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https://docs.opencv.org/4.1.0/d8/d34/group__cudaarithm__elem.html	[ "", null, "OpenCV 4.1.0 Open Source Computer Vision\nPer-element Operations\n\n## Functions\n\nvoid cv::cuda::abs (InputArray src, OutputArray dst, Stream &stream=Stream::Null())\nComputes an absolute value of each matrix element. More...\n\nvoid cv::cuda::absdiff (InputArray src1, InputArray src2, OutputArray dst, Stream &stream=Stream::Null())\nComputes per-element absolute difference of two matrices (or of a matrix and scalar). More...\n\nvoid cv::cuda::add (InputArray src1, InputArray src2, OutputArray dst, InputArray mask=noArray(), int dtype=-1, Stream &stream=Stream::Null())\nComputes a matrix-matrix or matrix-scalar sum. More...\n\nvoid cv::cuda::addWeighted (InputArray src1, double alpha, InputArray src2, double beta, double gamma, OutputArray dst, int dtype=-1, Stream &stream=Stream::Null())\nComputes the weighted sum of two arrays. More...\n\nvoid cv::cuda::bitwise_and (InputArray src1, InputArray src2, OutputArray dst, InputArray mask=noArray(), Stream &stream=Stream::Null())\nPerforms a per-element bitwise conjunction of two matrices (or of matrix and scalar). More...\n\nvoid cv::cuda::bitwise_not (InputArray src, OutputArray dst, InputArray mask=noArray(), Stream &stream=Stream::Null())\nPerforms a per-element bitwise inversion. More...\n\nvoid cv::cuda::bitwise_or (InputArray src1, InputArray src2, OutputArray dst, InputArray mask=noArray(), Stream &stream=Stream::Null())\nPerforms a per-element bitwise disjunction of two matrices (or of matrix and scalar). More...\n\nvoid cv::cuda::bitwise_xor (InputArray src1, InputArray src2, OutputArray dst, InputArray mask=noArray(), Stream &stream=Stream::Null())\nPerforms a per-element bitwise exclusive or operation of two matrices (or of matrix and scalar). More...\n\nvoid cv::cuda::cartToPolar (InputArray x, InputArray y, OutputArray magnitude, OutputArray angle, bool angleInDegrees=false, Stream &stream=Stream::Null())\nConverts Cartesian coordinates into polar. More...\n\nvoid cv::cuda::compare (InputArray src1, InputArray src2, OutputArray dst, int cmpop, Stream &stream=Stream::Null())\nCompares elements of two matrices (or of a matrix and scalar). More...\n\nvoid cv::cuda::divide (InputArray src1, InputArray src2, OutputArray dst, double scale=1, int dtype=-1, Stream &stream=Stream::Null())\nComputes a matrix-matrix or matrix-scalar division. More...\n\nvoid cv::cuda::exp (InputArray src, OutputArray dst, Stream &stream=Stream::Null())\nComputes an exponent of each matrix element. More...\n\nvoid cv::cuda::log (InputArray src, OutputArray dst, Stream &stream=Stream::Null())\nComputes a natural logarithm of absolute value of each matrix element. More...\n\nvoid cv::cuda::lshift (InputArray src, Scalar_< int > val, OutputArray dst, Stream &stream=Stream::Null())\nPerforms pixel by pixel right left of an image by a constant value. More...\n\nvoid cv::cuda::magnitude (InputArray xy, OutputArray magnitude, Stream &stream=Stream::Null())\nComputes magnitudes of complex matrix elements. More...\n\nvoid cv::cuda::magnitude (InputArray x, InputArray y, OutputArray magnitude, Stream &stream=Stream::Null())\n\nvoid cv::cuda::magnitudeSqr (InputArray xy, OutputArray magnitude, Stream &stream=Stream::Null())\nComputes squared magnitudes of complex matrix elements. More...\n\nvoid cv::cuda::magnitudeSqr (InputArray x, InputArray y, OutputArray magnitude, Stream &stream=Stream::Null())\n\nvoid cv::cuda::max (InputArray src1, InputArray src2, OutputArray dst, Stream &stream=Stream::Null())\nComputes the per-element maximum of two matrices (or a matrix and a scalar). More...\n\nvoid cv::cuda::min (InputArray src1, InputArray src2, OutputArray dst, Stream &stream=Stream::Null())\nComputes the per-element minimum of two matrices (or a matrix and a scalar). More...\n\nvoid cv::cuda::multiply (InputArray src1, InputArray src2, OutputArray dst, double scale=1, int dtype=-1, Stream &stream=Stream::Null())\nComputes a matrix-matrix or matrix-scalar per-element product. More...\n\nvoid cv::cuda::phase (InputArray x, InputArray y, OutputArray angle, bool angleInDegrees=false, Stream &stream=Stream::Null())\nComputes polar angles of complex matrix elements. More...\n\nvoid cv::cuda::polarToCart (InputArray magnitude, InputArray angle, OutputArray x, OutputArray y, bool angleInDegrees=false, Stream &stream=Stream::Null())\nConverts polar coordinates into Cartesian. More...\n\nvoid cv::cuda::pow (InputArray src, double power, OutputArray dst, Stream &stream=Stream::Null())\nRaises every matrix element to a power. More...\n\nvoid cv::cuda::rshift (InputArray src, Scalar_< int > val, OutputArray dst, Stream &stream=Stream::Null())\nPerforms pixel by pixel right shift of an image by a constant value. More...\n\nstatic void cv::cuda::scaleAdd (InputArray src1, double alpha, InputArray src2, OutputArray dst, Stream &stream=Stream::Null())\nadds scaled array to another one (dst = alphasrc1 + src2) More...\n\nvoid cv::cuda::sqr (InputArray src, OutputArray dst, Stream &stream=Stream::Null())\nComputes a square value of each matrix element. More...\n\nvoid cv::cuda::sqrt (InputArray src, OutputArray dst, Stream &stream=Stream::Null())\nComputes a square root of each matrix element. More...\n\nvoid cv::cuda::subtract (InputArray src1, InputArray src2, OutputArray dst, InputArray mask=noArray(), int dtype=-1, Stream &stream=Stream::Null())\nComputes a matrix-matrix or matrix-scalar difference. More...\n\ndouble cv::cuda::threshold (InputArray src, OutputArray dst, double thresh, double maxval, int type, Stream &stream=Stream::Null())\nApplies a fixed-level threshold to each array element. More...\n\n## § abs()\n\n void cv::cuda::abs ( InputArray src, OutputArray dst, Stream & stream = Stream::Null() )\n\n#include <opencv2/cudaarithm.hpp>\n\nComputes an absolute value of each matrix element.\n\nParameters\n src Source matrix. dst Destination matrix with the same size and type as src . stream Stream for the asynchronous version.\nSee also\nabs\n\n## § absdiff()\n\n void cv::cuda::absdiff ( InputArray src1, InputArray src2, OutputArray dst, Stream & stream = Stream::Null() )\n\n#include <opencv2/cudaarithm.hpp>\n\nComputes per-element absolute difference of two matrices (or of a matrix and scalar).\n\nParameters\n src1 First source matrix or scalar. src2 Second source matrix or scalar. dst Destination matrix that has the same size and type as the input array(s). stream Stream for the asynchronous version.\nSee also\nabsdiff\n\n## § add()\n\n void cv::cuda::add ( InputArray src1, InputArray src2, OutputArray dst, InputArray mask = noArray(), int dtype = -1, Stream & stream = Stream::Null() )\n\n#include <opencv2/cudaarithm.hpp>\n\nComputes a matrix-matrix or matrix-scalar sum.\n\nParameters\n src1 First source matrix or scalar. src2 Second source matrix or scalar. Matrix should have the same size and type as src1 . dst Destination matrix that has the same size and number of channels as the input array(s). The depth is defined by dtype or src1 depth. mask Optional operation mask, 8-bit single channel array, that specifies elements of the destination array to be changed. The mask can be used only with single channel images. dtype Optional depth of the output array. stream Stream for the asynchronous version.\nSee also\nadd\n\n## § addWeighted()\n\n void cv::cuda::addWeighted ( InputArray src1, double alpha, InputArray src2, double beta, double gamma, OutputArray dst, int dtype = -1, Stream & stream = Stream::Null() )\n\n#include <opencv2/cudaarithm.hpp>\n\nComputes the weighted sum of two arrays.\n\nParameters\n src1 First source array. alpha Weight for the first array elements. src2 Second source array of the same size and channel number as src1 . beta Weight for the second array elements. dst Destination array that has the same size and number of channels as the input arrays. gamma Scalar added to each sum. dtype Optional depth of the destination array. When both input arrays have the same depth, dtype can be set to -1, which will be equivalent to src1.depth(). stream Stream for the asynchronous version.\n\nThe function addWeighted calculates the weighted sum of two arrays as follows:\n\n$\\texttt{dst} (I)= \\texttt{saturate} ( \\texttt{src1} (I) \\texttt{alpha} + \\texttt{src2} (I)* \\texttt{beta} + \\texttt{gamma} )$\n\nwhere I is a multi-dimensional index of array elements. In case of multi-channel arrays, each channel is processed independently.\n\nSee also\naddWeighted\n\n## § bitwise_and()\n\n void cv::cuda::bitwise_and ( InputArray src1, InputArray src2, OutputArray dst, InputArray mask = noArray(), Stream & stream = Stream::Null() )\n\n#include <opencv2/cudaarithm.hpp>\n\nPerforms a per-element bitwise conjunction of two matrices (or of matrix and scalar).\n\nParameters\n src1 First source matrix or scalar. src2 Second source matrix or scalar. dst Destination matrix that has the same size and type as the input array(s). mask Optional operation mask, 8-bit single channel array, that specifies elements of the destination array to be changed. The mask can be used only with single channel images. stream Stream for the asynchronous version.\n\n## § bitwise_not()\n\n void cv::cuda::bitwise_not ( InputArray src, OutputArray dst, InputArray mask = noArray(), Stream & stream = Stream::Null() )\n\n#include <opencv2/cudaarithm.hpp>\n\nPerforms a per-element bitwise inversion.\n\nParameters\n src Source matrix. dst Destination matrix with the same size and type as src . mask Optional operation mask, 8-bit single channel array, that specifies elements of the destination array to be changed. The mask can be used only with single channel images. stream Stream for the asynchronous version.\n\n## § bitwise_or()\n\n void cv::cuda::bitwise_or ( InputArray src1, InputArray src2, OutputArray dst, InputArray mask = noArray(), Stream & stream = Stream::Null() )\n\n#include <opencv2/cudaarithm.hpp>\n\nPerforms a per-element bitwise disjunction of two matrices (or of matrix and scalar).\n\nParameters\n src1 First source matrix or scalar. src2 Second source matrix or scalar. dst Destination matrix that has the same size and type as the input array(s). mask Optional operation mask, 8-bit single channel array, that specifies elements of the destination array to be changed. The mask can be used only with single channel images. stream Stream for the asynchronous version.\n\n## § bitwise_xor()\n\n void cv::cuda::bitwise_xor ( InputArray src1, InputArray src2, OutputArray dst, InputArray mask = noArray(), Stream & stream = Stream::Null() )\n\n#include <opencv2/cudaarithm.hpp>\n\nPerforms a per-element bitwise exclusive or operation of two matrices (or of matrix and scalar).\n\nParameters\n src1 First source matrix or scalar. src2 Second source matrix or scalar. dst Destination matrix that has the same size and type as the input array(s). mask Optional operation mask, 8-bit single channel array, that specifies elements of the destination array to be changed. The mask can be used only with single channel images. stream Stream for the asynchronous version.\n\n## § cartToPolar()\n\n void cv::cuda::cartToPolar ( InputArray x, InputArray y, OutputArray magnitude, OutputArray angle, bool angleInDegrees = false, Stream & stream = Stream::Null() )\n\n#include <opencv2/cudaarithm.hpp>\n\nConverts Cartesian coordinates into polar.\n\nParameters\n x Source matrix containing real components ( CV_32FC1 ). y Source matrix containing imaginary components ( CV_32FC1 ). magnitude Destination matrix of float magnitudes ( CV_32FC1 ). angle Destination matrix of angles ( CV_32FC1 ). angleInDegrees Flag for angles that must be evaluated in degrees. stream Stream for the asynchronous version.\nSee also\ncartToPolar\n\n## § compare()\n\n void cv::cuda::compare ( InputArray src1, InputArray src2, OutputArray dst, int cmpop, Stream & stream = Stream::Null() )\n\n#include <opencv2/cudaarithm.hpp>\n\nCompares elements of two matrices (or of a matrix and scalar).\n\nParameters\n src1 First source matrix or scalar. src2 Second source matrix or scalar. dst Destination matrix that has the same size as the input array(s) and type CV_8U. cmpop Flag specifying the relation between the elements to be checked: CMP_EQ: a(.) == b(.) CMP_GT: a(.) > b(.) CMP_GE: a(.) >= b(.) CMP_LT: a(.) < b(.) CMP_LE: a(.) <= b(.) CMP_NE: a(.) != b(.) stream Stream for the asynchronous version.\nSee also\ncompare\n\n## § divide()\n\n void cv::cuda::divide ( InputArray src1, InputArray src2, OutputArray dst, double scale = 1, int dtype = -1, Stream & stream = Stream::Null() )\n\n#include <opencv2/cudaarithm.hpp>\n\nComputes a matrix-matrix or matrix-scalar division.\n\nParameters\n src1 First source matrix or a scalar. src2 Second source matrix or scalar. dst Destination matrix that has the same size and number of channels as the input array(s). The depth is defined by dtype or src1 depth. scale Optional scale factor. dtype Optional depth of the output array. stream Stream for the asynchronous version.\n\nThis function, in contrast to divide, uses a round-down rounding mode.\n\nSee also\ndivide\n\n## § exp()\n\n void cv::cuda::exp ( InputArray src, OutputArray dst, Stream & stream = Stream::Null() )\n\n#include <opencv2/cudaarithm.hpp>\n\nComputes an exponent of each matrix element.\n\nParameters\n src Source matrix. dst Destination matrix with the same size and type as src . stream Stream for the asynchronous version.\nSee also\nexp\n\n## § log()\n\n void cv::cuda::log ( InputArray src, OutputArray dst, Stream & stream = Stream::Null() )\n\n#include <opencv2/cudaarithm.hpp>\n\nComputes a natural logarithm of absolute value of each matrix element.\n\nParameters\n src Source matrix. dst Destination matrix with the same size and type as src . stream Stream for the asynchronous version.\nSee also\nlog\n\n## § lshift()\n\n void cv::cuda::lshift ( InputArray src, Scalar_< int > val, OutputArray dst, Stream & stream = Stream::Null() )\n\n#include <opencv2/cudaarithm.hpp>\n\nPerforms pixel by pixel right left of an image by a constant value.\n\nParameters\n src Source matrix. Supports 1, 3 and 4 channels images with CV_8U , CV_16U or CV_32S depth. val Constant values, one per channel. dst Destination matrix with the same size and type as src . stream Stream for the asynchronous version.\n\n## § magnitude() [1/2]\n\n void cv::cuda::magnitude ( InputArray xy, OutputArray magnitude, Stream & stream = Stream::Null() )\n\n#include <opencv2/cudaarithm.hpp>\n\nComputes magnitudes of complex matrix elements.\n\nParameters\n xy Source complex matrix in the interleaved format ( CV_32FC2 ). magnitude Destination matrix of float magnitudes ( CV_32FC1 ). stream Stream for the asynchronous version.\nSee also\nmagnitude\n\n## § magnitude() [2/2]\n\n void cv::cuda::magnitude ( InputArray x, InputArray y, OutputArray magnitude, Stream & stream = Stream::Null() )\n\n#include <opencv2/cudaarithm.hpp>\n\nThis is an overloaded member function, provided for convenience. It differs from the above function only in what argument(s) it accepts. computes magnitude of each (x(i), y(i)) vector supports only floating-point source\n\nParameters\n x Source matrix containing real components ( CV_32FC1 ). y Source matrix containing imaginary components ( CV_32FC1 ). magnitude Destination matrix of float magnitudes ( CV_32FC1 ). stream Stream for the asynchronous version.\n\n## § magnitudeSqr() [1/2]\n\n void cv::cuda::magnitudeSqr ( InputArray xy, OutputArray magnitude, Stream & stream = Stream::Null() )\n\n#include <opencv2/cudaarithm.hpp>\n\nComputes squared magnitudes of complex matrix elements.\n\nParameters\n xy Source complex matrix in the interleaved format ( CV_32FC2 ). magnitude Destination matrix of float magnitude squares ( CV_32FC1 ). stream Stream for the asynchronous version.\n\n## § magnitudeSqr() [2/2]\n\n void cv::cuda::magnitudeSqr ( InputArray x, InputArray y, OutputArray magnitude, Stream & stream = Stream::Null() )\n\n#include <opencv2/cudaarithm.hpp>\n\nThis is an overloaded member function, provided for convenience. It differs from the above function only in what argument(s) it accepts. computes squared magnitude of each (x(i), y(i)) vector supports only floating-point source\n\nParameters\n x Source matrix containing real components ( CV_32FC1 ). y Source matrix containing imaginary components ( CV_32FC1 ). magnitude Destination matrix of float magnitude squares ( CV_32FC1 ). stream Stream for the asynchronous version.\n\n## § max()\n\n void cv::cuda::max ( InputArray src1, InputArray src2, OutputArray dst, Stream & stream = Stream::Null() )\n\n#include <opencv2/cudaarithm.hpp>\n\nComputes the per-element maximum of two matrices (or a matrix and a scalar).\n\nParameters\n src1 First source matrix or scalar. src2 Second source matrix or scalar. dst Destination matrix that has the same size and type as the input array(s). stream Stream for the asynchronous version.\nSee also\nmax\n\n## § min()\n\n void cv::cuda::min ( InputArray src1, InputArray src2, OutputArray dst, Stream & stream = Stream::Null() )\n\n#include <opencv2/cudaarithm.hpp>\n\nComputes the per-element minimum of two matrices (or a matrix and a scalar).\n\nParameters\n src1 First source matrix or scalar. src2 Second source matrix or scalar. dst Destination matrix that has the same size and type as the input array(s). stream Stream for the asynchronous version.\nSee also\nmin\n\n## § multiply()\n\n void cv::cuda::multiply ( InputArray src1, InputArray src2, OutputArray dst, double scale = 1, int dtype = -1, Stream & stream = Stream::Null() )\n\n#include <opencv2/cudaarithm.hpp>\n\nComputes a matrix-matrix or matrix-scalar per-element product.\n\nParameters\n src1 First source matrix or scalar. src2 Second source matrix or scalar. dst Destination matrix that has the same size and number of channels as the input array(s). The depth is defined by dtype or src1 depth. scale Optional scale factor. dtype Optional depth of the output array. stream Stream for the asynchronous version.\nSee also\nmultiply\n\n## § phase()\n\n void cv::cuda::phase ( InputArray x, InputArray y, OutputArray angle, bool angleInDegrees = false, Stream & stream = Stream::Null() )\n\n#include <opencv2/cudaarithm.hpp>\n\nComputes polar angles of complex matrix elements.\n\nParameters\n x Source matrix containing real components ( CV_32FC1 ). y Source matrix containing imaginary components ( CV_32FC1 ). angle Destination matrix of angles ( CV_32FC1 ). angleInDegrees Flag for angles that must be evaluated in degrees. stream Stream for the asynchronous version.\nSee also\nphase\n\n## § polarToCart()\n\n void cv::cuda::polarToCart ( InputArray magnitude, InputArray angle, OutputArray x, OutputArray y, bool angleInDegrees = false, Stream & stream = Stream::Null() )\n\n#include <opencv2/cudaarithm.hpp>\n\nConverts polar coordinates into Cartesian.\n\nParameters\n magnitude Source matrix containing magnitudes ( CV_32FC1 or CV_64FC1 ). angle Source matrix containing angles ( same type as magnitude ). x Destination matrix of real components ( same type as magnitude ). y Destination matrix of imaginary components ( same type as magnitude ). angleInDegrees Flag that indicates angles in degrees. stream Stream for the asynchronous version.\n\n## § pow()\n\n void cv::cuda::pow ( InputArray src, double power, OutputArray dst, Stream & stream = Stream::Null() )\n\n#include <opencv2/cudaarithm.hpp>\n\nRaises every matrix element to a power.\n\nParameters\n src Source matrix. power Exponent of power. dst Destination matrix with the same size and type as src . stream Stream for the asynchronous version.\n\nThe function pow raises every element of the input matrix to power :\n\n$\\texttt{dst} (I) = \\fork{\\texttt{src}(I)^power}{if \\texttt{power} is integer}{\|\\texttt{src}(I)\|^power}{otherwise}$\n\nSee also\npow\n\n## § rshift()\n\n void cv::cuda::rshift ( InputArray src, Scalar_< int > val, OutputArray dst, Stream & stream = Stream::Null() )\n\n#include <opencv2/cudaarithm.hpp>\n\nPerforms pixel by pixel right shift of an image by a constant value.\n\nParameters\n src Source matrix. Supports 1, 3 and 4 channels images with integers elements. val Constant values, one per channel. dst Destination matrix with the same size and type as src . stream Stream for the asynchronous version.\n\n## § scaleAdd()\n\n static void cv::cuda::scaleAdd ( InputArray src1, double alpha, InputArray src2, OutputArray dst, Stream & stream = Stream::Null() )\ninlinestatic\n\n#include <opencv2/cudaarithm.hpp>\n\nadds scaled array to another one (dst = alpha*src1 + src2)\n\n## § sqr()\n\n void cv::cuda::sqr ( InputArray src, OutputArray dst, Stream & stream = Stream::Null() )\n\n#include <opencv2/cudaarithm.hpp>\n\nComputes a square value of each matrix element.\n\nParameters\n src Source matrix. dst Destination matrix with the same size and type as src . stream Stream for the asynchronous version.\n\n## § sqrt()\n\n void cv::cuda::sqrt ( InputArray src, OutputArray dst, Stream & stream = Stream::Null() )\n\n#include <opencv2/cudaarithm.hpp>\n\nComputes a square root of each matrix element.\n\nParameters\n src Source matrix. dst Destination matrix with the same size and type as src . stream Stream for the asynchronous version.\nSee also\nsqrt\n\n## § subtract()\n\n void cv::cuda::subtract ( InputArray src1, InputArray src2, OutputArray dst, InputArray mask = noArray(), int dtype = -1, Stream & stream = Stream::Null() )\n\n#include <opencv2/cudaarithm.hpp>\n\nComputes a matrix-matrix or matrix-scalar difference.\n\nParameters\n src1 First source matrix or scalar. src2 Second source matrix or scalar. Matrix should have the same size and type as src1 . dst Destination matrix that has the same size and number of channels as the input array(s). The depth is defined by dtype or src1 depth. mask Optional operation mask, 8-bit single channel array, that specifies elements of the destination array to be changed. The mask can be used only with single channel images. dtype Optional depth of the output array. stream Stream for the asynchronous version.\nSee also\nsubtract\n\n## § threshold()\n\n double cv::cuda::threshold ( InputArray src, OutputArray dst, double thresh, double maxval, int type, Stream & stream = Stream::Null() )\n\n#include <opencv2/cudaarithm.hpp>\n\nApplies a fixed-level threshold to each array element.\n\nParameters\n src Source array (single-channel). dst Destination array with the same size and type as src . thresh Threshold value. maxval Maximum value to use with THRESH_BINARY and THRESH_BINARY_INV threshold types. type Threshold type. For details, see threshold . The THRESH_OTSU and THRESH_TRIANGLE threshold types are not supported. stream Stream for the asynchronous version.\nSee also\nthreshold" ]	[ null, "https://docs.opencv.org/4.1.0/opencv-logo-small.png", null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.5427387,"math_prob":0.9287094,"size":21420,"snap":"2021-21-2021-25","text_gpt3_token_len":5558,"char_repetition_ratio":0.20932947,"word_repetition_ratio":0.70894754,"special_character_ratio":0.2608777,"punctuation_ratio":0.23320659,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9926062,"pos_list":[0,1,2],"im_url_duplicate_count":[null,null,null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2021-06-14T03:04:50Z\",\"WARC-Record-ID\":\"<urn:uuid:ab54af64-7d7a-4fa8-b497-bbef9ae67b53>\",\"Content-Length\":\"130561\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:24c66dde-8dc8-4855-b858-64455b7d20d3>\",\"WARC-Concurrent-To\":\"<urn:uuid:60d1c102-81c8-480e-9b0b-fc6106dd996e>\",\"WARC-IP-Address\":\"207.38.86.214\",\"WARC-Target-URI\":\"https://docs.opencv.org/4.1.0/d8/d34/group__cudaarithm__elem.html\",\"WARC-Payload-Digest\":\"sha1:UZWUDN6DFDKGABIKVWZ2KRR7ABKRUJU5\",\"WARC-Block-Digest\":\"sha1:GMT3Z7CUB7QOK65L3WQ46TD72PIGRWVX\",\"WARC-Identified-Payload-Type\":\"application/xhtml+xml\",\"warc_filename\":\"/cc_download/warc_2021/CC-MAIN-2021-25/CC-MAIN-2021-25_segments_1623487611320.18_warc_CC-MAIN-20210614013350-20210614043350-00433.warc.gz\"}"}
http://ac.yotesfan.com/33-PhysicsScript/index.php?id=33000B06	[ "File Completed.\n\n• id = 0x33000b06\n• num_script_data = 0x00000006\n• script_data (6)\n• 0 (3)\n• start_time_lo = 0x00000000\n• start_time_hi = 0x00000000\n• hook (13)\n• type = 0x0000000d\n• type_name = kCreateParticle\n• direction = 0x00000000\n• emitterInfoId = 0x320005ff\n• partIndex = 0x00000000\n• x = 0\n• y = 0\n• z = -0.80000001192093\n• qw = 1\n• qx = 0\n• qy = 0\n• qz = 0\n• emitterId = 0x00000000\n• 1 (3)\n• start_time_lo = 0x00000000\n• start_time_hi = 0x00000000\n• hook (13)\n• type = 0x0000000d\n• type_name = kCreateParticle\n• direction = 0x00000000\n• emitterInfoId = 0x320005fd\n• partIndex = 0x0000000f\n• x = 0\n• y = 0\n• z = 0\n• qw = 1\n• qx = 0\n• qy = 0\n• qz = 0\n• emitterId = 0x00000000\n• 2 (3)\n• start_time_lo = 0x00000000\n• start_time_hi = 0x00000000\n• hook (13)\n• type = 0x0000000d\n• type_name = kCreateParticle\n• direction = 0x00000000\n• emitterInfoId = 0x32000600\n• partIndex = 0x00000010\n• x = 0\n• y = 0\n• z = 0.89999997615814\n• qw = 1\n• qx = 0\n• qy = 0\n• qz = 0\n• emitterId = 0x00000000\n• 3 (3)\n• start_time_lo = 0x00000000\n• start_time_hi = 0x00000000\n• hook (13)\n• type = 0x0000000d\n• type_name = kCreateParticle\n• direction = 0x00000000\n• emitterInfoId = 0x320005fd\n• partIndex = 0x0000000c\n• x = 0\n• y = 0\n• z = 0\n• qw = 1\n• qx = 0\n• qy = 0\n• qz = 0\n• emitterId = 0x00000000\n• 4 (3)\n• start_time_lo = 0x00000000\n• start_time_hi = 0x00000000\n• hook (7)\n• type = 0x00000015\n• type_name = kSoundTweaked\n• direction = 0x00000000\n• soundId = 0x0a00047c\n• priority = 1\n• probability = 0.80000001192093\n• volume = 1\n• 5 (3)\n• start_time_lo = 0x00000000\n• start_time_hi = 0x40120000\n• hook (5)\n• type = 0x00000013\n• type_name = kCallPES\n• direction = 0x00000000\n• pesId = 0x33000b03\n• pause = 0" ]	[ null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.6260952,"math_prob":1.0000082,"size":1711,"snap":"2019-13-2019-22","text_gpt3_token_len":804,"char_repetition_ratio":0.31341535,"word_repetition_ratio":0.6147309,"special_character_ratio":0.60315603,"punctuation_ratio":0.020725388,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9877095,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2019-03-27T03:01:41Z\",\"WARC-Record-ID\":\"<urn:uuid:09db71fc-3b69-4c80-9377-e2cd9b6d5fc8>\",\"Content-Length\":\"11849\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:b9f30e26-e46a-4d2e-9b6f-6c31bf1611d7>\",\"WARC-Concurrent-To\":\"<urn:uuid:df93c8b5-6102-4b4e-bcfb-e88e387c29b6>\",\"WARC-IP-Address\":\"70.32.68.67\",\"WARC-Target-URI\":\"http://ac.yotesfan.com/33-PhysicsScript/index.php?id=33000B06\",\"WARC-Payload-Digest\":\"sha1:UPKGSOCGNRVMDTAQJWCAC4BILBYPNT6M\",\"WARC-Block-Digest\":\"sha1:2LFUA3YMWCWLU7TC23RV4OQFXKE6EUI4\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2019/CC-MAIN-2019-13/CC-MAIN-2019-13_segments_1552912207618.95_warc_CC-MAIN-20190327020750-20190327042750-00391.warc.gz\"}"}
http://www.test3.guru99.com/python-loops-while-for-break-continue-enumerate.html	[ "### What is Loop?\n\nLoops can execute a block of code number of times until a certain condition is met. Their usage is fairly common in programming. Unlike other programming language that have For Loop, while loop, dowhile, etc.\n\n### What is For Loop?\n\nFor loop is used to iterate over elements of a sequence. It is often used when you have a piece of code which you want to repeat \"n\" number of time.\n\n### What is While Loop?\n\nWhile Loop is used to repeat a block of code. Instead of running the code block once, It executes the code block multiple times until a certain condition is met.\n\nIn this tutorial, we will learn\n\n## How to use \"While Loop\"\n\nWhile loop does the exactly same thing what \"if statement\" does, but instead of running the code block once, they jump back to the point where it began the code and repeats the whole process again.\n\nSyntax\n\n```while expression\nStatement\n```\n\nExample:\n\n```#\n#Example file for working with loops\n#\ndef main():\nx=0\n#define a while loop\nwhile(x <4):\nprint(x)\nx = x+1\n\nif __name__ == \"__main__\":\nmain()```\n• Code Line 4: Variable x is set to 0\n• Code Line 7: While loop checks for condition x<4. The current value of x is 0. Condition is true. Flow of control enters into while Loop\n• Code Line 8: Value of x is printed\n• Code Line 9: x is incremented by 1. Flow of control goes back to line 7. Now the value of x is 1 which is less than 4. The condition is true, and again the while loop is executed. This continues till x becomes 4, and the while condition becomes false.\n\n## How to use \"For Loop\"\n\nIn Python, \"for loops\" are called iterators.\n\nJust like while loop, \"For Loop\" is also used to repeat the program.\n\nBut unlike while loop which depends on condition true or false. \"For Loop\" depends on the elements it has to iterate.\n\nExample:\n\n```#\n#Example file for working with loops\n#\ndef main():\nx=0\n#define a while loop\n#\twhile(x <4):\n#\t\tprint x\n#\t\tx = x+1\n\n#Define a for loop\nfor x in range(2,7):\nprint(x)\n\nif __name__ == \"__main__\":\nmain()```\n\nFor Loop iterates with number declared in the range.\n\nFor example,\n\nFor Loop for x in range (2,7)\n\nWhen this code is executed, it will print the number between 2 and 7 (2,3,4,5,6). In this code, number 7 is not considered inside the range.\n\nFor Loops can also be used for a set of other things and not just number. We will see thin in next section.\n\n## How to use For Loop for String\n\nIn this step, we will see how \"for loops\" can also be used for other things besides numbers.\n\nExample:\n\n```def main():\n#use a for loop over a collection\nMonths = [\"Jan\",\"Feb\",\"Mar\",\"April\",\"May\",\"June\"]\nfor m in Months:\nprint(m)\n\nif __name__ == \"__main__\":\nmain()```\n\nCode Line 3: We store the months (\"Jan, Feb , Mar,April,May,June\") in variable Months\n\nCode Line 4: We iterate the for loop over each value in Months. The current value of Months in stored in variable m\n\nCode Line 5: Print the month\n\n## How to use break statements in For Loop\n\nBreakpoint is a unique function in For Loop that allows you to break or terminate the execution of the for loop\n\nExample:\n\n```\ndef main():\n#use a for loop over a collection\n#Months = [\"Jan\",\"Feb\",\"Mar\",\"April\",\"May\",\"June\"]\n#for m in Months:\n#print m\n\n# use the break and continue statements\n\nfor x in range (10,20):\nif (x == 15): break\n#if (x % 2 == 0) : continue\nprint(x)\n\nif __name__ == \"__main__\":\nmain()```\nIn this example, we declared the numbers from 10-20, but we want that our for loop to terminate at number 15 and stop executing further. For that, we declare break function by defining (x==15): break, so as soon as the code calls the number 15 it terminates the program Code Line 10 declare variable x between range (10, 20)\n• Code Line 11 declare the condition for breakpoint at x==15,\n• Code Line 12 checks and repeats the steps until it reaches number 15\n• Code Line 13 Print the result in output\n\n## How to use \"continue statement\" in For Loop\n\nContinue function, as the name indicates, will terminate the current iteration of the for loop BUT will continue execution of the remaining iterations.\n\nExample\n\n```def main():\n#use a for loop over a collection\n#Months = [\"Jan\",\"Feb\",\"Mar\",\"April\",\"May\",\"June\"]\n#for m in Months:\n#print m\n\n# use the break and continue statements\n\nfor x in range (10,20):\n#if (x == 15): break\nif (x % 5 == 0) : continue\nprint(x)\n\nif __name__ == \"__main__\":\nmain()```\n\nContinue statement can be used in for loop when you want to fetch a specific value from the list.\n\nIn our example, we have declared value 10-20, but between these numbers we only want those number that are NOT divisible by 5 or in other words which don't give zero when divided by 5.\n\nSo, in our range (10,11, 12….19,20) only 3 numbers falls (10,15,20) that are divisible by 5 and rest are not.\n\nSo except number 10,15 & 20 the \"for loop\" will not continue and print out those number as output.\n\n• Code line 10 declare the variable x for range (10, 20)\n• Code line 12 declare the condition for x divided by 5=0 continue\n• Code line 13 print the result\n\n## How to use \"enumerate\" function for \"For Loop\"\n\nEnumerate function in \"for loop\" does two things\n\n• It returns the index number for the member\n• And the member of the collection that we are looking at\n\nExample:\n\nEnumerate function is used for the numbering or indexing the members in the list.\n\nSuppose, we want to do numbering for our month ( Jan, Feb, Marc, ….June), so we declare the variable i that enumerate the numbers while m will print the number of month in list.\n\n```def main():\n#use a for loop over a collection\nMonths = [\"Jan\",\"Feb\",\"Mar\",\"April\",\"May\",\"June\"]\nfor i, m in enumerate (Months):\nprint(i,m)\n\n# use the break and continue statements\n\n#for x in range (10,20):\n#if (x == 15): break\n#if (x % 5 == 0) : continue\n#print x\n\nif __name__ == \"__main__\":\nmain()```\n\nWhen code is executed the output of the enumerate function returns the months name with an index number like (0-Jan), (1- Feb), (2- March), etc.\n\n• Code Line 3 declares the list of months [ Jan, Feb,…Jun]\n• Code Line 4 declares variable i and m for For Loop\n• Code Line 5 will print the result and again enter the For Loop for the rest of the months to enumerate\n\n## Pratical Example\n\nLet see another example for For Loop to repeat the same statement over and again.\n\n Python loop Working Code for all exercises Code for while loop ```def main(): x=0 while (x<4): print x x= x+1 if __name__== \"__main__\": main() ``` For Loop Simple Example ```def main(): x=0 for x in range (2,7): print x if __name__== \"__main__\": main() ``` Use of for loop in string ```def main(): Months = [\"Jan\",\"Feb\",\"Mar\",\"April\",\"May\",\"June\"] for m in (Months): print m if __name__== \"__main__\": main() ``` Use break-statement in for loop ```def main(): for x in range (10,20): if (x == 15): break print x if __name__== \"__main__\": main() ``` Use of Continue statement in for loop ```def main(): for x in range (10,20): if (x % 5 == 0): continue print x if __name__== \"__main__\": main() ``` Code for \"enumerate function\" with \"for loop\" ```def main(): Months = [\"Jan\",\"Feb\",\"Mar\",\"April\",\"May\",\"June\"] for i, m in enumerate (Months): print i,m if __name__== \"__main__\": main() ```\n\n## How to use for loop to repeat the same statement over and again\n\nYou can use for loop for even repeating the same statement over and again. Here in the example we have printed out word \"guru99\" three times.\n\nExample: To repeat same statement number of times, we have declared the number in variable i (i in 123). So when you run the code as shown below, it prints the statement (guru99) that many times the number declared for our the variable in ( i in 123).\n\n```for i in '123':\nprint \"guru99\",i,```\n\nLike other programming languages, Python also uses a loop but instead of using a range of different loops it is restricted to only two loops \"While loop\" and \"for loop\".\n\n• While loops are executed based on whether the conditional statement is true or false.\n• For loops are called iterators, it iterates the element based on the condition set\n• Python For loops can also be used for a set of various other things (specifying the collection of elements we want to loop over)\n• Breakpoint is used in For Loop to break or terminate the program at any particular point\n• Continue statement will continue to print out the statement, and prints out the result as per the condition set\n• Enumerate function in \"for loop\" returns the member of the collection that we are looking at with the index number\n\nPython 2 Example\n\nAbove codes are Python 3 examples, If you want to run in Python 2 please consider following code.\n\n```# How to use \"While Loop\"\n#Example file for working with loops\n#\ndef main():\nx=0\n#define a while loop\nwhile(x <4):\nprint x\nx = x+1\n\nif __name__ == \"__main__\":\nmain()\n\n#How to use \"For Loop\"\n#Example file for working with loops\n#\ndef main():\nx=0\n#define a while loop\n#\twhile(x <4):\n#\t\tprint x\n#\t\tx = x+1\n\n#Define a for loop\nfor x in range(2,7):\nprint x\n\nif __name__ == \"__main__\":\nmain()\n\n#How to use For Loop for String\ndef main():\n#use a for loop over a collection\nMonths = [\"Jan\",\"Feb\",\"Mar\",\"April\",\"May\",\"June\"]\nfor m in Months:\nprint m\n\nif __name__ == \"__main__\":\nmain()\n\n#How to use break statements in For Loop\ndef main():\n#use a for loop over a collection\n#Months = [\"Jan\",\"Feb\",\"Mar\",\"April\",\"May\",\"June\"]\n#for m in Months:\n#print m\n\n# use the break and continue statements\nfor x in range (10,20):\nif (x == 15): break\n#if (x % 2 == 0) : continue\nprint x\n\nif __name__ == \"__main__\":\nmain()\n\n#How to use \"continue statement\" in For Loop\ndef main():\n#use a for loop over a collection\n#Months = [\"Jan\",\"Feb\",\"Mar\",\"April\",\"May\",\"June\"]\n#for m in Months:\n#print m\n\n# use the break and continue statements\nfor x in range (10,20):\n#if (x == 15): break\nif (x % 5 == 0) : continue\nprint x\n\nif __name__ == \"__main__\":\nmain()\n\n#How to use \"enumerate\" function for \"For Loop\"\ndef main():\n#use a for loop over a collection\nMonths = [\"Jan\",\"Feb\",\"Mar\",\"April\",\"May\",\"June\"]\nfor i, m in enumerate (Months):\nprint i,m\n\n# use the break and continue statements\n#for x in range (10,20):\n#if (x == 15): break\n#if (x % 5 == 0) : continue\n#print x\n\nif __name__ == \"__main__\":\nmain()\n```\n\nYOU MIGHT LIKE:" ]	[ null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.85124224,"math_prob":0.95039994,"size":10216,"snap":"2020-10-2020-16","text_gpt3_token_len":2690,"char_repetition_ratio":0.15452409,"word_repetition_ratio":0.302189,"special_character_ratio":0.3066758,"punctuation_ratio":0.12523365,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9784131,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2020-03-29T15:59:48Z\",\"WARC-Record-ID\":\"<urn:uuid:6ccc3761-24e4-4429-b844-ceb74f074543>\",\"Content-Length\":\"82859\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:1d358388-f50e-4b2d-8a5b-06f32b66575a>\",\"WARC-Concurrent-To\":\"<urn:uuid:2ba2ba12-d974-4777-ba8a-3456a1631a86>\",\"WARC-IP-Address\":\"72.52.251.71\",\"WARC-Target-URI\":\"http://www.test3.guru99.com/python-loops-while-for-break-continue-enumerate.html\",\"WARC-Payload-Digest\":\"sha1:UWFBD6G6C4D6INNVYZKKLJTU6UMYUGET\",\"WARC-Block-Digest\":\"sha1:O7AASH4C7IUTM2YYLPQHIHBT3G26XTIT\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2020/CC-MAIN-2020-16/CC-MAIN-2020-16_segments_1585370494349.3_warc_CC-MAIN-20200329140021-20200329170021-00036.warc.gz\"}"}
https://brilliant.org/problems/ramanujan-and-mahalanobis/	[ "# Ramanujan And Mahalanobis\n\nIn 1914, Ramanujan's friend P. C. Mahalanobis gave him a problem he had read in the English magazine Strand. The problem was to determine the number $x$ of a particular house on a street where the houses were numbered $1,2,3,\\ldots,n$. The house with number $x$ had the property that the sum of the house numbers to the left of it equaled the sum of the house numbers to the right of it. The problem specified that $50 < n < 500$.\n\nRamanujan quickly dictated a continued fraction for Mahalanobis to write down. The numerators and denominators of the convergents to that continued fraction gave all solutions $(n,x)$ to the problem $($not just the particular one where $50 < n < 500).$ Mahalanobis was astonished, and asked Ramanujan how he had found the solution.\n\nRamanujan responded, \"...It was clear that the solution should obviously be a continued fraction; I then thought, which continued fraction? And the answer came to my mind.\"\n\nThis is not the most illuminating answer! If we cannot duplicate the genius of Ramanujan, let us at least find the solution to the original problem. What is $x$?\n\n\nBonus: Which continued fraction did Ramanujan give Mahalanobis?\n\n×" ]	[ null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.9579956,"math_prob":0.96517724,"size":1256,"snap":"2020-45-2020-50","text_gpt3_token_len":282,"char_repetition_ratio":0.1341853,"word_repetition_ratio":0.096618354,"special_character_ratio":0.20382166,"punctuation_ratio":0.15789473,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9858723,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2020-11-25T10:28:10Z\",\"WARC-Record-ID\":\"<urn:uuid:43acf85b-ba65-4964-99ab-3d819ad93d87>\",\"Content-Length\":\"43145\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:dfea61f8-b93b-4f52-8e61-acf9bfd32079>\",\"WARC-Concurrent-To\":\"<urn:uuid:d4c3bdad-f25d-4a76-a427-8b9c0e42ba4f>\",\"WARC-IP-Address\":\"104.20.34.242\",\"WARC-Target-URI\":\"https://brilliant.org/problems/ramanujan-and-mahalanobis/\",\"WARC-Payload-Digest\":\"sha1:N7VKRUKQSPFMGZG5QZAGWDWN6UD7KC2D\",\"WARC-Block-Digest\":\"sha1:UPRIAN4XRK627VHMPXUOO4EVGUEWZROJ\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2020/CC-MAIN-2020-50/CC-MAIN-2020-50_segments_1606141182776.11_warc_CC-MAIN-20201125100409-20201125130409-00165.warc.gz\"}"}
https://www.basic-mathematics.com/basic-mathematics-worksheets.html	[ "# Basic mathematics worksheets\n\nA great source of basic mathematics worksheets, such as fractions worksheets, addition worksheets that you can print easily to practice and strengthen your basic mathematics skills.\n\nTeachers should feel free to print them and use them as quizzes, tests, or more practices. 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Or just go back and return to the same worksheet page and you will get a whole new worksheet!\n\nAddition worksheets generator:with carry or no carry\n\nAdding worksheets with one digit\nGenerate unlimited addition worksheets with one digit.\n\nAdding worksheets with two digits\nGenerate unlimited addition worksheets with two digits.\n\nAdding worksheets with three digits\nGenerate unlimited addition worksheets with three digits.\n\nSubtraction worksheets generator:with no borrowing\n\nSubtracting worksheets generator\nGenerate unlimited subtraction worksheets with one digit or two digits.\n\nFractions worksheets generator:\n\nAdding fractions worksheets\nGenerate unlimited worksheets when the fractions have the same denominators.\n\nAdding fractions worksheets: different denominator\nGenerate unlimited free worksheets when the fractions have different denominators.\n\nMultiplication of fractions\nGenerate unlimited free worksheets for multiplying fractions.\n\nDivision of fractions\nGenerate unlimited free worksheets for dividing fractions.\n\ninteger worksheets generator:\n\nInteger worksheets\nGenerate unlimited worksheets for adding, subtracting, and multiplying integers.\n\nYou may also want to check other popular topics in our site\n\nBasic mathematics games\nBasic mathematics games of all sorts to sharpen your math skills with unlimited practices\n\nBasic math calculator\nA wide variety of basic math calculators. Even a Pythagorean theorem calculator is here to help you easily solve for geometry problems!\n\n## Recent Articles", null, "1. ### Basic Math Review Game - Math adventure!\n\nMay 19, 19 09:20 AM\n\nBasic math review game - The one and only math adventure game online. Embark on a quest to solve math problems!\n\nRead More\n\nNew math lessons\n\nYour email is safe with us. We will only use it to inform you about new math lessons.", null, "Tough Algebra Word Problems.\n\nIf you can solve these problems with no help, you must be a genius!\n\n## Recent Articles", null, "1. ### Basic Math Review Game - Math adventure!\n\nMay 19, 19 09:20 AM\n\nBasic math review game - The one and only math adventure game online. Embark on a quest to solve math problems!\n\nRead More\n\nEverything you need to prepare for an important exam!\n\nK-12 tests, GED math test, basic math tests, geometry tests, algebra tests.", null, "Real Life Math Skills\n\nLearn about investing money, budgeting your money, paying taxes, mortgage loans, and even the math involved in playing baseball." ]	[ null, "https://www.basic-mathematics.com/objects/xrss.png.pagespeed.ic.nVmUyyfqP7.png", null, "https://www.basic-mathematics.com/images/xQUIZ.jpg.pagespeed.ic.G-H8wOb1YG.jpg", null, "https://www.basic-mathematics.com/objects/xrss.png.pagespeed.ic.nVmUyyfqP7.png", null, "https://www.basic-mathematics.com/images/xconsumer-math.png.pagespeed.ic.3jNfbNE_qG.png", null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.83263206,"math_prob":0.7477155,"size":3050,"snap":"2019-35-2019-39","text_gpt3_token_len":559,"char_repetition_ratio":0.23834537,"word_repetition_ratio":0.074324325,"special_character_ratio":0.16918033,"punctuation_ratio":0.11111111,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9922906,"pos_list":[0,1,2,3,4,5,6,7,8],"im_url_duplicate_count":[null,null,null,null,null,null,null,null,null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2019-08-18T09:54:58Z\",\"WARC-Record-ID\":\"<urn:uuid:7d598cbf-15c1-4fb3-a51c-332e34ec152b>\",\"Content-Length\":\"42809\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:6e1aabba-67ca-4337-b392-cc0f1c30f20a>\",\"WARC-Concurrent-To\":\"<urn:uuid:26f4b421-bda1-4b3a-ab20-309d15438900>\",\"WARC-IP-Address\":\"173.247.219.163\",\"WARC-Target-URI\":\"https://www.basic-mathematics.com/basic-mathematics-worksheets.html\",\"WARC-Payload-Digest\":\"sha1:66SX2X6QXUAKMXD63YUPF62JVPJ4CAY7\",\"WARC-Block-Digest\":\"sha1:BCOCSHHFZ2H22MIGSRIPT3RRWPTZEMMM\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2019/CC-MAIN-2019-35/CC-MAIN-2019-35_segments_1566027313747.38_warc_CC-MAIN-20190818083417-20190818105417-00002.warc.gz\"}"}
https://www.proprofs.com/quiz-school/story.php?title=year-8-maths-recall-3-algebra	[ "# Year 8 Maths 3.1 Algebra - Variables And Expressions\n\n10 Questions \| Total Attempts: 1177", null, "", null, "Settings", null, "", null, "For those of you who love all things mathematics especially graphs and algebra that measures the strength of materials then this is the quiz for you. If you are preparing for exams, and want to test your knowledge, try it out.\n\n• 1.\nWhat is the algebraic expression to represent m divided by 3\n• A.\n• B.\n• C.\n\n3m\n\n• D.\n\nM3\n\n• 2.\nWhat is the algebraic expression to represent two lots of p\n• A.\n• B.\n• C.\n• D.\n• 3.\nWhat is the algebraic expression to represent y less than twenty\n• A.\n\nY + 20\n\n• B.\n\nY - 20\n\n• C.\n\n20 + y\n\n• D.\n\n20 - y\n\n• 4.\nA ___________________ describes an unknown amount in an algebraic expression or equation. Can be represented by letters or symbols (e.g. a, y, ) and can be called pronumerals (which means 'in place of' a number).\n• A.\n\nCoefficient\n\n• B.\n\nConstant\n\n• C.\n\nEquation\n\n• D.\n\nExpression\n\n• E.\n\nTerm\n\n• F.\n\nVariable\n\n• 5.\nThe following are examples of ___________________ .\n• A.\n\nCoefficients\n\n• B.\n\nConstants\n\n• C.\n\nAn equation\n\n• D.\n\nAn expression\n\n• E.\n\nTerms\n\n• F.\n\nVariables\n\n• 6.\n8x means that x is multiplied 8 times. The number in front of the term x is called the _______________.\n• A.\n\nCoefficient\n\n• B.\n\nConstant\n\n• C.\n\nAn equation\n\n• D.\n\nAn expression\n\n• E.\n\nTerm\n\n• F.\n\nVariable\n\n• 7.\nA number that is written by itself is called a _______________. E.g. in the expression 3x + 6, the 6 is a _____________.\n• A.\n\nCoefficient\n\n• B.\n\nConstant\n\n• C.\n\nAn equation\n\n• D.\n\nAn expression\n\n• E.\n\nTerms\n\n• F.\n\nVariable\n\n• 8.\nTwo expressions joined together by an equals sign is E.g. 3x + 6 = 2x - 4.\n• A.\n\nCoefficients\n\n• B.\n\nConstants\n\n• C.\n\nAn equation\n\n• D.\n\nAn expression\n\n• E.\n\nTerms\n\n• F.\n\nVariables\n\n• 9.\nWhen two or more terms are added or subtracted together, it creates ___________________. E.g. 3x + 4y\n• A.\n\nCoefficients\n\n• B.\n\nConstants\n\n• C.\n\nAn equation\n\n• D.\n\nAn expression\n\n• E.\n\nTerms\n\n• F.\n\nVariables\n\n• 10.\nWhat is the algebraic expression to represent ten more than x?\n• A.\n\n10x\n\n• B.\n\nX - 10\n\n• C.\n\nX + 10\n\nRelated Topics", null, "Back to top" ]	[ null, "https://www.proprofs.com/quiz-school/images/story_settings_gear.png", null, "https://www.proprofs.com/quiz-school/images/story_settings_gear_color.png", null, "https://www.proprofs.com/quiz-school/loader.gif", null, "https://media.proprofs.com/images/QM/user_images/2362275/1548060184.jpg", null, "https://www.proprofs.com/quiz-school/img/top.png", null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.6456448,"math_prob":0.9845998,"size":651,"snap":"2021-43-2021-49","text_gpt3_token_len":167,"char_repetition_ratio":0.17310664,"word_repetition_ratio":0.1826923,"special_character_ratio":0.20430107,"punctuation_ratio":0.03773585,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9997689,"pos_list":[0,1,2,3,4,5,6,7,8,9,10],"im_url_duplicate_count":[null,null,null,null,null,null,null,1,null,null,null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2021-10-23T04:21:11Z\",\"WARC-Record-ID\":\"<urn:uuid:1cfea152-e04c-48b1-bd96-ef906841bef9>\",\"Content-Length\":\"269541\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:dde79682-0f5f-4471-af42-615671fa8e29>\",\"WARC-Concurrent-To\":\"<urn:uuid:cf4f700a-d13a-4f58-a757-4f739756aef9>\",\"WARC-IP-Address\":\"104.26.13.111\",\"WARC-Target-URI\":\"https://www.proprofs.com/quiz-school/story.php?title=year-8-maths-recall-3-algebra\",\"WARC-Payload-Digest\":\"sha1:47V5NM5QFPQWWVFZ7JWONZSNHCGKKYRG\",\"WARC-Block-Digest\":\"sha1:KX5WPEXBPKJJ4NKF3YGDSMEWR3J3EZBU\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2021/CC-MAIN-2021-43/CC-MAIN-2021-43_segments_1634323585561.4_warc_CC-MAIN-20211023033857-20211023063857-00601.warc.gz\"}"}
https://www.nmaps.net/10921	[ "#", null, "## into my soul", null, "Hover over the thumbnail for a full-size version.\n\nAuthor great_sea author:great_sea bitesized n-art rated 2005-07-23 2007-01-14 5 by 37 people. \\$into my soul#great_sea#none#00000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000\|12^36,108!12^48,96!12^54,90!12^66,78!12^72,72!12^84,60!12^96,54!12^102,48!12^120,36!12^126,30!12^30,96!12^42,96!12^48,84!12^54,78!12^66,72!12^78,66!12^84,48!12^96,48!12^108,42!12^114,36!12^78,42!12^66,48!12^48,60!12^30,78!12^24,78!12^42,72!12^54,66!12^60,60!12^84,54!12^96,42!12^102,36!12^108,30!12^96,30!12^78,36!12^66,36!12^48,42!12^42,60!12^36,60!12^36,54!12^48,42!12^60,36!12^66,30!12^72,24!12^48,30!12^42,30!12^30,42!12^30,42!12^24,30!12^36,30!12^42,36!12^60,48!12^84,36!12^96,30!12^102,30!12^84,54!12^72,66!12^60,66!12^36,84!12^18,96!12^42,78!12^78,54!12^78,54!12^36,42!12^42,54!12^24,66!0^24,120!0^42,120!0^42,114!0^48,102!0^66,90!0^84,78!0^90,72!0^102,60!0^108,54!0^126,42!0^132,36!0^132,36!0^120,48!0^102,66!0^78,84!0^42,114!0^30,126!0^54,120!0^72,108!0^90,90!0^108,72!0^126,60!0^138,48!0^150,48!0^132,66!0^108,78!0^96,102!0^66,138!0^36,150!0^54,162!0^72,144!0^90,126!0^114,102!0^126,84!0^150,60!0^168,42!0^174,42!0^150,36!0^126,54!0^96,90!0^36,138!0^24,162!0^54,144!0^72,132!0^96,102!0^120,84!0^168,54!0^192,36!0^192,36!0^186,54!0^162,72!0^132,96!0^102,126!0^60,150!0^36,186!0^60,186!0^90,162!0^108,138!0^132,120!0^162,96!0^192,60!0^204,48!0^216,42!0^222,42!0^216,60!0^186,90!0^126,132!0^54,192!0^24,210!0^24,216!0^30,180!0^72,150!0^156,72!0^156,42!0^162,36!0^174,36!0^198,30!0^216,36!0^198,30!0^162,30!0^144,30!12^24,126!12^36,114!12^60,102!12^72,96!12^102,84!12^120,60!12^132,42!12^162,30!12^114,48!12^108,60!12^90,78!12^66,108!12^36,132!12^18,138!12^42,132!12^72,96!12^90,72!12^114,48!12^144,30!12^162,24!12^156,42!12^108,72!12^72,102!12^42,126!12^36,132!12^42,132!12^72,120!12^108,90!12^78,84!12^72,84!12^84,102!12^102,66!12^138,36!12^174,36!12^174,54!12^126,72!12^120,60!12^120,72!12^96,102!12^78,90!12^60,132!12^36,138!12^60,138!12^102,114!12^120,90!12^156,66!12^198,30!12^186,36!12^126,72!12^96,108!12^60,144!12^30,174!12^60,174!12^96,150!12^126,102!12^156,78!12^186,60!12^204,48!12^192,60!12^138,102!12^120,126!12^66,162!12^42,180!12^42,192!12^90,174!12^120,144!12^138,120!12^174,84!12^216,48!12^228,42!12^174,72!12^126,126!12^66,174!12^54,186!12^48,210!12^66,204!12^108,162!12^138,120!12^174,84!12^240,36!12^216,60!12^132,150!12^60,192!12^90,204!12^120,192!12^168,120!12^252,42!12^258,54!12^186,150!12^96,222!12^48,270!12^126,246!12^222,132!12^306,66!0^54,210!0^60,210!0^138,150!0^174,120!0^36,90!0^48,90!0^72,78!0^90,66!0^120,42!0^120,42!0^78,66!0^72,78!0^42,90!0^30,96!0^78,72!0^114,48!0^48,246!0^72,228!0^132,174!0^186,126!0^258,72!12^90,78!12^90,72!12^84,66!12^96,54!12^102,48!0^246,24!0^210,78!0^132,150!0^42,222!0^48,228!0^162,168!0^258,114!0^312,78!0^288,72!0^246,96!0^108,192!0^60,222!0^54,252!0^48,264!0^186,210!0^288,132!0^360,78!0^360,66!0^240,114!0^174,162!0^48,252!0^24,270!0^144,180!0^186,126!0^240,84!0^276,48!0^186,96!0^114,156!0^60,174!0^60,162!0^114,108!0^132,90!0^150,102!0^144,126!0^156,78!0^156,96!0^162,84!0^126,96!0^168,90!0^144,108!0^102,108!0^66,156!0^66,156!0^78,150!0^102,144!0^132,228!0^102,168!0^162,144!0^192,114!0^216,84!0^126,162!0^78,216!0^162,210!0^210,174!0^216,78!0^228,120!0^210,210!0^198,138!0^252,78!0^318,36!0^318,36!0^192,156!0^138,204!0^72,258!0^42,300!0^90,276!0^192,222!0^246,186!0^306,132!0^432,72!0^402,60!0^216,168!0^210,174!0^192,174!0^228,138!0^270,120!0^294,102!0^210,186!0^174,198!0^162,192!0^174,186!0^198,192!0^168,180!0^126,216!0^108,222!0^108,210!0^90,258!0^90,246!0^102,246!0^138,252!0^162,228!0^156,186!0^174,138!0^186,96!0^222,90!0^198,126!0^204,96!0^198,90!0^174,84!0^132,72!0^180,108!0^174,108!0^144,72!0^180,84!0^204,78!0^216,66!0^198,66!0^204,66!0^234,60!0^234,66!0^228,90!0^216,180!0^258,162!0^252,126!0^246,132!0^288,138!0^306,108!0^312,96!0^252,66!0^264,90!0^264,90!0^216,102!0^246,90!0^282,42!0^282,42!0^318,60!0^360,36!0^360,36!0^330,60!0^330,108!0^312,102!0^294,42!0^294,30!0^276,48!0^258,24!0^276,48!0^276,78!0^330,72!0^342,48!0^330,36!0^354,30!0^360,48!0^342,78!0^240,186!0^144,258!0^72,294!0^54,312!0^108,294!0^186,252!0^264,204!0^324,150!0^372,90!0^396,48!0^408,36!0^414,54!0^402,84!0^312,162!0^276,204!0^156,288!0^102,324!0^54,324!0^78,318!0^186,288!0^264,228!0^330,156!0^384,66!0^366,78!0^258,162!0^234,204!0^216,246!0^216,258!0^60,336!0^42,336!0^78,348!0^138,336!0^210,318!0^264,300!0^306,240!0^330,198!0^378,114!0^384,54!0^384,54!0^390,120!0^270,300!0^210,342!0^102,342!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just a little artsy experiment here.\n\n## Other maps by this author", null, "", null, "", null, "", null, "", null, "", null, "Flux Jetsy Free Laptop (revised) Madman Rotor The Epic Part IX - Cloud Wrangler\n\nPages: (0)\n\n### i woulda\n\nproffered the exit switch instead of the drones...\n\nup to 5!\n\n### this is my\n\nsort of map 5aved!!!\n\nEh.\n\n### Woah\n\nCame accross this three years after I commented.\nNeat.\n\n### 5\n\nand that pushes it to 4.5 average\n\n5/5.\n\npretty cool\n\n### 4.5/5\n\nI think the drones ruin it. If the drones weren't there, it would be a 5. Truely a masterpiece(in the thumbnail)\n\n### 2/5\n\nYeah, it looks awesome in the thumbnail, i guess. ok.\n\n### 5/5\n\nYou're right, that is a true work of art.\n\nI'm tempted to go photoshop it into something and pass it off as nmy own work. :-P\n\n5/5\n\n### -\n\nyeah, it really does.\n\n### lag much?\n\nwow, that looks awesome in the thumbnail!" ]	[ null, "https://www.nmaps.net/static/logo.png", null, "http://lh3.ggpht.com/bBZvUzAEVBbe6wO6NhXUvVIBEBk5FdSeNp-rftHEgfL9Z4q-XhSs__phWkQn98Zd7wIs_6j5KkY_M_ZSZM_Rr_pH=s132", null, "http://lh3.ggpht.com/D7bKrnyliti6cRKcuhrxOz8Wb7LhNMbiGzRVx7Kas-sBKAJg-VtZ4HlRsnGJ3kYVFXmMBK_XMoKpb7yeEgnITIw=s132", null, "http://lh3.ggpht.com/8Y4WfWjONGInPre8kaIEQNg4WP9l1cgzf7rfY0ljyFe4urPu9EryqikP5R1Xfc2JCW5SElfaUd7vCyovVJ3fZw=s132", null, "http://lh3.ggpht.com/lkhN5vArq5Sruxgq2Ng6rngbskcqECaxpDNS1VDfznfXoaC-Ph-kHXJXmdTYXTDp_tJ6NdsA1OoC9hiC9zKaPfk=s132", null, "http://lh4.ggpht.com/oScPgTMzlEq4LKRN8fBw4eqLO_5wG8ddpQHNevm0ngQ_Pg1oHuQvZme6UAC75GjqudTeks9Uky9t-aM6ti752KM=s132", null, "http://lh4.ggpht.com/S-PezRKUd__8LLfnvcotXkTd_ndVNDGDTm5KruDDz4SXTNKs6Zh6ON7QY7CIb7oc4kv4neHQ8u3yalkzKh02RSs=s132", null, "http://lh6.ggpht.com/xuoGDHeKthksIRyg3ehqqy4SoN3JhdMDg0sDQYyg2nnpJ2ZQvhaKI3JlIXXlj1os8QyMszF_eZMVGgTYt7eSi1A=s132", null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.8851754,"math_prob":0.9997694,"size":845,"snap":"2020-34-2020-40","text_gpt3_token_len":278,"char_repetition_ratio":0.09988109,"word_repetition_ratio":0.18439716,"special_character_ratio":0.35502958,"punctuation_ratio":0.119791664,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.95105344,"pos_list":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16],"im_url_duplicate_count":[null,null,null,3,null,3,null,3,null,3,null,3,null,3,null,3,null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2020-09-19T08:14:24Z\",\"WARC-Record-ID\":\"<urn:uuid:51d5b4f7-27dc-4bde-960e-5b2c13dc0cfc>\",\"Content-Length\":\"22977\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:9170859d-9f91-4435-8819-2313593f2a53>\",\"WARC-Concurrent-To\":\"<urn:uuid:4c8261b3-1f6d-4cb6-944e-15efc750a300>\",\"WARC-IP-Address\":\"172.217.12.243\",\"WARC-Target-URI\":\"https://www.nmaps.net/10921\",\"WARC-Payload-Digest\":\"sha1:2TQXCZ2GT2BMDMPZ6W26A56I6CMLPCUU\",\"WARC-Block-Digest\":\"sha1:R53JDENGIHNSCWRYEGJU27L7ILFIHVLH\",\"WARC-Identified-Payload-Type\":\"application/xhtml+xml\",\"warc_filename\":\"/cc_download/warc_2020/CC-MAIN-2020-40/CC-MAIN-2020-40_segments_1600400191160.14_warc_CC-MAIN-20200919075646-20200919105646-00183.warc.gz\"}"}
https://ifelse.info/questions/1685/error-fixes	[ "# Error with fixes\n\n3\n\nHello everyone, is that I try to solve a problem with minor and major fixes and make their difference but ... the program defines well the major element but not the minor so is the arrangement\n\n2 5 3 7 12 9 8 5 4 10\n\nDefine the biggest one that in this case is 12\n\nThe minor defines it badly that it prints 10 and it is supposed to be 2\n\nIt is the following code\n\n`````` #include<iostream>\n#define MAX 10002\nusing namespace std;\n\nint arr[MAX];\nint mayor=arr, menor=arr;\nint diferencia=0;\n\nint main()\n{\nfor(int i=1; i<=10; i++)\n{\ncin>>arr[i];\n\nif(arr[i]>mayor)\n{\nmayor=arr[i];\n}\n\nif(arr[i]<menor);\n{\nmenor=arr[i];\n}\n}\ncout<<menor<<\" \"<<mayor;\n}\n``````\n\nIf someone can help me, I'd really appreciate it\n\nasked by carlos 29.09.2018 в 20:04\nsource\n\n3\n\nLet's go in parts:\n\n``````int arr[MAX];\nint mayor=arr, menor=arr;\n``````\n\nAt this time, the content of `arr` is unknown. It can be anything, because you have not initialized the array.\n\nSince we do not know that value, it turns out that `mayor` is already greater than any of the elements you put (and would not change in the iterations). O `menor` is less than any of your elements (and would not change in iterations).\n\nFor example, if `menor` starts with 0, none of the elements in your example would replace it. `menor` has to be so high that you are sure that you will find a lower value, `mayor` so low that you are sure that you will find a higher value.\n\n``````int diferencia=0;\nint main() {\nfor(int i=1; i<=10; i++)\n{\ncin>>arr[i];\n``````\n\nIf you do not need to save the data, you really do not need an array. If you make `int numero; cin >> numero;` it works exactly the same.\n\nNaturally, in the following lines you would have to change `arr[i]` by `numero` .\n\n`````` if(arr[i]>mayor) {\nmayor=arr[i];\n}\nif(arr[i]<menor);\n``````\n\nYou have a `;` , so the `if` ends on the top line and the block below always runs.\n\n`````` {\nmenor=arr[i];\n}\n}\ncout<<menor<<\" \"<<mayor;\n``````\n\nThe impression is complicated because it is confused with the data of the last line. I think that's part of the problem; it's much better: `cout << endl << \"Menor :\" << menor << endl << \"Mayor: \" << mayor << endl;`\n\n``````}\n``````\n\n3\n\nWhen you start to program it is very normal that some details escape but once you have time you notice certain details, your mistake is that you put a `;` where you should not go.\n\n`````` #include<iostream>\n#define MAX 10002\nusing namespace std;\n\nint arr[MAX];\nint mayor=arr, menor=arr;\nint diferencia=0;\n\nint main()\n{\nfor(int i=1; i<=10; i++)\n{\ncin>>arr[i];\n\nif(arr[i]>mayor)\n{\nmayor=arr[i];\n}\n\nif(arr[i]<menor); <---- precisamente aquí\n{\nmenor=arr[i];\n}\n}\ncout<<menor<<\" \"<<mayor;\n}\n``````\n\n0\n\nBasically the program has two errors. One has already been indicated and it is that you have cast a `;` after a `if` :\n\n``````if(arr[i]<menor);\n// ^\n``````\n\nThat semicolon causes the `if` to do absolutely nothing ... silly language details. To see it more clearly we could replace that semicolon with two keys:\n\n``````if(arr[i]<menor){ }\n{\nmenor = arr[i];\n}\n``````\n\nNow it looks better than that semicolon overrides the conditional.\n\nOn the other hand the initialization of `menor` is not correct.\n\nGlobal arrays are initialized automatically, not those that belong to a function. That is:\n\n``````int arr[MAX]; // Todos sus elementos valen 0\n// Pero únicamente porque es variable global!!!\n\nint main()\n{\nint arr2[MAX]; // Array no inicializado!!!\n}\n``````\n\nWell, the next thing you do after declaring the fix is to declare and initialize `mayor` and `menor` :\n\n``````int mayor=arr, menor=arr;\n``````\n\nAnd here the problems begin ... this initialization makes `mayor=menor=0` . If we subsequently give values to `arr` , the value of `mayor` and `menor` will not change automatically. Your starting value will be always 0.\n\nThis initialization poses a problem and is that for `menor` find the smallest number, it must be equal to or less than 0. Otherwise the program will not work.\n\nTo avoid this mess of minimums and maximums I would use pointers:\n\n``````int* mayor = arr;\nint* menor = arr;\n\nfor(int i=0; i<MAX; i++)\n{\ncin>>arr[i];\n\nif(arr[i] > mayor)\n{\nmayor = &arr[i];\n}\n\nif(arr[i] < menor)\n{\nmenor = &arr[i];\n}\n}\n\ncout << menor << ' ' << mayor;\n``````\n\nThis solution uses pointers instead of variables. Thus, `menor` does not store a value of its own, but will point to the smallest element (the same for `mayor` ).\n\nIf using pointers is not a possibility, then you should pay special attention to the initialization of `menor` and `mayor` . Pulling the standard the best is to use `std::numeric_limits` :\n\n``````int menor = std::numeric_limits<int>::max();\nint mayor = std::numeric_limits<int>::min();\n``````\n\nWhat we do here is initialize `menor` with the largest number that can be stored in a `int` , in this way any number that the user enters will be equal or smaller and, consequently, the algorithm will be able to detect it as smaller. In `mayor` , instead, we store the smallest number possible, so that any number that the user enters is equal or larger." ]	[ null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.7770735,"math_prob":0.92927533,"size":2243,"snap":"2021-43-2021-49","text_gpt3_token_len":549,"char_repetition_ratio":0.119249664,"word_repetition_ratio":0.010282776,"special_character_ratio":0.25991976,"punctuation_ratio":0.15869565,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.97732157,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2021-10-26T17:35:46Z\",\"WARC-Record-ID\":\"<urn:uuid:b996d80c-20b4-4851-a754-286f8c2568e9>\",\"Content-Length\":\"21391\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:3d781bd7-72c7-4c95-8ef2-4556371b3343>\",\"WARC-Concurrent-To\":\"<urn:uuid:d4194263-dcce-403f-84c0-47624fa1e194>\",\"WARC-IP-Address\":\"172.67.145.60\",\"WARC-Target-URI\":\"https://ifelse.info/questions/1685/error-fixes\",\"WARC-Payload-Digest\":\"sha1:GYAXFW2UQDUDQTU4MXADP24ORRDUM7DA\",\"WARC-Block-Digest\":\"sha1:IBM3QKWVNHZAIKJPGIYMEEMBBXHWSZDV\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2021/CC-MAIN-2021-43/CC-MAIN-2021-43_segments_1634323587915.41_warc_CC-MAIN-20211026165817-20211026195817-00188.warc.gz\"}"}
https://www.osapublishing.org/oe/fulltext.cfm?uri=oe-16-21-16659&id=172485	[ "## Abstract\n\nFrom first principles we develop figures of merit to determine the gain experienced by the guided mode and the lasing threshold for devices based on high-index-contrast waveguides. We show that as opposed to low-index-contrast systems, this quantity is not equivalent to the power confinement since in high-index-contrast structures the electric and magnetic field distributions cannot be related by proportionality constant. We show that with a slot waveguide configuration it is possible to achieve more gain than one would expect based on the power confinement in the gain media. Using the figures of merit presented here we optimize a slot waveguide geometry to achieve low-threshold lasing and discuss the fabrication tolerances of such a design.\n\n## 1. Introduction\n\nA critical photonic component yet to be demonstrated on a silicon-based platform is an electrically pumped device with optical gain for amplification or lasing. Achieving this optical gain in a silicon-based device is extremely challenging since silicon is an indirect band-gap semiconductor and therefore an inefficient photon source. Hybrid silicon devices based on direct band-gap III–V materials bonded to silicon photonic elements present an interim solution , however, the reliance on a wafer bonding step prohibits a high throughput fabrication process possible with a silicon-based process.\n\nOne possible configuration for silicon-based gain is the recently proposed slot waveguide design [2, 3]. In this configuration a low-index gain material such as Er-doped SiO2 or Erdoped Si3N4 can be inserted into one or multiple thin slots between two silicon rails. Electrical excitation of the gain material could be achieved by passing a tunneling current through the slot-region.\n\nOne key advantage of this configuration is the large optical field enhancement in the slot-region due to the boundary conditions imposed on the electric field normal the slot interface . Since the normal electric displacement (D=εE) must be continuous across the interface, the electric field in the slot waveguide is enhanced by the ratio of the dielectric constant of silicon to that of the slot material. In semiconductor materials this enhancement can be as large as one order of magnitude.\n\nTypically waveguide gain is assumed to be proportional to the percentage of the guided mode power which overlaps with the gain medium; however, this is not true for high-index contrast waveguides due to the large electric field discontinuities at dielectric interfaces. This discrepancy results from Fermi’s Golden Rule which states the electric field of an electromagnetic wave, not the power, determines the emission rate for an excited state (and consequently the modal gain).\n\nIn standard low-index-contrast waveguides, optical gain and power confinement are considered proportional based on the following arguments. For electromagnetic plane waves in homogenous media the magnetic field H can be written in terms of the electric field E and the impedance of the material according to:\n\n$H=cεn(êz×E),$\n\nwhere êz is a unit vector along the direction of propagation (which we have chosen to be the z-direction) and n is the index of refraction of the material. This is often written in the form relating the major components of the electric and magnetic fields (for a TM mode in this case):\n\n$Ey=−ωμ0βHx,$\n\nwhere β is the propagation constant defined as β≡2πn̄/λ. Based on these relationships the electric field energy, and waveguide power stored in a given region can be used interchangeably since they differ only by a constant. In this case the percentage of power overlapping the gain medium can be used to calculate the resulting modal gain, and it is often assumed that the same can be said for waveguide modes.", null, "Fig. 1. Fundamental TM modes at a wavelength of 1.5 µm for waveguides 500 nm wide and 600 nm tall. All modes are normalized to unit power. The high-index material (n=3.5) is outlined in black. The waveguides are clad with n=3.25 for (a) and (b) and n=1.5 for (c)–(f). The first and second columns show E y and $−ωμ0βHx$ respectively, plotted on the same color scale. The two fields become increasingly dissimilar as more electric field is concentrated at high-index-contrast boundaries.\n\nFor high-index-contrast waveguides, however, the linear relationships between the electric and magnetic fields (Eqs. (1) and (2)) do not hold since they must satisfy different boundary conditions. This is shown in Fig. 1. For low-index-contrast waveguides, Eqs. (1) and (2) are close approximations. Figures 1(a) and 1(b) shows the fundamental TM mode with an index difference of 0.25 between the core and cladding. We see close agreement in the magnitude and spatial profiles of these two fields. However, as the index-contrast is increased to 2.5 (Fig. 1(c) and (d)) there is a noticeable difference between E y and $−ωμ0βHx$. Notice that E y must be continuous across the dielectric interfaces to the left and right of the waveguide, and discontinuous across the top and bottom interfaces. The H x on the other hand must be continuous across all interfaces since the magnetic susceptibility is the same in all regions. This leads to noticeable differences between the electric and magnetic field magnitudes and profiles. This difference becomes dramatic when the peak of the electric field is placed at a dielectric discontinuity as is the case for slot waveguides (Figs. 1(e) and 1(f)).\n\nThe strong difference between the spatial distribution of the electric and magnetic fields in high index contrast is the reason that power confinement is no longer a relevant quantity when calculating gain in high-index-contrast waveguides. This issue is also present when calculating the sensitivity of waveguides to changes in refractive index. Several previous papers have overcome this problem by either determining waveguide sensitivity empirically or by introducing correction factors , however, there is little discussion as to the origin of the correction factors.\n\nIn this paper we derive from first principles the appropriate confinement factors and figures of merit for modal gain in high-index-contrast waveguides. We show explicitly that the modal gain is dependent on two quantities: the group velocity and the electric field energy confinement in the slot region. The lasing threshold however is independent of the group velocity and determined only by the electric field energy confinement. We also show that in some instances decreasing the width of the slot can decrease the lasing threshold despite the reduction of gain material. This counter-intuitive result can be understood as the increased emission rate for material in narrow slots overcoming the reduction in volume of gain material. Additionally we show that the percentage of power confined to the slot region (sometimes used as a confinement factor) can incorrectly predict the modal gain in high-index- contrast waveguides.\n\n## 2. Confinement factor for high-index-contrast waveguides\n\nTo rigorously calculate gain in high-index-contrast waveguides we derive from first principles a proportionality constant (known as a confinement factor Γ) which relates bulk material gain (g b) to the modal gain (g m) in a guiding structure:\n\n$Γ≡gmgb,$\n\nwhere g m and g b have units of inverse length. The bulk material gain can be determined from the magnitude of the electric field for a plane wave propagating along the z-direction through the gain medium:\n\n$∣E(z)∣2=∣E0∣2egbz.$\n\nTo calculate the effective confinement factor for a given waveguide mode profile, we begin with an expression for the electric field of a guided mode propagating along the z-direction. This can be written as the sum of all three vector components of the electric field using Einstein summation notation:\n\n$E(x,y,z)=êjEjoψj(x,y)ei(ωt−β˜z),$\n\nwhere ê is the polarization vector, Ψ is the cross sectional mode profile, and β̃ is the complex propagation constant defined as\n\n$β˜≡k0(n̅r+in̅i),$\n\nwhere k 0 is the angular wavenumber in free space, and n̄r and n̄i are the real and imaginary parts of the effective index respectively. By writing Eq. (5) in the same form as Eq. (4) we can see that the modal gain is determined by the complex propagation constant. From Eq. (6) we can then write the gain of the guided mode in terms of the imaginary part of the effective index:\n\n$gm=2Im{β˜}=2k0n̅i.$\n\nSimilarly we can write the bulk material gain in terms of the imaginary part of refractive index of the active gain material:\n\n$gb=2k0nAi.$\n\nExpressing gain in this form allows us to treat it as a perturbation to the refractive index of the waveguide. We can now calculate the modal gain by introducing a small imaginary part to the refractive index (Δn Ai) in a given active region (A) and solve for the complex propagation constant of the guided mode. This can be performed using variational methods :\n\n$Δβ˜=ω∬∞Δε˜∣E∣2dxdy12∬∞Re{E×H}∙êzdxdy,$\n\nwhere Δε̃=(n A+iΔn Ai)2. According to Eqs. (6)(8) this can be written in the form:\n\n$gm=[nAcε0∬A∣E∣2dxdy∬∞Re{E×H}∙êzdxdy]gb.$\n\nHere c is the speed of light in vacuum, the integral in the numerator is carried out only over the area of the active gain region (since this is where Δε̃≠0), and the integral in the denominator is carried out over the entire cross section of the mode. We recognize the term in the brackets as the proportionality constant in Eq. (3) and therefore we can express the confinement factor as:\n\n$Γ=nAcε0∬A∣E∣2dxdy∬∞Re{E×H}∙êzdxdy.$\n\nNote that as expected the stimulated emission rate (and thus the gain) depends on the intensity of the field which is proportional to \|E\|2. Since this term is normalized to unit power, the confinement factor can be thought of as the amount of intensity overlapping the gain medium per unit input power. Note that most previous derivations of this confinement factor err by incorrectly substituting the electric for magnetic field in attempts to simplify the expression. This is commonly written as :\n\n$12∬Re{E×H}∙êzdxdy=12βωμ0∬∣E∣2dxdy,$\n\nand thus Eq. (11) could be written as the percentage of power or intensity confined to the active region. However, as shown in Section 1, the expression in Eq. (12) is not valid for high-index-contrast waveguides since it is based on the relationship for plane waves in homogeneous media that $H=εcn̅(êz×E)$ .\n\nThe expression for the confinement factor (Eq. (9)) can be simplified into the product of two terms: one related to the group velocity, and the other related to the confinement of the energy density of the electric field. The energy stored per unit length (U/l) in a dielectric waveguide can be written as :\n\n$U/l=12∫∫∞ε∣E∣2dxdy.$\n\nNote that here we have neglected material dispersion when writing the stored energy per unit length. To account for material dispersion one should replace epsilon with d(ωε)/ in Eq. (11) [16, 17]. If silicon is the most dispersive material, the error introduced by making this approximation is less than 7% at a wavelength of 1.55 microns. The group velocity of the mode (v g) describes the speed with which energy flows through a given cross section. Therefore we can write the power flux through a given cross section of the waveguide as:\n\n$12∬Re{E×H}∙êzdxdy=vg12∬ε∣E∣2dxdy$\n\nUsing the definition of group index (n gc/v g) we substitute Eq. (14) into Eq. (11) and rewrite the confinement factor as:\n\n$Γ=ng∬Aε∣E∣2dxdynA∬∞ε∣E∣2dxdy≡ngnAγA.$\n\nWe see from the simplified expression for the confinement factor that the modal gain can be defined as the product of a term related to the group index and a term related to the confinement of the electric field energy density. The first term can be thought of as a confinement in time since increasing the real part of the group index relative to the bulk index has the effect of slowing the propagation of the guided mode. Therefore for a given waveguide length, light can spend more time in the gain media resulting in an enhancement of the modal gain per unit length. The second term represents the spatial confinement of the energy density to the active region of the waveguide which we define as:\n\n$γA≡∬Aε∣E∣2dxdy∬∞ε∣E∣2dxdy.$\n\nSince this term can be as large as 1 we see that the total confinement factor in Eq. (15) can be larger than 1 if the group index is larger than the bulk refractive index. This means that it is possible to achieve more gain per unit length in a guided structure than would be possible in bulk. This phenomenon has been noted before for modal absorption and results from the fact that light spends more time in a structure with a large group index, and thus interacts more with the gain material per unit length.\n\n## 3. Numerical verification of analytical results\n\nTo verify our expression for the confinement factor (Eq. (13)) we use numerical methods to calculate the relationship between material gain and modal gain and compare it to the analytical results. Since the material gain can be expressed in terms of an imaginary part of the dielectric constant, we can simulate material gain in a given waveguide region by adding an imaginary component to the dielectric constant and calculating the propagation constant of the guided mode using a finite difference mode solver. The imaginary part of this complex propagation constant can then be written in terms of the modal gain according to Eq. (7). This relationship between the modal gain and material gain is the definition of the confinement factor Eq. (3) and should match the derived expression Eq. (15).\n\nWe numerically calculate the modal gain according to the waveguide geometry and the corresponding fundamental TM mode as shown in Figure 2(a) and 2(b) respectively. The waveguide geometry consists of two high index rails 500 nm wide and 250 nm tall separated by a horizontal slot 50 nm tall. For simplicity we use a wavelength of 1.5 µm and 3.5 and 1.5 for the high and low refractive indices respectively. This is approximately the index contrast between Si and SiO2. Figure 2(b) shows the fundamental TM mode calculated using a Matlab-based finite difference mode solver. We simulate material gain by introducing an imaginary part of the dielectric constant to the low-index slot region. For each value of material gain we use the finite difference mode solver to calculate the complex propagation constant and calculate the corresponding modal gain according to Eq. (7).", null, "Fig. 2. Numerical study of modal gain. (a) Schematic of slot waveguide with gain material defined by an imaginary component of the refractive index confined to the slot region (pink). (b) Major field component of the fundamental TM mode for the same structure as (a) calculated using a finite difference mode solver. (c) Circles show the modal gain (g m) calculated from the complex effective index of the fundamental TM mode as determined using a finite difference mode solver. Material gain is added via the imaginary part of the refractive index in the slot. Dashed line shows the modal gain calculated according to Eq. (3) based on the confinement factor Γ determined from the zero-gain mode profile from Eq. (15). Dotted line shows the product of the power in the active gain region (P A) and the material gain. We see that the confinement factor proposed in this paper correctly predicts the modal gain simulated numerically, while the power confinement greatly underestimates the simulated modal gain.\n\nBy plotting the numerically calculated modal gain (g m) versus material gain (g b) in Fig. 1(c) we show excellent agreement with the analytically calculated confinement factor. The slope of the line g m versus g b (circles in Fig. 1(c)) represents the confinement factor according to Eq. (3). We also calculate the confinement factor Γ based on Eq. (15) and the calculated TM mode in Fig. 1(b). We plot Γg b as the dashed line in Fig. 1(c). As expected, our calculated confinement factor agrees very well with the relationship between the modal and material gain from numerical simulations. The difference between these two factors (0.4328 from Eq. (15) and 0.4368 from the numerical simulations) is less than 1%. To highlight the difference between this confinement factor and the confinement of power to the gain medium we also plot on this graph P A g b where we define the power in the active region as:\n\n$PA≡∬ARe{E×H}∙ezdxdy∬∞Re{E×H}∙ezdxdy.$\n\nWe see from Fig. 1(c) that the modal gain of a slot waveguide is substantially larger than would be expected from the percentage of power confined to the gain region. As discussed in Section 2, this enhanced modal gain results from both the large group index as well as the increased electric field energy density in the slot. The electric field energy density is underestimated using the H field (See Figs. 1(e) and 1(f)).\n\n## 4. Minimizing the lasing threshold\n\nOne of the primary interests in these structures is achieving lasing, therefore it is important to identify which factors aid in reaching the condition that the modal gain exceeds the modal loss. Since material gain was written as a positive imaginary part of the material’s refractive index, similarly, material loss can be written as a negative imaginary part of the refractive index. The modal loss can then be determined following the same derivation in Section 1. The result is that the modal loss (α m) is related to the bulk material loss (α b) by\n\n$αm=ngnbγbαb,$\n\nwhere n b is the refractive index of the bulk material and γ b is the energy density confinement in the material similar to Eq. (16). In general there will be several loss mechanisms which can be written as sum of terms of the form Eq. (18). To achieve lasing, the threshold condition requires the modal gain per unit length be greater than the modal loss per unit length:\n\n$ngγAnAgb−ng∑iγiniαi>0.$\n\nWe see immediately that the group index can be divided out of Eq. (19). This is because increasing n g increases the time it takes light to propagate through the waveguide which increases both the gain and loss per unit length equally.\n\nWe see from the lasing threshold condition (Eq. (19)) that increasing the group index will not help one reach the lasing threshold despite that fact that the gain per unit length increases. While above and below the lasing threshold the net modal gain depends on the confinement factor Γ, we see from Eq. (19) that the lasing threshold itself is determined only by the confinement of the electric field energy density (γ A). Although γ A is generally larger than the power confinement in the slot, it is always less than or equal to 1. Therefore the lowest lasing threshold for these structures is limited by the bulk material gain. An identical result can be derived by analyzing resonant cavities. In that case γ A relates the material to modal gain per unit time .\n\nBased on the threshold condition in Eq. (19) we can define a new figure of merit which represents how easily lasing can be achieved in a waveguide. Since in practice the modal loss α m is often an experimentally measured parameter with units of inverse length, we can substitute this measured quantity into Eq. (19) and rewrite the lasing threshold condition as:\n\n$gb>αmΓ.$\n\nThis quantity $αmΓ$ is a useful figure of merit since it is equal to the minimum bulk material gain needed to reach the lasing threshold in the waveguide. Because the group index cancels out in Eq. (18), this figure of merit can be directly compared for waveguides of different geometries to determine which can achieve lasing with the lowest material gain coefficient.\n\n## 5. Scaling behavior of gain versus slot thickness\n\nWhile narrow slots enjoy greater local field enhancement, they also contain less gain material. Therefore the important question arises as to how the total gain in these structures depends on the slot thickness. As the width of the slot becomes increasingly narrow, the magnitude of the electric field increases until it reaches its maximum determined by the difference between dielectric constants in the high and low index region. This results in a greater stimulated (and spontaneous) emission rate of the material in the slot region . According to the laser rate equations this should result in larger modal gain coefficients . However, as the slot becomes increasingly narrow the area of the active region decreases. Therefore although the gain material is “working harder” there is less matter contributing to the gain. Thus it is important to understand how these competing phenomena affect the lasing threshold.", null, "Fig. 3. (a). The spatial confinement factor γ A plotted as a function of slot thickness t, where the gain region is defined as the slot (pink region in inset) between the high-index rails (green). Narrow slots result greater emission rates of gain material while thicker slots provide more material which contributes to the gain. The peak in γ A near a slot width of 60 nm indicates the condition where the combination of enhanced emission rate and volume of gain material result in the lowest lasing threshold. (b). The total confinement factor (Γ) (squares) and power in the slot region (P A) (triangles) as a function of slot width. Dotted and dashed lines mark the slot widths which maximize Γ and P A respectively. The discrepancy between these two plots shows that the percentage of power in the gain media is not an accurate indication of either the magnitude or the optimal design for modal gain.\n\nTo minimize the lasing threshold we look for a maximum in the spatial confinement factor (γ A) as a function of slot thickness which is plotted in Fig. 3(a). The maximum near a slot width of 60 nm illustrates the important point that the tradeoff between emission rate (which increases as the slot is narrowed) and material volume (which decreases as the slot is narrowed) results in an optimal slot width for minimizing the lasing threshold. Initially, as the slot narrows from 120 nm, the increased emission rate more than compensates for the decrease in volume of gain material, and γ A increases. Near a thickness of about 50 nm the emission rate begins to saturate as it approaches its maximum value determined by the index contrast between the high and low index regions. After this point, further reduction of the slot thickness decreases the volume of material contributing to the gain without much enhancement of the emission rate, and the result is a sharp drop in γ A.\n\nAs a comparison to previous methods, we plot in Fig. 3(b) the relative power confined in the slot region according to Eq. (17) and show again that slot waveguides outperform expectations based on previous theoretical treatments. We see that both the magnitude of the modal gain and the optimal geometry are miscalculated using this method. The gain calculated numerically (in agreement with Γ) is nearly twice as large as would be expected based on the power confined to the slot mode. Additionally, the optimal slot width is miscalculated by more than 10% using power confinement.\n\n## 6. Optimizing slot waveguide geometry\n\nTo optimize the dimensions of a slot waveguide for an electrically pumped silicon laser, we apply the principles in the proceeding sections to achieve a waveguide design with a minimal lasing threshold. We estimate that the slot thickness should be no larger than 10 nm in order to achieve electrical injection via tunneling into an oxide-based gain media using bias voltages on the order of volts . Therefore we keep the slot thickness fixed at 10 nm and compute the confinement factors as we vary the height and width of the waveguide. Here we have used the refractive indices of Si (3.48) and SiO2 (1.46) as the high and low index material respectively, and a wavelength of 1.55 µm. We have assumed that gain only occurs in the slot region. Figures 4(b)4(d) show respectively the total confinement factor (Γ), the group index normalized to the slot index (n g/n A), and the energy density confinement factor (γ A) as a function of height and width of the waveguide.", null, "Fig. 4. Optimization of width and height of Si/SiO2/Si slot waveguide with a 10 nm thick slot assumed to contain a gain medium. (a) Schematic of slot waveguide. (b) Total confinement factor Γ, proportional to the total modal gain. (c) Group index n g divided by the slot index (1.46), which is responsible for the difference between the lasing threshold and modal gain. (d) Electric field energy confinement γ A, inversely proportional to the lasing threshold. The maximum total modal gain is marked by the square in (a). The white contour shows the region which corresponds to a 5% change from the maximum values of Γ and γ A.\n\nWe see in Fig. 4(b) that the maximum modal gain for a 10 nm thick slot occurs near a waveguide width of 940 nm and a height of 340 nm (marked by the square) and has a value Γ=0.336. Both the energy density and group index peak near a waveguide height of 340 nm. As the width of the waveguide is increased, the energy density in the slot region (and thus the net gain) increases. The group index, on the other hand, decreases with increasing waveguide width. These competing parameters result in a maximum Γ near a width of about 940 nm.\n\nIn contrast to the modal gain, with a fixed slot thickness, the minimal lasing threshold (determined by γ A) shows no well-defined optimum. This is because the lasing threshold will scale only with the energy density confinement. The value of γ A(Fig. 4(d)) increases monotonically with waveguide width and asymptotically approaches the value for an infinitely wide slab waveguide, which we calculate to have a maximum of γ A=0.137 for a waveguide height of 340 nm.\n\nWe see from Fig. 4 that the optimal device geometry is relatively insensitive to variation in waveguide dimensions. Around the optimal design, variations of approximately ±50 nm in the total height and width of the waveguide result in changes in the confinement factors of less than 5%. This allows the device performance to be relatively unaffected by size variations which can occur during fabrication.\n\n## 7. Discussion and conclusions\n\nWe have shown that some commonly applied metrics are not appropriate for determining gain in high-index-contrast waveguides, and from first principles developed several figures of merit to characterize waveguide structures for gain. In particular we have shown that the concept of power confinement to the gain region significantly miscalculates the gain experienced by the waveguide mode. Instead we have shown that the true confinement factor which determines gain per unit length results from the combination of group index and confinement of the electric field energy to the gain region. These terms can combine and in some cases exceed unity meaning that one can achieve greater gain per unit length than would be possible in the bulk material. The lasing threshold on the other hand only depends on the percentage of electric field energy in the gain region. To account for this we have introduced a new figure of merit to describe the suitability of a waveguide to achieve low-threshold lasing. This figure of merit is the experimentally measured propagation loss divided by the confinement factor introduced in this paper. The evaluation of this ratio determines the minimal material gain required to achieve lasing in the waveguide structure.\n\nAdditionally we have applied our analysis to the design of slot waveguide structures. We have shown that the lasing threshold has a minimum for a particular slot width and increases dramatically as the slot is made thinner. Also we have shown that gain characteristics of the waveguides are fairly insensitive to variations in overall waveguide dimensions.\n\nSince the confinement factors presented here were derived from perturbation theory, they can be applied to other phenomena in high-index-contrast waveguides including refractive index sensing. In deriving the confinement factors presented here we have studied gain as a perturbation of the imaginary part of the refractive index over a given region of the guided mode. The same formalism holds true for perturbations to the real part of the refractive index and therefore the confinement factors for gain presented in the paper can also be used as confinement factors for refractive index sensing [10, 18] and have shown good agreement with experiment .\n\nIn summary, this work provides the qualitative and quantitative analysis necessary in developing high-index-contrast waveguides for applications such as amplification and lasing.\n\n## Acknowledgments\n\nThe authors gratefully acknowledge Thomas L. Koch for his helpful discussions and Christina Manolatou for the use of her finite difference mode solver. Research support is gratefully acknowledged from the National Science Foundation Center on Materials and Devices for Information Technology Research (CMDITR), DMR-0120967, the National Science Foundation’s CAREER Grant No. 0446571, and the U.S. Air Force MURI program on “Electrically-Pumped Silicon-Based Lasers for Chip-Scale Nanophotonic Systems” supervised by Dr. Gernot Pomrenke.\n\n1. A. W. Fang, H. Park, O. Cohen, R. Jones, M. J. Paniccia, and J. E. Bowers, “Electrically pumped hybrid AlGaInAs-silicon evanescent laser,” Opt. Express 14, 9203-9210 (2006). [CrossRef] [PubMed]\n\n2. V. R. Almeida, Q. Xu, C. A. Barrios, and M. Lipson, “Guiding and confining light in void nanostructure,” Opt. Lett. 29, 1209–1211 (2004). [CrossRef] [PubMed]\n\n3. Q. Xu, V. R. Almeida, R. R. Panepucci, and M. Lipson, “Experimental demonstration of guiding and confining light in nanometer-size low-refractive-index material,” Opt. Lett. 29, 1626–1628 (2004). [CrossRef] [PubMed]\n\n4. C. A. Barrios and M. Lipson, “Electrically driven silicon resonant light emitting device based on slot-waveguide,” Opt. Express 13, 10092–10101 (2005). [CrossRef] [PubMed]\n\n5. F. Ning-Ning, J. Michel, and L. C. Kimerling, “Optical field concentration in low-index waveguides,” IEEE J. Quantum Electron . 42, 885–890 (2006).\n\n6. J. T. Robinson, C. Manolatou, C. Long, and M. Lipson, “Ultrasmall mode volumes in dielectric optical microcavities,” Phys. Rev. Lett. 95, 143901 (2005). [CrossRef] [PubMed]\n\n7. E. Burstein and C. Weisbuch, eds. Confined electrons and photons, (Plenum Press: New York, NY,1995) [CrossRef]\n\n8. T. D. Visser, H. Blok, B. Demeulenaere, and D. Lenstra, “Confinement factors and gain in optical amplifiers,” IEEE J. Quantum Electron . 33, 1763–1766 (1997). [CrossRef]\n\n9. C. A. Barrios, K. B. Gylfason, B. Sánchez, A. Griol, H. Sohlström, M. Holgado, and R. Casquel, “Slot-waveguide biochemical sensor,” Opt. Lett. 32, 3080–3082 (2007). [CrossRef] [PubMed]\n\n10. F. Dell’Olio and V. M. Passaro, “Optical sensing by optimized silicon slot waveguides,” Opt. Express 15, 4977–4993 (2007). [CrossRef] [PubMed]\n\n11. H. Kogelnik, Theory of optical waveguides, in Guided-wave optoelectronics, T. Tamir ed., (Springer Verlag: Berlin, 1990). p. 7.\n\n12. L. A. Coldren and S. W. Corzine, Diode lasers and photonic integrated circuits (J. Wiley & Sons, New York, NY, 1995).\n\n13. C. Pollock and M. Lipson, Integrated photonics (Kluwer Academic, Norwell, MA, 2003).\n\n14. J. Haes, B. Demeulenaere, R. Baets, D. Lenstra, T. D. Visser, and H. Blok, “Difference between te and tm modal gain in amplifying waveguides: Analysis and assessment of two perturbation approaches,” Opt. Quantum Electron. 29, 263–273 (1997). [CrossRef]\n\n15. J. D. Jackson, Classical electrodynamics. 3rd ed (John Wiley & Sons, Inc., Hoboken, NJ, 1999).\n\n16. H. A. Haus, Waves and fileds in optoelectronics (Prentice-Hall, Englewood Cliffs, NJ, 1984).\n\n17. L. D. Landau and E. M. Lifshitz, Electrodynamics of continuous media (Pergamon Press, Reading, MA, 1960).\n\n18. G. J. Veldhuis, O. Parriaux, H. J. W. M. Hoekstra, and P. V. Lambeck, “Sensitivity enhancement in evanescent optical waveguide sensors,” J. Lightwave Technol . 18, 677–682 (2000). [CrossRef]\n\n19. R. Perahia, O. Painter, V. Moreau, and R. Colombelli, “Design of quantum cascade lasers for intra-cavity sensing in the mid infrared,” (in preparation).\n\n20. A. E. Siegman, Lasers (University Science Books, Sausalito, CA,1986).\n\n21. J. T. Robinson, L. Chen, and M. Lipson, “On-chip gas detection in silicon optical microcavities,” Opt. Express 16, 4296–4301 (2008). [CrossRef] [PubMed]\n\n### References\n\n• View by:\n• \|\n• \|\n• \|\n\n1. A. W. Fang, H. Park, O. Cohen, R. Jones, M. J. Paniccia, and J. E. Bowers, “Electrically pumped hybrid AlGaInAs-silicon evanescent laser,” Opt. Express 14, 9203-9210 (2006).\n[Crossref] [PubMed]\n2. V. R. Almeida, Q. Xu, C. A. Barrios, and M. Lipson, “Guiding and confining light in void nanostructure,” Opt. Lett. 29, 1209–1211 (2004).\n[Crossref] [PubMed]\n3. Q. Xu, V. R. Almeida, R. R. Panepucci, and M. Lipson, “Experimental demonstration of guiding and confining light in nanometer-size low-refractive-index material,” Opt. Lett. 29, 1626–1628 (2004).\n[Crossref] [PubMed]\n4. C. A. Barrios and M. Lipson, “Electrically driven silicon resonant light emitting device based on slot-waveguide,” Opt. Express 13, 10092–10101 (2005).\n[Crossref] [PubMed]\n5. F. Ning-Ning, J. Michel, and L. C. Kimerling, “Optical field concentration in low-index waveguides,” IEEE J. Quantum Electron. 42, 885–890 (2006).\n6. J. T. Robinson, C. Manolatou, C. Long, and M. Lipson, “Ultrasmall mode volumes in dielectric optical microcavities,” Phys. Rev. Lett. 95, 143901 (2005).\n[Crossref] [PubMed]\n7. E. Burstein and C. Weisbuch, eds. Confined electrons and photons, (Plenum Press: New York, NY,1995)\n[Crossref]\n8. T. D. Visser, H. Blok, B. Demeulenaere, and D. Lenstra, “Confinement factors and gain in optical amplifiers,” IEEE J. Quantum Electron. 33, 1763–1766 (1997).\n[Crossref]\n9. C. A. Barrios, K. B. Gylfason, B. Sánchez, A. Griol, H. Sohlström, M. Holgado, and R. Casquel, “Slot-waveguide biochemical sensor,” Opt. Lett. 32, 3080–3082 (2007).\n[Crossref] [PubMed]\n10. F. Dell’Olio and V. M. Passaro, “Optical sensing by optimized silicon slot waveguides,” Opt. Express 15, 4977–4993 (2007).\n[Crossref] [PubMed]\n11. H. Kogelnik, Theory of optical waveguides, in Guided-wave optoelectronics, T. Tamir ed., (Springer Verlag: Berlin, 1990). p. 7.\n12. L. A. Coldren and S. W. Corzine, Diode lasers and photonic integrated circuits (J. Wiley & Sons, New York, NY, 1995).\n13. C. Pollock and M. Lipson, Integrated photonics (Kluwer Academic, Norwell, MA, 2003).\n14. J. Haes, B. Demeulenaere, R. Baets, D. Lenstra, T. D. Visser, and H. Blok, “Difference between te and tm modal gain in amplifying waveguides: Analysis and assessment of two perturbation approaches,” Opt. Quantum Electron. 29, 263–273 (1997).\n[Crossref]\n15. J. D. Jackson, Classical electrodynamics. 3rd ed (John Wiley & Sons, Inc., Hoboken, NJ, 1999).\n16. H. A. Haus, Waves and fileds in optoelectronics (Prentice-Hall, Englewood Cliffs, NJ, 1984).\n17. L. D. Landau and E. M. Lifshitz, Electrodynamics of continuous media (Pergamon Press, Reading, MA, 1960).\n18. G. J. Veldhuis, O. Parriaux, H. J. W. M. Hoekstra, and P. V. Lambeck, “Sensitivity enhancement in evanescent optical waveguide sensors,” J. Lightwave Technol. 18, 677–682 (2000).\n[Crossref]\n19. R. Perahia, O. Painter, V. Moreau, and R. Colombelli, “Design of quantum cascade lasers for intra-cavity sensing in the mid infrared,” (in preparation).\n20. A. E. Siegman, Lasers (University Science Books, Sausalito, CA,1986).\n21. J. T. Robinson, L. Chen, and M. Lipson, “On-chip gas detection in silicon optical microcavities,” Opt. Express 16, 4296–4301 (2008).\n[Crossref] [PubMed]\n\n#### 2006 (2)\n\nF. Ning-Ning, J. Michel, and L. C. Kimerling, “Optical field concentration in low-index waveguides,” IEEE J. Quantum Electron. 42, 885–890 (2006).\n\n#### 2005 (2)\n\nJ. T. Robinson, C. Manolatou, C. Long, and M. Lipson, “Ultrasmall mode volumes in dielectric optical microcavities,” Phys. Rev. Lett. 95, 143901 (2005).\n[Crossref] [PubMed]\n\n#### 2000 (1)\n\nG. J. Veldhuis, O. Parriaux, H. J. W. M. Hoekstra, and P. V. Lambeck, “Sensitivity enhancement in evanescent optical waveguide sensors,” J. Lightwave Technol. 18, 677–682 (2000).\n[Crossref]\n\n#### 1997 (2)\n\nJ. Haes, B. Demeulenaere, R. Baets, D. Lenstra, T. D. Visser, and H. Blok, “Difference between te and tm modal gain in amplifying waveguides: Analysis and assessment of two perturbation approaches,” Opt. Quantum Electron. 29, 263–273 (1997).\n[Crossref]\n\nT. D. Visser, H. Blok, B. Demeulenaere, and D. Lenstra, “Confinement factors and gain in optical amplifiers,” IEEE J. Quantum Electron. 33, 1763–1766 (1997).\n[Crossref]\n\n#### Baets, R.\n\nJ. Haes, B. Demeulenaere, R. Baets, D. Lenstra, T. D. Visser, and H. Blok, “Difference between te and tm modal gain in amplifying waveguides: Analysis and assessment of two perturbation approaches,” Opt. Quantum Electron. 29, 263–273 (1997).\n[Crossref]\n\n#### Blok, H.\n\nT. D. Visser, H. Blok, B. Demeulenaere, and D. Lenstra, “Confinement factors and gain in optical amplifiers,” IEEE J. Quantum Electron. 33, 1763–1766 (1997).\n[Crossref]\n\nJ. Haes, B. Demeulenaere, R. Baets, D. Lenstra, T. D. Visser, and H. Blok, “Difference between te and tm modal gain in amplifying waveguides: Analysis and assessment of two perturbation approaches,” Opt. Quantum Electron. 29, 263–273 (1997).\n[Crossref]\n\n#### Coldren, L. A.\n\nL. A. Coldren and S. W. Corzine, Diode lasers and photonic integrated circuits (J. Wiley & Sons, New York, NY, 1995).\n\n#### Colombelli, R.\n\nR. Perahia, O. Painter, V. Moreau, and R. Colombelli, “Design of quantum cascade lasers for intra-cavity sensing in the mid infrared,” (in preparation).\n\n#### Corzine, S. W.\n\nL. A. Coldren and S. W. Corzine, Diode lasers and photonic integrated circuits (J. Wiley & Sons, New York, NY, 1995).\n\n#### Demeulenaere, B.\n\nT. D. Visser, H. Blok, B. Demeulenaere, and D. Lenstra, “Confinement factors and gain in optical amplifiers,” IEEE J. Quantum Electron. 33, 1763–1766 (1997).\n[Crossref]\n\nJ. Haes, B. Demeulenaere, R. Baets, D. Lenstra, T. D. Visser, and H. Blok, “Difference between te and tm modal gain in amplifying waveguides: Analysis and assessment of two perturbation approaches,” Opt. Quantum Electron. 29, 263–273 (1997).\n[Crossref]\n\n#### Haes, J.\n\nJ. Haes, B. Demeulenaere, R. Baets, D. Lenstra, T. D. Visser, and H. Blok, “Difference between te and tm modal gain in amplifying waveguides: Analysis and assessment of two perturbation approaches,” Opt. Quantum Electron. 29, 263–273 (1997).\n[Crossref]\n\n#### Haus, H. A.\n\nH. A. Haus, Waves and fileds in optoelectronics (Prentice-Hall, Englewood Cliffs, NJ, 1984).\n\n#### Hoekstra, H. J. W. M.\n\nG. J. Veldhuis, O. Parriaux, H. J. W. M. Hoekstra, and P. V. Lambeck, “Sensitivity enhancement in evanescent optical waveguide sensors,” J. Lightwave Technol. 18, 677–682 (2000).\n[Crossref]\n\n#### Jackson, J. D.\n\nJ. D. Jackson, Classical electrodynamics. 3rd ed (John Wiley & Sons, Inc., Hoboken, NJ, 1999).\n\n#### Kimerling, L. C.\n\nF. Ning-Ning, J. Michel, and L. C. Kimerling, “Optical field concentration in low-index waveguides,” IEEE J. Quantum Electron. 42, 885–890 (2006).\n\n#### Kogelnik, H.\n\nH. Kogelnik, Theory of optical waveguides, in Guided-wave optoelectronics, T. Tamir ed., (Springer Verlag: Berlin, 1990). p. 7.\n\n#### Lambeck, P. V.\n\nG. J. Veldhuis, O. Parriaux, H. J. W. M. Hoekstra, and P. V. Lambeck, “Sensitivity enhancement in evanescent optical waveguide sensors,” J. Lightwave Technol. 18, 677–682 (2000).\n[Crossref]\n\n#### Landau, L. D.\n\nL. D. Landau and E. M. Lifshitz, Electrodynamics of continuous media (Pergamon Press, Reading, MA, 1960).\n\n#### Lenstra, D.\n\nJ. Haes, B. Demeulenaere, R. Baets, D. Lenstra, T. D. Visser, and H. Blok, “Difference between te and tm modal gain in amplifying waveguides: Analysis and assessment of two perturbation approaches,” Opt. Quantum Electron. 29, 263–273 (1997).\n[Crossref]\n\nT. D. Visser, H. Blok, B. Demeulenaere, and D. Lenstra, “Confinement factors and gain in optical amplifiers,” IEEE J. Quantum Electron. 33, 1763–1766 (1997).\n[Crossref]\n\n#### Lifshitz, E. M.\n\nL. D. Landau and E. M. Lifshitz, Electrodynamics of continuous media (Pergamon Press, Reading, MA, 1960).\n\n#### Lipson, M.\n\nJ. T. Robinson, C. Manolatou, C. Long, and M. Lipson, “Ultrasmall mode volumes in dielectric optical microcavities,” Phys. Rev. Lett. 95, 143901 (2005).\n[Crossref] [PubMed]\n\nC. Pollock and M. Lipson, Integrated photonics (Kluwer Academic, Norwell, MA, 2003).\n\n#### Long, C.\n\nJ. T. Robinson, C. Manolatou, C. Long, and M. Lipson, “Ultrasmall mode volumes in dielectric optical microcavities,” Phys. Rev. Lett. 95, 143901 (2005).\n[Crossref] [PubMed]\n\n#### Manolatou, C.\n\nJ. T. Robinson, C. Manolatou, C. Long, and M. Lipson, “Ultrasmall mode volumes in dielectric optical microcavities,” Phys. Rev. Lett. 95, 143901 (2005).\n[Crossref] [PubMed]\n\n#### Michel, J.\n\nF. Ning-Ning, J. Michel, and L. C. Kimerling, “Optical field concentration in low-index waveguides,” IEEE J. Quantum Electron. 42, 885–890 (2006).\n\n#### Moreau, V.\n\nR. Perahia, O. Painter, V. Moreau, and R. Colombelli, “Design of quantum cascade lasers for intra-cavity sensing in the mid infrared,” (in preparation).\n\n#### Ning-Ning, F.\n\nF. Ning-Ning, J. Michel, and L. C. Kimerling, “Optical field concentration in low-index waveguides,” IEEE J. Quantum Electron. 42, 885–890 (2006).\n\n#### Painter, O.\n\nR. Perahia, O. Painter, V. Moreau, and R. Colombelli, “Design of quantum cascade lasers for intra-cavity sensing in the mid infrared,” (in preparation).\n\n#### Parriaux, O.\n\nG. J. Veldhuis, O. Parriaux, H. J. W. M. Hoekstra, and P. V. Lambeck, “Sensitivity enhancement in evanescent optical waveguide sensors,” J. Lightwave Technol. 18, 677–682 (2000).\n[Crossref]\n\n#### Perahia, R.\n\nR. Perahia, O. Painter, V. Moreau, and R. Colombelli, “Design of quantum cascade lasers for intra-cavity sensing in the mid infrared,” (in preparation).\n\n#### Pollock, C.\n\nC. Pollock and M. Lipson, Integrated photonics (Kluwer Academic, Norwell, MA, 2003).\n\n#### Robinson, J. T.\n\nJ. T. Robinson, C. Manolatou, C. Long, and M. Lipson, “Ultrasmall mode volumes in dielectric optical microcavities,” Phys. Rev. Lett. 95, 143901 (2005).\n[Crossref] [PubMed]\n\n#### Siegman, A. E.\n\nA. E. Siegman, Lasers (University Science Books, Sausalito, CA,1986).\n\n#### Veldhuis, G. J.\n\nG. J. Veldhuis, O. Parriaux, H. J. W. M. Hoekstra, and P. V. Lambeck, “Sensitivity enhancement in evanescent optical waveguide sensors,” J. Lightwave Technol. 18, 677–682 (2000).\n[Crossref]\n\n#### Visser, T. D.\n\nJ. Haes, B. Demeulenaere, R. Baets, D. Lenstra, T. D. Visser, and H. Blok, “Difference between te and tm modal gain in amplifying waveguides: Analysis and assessment of two perturbation approaches,” Opt. Quantum Electron. 29, 263–273 (1997).\n[Crossref]\n\nT. D. Visser, H. Blok, B. Demeulenaere, and D. Lenstra, “Confinement factors and gain in optical amplifiers,” IEEE J. Quantum Electron. 33, 1763–1766 (1997).\n[Crossref]\n\n#### IEEE J. Quantum Electron (2)\n\nF. Ning-Ning, J. Michel, and L. C. Kimerling, “Optical field concentration in low-index waveguides,” IEEE J. Quantum Electron. 42, 885–890 (2006).\n\nT. D. Visser, H. Blok, B. Demeulenaere, and D. Lenstra, “Confinement factors and gain in optical amplifiers,” IEEE J. Quantum Electron. 33, 1763–1766 (1997).\n[Crossref]\n\n#### J. Lightwave Technol (1)\n\nG. J. Veldhuis, O. Parriaux, H. J. W. M. Hoekstra, and P. V. Lambeck, “Sensitivity enhancement in evanescent optical waveguide sensors,” J. Lightwave Technol. 18, 677–682 (2000).\n[Crossref]\n\n#### Opt. Quantum Electron. (1)\n\nJ. Haes, B. Demeulenaere, R. Baets, D. Lenstra, T. D. Visser, and H. Blok, “Difference between te and tm modal gain in amplifying waveguides: Analysis and assessment of two perturbation approaches,” Opt. Quantum Electron. 29, 263–273 (1997).\n[Crossref]\n\n#### Phys. Rev. Lett. (1)\n\nJ. T. Robinson, C. Manolatou, C. Long, and M. Lipson, “Ultrasmall mode volumes in dielectric optical microcavities,” Phys. Rev. Lett. 95, 143901 (2005).\n[Crossref] [PubMed]\n\n#### Other (9)\n\nE. Burstein and C. Weisbuch, eds. Confined electrons and photons, (Plenum Press: New York, NY,1995)\n[Crossref]\n\nH. Kogelnik, Theory of optical waveguides, in Guided-wave optoelectronics, T. Tamir ed., (Springer Verlag: Berlin, 1990). p. 7.\n\nL. A. Coldren and S. W. Corzine, Diode lasers and photonic integrated circuits (J. Wiley & Sons, New York, NY, 1995).\n\nC. Pollock and M. Lipson, Integrated photonics (Kluwer Academic, Norwell, MA, 2003).\n\nJ. D. Jackson, Classical electrodynamics. 3rd ed (John Wiley & Sons, Inc., Hoboken, NJ, 1999).\n\nH. A. Haus, Waves and fileds in optoelectronics (Prentice-Hall, Englewood Cliffs, NJ, 1984).\n\nL. D. Landau and E. M. Lifshitz, Electrodynamics of continuous media (Pergamon Press, Reading, MA, 1960).\n\nR. Perahia, O. Painter, V. Moreau, and R. Colombelli, “Design of quantum cascade lasers for intra-cavity sensing in the mid infrared,” (in preparation).\n\nA. E. Siegman, Lasers (University Science Books, Sausalito, CA,1986).\n\n### Cited By\n\nOSA participates in Crossref's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.\n\n### Figures (4)\n\nFig. 1. Fundamental TM modes at a wavelength of 1.5 µm for waveguides 500 nm wide and 600 nm tall. All modes are normalized to unit power. The high-index material (n=3.5) is outlined in black. The waveguides are clad with n=3.25 for (a) and (b) and n=1.5 for (c)–(f). The first and second columns show E y and $− ω μ 0 β H x$ respectively, plotted on the same color scale. The two fields become increasingly dissimilar as more electric field is concentrated at high-index-contrast boundaries.\nFig. 2. Numerical study of modal gain. (a) Schematic of slot waveguide with gain material defined by an imaginary component of the refractive index confined to the slot region (pink). (b) Major field component of the fundamental TM mode for the same structure as (a) calculated using a finite difference mode solver. (c) Circles show the modal gain (g m ) calculated from the complex effective index of the fundamental TM mode as determined using a finite difference mode solver. Material gain is added via the imaginary part of the refractive index in the slot. Dashed line shows the modal gain calculated according to Eq. (3) based on the confinement factor Γ determined from the zero-gain mode profile from Eq. (15). Dotted line shows the product of the power in the active gain region (P A ) and the material gain. We see that the confinement factor proposed in this paper correctly predicts the modal gain simulated numerically, while the power confinement greatly underestimates the simulated modal gain.\nFig. 3. (a). The spatial confinement factor γ A plotted as a function of slot thickness t, where the gain region is defined as the slot (pink region in inset) between the high-index rails (green). Narrow slots result greater emission rates of gain material while thicker slots provide more material which contributes to the gain. The peak in γ A near a slot width of 60 nm indicates the condition where the combination of enhanced emission rate and volume of gain material result in the lowest lasing threshold. (b). The total confinement factor (Γ) (squares) and power in the slot region (P A ) (triangles) as a function of slot width. Dotted and dashed lines mark the slot widths which maximize Γ and P A respectively. The discrepancy between these two plots shows that the percentage of power in the gain media is not an accurate indication of either the magnitude or the optimal design for modal gain.\nFig. 4. Optimization of width and height of Si/SiO2/Si slot waveguide with a 10 nm thick slot assumed to contain a gain medium. (a) Schematic of slot waveguide. (b) Total confinement factor Γ, proportional to the total modal gain. (c) Group index n g divided by the slot index (1.46), which is responsible for the difference between the lasing threshold and modal gain. (d) Electric field energy confinement γ A , inversely proportional to the lasing threshold. The maximum total modal gain is marked by the square in (a). The white contour shows the region which corresponds to a 5% change from the maximum values of Γ and γ A .\n\n### Equations (20)\n\n$H = c ε n ( e ̂ z × E ) ,$\n$E y = − ω μ 0 β H x ,$\n$Γ ≡ g m g b ,$\n$∣ E ( z ) ∣ 2 = ∣ E 0 ∣ 2 e g b z .$\n$E ( x , y , z ) = e ̂ j E j o ψ j ( x , y ) e i ( ω t − β ˜ z ) ,$\n$β ˜ ≡ k 0 ( n ̅ r + i n ̅ i ) ,$\n$g m = 2 Im { β ˜ } = 2 k 0 n ̅ i .$\n$g b = 2 k 0 n A i .$\n$Δ β ˜ = ω ∬ ∞ Δ ε ˜ ∣ E ∣ 2 d x d y 1 2 ∬ ∞ Re { E × H } ∙ e ̂ z d x d y ,$\n$g m = [ n A c ε 0 ∬ A ∣ E ∣ 2 d x d y ∬ ∞ Re { E × H * } ∙ e ̂ z d x d y ] g b .$\n$Γ = n A c ε 0 ∬ A ∣ E ∣ 2 d x d y ∬ ∞ Re { E × H * } ∙ e ̂ z d x d y .$\n$1 2 ∬ Re { E × H * } ∙ e ̂ z d x d y = 1 2 β ω μ 0 ∬ ∣ E ∣ 2 d x d y ,$\n$U / l = 1 2 ∫ ∫ ∞ ε ∣ E ∣ 2 d x d y .$\n$1 2 ∬ Re { E × H * } ∙ e ̂ z d x d y = v g 1 2 ∬ ε ∣ E ∣ 2 d x d y$\n$Γ = n g ∬ A ε ∣ E ∣ 2 d x d y n A ∬ ∞ ε ∣ E ∣ 2 d x d y ≡ n g n A γ A .$\n$γ A ≡ ∬ A ε ∣ E ∣ 2 d x d y ∬ ∞ ε ∣ E ∣ 2 d x d y .$\n$P A ≡ ∬ A Re { E × H * } ∙ e z d x d y ∬ ∞ Re { E × H * } ∙ e z d x d y .$\n$α m = n g n b γ b α b ,$\n$n g γ A n A g b − n g ∑ i γ i n i α i > 0 .$\n$g b > α m Γ .$" ]	[ null, "https://www.osapublishing.org/images/ajax-loader-big.gif", null, "https://www.osapublishing.org/images/ajax-loader-big.gif", null, "https://www.osapublishing.org/images/ajax-loader-big.gif", null, "https://www.osapublishing.org/images/ajax-loader-big.gif", null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.8992896,"math_prob":0.9191582,"size":28764,"snap":"2021-21-2021-25","text_gpt3_token_len":6347,"char_repetition_ratio":0.16842838,"word_repetition_ratio":0.029261,"special_character_ratio":0.21634682,"punctuation_ratio":0.10998339,"nsfw_num_words":1,"has_unicode_error":false,"math_prob_llama3":0.9843941,"pos_list":[0,1,2,3,4,5,6,7,8],"im_url_duplicate_count":[null,null,null,null,null,null,null,null,null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2021-05-13T09:35:55Z\",\"WARC-Record-ID\":\"<urn:uuid:12d141d9-fd48-4270-becb-cf06ca082852>\",\"Content-Length\":\"232865\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:4f8ad92a-8101-40a0-a668-4f1890cc83b2>\",\"WARC-Concurrent-To\":\"<urn:uuid:249b8fc0-a6c7-4a73-9045-ee311e5891ef>\",\"WARC-IP-Address\":\"38.95.177.60\",\"WARC-Target-URI\":\"https://www.osapublishing.org/oe/fulltext.cfm?uri=oe-16-21-16659&id=172485\",\"WARC-Payload-Digest\":\"sha1:6CG5TWFHYEC455L5OW4KUE6EDWDGZQT5\",\"WARC-Block-Digest\":\"sha1:K2CZRDADU5TIKK62VEITGPAIQEC2V6AH\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2021/CC-MAIN-2021-21/CC-MAIN-2021-21_segments_1620243990584.33_warc_CC-MAIN-20210513080742-20210513110742-00232.warc.gz\"}"}
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119.155.1.27\n)\n\n => Array\n(\n => 72.255.6.24\n)\n\n => Array\n(\n => 192.140.152.223\n)\n\n => Array\n(\n => 212.103.48.136\n)\n\n => Array\n(\n => 39.45.134.56\n)\n\n => Array\n(\n => 139.167.173.30\n)\n\n => Array\n(\n => 117.230.63.87\n)\n\n => Array\n(\n => 182.189.95.203\n)\n\n => Array\n(\n => 49.204.183.248\n)\n\n => Array\n(\n => 47.31.125.188\n)\n\n => Array\n(\n => 103.252.171.13\n)\n\n => Array\n(\n => 112.198.74.36\n)\n\n => Array\n(\n => 27.109.113.152\n)\n\n => Array\n(\n => 42.112.233.44\n)\n\n => Array\n(\n => 47.31.68.193\n)\n\n => Array\n(\n => 103.252.171.134\n)\n\n => Array\n(\n => 77.123.32.114\n)\n\n => Array\n(\n => 1.38.189.66\n)\n\n => Array\n(\n => 39.37.181.108\n)\n\n => Array\n(\n => 42.106.44.61\n)\n\n => Array\n(\n => 157.36.8.39\n)\n\n => Array\n(\n => 223.238.41.53\n)\n\n => Array\n(\n => 202.89.77.10\n)\n\n => Array\n(\n => 117.230.150.68\n)\n\n => Array\n(\n => 175.176.87.60\n)\n\n => Array\n(\n => 137.97.117.87\n)\n\n => Array\n(\n => 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=> 117.230.53.165\n)\n\n => Array\n(\n => 49.37.200.171\n)\n\n => Array\n(\n => 104.236.213.230\n)\n\n => Array\n(\n => 103.140.30.81\n)\n\n => Array\n(\n => 59.103.104.117\n)\n\n => Array\n(\n => 65.49.126.79\n)\n\n => Array\n(\n => 202.59.12.251\n)\n\n => Array\n(\n => 37.111.136.17\n)\n\n => Array\n(\n => 163.53.85.67\n)\n\n => Array\n(\n => 123.16.240.73\n)\n\n => Array\n(\n => 103.211.14.183\n)\n\n => Array\n(\n => 103.248.93.211\n)\n\n => Array\n(\n => 116.74.59.127\n)\n\n => Array\n(\n => 137.97.169.254\n)\n\n => Array\n(\n => 113.177.79.100\n)\n\n => Array\n(\n => 74.82.60.187\n)\n\n => Array\n(\n => 117.230.157.66\n)\n\n => Array\n(\n => 169.149.194.241\n)\n\n => Array\n(\n => 117.230.156.11\n)\n\n => Array\n(\n => 202.59.12.157\n)\n\n => Array\n(\n => 42.106.181.25\n)\n\n => Array\n(\n => 202.59.13.78\n)\n\n => Array\n(\n => 39.37.153.32\n)\n\n => Array\n(\n => 177.188.216.175\n)\n\n => Array\n(\n => 222.252.53.165\n)\n\n => Array\n(\n => 37.139.23.89\n)\n\n => Array\n(\n => 117.230.139.150\n)\n\n => Array\n(\n => 104.131.176.234\n)\n\n => Array\n(\n => 42.106.181.117\n)\n\n => Array\n(\n => 117.230.180.94\n)\n\n => Array\n(\n => 180.190.171.5\n)\n\n => Array\n(\n => 150.129.165.185\n)\n\n => Array\n(\n => 51.15.0.150\n)\n\n => Array\n(\n => 42.111.4.84\n)\n\n => Array\n(\n => 74.82.60.116\n)\n\n => Array\n(\n => 137.97.121.165\n)\n\n => Array\n(\n => 64.62.187.194\n)\n\n => Array\n(\n => 137.97.106.162\n)\n\n => Array\n(\n => 137.97.92.46\n)\n\n => Array\n(\n => 137.97.170.25\n)\n\n => Array\n(\n => 103.104.192.100\n)\n\n => Array\n(\n => 185.246.211.34\n)\n\n => Array\n(\n => 119.160.96.78\n)\n\n => Array\n(\n => 212.103.48.152\n)\n\n => Array\n(\n => 183.83.153.90\n)\n\n => Array\n(\n => 117.248.150.41\n)\n\n => Array\n(\n => 185.240.246.180\n)\n\n => Array\n(\n => 162.253.131.125\n)\n\n => Array\n(\n => 117.230.153.217\n)\n\n => Array\n(\n => 117.230.169.1\n)\n\n => Array\n(\n => 49.15.138.247\n)\n\n => Array\n(\n => 117.230.37.110\n)\n\n => Array\n(\n => 14.167.188.75\n)\n\n => Array\n(\n => 169.149.239.93\n)\n\n => Array\n(\n => 103.216.176.91\n)\n\n => Array\n(\n => 117.230.12.126\n)\n\n => Array\n(\n => 184.75.209.110\n)\n\n => Array\n(\n => 117.230.6.60\n)\n\n => Array\n(\n => 117.230.135.132\n)\n\n => Array\n(\n => 31.179.29.109\n)\n\n => Array\n(\n => 74.121.188.186\n)\n\n => Array\n(\n => 117.230.35.5\n)\n\n => Array\n(\n => 111.92.74.239\n)\n\n => Array\n(\n => 104.245.144.236\n)\n\n => Array\n(\n => 39.50.22.100\n)\n\n => Array\n(\n => 47.31.190.23\n)\n\n => Array\n(\n => 157.44.73.187\n)\n\n => Array\n(\n => 117.230.8.91\n)\n\n => Array\n(\n => 157.32.18.2\n)\n\n => Array\n(\n => 111.119.187.43\n)\n\n => Array\n(\n => 203.101.185.246\n)\n\n => Array\n(\n => 5.62.34.22\n)\n\n => Array\n(\n => 122.8.143.76\n)\n\n => Array\n(\n => 115.186.2.187\n)\n\n => Array\n(\n => 202.142.110.89\n)\n\n => Array\n(\n => 157.50.61.254\n)\n\n => Array\n(\n => 223.182.211.185\n)\n\n => Array\n(\n => 103.85.125.210\n)\n\n => Array\n(\n => 103.217.133.147\n)\n\n => Array\n(\n => 103.60.196.217\n)\n\n => Array\n(\n => 157.44.238.6\n)\n\n => Array\n(\n => 117.196.225.68\n)\n\n => Array\n(\n => 104.254.92.52\n)\n\n => Array\n(\n => 39.42.46.72\n)\n\n => Array\n(\n => 221.132.119.36\n)\n\n => Array\n(\n => 111.92.77.47\n)\n\n => Array\n(\n => 223.225.19.152\n)\n\n => Array\n(\n => 159.89.121.217\n)\n\n => Array\n(\n => 39.53.221.205\n)\n\n => Array\n(\n => 193.34.217.28\n)\n\n => Array\n(\n => 139.167.206.36\n)\n\n)\n```\nHunt and Go Seek: Save More, Spend Less\n << Back to all Blogs Login or Create your own free blog Layout: Blue and Brown (Default) Author's Creation\nHome > Hunt and Go Seek", null, "", null, "", null, "# Hunt and Go Seek\n\nJuly 29th, 2010 at 11:06 am\n\nIt's now impossible to walk inside my fenced vegetable garden without treading on squash vines. My tomato plants are taller than I am. (I'm 5'4\".)\n\nYesterday, I ventured inside to tie up a few tomato branches that were being pulled down by a squash vine.\n\nI discovered a giant cucumber hidden under all those leaves, as well as two over-sized zucchini I hadn't seen before. It's kind of like a vegetarian version of the game Hide and Go Seek.\n\nWell, you know what that means....zucchini bread! I also picked another medium-sized tomato and a few string beans.\n\nHere's a rundown of what's growing:\n\nTomatoes: Lots of green tomatoes, but thus far, just a few mid-sized tomatoes that had ripened. I could have sworn that I bought cherry tomato plants, but it appears all 5 of my plants produce a mid-sized variety of tomato.\n\nZucchini: This has been my best year of the last 3 for zucchini. I made a special effort to hand pollinate this year and i'm getting a lot of zucchini.\n\nSpaghetti Squash: I think these may be able to be picked soon. They went from a greenish-white color when formed to a pale yellow now. But I'll have to check online as I've never grown these before. But I have 4, maybe 5 loaf-sized squashes...HUGE. Can't wait to try them!\n\nAcorn squash: I just spotted one large one yesterday, but that's all so far. Guess they take a while to grow.\n\nString beans I usually grow bush beans and get plenty of beans, but this year i decided to try pole beans. I have 2 tripod structures made of 6 foot high branches and these have been absolutely covered by the beans, but so far, lots and lots of leaves (I haven't fertilized) and very few beans. Disappointing.\n\nPotatoes: I had great fun with potatoes last year, so this year i planted both red and russets. They appear to be dying back already; this is supposed to happen by end of August, after which time you can dig up the potatoes. I expect to have a great crop.\n\nOrnamental gourds I planted a row of these but only 2 plants came up, I think becus I was stepping in this area while weeding and forgot I'd planted something here along the fenceline. I only noticed one small gourd; let's hope more will come.\n\nCucumbers Last year I had maybe 4 plants and I was awash in cucumbers. I decided to pare it down to just 2 plants this year, and now it seems I don't have ENOUGH cucumbers. I've only gotten maybe 3 or 4 at this point.\n\nBell pepper I've never had huge successes with bell peppers. It seems to take so long for them to really take off. Haven't gotten any peppers yet, but the plants themselves finally seem healthier, with more foliage. Of course, they're being shaded out by the squash vines, which are hard to contain and control.\n\nI also planted some sort of vining annual flower, just to add some color and attract pollinators, but while they also got off to a slow start, they have yet to flower. Just green....\n\nI'm off to a focus group this a.m. about an hour's drive from here. I'll return mid-day with \\$125 cash in my pocket.\n\nI want to lay my new soaker hose in the garden and also head over to friend's house to continue cleanup work in her garage/house and start getting ready for her garage sale. She's basically cleaning out a lifetime of accumulated stuff. I've already gotten 2 nice lunches out of it (a bison burger at the local diner, and a shrimp caesar salad and dessert at a good Italian place).\n\nI also need to mow the lawn!\n\n### 5 Responses to “Hunt and Go Seek”\n\n1. NJDebbie Says:\n\nCan you post a picture of your garden? I love to see it!\n\n2. creditcardfree Says:\n\nWow! Your garden is flourishing. I have some gourd plants that volunteered from my compost last year, they are taking over my garden and trying to escape the fence!\n\n3. Ima saver Says:\n\nWe have about a million tomatoes. We are giving them out to everyone!\n\n4. Jerry Says:\n\nI swear I haven't even thought of zucchini bread in YEARS, and your post just leads my mouth to start watering thinking about it! Yum...\nUnrelated - I wish we had some insurance of a good bison burger joint around here, but there just aren't a lot of American Bison running around this part of Europe. I want a bison burger!\nJerry\n\n5. PatientSaver Says:\n\nNJ Debbie, you can see photos of my garden at my gardening blog:" ]	[ null, "https://www.savingadvice.com/blogs/images/search/top_left.php", null, "https://www.savingadvice.com/blogs/images/search/top_right.php", null, "https://www.savingadvice.com/blogs/images/search/bottom_left.php", null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.97403306,"math_prob":0.99828994,"size":4040,"snap":"2020-24-2020-29","text_gpt3_token_len":1021,"char_repetition_ratio":0.08845391,"word_repetition_ratio":0.057771664,"special_character_ratio":0.25222772,"punctuation_ratio":0.1216686,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9990087,"pos_list":[0,1,2,3,4,5,6],"im_url_duplicate_count":[null,null,null,null,null,null,null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2020-06-04T00:42:54Z\",\"WARC-Record-ID\":\"<urn:uuid:15701482-b0ca-4b57-9edf-2e04c080eb26>\",\"Content-Length\":\"134665\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:a2cc886c-37d4-4c04-b5d6-583b477eaa39>\",\"WARC-Concurrent-To\":\"<urn:uuid:0a12e889-c7fc-4005-b817-57a5560c464c>\",\"WARC-IP-Address\":\"173.231.200.26\",\"WARC-Target-URI\":\"https://patientsaver.savingadvice.com/2010/07/29/hunt-and-go-seek_60885/\",\"WARC-Payload-Digest\":\"sha1:JBXHBUANBOKPGW47DKXUMLA2FHPCJGJ6\",\"WARC-Block-Digest\":\"sha1:BGBJVMJLFEHAYRVUKAQAJABNPRAK7NOL\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2020/CC-MAIN-2020-24/CC-MAIN-2020-24_segments_1590347436828.65_warc_CC-MAIN-20200604001115-20200604031115-00080.warc.gz\"}"}
http://rin.io/spec/	[ "Spectrum of a Ring\n\nApproaching Scheme Theory as a Beginner\n\nThank you to James Tao for explaining the following, geometrically intuitive, interpretation to me.\n\nA scheme is the result of patching affine schemes together, an affine scheme is any locally ringed space that is equivalent to Spec of a ring. To make an analogy, a manifold is the result of patching together copies of $\\mathbb{R}^n$, in a way that preserves the smooth structure.\n\nAnd the smooth structure of $\\mathbb{R}^n$ is essentially the fact that it is a sheaf: over any open set, we have the space of smooth functions on that set, and these functions restrict and glue nicely, etc.\n\nSo when you go from $\\mathbb{R}^n$ to “smooth manifolds,” you are creating spaces that locally look like $\\mathbb{R}^n$, respect the same kind of smooth structure (as captured by the fact that smooth functions are defined on open sets in a way independent of the patching), yet have some global structure that might prevent them from just being the same as $\\mathbb{R}^n$.\n\nIf you care about algebraic (polynomial / rational) functions defined on algebraic sets (zeroes of systems of polynomials), then you can do the same program! The “basic patch” is “affine scheme,” and you patch affine schemes together to get a scheme.\n\nThe category of affine schemes is equivalent to $\\text{Ring}^{op}$. In other words, telling you how to map forward is the same as telling you how to pull back functions. However, this is NOT true for the larger category of schemes. $\\text{Hom}_{Ring}(A, \\Gamma(\\mathcal{O}_B)) = \\text{Hom}_{Schemes} (B, \\text{Spec } A)$ (note that $\\Gamma(\\mathcal{O}_B)$ := the global sections of the structure sheaf $\\mathcal{O}_B$).\n\nIn particular, given a scheme, it’s unclear how to get a ring that encodes all the information in that scheme. Consider, for example, projective space, whose only global sections are constants, yet the field $k$ of constant functions certainly doesn’t single out any projective space!\n\nWhat are these affine Spec R things?\n\nWe have a functor Spec from Ring to Schemes:\n\n$\\text{Ring} \\xrightarrow{\\text{Spec}} \\text{Schemes}$\n\nSchemes look like Fun(Ring, Set). So Spec sends a ring to a functor from Ring to Set.\n\n$\\text{Ring} \\xrightarrow{\\text{Spec}} (\\text{Ring} \\to \\text{Set})$\n\nHonestly, whenever I see “scheme” I replace it mentally with “algebraic curve,” but this is just because I don’t really get what a scheme is yet.\n\n$R \\xrightarrow{\\text{Spec} (-)} \\text{Spec(R)}$\n\n$R \\xrightarrow{\\text{Spec} (-)} (S \\mapsto \\text{hom}(R,S))$\n\n$\\text{Spec}$ $R$ $S$ = $\\text{Hom}_{\\text{Ring}}(R, S)$\n\nLet say we have:\n\n1. a ring $\\mathbb{Z}[t]$ (this is just the ring of polynomials with $t$ as a variable and coefficients in $\\mathbb{Z}$)\n2. an arbitrary ring $S$.\n\nWhat is $\\text{hom}(\\mathbb{Z}[t], S)$?\n\nNotice that we’re in Ring, so the members of $\\text{hom}(\\mathbb{Z}[t], S)$ must be Ring homomorphisms. Let’s say we’re looking at a ring homomorphism $\\phi: \\mathbb{Z}[t] \\to S$.\n\nI learned from Cris Moore that demanding examples is a useful practice, so: What is $\\phi(7t^2-4t+3)$ in simpler terms?\n\n$\\phi$ is a ring homomorphism, so we know that $\\phi(1) = 1_s$. This implies that $\\phi(n) = n_s$.\n\n$\\phi(7t^2-4t+3) = \\phi(7t^2) – \\phi(4t) + \\phi(3)$\n$= \\phi(t)(\\phi(7t) – \\phi(4)) + \\phi(3)$\n$= \\phi(t) (\\phi(t) 7_s – 4_s) + 3_s$\n\n$\\phi(t)$ is not determined! We can pick it to be any element of S!\n\n$\\text{hom}(\\mathbb{Z}[t], S) \\simeq \\{\\phi(t) \\in S\\} = S$\n\nIn other words, $\\text{Spec}(\\mathbb{Z}[t])$ is a forgetful functor from $S$ as an object in $\\text{Ring}$ to its underlying set.\n\nExercise: show that $\\mathbb{Z}[t]$ is an initial object in an appropriate category, and that the trivial ring is the terminal object in the appropriate category.\n\nBut what is Spec?\n\nIt’s the spectrum of a ring! Intuitively, $\\text{Spec} R$ is the geometric object canonically associated to $R$.\n\nFor example:\n\n• $\\text{Spec}(\\mathbb{Z})$ is “The Point”\n• $\\text{Spec}(\\mathbb{Z}[x,y]/ (x^2 + y^2 – 1))$ is “The Circle”\n• $\\text{Spec}(\\mathbb{Z}[x,x^{-1}]$ is “Pairs of Invertible Elements” (e.g. a field with the origin removed)\n\nLet’s look at $\\text{Spec}(\\mathbb{Z}[x,y]/ (x^2 + y^2 – 1))(S)$ when $S$ is $\\mathbb{R}$, it’s obviously the circle BUT IT WORKS FOR (almost) ANY S. The concept of there being a canonical geometry associated to every ring is very exciting!\n\nMurky Afterthoughts\n\nWhen people say “elliptic curve” they might mean “a family of elliptic curves”, this is commonly written $C \\to \\text{Spec} R$ where $R$ is the underlying coefficient ring.\n\nFor example, $y^2=4x^3+ax+b \\mapsto a, b \\in R$ is the family of elliptic curves of the form $y^2=4x^3+ax+b$ over the coefficient ring $R$.\n\nI’m trying to figure out what happens when $S$ is not something nice like $\\mathbb{R}$ or $\\mathbb{C}$. What is a circle in the coefficient ring of an elliptic curve?\n\nThanks to Aaron Mazel-Gee for walking me through the concept of $\\text{Spec}$ (all errors in this post are mine and not his).\n\nOne thought on “Spectrum of a Ring”\n\n1.", null, "Josh G says:\n\nI think you do yourself a disservice by jumping right from “algebraic curves” to the viewpoint of (affine) schemes as functors. To make the connection more concrete, one should first understand the connection between varieties (or even just algebraic curves) and their coordinate rings, and then understand how Spec of the coordinate ring recovers the Zariski topology on the original variety. Then one can take the more abstract viewpoint, whereby one wishes to treat not just integral domains over a field, but any ring as the coordinate ring of some “geometric object”, namely, a scheme. And then one can generalize from schemes as the categorical duals of rings to the “functor of points” viewpoint. Taking this route will give you a much more intuitive, geometric picture of what is going on (and should enable you to connect your intuitive geometric picture of algebraic curves to the highly abstract picture of schemes as functors.)\n\nFrom this viewpoint, it is clear why $hom(\\mathbb{Z}[t], S)$ should just be $S$. Imagine every ring is the ring of “well-behaved” functions on some geometric object. Then certainly $\\mathbb{Z}[t]$ is the ring of polynomial functions on the line (where, here, we haven’t specified which line – real, complex, or what have you – but that’s okay because we’re only considering polynomials with integer coefficients, which make sense on the line over any ring). And $S$ is the ring of well-behaved functions on some geometric object which we’ll call $Spec(S)$. Since rings are categorically dual to schemes, $hom(\\mathbb{Z}[t], S)$ should be dual to $hom(Spec(S), line)$. But what are maps from $Spec(S)$ to the line? They are exactly the (well-behaved) functions on $Spec(S)$ – i.e., the elements of $S$!" ]	[ null, "http://1.gravatar.com/avatar/d74e21661477efeec5c02cb24f4f58f3", null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.8649725,"math_prob":0.998351,"size":6733,"snap":"2019-43-2019-47","text_gpt3_token_len":1884,"char_repetition_ratio":0.13048,"word_repetition_ratio":0.0037807184,"special_character_ratio":0.2735779,"punctuation_ratio":0.09337349,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.99977857,"pos_list":[0,1,2],"im_url_duplicate_count":[null,null,null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2019-10-19T21:26:26Z\",\"WARC-Record-ID\":\"<urn:uuid:af77db10-b27d-4349-a438-4308792509d8>\",\"Content-Length\":\"39790\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:11defd26-eb5e-4f5b-b06c-af668a2e3563>\",\"WARC-Concurrent-To\":\"<urn:uuid:1b4e98cf-2758-4c26-ab6b-a80174191c48>\",\"WARC-IP-Address\":\"162.243.159.36\",\"WARC-Target-URI\":\"http://rin.io/spec/\",\"WARC-Payload-Digest\":\"sha1:NQNWOIP4GONIB46KQPCOF6NS5NOYG663\",\"WARC-Block-Digest\":\"sha1:Q4NVG5YL4NLLY7UNZXFIGZISY5XHVCYL\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2019/CC-MAIN-2019-43/CC-MAIN-2019-43_segments_1570986697760.44_warc_CC-MAIN-20191019191828-20191019215328-00131.warc.gz\"}"}
https://encyclopedia2.thefreedictionary.com/electron-volt	[ "# electron-volt\n\nAlso found in: Dictionary, Medical, Wikipedia.\nRelated to electron-volt: kiloelectron volt\n\n## electron-volt,\n\nabbr. eV, unit of energy used in atomic and nuclear physics; 1 electron-volt is the energy transferred in moving a unit chargecharge,\nproperty of matter that gives rise to all electrical phenomena (see electricity). The basic unit of charge, usually denoted by e, is that on the proton or the electron; that on the proton is designated as positive (+e\n, positive or negative and equal to that charge on the electron, through a potentialpotential, electric,\nwork per unit of electric charge expended in moving a charged body from a reference point to any given point in an electric field (see electrostatics).\ndifference of 1 volt. The maximum energy of a particle acceleratorparticle accelerator,\napparatus used in nuclear physics to produce beams of energetic charged particles and to direct them against various targets. Such machines, popularly called atom smashers, are needed to observe objects as small as the atomic nucleus in studies of its\nis usually expressed in multiples of the electron-volt, such as million electron-volts (MeV) or billion electron-volts (GeV). Because mass is a form of energy (see relativityrelativity,\nphysical theory, introduced by Albert Einstein, that discards the concept of absolute motion and instead treats only relative motion between two systems or frames of reference.\n), the masses of elementary particleselementary particles,\nthe most basic physical constituents of the universe. Basic Constituents of Matter\n\nMolecules are built up from the atom, which is the basic unit of any chemical element. The atom in turn is made from the proton, neutron, and electron." ]	[ null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.8918005,"math_prob":0.9261045,"size":2110,"snap":"2020-24-2020-29","text_gpt3_token_len":441,"char_repetition_ratio":0.13770181,"word_repetition_ratio":0.046296295,"special_character_ratio":0.2042654,"punctuation_ratio":0.18780488,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9563636,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2020-07-10T20:20:39Z\",\"WARC-Record-ID\":\"<urn:uuid:1e62b738-6f62-4614-8642-a2dc41ec9e4a>\",\"Content-Length\":\"49058\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:e8af5a42-4dd2-4091-8d7c-8256c8a6f058>\",\"WARC-Concurrent-To\":\"<urn:uuid:ed88e89d-0c2b-4c97-b380-0cf8166a6efe>\",\"WARC-IP-Address\":\"91.204.210.226\",\"WARC-Target-URI\":\"https://encyclopedia2.thefreedictionary.com/electron-volt\",\"WARC-Payload-Digest\":\"sha1:537CPKRR4BIFO43BFKIU46CVNRBR56Q5\",\"WARC-Block-Digest\":\"sha1:AQFYDLWNE4YYYVLHSQZXDYZK5TGPQP4A\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2020/CC-MAIN-2020-29/CC-MAIN-2020-29_segments_1593655911896.73_warc_CC-MAIN-20200710175432-20200710205432-00119.warc.gz\"}"}
https://www.varsitytutors.com/hotmath/hotmath_help/topics/multiplying-and-dividing-negatives.html	[ "# Multiplying and Dividing with Negatives\n\n## Multiplying with Negatives\n\nThere's a simple rule here:\n\n• (positive) $×$ (positive) = (positive)\n• (positive) $×$ (negative) = (negative)\n• (negative) $×$ (positive) = (negative)\n• (negative) $×$ (negative) = (positive)\n\nExample :\n\n• $\\left(3\\right)\\left(7\\right)=21$\n• $\\left(3\\right)\\left(-7\\right)=-21$\n• $\\left(-3\\right)\\left(7\\right)=-21$\n• $\\left(-3\\right)\\left(-7\\right)=21$\n\nImportant Thing To Notice:\n\nSince a positive times a positive is positive, and a negative times a negative is also positive, it follows that any number times itself is either zero or positive.\n\n• $\\left(12\\right)\\left(12\\right)=144$\n• $\\left(-12\\right)\\left(-12\\right)=144$\n\n## Dividing with Negatives\n\nThe same rule works with division:\n\n• (positive) $÷$ (positive) = (positive)\n• (positive) $÷$ (negative) = (negative)\n• (negative) $÷$ (positive) = (negative)\n• (negative) $÷$ (negative) = (positive)\n\nExample :\n\n• $56÷7=8$\n• $56÷\\left(-7\\right)=-8$\n• $-56÷7=-8$\n• $-56÷\\left(-7\\right)=8$" ]	[ null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.8821308,"math_prob":0.9999807,"size":661,"snap":"2022-27-2022-33","text_gpt3_token_len":164,"char_repetition_ratio":0.3607306,"word_repetition_ratio":0.37894738,"special_character_ratio":0.25869894,"punctuation_ratio":0.08988764,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":1.0000055,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2022-08-09T00:52:56Z\",\"WARC-Record-ID\":\"<urn:uuid:68502467-f93d-4077-aad6-1c4639b23e68>\",\"Content-Length\":\"81726\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:54519b5d-9e8b-4ad8-871f-f27f009e0376>\",\"WARC-Concurrent-To\":\"<urn:uuid:948ce44d-043d-4c6d-980c-deb396f5564d>\",\"WARC-IP-Address\":\"18.67.65.107\",\"WARC-Target-URI\":\"https://www.varsitytutors.com/hotmath/hotmath_help/topics/multiplying-and-dividing-negatives.html\",\"WARC-Payload-Digest\":\"sha1:2BOAA46GKCKMAZ26PFX3NUQOVIGUU4TV\",\"WARC-Block-Digest\":\"sha1:TAOJCQLVXY4PUPZ7Z2YJQ4EIL2RTBSFD\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2022/CC-MAIN-2022-33/CC-MAIN-2022-33_segments_1659882570879.37_warc_CC-MAIN-20220809003642-20220809033642-00456.warc.gz\"}"}
http://cpc.cs.qub.ac.uk/summaries/ADGB_v1_0.html	[ "", null, "Computer Physics Communications Program Library", null, "Programs in Physics & Physical Chemistry", null, "[Licence\| Download \| New Version Template] adgb_v1_0.gz(5 Kbytes) Manuscript Title: Algebraic and numerical manipulation of the even-power-series central potentials by means of the hypervirial theorems technique. Authors: Th.E. Liolios, M.E. Grypeos Program title: HVTDER-HVTPAS Catalogue identifier: ADGB_v1_0Distribution format: gz Journal reference: Comput. Phys. Commun. 105(1997)254 Programming language: Derive, Pascal. Computer: IBM PC/AT. RAM: 640K words Word size: 16 Keywords: Hypervirial, Algebraic solution of Schroedinger eigenvalue problem, Binding energies, Expectation values, Differential equations, Nuclear physics, Theoretical methods, General purpose. Classification: 4.3, 17.16. Nature of problem:Algebraic and numerical calculation of the energy eigenvalues Enl of the 3D Schroedinger equation as well as of the expectation values nl, nl, nl for the class of central potentials V(r) = -D + Sigma (Vk bk r(2k+2)) k=0 to infinity Solution method:A combination of the Quantum Mechanical Hypervirial theorems and the Hellman-Feynman theorem. Restrictions:The central potential V(r) must be harmonic at short distances. Running time:5s for HVTDER and 1s for HVTPAS.\n Disclaimer \| ScienceDirect \| CPC Journal \| CPC \| QUB" ]	[ null, "http://cpc.cs.qub.ac.uk/icons/E.gif", null, "http://cpc.cs.qub.ac.uk/icons/sd.jpg", null, "http://cpc.cs.qub.ac.uk/icons/CPCjournal.gif", null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.64154845,"math_prob":0.6511577,"size":1312,"snap":"2019-13-2019-22","text_gpt3_token_len":359,"char_repetition_ratio":0.10091743,"word_repetition_ratio":0.0,"special_character_ratio":0.24009146,"punctuation_ratio":0.19491525,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9733992,"pos_list":[0,1,2,3,4,5,6],"im_url_duplicate_count":[null,null,null,null,null,null,null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2019-03-23T16:28:26Z\",\"WARC-Record-ID\":\"<urn:uuid:878aa344-e95c-4f54-9c8f-18f8e3763b24>\",\"Content-Length\":\"5139\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:208361c8-c18d-4666-a7a7-eb7952bcb15a>\",\"WARC-Concurrent-To\":\"<urn:uuid:5d190db2-f7b1-4deb-930f-f99a17a96130>\",\"WARC-IP-Address\":\"143.117.100.34\",\"WARC-Target-URI\":\"http://cpc.cs.qub.ac.uk/summaries/ADGB_v1_0.html\",\"WARC-Payload-Digest\":\"sha1:7POTNRXSRYUOBLT7FZTTWU2HXCBN36XD\",\"WARC-Block-Digest\":\"sha1:5UFTJ4ZQAVLTPPFIHFB67X7UH4AQSUO4\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2019/CC-MAIN-2019-13/CC-MAIN-2019-13_segments_1552912202889.30_warc_CC-MAIN-20190323161556-20190323183556-00153.warc.gz\"}"}
http://koreascience.or.kr/article/JAKO200934152712920.page	[ "# TRANSFORMATION OF DIMENSIONLESS HEAT DIFFUSION EQUATION FOR THE SOLUTION OF DYNAMIC DOMAIN IN PHASE CHANGE PROBLEMS\n\n• Ashraf, Muhammad (DEPT OF MATH, COMSATS INST OF INFORMATION TECHNOLOGY) ;\n• Avila, R. (THERMAL FLUID LAB, FAC OF MECH ENG, NATL AUTONOMOUS UNIV OF MEXICO (UNAM)) ;\n• Raza, S. S. (GLOBAL CHANGE IMPACT STUDY CENTER)\n• Received : 2009.01.26\n• Accepted : 2009.02.26\n• Published : 2009.03.25\n\n#### Abstract\n\nIn the present work transformation of dimensionless heat diffusion equation for the solution of moving boundary problems have been formulated. The formulation is based on 1-D, 2-D and 3-D, unsteady heat diffusion equations. These equations are rst turned int dimensionless form by using dimensionless quantities and their transformation was formulated in liquid and solid phases. The salient feature of this work is that during the transformation of dimensionless heat diffusion equation there arises a convective term $\\tilde{v}$ which is responsible for the motion of interface in liquid as well as solid phase. In the transformed heat equation, a correction factor $\\beta$ also arises naturally which gives the correct transformed flux at interface.\n\n#### References\n\n1. M. N. Ozisik, Boundary Value Problems of Heat Conductions, Seraton, Pennsylvania International Textbook Company, 1968\n2. J. Crank, Free and Moving Boundary Problems, New York Claredon Press, 1984.\n3. A. Rathjen and L. M. Jiji, Tarnsient Heat Transfer in Fins Undergoing Phase Transformation, Fourth International Heat Transfer Conference, Paris-Versailles,2, 1970.\n4. E. M. Ronquist and A. T. Patera, 'A Legender Spectral Element Method for the Stefan Problem', International Journal for Numerical Methods in Engineering, Vol,24, 2273-2299 (1987) https://doi.org/10.1002/nme.1620241204\n5. C. H. Li, D. R.. Jenkins, 'The effect of natural convection in solidification in tall tapered feeders', ANZIAM J. 44(E) ppC496-C511,2003. https://doi.org/10.21914/anziamj.v44i0.693\n6. R. M. Furzeland, 'A comparative study of numerical methods for moving boundary Problems', J. Inst. Math. Applic., 26, 411-429(1980). https://doi.org/10.1093/imamat/26.4.411\n7. D. R. Lynch and K. O'Neill, 'Continuously deforming finite elements for the solution of parabolic problems, with and without phase change', Int. J. numer. Methods eng., 17, 81-96 (1981) https://doi.org/10.1002/nme.1620170107\n8. A. T. Patera, 'A finite element/Green's function embedding technique applied to one dimensional change of phase heat transfer', Numer. Heat Transfer, 7, 241-247 (1984).\n9. Kees Vuik and Fred Vermolen, A conserving discretization for the free boundary in a two-dimensional Stefan problem J. Comp. Phys., 141, pp. 1-21, 1998. https://doi.org/10.1006/jcph.1998.5900\n10. C. Vuik and G. Segal and F.J. Vermolen A conserving discretization for a Stefan problem with an interface reaction at the free boundary Comput. Visual Sci., 3, pp. 109-114, 2000. https://doi.org/10.1007/s007910050058\n11. N.C.W. Kuijpers, F.J. Vermolen, K. Vuik, and S. van der Zwaag A model of the beta-AlFeSi to alpha-Al(FeMn)Si transformation in Al-Mg-Si alloys Materials Transactions, 44, pp. 1448-1456, 2003 https://doi.org/10.2320/matertrans.44.1448\n12. F.J. Vermolen and C. Vuik Solution of vector Stefan problems with cross-diffusion Journal of Computational and Applied Mathematics, 176, pp. 179-201,2005 https://doi.org/10.1016/j.cam.2004.07.011\n13. F.J. Vermolen, C. Vuik, E. Javierre and S. van der Zwaag Review on some Stefan Problems for Particle Dissolution Nonlinear Analysis: Modelling and Control, 10, pp. 257-292, 2005\n14. E. Javierre and C. Vuik and F.J. Vermolen and S. van der Zwaag A comparison of models for one-dimensional Stefan problems J. Comp. Appl. Math., 192, pp. 445-459, 2006. https://doi.org/10.1016/j.cam.2005.04.062" ]	[ null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.6730689,"math_prob":0.6760837,"size":3700,"snap":"2021-04-2021-17","text_gpt3_token_len":1117,"char_repetition_ratio":0.11390693,"word_repetition_ratio":0.017208412,"special_character_ratio":0.3235135,"punctuation_ratio":0.25738916,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9555996,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2021-04-17T13:20:45Z\",\"WARC-Record-ID\":\"<urn:uuid:72a972f1-5bb8-443c-9a09-c07f271ff3aa>\",\"Content-Length\":\"49844\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:56ee370f-0bad-4d78-a63f-03bcd27120d0>\",\"WARC-Concurrent-To\":\"<urn:uuid:18fa91a8-64d1-4235-b92d-a22d527ba159>\",\"WARC-IP-Address\":\"203.250.218.22\",\"WARC-Target-URI\":\"http://koreascience.or.kr/article/JAKO200934152712920.page\",\"WARC-Payload-Digest\":\"sha1:JMTS7WNHIF3DHNNWRKE54FRC5EHCF3WJ\",\"WARC-Block-Digest\":\"sha1:MK5XOCEPWEFP3R36S4HHYF7AML3EDKU4\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2021/CC-MAIN-2021-17/CC-MAIN-2021-17_segments_1618038119532.50_warc_CC-MAIN-20210417102129-20210417132129-00364.warc.gz\"}"}
https://www.mathlearnit.com/what-is-501-39-as-a-decimal	[ "# What is 501/39 as a decimal?\n\n## Solution and how to convert 501 / 39 into a decimal\n\n501 / 39 = 12.846\n\nAs a refresher, we encourage you to check out our fractions section of the website. Also check out our introduction to decimal numbers article. Fractions can also be represented as decimal numbers which is a number with a whole part and a fractional part. In this section, we show you how to calculate the solution to such problems on your own.\n\n• Step 1 (only step): Divide 501 by 39 to get the number as a decimal. 501 / 39 = 12.846. That's all there is to it. After all, a fraction is another way of expressing a division.\nIt is often easier to use a calculator to solve these types of problems, but it is also possible to do it by hand. If you want to practice doing it by hand, you may want to double check your solution with a calculator afterward.\n\n### When is this useful?\n\nSometimes it is more useful to convey a number as a decimal number rather than a fraction or a percentage. In those cases, this type of conversion is helpful." ]	[ null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.9556992,"math_prob":0.9883786,"size":1219,"snap":"2022-05-2022-21","text_gpt3_token_len":293,"char_repetition_ratio":0.21234567,"word_repetition_ratio":0.041666668,"special_character_ratio":0.25840855,"punctuation_ratio":0.09448819,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9885029,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2022-05-16T11:10:55Z\",\"WARC-Record-ID\":\"<urn:uuid:a552121a-37b3-445d-84ce-ab237d483f4f>\",\"Content-Length\":\"30351\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:9846aa11-d7d1-404d-a4b7-63cda5d6680a>\",\"WARC-Concurrent-To\":\"<urn:uuid:0e123102-36b0-4a73-b9ee-06105fab8bad>\",\"WARC-IP-Address\":\"172.67.223.70\",\"WARC-Target-URI\":\"https://www.mathlearnit.com/what-is-501-39-as-a-decimal\",\"WARC-Payload-Digest\":\"sha1:ZQ3XCMMRG75PHKYX774ZB75WGPE43DRX\",\"WARC-Block-Digest\":\"sha1:LFI22MXGZ5SBNBVWYLII3ERF3HX5KG4B\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2022/CC-MAIN-2022-21/CC-MAIN-2022-21_segments_1652662510117.12_warc_CC-MAIN-20220516104933-20220516134933-00388.warc.gz\"}"}
http://www.wiredchemist.com/chemistry/instructional/an-introduction-to-chemistry/matter-at-the-atomic-level/electronic-structure	[ "## Electronic Structure\n\nThe most mysterious part of the atom is the electrons. The nucleus is relatively easy to comprehend because we think of it as a core of positive charge. But the electrons have a lot more space (remember the analogy with the football field) and it is natural to inquire about things like--are the electrons moving or are they stationary? If the electrons are moving, how do they move, and finally, what are electrons? These were questions that occupied the minds of the physicists at the beginning of the 20th century, a period that was probably one of the most interesting and influential in the last 100 years of science. This period was alive with other important discoveries and theories. One of the most important set of observations concerns the nature of light. At the time of the great physicist Isaac Newton, the light that allows us to see was thought to consist of tiny particles. It was said that when these particles hit the retina of the eye they induce sight.\n\nAround the turn of the 20th century an experiment produced a result that could not be explained by the particle theory. This experiment is illustrated in Figure 10 where a the light from a light bulb is shown passing through two slits and then striking a screen. If light consists of particles, the expected result was that two images of the slits would appear on the screen--the particles would pass through the slits and strike the screen and cause two bands of light. In fact, the experiment showed that around the two bands of light there were less intense bands that appear to be images of the \"original\" bands. This result was explained by the suggestion that light actually was a wave, or, at least, has some of the properties of a wave.", null, "Figure 10. The double slit experiment.\n\nBecause waves are so important in science we should take a minute to define some of their characteristics. Imagine standing alongside a very still pond. In the tree above you a bird drops a seed that strikes the water in the pond and sends out a set of ripples, or waves.\n\nThese waves actually have the shape of what in mathematics is called a sine (or cosine) wave (see Figure 11). The speed at which the waves move outward is their velocity. The distance between the waves is the wavelength, usually designated by the Greek symbol lambda (λ or l). Another characteristic of waves is their frequency, which is the number of wave crests, or troughs, that pass a particular point in the water each second. The frequency (n, Greek ν) is designated as cycles per second, one cycle being one complete motion from the top of a wave to the bottom of the trough and up again to the top of the wave. The unit cycles per second is also called Hertz (abbreviated Hz) in honor of the physicist Heinrich Hertz who did much of the early experimentation on wave motion.", null, "Figure 11. A sine wave.\n\nNow let us return to the double slit experiment. If light has a wave motion, we should be able to use the pebble in the water analogy (scientists frequently use analogy to produce explanations of observations). When a pebble hits a body of water, it will produce a set of waves. Imagine what will happen if these waves encounter a barrier that has two openings. These openings will initiate a new set of waves. Of course, these waves are just the off-spring of the waves created by the pebble so they have the same wavelength and frequency and speed. The waves in turn may \"interfere\" with one another.\n\nFigure 12 shows that some waves will be \"in phase\" and will add to produce a larger wave; others will be exactly \"out of phase\" and crests will cancel troughs to produce no wave at all. These are analogous to the bright spots on the screen of the double-slit experiment and, thus, the experiment has been explained by constructive and destructive interference of the waves.\n\nFigure 12. The Double Slit Experiment - Video\n\nOf course, you will now be thinking, \"Okay, I can buy this stuff about the waves, but aren't we talking about light? Light is not like water, so what is it that is moving up and down?\" This is a question that baffled scientists for many years and produced models like the one that proposed that light travels through a universal medium called the \"ether\". Eventually, however, physicists agreed that light is perpendicular oscillating electric and magnetic fields. Light has a constant speed, but it can have a variety of wavelengths and frequencies. Indeed, these three characteristics of light are related by a simple equation:\n\nc = λn\n\nwhich tells that if c is a constant and if λ increases n must decrease. In mathematical terms we say that λ and n are inversely proportional.\n\nIn any event, the term light is now used to mean any electromagnetic radiation--from radio waves, which have long wavelengths of several meters, to visible light, with its wavelengths of about 10-5 meters to x-rays which have wavelengths of 10-8 meters. The electromagnetic \"spectrum\" is shown in Figure 13.", null, "Figure 13. The electromagnetic spectrum.\n\nJust when everyone seemed to agree about the nature of light, Einstein explained the photoelectric effect (which we won't describe except to say that it involves the velocity with which electrons leave the surface of metals when the metal is exposed to light) by proposing that light consists of little bundles of energy called photons. In essence, this is a bit of a return to a particle theory for light because we now (according to the photon theory) think of light as little packages of electromagnetic radiation. Einstein, moreover, made use of some work that Max Planck had done earlier and showed that these packages of light have an energy given by another simple relationship:\n\nE = hn\n\nwhere E is the energy of a photon, n is the frequency of the wave, and h is Planck's constant ( a number, 6.627 x 10-34 J-s).\n\nWe can now turn to our first model of the electronic structure of atoms. Niels Bohr, a Danish physicist, proposed that a hydrogen atom, which has only one electron and one proton, could be thought of as having the electron on one of a series of circular tracks. The electron according to Bohr must move in one circle around the nucleus and if it is undisturbed the electron must remain on that circle or track. The interesting, and very revolutionary thing about Bohr's model is that he proposed that the angular momentum, which is similar to the energy of the electron, can have only certain values. In fact, he proposed that this angular momentum can have values that are integral values of Planck's constant divided by 21. In other words,\n\nangular momentum = nh/21\n\nwhere n is an integer (1, 2, 3, 4, ...). This constituted a major break with classical physics and was the beginning of the quantum era of science. To be sure that you understand this notion, notice that the angular momentum cannot be 1.134h/21, or 34.5h/21, but it can be 1h/21 or 2h/21 or 55h/21. That is, the angular momentum can have only certain values and not others. The angular momentum is said to be quantized (from the Latin quantus, \"how great\"). Although we cannot derive the consequences here, it turns out that quantization of the angular momentum also leads to quantization of the radius of the tracks and the energy of the atom.\n\nThe first three orbits, or tracks, of the Bohr model and the energies of each are shown in Figure 14. If the electron is in the first orbit, it is characterized by the quantum number n = 1, and it has an energy of -21 x 10-19 joules. If it is in the second orbit, n = 2, and the energy is -5 x 10-19 joules. Notice again that the energy of the electron cannot be -10 x 10-19 joules; it can have only those values dictated by the quantization of the angular momentum.", null, "Figure 14. The first three Bohr orbits and their energies.\n\nThe Bohr model was an immediate success because it was able to successfully predict the ionization energy of the hydrogen atom and to predict with very good precision the wavelengths of light that are given off when hydrogen is heated in an electric arc. However, the success of the model was short-lived. It could not predict the ionization energies of other atoms, and there are features of the line spectrum of even hydrogen that it could not explain. Thus, within a few years of Bohr's success, the search was on for another model of electronic structure. The model that was finally proposed by several physicists and that has remained the model used by chemists and physicists today is referred to as the wave model, wave mechanics, or quantum mechanics. This model incorporates both the wave ideas formulated for light and the quantum ideas proposed by Einstein, Planck, and Bohr.\n\nBasically, the wave model assumes that an electron has some characteristics of electromagnetic radiation--that it can be represented, in other words, by a wave. This model is extremely mathematical in nature, and, moreover, contains a number of features that are almost totally outside of the realm of our everyday experiences. For example, it turns out that the amplitude (actually it is the square of the amplitude) of the wave that represents an electron is proportional to the probabiity of finding the electron. Suppose that a fictitious electron confined to a line has the wave shown in Figure 15(a). The square of the amplitude of the wave is shown in Figure 15(b). Notice that the negative part of the wave (on the right hand side of part (a)) becomes positive when the amplitude is squared (the square of -2 is 4).", null, "Figure 15. A wave for an electron confined to a line.\n\nThus, if we want to know where we can most likely find the electron we look for that region of the line where the amplitude squared is greatest. In this case, it occurs at 1/4 d and 3/4 d. In the middle of the line the probability of finding the electron is zero. This notion that we can never be sure of the location of the electron, but can only deal in probabilities is somewhat like passing the scene of an accident involving a large truck and a small car. You cannot tell whether anyone was injured but you would be willing to bet that there is a greater probability that the person in the small car was injured than the person in the truck.\n\nAnother consequence of the wave model is that the energy of the electron is quantized, just as was true in the Bohr model. Although the wave model is superior to the Bohr model in many respects, it also cannot be solved exactly for the energies of electrons in atoms more complicated than hydrogen. However, there are approximations that can be used in the wave model that give very good estimates of the relative energies of the electrons not only in a variety of atoms, but even in complicated molecules.\n\nThe wave model is also similar to the Bohr model in that it uses quantum numbers to characterize an electron. Rather than the one quantum number that was used in the Bohr model, three are required in the wave model. These numbers are given names such as n (the principal quantum number), λ (the angular momentum quantum number), and m (the magnetic quantum number). Each of the numbers can have only certain integral values, and each reveals something different about the energy and the location of the electron. Every unique set of these three quantum numbers is called an orbital. For example, if n = 2, λ = 1, and m = 0, this set of numbers is referred to as a 2p orbital (the 2 comes from the value of n, the p from the value of λ). Table 3 lists some of the allowed sets of n, λ, and m for values of n equal to 1 and 2.", null, "Table 3. Allowed sets (orbitals) of n, λ, and m.\n\nEvery electron in an atom must, according to the wave model, have a set of n, λ, and m values. Moreover, only two electrons can have the same values of n, λ, and m. To make matters more complicated, two electrons that have the same values of n, λ, and m, must differ in their value of another quantum number (a result of relativity theory, not quantum theory), the spin quantum number, s. The spin quantum number is a strange one--it can have only two values, and these are half-integral. The allowed values of s are +1/2 and -1/2. Now if we include all four of the quantum numbers, the Pauli exclusion principle says that each electron in an atom must have a unique set of four quantum numbers.\n\nIf you are feeling a little confused by all of this quantum number mumbo-jumbo, consider this analogy. Think of an empy apartment house with a bunch of rooms, each of the rooms is one apartment and each has an address. The landlord for this apartment has set an unusual and probably illegal rule: no more than two people can occupy each apartment and those people must be of opposite sex. In our analogy, each apartment is an orbital and the people that rent the apartments are the electrons. The sex, male or female, of the people is analogous to the spin of the electrons. If we assume that there is only one set of stairs, and no elevator, it will even be true that each apartment (orbital) has a different location and that a different amount of energy will be required to get to each. Moreover, the floor that each apartment is on is somewhat analogous to the value of n, the principal quantum number; that is, the higher the apartment the greater the value of n. What our analogy does not convey is the probabilistic nature of the electron. In our analogy we know exactly where each person is at any particular time; we also know the exact confines of each apartment. In the strange world of wave mechanics, we know neither of these with certainty. We can only say that a particular electron is in an orbital and that it therefore has such and such a probability of being at a certain point in space (for example, at some point 2 Å from the nucleus).\n\nIn spite of the uncertainty about these orbitals, they are always represented in general chemistry texts as solid figures. Some authors are careful to indicate that these solid figures represent an area of space within which the electron can be found 95% of the time. In the orbital representations shown below (Figures 16-18) you will notice that s orbitals are spherical, p-orbitals have two lobes, and d-orbitals are even more complex. You will also notice that p and d orbitals have other designations that come from their values of m. These designations indicate (in most cases) something about on what axes the orbitals have their maximum probabilities. We have also shown several different types of representations for a p-orbital, to give you the chance to see how authors differ in their feelings about how best to represent them. Notice also that p and d-orbitals are sometimes shown with different colors for alternating lobes. These colors indicate the different signs of the equation for the wave responsible for the orbital (see Figure 15 to remind yourself that waves can have positive and negative parts).", null, "Figure 16. Representations of s and p-orbitals.", null, "Figure 17. Some alternative representations of p-orbitals.\n\nThe different colors and + and - signs indicate the sign of the mathematical equation that describes the orbital.", null, "Figure 18. Photographs of models representing the d-orbitals.\n\nCarefully examine the orientation of the orbitals relative to the x, y, and z axes.\n\nFinally, we show in Figure 19 an energy level diagram for a fluorine atom with four orbitals completely filled and one orbital, a 2p orbital, half-filled. This is called the electron configuration for an atom and is usually designated as:\n\nF 1s22s22p5", null, "Figure 19. Energy level diagram for fluorine.\n\nThis energy level diagram also suggests that electrons can be \"bumped up\" into higher energy levels if they somehow are subjected to the right amount of energy. For example, the electron in the half-filled p orbital could be bumped up to the 3s orbital. This would require an amount of energy exactly equal to the difference in energy between the 2p and 3s orbitals. This amount of energy can be supplied by exposing the atom to electromagnetic radiation (for example ultraviolet light). The \"bumping up\" process is called excitation and the resulting atom is said to be excited. After some time has passed the atom will try to \"relax\" by losing energy to go back to its ground state. When this happens, the same amount of energy will be lost, and a photon will be emitted. It is this excitation/relaxation process that is responsible for most of modern spectroscopy: the measurement of the amount or type of a substance or atom present by irradiating the sample with electromagnetic radiation." ]	[ null, "http://www.wiredchemist.com/files/shared/images/figure10.gif", null, "http://www.wiredchemist.com/files/shared/images/figure11.gif", null, "http://www.wiredchemist.com/files/shared/images/figure13.gif", null, "http://www.wiredchemist.com/files/shared/images/figure14.gif", null, "http://www.wiredchemist.com/files/shared/images/figure15.gif", null, "http://www.wiredchemist.com/files/shared/images/table3.gif", null, "http://www.wiredchemist.com/files/shared/images/figure16.gif", null, "http://www.wiredchemist.com/files/shared/images/figure17.gif", null, "http://www.wiredchemist.com/files/shared/images/figure18.gif", null, "http://www.wiredchemist.com/files/shared/images/figure19.gif", null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.9529574,"math_prob":0.9751903,"size":16492,"snap":"2021-31-2021-39","text_gpt3_token_len":3545,"char_repetition_ratio":0.15581028,"word_repetition_ratio":0.0075601377,"special_character_ratio":0.21258792,"punctuation_ratio":0.092449926,"nsfw_num_words":2,"has_unicode_error":false,"math_prob_llama3":0.9832511,"pos_list":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20],"im_url_duplicate_count":[null,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\":\"2021-09-27T13:25:42Z\",\"WARC-Record-ID\":\"<urn:uuid:07f947ab-4d58-4d31-80ac-e8f5581f28d9>\",\"Content-Length\":\"27618\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:afc5881d-a0f3-4a5f-98ae-7cc184d33a12>\",\"WARC-Concurrent-To\":\"<urn:uuid:c9cef9f3-862e-43c4-a4b5-ffc987d4c3d5>\",\"WARC-IP-Address\":\"192.252.154.32\",\"WARC-Target-URI\":\"http://www.wiredchemist.com/chemistry/instructional/an-introduction-to-chemistry/matter-at-the-atomic-level/electronic-structure\",\"WARC-Payload-Digest\":\"sha1:LYZ3BODYPLEJ66NRUV6XJLRGCDFW7X4Y\",\"WARC-Block-Digest\":\"sha1:LQ4NEBKTUWVNISTIHOE4XTCJZIFDNMXH\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2021/CC-MAIN-2021-39/CC-MAIN-2021-39_segments_1631780058450.44_warc_CC-MAIN-20210927120736-20210927150736-00626.warc.gz\"}"}
https://askfilo.com/math-question-answers/how-many-spherical-bullets-can-be-made-out-of-a-solid-cube-of-lead-whose-edge	[ "", null, "The world’s only live instant tutoring platform", null, "Question\n\n# How many spherical bullets can be made out of a solid cube of lead whose edge measures 44 cm, each bullet being 4 cm in diameter?\n\n## Solutions\n\n(1)", null, "In the given problem, we have a lead cube which is remolded into small spherical bullets.\nHere, edge of the cube (s) = 44 cm\nDiameter of the small spherical bullets (d) = 4 cm\nNow, let us take the number of small bullets be x\nSo, the total volume of x spherical bullets is equal to the volume of the lead cube.\nTherefore, we get,\nVolume of the x bullets = volume of the cube\n\nx =2541\nTherefore, 2541 small bullets can be made from the given lead cube.", null, "70", null, "Connect with 50,000+ expert tutors in 60 seconds, 24X7" ]	[ null, "https://askfilo.com/images/logo.svg", null, "https://askfilo.com/images/icons/navbar.svg", null, "https://askfilo.com/images/icons/dropdown-arrow.svg", null, "https://askfilo.com/images/icons/upvote.svg", null, "https://askfilo.com/images/icons/tutors.svg", null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.7040767,"math_prob":0.9819167,"size":426,"snap":"2022-40-2023-06","text_gpt3_token_len":118,"char_repetition_ratio":0.2843602,"word_repetition_ratio":0.18461539,"special_character_ratio":0.23943663,"punctuation_ratio":0.065789476,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9908677,"pos_list":[0,1,2,3,4,5,6,7,8,9,10],"im_url_duplicate_count":[null,null,null,null,null,null,null,null,null,null,null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2023-01-28T03:11:55Z\",\"WARC-Record-ID\":\"<urn:uuid:d61b30d6-7fbe-4f73-b3b3-f829d81fe930>\",\"Content-Length\":\"87306\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:0427a287-bcda-4b5b-b6d2-8c18caf5cd96>\",\"WARC-Concurrent-To\":\"<urn:uuid:8a8ef347-0dd9-4d7e-850d-173eafaccff1>\",\"WARC-IP-Address\":\"34.117.94.82\",\"WARC-Target-URI\":\"https://askfilo.com/math-question-answers/how-many-spherical-bullets-can-be-made-out-of-a-solid-cube-of-lead-whose-edge\",\"WARC-Payload-Digest\":\"sha1:3EQJQLPJ5NYYA4UP2PTUR345QX37QKCS\",\"WARC-Block-Digest\":\"sha1:MYMHAKHZV37ZRC7S4LVKQGT4JLUC6U7Z\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2023/CC-MAIN-2023-06/CC-MAIN-2023-06_segments_1674764499470.19_warc_CC-MAIN-20230128023233-20230128053233-00727.warc.gz\"}"}
https://cs.stackexchange.com/questions/6173/given-many-partial-orders-check-them-for-consistency-and-report-any-that-are-no	[ "# Given many partial orders, check them for consistency and report any that are not consistent\n\nInputs. I am given a finite set $S$ of symbols. I know there should exist some total order $<$ on $S$, but I'm not given this ordering and it could be anything.\n\nI am also given a collection of assertions. Each assertion takes the form $s_1<s_2<\\cdots<s_m$, where $s_1,\\dots,s_m$ form a subset of the symbols of $S$. The assertion probably won't mention all of the symbols of $S$, just a subset. Each assertion will probably cover a different subset.\n\nWarmup problem. The starter problem is: Given $n$ assertions, identify whether they are all internally self-consistent, i.e., whether there exists a total order on $S$ that is consistent with all of the assertions, and if so, output an example of such a total order.\n\nThe real problem. In practice, a few assertions might be faulty. Almost all of them should be correct, though. So, the real problem is: if the assertions are not all internally self-consistent, find a minimal subset of assertions to label as \"probably-erroneous\", such that if you remove the probably-erroneous assertions, the remainder are all self-consistent.\n\nWhat I know. I know how to solve the warmup problem (just compute the transitive closure of the union of the partial orders given by each assertion, and check that the result is antisymmetric; or, in other words, create a graph with $S$ as vertex set and an edge $s\\to t$ if $s<t$ appears in any assertion, then check for cycles). However, I don't know how to solve the real problem. Any ideas?\n\nReal-world parameters. In the application domain where I've run into this, $S$ might have up to a few hundred symbols, and I might have up to a few thousand assertions, with each assertion typically mentioning dozens of symbols.\n\n• I suppose this is implied, but you are looking for an efficient algorithm for this problem, not just any (brute-force) algorithm, right? Oct 19, 2012 at 18:10\n• Correct, I'm looking for a reasonably efficient algorithm. I would be satisfied with an approximation algorithm (one that may not find the exact-minimum set of \"possibly-erroneous\" assertions, if the size of the set it outputs is usually not too much larger than the minimum).\n– D.W.\nOct 20, 2012 at 1:41\n\nThis sounds like weighted FAS (Feedback Arc Set). Find a minimal (weight) feedback arc set in a directed graph and report the assertions that contributed to the wrongly directed arcs.\n\nFor each pair of symbols s and t, there is an arc saying how many assertions have s < t and how many assertions have t < s.\n\nThis is a known NP-complete problem.\n\nOn a different note, your problem is a classic social choice problem where each of the \"assertions\" are votes.\n\n• Brilliant! Algorithms for weighted FAS do sound like just the sort of thing I was looking for. This search term should be a great start to help me look for approximation algorithms for FAS. Thank you! Do you have any advice regarding approximation algorithms or heuristics for weighted FAS that are good candidates to try, for a practical application? (Simple schemes would be nice.)\n– D.W.\nOct 20, 2012 at 1:51\n• This is indeed an interesting question, and as Jonas says, it resembles FAS and has relations to social choice theory. However, it is a generalization to FAS, and I believe that the problem is potentially much harder than FAS. To say that it generalizes FAS, you simply let S be the vertices of a graph and your linear orders the set of edges (each order has size 1). Now, FAS was (relatively) recently shown to be FPT, i.e., solvable in time f(k)*poly(n), but this was a long standing open problem. I am afraid this problem is not easily solvable. (k is the number of orders you must delete.) Oct 27, 2012 at 15:23\n\nI thought I'd jot down the two approaches that have occurred to me, but maybe you can do better.\n\nApproach 1. Here's a randomized procedure that, given an ordering of the assertions, will emit a candidate set of assertions labelled as \"probably-erroneous\" such that the remainder are all self-consistent. We can repeat the procedure many times, and keep the best output (the one with the smallest number of assertions labelled as \"probably-erroneous\").\n\nThe procedure:\n\n• Step 1. Randomly permute the assertions. Create an empty graph $G$ with vertex set $S$ and with no edges\n• Step 2. For each assertion, in the order chosen in step 1, do the following:\n\n• Suppose the assertion is $s_1<s_2<\\cdots<s_m$. Test whether adding the edges $s_1\\to s_2, \\dots, s_{m-1} \\to s_m$ would create a cycle in $G$.\n• If this would create a cycle, label this assertion as \"probably-erroneous\", and don't add the edges to $G$.\n• If this would not create a cycle, add the edges to $G$.\n\nApproach 2. Create a $\|S\|\\times\|S\|$ matrix $V[\\cdot,\\cdot]$, where $V[s,t]$ counts the number of \"votes\" for the claim $s<t$. Initialize all counts in the matrix to zero.\n\nTreat the assertion $s_1<s_2<\\cdots<s_m$ as a collection of $m(m-1)/2$ votes for the claims $s_1<s_2$, $s_1<s_3$, $s_2<s_3$, etc. Process each assertion, incrementing the counts in $V$ appropriately. This gives you the full matrix $V$.\n\nNow look through $V$ to check for symbols $s,t$ such that $V[s,t]>0$ and $V[t,s]>0$ and such that the quantity $V[t,s]/(V[s,t]+V[t,s])$ is minimized. Conclude that $s<t$ is correct, and label all assertions which contradict this (which voted for $t<s$) as \"probably-erroneous\". Remove these assertions from the set of assertions, and re-start the entire procedure, until the assertions that remain are all self-consistent.\n\nThis procedure is not guaranteed to terminate with a correct answer (e.g., consider the five assertions $A<P$, $P<Z$, $A<Q$, $Q<Z$, $Z<A$), but it's a heuristic that might be worth trying.\n\n• Again, are you looking for (a) a known correct algorithm, regardless of cost; (b) a known correct algorithm which is optimal or significantly better than brute-force; or (c) an efficient heuristic that does pretty well most of the time? Oct 19, 2012 at 18:57\n• Yes. :-) I'll take whatever I can get. The only hard requirement is that it be efficient (feasible to run within, say, a few minutes or hours). An efficient heuristic that does pretty well most of the time might be perfectly satisfactory. Of course, if you can do better than that, so much the better! (Obviously, there is a trivial solution that is guaranteed-correct but is very slow: simply enumerate all $2^n$ subsets of assertions. I'm ruling that one out because it's too slow to run in practice.) Sorry to be a bit vague and open-ended!\n– D.W.\nOct 20, 2012 at 1:46" ]	[ null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.93385804,"math_prob":0.9745148,"size":1702,"snap":"2022-05-2022-21","text_gpt3_token_len":399,"char_repetition_ratio":0.14310955,"word_repetition_ratio":0.02112676,"special_character_ratio":0.22972973,"punctuation_ratio":0.13753581,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9977249,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2022-05-17T10:37:24Z\",\"WARC-Record-ID\":\"<urn:uuid:75570d70-4f76-46a0-ae1d-bf56fc1fcb6d>\",\"Content-Length\":\"244995\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:858baecb-a481-4259-8017-0c0e36bbbae0>\",\"WARC-Concurrent-To\":\"<urn:uuid:5e063af7-090e-4aaa-8b71-3859317de5d1>\",\"WARC-IP-Address\":\"151.101.1.69\",\"WARC-Target-URI\":\"https://cs.stackexchange.com/questions/6173/given-many-partial-orders-check-them-for-consistency-and-report-any-that-are-no\",\"WARC-Payload-Digest\":\"sha1:ZRBOJOLLJJSPSKMXJ5EBFJRR2H5S3U4X\",\"WARC-Block-Digest\":\"sha1:SOADA3RU5KX7HBNGBK6UQVPL5VBMWZRM\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2022/CC-MAIN-2022-21/CC-MAIN-2022-21_segments_1652662517245.1_warc_CC-MAIN-20220517095022-20220517125022-00332.warc.gz\"}"}
https://mathematica.stackexchange.com/questions/149129/ndsolve-error-control-of-an-odes-system	[ "# NDSolve Error control of an ODEs system\n\nHaving studied the answers to this question on how to control the error of NDSolve, I then tried to apply it to the following system of second-order nonlinear ODEs:\n\nlowsol = With[{μ = 1/3},\nNDSolve[{x''[t] == -Surd[x[t]^(Numerator@μ), Denominator@μ] +\nSurd[(y[t] - x[t])^(Numerator@μ), Denominator@μ],\ny''[t] == -Surd[(y[t] - x[t])^(Numerator@μ), Denominator@μ],\ny == 0, y' == 0, x == 0,\nx' == Sqrt[2 - y'^2 - ((Surd[(x)^(Numerator@(μ + 1)),\nDenominator@(μ + 1)]) + (Surd[(y - x)^(Numerator@(μ + 1)),\nDenominator@(μ + 1)]))/(μ + 1)]}, {x, y}, {t, 0, 10},\nMethod -> \"ExplicitRungeKutta\", WorkingPrecision -> 61,\nInterpolationOrder -> All, MaxSteps -> 2^6,\nStartingStepSize -> 1^-8, MaxStepSize -> 1^-4]] // Timing\n\n\nwhich returns a pair of Interpolating Function. The system has a first integral of motion, which is set by its energy $E$, given by: \\begin{equation} E=\\frac{x'^2+y'^2}{2}+V(x,y) \\end{equation}\n\nI then tried to plot this energy and see if it remains a constant in the following way:\n\nWith[{μ = 1/3},\nPlot[{(((x'[t])^2 + (y'[t])^2)/2 + (3/4)((x[t])^(4/3) +\n(y[t] - x[t])^(4/3)))} /. lowsol // RealExponent // Evaluate, {t, 0, 10},\nPlotRange -> All]]\n\n\nbut what was returned, is the following error:\n\nReplaceAll::rmix: Elements of {513.969,\n{{x->InterpolatingFunction[{{0,10.0000000000000000000000000000000000000000000000\n0000000000000}},{5,1,7,{327114},\n{4},{InterpolatingFunction[<<5>>]},0,0,0,Automatic,{},{},False},\n{{0,5.08686050006504129509726404275435305818918558066420099758241410^-\n20,<<47>>,5.89055934479967348798377979964187966100374717687094611055214710^-\n15,<<327064>>}},\n{{0,1.414213562373095048801688724209698078569671875376948073176680},<<49>>,\n<<327064>>},{{{<<327115>>},\n{<<327115>>}}}],y->InterpolatingFunction[{{0,10.00000000000000000000000000000000\n000000000000000000000000000}},<<3>>,{{{<<327115>>},{<<327115>>}}}]}}} are a\nmixture of lists and nonlists.\n\n\nwhich states that there are elements which are a mixture of lists and nonlists. I would like to ask the following:\n\n• Why did this error occur and how can I avoid it?\n• How is it possible to reduce the computation time? In the question linked above, the computation time is significantly less than the one I need to acquire the results for my system.\n• Is it possible to do the same for a number of initial conditions? I tried ParametricNDSolve but the plotting issue would remain.\n\nUpdate\n\nAfter running the piece of code provided by @zhk, I found out that the numerical integration returns a huge error estimate for t=10^4 time units as it can be seen below:", null, "How can I reduce this error?\n\n• You're setting lowsol equal to the result of Timing. You probably want to set it equal to the result of NDSolve. Try moving it inside the With: With[{..}, lowsol = NDSolve[..]] // Timing. – Michael E2 Jun 26 '17 at 23:04\n• @MichaelE2 I will try it now and let you know. – Mitscaype Jun 27 '17 at 9:05\n• It seems like you should be using the Projection or SymplecticPartitionedRungeKutta methods for NDSolve, which explicitly check that the invariant is conserved during the integration process. But I'm not getting that to work either. – KraZug Jun 27 '17 at 9:43\n• @KraZug I read about those methods on the documentation but I was not able to make it run, though I have not idea why. – Mitscaype Jun 27 '17 at 9:46\n• I integrate it with SPRK, but the invariant is not conserved: \\[Mu] = 1/3; ode1 = x''[t] == -x[t]^\\[Mu] + (-x[t] + y[t])^\\[Mu]; ode2 = y''[t] == -(-x[t] + y[t])^\\[Mu]; ics = {y == 0, y' == 0, x == 0, x' == Sqrt}; invariant = (x'[t]^2 + y'[t]^2)/ 2 + (3/4)(x[t]^(4/3) + (y[t] - x[t])^(4/3)); projerksol = NDSolve[Flatten[{ode1, ode2, ics}], {x, y}, {t, 0, 1}, Method -> {\"SymplecticPartitionedRungeKutta\", \"DifferenceOrder\" -> 2, \"PositionVariables\" -> {x, y}}, StartingStepSize -> 0.001][]; Plot[Im[invariant /. projerksol], {t, 0, 1}] – KraZug Jun 27 '17 at 9:48\n\nSomething like this?\n\nμ = 1/3;\n\node1 = x''[t] == -x[t]^μ + (-x[t] + y[t])^μ;\n\node2 = y''[t] == -(-x[t] + y[t])^μ;\n\nlowsol = NDSolve[{ode1, ode2, y == 0, y' == 0, x == 0,\nx' == Sqrt[2 - (x^(1 + μ) + (-x + y)^(1 + μ))/(1 + μ) -\ny'^2]}, {x, y}, {t, 0, 10}, Method -> {\"FixedStep\", Method -> \"ExplicitEuler\"}];\n\nxy = Flatten /@Table[{t, Re[{(((x'[t])^2 + (y'[t])^2)/\n2 + (3/4)((x[t])^(4/3) + (y[t] - x[t])^(4/3)))} /. First@lowsol]}, {t, 0, 10, 0.1}];\n\nListLinePlot[xy, PlotRange -> All]", null, "• Looks good. Perhaps Flatten was what I was looking for? But why \"ExplicitEuler\"? Also, from the plot I can see an error estimate of 10^-3 for the first 10 time steps. I would like to bound it under 10^-4 since I am running 10^4 time steps integration. How can I apply the options: WorkingPrecision -> 61, InterpolationOrder -> All, MaxSteps -> 2^6, StartingStepSize -> 1^-8, MaxStepSize -> 1*^-4 as above? – Mitscaype Jun 26 '17 at 17:27" ]	[ null, "https://i.stack.imgur.com/cBxPE.jpg", null, "https://i.stack.imgur.com/bDOvx.jpg", null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.6890556,"math_prob":0.99489456,"size":3351,"snap":"2020-24-2020-29","text_gpt3_token_len":1210,"char_repetition_ratio":0.113235734,"word_repetition_ratio":0.04,"special_character_ratio":0.4756789,"punctuation_ratio":0.19271624,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.999291,"pos_list":[0,1,2,3,4],"im_url_duplicate_count":[null,2,null,2,null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2020-07-15T18:59:17Z\",\"WARC-Record-ID\":\"<urn:uuid:601ae9bf-6f28-487b-844f-bf31c7f76865>\",\"Content-Length\":\"153607\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:9c3702a2-258f-4f3d-bf78-a6bd94af3249>\",\"WARC-Concurrent-To\":\"<urn:uuid:ebc8d60b-b998-45ab-a56e-8c1c03684d04>\",\"WARC-IP-Address\":\"151.101.193.69\",\"WARC-Target-URI\":\"https://mathematica.stackexchange.com/questions/149129/ndsolve-error-control-of-an-odes-system\",\"WARC-Payload-Digest\":\"sha1:LA7DBIIWJKCG2OPCJSY2VH7FVCX6FBUM\",\"WARC-Block-Digest\":\"sha1:7K7OLYIZHA7SWKE4IH5EXDWDKNHRDM2I\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2020/CC-MAIN-2020-29/CC-MAIN-2020-29_segments_1593657170639.97_warc_CC-MAIN-20200715164155-20200715194155-00174.warc.gz\"}"}
https://michigan.alumni.columbia.edu/past_events	[ "WHERE\n\nWHERE\n\nWHERE\n\nWHERE\n\nWHERE\n\nWHERE\n\nWHERE\n\nWHERE\n\nWHERE\n\nWHERE\n\nWHERE\n\nWHERE\n\nWHERE\n\nWHERE\n\nWHERE\n\nWHERE\n\nWHERE\n\nWHERE\n\nWHERE\n\nWHERE\n\nWHERE\n\nWHERE\n\nWHERE\n\nWHERE\n\nWHERE\n\nWHERE\n\nWHERE\n\nWHERE\n\nWHERE\n\nWHERE\n\nWHERE\n\nWHERE\n\nWHERE\n\nWHERE" ]	[ null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.79574084,"math_prob":0.91556364,"size":5138,"snap":"2020-45-2020-50","text_gpt3_token_len":1600,"char_repetition_ratio":0.19633813,"word_repetition_ratio":0.12988506,"special_character_ratio":0.32288828,"punctuation_ratio":0.18933824,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":1.00001,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2020-11-30T04:34:53Z\",\"WARC-Record-ID\":\"<urn:uuid:846fc91f-2bbc-4c67-986d-d463243815c0>\",\"Content-Length\":\"53913\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:06cfb9c7-7b6e-4b0b-a217-01aa70d04459>\",\"WARC-Concurrent-To\":\"<urn:uuid:8a277b63-b0e4-410a-b1f5-cc66774202c6>\",\"WARC-IP-Address\":\"23.63.249.216\",\"WARC-Target-URI\":\"https://michigan.alumni.columbia.edu/past_events\",\"WARC-Payload-Digest\":\"sha1:SRICAOEPDX7NUXER2I5OVOPULLUIYEOC\",\"WARC-Block-Digest\":\"sha1:CC7A6ND777RID4PUPD6Q6C4XSICOIIUJ\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2020/CC-MAIN-2020-50/CC-MAIN-2020-50_segments_1606141205147.57_warc_CC-MAIN-20201130035203-20201130065203-00287.warc.gz\"}"}
https://www.jpost.com/middle-east/hizbullah-threatens-petrochemical-installations	[ "(function (a, d, o, r, i, c, u, p, w, m) { m = d.getElementsByTagName(o), a[c] = a[c] \|\| {}, a[c].trigger = a[c].trigger \|\| function () { (a[c].trigger.arg = a[c].trigger.arg \|\| []).push(arguments)}, a[c].on = a[c].on \|\| function () {(a[c].on.arg = a[c].on.arg \|\| []).push(arguments)}, a[c].off = a[c].off \|\| function () {(a[c].off.arg = a[c].off.arg \|\| []).push(arguments) }, w = d.createElement(o), w.id = i, w.src = r, w.async = 1, w.setAttribute(p, u), m.parentNode.insertBefore(w, m), w = null} )(window, document, \"script\", \"https://95662602.adoric-om.com/adoric.js\", \"Adoric_Script\", \"adoric\",\"9cc40a7455aa779b8031bd738f77ccf1\", \"data-key\");\nvar domain=window.location.hostname; var params_totm = \"\"; (new URLSearchParams(window.location.search)).forEach(function(value, key) {if (key.startsWith('totm')) { params_totm = params_totm +\"&\"+key.replace('totm','')+\"=\"+value}}); var rand=Math.floor(10*Math.random()); var script=document.createElement(\"script\"); script.src=`https://stag-core.tfla.xyz/pre_onetag?pub_id=34&domain=\\${domain}&rand=\\${rand}&min_ugl=0\\${params_totm}`; document.head.append(script);" ]	[ null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.93120944,"math_prob":0.96919554,"size":539,"snap":"2023-14-2023-23","text_gpt3_token_len":113,"char_repetition_ratio":0.1271028,"word_repetition_ratio":0.0,"special_character_ratio":0.17254174,"punctuation_ratio":0.08235294,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9789482,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2023-06-05T23:52:57Z\",\"WARC-Record-ID\":\"<urn:uuid:5297ea90-07ef-4712-b5b6-376b6768927c>\",\"Content-Length\":\"78057\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:e4518b02-971e-471d-8a22-f6405cb11552>\",\"WARC-Concurrent-To\":\"<urn:uuid:60f9212e-589c-4139-ac87-0a5ca385b9bc>\",\"WARC-IP-Address\":\"159.60.130.79\",\"WARC-Target-URI\":\"https://www.jpost.com/middle-east/hizbullah-threatens-petrochemical-installations\",\"WARC-Payload-Digest\":\"sha1:IP33OUDF5ZCLZRZ7JSS3A562QCDMXFSQ\",\"WARC-Block-Digest\":\"sha1:HLXTDFXM2SWMFNN7TXZHAGONLKLZF3ZA\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2023/CC-MAIN-2023-23/CC-MAIN-2023-23_segments_1685224652184.68_warc_CC-MAIN-20230605221713-20230606011713-00464.warc.gz\"}"}
https://numbermatics.com/n/133632/	[ "# 133632\n\n## 133,632 is an even composite number composed of three prime numbers multiplied together.\n\nWhat does the number 133632 look like?\n\nThis visualization shows the relationship between its 3 prime factors (large circles) and 60 divisors.\n\n133632 is an even composite number. It is composed of three distinct prime numbers multiplied together. It has a total of sixty divisors.\n\n## Prime factorization of 133632:\n\n### 29 × 32 × 29\n\n(2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 3 × 3 × 29)\n\nSee below for interesting mathematical facts about the number 133632 from the Numbermatics database.\n\n### Names of 133632\n\n• Cardinal: 133632 can be written as One hundred thirty-three thousand, six hundred thirty-two.\n\n### Scientific notation\n\n• Scientific notation: 1.33632 × 105\n\n### Factors of 133632\n\n• Number of distinct prime factors ω(n): 3\n• Total number of prime factors Ω(n): 12\n• Sum of prime factors: 34\n\n### Divisors of 133632\n\n• Number of divisors d(n): 60\n• Complete list of divisors:\n• Sum of all divisors σ(n): 398970\n• Sum of proper divisors (its aliquot sum) s(n): 265338\n• 133632 is an abundant number, because the sum of its proper divisors (265338) is greater than itself. Its abundance is 131706\n\n### Bases of 133632\n\n• Binary: 1000001010000000002\n• Hexadecimal: 0x20A00\n• Base-36: 2V40\n\n### Squares and roots of 133632\n\n• 133632 squared (1336322) is 17857511424\n• 133632 cubed (1336323) is 2386334966611968\n• The square root of 133632 is 365.5571090815\n• The cube root of 133632 is 51.1254122271\n\n### Scales and comparisons\n\nHow big is 133632?\n• 133,632 seconds is equal to 1 day, 13 hours, 7 minutes, 12 seconds.\n• To count from 1 to 133,632 would take you about thirteen hours.\n\nThis is a very rough estimate, based on a speaking rate of half a second every third order of magnitude. If you speak quickly, you could probably say any randomly-chosen number between one and a thousand in around half a second. Very big numbers obviously take longer to say, so we add half a second for every extra x1000. (We do not count involuntary pauses, bathroom breaks or the necessity of sleep in our calculation!)\n\n• A cube with a volume of 133632 cubic inches would be around 4.3 feet tall.\n\n### Recreational maths with 133632\n\n• 133632 backwards is 236331\n• 133632 is a Harshad number.\n• The number of decimal digits it has is: 6\n• The sum of 133632's digits is 18\n• More coming soon!\n\n## Link to this page\n\nHTML: To link to this page, just copy and paste the link below into your blog, web page or email.\n\nBBCODE: To link to this page in a forum post or comment box, just copy and paste the link code below:\n\n## Cite this page\n\nMLA style:\n\"Number 133632 - Facts about the integer\". Numbermatics.com. 2021. Web. 23 April 2021.\n\nAPA style:\nNumbermatics. (2021). Number 133632 - Facts about the integer. Retrieved 23 April 2021, from https://numbermatics.com/n/133632/\n\nChicago style:\nNumbermatics. 2021. \"Number 133632 - Facts about the integer\". https://numbermatics.com/n/133632/\n\nThe information we have on file for 133632 includes mathematical data and numerical statistics calculated using standard algorithms and methods. We are adding more all the time. If there are any features you would like to see, please contact us. Information provided for educational use, intellectual curiosity and fun!\n\nKeywords: Divisors of 133632, math, Factors of 133632, curriculum, school, college, exams, university, Prime factorization of 133632, STEM, science, technology, engineering, physics, economics, calculator, one hundred thirty-three thousand, six hundred thirty-two." ]	[ null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.85781825,"math_prob":0.97535604,"size":2797,"snap":"2021-04-2021-17","text_gpt3_token_len":751,"char_repetition_ratio":0.13104188,"word_repetition_ratio":0.06167401,"special_character_ratio":0.3350018,"punctuation_ratio":0.16111112,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9814957,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2021-04-23T07:28:19Z\",\"WARC-Record-ID\":\"<urn:uuid:3fd7b2cf-c405-4742-b7bd-7bb17970b751>\",\"Content-Length\":\"21089\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:300893f6-64d4-4165-b520-1be45b5cb4b5>\",\"WARC-Concurrent-To\":\"<urn:uuid:10ee87ed-20e4-44a6-8a6d-551209eb86ec>\",\"WARC-IP-Address\":\"72.44.94.106\",\"WARC-Target-URI\":\"https://numbermatics.com/n/133632/\",\"WARC-Payload-Digest\":\"sha1:D2SFWNDFKUXUYNMDBA5H2V3CZWL3WWVQ\",\"WARC-Block-Digest\":\"sha1:NLQHHPZDN5RSCV5KETE66FMOEUG5UBHA\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2021/CC-MAIN-2021-17/CC-MAIN-2021-17_segments_1618039568689.89_warc_CC-MAIN-20210423070953-20210423100953-00128.warc.gz\"}"}
http://www.stumblingrobot.com/2016/04/29/prove-two-lines-r3-intersect-determine-points-intersection/	[ "Home » Blog » Prove that two lines in R3 intersect and determine the points of intersection\n\nProve that two lines in R3 intersect and determine the points of intersection\n\nLet", null, "Consider the lines", null, "and", null, "where", null, "is the line through", null, "parallel to", null, "and", null, "is the line through", null, "parallel to", null, ". Prove that these two lines intersect and determine the point of intersection.\n\nProof. We have", null, "Then if", null, "and", null, "intersect we must have", null, "such that", null, "From the second equation we have", null, "which implies", null, ". Plugging this into the first equation we then have", null, "which implies", null, ". Therefore,", null, ". Then we check these values of", null, "satisfy the third equation,", null, "So, the point of intersection is", null, "" ]	[ null, "http://www.stumblingrobot.com/wp-content/ql-cache/quicklatex.com-a30a21c92c2829f3e580f4291784a8f5_l3.png", null, "http://www.stumblingrobot.com/wp-content/ql-cache/quicklatex.com-07a544e869d82a44515d00b9f7a7d3b3_l3.png", null, "http://www.stumblingrobot.com/wp-content/ql-cache/quicklatex.com-a9e2757b7ad6fc34782a6eca14677d52_l3.png", null, "http://www.stumblingrobot.com/wp-content/ql-cache/quicklatex.com-07a544e869d82a44515d00b9f7a7d3b3_l3.png", null, "http://www.stumblingrobot.com/wp-content/ql-cache/quicklatex.com-d51820f063062684c2ebd4d6fddfebc9_l3.png", null, "http://www.stumblingrobot.com/wp-content/ql-cache/quicklatex.com-4d51fa2434eb633d837f2dc1f7ed3e19_l3.png", null, "http://www.stumblingrobot.com/wp-content/ql-cache/quicklatex.com-a9e2757b7ad6fc34782a6eca14677d52_l3.png", null, "http://www.stumblingrobot.com/wp-content/ql-cache/quicklatex.com-c939ebf714d017898a763399989c5802_l3.png", null, "http://www.stumblingrobot.com/wp-content/ql-cache/quicklatex.com-17b8540ee4b2972f216c26316ac97287_l3.png", null, "http://www.stumblingrobot.com/wp-content/ql-cache/quicklatex.com-227e5fac5450ecf6c4af9b214ce90ae6_l3.png", null, "http://www.stumblingrobot.com/wp-content/ql-cache/quicklatex.com-07a544e869d82a44515d00b9f7a7d3b3_l3.png", null, "http://www.stumblingrobot.com/wp-content/ql-cache/quicklatex.com-a9e2757b7ad6fc34782a6eca14677d52_l3.png", null, "http://www.stumblingrobot.com/wp-content/ql-cache/quicklatex.com-57ef7c07f1bc3690ea1cf19213338e9d_l3.png", null, "http://www.stumblingrobot.com/wp-content/ql-cache/quicklatex.com-28a63579058e112cd272172803ffe302_l3.png", null, "http://www.stumblingrobot.com/wp-content/ql-cache/quicklatex.com-c911d93ecd3db78f71117d4c019eb379_l3.png", null, "http://www.stumblingrobot.com/wp-content/ql-cache/quicklatex.com-77e4c496f28649aa8e1c1ca15a17c1be_l3.png", null, "http://www.stumblingrobot.com/wp-content/ql-cache/quicklatex.com-ba3be510c6204d92d8d9eb614e50aba0_l3.png", null, "http://www.stumblingrobot.com/wp-content/ql-cache/quicklatex.com-af1c246d305873489d629993433c1620_l3.png", null, "http://www.stumblingrobot.com/wp-content/ql-cache/quicklatex.com-6b9dea2c574d04e6374a7733f564e3d8_l3.png", null, "http://www.stumblingrobot.com/wp-content/ql-cache/quicklatex.com-23c7b4d76f5f0ca8f266175bfc77557b_l3.png", null, "http://www.stumblingrobot.com/wp-content/ql-cache/quicklatex.com-242c8e068b5cd7da7f827441542f17fb_l3.png", null, "http://www.stumblingrobot.com/wp-content/ql-cache/quicklatex.com-7aa958ab8e2624f74ddf7f6babf0f8f2_l3.png", null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.9597129,"math_prob":0.997774,"size":460,"snap":"2019-43-2019-47","text_gpt3_token_len":93,"char_repetition_ratio":0.1491228,"word_repetition_ratio":0.050632913,"special_character_ratio":0.19782609,"punctuation_ratio":0.10227273,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9996617,"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],"im_url_duplicate_count":[null,2,null,6,null,6,null,6,null,null,null,4,null,6,null,null,null,2,null,2,null,6,null,6,null,5,null,2,null,2,null,2,null,2,null,6,null,2,null,null,null,2,null,2,null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2019-10-22T09:20:28Z\",\"WARC-Record-ID\":\"<urn:uuid:6c176723-3f18-4d95-876d-f77bfe6a707d>\",\"Content-Length\":\"61666\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:a2999c30-943e-41dc-8575-31230df8b750>\",\"WARC-Concurrent-To\":\"<urn:uuid:e57e030e-13d3-487f-8e5e-f13d13ee0156>\",\"WARC-IP-Address\":\"194.1.147.70\",\"WARC-Target-URI\":\"http://www.stumblingrobot.com/2016/04/29/prove-two-lines-r3-intersect-determine-points-intersection/\",\"WARC-Payload-Digest\":\"sha1:TY2RCYIOEFLVPEXEF3INWQ2HO74SUG5E\",\"WARC-Block-Digest\":\"sha1:KFTQHNLABJ3ONX3YYFYTOHHPYOT7DFN6\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2019/CC-MAIN-2019-43/CC-MAIN-2019-43_segments_1570987813307.73_warc_CC-MAIN-20191022081307-20191022104807-00000.warc.gz\"}"}
https://www.davemanuel.com/investor-dictionary/return-on-investment-roi/	[ "## Definition of Return on Investment (ROI)\n\nWhat does Return on Investment, or ROI, mean? What is the definition of Return on Investment?\n\nWill an investment (of any kind) yield positive returns? And if so, how much?\n\nTo answer these questions, you need to calculate the ROI (Return on Investment).", null, "The simplest way to explain Return on Investment is to present you with the formula that is used to calculate ROI:\n\n(Return From Investment - Initial Cost of Investment) / Initial Cost of Investment\n\nSo, let's say that you invested \\$10,000 into a venture and ended up receiving \\$15,000 back (your initial \\$10,000 investment, plus \\$5,000 in profit).\n\nTo figure out your ROI, simply substitute these values into the equation given above:\n\n(\\$15,000 - \\$10,000) / \\$10,000\n\n\\$5,000 / \\$10,000 = .5\n\nThis would mean that you saw a ROI of 50%, which would be a \"positive return on investment.\"\n\nLet's say that you ended up receiving just \\$7,500 of your original \\$10,000 investment back.\n\n(\\$7,500 - \\$10,000) / \\$10,000\n\n-\\$2,500 / \\$10,000 = -.25\n\nThis would mean that you saw a ROI of -25%, which would be a \"negative return on investment\".\n\nThis is the simplest definition of the term \"Return on Investment\".\n\n--\n\nDavemanuel.com Articles That Mention Return on Investment (ROI):\n\nNone" ]	[ null, "https://www.davemanuel.com/images/definition_roi.jpg", null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.93070424,"math_prob":0.9524415,"size":1252,"snap":"2019-51-2020-05","text_gpt3_token_len":311,"char_repetition_ratio":0.20352565,"word_repetition_ratio":0.04784689,"special_character_ratio":0.31309903,"punctuation_ratio":0.15769231,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9935202,"pos_list":[0,1,2],"im_url_duplicate_count":[null,5,null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2020-01-20T11:47:24Z\",\"WARC-Record-ID\":\"<urn:uuid:92aee977-5b8c-4313-9832-ee2e7b89313d>\",\"Content-Length\":\"24154\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:610e6480-5668-4254-b6fb-473eb09d5c68>\",\"WARC-Concurrent-To\":\"<urn:uuid:37ac8bc7-2e42-44e3-85fe-7f2fb5d3df5b>\",\"WARC-IP-Address\":\"64.64.9.184\",\"WARC-Target-URI\":\"https://www.davemanuel.com/investor-dictionary/return-on-investment-roi/\",\"WARC-Payload-Digest\":\"sha1:RLRRMFVV74QG7D72EOLF3NBFQO3H64GA\",\"WARC-Block-Digest\":\"sha1:6P36BKW3KW3STW6KIMM46W7WUFTLOFJL\",\"WARC-Identified-Payload-Type\":\"application/xhtml+xml\",\"warc_filename\":\"/cc_download/warc_2020/CC-MAIN-2020-05/CC-MAIN-2020-05_segments_1579250598726.39_warc_CC-MAIN-20200120110422-20200120134422-00305.warc.gz\"}"}
https://homework.cpm.org/category/CC/textbook/cca/chapter/7/lesson/7.2.2/problem/7-99	[ "", null, "", null, "### Home > CCA > Chapter 7 > Lesson 7.2.2 > Problem7-99\n\n7-99.\n\nIf $f(x)=3(2)^x$, find the value of the expressions in parts (a) through (c) below. Then complete parts (d) through (f).\n\n1. $f(−1)$\n\nSubstitute $−1$ for $x$ and simplify.\n\n$f\\left(−1\\right) = 3\\left(2\\right)^{−1}$\n\n$=3\\cdot\\frac{1}{2}$\n\n$\\frac{3}{2}$\n\n1. $f(0)$\n\nRefer to part (a).\n\n1. $f(1)$\n\nRefer to part (a).\n\n1. What value of $x$ gives $f\\left(x\\right) = 12$?\n\nSubstitute $12$ for $f\\left(x\\right)$ and solve for $x$.\n\n$x = 2$\n\n2. Where does the graph of this function cross the $x$-axis? The $y$-axis?\n\nA function crosses the $x$-axis at $y = 0$, and a function crosses the $y$-axis at $x = 0$.\n\n3. If $g\\left(x\\right)=\\frac{1}{3x}$, find $f(x)·g(x)$.\n\n$\\text{Substitute }3(2)^{x}\\text{ for }f(x)\\text{ and }\\frac{1}{3x}\\text{ for }g(x).$\n\nUse the eTool below to find the value of each expression in parts (a) through (c).\nClick on the link at right for the full eTool version: 7-99 HW eTool" ]	[ null, "https://homework.cpm.org/dist/7d633b3a30200de4995665c02bdda1b8.png", null, 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https://bmcmedresmethodol.biomedcentral.com/articles/10.1186/s12874-016-0189-0	[ "# Dose-response meta-analysis of differences in means\n\n## Abstract\n\n### Background\n\nMeta-analytical methods are frequently used to combine dose-response findings expressed in terms of relative risks. However, no methodology has been established when results are summarized in terms of differences in means of quantitative outcomes.\n\n### Methods\n\nWe proposed a two-stage approach. A flexible dose-response model is estimated within each study (first stage) taking into account the covariance of the data points (mean differences, standardized mean differences). Parameters describing the study-specific curves are then combined using a multivariate random-effects model (second stage) to address heterogeneity across studies.\n\n### Results\n\nThe method is fairly general and can accommodate a variety of parametric functions. Compared to traditional non-linear models (e.g. E max, logistic), spline models do not assume any pre-specified dose-response curve. Spline models allow inclusion of studies with a small number of dose levels, and almost any shape, even non monotonic ones, can be estimated using only two parameters. We illustrated the method using dose-response data arising from five clinical trials on an antipsychotic drug, aripiprazole, and improvement in symptoms in shizoaffective patients. Using the Positive and Negative Syndrome Scale (PANSS), pooled results indicated a non-linear association with the maximum change in mean PANSS score equal to 10.40 (95 % confidence interval 7.48, 13.30) observed for 19.32 mg/day of aripiprazole. No substantial change in PANSS score was observed above this value. An estimated dose of 10.43 mg/day was found to produce 80 % of the maximum predicted response.\n\n### Conclusion\n\nThe described approach should be adopted to combine correlated differences in means of quantitative outcomes arising from multiple studies. Sensitivity analysis can be a useful tool to assess the robustness of the overall dose-response curve to different modelling strategies. A user-friendly R package has been developed to facilitate applications by practitioners.\n\n## Background\n\nThe identification and characterization of dose-response relationships is an essential part of the analysis in many scientific disciplines such as toxicology, pharmacology, and epidemiology. This is particularly important in the development and testing of new compounds (e.g. a new drug, pharmaceutical treatment) where trials at different stages aim to evaluate the efficacy of increasing levels of dosage (Phase II-III trials) or to derive a dose-response curve for selection of optimal doses (Phase IV trials) [1, 2].\n\nRandomized clinical trials often investigate a continuous outcome variable, such as the efficacy or safety of a drug, reporting the change from baseline of a medical score, or the final value of a clinical measurement. The dose-response results are typically summarized by dose-specific means and standard deviations . Measures of effect are expressed in terms of mean or standardized mean differences using a dose level, usually the placebo group, as referent . Over the last few years methodological research focused on developing and improving methods for performing dose-response analysis in a single study [4, 5]. A conclusive result is hardly obtained by a single investigation and there is often the need to synthesize information collected from multiple studies. In such a case meta-analytic methods can be used to define an overall relation or to investigate heterogeneity across study findings.\n\nA method for pooling aggregated dose-response data where the outcome is a log relative risk was originally presented by Greenland and Longnecker in 1992 . Since then, several papers have refined and covered specific aspects of the methodology such as model specification [7, 8], modeling strategies [9, 10], and software implementation [11, 12]. Other methodological articles extended the approach for continuous outcome but in the case where individual patient data are available, mainly in the context of time-series environmental studies .\n\nOnly a few alternatives have been proposed to pool aggregated dose-response data where the findings are summarized by differences in means. Davis and Chen in 2004 described a methodology for summarizing dose-response curves of first and second generation antipsychotics in schizoaffective patients. The authors reconstructed drug-specific dose-response curves and conducted a meta-analysis to compare the effectiveness of medium vs high dosages. A common alternative to analyze the drug effect consists of fitting a random-intercept E max model, where the random component accounts for heterogeneity in placebo effect across trials . Heterogeneity, however, may be related to other study characteristics rather than differences in placebo response such as implementation, participants, intervention, and outcome definition. Thomas et al. adopted hierarchical Bayesian models to summarize and describe, independently, the distribution of study-specific model parameters derived from an E max model.\n\nThe mentioned strategies assumed pre-specified models that do not allow for non-monotonic curves which may occur in practice , as in case of dose-response data of antipsychotics. In addition, fitting study-specific sigmoidal curves such as the E max model requires that the single studies have assessed at least three dose levels in order to estimate model parameters. Discarding studies not providing enough data points represents a loss of information and may introduce bias.\n\nThe aim of this paper is to formalize and propose a general and flexible methodology to pool dose-response relations from aggregated data where the changes in the distribution of the quantitative outcome are expressed in terms of differences in means. We first present the data necessary for a dose-response meta-analysis and derive formulas for obtaining effect sizes and their variance/covariance structure. We describe flexible dose-response models with particular emphasis on regression splines. The method is then applied to dose-response data from clinical trials on use of aripiprazole and symptoms improvement in schizoaffective patients.\n\n## Methods\n\n### Dose-response data\n\nThe notation and data required for a dose-response meta-analysis for a generic study are displayed in Table 1. We consider I studies indexed by i=1,…,I reporting the results of a common treatment at different dose levels x ij ,j=1,…,J i , where x 0i =0 indicates the control or placebo group in the i-th study. The study-specific results typically consist of dose-specific means of an outcome variable, Y ij , that measures the efficacy of the j-th dose in the i-th study . Additional information about the number of patients allocated in each treatment, n ij , and the sample standard deviations of Y ij , s d ij , is generally reported or obtained from the study-specific results.\n\n### Effect sizes and their variance/covariance\n\nA common way to reduce heterogeneity in placebo response is to compute the effect size (or treatment effect) as difference between dose-specific means and placebo mean. In case all studies measure the outcome on a common and interpretable scale, the difference can be based on the absolute scale\n\n$$d_{ij} = \\bar{Y}_{ij} - \\bar{Y}_{i0}, \\quad j = 1,\\dots, J_{i}, \\; i = 1, \\dots, I$$\n(1)\n\nAssuming common study-specific population standard deviations, the variance of d ij is defined as\n\n$$\\text{Var} \\left(d_{ij} \\right) = \\frac{n_{ij}+ n_{i0}}{n_{ij} n_{i0}} s_{p_{i}}^{2}, \\quad j = 1,\\dots, J_{i}, \\; i = 1, \\dots, I$$\n(2)\n\nwhere $$s_{p_{i}}^{2} = \\sum _{j = 0}^{J_{i}} \\left (n_{ij} - 1 \\right) sd_{ij}^{2} / \\sum _{j = 0}^{J_{i}} \\left (n_{ij} - 1 \\right)$$ is the square of the pooled standard deviation for the i-th study. Since the study-specific mean differences d ij use the same referent values, $$\\bar {Y}_{i0}$$, they cannot be regarded as independent. The covariance term is defined as\n\n$$\\text{Cov} \\left(d_{ij}, d_{ij^{'}} \\right) = \\text{Var} \\left(\\bar{Y}_{j0} \\right) = \\frac{s_{i0}^{2}}{n_{i0}}, \\quad j \\ne j^{'}, \\; i = 1,\\dots, I$$\n(3)\n\nIn case the outcome is measured on different scales the effect sizes can be based on standardized mean differences\n\n$$d_{ij}^{} = \\frac{\\bar{Y}_{ij} - \\bar{Y}_{i0}}{s_{p_{i}}}, \\quad j = 1,\\dots, J_{i}, \\; i = 1, \\dots, I$$\n(4)\n\nwith\n\n$$\\begin{array}{{20}l} \\text{Var} \\left(d_{ij}^{} \\right) &= \\frac{1}{n_{ij}} + \\frac{1}{n_{i0}} + \\frac{{d_{ij}^{}}^{2}}{2 \\sum_{j=0}^{J_{i}} n_{ij}}, \\quad j = 1,\\dots, J_{i},\\\\ i &= 1, \\dots, I \\\\ \\text{Cov} \\left(d_{ij}^{}, d_{ij^{'}}^{} \\right) &= \\frac{1}{n_{i0}} + \\frac{d_{ij}^{}d_{ij^{'}}^{}}{2 \\sum_{j=0}^{J_{i}} n_{ij}}, \\quad j \\ne j^{'}, \\; i = 1,\\dots, I \\end{array}$$\n(5)\n\n### Dose-response analysis\n\nThe chosen effect sizes and the corresponding (co)variances are used to estimate the study-specific dose-response curves. The dose-response curves characterize the relative efficacy of the dose under investigation using the placebo effect as referent (i.e. the relative efficacy for the placebo is zero by definition). The dose-response models are expressed through the parametric model f, which specifies how the effect size varies according to the dose values. The functional relationship f is parametrized in terms of θ i , the p×1 vector of dose-response coefficients. We consider the case of mean differences, d ij , but the same principles apply for standardized mean differences, $$d^{}_{ij}$$. The study-specific curves can be written as\n\n$$\\boldsymbol{d}_{i} = f\\left(\\boldsymbol{x}_{i}, \\boldsymbol{\\theta}_{i} \\right) + \\boldsymbol{\\varepsilon}_{i}, \\;\\;\\; \\boldsymbol{\\varepsilon}_{i} \\sim N \\left(\\boldsymbol{0}, \\boldsymbol{\\hat \\Sigma}_{i} \\right), \\quad i = 1, \\dots, I$$\n(6)\n\n$$\\boldsymbol {\\hat \\Sigma }_{i}$$ is the covariance matrix of the residual error term, with Var(d ij ) along the diagonal and $$\\text {Cov} \\left (d_{ij}, d_{ij^{'}} \\right)$$ off-diagonal.\n\nSeveral alternatives are available to model the dose-response curve (i.e. for the choice of f). Table 2 shows the definition of 4 types of models ordered according to the number of parameters, ranging from 1 for a linear model up to 3 for a logistic model. See Bretz et al. for a comprehensive description .\n\nThe most common choice in dose findings is the use of the E max model which is expressed in terms of three parameters: the maximum effect (θ 1i ), the dose to produce half of the maximum effect (θ 2i ) and the steepness of the curve (θ 3i ) . As other non-linear models, the E max model assumes a specific shape that does not allow for non-monotonic curve and its estimation requires at least three non reference dose levels. Quadratic models are defined by only p=2 coefficients but may poorly fit at extreme dose values . Other non-linear models such as logistic and sigmoidal models, are commonly defined by p≥3 coefficients so that study-specific aggregated data may not be sufficient to estimate the parameters.\n\nWe propose the use of regression splines to flexibly model the dose of interest. Splines represent a family of smooth functions that can describe a wide range of curves (i.e. U-shaped, J-shaped, S-shaped, threshold) . The curves consist of piecewise polynomials over consecutive intervals defined by k knots. Their use may facilitate curve fitting since many non-linear curves can be examined by estimating only a small number of coefficients. For instance, a restricted cubic spline model with three knots k=(k 1,k 2,k 3) is defined only in terms of p=2 coefficients \n\n$$\\mathrm{E} \\left[ \\boldsymbol{d}_{i} \| \\boldsymbol{x}_{i} \\right] = \\theta_{1i} \\boldsymbol{x}_{1i} + \\theta_{2i} \\boldsymbol{x}_{2i}$$\n(7)\n\nwith two transformations defined as\n\n$$\\begin{array}{{20}l} x_{1} &= x \\\\ x_{2} &= \\frac{\\left(x - k_{1} \\right)_{+}^{3} - \\frac{k_{3} - k_{1}}{k_{3} - k_{2}} \\left(x - k_{2} \\right)_{+}^{3} + \\frac{k_{2} - k_{1}}{k_{3} - k_{2}} \\left(x - k_{3} \\right)_{+}^{3}}{ (k_{3} - k_{1})^{2}} \\end{array}$$\n(8)\n\nwhere the ‘+’ notation, with u +=u if u≥0 and u +=0 otherwise, has been used.\n\nAn alternative flexible approach to model the dose-response association is represented by fractional polynomials. In particular, a dose-response model based on fractional polynomial of order two can be written as in Eq. 6 with the two transformations defined as\n\n$$\\begin{array}{{20}l} x_{1} &= x^{p_{1}} \\text{ and } x_{2} = x^{p_{2}} \\text{ if $$p_{1} \\neq p_{2}$$} \\\\ x_{1} &= x^{p_{1}} \\text{ and } x_{2} = {x}^{p_{1}} \\log({x}) \\text{ if $$p_{1} = p_{2}$$ } \\end{array}$$\n(9)\n\nfor each combination of p 1 and p 2 in the predefined set of values {−2,−1,−0.5,0,0.5,1,2,3}; for p=0, x p becomes log(x). The best fitting fractional polynomial is typically chosen based on the Akaike’s Information Criterion .\n\nOnce the functional relation f has been selected, generalized least square estimation can be performed to efficiently estimates the dose-response coefficients $$\\boldsymbol {\\hat \\theta }_{j}$$ and the corresponding (co)variance matrix $$\\boldsymbol {\\hat V}_{j}$$, by minimizing\n\n$$\\left(\\boldsymbol{d}_{i} - f\\left(\\boldsymbol{x}_{i}, \\boldsymbol{\\theta}_{i} \\right) \\right)^{T} \\boldsymbol{\\hat \\Sigma}_{i}^{-1} \\left(\\boldsymbol{d}_{i} - f\\left(\\boldsymbol{x}_{i}, \\boldsymbol{\\theta}_{i} \\right) \\right)$$\n(10)\n\nthat generally requires numerical optimization algorithms. If f is a linear combination of the parameters θ i , as in the case of regression splines and fractional polynomials, the close solution can be written as\n\n$$\\begin{array}{{20}l} \\boldsymbol{\\hat \\theta}_{i} &= \\left(\\boldsymbol{X}_{i}^{T} \\boldsymbol{\\hat \\Sigma}_{i}^{-1} \\boldsymbol{X}_{i} \\right)^{-1} \\boldsymbol{X}_{i}^{T} \\boldsymbol{\\hat \\Sigma}_{i}^{-1} \\boldsymbol{d}_{i} \\\\ \\boldsymbol{\\hat V}_{i} &= \\text{Var} \\left(\\boldsymbol{\\hat \\theta}_{i} \\right) = \\left(\\boldsymbol{X}_{i}^{T} \\boldsymbol{\\hat \\Sigma}_{i}^{-1} \\boldsymbol{X}_{i} \\right)^{-1} \\end{array}$$\n(11)\n\nwhere X i indicates the J i ×p design matrix in the i-th study.\n\n### Meta-analysis\n\nThe estimated study-specific dose-response coefficients $$\\boldsymbol {\\hat \\theta }_{i}$$ and the accompanying (co)variance matrices $$\\boldsymbol {\\hat V}_{i}$$ are combined by means of multivariate meta-analysis\n\n$$\\boldsymbol{\\hat \\theta}_{i} \\sim N \\left(\\boldsymbol{\\theta}, \\boldsymbol{\\hat V}_{i} + \\boldsymbol{\\Psi} \\right)$$\n(12)\n\nA fixed-effects model assumes no statistical heterogeneity among study results, i.e. differences in the dose-response coefficients are only related to sampling error. The assumption of homogeneity may not hold in practice, unless it is known that the studies are performed in a similar way and are sampled from the same population . The Cochran’s Q test is typically used to test statistical heterogeneity across studies (H 0:Ψ=0) . Selected studies, however, will typically differ with respect to study design and implementation, selection of participants, and type of analyses. A certain degree of heterogeneity is expected and should be taken into account in the analysis. A random-effects model allows the dose-response coefficients, θ i , to vary across studies. Statistical heterogeneity is captured by the between-studies variance Ψ while θ represents the mean of the distribution of dose-response coefficients and an estimate, $$\\boldsymbol {\\hat \\theta }$$, can be obtained using (restricted) maximum likelihood estimation .\n\nAs a final result, the pooled dose-response curve can be presented in either a graphical or tabular form by predicting the mean differences of the outcome for a set of x dose values\n\n$$\\mathrm{E} [ \\boldsymbol{\\hat{d}} \| \\boldsymbol{x} ] = f\\left(\\boldsymbol{X}, \\boldsymbol{\\hat \\theta} \\right)$$\n(13)\n\nwith an approximate (1−α/2)% confidence interval (CI), that in case f is a linear combination of θ can be expressed as\n\n$$\\mathrm{E} [ \\boldsymbol{\\hat{d}} \| \\boldsymbol{x} ] \\mp z_{\\frac{\\alpha}{2}} \\sqrt{\\text{diag} \\left(f\\left(\\boldsymbol{X}, \\boldsymbol{\\hat \\theta} \\right)^{T} \\text{Cov}\\left(\\boldsymbol{\\hat \\theta} \\right) f\\left(\\boldsymbol{X}, \\boldsymbol{\\hat \\theta} \\right) \\right)}$$\n(14)\n\nwhere z α/2 is the α/2-th quantile of a standard normal distribution.\n\n### Dose findings\n\nOnce the pooled dose-response curve has been estimated, it may be of interest to determine a set of target doses, i.e. doses associated with prespecified outcome effects. In development of new compounds it is often important to select an optimal dose which is almost as effective as the maximum effective dose but has less undesired side effects, which often occur at high dosages. Suppose one wants to determine which is the lowest dose (ED γ ) to produce an almost complete effect, e.g. γ % of the observed maximum predicted response.\n\nThe ED γ can be determined as\n\n$$\\mathrm{\\widehat{ED}}_{\\gamma} = \\underset{x \\in \\left(0, x_{\\text{max}} \\right]}{\\operatorname{argmax}} \\left\\{ \\frac{\\mathrm{E} [ {\\hat{d}} \| {x} ] }{\\mathrm{E} [ {\\hat{d}} \| {{x_{\\text{max}}}} ]} \\geq \\gamma \\right\\}$$\n(15)\n\nwhere x max is the dose corresponding to the maximum predicted outcome.\n\nAn important step when presenting results from dose findings analysis is to accompany the previous estimates with a measure of precision, typically confidence intervals. Pinheiro et al. proposed the use a parametric bootstrap approach based on the asymptotic normal distribution of $$\\boldsymbol {\\hat \\theta }$$, the pooled estimate of the dose-response coefficients. The approach consists in re-sampling the dose-response coefficients θ from its approximate normal distribution and derive the distribution of $$\\mathrm {\\widehat {ED}}_{\\gamma }$$ based on the samples. Approximated confidence intervals for $$\\mathrm {\\widehat {ED}}_{\\gamma }$$ can be constructed using percentiles of the sampling distribution.\n\n## Results\n\nTo illustrate the methodology we examined the dose-response relation between aripiprazole, a second-generation antipsychotic, and symptoms improvement in schizoaffective patients. We updated the search strategy presented in a previous review by Davis and Chen by searching the Medline, International Pharmaceutical Abstracts, CINAHL, and the Cochrane Database of Systematic Reviews. To reduce the exclusion of unpublished papers, additional sources including Food and Drug administration website, data from Cochrane reviews, poster presentations and conference abstracts were also searched. All random-assignment, double-blind, controlled clinical trials of schizoaffective patients providing dose-response results for at least two non-zero dosages of aripiprazole were eligible.\n\nFive studies met the inclusion criteria and were included in the analysis. All the studies reported mean changes from baseline as main outcome variable, using the Positive and Negative Syndrome Scale (PANSS). The PANSS scale is an ordinal score derived from 30 items ranging from 1 to 7. Computations of ratios such as percentage changes are not directly applicable and may lead to erroneous results [33, 34]. To address this issue, the theoretical minimum (i.e. 30) needs to be subtracted from the original score . Information about the means, the number of patients assigned to each treatment, and the standard deviation was available from the published data. Because all the studies measured the outcome variable on the same scale, we computed PANSS mean differences as effect sizes. Data are reported in Table 3.\n\nWe used the trial by Cutler et al. to illustrate the steps required for estimating the dose-response curve for a single study. For example, the difference in mean PANSS comparing the dose of 2 mg/day relative to 0 mg/day is d 11=8.23−5.3=2.93 mg/day. Its variance is $$\\text {Var}(d_{11}) = (85+92)/(85\\times 92)\\times s_{p_{1}}^{2}$$ = 7.59, where $$s_{p_{1}}^{2}=119,419/356=335.4$$. The covariance of this difference in means PANSS is 18.312/85=3.94. The variance/covariance structure associated with the vector of differences in means for this trial d 1 can be presented in a matrix form\n\n$$\\begin{array}{@{}rcl@{}} \\boldsymbol{\\hat \\Sigma_{1}} =\\left[ \\begin{array}{lll} 7.59 & & \\\\ 3.94 & 7.72 & \\\\ 3.94 & 3.94 & 7.52 \\\\ \\end{array}\\right] \\end{array}$$\n\nTo estimate the dose-response curve we need first to specify the model f. We characterized the dose-response relation using a restricted cubic spline model with three knots located at the 10th, 50th, and 90th percentiles (0, 10, and 30 mg/day) of the overall dose distribution (p = 2). The restricted cubic spline dose-response model is defined as in Eq. 7. Efficient estimates of the dose-response coefficients and (co)variance matrix were obtained by generalized least square estimation\n\n$$\\begin{array}{*{20}l} \\boldsymbol{\\hat \\theta}_{1} &= (1.215, -5.738)^{T} \\\\ \\boldsymbol{\\hat V}_{1} &= \\text{Var} \\left(\\boldsymbol{\\hat \\theta}_{1} \\right) =\\left[ \\begin{array}{ll} 0.49 & \\\\ -3.65 & 31.64 \\\\ \\end{array}\\right] \\end{array}$$\n(16)\n\nWe applied the same procedure to the other studies included in the meta-analysis in order to obtain the study-specific $$\\boldsymbol {\\hat \\theta }_{i}$$ and $$\\boldsymbol {\\hat {V}}_{i}$$, i=1,…,5 (Table 4). The study-specific predicted curves are presented in Fig. 1. Under a random-effects model, restricted maximum likelihood estimates were\n\n\\begin{aligned} \\boldsymbol{\\hat \\theta} &= (0.937, -1.156)^{T} \\\\ \\widehat{\\text{Cov}} \\left(\\boldsymbol{\\hat \\theta} \\right) &= \\left[\\begin{array}{cc} 0.03 & \\\\ -0.05 & 0.10 \\\\ \\end{array}\\right] \\end{aligned}\n(17)\n\nA p-value < 0.001 for the multivariate Wald-type test H 0:θ=0 provided strong evidence against the null hypothesis of no relation between different doses of aripiprazole and mean change PANSS score. The Q test (Q=3.5, p-value = 0.899) did not detect substantial statistical heterogeneity across studies.\n\nTo communicate results of the pooled dose-response analysis, we can estimate the pooled mean differences in PANSS scores using 0 mg/day as referent as 0.937x 1−1.156x 2, together with the corresponding 95 % confidence interval for a generic dose x of interest as following\n\n$$\\left(0.937 x_{1} -1.156 x_{2} \\right) \\mp 1.96 \\sqrt{0.03 {x_{1}^{2}} + 0.1 {x_{2}^{2}} - 0.1x_{1}x_{2}}$$\n\nwhere x 1 and x 2 are defined as in Eq. 8. For instance, the model-based predicted mean changes in PANSS score compared to placebo were 4.52 (95 % CI: 2.96, 6.08) for 5 mg/day, 8.08 (95 % CI: 5.43, 10.73) for 10 mg/day, 9.95 (95 % CI: 6.97, 12.94) for 15 mg/day, 10.38 (95 % CI: 7.49, 13.27) for 20 mg/day, 9.84 (95 % CI: 6.86, 12.83) for 25 mg/day, and 8.83 (95 % CI: 5.11, 12.54) for 30 mg/day. The pooled predicted dose-response curve together with the confidence intervals and the model mean differences is provided in Fig. 2.\n\nThe results indicated a statistically significant positive association between increasing doses of aripiprazole and the mean change in PANSS score with the maximum value of 10.39 (95 % CI: 7.48, 13.30) observed at x max = 19.32 mg/day. The model suggested a slight decrease in the predicted mean PANSS score for dosages greater than 20 mg/day. The estimated dose to produce 50 % and 80 % of the predicted maximum effect were $$\\mathrm {\\widehat {ED}}_{50} =$$ 5.82 mg/day (95 % CI: 5.10, 8.58) and $$\\mathrm {\\widehat {ED}}_{80} =$$ 10.43 mg/day (95 % CI: 9.02, 16.73).\n\n### Sensitivity analysis\n\nA sensitivity analysis is often required to evaluate the robustness of the pooled dose-response curve. In the spline model, for example, the location of the knots may affect the shape of the dose-response curve. Therefore we considered alternative knots locations including different combinations of the 10th, 25th, 50th, 75th and 90th percentiles of the overall dose distribution (0, 0.5, 10, 18.75, and 30 mg/day). A graphical comparison is presented in the left panel of Fig. 3. The alternative curves roughly described the same dose-response shape with no substantial variation, all indicating an increase in the mean change PANSS score up to 20 mg/day of aripiprazole. We can assess whether there is an increasing trend above 20 mg/day by simply re-defining x 2 equal to (x−20)+ in Eq. 8; this approach is known as piecewise linear model. The rate of change in the PANSS mean differences was negative and not statistically significant (θ 1+θ 2=-0.284, p = 0.18) after 20 mg/day of aripiprazole.\n\nTo evaluate the sensitivity of the dose-response curve to the choice of the parametric model f adopted, instead, we considered three alternatives: fractional polynomials; quadratic; and E max. Since two studies only had two non-referent doses, the study-specific (sigmoidal) E max models as described in Table 2 cannot be estimated. A common solution is to fix the steepness of the curve θ 3 to be 1, also referred to as hyperbolic E max .\n\nThe “best” fractional polynomials (p 1=0.5, p 2=1) provided overall a similar dose-response curve when compared to the spline model, with slightly higher value for the maximum predicted response (right panel of Fig. 3). The hyperbolic E max had substantially higher predicted mean differences for low values of the dose. The non-linear model assumes a specific hyperbolic dose-response curve that did not seem to fit the data and may be dependent from the choice of fixing θ 3 to be 1. The dose-response curve described by the quadratic model fall in between the spline and the hyperbolic E max curves.\n\n## Discussion\n\nIn this paper we proposed a statistical method to combine differences in means of quantitative outcomes. The method consists of dose-response models estimated within each study (first stage) and an overall curve obtained by pooling study-specific dose-response coefficients (second stage). The covariance among study-specific mean differences is taken into account in the first stage analysis using generalized least square estimators, while statistical heterogeneity across studies is allowed by multivariate random-effects model in the second stage.\n\nOne major strength of the proposed method is that it is fairly general and can accommodate different modeling strategies, including non-linear ones described by Pinheiro et al. . Non-linear models, however, are defined by at least three or four parameters, and hence require an equal number of dose levels for each single study included in the analysis. Given that some studies may have investigated a lower number of dose levels, exclusion of these studies may result in substantial loss of information. In addition, many non-linear models assume a specific behaviour (e.g. monotonicity) requiring a strong a priori information about the dose-response curve. The choice of the parametric model is critical, since it highly influences the final results . Indeed, the selection of the dose-response model should be informed by subject-matter knowledge as well as understanding of the research questions at hand. We presented the use of regression splines as a flexible tool for modeling any quantitative exposure. The major advantage is that a variety of curves, even non monotonic ones, can be estimated using only two parameters. It is considered to be closed to non-parametric regression, since no major assumptions about the shape of the curve are needed . A possible alternative is the use of fractional polynomials. In comparing the two strategies, we did not find important differences between the two strategies and concluded that both are useful tools to characterize a (non-linear) dose-response curve. Nonetheless a sensitivity analysis is generally required to evaluate the robustness of the combined results.\n\nA possible limitation of the proposed methodology is that it requires information about dose-specific means and standard deviations. Studies providing other summary measures, such as dose-specific medians, would not be included the analysis. The dose-response analysis presented in Eq. 6 is based on the asymptotic normal distribution of the conditional mean effect size. Extension of the introduced methodology to percentiles is not straightforward and may represent an interesting topic of future research.\n\nAn additional limitation of aggregated dose-response data is that supplementary information for approximating the covariance terms may not be available. Articles may report directly mean or standardized mean differences and standard errors for non-referent dose groups. Whenever the standard deviation for the outcome variable in the control group ($${s_{i0}^{2}}$$) cannot be obtained, it may be approximated using the pooled standard deviation based on the non-referent dose levels ($$s_{p_{ij}}^{2}$$). Alternatively a specific value may be imputed and a sensitivity analysis can be performed to evaluate how the results of the meta-analysis vary for different values of $${s_{0j}^{2}}$$. Further limitations relate to the general application of meta-analysis based on aggregated data. These include restrictions in subgroup analyses, the impossibility of assessing the appropriateness of individual analyses, and to harmonize variable definitions and analyses for reducing the extent of heterogeneity, as well as specific biases such as aggregation (or ecological) bias in meta-regression models. Meta-analysis of individual patient data, however, are often difficult to undertake especially for the availability of individual data, so that usage of aggregated data may represent the only alternative . Specific to aggregated dose-response data, different dose references and exposure range may complicate the analysis. The presented methodology assume that all the selected studies share a common dose-response model. Important departure from this assumption may limit and/or impact the pooling of individual dose-response coefficients. An alternative methods has been proposed based on a series of univariate meta-analyses of effect sizes for a pre-specified grid of dose-levels . Further work is needed to analyze this possibility and potential advantages. Depending on the extent of heterogeneity of the dose-response curves, however, it may not be opportune to pool study-specific results, and meta-regression or stratified analyses should be performed .\n\nIn our application, we considered the effectiveness of increasing dosages of aripiprazole in shizoaffective patients. We described the steps needed to obtain the overall dose-response curve and to present it in a graphical form. We observed a non-linear association with the maximum efficacy corresponding to aripiprazole 19.32 mg/day. An estimated dose of 10.46 mg/day, however, may be sufficient to obtain 80 % of the maximum effect, which may be relevant for avoiding possible undesired side effects. Sensitivity analysis showed similar results as compared to fractional polynomials. The E max model presented higher drug efficacy for low dosages. Compared to the previous models, the E max model did not seem to fit properly the data at low dosages.\n\n## Conclusions\n\nWe described an approach to combine differences in means of a quantitative outcome contrasting different dose levels relative to a placebo in randomized trials. The general framework of the proposed methodology can include a variety of flexible models. Sensitivity analysis can be a useful tool to assess the stability of the overall dose-response curve to different modelling strategies. Although the method was presented for the analysis of randomized trials, it may be extended to observational studies where mean differences are further adjusted for potential confounders. Future work is needed to evaluate the properties of the statistical model and validity of the underlying assumptions. 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Lyman GH, Kuderer NM. The strengths and limitations of meta-analyses based on aggregate data. BMC Med Res Methodol. 2005; 5(1):14.\n\n37. Sauerbrei W, Royston P. A new strategy for meta-analysis of continuous covariates in observational studies. Stat Med. 2011; 30(28):3341–360.\n\n38. Greenland S. Invited commentary: a critical look at some popular meta-analytic methods. Am J Epidemiol. 1994; 140(3):290–6.\n\n39. Crippa A, Orsini N. Multivariate dose-response meta-analysis: the dosresmeta r package. J Stat Softw. 2016. In Press.\n\n## Acknowledgements\n\nWe are grateful to Dr. Stefan Leucht and John M. Davis for providing the data and raising the methodological question under study.\n\n### Funding\n\nThis work was supported by Karolinska Institutet’s funding for doctoral students (KID-funding) (AC) and by a Young Scholar Award from the Karolinska Institutet’s Strategic Program in Epidemiology (SfoEpi) (NO).\n\n### Availability of data and materials\n\nThe data on the effectiveness of aripiprazole are publicly available and also contained in dosresmeta R package on github (https://github.com/alecri/dosresmeta/blob/master/data/ari.rda).\n\n### Authors’ contributions\n\nAC developed the methods and prepared a draft. NO provided critical reviews, corrections and revisions. Both authors read and approved the final version of the manuscript.\n\n### Authors’ information\n\nAC is a PhD student in Epidemiology and Biostatistics. Dr. NO is Associate Professor of Medical Statistics.\n\n### Competing interests\n\nThe authors declare that they have no competing interest.\n\nNot applicable.\n\nNot applicable.\n\n## Author information\n\nAuthors\n\n### Corresponding author\n\nCorrespondence to Alessio Crippa.\n\n## Rights and permissions", null, "" ]	[ null, "https://bmcmedresmethodol.biomedcentral.com/track/article/10.1186/s12874-016-0189-0", null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.8786023,"math_prob":0.9495405,"size":32285,"snap":"2022-40-2023-06","text_gpt3_token_len":7314,"char_repetition_ratio":0.14181717,"word_repetition_ratio":0.030500632,"special_character_ratio":0.22889887,"punctuation_ratio":0.120242506,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.99343127,"pos_list":[0,1,2],"im_url_duplicate_count":[null,4,null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2023-02-02T22:16:14Z\",\"WARC-Record-ID\":\"<urn:uuid:293907a8-b335-439a-9c40-4da2582ea30f>\",\"Content-Length\":\"326822\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:e87428fb-e2de-41d2-a0e3-9d2bcb5b1f8a>\",\"WARC-Concurrent-To\":\"<urn:uuid:52568d72-78fc-4c2c-a0c5-d795c38ac2de>\",\"WARC-IP-Address\":\"146.75.32.95\",\"WARC-Target-URI\":\"https://bmcmedresmethodol.biomedcentral.com/articles/10.1186/s12874-016-0189-0\",\"WARC-Payload-Digest\":\"sha1:2ZZKRZRUPCZDVG7NXLPL2GGDA2NFHXVU\",\"WARC-Block-Digest\":\"sha1:QOE35ADBLAHEEITY3EYJ3LN6SEYDCAKQ\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2023/CC-MAIN-2023-06/CC-MAIN-2023-06_segments_1674764500041.18_warc_CC-MAIN-20230202200542-20230202230542-00276.warc.gz\"}"}
https://stats.stackexchange.com/questions/18975/proof-for-a-binomial-equation/18981	[ "# Proof for a binomial equation\n\nI'm trying to formally write out the problem: Light bulb color problem when I come up with the following equation\n\n$$\\sum_{i=0}^{k-1}{C}_{2k-1}^{2k-1-i}{p}^{2k-1-i}{(1-p)}^{i}=\\sum_{i=0}^{k-1}{C}_{2k}^{2k-i}{p}^{2k-i}{(1-p)}^{i}+\\frac{1}{2}{C}_{2k}^{k}{p}^{k}{(1-p)}^{k}$$\n\nwhere it is supposed to hold for all $k\\geq1$. Let's forget the original light bulb problem for a while. If we just want to prove the above equation, how shall we do it? I tried induction and it seemed difficult and baffling... Is there any idea or perhaps tricks one can use to establish it? What is the general approach for establishing these kind of complex combinatorial equations? (By the way, I know from the light bulb problem that it must be true for $0\\leq p\\leq1$, but does it also hold for any $p$ beyond that domain?)\n\nEdit: I vaguely remember there's a theorem or something that says if the above equation holds for any $0\\leq p\\leq1$ then it must hold for any $p\\in\\mathbb{R}$. Does any one know what that theorem is?\n\n• How would you interpret $C_{2k}^k$? What would selecting $2k$ items from $k$ items mean? Should this term not be $0$? – varty Nov 26 '11 at 5:06\n• @varty:$C_{2k}^k$ is the situation when there're equal numbers of flashes of red and blue (where each color has k flashes). Left hand side is the probability you guess correctly under best strategy when you decide to observe $2k-1$ times; RHS is the probability you when you decide to observe $2k$ times. – Eric Nov 26 '11 at 5:49\n• Are you using $C_{2k}^k$ to mean the number of ways we select $k$ items from $2k$ items? – varty Nov 26 '11 at 5:53\n• @varty: This is the alternative notation to ${2k \\choose k}$, used for instance in French textbooks. – Xi'an Nov 26 '11 at 7:41\n\nAbout the combinatorial question: the proof follows from the identity $${2k-1 \\choose i-1} + {2k-1 \\choose i} = {2k \\choose i}$$ Thus $$\\sum_{i=0}^{k-1} {2k\\choose i} p^{2k-i} (1-p)^i = \\sum_{i=0}^{k-1} {2k-1 \\choose i} p^{2k-i} (1-p)^i + \\sum_{i=0}^{k-1} {2k-1 \\choose i-1} p^{2k-i} (1-p)^i$$ which implies $$\\sum_{i=0}^{k-1} {2k\\choose i} p^{2k-i} (1-p)^i = \\sum_{i=0}^{k-2} {2k-1 \\choose i} p^{2k-1-i} (1-p)^i + {2k-1 \\choose k-1} p^{k+1} (1-p)^{k-1}$$ and $$\\frac{1}{2}{2k\\choose k} = {2k-1\\choose k-1}$$ implying that $${2k-1 \\choose k-1} p^{k+1} (1-p)^{k-1} + \\frac{1}{2}{2k\\choose k} p^{k} (1-p)^{k} = {2k-1\\choose k-1 } p^{k} (1-p)^{k-1}$$ which establishes your identity.\n\nAbout your \"Edit\" question, I think you mean the theorem in complex calculus that states that, if an analytic function is constant over an interval, it is constant everywhere.\n\n• Even better, for any fixed $k$ it's a univariate polynomial (in $p$) which has infinitely many zeroes, and thus is zero. No complex calculus necessary! – Erik P. Dec 1 '11 at 5:07\n\nA coin has a chance $p$ of landing heads. Among an odd number ($2k-1$) of tosses, what are the chances there are more heads than tails?\n\nLet's look at two solutions.\n\n1. The number of heads is $k$, $k+1$, $k+2$, $...$, or $2k-1$. Sum the chances of each of these events.\n\n2. Toss the coin one more time!\n\nIf the number of heads is now $k+1$, $k+2$, $...$, or $2k$, then originally there were at least $k$ heads. Sum these chances.\n\nIf the number of heads is now exactly $k$, it's uncertain whether there were $k-1$ or $k$ heads among the original $2k-1$ tosses. It all depends on whether the last toss was a head: there were $k$ heads originally if and only if the last toss was a tail. Do the following: randomly write down all $2k$ results--of which $k$ are heads and $k$ are tails, by assumption. One of them represents the last toss, but which? We don't know, but we do know it has equal chances of being in any of the $2k$ positions. Since $k$ of those are tails, there is a $k/(2k)=1/2$ chance the last toss was a tail.\n\nIn solution (1), the chance of exactly $i$ tails, equivalent to $2k-1-i$ heads, is given by the Binomial Distribution for $2k-1$ tosses,\n\n$${\\Pr}_{2k-1}(2k-1-i) = \\binom{2k-1}{i}p^{2k-1-i}(1-p)^i = C_{2k-1}^{2k-1-i}p^{2k-1-i}(1-p)^i.$$\n\nThe sum goes from $i=0$ to $i=k-1$. That's the left hand side of the equation in the question.\n\nIn solution (2), the chance of exactly $i$ tails is equivalent to $2k-i$ heads, given by\n\n$${\\Pr}_{2k}(2k-i) = \\binom{2k}{i}p^{2k-i}(1-p)^i = C_{2k}^{2k-i}p^{2k-i}(1-p)^i.$$\n\nTotaling these for $i=0, 1, \\ldots, k-1$ gives the summation to the right of the equation in the question. The chance that the last toss is a tail and all $2k$ tosses are heads is the product\n\n$${\\Pr}_{2k}\\left(\\text{last toss is tail}\\,\|\\,k\\text{ heads}\\right){\\Pr}_{2k}\\left(k\\text{ heads}\\right) = \\frac{1}{2}\\binom{2k}{k}p^{2k-k}(1-p)^k = \\frac{1}{2}C_{2k}^kp^k(1-p)^k.$$\n\nThis is the extra term at the right of the equation. Solution (2) therefore produces the formula on the right hand side.\n\nThus, the equation holds for all $k$ because it gives two ways of expressing the chance of observing a majority of heads in any odd number of tosses.\n\nThe general approach exemplified here is to count the same thing in two different ways. It's a powerful method, advocated particularly by combinatorialists (people who specialize in counting things). Many complicated formulas turn out to have simple, illuminating proofs of this nature. For details, see Richard Stanley's Enumerative Combinatorics. (Volume 1 is available on the Web--Google it.)\n\nLet's address the last part of the question. Both sides of the equation are obviously polynomials in $p$. Their difference is a polynomial of degree at most $2k$. It is zero for all valid probabilities $p$. There are more than $2k+1$ probabilities (there are infinitely many of them in the interval from $0$ to $1$). Since a polynomial of degree $n$ is completely determined by its values at any $n+1$ numbers, the difference is the zero function--and in particular equals zero for any number $p$." ]	[ null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.900793,"math_prob":0.9998104,"size":3061,"snap":"2019-51-2020-05","text_gpt3_token_len":927,"char_repetition_ratio":0.12266929,"word_repetition_ratio":0.034343433,"special_character_ratio":0.30872265,"punctuation_ratio":0.10746269,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9999559,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2019-12-15T08:40:05Z\",\"WARC-Record-ID\":\"<urn:uuid:98fd52a2-8d01-483b-b66f-5d6fa3952520>\",\"Content-Length\":\"147477\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:47526274-7907-40a2-954b-7ab51a12b92a>\",\"WARC-Concurrent-To\":\"<urn:uuid:e1b6071f-f053-4269-b412-21a8aadbb69a>\",\"WARC-IP-Address\":\"151.101.1.69\",\"WARC-Target-URI\":\"https://stats.stackexchange.com/questions/18975/proof-for-a-binomial-equation/18981\",\"WARC-Payload-Digest\":\"sha1:CORAJUXHS2UDBELDF7XSZU7XP3M3PQAL\",\"WARC-Block-Digest\":\"sha1:FHJTSYDWXM6D7L2CCCSPQR3SRDOVLYDN\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2019/CC-MAIN-2019-51/CC-MAIN-2019-51_segments_1575541307797.77_warc_CC-MAIN-20191215070636-20191215094636-00476.warc.gz\"}"}
https://fatareforms.org/5358/why-are-intersecting-lines-always-coplanar/	[ "# Why are intersecting lines always coplanar?\n\nTwo intersecting lines are always coplanar.\n\nIntersecting: The two lines are coplanar (meaning that they lie on the same plane) and intersect at a single point. Parallel: The two lines are coplanar but never intersect because they travel through different points, while their direction vectors are scalar multiples of one another.\n\nBeside above, what are two coplanar lines that dont intersect? Two lines are parallel lines if they are coplanar and do not intersect. Lines that are not coplanar and do not intersect are called skew lines. Two planes that do not intersect are called parallel planes.\n\nBesides, are intersecting lines non coplanar?\n\nFor two intersecting noncoplanar lines, take the point of intersection, and one other point on each line. Because the lines intersect, the three points are not colinear. That means each line has two points in that plane, so each line is in that plane, therefore they are coplanar.\n\nWhat does the intersection of two coplanar lines form?\n\nIf the lines are coplanar, they either intersect (in a single point), or are the same line (colinear) or are parallel (no intersection). If you know they intersect (perhaps from the context of the question), you can immediately look for the single point.\n\n### Are parallel lines congruent?\n\nIf two parallel lines are cut by a transversal, the corresponding angles are congruent. If two lines are cut by a transversal and the corresponding angles are congruent, the lines are parallel. Interior Angles on the Same Side of the Transversal: The name is a description of the “location” of the these angles.\n\n### Are 2 points always coplanar?\n\nCoplanar points: A group of points that lie in the same plane are coplanar. Any two or three points are always coplanar.\n\n### Are skew lines coplanar?\n\nIn three-dimensional geometry, skew lines are two lines that do not intersect and are not parallel. Two lines that both lie in the same plane must either cross each other or be parallel, so skew lines can exist only in three or more dimensions. Two lines are skew if and only if they are not coplanar.\n\n### Are vertical angles congruent?\n\nWhen two lines intersect to make an X, angles on opposite sides of the X are called vertical angles. These angles are equal, and here’s the official theorem that tells you so. Vertical angles are congruent: If two angles are vertical angles, then they’re congruent (see the above figure).\n\n### Can two planes meet in exactly one point?\n\nTwo planes intersect in exactly one point. Two intersecting lines are always coplanar.\n\n### Are perpendicular lines coplanar?\n\nPerpendicular Lines. Coplanar lines that are not parallel must intersect or cross each other. They can intersect at any angle, but when the lines intersect at exactly 90° they are perpendicular lines. Perpendicular lines create four right angles at their point of intersection.\n\n### Are 4 points coplanar?\n\nCoplanar points are three or more points which all lie in the same plane. Any set of three points in space is coplanar. A set of four points may be coplanar or may be not coplanar.\n\n### How many lines are determined by two points?\n\nSince we have given two points. And we know the definition of line which states that ” A line is formed by two points and it can be extended from both the sides.” So, From two points, there must be 1 line . Hence, one line is determined by two points.\n\n### What is a perpendicular line?\n\nIn elementary geometry, the property of being perpendicular (perpendicularity) is the relationship between two lines which meet at a right angle (90 degrees). A line is said to be perpendicular to another line if the two lines intersect at a right angle.\n\n### Can two lines be coplanar?\n\nTwo lines in three-dimensional space are coplanar if there is a plane that includes them both. This occurs if the lines are parallel, or if they intersect each other. Two lines that are not coplanar are called skew lines.\n\n### What is the symbol for intersecting lines?\n\nTwo lines that intersect and form right angles are called perpendicular lines. The symbol ⊥ is used to denote perpendicular lines.\n\n### Are all parallel lines coplanar?\n\nIt’s important to know about parallel lines because you’re going to use it through all parts of Geometry. Technically parallel lines are two coplanar which means they share the same plane or they’re in the same plane that never intersect. And you can identify parallel lines by having the same number of arrows.\n\n### What are two coplanar lines cut by a transversal?\n\nIf two coplanar lines are cut by a transversal so that a pair of corresponding angles are congruent, then the two lines are parallel. You can use the Converse of the Corresponding Angles Postulate to show that two lines are parallel.\n\n### Do 2 intersecting lines determine a plane?\n\n“If two lines intersect, then exactly one plane contains the lines.” “If two lines intersect, then they intersect in exactly one point.” and three noncollinear points define a plane." ]	[ null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.92944056,"math_prob":0.956971,"size":5325,"snap":"2022-05-2022-21","text_gpt3_token_len":1173,"char_repetition_ratio":0.22232664,"word_repetition_ratio":0.018599562,"special_character_ratio":0.2026291,"punctuation_ratio":0.10752688,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.99492717,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2022-05-16T11:32:59Z\",\"WARC-Record-ID\":\"<urn:uuid:9006e41b-42b7-4bc0-a1c0-821dfb97497f>\",\"Content-Length\":\"40642\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:5a4a6e17-6111-40fb-827d-32d4a7408f03>\",\"WARC-Concurrent-To\":\"<urn:uuid:61dd2359-212c-439b-a73e-693b6c3e08b1>\",\"WARC-IP-Address\":\"104.21.2.45\",\"WARC-Target-URI\":\"https://fatareforms.org/5358/why-are-intersecting-lines-always-coplanar/\",\"WARC-Payload-Digest\":\"sha1:RISOUV5KSQRW76OWQGXIT2NCWMTXIGFT\",\"WARC-Block-Digest\":\"sha1:3XYSOK2E3RFTYNYUEEKRZL32NUOKBHJY\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2022/CC-MAIN-2022-21/CC-MAIN-2022-21_segments_1652662510117.12_warc_CC-MAIN-20220516104933-20220516134933-00236.warc.gz\"}"}
https://qchu.wordpress.com/2009/11/15/the-many-faces-of-schur-functions/?replytocom=448	[ "Feeds:\nPosts\n\n## The many faces of Schur functions\n\nThe last time we talked about symmetric functions, I asked whether the vector space", null, "$\\mathcal{R}$ could be turned into an algebra, i.e. equipped with a nice product. It turns out that the induced representation allows us to construct such a product as follows:\n\nGiven representations", null, "$\\rho_1, \\rho_2$ of", null, "$S_n, S_m$, their tensor product", null, "$\\rho_1 \\otimes \\rho_2$ is a representation of the direct product", null, "$S_n \\times S_m$ in a natural way. Now,", null, "$S_n \\times S_m$ injects naturally into", null, "$S_{n+m}$, which gives a new representation", null, "$\\rho = \\text{Ind}_{S_n \\times S_m}^{S_{n+m}} \\rho_1 \\otimes \\rho_2$.\n\nThe character of this representation is called the induction product", null, "$\\rho_1 * \\rho_2$ of the characters of", null, "$\\rho_1, \\rho_2$, and with this product", null, "$\\mathcal{R}$ becomes a graded commutative algebra. (Commutativity and associativity are fairly straightforward to verify.) It now remains to answer the first question: does there exist an algebra homomorphism", null, "$\\phi : \\Lambda \\to \\mathcal{R}$? And can we describe the inner product on", null, "$\\Lambda$ coming from the inner product on", null, "$\\mathcal{R}$?\n\nTo answer this question we’ll introduce perhaps the most important class of symmetric functions, the Schur functions", null, "$s_{\\lambda}$.\n\nN.B. I’ll be skipping even more proofs than usual today, partly because they require the development of a lot of machinery I haven’t described and partly because I don’t understand them all that well. Again, good references are Sagan or EC2.\n\nSchur-Weyl duality\n\nOne way that the Schur functions naturally occur is in the description of Schur-Weyl duality. To the extent that I understand this, it goes as follows. On", null, "$\\underbrace{\\mathbb{C}^n \\otimes ... \\otimes \\mathbb{C}^n}_{k \\text{ times}}$ there are two commuting representations: one of", null, "$GL_n(\\mathbb{C})$ which acts on all the factors simultaneously and one of", null, "$S_k$ which permutes the factors. An important property of these representations is that they are mutual centralizers, i.e. each is the largest group of transformations which commutes with the other. It then turns out that the above representation decomposes as", null, "$\\displaystyle \\bigoplus_{\\lambda} \\pi^{\\lambda} \\otimes \\rho_n^{\\lambda}$\n\nwhere", null, "$\\lambda$ runs through all partitions with", null, "$k$ boxes and at most", null, "$n$ rows,", null, "$\\pi^{\\lambda}$ is the irreducible representation of", null, "$S_k$ associated to the partition", null, "$\\lambda$, and", null, "$\\rho_n^{\\lambda}$ is a corresponding irreducible representation of", null, "$GL_n(\\mathbb{C})$.\n\nWe already know of two of these representations, the symmetric algebra", null, "$S^k(\\mathbb{C}^n)$ and the exterior algebra", null, "$E^k(\\mathbb{C}^n)$. It turns out that the symmetric algebra corresponds to the partition", null, "$\\lambda = (k)$ (one row) and the exterior algebra corresponds to the partition", null, "$\\lambda = (1^k)$ (one column), and it’s not hard to see that the former is always there and the latter is only there as long as", null, "$k \\ge n$. As we saw previously, the characters of these representations can be described in terms of eigenvalues. This turns out to be true more generally.\n\nProposition: Suppose", null, "$\\phi$ is a representation", null, "$GL_n(\\mathbb{C}) \\to GL_m(\\mathbb{C})$ which is homogeneous, i.e. it respects scalar multiplication, and polynomial, i.e. every entry of the output", null, "$\\phi(A)$ is a polynomial function of the entries of the input", null, "$A$. Then the character of", null, "$\\phi$ is a symmetric function of the eigenvalues of", null, "$A$.\n\nThe restriction to homogeneous polynomial representations is to avoid discontinuity and powers of the representation", null, "$A \\mapsto \\det (A)$. Anyway, the above result is not too hard to see: since the diagonalizable matrices are dense and characters are both continuous and invariant under conjugation, the character must depend only on the multiset of eigenvalues, and since the representation is homogeneous and polynomial, the character must be as well.\n\nI am not familiar with the explicit description of the representations", null, "$\\rho_n^{\\lambda}$, but the important point is that these representations are well-defined with respect to Schur-Weyl duality.\n\nDefinition 1: The Schur polynomial", null, "$s_{\\lambda}(x_1, ... x_n)$ is the value of the character of", null, "$\\rho_n^{\\lambda}$ evaluated at the diagonal matrix with entries", null, "$x_1, ... x_n$.\n\nFor example,", null, "$s_{(k)}(x_1, ... x_n) = h_k(x_1, ... x_n)$ (the symmetric algebra) and", null, "$s_{(1^k)}(x_1, ... x_n) = e_k(x_1, ... x_n)$ (the exterior algebra). Note that if", null, "$\\lambda$ has more than", null, "$n$ rows then", null, "$s_{\\lambda}(x_1, ... x_n)$ is not defined.\n\nOf course, since I haven’t told you how to write these functions down, this is a rather unsatisfying definition, but at least it has a concrete tie to an important representation-theoretic concept. Now, it turns out that", null, "$s_{\\lambda}(x_1, ... x_n, 0) = s_{\\lambda}(x_1, ... x_n)$, so the Schur functions", null, "$s_{\\lambda} \\in \\Lambda$ are well-defined as symmetric functions.\n\nSchur-Weyl duality turns out to imply a strong relationship between these characters and the characters of the symmetric group which generalizes the relationship the complete homogeneous symmetric functions and elementary symmetric functions have to the trivial and sign representations.\n\nDefinition 2: Let", null, "$\\chi^{\\lambda}(\\pi)$ denote the character of the representation of", null, "$S_n$ corresponding to", null, "$\\lambda$ evaluated at a permutation", null, "$\\pi$. Then", null, "$\\displaystyle s_{\\lambda} = \\frac{1}{n!} \\sum_{\\pi \\in S_n} \\chi^{\\lambda}(\\pi) p_{\\pi}$.\n\nDeterminantal formula\n\nThe Weyl character formulas imply the following determinantal formula for", null, "$s_{\\lambda}$. First, some notation. Define", null, "$a_{\\lambda}(x_1, ... x_n) = \\det \\left\| \\begin{array}{cccc} x_1^{n-1 + \\lambda_1} & x_2^{n-1 + \\lambda_1} & \\hdots & x_n^{n-1 + \\lambda_1} \\\\ x_1^{n-2 + \\lambda_2} & x_2^{n-2 + \\lambda_2} & \\hdots & x_n^{n-2 + \\lambda_2} \\\\ \\vdots & \\vdots & \\ddots & \\vdots \\\\ x_1^{\\lambda_n} & x_2^{\\lambda_n} & \\hdots & x_n^{\\lambda_n} \\end{array} \\right\|$.\n\nIn particular", null, "$a_{0^n}(x_1, ... x_n)$ is the Vandermonde determinant", null, "$\\prod_{i < j} (x_i - x_j)$, and for any other partition", null, "$\\lambda$ the Vandermonde determinant divides it. (One could imagine these determinants appearing in the discussion of polynomial interpolation when certain coefficients are restricted to be zero, but I have never heard anyone talk about Schur functions this way.)\n\nDefinition 3:", null, "$\\displaystyle s_{\\lambda}(x_1, ... x_n) = \\frac{a_{\\lambda}(x_1, ... x_n)}{a_{0^n}(x_1, ... x_n)}$.\n\nThis definition has the advantage that it does not refer directly to representation theory, and it is also relatively straightforward to do computations with for small cases.\n\nThere are two other determinantal formulas for the Schur functions which are useful in studying, for example, the cohomology ring of a Grassmannian, but they can wait.\n\nThe characteristic map\n\nWe now define the characteristic map", null, "$\\text{ch} : \\mathcal{R} \\to \\Lambda$ as follows: if", null, "$f$ is a class function on", null, "$S_n$, then define", null, "$\\displaystyle \\text{ch } f = \\frac{1}{n!} \\sum_{\\pi \\in S_n} f(\\pi) p_{\\pi}$\n\nand extend by linearity. As we saw above,", null, "$\\text{ch } \\chi^{\\lambda} = s_{\\lambda}$, so this is a natural definition from that perspective.\n\nProposition:", null, "$\\text{ch } f * g = (\\text{ch } f)(\\text{ch } g)$. In other words,", null, "$\\text{ch}$ defines a ring homomorphism from", null, "$\\mathcal{R}$ with the induction product to", null, "$\\Lambda$.\n\nThe main technical detail of this proof is Frobenius reciprocity, but the point is that we have now found the relationship between", null, "$\\mathcal{R}$ and", null, "$\\Lambda$ that we were looking for. With a little more work one can show that", null, "$\\text{ch}$ is bijective, and it follows that the", null, "$s_{\\lambda}$ form a basis of the symmetric functions.\n\nThe characteristic map takes the usual inner product on", null, "$\\mathcal{R}$ to the unique inner product on symmetric functions satisfying", null, "$\\left< s_{\\lambda}, s_{\\mu} \\right> = \\delta_{\\lambda \\mu}$.\n\nThe study of this inner product was initiated by Philip Hall and clarifies a number of results in symmetric function theory; see EC2, for example. But I haven’t digested this point enough to say anything meaningful.\n\n### 7 Responses\n\n1. on April 26, 2019 at 7:15 am \| Reply", null, "Andrei Sipoș\n\n“One could imagine these determinants appearing in the discussion of polynomial interpolation when certain coefficients are restricted to be zero, but I have never heard anyone talk about Schur functions this way.”\n\nI just wrote a paper where I exploit exactly this feature of Schur functions 😀 Your post has been helpful in figuring it out, thanks\n\nhttps://arxiv.org/abs/1904.10284\n\n2. […] Now finally we are back to The Jacobi-Trudi identities again. This time I did see semistandard Young tableau differs from standard ones in the way that integers are only weakly increasing along rows. Now I guess means a Young diagram. Equivalent definitions: , . So there exists one problem, I still don’t get what and is. Let me check it out. After checking wiki, is complete homogeneous symmetric polynomials and is elementary symmetric polynomials. Ah, now I understand it. Let me play around why , . Umm, that’s basically because the integers weakly increase along the row and strictly increase along the column, yeah, I see the reason. Let’s move on the proof of equivalence of definitions. While reading the proof, I suddenly found the", null, "$det$ part is related to Qiaochu’s previous post The many faces of Schur functions. […]\n\n3. on November 23, 2009 at 5:08 am \| Reply", null, "Spanferkel\n\nA good book (but not very easy) is also\n\nBannai, Eiichi; Ito, Tatsuro: Algebraic combinatorics. I. Association schemes. The Benjamin/Cummings Publishing Co., Inc., Menlo Park, CA, 1984. xxiv+425 pp. ISBN: 0-8053-0490-8\n\nMR882540 (87m:05001) 05-01\nhttp://www.ams.org/mathscinet-getitem?mr=882540\n\n4. on November 22, 2009 at 5:39 pm \| Reply", null, "su\n\nOk great thanks!!\n\n5. on November 20, 2009 at 12:57 am \| Reply", null, "su\n\nHey,\n• on November 20, 2009 at 9:33 am \| Reply", null, "Qiaochu Yuan" ]	[ 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, "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.rdocumentation.org/packages/ggstatsplot/versions/0.0.10/topics/t1way_ci	[ "# t1way_ci\n\n0th\n\nPercentile\n\n##### #' @title A heteroscedastic one-way ANOVA for trimmed means with confidence interval for effect size.\n\nCustom function to get confidence intervals for effect size measure for robust ANOVA.\n\nKeywords\ninternal\n##### Usage\nt1way_ci(data, x, y, tr = 0.1, nboot = 100, conf.level = 0.95,\nconf.type = \"norm\", ...)\n##### Arguments\ndata\n\nA dataframe (or a tibble) from which variables specified are to be taken. A matrix or tables will not be accepted.\n\nx\n\nThe grouping variable from the dataframe data.\n\ny\n\nThe response (a.k.a. outcome or dependent) variable from the dataframe data.\n\ntr\n\nTrim level for the mean when carrying out robust tests. If you get error stating \"Standard error cannot be computed because of Winsorized variance of 0 (e.g., due to ties). Try to decrease the trimming level.\", try to play around with the value of tr, which is by default set to 0.1. Lowering the value might help.\n\nnboot\n\nNumber of bootstrap samples for computing confidence interval for the effect size (Default: 100).\n\nconf.level\n\nScalar between 0 and 1. If unspecified, the defaults return 95% lower and upper confidence intervals (0.95).\n\nconf.type\n\nA vector of character strings representing the type of intervals required. The value should be any subset of the values \"norm\", \"basic\", \"perc\", \"bca\". For more, see ?boot::boot.ci.\n\n...\n\nArguments passed on to boot::boot\n\ndata\n\nThe data as a vector, matrix or data frame. If it is a matrix or data frame then each row is considered as one multivariate observation.\n\nstatistic\n\nA function which when applied to data returns a vector containing the statistic(s) of interest. When sim = \"parametric\", the first argument to statistic must be the data. For each replicate a simulated dataset returned by ran.gen will be passed. In all other cases statistic must take at least two arguments. The first argument passed will always be the original data. The second will be a vector of indices, frequencies or weights which define the bootstrap sample. Further, if predictions are required, then a third argument is required which would be a vector of the random indices used to generate the bootstrap predictions. Any further arguments can be passed to statistic through the … argument.\n\nR\n\nThe number of bootstrap replicates. Usually this will be a single positive integer. For importance resampling, some resamples may use one set of weights and others use a different set of weights. In this case R would be a vector of integers where each component gives the number of resamples from each of the rows of weights.\n\nsim\n\nA character string indicating the type of simulation required. Possible values are \"ordinary\" (the default), \"parametric\", \"balanced\", \"permutation\", or \"antithetic\". Importance resampling is specified by including importance weights; the type of importance resampling must still be specified but may only be \"ordinary\" or \"balanced\" in this case.\n\nstype\n\nA character string indicating what the second argument of statistic represents. Possible values of stype are \"i\" (indices - the default), \"f\" (frequencies), or \"w\" (weights). Not used for sim = \"parametric\".\n\nstrata\n\nAn integer vector or factor specifying the strata for multi-sample problems. This may be specified for any simulation, but is ignored when sim = \"parametric\". When strata is supplied for a nonparametric bootstrap, the simulations are done within the specified strata.\n\nL\n\nVector of influence values evaluated at the observations. This is used only when sim is \"antithetic\". If not supplied, they are calculated through a call to empinf. This will use the infinitesimal jackknife provided that stype is \"w\", otherwise the usual jackknife is used.\n\nm\n\nThe number of predictions which are to be made at each bootstrap replicate. This is most useful for (generalized) linear models. This can only be used when sim is \"ordinary\". m will usually be a single integer but, if there are strata, it may be a vector with length equal to the number of strata, specifying how many of the errors for prediction should come from each strata. The actual predictions should be returned as the final part of the output of statistic, which should also take an argument giving the vector of indices of the errors to be used for the predictions.\n\nweights\n\nVector or matrix of importance weights. If a vector then it should have as many elements as there are observations in data. When simulation from more than one set of weights is required, weights should be a matrix where each row of the matrix is one set of importance weights. If weights is a matrix then R must be a vector of length nrow(weights). This parameter is ignored if sim is not \"ordinary\" or \"balanced\".\n\nran.gen\n\nThis function is used only when sim = \"parametric\" when it describes how random values are to be generated. It should be a function of two arguments. The first argument should be the observed data and the second argument consists of any other information needed (e.g. parameter estimates). The second argument may be a list, allowing any number of items to be passed to ran.gen. The returned value should be a simulated data set of the same form as the observed data which will be passed to statistic to get a bootstrap replicate. It is important that the returned value be of the same shape and type as the original dataset. If ran.gen is not specified, the default is a function which returns the original data in which case all simulation should be included as part of statistic. Use of sim = \"parametric\" with a suitable ran.gen allows the user to implement any types of nonparametric resampling which are not supported directly.\n\nmle\n\nThe second argument to be passed to ran.gen. Typically these will be maximum likelihood estimates of the parameters. For efficiency mle is often a list containing all of the objects needed by ran.gen which can be calculated using the original data set only.\n\nsimple\n\nlogical, only allowed to be TRUE for sim = \"ordinary\", stype = \"i\", n = 0 (otherwise ignored with a warning). By default a n by R index array is created: this can be large and if simple = TRUE this is avoided by sampling separately for each replication, which is slower but uses less memory.\n\nparallel\n\nThe type of parallel operation to be used (if any). If missing, the default is taken from the option \"boot.parallel\" (and if that is not set, \"no\").\n\nncpus\n\ninteger: number of processes to be used in parallel operation: typically one would chose this to the number of available CPUs.\n\ncl\n\nAn optional parallel or snow cluster for use if parallel = \"snow\". If not supplied, a cluster on the local machine is created for the duration of the boot call.\n\n• t1way_ci\n##### Examples\n# NOT RUN {\nset.seed(123)\nggstatsplot:::t1way_ci(\ndata = dplyr::filter(ggplot2::msleep, vore != \"insecti\"),\nx = vore,\ny = brainwt,\ntr = 0.05,\nnboot = 50,\nconf.level = 0.99,\nconf.type = \"perc\"\n)\n# }\n# NOT RUN {\n# }\n\nDocumentation reproduced from package ggstatsplot, version 0.0.10, License: GPL-3 \| file LICENSE\n\n### Community examples\n\nLooks like there are no examples yet." ]	[ null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.7840262,"math_prob":0.97307664,"size":6829,"snap":"2021-04-2021-17","text_gpt3_token_len":1604,"char_repetition_ratio":0.10564102,"word_repetition_ratio":0.008826125,"special_character_ratio":0.22287305,"punctuation_ratio":0.1359447,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.984382,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2021-01-19T18:58:52Z\",\"WARC-Record-ID\":\"<urn:uuid:79ee1ee6-4ae2-4654-a23a-272787317c22>\",\"Content-Length\":\"21928\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:ee26e45d-170a-461d-8625-5ce4c778875d>\",\"WARC-Concurrent-To\":\"<urn:uuid:87806ca1-f369-40d6-8aab-37b899755a8a>\",\"WARC-IP-Address\":\"52.20.214.25\",\"WARC-Target-URI\":\"https://www.rdocumentation.org/packages/ggstatsplot/versions/0.0.10/topics/t1way_ci\",\"WARC-Payload-Digest\":\"sha1:SGOKY725BUTDMTNLYYTMVEWF57OZT4NL\",\"WARC-Block-Digest\":\"sha1:MAE53JPF2AA367F5X2E3JQB2DJSMKWYZ\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2021/CC-MAIN-2021-04/CC-MAIN-2021-04_segments_1610703519600.31_warc_CC-MAIN-20210119170058-20210119200058-00663.warc.gz\"}"}
https://usethinkscript.com/threads/making-money-in-a-choppy-market.3765/	[ "", null, "# Making Money in a Choppy Market\n\n##### Member\nVIP\nVIP Enthusiast\nIntro:\n\nIn this choppy, super volatile trading environment I decided it was time to look for ways to smooth out the risk/reward ratios. Whether or not you believe we are in the middle of a sector rotation we can all agree things are getting a little sketchy in the market. Using two prebuilt thinkorswim indicators I built a scan that gives us low volatility combined with a high probability for above average returns.\n\nThe Conditions:\n• 7 day RSI > 60\n• 60 day RSI > 50\n• BollingerBandwidth < 10\nThe logic:\n• RSI is a measure of momentum, many people view a high RSI as being overbought. I agree when there is a spike in price. I disagree when the price is slowly trending up. A slowly upward trending stock can hold high levels of RSI for extended periods of time. Consider the stock MCO during the periods of 11/2019-2/2020. The 14 day RSI of MCO was above 70 most of that period. During that same time the stock ran 27%.\n• The BollingerBandwidth is the measure of volatility. High values imply high volatility, whereas low values imply low volatility. To my example, during 11/2019 MCO held the BollingerBandwidth value of less than 12.\nThe Example:", null, "Caveats:\n\n• It should be a given but cherry pick the best charts for best results\n\nThe Scan: https://tos.mx/JJUDFQz\n\nLast edited:\n•", null, "•", null, "MerryDay and bspratt22\n\n#### bspratt22\n\n##### New member\nagree with your premise on current market conditions; it is sketchy and might be the beginning of reversal but we still trade the charts right? My question is your entry. Are you in on the median line or exactly what is your entry?\n\n##### Member\nVIP\nVIP Enthusiast\n•", null, "BenTen\n\n#### APOT7\n\n##### Member\nVIP\nCan you share the lower study of the RSI 7day, 14,60, 120? As well as the BollingerBandwidth band simple?\n\n##### Member\nVIP\nVIP Enthusiast\nCan you share the lower study of the RSI 7day, 14,60, 120? As well as the BollingerBandwidth band simple?\n\nJust search for bollingerbandwidth in tos. Here is the multiple RSI\n\nCode:\n``````declare lower;\n\ninput length = 7;\ninput length1 = 14;\ninput length2 = 60;\ninput length3 = 120;\ninput over_Bought = 70;\ninput over_Sold = 30;\ninput price = close;\ninput averageType = AverageType.WILDERS;\n\ndef NetChgAvg = MovingAverage(averageType, price - price, length);\ndef TotChgAvg = MovingAverage(averageType, AbsValue(price - price), length);\n\ndef NetChgAvg1 = MovingAverage(averageType, price - price, length1);\ndef TotChgAvg1 = MovingAverage(averageType, AbsValue(price - price), length1);\n\ndef NetChgAvg2 = MovingAverage(averageType, price - price, length2);\ndef TotChgAvg2 = MovingAverage(averageType, AbsValue(price - price), length2);\n\ndef NetChgAvg3 = MovingAverage(averageType, price - price, length3);\ndef TotChgAvg3 = MovingAverage(averageType, AbsValue(price - price), length3);\n\ndef ChgRatio = if TotChgAvg != 0 then NetChgAvg / TotChgAvg else 0;\ndef ChgRatio1 = if TotChgAvg1 != 0 then NetChgAvg1 / TotChgAvg1 else 0;\ndef ChgRatio2 = if TotChgAvg2 != 0 then NetChgAvg2 / TotChgAvg2 else 0;\ndef ChgRatio3 = if TotChgAvg3 != 0 then NetChgAvg3 / TotChgAvg3 else 0;\n\nplot OverSold = over_Sold;\nplot OverBought = over_Bought;\nOverSold.SetDefaultColor(Color.DARK_GRAY);\nOverBought.SetDefaultColor(Color.DARK_GRAY);\n\nplot RSI = 50 * (ChgRatio + 1);\nplot RSI1 = 50 * (ChgRatio1 + 1);\nplot RSI2 = 50 * (ChgRatio2 + 1);\nplot RSI3 = 50 * (ChgRatio3 + 1);\n\nRSI.SetDefaultColor(Color.GREEN);\nRSI1.SetDefaultColor(Color.MagenTA);\nRSI2.SetDefaultColor(Color.YeLLOW);\nRSI3.SetDefaultColor(Color.BLUE);\n\nAddLabel(RSI, Concat(\"7RSI = \", RSI), color.GREEN);\nAddLabel(RSI1, Concat(\"14RSI = \", RSI1), color.MAGENTA);\nAddLabel(RSI2, Concat(\"60RSI = \", RSI2), color.YELLOW);\nAddLabel(RSI3, Concat(\"120RSI = \", RSI3), color.BLUE);``````\n\n•", null, "APOT7 and BenTen\n\n•", null, "" ]	[ null, "https://www.facebook.com/tr", null, "https://i.imgur.com/NGf46Xw.png", null, "data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7", null, "data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7", null, "data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7", null, "data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7", null, "data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7", null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.9200561,"math_prob":0.952578,"size":1840,"snap":"2021-43-2021-49","text_gpt3_token_len":451,"char_repetition_ratio":0.09150327,"word_repetition_ratio":0.44025156,"special_character_ratio":0.22445652,"punctuation_ratio":0.0952381,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9718108,"pos_list":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"im_url_duplicate_count":[null,null,null,1,null,null,null,null,null,null,null,null,null,null,null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2021-12-07T00:50:59Z\",\"WARC-Record-ID\":\"<urn:uuid:83069486-5d6b-4e13-8704-9d04090b602a>\",\"Content-Length\":\"88730\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:efd2e451-cac1-4a30-bab2-7e8f17692a06>\",\"WARC-Concurrent-To\":\"<urn:uuid:fd405cc0-bcfb-4b03-a904-45a62d1316db>\",\"WARC-IP-Address\":\"172.67.206.80\",\"WARC-Target-URI\":\"https://usethinkscript.com/threads/making-money-in-a-choppy-market.3765/\",\"WARC-Payload-Digest\":\"sha1:PQNHVTIDNHO5HSI22SBSX2QSA36VHRVW\",\"WARC-Block-Digest\":\"sha1:CHH3JGHYM6MPDEFPAWJMKD5X36NOQTAC\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2021/CC-MAIN-2021-49/CC-MAIN-2021-49_segments_1637964363327.64_warc_CC-MAIN-20211206224536-20211207014536-00055.warc.gz\"}"}
https://kdaher.com.br/categorias/botas	[ "", null, "# Botas\n\nR\\$459,00\n\nR\\$229,50\n\n6 x R\\$38,25\n\nR\\$359,00\n\nR\\$179,50\n\n6 x R\\$29,92\n\nR\\$476,00\n\nR\\$238,00\n\n6 x R\\$39,67\n\nR\\$456,00\n\nR\\$228,00\n\n6 x R\\$38,00\n\nR\\$449,00\n\nR\\$224,50\n\n6 x R\\$37,42\n\nR\\$486,00\n\nR\\$243,00\n\n6 x R\\$40,50\n\nR\\$396,00\n\nR\\$198,00\n\n6 x R\\$33,00\n\nR\\$449,00\n\nR\\$224,50\n\n6 x R\\$37,42\n\nR\\$359,00\n\nR\\$179,50\n\n6 x R\\$29,92\n\nR\\$449,00\n\nR\\$224,50\n\n6 x R\\$37,42\n• ##### Bota Eduarda K. Daher\n\nR\\$475,00\n\nR\\$237,50\n\n6 x R\\$39,58", null, "" ]	[ null, "https://www.facebook.com/tr", null, "https://kdaher.com.br/skin/frontend/mgstheme/claue/images/loader.svg", null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.5098286,"math_prob":0.9999876,"size":669,"snap":"2020-24-2020-29","text_gpt3_token_len":361,"char_repetition_ratio":0.26766917,"word_repetition_ratio":0.18103448,"special_character_ratio":0.5695067,"punctuation_ratio":0.19909503,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9519898,"pos_list":[0,1,2,3,4],"im_url_duplicate_count":[null,null,null,5,null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2020-07-10T18:07:58Z\",\"WARC-Record-ID\":\"<urn:uuid:4cfaf4d9-8fd4-493b-9bf3-a1d66ae25885>\",\"Content-Length\":\"127925\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:4f37e4ac-1471-4ee0-abb1-4084def787b5>\",\"WARC-Concurrent-To\":\"<urn:uuid:f1622dd8-ee41-45bf-85e9-30a1f1446903>\",\"WARC-IP-Address\":\"207.246.71.6\",\"WARC-Target-URI\":\"https://kdaher.com.br/categorias/botas\",\"WARC-Payload-Digest\":\"sha1:SR4TH5G6JSBQNRLG46QEXEBE6DBWMWSH\",\"WARC-Block-Digest\":\"sha1:CAHWAUTOANITX2PDBLGUP7SL2DRSTB4E\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2020/CC-MAIN-2020-29/CC-MAIN-2020-29_segments_1593655911896.73_warc_CC-MAIN-20200710175432-20200710205432-00411.warc.gz\"}"}
https://forums.wolfram.com/mathgroup/archive/2005/Oct/msg00521.html	[ "", null, "", null, "", null, "", null, "", null, "", null, "", null, "Re: Mathematica not simplifying Laplace transforms\n\n• To: mathgroup at smc.vnet.net\n• Subject: [mg61388] Re: Mathematica not simplifying Laplace transforms\n• From: \"roger\" <rogerbrent at gmail.com>\n• Date: Tue, 18 Oct 2005 02:44:29 -0400 (EDT)\n• References: <[email protected]><divhkr\\$gco\\[email protected]>\n• Sender: owner-wri-mathgroup at wolfram.com\n\n```Thanks, I tried that but it still doesnt simplify\n\nLaplaceTransform[Exp[t]*f[t], t, s, Assumptions -> t > 0]\n\nto\n\nLaplaceTransform[f[t], t, -1 + s]\n\nThe v4 documentation says it should do that, so I'm thinking they\nchanged this behaviour in v5.\nAnyone have any ideas?\n\n```\n\n• Prev by Date: Matrices and Conditional Statements\n• Next by Date: Re: surface fitting question\n• Previous by thread: Re: Mathematica not simplifying Laplace transforms\n• Next by thread: Re: Mathematica not simplifying Laplace transforms" ]	[ null, "https://forums.wolfram.com/mathgroup/images/head_mathgroup.gif", null, "https://forums.wolfram.com/mathgroup/images/head_archive.gif", null, "https://forums.wolfram.com/mathgroup/images/numbers/2.gif", null, "https://forums.wolfram.com/mathgroup/images/numbers/0.gif", null, "https://forums.wolfram.com/mathgroup/images/numbers/0.gif", null, "https://forums.wolfram.com/mathgroup/images/numbers/5.gif", null, "https://forums.wolfram.com/mathgroup/images/search_archive.gif", null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.7535389,"math_prob":0.4387586,"size":781,"snap":"2021-43-2021-49","text_gpt3_token_len":244,"char_repetition_ratio":0.14671814,"word_repetition_ratio":0.07920792,"special_character_ratio":0.29577464,"punctuation_ratio":0.21333334,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.96187186,"pos_list":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"im_url_duplicate_count":[null,null,null,null,null,null,null,null,null,null,null,null,null,null,null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2021-11-30T06:08:33Z\",\"WARC-Record-ID\":\"<urn:uuid:2db14a14-5817-4261-ad94-ae011995bebf>\",\"Content-Length\":\"44710\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:ace998a7-7093-4442-a916-595c7561d29a>\",\"WARC-Concurrent-To\":\"<urn:uuid:c30c6d95-e81a-4283-bc6f-c787aea90619>\",\"WARC-IP-Address\":\"140.177.205.73\",\"WARC-Target-URI\":\"https://forums.wolfram.com/mathgroup/archive/2005/Oct/msg00521.html\",\"WARC-Payload-Digest\":\"sha1:6V5XTCWROFBN4UHMJO66NN62ZYRTOOVZ\",\"WARC-Block-Digest\":\"sha1:RFYTNOWQOS6H56JVN5XFOASMBZ7A5IW3\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2021/CC-MAIN-2021-49/CC-MAIN-2021-49_segments_1637964358953.29_warc_CC-MAIN-20211130050047-20211130080047-00364.warc.gz\"}"}
http://prosib.info/1-digit-multiplication-worksheets/	[ "2 digit by 1 multiplication worksheets 3 addition without regrouping and word problem work.\n\nfree collection of multiplication worksheets grade 4 2 digit by 1 download them pdf no regroupin.\n\nsmall size two digits multiplication worksheets 2 digit with answers by 1 pdf no regrouping multiplic.\n\n3 digit multiplication worksheet 1 multiplier 4 by worksheets pdf.\n\n4 digit by 1 multiplication worksheets unique inspiring math 2 subtraction with new long worksheet.\n\n4 digit by 1 multiplication worksheets amazing math anchor charts class 3 2 pdf digi.\n\nmultiplication worksheets 2 digit by 1 times 4 word problems math ri.\n\nmultiplication worksheets long 2 digit by 1 grade 4 for free multiplicatio.\n\n3 by 2 multiplication worksheets 1 digit worksheet up grade coloring x arrays pdf workshe.\n\n2 by 1 digit multiplication worksheets best solutions of free printable no regrouping long and multiplica.\n\n3 digit times 1 multiplication worksheets best of math riddle digits multiplying by 2 digi.\n\n2 digit by 1 multiplication word problems worksheets practice one.\n\nbrilliant ideas of 4 digit by 1 multiplication worksheets 3 digits times awesome collection wor.\n\n2 digit by 1 multiplication worksheet worksheets for all free library download and print on 4th grade 8 per page a large mat.\n\n4 digit by 1 multiplication worksheets lovely division problems grade math 3 new long 5 pdf d.\n\ntwo digit by 1 ation worksheets inspirational double brad multiplication free and 2.\n\n2 digit by 1 multiplication worksheets fresh free without regrouping.\n\ngit multiplications grade math year 2 multiplication worksheets times 1 free printable by various lattice maths multi and digit mu.\n\ntwo digit by 1 multiplication worksheets hard 2 problems in this grade 3 math 4th worksh.\n\ntwo digit multiplication worksheet free worksheets library download and print on 2 by 1 grade 4 3 6.\n\nmath worksheets grade 2 multiplication worksheet printable free digit x 1 by on graph paper m.\n\n3 digit by 1 multiplication worksheets the best image collection download and share 2 mult.\n\n1 digit multiplication worksheets task cards printable 2 times 3 by no regrouping car.\n\nmultiplication worksheet freebie 2 digit by 1 worksheets pdf grade original.\n\ncollection of multiplication worksheet 2 digit by 1 download worksheets horizontal them and t multi 3 word proble.\n\n2 digit multiplication worksheets printable by 1 pdf grade 3 multiplicat.\n\nmultiplication worksheets grade 2 multiplying digit by 1 numbers a long practice 4 colo.\n\ngrade 4 multiplication worksheets free library download and print on 1 digit by 2 without regrouping traditional works.\n\nmultiplication worksheets 2 digit by 1 cupcakes and pdf original.\n\n2 digit by 1 multiplication worksheets grade two word 3 coloring works.\n\nworksheets 2 digit by 1 multiplication x math worksheet on grid paper word grade horizon.\n\nhorizontal multiplication worksheet worksheets 4 digits times 1 digit 2 by pdf." ]	[ null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.73039883,"math_prob":0.95582575,"size":2901,"snap":"2019-35-2019-39","text_gpt3_token_len":610,"char_repetition_ratio":0.38660684,"word_repetition_ratio":0.07173913,"special_character_ratio":0.20234402,"punctuation_ratio":0.06451613,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9977079,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2019-09-19T08:29:48Z\",\"WARC-Record-ID\":\"<urn:uuid:30ae9f0d-ee94-4d6e-967a-e5da9c61a145>\",\"Content-Length\":\"64866\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:2ad1bcaa-d8ad-4658-a4ea-455a08f55149>\",\"WARC-Concurrent-To\":\"<urn:uuid:a33821b5-ff4e-455e-9842-50064cc35bc7>\",\"WARC-IP-Address\":\"104.27.186.84\",\"WARC-Target-URI\":\"http://prosib.info/1-digit-multiplication-worksheets/\",\"WARC-Payload-Digest\":\"sha1:OVOFPH2KZECMTYOFVLJPOT4LDVSM4JF2\",\"WARC-Block-Digest\":\"sha1:TCZIODFQSHS2YC3USA5JVGUFN76IBNWH\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2019/CC-MAIN-2019-39/CC-MAIN-2019-39_segments_1568514573465.18_warc_CC-MAIN-20190919081032-20190919103032-00418.warc.gz\"}"}
https://akiboy96.medium.com/community-detection-network-analysis-of-the-stock-market-in-python-r-part-3-ca4e0fbb8ca9	[ "# Building your Long Term Portfolio using Unsupervised ML with Community Detection Analysis in Python & R\n\nTo build the portfolio based on the correlation of stock prices in the S&P 500, we will use network centrality as a measure to understand how the stocks are better integrated with one and other.\n\n`stocks_cross_corr, _, _ = calculate_corr(df_stock_prices,1, len(df_stock_prices), 'pearson')stocks_cross_corr = stocks_cross_corrcor_thresold = 0.7G = build_graph(stocks_cross_corr, cor_thresold)partition = community.best_partition(G)modularity = community.modularity(partition, G)values = [partition.get(node) for node in G.nodes()]plt.figure(figsize=(10,10))nx.draw_spring(G, cmap = plt.get_cmap('jet'), node_color = values, node_size=30, with_labels=False)print(modularity)print(\"Total number of Communities=\", len(G.nodes()))dict_betwenness_centrality = nx.betweenness_centrality(G)dict_degree_centrality = nx.degree_centrality(G)dict_closeness_centrality = nx.closeness_centrality(G)dict_eigenvector_centrality = nx.eigenvector_centrality(G)print(\"dict_degree_centrality: \", dict_degree_centrality)print(\"dict_closeness_centrality: \", dict_closeness_centrality)print(\"dict_eigenvector_centrality: \", dict_eigenvector_centrality)print(\"dict_betweenness_centrality: \", dict_betwenness_centrality)`\n\n## Once the four types of centralities are obtained we will build our own portfolio determining algorithm using these centralities in an equal weightage sense. To do that we combine the dictionary data structure of our 4 centralties and use a for loop to iterate through each stock and add the four different centralities to create a final centrality metric. Since we are using equal weightage the formula would use a simple sum and not be any assigned any weights.\n\n`#Portfolio Formula: c_dict = dict([(k, [dict_betwenness_centrality[k], dict_eigenvector_centrality[k], dict_degree_centrality[k], dict_closeness_centrality[k] ]) for k in dict_betwenness_centrality])#print(c_dict) C_total = {}for key in c_dict: C_total[key] = sum(c_dict[key]) print(\"The Centrality total for stocks are:\", C_total) newDict = dict(filter(lambda elem: elem > 0, C_total.items()))print(\"Stocks greater than 0.3 centrality are\",newDict)print(len(newDict))df_centrality = pd.DataFrame(list(newDict.items()),columns = ['Symbol','Centrality']) df_centrality.sort_values(by='Centrality', ascending=False)#df_centrality.head(20)#type(df_centrality['Centrality'])df_centrality.to_csv('centrality_of_stocks_0.7cor.csv',index=False)`\n`# This R environment comes with many helpful analytics packages installed# It is defined by the kaggle/rstats Docker image: https://github.com/kaggle/docker-rstats# For example, here's a helpful package to loadlibrary(tidyverse)library(tidyquant)library(jsonlite)library(tidyverse)library(readr)library(igraph)library(dplyr)library(lubridate)library(data.table)library(Quandl) # metapackage of all tidyverse packages# Input data files are available in the read-only \"../input/\" directory# For example, running this (by clicking run or pressing Shift+Enter) will list all files under the input directorylist.files(path = \"../input\")# You can write up to 5GB to the current directory (/kaggle/working/) that gets preserved as output when you create a version using \"Save & Run All\" # You can also write temporary files to /kaggle/temp/, but they won't be saved outside of the current session`\n`#DataLoad stockprices <- read.csv(\"../input/usstockprices/stocks_price_final.csv\")centrality0.5 <- read.csv(\"../input/centrality-of-stocks-network-analysis/centrality_of_stocks_0.5cor.csv\")centrality0.6 <- read.csv(\"../input/centrality-of-stocks-network-analysis/centrality_of_stocks_0.6cor.csv\")centrality0.7 <- read.csv(\"../input/centrality-of-stocks-network-analysis/centrality_of_stocks_0.7cor.csv\")`\n`#Merges Centtraility values with other potential useful information like market cap and sector for further filteringstocks <- stockprices[!duplicated(stockprices\\$symbol), ] # removes duplicate symbolsIndex0.5cor <- merge(stocks, centrality0.5, by.x = \"symbol\", by.y = \"Symbol\")Index0.6cor <- merge(stocks, centrality0.6, by.x = \"symbol\", by.y = \"Symbol\")Index0.7cor <- merge(stocks, centrality0.7, by.x = \"symbol\", by.y = \"Symbol\")#head(Index0.6cor)IndexGet0.5 <- as.character(Index0.5cor\\$symbol)IndexGet0.6 <- as.character(Index0.6cor\\$symbol)IndexGet0.7 <- as.character(Index0.7cor\\$symbol)#Portfolio Testing##0.5 correlation centrality pricesstockindex0.5 <- tq_get(IndexGet0.5, get=\"stock.prices\", from = \"2015-07-01\",warnings = FALSE, stringsAsFactors = FALSE) %>% group_by(symbol) %>% tq_transmute(select=adjusted, mutate_fun=periodReturn, period=\"monthly\", col_rename = \"monthly_return\")stockindex0.5##0.6 correlation centrality pricesstockindex0.6 <- tq_get(IndexGet0.6, get=\"stock.prices\", from = \"2015-07-01\",warnings = FALSE, stringsAsFactors = FALSE) %>% group_by(symbol) %>% tq_transmute(select=adjusted, mutate_fun=periodReturn, period=\"monthly\", col_rename = \"monthly_return\")##0.7 correlation centrality pricesstockindex0.7 <- tq_get(IndexGet0.7, get=\"stock.prices\", from = \"2015-07-01\",warnings = FALSE, stringsAsFactors = FALSE) %>% group_by(symbol) %>% tq_transmute(select=adjusted, mutate_fun=periodReturn, period=\"monthly\", col_rename = \"monthly_return\")##Base Portfolio to compare baseline_returns_monthly <- \"SPY\" %>% tq_get(get = \"stock.prices\", from = \"2015-07-01\", warnings = FALSE,stringsAsFactors = FALSE) %>% tq_transmute(select = adjusted, mutate_fun = periodReturn, period = \"monthly\", col_rename = \"spy_monthly_return\")baseline_returns_monthly`\n`portfolio_returns_monthly0.5 <- stockindex0.5 %>% tq_portfolio(assets_col = symbol, returns_col = monthly_return, col_rename = \"portfolio-monthly\")portfolio_returns_monthly0.6 <- stockindex0.6 %>% tq_portfolio(assets_col = symbol, returns_col = monthly_return, col_rename = \"portfolio-monthly\")portfolio_returns_monthly0.7 <- stockindex0.7 %>% tq_portfolio(assets_col = symbol, returns_col = monthly_return, col_rename = \"portfolio-monthly\")#Portfolio Comparestock0.5indexVSSPY <- left_join(portfolio_returns_monthly0.5, baseline_returns_monthly, by = \"date\")stock0.6indexVSSPY <- left_join(portfolio_returns_monthly0.6, baseline_returns_monthly, by = \"date\")stock0.7indexVSSPY <- left_join(portfolio_returns_monthly0.7, baseline_returns_monthly, by = \"date\")#stock0.5indexVSSPYggplot(stock0.5indexVSSPY) + geom_line(aes(x = `date`, y = `portfolio-monthly`), color = \"blue\")+ geom_line(aes(x = `date`, y = `spy_monthly_return`), color = \"red\")ggplot(stock0.6indexVSSPY) + geom_line(aes(x = `date`, y = `portfolio-monthly`), color = \"blue\")+ geom_line(aes(x = `date`, y = `spy_monthly_return`), color = \"red\")ggplot(stock0.7indexVSSPY) + geom_line(aes(x = `date`, y = `portfolio-monthly`), color = \"blue\")+ geom_line(aes(x = `date`, y = `spy_monthly_return`), color = \"red\")`\n`##PLAYGROUND - LETS DO TRIAL AND ERROR HERE TO BEAT THE SPY GRAPH in RETURNScentrality_filter <- Index0.6cor#filter(Centrality > 0.8 & Centrality < 0.5)centrality_filterIndexGet0.6 <- as.character(centrality_filter\\$symbol)stockindex0.6 <- tq_get(IndexGet0.6, get=\"stock.prices\", from = \"2015-07-01\",warnings = FALSE, stringsAsFactors = FALSE) %>% group_by(symbol) %>% tq_transmute(select=adjusted, mutate_fun=periodReturn, period=\"monthly\", col_rename = \"monthly_return\")portfolio_returns_monthly0.6 <- stockindex0.6 %>% tq_portfolio(assets_col = symbol, returns_col = monthly_return, col_rename = \"portfolio-monthly\")stock0.6indexVSSPY <- left_join(portfolio_returns_monthly0.6, baseline_returns_monthly, by = \"date\")ggplot(stock0.6indexVSSPY) + geom_line(aes(x = `date`, y = `portfolio-monthly`), color = \"blue\")+ geom_line(aes(x = `date`, y = `spy_monthly_return`), color = \"red\")`\n\nThis is the code I wrote to achieve a portfolio that could beat the SPY returns over a 5 year period from July 1st 2015 to July 22nd 2020\n\n`centrality_filter <- Index0.5cor %>%filter((sector == \"Health Care\" & Centrality > 0.4) \| (sector == \"Technology\" & Centrality > 0.42) \| (sector == \"Consumer Services\" & Centrality > 0.49) \| (sector == \"Finance\" & Centrality > 0.5) \| (sector == \"Transportation\" & Centrality > 0.5) \| (sector == \"Capital Goods\" & Centrality > 0.35) \| (sector == \"Miscellaneous\" & Centrality > 0.5) \| (sector == \"Basic Industries\" & Centrality > 0.39) \| (sector == \"Public Utilities\" & Centrality > 0.36) \| (sector == \"Consumer Durables\" & Centrality > 0.3)\| (sector == \"Consumer Non-Durables\" & Centrality > 0.25))centrality_filterIndexGet0.5 <- as.character(centrality_filter\\$symbol)stockindex0.5 <- tq_get(IndexGet0.5, get=\"stock.prices\", from = \"2015-07-01\", till = \"2020-07-22\",warnings = FALSE, stringsAsFactors = FALSE) %>% group_by(symbol) %>% tq_transmute(select=adjusted, mutate_fun=periodReturn, period=\"monthly\", col_rename = \"monthly_return\")portfolio_returns_monthly0.5 <- stockindex0.5 %>% tq_portfolio(assets_col = symbol, returns_col = monthly_return, col_rename = \"portfolio-monthly\")stock0.5indexVSSPY <- left_join(portfolio_returns_monthly0.5, baseline_returns_monthly, by = \"date\")stock0.5indexVSSPY\\$`portfolio-monthly` <- stock0.5indexVSSPY\\$`portfolio-monthly` * 100stock0.5indexVSSPY\\$spy_monthly_return <- stock0.5indexVSSPY\\$spy_monthly_return * 100stock0.5indexVSSPY <- stock0.5indexVSSPY[-c(14), ]returns <- ggplot(stock0.5indexVSSPY) + geom_line(aes(x = `date`, y = `portfolio-monthly`), color = \"blue\")+ geom_line(aes(x = `date`, y = `spy_monthly_return`), color = \"red\") + ggtitle(\"Portfolio vs S&P 500 Returns over 5 years\") + labs(y=\"Returns Percenatge\", x = \"Date\") plot(returns)`\n\nThe centralities as shown in the above code were chosen based on my predictions for future trends in terms of what sectors will do well and become important to society as a service.\n\n--\n\n--\n\n--\n\n## More from Aakash Kedia\n\nCS, music and football in no particular order\n\nLove podcasts or audiobooks? Learn on the go with our new app.\n\n## How Much Does It Cost to Make an App Like Postmates?", null, "## Run MySQL and phpMyAdmin using Docker", null, "## How I got my (functional) team to collaborate with many Scrum teams", null, "## 10 Mistakes When Sizing Cloud Resources", null, "## Ease of Building UI Elements in Unity", null, "## Error Code ssl_error_no_cypher_overlap Firefox 34", null, "## Overview of Inverse Kinematics", null, "", null, "## Aakash Kedia\n\nCS, music and football in no particular order\n\n## Using ML to Predict Search Volume", null, "## Time Series Data Components with Microsoft excel", null, "## Computing Forecast Error: Should I divide by forecast or demand?", null, "## Machine Learning Applications in Climate Prediction Analytics", null, "" ]	[ null, "https://miro.medium.com/focal/112/112/50/50/1YJiTqVZpFMLHfXCFUd0AXA.png", null, "https://miro.medium.com/focal/112/112/50/50/1C-oLJBr9VbYfx-9Qwp7dlw.png", null, "https://miro.medium.com/focal/112/112/50/50/0IIUmx40Gurletic7", null, "https://miro.medium.com/focal/112/112/50/50/0l8lw5gblqAVgKfqy", null, "https://miro.medium.com/focal/112/112/50/50/10QwUt1UA4tfdeZtlheooFQ.png", null, "https://miro.medium.com/focal/112/112/50/50/03DK2qHJg-421FtTp.png", null, "https://miro.medium.com/focal/112/112/50/50/19JorUgXUcaJEz7Q3AhDjow.jpeg", null, "https://miro.medium.com/fit/c/176/176/0emaZ2wHvCzhl_Apj", null, "https://miro.medium.com/focal/112/112/50/50/1db4rgvcJW9yqGe-IAn1moA.png", null, "https://miro.medium.com/focal/112/112/50/50/0jNRwJE_erN32nwhL", null, "https://miro.medium.com/focal/112/112/50/50/148XsEtETn_lYfsEUe6KOFQ.png", null, "https://miro.medium.com/focal/112/112/50/50/0Ibv4SFoStHNEpXoa", null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.68728596,"math_prob":0.88684905,"size":13012,"snap":"2022-27-2022-33","text_gpt3_token_len":3483,"char_repetition_ratio":0.16190037,"word_repetition_ratio":0.14447236,"special_character_ratio":0.26344913,"punctuation_ratio":0.15738499,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9673924,"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],"im_url_duplicate_count":[null,1,null,1,null,1,null,1,null,1,null,1,null,1,null,3,null,1,null,1,null,1,null,1,null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2022-06-28T02:45:05Z\",\"WARC-Record-ID\":\"<urn:uuid:05aaea61-a06e-4aee-8a62-493c938af459>\",\"Content-Length\":\"182436\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:5d0ef349-5123-445f-85a5-9a8986af5de2>\",\"WARC-Concurrent-To\":\"<urn:uuid:02286087-9246-40a2-9b1c-98991724b094>\",\"WARC-IP-Address\":\"162.159.153.4\",\"WARC-Target-URI\":\"https://akiboy96.medium.com/community-detection-network-analysis-of-the-stock-market-in-python-r-part-3-ca4e0fbb8ca9\",\"WARC-Payload-Digest\":\"sha1:URGYPIDHX7UYIMTEQM34KLXCGTKKAQP6\",\"WARC-Block-Digest\":\"sha1:EBWUR72EACQ4RYT6ETT2KM7OFM4AGOZM\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2022/CC-MAIN-2022-27/CC-MAIN-2022-27_segments_1656103347800.25_warc_CC-MAIN-20220628020322-20220628050322-00285.warc.gz\"}"}
https://michelanders.blogspot.com/2012/02/3d-convex-hull-in-python.html	[ "## Sunday, 19 February 2012\n\n### 3D Convex hull in Python\n\nIn this article I present a present a reimplementation in pure Python of Joseph O'Rourke's incremental 3D convex hull algorithm from his book Computational Geometry in C.\n\n## A convex hull in pure Python\n\nThis is the second, rather off topic, article on computational geometry in this blog. The previous article presented an implementation of the marching tetrahedrons algorithm.\nThe goal of this article is to provide object oriented, pythonic code to compute the convex hull of a collection of 3D points. The code is contained in a single Python module that may be downloaded from GitHub.\nA sample of how to use this module is shown below, where we create a a roughly spherical cloud of points, calculate its convex hull and print this hull in STL format to stdout. The resulting object is shown in the image (as seen in Blender).\n```from random import random\nfrom chull import Vector,Hull\n\nsphere=[]\nfor i in range(2000):\nx,y,z = 2random()-1,2random()-1,2random()-1\nif xx+yy+zz < 1.0:\nsphere.append(Vector(x,y,z))\nh=Hull(sphere)\nh.Print()\n```\n\n## Difference from the original implementation\n\nThe original code restricted the coordinates of points to integers, here there is no such restriction. This might result in errors if the coordinates are large.\n\n1.", null, "When I try to open ti with the GNU Meshlab I get an error saying \"Premature end of file\". What is that supposed to mean?\n\n2.", null, "Hi,\nthank you very much for this piece of code! I am using it succesfully.\nThere is only one strange behaviour I couldnt yet figure out why it happens: Sometimes the code doesnt finish saying\n\n338 # e interior: mark for deletion\n339 e.delete = REMOVED;\n341 # e border: make a new face\n342 e.newface = self.MakeConeFace( e, p )\n\nAttributeError: 'NoneType' object has no attribute 'visible'\n\nin the constructor when calling (Hull(myvectors)).\n\nDo you have an idea about how I could avoid this? For a some selection of my data points the code works, for some this error occurs and I cant figure out what the pattern is.\n\nThanks again!\n\n3.", null, "Thank you for sharing your code!\nit would be great if there was an easy build-in method available, which can calculate the surface area of the resulting convex polyhedron.\n\n4.", null, "Great code! I've used it to create an STL file for a 3D shape that contains concavities and convexity. Unfortunately, the code seems to fill in those concavities to create a convex shape. I'm not a Python expert - is there a (simple?) way to modify this code to not get rid of the concavities?\n\n5.", null, "Thank you very much, this code was very useful for me to represent some data :)\n\nI added a little function:\n\ndef PrintFile(self, filename='tmpshape.stl'):\nf1 = open(filename, 'w+')\nf1.write(\"solid points\\n\")\nfor f in self.faces:\nf1.write(str(f) + '\\n')\nf1.write(\"endsolid points\\n\")\nf1.close()\n\nto print directly to a stl file which was handy for me." ]	[ null, "https://img1.blogblog.com/img/blank.gif", null, "https://lh3.googleusercontent.com/zFdxGE77vvD2w5xHy6jkVuElKv-U9_9qLkRYK8OnbDeJPtjSZ82UPq5w6hJ-SA=s35", null, "https://img1.blogblog.com/img/blank.gif", null, "https://img1.blogblog.com/img/blank.gif", null, "https://3.bp.blogspot.com/_7vTskwlNObE/TCsw9bM8l4I/AAAAAAAAClw/d8ik6xqnXr4/s35/iconor0s.jpg", null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.8743726,"math_prob":0.6741904,"size":2452,"snap":"2019-13-2019-22","text_gpt3_token_len":605,"char_repetition_ratio":0.09068628,"word_repetition_ratio":0.0,"special_character_ratio":0.24429038,"punctuation_ratio":0.13111547,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9610441,"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\":\"2019-03-24T18:35:24Z\",\"WARC-Record-ID\":\"<urn:uuid:427bf892-d893-4940-a9d6-20a9816577d8>\",\"Content-Length\":\"131178\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:30151b0e-3ddb-4cd0-8c5e-da186bb5c6d2>\",\"WARC-Concurrent-To\":\"<urn:uuid:6d7fad48-7df1-40cb-99ad-797b51b7e8c3>\",\"WARC-IP-Address\":\"172.217.5.225\",\"WARC-Target-URI\":\"https://michelanders.blogspot.com/2012/02/3d-convex-hull-in-python.html\",\"WARC-Payload-Digest\":\"sha1:S4ZZALC2X3W72D57ZMFFPQQBDA4UWIV3\",\"WARC-Block-Digest\":\"sha1:L5L5NWTR27NHIATFR6VOFXFUGMNXUGY2\",\"WARC-Identified-Payload-Type\":\"application/xhtml+xml\",\"warc_filename\":\"/cc_download/warc_2019/CC-MAIN-2019-13/CC-MAIN-2019-13_segments_1552912203464.67_warc_CC-MAIN-20190324165854-20190324191854-00285.warc.gz\"}"}
https://www.yaclass.in/p/science-state-board/class-9/organisation-of-tissues-draft-5860/plant-tissues-draft-5861/re-fe3207bd-325f-4375-89d5-85f4f6d80c99	[ "### Theory:\n\nOnion root-meristematic tissue experiment:\nOnion bulb experiment - presence of apical meristem\n• Take two jars (jar $$1$$ & jar $$2$$) and two onion bulbs with roots protruding from them. Pour water into the jars until the jars are nearly full.\n• Place the two onions with their roots immersed in the water into each of the jars as shown in the picture.\n• Measure the length of the roots on day $$1$$, $$2$$ and $$3$$.\n• After a few days we observe that both the roots have grown to a certain length.\n• Take onion bulb out of the jar $$2$$ and cut the tip of the root by about $$1$$ cm. Place it again in the jar and observe the growth of roots in both jar $$1$$ and jar $$2$$ for another $$5$$ days. Mark the jar of the root-trimmed onion bulb.\nNote the measurement of the length of the roots in the below table before cutting the tip of the root:\n\n Day $$1$$ Day $$2$$ Day $$3$$ Jar $$1$$ Jar $$2$$\n\nNote the measurement of the length of the roots in the below table after cutting the tip of the root:\n\n Day $$1$$ Day $$2$$ Day $$3$$ Day $$4$$ Day $$5$$ Jar$$1$$ Jar $$2$$\nQuestions:\n1. Is there a difference in length of roots between the root-trimmed onion (in jar $$2$$) and the un-trimmed one (in jar $$1$$)? Give reasons.\n2. Do the roots continue growing even after we have removed their tip?\nAnswers:\n1. Yes, there will be a difference in the length of the roots. The root-trimmed onion's root (in jar $$2$$) is shorter in length than the un-trimmed one (in jar $$1$$). Tips of the plant roots contain apical meristematic tissue, and is responsible for the elongation or length of the roots. As the tip of the onion roots in jar $$2$$ is cut, apical meristematic tissue is removed from it. There will not be any observable growth in the roots of the onion in jar $$2$$ . So the length of the roots of the onion in jar $$2$$ is lesser than the length of the roots of the onion in jar$$1$$.\n2. No, the roots won't grow after their tips are removed. Tips of the plant roots contain apical meristematic tissue, and is responsible for the elongation or length of the roots. As the tip of the onion roots in jar $$2$$ is cut, apical meristematic tissue is removed from it. So there will not be any growth in the roots of the onion in jar $$2$$ after the tips are cut." ]	[ null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.9089579,"math_prob":0.99952567,"size":2268,"snap":"2021-21-2021-25","text_gpt3_token_len":591,"char_repetition_ratio":0.22084805,"word_repetition_ratio":0.30679157,"special_character_ratio":0.28791887,"punctuation_ratio":0.07375271,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.994799,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2021-06-19T02:22:41Z\",\"WARC-Record-ID\":\"<urn:uuid:7a80233a-eb26-4781-9721-7cccedac103e>\",\"Content-Length\":\"37331\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:aa09ac09-4230-4486-b947-2a48ffb6cc14>\",\"WARC-Concurrent-To\":\"<urn:uuid:e0f9546f-fdd5-4364-990c-036db215fb42>\",\"WARC-IP-Address\":\"52.172.204.196\",\"WARC-Target-URI\":\"https://www.yaclass.in/p/science-state-board/class-9/organisation-of-tissues-draft-5860/plant-tissues-draft-5861/re-fe3207bd-325f-4375-89d5-85f4f6d80c99\",\"WARC-Payload-Digest\":\"sha1:CO7F36WWF3QTMLILUUYTAPMASUZLOG7J\",\"WARC-Block-Digest\":\"sha1:HDG7SQ7W6KMXMFD7II5II5GWYUA2YSVO\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2021/CC-MAIN-2021-25/CC-MAIN-2021-25_segments_1623487643380.40_warc_CC-MAIN-20210619020602-20210619050602-00321.warc.gz\"}"}
https://jov.arvojournals.org/article.aspx?articleid=2772046	[ "", null, "jov\nOpen Access\nMethods \| December 2020\nA new toolbox to distinguish the sources of spatial memory error\nAuthor Affiliations\n• John P. Grogan\nNuffield Department of Clinical Neurosciences, University of Oxford, Oxford, UK\[email protected]\n• Sean J. Fallon\nPopulation Health Sciences, University of Bristol, Bristol, UK\[email protected]\n• Nahid Zokaei\nDepartment of Experimental Psychology, University of Oxford, Oxford, UK\nOxford Centre for Human Brain Activity, University of Oxford, Oxford, UK\[email protected]\n• Masud Husain\nNuffield Department of Clinical Neurosciences, University of Oxford, Oxford, UK\nDepartment of Experimental Psychology, University of Oxford, Oxford, UK\[email protected]\n• Elizabeth J. Coulthard\nTranslational Health Sciences, University of Bristol, Bristol, UK\nNorth Bristol NHS Trust, Bristol, UK\[email protected]\n• Sanjay G. Manohar\nNuffield Department of Clinical Neurosciences, University of Oxford, Oxford, UK\nOxford Centre for Human Brain Activity, University of Oxford, Oxford, UK\[email protected]\nJournal of Vision December 2020, Vol.20, 6. doi:https://doi.org/10.1167/jov.20.13.6\n• Views\n• PDF\n• Share\n• Tools\n×\n###### This feature is available to authenticated users only.\n• Get Citation\n\nJohn P. Grogan, Sean J. Fallon, Nahid Zokaei, Masud Husain, Elizabeth J. Coulthard, Sanjay G. Manohar; A new toolbox to distinguish the sources of spatial memory error. Journal of Vision 2020;20(13):6. doi: https://doi.org/10.1167/jov.20.13.6.\n\n© ARVO (1962-2015); The Authors (2016-present)\n\n×\n• Supplements\nAbstract\n\nStudying the sources of errors in memory recall has proven invaluable for understanding the mechanisms of working memory (WM). While one-dimensional memory features (e.g., color, orientation) can be analyzed using existing mixture modeling toolboxes to separate the influence of imprecision, guessing, and misbinding (the tendency to confuse features that belong to different memoranda), such toolboxes are not currently available for two-dimensional spatial WM tasks.\n\nHere we present a method to isolate sources of spatial error in tasks where participants have to report the spatial location of an item in memory, using two-dimensional mixture models. The method recovers simulated parameters well and is robust to the influence of response distributions and biases, as well as number of nontargets and trials.\n\nTo demonstrate the model, we fit data from a complex spatial WM task and show the recovered parameters correspond well with previous spatial WM findings and with recovered parameters on a one-dimensional analogue of this task, suggesting convergent validity for this two-dimensional modeling approach. Because the extra dimension allows greater separation of memoranda and responses, spatial tasks turn out to be much better for separating misbinding from imprecision and guessing than one-dimensional tasks.\n\nCode for these models is freely available in the MemToolbox2D package and is integrated to work with the commonly used MATLAB package MemToolbox.\n\nIntroduction\nWorking memory (WM) is typically measured by providing a person with a set of stimuli to remember and then probing their memory after a delay of a few seconds. For example, they may be asked which direction of one of a set of colored arrows pointed. The types of errors people make on these tasks provide important information about the processes involved in retaining information in WM over and above simply looking at accuracy or overall error (Ma, Husain, & Bays, 2014). One type of error occurs when people remember which features were presented but fail to remember the way in which they were combined, such as reporting the orientation of a different-colored arrow. This might indicate that people misbind the features. A second type of error involves knowing the features of an object only approximately, leading to imprecision. A third type of error arises if everything is forgotten and responses are essentially guesses. Importantly, these three types of errors may each reflect distinct processes affecting WM.\nAlthough these processes can be distinguished when participants make binary judgments about the items in memory (Parra et al., 2010; Stark, Yassa, & Stark, 2010), recent studies have allowed greater sensitivity to these types of error by using continuous report, in which participants reproduce the feature they had to remember (Bays, Catalao, & Husain, 2009; Zokaei, Burnett Heyes, Gorgoraptis, Budhdeo, & Husain, 2015). This data-rich method allows us to separate the errors people make into their component parts. The technique of mixture modeling (Bays et al., 2009; Bays & Husain, 2008; Zhang & Luck, 2008) is now commonly used if the report domain is one-dimensional and circular, for example, line orientation or color on a wheel.\nThese methods have proven invaluable in understanding memory mechanisms, demonstrating, for example, that interstimulus distance affects misbinding and not precision (Emrich & Ferber, 2012), increasing set size affects misbinding and precision more than random guessing (Bays et al., 2009; Pertzov, Manohar, & Husain, 2017), and decreasing sample duration increases random guessing but not imprecision or misbinding (Bays et al., 2009). Importantly, this has also allowed mechanistic understandings in clinical populations where the different sources can be selectively impaired (Rolinski et al., 2016; Zokaei et al., 2014).\nHowever, many working memory tasks are two-dimensional (2D; e.g., spatial location; Corsi, 1972), and extending the mixture modeling technique to work on such 2D data would allow researchers to decompose errors in their component sources. While some studies used binary report (Bays & Husain, 2008), continuous-report measures of spatial memory have emerged more recently, in which participants move a probe to its original location (Pertzov, Dong, Peich, & Husain, 2012). This can be accomplished by participants dragging or pointing to locations on a touchscreen computer, allowing precise spatial WM measures in patients with disease (Liang et al., 2016; Pertzov et al., 2013; Zokaei, Nour, et al., 2019).\nTo examine the types of errors made, previous analyses employed simple measures based on the distance of each response from the other items in the display (Pertzov et al., 2012). The most basic method includes all sources of errors, being simply the mean distance from the true location of the target to where a participant recalls it to have been. This distance-to-target can be compared against the distance from the response to the nearest stimulus that had appeared in the memory array (nearest-neighbor distance) (Pertzov et al., 2013). Such a measure removes the influence of misbinding, giving a measure of imprecision. However, this metric still conflates errors of imprecision (i.e., knowing the location only approximately) with errors due to pure guessing. Further analysis methods to take guessing into account (Pertzov et al., 2013) involve counting the number of responses that occur within a certain distance of a nontarget (swap errors), but this does not consider a person's precision or the distance between the stimuli on each trial. These simple behavioral metrics each attempt to separate out different sources of errors, but the only way to validate them has been by comparison of results with those from one-dimensional (1D) circular tasks.\nOne study has applied mixture modeling to 2D working memory tasks (Schneegans & Bays, 2016) and found that misbinding increased with set size, while random guessing was negligible, and that the pattern of target errors, nontarget responses, and reaction times was in keeping with a continuous-resource model, not a discrete-slot model. They also found that misbinding, but not random guessing, contributed to errors, which may highlight an advantage of spatial data; the extra dimension of separation between items leads to less overlap in response distributions centered on the items, giving greater ability to distinguish error components. This article demonstrated the usefulness of mixture modeling for spatial tasks, but to our knowledge, there is no available package for applying these models to 2D data or an investigation into the suitability and limitations of such an approach.\nHere, we present 2D mixture models (adapted from a previous 1D mixture model package MemToolbox; Suchow, Brady, Fougnie, & Alvarez, 2013; MemToolbox.org) that allow us to fit a person's imprecision of location memories, along with the proportion of random guesses and nontarget responses (misbinding), taking into account stimulus locations and any biases in responses that may occur. This article serves as an introduction to the concepts behind the models, the practicalities of fitting the models, and issues that may arise when working with 2D data, and it provides examples of fitting the models to human data. The Discussion section provides an overview of the limitations and advantages of the modeling approach raised during specific tests in the Results and Method sections, and readers may consult the Contents to determine which sections are useful to them when working with these models. The new MemToolbox2D is available at https://doi.org/10.5281/zenodo.3752705 (Grogan et al., 2019).\nMethod\nModel\nThe 1D mixture models split errors into three components (Bays et al., 2009) (Figure 1). Throughout the article, we shall focus on this three-component model (the misbinding model), although the approach is the same for related models such as the two-component mixture model (without misbinding; Zhang & Luck, 2008), and all such models have been adapted for 2D. The misbinding model has three sources of errors: imprecision, misbinding, and random guessing:\n\\begin{eqnarray}P\\left( {\\hat \\theta {\\rm{\\;}}} \\right){\\rm{\\;}} &=& {\\rm{\\;}}\\alpha {\\phi _\\kappa }\\left( {\\hat \\theta {\\rm{\\;}} - {\\rm{\\;}}\\theta } \\right) + {\\rm{\\;}}\\beta \\frac{1}{m}\\mathop \\sum \\limits_i^m {\\phi _\\kappa }\\left( {\\hat \\theta {\\rm{\\;}} - {\\rm{\\;}}{\\varphi _i}} \\right) \\nonumber\\\\ &&+ {\\rm{\\;}}\\gamma \\frac{1}{{2\\pi }},\\end{eqnarray}\n(1)\nwhere $$P( {\\hat \\theta \\;} )$$ is the probability of finding a response orientation $$\\hat \\theta$$, θ is the orientation of the target stimulus, φκ is the von Mises distribution (circular analogue of the Gaussian distribution), ϕi is the orientation of the nontarget stimulus i, m is the number of nontarget stimuli, and guessing is uniform over the entire circle (2π). The parameters α, β, γ, and κ control the proportion of target responding, nontarget responding, guessing, and the concentration of the von Mises distribution, respectively. The spread of the distributions of target and nontarget responses is assumed to be the same. As α, β, and γ must sum to 1, α is not included as a free parameter in the fitting. The three free parameters (β, γ, κ) are estimated using maximum likelihood methods.\nFigure 1.\n\nOne-dimensional and 2D misbinding models. The top row shows a 1D task. Fictive stimuli and responses are shown (left) where the target (blue) and nontarget (red) were always shown at the same orientations. The responses column shows sample responses centered on the target with some imprecision (blue), around the nontarget with the same imprecision (red), and randomly distributed guesses (green). The PDF of responses (right) shows peaks around those locations with heights proportional to the number of target and nontarget responses, widths proportional to the imprecision of the memories, and a background guessing rate proportional to the height of the horizontal line. The bottom row shows a similar setup for a 2D task where the target and nontarget were always shown in the same location on a 2D screen (left), and the PDF is now 2D (right), which shows peaks of responses (upper part) around those locations (bottom part) with a 2D width (dashed circle) and uniform guessing across the entire screen (green dots).\nFigure 1.\n\nOne-dimensional and 2D misbinding models. The top row shows a 1D task. Fictive stimuli and responses are shown (left) where the target (blue) and nontarget (red) were always shown at the same orientations. The responses column shows sample responses centered on the target with some imprecision (blue), around the nontarget with the same imprecision (red), and randomly distributed guesses (green). The PDF of responses (right) shows peaks around those locations with heights proportional to the number of target and nontarget responses, widths proportional to the imprecision of the memories, and a background guessing rate proportional to the height of the horizontal line. The bottom row shows a similar setup for a 2D task where the target and nontarget were always shown in the same location on a 2D screen (left), and the PDF is now 2D (right), which shows peaks of responses (upper part) around those locations (bottom part) with a 2D width (dashed circle) and uniform guessing across the entire screen (green dots).", null, "To adapt the model for 2D data, several changes are needed. First, 2D coordinates are used in place of angles, which means that the von Mises distribution is replaced with a bivariate Gaussian distribution ψσ with standard deviation σ and zero covariance. Second, a distribution for the random guesses must be chosen. For simplicity, we begin by assuming that they are drawn from a uniform distribution over the entire area of the screen (A). This gives the following response density function:\n\\begin{eqnarray}P\\;\\left( {{\\bf{\\hat \\theta }}{\\rm{\\;}}} \\right) &=& {\\rm{\\;\\alpha \\;}}{{\\rm{\\psi }}_{\\rm{\\sigma }}}\\left( {{\\bf{\\hat \\theta }}\\; - {\\bf{\\theta }}} \\right) + {\\rm{\\;\\beta }}\\frac{1}{m}\\mathop \\sum \\limits_i^m {{\\rm{\\psi }}_{\\rm{\\sigma }}}\\left( {{\\bf{\\hat \\theta }}{\\rm{\\;}} - {\\rm{\\;}}{\\varphi _i}} \\right){\\rm{\\;}} \\nonumber\\\\ &&+ {\\rm{\\;\\gamma }}\\frac{1}{A},\\end{eqnarray}\n(2)\nwhere here, θ and ϕ are vectors indicating locations on the screen, and again, α, β, and γ sum to 1, which together with σ yield three free parameters.\nThe model will operate in any spatial units, as long as A is changed to reflect this. For the examples provided here, we will use 1366768 as the screen area and pixels as units. The dimensions of the screen will dictate the scale of the errors in the task (i.e., on a 40 cm × 30 cm screen, the errors will be under 50 cm) and thus the scale of the imprecision parameter σ. The models take in a dimensions argument, which sets the screen dimensions for simulations and fitting. In other words, it implicitly sets the units for the model. Any appropriate scaling values are possible (e.g., 960540 pixels, 4626 cm, 3017 visual degrees). This allows you to convert data collected on different screen sizes into standard units (e.g., visual degrees or percentage of screen size), for example, to correct for any influence that different screen sizes may have on the data. A rectangular screen is assumed, so if a different shape screen is used (e.g., a circular window on a screen), the model will need to be modified to account for this. Different dimensions can be supplied for simulating and fitting the data (e.g., if the stimuli may not appear too close to the edges of the screen but possible response locations are not so constrained). This can also be achieved by using the method described in the “Effect of stimulus separation on recovery” section below.\nPrevious 1D models used a circular space, meaning that responses wrapped around the domain, without any edges. In contrast, the 2D spatial tasks involve a finite space defined by the screen or screen region. When simulating the model, any responses that fall outside of the screen when drawn from the normal distributions are replaced by new samples drawn from the same distribution.\nWe use the framework for 1D models provided in MemToolbox (Suchow et al., 2013; MemToolbox.org) to provide a new toolbox (MemToolbox2D; Grogan et al., 2019; https://doi.org/10.5281/zenodo.3752705) for modeling 2D data that integrates with the existing 1D MemToolbox. New functions were created to run the fitting, display plots, correct for biases, and simulate the models. Thus, MemToolbox2D will not interfere with the functions of MemToolbox.\nSimulations\nFirst, we provide simulations to show the behavior of the model during fitting and some of the issues that may impact the quality of the fits, including true parameter values, number of trials, number of nontargets, response distributions and biases, stimulus constraints, and screen edges.\nFor all simulations, unless otherwise specified, we simulated a task in which an agent views three stimuli. The three stimuli were randomly placed on the screen, and one was randomly chosen to be the “target,” with the other two being “nontargets.” The agent then had to place the target stimulus in the location they remembered it occurring before. This is similar to a spatial WM task used previously (Pertzov et al., 2013). We simulated 100 trials. The type of response made on each trial was determined by the parameters used for the simulations: γ controlled the proportion of responses that were random guesses, β the proportion that were misbinds (centered on a nontarget location), α (α = 1 – γ – β) the proportion that were target responses, and σ the imprecision (standard deviation) of target and misbinding responses. We then fit the misbinding model to these simulated data to see how accurately the true parameters could be recovered.\nA large sweep of parameters was used for these simulations, ranging from almost no to almost all guessing (γ ϵ [0.01, .98]), almost no misbinding to almost all misbinding (β ϵ [0.01, 0.98]), and low to high imprecision (σ ϵ [0.1, 100]). Eleven values of each parameter were chosen, evenly spaced across these ranges, and all combinations used, with the caveat that γ and β summed to 1 or less, and we simulated each parameter combination 100 times. This parameter sweep was chosen to test how well the model can recover true parameters across a large range, including values that represent poor performance due to high guessing, misbinding, and imprecision, either in isolation or combination. These values are unlikely to occur in real data as few people will guess or misbind on all trials.\nFor each simulation at a particular set of parameters, the 100 simulated trials were fit by maximum likelihood estimation. The parameter fits account for the position of each of the items, as well as the response, on each trial.\nTo show the accuracy of parameter recovery, we plot the difference between recovered and true parameters (parameter recovery error) against the values of the true parameters; zero indicate perfect parameter recovery, and larger (positive or negative) values indicate poorer recovery. These recovery errors are plotted for each parameter, marginalized over the other parameter values.\nWhen comparing different models or methods, we use the Bayesian information criterion (BIC; Schwarz, 1978), which adds a penalty for the number of trials and parameters to the negative log-likelihood of the model fit; smaller values indicate better-fitting models.\nCode for all the simulations here is available from https://doi.org/10.5281/zenodo.3752763 (Grogan, 2019).\nHuman data\nTo illustrate how the models work with real human data and to examine how they cope with more complex task designs, we fit previously unpublished data collected from an Ignore/Update spatial WM task (i.e., a 2D task) and compared the pattern of recovered parameters to those found on a 1D orientation analogue of this task (Fallon, Mattiesing, Dolfen, Manohar, & Husain, 2018; Fallon, Mattiesing, Muhammed, Manohar, & Husain, 2017). New data were collected from 49 healthy older adults (mean age = 71.4 years, SD = 5.5, 18 females, 31 males). Ethical approval was granted by South West–Central Bristol NHS REC (16/SW/0205). Participants gave written informed consent, and the study was conducted in line with the Declaration of Helsinki.\nIn brief, the task involved remembering a pair of abstract shapes (Figure 2) that were shown with a “T” cue in the screen center, indicating they were “test shapes”, one of which would be tested later in the trial. These could be shown at Time 1 and/or Time 2. On some trials, nontest shapes (no “T” shown in center) could be shown either at Time 2, which would not be tested, and were not to be remembered. Participants were instructed to remember the most recent test shapes they saw. They were tested with a test shape and a nontest shape (either distractor or novel shape) presented on the screen. Participants were asked first to select the shape they recalled to be in the (most recent) test shape pair by touching it, rather than the nontest shape that was presented with it. Then they were requested to drag the test shape to where they remembered it to have been shown originally.\nFigure 2.\n\nProcedure of the spatial WM task. Participants were presented a pair of test shapes (with a “T” at fixation), which could be either preceded or followed by a blank screen (Maintain T2 and T1, respectively) or by a second pair of nontest shapes (Update or Ignore, respectively). They were asked to remember the most recent pair of test shapes. At the test, one test shape and one nontest shape (novel item or distractor) were shown, and participants chose the item they remembered to have been a test shape and dragged it to where it appeared previously.\nFigure 2.\n\nProcedure of the spatial WM task. Participants were presented a pair of test shapes (with a “T” at fixation), which could be either preceded or followed by a blank screen (Maintain T2 and T1, respectively) or by a second pair of nontest shapes (Update or Ignore, respectively). They were asked to remember the most recent pair of test shapes. At the test, one test shape and one nontest shape (novel item or distractor) were shown, and participants chose the item they remembered to have been a test shape and dragged it to where it appeared previously.", null, "Overall, there were four different conditions, each occurring with the same frequency (Figure 2):\n• Ignore trials: Consisting of test shapes at Time 1 followed by nontest shapes at Time 2\n• Maintain long delay trials (T1): Test shapes at Time 1 followed by a long delay\n• Update trials: Consisting of test shapes at Time 1 followed by new test shapes at Time 2 (so now working memory of test shapes at Time 1 had to be updated to instead store the test shapes at Time 2)\n• Maintain short delay trials (T2): Test shapes at Time 2 followed by a short delay.\nThere were two key factors in this design. First, there were irrelevant stimuli—either ones that had to be ignored (Ignore condition) or old items stored in memory that now had to be displaced by new ones (Update condition). Then there were two conditions that controlled for different maintenance durations in the Ignore and Update conditions, by simply having either a long (T1) or short (T2) delay.\nNote that the ignore and update trials have three nonprobed items (one nonprobed test shape plus two distractors), while T1 and T2 trials only have one nonprobed test shape; the model accounts for this by averaging the likelihoods of misbinding to each nonprobed shape together (“m” in Equation 2). Other models are available that allow for two sets of nonprobed shapes, for example, with separate misbinding parameters for the distractors and the nonprobed test shape.\nThe task was performed on Dell laptops using a mouse, at a viewing distance of approximately 40 cm. After 11 practice trials, there were three blocks of 40 trials, giving 120 trials in total (30 per condition, random order).\nResults\nHere we present several simulations showing how 2D models compare to 1D models and investigating factors influencing the accuracy of parameter recovery.\nAccuracy of recovering simulated parameters\nTo see what is gained from 2D responses over 1D responses, we compared how well the fitted parameters relate to the simulation parameters used to generate the simulated data. We simulated and fit 1D and 2D versions of the misbinding model. The 1D models fit the κ parameter for the von Mises distribution concentration, which is converted into standard deviation (σ) for comparison with the standard deviation parameter in the 2D models.\nFirst, we looked for recovery error and biases, using 100 iterations across the parameter sweep. Both models recovered parameters that correlated highly with the true parameters (Table 1). Figure 3 shows the difference between simulated (true) parameters and fitted (recovered) parameters across the parameter space; the 2D model has smaller deviations from zero, meaning better parameter recovery (paired-sample t tests for each parameter, p < 0.0001).\nTable 1.\n\nSpearman correlation coefficients between each true parameter and the recovered parameters for the 1D and 2D misbinding models. *p < 10−10.\nTable 1.\n\nSpearman correlation coefficients between each true parameter and the recovered parameters for the 1D and 2D misbinding models. p < 10−10.", null, "Figure 3.\n\nMean difference between recovered and simulated parameters (parameter recovery error). This is shown for 1D (top row) and 2D (bottom row) misbinding models. Dashed line shows the y = 0 line, which is where there is no difference between the simulation and recovered parameters for any simulation parameter value. Shaded area shows standard deviation. Each column shows one parameter value, marginalized across all other parameters from the parameter sweep. The 2D model performs better than the 1D model (t tests: p < 0.0001; see Supplementary Figures S4S6 for 1D models).\nFigure 3.\n\nMean difference between recovered and simulated parameters (parameter recovery error). This is shown for 1D (top row) and 2D (bottom row) misbinding models. Dashed line shows the y = 0 line, which is where there is no difference between the simulation and recovered parameters for any simulation parameter value. Shaded area shows standard deviation. Each column shows one parameter value, marginalized across all other parameters from the parameter sweep. The 2D model performs better than the 1D model (t tests: p < 0.0001; see Supplementary Figures S4S6 for 1D models).", null, "We next performed a parameter sweep over β, γ, and σ, simulating each parameter set 1,000 times, and plotted the simulated and recovered parameters at various slices through this parameter space to see the effect of increasing one source of error while holding the others equal. The median recovered parameters are close to the simulated values (Supplementary Figures S1S3 for 2D simulations and S4S6 for 1D simulations), although there is greater deviation from the true parameters when the simulated participant generates more inaccurate responses due to higher imprecision, guessing, and misbinding.\nThe recovered parameters show trade-offs within the 2D model, as there are within the 1D model, because responses remote from any stimuli can be explained either as random guesses or by a participant having very imprecise spatial memory. Thus, the recovered imprecision and guessing parameters negatively correlate (Figure 4) as do guessing and misbinding, but misbinding and imprecision are positively correlated. These correlations between recovered parameters are weaker in the 2D model (γ and β: r = –.3817, σ and γ: r = –.1548, β and σ: r = .1208; all p < 0.000001) than in the 1D model (γ and β: r = –.6799, γ and σ: r = –.6563, β and σ: r = .3314; all p < 0.000001). In the 1D model, imprecise responses are very likely to run into other nontargets due to the circular 1D space, whereas in the 2D model, this is less common because the locations differ in two dimensions, and the boundaries absorb imprecise responses without them wrapping around to the other side. Please note these trade-offs are within the recovered parameters, due to the dependence of them in the model, and do not mean there are trade-offs between the “true” imprecision, misbinding, and guessing that occur within people's working memory.\nFigure 4.\n\nHeatmaps showing the recovered parameter trade-offs within the misbinding model in 1D (top) and 2D (bottom) space. The plots show the recovered parameter values at 10,000 MCMC posterior samples from fitting the model to simulated data (β = .3, γ = .3, σ = 30). Guessing correlates negatively with misbinding and imprecision, and misbinding and imprecision are positively correlated in both 1D and 2D, but in all cases, the correlations are weaker in the 2D model, shown here by more circular, less elliptical, patterns.\nFigure 4.\n\nHeatmaps showing the recovered parameter trade-offs within the misbinding model in 1D (top) and 2D (bottom) space. The plots show the recovered parameter values at 10,000 MCMC posterior samples from fitting the model to simulated data (β = .3, γ = .3, σ = 30). Guessing correlates negatively with misbinding and imprecision, and misbinding and imprecision are positively correlated in both 1D and 2D, but in all cases, the correlations are weaker in the 2D model, shown here by more circular, less elliptical, patterns.", null, "Factors affecting recovery accuracy\nParameter recovery can be affected by several aspects of the task, and many of these are under the experimenter's control. Here we show how factors such as number of trials, number of nontargets, and constraints on stimulus placement affect recovery accuracy from simulations. Experimenters can thus design tasks taking these factors into account when considering their model fitting and may find it useful to simulate their tasks before running them to test their recovery accuracy.\nBetter estimates with more trials\nThe more trials available for fitting, the more accurate the parameter recovery (Figure 5). This was seen for both 1D and 2D misbinding models (linear regression: p < 10−10). As it is not always possible to have large numbers of trials, recovering simulated parameters can be useful to experimenters for assessing the accuracy of model fitting in their study.\nFigure 5.\n\nThe absolute recovery error for different numbers of trials for 1D (top) and 2D (bottom) misbinding models. More trials (warmer colors—see color bar) leads to more accurate parameters (lower absolute recovery errors) in 1D and 2D models (linear regression: p < 10−10). Shading shows standard error of the mean.\nFigure 5.\n\nThe absolute recovery error for different numbers of trials for 1D (top) and 2D (bottom) misbinding models. More trials (warmer colors—see color bar) leads to more accurate parameters (lower absolute recovery errors) in 1D and 2D models (linear regression: p < 10−10). Shading shows standard error of the mean.", null, "More nontargets worsen parameter recovery\nIncluding nontarget stimuli is useful to assess misbinding between stimuli (Bays et al., 2009). Increasing the number of nontargets (from one to six, same parameter sweep, 100 iterations) increased the error in parameter estimation (Figure 6; linear regression: p < 10−10). These differences are larger in the 1D model because with more nontargets, imprecise target responses or random guesses are more likely to end up close to a nontarget and be classed as misbinds, which acts to decrease imprecision estimates especially. In the 2D model, this is less likely as the items are separated along two dimensions and the responses do not wrap around the edges, instead being bounded by the screen edges.\nFigure 6.\n\nAbsolute parameter recovery errors for different numbers of nontargets. More nontargets (cooler colors) leads to higher absolute recovery errors in 1D and 2D models (linear regression: p < 10−10). Shading shows standard error.\nFigure 6.\n\nAbsolute parameter recovery errors for different numbers of nontargets. More nontargets (cooler colors) leads to higher absolute recovery errors in 1D and 2D models (linear regression: p < 10−10). Shading shows standard error.", null, "Effect of stimulus separation on recovery\nAll of the 2D simulations presented here have used entirely random (uniform) generation of stimulus locations to match the 1D simulations and many 1D orientation tasks. However, many spatial WM tasks constrain the stimuli so that they cannot appear within a certain distance of other stimuli, the edges of the screen, or the center of the screen (if probe items are presented there; e.g., Pertzov et al., 2012).\nTo examine how such constraints affect parameter recovery, we simulated two different tasks, one where stimuli appeared uniformly randomly across the entire screen (no constraints) and one where stimuli were constrained to not appear within 88 pixels of each other or the screen center, or 29 pixels from the screen edge. These distances correspond to 3 and 1 visual degrees, respectively, assuming a viewing distance of 40 cm and 42 pixels per centimeter. The entire parameter sweep was simulated 100 times for each task, which were fit with the misbinding model. The errors in the recovered parameters from the two tasks are shown in Figure 7. Paired-sample t tests revealed that parameters α, β, and γ had significantly smaller recovery errors when stimulus constraints were used (p < 10−10), while σ only showed a trend effect (p = 0.0615), but absolute recovery errors were significantly smaller for σ and γ (p = 0.0071, p = 9.6457 × 10−8) but not α and β (p = 0.4405, p = 0.0610). These statistics suggest that constraining stimuli can improve the recovery accuracy, although the benefits are not huge. This is likely due to the constraints reducing overlap of items (and thus responses) and making distributions centered on items less likely to be cut off by the screen edges.\nFigure 7.\n\nParameter recovery errors (top) and absolute recovery errors (bottom) for the 2D model in simulated tasks with and without stimulus constraints. Adding in constraints to the stimulus locations leads to smaller errors in parameter recovery (orange line; see text for statistics). Shading shows standard error.\nFigure 7.\n\nParameter recovery errors (top) and absolute recovery errors (bottom) for the 2D model in simulated tasks with and without stimulus constraints. Adding in constraints to the stimulus locations leads to smaller errors in parameter recovery (orange line; see text for statistics). Shading shows standard error.", null, "Response biases and distributions\nWhen people perform these types of tasks, their responses may not be uniformly distributed across the screen for several reasons. First, often the stimuli themselves may not be uniformly distributed, for example, if there are areas they cannot appear in (see stimulus separation section above). Task constraints can give rise to biased distributions either through veridical memory or via strategies for guesses. Individuals may also sometimes show further biases to prefer one side of the screen, the top or the bottom half, or a radial bias where responses shrink toward the center of the screen. Many of these will depend on the experiment and stimulus characteristics, so here we shall demonstrate how two kinds of biases (constant and proportional biases) affect the model and how experimenters may account for them.\nConstant biases\nFirst, we simulated responses that have a constant directional bias by subtracting a fixed translation (from 20 to 200 pixels, step size = 36) to X- and Y-coordinates of all responses (and reapplying the screen boundary via edge constraining). The error of the fitted parameters rises as this bias increases (Figure 8). MemToolbox contains a function to correct for biases in 1D models, which we have adapted to work in 2D models. This fits a constant translation bias term for X- and Y-dimensions, thus improving fits as shown by smaller deviations from the true parameters (Figure 8, paired t tests: p < 10−8). However, when the bias is sufficiently large, responses are more likely to hit the edge of the screen (and, in this case, are edge-constrained to lie on the screen edge instead), which hinders accurate recovery.\nFigure 8.\n\nParameter recovery errors from simulations with a constant translation bias subtracted (top) and corrected for (bottom). The color bar shows bias size (from 20 to 200 pixels). The bias correction works well for biases up to about 150 pixels in size. Shading shows standard error.\nFigure 8.\n\nParameter recovery errors from simulations with a constant translation bias subtracted (top) and corrected for (bottom). The color bar shows bias size (from 20 to 200 pixels). The bias correction works well for biases up to about 150 pixels in size. Shading shows standard error.", null, "Proportional biases\nPeople may be unlikely to shift all responses in the same direction by the same amount, especially if this leads to many responses landing on the screen edges, so we also simulated a proportional bias toward the right-hand edge. Here, a bias parameter controls the proportion of the distance to the edge that responses are shifted; positive values shift it toward the edge, and negative values shift away (zero means no shifting). This is perhaps more likely as a response bias in people and can also be corrected by the toolbox (Figure 9; paired t tests: p < 10−15).\nFigure 9.\n\nDifference plot of recovered and simulated parameter values for different proportional biases toward the right-hand screen edge. Data with (bottom) and without (top) the bias correction for different proportional biases (color bar). Positive biases mean responses move toward the right-hand screen edge, and negative biases move responses away from this edge (responses do not move vertically). The bias correction improves the fits for all parameters. Shading shows standard error.\nFigure 9.\n\nDifference plot of recovered and simulated parameter values for different proportional biases toward the right-hand screen edge. Data with (bottom) and without (top) the bias correction for different proportional biases (color bar). Positive biases mean responses move toward the right-hand screen edge, and negative biases move responses away from this edge (responses do not move vertically). The bias correction improves the fits for all parameters. Shading shows standard error.", null, "Next, we applied a proportional radial bias that either shrinks all responses toward the screen center (positive values) or expands them out (negative values) by a proportion of the distance the response is from the screen. Biases mainly affect the imprecision parameter (Figure 10), although the guess parameter is affected by an expanding radial bias. We adapted the bias function to account for radial biases, which improves parameter recovery (paired t tests: p < 10−12), especially for the imprecision parameter, although does not fully correct large guess parameters when there is a large expanding bias.\nFigure 10.\n\nDifference plot of recovered and simulated parameter values for different radial biases. Data with (bottom) and without (top) the bias correction for different radial biases (color bar). Positive radial biases mean points move toward the center of the screen, and negative biases move points away from the center. The bias correction improves the fits for all parameters, although large guess rate parameters are not fully corrected when there is a large expanding bias. Shading shows standard error.\nFigure 10.\n\nDifference plot of recovered and simulated parameter values for different radial biases. Data with (bottom) and without (top) the bias correction for different radial biases (color bar). Positive radial biases mean points move toward the center of the screen, and negative biases move points away from the center. The bias correction improves the fits for all parameters, although large guess rate parameters are not fully corrected when there is a large expanding bias. Shading shows standard error.", null, "This function also allows us to test whether there are radial biases toward points other than the center. For instance, in trials with a single target and nontarget, it is possible that rather than the nontarget “swapping” locations with the target, it exerts a “pull” on the response locations. Note that this contribution to error is separable from misbinding, as the responses will be centered somewhere between targets and nontargets, rather than being centered on the nontarget in the case of misbinding. We can test for this by using the radial bias function and supplying the nontarget location as the bias coordinate for each trial; if we simulate such a bias, this function corrects for the bias and gives much more accurate parameter recovery (Figure 11; paired t tests: p < 10−15). For this simulation, we used one nontarget and the standard mixture model (i.e., no misbinding). We also fit the misbinding model to these nontarget-biased data, which did not fit the data as well as the standard mixture model with the radial nontarget bias (mean BIC = 2,720 vs. 2,437), suggesting this pattern of nontarget “pull” is distinguishable from misbinding in simulated data.\nFigure 11.\n\nParameter recovery errors with a radial bias toward the nontarget location on each trial. Data shown without (top) and with (bottom) bias correction. The bias correction reduced the errors in parameter recovery. This simulation had β = 0 (i.e., no misbinding). Shading shows standard error.\nFigure 11.\n\nParameter recovery errors with a radial bias toward the nontarget location on each trial. Data shown without (top) and with (bottom) bias correction. The bias correction reduced the errors in parameter recovery. This simulation had β = 0 (i.e., no misbinding). Shading shows standard error.", null, "Response sampling for guess distributions\nPeople may not (and indeed probably do not) have a uniform guess rate in these tasks but instead may have complex distributions that include some of the biases mentioned above but may have others we have not thought of. One way of dealing with this is to use the participants’ own responses to determine the “guessing” distribution. To achieve this, a probability density function (PDF) can be built from all of a participant's response locations over an experiment (across all conditions for a participant's data). Drawing from this distribution samples a response taken from a random trial, which, if the trials are independent, should be unrelated to the current trial's stimuli and thus can constitute a random guess constrained by the person's own response biases.\nWe implemented this in a model that uses a kernel smoothing density function to build a PDF of the person's responses over all trials, which is then used to estimate the probability of each trial's response over the entire experiment. MATLAB's ksdensity function was used to build the response PDF, which estimates the default bandwidth of the smoothing function from the data using Scott's rule. The default bandwidth can be optionally overridden to supply your own bandwidth in the response sampling function.\nThis model cannot easily be tested with simulations as people have nonrandom distributions of responses that we cannot predict, so we tested this model using cross-validation from the healthy participants’ data fitting (see later section or the Method section for details). We fit 90% of all the trials (combining all conditions) with the misbinding model with and without response sampling and then generated the likelihood of the responses for the remaining 10% of trials using the fitted parameters. We repeated this 100 times for each person and took the mean likelihood for each person. Crucially, the response sampling only included the 90% of training trials, not the 10% of validation trials. If the response sampling model accounts for people’s response distributions and therefore gives better fits, it should lead to larger likelihoods for the out-of-sample validation trials than the uniform guessing model, whereas if it is instead simply overfitting the participants’ responses, then out-of-sample probabilities will be lower than the uniform guessing model. The response sampling model gave higher likelihoods than the uniform guessing model for the validation trials (t test: p = 4.8265 × 10−10), suggesting it is not simply overfitting the data but that drawing guesses from the distribution of all responses can give more accurate estimates of guess rates.\nChange detection\nThe models are also able to fit data from change detection tasks similar to a two-alternative forced-choice (2AFC) paradigm. Change detection tasks probe the ability to detect changes to stimuli's features, such as location. A stimulus is probed, shown displaced from its original location by some distance, and participants must decide if the location is the same or different. The paradigm is different from many 2AFC tasks used for signal detection, because often all probes have some displacement, however small, and the task is used not to measure people's ability to separate changed from unchanged items but to measure the minimum detectable changes in feature memory. These tasks give much less information on each trial about the source of error, but with sufficient data, corresponding models can still be fit. These models are adapted from the 2AFC models in the original 1D MemToolbox (Suchow et al., 2013).\nThe model is based on assumptions about the spatial pattern of erroneous acceptances of a displaced probe caused by the different underlying error sources. A person with perfect memory would reject all displaced probes, but as the error sources increase, they give different patterns of acceptances. Imprecision will increase acceptance closer to the target, misbinding will increase acceptance close to the nontargets, and guessing will increase acceptance uniformly. We also assume that if someone incorrectly accepts a probe of a certain displacement as unchanged, they would also have accepted smaller displacements on that particular trial. Taken together, these assumptions lead to us integrating over the continuous probability density function to estimate the probability of correctly rejecting a displaced probe, which we calculate as the probability of generating a response that is closer to the target than the current probe is. Note that this approach assumes a nonzero displacement of each item; otherwise, this integration will be affected mainly by the area on the far side of the target to the probe.\nFigure 12 shows, when a range of different probe (and nontarget) distances is considered, different patterns of acceptance are seen: Increasing imprecision increases the acceptance of probes closer to the target, increasing guessing increases acceptance similarly across the entire circle, and increasing misbinding increases acceptance closer to nontargets.\nFigure 12.\n\nIllustrative examples of mixture modeling approach to change detection. We show 1D examples for ease, although the same principles apply for 2D (see Supplementary Figure S10). The top row shows how each error component affects the proportion of generative responses that would fall at or beyond a probed distance of 60 degrees (arbitrary y-units). The second row shows the values of the parameters used for those simulations, along with the summed probabilities of all responses that occur at ≥ 60 degrees from the target. Increasing each parameter increases the probability of incorrectly accepting this probe as unchanged. The bottom row shows the probability of accepting a probe as unchanged as a function of the probe distance (or probe-nontarget distance for misbinding). This shows that increasing imprecision increases the spread of these same responses around the target, increasing guessing affects all orientations similarly, and increasing misbinding affects responses in relation to their distance to the distractor (if we plotted probe-targ for misbinding, it would look like the guessing panel).\nFigure 12.\n\nIllustrative examples of mixture modeling approach to change detection. We show 1D examples for ease, although the same principles apply for 2D (see Supplementary Figure S10). The top row shows how each error component affects the proportion of generative responses that would fall at or beyond a probed distance of 60 degrees (arbitrary y-units). The second row shows the values of the parameters used for those simulations, along with the summed probabilities of all responses that occur at ≥ 60 degrees from the target. Increasing each parameter increases the probability of incorrectly accepting this probe as unchanged. The bottom row shows the probability of accepting a probe as unchanged as a function of the probe distance (or probe-nontarget distance for misbinding). This shows that increasing imprecision increases the spread of these same responses around the target, increasing guessing affects all orientations similarly, and increasing misbinding affects responses in relation to their distance to the distractor (if we plotted probe-targ for misbinding, it would look like the guessing panel).", null, "Radial and translation biases can be included in the models as in the continuous case. As the task is change detection rather than a signal detection task (as the probe is not the same as the target, and even if it were would likely not be perceived so due to memory decay), the aim is not to measure a person's ability to separate out signals from nonsignals but rather to estimate these underlying error sources. The estimates of imprecision, guessing, and misbinding are based on the effect of probe distance on acceptance of changes. Under this framework, a person who accepts probes closer to the target does so because of less imprecise memory, and a person who accepts more probes regardless of distance does so because of increased guessing, not because of a greater propensity to accept changes. It is possible to estimate a response bias separately by averaging the proportion of acceptances.\nWe ran simulations and fittings for 1D and 2D versions of this. The standard mixture model (without misbinding) was used because fitting 2AFC data is slow due to the more complex probability density taking into account the nontarget locations for each trial.\nThe parameter recovery works well in comparison to the 1D model (Figure 13), especially at higher imprecision values, but as expected is considerably less accurate than the continuous report fitting (Figure 3).\nFigure 13.\n\nParameter recovery errors for the 1D and 2D two-alternative forced-choice simulations. The 2D model (bottom) is more accurate than the 1D model (top), although both are far less accurate than with continuous report data (see Figure 3). Shading shows the standard error of the mean.\nFigure 13.\n\nParameter recovery errors for the 1D and 2D two-alternative forced-choice simulations. The 2D model (bottom) is more accurate than the 1D model (top), although both are far less accurate than with continuous report data (see Figure 3). Shading shows the standard error of the mean.", null, "We have developed further models, including one with a covariance parameter for the imprecision, one with separate precision parameters for targets and nontarget responses, and one with separate parameters for misbinding to two types of nontargets (e.g., the other test shape that was not probed and the nontest shapes). The latter can also be used to assess misbinding toward previous targets/responses if desired. We have also adapted all the other models present in MemToolbox (Suchow et al., 2013), including the variable precision models (Fougnie, Suchow, & Alvarez, 2012; van den Berg, Shin, Chou, George, & Ma, 2012), ensemble integration model (Brady & Alvarez, 2011), continuous resource model (Wilken & Ma, 2004), exponential decay model (Zhang & Luck, 2009), and slot model (Zhang & Luck, 2008).\nWe have presented only the maximum likelihood fits here, but MemToolbox2D also contains adapted functions for performing Bayesian model fitting using the Markov chain Monte Carlo (MCMC) method. These fits take longer to run, which is why we used maximum likelihood for the analyses presented here. This method also allows hierarchical model fitting across participants.\nWe have also adapted several other functions from MemToolbox (Suchow et al., 2013) such as the plotting functions for visualizing the data and results of the fits.\nEmpirical data fitting\nThe 2D models can be fit to simple spatial WM tasks (e.g., Pertzov et al., 2012, 2013) but can also be used for more complex tasks to compare between multiple conditions. Here we fit the models to data collected from people performing a spatial ignore/update WM task (see “Human data” section for details) to show how the models can be useful for more complex experimental designs too.\nPeople show biases and nonuniform guesses\nTo see which model fit the data best, we fit several models to the behavioral data (all conditions). First, we fit the misbinding model and the standard mixture model (no misbinding), and the misbinding model fit best (BIC: 2,984 vs. 3,051). Next we looked at whether the bias functions improved the fit by including each bias in the misbinding model. The constant bias did not improve the fit (BIC = 2,984), while the radial bias did (BIC = 2,966). Adding response sampling improved the misbinding model (BIC = 2,974), as did combining the misbinding model with the radial bias (BIC = 2,958). Therefore, the misbinding model with a radial bias and response sampling was the best-fitting model. The parameters from this fitting are shown in the Supplementary Materials and Supplementary Figure S7 and show that most participants were biased away from the screen center and that these biases were consistent within people across conditions (Pearson correlations: ρ > .65, p < 0.0001).\nConvergent validity of parameter recovery\nTo compare convergent validity between the 1D and 2D modeling approaches, we compared the fitted parameters from the 2D misbinding model to those reported on the 1D version of this task (Fallon et al., 2018). Histograms of the recovered parameters are shown in Figure 14 and show that while imprecision is approximately normally distributed, the other parameters are skewed, being bounded by 0 and 1. We did not logit transform the α, β, and γ parameters for the comparison to the 1D task as the 1D analysis did not transform parameters (Fallon et al., 2018).\nFigure 14.\n\nHistograms of the recovered parameters from the ignore/update task, across all conditions.\nFigure 14.\n\nHistograms of the recovered parameters from the ignore/update task, across all conditions.", null, "Repeated-measures analysis of variance (ANOVA) with delay and irrelevant stimuli as factors and a random effect of participant were used to analyze the fitted parameters (Figure 15). The analyses showed the following:\n• Delay duration increased the imprecision (F(1, 144) = 53.7751, p = 1.4959 × 10−11), guessing (F(1, 144) = 14.4042, p = 2.1660 × 10−4), and misbinding (F(1, 144) = 3.9560, p = 0.0486).\n• Irrelevant stimuli increased misbinding (F(1, 144) = 19.9033, p = 1.6302 × 10−5) and guessing (F(1, 144) = 19.8437, p = 1.6755 × 10−5) but did not affect imprecision (F(1, 144) = 1.3353, p = 0.2498).\n• There was a significant interaction of delay and irrelevant information for guessing (F(1, 144) = 4.0163, p = 0.0469), although it was only weakly significant and was due to greater guessing in the ignore condition.\n• There were no other interactions (p > .1), suggesting that ignore and update trials both affected imprecision and misbinding similarly.\nFigure 15.\n\nMean fitted parameter values for each trial type from the healthy participants. Delay increased imprecision (σ; top left; blue vs. orange; see Figure 2 for task explanation), guessing (γ), and misbinding (β; see text for stats); irrelevant stimuli increased misbinding and guessing but not imprecision (ignore and update vs. T1 and T2); and there was an interaction of delay and irrelevant stimuli only for guessing. Standard error bars.\nFigure 15.\n\nMean fitted parameter values for each trial type from the healthy participants. Delay increased imprecision (σ; top left; blue vs. orange; see Figure 2 for task explanation), guessing (γ), and misbinding (β; see text for stats); irrelevant stimuli increased misbinding and guessing but not imprecision (ignore and update vs. T1 and T2); and there was an interaction of delay and irrelevant stimuli only for guessing. Standard error bars.", null, "These main effects are similar to the orientation version of this task, although they differ regarding the interaction of delay and irrelevant information. The 1D version found such an interaction in misbinding (Fallon et al., 2018), while here we found it for guessing; in both tasks, the errors were highest in the ignore condition. Potential reasons for this will be discussed later. The similarity in the rest of these results suggests convergent validity of the 2D modeling technique for real human data.\nTo establish whether the different results in 1D and 2D tasks are due to differences in the modeling, we examined how well each model can distinguish the parameters for the different conditions. We simulated 1D and 2D versions of the task using the mean recovered parameters of each condition from the 1D task (Fallon et al., 2018) and fit the misbinding models using fmincon and inverted the Hessian matrices to get the covariance between parameters. We then used MATLAB's linear hypothesis test to examine whether the recovered parameters for the different conditions were actually from different distributions of parameters (alternate hypothesis) or from the same distribution (null hypothesis).\nWe performed 1,000 iterations of this and counted how often the 1D and 2D models returned p ≤ .05. If the 2D model is better at distinguishing between parameters due to the lower covariance between them, it will be more likely to return p ≤ .05 in these linear hypothesis tests. The 2D model found a significant difference between the conditions more often (mean = 29.1% vs. 19.2%). This suggests that these parameters are less confusable (i.e., more distinguishable) in the 2D model than the 1D model. The 2D model was particularly better than the 1D model for comparing the ignore condition with update and T2 conditions, which could mean that the 1D task confused parameters between these conditions, leading to a different interaction being found in the orientation task.\nTo illustrate this, we generated posterior MCMC samples from fitting to data simulated using the mean parameters from the 1D orientation task and plotted the parameters against each other (Figure 16). There was greater separation between the parameters of the different conditions in the 2D model than the 1D model, suggesting that the 2D model is better at distinguishing between parameters of different conditions, within the range recovered from human data.\nFigure 16.\n\nHeatmaps showing the posterior MCMC samples for recovering the mean parameters from Fallon et al. (2018). Each color is a different condition (see legend; Ig = ignore, T1 = long delay, Up = update, T2 = long delay), and the rows show 1D and 2D models, respectively. The γ and β parameters have been arcsine transformed. Both models were simulated using the same parameters. The 2D model has better separation of the different conditions’ parameters, as shown by less overlap between the colors.\nFigure 16.\n\nHeatmaps showing the posterior MCMC samples for recovering the mean parameters from Fallon et al. (2018). Each color is a different condition (see legend; Ig = ignore, T1 = long delay, Up = update, T2 = long delay), and the rows show 1D and 2D models, respectively. The γ and β parameters have been arcsine transformed. Both models were simulated using the same parameters. The 2D model has better separation of the different conditions’ parameters, as shown by less overlap between the colors.", null, "Validating the behavioral metrics\nWe wanted to see whether the behavioral metrics were actually measuring the types of errors purported. We calculated those behavioral metrics on simulated data and looked at which metrics correlated with the true parameters and compared this against the accuracy of parameter recovery. These simulations used the same stimulus separation constraints as in the “Stimulus separation” section to match the experiments that commonly use these metrics, although running them without such constraints did not change the pattern of results.\nWe examined several commonly used behavioral metrics. Target distance is the Euclidean distance from responses to targets, while nearest-neighbor distance is the Euclidean distance from responses to their nearest stimulus (Pertzov et al., 2013). The former will contain imprecision, guessing, and misbinding influences, while the latter has attempted to remove the influence of misbinding and thus should contain mainly imprecision and guessing. The difference between these two represents the influence of misbinding. A further measure to capture misbinding, termed swap errors (Pertzov et al., 2012), is the proportion of responses occurring within a certain fixed distance (1.5 visual degrees) of a nontarget location. This measure is also confounded by imprecision since imprecise responses to the target location are more likely to end up close to a nontarget by chance, compared to precise responses. We can correct for this by calculating the proportion of possible locations with that specific distance from the target that would have been classed as “swap errors” and subtracting that. Swap errors can also be calculated using the mean target distance as the threshold, effectively scaling the criterion by a participant's average accuracy. This measure can likewise be corrected for chance in the same way.\nWe correlated the mean values of each of these metrics for simulated data with each of the true parameters (parameter sweep from previous simulations, 10 iterations). All the metrics correlated with multiple true parameters, so we present here only the metric most strongly correlated with each parameter (see Supplementary Figure S9 and Supplementary Table S1 for full correlation matrix between metrics and parameters).\nTarget distance correlated strongly and negatively with the proportion of target responding (Figure 17; Spearman correlations: r = –.9687, p < 10−10), nearest-neighbor distance positively with guess rate (r = .9279, p < 10−10), and the difference between these positively with misbinding (r = .8589, p < 10−10). There were no metrics that correlated strongly with imprecision, although swap errors (fixed threshold) had a medium negative correlation (r = –.5519, p < 10−10) while the nearest-neighbor distance (assumed to be a good measure of imprecision) had a weak positive correlation with it (r = .2931, p < 10−10), so both of these are presented. These metrics were all less strongly correlated with the true parameters than the recovered parameters were, although this may be an artifact of recovering the parameters with the same model used for simulation, as this makes it unlikely the behavioral metrics can perform better.\nFigure 17.\n\nRelationship between the true parameters (x-axis) and the best behavioral metrics (top) and recovered parameters (bottom). All correlations were significant (p < 10−10). Two metrics were compared against the imprecision (σ) parameter. The recovered parameters had stronger correlations than the best behavioral metrics (see text for statistics).\nFigure 17.\n\nRelationship between the true parameters (x-axis) and the best behavioral metrics (top) and recovered parameters (bottom). All correlations were significant (p < 10−10). Two metrics were compared against the imprecision (σ) parameter. The recovered parameters had stronger correlations than the best behavioral metrics (see text for statistics).", null, "To investigate this idea, we compared the split-half reliability of these four behavioral metrics and the recovered parameters for these behavioral data (pooling across all four conditions, 100 iterations of random splitting). All measures showed significant split-half reliability (Spearman correlations: p < 0.0001) with varying strengths. The swap errors metric was only weakly correlated with itself (ρ = 0.2068), while the imprecision parameter was strongly correlated (ρ = 0.7160). Target distance was slightly more reliable (ρ = 0.7955) than the recovered α parameter (ρ = 0.7811), but nearest-neighbor distance (ρ = 0.8196) and target minus nearest-neighbor distance (ρ = 0.7450) were better than γ (ρ = 0.3347) and β (ρ = 0.6278). This suggests that some of the behavioral metrics have good split-half reliability, and these are the ones that are most strongly correlated with the true α, β, and γ parameters in the simulations above. However, these simulations and correlations suggest none of the behavioral metrics currently used are good estimators of memory imprecision, and the ones currently available have low reliability. They also suggest that guessing and misbinding estimation in the model has poor split-half reliability, which could be due to the low guessing and misbinding parameters used for these simulations (recovered from the behavioral data), meaning few trials contain a guess or misbind, which leads to poorer estimation in the split-halves.\nWe can also look at convergent validity by comparing the outputs of ANOVA on the behavioral metrics in the 2D human data to those previously reported on the 1D circular analogue task, as we did earlier with the 2D modeling outputs. These showed that delay increased target distance (p < 0.0001), nearest-neighbor distance (p < 0.0001), and the difference between these two (p = 0.0077), while irrelevant distractors increased swap errors (p = 0.0001), target distance (p < 0.0001), and target minus nearest-neighbor distance (p < 0.0001)—but not the nearest-neighbor distance itself (p = 0.6066). There were no interactions (p > 0.1). While target distance and target minus nearest-neighbor distance patterns are similar to α and β (in the 1D data and our 2D data), the nearest-neighbor distance effects (delay but not irrelevance) were more similar to that seen in imprecision than guessing. This may not be surprising as the correlations (Figure 17) showed nearest-neighbor distance was correlated with both imprecision and guessing. This suggests that while nearest-neighbor distance is reliable, it is not a pure measure of one type of error source.\nDiscussion\nWe have developed a 2D mixture modeling toolbox for tasks such as spatial WM tasks. The models are able to recover parameters from simulations with good accuracy across a large parameter space (Figure 3) in line with the standard 1D mixture models and have less interdependent parameters (Figure 4). The accuracy rises as the number of trials increases (Figure 5) or the number of nontargets decreases (Figure 6). Behavioral data from a complex spatial WM task showed a similar pattern of results to a 1D orientation analogue of the 2D task, giving convergent validity (Figure 15). The 2D modeling provided a better estimate of true parameters of simulations than commonly used behavioral metrics (Figure 17). This suggests that the 2D mixture modeling approach is valid for analysis of data from complex spatial WM tasks and is more sensitive than previously available behavioral metrics. The models are provided in the MemToolbox2D package (Grogan et al., 2019; https://doi.org/10.5281/zenodo.3752705), compatible with the MemToolbox package for modeling 1D tasks (Suchow et al., 2013; MemToolbox.org).\nThe fact that the 2D models outperform 1D models in the simulations can be understood when examining the correlations of the different parameters from the posterior samples (Figure 4). The 2D model's parameters are less strongly (albeit still significantly) associated with each other than the 1D model's parameters. Mixture modeling can give accurate and reliable estimation of parameters in 1D tasks, helped by the fact that responses are often tightly clustered around stimuli locations (i.e., low imprecision). However, as imprecision increases, responses become more and more likely to land near other stimuli and be classed as misbinds; due to the circular space, if you move far enough away from a target, you will hit a nontarget. In the 2D task, while increasing imprecision does increase the chance of hitting a nontarget, this is overall much lower as there are two uncorrelated dimensions of separation between stimuli, and the responses do not wrap around the screen edges, instead being contained by them. This means that you are not guaranteed to hit a nontarget as you move further from a target, unlike in the circular task—sometimes you will hit a screen edge instead. These factors mean that the 2D models are better able to distinguish imprecision, guessing, and misbinding; these differences are due to differences in the task setup rather than the models themselves.\nThe simulations presented here used uniformly random item locations and performed well, but adding in stimulus constraints such as minimum distances to other items, screen edges, and screen center improved the parameter recovery, particularly for guessing and misbinding. Such constraints mean that the probability density functions for target and misbinding responses have less overlap and are cut off less by the screen edges, leading to better estimation of the probabilities of each type of response. This analysis also demonstrates how important simulation can be in task design, allowing the experimenter to try out different stimulus constraints, number of trials, and numbers of nontargets to optimize their task for the recovery of parameters of interest.\nAlternatively, the 2D modeling was shown to be more sensitive during simulations and to have less interdependent parameters (i.e., weaker correlations between recovered parameters) than the 1D modeling, and simulations from the fitted parameters showed the 2D model was better at distinguishing between parameters from different conditions. The different pattern of responses could therefore be due to more accurate parameter recovery in 2D tasks. Finally, and most interestingly, it is possible that the different results reflect differences between the way locations and orientations are processed in WM. There is a proposal that location plays a fundamentally distinct role in WM, such that all other features of an item (e.g., orientation, color, shape) are bound to location (Schneegans & Bays, 2017). It is possible that orientations are misbound more when distracting information is to be ignored, but locations are not as they are the feature to which others are bound.\nOther spatial WM tasks have reported an interaction of delay and number of items (Pertzov et al., 2012), which was present in the overall target distance but not in the nearest-neighbor distance (i.e., when controlling for misbinding). This was taken to mean that larger memory loads for longer durations lead to more misbinding errors but did not affect the imprecision of the memory. These behavioral metrics have been used in many spatial WM tasks (Liang et al., 2016; Pertzov et al., 2013; Zokaei, Nour, et al., 2019) and the current modeling approach provides a way to examine the validity of those metrics for estimating different sources of error.\nComparing the true parameters of simulations against traditional behavioral measures for this task gave a sensible pattern of results. Mean target distance was associated with imprecision, guessing, and misbinding. Nearest-neighbor distance (which removes the influence of misbinding to give a better estimate of imprecision) had no association with misbinding but instead associated mainly with guessing and a little with imprecision. The difference between these two was a good estimate of misbinding. These relationships were all expected and assumed in previous studies relying on them (Pertzov et al., 2012, 2013).\nHowever, imprecision was not strongly associated with any behavioral metric, suggesting it independently quantifies a new aspect of performance that is not well captured by existing measures. Imprecision had medium and weak associations with the proportion of swap errors and nearest-neighbor distance, respectively. The behavioral measures were less strongly correlated with “true” parameters used for simulations than recovered parameters were, but this is not surprising as the same model was used for simulation and fitting. While the simulations here show that the models can accurately recover simulated parameters, people likely use a range of behavioral heuristics not captured by the models and functions provided here. Split-half reliability and convergent validity checks showed that while target selection and imprecision could be reliably measured by the metrics, the metrics associated with misbinding and guessing were less reliable and valid measures.\nTwo-dimensional mixture modeling has been used previously (Schneegans & Bays, 2016) on a task in which stimuli appeared within an annulus of 5° width, thus having two dimensions, in contrast to tasks where stimuli locations appear on a ring (i.e., with no width) (Rajsic & Wilson, 2014; Schneegans & Bays, 2018), meaning the 2D coordinate data can be treated as a 1D orientation and modeled with the existing 1D model toolboxes (Suchow, Brady, Fougnie, & Alvarez, 2013). To account for this spatial constraint, the authors built in priors to their model, such that the bivariate normal distributions around targets and nontargets and the uniform distribution were convolved with an estimated distribution of their response eccentricities (which closely matched the annulus location and width). This method illustrates the flexibility of mixture modeling for 2D tasks with stimuli or response constraints, as well as the usefulness of incorporating priors into the model. This method also bears similarity to the response sampling method presented here, although applied to not only the guessing distribution but also the target and nontarget distributions, which could be useful for future researchers—the toolbox provided here is adaptable, and an example model outline is provided to show how new or modified models may be created.\nThere are some limitations with adapting the models to two dimensions. Previous 1D tasks often use a circular space (e.g., orientation or color), and this unbounded space works well because of the lack of edge effects. In contrast, 2D tasks using screens for stimuli presentation have a bounded space, meaning that responses do not “wrap around” as they do in a circular space. In this case, assumptions must be made regarding behavior close to the boundary and in the generative models that may place items outside of the boundaries. The generative models implemented here are likely not the true process that occurs when people recall items near the edge of the display, but it seems a reasonable simplification that still produces a well-fitting model.\nIn reality, people may have repelling biases away from edges or anchoring or landmark effects close to them. The bias functions in the MemToolbox2D package can test for some of these, and more complicated models can be developed that include these, and other, effects. We also provide a “response sampling” method that builds a response density function over all the responses from a person and can be used to sample the guesses from. This is an alternative to assuming a uniform guessing distribution. There are myriad potential biases and guessing distributions, and the MemToolbox2D package is provided with a permissive license to allow modification of the code here, as was the original MemToolbox package (Suchow et al., 2013).\nWhile these models were designed for WM tasks, they can be used for other types of tasks, including long-term memory tasks for spatial locations (e.g., Zokaei, Čepukaitytė, et al., 2019). There is also no reason why the models cannot be extended into the third dimension, should experiments require it, for example, in virtual reality navigation tasks.\nMemToolbox2D is provided as free code available from https://doi.org/10.5281/zenodo.3752705 (Grogan et al., 2019).\nConclusion\nWe provide here a new toolbox (MemToolbox2D; Grogan et al., 2019; https://doi.org/10.5281/zenodo.3752705) for analyzing 2D spatial WM tasks. Simulations showed it is accurate at recovering parameters over a wide range of values, number of trials, and number of nontargets and can take into account response distributions and biases. It provides a more accurate measure of true parameter values than previously used behavioral metrics on such tasks and may be better at separating out codependent parameter values because of the second dimension of separation present in the data.\nAcknowledgments\nSupported by the MRC (MR/P00878X) and BRACE (BRI17/15) (JG), MRC (MR/P00878X) (SM), and Wellcome Trust Principal Research Fellowship and the NIHR Oxford Biomedical Research Centre (MH).\nCommercial relationships: none.\nCorresponding author: John P. Grogan.\nEmail: [email protected].\nAddress: Nuffield Department of Clinical Neurosciences, University of Oxford, Oxford, UK.\nReferences\nBays, P. M., Catalao, R. F. G., & Husain, M. (2009). The precision of visual working memory is set by allocation of a shared resource. Journal of Vision, 9(10), 7, https://doi.org/10.1167/9.10.7.\nBays, P. M., & Husain, M. (2008). Dynamic shifts of limited working memory resources in human vision. 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Journal of Vision, 4(12), 11, https://doi.org/10.1167/4.12.11.\nZhang, W., & Luck, S. J. (2008). Discrete fixed-resolution representations in visual working memory. Nature, 453(7192), 233–235, https://doi.org/10.1038/nature06860.\nZhang, W., & Luck, S. J. (2009). Sudden death and gradual decay in visual working memory. Psychological Science, 20(4), 423–428, https://doi.org/10.1111/j.1467-9280.2009.02322.x.\nZokaei, N., Burnett Heyes, S., Gorgoraptis, N., Budhdeo, S., & Husain, M. (2015). Working memory recall precision is a more sensitive index than span. Journal of Neuropsychology, 9, 319–329, https://doi.org/10.1111/jnp.12052.\nZokaei, N., Čepukaitytė, G., Board, A. G., Mackay, C. E., Husain, M., & Nobre, A. C. (2019). Dissociable effects of the apolipoprotein-E (APOE) gene on short- and long-term memories. Neurobiology of Aging, 73, 115–122, https://doi.org/10.1016/j.neurobiolaging.2018.09.017.\nZokaei, N., Mcneill, A., Proukakis, C., Beavan, M., Jarman, P., Korlipara, P., & Husain, M. (2014). Visual short-term memory deficits associated with GBA mutation and Parkinson's disease. Brain, 137(8), 2303–2311, https://doi.org/10.1093/brain/awu143.\nZokaei, N., Nour, M. M., Sillence, A., Drew, D., Adcock, J., Stacey, R., & Husain, M. (2019). Binding deficits in visual short-term memory in patients with temporal lobe lobectomy. Hippocampus, 29(2), 63–67, https://doi.org/10.1002/hipo.22998.\nFigure 1.\n\nOne-dimensional and 2D misbinding models. The top row shows a 1D task. Fictive stimuli and responses are shown (left) where the target (blue) and nontarget (red) were always shown at the same orientations. The responses column shows sample responses centered on the target with some imprecision (blue), around the nontarget with the same imprecision (red), and randomly distributed guesses (green). The PDF of responses (right) shows peaks around those locations with heights proportional to the number of target and nontarget responses, widths proportional to the imprecision of the memories, and a background guessing rate proportional to the height of the horizontal line. The bottom row shows a similar setup for a 2D task where the target and nontarget were always shown in the same location on a 2D screen (left), and the PDF is now 2D (right), which shows peaks of responses (upper part) around those locations (bottom part) with a 2D width (dashed circle) and uniform guessing across the entire screen (green dots).\nFigure 1.\n\nOne-dimensional and 2D misbinding models. The top row shows a 1D task. Fictive stimuli and responses are shown (left) where the target (blue) and nontarget (red) were always shown at the same orientations. The responses column shows sample responses centered on the target with some imprecision (blue), around the nontarget with the same imprecision (red), and randomly distributed guesses (green). The PDF of responses (right) shows peaks around those locations with heights proportional to the number of target and nontarget responses, widths proportional to the imprecision of the memories, and a background guessing rate proportional to the height of the horizontal line. The bottom row shows a similar setup for a 2D task where the target and nontarget were always shown in the same location on a 2D screen (left), and the PDF is now 2D (right), which shows peaks of responses (upper part) around those locations (bottom part) with a 2D width (dashed circle) and uniform guessing across the entire screen (green dots).", null, "Figure 2.\n\nProcedure of the spatial WM task. Participants were presented a pair of test shapes (with a “T” at fixation), which could be either preceded or followed by a blank screen (Maintain T2 and T1, respectively) or by a second pair of nontest shapes (Update or Ignore, respectively). They were asked to remember the most recent pair of test shapes. At the test, one test shape and one nontest shape (novel item or distractor) were shown, and participants chose the item they remembered to have been a test shape and dragged it to where it appeared previously.\nFigure 2.\n\nProcedure of the spatial WM task. Participants were presented a pair of test shapes (with a “T” at fixation), which could be either preceded or followed by a blank screen (Maintain T2 and T1, respectively) or by a second pair of nontest shapes (Update or Ignore, respectively). They were asked to remember the most recent pair of test shapes. At the test, one test shape and one nontest shape (novel item or distractor) were shown, and participants chose the item they remembered to have been a test shape and dragged it to where it appeared previously.", null, "Figure 3.\n\nMean difference between recovered and simulated parameters (parameter recovery error). This is shown for 1D (top row) and 2D (bottom row) misbinding models. Dashed line shows the y = 0 line, which is where there is no difference between the simulation and recovered parameters for any simulation parameter value. Shaded area shows standard deviation. Each column shows one parameter value, marginalized across all other parameters from the parameter sweep. The 2D model performs better than the 1D model (t tests: p < 0.0001; see Supplementary Figures S4S6 for 1D models).\nFigure 3.\n\nMean difference between recovered and simulated parameters (parameter recovery error). This is shown for 1D (top row) and 2D (bottom row) misbinding models. Dashed line shows the y = 0 line, which is where there is no difference between the simulation and recovered parameters for any simulation parameter value. Shaded area shows standard deviation. Each column shows one parameter value, marginalized across all other parameters from the parameter sweep. The 2D model performs better than the 1D model (t tests: p < 0.0001; see Supplementary Figures S4S6 for 1D models).", null, "Figure 4.\n\nHeatmaps showing the recovered parameter trade-offs within the misbinding model in 1D (top) and 2D (bottom) space. The plots show the recovered parameter values at 10,000 MCMC posterior samples from fitting the model to simulated data (β = .3, γ = .3, σ = 30). Guessing correlates negatively with misbinding and imprecision, and misbinding and imprecision are positively correlated in both 1D and 2D, but in all cases, the correlations are weaker in the 2D model, shown here by more circular, less elliptical, patterns.\nFigure 4.\n\nHeatmaps showing the recovered parameter trade-offs within the misbinding model in 1D (top) and 2D (bottom) space. The plots show the recovered parameter values at 10,000 MCMC posterior samples from fitting the model to simulated data (β = .3, γ = .3, σ = 30). Guessing correlates negatively with misbinding and imprecision, and misbinding and imprecision are positively correlated in both 1D and 2D, but in all cases, the correlations are weaker in the 2D model, shown here by more circular, less elliptical, patterns.", null, "Figure 5.\n\nThe absolute recovery error for different numbers of trials for 1D (top) and 2D (bottom) misbinding models. More trials (warmer colors—see color bar) leads to more accurate parameters (lower absolute recovery errors) in 1D and 2D models (linear regression: p < 10−10). Shading shows standard error of the mean.\nFigure 5.\n\nThe absolute recovery error for different numbers of trials for 1D (top) and 2D (bottom) misbinding models. More trials (warmer colors—see color bar) leads to more accurate parameters (lower absolute recovery errors) in 1D and 2D models (linear regression: p < 10−10). Shading shows standard error of the mean.", null, "Figure 6.\n\nAbsolute parameter recovery errors for different numbers of nontargets. More nontargets (cooler colors) leads to higher absolute recovery errors in 1D and 2D models (linear regression: p < 10−10). Shading shows standard error.\nFigure 6.\n\nAbsolute parameter recovery errors for different numbers of nontargets. More nontargets (cooler colors) leads to higher absolute recovery errors in 1D and 2D models (linear regression: p < 10−10). Shading shows standard error.", null, "Figure 7.\n\nParameter recovery errors (top) and absolute recovery errors (bottom) for the 2D model in simulated tasks with and without stimulus constraints. Adding in constraints to the stimulus locations leads to smaller errors in parameter recovery (orange line; see text for statistics). Shading shows standard error.\nFigure 7.\n\nParameter recovery errors (top) and absolute recovery errors (bottom) for the 2D model in simulated tasks with and without stimulus constraints. Adding in constraints to the stimulus locations leads to smaller errors in parameter recovery (orange line; see text for statistics). Shading shows standard error.", null, "Figure 8.\n\nParameter recovery errors from simulations with a constant translation bias subtracted (top) and corrected for (bottom). The color bar shows bias size (from 20 to 200 pixels). The bias correction works well for biases up to about 150 pixels in size. Shading shows standard error.\nFigure 8.\n\nParameter recovery errors from simulations with a constant translation bias subtracted (top) and corrected for (bottom). The color bar shows bias size (from 20 to 200 pixels). The bias correction works well for biases up to about 150 pixels in size. Shading shows standard error.", null, "Figure 9.\n\nDifference plot of recovered and simulated parameter values for different proportional biases toward the right-hand screen edge. Data with (bottom) and without (top) the bias correction for different proportional biases (color bar). Positive biases mean responses move toward the right-hand screen edge, and negative biases move responses away from this edge (responses do not move vertically). The bias correction improves the fits for all parameters. Shading shows standard error.\nFigure 9.\n\nDifference plot of recovered and simulated parameter values for different proportional biases toward the right-hand screen edge. Data with (bottom) and without (top) the bias correction for different proportional biases (color bar). Positive biases mean responses move toward the right-hand screen edge, and negative biases move responses away from this edge (responses do not move vertically). The bias correction improves the fits for all parameters. Shading shows standard error.", null, "Figure 10.\n\nDifference plot of recovered and simulated parameter values for different radial biases. Data with (bottom) and without (top) the bias correction for different radial biases (color bar). Positive radial biases mean points move toward the center of the screen, and negative biases move points away from the center. The bias correction improves the fits for all parameters, although large guess rate parameters are not fully corrected when there is a large expanding bias. Shading shows standard error.\nFigure 10.\n\nDifference plot of recovered and simulated parameter values for different radial biases. Data with (bottom) and without (top) the bias correction for different radial biases (color bar). Positive radial biases mean points move toward the center of the screen, and negative biases move points away from the center. The bias correction improves the fits for all parameters, although large guess rate parameters are not fully corrected when there is a large expanding bias. Shading shows standard error.", null, "Figure 11.\n\nParameter recovery errors with a radial bias toward the nontarget location on each trial. Data shown without (top) and with (bottom) bias correction. The bias correction reduced the errors in parameter recovery. This simulation had β = 0 (i.e., no misbinding). Shading shows standard error.\nFigure 11.\n\nParameter recovery errors with a radial bias toward the nontarget location on each trial. Data shown without (top) and with (bottom) bias correction. The bias correction reduced the errors in parameter recovery. This simulation had β = 0 (i.e., no misbinding). Shading shows standard error.", null, "Figure 12.\n\nIllustrative examples of mixture modeling approach to change detection. We show 1D examples for ease, although the same principles apply for 2D (see Supplementary Figure S10). The top row shows how each error component affects the proportion of generative responses that would fall at or beyond a probed distance of 60 degrees (arbitrary y-units). The second row shows the values of the parameters used for those simulations, along with the summed probabilities of all responses that occur at ≥ 60 degrees from the target. Increasing each parameter increases the probability of incorrectly accepting this probe as unchanged. The bottom row shows the probability of accepting a probe as unchanged as a function of the probe distance (or probe-nontarget distance for misbinding). This shows that increasing imprecision increases the spread of these same responses around the target, increasing guessing affects all orientations similarly, and increasing misbinding affects responses in relation to their distance to the distractor (if we plotted probe-targ for misbinding, it would look like the guessing panel).\nFigure 12.\n\nIllustrative examples of mixture modeling approach to change detection. We show 1D examples for ease, although the same principles apply for 2D (see Supplementary Figure S10). The top row shows how each error component affects the proportion of generative responses that would fall at or beyond a probed distance of 60 degrees (arbitrary y-units). The second row shows the values of the parameters used for those simulations, along with the summed probabilities of all responses that occur at ≥ 60 degrees from the target. Increasing each parameter increases the probability of incorrectly accepting this probe as unchanged. The bottom row shows the probability of accepting a probe as unchanged as a function of the probe distance (or probe-nontarget distance for misbinding). This shows that increasing imprecision increases the spread of these same responses around the target, increasing guessing affects all orientations similarly, and increasing misbinding affects responses in relation to their distance to the distractor (if we plotted probe-targ for misbinding, it would look like the guessing panel).", null, "Figure 13.\n\nParameter recovery errors for the 1D and 2D two-alternative forced-choice simulations. The 2D model (bottom) is more accurate than the 1D model (top), although both are far less accurate than with continuous report data (see Figure 3). Shading shows the standard error of the mean.\nFigure 13.\n\nParameter recovery errors for the 1D and 2D two-alternative forced-choice simulations. The 2D model (bottom) is more accurate than the 1D model (top), although both are far less accurate than with continuous report data (see Figure 3). Shading shows the standard error of the mean.", null, "Figure 14.\n\nHistograms of the recovered parameters from the ignore/update task, across all conditions.\nFigure 14.\n\nHistograms of the recovered parameters from the ignore/update task, across all conditions.", null, "Figure 15.\n\nMean fitted parameter values for each trial type from the healthy participants. Delay increased imprecision (σ; top left; blue vs. orange; see Figure 2 for task explanation), guessing (γ), and misbinding (β; see text for stats); irrelevant stimuli increased misbinding and guessing but not imprecision (ignore and update vs. T1 and T2); and there was an interaction of delay and irrelevant stimuli only for guessing. Standard error bars.\nFigure 15.\n\nMean fitted parameter values for each trial type from the healthy participants. Delay increased imprecision (σ; top left; blue vs. orange; see Figure 2 for task explanation), guessing (γ), and misbinding (β; see text for stats); irrelevant stimuli increased misbinding and guessing but not imprecision (ignore and update vs. T1 and T2); and there was an interaction of delay and irrelevant stimuli only for guessing. Standard error bars.", null, "Figure 16.\n\nHeatmaps showing the posterior MCMC samples for recovering the mean parameters from Fallon et al. (2018). Each color is a different condition (see legend; Ig = ignore, T1 = long delay, Up = update, T2 = long delay), and the rows show 1D and 2D models, respectively. The γ and β parameters have been arcsine transformed. Both models were simulated using the same parameters. The 2D model has better separation of the different conditions’ parameters, as shown by less overlap between the colors.\nFigure 16.\n\nHeatmaps showing the posterior MCMC samples for recovering the mean parameters from Fallon et al. (2018). Each color is a different condition (see legend; Ig = ignore, T1 = long delay, Up = update, T2 = long delay), and the rows show 1D and 2D models, respectively. The γ and β parameters have been arcsine transformed. Both models were simulated using the same parameters. The 2D model has better separation of the different conditions’ parameters, as shown by less overlap between the colors.", null, "Figure 17.\n\nRelationship between the true parameters (x-axis) and the best behavioral metrics (top) and recovered parameters (bottom). All correlations were significant (p < 10−10). Two metrics were compared against the imprecision (σ) parameter. The recovered parameters had stronger correlations than the best behavioral metrics (see text for statistics).\nFigure 17.\n\nRelationship between the true parameters (x-axis) and the best behavioral metrics (top) and recovered parameters (bottom). All correlations were significant (p < 10−10). Two metrics were compared against the imprecision (σ) parameter. The recovered parameters had stronger correlations than the best behavioral metrics (see text for statistics).", null, "Table 1.\n\nSpearman correlation coefficients between each true parameter and the recovered parameters for the 1D and 2D misbinding models. p < 10−10.\nTable 1.\n\nSpearman correlation coefficients between each true parameter and the recovered parameters for the 1D and 2D misbinding models. *p < 10−10.", null, "×\n\nThis PDF is available to Subscribers Only" ]	[ null, "https://jov.arvojournals.org/UI/app/images/arvo_journals_logo-white.png", null, "https://jov.arvojournals.org/Images/grey.gif", null, "https://jov.arvojournals.org/Images/grey.gif", null, "https://jov.arvojournals.org/Images/grey.gif", null, "https://jov.arvojournals.org/Images/grey.gif", null, "https://jov.arvojournals.org/Images/grey.gif", null, "https://jov.arvojournals.org/Images/grey.gif", null, "https://jov.arvojournals.org/Images/grey.gif", null, "https://jov.arvojournals.org/Images/grey.gif", null, "https://jov.arvojournals.org/Images/grey.gif", null, "https://jov.arvojournals.org/Images/grey.gif", null, "https://jov.arvojournals.org/Images/grey.gif", null, "https://jov.arvojournals.org/Images/grey.gif", null, "https://jov.arvojournals.org/Images/grey.gif", null, "https://jov.arvojournals.org/Images/grey.gif", null, "https://jov.arvojournals.org/Images/grey.gif", null, "https://jov.arvojournals.org/Images/grey.gif", null, "https://jov.arvojournals.org/Images/grey.gif", null, "https://jov.arvojournals.org/Images/grey.gif", null, "https://jov.arvojournals.org/Images/grey.gif", null, "https://jov.arvojournals.org/Images/grey.gif", null, "https://jov.arvojournals.org/Images/grey.gif", null, "https://jov.arvojournals.org/Images/grey.gif", null, "https://jov.arvojournals.org/Images/grey.gif", null, "https://jov.arvojournals.org/Images/grey.gif", null, "https://jov.arvojournals.org/Images/grey.gif", null, "https://jov.arvojournals.org/Images/grey.gif", null, "https://jov.arvojournals.org/Images/grey.gif", null, "https://jov.arvojournals.org/Images/grey.gif", null, "https://jov.arvojournals.org/Images/grey.gif", null, "https://jov.arvojournals.org/Images/grey.gif", null, "https://jov.arvojournals.org/Images/grey.gif", null, "https://jov.arvojournals.org/Images/grey.gif", null, "https://jov.arvojournals.org/Images/grey.gif", null, "https://jov.arvojournals.org/Images/grey.gif", null, "https://jov.arvojournals.org/Images/grey.gif", null, "https://jov.arvojournals.org/Images/grey.gif", null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.8218702,"math_prob":0.91623265,"size":4860,"snap":"2021-31-2021-39","text_gpt3_token_len":1259,"char_repetition_ratio":0.13200165,"word_repetition_ratio":0.010568032,"special_character_ratio":0.27283952,"punctuation_ratio":0.14498934,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.96414965,"pos_list":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74],"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,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],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2021-09-16T10:02:28Z\",\"WARC-Record-ID\":\"<urn:uuid:0b8fc983-d49f-4183-b661-671156f0782a>\",\"Content-Length\":\"456867\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:0e8c3b0f-8109-4a10-a799-ac02b6c97099>\",\"WARC-Concurrent-To\":\"<urn:uuid:bfd9e1da-76c5-4b89-9a4d-392b89368620>\",\"WARC-IP-Address\":\"52.191.96.132\",\"WARC-Target-URI\":\"https://jov.arvojournals.org/article.aspx?articleid=2772046\",\"WARC-Payload-Digest\":\"sha1:2D4KGTIJ7FOQMUOBKTPTFURKDEIHGOZN\",\"WARC-Block-Digest\":\"sha1:N54PVLRKUGPVFY4RW634VCAER6Z54PAX\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2021/CC-MAIN-2021-39/CC-MAIN-2021-39_segments_1631780053493.41_warc_CC-MAIN-20210916094919-20210916124919-00127.warc.gz\"}"}
http://www-math.ucdenver.edu/~wcherowi/courses/m4408/gtaln7.html	[ "## Graph Theory Lecture Notes 7\n\n### Counting Spanning Trees\n\nTwo labeled trees are the same if their edge-sets are identical.\n\nThe number of spanning trees of Kn is nn-2, for n", null, "2, this is Cayley's Formula.\n\nPrüfer Encoding\n\nA Prüfer sequence of length n-2, for n", null, "2, is any sequence of integers between 1 and n, with repetitions allowed.\n\nA labeled tree gives a Prüfer sequence by:\n\n• Repeat n-2 times:\n• Pick the 1-valent vertex with the smallest label, call it v.\n• Put the label of v's neighbor in the output sequence.\n• Remove v from the tree.\nPrüfer Decoding\n\nA Prüfer sequence determines a labeled tree by:\n\n• Let L be the ordered list of numbers 1, 2, ..., n. Let P be the Prüfer sequence. Start with n labeled isolated vertices.\n• Repeat n-2 times:\n• Let k be the smallest number in L which is not in P.\n• Let j be the first number in P.\n• Add the edge kj to the graph.\n• Remove k from L and the first number in P.\n• When this is completed there will be two numbers left in L, add the edge corresponding to these two numbers.\nPrüfer coding and decoding are inverse operations, that means that there is a one-to-one correspondence between labeled trees with n vertices and Prüfer sequences of length n-2.\n\nCayley's formula now follows from counting Prüfer sequences of length n-2.\n\n### Finding Minimum Spanning Trees\n\nLet G be a connected graph with weights assigned to its edges. We wish to find the spanning tree with the smallest sum of weights.\n\nPrim's Algorithm: Grow a tree by picking the frontier edge with the smallest weight.\n\nThis is an example of a greedy algorithm.\n\n### Finding the Shortest Path\n\nLet G be a connected graph with weights assigned to its edges. We wish to find for any two vertices s and t, the path from s to t whose total edge-weight is minimum.\n\nIf the weights are all equal, the problem reduces to finding the s-t path of shortest length. This can be done by using a breadth-first search starting at s, and ending when t is reached.\n\nWhen the weights are non-negative, but not all equal we can use Dijkstra's algorithm (for situations where negative weights are used, Floyd's algorithm can be used).\n\nDijkstra's Algorithm\n\nIn this tree growing algorithm, vertices that are added to the tree are labeled with their \"distance\" from the starting vertex s. For a given vertex x, this label will be denoted dist[x]. Start by setting dist[s]:=0. At each step of the tree growing process, for each frontier edge e, calculate P(e) = dist[x] + wt(e), where x is the vertex of e that is in the tree (and therefore has been labeled) and wt(e) is the weight of edge e. Choose the frontier edge with the smallest P(e) value to add to the tree and label the new vertex with P(e)." ]	[ null, "http://www-math.ucdenver.edu/~wcherowi/courses/gif/ge.gif", null, "http://www-math.ucdenver.edu/~wcherowi/courses/gif/ge.gif", null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.8855506,"math_prob":0.9860477,"size":1416,"snap":"2022-40-2023-06","text_gpt3_token_len":336,"char_repetition_ratio":0.12181303,"word_repetition_ratio":0.016194332,"special_character_ratio":0.22951977,"punctuation_ratio":0.10508475,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9988491,"pos_list":[0,1,2,3,4],"im_url_duplicate_count":[null,null,null,null,null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2022-10-02T19:22:27Z\",\"WARC-Record-ID\":\"<urn:uuid:a93b1652-5d7b-41aa-99a8-553b7245c9eb>\",\"Content-Length\":\"3513\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:dd1b447c-e44c-4e2c-a479-537a86706097>\",\"WARC-Concurrent-To\":\"<urn:uuid:c81fe538-c643-4801-934c-09b2d61ea7d0>\",\"WARC-IP-Address\":\"132.194.174.10\",\"WARC-Target-URI\":\"http://www-math.ucdenver.edu/~wcherowi/courses/m4408/gtaln7.html\",\"WARC-Payload-Digest\":\"sha1:WPJ4RYKBJVX3IIAZFNCWK4YK7XQUNWMI\",\"WARC-Block-Digest\":\"sha1:VVVFSNBF5MGQZY6OI5BQRVGXA3QHEHRS\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2022/CC-MAIN-2022-40/CC-MAIN-2022-40_segments_1664030337339.70_warc_CC-MAIN-20221002181356-20221002211356-00362.warc.gz\"}"}