu, Wei; Jiang, Yaolin; Chen, Yuxian\n2016\nSymplectic schemes for telegraph equations. Zbl 1363.65216\nLu, Yi; Jiang, Yaolin\n2016\nApproximation algorithms for minimum weight partial connected set cover problem. Zbl 1360.90224\nLiang, Dongyue; Zhang, Zhao; Liu, Xianliang; Wang, Wei; Jiang, Yaolin\n2016\nDynamics of a modified predator-prey system to allow for a functional response and time delay. Zbl 1403.92256\nLiu, Wei; Jiang, Yaolin\n2016\nGlobal Mittag-Leffler stability of coupled system of fractional-order differential equations on network. Zbl 1410.34025\nLi, Hong-Li; Jiang, Yao-Lin; Wang, Zuolei; Zhang, Long; Teng, Zhidong\n2015\nAnti-synchronization and intermittent anti-synchronization of two identical hyperchaotic Chua systems via impulsive control. Zbl 1345.34095\nLi, Hong-Li; Jiang, Yao-Lin; Wang, Zuo-Lei\n2015\nA new parareal waveform relaxation algorithm for time-periodic problems. Zbl 1308.65131\nSong, Bo; Jiang, Yao-Lin\n2015\nOn the uniqueness and perturbation to the best rank-one approximation of a tensor. Zbl 1317.65111\nJiang, Yao-Lin; Kong, Xu\n2015\nLie group analysis method for two classes of fractional partial differential equations. Zbl 1440.35340\nChen, Cheng; Jiang, Yao-Lin\n2015\nStructure-preserving model order reduction by general orthogonal polynomials for integral-differential systems. Zbl 1307.93093\nYuan, Jia-Wei; Jiang, Yao-Lin; Xiao, Zhi-Hua\n2015\nArnoldi-based model reduction for fractional order linear systems. Zbl 1312.93025\nJiang, Yao-Lin; Xiao, Zhi-Hua\n2015\n$$H_2$$ order-reduction for bilinear systems based on Grassmann manifold. Zbl 1395.93136\nXu, Kang-Li; Jiang, Yao-Lin; Yang, Zhi-Xia\n2015\nStructure-preserving model order reduction based on Laguerre-SVD for coupled systems. Zbl 1333.93065\nQi, Zhen-Zhong; Jiang, Yao-Lin; Xiao, Zhi-Hua\n2015\nA windowing waveform relaxation method for time-fractional differential equations. Zbl 1489.65110\nDing, Xiao-Li; Jiang, Yao-Lin\n2015\nAnalysis of a new parareal algorithm based on waveform relaxation method for time-periodic problems. Zbl 1308.65117\nSong, Bo; Jiang, Yao-Lin\n2014\nWaveform relaxation method for fractional differential-algebraic equations. Zbl 1305.26017\nDing, Xiao-Li; Jiang, Yao-Lin\n2014\nOn model reduction of $$K$$-power bilinear systems. Zbl 1290.93031\nWang, Xiao-Long; Jiang, Yao-Lin\n2014\nDimension reduction for second-order systems by general orthogonal polynomials. Zbl 1298.93100\nXiao, Zhi-Hua; Jiang, Yao-Lin\n2014\nModel order reduction based on general orthogonal polynomials in the time domain for coupled systems. Zbl 1291.93062\nQi, Zhen-Zhong; Jiang, Yao-Lin; Xiao, Zhi-Hua\n2014\nA note on the $$H^1$$-convergence of the overlapping Schwarz waveform relaxation method for the heat equation. Zbl 1295.65091\nZhang, Hui; Jiang, Yao-Lin\n2014\nA simple synchronization scheme of Genesio-Tesi system based on the back-stepping design. Zbl 1284.34095\nWang, Zuolei; Jiang, Yaolin\n2014\nDynamic behaviors of Holling type II predator-prey system with mutual interference and impulses. Zbl 1418.92106\nLi, Hongli; Zhang, Long; Teng, Zhidong; Jiang, Yaolin\n2014\nAnalytical solutions for the multi-term time-space fractional advection-diffusion equations with mixed boundary conditions. Zbl 1260.35241\nDing, Xiao-Li; Jiang, Yao-Lin\n2013\nWaveform relaxation methods for fractional differential equations with the Caputo derivatives. Zbl 1259.65113\nJiang, Yao-Lin; Ding, Xiao-Li\n2013\nAnalysis of two parareal algorithms for time-periodic problems. Zbl 1283.65077\nGander, Martin J.; Jiang, Yao-Lin; Song, Bo; Zhang, Hui\n2013\nOn the determinant evaluation of quasi penta-diagonal matrices and quasi penta-diagonal Toeplitz matrices. Zbl 1286.65057\nJiang, Yao-Lin; Jia, Ji-Teng\n2013\nTwo-sided projection methods for model reduction of MIMO bilinear systems. Zbl 1305.93046\nWang, Xiao-Long; Jiang, Yao-Lin\n2013\nWaveform relaxation methods for fractional functional differential equations. Zbl 1312.34011\nDing, Xiao-Li; Jiang, Yao-Lin\n2013\nSymbolic algorithm for solving cyclic penta-diagonal linear systems. Zbl 1269.65027\nJia, Ji-Teng; Jiang, Yao-Lin\n2013\nComputation of universal unfolding of the double-zero bifurcation in $$Z_2$$-symmetric systems by a homological method. Zbl 1282.34047\nPeng, Guojun; Jiang, Yao Lin\n2013\nOn structured variants of modified HSS iteration methods for complex Toeplitz linear systems. Zbl 1289.65052\nChen, Fang; Jiang, Yaolin; Liu, Qingquan\n2013\nA note on the ranks of $$2\\times 2\\times 2$$ and $$2\\times 2\\times 2\\times 2$$ tensors. Zbl 1282.15004\nKong, Xu; Jiang, Yao-Lin\n2013\n...and 48 more Documents\nall top 5\n\n### Cited by 1,032 Authors\n\n 102 Jiang, Yaolin 15 Li, Hongli 14 Wu, Shulin 12 Jia, Jiteng 12 Wang, Xiaolong 12 Xu, Kangli 11 Ding, Xiaoli 11 Xiao, Zhihua 11 Zhang, Guofeng 10 Zhang, Long 9 Hu, Cheng 9 Jiang, Haijun 9 Ma, Changfeng 9 Teng, Zhi-dong 8 Băleanu, Dumitru I. 8 Bhrawy, Ali Hassan 8 Dehghan Takht Fooladi, Mehdi 7 Assari, Pouria 7 Cao, Jinde 7 Kong, Xu 6 Al-saedi, Ahmed Eid Salem 6 Gander, Martin Jakob 6 Qi, Zhen-Zhong 6 Song, Bo 5 Abdullah, Farah Aini 5 Chen, Cheng 5 Fan, Zhencheng 5 Huang, Pengzhan 5 Kao, Yonggui 5 Kong, Qiongxiang 5 Li, Changpin 5 Liang, Zhaozheng 5 Miao, Zhen 5 Nian, Fuzhong 5 Rihan, Fathalla A. 5 Wang, Zhen 5 Yang, Ping 5 Zhu, Muzheng 4 Azar, Ahmad Taher 4 Cao, Yang 4 Ezz-Eldien, Samer S. 4 Ismail, Ahmad Izani Md. 4 Li, Sumei 4 Machado, José António Tenreiro 4 Mohd, Mohd Hafiz 4 Moustafa, Mahmoud 4 Ouannas, Adel 4 Song, Qiankun 4 Wang, Xingyuan 4 Wang, Zuolei 4 Xie, Yingkang 4 Yang, Xujun 4 Yang, Yunbo 4 Zhang, Naimin 3 Ali Akbar, M. 3 Attili, Basem S. 3 Bai, Zhongzhi 3 Baishya, Chandrali 3 Banerjee, Malay 3 Chen, Chunyue 3 Chen, Haibao 3 Çibik, Aytekin Bayram 3 Eghtesad, Mohammad 3 Elsonbaty, Amr R. 3 Göttlich, Simone 3 Hayat, Tasawar 3 Hu, Hongxiao 3 Huang, Ting-Zhu 3 Jiang, Daqing 3 Jiang, Meiqun 3 Kaya, Songul 3 Khojasteh Salkuyeh, Davod 3 Li, Chuandong 3 Li, Yan 3 Liu, Qun 3 Liu, Zhongyun 3 Mokhtary, Payam 3 Muhammadhaji, Ahmadjan 3 Nieto Roig, Juan Jose 3 Peng, Guojun 3 Shao, Xinhui 3 Sun, Wei 3 Vaidyanathan, Sundarapandian 3 Vakilzadeh, Mohsen 3 Vatankhah, Ramin 3 Wang, Fei 3 Wang, Zhaohong 3 Wu, Zhaoyan 3 Xiong, Junlin 3 Xu, Liguang 3 Yang, Jun-Man 3 Yang, Xi 3 Yang, Yongqing 3 Yu, Lanlin 3 Yuan, Jiawei 3 Zhang, Fengrong 3 Zhang, Yulin 3 Ziar, Toufik 2 Abdelkawy, Mohamed A. 2 Achar, Sindhu J. ...and 932 more Authors\nall top 5\n\n### Cited in 165 Serials\n\n 56 Applied Mathematics and Computation 30 Journal of the Franklin Institute 30 Chaos, Solitons and Fractals 27 Advances in Difference Equations 24 Journal of Computational and Applied Mathematics 20 Computers & Mathematics with Applications 18 Numerical Algorithms 16 International Journal of Systems Science. Principles and Applications of Systems and Integration 15 Nonlinear Dynamics 14 International Journal of Computer Mathematics 13 International Journal of Bifurcation and Chaos in Applied Sciences and Engineering 10 Applied Mathematical Modelling 10 Mathematical Problems in Engineering 9 Physica A 9 Computational and Applied Mathematics 8 International Journal of Control 8 Abstract and Applied Analysis 8 Journal of Applied Mathematics and Computing 7 Applied Numerical Mathematics 7 Communications in Nonlinear Science and Numerical Simulation 7 AIMS Mathematics 6 Mathematics and Computers in Simulation 6 Applied Mathematics Letters 6 Journal of Scientific Computing 6 Fractional Calculus & Applied Analysis 6 Mathematical and Computer Modelling of Dynamical Systems 6 Nonlinear Analysis. Modelling and Control 5 Journal of Computational Physics 5 SIAM Journal on Scientific Computing 5 Complexity 5 Journal of Mathematical Chemistry 5 Discrete Dynamics in Nature and Society 4 International Journal of Modern Physics B 4 BIT 4 SIAM Journal on Matrix Analysis and Applications 4 International Journal of Biomathematics 3 Mathematical Methods in the Applied Sciences 3 Automatica 3 Acta Applicandae Mathematicae 3 Neural Networks 3 Linear Algebra and its Applications 3 Numerical Linear Algebra with Applications 3 Mathematical Biosciences and Engineering 3 Advances in Mathematical Physics 3 Asian Journal of Control 2 Computer Methods in Applied Mechanics and Engineering 2 Journal of Mathematical Analysis and Applications 2 Wave Motion 2 Calcolo 2 Numerical Functional Analysis and Optimization 2 Numerische Mathematik 2 Systems & Control Letters 2 Stochastic Analysis and Applications 2 Journal of Computational Mathematics 2 Physica D 2 ETNA. Electronic Transactions on Numerical Analysis 2 Advances in Computational Mathematics 2 Journal of Difference Equations and Applications 2 Differential Equations and Dynamical Systems 2 International Journal of Nonlinear Sciences and Numerical Simulation 2 Dynamics of Continuous, Discrete & Impulsive Systems. Series A. Mathematical Analysis 2 Discrete and Continuous Dynamical Systems. Series B 2 Computational Methods in Applied Mathematics 2 Journal of Applied Mathematics 2 Acta Mathematica Scientia. Series B. (English Edition) 2 Journal of Dynamical Systems and Geometric Theories 2 Boundary Value Problems 2 Waves in Random and Complex Media 2 Journal of Nonlinear Science and Applications 2 Advances in Applied Mathematics and Mechanics 2 Symmetry 2 Journal of Applied Analysis and Computation 2 Journal of Applied Nonlinear Dynamics 2 East Asian Journal on Applied Mathematics 2 Journal of Function Spaces 2 International Journal of Applied and Computational Mathematics 1 Applicable Analysis 1 Computers and Fluids 1 Indian Journal of Pure & Applied Mathematics 1 International Journal of Theoretical Physics 1 Linear and Multilinear Algebra 1 Mathematical Biosciences 1 Nonlinearity 1 Reports on Mathematical Physics 1 Rocky Mountain Journal of Mathematics 1 ZAMP. Zeitschrift für angewandte Mathematik und Physik 1 Mathematics of Computation 1 Bulletin of Mathematical Biology 1 Journal of Differential Equations 1 Journal of the Korean Mathematical Society 1 Journal of Optimization Theory and Applications 1 Kybernetika 1 Mathematische Nachrichten 1 Proceedings of the Edinburgh Mathematical Society. Series II 1 Quaestiones Mathematicae 1 SIAM Journal on Numerical Analysis 1 Optimal Control Applications & Methods 1 Applied Mathematics and Mechanics. (English Edition) 1 Chinese Annals of Mathematics. Series B 1 Acta Mathematicae Applicatae Sinica. English Series ...and 65 more Serials\nall top 5\n\n### Cited in 39 Fields\n\n 231 Numerical analysis (65-XX) 229 Ordinary differential equations (34-XX) 149 Systems theory; control (93-XX) 108 Biology and other natural sciences (92-XX) 107 Partial differential equations (35-XX) 57 Dynamical systems and ergodic theory (37-XX) 56 Real functions (26-XX) 41 Linear and multilinear algebra; matrix theory (15-XX) 26 Fluid mechanics (76-XX) 15 Integral equations (45-XX) 15 Operator theory (47-XX) 14 Calculus of variations and optimal control; optimization (49-XX) 12 Special functions (33-XX) 11 Difference and functional equations (39-XX) 10 Mechanics of deformable solids (74-XX) 10 Statistical mechanics, structure of matter (82-XX) 8 Probability theory and stochastic processes (60-XX) 7 Computer science (68-XX) 7 Optics, electromagnetic theory (78-XX) 7 Operations research, mathematical programming (90-XX) 7 Information and communication theory, circuits (94-XX) 6 Combinatorics (05-XX) 6 Game theory, economics, finance, and other social and behavioral sciences (91-XX) 5 Integral transforms, operational calculus (44-XX) 4 Approximations and expansions (41-XX) 4 Harmonic analysis on Euclidean spaces (42-XX) 4 Classical thermodynamics, heat transfer (80-XX) 3 Number theory (11-XX) 3 Global analysis, analysis on manifolds (58-XX) 3 Statistics (62-XX) 3 Mechanics of particles and systems (70-XX) 2 Algebraic geometry (14-XX) 2 Nonassociative rings and algebras (17-XX) 2 Functional analysis (46-XX) 2 Quantum theory (81-XX) 1 Functions of a complex variable (30-XX) 1 Differential geometry (53-XX) 1 Geophysics (86-XX) 1 Mathematics education (97-XX)" ]	[ null, "https://zbmath.org/static/feed-icon-14x14.png", null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.64914876,"math_prob":0.6274641,"size":46138,"snap":"2022-40-2023-06","text_gpt3_token_len":13291,"char_repetition_ratio":0.21615295,"word_repetition_ratio":0.4,"special_character_ratio":0.27335384,"punctuation_ratio":0.18415102,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.95523965,"pos_list":[0,1,2],"im_url_duplicate_count":[null,null,null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2022-09-30T07:38:49Z\",\"WARC-Record-ID\":\"<urn:uuid:8af3f61d-c976-47e7-bfb3-8aa8558cf015>\",\"Content-Length\":\"503761\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:a274bf6f-f62a-405d-9f3a-2f3d6d24625d>\",\"WARC-Concurrent-To\":\"<urn:uuid:074c0ebc-2393-48c7-8e84-cc6337819eff>\",\"WARC-IP-Address\":\"141.66.194.2\",\"WARC-Target-URI\":\"https://zbmath.org/authors/?q=ai%3Ajiang.yaolin\",\"WARC-Payload-Digest\":\"sha1:DCDVU2D2VMKBHF3BNDO5U7JG5LTRREVK\",\"WARC-Block-Digest\":\"sha1:DSCTUORZM4FC5ZQR4K5BHYHMM4KBVY6O\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2022/CC-MAIN-2022-40/CC-MAIN-2022-40_segments_1664030335444.58_warc_CC-MAIN-20220930051717-20220930081717-00317.warc.gz\"}"}
https://socratic.org/questions/how-do-you-solve-the-following-linear-system-y-3x-4-y-3x-2	[ "# How do you solve the following linear system: y = 3x+4 , y-3x=-2 ?\n\nJul 8, 2016\n\nThe given pair of equations has no solution(s).\n\n#### Explanation:\n\nIf\n$\\textcolor{w h i t e}{\\text{XXX}} y = 3 x + 4$\nand\n$\\textcolor{w h i t e}{\\text{XXX}} y - 3 x = - 2$\n\nSubstituting (from ) $3 x + 4$ in place of $y$ into \n$\\textcolor{w h i t e}{\\text{XXX}} \\cancel{3 x} + 4 - \\cancel{3 x} = - 2$\n\nSince $4 \\ne - 2$\nthese equations are inconsistent.\n\nJul 8, 2016\n\nThe system has no soln. , or, the Soln. Set $= \\phi .$\nSubstituting the value of $y$, obtained from the first eqn., in the second eqn., we get,\n$\\left(3 x + 4\\right) - 3 x = - 2 ,$ i.e., $4 = - 2$, which is an impossible result!\nHence, the Soln. Set $= \\phi .$" ]	[ null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.6626739,"math_prob":0.9999902,"size":493,"snap":"2019-43-2019-47","text_gpt3_token_len":148,"char_repetition_ratio":0.096114516,"word_repetition_ratio":0.024691358,"special_character_ratio":0.28803244,"punctuation_ratio":0.16666667,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9998989,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2019-11-21T10:32:54Z\",\"WARC-Record-ID\":\"<urn:uuid:663fcad9-d6c6-453c-bb67-a16315749e87>\",\"Content-Length\":\"35363\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:dcffe1bc-8f45-4402-91fe-66883bed11c3>\",\"WARC-Concurrent-To\":\"<urn:uuid:867c37dd-2705-4813-9756-7532a4ea248c>\",\"WARC-IP-Address\":\"54.221.217.175\",\"WARC-Target-URI\":\"https://socratic.org/questions/how-do-you-solve-the-following-linear-system-y-3x-4-y-3x-2\",\"WARC-Payload-Digest\":\"sha1:EJIJHQOLBVVCXAXEU24HK6XQFXYR3NH3\",\"WARC-Block-Digest\":\"sha1:A3VNV7ZUXACP5TDKK2NM6QMVZHV3JCHA\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2019/CC-MAIN-2019-47/CC-MAIN-2019-47_segments_1573496670770.21_warc_CC-MAIN-20191121101711-20191121125711-00111.warc.gz\"}"}
https://afontaine.dev/blog/advent-of-code-days-4-5-6	[ "# Advent of Code: Days 4, 5, 6\n\nAndrew Fontaine <[email protected]>\n\nI’ve been busy so this is going to be a bit longer but also contain 3 whole solutions!\n\n## Day four\n\nDiving deeper and deep, we come across a giant squid! To distract it, we decide to play bingo.\n\nAn important rule to this game of bingo is that diagonals don’t count, and don’t need to be considered. I am given a set of calls and boards, and need to figure out which board will win first.\n\nStep one is parsing the input. The first line contains the calls for the game, and the rest of the file consists of separate boards. As I want to be able to keep track if whether or not a number has been called, I make a type to hold that information. Then, I split up the file to the different boards:\n\n``````exception Empty\n\ntype i = Marked of int \| Unmarked of int\n\n\| [] -> [ [ line ] ]\n\| last :: rest -> (line :: last) :: rest\n\nlet parse_line line =\nString.split_on_char ' ' line\n\|> List.filter (fun s -> s <> \"\")\n\|> List.map (fun s -> Unmarked (int_of_string s))\n\nlet input =\n\| [] -> raise Empty\n\| calls :: rest ->\nlet boards =\nList.fold_left\n(fun b l ->\nmatch l with\n\| \"\" \| \"\\n\" -> [] :: b\n\| line -> add_to_board (parse_line line) b)\n[ [] ] rest\n\|> List.filter (fun x -> x <> [])\nin\nlet numbers = List.map int_of_string (String.split_on_char ',' calls) in\n(numbers, boards)``````\n\nMy type `i` indicates whether or not a number has been marked. Because ocaml requires exhaustive pattern matching, I have to make sure my list of lines from the file is not empty. Then, I pull the first line out with pattern matching (noted as `calls` here), and fold over the rest of the list to construct the boards. There are a few empty lines in there that need to be filtered out. Finally, I split the calls on commas, and parse those strings into proper numbers.\n\nPart one just requires I play the game on all the boards. There are some small helper functions I need first:\n\n``````let check_wins board =\nlet row_win =\nList.exists\n(fun row ->\nList.for_all\n(fun m -> match m with Marked _ -> true \| Unmarked _ -> false)\nrow)\nboard\nin\nlet column_win =\nList.exists\n(fun i ->\nList.for_all\n(fun row ->\nmatch List.nth row (i - 1) with Marked _ -> true \| Unmarked _ -> false)\nboard)\n(List.init (List.length (List.nth board 0)) (fun x -> x + 1))\nin\nrow_win \|\| column_win\n\nlet mark i board =\nList.map\n(fun row ->\nList.map\n(fun x -> match x with Unmarked y when y = i -> Marked y \| y -> y)\nrow)\nboard``````\n\n`check_wins` checks to see if any row or column has been completely marked, and `mark` updates the board to set a number to `Marked` if it is found.\n\nAll that is left is to play and compute the score:\n\n``````let compute_score board call =\nlet sum =\nList.fold_left\n(fun sum row ->\nList.fold_left\n(fun s x -> match x with Marked _ -> s \| Unmarked y -> s + y)\nsum row)\n0 board\nin\ncall * sum\n\nlet rec play boards = function\n\| [] -> -100\n\| call :: rest -> (\nlet marks = mark call in\nlet b = List.map marks boards in\nmatch List.find_opt check_wins b with\n\| None -> play b rest\n\| Some board -> compute_score board call)\n\nlet p1 =\nlet calls, boards = input in\nplay boards calls \|> string_of_int``````\n\n`play` recursively iterates over the called numbers, marking boards until a winner is found. It returns the score of `-100` if I run out of numbers make it obvious (while still an integer) that something went wrong. While not the best error handling method, it works for such a small script. For more complicated problems later on, I plan on looking to Vladimir Keleshev’s Composable Error Handling in OCaml for guidance.\n\nOnce a winner is found, the score is computed. ⭐ one done!\n\n### Part two\n\nPart two is a small extension to part one, which suggests that I should pick the board that will win last to ensure the squid wins and won’t crush us for losing. As such, it only requires some small extension.\n\n``````\nlet rec play boards = function\n\| [] -> -100\n\| call :: rest -> (\nlet marks = mark call in\nlet new_boards = List.map marks boards in\nmatch new_boards with\n\| [] -> -100\n\| [b] -> if check_wins b then compute_score b call else play [b] rest\n\| boards -> let losers = List.filter (fun b -> not (check_wins b)) boards in\nplay losers rest)\n\nlet p2 =\nlet calls, boards = input in\nplay boards calls \|> string_of_int\n``````\n\nInstead of stopping as soon as a winner is found, I continue until only one board is left to win.\n\nThat’s day four ✔️\n\n## Day 5\n\nEvery year involves at least one problem about interesting lines. Every one of them is a pain to solve.\n\nThis year, the submarine is avoiding large, opaque clouds spewing out of hydrothermal vents on the ocean floor. Fortunately for us, the vents exist in extremely straight lines.\n\nI first attempted to do this the mathematically clever way using the determinant of the two lines and solving for their intersection point, if any, but that turned into a total bust that I could not puzzle out and so I shall speak no more of it.\n\nInstead, it was simply easier to walk the line segments and remember all the points I’ve been.\n\nOCaml handles generic data structures such as maps and sets via a functor, which is a function that takes and returns a module instead of normal values. To make a set module, the functor `Set.Make` is used. `Set.Make` takes a module that specifies a type, `t`, and a function to compare values, `compare`. This is a module type named `OrderedType`, where `t` is the type to order, and `compare` lets us order them. I set up a set to track the points I’ve been as so:\n\n``````type point = Point of int * int\n\nmodule PointSet = Set.Make (struct\ntype t = point\n\nlet compare (Point (x1, y1)) (Point (x2, y2)) =\nif x1 < x2 then -1\nelse if x1 = x2 then if y1 < y2 then -1 else if y1 = y2 then 0 else 1\nelse 1\nend)``````\n\n`compare` returns `-1` if the first point has a smaller `x` value, `1` if a larger `x` value, and defers to the `y` value if equal. If both are equal, `0` is returned. This is how the set knows if it already has the given point.\n\nI also need a line type to keep track of all my points:\n\n``type line = Line of point * point``\n\nAs the first problem only requires I deal with horizontal and vertical lines, that should be quick to set up:\n\n``````let is_vertical (Line (Point (_, y1), Point (_, y2))) = y1 = y2\n\nlet is_horizontal (Line (Point (x1, _), Point (x2, _))) = x1 = x2\n\nlet is_p1 line = is_vertical line \|\| is_horizontal line``````\n\nAll that’s left is to walk the lines:\n\n``````let walk_line acc (Line (Point (x1, y1), Point (x2, y2))) =\nlet points =\nif x1 = x2 then\nList.init (abs (y2 - y1) + 1) (fun y -> Point (x1, min y1 y2 + y))\nelse\nList.init (abs (x2 - x1) + 1) (fun x -> Point (min x1 x2 + x, y1))\nin\nList.fold_left\n(fun (s1, s2) point ->\nif PointSet.mem point s1 then (s1, PointSet.add point s2)\nacc points\n\nlet walk_lines lines =\nList.fold_left walk_line (PointSet.empty, PointSet.empty) lines\n\nlet p1 =\nlet _, points = input \|> List.filter is_p1 \|> walk_lines in\npoints \|> PointSet.elements \|> List.length \|> string_of_int``````\n\n`walk_line` makes a list covering every point contained in the line, and then adds them to the visited set. If the point is already in the visited set, it is added to the final set, as it matches our criteria.\n\n`walk_lines` iterates over all the lines, until the final set of points is found.\n\n⭐ one done!\n\n### Part two\n\nPart two, again, is only a small extension of part one. It demands we include the diagonal lines to check as well. Simple enough to add to `walk_line`\n\n``````let walk_line acc (Line (Point (x1, y1), Point (x2, y2))) =\nlet points =\nif x1 = x2 then\nList.init (abs (y2 - y1) + 1) (fun y -> Point (x1, min y1 y2 + y))\nelse if y1 = y2 then\nList.init (abs (x2 - x1) + 1) (fun x -> Point (min x1 x2 + x, y1))\nelse\nList.init\n(abs (x2 - x1) + 1)\n(fun z ->\nlet x = if x1 < x2 then x1 + z else x1 - z in\nlet y = if y1 < y2 then y1 + z else y1 - z in\nPoint (x, y))\nin\nList.fold_left\n(fun (s1, s2) point ->\nif PointSet.mem point s1 then (s1, PointSet.add point s2)\nacc points``````\n\nThe `else` expression now knows how to follow along a diagonal! Then to just run `walk_lines` on the whole dataset and count up the points:\n\n``````let p2 =\nlet _, points = input \|> walk_lines in\npoints \|> PointSet.elements \|> List.length \|> string_of_int``````\n\nThat’s day five ✔️\n\n## Day six\n\nDay six requires we do some biological work by counting fish.\n\nPart one is simple enough, so let’s write some code for it:\n\n``````let rec grow_fish old_fish new_fish = function\n\| [] -> List.rev_append old_fish new_fish\n\| h :: rest ->\nif h = 0 then grow_fish (6 :: old_fish) (8 :: new_fish) rest\nelse grow_fish ((h - 1) :: old_fish) new_fish rest\n\nlet p1_sol input count =\nList.init count (fun x -> x + 1)\n\|> List.fold_left (fun fish _ -> grow_fish [] [] fish) input\n\nlet p1 =\nlet sol = p1_sol input 80 \|> List.length in\nInt.to_string sol``````\n\nFor 80 iterations, I go through the list of fish, decrementing the days til spawn. Once it hits 0, I bump it back up to 6 and add a new fish with a countdown starting at 8. All that’s left is to count up the fish!\n\n⭐ one done!\n\n### Part two\n\nI was hopeful I could just run my code for 256 days, but soon realized why it was a “part two”: bottlenecks and stack overflows.\n\nThe list was growing so much that my original solution to part one (not posted) would blow up with a stack overflow. The one posted above is tail-call optimized, as I thought it would be my only issue. Turns out, it was also growing so much that iterating through the list for ever day took ages.\n\nBack to the drawing board.\n\nI had noticed a lot of the fish ended up on the same cycle. There were several fish that each had 0 to 8 days left on their spawn timers, which got me thinking… if I could group the fish, I wouldn’t have to add to the list and instead just increment the count!\n\nI initially started off trying to make a map to handle this, but as I don’t quite understand how ocaml’s maps worked, it was easier for me to use a list of tuples instead. I also needed to be able to update both the key and the value, so a tuple felt better.\n\n``````let update x up list = up (List.find_opt (fun (y, _) -> x = y) list)\n\nlet dedup list =\nList.fold_left\n(fun deduped (x, z) ->\nupdate x\n(function\n\| None -> (x, z) :: deduped\n\| Some (_, y) -> (x, y + z) :: List.filter (fun (a, _) -> x <> a) deduped)\ndeduped)\n[] list\n\nlet grow_fish_2 fishes _ =\nList.fold_left\n(fun fish (x, y) ->\nmatch x with 0 -> (6, y) :: (8, y) :: fish \| z -> (z - 1, y) :: fish)\n[] fishes\n\|> dedup\n\nlet p2_sol (input : int list) count =\nlet fishes =\nList.fold_left\n(fun fishes x ->\nupdate x\n(function\n\| None -> (x, 1) :: fishes\n\| Some (_, y) ->\n(x, y + 1) :: List.filter (fun (y, _) -> x <> y) fishes)\nfishes)\n[] input\nin\nList.init count (fun x -> x + 1)\n\|> List.fold_left grow_fish_2 fishes\n\|> List.fold_left (fun sum (_, x) -> sum + x) 0``````\n\n`update` replicates map’s `update` function, where you provide a function that takes an option type. The option type either contains the value the key is pointing to, or nothing, and you must handle both cases. `dedup` takes all the elements and buckets and merges the matching ones together again.\n\n`grow_fish_2`, then, goes over the list of fish, decrementing the number of days til they spawn. If they are 0, the number gets bumped back up to 6, and a new tuple of fish is added. This tuple starts at 8, their spawn countdown, and the same number of fish as was in the first bucket. Once that is complete, I `dedup` the buckets, as it was easier to `dedup` the list once the fish spawning was done.\n\nI repeat this for the 256 days to get… `1.73e12` fish!\n\nThat’s a lot of fish!\n\nI’m trying to stay on top of these posts this year, but the weekends are hectic, as usual. I hope to not have to cram 3 days of updates in one day again though!\n\n#### Want to discuss this post?\n\nReach out via email to ~afontaine/[email protected], and be sure to follow the mailing list etiquette.\n\nOther posts" ]	[ null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.8613242,"math_prob":0.94363475,"size":11749,"snap":"2023-14-2023-23","text_gpt3_token_len":3283,"char_repetition_ratio":0.11375053,"word_repetition_ratio":0.12435233,"special_character_ratio":0.29942974,"punctuation_ratio":0.1243397,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9834615,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2023-03-25T11:16:24Z\",\"WARC-Record-ID\":\"<urn:uuid:6ee5dbde-eae1-4a0d-9034-c30ecb9e70cd>\",\"Content-Length\":\"22473\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:dc9e684f-8fb9-4596-8660-76a179f794ac>\",\"WARC-Concurrent-To\":\"<urn:uuid:f7f0645c-d4a0-4d4f-a48a-36508788595f>\",\"WARC-IP-Address\":\"159.89.120.242\",\"WARC-Target-URI\":\"https://afontaine.dev/blog/advent-of-code-days-4-5-6\",\"WARC-Payload-Digest\":\"sha1:TQ3MIRGV7TYMF5SQAO2MUHRYT4WWA66P\",\"WARC-Block-Digest\":\"sha1:LHNOR2NVIPEOL6QTASSBSNXVUYBIPN3R\",\"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-20230325095252-20230325125252-00458.warc.gz\"}"}
https://www.javaexercise.com/python/find-element-in-python-list	[ "", null, "# Python Program to Find an Element in Python List\n\nTo find an element in a list, python provides several built-in ways such as in operator, count() function, and index() function. Apart from that, we will use custom code as well to find an element. We will learn to use all these in our examples to find an element in the list. So, let's get started.\n\n## Find an element in Python List\n\nThis is an iterative approach where we use a loop to traverse each element of the list and then check via an if statement. If the element is matched, then display found message else, display not found. See the below python code.\n\n``````# Python program to find an element in a list\n\n# Take a list\nlist1 = [45, 11, 15, 9, 56, 17]\nprint(list1)\n\nval = 15\nflag = 0\nfor i in list1:\nif i == val:\nflag = 1\n\nif flag:\nprint(\"Element exists\")\nelse:\nprint(\"Element does not exist\")``````\n\nOutput:\n\nElement exists\n\n## Find an element in List by using the count() method in Python\n\nWe can use the count() method to check whether the element is present in the list or not. The count() method returns the frequency of an element in the list. So, if the count is greater than 0 then it means the element is present in the list. See the python code.\n\n``````# Python program to find an element in a list\n\n# Take a list\nlist1 = [45, 11, 15, 9, 56, 17]\nprint(list1)\n\nval = 15 # Element to find\n\n# Count method returns frequency of a number\nresult = list1.count(val)\n\nif result:\nprint(\"Element exists\")\nelse:\nprint(\"Element does not exist\")``````\n\nOutput:\n\nElement exists\n\n## Find an element in List by using an in operator in Python\n\nWe can use an in operator to check whether the specified element is present in a list. It is a membership operator that works for all iterable like list, set, tuple, etc. See the python code.\n\n``````# Python program to find an element in a list\n\n# Take a list\nlist1 = [45, 11, 15, 9, 56, 17]\nprint(list1)\n\nval = 15\n\n# Python in operator\nif val in list1:\nprint(\"Element exists\")\nelse:\nprint(\"Element does not exist\")``````\n\nOutput:\n\nElement exists\n\n## Find an element in List by using index() Method in Python\n\nPython index() method is used to find the index value of an element in the iterable(list, set, tuple, etc). We can use it to find an element in the list. This method returns an error if the element is not found, then you should use it with the try-catch statement. See the python code.\n\n``````# Python program to find an element in a list\n\n# Take a list\nlist1 = [45, 11, 15, 9, 56, 17]\nprint(list1)\n\nval = 15\n\ntry:\nresult = list1.index(val)\nexcept ValueError:\nprint(\"Element does not exist\")\nelse:\nprint(\"Element exists\") ``````\n\nOutput:\n\nElement exists\n\nif you don't use the try-catch statement with the index() method then it throws an error :\n\nresult = list1.index(111)\n\nValueError: 111 is not in list\n\nTrending" ]	[ null, "https://www.javaexercise.com/img/javaexercise.png", null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.76129,"math_prob":0.83724815,"size":2458,"snap":"2022-27-2022-33","text_gpt3_token_len":652,"char_repetition_ratio":0.18459658,"word_repetition_ratio":0.3178808,"special_character_ratio":0.28559804,"punctuation_ratio":0.13878328,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9768585,"pos_list":[0,1,2],"im_url_duplicate_count":[null,null,null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2022-08-11T11:18:47Z\",\"WARC-Record-ID\":\"<urn:uuid:1770181c-e57e-406e-81a9-f19998c3ad5b>\",\"Content-Length\":\"36881\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:bc374fa1-7ed7-4be2-bdaf-0be6ac075142>\",\"WARC-Concurrent-To\":\"<urn:uuid:60971493-7437-4a89-8c49-3f458551c267>\",\"WARC-IP-Address\":\"172.67.212.251\",\"WARC-Target-URI\":\"https://www.javaexercise.com/python/find-element-in-python-list\",\"WARC-Payload-Digest\":\"sha1:4AF2KOED5GAWRHMD72DCRHQOHVPXICPD\",\"WARC-Block-Digest\":\"sha1:OPMAQPIUWHYEYMADUYF4VS2WT4FBC2HE\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2022/CC-MAIN-2022-33/CC-MAIN-2022-33_segments_1659882571284.54_warc_CC-MAIN-20220811103305-20220811133305-00500.warc.gz\"}"}
https://richelbilderbeek.nl/CppRescale.htm	[ "# (C++) Rescale\n\nRescale is a math code snippet to rescale a double, 1D std::vector or 2D std::vector from a certain range to a new range.\n\n## Rescale on a double\n\n ``` #include #include //From http://www.richelbilderbeek.nl/CppRescale.htm const double Rescale( const double value, const double oldMin, const double oldMax, const double newMin, const double newMax) { assert(value >= oldMin); assert(value <= oldMax); const double oldDistance = oldMax - oldMin; //At which relative distance is value on oldMin to oldMax ? const double distance = (value - oldMin) / oldDistance; assert(distance >= 0.0); assert(distance <= 1.0); const double newDistance = newMax - newMin; const double newValue = newMin + (distance * newDistance); assert(newValue >= newMin); assert(newValue <= newMax); return newValue; } ```\n\n## Rescale on a 1D std::vector\n\n ``` #include #include #include //From http://www.richelbilderbeek.nl/CppRescale.htm const std::vector Rescale( std::vector v, const double newMin, const double newMax) { const double oldMin = std::min_element(v.begin(),v.end()); const double oldMax = std::max_element(v.begin(),v.end()); typedef std::vector::iterator Iter; Iter i = v.begin(); const Iter j = v.end(); for ( ; i!=j; ++i) { i = Rescale(i,oldMin,oldMax,newMin,newMax); } return v; } ```\n\n## Rescale on a 2D std::vector\n\n ``` #include #include #include //From http://www.richelbilderbeek.nl/CppRescale.htm const std::vector > Rescale( std::vector > v, const double newMin, const double newMax) { const double oldMin = MinElement(v); const double oldMax = MaxElement(v); typedef std::vector >::iterator RowIter; RowIter y = v.begin(); const RowIter maxy = v.end(); for ( ; y!=maxy; ++y) { typedef std::vector::iterator ColIter; ColIter x = y->begin(); const ColIter maxx = y->end(); for ( ; x!=maxx; ++x) { x = Rescale(x,oldMin,oldMax,newMin,newMax); } } return v; } ```", null, "" ]	[ null, "https://richelbilderbeek.nl/valid-xhtml10.png", null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.6886252,"math_prob":0.9473462,"size":398,"snap":"2023-14-2023-23","text_gpt3_token_len":106,"char_repetition_ratio":0.15228426,"word_repetition_ratio":0.16666667,"special_character_ratio":0.23115578,"punctuation_ratio":0.13157895,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.992077,"pos_list":[0,1,2],"im_url_duplicate_count":[null,null,null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2023-05-30T10:41:49Z\",\"WARC-Record-ID\":\"<urn:uuid:a1bb228b-58d3-44ee-8d3d-0b32f6c11aea>\",\"Content-Length\":\"11036\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:46971dc4-a891-4473-b1fe-de11d2f46e1d>\",\"WARC-Concurrent-To\":\"<urn:uuid:9b89e92f-788b-434d-91c2-dae7fe760c81>\",\"WARC-IP-Address\":\"46.30.213.109\",\"WARC-Target-URI\":\"https://richelbilderbeek.nl/CppRescale.htm\",\"WARC-Payload-Digest\":\"sha1:5X45JAWMAARBGCYFQ5WZRU2ZZFHFPFZM\",\"WARC-Block-Digest\":\"sha1:L5V65SPOPGALLWSDZAP3VZBX6WJ7K3GM\",\"WARC-Identified-Payload-Type\":\"application/xhtml+xml\",\"warc_filename\":\"/cc_download/warc_2023/CC-MAIN-2023-23/CC-MAIN-2023-23_segments_1685224645595.10_warc_CC-MAIN-20230530095645-20230530125645-00498.warc.gz\"}"}
https://aleph0.blog/2022/10/29/tilings-and-integer-programming/	[ "# Tilings and Integer Programming\n\nThis is the story behind sequence A355477 from the Online Encyclopedia of Integer Sequences.\n\nLet’s say you have an unlimited supply of “zig-zag” Tetris pieces:\n\nHow many can you pack into an n x n square?\n\nFor example, if you’re trying to fill a 4×4 square, then you can fit three such pieces:\n\nbut, try as you might, you’ll never get a fourth piece in there (without any of the pieces overlapping or poking out the sides of the square). What about a 5×5 square, a 6×6 square, etc.?\n\nOur goal is to answer this question (reasonably) efficiently to find, for example, that this packing of a 16×16 square with 60 tiles is optimal:\n\nIt turns out that we know such optimal packings for squares up to 21 x 21, as well as a handful of larger values, but the problem is not completely solved yet. The rest of this post explains what’s known and how we know it.\n\n#### Set Packing\n\nThis tiling problem is a nice example of the “Maximum Set Packing” problem, which is a classic optimization problem that has many applications. Specifically, if we label each square with a letter:\n\nthen any placement of a tile corresponds to a 4-element set of squares. For example, the three tiles in Figure 1 correspond to the sets:\n\n$$\\begin{array}\\\\ S_1 &=& \\{B, C, E, F\\} \\\\ S_2 &=& \\{D, G, H, K\\} \\\\ S_3 &=& \\{I, J, N, O\\} \\end{array}.$$\n\nAnd, in addition to these three sets, there are 21 other sets corresponding to other placements of zig-zag tiles (e.g.,", null, "$\\{J, K, O, P\\}$ for a tile in the bottom-right corner, etc.) that we didn’t end up using in Figure 1, for a total of 24 possible placements of tiles within a 4×4 square.\n\nThe goal of the set packing problem is to find a collection of non-overlapping sets (from this family of 24 sets) that is as large as possible, which is equivalent to trying to squeeze as many non-overlapping tiles as possible into the square.\n\nUnfortunately, the maximum set packing problem is NP-complete, so we’re not going to find an efficient algorithm that works in all cases, but there is still hope: Integer Programming.\n\n#### Integer Programming\n\nInteger programming is also a very hard computation problem, but there are excellent “mixed integer programming solvers” that work very well in practice on set-packing/tiling problems like these, so we’re going to take advantage of that as we try to understand this tiling question.\n\nLet’s return to the 4×4 tiling example we looked at above and see how it can be transformed into an “integer programming” problem. First, it will be convenient to express this problem as matrix with 16 columns (each corresponding to one of the cells in the 4×4 square) and with rows corresponding to each possible placement of a tile. For example, these four placements of tiles:\n\ncorrespond to the first four rows of the matrix", null, "$X$:\n\n$$X = \\begin{matrix}A & B & C & D & E & F & G & H & I & J &K & L & M & N & O &P \\\\ \\hline 1 & 1 & 0 & 0 & 0 & 1 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\ 0 & 1 & 1 & 0 & 0 & 0 & 1 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\ 0 & 0 & 0 & 0 & 1 & 1 & 0 & 0 & 0 & 1 & 1 & 0 & 0 & 0 & 0 & 0 \\\\ 0 & 0 & 0 & 0 & 0 & 1 & 1 & 0 & 0 & 0 & 1 & 1 & 0 & 0 & 0 & 0 \\\\ \\vdots & \\vdots & \\vdots & \\vdots & \\vdots & \\vdots & \\vdots & \\vdots & \\vdots & \\vdots & \\vdots & \\vdots & \\vdots & \\vdots & \\vdots & \\vdots \\end{matrix}$$\n\nThe complete matrix has", null, "$24$ rows corresponding to all the different possible placements/orientations of the zig-zag tile.\n\nNow, the tiling problem is equivalent to finding as many rows of", null, "$X$ as possible that have at most one", null, "$1$ in each column. This is easily expressed as an integer programming problem as follows:\n\nDefine binary variables\n\n$$s_1, \\ldots, s_n \\in \\{0, 1\\}$$\n\nwhere", null, "$n$ is the number of rows in the matrix", null, "$X$, and then\n\n$$\\textrm{maximize } s_1 + \\cdots + s_n$$\n\n$$\\textrm{subject to}$$\n\n$$\\vec{s} \\cdot X \\leq 1$$\n\nIn code (using the ‘mip’ package for python) this would look something like:\n\ndef maxSetPacking(X):\n# Find the largest subset of disjoint rows of X\nm = mip.Model(sense=mip.MAXIMIZE)\nm.objective = S.sum()\nm += (np.dot(S, X) <= 1)\nm.optimize()\nreturn np.array([s.x for s in S]).round().astype(bool)\n\nThe result of this optimization is a boolean vector", null, "$\\vec{s}$ indicating which of the rows of", null, "$X$ form an optimal set covering.\n\nUnleashing our Integer Programming solver on our tiling problem, we can find optimal packings of an", null, "$n \\times n$ square for various values of", null, "$n$. Here are examples of optimal packings for", null, "$n \\leq 18$:\n\n#### Hmmmm… that’s odd\n\nHere’s one interesting pattern that emerges from the optimal packings shown above: whenever", null, "$n$ is odd, then the optimal packing contains exactly $$\\left( \\frac{n-1}{2}\\right)^2$$ tiles. And, in fact, this is true for all odd", null, "$n$. Here’s a simple proof, using the", null, "$7 \\times 7$ case as an example:\n\nFirst off, we can simply arrange the tiles in an interlocking", null, "$3 \\times 3$ array to fit", null, "$9$ tiles into the", null, "$7 \\times 7$ grid:\n\nAnd, in general, the same approach will allow $$\\left( \\frac{n-1}{2}\\right)^2$$ tiles to fit within an", null, "$n \\times n$ square whenever", null, "$n$ is odd, so we have:\n\n$$a(n) \\geq \\left( \\frac{n-1}{2}\\right)^2$$\n\nNow we have to show that no more than", null, "$9$ tiles can fit into an", null, "$7 \\times 7$ grid. To prove this, take the", null, "$7 \\times 7$ grid and place an “X” in the cells that are in both an even-numbered row and an even-numbered column:\n\nNow observe that no matter where a tile is placed it will cover one of the “X”s, for example:\n\nSince there are only", null, "$9$ “X”s, no more than", null, "$9$ tiles can be placed in the grid. (Otherwise, at least two of them would have to share an “X”, which would mean they overlap.)\n\nTogether with the previous bound, this shows that $$a(n) = \\left( \\frac{n-1}{2}\\right)^2$$ whenever", null, "$n$ is odd, so we completely understand the odd-numbered terms of this sequence.\n\n#### Getting even…\n\nThe even-numbered terms, however, seem more subtle. The integer programming approach handily deals with all cases up to", null, "$n=20$, but gets a bit stuck on", null, "$n = 22$. It finds the following packing with", null, "$116$ tiles:\n\nbut (after running for days) it can’t quite rule out the possibility that there’s an even better solution with", null, "$117$ tiles. That being said, we do know that there’s no solution with more than", null, "$117$ tiles. This is because the integer programming solver uses a “primal-dual” algorithm which, roughly, means that as it’s performing the search it’s keeping track not only of the best solution it’s found so far (which is a lower-bound for the value of", null, "$a(n)$) but also the best solution to the “dual” version of the problem (which, it turns out, is an upper-bound for the value of", null, "$a(n)$). And the search did find a “dual” solution that was strictly smaller than", null, "$118$, so we know that", null, "$a(22) \\leq 117$.\n\nSo, even in cases when the algorithm doesn’t find the optimal solution, it can give us useful bounds on the optimal solution. This is another reason why the integer programming approach is particularly appealing for this kind of tiling problem.\n\nAfter", null, "$n = 22$, the next “hard” case is", null, "$n = 26$, where we know that $$163 \\leq a(26) \\leq 165,$$ but we don’t know the exact value. Similarly for other even", null, "$n \\geq 28$ we have upper and bounds, but don’t (yet) know the exact values.\n\n#### Going Further\n\nUndoubtedly, as computers get faster and as even better integer programming solvers are developed, we’ll come to learn the true values of", null, "$a(22), a(26)$ and other terms of A355477. (Some readers may even have access to “industrial grade” (mixed) integer programming solvers that can already solve the instances that stumped the open-source tools I used…) And perhaps there are some clever mathematical insights that will lead to a better understanding of the even terms of this series. 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https://www.colorhexa.com/1a3936	[ "# #1a3936 Color Information\n\nIn a RGB color space, hex #1a3936 is composed of 10.2% red, 22.4% green and 21.2% blue. Whereas in a CMYK color space, it is composed of 54.4% cyan, 0% magenta, 5.3% yellow and 77.6% black. It has a hue angle of 174.2 degrees, a saturation of 37.3% and a lightness of 16.3%. #1a3936 color hex could be obtained by blending #34726c with #000000. Closest websafe color is: #333333.\n\n• R 10\n• G 22\n• B 21\nRGB color chart\n• C 54\n• M 0\n• Y 5\n• K 78\nCMYK color chart\n\n#1a3936 color description : Very dark desaturated cyan.\n\n# #1a3936 Color Conversion\n\nThe hexadecimal color #1a3936 has RGB values of R:26, G:57, B:54 and CMYK values of C:0.54, M:0, Y:0.05, K:0.78. Its decimal value is 1718582.\n\nHex triplet RGB Decimal 1a3936 `#1a3936` 26, 57, 54 `rgb(26,57,54)` 10.2, 22.4, 21.2 `rgb(10.2%,22.4%,21.2%)` 54, 0, 5, 78 174.2°, 37.3, 16.3 `hsl(174.2,37.3%,16.3%)` 174.2°, 54.4, 22.4 333333 `#333333`\nCIE-LAB 21.624, -12.396, -1.694 2.555, 3.412, 4.014 0.256, 0.342, 3.412 21.624, 12.511, 187.78 21.624, -11.941, -0.416 18.472, -7.637, 0.047 00011010, 00111001, 00110110\n\n# Color Schemes with #1a3936\n\n• #1a3936\n``#1a3936` `rgb(26,57,54)``\n• #391a1d\n``#391a1d` `rgb(57,26,29)``\nComplementary Color\n• #1a3927\n``#1a3927` `rgb(26,57,39)``\n• #1a3936\n``#1a3936` `rgb(26,57,54)``\n• #1a2d39\n``#1a2d39` `rgb(26,45,57)``\nAnalogous Color\n• #39271a\n``#39271a` `rgb(57,39,26)``\n• #1a3936\n``#1a3936` `rgb(26,57,54)``\n• #391a2d\n``#391a2d` `rgb(57,26,45)``\nSplit Complementary Color\n• #39361a\n``#39361a` `rgb(57,54,26)``\n• #1a3936\n``#1a3936` `rgb(26,57,54)``\n• #361a39\n``#361a39` `rgb(54,26,57)``\n• #1d391a\n``#1d391a` `rgb(29,57,26)``\n• #1a3936\n``#1a3936` `rgb(26,57,54)``\n• #361a39\n``#361a39` `rgb(54,26,57)``\n• #391a1d\n``#391a1d` `rgb(57,26,29)``\n• #020404\n``#020404` `rgb(2,4,4)``\n• #0a1615\n``#0a1615` `rgb(10,22,21)``\n• #122725\n``#122725` `rgb(18,39,37)``\n• #1a3936\n``#1a3936` `rgb(26,57,54)``\n• #224b47\n``#224b47` `rgb(34,75,71)``\n• #2a5c57\n``#2a5c57` `rgb(42,92,87)``\n• #326e68\n``#326e68` `rgb(50,110,104)``\nMonochromatic Color\n\n# Alternatives to #1a3936\n\nBelow, you can see some colors close to #1a3936. Having a set of related colors can be useful if you need an inspirational alternative to your original color choice.\n\n• #1a392e\n``#1a392e` `rgb(26,57,46)``\n• #1a3931\n``#1a3931` `rgb(26,57,49)``\n• #1a3933\n``#1a3933` `rgb(26,57,51)``\n• #1a3936\n``#1a3936` `rgb(26,57,54)``\n• #1a3939\n``#1a3939` `rgb(26,57,57)``\n• #1a3739\n``#1a3739` `rgb(26,55,57)``\n• #1a3439\n``#1a3439` `rgb(26,52,57)``\nSimilar Colors\n\n# #1a3936 Preview\n\nText with hexadecimal color #1a3936\n\nThis text has a font color of #1a3936.\n\n``<span style=\"color:#1a3936;\">Text here</span>``\n#1a3936 background color\n\nThis paragraph has a background color of #1a3936.\n\n``<p style=\"background-color:#1a3936;\">Content here</p>``\n#1a3936 border color\n\nThis element has a border color of #1a3936.\n\n``<div style=\"border:1px solid #1a3936;\">Content here</div>``\nCSS codes\n``.text {color:#1a3936;}``\n``.background {background-color:#1a3936;}``\n``.border {border:1px solid #1a3936;}``\n\n# Shades and Tints of #1a3936\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, #010303 is the darkest color, while #f5fafa is the lightest one.\n\n• #010303\n``#010303` `rgb(1,3,3)``\n• #081110\n``#081110` `rgb(8,17,16)``\n• #0e1e1c\n``#0e1e1c` `rgb(14,30,28)``\n• #142c29\n``#142c29` `rgb(20,44,41)``\n• #1a3936\n``#1a3936` `rgb(26,57,54)``\n• #204643\n``#204643` `rgb(32,70,67)``\n• #265450\n``#265450` `rgb(38,84,80)``\n• #2c615c\n``#2c615c` `rgb(44,97,92)``\n• #336f69\n``#336f69` `rgb(51,111,105)``\n• #397c76\n``#397c76` `rgb(57,124,118)``\n• #3f8a83\n``#3f8a83` `rgb(63,138,131)``\n• #45978f\n``#45978f` `rgb(69,151,143)``\n• #4ba59c\n``#4ba59c` `rgb(75,165,156)``\n• #53b1a7\n``#53b1a7` `rgb(83,177,167)``\n• #60b7ae\n``#60b7ae` `rgb(96,183,174)``\n• #6ebdb5\n``#6ebdb5` `rgb(110,189,181)``\n• #7bc3bc\n``#7bc3bc` `rgb(123,195,188)``\n• #89c9c3\n``#89c9c3` `rgb(137,201,195)``\n• #96cfca\n``#96cfca` `rgb(150,207,202)``\n• #a4d5d1\n``#a4d5d1` `rgb(164,213,209)``\n• #b1dcd7\n``#b1dcd7` `rgb(177,220,215)``\n• #bfe2de\n``#bfe2de` `rgb(191,226,222)``\n• #cce8e5\n``#cce8e5` `rgb(204,232,229)``\n• #daeeec\n``#daeeec` `rgb(218,238,236)``\n• #e7f4f3\n``#e7f4f3` `rgb(231,244,243)``\n• #f5fafa\n``#f5fafa` `rgb(245,250,250)``\nTint Color Variation\n\n# Tones of #1a3936\n\nA tone is produced by adding gray to any pure hue. In this case, #272c2c is the less saturated color, while #00534b is the most saturated one.\n\n• #272c2c\n``#272c2c` `rgb(39,44,44)``\n• #242f2e\n``#242f2e` `rgb(36,47,46)``\n• #203331\n``#203331` `rgb(32,51,49)``\n• #1d3633\n``#1d3633` `rgb(29,54,51)``\n• #1a3936\n``#1a3936` `rgb(26,57,54)``\n• #173c39\n``#173c39` `rgb(23,60,57)``\n• #143f3b\n``#143f3b` `rgb(20,63,59)``\n• #10433e\n``#10433e` `rgb(16,67,62)``\n• #0d4640\n``#0d4640` `rgb(13,70,64)``\n• #0a4943\n``#0a4943` `rgb(10,73,67)``\n• #074c45\n``#074c45` `rgb(7,76,69)``\n• #044f48\n``#044f48` `rgb(4,79,72)``\n• #00534b\n``#00534b` `rgb(0,83,75)``\nTone Color Variation\n\n# Color Blindness Simulator\n\nBelow, you can see how #1a3936 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":0.54192096,"math_prob":0.739827,"size":3684,"snap":"2019-13-2019-22","text_gpt3_token_len":1674,"char_repetition_ratio":0.12472826,"word_repetition_ratio":0.011070111,"special_character_ratio":0.5646037,"punctuation_ratio":0.23756906,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9910578,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2019-03-25T06:05:30Z\",\"WARC-Record-ID\":\"<urn:uuid:fb88859a-494e-4650-969c-02ca3d2223c6>\",\"Content-Length\":\"36365\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:e6c7f9ea-7eb4-438e-9b8d-22ae13a96b8c>\",\"WARC-Concurrent-To\":\"<urn:uuid:bf80685c-02ca-41f6-a440-edb85b65d1e4>\",\"WARC-IP-Address\":\"178.32.117.56\",\"WARC-Target-URI\":\"https://www.colorhexa.com/1a3936\",\"WARC-Payload-Digest\":\"sha1:SPTCA3RD3JPUZOS5NWIOAXCDLBCTHEW2\",\"WARC-Block-Digest\":\"sha1:TSDMHWZZ5ENKFIZXVQF7XRXP2BEXFWD3\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2019/CC-MAIN-2019-13/CC-MAIN-2019-13_segments_1552912203755.18_warc_CC-MAIN-20190325051359-20190325073359-00323.warc.gz\"}"}
http://www.sport-tx.com/w_general.html	[ "< id=\"6e2kg\">< id=\"6e2kg\">< id=\"6e2kg\">< id=\"6e2kg\">< id=\"6e2kg\">< id=\"6e2kg\">< id=\"6e2kg\">< id=\"6e2kg\">< id=\"6e2kg\">< id=\"6e2kg\">< id=\"6e2kg\">< id=\"6e2kg\">< id=\"6e2kg\">< id=\"6e2kg\">< id=\"6e2kg\">< id=\"6e2kg\">< id=\"6e2kg\">< id=\"6e2kg\">< id=\"6e2kg\">< id=\"6e2kg\">< id=\"6e2kg\">< id=\"6e2kg\">< id=\"6e2kg\">< id=\"6e2kg\">< id=\"6e2kg\">< id=\"6e2kg\">< id=\"6e2kg\">< id=\"6e2kg\">< id=\"6e2kg\">< id=\"6e2kg\">< id=\"6e2kg\">< id=\"6e2kg\">< id=\"6e2kg\">< id=\"6e2kg\">< id=\"6e2kg\">< id=\"6e2kg\">< id=\"6e2kg\">< id=\"6e2kg\">< id=\"6e2kg\">< id=\"6e2kg\">< id=\"6e2kg\">< id=\"6e2kg\">< id=\"6e2kg\">< id=\"6e2kg\">< id=\"6e2kg\">< id=\"6e2kg\">< id=\"6e2kg\">< id=\"6e2kg\">< id=\"6e2kg\">< id=\"6e2kg\"> < id=\"6e2kg\">< id=\"6e2kg\">< id=\"6e2kg\">< id=\"6e2kg\">< id=\"6e2kg\">< id=\"6e2kg\">< id=\"6e2kg\">< id=\"6e2kg\">< id=\"6e2kg\">< id=\"6e2kg\">< id=\"6e2kg\">< id=\"6e2kg\">< id=\"6e2kg\">< 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https://answers.launchpad.net/yade/+question/690942	[ "# Concrete cube dimension, aggregate particle and peri3dController\n\nAsked by Faqih Maarif on 2020-05-25\n\nDear All,\n\nI am new to YADE. I have studied some code from this forum. There are some of the questions related to concrete cube testing.\n\n1. How to determine the dimensions of the concrete cube (150x150mm) in the modeling? In the code, there is an initial size = 1.2, but apparently, this is not the cube dimension.\n\n2. How to determine the variation of aggregate granules if specified;\na. 0-4mm: 40%\nb. 4-8 mm: 22%\nc. 8-16mm: 38%\n\n3. How to determine the value (goal, xxPath, yyPath, zzPath, zxPath, xyPath), I'm confused with the determination of the numbers . I have read it in YADE Book (pg.280), but i dont understand about it.\n\ngoal=(20e-4,-6e-4,0, -2e6,3e-4,2e6)\n# the prescribed path (step,value of stress/strain) can be defined in absolute values\nxxPath=[(465,5e-4),(934,-5e-4),(1134,10e-4)],\n# or in relative values\nyyPath=[(2,4),(7,-2),(11,0),(14,4)],\n# if the goal value is 0, the absolute stress/strain values are always considered (step values remain relative)\nzzPath=[(5,-1e7),(10,0)],\n# if ##Path is not explicitly defined, it is considered as linear function between (0,0) and (nSteps,goal)\n# as in yzPath and xyPath\n# the relative values are really relative (zxPath gives the same - except of the sign from goal value - result as yyPath)\nzxPath=[(4,2),(14,-1),(22,0),(28,2)],\nxyPath=[(1,1),(2,-1),(3,1),(4,-1),(5,1)],\n\nRegards,\nFaqih Ma’arif\n\n**************************************************************************************************************************************************************************************************************\nthe complete code as follows:\n\n# peri3dController_example1.py\n# script, that explains funcionality and input parameters of Peri3dController\n\nimport string\n\n# create some material\n#O.materials.append(CpmMat(neverDamage=True,young=25e9,frictionAngle=.7,poisson=.2,sigmaT=3e6,epsCrackOnset=1e-4,relDuctility=30))\nO.materials.append(CpmMat(neverDamage=True,young=25e9,frictionAngle=.7,poisson=.2,sigmaT=3e6,epsCrackOnset=1e-4,relDuctility=30))\n\n# create periodic assembly of particles\ninitSize=1.2 #old\n\nsp.toSimulation()\n\n# plotting\n#plot.live=False\nplot.plots={'progress':('sx','sy','sz','syz','szx','sxy',),'progress_':('ex','ey','ez','eyz','ezx','exy',)}\nprogress=p3d.progress,progress_=p3d.progress,\nsx=p3d.stress,sy=p3d.stress,sz=p3d.stress,\nsyz=p3d.stress,szx=p3d.stress,sxy=p3d.stress,\nex=p3d.strain,ey=p3d.strain,ez=p3d.strain,\neyz=p3d.strain,ezx=p3d.strain,exy=p3d.strain,\n)\n\n# in how many time steps should be the goal state reached\nnSteps=4000 #new\n\nO.dt=PWaveTimeStep()/2\nEnlargeFactor=1.5\nEnlargeFactor=1.0\nO.engines=[\nForceResetter(),\nInsertionSortCollider([Bo1_Sphere_Aabb(aabbEnlargeFactor=EnlargeFactor,label='bo1s')]),\nInteractionLoop(\n[Ig2_Sphere_Sphere_ScGeom(interactionDetectionFactor=EnlargeFactor,label='ig2ss')],\n[Ip2_CpmMat_CpmMat_CpmPhys()],[Law2_ScGeom_CpmPhys_Cpm()]),\nNewtonIntegrator(),\nPeri3dController(\ngoal=(20e-4,-6e-4,0, -2e6,3e-4,2e6), # Vector6 of prescribed final values (xx,yy,zz, yz,zx,xy)\nstressMask=0b101100, # prescribed ex,ey,sz,syz,ezx,sxy; e..strain; s..stress\nnSteps=nSteps, # how many time steps the simulation will last\n# after reaching nSteps do doneHook action\ndoneHook='print \"Simulation with Peri3dController finished.\"; O.pause()',\n\n# the prescribed path (step,value of stress/strain) can be defined in absolute values\nxxPath=[(465,5e-4),(934,-5e-4),(1134,10e-4)],\n# or in relative values\nyyPath=[(2,4),(7,-2),(11,0),(14,4)],\n# if the goal value is 0, the absolute stress/strain values are always considered (step values remain relative)\nzzPath=[(5,-1e7),(10,0)],\n# if ##Path is not explicitly defined, it is considered as linear function between (0,0) and (nSteps,goal)\n# as in yzPath and xyPath\n# the relative values are really relative (zxPath gives the same - except of the sign from goal value - result as yyPath)\nzxPath=[(4,2),(14,-1),(22,0),(28,2)],\nxyPath=[(1,1),(2,-1),(3,1),(4,-1),(5,1)],\n# variables used in the first step\nlabel='p3d'\n),\n]\n\nO.step()\nbo1s.aabbEnlargeFactor=ig2ss.interactionDetectionFactor=1.\nO.run(); #O.wait()\n\nplot.plot(subPlots=False)\n\n**************************************************************************************************************************************************************************************************************\n\n## Question information\n\nLanguage:\nEnglish Edit question\nStatus:\nSolved\nFor:\nAssignee:\nNo assignee Edit question\nSolved by:\nJan Stránský\nSolved:\n2020-05-26\nLast query:\n2020-05-26\n2020-05-26\n Bruno Chareyre (bruno-chareyre) said on 2020-05-25: #1\n\nHi,\n\n> there is an initial size = 1.2, but apparently, this is not the cube dimension\n\nWhy not?\n\n> How to determine the value (goal, xxPath, yyPath, zzPath, zxPath, xyPath)\n\nCould you explain the loading path you would like to impose? Then maybe understanding this part will be easier.\n\nRegards\n\nBruno\n\n Jan Stránský (honzik) said on 2020-05-25: #2\n\nHello,\n\n> I am new to YADE\n\nwelcome :-)\n\n> There are some of the questions ...\n\nnext time please open separate \"ask a question\" for each question \n\n> 1...\n> peri3dController\n> How to determine the dimensions of the concrete cube\n> there is an initial size = 1.2, but apparently, this is not the cube dimension.\n\nthere is no \"cube\", the simulation is periodic, simulating infinite space.\nTo get dimension of a periodic cell (which is in no way related to physical dimensions - as it is infinite), you can use O.cell.size .\nThe initSize is passed to randomPeriPack. It creates a loose packing using makeCloud using the initSize size, then compress it. The parameter basically control number of particles in the simulation.\n\nHave a look at yade/examples/concrete for simulation of \"physical\" samples.\n\n> 2. How to determine the variation of aggregate granules if specified;\n\nIt is not possible \"as is\", randomPariPack just supports uniform distribution (defined by radius and rRelFuzz parameters).\nTo use defined particle size distribution, one option is to \"rewrite\" the randomPeriPack function with adjusted makeCloud call.\n\nit is better to use links to online documentation, like \n\n> 3. How to determine the value (goal, xxPath, yyPath, zzPath, zxPath, xyPath), I'm confused with the determination of the numbers . I have read it in YADE Book (pg.280), but i dont understand about it.\n\nPlease describe more what you do not understand.\nThe path arguments are just a list of (time,value) pair, i.e. what value is prescribed at what time/iter.\nIn certain cases, the values may be relative (e.g. the time on scale [0,1], internally rescaling the values with nSteps argument)\nThe prescribed value is linearly interpolated between these defined points.\nHave a look at comments in the example you have posted and compare them with the simulation results.\n\nA note about modeling of concrete using periodic simulations: in case of strain localization, the periodicity has influence on cracks/strain localized area and therefore influence on overall response.\nWhich is not a problem \"by default\", one just need to (should) have this in mind while using it.\n\ncheers\nJan\n\n Faqih Maarif (faqih07) said on 2020-05-26: #3\n\nDear All\nFirst of all, permission, I say thank you for an enlightening answer, and I would like to apologize for not obeying the rules.\n\nThank you also for the answers to the concrete cube's dimensions and the variation of aggregate particles. I will ask separately as a rule in the forum.\n\nMy cases are the following:\nI have tested the compressive strength of the concrete cube (150mmx150mm), and I will be modeling in YADE. In modeling, I will consider stress and strain periodically. I am still confused about how to determine the (xxPath, yyPath,zzPath,zxPath,xyPath), as written below.\n\nxxPath=[(465,5e-4),(934,-5e-4),(1134,10e-4)], #old\nyyPath=[(2,4),(7,-2),(11,0),(14,4)],\nzzPath=[(5,-1e7),(10,0)],\nzxPath=[(4,2),(14,-1),(22,0),(28,2)],\nxyPath=[(1,1),(2,-1),(3,1),(4,-1),(5,1)],\n\nRegards,\nFaqih Ma’arif", null, "Jan Stránský (honzik) said on 2020-05-26: #4\n\n> I have tested the compressive strength of the concrete cube (150mmx150mm)\n> In modeling, I will consider stress and strain periodically.\n\nStrain field in the post-peak range in uniaxial compression is not uniform, so using periodicity needs some caution (!)\n\n> I am still confused about how to determine the (xxPath, yyPath,zzPath,zxPath,xyPath), as written below.\n\nthe easiest is not to bother with paths at all :-) by default they are linear from zero to the goal value, which should be OK for uniaxial test - just define appropriate goal and stressMask.\n\ncheers\nJan\n\n Faqih Maarif (faqih07) said on 2020-05-26: #5\n\nDear\nMr. Jan and Mr. Bruno\n\nThank you very much for your attention and cooperation" ]	[ null, "https://answers.launchpad.net/@@/favourite-yes", null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.82947075,"math_prob":0.9073839,"size":6943,"snap":"2020-24-2020-29","text_gpt3_token_len":1830,"char_repetition_ratio":0.10376135,"word_repetition_ratio":0.26801407,"special_character_ratio":0.2529166,"punctuation_ratio":0.14864865,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9605663,"pos_list":[0,1,2],"im_url_duplicate_count":[null,null,null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2020-07-14T00:43:44Z\",\"WARC-Record-ID\":\"<urn:uuid:8997e860-37cd-4173-84de-87ce282a759b>\",\"Content-Length\":\"42401\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:e6ef6d13-7a83-427d-a07d-a7f1524d522c>\",\"WARC-Concurrent-To\":\"<urn:uuid:9943ff83-ca61-4b21-a54c-60eff9661d5e>\",\"WARC-IP-Address\":\"91.189.89.225\",\"WARC-Target-URI\":\"https://answers.launchpad.net/yade/+question/690942\",\"WARC-Payload-Digest\":\"sha1:JKNZU727V43GZQWP4LS5IB77472GQVPE\",\"WARC-Block-Digest\":\"sha1:QG2BU5N7J26AUOUQQ5R3Q6WMWMYAEXP4\",\"WARC-Identified-Payload-Type\":\"application/xhtml+xml\",\"warc_filename\":\"/cc_download/warc_2020/CC-MAIN-2020-29/CC-MAIN-2020-29_segments_1593657147031.78_warc_CC-MAIN-20200713225620-20200714015620-00421.warc.gz\"}"}
https://www.tgmarinho.com/high-order-functions-%E2%80%94-easy-mode/	[ "# High Order Functions — Easy Mode\n\n## I intend to explain a little bit about High Order Functions with Javascript.\n\nI intend to explain a little bit about High Order Functions with Javascript.\n\nBy the way, A higher order function is a function that takes a function as an argument, or returns a function. This is it, no big deal.\n\nThis one allows JS to be more flexible and pluggable.\n\nLook at code bellow, it is easy, I declare three constants which receive a function that displays to the user an alert message that awaits for an answer.\n\n``const fnName = () => prompt('Tell me your name.') ``\n``const fnMiddle = () => prompt('Tell me your middle name.') ``\n``const fnLastName = () => prompt('Tell me your last name.')``\n\nThen, if you run fnName() in the console, it will appear a small window (modal) like that:\n\nWhen you type your name and click OK, then it finalizes the process.\n\nSo, now I have bellow a new function that receives functions, look it:\n\n``const withNameComplete = (...funcs) => funcs.map(func => func()).join(' ')``\n\nThankfully ES06++ I may use this: …funcs (destructuring arguments), receive a lot of arguments inside of Array, then I have an array of functions, though.\n\nIn the code above the const withNameComplete receive another function that makes anything for us. But I have a complexity with Map in addition to HOC.\n\nThis function can receives a lot of arguments like: withNameComplete(a,b,c,d,e,f), and I have Array(a,b,c,d,e,f) when I use …funcs like my example above.\n\n``````// example declare 3 funcs and I use it passing to withNameComplete\n\nconst fnName = () => prompt('Tell me your name')\nconst fnMiddle = () => prompt('Tell me your middle name')\nconst fnLastName = () => prompt('Tell me your lastname')\n\nconst withNameComplete = (…funcs) => console.log(funcs);\n\nwithNameComplete(fnName, fnMiddle, fnLastName)``````\n\nThe code above produces that:\n\nAbove I'm consoling an array with three functions as I said.\n\nSo, all I need to do is calling each function inside of the array.\n\nNow I have a bunch of alternatives, first I use Map in the second way I use Reduce. Please, If you have a better solution I would love knowing, share your knowledge as well.\n\nFirst solution:\n\n``````const fnName = () => prompt('tell me your name')\nconst fnMiddle = () => prompt('tell me your middle name')\nconst fnLastName = () => prompt('tell me your lastname')\n\n// using Map\nconst withNameComplete = (...funcs) => funcs.map(func => func()).join(' ')\n\nconsole.log(withNameComplete(fnName, fnMiddle, fnLastName))``````\n\nYou can see in line eight I calling the function passing others function, but when I put these arguments I can't call it, only pass. for example, I can't do that, If you will occur an error:\n\n``````const fnName = () => prompt('tell me your name')\nconst fnMiddle = () => prompt('tell me your middle name')\nconst fnLastName = () => prompt('tell me your lastname')\n\n// using Map\nconst withNameComplete = (...funcs) => funcs.map(func => func()).join(' ')\n\nconsole.log(withNameComplete(fnName(), fnMiddle(), fnLastName()))``````\n\nUsing Map from ES06 I may iterate each function inside of Array func and execute, it will return a new array (thanks immutability, it is another post maybe) and then I use the function join(' ') that extract the values of the new array and put in a String.\n\nSecond solution:\n\nThis way I will use Reduce:\n\n``````const fnName = () => prompt('Tell me your name')\nconst fnMiddle = () => prompt('Tell me your middle name')\nconst fnLastName = () => prompt('Tell me your lastname')\n\n// using Reduce\nconst withNameComplete = (...funcs) => funcs.reduce((acc, cv) => \\`\\${acc} \\${cv()}\\`.trimStart(), '')\n\nconsole.log(withNameComplete(fnName, fnMiddle, fnLastName))``````\n\nCheck it out, only changed map to reduce and I have using template string to concatenate strings and using the function trimStart for removing white space of initial String because accumulator starts with this one value.\n\nWith the function reduce the return of method is the value accumulated (acc) plus current value (cv). How I am using template string then it is hard to realize, though.\n\nFinally, HOC, Array.Map and Array.Reduce are subjects hard to understand in begin, but with the theory and practice you gotcha it easily, don't give up, keep studying is the best way.\n\nSo, again, if you have a better solution I would love knowing.\n\nAll in one:\n\n``````// Training HOF => High Order Functions\n\nconst fnName = () => prompt('tell me your name')\nconst fnMiddle = () => prompt('tell me your middle name')\nconst fnLastName = () => prompt('tell me your lastname')\n\n// Debuging\n//const withNameComplete = (...funcs) => console.log(funcs)\n\n// using Map\n//const withNameComplete = (...funcs) => funcs.map(func => func()).join(' ')\n\n// using Reduce\n\nconst withNameComplete = (...funcs) => funcs.reduce((acc, cv, index, array) => \\`\\${acc} \\${cv()}\\`.trimStart(), '')\n\nconsole.log(withNameComplete(fnName, fnMiddle, fnLastName))``````\n\nLast but not least, whether you do you wanna see something cool and advance about HOC and Reduce, take a look this: https://github.com/acdlite/recompose/blob/master/src/packages/recompose/compose.js" ]	[ null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.7585449,"math_prob":0.85920286,"size":5246,"snap":"2019-51-2020-05","text_gpt3_token_len":1240,"char_repetition_ratio":0.18237314,"word_repetition_ratio":0.25665858,"special_character_ratio":0.26401067,"punctuation_ratio":0.14994934,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9510637,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2020-01-19T08:38:54Z\",\"WARC-Record-ID\":\"<urn:uuid:07603b25-be08-401f-83d6-8f5a02adf402>\",\"Content-Length\":\"27607\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:8308cd17-7fae-4967-8f80-0463f805df4e>\",\"WARC-Concurrent-To\":\"<urn:uuid:26b6486e-f487-4e96-8657-af4e2c67f827>\",\"WARC-IP-Address\":\"157.245.130.6\",\"WARC-Target-URI\":\"https://www.tgmarinho.com/high-order-functions-%E2%80%94-easy-mode/\",\"WARC-Payload-Digest\":\"sha1:SJVGZ2XUVYIM3ZJ3NW35A2OQANXY6EK2\",\"WARC-Block-Digest\":\"sha1:MNFH3TN2W6IS32TWS2I6AYMJE4LCVMEU\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2020/CC-MAIN-2020-05/CC-MAIN-2020-05_segments_1579250594333.5_warc_CC-MAIN-20200119064802-20200119092802-00441.warc.gz\"}"}
https://terrytao.wordpress.com/tag/bolzano-weierstrass-theorem/	[ "You are currently browsing the tag archive for the ‘Bolzano-Weierstrass theorem’ tag.\n\nNonstandard analysis is a mathematical framework in which one extends the standard mathematical universe", null, "${{\\mathfrak U}}$ of standard numbers, standard sets, standard functions, etc. into a larger nonstandard universe", null, "${{}^* {\\mathfrak U}}$ of nonstandard numbers, nonstandard sets, nonstandard functions, etc., somewhat analogously to how one places the real numbers inside the complex numbers, or the rationals inside the reals. This nonstandard universe enjoys many of the same properties as the standard one; in particular, we have the transfer principle that asserts that any statement in the language of first order logic is true in the standard universe if and only if it is true in the nonstandard one. (For instance, because Fermat’s last theorem is known to be true for standard natural numbers, it is automatically true for nonstandard natural numbers as well.) However, the nonstandard universe also enjoys some additional useful properties that the standard one does not, most notably the countable saturation property, which is a property somewhat analogous to the completeness property of a metric space; much as metric completeness allows one to assert that the intersection of a countable family of nested closed balls is non-empty, countable saturation allows one to assert that the intersection of a countable family of nested satisfiable formulae is simultaneously satisfiable. (See this previous blog post for more on the analogy between the use of nonstandard analysis and the use of metric completions.) Furthermore, by viewing both the standard and nonstandard universes externally (placing them both inside a larger metatheory, such as a model of Zermelo-Frankel-Choice (ZFC) set theory; in some more advanced set-theoretic applications one may also wish to add some large cardinal axioms), one can place some useful additional definitions and constructions on these universes, such as defining the concept of an infinitesimal nonstandard number (a number which is smaller in magnitude than any positive standard number). The ability to rigorously manipulate infinitesimals is of course one of the most well-known advantages of working with nonstandard analysis.\n\nTo build a nonstandard universe", null, "${{}^* {\\mathfrak U}}$ from a standard one", null, "${{\\mathfrak U}}$, the most common approach is to take an ultrapower of", null, "${{\\mathfrak U}}$ with respect to some non-principal ultrafilter over the natural numbers; see e.g. this blog post for details. Once one is comfortable with ultrafilters and ultrapowers, this becomes quite a simple and elegant construction, and greatly demystifies the nature of nonstandard analysis.\n\nOn the other hand, nonprincipal ultrafilters do have some unappealing features. The most notable one is that their very existence requires the axiom of choice (or more precisely, a weaker form of this axiom known as the boolean prime ideal theorem). Closely related to this is the fact that one cannot actually write down any explicit example of a nonprincipal ultrafilter, but must instead rely on nonconstructive tools such as Zorn’s lemma, the Hahn-Banach theorem, Tychonoff’s theorem, the Stone-Cech compactification, or the boolean prime ideal theorem to locate one. As such, ultrafilters definitely belong to the “infinitary” side of mathematics, and one may feel that it is inappropriate to use such tools for “finitary” mathematical applications, such as those which arise in hard analysis. From a more practical viewpoint, because of the presence of the infinitary ultrafilter, it can be quite difficult (though usually not impossible, with sufficient patience and effort) to take a finitary result proven via nonstandard analysis and coax an effective quantitative bound from it.\n\nThere is however a “cheap” version of nonstandard analysis which is less powerful than the full version, but is not as infinitary in that it is constructive (in the sense of not requiring any sort of choice-type axiom), and which can be translated into standard analysis somewhat more easily than a fully nonstandard argument; indeed, a cheap nonstandard argument can often be presented (by judicious use of asymptotic notation) in a way which is nearly indistinguishable from a standard one. It is obtained by replacing the nonprincipal ultrafilter in fully nonstandard analysis with the more classical Fréchet filter of cofinite subsets of the natural numbers, which is the filter that implicitly underlies the concept of the classical limit", null, "${\\lim_{{\\bf n} \\rightarrow \\infty} a_{\\bf n}}$ of a sequence when the underlying asymptotic parameter", null, "${{\\bf n}}$ goes off to infinity. As such, “cheap nonstandard analysis” aligns very well with traditional mathematics, in which one often allows one’s objects to be parameterised by some external parameter such as", null, "${{\\bf n}}$, which is then allowed to approach some limit such as", null, "${\\infty}$. The catch is that the Fréchet filter is merely a filter and not an ultrafilter, and as such some of the key features of fully nonstandard analysis are lost. Most notably, the law of the excluded middle does not transfer over perfectly from standard analysis to cheap nonstandard analysis; much as there exist bounded sequences of real numbers (such as", null, "${0,1,0,1,\\ldots}$) which do not converge to a (classical) limit, there exist statements in cheap nonstandard analysis which are neither true nor false (at least without passing to a subsequence, see below). The loss of such a fundamental law of mathematical reasoning may seem like a major disadvantage for cheap nonstandard analysis, and it does indeed make cheap nonstandard analysis somewhat weaker than fully nonstandard analysis. But in some situations (particularly when one is reasoning in a “constructivist” or “intuitionistic” fashion, and in particular if one is avoiding too much reliance on set theory) it turns out that one can survive the loss of this law; and furthermore, the law of the excluded middle is still available for standard analysis, and so one can often proceed by working from time to time in the standard universe to temporarily take advantage of this law, and then transferring the results obtained there back to the cheap nonstandard universe once one no longer needs to invoke the law of the excluded middle. Furthermore, the law of the excluded middle can be recovered by adopting the freedom to pass to subsequences with regards to the asymptotic parameter", null, "${{\\bf n}}$; this technique is already in widespread use in the analysis of partial differential equations, although it is generally referred to by names such as “the compactness method” rather than as “cheap nonstandard analysis”.\n\nBelow the fold, I would like to describe this cheap version of nonstandard analysis, which I think can serve as a pedagogical stepping stone towards fully nonstandard analysis, as it is formally similar to (though weaker than) fully nonstandard analysis, but on the other hand is closer in practice to standard analysis. As we shall see below, the relation between cheap nonstandard analysis and standard analysis is analogous in many ways to the relation between probabilistic reasoning and deterministic reasoning; it also resembles somewhat the preference in much of modern mathematics for viewing mathematical objects as belonging to families (or to categories) to be manipulated en masse, rather than treating each object individually. (For instance, nonstandard analysis can be used as a partial substitute for scheme theory in order to obtain uniformly quantitative results in algebraic geometry, as discussed for instance in this previous blog post.)\n\nMany structures in mathematics are incomplete in one or more ways. For instance, the field of rationals", null, "${{\\bf Q}}$ or the reals", null, "${{\\bf R}}$ are algebraically incomplete, because there are some non-trivial algebraic equations (such as", null, "${x^2=2}$ in the case of the rationals, or", null, "${x^2=-1}$ in the case of the reals) which could potentially have solutions (because they do not imply a necessarily false statement, such as", null, "${1=0}$, just using the laws of algebra), but do not actually have solutions in the specified field.\n\nSimilarly, the rationals", null, "${{\\bf Q}}$, when viewed now as a metric space rather than as a field, are also metrically incomplete, beause there exist sequences in the rationals (e.g. the decimal approximations", null, "${3, 3.1, 3.14, 3.141, \\ldots}$ of the irrational number", null, "${\\pi}$) which could potentially converge to a limit (because they form a Cauchy sequence), but do not actually converge in the specified metric space.\n\nA third type of incompleteness is that of logical incompleteness, which applies now to formal theories rather than to fields or metric spaces. For instance, Zermelo-Frankel-Choice (ZFC) set theory is logically incomplete, because there exist statements (such as the consistency of ZFC) which could potentially be provable by the theory (because it does not lead to a contradiction, or at least so we believe, just from the axioms and deductive rules of the theory), but is not actually provable in this theory.\n\nA fourth type of incompleteness, which is slightly less well known than the above three, is what I will call elementary incompleteness (and which model theorists call the failure of the countable saturation property). It applies to any structure that is describable by a first-order language, such as a field, a metric space, or a universe of sets. For instance, in the language of ordered real fields, the real line", null, "${{\\bf R}}$ is elementarily incomplete, because there exists a sequence of statements (such as the statements", null, "${0 < x < 1/n}$ for natural numbers", null, "${n=1,2,\\ldots}$) in this language which are potentially simultaneously satisfiable (in the sense that any finite number of these statements can be satisfied by some real number", null, "${x}$) but are not actually simultaneously satisfiable in this theory.\n\nIn each of these cases, though, it is possible to start with an incomplete structure and complete it to a much larger structure to eliminate the incompleteness. For instance, starting with an arbitrary field", null, "${k}$, one can take its algebraic completion (or algebraic closure)", null, "${\\overline{k}}$; for instance,", null, "${{\\bf C} = \\overline{{\\bf R}}}$ can be viewed as the algebraic completion of", null, "${{\\bf R}}$. This field is usually significantly larger than the original field", null, "${k}$, but contains", null, "${k}$ as a subfield, and every element of", null, "${\\overline{k}}$ can be described as the solution to some polynomial equation with coefficients in", null, "${k}$. Furthermore,", null, "${\\overline{k}}$ is now algebraically complete (or algebraically closed): every polynomial equation in", null, "${\\overline{k}}$ which is potentially satisfiable (in the sense that it does not lead to a contradiction such as", null, "${1=0}$ from the laws of algebra), is actually satisfiable in", null, "${\\overline{k}}$.\n\nSimilarly, starting with an arbitrary metric space", null, "${X}$, one can take its metric completion", null, "${\\overline{X}}$; for instance,", null, "${{\\bf R} = \\overline{{\\bf Q}}}$ can be viewed as the metric completion of", null, "${{\\bf Q}}$. Again, the completion", null, "${\\overline{X}}$ is usually much larger than the original metric space", null, "${X}$, but contains", null, "${X}$ as a subspace, and every element of", null, "${\\overline{X}}$ can be described as the limit of some Cauchy sequence in", null, "${X}$. Furthermore,", null, "${\\overline{X}}$ is now a complete metric space: every sequence in", null, "${\\overline{X}}$ which is potentially convergent (in the sense of being a Cauchy sequence), is now actually convegent in", null, "${\\overline{X}}$.\n\nIn a similar vein, we have the Gödel completeness theorem, which implies (among other things) that for any consistent first-order theory", null, "${T}$ for a first-order language", null, "${L}$, there exists at least one completion", null, "${\\overline{T}}$ of that theory", null, "${T}$, which is a consistent theory in which every sentence in", null, "${L}$ which is potentially true in", null, "${\\overline{T}}$ (because it does not lead to a contradiction in", null, "${\\overline{T}}$) is actually true in", null, "${\\overline{T}}$. Indeed, the completeness theorem provides at least one model (or structure)", null, "${{\\mathfrak U}}$ of the consistent theory", null, "${T}$, and then the completion", null, "${\\overline{T} = \\hbox{Th}({\\mathfrak U})}$ can be formed by interpreting every sentence in", null, "${L}$ using", null, "${{\\mathfrak U}}$ to determine its truth value. Note, in contrast to the previous two examples, that the completion is usually not unique in any way; a theory", null, "${T}$ can have multiple inequivalent models", null, "${{\\mathfrak U}}$, giving rise to distinct completions of the same theory.\n\nFinally, if one starts with an arbitrary structure", null, "${{\\mathfrak U}}$, one can form an elementary completion", null, "${{}^* {\\mathfrak U}}$ of it, which is a significantly larger structure which contains", null, "${{\\mathfrak U}}$ as a substructure, and such that every element of", null, "${{}^* {\\mathfrak U}}$ is an elementary limit of a sequence of elements in", null, "${{\\mathfrak U}}$ (I will define this term shortly). Furthermore,", null, "${{}^* {\\mathfrak U}}$ is elementarily complete; any sequence of statements that are potentially simultaneously satisfiable in", null, "${{}^* {\\mathfrak U}}$ (in the sense that any finite number of statements in this collection are simultaneously satisfiable), will actually be simultaneously satisfiable. As we shall see, one can form such an elementary completion by taking an ultrapower of the original structure", null, "${{\\mathfrak U}}$. If", null, "${{\\mathfrak U}}$ is the standard universe of all the standard objects one considers in mathematics, then its elementary completion", null, "${{}^* {\\mathfrak U}}$ is known as the nonstandard universe, and is the setting for nonstandard analysis.\n\nAs mentioned earlier, completion tends to make a space much larger and more complicated. If one algebraically completes a finite field, for instance, one necessarily obtains an infinite field as a consequence. If one metrically completes a countable metric space with no isolated points, such as", null, "${{\\bf Q}}$, then one necessarily obtains an uncountable metric space (thanks to the Baire category theorem). If one takes a logical completion of a consistent first-order theory that can model true arithmetic, then this completion is no longer describable by a recursively enumerable schema of axioms, thanks to Gödel’s incompleteness theorem. And if one takes the elementary completion of a countable structure, such as the integers", null, "${{\\bf Z}}$, then the resulting completion", null, "${{}^* {\\bf Z}}$ will necessarily be uncountable.\n\nHowever, there are substantial benefits to working in the completed structure which can make it well worth the massive increase in size. For instance, by working in the algebraic completion of a field, one gains access to the full power of algebraic geometry. By working in the metric completion of a metric space, one gains access to powerful tools of real analysis, such as the Baire category theorem, the Heine-Borel theorem, and (in the case of Euclidean completions) the Bolzano-Weierstrass theorem. By working in a logically and elementarily completed theory (aka a saturated model) of a first-order theory, one gains access to the branch of model theory known as definability theory, which allows one to analyse the structure of definable sets in much the same way that algebraic geometry allows one to analyse the structure of algebraic sets. Finally, when working in an elementary completion of a structure, one gains a sequential compactness property, analogous to the Bolzano-Weierstrass theorem, which can be interpreted as the foundation for much of nonstandard analysis, as well as providing a unifying framework to describe various correspondence principles between finitary and infinitary mathematics.\n\nIn this post, I wish to expand upon these above points with regard to elementary completion, and to present nonstandard analysis as a completion of standard analysis in much the same way as, say, complex algebra is a completion of real algebra, or real metric geometry is a completion of rational metric geometry.", null, "Usman Nizami on Applets", null, "Anonymous on 247B, Notes 3: pseudodifferent…", null, "Xiaoyan Su on 247B, Notes 3: pseudodifferent…", null, "Rex on 247B, Notes 4: almost everywhe…", null, "Rex on 247B, Notes 4: almost everywhe…", null, "Terence Tao on 254A, Notes 1: Local well-pose…", null, "Terence Tao on The completeness and compactne…", null, "Terence Tao on 247B, Notes 3: pseudodifferent…", null, "Terence Tao on 247B, Notes 4: almost everywhe…", null, "John Mangual on 247B, Notes 4: almost everywhe…", null, "Liam Shakib Llamazar… on 254A, Notes 1: Local well-pose…", null, "Alan Chang on 247B, Notes 4: almost everywhe…", null, "Anonymous on 247B, Notes 1: Restriction…", null, "Anonymous on 247B, Notes 4: almost everywhe…", null, "Anonymous on 247B, Notes 1: Restriction…" ]	[ null, "https://s0.wp.com/latex.php", null, "https://s0.wp.com/latex.php", null, "https://s0.wp.com/latex.php", null, "https://s0.wp.com/latex.php", null, "https://s0.wp.com/latex.php", null, "https://s0.wp.com/latex.php", null, "https://s0.wp.com/latex.php", null, "https://s0.wp.com/latex.php", null, "https://s0.wp.com/latex.php", null, "https://s0.wp.com/latex.php", null, "https://s0.wp.com/latex.php", null, "https://s0.wp.com/latex.php", null, "https://s0.wp.com/latex.php", null, 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https://www.freethesaurus.com/Holomorphic	[ "# analytic\n\n(redirected from Holomorphic)\nAlso found in: Dictionary, Medical, Encyclopedia, Wikipedia.\nGraphic Thesaurus 🔍\n Display ON Animation ON\n Legend Synonym Antonym Related\n\n## Synonyms for analytic\n\n### of a proposition that is necessarily true independent of fact or experience\n\n#### Antonyms\n\nReferences in periodicals archive ?\nLIU, Modified Roper-Suffridge operator for some holomorphic mappings, Front.\nSkrypnik, \"On holomorphic solutions of the Darwin equations of motion of point charges,\" Ukrainian Mathematical Journal, vol.\n(Idea of the proof) by the standard holomorphic coordinate changes, r(w) has the Taylor series expansion as in (8).\nWe denote the Smirnov class by [N.sub.](U), which consists of all holomorphic functions f on U such that log(1 + [absolute value of (f(z))]) [less than or equal to] Q[[phi]](z) (z [member of] U) for some [phi] [member of] [L.sup.1](T), [phi] [greater than or equal to] 0, where the right side denotes the Poisson integral of [phi] on U.\nHolland and Walsh characterized holomorphic Bloch space in D in terms of weighted Euclidian Lipschitz functions of indices (1/2,1/2).\nin [epsilon] with coefficients [a.sub.i](z) in the ring O(r) of holomorphic functions on [D.sub.r], continuous in its closure, satisfying\nShafikov, \"Analytic continuation of holomorphic mappings from nonminimal hypersurfaces,\" Indiana University Mathematics Journal, vol.\nwhere a tangent space index A = 1, ..., 6 has been split into a holomorphic index i = 1,2,3 and an antiholomorphic index [bar.i] = 1,2,3.\nIf f: D \\ E c O is holomorphic and bounded, then f has a unique holomorphic extension to D.\nConsider the holomorphic function F(z) := [??](z) - z defined on [B.sub.[delta]].\nIf P (or F) is parallel, then the holomorphic distribution H is intergrable.\nOn [R.sup.2], a 1-quasiconformal mapping is holomorphic or antiholomorphic.\nIf D is a non-empty simply connected open subset of the complex plane C which is not all of C, then there exists a biholomorphic (bijective and holomorphic) mapping f from D onto the open unit disk U = {z [member of] C :[absolute value of z] <1} (Krantz, 1999, Section 6.4.3, p.\nLet [A.sup..sub.n[zeta]] = {f [member of] H(U x [bar.U]), f(z, [zeta]) = z + [a.sub.n+1]([zeta])[z.sup.n+1] + **, z [member of] U, [zeta] [member of] [bar.U]}, with [A.sup..sub.1[zeta]] = [A.sup.*.sub.[zeta]], where [a.sub.k]([zeta]) are holomorphic functions in [bar.U] for k [greater than or equal to] 2, and\nwith holomorphic coefficients [a.sub.k] (t, z) on some domain D [subset] C with respect to t and near the origin in C with respect to z.\nSite: Follow: Share:\nOpen / Close" ]	[ null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.8262012,"math_prob":0.98134655,"size":2261,"snap":"2021-21-2021-25","text_gpt3_token_len":688,"char_repetition_ratio":0.15152858,"word_repetition_ratio":0.005586592,"special_character_ratio":0.29102167,"punctuation_ratio":0.18811882,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9948972,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2021-05-12T15:15:28Z\",\"WARC-Record-ID\":\"<urn:uuid:23c6c07f-e303-412b-9f8f-8a4bba77933a>\",\"Content-Length\":\"62377\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:ff789c30-af66-4257-908c-28ab3311ef96>\",\"WARC-Concurrent-To\":\"<urn:uuid:6779f6e6-e532-4336-b5c7-034222ab67d0>\",\"WARC-IP-Address\":\"45.35.33.114\",\"WARC-Target-URI\":\"https://www.freethesaurus.com/Holomorphic\",\"WARC-Payload-Digest\":\"sha1:BSWCCBQDNP622IWCUYLDNWRZ4SJ2WAFS\",\"WARC-Block-Digest\":\"sha1:MCNGG4AYGQW7ATBUP2T7W4B6X5DM5CSN\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2021/CC-MAIN-2021-21/CC-MAIN-2021-21_segments_1620243990929.24_warc_CC-MAIN-20210512131604-20210512161604-00547.warc.gz\"}"}
https://planetmath.org/polynomialfunctionalcalculus	[ "# polynomial functional calculus\n\nLet $\\mathcal{A}$ be an unital associative algebra over $\\mathbb{C}$ with identity element", null, "", null, "$e$ and let $a\\in\\mathcal{A}$.\n\nThe polynomial functional calculus is the most basic form of a functional calculus. It allows the expression\n\n $\\displaystyle p(a)$\n\nto make sense as an element of $\\mathcal{A}$, for any polynomial", null, "", null, "", null, "", null, "$p:\\mathbb{C}\\longrightarrow\\mathbb{C}$.\n\nThis is achieved in the following natural way: for any polynomial $p(\\lambda):=\\sum c_{n}\\,\\lambda^{n}$ we the element $p(a):=\\sum c_{n}\\,a^{n}\\in\\mathcal{A}$.\n\n## 1 Definition\n\nRecall that the set of polynomial functions in $\\mathbb{C}$, denoted by $\\mathbb{C}[\\lambda]$, is an associative algebra over $\\mathbb{C}$ under pointwise operations and is generated by the constant polynomial $1$ and the variable", null, "", null, "$\\lambda$ (corresponding to the identity function in $\\mathbb{C}$).\n\nConsider the algebra homomorphism $\\pi:\\mathbb{C}[\\lambda]\\longrightarrow\\mathcal{A}$ such that $\\pi(1)=e$ and $\\pi(\\lambda)=a$. This homomorphism is denoted by\n\n $\\displaystyle p\\longmapsto p(a)$\n\nand it is called the polynomial functional calculus for $a$.\n\nIt is clear that for any polynomial $p(\\lambda):=\\sum c_{n}\\lambda^{n}$ we have $p(a)=\\sum c_{n}\\,a^{n}$.\n\n## 2 Spectral Properties\n\nWe will denote by $\\sigma(x)$ the spectrum (http://planetmath.org/Spectrum) of an element $x\\in\\mathcal{A}$.\n\nLet $\\mathcal{A}$ be an unital associative algebra over $\\mathbb{C}$ and $a$ an element in $\\mathcal{A}$. For any polynomial $p$ we have that\n\n $\\displaystyle\\sigma(p(a))=p(\\sigma(a))$\n\n: Let us first prove that $\\sigma(p(a))\\subseteq p(\\sigma(a))$. Suppose $\\widetilde{\\lambda}\\in\\sigma(p(a))$, which means that $p(a)-\\widetilde{\\lambda}e$ is not invertible", null, "", null, ". Now consider the polynomial in $\\mathbb{C}$ given by $q:=p-\\widetilde{\\lambda}$. It is clear that $q(a)=p(a)-\\widetilde{\\lambda}e$, and therefore $q(a)$ is not invertible. Since $\\mathbb{C}$ is algebraically closed", null, "", null, "(http://planetmath.org/FundamentalTheoremOfAlgebra), we have that\n\n $\\displaystyle q(\\lambda)=(\\lambda-\\lambda_{1})^{n_{1}}\\cdots(\\lambda-\\lambda_{% k})^{n_{k}}$\n\nfor some $\\lambda_{1},\\dots,\\lambda_{k}\\in\\mathbb{C}$ and $n_{1},\\dots,n_{k}\\in\\mathbb{N}$. Thus, we can also write a similar product", null, "", null, "", null, "for $q(a)$ as\n\n $\\displaystyle q(a)=(a-\\lambda_{1}e)^{n_{1}}\\cdots(a-\\lambda_{k}e)^{n_{k}}$\n\nNow, since $q(a)$ is not invertible we must have that at least one of the factors $(a-\\lambda_{i}e)$ is not invertible, which means that for that particular $\\lambda_{i}$ we have $\\lambda_{i}\\in\\sigma(a)$. But we also have that $q(\\lambda_{i})=0$, i.e. $p(\\lambda_{i})=\\widetilde{\\lambda}$, and hence $\\widetilde{\\lambda}\\in p(\\sigma(a))$.\n\nWe now prove the inclusion $\\sigma(p(a))\\supseteq p(\\sigma(a))$. Suppose $\\widetilde{\\lambda}\\in p(\\sigma(a))$, which means that $\\widetilde{\\lambda}=p(\\lambda_{0})$ for some $\\lambda_{0}\\in\\sigma(a)$. The polynomial $p-\\widetilde{\\lambda}$ has a zero at $\\lambda_{0}$, hence there is a polynomial $d$ such that\n\n $\\displaystyle p(\\lambda)-\\widetilde{\\lambda}=d(\\lambda)(\\lambda-\\lambda_{0})\\,% ,\\qquad\\qquad\\lambda\\in\\mathbb{C}$\n\nThus, we can also write a similar product for $q(a)$ as\n\n $\\displaystyle p(a)-\\widetilde{\\lambda}e=d(a)(a-\\lambda_{0}e)$\n\nIf $p(a)-\\widetilde{\\lambda}e$ was invertible, then we would see that $a-\\lambda_{0}e$ had a left (http://planetmath.org/InversesInRings) and a right inverse", null, "", null, "(http://planetmath.org/InversesInRings), thus being invertible. But we know that $\\lambda_{0}\\in\\sigma(a)$, hence we conclude that $p(a)-\\widetilde{\\lambda}e$ cannot be invertible, i.e. $\\widetilde{\\lambda}\\in\\sigma(p(a))$. $\\square$\n\nTitle polynomial functional calculus PolynomialFunctionalCalculus 2013-03-22 18:48:23 2013-03-22 18:48:23 asteroid (17536) asteroid (17536) 8 asteroid (17536) Feature msc 46H30 msc 47A60 FunctionalCalculus ContinuousFunctionalCalculus2 BorelFunctionalCalculus polynomial spectral mapping theorem" ]	[ null, "http://mathworld.wolfram.com/favicon_mathworld.png", null, "http://planetmath.org/sites/default/files/fab-favicon.ico", null, "http://mathworld.wolfram.com/favicon_mathworld.png", null, "http://planetmath.org/sites/default/files/fab-favicon.ico", null, "http://planetmath.org/sites/default/files/fab-favicon.ico", null, "http://planetmath.org/sites/default/files/fab-favicon.ico", null, "http://mathworld.wolfram.com/favicon_mathworld.png", null, "http://planetmath.org/sites/default/files/fab-favicon.ico", null, "http://planetmath.org/sites/default/files/fab-favicon.ico", null, "http://planetmath.org/sites/default/files/fab-favicon.ico", null, "http://mathworld.wolfram.com/favicon_mathworld.png", null, "http://planetmath.org/sites/default/files/fab-favicon.ico", null, "http://mathworld.wolfram.com/favicon_mathworld.png", null, "http://planetmath.org/sites/default/files/fab-favicon.ico", null, "http://planetmath.org/sites/default/files/fab-favicon.ico", null, "http://mathworld.wolfram.com/favicon_mathworld.png", null, "http://planetmath.org/sites/default/files/fab-favicon.ico", null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.9019902,"math_prob":0.9999846,"size":2580,"snap":"2019-35-2019-39","text_gpt3_token_len":598,"char_repetition_ratio":0.14751554,"word_repetition_ratio":0.05955335,"special_character_ratio":0.22751938,"punctuation_ratio":0.12938596,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":1.000008,"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,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],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2019-09-18T20:17:11Z\",\"WARC-Record-ID\":\"<urn:uuid:485d3bb7-84f2-477f-8453-92326d2c0d4d>\",\"Content-Length\":\"37812\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:20af1357-c4a0-4891-8b49-5299ceb98cb0>\",\"WARC-Concurrent-To\":\"<urn:uuid:37ee203c-6ac6-471c-a070-2a9e085fa137>\",\"WARC-IP-Address\":\"129.97.206.129\",\"WARC-Target-URI\":\"https://planetmath.org/polynomialfunctionalcalculus\",\"WARC-Payload-Digest\":\"sha1:WB3VH63QXMU76N3OQPG5YTKRFMSYVEDB\",\"WARC-Block-Digest\":\"sha1:U3OYOPTGEQK4UPAOODVUECMFHSFOY3EU\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2019/CC-MAIN-2019-39/CC-MAIN-2019-39_segments_1568514573331.86_warc_CC-MAIN-20190918193432-20190918215432-00138.warc.gz\"}"}