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Infi-MM

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InfiMM-WebMath-40B

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[ "FITS Image", null, "HTTP dir\n```TELESCOP= 'ASCA ' / Telescope (mission) name\nOBSERVER= 'DR. KAZUO MAKISHIMA' / Principal Investigator\nOBJECT = 'MAFFEI_1' / Name of observed object\nRA_PNT = 39.166748 / File average value of RA (degrees)\nDEC_PNT = 59.752846 / File average value of DEC (degrees)\nPA_PNT = 304.046173 / File average value of ROLL (degrees)\nRA__MEAN= 39.173183 / Mean pointing RA (degrees)\nDEC_MEAN= 59.730816 / Mean pointing DEC (degrees)\nDATE-OBS= '11/09/95' / Start date of observations\nTIME-OBS= '14:43:46' / Start time of observations\nDATE-END= '12/09/95' / End date of observations\nTIME-END= '14:09:52' / End time of observations\nTSTART = 8.498455461244699E+07 / time start\nTSTOP = 8.506807461244699E+07 / time stop\nTELAPSE = 8.352000000000000E+04 / elapsed time\nONTIME = 2.669462230165303E+04 / On-source time\nEXPOSURE= 2.669462230165303E+04 / Exposure time\nMJD-OBS = 4.997161753023665E+04 / MJD of data start time\nFILIN001= 'ad63004000s000102h.evt.gz' / Input file name\nFILIN002= 'ad63004000s000202m.evt.gz' / Input file name\nFILIN003= 'ad63004000s100102h.evt.gz' / Input file name\nFILIN004= 'ad63004000s100202m.evt.gz' / Input file name\nCOMMENT -------------------------\nCOMMENT BITPIX compressed from 32 to 8.\nCOMMENT Pixel values greater than 255 was set to 255.\nCOMMENT This file was made by Ken Ebisawa (ISAS)\nCOMMENT for JUDO-ASCA.\n```" ]

[ null, "http://darts.isas.jaxa.jp/pub/judo2/meta_info_page/image/ASCA_SIS/63004000.png", null ]

[ "# Inspired By Sophie Germain\n\n$y=x^6+27$\n\nFind the sum of all positive integers $x$ and $y$ such that $(x,y)$ satisfies the equation above and $y$ is the product of exactly two primes (including multiplicity).\n\n×" ]

[ null ]

[ "", null, "Rendering Functions\n\nRendering functions to draw images, plots etc. More...\n\n## Functions\n\nvoid image (const array &in, const char *title=NULL)\nRenders the input array as an image to the window. More...\n\nvoid plot3 (const array &in, const char *title=NULL)\nRenders the input array as an 3d line plot to the window. More...\n\nvoid plot (const array &in, const char *const title=NULL)\nRenders the input arrays as a 2D or 3D plot to the window. More...\n\nvoid plot (const array &X, const array &Y, const array &Z, const char *const title=NULL)\nRenders the input arrays as a 3D plot to the window. More...\n\nvoid plot (const array &X, const array &Y, const char *const title=NULL)\nRenders the input arrays as a 2D plot to the window. More...\n\nvoid scatter (const array &in, const af::markerType marker=AF_MARKER_POINT, const char *const title=NULL)\nRenders the input arrays as a 2D or 3D scatter-plot to the window. More...\n\nvoid scatter (const array &X, const array &Y, const array &Z, const af::markerType marker=AF_MARKER_POINT, const char *const title=NULL)\nRenders the input arrays as a 3D scatter-plot to the window. More...\n\nvoid scatter (const array &X, const array &Y, const af::markerType marker=AF_MARKER_POINT, const char *const title=NULL)\nRenders the input arrays as a 2D scatter-plot to the window. More...\n\nvoid scatter3 (const array &P, const af::markerType marker=AF_MARKER_POINT, const char *const title=NULL)\nRenders the input arrays as a 3D scatter-plot to the window. More...\n\nvoid hist (const array &X, const double minval, const double maxval, const char *const title=NULL)\nRenders the input array as a histogram to the window. More...\n\nvoid surface (const array &S, const char *const title=NULL)\nRenders the input arrays as a 3D surface plot to the window. More...\n\nvoid surface (const array &xVals, const array &yVals, const array &S, const char *const title=NULL)\nRenders the input arrays as a 3D surface plot to the window. More...\n\nvoid vectorField (const array &points, const array &directions, const char *const title=NULL)\nRenders the input arrays as a 2D or 3D vector field plot to the window. More...\n\nvoid vectorField (const array &xPoints, const array &yPoints, const array &zPoints, const array &xDirs, const array &yDirs, const array &zDirs, const char *const title=NULL)\nRenders the input arrays as a 3D vector field plot to the window. More...\n\nvoid vectorField (const array &xPoints, const array &yPoints, const array &xDirs, const array &yDirs, const char *const title=NULL)\nRenders the input arrays as a 2D vector field plot to the window. More...\n\nAFAPI af_err af_draw_image (const af_window wind, const af_array in, const af_cell *const props)\nC Interface wrapper for drawing an array as an image. More...\n\nAFAPI af_err af_draw_plot (const af_window wind, const af_array X, const af_array Y, const af_cell *const props)\nC Interface wrapper for drawing an array as a plot. More...\n\nAFAPI af_err af_draw_plot3 (const af_window wind, const af_array P, const af_cell *const props)\nC Interface wrapper for drawing an array as a plot. More...\n\nAFAPI af_err af_draw_plot_nd (const af_window wind, const af_array P, const af_cell *const props)\nC Interface wrapper for drawing an array as a 2D or 3D plot. More...\n\nAFAPI af_err af_draw_plot_2d (const af_window wind, const af_array X, const af_array Y, const af_cell *const props)\nC Interface wrapper for drawing an array as a 2D plot. More...\n\nAFAPI af_err af_draw_plot_3d (const af_window wind, const af_array X, const af_array Y, const af_array Z, const af_cell *const props)\nC Interface wrapper for drawing an array as a 3D plot. More...\n\nAFAPI af_err af_draw_scatter (const af_window wind, const af_array X, const af_array Y, const af_marker_type marker, const af_cell *const props)\nC Interface wrapper for drawing an array as a plot. More...\n\nAFAPI af_err af_draw_scatter3 (const af_window wind, const af_array P, const af_marker_type marker, const af_cell *const props)\nC Interface wrapper for drawing an array as a plot. More...\n\nAFAPI af_err af_draw_scatter_nd (const af_window wind, const af_array P, const af_marker_type marker, const af_cell *const props)\nC Interface wrapper for drawing an array as a plot. More...\n\nAFAPI af_err af_draw_scatter_2d (const af_window wind, const af_array X, const af_array Y, const af_marker_type marker, const af_cell *const props)\nC Interface wrapper for drawing an array as a 2D plot. More...\n\nAFAPI af_err af_draw_scatter_3d (const af_window wind, const af_array X, const af_array Y, const af_array Z, const af_marker_type marker, const af_cell *const props)\nC Interface wrapper for drawing an array as a 3D plot. More...\n\nAFAPI af_err af_draw_hist (const af_window wind, const af_array X, const double minval, const double maxval, const af_cell *const props)\nC Interface wrapper for drawing an array as a histogram. More...\n\nAFAPI af_err af_draw_surface (const af_window wind, const af_array xVals, const af_array yVals, const af_array S, const af_cell *const props)\nC Interface wrapper for drawing array's as a surface. More...\n\nAFAPI af_err af_draw_vector_field_nd (const af_window wind, const af_array points, const af_array directions, const af_cell *const props)\nC Interface wrapper for drawing array's as a 2D or 3D vector field. More...\n\nAFAPI af_err af_draw_vector_field_3d (const af_window wind, const af_array xPoints, const af_array yPoints, const af_array zPoints, const af_array xDirs, const af_array yDirs, const af_array zDirs, const af_cell *const props)\nC Interface wrapper for drawing array's as a 3D vector field. More...\n\nAFAPI af_err af_draw_vector_field_2d (const af_window wind, const af_array xPoints, const af_array yPoints, const af_array xDirs, const af_array yDirs, const af_cell *const props)\nC Interface wrapper for drawing array's as a 2D vector field. More...\n\n## Detailed Description\n\nRendering functions to draw images, plots etc.\n\n## ◆ af_draw_hist()\n\n AFAPI af_err af_draw_hist ( const af_window wind, const af_array X, const double minval, const double maxval, const af_cell *const props )\n\nC Interface wrapper for drawing an array as a histogram.\n\nParameters\n [in] wind is the window handle [in] X is the data frequency af_array [in] minval is the value of the minimum data point of the array whose histogram(X) is going to be rendered. [in] maxval is the value of the maximum data point of the array whose histogram(X) is going to be rendered. [in] props is structure af_cell that has the properties that are used for the current rendering.\nReturns\nAF_SUCCESS if rendering is successful, otherwise an appropriate error code is returned.\nNote\nX should be a vector.\n\n## ◆ af_draw_image()\n\n AFAPI af_err af_draw_image ( const af_window wind, const af_array in, const af_cell *const props )\n\nC Interface wrapper for drawing an array as an image.\n\nParameters\n [in] wind is the window handle [in] in is an af_array [in] props is structure af_cell that has the properties that are used for the current rendering.\nReturns\nAF_SUCCESS if rendering is successful, otherwise an appropriate error code is returned.\nNote\nin should be 2d array or 3d array with 3 channels.\n\n## ◆ af_draw_plot()\n\n AFAPI af_err af_draw_plot ( const af_window wind, const af_array X, const af_array Y, const af_cell *const props )\n\nC Interface wrapper for drawing an array as a plot.\n\nParameters\n [in] wind is the window handle [in] X is an af_array with the x-axis data points [in] Y is an af_array with the y-axis data points [in] props is structure af_cell that has the properties that are used for the current rendering.\nReturns\nAF_SUCCESS if rendering is successful, otherwise an appropriate error code is returned.\nNote\nX and Y should be vectors.\n\n## ◆ af_draw_plot3()\n\n AFAPI af_err af_draw_plot3 ( const af_window wind, const af_array P, const af_cell *const props )\n\nC Interface wrapper for drawing an array as a plot.\n\nParameters\n [in] wind is the window handle [in] P is an af_array or matrix with the xyz-values of the points [in] props is structure af_cell that has the properties that are used for the current rendering.\nReturns\nAF_SUCCESS if rendering is successful, otherwise an appropriate error code is returned.\nNote\nP should be a 3n x 1 vector or one of a 3xn or nx3 matrices.\n\n## ◆ af_draw_plot_2d()\n\n AFAPI af_err af_draw_plot_2d ( const af_window wind, const af_array X, const af_array Y, const af_cell *const props )\n\nC Interface wrapper for drawing an array as a 2D plot.\n\nParameters\n [in] wind is the window handle [in] X is an af_array with the x-axis data points [in] Y is an af_array with the y-axis data points [in] props is structure af_cell that has the properties that are used for the current rendering.\nReturns\nAF_SUCCESS if rendering is successful, otherwise an appropriate error code is returned.\nNote\nX and Y should be vectors.\n\n## ◆ af_draw_plot_3d()\n\n AFAPI af_err af_draw_plot_3d ( const af_window wind, const af_array X, const af_array Y, const af_array Z, const af_cell *const props )\n\nC Interface wrapper for drawing an array as a 3D plot.\n\nParameters\n [in] wind is the window handle [in] X is an af_array with the x-axis data points [in] Y is an af_array with the y-axis data points [in] Z is an af_array with the z-axis data points [in] props is structure af_cell that has the properties that are used for the current rendering.\nReturns\nAF_SUCCESS if rendering is successful, otherwise an appropriate error code is returned.\nNote\nX, Y and Z should be vectors.\n\n## ◆ af_draw_plot_nd()\n\n AFAPI af_err af_draw_plot_nd ( const af_window wind, const af_array P, const af_cell *const props )\n\nC Interface wrapper for drawing an array as a 2D or 3D plot.\n\nParameters\n [in] wind is the window handle [in] P is an af_array or matrix with the xyz-values of the points [in] props is structure af_cell that has the properties that are used for the current rendering.\nReturns\nAF_SUCCESS if rendering is successful, otherwise an appropriate error code is returned.\nNote\nin must be 2d and of the form [n, order], where order is either 2 or 3. If order is 2, then chart is 2D and if order is 3, then chart is 3D.\n\n## ◆ af_draw_scatter()\n\n AFAPI af_err af_draw_scatter ( const af_window wind, const af_array X, const af_array Y, const af_marker_type marker, const af_cell *const props )\n\nC Interface wrapper for drawing an array as a plot.\n\nParameters\n [in] wind is the window handle [in] X is an af_array with the x-axis data points [in] Y is an af_array with the y-axis data points [in] marker is an af_marker_type enum specifying which marker to use in the scatter plot [in] props is structure af_cell that has the properties that are used for the current rendering.\nReturns\nAF_SUCCESS if rendering is successful, otherwise an appropriate error code is returned.\nNote\nX and Y should be vectors.\n\n## ◆ af_draw_scatter3()\n\n AFAPI af_err af_draw_scatter3 ( const af_window wind, const af_array P, const af_marker_type marker, const af_cell *const props )\n\nC Interface wrapper for drawing an array as a plot.\n\nParameters\n [in] wind is the window handle [in] P is an af_array or matrix with the xyz-values of the points [in] marker is an af_marker_type enum specifying which marker to use in the scatter plot [in] props is structure af_cell that has the properties that are used for the current rendering.\nReturns\nAF_SUCCESS if rendering is successful, otherwise an appropriate error code is returned.\n\n## ◆ af_draw_scatter_2d()\n\n AFAPI af_err af_draw_scatter_2d ( const af_window wind, const af_array X, const af_array Y, const af_marker_type marker, const af_cell *const props )\n\nC Interface wrapper for drawing an array as a 2D plot.\n\nParameters\n [in] wind is the window handle [in] X is an af_array with the x-axis data points [in] Y is an af_array with the y-axis data points [in] marker is an af_marker_type enum specifying which marker to use in the scatter plot [in] props is structure af_cell that has the properties that are used for the current rendering.\nReturns\nAF_SUCCESS if rendering is successful, otherwise an appropriate error code is returned.\nNote\nX and Y should be vectors.\n\n## ◆ af_draw_scatter_3d()\n\n AFAPI af_err af_draw_scatter_3d ( const af_window wind, const af_array X, const af_array Y, const af_array Z, const af_marker_type marker, const af_cell *const props )\n\nC Interface wrapper for drawing an array as a 3D plot.\n\nParameters\n [in] wind is the window handle [in] X is an af_array with the x-axis data points [in] Y is an af_array with the y-axis data points [in] Z is an af_array with the z-axis data points [in] marker is an af_marker_type enum specifying which marker to use in the scatter plot [in] props is structure af_cell that has the properties that are used for the current rendering.\nReturns\nAF_SUCCESS if rendering is successful, otherwise an appropriate error code is returned.\nNote\nX, Y and Z should be vectors.\n\n## ◆ af_draw_scatter_nd()\n\n AFAPI af_err af_draw_scatter_nd ( const af_window wind, const af_array P, const af_marker_type marker, const af_cell *const props )\n\nC Interface wrapper for drawing an array as a plot.\n\nParameters\n [in] wind is the window handle [in] P is an af_array or matrix with the xyz-values of the points [in] marker is an af_marker_type enum specifying which marker to use in the scatter plot [in] props is structure af_cell that has the properties that are used for the current rendering.\nReturns\nAF_SUCCESS if rendering is successful, otherwise an appropriate error code is returned.\nNote\nin must be 2d and of the form [n, order], where order is either 2 or 3. If order is 2, then chart is 2D and if order is 3, then chart is 3D.\n\n## ◆ af_draw_surface()\n\n AFAPI af_err af_draw_surface ( const af_window wind, const af_array xVals, const af_array yVals, const af_array S, const af_cell *const props )\n\nC Interface wrapper for drawing array's as a surface.\n\nParameters\n [in] wind is the window handle [in] xVals is an af_array with the x-axis data points [in] yVals is an af_array with the y-axis data points [in] S is an af_array with the z-axis data points [in] props is structure af_cell that has the properties that are used for the current rendering.\nReturns\nAF_SUCCESS if rendering is successful, otherwise an appropriate error code is returned.\nNote\nX and Y should be vectors. S should be a 2D array\n\n## ◆ af_draw_vector_field_2d()\n\n AFAPI af_err af_draw_vector_field_2d ( const af_window wind, const af_array xPoints, const af_array yPoints, const af_array xDirs, const af_array yDirs, const af_cell *const props )\n\nC Interface wrapper for drawing array's as a 2D vector field.\n\nParameters\n [in] wind is the window handle [in] xPoints is an af_array with the x-axis points [in] yPoints is an af_array with the y-axis points [in] xDirs is an af_array with the x-axis directions [in] yDirs is an af_array with the y-axis directions [in] props is structure af_cell that has the properties that are used for the current rendering.\nReturns\nAF_SUCCESS if rendering is successful, otherwise an appropriate error code is returned.\nNote\nall the af_array inputs should be vectors and the same size\n\n## ◆ af_draw_vector_field_3d()\n\n AFAPI af_err af_draw_vector_field_3d ( const af_window wind, const af_array xPoints, const af_array yPoints, const af_array zPoints, const af_array xDirs, const af_array yDirs, const af_array zDirs, const af_cell *const props )\n\nC Interface wrapper for drawing array's as a 3D vector field.\n\nParameters\n [in] wind is the window handle [in] xPoints is an af_array with the x-axis points [in] yPoints is an af_array with the y-axis points [in] zPoints is an af_array with the z-axis points [in] xDirs is an af_array with the x-axis directions [in] yDirs is an af_array with the y-axis directions [in] zDirs is an af_array with the z-axis directions [in] props is structure af_cell that has the properties that are used for the current rendering.\nReturns\nAF_SUCCESS if rendering is successful, otherwise an appropriate error code is returned.\nNote\nall the af_array inputs should be vectors and the same size\n\n## ◆ af_draw_vector_field_nd()\n\n AFAPI af_err af_draw_vector_field_nd ( const af_window wind, const af_array points, const af_array directions, const af_cell *const props )\n\nC Interface wrapper for drawing array's as a 2D or 3D vector field.\n\nParameters\n [in] wind is the window handle [in] points is an af_array with the points [in] directions is an af_array with the directions [in] props is structure af_cell that has the properties that are used for the current rendering.\nReturns\nAF_SUCCESS if rendering is successful, otherwise an appropriate error code is returned.\nNote\npoints and directions should have the same size and must be 2D. The number of rows (dim 0) determines are number of points and the number columns determines the type of plot. If the number of columns are 2, then the plot is 2D and if there are 3 columns, then the plot is 3D.\nall the af_array inputs should be vectors and the same size\n\n## ◆ hist()\n\n void hist ( const array & X, const double minval, const double maxval, const char *const title = NULL )\n\nRenders the input array as a histogram to the window.\n\nParameters\n [in] X is the data frequency array [in] minval is the value of the minimum data point of the array whose histogram(X) is going to be rendered. [in] maxval is the value of the maximum data point of the array whose histogram(X) is going to be rendered. [in] title parameter is used when this function is called in grid mode\nNote\nX should be a vector.\nExamples\ngraphics/histogram.cpp.\n\n## ◆ image()\n\n void image ( const array & in, const char * title = NULL )\n\nRenders the input array as an image to the window.\n\nParameters\n [in] in is an array [in] title parameter is used when this function is called in grid mode\nNote\nin should be 2d array or 3d array with 3 channels.\nExamples\ncomputer_vision/fast.cpp, computer_vision/harris.cpp, computer_vision/susan.cpp, graphics/conway.cpp, graphics/conway_pretty.cpp, graphics/histogram.cpp, and image_processing/confidence_connected_components.cpp.\n\n## ◆ plot() [1/3]\n\n void plot ( const array & in, const char *const title = NULL )\n\nRenders the input arrays as a 2D or 3D plot to the window.\n\nParameters\n [in] in is an array with the data points [in] title parameter is used when this function is called in grid mode\nNote\nin must be 2d and of the form [n, order], where order is either 2 or 3. If order is 2, then chart is 2D and if order is 3, then chart is 3D.\nExamples\ngraphics/plot2d.cpp, and graphics/plot3.cpp.\n\n## ◆ plot() [2/3]\n\n void plot ( const array & X, const array & Y, const array & Z, const char *const title = NULL )\n\nRenders the input arrays as a 3D plot to the window.\n\nParameters\n [in] X is an array with the x-axis data points [in] Y is an array with the y-axis data points [in] Z is an array with the z-axis data points [in] title parameter is used when this function is called in grid mode\nNote\nX, Y and Z should be vectors.\n\n## ◆ plot() [3/3]\n\n void plot ( const array & X, const array & Y, const char *const title = NULL )\n\nRenders the input arrays as a 2D plot to the window.\n\nParameters\n [in] X is an array with the x-axis data points [in] Y is an array with the y-axis data points [in] title parameter is used when this function is called in grid mode\nNote\nX and Y should be vectors.\n\n## ◆ plot3()\n\n void plot3 ( const array & in, const char * title = NULL )\n\nRenders the input array as an 3d line plot to the window.\n\nParameters\n [in] in is an array [in] title parameter is used when this function is called in grid mode\nNote\nin should be 1d array of size 3n or 2d array with (3 x n) or (n x 3) channels.\n\n## ◆ scatter() [1/3]\n\n void scatter ( const array & in, const af::markerType marker = AF_MARKER_POINT, const char *const title = NULL )\n\nRenders the input arrays as a 2D or 3D scatter-plot to the window.\n\nParameters\n [in] in is an array with the data points [in] marker is an markerType enum specifying which marker to use in the scatter plot [in] title parameter is used when this function is called in grid mode\nNote\nin must be 2d and of the form [n, order], where order is either 2 or 3. If order is 2, then chart is 2D and if order is 3, then chart is 3D.\nExamples\ngraphics/plot2d.cpp.\n\n## ◆ scatter() [2/3]\n\n void scatter ( const array & X, const array & Y, const af::markerType marker = AF_MARKER_POINT, const char *const title = NULL )\n\nRenders the input arrays as a 2D scatter-plot to the window.\n\nParameters\n [in] X is an array with the x-axis data points [in] Y is an array with the y-axis data points [in] marker is an markerType enum specifying which marker to use in the scatter plot [in] title parameter is used when this function is called in grid mode\nNote\nX and Y should be vectors.\n\n## ◆ scatter() [3/3]\n\n void scatter ( const array & X, const array & Y, const array & Z, const af::markerType marker = AF_MARKER_POINT, const char *const title = NULL )\n\nRenders the input arrays as a 3D scatter-plot to the window.\n\nParameters\n [in] X is an array with the x-axis data points [in] Y is an array with the y-axis data points [in] Z is an array with the z-axis data points [in] marker is an markerType enum specifying which marker to use in the scatter plot [in] title parameter is used when this function is called in grid mode\nNote\nX, Y and Z should be vectors.\n\n## ◆ scatter3()\n\n void scatter3 ( const array & P, const af::markerType marker = AF_MARKER_POINT, const char *const title = NULL )\n\nRenders the input arrays as a 3D scatter-plot to the window.\n\nParameters\n [in] P is an af_array or matrix with the xyz-values of the points [in] marker is an markerType enum specifying which marker to use in the scatter plot [in] title parameter is used when this function is called in grid mode\n\n## ◆ surface() [1/2]\n\n void surface ( const array & S, const char *const title = NULL )\n\nRenders the input arrays as a 3D surface plot to the window.\n\nParameters\n [in] S is an array with the z-axis data points [in] title parameter is used when this function is called in grid mode\nNote\nS should be a 2D array\nExamples\ngraphics/surface.cpp.\n\n## ◆ surface() [2/2]\n\n void surface ( const array & xVals, const array & yVals, const array & S, const char *const title = NULL )\n\nRenders the input arrays as a 3D surface plot to the window.\n\nParameters\n [in] xVals is an array with the x-axis data points [in] yVals is an array with the y-axis data points [in] S is an array with the z-axis data points [in] title parameter is used when this function is called in grid mode\nNote\nX and Y should be vectors or 2D arrays S should be s 2D array\n\n## ◆ vectorField() [1/3]\n\n void vectorField ( const array & points, const array & directions, const char *const title = NULL )\n\nRenders the input arrays as a 2D or 3D vector field plot to the window.\n\nParameters\n [in] points is an array with the points [in] directions is an array with the directions at the points [in] title parameter is used when this function is called in grid mode\nNote\npoints and directions should have the same size and must be 2D. The number of rows (dim 0) determines are number of points and the number columns determines the type of plot. If the number of columns are 2, then the plot is 2D and if there are 3 columns, then the plot is 3D.\nExamples\ngraphics/field.cpp.\n\n## ◆ vectorField() [2/3]\n\n void vectorField ( const array & xPoints, const array & yPoints, const array & xDirs, const array & yDirs, const char *const title = NULL )\n\nRenders the input arrays as a 2D vector field plot to the window.\n\nParameters\n [in] xPoints is an array with the x-coordinate points [in] yPoints is an array with the y-coordinate points [in] xDirs is an array with the x-coordinate directions at the points [in] yDirs is an array with the y-coordinate directions at the points [in] title parameter is used when this function is called in grid mode\nNote\nAll the array inputs must be vectors and must have the size sizes.\n\n## ◆ vectorField() [3/3]\n\n void vectorField ( const array & xPoints, const array & yPoints, const array & zPoints, const array & xDirs, const array & yDirs, const array & zDirs, const char *const title = NULL )\n\nRenders the input arrays as a 3D vector field plot to the window.\n\nParameters\n [in] xPoints is an array with the x-coordinate points [in] yPoints is an array with the y-coordinate points [in] zPoints is an array with the z-coordinate points [in] xDirs is an array with the x-coordinate directions at the points [in] yDirs is an array with the y-coordinate directions at the points [in] zDirs is an array with the z-coordinate directions at the points [in] title parameter is used when this function is called in grid mode\nNote\nAll the array inputs must be vectors and must have the size sizes." ]

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[ "# Inorder Successor in Binary Search Tree\n\nIn Binary Tree, Inorder successor of a node is the next node in Inorder traversal of the Binary Tree. Inorder Successor is NULL for the last node in Inoorder traversal.\nIn Binary Search Tree, Inorder Successor of an input node can also be defined as the node with the smallest key greater than the key of input node. So, it is sometimes important to find next node in sorted order.", null, "In the above diagram, inorder successor of 8 is 10, inorder successor of 10 is 12 and inorder successor of 14 is 20.\n\nMethod 1 (Uses Parent Pointer)\nIn this method, we assume that every node has parent pointer.\n\nThe Algorithm is divided into two cases on the basis of right subtree of the input node being empty or not.\n\nInput: node, root // node is the node whose Inorder successor is needed.\noutput: succ // succ is Inorder successor of node.\n\n1) If right subtree of node is not NULL, then succ lies in right subtree. Do following.\nGo to right subtree and return the node with minimum key value in right subtree.\n2) If right sbtree of node is NULL, then succ is one of the ancestors. Do following.\nTravel up using the parent pointer until you see a node which is left child of it’s parent. The parent of such a node is the succ.\n\nImplementation\nNote that the function to find InOrder Successor is highlighted (with gray background) in below code.\n\n## C\n\n `#include ` `#include ` `  `  `/* A binary tree node has data, pointer to left child ` `   ``and a pointer to right child */` `struct` `node ` `{ ` `    ``int` `data; ` `    ``struct` `node* left; ` `    ``struct` `node* right; ` `    ``struct` `node* parent; ` `}; ` ` `  `struct` `node * minValue(``struct` `node* node);  ` ` `  `struct` `node * inOrderSuccessor(``struct` `node *root, ``struct` `node *n) ` `{ ` `  ``// step 1 of the above algorithm  ` `  ``if``( n->right != NULL ) ` `    ``return` `minValue(n->right); ` ` `  `  ``// step 2 of the above algorithm ` `  ``struct` `node *p = n->parent; ` `  ``while``(p != NULL && n == p->right) ` `  ``{ ` `     ``n = p; ` `     ``p = p->parent; ` `  ``} ` `  ``return` `p; ` `} ` ` `  `/* Given a non-empty binary search tree, return the minimum data   ` `    ``value found in that tree. Note that the entire tree does not need ` `    ``to be searched. */` `struct` `node * minValue(``struct` `node* node) { ` `  ``struct` `node* current = node; ` `  `  `  ``/* loop down to find the leftmost leaf */` `  ``while` `(current->left != NULL) { ` `    ``current = current->left; ` `  ``} ` `  ``return` `current; ` `} ` ` `  `/* Helper function that allocates a new node with the given data and  ` `    ``NULL left and right pointers. */` `struct` `node* newNode(``int` `data) ` `{ ` `  ``struct` `node* node = (``struct` `node*) ` `                       ``malloc``(``sizeof``(``struct` `node)); ` `  ``node->data   = data; ` `  ``node->left   = NULL; ` `  ``node->right  = NULL; ` `  ``node->parent = NULL; ` `   `  `  ``return``(node); ` `} ` ` `  `/* Give a binary search tree and a number, inserts a new node with     ` `    ``the given number in the correct place in the tree. Returns the new ` `    ``root pointer which the caller should then use (the standard trick to  ` `    ``avoid using reference parameters). */` `struct` `node* insert(``struct` `node* node, ``int` `data) ` `{ ` `  ``/* 1. If the tree is empty, return a new, ` `      ``single node */` `  ``if` `(node == NULL) ` `    ``return``(newNode(data)); ` `  ``else` `  ``{ ` `    ``struct` `node *temp;   ` ` `  `    ``/* 2. Otherwise, recur down the tree */` `    ``if` `(data <= node->data) ` `    ``{     ` `         ``temp = insert(node->left, data); ` `         ``node->left  = temp; ` `         ``temp->parent= node; ` `    ``} ` `    ``else` `    ``{ ` `        ``temp = insert(node->right, data); ` `        ``node->right = temp; ` `        ``temp->parent = node; ` `    ``}     ` `  `  `    ``/* return the (unchanged) node pointer */` `    ``return` `node; ` `  ``} ` `}  ` `  `  `/* Driver program to test above functions*/` `int` `main() ` `{ ` `  ``struct` `node* root = NULL, *temp, *succ, *min; ` ` `  `  ``//creating the tree given in the above diagram ` `  ``root = insert(root, 20); ` `  ``root = insert(root, 8); ` `  ``root = insert(root, 22); ` `  ``root = insert(root, 4); ` `  ``root = insert(root, 12); ` `  ``root = insert(root, 10);   ` `  ``root = insert(root, 14);     ` `  ``temp = root->left->right->right; ` ` `  `  ``succ =  inOrderSuccessor(root, temp); ` `  ``if``(succ !=  NULL) ` `    ``printf``(````\" Inorder Successor of %d is %d \"````, temp->data, succ->data);     ` `  ``else` `    ``printf``(````\" Inorder Successor doesn't exit\"````); ` ` `  `  ``getchar``(); ` `  ``return` `0; ` `} `\n\n## Java\n\n `// Java program to find minimum value node in Binary Search Tree ` ` `  `// A binary tree node ` `class` `Node { ` ` `  `    ``int` `data; ` `    ``Node left, right, parent; ` ` `  `    ``Node(``int` `d) { ` `        ``data = d; ` `        ``left = right = parent = ``null``; ` `    ``} ` `} ` ` `  `class` `BinaryTree { ` ` `  `    ``static` `Node head; ` ` `  `    ``/* Given a binary search tree and a number,  ` `     ``inserts a new node with the given number in  ` `     ``the correct place in the tree. Returns the new  ` `     ``root pointer which the caller should then use  ` `     ``(the standard trick to avoid using reference  ` `     ``parameters). */` `    ``Node insert(Node node, ``int` `data) { ` ` `  `        ``/* 1. If the tree is empty, return a new,      ` `         ``single node */` `        ``if` `(node == ``null``) { ` `            ``return` `(``new` `Node(data)); ` `        ``} ``else` `{ ` ` `  `            ``Node temp = ``null``; ` `             `  `            ``/* 2. Otherwise, recur down the tree */` `            ``if` `(data <= node.data) { ` `                ``temp = insert(node.left, data); ` `                ``node.left = temp; ` `                ``temp.parent = node; ` ` `  `            ``} ``else` `{ ` `                ``temp = insert(node.right, data); ` `                ``node.right = temp; ` `                ``temp.parent = node; ` `            ``} ` ` `  `            ``/* return the (unchanged) node pointer */` `            ``return` `node; ` `        ``} ` `    ``} ` ` `  `    ``Node inOrderSuccessor(Node root, Node n) { ` ` `  `        ``// step 1 of the above algorithm  ` `        ``if` `(n.right != ``null``) { ` `            ``return` `minValue(n.right); ` `        ``} ` ` `  `        ``// step 2 of the above algorithm ` `        ``Node p = n.parent; ` `        ``while` `(p != ``null` `&& n == p.right) { ` `            ``n = p; ` `            ``p = p.parent; ` `        ``} ` `        ``return` `p; ` `    ``} ` ` `  `    ``/* Given a non-empty binary search tree, return the minimum data   ` `     ``value found in that tree. Note that the entire tree does not need ` `     ``to be searched. */` `    ``Node minValue(Node node) { ` `        ``Node current = node; ` ` `  `        ``/* loop down to find the leftmost leaf */` `        ``while` `(current.left != ``null``) { ` `            ``current = current.left; ` `        ``} ` `        ``return` `current; ` `    ``} ` ` `  `    ``// Driver program to test above functions ` `    ``public` `static` `void` `main(String[] args) { ` `        ``BinaryTree tree = ``new` `BinaryTree(); ` `        ``Node root = ``null``, temp = ``null``, suc = ``null``, min = ``null``; ` `        ``root = tree.insert(root, ``20``); ` `        ``root = tree.insert(root, ``8``); ` `        ``root = tree.insert(root, ``22``); ` `        ``root = tree.insert(root, ``4``); ` `        ``root = tree.insert(root, ``12``); ` `        ``root = tree.insert(root, ``10``); ` `        ``root = tree.insert(root, ``14``); ` `        ``temp = root.left.right.right; ` `        ``suc = tree.inOrderSuccessor(root, temp); ` `        ``if` `(suc != ``null``) { ` `            ``System.out.println(``\"Inorder successor of \"` `+ temp.data +  ` `                                                      ``\" is \"` `+ suc.data); ` `        ``} ``else` `{ ` `            ``System.out.println(``\"Inorder successor does not exist\"``); ` `        ``} ` `    ``} ` `} ` ` `  `// This code has been contributed by Mayank Jaiswal `\n\n## Python\n\n `# Python program to find the inroder successor in a BST ` ` `  `# A binary tree node  ` `class` `Node: ` ` `  `    ``# Constructor to create a new node ` `    ``def` `__init__(``self``, key): ` `        ``self``.data ``=` `key  ` `        ``self``.left ``=` `None` `        ``self``.right ``=` `None` ` `  `def` `inOrderSuccessor(root, n): ` `     `  `    ``# Step 1 of the above algorithm ` `    ``if` `n.right ``is` `not` `None``: ` `        ``return` `minValue(n.right) ` ` `  `    ``# Step 2 of the above algorithm ` `    ``p ``=` `n.parent ` `    ``while``( p ``is` `not` `None``): ` `        ``if` `n !``=` `p.right : ` `            ``break`  `        ``n ``=` `p  ` `        ``p ``=` `p.parent ` `    ``return` `p ` ` `  `# Given a non-empty binary search tree, return the  ` `# minimum data value found in that tree. Note that the ` `# entire tree doesn't need to be searched ` `def` `minValue(node): ` `    ``current ``=` `node ` ` `  `    ``# loop down to find the leftmost leaf ` `    ``while``(current ``is` `not` `None``): ` `        ``if` `current.left ``is` `None``: ` `            ``break` `        ``current ``=` `current.left ` ` `  `    ``return` `current ` ` `  ` `  `# Given a binary search tree and a number, inserts a ` `# new node with the given number in the correct place ` `# in the tree. Returns the new root pointer which the ` `# caller should then use( the standard trick to avoid  ` `# using reference parameters) ` `def` `insert( node, data): ` ` `  `    ``# 1) If tree is empty then return a new singly node ` `    ``if` `node ``is` `None``: ` `        ``return` `Node(data) ` `    ``else``: ` `        `  `        ``# 2) Otherwise, recur down the tree ` `        ``if` `data <``=` `node.data: ` `            ``temp ``=` `insert(node.left, data) ` `            ``node.left ``=` `temp  ` `            ``temp.parent ``=` `node ` `        ``else``: ` `            ``temp ``=` `insert(node.right, data) ` `            ``node.right ``=` `temp  ` `            ``temp.parent ``=` `node ` `         `  `        ``# return  the unchanged node pointer ` `        ``return` `node ` ` `  ` `  `# Driver progam to test above function ` ` `  `root  ``=` `None` ` `  `# Creating the tree given in the above diagram  ` `root ``=` `insert(root, ``20``) ` `root ``=` `insert(root, ``8``); ` `root ``=` `insert(root, ``22``); ` `root ``=` `insert(root, ``4``); ` `root ``=` `insert(root, ``12``); ` `root ``=` `insert(root, ``10``);   ` `root ``=` `insert(root, ``14``);     ` `temp ``=` `root.left.right.right  ` ` `  `succ ``=` `inOrderSuccessor( root, temp) ` `if` `succ ``is` `not` `None``: ` `    ``print` ```\" Inorder Successor of %d is %d \"``` ` ` `            ``%``(temp.data , succ.data) ` `else``: ` `    ``print` ```\" Inorder Successor doesn't exist\"``` ` `  `# This code is contributed by Nikhil Kumar Singh(nickzuck_007) `\n\nOutput of the above program:\nInorder Successor of 14 is 20\n\nTime Complexity: O(h) where h is height of tree.\n\nMethod 2 (Search from root)\nParent pointer is NOT needed in this algorithm. The Algorithm is divided into two cases on the basis of right subtree of the input node being empty or not.\n\nInput: node, root // node is the node whose Inorder successor is needed.\noutput: succ // succ is Inorder successor of node.\n\n1) If right subtree of node is not NULL, then succ lies in right subtree. Do following.\nGo to right subtree and return the node with minimum key value in right subtree.\n2) If right sbtree of node is NULL, then start from root and us search like technique. Do following.\nTravel down the tree, if a node’s data is greater than root’s data then go right side, otherwise go to left side.\n\n `struct` `node * inOrderSuccessor(``struct` `node *root, ``struct` `node *n) ` `{ ` `    ``// step 1 of the above algorithm ` `    ``if``( n->right != NULL ) ` `        ``return` `minValue(n->right); ` ` `  `    ``struct` `node *succ = NULL; ` ` `  `    ``// Start from root and search for successor down the tree ` `    ``while` `(root != NULL) ` `    ``{ ` `        ``if` `(n->data < root->data) ` `        ``{ ` `            ``succ = root; ` `            ``root = root->left; ` `        ``} ` `        ``else` `if` `(n->data > root->data) ` `            ``root = root->right; ` `        ``else` `           ``break``; ` `    ``} ` ` `  `    ``return` `succ; ` `} `\n\nThanks to R.Srinivasan for suggesting this method.\n\nTime Complexity: O(h) where h is height of tree.\n\nPlease write comments if you find anything incorrect, or you want to share more information about the topic discussed above.\n\nThis article is attributed to GeeksforGeeks.org\n\ncode\n\nload comments" ]

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[ "# Efficiency improvement of the fixed-complexity sphere decoder.\n\n1. Introduction\n\nMultiple-input multiple-output (MIMO) spatial multiplexing techniques linearly increase the channel capacity without requiring additional spectral resources which are not only expensive but also scarce . At the transmitter side, signals are transmitted simultaneously from sufficiently-separated antennas, whereas at the receiver side signals are recovered by means of detection algorithms.\n\nThe maximum-likelihood detector (MLD) is the optimum receiver for MIMO spatial multiplexing systems . Nevertheless, MLD becomes infeasible for systems employing large number of transmit antennas and high order modulations due to its exponential complexity increase. Therefore, suboptimal detection schemes were proposed in the literature to achieve a tradeoff between performance and complexity as below.\n\nLinear detection schemes including zero-forcing (ZF) and minimum mean square error (MMSE) treat the received signal by the pseudo-inverse and a regularized pseudo-inverse of the channel matrix, respectively. Although linear detection schemes are simple in terms of computational complexity, they lead to degradation in the performance due to the independent processing of the transmitted signals .\n\nSuccessive interference cancellation (SIC) schemes detect signals iteratively, where already-detected signals are subtracted out from the received vector leading to a system with fewer interferers. ZF and MMSE VBLAST (vertical Bell Laboratories layered space-time) detection schemes , which are the first proposed SIC schemes, outperform linear algorithms. Nonetheless, VBLAST detection schemes have high computational complexity because of requiring a matrix pseudo-inversion at each iterative detection stage. Low complexity SIC detection algorithms based on the QR decomposition (QRD) of the channel matrix were proposed in the literature . Although QRD-based SIC schemes are favorable in terms of computational complexity, their performance is still far from the optimum performance of the MLD.\n\nSeveral detection algorithms have been proposed in the literature achieving quasi-ML performance while requiring lower computational complexity. Among these suboptimal algorithms, sphere decoding (SD) achieves quasi-ML performance with polynomial average computational complexity . It is shown; however, that SD has variable complexity which depends on the channel condition and the instantaneous noise power, where the worst-case complexity of SD is consequently comparable with that of MLD . That is, the extreme-case complexity of SD is exponential. Also, in terms of implementation complexity, SD is inefficient due to its sequential nature in the tree search stage that limits the possibility of pipelining, where consequently the detection latency is increased .\n\nTo overcome the two aforementioned drawbacks of SD, fixed-complexity sphere decoder (FSD) was proposed in . Analysis on the error performance of the FSE was introduced in . The main idea of the FSD is to perform the search over a fixed number of hypotheses of the transmitted vector independently of the noise power and the channel condition. The complexity of FSD is, therefore, fixed and known prior to the tree search phase of the detection algorithm. Moreover, the hypotheses are tested in parallel leading to reduction in the algorithm latency, i.e., increase in the detector throughput. To achieve a quasi-ML performance, FSD requires a specific signal ordering in which signals suffering from the highest noise amplification are detected in levels where all signal candidates are retained. As a consequence, weak signals don't affect the performance of the detection algorithm. The conventional VBLAST algorithm is, therefore, used to obtain accurate signal ordering . Despite that VBLAST ordering algorithm obtains a precise signal ordering, its complexity is shown to be O([N.sup.4]), with N as the number of transmit antennas. This is computationally heavy when signal ordering is frequently performed .\n\nOur original contributions: In this paper, we propose two schemes to reduce the complexity of FSD algorithm in the ordering and tree-search phases as follows :\n\n* QRD-based FSD signal ordering: In the ordering stage, we propose to embed the signal ordering required by the FSD in the QRD. Thus, the proposed ordering scheme requires a few additional complex multiplications and additions compared to the unsorted QRD algorithm with negligible degradation in the performance. Therefore, the proposed ordering scheme requires only a small fraction of the computational effort required by the conventional FSD-VBLAST sorting algorithm. ZF and MMSE performance-based criteria are used in the proposed FSD-ZF-SQRD and FSD-MMSE-SQRD, respectively, where SQRD refers to sorted QR decomposition.\n\n* Analysis of the computational complexity: We give analytical formulas for the computational complexities of the conventional and proposed ordering schemes.\n\n* Threshold-based complexity reduction: In the tree-search stage, based on the reliability of the weakest received signal, i.e., signal with the lowest received signal to noise ratio (SNR), the number of retained symbol replicas is controlled. Thus, when the received SNR of the weakest signal is larger than a pre-defined threshold, unnecessary computations by the FSD are avoided while achieving a quasi-ML performance.\n\nThe rest of this paper is organized as follows. In section II, we introduce the system model and review the FSD. In section III, we introduce the proposed FSD-ZF-SQRD and FSD-MMSE-SQRD schemes, and in section IV we derive the formulas for the computational complexities of the introduced sorting algorithms. In section V, threshold-based complexity reduction scheme is proposed. Simulation results are shown in section VI, and conclusions are drawn in section VII.\n\n2. System Description and Review of the FSD\n\nWe consider a MIMO multiplexing system employing N transmit and M receive antennas, where M [greater than or equal to] N. Under the assumption of narrowband flat-fading channel, the received vector r [member of] [C.sup.Mx1] is given by:\n\nr = Hx + n, (1)\n\nwhere x [member of] [C.sup.Nx1] is the transmitted vector whose elements are drawn from the modulation set [OMEGA] satisfying E[xx*] = [I.sub.N], where [I.sub.N] is the NxN identity matrix. n [member of] [C.sup.Mx1] is the additive white Gaussian noise with zero mean and unity variance. H [member of] [C.sup.MxN] is the channel matrix whose row-column element [h.sub.i,j] the complex channel coefficient between the j-th transmit and i-th receive antennas, is modeled as circularly symmetric complex Gaussian random variable with zero mean and unity variance.\n\nWorking on x, the matrix H generates the lattice\n\n[DELTA](H) = {Hx :x[member of][[OMEGA].sup.N]}, (2)\n\nwhere the columns of H, i.e., ([h.sub.1], [h.sub.2], ..., [h.sub.N]), are the bases of the lattice. The received vector r is then represented as the lattice point Hx perturbed by the noise vector n. As a consequence, the MLD finds the closest lattice point Hx to the received vector r, where x is the estimate of the transmitted vector x. That is,\n\n[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3)\n\nSD restricts the search in (3) to the lattice points that reside in the hyper-sphere of radius d and centred at the received vector r. Therefore,\n\n[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (4)\n\nTo perform the search in (4) successively, the channel matrix H is factorized into the multiplication of a unitary matrix Q and an upper triangular matrix R.\n\n[FIGURE 1 OMITTED]\n\nTherefore, the search problem in (4) is simplified to:\n\n[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (5)\n\nwhere y = [Q.sup.H] r. The accumulative metric in (5) is then calculated successively, where the metric at the N-th detection level is given by:\n\n[E.sub.N] = [([y.sub.N] - [R.sub.N,N][[??].sub.N]).sup.2], (6)\n\nand the accumulative metric at the (N-1)-th detection level is given as follows:\n\n[E.sub.N-1] = [E.sub.N] +([y.sub.N-1] - [R.sub.N-1,N-1] [[??].sub.N-1] - [R.sub.N-1,N][x.sub.N]).sup.2], (7)\n\nand so on.\n\nAlthough the average complexity of SD is known to be polynomial, its extreme-case complexity is believed to be exponential, particularly for high noise power and when the channel matrix is ill-conditioned. Also, due to the sequential nature of the SD, it can't be pipelined leading to high latency in the detection algorithm.\n\nTo overcome the aforementioned problems of the SD, FSD was proposed by Barbero et al. FSD achieves a quasi-ML performance by performing the following two-stage tree search:\n\n* Fixed expansion stage: In the first p detection levels, a full expansion of the retained nodes is performed, where all the resulting branches are retained for the next levels.\n\n* Single expansion stage: All retained branches in the precedent level are extended independently to all possible nodes. The accumulative metrics of the resulting branches are calculated as in (5), and only the branch with the smallest accumulative metric is retained for the next level. This can be done using Schnorr-Euchner strategy , where only the metric of the node with the smallest accumulative metric is calculated.\n\nFig. 1 shows an example of the FSD for quadratic phase shift keying (QPSK) signalling, N = 3, and p = 1. Each node represents a constellation point belonging to the QPSK constellation set [OMEGA] = At the 3rd level, all the symbol replica candidates for [x.sub.3] are retained for the next detection levels, and the metric of each node, given inside the node in Fig. 1, is calculated as in (6). At the 2nd level, each retained node at the precedent level is extended independently from others, and the best child node is selected such that the accumulative metric is minimized. This strategy is repeated up to the 1st level, where the node with the smallest accumulative metric and its connected nodes are considered as the final estimate of the transmitted vector. For the example in Fig. 1, [??] = and is indicated by the thick branches and nodes.\n\nBecause all possible symbol candidates are retained in the first p levels, the reliability of signals does not affect the final detection performance. Therefore, signals with the least robustness are detected in the full expansion stage. On the other hand, in the remaining (N - p) levels, signals are sorted based on their reliability, where signals with the least noise amplification are detected first.\n\nIn the conventional FSD, VBLAST signal sorting is iteratively used to decide the order in which signals are detected. Although reduced complexity VBLAST ordering schemes were proposed in the literature , it is still complex compared to other QRD-based signal sorting schemes.\n\nIn the following section, we propose a low-complexity ordering scheme that achieves the required performance while reducing the computational complexity. Also, computational complexity comparison among different ordering schemes is performed.\n\n3. Proposed QRD-based FSD Signal Ordering Schemes\n\nIn the pre-detection stage, the channel matrix is decomposed into the multiplication of a unitary matrix Q and an upper triangular matrix R. Due to the structure of the upper triangular matrix R, the detection is performed successively starting by the last component of the vector x, i.e., [x.sub.N]. In the conventional QR-decomposition, the diagonal elements of R are obtained in opposite order of signal detection. In the sorted QR-decomposition (SQRD), the columns of the channel matrix are reordered prior to each orthogonalization step, where [R.sub.1,1] is minimized over all the columns of the channel matrix, followed by [R.sub.2,2], and so on. Unfortunately, the SQRD does not sort signals in the order required by FSD.\n\nIn FSD, the efficient VBLAST algorithm is used to obtain the signal ordering prior to the QR-decomposition of the channel matrix. Despite that FSD achieves a quasi-ML performance using VBLAST ordering, the computational effort required by the ordering stage becomes exhaustive as the channel varies rapidly, where ordering is performed frequently.\n\nThe main idea of the proposed signal ordering algorithm is to obtain the (p+1)th minimum diagonal element of R prior to the orthogonalization step. Now, let p = 1, then [R.sub.1,1] is simply the norm of the channel matrix column whose order is the second in the norm sense. So, the column of H with the 2nd minimum norm is permuted with the first column. Thereafter, [R.sub.2,2] is computed in the same manner by considering only the remaining (N-1) columns, etc. At the last iteration, the order of signal detection is same as that required by FSD algorithm.\n\nFig. 2 depicts an example of the proposed FSD-SQRD for p = 1, where p is the permutation vector. At the first iteration, the column with the 2nd minimum norm is permuted with the first column of H and the remaining (N-1) vectors are orthogonalized. In the next iteration, the orthogonalized column with the 2nd minimum norm is permuted with the first remaining orthogonalized column. At the [(N - 1).sup.th] iteration, the remaining two columns correspond to the strongest and the weakest signals, i.e., the largest and the smallest received signal to noise ratio (SNR), respectively. Thus, the selected column corresponds to the signal with the largest SNR and the remaining one corresponds to the weakest signal. By doing so, the required order by FSD is obtained.\n\nThe introduced FSD signal sorting scheme is applied with the ZF and the MMSE performance-based criteria in the proposed FSD-ZF-SQRD and FSD-MMSE-SQRD algorithms, respectively. To apply the FSD-MMSE-SQRD, the channel matrix is extended to take into consideration the noise effect; that is,\n\n[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (8)\n\nTable 1 gives a detailed algorithmic description of the proposed FSD-MMSE-SQRD algorithm where the input n equals (p + 1). [0.sup.N] is the NXNmatrix whose elements are all zeros, and mink is the k-th minimum. The FSD-ZF-SQRD is obtained by simply replacing [H.sub.ext] by H, and line (7) in Table 1 by the following line.\n\n(7) Exchange columns i and ki in Q, R, norm, and p.\n\n4. Analysis on the Computational Complexity\n\nIn this section, the computational complexities of the introduced signal ordering schemes are investigated. The complex floating points (flops) are given as a function of N and M, where the complex addition and multiplication operations are counted as one and three flops, respectively. Other operations are tracked back to the complex operations. For instance, real addition requires half a flop.\n\nTable 2 gives the formulas for the computational complexities of the aforementioned signal sorting schemes. Hassibi's sorting scheme is a low complexity VBLAST algorithm used herein for comparison and the assorted-ZF-QRD refers to the zero-forcing QRD without signal ordering.\n\n[FIGURE 2 OMITTED]\n```Table 1. The proposed FSD-MMSE-SQRD algorithm.\n\nInput: R = [0.sub.N], Q = [H.sub.ext], p = [1,2, ... N], n\n\n1: for i = 1 to A^ do\n2: [norm.sub.i] = [||[q.sub.i]||.sup.2]\n3: end for\n4: for i = 1 to N do\n5: k = min (\". Ar - i + 1)\n6: [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]\n7: Exchange columns i and [k.sub.i] in R, norm, and p, and in\nthe first (M + i -1) rows of Q\n8: [R.sub.i,i]\n9: [q.sub.i] = [q.sub.i]/[R.sub.i,i]\n10: for j = i + 1 to N do\n11: [R.sub.i,j] = [q.sup.H.sub.i] [q.sub.j]\n12: [q.sub.i] = [q.sub.j] - [R.sub.i,j] * [q.sub.i]\n13: [norm.sub.j] = [norm.sub.j] - [R.sup.2.sub.i,j]\n14: end for\n15: end for\nOutput: Q. R, p\n```\n\n[FIGURE 3 OMITTED]\n\nFig. 3 shows the computational complexities of the proposed algorithms with the conventional ones. Clearly, the proposed algorithms require much lower computational efforts compared to Hassibi's scheme. For instance, in 8X8 MIMO system, the proposed FSD-ZF-SQRD and FSD-MMSE-SQRD only require 19.5% and 26.3%, respectively, of the computational efforts required by Hassibi's scheme. The proposed signal sorting algorithms require only 1.25 x ([N.sup.2] - N) additional flops compared to the assorted algorithms, i.e., without signal ordering.\n\nThe complexity of the conventional norm sorting FSD-norm is considered to be equivalent to that of the assorted QRD algorithm.\n\n5. Threshold-based Complexity Reduction for the FSD\n\nThe first detected signal in the FSD, i.e., [x.sub.N], is the signal that suffers the largest noise amplification. Therefore, all the remaining (N-1) signals are more robust than the first ranked signal. Thus, if xN experiences low noise amplification, the number of retained symbol replica candidates can be reduced without affecting the final performance. In contrast to the common adaptive symbol replica selection proposed in , FSD can have more reliable decision about the number of retained symbol replicas because its decision is based on the reliability of the signal suffering the largest noise amplification, whereas the algorithm in does not take into consideration the weak signals.\n\nDue to the structure of the matrix R, the signal [x.sb.N] is interference-free, therefore, the robustness of the signal is directly proportional to |[R.sub.N,N]|. Thus, based on the value of |[R.sub.N,N]|, we decide the number of retained symbol replicas at the first full expansion stage. To accomplish that, we introduce three threshold vectors and the corresponding number of retained symbol replicas for the first full expansion level as follows:\n\n[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (9)\n\n[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (10)\n\nand\n\n[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (11)\n\nDue to the fact that\n\nP{[R.sub.N,N] [less than or equal to] 0.4} <P {[R.sub.N,N] [less than or equal to] 0.5} <P {[R.sub.N,N] [less than or equal to] 0.75}, (12)\n\nmore complexity reduction can be achieved when the threshold vector [p.sub.3] is used as compared to the case of using the threshold vector [P.sub.2] or [P.sub.1]. This is because the three aforementioned probabilities in (12) are the cases in which the proposed approach fails to achieve any complexity reduction using [P.sub.3], [P.sub.2], and [P.sub.1], respectively. Note that we obtained the threshold vectors using intensive simulations. In practice, the channel models are usually known, which indicates that an offline optimization of these threshold vectors is possible. Another alternative is to compute the probability density function (pdf) of [R.sub.N,N] and set the threshold vector as a trade-off between complexity reduction and performance degradation. The later option is suitable for systems that have preambles. These preambles can be used in the training phase to obtain optimized values of the threshold vector.\n\n6. Simulation Results and Discussions\n\nIn this section, we investigate the bit error rate (BER) performance of the FSD in 4X4 MIMO spatial multiplexing system. Transmitted symbols are modulated using 4-QAM, 16-QAM, or 64-QAM mappers. The channel state information (CSI) is considered to be perfectly known at the receiver and unknown at the transmitter.\n\nFig. 4 shows the BER performance of the FSD for different signal ordering schemes in a 4x4 MIMO system using 4-QAM modulation. Without signal ordering, the performance of FSD is highly degraded and the diversity order, i.e., the slope of the BER curve, is lower than that of SD. When applying FSD-norm signal ordering, the BER performance is improved by approximately 2.7dB at BER of 10-4 whereas no improvement in the diversity order is achieved. On the other hand, when FSD-VBLAST ordering is applied, FSD algorithm achieves the same diversity order of SD with a 0.2dB performance lagging at BER = [10.sup.-4]. When the proposed FSD-ZF-SQRD signal ordering is employed, FSD attains the diversity order of SD with a degradation of less than 0.6dB compared to SD at BER of [10.sup.-4]. At the same target BER, FSD lags the optimum performance by only 0.4dB when the proposed FSD-MMSE-SQRD sorting algorithm is employed. This small degradation in the BER performance is tolerable considering the low computational complexity of the proposed algorithms compared with the conventional FSD-VBLAST signal ordering. Due to space limitation, we consider the proposed FSD-ZF-SQRD for the following comparisons.\n\nFig. 5 depicts the performance of the FSD employing the proposed FSD-ZF-SQRD algorithm using different QAM schemes. At BER of 10-4, a degradation of 0.6dB, 0.3dB, and 0.25dB are remarked in the BER performance of the FSD when employing FSD-ZF-SQRD in a 4x4 system using 4-QAM, 16-QAM, and 64-QAM, respectively. That is, as the modulation order increases, the degradation due to the use of the proposed FSD-SQRD algorithms vanishes compared with the performance of FSD employing the complex FSD-VBLAST sorting scheme.\n\n[FIGURE 4 OMITTED]\n\n[FIGURE 5 OMITTED]\n\n[FIGURE 6 OMITTED]\n\nFig. 6 shows the BER performance of the FSD when the number of retained symbol replicas at the first detection level is controlled by the threshold vectors [P.sub.1], [P.sub.2], or [P.sub.3]. In the case of using [P.sub.1], the complexity reduction does not have any effect on the achieved performance of FSD for the different applied modulation schemes. By contrast, when the threshold vector [P.sub.2] is used, the performance of FSD using 4-QAM is degraded at medium Eb/No values, whereas the performance is still intact when 16-QAM or 64-QAM is used.\n\nTherefore, the vector [P.sub.3], which is tighter than [P.sub.1] and [P.sub.2], is not used with 4-QAM signalling scheme. In the case of using [P.sub.3], we remark a slight degradation in the BER performance when 16-QAM or 64-QAM is used. For instance, at BER of [10.sup.-3], the maximum degradation is about 0.5dB and 0.3dB for 16-QAM and 64-QAM, respectively. This degradation in the performance is tolerable due to the remarkable reduction in the computational complexity.\n\nTable 3 gives the achieved percentage of reduction in the computational complexity of FSD for different modulation schemes and threshold vectors. When the threshold vector [P.sub.1] is used, about 12% of complexity reduction is achieved for all modulation schemes. On the other side, when [P.sub.2] is used, the reduction in the complexity of FSD increases as the modulation order increases attaining 31.1% reduction in the case of 64-QAM. In the case of [P.sub.3], 37.14% and 46.33% of the computational efforts required by the conventional FSD are avoided when 16-QAM and 64-QAM are used, respectively.\n\nWe conclude that for high modulation orders, the set of candidates for each symbol becomes large. Therefore, when we apply a high complexity reduction, e.g., using [P.sub.3], there will still be enough retained candidates to obtain a quasi-ML solution. However, in the case of low modulation orders, e.g., QPSK, when the threshold vector [P.sub.3] is applied, there will be a few retained candidates at each detection level. These retained candidates might not be sufficient to obtain a quasi-ML performance.\n\n7. Conclusions\n\nIn this paper, we proposed two techniques to reduce the computational complexity of FSD in the ordering and tree search stages, respectively. In the ordering stage, we proposed QRD-based FSD signal ordering scheme (FSD-SQRD) that leads to quasi-ML performance while requiring only a fraction of the computational complexity of the conventional FSD-VBLAST ordering algorithm. The proposed ordering algorithm is then extended to the MMSE case, where better performance is obtained. In the tree-search stage, we introduced a complexity reduction approach for FSD algorithm based on the reliability of the signal with the lowest received SNR value. That is, if the lowest SNR of the received signals is higher than a pre-defined threshold, the number of retained symbol replicas is reduced leading to the reduction in the computational complexity of the FSD algorithm with negligible degradation in the BER performance. The proposed improvements for FSD remarkably reduce the computational efforts required to achieve quasi-ML performance. As a consequence, FSD can be considered as a prominent detection scheme for next generation communication systems.\n\nReceived November 25, 2010; revised January 14, 2011; accepted February 5, 2011; published March 31, 2011\n\nReferences\n\n E. Telatar, \"Capacity of multi-antenna Gaussian channels,\" European Transactions on Telecommunications, vol. 10, pp. 585-595, Dec. 1999. Article (CrossRef Link)\n\n W. Van Etten, \"Maximum likelihood receiver for multiple channel transmission systems,\" IEEE Transactions on Communications, pp. 276-283, Feb. 1976. Article (CrossRef Link)\n\n Q. H. Spencer et al., \"An introduction to the multi-user MIMO downlink,\" IEEE Communications Magazine, vol. 42, no. 10, pp. 60-67, Oct. 2004. Article (CrossRef Link)\n\n D. Shiu and J. Kahn, \"Layered space-time codes for wireless communications using multiple transmit antennas,\" in Proc. the International Conference on Communications, Vancouver, Canada, pp. 436-440, Jun. 1999. Article (CrossRef Link)\n\n D. Wubben et al., \"Efficient algorithm for decoding layered space-time codes,\" Electronics Letters, vol. 37, no. 22, pp. 1348-1350, Oct. 2001. Article (CrossRef Link)\n\n D. Wubben et al., \"MMSE extension of V-BLAST based on sorted QR decomposition,\" in Proc. IEEE Vehicular Technology Conference, Lake Buena Vista, USA, pp. 508-512, Oct. 2003. Article (CrossRef Link)\n\n E. Agrell et al., \"Closest point search in lattices,\" IEEE Transactions on Information Theory, vol. 48, no. 8, pp. 2201-2214, Nov. 2002. Article (CrossRef Link)\n\n B. Hassibi and H. Vikalo, \"On the expected complexity of sphere decoding,\" in Proc. Asilomar Conference on Signals, Systems and Computers, Pacific Grove, USA, pp. 1051-1055, Nov. 2001. Article (CrossRef Link)\n\n J. Jalden and B. Ottersten, \"On the complexity of sphere decoding in digital communications,\" IEEE Transactions on Signal Processing, vol. 53, no. 4, pp. 1474-1484, Apr. 2005. Article (CrossRef Link)\n\n L. Barbero and J. Thompson, \"Fixing the complexity of the sphere decoder for MIMO detection,\" IEEE Trans. on Wireless Communications, vol. 7, no. 6, pp. 2131-2142, Jun. 2008. Article (CrossRef Link)\n\n L. Barbero and J. Thompson, \"Extending a fixed-complexity sphere decoder to obtain likelihood information for turbo-MIMO systems,\" IEEE Trans. on Vehicular Technology, vol. 57, no. 5, pp. 2804-2814, Sep. 2008. Article (CrossRef Link)\n\n J. Jalden et al., \"Full diversity detection in MIMO systems with a fixed-complexity sphere decoder,\" in Proc. of IEEE International Conference on Acoustics, Speech, and Signal Processing, Honolulu, USA, pp. 49-52, Apr. 2007. Article (CrossRef Link)\n\n J. Jalden et al., \"The error probability of the fixed-complexity sphere decoder,\" IEEE Trans. On Signal Processing, vol. 57, no. 7, pp. 2711-2720, Jul. 2009. Article (CrossRef Link)\n\n P. Wolniansky et al., \"V-BLAST: An architecture for realizing very high data rates over the rich-scattering wireless channel,\" in Proc. URSI International Symposium on Signals, Systems, and Electronics, Pisa, Italy, pp. 295-300, Oct. 1998. Article (CrossRef Link)\n\n B. Hassibi, \"An efficient square-root algorithm for BLAST,\" in Proc. IEEE International Conf. Acoustics, Speech, Signal Processing, Istanbul, Turkey, pp. 737-740, Jun. 2000. Article (CrossRef Link)\n\n M. Mohaisen and K.H. Chang, \"On improving the efficiency of the fixed-complexity sphere decoder,\" in Proc. IEEE VTC--Fall, Sep. 2009, Session 3B-1. Article (CrossRef Link)\n\n C. Schnorr and M. Euchner, \"Lattice basis reduction: Improved practical algorithms and solving subset sum problems,\" Mathematical Programming, vol. 66, pp. 181-199, Sep. 1994. Article (CrossRef Link)\n\n H. Zhu, Z. Lei and F. P. S. Chin, \"An improved square-root algorithm for BLAST,\" IEEE Signal Processing Letters, vol. 11, no. 9, pp. 772-775, Sep. 2004. Article (CrossRef Link)\n\n K. J. Kim et al., \"AQRD-M/Kalman filter based detection and channel estimation algorithm for MIMO-OFDM systems,\" IEEE Transactions on Wireless Communications, vol. 4, no. 2, pp. 710-721, Mar. 2005. Article (CrossRef Link)\n\nDOI: 10.3837/tiis.2011.03.002\n\nManar Mohaisen and KyungHi Chang\n\nThe Graduate School of IT & T, Inha University 402-751 Incheon, Republic of Korea [e-mail: [email protected], [email protected]]\n\nThis work was supported by the National Research Foundation of Korea(NRF) grant funded by the Korea government(MEST) (No. R01-2008-000-20333-0). Part of this paper is presented at IEEE VTC-Fall, 2009.\n\nManar Mohaisen received a B.Eng. in electrical engineering from the University of Gaza (IUG), Gaza, Palestine, in 2001. From 2001 to 2004, he was with the Palestinian Telecommunications Company--JAWWAL, Gaza, Palestine, where he worked as an operation and maintenance engineer, and then as a cell planning engineer. He received his M.S. degree in communication and signal processing from the University of Nice-Sophia Antiplois, Sophia Antipolis, France, in 2005. From March to September 2005, he followed an internship at IMRA Europe Co., Sophia Antipolis, France, as a part of his M.S. degree, where he worked on noise reduction in car environments. In February 2010, he obtained a Ph.D. degree from the Graduate School of Information Technology and Telecommunication, Inha University, Incheon, Korea. Since September 2010, he is with the School of Information Technology Engineering, Korea University of Technology and Education (KUT), Cheonan, where he is an assistant professor. His research interests include 3GPP LTE/LTE-A systems, detection schemes for spatial multiplexing MIMO systems, and precoding techniques for multiuser MIMO systems.\n\nKyungHi Chang received his B.S. and M.S. degrees in electronics engineering from Yonsei University, Seoul, Korea, in 1985 and 1987, respectively. He received his Ph.D. degree in electrical engineering from Texas A&M University, College Station, Texas, in 1992. From 1989 to 1990, he was with the Samsung Advanced Institute of Technology (SAIT) as a member of research staff and was involved in digital signal processing system design. From 1992 to 2003, he was with the Electronics and Telecommunications Research Institute (ETRI) as a principal member of technical staff. During this period, he led the design teams working on the WCDMA UE modem and 4G radio transmission technology (RTT). He is currently with the Graduate School of Information Technology and Telecommunications, Inha University, where he has been a professor since 2003. His current research interests include RTT design for IMT-Advanced system, 3GPP LTE and M-WiMAX system design, cognitive radio, cross-layer design, cooperative relaying systems, RFID/USN, and mobile ad-hoc networks. Dr. Chang has served as a senior member of IEEE since 1998, and as an editor-in-chief of the Korean Institute of Communication Sciences (KICS) proceedings during 2007~2009. Currently, he is an editor-in-chief of the KICS Journal A. He has also served as an editor of ITU-R TG8/1 IMT.MOD, and he is currently an international IT standardization expert of the Telecommunications Technology Association (TTA). He is an recipient of the LG Academic Awards (2006), Haedong Best Paper Awards (2007), IEEE ComSoc Best Paper Awards (2008), and Haedong Academic Awards (2010).\n```Table 2. Computational complexities of the sorting schemes.\n\nOrdering scheme Complexity (flops)\n\nHassibi's 6[N.sup.3] + 12[N.sup.2] M + 40/3 [N.sup.2]\n+ 4NM + 39/2 n - 7M - 30\nFSD-MMSE-SQRD 4/3 [N.sup.3] + 4[N.sup.2] M + 5/4 [N.sup.2]\n- NM - 13/12 N\nFSD-ZF-SQRD 4[N.sup.2] M + 3/4 [N.sup.2] -NM - 3/4 N\nAssorted ZF-QRD 4[N.sup.2] M - 1/2 [N.sup.2] - NM + 1/2 N\n\nTable 3. Achieved reduction in the complexity of FSD by the\nproposed threshold-based approach in a 4x4 MIMO system.\n\nReduced Computational Complexity (%)\n\nThreshold 4-QAM 16-QAM 64-QAM\n\n[P.sub.1] 12.04 12.08 12.09\n[P.sub.2] 28.03 30.52 31.10\n[P.sub.3] Not applied 37.14 46.33\n```\nCOPYRIGHT 2011 KSII, the Korean Society for Internet Information\nNo portion of this article can be reproduced without the express written permission from the copyright holder.\nAuthor:", null, "Printer friendly", null, "Cite/link", null, "Email", null, "Feedback Mohaisen, Manar; Chang, KyungHi KSII Transactions on Internet and Information Systems Report Mar 1, 2011 5132 A bandwidth adaptive path selection scheme in IEEE 802.16 relay networks. Interference-aware downlink resource management for OFDMA femtocell networks. Algorithms Coding theory MIMO communications Signal processing" ]

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[ "• MOHSEN SARBISHEI\n\nArticles written in Pramana – Journal of Physics\n\n• A Bohmian approach to the perturbations of non-linear Klein--Gordon equation\n\nIn the framework of Bohmian quantum mechanics, the Klein--Gordon equation can be seen as representing a particle with mass m which is guided by a guiding wave $\\phi(x)$ in a causal manner. Here a relevant question is whether Bohmian quantum mechanics is applicable to a non-linear Klein--Gordon equation? We examine this approach for $\\phi_{4}(x)$ and sine-Gordon potentials. It turns out that this method leads to equations for quantum states which are identical to those derived by field theoretical methods used for quantum solitons. Moreover, the quantum force exerted on the particle can be determined. This method can be used for other non-linear potentials as well.\n\n• # Editorial Note on Continuous Article Publication\n\nPosted on July 25, 2019" ]

[ null ]

[ "Table 4α   Error limits for measurement current transformers\n\n Class % current error at the given proportion of rated current shown below % phase error at the given proportion of the  rated current shown below 2.0* 1.2 1.00 0.50 0.20 0.10       0.05 2.0* 1.2 1.0 0.5 0.2 0.1 0.05 0.1 0.1 0.1 0.2 0.25 5 5 8 10 0.2 0.2 0.2 0.35 0.50 10 10 15 20 0.5 0.5 0.5 0.75 1.00 30 30 45 60 1.0 1.0 1.0 1.5 2.00 60 60 - 90 120 - 3.0 3.0 3.0 - -       - - _ 120 - 120 - - - 0.1 0.1 0.1 0.2 0.25 0.4 5 - 5 8 10 15 0.2 ext 0.2 0.2 0.35 0.50 0.75 10 - 10 15 20 30 0.5 ext 0.5 0.5 0.75 1.00 1.5 30 - 30 45 60 90 1.0 ext 1.0 1.0 1.5 2.00 60 - 60 - 90 120 - 3.0 ext 3.0 - - 3.0 - - - 120 - - 120 - - -\n\n*ext = 200 %\n\nTable 4b  Error limits for protection current transformers\n\n Accuracy Class +/- percentage Current ratio error +/- Phase error (minutes) % Current 5 20 100 120 5 20 100 120 0.1 0.4 0.2 0.1 0.1 15 8 5 5 0.2 0.75 0.35 0.2 0.2 30 15 10 10 0.5 1.5 0.75 0.5 0.5 90 45 30 30 1.0 3 1.5 1.0 1.0 180 90 60 60\n\nTotal error for nominal error limit current and nominal load is five per cent for 5P and 5Ρ ext CTs and ten per cent for 10P and 10P ext CTs.\n\nThe cross-sectional area of metal and the saturation flux density are sometimes difficult to obtain.\n\nThe latter can be taken as equal to 100 000 lines/Cm2, which is a typical value for modern transformers. To use the formula, V is determined from eqn. 4.1 and Bmax. is then calculated using eqn. 2. If Bmax.\n\nExceeds the saturation density, there could be appreciable errors in the secondary current and the CT selected would not be appropriate.\n\nExample 1.\n\nAssume that a CT with a ratio of 2000/5 is available, having a steel core of high permeability, a cross-sectional area of 3.25 In cm2 and a secondary winding with a resistance of 0.31 Ω. The impedance of the relays, including connections, is 2 Ω. Determine whether the CT would be saturated by a fault of 35 000 A at 50 Hz.\n\nSolution\n\nIf the CT is not saturated, then the secondary current, IL, is\n\n35 000x 5/2000=87.5 A. N= 2000/5 = 400 turns\n\nAnd Vs=87.5x (0.31+2) =202.1 V. Using eqn. 4.2, Bmax, can now be calculated:\n\nBmax = 202.1X108/4.44X50X3.25X400=70 030 lines/ cm2\n\nSince the transformer in this example has a steel core of high permeability, this relatively low value of flux density should not result in saturation.\n\nUsing the magnetization curve\n\nTypical CT excitation curves which are supplied by manufacturers state the r.m.s. current obtained on applying an r.m.s. voltage to the secondary winding, with the primary winding open-circuited.\n\nThe curves give the magnitude of the excitation current required order to obtain a specific secondary voltage.\n\nThe method consists of producing a curve which shows the relationship between the primary and secondary currents for one tap and specified load conditions, such as shown in Figure 4.9.\n\nStarting with any value of secondary current, and with the help of the magnetisation curves, the value of the corresponding primary current can be determined. The process is summarized in the following steps:\n\n(a) Assume a value for IL.\n\n(b) Calculate Vs in accordance with eqn. 4.1.\n\n(c)  Locate the value of Vs on the curve for the tap selected, and find the associated value of the magnetization current, Ie.\n\n(d) Calculate IH/n (=IL + Ie) and multiply this value by n to refer it to the primary side of the CT.\n\n(e)  This provides one point on the curve of IL against IH, and the process is then repeated to obtain other values of IL and the resultant values of IH. By joining the points together the curve of IL against IH  is obtained.", null, "", null, "Figure 4.9 using the magnetization curve\n\na- assume a value for IL.\n\nb-Vs = IL (ZL +ZC+ZB)\n\nc - find Ie from the curve\n\nd - IH=n(I1,+Ie )\n\ne - draw the point on the curve\n\nThis method incurs an error in calculating IH /n by adding Ie and IL together arithmetically and not vectorially, which implies not taking account of the load angle and the magnetizations branch of the equivalent circuit. However, this error is not great and the simplifica­tion snakes it easier to carry out the calculations.\n\nAfter construction, the curve should be checked to confirm that the maximum primary fault current is within the transformer saturation zone. If not, then it will be necessary to repeat the process, changing the tap until the fault current is within the linear part of the characteristic.\n\nIn practice it is not necessary to draw the complete curve because it is sufficient to take the known fault current and refer to the secondary winding, assuming that there is no saturation for the tap selected.\n\nThis converted value can be taken as IL initially for the process described earlier. If the tap is found to be suitable after finishing the calculations, then a value of IH can be obtained which is closer to the fault current.\n\nAccuracy classes established by the ANSI standards\n\nThe ANSI accuracy class of a CT (Standard C57.13) is described by two symbols — a letter and a nominal voltage; these define the capability of the CT.\n\nC indicates that the transformation ratio can be calculated, and T indicates that the transformation ratio can be determined by means of tests. The classification C includes those CTs with uniformly distributed windings and other CTs with a dispersion flux which has a negligible effect on the ratio, within defined limits.\n\nThe classification T includes those CTs with a dispersion flux which considerably affects the transformation ratio.\n\nFor example, with a CT of class C—100 the ratio can be calculated, and the error should not exceed ten per cent if the secondary current does not go outside the range of 1 to 20 times the nominal current and if the load does not exceed 1Ω (1Ω x 5 Ax 20=100 V) at a minimum power factor of 0.5.\n\nThese accuracy classes are only applicable for complete windings. When considering a winding provided with taps, each tap will have a voltage capacity proportionally smaller, and in consequence it can only feed a portion of the load without exceeding the ten per cent error limit. The permissible load is defined as ZB= (NP  Vc) / 100, where ZB, is the permissible load for a given tap of the CT, NP, is the fraction of the total number of turns being used and Vc is the ANSI voltage capacity for the complete CT.\n\n2.6 DC saturation\n\nUp to now, the behavior of a CT has been discussed in terms of a steady state, without considering the DC transient component of the\n\nDC saturation is particularly significant in complex protection schemes since, in the case of external faults, high fault currents circulate through the CTs.\n\nIf saturation occurs in different CTs associated with a particular relay arrangement, this could result in the circulation of unbalanced secondary currents which would cause the system to malfunction.\n\n2.7 Precautions when working with CTs\n\nWorking with CTs associated with energized network circuits can be extremely hazardous. In particular, opening the secondary circuit of a CT could result in dangerous over voltages which might harm operational staff or lead to equipment being damaged, because the current transformers are designed to be used in power circuits which have impedance much greater than their own.\n\nAs a consequence, when secondary circuits are left open, the equivalent primary-circuit impedance is almost unaffected but a high voltage will be developed by the primary current passing through the magnetizing impedance Thus, secondary circuits associated with CTs must always he kept in a closed condition or short-circuited in order to prevent these adverse situations occurring. To illustrate this, an example is given next using typical data for a CT and a 13.2 kV feeder.\n\nNext" ]

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[ "Faq\n\n# How to get volume in autocad?\n\n1. Enter massprop on the command line.\n2. Select the solid object.\n3. Press ENTER.\n\nCorrespondingly, how do you calculate volume? To find the volume of a box, simply multiply length, width, and height — and you’re good to go! For example, if a box is 5×7×2 cm, then the volume of a box is 70 cubic centimeters.\n\nQuick Answer, how do you find the volume of an irregular shape in AutoCAD 3D?\n\nFrequent question, how do you find the volume of a 3D model?\n\nSubsequently, how does AutoCAD calculate density? Here’s a method that uses standard AutoCAD features for addressing these massprop limitations. To calculate the mass or other mass properties of a collection of solid objects simply use the union function to combine all the solids into one object then multiply the massprop results by the density of the material.Volume is the measure of the 3-dimensional space occupied by matter, or enclosed by a surface, measured in cubic units. The SI unit of volume is the cubic meter (m3), which is a derived unit. Liter (L) is a special name for the cubic decimeter (dm3).\n\n## How do you find volume from area?\n\nVolume is a measure of capacity and is measure in cubic units. To calculate the volume of a rectangular prism, multiply the area of the base (length × width) times height. Example 1: Compute the volume of a square prism with a base area of 25 square feet and a height of 9 feet.\n\n## How do I calculate area in AutoCAD?\n\nRight-click and choose Properties. Properties of the selected object, including the area, are displayed. Note: Unlike individual lines, polylines are considered closed objects and always have a calculated area. Or at the Command prompt, type aa (AREA).\n\n## What is volume in 3D shapes?\n\nVolume of a 3-d shape is defined as the total space enclosed/occupied by any 3-dimensional object or solid shape. It also can be defined as the number of unit cubes that can be fit into the shape. The SI unit of volume is cubic meters.\n\n## How do you find the volume of a 3D rectangle?\n\nTo find the volume of a rectangular prism, multiply its 3 dimensions: length x width x height. The volume is expressed in cubic units.\n\n## How do I use Massprop in AutoCAD?\n\nMASSPROP Command To use this command, type “MASSPROP” on the command line and press Enter. Then select the object from the drawing area and press Enter again. You will see that a list will pop up above the command line with all of the object’s properties, as shown in the image below. Mass properties of a 3D solid.\n\n## Can you view 3D in AutoCAD LT?\n\nIn AutoCAD LT, you can open and view 3D models created in the full version of AutoCAD, though you can’t create new ones or edit them, other than to move, copy, or delete them. AutoCAD LT is a 2D drafting program, it doesn’t have much visualization or presentation capabilities.\n\n## What are the two options for creating splines in AutoCAD?\n\nA 1-degree spline results in a line; there is no bend. A 2-degree spline results in a parabola; there can be only one bend. A 3-degree spline results in a cubic Bezier curve; there can be two bends.\n\n## What is the tool used to measure volume?\n\nLiquid volume is usually measured using either a graduated cylinder or a buret. As the name implies, a graduated cylinder is a cylindrical glass or plastic tube sealed at one end, with a calibrated scale etched (or marked) on the outside wall.\n\n## What is the basic unit of volume?\n\nThe base unit of volume is liter. Another base unit of volume is the cubic meter." ]

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[ "", null, "", null, "", null, "Chapter 11.7, Problem 9E\n\nChapter\nSection\nTextbook Problem\n\nTest the series for convergence or divergence. ∑ n = 0 ∞ ( − 1 ) n π 2 n ( 2 n ) !\n\nTo determine\n\nTo test:\n\nThe series for convergence or divergence.\n\nExplanation\n\n1) Concept:\n\nSeries that involves factorials or other products (including a constant raised to the nth power) are often conveniently tested using the ratio test.\n\nRatio test:\n\nIf limnan+1an=L, then n=1an is,\n\ni. Absolutely convergent hence convergent if, L<1\n\nii. Divergent if L>1 or L=\n\niii. Test is inconclusive if, L=1\n\n2) Given:\n\nn=0-1nπ2n(2n)!\n\n3) Calculation:\n\nHere,\n\nn=0-1nπ2n(2n)!\n\nThe series involves a constant raised to nth term and factorial term therefore, the ratio test should be use.\n\nConsider,\n\nlimnan+1an=limn-1\n\nStill sussing out bartleby?\n\nCheck out a sample textbook solution.\n\nSee a sample solution\n\nThe Solution to Your Study Problems\n\nBartleby provides explanations to thousands of textbook problems written by our experts, many with advanced degrees!\n\nGet Started\n\nFactoring Factor the expression completely. 36. w3 3w2 4w + 12\n\nPrecalculus: Mathematics for Calculus (Standalone Book)\n\nWhat is the lowest score in the following distribution?\n\nEssentials of Statistics for The Behavioral Sciences (MindTap Course List)\n\nChoose the type of function that models each of the following graphs.\n\nMathematical Applications for the Management, Life, and Social Sciences\n\nUse sets toshow that 06=0.\n\nMathematical Excursions (MindTap Course List)\n\n2,339+118+3,650+8,770+81+6=\n\nContemporary Mathematics for Business & Consumers", null, "" ]

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[ "×\nGet Full Access to Physics: Principles With Applications - 6 Edition - Chapter 10 - Problem 11p\nGet Full Access to Physics: Principles With Applications - 6 Edition - Chapter 10 - Problem 11p\n\n×\n\n# (II) The gauge pressure in each of the four tires of an", null, "ISBN: 9780130606204 3\n\n## Solution for problem 11P Chapter 10\n\nPhysics: Principles with Applications | 6th Edition\n\n• Textbook Solutions\n• 2901 Step-by-step solutions solved by professors and subject experts\n• Get 24/7 help from StudySoup virtual teaching assistants", null, "Physics: Principles with Applications | 6th Edition\n\n4 5 1 297 Reviews\n17\n3\nProblem 11P\n\nProblem 11P\n\n(II) The gauge pressure in each of the four tires of an automobile is 240 kPa. If each tire has a \"footprint\" of 190 cm2 (area touching the ground), estimate the mass of the car.\n\nStep-by-Step Solution:\n\nSolution 11P:\n\nWe have to determine the mass of the car using the value of pressure and the area given problem.\n\nStep 1 of 2\n\nConcept:\n\nPressure", null, "on the body is defined as the net force", null, "applied per unit cross-sectional area", null, "of the body. Mathematically,", null, "Step 2 of 2\n\n##### ISBN: 9780130606204\n\nSince the solution to 11P from 10 chapter was answered, more than 708 students have viewed the full step-by-step answer. This textbook survival guide was created for the textbook: Physics: Principles with Applications, edition: 6. The full step-by-step solution to problem: 11P from chapter: 10 was answered by , our top Physics solution expert on 03/03/17, 03:53PM. The answer to “(II) The gauge pressure in each of the four tires of an automobile is 240 kPa. If each tire has a \"footprint\" of 190 cm2 (area touching the ground), estimate the mass of the car.” is broken down into a number of easy to follow steps, and 35 words. This full solution covers the following key subjects: ball, moving, done, left, move. This expansive textbook survival guide covers 35 chapters, and 3914 solutions. Physics: Principles with Applications was written by and is associated to the ISBN: 9780130606204.\n\nUnlock Textbook Solution" ]

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[ "", null, "# Ignore zero values in range for user defined function\n\n• Hello,\n\nI imported in my workbook a spline function that someone else has made, MySpline. MySpline takes two column ranges of equal lengths, one as\n\nx-values you could say andone as y values, and one new x-value to be interpolated.\n\nNow I am trying to make a template, so the input ranges will be fixed, fx A3:A13 and B3:B13. Users will insert their values in these two columns.\n\nHowever, the inputs may vary. Fx let's say that one user only has data that fill A3:A9 and B3:B9. In that case, the remaining cells will be\n\nempty and MySpline cannot handle that.\n\nSo I've tried different things. I've tried changing the VBA code of MySpline with something like\n\nCode\n``NewRange = InputRange(InputRange <> 0)``\n\nI've tried the same following the method poste dby Kenneth here: VBA code to select cells with value greater than zero from a range.\n\nI've tried\n\nCode\n``MySpline(\\$A\\$3:INDEX(A:A,COUNT(A:A)),\\$B\\$3:INDEX(B:B,COUNT(B:B)),C3)``\n\nHowever, there are some headers at A1 and A2, so\n\nCode\n``INDEX(A:A,COUNT(A:A)) ``\n\ndoesn't give the correct last value in the column. It would give the value in A11.\n\nI've tried\n\nCode\n``MySpline(\\$A\\$3:LOOKUP(2,1/(\\$A\\$3:\\$A\\$13<>\"\"),\\$A\\$3:\\$A\\$13)),\\$B\\$3:LOOKUP(2,1/(\\$B\\$3:\\$B\\$13<>\"\"),\\$B\\$3:\\$B\\$13)),B3``\n\nBut I figured out that INDEX and LOOKUP don't work together.\n\nI've tried making the cells be empty (see Return empty cells and here - the part with the \"suicide cells\") for some reason the \"suicide cell function\" doesn't work for me.\n\nNow I've run out of ideas...", null, "Is there anyone who could help? Either with a better idea or improving my previous ideas.\n\nI can do this with a button and VBA, however I would prefer a solution without no user interaction other than the input, such that the user inputs some data, MySpline does the math.\n\nNB: I am not exactly sure if this is the right place to post this and if I'm using the \"code\" function correctly as I am new here...\n\n• Hi, I can unfortunately not do that", null, "I tried to write a detailed description hoping it would describe enough\n\n• You could use:\n\n\\$A\\$3:INDEX(\\$A\\$3:\\$A\\$13,COUNT(\\$A\\$3:\\$A\\$13))\n\nRory\nTheory is when you know something, but it doesn’t work. Practice is when something works, but you don’t know why. Programmers combine theory and practice: nothing works and they don’t know why\n\n## Participate now!\n\nDon’t have an account yet? Register yourself now and be a part of our community!" ]

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[ "# Tag Info\n\n37\n\nYour question presupposes that people were excited about the Jones polynomial because it would help them to classify/distinguish knots. In fact, I suspect the interest came from the fact that this knot invariant was originally defined using operator algebras (rather than in the more combinatorial way people usually define it now). Operator algebras had grown ...\n\n36\n\nTo strengthen Sam Nead's answer, note that it is trivial to compute the Jones polynomial from the Khovanov homology. It is known that computing (or even approximating) the Jones polynomial is #P-hard: Kuperberg, Greg, How hard is it to approximate the Jones polynomial?, Theory Comput. 11 (2015), 183–219. http://arxiv.org/pdf/0908.0512.pdf Vertigan, Dirk, ...\n\n35\n\nYes, forcing can add fundamentally new knots, not equivalent to any ground model knot. Indeed, whenever you extend the set-theoretic universe to add new reals, then you must also have added fundamentally new knots. Theorem. If $V\\subset W$ are two models of set theory with the same ordinals and different reals (for example, any forcing extension with new ...\n\n30\n\nA quick skim of the paper you linked to finds (on paragraph two of page two) comments by Bar-Natan suggesting that his improvement should take time $\\exp({\\sqrt{n}})$, beating the naive algorithm (taking time $\\exp(n)$). That is, the improvement hopefully makes an exponential algorithm subexponential. He is not claiming a sublinear algorithm.\n\n25\n\nThe evaluation of the Jones polynomial at $e^{i\\pi/3}$ is related to the number of 3-colourings $tri(K)$ of $K$ (see also here) as well as to the topology of the branched double cover $\\Sigma(K)$: $$tri(K) = 3\\left|V^2_K(e^{i\\pi/3})\\right| = 3^{\\dim H_1(\\Sigma(K);\\mathbb{Z}/3\\mathbb{Z})+1}$$ This was proved by Przytycki in this paper (Theorem 1.13) and ...\n\n25\n\nAs a historical note (others may have had a different perspective - I was a graduate student when the Jones polynomial made its appearance), when it came out there was some mild excitement because the Jones polynomial could distinguish some pairs of knots which the previous invariants (Alexander polynomial) could not. There was some excitement that the ...\n\n23\n\nThis is a question that I remember worrying about when I first started learning about knot theory. Older books have a tendency to skim over this point rather lightly, perhaps because the resolution of the question seems to involve techniques that have little to do with standard knot theory. One book that doesn't avoid treating this issue squarely is Burde ...\n\n23\n\nNo, there cannot exist infinitely many alternating knots with the same Alexander polynomial. To see why, suppose for the contrary that $K$ belongs to an infinite family $\\{K_n\\}_{n\\in\\mathbb{Z}}$ of alternating knots with $\\Delta_{K_n}(t)=\\Delta_K(t)$. Immediately we have $$\\det(K_n ) = |\\Delta_{K_n} (−1)| = |\\Delta_K (−1)| = \\det(K ).$$ for all $n$. ...\n\n21\n\nThis is a result of Lickorish, in his paper \"A representation of orientable combinatorial 3-manifolds\". The paper is only eleven pages, and is very readable. In his proof, Lickorish rediscovers some ideas first investigated by Max Dehn 40 years earlier. Lickorish's theorem was unexpected at the time; I don't think that Witten's remark is really ...\n\n21\n\nThis isn't directly what you ask, but it's also worth noting that unknot detection is in $\\text{NP} \\cap \\text{co-NP}$, that is, there are polynomial-checkable certificates that will show that either a knot is the unknot or that the knot is not the unknot. The $\\text{NP}$ certificate uses normal surface theory: Ian Agol, Joel Hass, William Thurston, The ...\n\n21\n\nThe statement that for arbitrary K in $S^3$, if for some $n \\ge 2$, the n-fold cyclic branched cover is $S^3$ (or in some versions, a homotopy 3-sphere) then K is the unknot, was known as the Smith conjecture. It was proved around 1979, combining work of many authors: Thurston, Meeks-Yau, Gordon-Litherland, Bass, Shalen, and perhaps others. Its proof was ...\n\n21\n\nYes, this is done in Penney, D.E., Generalized Brunnian links, Duke Math. J. 36, 31-32 (1969). ZBL0176.22201. Call a link $(n,k)$-Brunnian if it has $n$ components, and every sublink with $m$ components: does not split when $k<m\\le n$; completely splits when $1\\le m\\le k$. Then it is shown in the above paper how to use iterated commutators to build an $... 20 I'm very curious where this came up. In any case, the answer to the first question is yes, it does distinguish these trefoils; you found the minimal representatives. Let$a_0,\\dots,a_{N-1}$be the roots of unity that are visited along the knot, in (cyclic) order. Suppose we have a minimal representative for some non-trivial knot. Then we cannot have$|...\n\n20\n\nIt wasn't the properties of knots, but rather the hydrodynamical properties of closed fluids. It stems from the most basic facts in fluid mechanics: Kelvin proved (assuming inviscid flows) that a closed curve $C$ of fluid particles (velocity field $u$) has its circulation $\\oint_Cu\\cdot dl$ independent of time. His theorem isn't true if the curve is fixed in ...\n\n20\n\nThere have been some topological applications of the Jones polynomial and its various generalizations. I believe that these applications increased the interest in these invariants by topologists. One application was to the Tait conjectures. Jones used his polynomial to give lower bounds on the bridge number of links; see Proposition 15.6 of Jones, V. F. R....\n\n19\n\n(Since this is just a string of references, I do not believe this constitutes a 'real answer' but it is too long for a comment, so I'm placing it in the answer field. Editors, please feel free to correct my etiquette.) As for a general introduction or survey article, you might also look at these: \"An introduction to Heegaard Floer homology\" by Ozsvath and ...\n\n18\n\nI did not see any mention of this preprint by Marc Lackenby, probably because the question is quite old : A polynomial upper bound on Reidemeister moves http://arxiv.org/abs/1302.0180 Building on the techniques introduced by Dynnikov and adding some normal surface theory, he shows that any unknot can be unknotted using only a polynomial number of ...\n\n18\n\nI would recommend you to look at Jones' survey paper from 1986, entitled A New Knot Polynomial and Von Neumann Algebras. It is very readable. Let me try to make a brief summary, though. The basic object you start with is a $II_1$ factor. This is a von Neumann algebra $M$ with trivial center $Z(M)\\simeq \\mathbb{C}$, possessing a faithful trace $\\tau : M \\... 18 This is not a definitive answer to your query, but you might be interested in the model explored in the paper, \"Localization of Breakage Points in Knotted Strings,\" Piotr Pieranski, Sandor Kasas, Giovanni Dietler, Jacques Dubochet, and Andrzej Stasiak, New Journal of Physics, vol. 3, June 2001, p. 1-13 (journal link). They model a knot by a curve with ... 18 Here is a figure of a (4,2)-Brunnian link (in the terminology of Mark Grant's answer): And here is an image of a (5,3)-Brunnian link: These are taken from G.C. Shephard's 2006 article \"Interlinked Loops\". He was not aware of the work of Debrunner and Penney at that time as he states essentially this MO question as an open problem. However, the 2009 ... 17 Such a knot would yield a counterexample to one of two important conjectures in the area. A preliminary definition: a slice knot is homotopically ribbon if the inclusion of the knot into the slice disk complement induces a surjection on the fundamental group. It's easy to see that a ribbon knot is homotopically ribbon; note that homotopically ribbon doesn't ... 17 Le and Murakami (HERE and HERE) discovered several previously unknown relations between multiple zeta values through the study of quantum invariants of knots. Further relations were later discovered through knot theory by Takamuki, and by Ihara and Takamuki. These relations stem from the fact that the Kontsevich invariant extends to an invariant not of ... 17 A proof may be given along the lines of the proof of the Jordan Curve theorem by Doyle (see this answer). This uses the fundamental group and a variation on Van Kampen, but not homology. So this probably still won't be satisfying to you, but it doesn't use Alexander duality (which is the normal way to do this). By removing a point from the knot, we may ... 16 The relationship between operator algebras and braids is fairly straightforward to explain, and is nicely written up in many places (e.g. in Kauffman's Knots and Physics). Jones studied representations of the braid group$B_n$into the Temperley-Lieb algebra$TL_n$. The existence of such a representation is not so surprising (the following explanation is ... 16 Revision: For$n>6$, there is no embedding of$\\mathcal{S}_n \\hookrightarrow \\mathcal{S}_{n+1}$. First, recall that there is an extension$\\mathbb{Z}/2\\mathbb{Z} \\to \\mathcal{S}_n \\to Mod(S_{0,n})$, where$Mod(S_{0,n})$is the (orientation preserving) mapping class group of the$n$-punctured sphere. Inside$Mod(S_{0,n})$, there is a subgroup isomorphic ... 16 Here is one reason not to expect such a relationship (although I'm not sure if it can be completed to a proof). The Jones polynomial$J_\\sigma$(roughly) comes from taking the trace of a linear map$A_\\sigma$associated to the braid$\\sigma$, so the question (roughly) asks about relations between$Tr(A_\\sigma)$,$Tr(A_\\tau)$, and$Tr(A_\\sigma A_\\tau)$. Let ... 16 This is just a comment. The same week (!) when Dylan asked this question, we received at our department a message from a non-professional mathematician who wrote a computer program that tries to simplify knots using level moves. (A \"level move\" is like an under move, but there can be more strands lying below the arc that you move.) He says that he tried all ... 16 This paper from earlier this year (Jan 18, to be precise) proves the existence of$\\mathbb{Z}/n\\mathbb{Z}$-torsion for$n\\le 8$and$\\mathbb{Z}/2^s\\mathbb{Z}$-torsion for$s\\le23$. It also states at the beginning of Section 3.4: Until now, no knot or link with torsion larger than$\\mathbb{Z}/8\\mathbb{Z}$was known. I believe this is the state of art. ... 15 A non-trivial (and much more general) result of Gabai implies that$g_n(K)=g_1(K)$for all$n$. This is encapsulated in the phrase Gromov norm equals Thurston norm\". Roughly, the Gromov norm represents the minimal genus of an immersed Seifert surface, whereas the Thurston norm represents the minimal genus embedded Seifert surface. It follows that the ... 15 If by the symmetry group of a knot you mean the group of isometries of$S^3$leaving the knot invariant, then this can only be cyclic or dihedral, apart from the special case of torus knots which can have an$O(2)\\$ group of symmetries. By restricting symmetries to the knot itself one gets a homomorphism from the symmetry group of the knot to the symmetry ...\n\nOnly top voted, non community-wiki answers of a minimum length are eligible" ]

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[ "## A Lagrangian approach for density-dependent incompressible Navier-Stokes equations\n\nIn this talk, we  investigate the Cauchy problem for  the inhomogeneous Navier-Stokes equations in the whole n-dimensional space. Under some smallness assumption on the data, we show the existence of global-in-time unique solutions in a critical functional framework. The initial density is required to belong  to the multiplier space of the homogeneous Besov space B^{n/p-1}_{p,1}(\\R^n). In particular, piecewise constant initial densities are admissible data provided the jump at the interface is small enough, and generate global unique solutions with piecewise constant densities.\nUsing Lagrangian coordinates is the key to our results as it enables us to solve the system by means of the basic contraction mapping theorem. As a consequence, conditions for uniqueness are the same as for existence.\n\nThis is a joint work with P.B. Mucha (Varsaw University).\n\nTo reports list" ]

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[ "", null, "# 1240 in words\n\n1240 in words is written as One Thousand Two Hundred and Forty. 1240 represents the count or value. The article on Counting Numbers can give you an idea about counting. The number 1240 is used in expressions that relate to money, distance, length, year and others. Let us consider an example for 1240. “The cost of 3 kgs of Apples is One Thousand Two Hundred and Forty rupees.” Another example is, “The people gathered at a marriage were One Thousand Two Hundred and Forty.”\n\n 1240 in words One Thousand Two Hundred and Forty One Thousand Two Hundred and Forty in Numerical Form 1240\n\n## 1240 in English Words", null, "## How to Write 1240 in Words?\n\nWe can convert 1240 to words using a place value chart. The number 1240 has 4 digits, so let’s make a chart that shows the place value up to 4 digits.\n\n Thousands Hundreds Tens Ones 1 2 4 0\n\nThus, we can write the expanded form as:\n\n1 × Thousand + 2 × Hundred + 4 × Ten + 0 × One\n\n= 1 × 1000 + 2 × 100 + 4 × 10 + 0 × 1\n\n= 1240\n\n= One Thousand Two Hundred and Forty.\n\n1240 is the natural number that is succeeded by 1239 and preceded by 1241.\n\n1240 in words – One Thousand Two Hundred and Forty.\n\nIs 1240 an odd number? – No.\n\nIs 1240 an even number? – Yes.\n\nIs 1240 a perfect square number? – No.\n\nIs 1240 a perfect cube number? – No.\n\nIs 1240 a prime number? – No.\n\nIs 1240 a composite number? – Yes.\n\n## Solved Example\n\nQuestion:\n\nWrite the number 1240 in expanded form\n\nSolution:\n\n1240 = 1 × Thousands + 2 × Hundreds + 4 × Tens + 0 × Ones\n\n= 1 × 1000 + 2 × 100 + 4 × 10 + 0 × 1\n\n= 1000 + 200 + 40 + 0\n\n= 1000 + 200 + 40\n\n## Frequently Asked Questions on 1240 in words\n\nQ1\n\n### How to write 1240 in words?\n\n1240 in words is written as One Thousand Two Hundred and Forty.\nQ2\n\n### State if True or False. 1240 is divisible by 2?\n\nTrue. 1240 is divisible by 2.\nQ3\n\n### Is 1240 divisible by 10?\n\nYes. 1240 is divisible by 10.\nTest your Knowledge on 1240 in Words" ]

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[ "## Sunday, November 29, 2015\n\n### HP Prime: Approximate Length of Daylight\n\nI hope you had a great weekend and a great Thanksgiving.  For those of you in the United States, hopefully Friday was peaceful and not crazy.\n\nHP Prime: Approximating the Length of Daylight in Hours\n\nThis time, we are taking a slightly more complex formula.   The inputs are:\n\nlat = Earth's latitude of the observer.  (from -90° (South) to 90° (North))\n\nday = the number of days since the December solstice (around December 21-22).  Note that is different from many approximate formulas which used the vernal equinox as a starting point.\n\nThe formula used is a simplification of the Final Formula presented by Herbert Glarner (http://www.gandraxa.com/).  For the complete article and the derivation, please click on link:\n\nhttp://www.gandraxa.com/length_of_day.xml    (Retrieved November 23, 2015)\n\nThe algorithm used in the DAYLIGHT is:\n\nWith the inputs lat and day:\n\nm = 1 – tan(lat°) * tan(23.439° * cos(480/487 * day))\n\nIf m > 2, let m = 2.   If m < 0, let m = 0.\n\nThen:\n\nb = acos(1 – m)/180 *24\n\nWhere b is the length of daylight in hours.\n\nI adjusted the formula to allow for all inputs to be in degrees.  Glamer had mixed inputs for the trigonometric functions.\n\nHere is the program:\n\nProgram DAYLIGHT:\n\nEXPORT DAYLIGHT(lat,day)\nBEGIN\n// latitude, days from\n// December solstice\nLOCAL m,b,a:=HAngle;\nHAngle:=1; // Degrees\nm:=1-TAN(lat)*TAN(23.439*COS(\n480/487*day));\nIF m<0 THEN\nm:=0;\nEND;\nIF m>2 THEN\nm:=2;\nEND;\nb:=ACOS(1-m)/180*24; // hours\nRETURN b;\nHAngle:=a;\n// www.grandraxa.com\nEND;\n\nExamples:\n\nLet’s assume a 365 day and the December solstice was December 22.\n\nLatitude:  -60°, March 1  (day = 69)\nResult:  Approx. 16.67808 hours\n\nLatitude: 54°,  July 24  (day = 214)\nResult:  Approx. 16.03426 hours\n\nFor another approximate formula, I wrote on one for the HP 35S here:  http://edspi31415.blogspot.com/2013/05/hp-35s-approximate-length-of-sunlight.html\n\nHave a great day,\n\nEddie\n\nThis blog is property of Edward Shore.  2015\n\n## Wednesday, November 18, 2015\n\n### Quick Tips for the Casio fx-115ES Plus and fx-991EX Classwiz\n\nQuick Tips for the Casio fx-115ES Plus and fx-991EX Classwiz\n\nThis applies to other similar and earlier Casio calculators.   Please consult your manual if you have a different model.   This refers to Casio models that have textbook entry and output, such as the fx-115ES (Plus), fx-991ES (Plus), fx-991EX Classwiz, and fx-570EX Classwiz.\n\nIn the Math input mode, the Casio attempts to return an exact answer (fractions, terms of π, terms of square roots).  If you want to get an approximate answer from the get go, all that is needed is to press [SHIFT], [ = ] (≈).\n\nUsing a Formula\n\nSteps:\n\n2.  Press [CALC].\n3.  Enter a value for each variable prompted, then press [ = ].\n4.  For the Classwiz models:  You can scroll up and down between variables.\n\nExample:\n\n(A^2 + B^2)^(1/3)\n\nA = 3, B = 4, result 2.924017738\nA = 5, B = 10, result 5\n\nSolve f(X) = 0\n\nSteps:\n\n1.  Enter f(X).  On the Classwiz series, you can use the [ x ] button.  There is no need to enter the “=0”.\n\n2.  Press [SHIFT], [ CALC ] (SOLVE)\n\n3.  Enter a guess and press [ = ].\n\nExample:\n\nX sin(X) – 1 with guess X = π, result:  X = 2.772604708\n\nTip:  I prefer to use X, but you should be able to use any of the other variables available (A, B, C, D, E*, F*, Y, M).  *E and F are available on later models.\n\nSolve f(X) = g(X)\n\nSteps:\n\n1.  Enter f(X).  On the Classwiz series, you can use the [ x ] button.\n\n2.  Press [ALPHA], [CALC] ( = ) for the equals symbol.  This is very important.   Then enter g(X).\n\n3.  Press [SHIFT], [ CALC ] (SOLVE)\n\n4.  Enter a guess and press [ = ] (the equals key).\n\nExample:\n\nln(X) = X^2 – 2 with guess X = 1, result X = 1.564462259\n\nUnit Conversion\n\n1.  Enter a number that is needed to be converted.\n\n2.  Depending on what version for Casio you have:\n\nFor the fx-991EX Classwiz (I think this applies to the fx-570EX Classwiz as well):  Press [SHIFT] [ 8 ] (CONV).  Select a category and select a conversion.\n\nFor the fx-115ES PLUS and earlier models (non-Classwiz models):  Press [SHIFT] [ 8 ] (CONV) and enter a code.  For the fx-115ES PLUS, the conversions are listed both in the manual and the hard slide on case.\n\nHere is a sample of the conversions offered on the fx-115ES PLUS:\n\n 01 in → cm 19 km/h → m/s 35 lbf/in^2 → kPa 02 cm → in 20 m/s → km/h 36 kPa → lb/in^2 07 mi → km 21 oz → g 37 °F → °C 08 km → mi 22 g → oz 38 °C → °F 23 lb → kg 24 kg → lb\n\n3.  Press [ = ].\n\nPlease be aware the older models may not have the conversion function.\n\nFactoring an Integer\n\n‘1.  Enter an integer then press [ = ].\n\n‘2.  Press [SHIFT], [ ° ‘ ‘’ ] (FACT)\n\nExamples:\n\n188 = 2^2 * 47\n\n2506 = 2 * 7 * 179\n\nPlease be aware the older models may not have the factoring function.\n\nCalculus\n\nRemember for derivative (d/dx), integral ( ∫ ), sum ( Σ ), and product ( Π )*, the variable used is X.  *Product may not be available on all models.\n\nInverse and Determinant of a Matrix\n\nFor these calculators, matrices are a separate mode.  I assume that you know how to edit and define matrices.\n\nCasio fx-115 ES Plus and Non-Classwiz Modes that have a Matrix mode:\n\nMatrices are mode 6 (at least for the fx-115EX Plus.)\n\nInverse:\n\n[SHIFT] [ 4 ] (MATRIX), choose 3, 4, or 5 for Matrix A, B, or C respectively, [ x^-1 ], [ = ].\n\nDeterminant:\n\n[SHIFT] [ 4 ] (MATRIX), 7 for det, [SHIFT] [ 4 ] (MATRIX), choose 3, 4, or 5 for Matrix A, B, or C respectively, [ = ].\n\nCasio fx-991EX Classwiz and other Classwiz models:\n\nMatrices are mode 4 (matrix icon).\n\nInverse:\n\n[OPTN], choose 3, 4, 5, or 6 for Matrix A, B, C, or D respectively, [ x^-1 ], [ = ]\n\nDeterminant:\n\n[OPTN], [ down ], 2 for det, [OPTN], choose 3, 4, 5, or 6 for Matrix A, B, C, or D respectively, [ = ].\n\nHope you find these tips helpful.  Have a great day!\n\nEddie\n\nThis blog is property of Edward Shore.  2015.\n\n## Sunday, November 15, 2015\n\n### The Series ( ((((0 + 1)^-1 + 1)^-1 + 1)^-1 + 1)^-1 + 1)^-1 + … + 1)^-1 and Fibonacci Numbers\n\nThe Series ( ((((0 + 1)^-1 + 1)^-1 + 1)^-1 + 1)^-1 + 1)^-1 + … + 1)^-1 and Fibonacci Numbers\n\nDefine the series t as:\n\nt = ( ((((0 + 1)^-1 + 1)^-1 + 1)^-1 + 1)^-1 + 1)^-1 + … + 1)^-1    (an infinite amount of terms)\n\nThis is a sum that can’t be easily stated in summation statement (Σ f(x)).\n\nOn the HP Prime, I programmed this as:\n\nEXPORT TEST1112(n)\nBEGIN\nLOCAL k, t:=0;\nFOR k FROM 1 TO n DO\nt:=(t+1)^-1;\nEND;\nRETURN t;\nEND;\n\nThe result seems to converge at 0.6180339785 when n ≥ 27.  Note that 0.6180339785 = ϕ – 1, where ϕ is the Golden Ratio ( ϕ = (√5 + 1)/2)\n\nFibonacci Gets Involved\n\nNote that:\n\n k = t = 1 1 2 (1 + 1)^-1 = 1/2 3 (1 + 1/2)^-1 = (3/2)^-1 = 2/3 4 (1 + 2/3)^-1 = (5/3)^-1 = 3/5 5 (1 + 3/5)^-1 = (8/5)^-1 = 5/8 6 (1 + 5/8)^-1 = (13/8)^-1 = 8/13 7 (1 + 8/13)^-1 = (21/13)^-1 = 13/21\n\nWe get a sequence of terms {1, 1/2, 2/3, 3/5, 5/8, 8/13, 13/21, 21/34, 34/55, 55/89, 89/144, …} where each term takes the fraction a/b, a is the kth Fibonacci number and b is the (k+1)th Fibonacci number.  Can we show that this sequence of partial sums is convergent?\n\nEach partial sums of the series takes the form F_k / F_k+1 where F is the Fibonacci number.\n\nThe closed formula for the Fibonacci number is:\n\nF_k = ( ϕ^k – α^k )/√5 , where ϕ  = (1 + √5)/2 and α = (1 - √5)/2.\n\nThen:\n\nF_k / F_k+1\n=( ϕ^k – α^k )/√5 * √5/( ϕ^k+1 – α^k+1)\n=( ϕ^k – α^k) / ( ϕ^k+1 – α^k+1 )\n= ( ϕ^k / ϕ^k+1) * ( (1 – (α/ϕ)^k) / (1 – (α/ϕ)^k+1) )\n= 1/ϕ * ( (1 – (α/ϕ)^k) / (1 – (α/ϕ)^k+1) )\n\nNote that α/ϕ = (1 - √5)/(1 + √5 ) ≈ -0.38197 < 1\n\nAs k → ∞,  α/ϕ → 0.\n\nHence,\n\nlim k → ∞ (F_k / F_k+1)\n= lim k → ∞ (1/ϕ * ( (1 – (α/ϕ)^k) / (1 – (α/ϕ)^k+1) ) )\n= 1/ϕ\n\nSimplifying:\n\n1/ϕ\n= 2/(1 + √5)\n= 2*(1 - √5) / ((1 + √5)*(1 - √5))\n= 2*(1 - √5)/-4\n= (√5 – 1)/2\n= √5/2 – 1/2\n\n√5/2 – 1/2 + (1/2 – 1/2)\n= (√5 + 1)/2 – 1\n= ϕ – 1\n\nSince the sequence of partial sums converge to ϕ – 1, the series\n\nt = ( ((((0 + 1)^-1 + 1)^-1 + 1)^-1 + 1)^-1 + 1)^-1 + … + 1)^-1\n\nconverges to ϕ – 1.\n\nThis blog is property of Edward Shore.  2015.\n\n## Monday, November 9, 2015\n\n### HP Prime Geometry App Tutorial Part 8: Rotating Triangles\n\nHP Prime Geometry App Tutorial Part 8:  Rotating Triangles\n\nToday’s lesson will show how to rotate a triangle given an angle.  The angle is in a counter-clockwise direction.\n\nFor the purpose of this lesson, we will set the HP Prime in Degrees mode.\n\nDraw the Triangle\n\n1.  Set the calculator to Degrees mode.  Draw a triangle with vertices (4,4), (4, -4), and (8, 0).\n\nRotate the Triangle\n\n2.  Press (Cmds), 7 for Transformation, 3 for Rotation.\n3.  Select the reference point.  In our example, let’s make this point (0,0) and press [ Enter ].\n4.  You are asked for an angle.  Enter 90 for 90°.  Press [ Enter ].\n\nMore Rotation\n\n5.  Repeat steps 2 through 4, for angles of 180 and 270.\n\nThank you.  Until next time, have a great day!\n\nEddie\n\nThis blog is property of Edward Shore.  2015.\n\n### HP Prime Geometry App Tutorial Part 7: Conic Sections and Equations\n\nHP Prime Geometry App Tutorial Part 7: Conic Sections and Equations\n\nIn today’s lesson we will work conic sections and showing the equations for each of the conic sections.\n\nDrawing an Ellipse and Display its Equation\n\nCommand:  ellipse(focus point 1, focus point 2, point on the ellipse)\n\nDraw an ellipse with foci (1,4) and (5,4) and the point on the ellipse on (5,6).\n\n1.  Press (Cmds), 5 for Curve, 5 for Ellipse.\n2.  Go to the point (1,4) and press [ Enter ].\n3.  Go to the point (5,4) and press [ Enter ].\n4.  Go to the point (5,6) and press [ Enter ].\n\nThe next step is to display its equation.\n\n5.  Press (Cmds), 8 for Cartesian, 5 for Equation of.\n6.  Select the ellipse and press [ Enter ].  In this example the equation is named GD.\n7.  To see the whole equation, press [ CAS ] and execute the equation(GD) command.\n8.  Press the up button to highlight the equation and press (Show).  Scroll to see the entire equation.\n\nDrawing a Hyperbola and Display its Equation\n\nCommand:  hyperbola(focus point 1, focus point 2, point on the hyperbola)\n\nDrawn a hyperbola with foci (-2, -2) and (-4, 2) with point (-9, 2).\n\n1.  Press (Cmds), 5 for Curve, 6 for Hyperbola.\n2.  Go to the point (-2,-2) and press [ Enter ].\n3.  Go to the point (-4,-2) and press [ Enter ].\n4.  Go to the point (-9,2) and press [ Enter ].\n\nThe next step is to display its equation.\n\n5.  Press (Cmds), 8 for Cartesian, 5 for Equation of.\n6.  Select the ellipse and press [ Enter ].  In this example the equation is named GD.\n7.  To see the whole equation, press [ CAS ] and execute the equation(GD) command.\n8.  Press the up button to highlight the equation and press (Show).  Scroll to see the entire equation.\n\nDrawing a Parabola and Display its Equation\n\nCommand:  parabola(focus point, directrix line)\n\nDraw a parabola with focus point (0,-5) and (0,3) is on the directrix line.\n\n1.  Press (Cmds), 5 for Curve, 7 for Parabola.\n2.  Go to the point (0,-5) and press [ Enter ].\n3.  Go to the point (0,3) and press [ Enter ].\n\nThe next step is to display its equation.\n\n4.  Press (Cmds), 8 for Cartesian, 5 for Equation of.\n5.  Select the ellipse and press [ Enter ].  In this example the equation is named GC.\n6.  To see the whole equation, press [ CAS ] and execute the equation(GC) command.\n7.  Press the up button to highlight the equation and press (Show).  Scroll to see the entire equation.\n\nOn the next part, we will work with rotating geometric objects.   Thank you and see you next time,\n\nEddie\n\nThis blog is property of Edward Shore – 2015.\n\n### HP Prime Geometry App Tutorial Part 6: Polygons\n\nHP Prime Geometry App Tutorial Part 6:  Polygons\n\nDrawing a Regular Polygon\n\n1.  Clear the Plot screen.  Press (Cmds), 4 for Polygon, 9 for Regular Polygon.\n2.  Select one corner and press [ Enter ].\n3.  Select another corner point and press [ Enter ].\n4.  Enter the number of sides of the regular polygon.\n\nDrawing a General Polygon\n\n1.  Press (Cmds), 4 for Polygon, 8 for Polygon.\n2.  Start with a corner point and press [ Enter ].  Keep going choosing corner points and pressing [ Enter ].\n3.  When you finished plotting the final corner point, press [ Enter ] one last time.  This connects the last corner point with the first corner point.   Hence, it is like go to the final point and press [ Enter ] twice.\n\nFinding Areas of the Polygon\n\nTo recall, press (Cmds), 9 for Measure, 5 for Area.  Select a polygon (when selected they turned red) and press [ Enter ].\n\nNext time, we will work with conic sections.   Have a great day,\n\nEddie\n\nThis blog is property of Edward Shore – 2015.\n\n## Sunday, November 8, 2015\n\n### Speed Test: Casio fx-991EX Classwiz vs Canon F-792SGA\n\nSpeed Test:  Casio fx-991EX Classwiz vs Canon F-792SGA\n\nEddie\n\nThis blog is property of Edward Shore.  2015\n\n### HP Prime Geometry App Tutorial Part 5: Plotting Functions and Differential Equations\n\nHP Prime Geometry App Tutorial Part 5:  Plotting Functions and Differential Equations\n\nThe Geometry App can plot functions, parametric functions, polar functions, sequences, implicit statements, slope-field and ordinary differential equations, lists, and designate sliders.\n\nIn this lesson, we will demonstrate three types of plots.   For the purpose of the tutorial, clear the Plot screen before each example.\n\nPlotting Functions ( y = f(x))\n\nPlot y = e^x + 1.\n\n1.  Press (Cmds), 6 for Plot, 1 for Function.\n2.  Type e^(x) + 1.  Press (OK).\n\nUse the lowercase x.  The format is plotfunc( y(x) ).\n\nPlot an Ordinary Differential Equation  (y’ = dy/dx = f(x,y))\n\nPlot y’ = y*e^x +1 with the initial condition (1,1).\n\n1.  Press (Cmds), 6 for Plot, 7 for ODE.\n2.  Type y*e^(x)+1.   Note that x and y are in lowercase.\n3.  Press [ , ] and type [x,y].   Here you designate which variable is independent and which is dependent.\n4.  Press [ , ] and type  [1,1].  This is your initial conditions.  Press (OK),\n\nThe entire format is plotode( f(x,y),  [x, y], [x0, y0])\n\nPlot a Parametric Equation  ( x(t) + i*y(t) where i = √-1)\n\nPlot x = 3t + 1, y = 2t – 1.   The format to be used is (3t + 1) + i*(2t -1).\n\n‘1. Press (Cmds), 6 for Plot, 2 for Parametric.\n‘2. Type (3t + 1) + i*(2t -1) and press ( OK ).\n\nThe entire format is plotparam( x(t) + i*y(t), var = tbeg..tend, tstep=step).  The t is in lowercase and the last two arguments, interval and step, are optional.\n\nIn Part 6 we’ll be plotting and working with polygons.   Thanks and take care,\n\nEddie\n\nThis blog is property of Edward Shore.  2015.\n\n### HP Prime Geometry App Tutorial Part 4: Tangent Lines and Changing the Color of Objects\n\nHP Prime Geometry App Tutorial Part 4:  Tangent Lines and Changing the Color of Objects\n\nPlotting Lines Tangent to a Circle’s Point\n\n1.  Start with a clear plot screen.   Draw a circle:  the location of its center and size is up to you.\n2.  Press (Cmds), 3 for Line, 6 for Tangent.\n3.  Select a point on the circle and press [ Enter ].  The point is designated as object GD.\n4.  If you wish, display the coordinates of GD.  See Part 3 as a refresher.\n\nPer HP Prime’s help:  The tangent command draws one or more tangent lines through a specified point to the circle and the line segment connecting it to the radius.\n\nColoring the Tangent Lines\n\nThe following steps is how to color tangent lines.  By default, geometric objects are colored black.  Adopt this procedure for any geometric object you want to designate as a specific color.\n\n1.  Move to the tangent lines so that they turn red.   The (Optns) soft menu appears.  It is key that the (Optns) menu is available.\n2.  Press (Optns), 1 for Choose Color, choose any color you want.  If the pictures, I have selected blue.  In order to see the different color, move the cursor away from the tangent lines.\n\nWhat does Part 5 mean?  It’s time to plot some functions!  That is next time.\n\nEddie\n\nThis blog is property of Edward Shore.  2015\n\n### HP Prime Geometry App Tutorial Part 3: Lines and Line Segments\n\nHP Prime Geometry App Tutorial Part 3:  Lines and Line Segments\n\nWith Parts 1 and 2, we will start with a clear plot screen, with a Plot window of XRange = [ -16, 16 ], YRange = [ -11, 10.9 ], ticks are at 1.  This lesson will focus on drawing line segments, parallel, and perpendicular lines.\n\nMidpoint on a Line\n\nFor this part, put the line segment anywhere you wish.\n\n1.  Press (Cmds), 3 for Line, 1 for Segment.  You are prompted to select on the segment’s end points.  Press [ Enter ] to select the end point.\n2. Select the other end of segment and press [ Enter ].  The segment is designated as object GC.\n3.  For the Midpoint, press (Cmds), 2 for Point, 3 for Midpoint.  Select the line segment you have just drawn.  Remember, when you cursor over objects that you are about to select, that object turns red.   Press [ Enter ].   The midpoint is plotted, as object GD.\n\nNote that the coordinates the midpoint are not displayed.  Learn how to display the coordinates in the next segment.\n\nDisplay a Point’s Coordinates\n\nThere are two ways to display a point’s coordinates.  Try both methods and see which method works better for you.  You can use these methods of displaying the coordinates for any point.\n\nFor this exercise, we’ll concentrate on the midpoint (object GD).\n\nThe Num Screen Method\n\n1.  Press the [ Num ] key.\n2.  Select a blank line, press (Cmds), 1 for Cartesian, 4 for Coordinates.\n3.  Type GD and press [ Enter ].\n4.  Return the plot screen by pressing [ Plot ].\n\nThe Plot Screen Method\n\n1.  Press (Cmds), 8 for Cartesian, 4 for Coordinates.\n2.  Select the point of interest, press [ Enter ].\n\nEither way, the coordinates of the point are displayed on top of the screen.\n\nFor the next part of the lesson, please clear the plot screen.\n\nDraw Parallel and Perpendicular Lines\n\n1.  Draw a line.  To recall, press (Cmds), 3 for Line, 3 for Line.  Place the line anywhere you wish, at any angle you wish.  In the pictures that are displayed, I just chose to place a horizontal line for demonstration purposes (boring, huh?)\n2.  Parallel Line:   Press (Cmds), 3 for Line, 4 for // (Parallel) .  Select the line you drew and press [ Enter ].  Place the parallel line with a second [ Enter ].   A parallel line is drawn.\n3.  Perpendicular Line:  Press (Cmds), 3 for Line, 5 for ⊥  (Perpendicular).  Select either of the lines and press [ Enter ].  Place the perpendicular lines by pressing [ Enter ].\n\nIn Part 4 we will work with drawing tangent lines and how to color objects.\n\nThis blog entry is coming from me enjoying a pumpkin vanilla latte at The Coffee Bean & Tea Leaf in Monrovia, CA.   Hope you day is wonderful and see you next time!\n\nEddie\n\nThis blog is property of Edward Shore. 2015.\n\n## Friday, November 6, 2015\n\n### HP Prime Geometry App Tutorial Part 2: Triangles\n\nHP Prime Geometry App Tutorial Part 2:  Triangles\n\nLike Part 1, we’ll use a Plot window of XRange = [ -16, 16 ], YRange = [ -11, 10.9 ], ticks are at 1.  For this lesson, we are going to focus on triangles.  Start with a clear Plot Screen ([Shift], [ Esc ] (Clear)).\n\nDrawing a Triangle\n\n1.  On the plot screen, press the soft key (Cmds), then select 4 for Polygon and 1 for Triangle.\n2.  Select the first vertex (corner point) and press [ Enter ].  For this lesson, put the triangle wherever you want.\n3.  Place the second and third vertex.  Press [ Enter ] after each point.  The vertices are labeled A, B, and C.\n\nFinding an Angle\n\nThe steps will demonstrate how to find the angle.  Be aware that the direction you enter the vertexes will determine the sign of the angle.\n\n1.  Press (Cmds), 9 for Measure, 6 for Angle.\n2.  Select one vertex, press [Enter].  Follow the triangle to select the second vertex (where the angle will be measured), press [ Enter ].  Follow that with a third vertex and press [Enter].  The angle displayed on top of the screen.\n\nResizing the Triangle\n\nClick on one of the points of the points.  Then drag the point with the arrow pad.  When satisfied, press [ Enter ].\n\nFor the last part, clear the screen.\n\nReflect a Triangle – reflect about a point\n\n1.  Clear the screen.  ([ Shift ] [ Esc ] (Clear))\n2.  Draw a triangle with the vertices (-6, -4), (-6, 4), and (-12, 0).\n3.  Press (Cmds), 7 for Transform, 2 for Reflection.\n4.  You will be promoted for a reflection point.  For this exercise, set the point at (0,0).\n5.  Select the triangle (scroll until the triangle turns red) and press [ Enter ].\n\nIn Part 3, we’re going to work with lines and line segments.  Until next time, have a great day!\n\nEddie\n\nThis blog is property of Edward Shore.  2015.\n\n### Σ(1 / (a^n)) from n=1 to m\n\nΣ(1 / (a^n)) from n=1 to m This blog entry covers the sum of the series: Σ[1 / (a^n), n=1 to m] with n and m positive integers Specific Cas..." ]

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[ "Arithmetic circuits with restrictions on occurrence of pairs of variables\n\nI am curious if the following model was studied or has some obvious lower bounds:\n\nWe want to compute a polynomial $P(x_1,x_2, \\dots , x_n)$. Suppose we have a graph G on $n$ nodes that we are going to call restriction graph. Now we will call a monomial $x_{i_1} \\times x_{i_2} \\times \\dots \\times x_{i_d}$ valid if no two variables form an edge in the restriction graph $G$:\n\n$$x_{i_1} \\times x_{i_2} \\times \\dots \\times x_{i_d} \\text{ is valid } \\iff \\forall a,b: \\{x_{i_{a}},x_{i_b}\\} \\not\\in E(G)$$\n\nA polynomial is valid if all of its monomials are valid. We will call a circuit $G$-restricted if all the gates in $G$ compute valid polynomials. We will be only interested in $G$-restricted circuits for valid polynomials.\n\nThis model is a strict generalization of multilinear circuits: we can take $G$ to be a collection of loops. My questions are:\n\n• Are there any obvious lower bounds for $G$-restricted circuits for explicit polynomials (and explicit $G$)?" ]

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[ "# Stochastic differential equations for eigenvalues and eigenvectors of a $G$-Wishart process with drift\n\n• H. Boutabia\nWe propose a system of G-stochastic differential equations for the eigenvalues and eigenvectors of the $G$-Wishart process defined according to a $G$-Brownian motion matrix as in the classical case. Since we do not necessarily have the independence between the entries of the $G$-Brownian motion matrix, we assume in our model that their quadratic covariations are zero. An intermediate result, which states that the eigenvalues never collide is also obtained. This extends Bru’s results obtained for the classical Wishart process (1989).\nBoutabia, H., S. Meradji, and S. Stihi. “Stochastic Differential Equations for eigenvalues and Eigenvectors of a $G$-Wishart Process With Drift”. Ukrains’kyi Matematychnyi Zhurnal, Vol. 71, no. 4, Apr. 2019, pp. 502-15, http://umj.imath.kiev.ua/index.php/umj/article/view/1454." ]

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[ "Skip to content\n• Review\n• Open Access\n\n# Use of pathway information in molecular epidemiology\n\nHuman Genomics20094:21\n\nhttps://doi.org/10.1186/1479-7364-4-1-21\n\n• Received: 23 June 2009\n• Accepted: 23 June 2009\n• Published:\n\n## Abstract\n\nCandidate gene studies are generally motivated by some form of pathway reasoning in the selection of genes to be studied, but seldom has the logic of the approach been carried through to the analysis. Marginal effects of polymorphisms in the selected genes, and occasionally pairwise gene-gene or gene-environment interactions, are often presented, but a unified approach to modelling the entire pathway has been lacking. In this review, a variety of approaches to this problem is considered, focusing on hypothesis-driven rather than purely exploratory methods. Empirical modelling strategies are based on hierarchical models that allow prior knowledge about the structure of the pathway and the various reactions to be included as 'prior covariates'. By contrast, mechanistic models aim to describe the reactions through a system of differential equations with rate parameters that can vary between individuals, based on their genotypes. Some ways of combining the two approaches are suggested and Bayesian model averaging methods for dealing with uncertainty about the true model form in either framework is discussed. Biomarker measurements can be incorporated into such analyses, and two-phase sampling designs stratified on some combination of disease, genes and exposures can be an efficient way of obtaining data that would be too expensive or difficult to obtain on a full candidate gene sample. The review concludes with some thoughts about potential uses of pathways in genome-wide association studies.\n\n## Keywords\n\n• colorectal cancer\n• complex diseases\n• folate\n• gene-environment interactions\n• gene-gene interactions\n\n## Introduction\n\nMolecular epidemiology has advanced from testing associations of disease with single polymorphisms, to exhaustive examination of all polymorphisms in a candidate gene using haplotype tagging single nucleotide polymorphisms (SNPs), to studying increasing numbers of candidate genes simultaneously. Often, gene-environment and gene-gene interactions are considered at the same time. As the number of main effects and interactions proliferate, there is a growing need for a more systematic approach to model development .\n\nIn recognition of this need, the American Association for Cancer Research held a special conference in May 2007, bringing together experts in epidemiology, genetics, statistics, computational biology, systems biology, toxicology, bioinformatics and other fields to discuss various multidisciplinary approaches to this problem.\n\nA broad range of exploratory methods have been developed recently for identifying interactions, such as neural nets, classification and regression trees, multi-factor dimension reduction, random forests, hierarchical clustering, etc. Our focus here, however, is instead on hypothesis-driven methods based on prior understanding about the structure of biological pathways postulated to be relevant to a particular disease. Our primary purpose is to contrast mechanistic and empirical methods and explore ways of combining the two.\n\n### The folate pathway as an example\n\nFolate metabolism provides a rich example to illustrate these challenges. Folate has been implicated in colorectal cancer, coronary heart disease and neural tube defects, [10, 11] among other conditions. Several steps in the metabolism of folate could be involved in these various diseases (Figure 1) and could have quite different effects. The pathway is complex, involving 19 enzymes or carrier proteins, with various feedback loops and two main cycles, the folate and the methionine cycles. The former is involved in pyrimidine synthesis through the action of thymidylate synthase (TS), potentially leading to uracil misincorporation into DNA and subsequent DNA damage and repair or misrepair. The latter is involved in DNA methylation through the conversion of S-adenosyl methionine (SAM) to S-adenosyl homocysteine (SAH) by DNA-methyltransferase (DNMT). These two mechanisms in particular have been suggested as important links between folate and carcinogenesis, although other possibilities include purine synthesis (via the aminoimidazole-carboxamide ribonucleotide transferase [AICART] reaction) and homocysteine itself. Because polymorphisms that tend to increase one of these effects may decrease others, their effects on disease endpoints can be quite different, depending on which part of the pathway is more important. A detailed mathematical model for this system has been developed by Nijhout et al. [12, 13] and Reed et al., [14, 15] based on the equilibrium solution to a set of linked ordinary differential equations for Michaelis-Menten kinetics and implemented in software available at http://metabolism.math.duke.edu/.", null, "Figure 1 Biochemical diagram of folate metabolism (reproduced with permission from Reed et al.). AICART, aminoimidazolecarboxamide ribonucleotide transferase; BHMT, betaine-homocysteine methyltransferase; CBS, cystathionine b-synthase; DHFR, dihydrofolate reductase; DNMT, DNA-methyltransferase; dTMP, thymidine monophosphate; FTD, 10-formyltetrahydrofolate dehydrogenase; FTS, 10-formyltetrahydrofolate synthase; GAR, glycinamide ribotid; G-NMT, glycine N-methyltransferase; HCOOH; formic acid; H2C = O, formaldehyde; HCY, homocysteine; MAT, methionine adenosyl transferase; MS, methionine synthase; MTCH, 5,10-methylenetetrahydrofolate cyclohydrolase; MTD, 5,10-methylenetetrahydrofolate dehydrogenase; MTHFR, 5,10-methylenetetrahydrofolate reductase; NE, nav-enzymatic; PGT, phosphoribosyl glycinamide transferase; SAH, S-adenosylhomocysteine; SAHH, SAH hydrolase; SAM, S-adenosylmethionine; SHMT, serine hydroxymethyltransferase; THF, tetrahydrofolate; 5m-THF, 5-methylTHF; 5,10-CH2-THF, 5,10-methyleneTHF; 10f-THF, 10-formylTHF; TS, thymidylate synthase.\nTo illustrate the various approaches, we simulated some typical data in the form that might be available from a molecular epidemiology study -- specifically data on genetic variants, various environmental exposures, a disease outcome or clinical trait, and, possibly, biomarker measurements on some or all subjects. We began with a population of 10,000 individuals with randomly generated values of intracellular folate E1 (the total tetrahydrofolate [THF] concentration in the six compartments forming the closed loop shown on the left-hand side of Figure 1), methionine intake E2 (METin, log-normally distributed) and 14 of the key genes G shown in Figure 1. For each gene, a person-specific value of the corresponding V max was sampled from log-normal distributions with genotype-specific geometric means (GMs = 0.6, 0.8, or 1.1 times the overall GM) and common geometric standard deviations (GSD = 1.1) and K m appropriate for that enzyme (see Table 1 in Reed et al. for reference values for V max and K m for each gene). The differential equations were then evaluated to determine the steady-state solutions for ten intermediate metabolite concentrations and 14 reaction rates for each individual, based on their specific environmental variables and enzyme activity rates. The probability of disease was calculated under a logistic model for each of four scenarios for the causal biological mechanism -- homocysteine concentration, the rate of DNA methylation reactions and the rates of purine and pyrimidine synthesis -- and a binary disease indicator Y was sampled with the corresponding probability. Only the data on (Y, E, G) were retained from the first 500 cases and 500 controls for the first level of the epidemiological analysis. For some analyses, we also simulated biomarkers on stratified subsamples of these subjects, as will be described later. Various summaries of the correlations among the (X, E, G) values for the remaining 9,000 subjects were deposited into what we shall call the 'external database' for use in constructing priors, as described below (no individual Y data were used for this purpose).\n\nTable 1 shows the univariate associations of each gene with disease under each assumption about the causal risk factor. In these simulations, only one of these was taken as causal at a time, each scaled with the relative risk coefficient β = 2.0 per standard deviation of the respective risk factor. When homocysteine concentration was taken as the causal factor, the strongest association was with genetic variation in the cystathionine b-synthase (CBS) and S-adenosylhomocysteine hydrolase (SAHH) genes. The remaining three columns relate to various reaction rates as causal mechanisms. For pyrimidine synthesis (characterised here by the TS reaction rate), the strongest influence was seen for genetic variation in TS and the 5,10-methyleneTHF dehydrogenase (MTD) gene. For purine synthesis (reflected in the AICART reaction rate), the strongest associations were with genetic variation in the phosphoribosyl glycinamide transferase (PGT) gene and somewhat weaker for MTD and 5,10-methyleneTHF cyclohydrolase (MTCH) and serine hydroxymethyltransferase (SHMT) genes; interestingly, the disease risk is not particularly related to the AICART genotype itself. When DNA methylation (reflected by the DNMT reaction rate) was assumed to be causal, none of the genetic associations were as strong as for the other three causal mechanisms, the strongest being with the 5,10-methyleneTHF reductase (MTHFR) gene, SAHH and MTD. Genetic variation in DNMT was not explicitly simulated, but the reaction rates for this enzyme were identical to those for methionine adenosyl transferase (MAT-II) and SAHH, reflecting a rate-limiting step. Thus, genetic variation in MAT-II had no effect on risk, the reaction rate being driven entirely by SAHH. Other rate-limited combinations included dihydrofolate reductase (DHFR) with TS, MTD with MTCH, and PGT with AICART. Methionine intake was the strongest environmental exposure factor for the simulation with homocysteine as the causal mechanism, whereas intracellular folate had a stronger effect under the other three mechanisms.\n\n### Mechanistic vs empirical models\n\nFor the four highlighted simulations, we also conducted multiple logistic regressions in a stepwise manner, offering methionine, folate, the 14 genotypes and all 91 pairwise G × G and 28 G × E interactions (Table 2). These are difficult to interpret, however, owing to the large numbers of comparisons and unstable regression coefficients, particularly in the models that include interaction terms. In an attempt to gain greater insight into mechanisms, attention will now be turned to more pathway-driven modelling approaches, based on hierarchical or mechanistic models. The former extend the standard logistic models summarised in Table 2 by the addition of 'prior covariates' incorporating knowledge about the relative risk coefficients predicted by the pathway. The latter attempt to model the pathways explicitly, using simplified versions of physiologically based pharmacokinetic (PBPK) models, thereby requiring stronger assumptions about reaction dynamics and population distributions of rate parameters.\n\n### Hierarchical models for disease-pathway associations\n\nIn the first level, the epidemiological data are fitted using a conventional 'empirical' model for the main effects and interactions among the various input genotypes and exposures, here denoted generically as X = (X ip )p = 1 ... p= (E, G, G × G, G × E, G × G × E, ...); for example, a logistic regression model of the form\n(1)\n\nthe sum being taken over the range of terms included in the X vector. Note that all possible effects of some predetermined complexity (eg all main effects and two-way, or perhaps higher order, interactions possibly limited to subsets relevant to the hypothesised pathway structure) are included, rather than using some form of model selection, as was done in the stepwise analyses summarised in Table 2.\n\nIn the second-level model, each of the regression coefficients from Eq. (1) is in turn regressed on a vector Z p = (Z pv )v = 1 ... Vof 'prior covariates' describing characteristics of the corresponding terms in X; for example,\n${\\beta }_{p}~N\\left({\\pi }_{0}+\\sum _{v=1}^{v}{\\pi }_{v}{Z}_{pv},{\\sigma }^{2}\\right)$\n(2)\n\nThere are many possibilities for what could be included in the set of prior covariates, ranging from indicator variables for which of several pathways each gene might act in, in silico predictions of the functional significance of polymorphisms in each gene, [18, 19] or genomic annotation from formal ontologies . Summaries of the effects of genes on expression levels ('genetical genomics') or of associations of genes with relevant biomarkers might also be used as prior covariates. Rebbeck et al. provide a good review of available tools that could be used for constructing prior covariates.\n\nAlternatively, one could model the variances, for example:\n${\\beta }_{p}~N\\left(\\pi \\text{'}{\\mathsf{\\text{Z}}}_{p},{\\sigma }^{2}\\mathsf{\\text{exp}}\\left(\\phi \\text{'}{\\mathsf{\\text{Z}}}_{p}\\right)\\right)$\n(3)\n\nFor example, suppose the X vector comprised effects for different polymorphisms within each gene and one had some prior predictors of the effects of each polymorphism (eg in silico predictions of functional effects or evolutionary conservation) and other predictors of the general effects of genes (eg their roles in different pathways or the number of other genes that they are connected to in a pathway). Then, it might be appropriate to include the former in the π ' Z part of the model for the means, and the latter in the φ ' Z part of the model for the variances.\n\nSo far, the second-level models have assumed an independence prior for each of the regression coefficients; but now, suppose we have some prior information about the relationships among the genes, such as might come from networks inferred from gene co-expression data. Let A = (A pq )p, q = 1 ... Pdenote a matrix of prior connectivities between pairs of genes -- for example, taking the value 1 if the two are adjacent (connected) in some network or 0 otherwise. Then, one might replace the independence prior of Eq. (2) by a multivariate prior of the form:\n$\\beta ~{\\mathsf{\\text{N}}}_{p}\\left(\\pi \\text{'}\\mathsf{\\text{Z}},{\\sigma }^{2}\\left(\\mathsf{\\text{I}}\\phantom{\\rule{0.3em}{0ex}}-\\phantom{\\rule{0.3em}{0ex}}\\rho {\\mathsf{\\text{A)}}}^{-1}\\right)$\n\nThis is known as the conditional autoregressive model, and is widely used in spatial statistics . Sample WinBUGS code to implement these and other models described below are available in an online supplement.\n\nIn applications to the folate simulation, we tried two variants of this model. First, we considered three prior covariates in Z: an indicator for whether a gene is involved in the methionine cycle; whether it is involved in the folate cycle; and the number of other genes it is connected to in the entire network (a measure of the extent to which it might have a critical role as a 'hub' gene). The A matrix was specified in terms of whether a pair of genes had a metabolite in common, either as substrate or product.\n\nTable 3 summarises the results of several models, including these three prior covariates in the means or variance model, as well as the connectivities in the covariance model. As would be expected, in the zero mean model, all the significant parameter estimates were shrunk towards zero because of the large number of genes with no true effect in the model. In general, none of the prior covariates significantly predicted the means. The estimates of the βs in all these models were much closer to the simple maximum likelihood estimates (the first column), however, and their standard errors were generally somewhat smaller, indicating the 'borrowing of strength' from each other. In the model with covariates in the prior variances, however, the number of connections for each gene was significantly associated with the variance. In the final model, with correlations between genes being given by indicators for whether they were connected in the graph, the posterior distribution for the parameter ρ is constrained by the requirement that the covariance matrix be positive definite, but showed strong evidence of gene-gene correlations following the pattern given by the connectivities in Figure 1. The generally weak effects of prior covariates in these models may simply reflect the crudeness of these classifications. Below, we will revisit these models with more informative covariates based on the quantitative predictions of the differential equations model.\n\n### Mechanistic models\n\nWhereas hierarchical models are generally applicable whenever one has external information about the genes and exposures available, in some circumstances the dynamics of the underlying biological process may be well enough understood to support mechanistic modelling. These are typically based on systems of ordinary differential equations (ODEs) describing each of the intermediate nodes in a graphical model as deterministic quantities given by their various inputs (exposures or previous substrates) with reaction rates determined by genotypes (Figure 2). For example, in a sequence j = 1, ..., J of linear kinetic steps, with conversion from metabolite M j to M j+1 at rate λj and removal at rate μ j , the instantaneous concentration is given by the differential equation:\n$\\frac{d{M}_{j}}{dt}=\\left({\\lambda }_{j-\\mathsf{\\text{1}}}{M}_{j-1}-\\left({\\lambda }_{j}+{\\mu }_{j}\\right){M}_{j}$\n(4)", null, "Figure 2 Schematic representation of simplified one-compartment model.\nleading to the equilibrium solution for the final metabolite M J as:\n${M}_{J}\\left(E,\\mathbf{G}\\right)=E×\\prod _{j=1}^{J}\\left(\\frac{{\\lambda }_{j\\mathsf{\\text{-1}}}\\left({G}_{j-1}^{\\left(\\lambda \\right)}\\right)}{{\\lambda }_{j}\\left({G}_{j}^{\\left(\\lambda \\right)}\\right)+{\\mu }_{j}\\left({G}_{j}^{\\left(\\mu \\right)}\\right)}\\right)$\nwhere X0 denotes the concentration of exposure E. This predicted equilibrium concentration of the final metabolite in the graph is then treated as a covariate in a logistic model for the risk of disease:\nIf sufficient external knowledge about the genotype-specific reaction rates is available, these could be treated as fixed constants, but more likely they would need to be estimated jointly with the βs in the disease model using a non-linear fitting program. More sophisticated non-linear models are possible -- for example, incorporating Michaelis-Menten kinetics by replacing each of the λM terms in Eq. (4) by expressions of the form:\n$\\frac{{V}_{max}^{\\lambda j}\\left({G}_{j}\\right){M}_{j}}{{M}_{j}+{K}_{m}^{\\lambda j}\\left({G}_{j}\\right)}$\nand similarly for the μM terms. The resulting equilibrium solutions for MJ(E, G) are now more complex solutions to a polynomial equation. For example, with only a single intermediate metabolite with one activation rate λ and one detoxification rate μ, the solution becomes:\n$M=E\\frac{\\lambda /\\mu }{1+\\left(\\left(1/{K}_{m}^{\\lambda }\\right)-\\left(\\lambda /\\mu \\right)/{K}_{m}^{\\mu }\\right)E}$\n\nwhere $\\lambda ={V}_{max}^{\\lambda }\\left({G}_{1}\\right)/{K}_{m}^{\\lambda }$ and $\\mu ={V}_{max}^{\\mu }\\left({G}_{2}\\right)/{K}_{m}^{\\mu }$denote the low-dose slopes of the two reactions. These solutions can be either upwardly or downwardly curvilinear in E, depending on whether the term in parentheses is positive or negative (basically, whether the creation of the intermediate exceeds the rate at which it can be removed). For the fitted values in the application below (third block of Table 4), the dose-response relationship for M|E is upwardly curved for all genotype combinations (not shown).\n\nA more realistic and more flexible model would allow for stochastic variation in the reaction rates λ ij and μ ij for each individual i conditional on their genotypes G ij ; for example, ${\\lambda }_{ij}~LN\\left({\\stackrel{̄}{\\lambda }}_{j}\\left({G}_{ij}\\right),{\\sigma }_{j}^{2}\\right)$ and likewise for μ ij or similarly for their corresponding V max and K m . The population genotype-specific rates are, in turn, assumed to have log-normal prior distributions ${\\stackrel{̄}{\\lambda }}_{j}\\left(g\\right)~LN\\left({\\stackrel{̄}{\\stackrel{̄}{\\lambda }}}_{j}{\\omega }_{j}^{2}\\right)$ (and similarly for the μs), with vague priors on the population means ${\\stackrel{̄}{\\stackrel{̄}{\\lambda }}}_{j}$, inter-individual variances ${\\sigma }_{j}^{2}$ and between-genotype variances ${\\omega }_{j}^{2}$. The individual data might be further supplemented by available biomarker measurements B ij of either the enzyme activity levels or intermediate metabolite concentrations, modelled as ${B}_{ij}~LN\\left({\\lambda }_{ij},{\\omega }^{2}\\right)$and ${B}_{ij}~LN\\left({M}_{ij},{\\omega }^{2}\\right)$respectively.\n\nThe WinBUGS software has an add-in called PKBUGS, which implements a Bayesian analysis of population pharmacokinetic parameters . More complex models can, in principle, be fitted using the add-in WBDIFF http://www.winbugs-development.org.uk/wbdiff.html, which allows user-specified differential equations as nodes in a Bayesian graphical model.\n\nTo illustrate the approach, we consider a highly simplified model with only a single intermediate metabolite M (homocysteine). We assume this is created at linear rate λ determined by SAHH and removed at linear rate μ determined by CBS. The ratios of λ and μ between genotypes are estimated jointly with β. The first two lines of Table 4 provide the results of fitting the linear kinetics model, with and without inter-individual variability in the two rate parameters. Although, of course, many other genes are involved in the simulated model, the estimated homocysteine concentrations M are strongly predictive of disease, and both genes have highly significant effects on their respective rates. Allowing additional random variability in these rates slightly increased the population average genetic effects. For the Michaelis-Menten models, we allowed the V max s to depend on genotype, while keeping the K m s fixed. Not all the parameters can be independently estimated, but only the ratios μ00 and ${K}_{m}^{\\mu }/{K}_{m}^{\\lambda }$, along with the genetic rate ratios λ10 and μ10. Allowing the V max s and K m s to vary between subjects leads to some instability, but did not substantially alter the population mean parameter estimates. Adding in biomarker measurements B i as surrogates for M i for even a subset of subjects, as described below, substantially improved the precision of estimation of all the model parameters (results not shown).\n\n### Combining mechanistic and statistical models\n\nSuch an approach is likely to be impractical for complex looped pathways like folate, however. In this case, one might use the results of a preliminary exploratory or hierarchical model to simplify the pathway to a few key rate-limiting steps, so as to yield a simpler unidirectional model for which the differential equation steady-state solutions can be obtained in closed form.\n\nRather than taking M(E, G) as a deterministic node in the mechanistic modelling approach described above, a fully Bayesian treatment would use stochastic differential equations to derive Pr(M|E, G). For example, suppose one postulated that the rate of change dM/dt depends on the rate at which it is created as a constant rate λ(G1)E and the rate at which it is removed at rate μ(G2)M. (Of course, the exposures E could be time dependent, in which case one would be interested in the long-term average of M rather than its steady state, but in most epidemiological studies there is little information available on short-term variation in exposures, so the following discussion is limited to the case of time-constant exposures.) Consider a discrete number of molecules and let p m (t) = Pr(M = m|T = t). Then, the resulting stochastic differential equation becomes:\n$\\frac{d{p}_{m}}{dt}=-\\left(\\lambda E+\\mu m\\right){p}_{m}+\\lambda E{p}_{m-1}+\\mu \\left(m+1\\right){p}_{m+1}$\nThe solution turns out to be simply a Poisson distribution for m with mean E(m) = λE/μ. This suggests as a distribution for continuous metabolite concentrations M in some volume of size N:\n$p\\left(M\\right)=N{e}^{-\\lambda EN/\\mu }{\\left(\\lambda EN/\\mu \\right)}^{NM}/\\Gamma \\left(NM+1\\right)$\n\nwhere N now controls the dispersion of the distribution. More complex solutions for Michaelis-Menten kinetics with a finite number of binding sites have been provided by Kou et al., who showed that the classical solutions still held in expectation, but other properties -- like the distribution of waiting times in various binding states -- were different, appearing to demonstrate a non-Markov memory phenomenon, particularly at high substrate concentrations. Further stochastic variability arises from fluctuations in binding affinity due to continual changes in enzyme conformation .\n\nTo illustrate the general idea, we fitted this simplified version of the model, treating λ and μ as fixed genotype-specific population values, yielding the estimates shown in the last line of Table 4. The dispersion parameter N cannot be estimated, but the results for other parameters are relatively insensitive to this choice; the results in Table 4 are based on either a fixed value N = 10 or using an informative Γ(100,1) prior; as N gets very large, the estimates converge to those in the first line for linear kinetics with fixed genotype-specific λ and μ.\n\nFor more complex models, for which analytic solution of the differential equations may be intractable, the technique of approximate Bayesian computation may be helpful. The basic idea is, at each Markov chain Monte Carlo cycle, to simulate data from the differential equations model using the current and proposed estimates of model parameters and evaluate the 'closeness' of the simulated data to the observed data in terms of some simple statistics. This is then used to decide whether to accept or reject the proposed new estimates, rather than having to compute the likelihood itself.\n\nA simpler approach uses the output of a PBPK simulation model as prior covariates in a hierarchical model. Let Z ge = E[M(G g , E e )] denote the predicted steady-state concentrations of the final metabolite from a differential equations model for a particular combination of genes and/or exposures (thus, Z gg' might represent the predicted effect of a G × G interaction between genes g and g'). As discussed above, other Zs could comprise variances of predicted Ms across a range of input values as a measure of the sensitivity of the output to variation in that particular combination of inputs. Z ge could also be a vector of several different predicted metabolite concentrations if there were multiple hypotheses about which was the most aetiologically relevant.\n\nFor the folate application, the Z matrix was obtained by correlating the simulated intermediate phenotypes v (reaction rates or metabolite concentrations) with the 14 genotypes, 91 G × G and 28 G × E interaction terms. The resulting correlation coefficients for the four simulated causal variables were then used as a vector of in silico prior covariates Z p = (Z pv )v= 1..4 for the relative risk coefficients β p . The full set of correlations Z pv across all ten metabolites and nine non-redundant velocities were also used to compute an adjacency matrix as A pq = corr v (Z pv , Z qv ), representing the extent to which a pair of genes had similar effects across the whole range of intermediate phenotypes. The effects of these in silico covariates (Table 5) were substantially stronger than for the simple indicator variables illustrated earlier. In each simulation, the prior covariate corresponding to the causal variable was the strongest predictor of the genetic main effects.\n\n### Designs incorporating biomarkers\n\nUltimately, it may be helpful to incorporate various markers of the internal workings of a postulated pathway, perhaps in the form of biomarker measurements of intermediate metabolites, external bioinformatic knowledge about the structure and parameters of the network, or toxicological assays of the biological effects of the agents under study. For example, in a multi-city study of air pollution, we are applying stored particulate samples from each city to cell cultures with a range of genes experimentally knocked down to assess their genotype-specific biological activities. We will then incorporate these measurements directly into the analysis of G × E interactions in epidemiological data . See Thomas, Thomas et al., Conti et al. and Parl et al. for further discussion about approaches to incorporating biomarkers and other forms of biological knowledge into pathway-driven analyses.\n\nTypically biomarker measurements are difficult to obtain and are only feasible to collect on a subset of a large epidemiological study. While one might consider using a simple random sample for this purpose, greater efficiency can often be obtained by stratified sampling. Suppose the parent study is a case-control study with exposure information and DNA already obtained. One might then consider sampling on the basis of some combination of disease status, exposure and the genotypes of one or more genes thought to be particularly important for the intermediate phenotype(s) for which biomarkers are to be obtained. The optimal design would require knowledge of the true model (which, of course, is unknown), but a balanced design, selecting the subsample so as to obtain equal numbers in the various strata defined by disease and predictors is often nearly optimal . The analysis can then be conducted by full maximum likelihood, integrating the biomarkers for unmeasured subjects over their distribution (given the available genotype, exposure and disease data) or by some form of multiple imputation, quasi-likelihood or MCMC methods. Here, the interest is not in the association of disease with the biomarker B itself, but rather with the unobserved intermediate phenotype M it is a surrogate for. The disease model is thus of the form Pr(Y|M), with a latent process model for Pr(M|G, E) and a measurement model for Pr(B|M).\n\nAgain, using the folate simulation as the example, we simulated biomarkers for samples of ten or 25 individuals selected at random from each of the eight cells defined by disease status, the MTHFR genotype and high or low folate intake. A measurement B of either homocysteine concentration or the TS enzyme activity level was assumed to be normally distributed around their simulated equilibrium concentrations with standard deviations 10 per cent of that the true long-term average concentrations.\n\nThese data were analysed within a conventional measurement error framework [41, 42] by treating the true long-term average values of homocysteine or TS activity as a latent variable X in a model given by the following equations:\n\nFor joint analyses of homocysteine and TS activity measurements, M and B were assumed to be bivariate normally distributed with M ~ N2(X'A, Σ) and B ~ N2(M, T), and Y as having a multiple logistic dependence on M. Only the main effects of the 14 genes and two environmental factors were included in X for this analysis. While the model can be fitted by maximum likelihood, it is convenient to use MCMC methods, which more readily deal with arbitrary patterns of missing B data. Thus, it is not essential for the different biomarkers to be measured on the same subset of subjects, but some overlap is needed to estimate the covariances Σ12 and T12. More complex mechanistic models could, of course, be used in place of the regression model M|X. For this model to be identifiable, however, it is essential that distinct biomarkers be available for each of the intermediate phenotypes included in the disease model.\n\nEstimates of the effects of both homocysteine and TS enzyme activity were highly significant in univariate analyses, even though the simulated causal variable is homocysteine. In bivariate analyses, however, the TS effect became non-significant, owing to the strong positive correlation (rΣ = 0.45; 95 per cent confidence interval [CI] 0.21, 0.71) between the residuals of M, while correlation between the residuals of the measurement errors was not significant (rT = 0.34; 95% CI -0.12, +0.63). Although the standard errors varied strongly with subsample size, stratified sampling did not seem to improve the precision of the estimates. The reason for this appears to be that the biased sampling is not properly allowed for in the Bayesian analysis. Further work is needed to explore whether incorporating the sampling fractions into a conditional likelihood would yield more efficient estimators in the stratified designs.\n\n### Dealing with reverse causation: Mendelian randomisation\n\nThe foregoing development assumes that the bio-marker measurement B or the underlying phenotype M of which it is a measurement is not affected by the disease process. While this may be a reasonable assumption in a cohort or nested case-control study where biomarker measurements are made on stored specimens obtained at entry to the cohort rather than after the disease has already occurred, it is a well known problem (known as 'reverse causation') in case-control studies. In this situation, one might want to restrict biomarker measurements only to controls and use marginal likelihood or imputation to deal with the unmeasured biomarkers for cases. Alternatively, one might consider using case measurements in a model that includes terms for differential error in the measurement model, Pr(B|M, Y).\n\nThese ideas have been formalised in literature known as 'Mendelian randomisation' (MR), sometimes referred to as 'Mendelian deconfounding' . Here, the focus of attention is not the genes themselves, but intermediate phenotypes (M) as risk factors for disease. The genes that influence M are treated as 'instrumental variables' (IVs) in an analysis that indirectly infers the M-Y relationship from separate analyses of the G-M and G-Y relationships. The appeal of the approach is that uncontrolled confounding and reverse causation are less likely to distort these relationships than they are to distort the M-Y relationship if studied directly. In essence, the idea of imputing M values using G as an IV in a regression of Y on E(M|G) is a form of MR argument. Nevertheless, the approach is not without its pitfalls, both as a means of testing the null hypothesis of no causal connection between M and Y and as a means of estimating the magnitude of its effect. Particularly key is the assumption that the effect of G on Y is mediated solely through M. For complex pathways, the simple MR approach is unlikely to be of much help, but the idea of using samples free of reverse causation to learn about parts of the model from biomarker measurements and incorporating these into the analysis of a latent variable model is promising.\n\nTo illustrate these methods, consider the scenario where homocysteine is the causal variable for disease. The logistic regression of disease directly on homocysteine yields a logRR coefficient β of 2.57 (SE 0.22) per SD change of homocysteine (Table 7). This estimate is, however, potentially subject to confounding and reverse causation, and indeed in this simulation we generated an upward bias in B|M of 50 per cent of the SD of M, which produced a substantial overestimate of the simulated β = 2. An MR estimate could in principle be obtained by using any of the genes in the pathway as an IV, MTHRF being the most widely studied. The regression of homocysteine on MTHFR yields a regression coefficient of α = 0.216 (0.079) and a logistic regression of disease on MTHFR yields a regression coefficient of γ = 0.112 (0.142), to produce an MR estimate of β = γ/α = 0.52 (0.68). Since MTHRF is only a relatively weak predictor of homocysteine concentrations in this simulation, however, it is a poor instrumental variable, as reflected in the large SE of the ratio estimate. Several other genes, exposures and interactions have much stronger effects on both homocysteine and disease risk -- notably, SAHH and CBS, which yield significant MR estimates, 1.27 (0.33) and 1.09 (0.20), respectively. These differences between estimates using different IVs and their underestimation of the simulated β suggest that simple Mendelian randomisation is inadequate to deal with complex pathways.\n\nA stepwise multiple regression model for $\\stackrel{^}{M}=E\\left(B|\\mathsf{\\text{G}}\\right)$ included 13 main effects and G × G interactions and attained an R2 of 0.43. Treating these predicted homocysteine concentrations as the covariate yielded a single imputation estimate of the log RR for disease of 1.32 (0.16), only slightly less precise than that from the logistic regression of disease directly on the measured values. While robust to uncontrolled confounding, this approach is not robust to reverse causation or misspecification of the prediction model; for example, it fails to include any exposure effects, which we have excluded to avoid distortion by reverse causation. More importantly, it also assumes that the entire effect of the predictors is mediated through homocysteine; this is true for this simulation, but is unlikely to be in practice. While not quite as downwardly biased as the Mendelian randomisation estimates (resulting from the improved prediction of B|G), the incompleteness of the model has still produced some underestimation.\n\nSince we have simulated the case where the biomarker measurements are distorted by disease status, one might consider one of two alternative single imputation analyses. If both cases and controls have biomarker measurements available, one might include disease status in a model for $\\stackrel{^}{M}=E\\left(B|\\mathbf{G},Y\\right)={\\mathbf{\\alpha }}^{\\text{'}}\\mathbf{G}+\\delta Y$, and then set Y = 0 in the fitted regression in order to estimate the predisease values for the cases. Alternatively, one could fit the model for $\\stackrel{^}{M}=E\\left(B|\\mathbf{G}\\right)$ using data only from controls and then apply the fitted model to all subjects, cases and controls. In either case, one would use only the predicted values for all subjects, not the actual biomarker measurements for those having them. In these simulated data, these approaches yield log RR estimates of 1.28 (0.20) and 1.31 (0.20), respectively. Either of these approaches avoids the circularity of using disease status to predict B|G, Y and then using it again in the regression of Y on $\\stackrel{^}{M}=E\\left(B|\\mathbf{G},Y\\right)$. While the first approach uses more of the data, it requires a stronger assumption that the effect of Y on B is correctly specified, including possible interactions with G. In this simulation, the estimate of δ is 1.33 (0.06), substantially biased away from the simulated value of 0.50 because it includes some of the causal effect of X on Y. A fully Bayesian analysis jointly estimates the bias term δY in the full model ${p}_{\\alpha }{\\left(M|E,G\\right)}_{{p}_{\\beta }}{{\\left(Y|M\\right)}_{p}}_{{}_{\\gamma ,\\delta }}\\left(B|M,Y\\right)$. In this simulation, the fully Bayesian analysis yielded an estimate of β = 2.95 (0.22) and δ = -0.02 (1.02). Obviously, δ is so poorly estimated and β so overestimated that this approach appears to suffer from problems of identifiability that require further investigation.\n\nIn the Colon Cancer Family Registries, we have pre-disease biospecimens on several hundred relatives of probands who were initially unaffected and subsequently became cases themselves. In a currently ongoing substudy of biomarkers for the folate pathway, it will be possible to use these samples to estimate the effect of reverse causation directly. Of course, it would have been even more informative to have both pre- and post-diagnostic biomarker measurements on incident cases to model reverse causation more accurately.\n\n### Incorporating external information: Ontologies\n\nThere are now numerous databases available that catalogue various types of genomic information. The Kyoto Encyclopedia of Genes and Genomes (KEGG) is perhaps the most familiar of these for knowledge about the structure of pathways and the parameters of each step therein. Others include the Gene Ontology, Biomolecular Interaction Network Database, Reactome, PANTHER, Ingenuity Pathway Analysis, BioCARTA, GATHER, DAVID and the Human Protein Reference Database, (see, for example, Meier and Gehring, Thomas et al. and Werner for reviews). Literature mining is emerging as another tool for this purpose, although potentially biased by the vagaries of research and publication trends. Such repositories form part of a system for organising knowledge known as an 'ontology' . Representation of our knowledge via an ontology may provide a more useful and broadly informative platform to generate system-wide hypotheses about how variation in human genes ultimately impacts on organism-level phenotypes via the underlying pathway or complex system. Since the biological and environmental knowledge relevant to most diseases spans many research fields, each with specific theories guiding ongoing research, expertise across the entire system by one individual scientist is limited. While the information that contributes to each knowledge domain may contain uncertainties and sources of error stemming from the underlying experiments and studies, biases in the selection of genes and pathways chosen to be included and lack of comparability across terms and databases, an ontology as a whole can generate hypotheses and links across research disciplines that may only arise when information is integrated from several disciplines across the entire span of suspected disease aetiology. An ontology should not be taken as the truth, but rather as the current representation of knowledge that can, and should, be updated as new findings arise and hypotheses are tested. Evaluation of the accuracy of ontologies is an active research area.\n\nIn our folate simulation, we considered three prior covariates for Z in Table 3. The creation of these priors followed directly from the network representation given in Figure 1, obtained from a previously published article representing one research group's interpretation of the folate pathway . An ontology, such as Gene Ontology (GO), has the potential advantage of allowing for the construction of prior covariates across a range of biological mechanisms. For example, a very refined biological process captured by the GO term folic acid and derivative biosynthetic process indicates two genes (MTCH and MS) from our example set of genes. A more general term, methionine biosynthetic process, identifies three genes (MTCH, MTHFR and MS). Finally, a broad process, such as one-carbon compound metabolic process, identifies five genes (SAHH, DHFR, MAT-II, MTCH and SHMT). Since an ontology has a hierarchical structure in a easily computable format, one may consider more quantitative approaches in generating prior covariates, such as the distance between two genes in the ontology. Across the full range of 184 GO terms involving one or more of these 14 genes, positively correlated sets include (MTHFR, MTCH, MS), (MTD, CBS), (FTD, DHFR) and (AICART, TS), while PGT and MTCH are negatively correlated. Figure 3 represents these correlations using a complete agglomerative clustering.", null, "Figure 3 Hierarchical clustering of folate genes based on 184 GO terms.\n\nAlthough both approaches to building prior covariates, via either the visual interpretation of a network or the use of Gene Ontology, use knowledge of biological mechanisms, they lack a formal link of these mechanisms to disease risk or organism-level phenotypes. Such links may be critical when generating hypotheses or informing statistical analyses using biological mechanisms. Many publicly available ontologies provide a vast amount of structural information on various bio-logical processes, but interpretation or weighting of the importance of those processes in relation to specific phenotypes will only come when ontologies from biological domains are linked to ontologies characterising phenotypes. As one example, Thomas et al. created a novel ontological representation linking smoking-related phenotypes and response to smoking cessation treatments with the underlying biological mechanisms, mainly nicotine metabolism. Most of the ontological concepts created for this specific ontology were created using concept definitions from existing ontologies, such as SOPHARM and Gene Ontology. This ontology was used in Conti et al. to demonstrate the use in pathway analysis as a systematic way of eliciting priors for a hierarchical model. Specifically, the ontology was used to generate quantitative priors to reduce the space of potential models and to inform subsequent analysis via a Bayesian model selection approach.\n\n### Dealing with uncertainty in pathway structure\n\nA more general question is how to deal with model uncertainty in any of these modelling strategies. The general hierarchical modelling strategy was first extended by Conti et al. to deal with uncertainty about the set of main effects and interactions to be included in X using stochastic search variable selection . Specifically, they replaced the second-level model by a pair of models, a logistic regression for the probability that β p = 0 and a linear regression of the form of Eq. (2) for the expected values of the coefficient, given that it was not zero. In turn, the pair of second-level models inform the probability that any given term will be included in the model at the current iteration of the stochastic search. Thus, over the course of MCMC iterations, variables are entered and removed, and one can then estimate the posterior probability or Bayes factor (1) for each factor or possible model (2), for whether each factor has a non-zero β averaging over the set of other variables in the model, or (3) the posterior mean of each β, given that it is non-zero. Other alternatives include the Lasso prior, which requires only a single hyperparameter to accomplish both shrinkage and variable selection in a natural way, and the elastic net, which combines the Lasso and normal priors and can be implemented in a hierarchical fashion combining variable selection at lower levels (eg among SNPs within a pathway) and shrinkage at higher levels (eg between genes within a pathway or between pathways) (Chen et al. Presented at the Eastern North American Region Meeting of the Biometric Society; San Antonio, TX: February 2009).\n\nIn an analysis, utilising the methods described by Conti et al., of the simulated data when homocysteine is the causal variable (Table 5, first column) and incorporating an exchangeable prior structure in which all genes are treated equally (ie intercept only in the prior covariate matrix, Z), the posterior probabilities of including the two modestly significant genes TS and FTD are 0.57 and 0.48, respectively. By contrast, when the prior covariate matrix is derived from the 'external database' from the simulation model and is thus more informative of the underlying mechanism, these posterior probabilities change to 0.84 and 0.14, respectively. These changes in the posterior probabilities of inclusion reflect the covariate values for these genes in relation to homocysteine concentration and the AICART reaction velocity (the two prior covariates with the largest estimated second-level effects). In the case of TS, the velocities for these covariates are large, resulting in an increase in the posterior probability of inclusion. By contrast, for FTD these values are much smaller and there is a subsequent decrease.\n\nFor mechanistic models, the 'topology' of the model Λ and the corresponding vector of model parameters θΛ are treated as unknown quantities, about which we might have some general prior knowledge in the form of the 'ontology' Z. In the microarray analysis world, Bayesian network analysis has emerged as a powerful technique for inferring the structure of a complex network of genes . Might such a technique prove helpful for epidemiological analysis?\n\nOne promising approach is 'logic regression', which considers a set of tree-structured models relating measurable inputs (genes and exposures) to a disease trait through a network of unobserved intermediate nodes representing logical operators (AND, OR, XOR etc) . To allow for uncertainty about model form, a MCMC method is used to update the structure of the graphical model by adding, deleting, moving or changing the types of the intermediate nodes . Although appealing as a way of representing the biochemical pathways, logic regression does not exploit any external information about the form of network. It also treats all intermediate nodes as binary, so it is more suitable for modelling regulatory than metabolic pathways where the intermediate nodes would represent continuous metabolite concentrations.\n\nTo overcome some of these difficulties, we relaxed the restriction to binary nodes, parameterising the model as:\n${M}_{j}={\\theta }_{j1}{M}_{pj1}+{\\theta }_{n2}{M}_{pj2}+\\left(1-{\\theta }_{j\\mathsf{\\text{1}}}-{\\theta }_{j\\mathsf{\\text{2}}}\\right){M}_{pj1}{M}_{pj2}$\n(5)\n\nWhen both input nodes (the 'parents' pj = [pj1, pj2]) are binary, various combinations of θs will yield the full range of possible logical operators (eg AND = [0,0], OR = [1,1]), but this framework allows great flexibility in modelling interactions between continuous nodes, while remaining identifiable. The Ms are treated as deterministic nodes, so the final metabolite concentration M J (E, G; Λ, θ) can be calculated via a simple recursion. The disease risk is assumed to have a logistic dependence on M J . Prior knowledge about the topology can be incorporated by use of a measure of similarity of each fitted network to the postulated true network (eg the proportion of connections in the true graph which are represented in the fitted one, minus the number of connections in the fitted graph which are not represented in the true one). In the spirit of Monte Carlo logic regression, the topology of the graph is modified by proposing to add or delete nodes or to move a connection between them using the Metropolis-Hastings algorithm . Finally, the model parameters are updated conditional on the current model form. By post-processing the resulting set of graphs, various kinds of inference can be drawn, such as the posterior probability that a given input appears in the fitted graphs, that a pair of inputs is represented by a node in the graph, or the marginal effect of any input or combination of inputs on the disease risk. In small simulations, we demonstrated that the model could correctly identify the true network structure (or logically equivalent ones) and estimate the parameters well, while not identifying any incorrect models. In an application to data on ten candidate genes from the Children's Health Study, we were able to replicate the interactions found by a purely exploratory technique and identified several alternative networks with comparable Bayes factors.\n\nThe folate pathway poses difficulties for mechanistic modelling because it is not a directed acyclic graph (DAG); although each arrow in Figure 1 is directed, the graph contains numerous cycles (feedback loops), making direct computation of probabilities difficult. In some instances, such cycles can be treated as single composite nodes with complex deterministic or stochastic laws, thereby rendering the remainder of the graph acyclic, but when there are many interconnected cycles, as in the folate pathway, such decomposition may be difficult or impossible to identify. Might it be possible, however, to identify a simpler DAG that captures the key behaviour of the network? Since any DAG would be an oversimplification and there could be many such DAGs that provide a reasonable approximation, the problem of model uncertainty is important.\n\nA further extension of the Baurley et al. approach to the folate simulation will now be summarised. As in their approach, we assume that each node has exactly two inputs, but now distinguish three basic types of nodes, G × G, G × M (or G × E) and M × M. G × G nodes are treated as logical operators, yielding a binary output as high or low risk. G × M and G × E nodes represent intermediate metabolite concentrations, treated as continuous variables with deterministic values given by Michaelis-Menten kinetics with rate parameters V max (G) and K m . M × M nodes are regression expressions yielding a continuous output variable with the mean parameterised as in Eq. (5). Disease risk is assumed to have a logistic dependence on one or more of the Zs. Finally, each measured biomarker B is assumed to be log-normally distributed around one of the Ms, with some measurement error variance. Rather than treating the intermediate nodes as deterministic, the likelihood of the entire graph is now calculated by peeling over possible states of all the intermediate nodes.\n\nFigure 4 shows the topologies discovered by the MCMC search. The largest Bayes factors are obtained when using no prior topologies. With a prior topology, essentially the same networks are found, with somewhat different Bayes factors.", null, "Figure 4 Top-ranking topologies without incorporating priors: left, gene only; right, genes and exposures. With no priors, the two topologies have posterior probabilities 3.9 per cent and 2.3 per cent, respectively. Using a topology derived by hierarchical clustering of the A matrix from simulated data, the top-ranked gene-only topology was identical to that shown on the left, with posterior probability of 9.5 per cent. Using the GO topology shown in Figure 3, the same genes were included, but reordered as (((MTHR, SAHH), MTD), SHMT)with a posterior probability of 6.4 per cent.\n\n### Pathways in a genome-wide context\n\nGenome-wide association studies (GWAS) are generally seen as 'agnostic' -- the antithesis of hypothesis-driven pathway-based studies. Aside from the daunting computational challenge, their primary goal is, after all, the discovery of novel genetic associations, possibly in genes with unknown function or even with genomic variation in 'gene desert' regions not known to harbour genes. How, then, could one hope to incorporate prior knowledge in a GWAS? The response has generally been to wait until the GWAS has been completed (after a multi-stage scan and independent replication) and then conduct various in vitro functional studies of the novel associations before attempting any pathway modelling.\n\nThe idea of incorporating prior knowledge from genomic annotation databases or other sources as a way of improving the power of a genome-wide scan for discovery has, however, been suggested by several authors. Roeder et al., Saccone et al., Wakefield and Whittemore introduced variants of a weighted false discovery rate, while Lewinger et al. and Chen and Witte described hierarchical modelling approaches for this purpose. These could be applied at any stage of a GWAS to improve the prioritisation of variants to be taken forward to the next stage. For example, Sebastiani et al. used a Bayesian test to incorporate external information for prioritising SNP associations from the first stage of a GWAS using pooled DNA, to be subsequently tested using individual genotyping. Roeder et al. originally suggested the idea of exploiting external information in the context of using a prior linkage scan to focus attention in regions of the genome more likely to harbour causal variants, but subsequent authors have noted that various other types of information, such as linkage disequilibrium, functional characterisation or evolutionary conservation, could be included as predictors. An advantage of hierarchical modelling is that multiple sources can be readily incorporated in a flexible regression framework, whereas the weighted FDR requires a priori choice of a specific weighting scheme.\n\nA recent trend has been the incorporation of pathway inference in genome-wide association scans, [75, 8389] borrowing ideas from the extensive literature on network analysis of gene expression array data [90, 91]. Currently, the most widely used tool for this purpose is gene set enrichment analysis, which in GWAS applications aims to test whether groups of genes in a common pathway tend to rank higher in significance. Several published applications have yielded novel insights using this approach, although others have found that no specific pathway outranks the most significant single markers, [89, 97, 98] suggesting that the approach may not be ideal for all complex diseases. Many other empirical approaches have been used in the gene-expression field, including Bayesian network analysis, [69, 99, 100] neural networks, support vector machines and a variety of other techniques from the fields of bioinformatics, computational or systems biology and machine learning . Most of these are empirical, although in the sense of trying to reconstruct the unknown network structure from observational data, rather than using a known network to analyse the observational data. It is less obvious how such methods could be applied to mining single-marker associations from a GWAS, but they could be helpful in mining G × G interactions. Even simple analyses of GWAS data can be computationally demanding, particularly if all possible G × G interactions are to be included, and analyses incorporating pathway information is likely to be even more daunting. Recent developments in computational algorithms for searching high-dimensional spaces and parallel cluster computing implementations may, however, make this feasible.\n\nRecently, several authors have undertaken analyses of the association of genome-wide expression data with genome-wide SNP genotypes in search of patterns of genetic control that would identify cis- and trans-activating factors and master regulatory regions. Ultimately, one could foresee using networks inferred from gene expression directly as priors in a hierarchical modelling analysis for GWAS data, or a joint analysis of the two phenotypes, but this has yet to be attempted. Other novel technologies, such as whole-genome sequencing, metabolomics, proteomics and so on may provide other types of data that will inform pathway-based analysis on a genome-wide scale.\n\n## Conclusions\n\nAs in any other form of statistical modelling, the analyst should be cautious in interpretation. An pointed out by Jansen: \n\n'So, the modeling of the interplay of many genes -- which is the aim of complex systems biology -- is not without danger. Any model can be wrong (almost by definition), but particularly complex (overparameterized) models have much flexibility to hide their lack of biological relevance' [emphasis added].\n\nA good fit to a particular model does not, of course, establish the truth of the model. Instead, the value of models, whether descriptive or mechanistic, lies in their ability to organise a range of hypotheses into a systematic framework in which simpler models can be tested against more complex alternatives. The usefulness of the Armitage-Doll multistage model of carcinogenesis, for example, lies not in our belief that it is a completely accurate description of the process, but rather in its ability to distinguish whether a carcinogen appears to act early or late in the process or at more than one stage. Similarly, the importance of the Moolgavkar-Knudson two-stage clonal-expansion model lies in its ability to test whether a carcinogen acts as an 'initiator' (ie on the mutation rates) or a 'promoter' (ie on proliferation rates). Such inferences can be valuable, even if the model itself is an incomplete description of the process, as must always be the case.\n\nAlthough mechanistic models do make some testable predictions about such things as the shape of the dose-response relationship and the modifying effects of time-related variables, testing such patterns against epidemiological data tends to provide only weak evidence in support of the alternative models, and only within the context of all the other assumptions involved. Generally, comparisons of alternative models (or specific sub-models) can only be accomplished by direct fitting. Visualisation of the fit to complex epidemiological datasets can be challenging. Any mechanistic interpretations of model fits should therefore consider carefully the robustness of these conclusions to possible misspecification of other parts of the model.\n\n## Declarations\n\n### Acknowledgements\n\nThis work was supported in part by NIH grants R01-CA92562, P50-ES07048, R01-CA112237 and U01-ES015090 (D.C.T., D.V.C., J.B.), R01-CA105437, R01-CA105145, R01-CA59045 (C.M.U.) and NSF grants DMS-0616710 and DMS-0109872 (F.N., M.R.). The authors are particularly grateful to Wei Liang and Fan Yang for programming support.\n\n## Authors’ Affiliations\n\n(1)\nDepartment of Preventive Medicine, University of Southern California, 1540 Alcazar St., CHP-220, Los Angeles, CA 90089-9011, USA\n(2)\nDepartment of Biology, Duke University, Durham, NC, USA\n(3)\nDepartment of Mathematics, Duke University, Durham, NC, USA\n(4)\nFred Hutchison Cancer Research Center, 1100 Fairview Avenue N., PO Box 19024, Seattle, WA 98109-1024, USA\n\n## References\n\nAdvertisement", null, "" ]

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[ "# Gross to Net Calculation in Gen2\n\n• Question\n• Updated 5 years ago\nHow do I create a gross to net formula with in Replicon?\n\n(Originally posted by anna)\n\nArchived Post, Official Rep\n\n• 2132 Posts\n\nPosted 5 years ago\n\nArchived Post, Official Rep\n\n• 2132 Posts\nHi Anna,\n\nThe \"gross to net\" formula instructs Replicon how to calculate backwards from the gross (total expense amount) to the net (pre-tax amount) for an individual expense. Providing this formula will ensure expense amounts are correct, whether the user enters the net or the gross amount for a specific expense.\n\nBelow is the expansion of the formula on how gross to net is calculated.\n\nConsidering that tax is 7% of Net\n\nGross = Tax + Net\n\nTax = 0.07*Net\n\nSubstituting Tax with the formula mentioned above,\n\nGross = 0.07*net+Net\n\nGross = Net(0.07+1)\n\nTherefore,\n\nGross = Net * 1.07\n\nNet = Gross/1.07\n\nNote:\nhttp://www.replicon.com/customer-zone/kb-1000426\n\nSteps to add the above formula are as follows:-" ]

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[ "### We found 23 resources with the keyterm graph paper\n\nVideos (Over 2 Million Educational Videos Available)", null, "6:09\nThree-Dimensional Coordinates and the...", null, "2:45\nPopulation Genetics", null, "2:54\nPhonics Song 2 (new version)\nOther Resource Types ( 23 )\nLesson Planet\n\n#### Graph Paper Patterns\n\nFor Teachers 3rd - 8th\nYoung artists divide a piece of graph paper into sections using rectangles, squares, and triangles. They then fill each section with patterns of shape and color. Elementary graders describe how their patterns are organized. Secondary...\nLesson Planet\n\n#### Graph Paper\n\nFor Teachers 2nd - 8th\nIn this blank graph paper worksheet, students use the gridded sections marked off with the numbers 1-10 on the x and the y axis as directed by their teachers.\nLesson Planet\n\n#### The Koch Snowflake: Measuring and Area\n\nFor Students 4th - 8th\nIn this snowflakes worksheet, students cut out the given snowflakes and place them on graph paper to solve problems such as area and measurement. Students do this for 6 snowflakes.\nLesson Planet\n\n#### Co-ordinate Pictures\n\nFor Students 4th - 8th\nIn this co-ordinates learning exercise, students use each row of co-ordinates to create a shape on graph paper. Students complete 18 tasks total.\nLesson Planet\n\n#### Danger Mouse\n\nFor Students 5th - 8th\nIn this graphing and word problems worksheet, students plot and connect points on a graph to create a picture and complete math word problems. Students complete 10 pages of problems.\nLesson Planet\n\n#### Graph Paper 2\n\nFor Teachers 2nd - 12th\nIn this blank graph paper worksheet, pupils use the gridded sections that may be marked off on the x and the y axis as directed by their teachers.\nLesson Planet\n\n#### Halloween Graph Paper\n\nFor Students 5th\nIn this Halloween graphing worksheet, 5th graders enhance their math skills by using the graph to help them collect and organize data.\nLesson Planet\n\n#### Halloween Graph Paper\n\nFor Students 2nd - 8th\nIn this Halloween-themed math worksheet, students use graph paper for math activities. Students may use this to align numbers correctly or to create graphs.\nLesson Planet\n\n#### 1/2 Inch Graph Paper\n\nFor Students 4th - 5th\nIn this math worksheet, students use the 1/2 inch graph paper for any math or art purpose. Students can use this to align numbers correctly or to create graphs.\nLesson Planet\n\n#### Ocean Mapping\n\nFor Teachers 6th - 9th\nStudents create an ocean floor out of a shoe box, clay, and graph paper. In this ocean mapping lesson plan, students trade shoe boxes to view how no piece of the ocean floor is alike.\nLesson Planet\n\n#### Graph Paper\n\nFor Students 3rd - 6th\nIn this graph paper activity, students receive a blank piece of graph paper that they can complete graphs on. Students can have an unlimited amount of sheets.\n1 In 1 Collection\nLesson Planet\n\n#### Halloween Graph Paper\n\nFor Students 4th - 6th\nIn this graph paper worksheet, students improve their graphing skills using the graph paper. The paper is themed for a Halloween math activity.\nLesson Planet\n\n#### Centimetre Graph Paper\n\nFor Students 4th - 5th\nIn this math worksheet, students use the blank graph paper for any art or math purpose. The intervals marked are centimeters. There are no directions for the use of this paper.\nLesson Planet\n\n#### 3/8 Inch Graph Paper\n\nFor Students 5th - 6th\nIn this math worksheet, students use the blank 3/8 inch graph paper for any math or art purpose. There are no instructions on the page.\nLesson Planet\n\n#### Centimetre Graph Paper\n\nFor Students 5th - 6th\nIn this math worksheet, students use the blank centimetre graph paper for any math or art purpose. Students may also use this to help in graphing or proper alignment of numbers.\nLesson Planet\n\n#### 1/4 Inch Graph Paper\n\nFor Students 5th - 6th\nIn this math worksheet, students use the blank 1/4 inch graph paper for any math or art purpose. This would be difficult to use for calculations as it is small but students could create graphs on this page.\nLesson Planet\n\n#### Jelly Bean Count\n\nFor Teachers Pre-K - 1st\nStudents study graphs. In this mathematics lesson, students count and make a graph use jelly beans by sorting the jelly beans into colors. Students write the amount of each color under the color written on the graph paper and color that...\nLesson Planet\n\n#### Graph Paper\n\nFor Teachers 8th - 10th\nPupils are given pre-made graph paper. In this algebra lesson, students are given copies of graph paper that can be used to plot points on a coordinate plane already measured out.\nLesson Planet\n\n#### Half Time\n\nFor Students 1st\nIn this shapes worksheet, 1st graders study and analyze how to cut three different shapes on dotted lines in half, two different ways on each line.\nLesson Planet\n\n#### Graph Paper\n\nFor Students 1st - 12th\nIn this literacy worksheet, learners are asked to complete the graphing activities using the newly created worksheets. The sheet itself is a guide for teachers to create graph paper styles.\nLesson Planet\n\n#### Asymmetric Graph Paper\n\nFor Students 1st - 12th\nIn this literacy worksheet, students use the created asymmetrical graph paper for class activities. The sheet is intended to be a teacher's guide.\nLesson Planet\n\n#### Axis Graph Paper\n\nFor Students 1st - 12th\nIn this literacy worksheet, students use the axis graph paper that is created using this worksheet. The sheet is a teacher's guide.\nLesson Planet\n\n#### Quarter-inch Grid\n\nFor Students 6th\nIn this learning tool activity, 6th graders can draw or depict information on a blank quarter-inch grid. There is a space at the top for name and date." ]

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[ "# Kerodon\n\n$\\Newextarrow{\\xRightarrow}{5,5}{0x21D2}$\n\n### 2.5.5 Digression: The Homology of Simplicial Sets\n\nAmong the most useful invariants studied in algebraic topology are the singular homology groups $\\mathrm{H}_{\\ast }(X; \\operatorname{\\mathbf{Z}})$ of a topological space $X$. These are defined as the homology groups of the singular chain complex\n\n$\\cdots \\xrightarrow {\\partial } \\mathrm{C}_{3}(X; \\operatorname{\\mathbf{Z}}) \\xrightarrow {\\partial } \\mathrm{C}_{2}(X; \\operatorname{\\mathbf{Z}}) \\xrightarrow {\\partial } \\mathrm{C}_1(X; \\operatorname{\\mathbf{Z}}) \\xrightarrow {\\partial } \\mathrm{C}_0(X; \\operatorname{\\mathbf{Z}}),$\n\nwhere $\\mathrm{C}_ n(X; \\operatorname{\\mathbf{Z}})$ denotes the free abelian group generated by the set $\\operatorname{Hom}_{\\operatorname{Top}}( | \\Delta ^ n |, X )$ of singular $n$-simplices of $X$, and the boundary operator $\\partial$ is given by the formula\n\n$\\partial : \\mathrm{C}_ n(X; \\operatorname{\\mathbf{Z}}) \\rightarrow \\mathrm{C}_{n-1}(X; \\operatorname{\\mathbf{Z}}) \\quad \\quad \\partial (\\sigma ) = \\sum _{i = 0}^{n} (-1)^{i} d_ i(\\sigma ).$\n\nWe can therefore view the passage from the topological space $X$ to its homology $\\mathrm{H}_{\\ast }(X; \\operatorname{\\mathbf{Z}})$ as proceeding in four stages:\n\n• We first extract from the topological space $X$ its singular simplicial set $\\operatorname{Sing}_{\\bullet }(X)$ (Construction 1.1.7.1).\n\n• We then replace $\\operatorname{Sing}_{\\bullet }(X)$ by the simplicial abelian group $\\operatorname{\\mathbf{Z}}[ \\operatorname{Sing}_{\\bullet }(X) ]$, carrying each object $[n] \\in \\operatorname{{\\bf \\Delta }}^{\\operatorname{op}}$ to the free abelian group $\\operatorname{\\mathbf{Z}}[ \\operatorname{Sing}_{n}(X) ]$ generated by the set $\\operatorname{Sing}_{n}(X)$.\n\n• We next regard the abelian groups $\\{ \\operatorname{\\mathbf{Z}}[ \\operatorname{Sing}_{n}(X) ] \\} _{n \\geq 0}$ as the terms of a chain complex $(C_{\\ast }(X; \\operatorname{\\mathbf{Z}}), \\partial )$, where the differential $\\partial$ is given by the alternating sum of the face maps of the simplicial abelian group $\\operatorname{\\mathbf{Z}}[ \\operatorname{Sing}_{\\bullet }(X) ]$.\n\n• For each integer $n$, we define $\\mathrm{H}_{n}(X; \\operatorname{\\mathbf{Z}})$ to be the $n$th homology group of the chain complex $(C_{\\ast }(X; \\operatorname{\\mathbf{Z}}), \\partial )$ (Definition 2.5.1.4).\n\nIn other words, the functor $X \\mapsto \\mathrm{H}_{n}(X; \\operatorname{\\mathbf{Z}})$ factors as a composition\n\n$\\operatorname{Top}\\xrightarrow {\\operatorname{Sing}_{\\bullet }} \\operatorname{Set_{\\Delta }}\\xrightarrow { \\operatorname{\\mathbf{Z}}[-]} \\operatorname{Ab_{\\Delta }}\\xrightarrow {\\mathrm{C}_{\\ast }} \\operatorname{Ch}(\\operatorname{\\mathbf{Z}}) \\xrightarrow { \\mathrm{H}_{n}} \\operatorname{ Ab },$\n\nwhere $\\operatorname{Ab_{\\Delta }}$ denotes the category of simplicial abelian groups and $\\mathrm{C}_{\\ast }: \\operatorname{Ab_{\\Delta }}\\rightarrow \\operatorname{Ch}(\\operatorname{\\mathbf{Z}})$ is given by the following:\n\nConstruction 2.5.5.1 (The Moore Complex). Let $A_{\\bullet }$ be a semisimplicial abelian group (Variant 1.1.1.6). For each $n \\geq 1$, we define a group homomorphism $\\partial : A_{n} \\rightarrow A_{n-1}$ by the formula\n\n$\\partial (\\sigma ) = \\sum _{i = 0}^{n} (-1)^{i} d_ i(\\sigma ),$\n\nwhere $d_{i}: A_{n} \\rightarrow A_{n-1}$ is the $i$th face map (Notation 1.1.1.8). For $n \\geq 2$ and $\\sigma \\in A_{n}$, we compute\n\n\\begin{eqnarray*} \\partial ^2( \\sigma ) & = & \\partial ( \\sum _{i = 0}^{n} (-1)^{i} d_ i(\\sigma ) ) \\\\ & = & \\sum _{i = 0}^{n} \\sum _{j = 0}^{n-1} (-1)^{i+j} (d_{j} d_ i)(\\sigma ) \\\\ & = & 0 \\end{eqnarray*}\n\nwhere the final equality follows from the identity $d_{i} \\circ d_{j} = d_{j-1} \\circ d_{i}$ for $0 \\leq i < j \\leq n$ (see Exercise 1.1.1.10). We let $\\mathrm{C}_{\\ast }(A)$ denote the chain complex of abelian groups given by\n\n$\\mathrm{C}_{n}(A) = \\begin{cases} A_{n} & \\text{ if } n \\geq 0 \\\\ 0 & \\text{otherwise,} \\end{cases}$\n\nwhere the differential is given by $\\partial$. We will refer to $\\mathrm{C}_{\\ast }(A)$ as the Moore complex of the semisimplicial abelian group $A_{\\bullet }$.\n\nIf $A_{\\bullet }$ is a simplicial abelian group, we let $\\mathrm{C}_{\\ast }(A)$ denote the Moore complex of the semisimplicial abelian group underlying $A_{\\bullet }$ (Remark 1.1.1.7).\n\nDefinition 2.5.5.2 (Homology of Simplicial Sets). Let $S_{\\bullet }$ be a simplicial set and let $\\operatorname{\\mathbf{Z}}[ S_{\\bullet } ]$ denote the simplicial abelian group freely generated by $S_{\\bullet }$. We let $\\mathrm{C}_{\\ast }( S; \\operatorname{\\mathbf{Z}})$ denote the Moore complex of $\\operatorname{\\mathbf{Z}}[ S_{\\bullet } ]$. We will refer to $\\mathrm{C}_{\\ast }(S; \\operatorname{\\mathbf{Z}})$ as the chain complex of $S_{\\bullet }$. For each integer $n$, we denote the $n$th homology group of $\\mathrm{C}_{\\ast }(S; \\operatorname{\\mathbf{Z}})$ by $\\mathrm{H}_{n}(S; \\operatorname{\\mathbf{Z}})$ and refer to it as the $n$th homology group of $X$ (with coefficients in $\\operatorname{\\mathbf{Z}}$).\n\nExample 2.5.5.3. Let $X$ be a topological space. Then the singular chain complex $\\mathrm{C}_{\\ast }(X; \\operatorname{\\mathbf{Z}})$ is the chain complex of the singular simplicial set $\\operatorname{Sing}_{\\bullet }(X)$. In particular, the homology groups of the simplicial set $\\operatorname{Sing}_{\\bullet }(X)$ are the usual singular homology groups of the topological space $X$.\n\nExample 2.5.5.4. Let $S_{\\bullet } = \\Delta ^{0}$ be the standard $0$-simplex. Then $S_{\\bullet }$ is a simplicial set having a single simplex of each dimension. Consequently, the chain complex $\\mathrm{C}_{\\ast }( S; \\operatorname{\\mathbf{Z}})$ is given by $\\operatorname{\\mathbf{Z}}$ in each nonnegative degree. For $n > 0$, the differential $\\operatorname{\\mathbf{Z}}\\simeq \\mathrm{C}_{n}(S; \\operatorname{\\mathbf{Z}}) \\xrightarrow {\\partial } \\mathrm{C}_{n-1}(S;\\operatorname{\\mathbf{Z}}) \\simeq \\operatorname{\\mathbf{Z}}$ is given by multiplication by the integer\n\n$\\sum _{i = 0}^{n} (-1)^{i} = \\begin{cases} 0 & \\text{ if n is odd } \\\\ 1 & \\text{ if n is even, } \\end{cases}$\n\nas indicated in the diagram\n\n$\\cdots \\rightarrow \\operatorname{\\mathbf{Z}}\\xrightarrow {0} \\operatorname{\\mathbf{Z}}\\xrightarrow {1} \\operatorname{\\mathbf{Z}}\\xrightarrow {0} \\operatorname{\\mathbf{Z}}\\xrightarrow {1} \\operatorname{\\mathbf{Z}}\\xrightarrow {0} \\operatorname{\\mathbf{Z}}\\xrightarrow {1} \\operatorname{\\mathbf{Z}}\\xrightarrow {0} \\operatorname{\\mathbf{Z}}.$\n\nIt follows that the homology groups of $S_{\\bullet }$ are given by\n\n$\\mathrm{H}_{n}( S; \\operatorname{\\mathbf{Z}}) = \\begin{cases} \\operatorname{\\mathbf{Z}}& \\text{ if n = 0 } \\\\ 0 & \\text{ otherwise.} \\end{cases}$\n\nNote that although the homology of the simplicial set $S_{\\bullet } = \\Delta ^0$ is concentrated in degree zero, the chain complex $\\mathrm{C}_{\\ast }(S; \\operatorname{\\mathbf{Z}})$ is not. Essentially, this is because $S_{\\bullet }$ has degenerate simplices in each dimension $n > 0$ which do not contribute to its homology. This is a special case of a more general phenomenon.\n\nNotation 2.5.5.5. Let $A_{\\bullet }$ be a simplicial abelian group. For each $n \\geq 0$, let $\\mathrm{D}_{n}(A)$ denote the subgroup of $\\mathrm{C}_{n}(A) = A_ n$ generated by the images of the degeneracy operators $\\{ s_{i}: A_{n-1} \\rightarrow A_ n \\} _{0 \\leq i \\leq n-1}$. By convention, we set $\\mathrm{D}_{n}(A) = 0$ for $n < 0$.\n\nProposition 2.5.5.6. Let $A_{\\bullet }$ be a simplicial abelian group. For every positive integer $n$, the boundary operator $\\partial : \\mathrm{C}_{n}(A) \\rightarrow \\mathrm{C}_{n-1}(A)$ carries the subgroup $\\mathrm{D}_{n}(A)$ into $\\mathrm{D}_{n-1}(A)$. Consequently, we can regard $\\mathrm{D}_{\\ast }(A)$ as a subcomplex of the Moore complex $\\mathrm{C}_{\\ast }(A)$.\n\nProof. Choose an element $\\sigma \\in \\mathrm{D}_{n}(A)$; we wish to show that $\\partial (\\sigma )$ belongs to $\\mathrm{D}_{n-1}(A)$. Without loss of generality, we may assume that $\\sigma = s_{i}(\\tau )$ for some $0 \\leq i \\leq n-1$ and some $\\tau \\in A_{n-1}$. We now compute\n\n\\begin{eqnarray*} \\partial (\\sigma ) & = & \\sum _{ j =0 }^{n} (-1)^{j} d_ i(\\sigma ) \\\\ & = & (\\sum _{j=0}^{i-1} (-1)^{j} d_ j s_ i \\tau ) + (-1)^{i} d_ i s_ i \\tau + (-1)^{i+1} d_{i+1} s_ i \\tau + (\\sum _{j = i+2}^{n} (-1)^{j} d_ j s_ i \\tau ) \\\\ & = & (\\sum _{j < i} (-1)^{j} s_{i-1} d_ j \\tau ) + (-1)^{i} \\tau + (-1)^{i+1} \\tau + (\\sum _{j = i+2}^{n} (-1)^{j} s_{i} d_{j-1}\\tau ) \\\\ & \\in & \\operatorname{im}( s_{i-1} ) + \\operatorname{im}( s_ i ) \\\\ & \\subseteq & \\mathrm{D}_{n-1}(A). \\end{eqnarray*}\n$\\square$\n\nConstruction 2.5.5.7 (The Normalized Moore Complex: First Construction). Let $A_{\\bullet }$ be a simplicial abelian group. We let $\\mathrm{N}_{\\ast }(A)$ denote the chain complex given by the quotient $\\mathrm{C}_{\\ast }(A) / \\mathrm{D}_{\\ast }(A)$, where $\\mathrm{C}_{\\ast }(A)$ is the Moore complex of Construction 2.5.5.1 and $\\mathrm{D}_{\\ast }(A) \\subseteq \\mathrm{C}_{\\ast }(A)$ is the subcomplex of Proposition 2.5.5.6. We will refer to $\\mathrm{N}_{\\ast }(A)$ as the normalized Moore complex of the simplicial abelian group $A_{\\bullet }$.\n\nPut more informally, the normalized Moore complex $\\mathrm{N}_{\\ast }(A)$ of a simplicial abelian group $A_{\\bullet }$ is obtained the Moore complex $\\mathrm{C}_{\\ast }(A)$ by forming the quotient by degenerate simplices of $A_{\\bullet }$.\n\nRemark 2.5.5.8. By taking Construction 2.5.5.7 as our definition of the chain complex $\\mathrm{N}_{\\ast }(A)$, we have adopted the perspective that $\\mathrm{N}_{\\ast }(A)$ is a quotient of the Moore complex $\\mathrm{C}_{\\ast }(A)$. However, it can also be realized as a subcomplex of the Moore complex $\\mathrm{C}_{\\ast }(A)$: see Construction 2.5.6.16 and Proposition 2.5.6.19.\n\nConstruction 2.5.5.9 (The Normalized Chain Complex of a Simplicial Set). Let $S_{\\bullet }$ be a simplicial set and let $\\operatorname{\\mathbf{Z}}[ S_{\\bullet } ]$ be the simplicial abelian group freely generated by $S_{\\bullet }$. We let $\\mathrm{N}_{\\ast }(S; \\operatorname{\\mathbf{Z}})$ denote the normalized Moore complex of $\\operatorname{\\mathbf{Z}}[ S_{\\bullet } ]$. This chain complex can be described more concretely as follows:\n\n• For each integer $n \\geq 0$, we can identify $\\mathrm{N}_{n}(S)$ with the free abelian group generated by the set $S_{n}^{\\mathrm{nd}}$ of nondegenerate $n$-simplices of $S_{\\bullet }$.\n\n• The boundary map $\\partial : \\mathrm{N}_{n}(S) \\rightarrow \\mathrm{N}_{n-1}(S)$ is given by the formula\n\n$\\partial (\\sigma ) = \\sum _{i=0}^{n} (-1)^{i} \\begin{cases} d_ i(\\sigma ) & \\text{ if d_ i(\\sigma ) is nondegenerate } \\\\ 0 & \\text{otherwise.} \\end{cases}$\n\nWe will refer to $\\mathrm{N}_{\\ast }(S; \\operatorname{\\mathbf{Z}})$ as the normalized chain complex of the simplicial set $S_{\\bullet }$.\n\nExample 2.5.5.10. Let $S_{\\bullet } = \\Delta ^0$ be the standard $0$-simplex. Then the normalized chain complex $\\mathrm{N}_{\\ast }(S; \\operatorname{\\mathbf{Z}})$ can be identified with abelian group $\\operatorname{\\mathbf{Z}}$, regarded as a chain complex concentrated in degree zero. Note that the calculation of Example 2.5.5.4 shows that the quotient map $\\mathrm{C}_{\\ast }(S; \\operatorname{\\mathbf{Z}}) \\twoheadrightarrow \\mathrm{N}_{\\ast }(S; \\operatorname{\\mathbf{Z}})$ induces an isomorphism on homology.\n\nExample 2.5.5.10 is a special case of the following:\n\nProposition 2.5.5.11. For every simplicial abelian group $A_{\\bullet }$, the quotient map $\\mathrm{C}_{\\ast }(A) \\twoheadrightarrow \\mathrm{N}_{\\ast }(A)$ is a quasi-isomorphism of chain complexes: that is, it induces an isomorphism on homology groups.\n\nRemark 2.5.5.12. In the situation of Proposition 2.5.5.11, an even stronger statement holds: the quotient map $\\mathrm{C}_{\\ast }(A) \\twoheadrightarrow \\mathrm{N}_{\\ast }(A)$ is a chain homotopy equivalence (Definition 2.5.0.5).\n\nWe will give the proof of Proposition 2.5.5.11 in §2.5.6 (see Proposition 2.5.6.22).\n\nExample 2.5.5.13. Let $S_{\\bullet }$ be a simplicial set. It follows from Proposition 2.5.5.11 that the quotient map $\\mathrm{C}_{\\ast }( S; \\operatorname{\\mathbf{Z}}) \\twoheadrightarrow \\mathrm{N}_{\\ast }( S; \\operatorname{\\mathbf{Z}})$ induces an isomorphism on homology. In particular, the homology groups $\\mathrm{H}_{\\ast }(S; \\operatorname{\\mathbf{Z}})$ of the simplicial set $S_{\\bullet }$ (in the sense of Definition 2.5.5.2) can be computed by means of the normalized chain complex $\\mathrm{N}_{\\ast }(S; \\operatorname{\\mathbf{Z}})$. This has various practical advantages. For example, if $S_{\\bullet }$ is a simplicial set of dimension $\\leq d$, then the chain complex $\\mathrm{N}_{\\ast }(S; \\operatorname{\\mathbf{Z}})$ is concentrated in degrees $\\leq d$. It follows that the homology groups $\\mathrm{H}_{\\ast }(S; \\operatorname{\\mathbf{Z}})$ are also concentrated in degrees $\\leq d$, which is not immediately obvious from the definition (note that the chain complex $\\mathrm{C}_{\\ast }(S; \\operatorname{\\mathbf{Z}})$ is never concentrated in degrees $\\leq d$, except in the trivial case where $S_{\\bullet }$ is empty).\n\nExample 2.5.5.14. Let $S_{\\bullet } = \\operatorname{N}_{\\bullet }(Q)$ be the nerve of a partially ordered set $Q$. Suppose that $Q$ has a least element $e$, which determines a map of simplicial sets $i: \\Delta ^{0} \\rightarrow S_{\\bullet }$ which is right inverse to the projection map $q: S_{\\bullet } \\rightarrow \\Delta ^{0}$. Passing to normalized chain complexes, we obtain chain maps\n\n$\\widehat{i}: \\operatorname{\\mathbf{Z}} \\simeq \\mathrm{N}_{\\ast }(\\Delta ^0; \\operatorname{\\mathbf{Z}}) \\hookrightarrow \\mathrm{N}_{\\ast }( S_{\\bullet }; \\operatorname{\\mathbf{Z}}) \\quad \\quad \\widehat{q}: \\mathrm{N}_{\\ast }(S_{\\bullet }; \\operatorname{\\mathbf{Z}}) \\rightarrow \\mathrm{N}_{\\ast }( \\Delta ^{0}; \\operatorname{\\mathbf{Z}}) \\simeq \\operatorname{\\mathbf{Z}}.$\n\nWe claim that $\\widehat{i}$ and $\\widehat{q}$ are chain homotopy inverse to one another. In one direction, this is clear: the composition $\\widehat{q} \\circ \\widehat{i}$ is equal to the identity. We complete the proof by constructing a chain homotopy from the composite map $\\widehat{i} \\circ \\widehat{q}$ to the identity $\\operatorname{id}$ on $\\mathrm{N}_{\\ast }(S_{\\bullet }; \\operatorname{\\mathbf{Z}})$. This chain homotopy is given by a collection of maps $h_{m}: \\mathrm{N}_{m}( S; \\operatorname{\\mathbf{Z}}) \\rightarrow \\mathrm{N}_{m+1}( S; \\operatorname{\\mathbf{Z}})$, given on nondegenerate simplices by the construction\n\n$( q_0 < q_1 < \\cdots < q_ m ) \\mapsto \\begin{cases} 0 & \\text{ if } q_0 = e \\\\ ( e < q_0 < q_1 < \\cdots < q_ m ) & \\text{ otherwise. } \\end{cases}$\n\nIn particular, if $Q$ is a partially ordered set with a least element, then the homology groups of the nerve $S_{\\bullet } = \\operatorname{N}_{\\bullet }(Q)$ are given by\n\n$\\mathrm{H}_{\\ast }(S; \\operatorname{\\mathbf{Z}}) = \\begin{cases} \\operatorname{\\mathbf{Z}}& \\text{ if \\ast = 0} \\\\ 0 & \\text{ otherwise. }\\end{cases}$\n\nVariant 2.5.5.15 (Relative Chain Complexes). Let $S_{\\bullet }$ be a simplicial set and let $S'_{\\bullet } \\subseteq S_{\\bullet }$ be a simplicial subset. Then we can identify the free simplicial abelian group $\\operatorname{\\mathbf{Z}}[ S'_{\\bullet } ]$ with a simplicial subgroup of $\\operatorname{\\mathbf{Z}}[ S_{\\bullet } ]$. We let $\\mathrm{C}_{\\ast }( S,S'; \\operatorname{\\mathbf{Z}})$ and $\\mathrm{N}_{\\ast }(S,S'; \\operatorname{\\mathbf{Z}})$ denote the Moore complex and normalized Moore complex of the simplicial abelian group $\\operatorname{\\mathbf{Z}}[ S_{\\bullet } ] / \\operatorname{\\mathbf{Z}}[ S'_{\\bullet } ]$. By virtue of Proposition 2.5.5.11, these complexes have the same homology groups, which we denote by $\\mathrm{H}_{\\ast }(S,S'; \\operatorname{\\mathbf{Z}})$ and refer to as the relative homology groups of the pair $(S'_{\\bullet } \\subseteq S_{\\bullet })$." ]

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[ "# Tutorial Aims:\n\nAll the files you need to complete this tutorial can be downloaded from this Github repository. Click on `Clone/Download/Download ZIP` and unzip the folder, or clone the repository to your own GitHub account.\n\nThis tutorial is based on work by Max Farrell - you can find Max’s original tutorial here which includes an explanation about how `Stan` works using simulated data, as well as information about model verification and comparison.\n\n# 1. Learn about `Stan`\n\nBayesian modelling like any statistical modelling can require work to design the appropriate model for your research question and then to develop that model so that it meets the assumptions of your data and runs. You can check out the Coding Club tutorial on /tutorials/model-design/index.html, and Bayesian Modelling in `MCMCglmm` for key background information on model design and Bayesian statistics.\n\nStatistical models can be fit in a variety of packages in `R` or other statistical languages. But sometimes the perfect model that you can design conceptually is very hard or impossible to implement in a package or programme that restricts the distributions and complexity that you can use. This is when you may want to move to a statistical programming language such as `Stan`.\n\n`Stan` is a new-ish language that offers a more comprehensive approach to learning and implementing Bayesian models that can fit complex data structures. A goal of the `Stan` development team is to make Bayesian modelling more accessible with clear syntax, a better sampler (sampling here refers to drawing samples out of the Bayesian posterior distribution), and integration with many platforms and including `R`, `RStudio`, `ggplot2`, and `Shiny`.\n\nIn this introductory tutorial we’ll go through the iterative process of model building starting with a linear model. In our advanced `Stan` tutorial we will explore more complex model structures.\n\nFirst, before building a model you need to define your question and get to know your data. Explore them, plot them, calculate some summary statistics.\n\nOnce you have a sense of your data and what question you want to answer with your statistical model, you can begin the iterative process of building a Bayesian model:\n\n2. Choose priors (Informative? Not? Do you have external data you could turn into a prior?)\n3. Sample the posterior distribution.\n4. Inspect model convergence (traceplots, rhats, and for Stan no divergent transitions - we will go through these later in the tutorial)\n5. Critically assess the model using posterior predictions and checking how they compare to your data!\n6. Repeat…\n\nIt’s also good practice to simulate data to make sure your model is doing what you think it’s doing, as a further way to test your model!\n\n# 2. Data\n\nFirst, let’s find a dataset where we can fit a simple linear model. The National Snow and Ice Data Center provides loads of public data that you can download and explore. One of the most prominent climate change impacts on planet earth is the decline in annual sea ice extent in the Northern Hemisphere. Let’s explore how sea ice extent is changing over time using a linear model in Stan.\n\nSet your working directory to the folder where you’ve saved the data by either clicking on `Session/Set working directory/Choose directory` or running the code `setwd(\"your-file-path\")` with your own filepath inside. Now, let’s load the data:\n\n``````# Adding stringsAsFactors = F means that numeric variables won't be\n# read in as factors/categorical variables\nseaice <- read.csv(\"seaice.csv\", stringsAsFactors = F)\n``````\n\nLet’s have a look at the data:\n\n``````head(seaice)\n``````\n\nIf for some reason the column names are not read in properly, you can change column names using:\n\n``````colnames(seaice) <- c(\"year\", \"extent_north\", \"extent_south\")\n``````\n\nWhat research question can we ask with these data? How about the following:\n\nResearch Question: Is sea ice extent declining in the Northern Hemisphere over time?\n\nTo explore the answer to that question, first we can make a figure.\n\n``````plot(extent_north ~ year, pch = 20, data = seaice)\n``````", null, "Figure 1. Change in sea ice extent in the Northern Hemisphere over time.\n\nNow, let’s run a general linear model using `lm()`.\n\n``````lm1 <- lm(extent_north ~ year, data = seaice)\nsummary(lm1)\n``````\n\nWe can add that model fit to our plot:\n\n``````abline(lm1, col = 2, lty = 2, lw = 3)\n``````", null, "Figure 2. Change in sea ice extent in the Northern Hemisphere over time (plus linear model fit).\n\nLet’s remember the equation for a linear model:\n\ny = α + β∗x + error\n\nIn `Stan` you need to specify the equation that you are trying to model, so thinking about that model equation is key!\n\nWe have the answer to our question perhaps, but the point of this tutorial is to explore using the programming language `Stan`, so now let’s try writing the same model in Stan.\n\n## Preparing the data\n\nLet’s rename the variables and index the years from 1 to 39. One critical thing about Bayesian models is that you have to describe the variation in your data with informative distributions. Thus, you want to make sure that your data do conform to those distributions and that they will work with your model. In this case, we really want to know is sea ice changing from the start of our dataset to the end of our dataset, not specifically the years 1979 to 2017 which are really far from the year 0. We don’t need our model to estimate what sea ice was like in the year 500, or 600, just over the duration of our dataset. So we set up our year data to index from 1 to 30 years.\n\n``````x <- I(seaice\\$year - 1978)\ny <- seaice\\$extent_north\nN <- length(seaice\\$year)\n``````\n\nWe can re-run that linear model with our new data.\n\n``````lm1 <- lm(y ~ x)\nsummary(lm1)\n``````\n\nWe can also extract some of the key summary statistics from our simple model, so that we can compare them with the outputs of the `Stan` models later.\n\n``````lm_alpha <- summary(lm1)\\$coeff # the intercept\nlm_beta <- summary(lm1)\\$coeff # the slope\nlm_sigma <- sigma(lm1) # the residual error\n``````\n\nNow let’s turn that into a dataframe for inputting into a `Stan` model. Data passed to Stan needs to be a list of named objects. The names given here need to match the variable names used in the models (see the model code below).\n\n``````stan_data <- list(N = N, x = x, y = y)\n``````\n\n## Libraries\n\nPlease make sure the following libraries are installed (these are the libraries for this and the next `Stan` tutorial). `rstan` is the most important, and requires a little extra if you dont have a C++ compiler.\n\nYou can find detailed instructions here.\n\n``````library(rstan)\nlibrary(gdata)\nlibrary(bayesplot)\n``````\n\n# 3. Our first `Stan` program\n\nWe’re going to start by writing a linear model in the language `Stan`. This can be written in your R script, or saved seprately as a `.stan` file and called into `R`.\n\nA `Stan` program has three required “blocks”:\n\n1. “data” block: where you declare the data types, their dimensions, any restrictions (i.e. upper = or lower = , which act as checks for `Stan`), and their names. Any names you give to your `Stan` program will also be the names used in other blocks.\n2. “parameters” block: This is where you indicate the parameters you want to model, their dimensions, restrictions, and name. For a linear regression, we will want to model the intercept, any slopes, and the standard deviation of the errors around the regression line.\n3. “model” block: This is where you include any sampling statements, including the “likelihood” (model) you are using. The model block is where you indicate any prior distributions you want to include for your parameters. If no prior is defined, `Stan` uses default priors with the specifications `uniform(-infinity, +infinity)`. You can restrict priors using upper or lower when declaring the parameters (i.e. `lower = 0`> to make sure a parameter is positive). You can find more information about prior specification here.\n\nSampling is indicated by the `~` symbol, and `Stan` already includes many common distributions as vectorized functions. You can check out the manual for a comprehensive list and more information on the optional blocks you could include in your `Stan` model.\n\nThere are also four optional blocks:\n\n• “functions”\n• “transformed data”\n• “transformed parameters”\n• “generated quantities”\n\nComments are indicated by `//` in Stan. The `write(\"model code\", \"file_name\")` bit allows us to write the Stan model in our R script and output the file to the working directory (or you can set a different file path).\n\n``````write(\"// Stan model for simple linear regression\n\ndata {\nint < lower = 1 > N; // Sample size\nvector[N] x; // Predictor\nvector[N] y; // Outcome\n}\n\nparameters {\nreal alpha; // Intercept\nreal beta; // Slope (regression coefficients)\nreal < lower = 0 > sigma; // Error SD\n}\n\nmodel {\ny ~ normal(alpha + x * beta , sigma);\n}\n\ngenerated quantities {\n} // The posterior predictive distribution\",\n\n\"stan_model1.stan\")\n``````\n\nFirst, we should check our `Stan` model to make sure we wrote a file.\n\n``````stanc(\"stan_model1.stan\")\n``````\n\nNow let’s save that file path.\n\n``````stan_model1 <- \"stan_model1.stan\"\n``````\n\nHere we are implicitly using `uniform(-infinity, +infinity)` priors for our parameters. These are also known as “flat” priors. Weakly informative priors (e.g. `normal(0, 10)` are more restricted than flat priors. You can find more information about prior specification here.\n\n# 4. Running our `Stan` model\n\nStan programs are complied to `C++` before being used. This means that the C++ code needs to be run before R can use the model. For this you must have a `C++` compiler installed (see this wiki if you don’t have one already). You can use your model many times per session once you compile it, but you must re-compile when you start a new `R` session. There are many `C++` compilers and they are often different across systems. If your model spits out a bunch of errors (unintelligible junk), don’t worry. As long as your model can be used with the `stan()` function, it compiled correctly. If we want to use a previously written `.stan` file, we use the `file` argument in the `stan_model()` function.\n\nWe fit our model by using the `stan()` function, and providing it with the model, the data, and indicating the number of iterations for warmup (these iterations won’t be used for the posterior distribution later, as they were just the model “warming up”), the total number of iterations, how many chains we want to run, the number of cores we want to use (`Stan` is set up for parallelization), which indicates how many chains are run simultaneously (i.e., if you computer has four cores, you can run one chain on each, making for four at the same time), and the thinning, which is how often we want to store our post-warmup iterations. “thin = 1” will keep every iteration, “thin = 2” will keep every second, etc…\n\n`Stan` automatically uses half of the iterations as warm-up, if the `warmup =` argument is not specified.\n\n``````fit <- stan(file = stan_model1, data = stan_data, warmup = 500, iter = 1000, chains = 4, cores = 2, thin = 1)\n``````\n\n## Accessing the contents of a `stanfit` object\n\nResults from `stan()` are saved as a `stanfit` object (S4 class). You can find more details in the `Stan` vignette: [https://cran.r-project.org/web/packages/rstan/vignettes/stanfit-objects.html].\n\nWe can get summary statistics for parameter estimates, and sampler diagnostics by executing the name of the object:\n\n``````fit\n``````", null, "What does the model output show you? How do you know your model has converged? Can you see that text indicating that your C++ compiler has run?\n\nFrom this output we can quickly assess model convergence by looking at the `Rhat` values for each parameter. When these are at or near 1, the chains have converged. There are many other diagnostics, but this is an important one for Stan.\n\nWe can also look at the full posterior of our parameters by extracting them from the model object. There are many ways to view the posterior.\n\n``````posterior <- extract(fit)\nstr(posterior)\n``````\n\n`extract()` puts the posterior estimates for each parameter into a list.\n\nLet’s compare to our previous estimate with “lm”:\n\n``````plot(y ~ x, pch = 20)\n\nabline(lm1, col = 2, lty = 2, lw = 3)\nabline( mean(posterior\\$alpha), mean(posterior\\$beta), col = 6, lw = 2)\n``````", null, "", null, "Figure 3. Change in sea ice extent in the Northern Hemisphere over time (comparing a `Stan` linear model fit and a general `lm` fit).\n\nThe result is identical to the `lm` output. This is because we are using a simple model, and have put non-informative priors on our parameters.\n\nOne way to visualize the variability in our estimation of the regression line is to plot multiple estimates from the posterior.\n\n``````for (i in 1:500) {\nabline(posterior\\$alpha[i], posterior\\$beta[i], col = \"gray\", lty = 1)\n}\n``````\n``````plot(y ~ x, pch = 20)\n\nfor (i in 1:500) {\nabline(posterior\\$alpha[i], posterior\\$beta[i], col = \"gray\", lty = 1)\n}\n\nabline(mean(posterior\\$alpha), mean(posterior\\$beta), col = 6, lw = 2)\n``````", null, "Figure 4. Change in sea ice extent in the Northern Hemisphere over time (`Stan` linear model fits).\n\n# 5. Changing our priors\n\nLet’s try again, but now with more informative priors for the relationship between sea ice and time. We’re going to use normal priors with small standard deviations. If we were to use normal priors with very large standard deviations (say 1000, or 10,000), they would act very similarly to uniform priors.\n\n``````\nwrite(\"// Stan model for simple linear regression\n\ndata {\nint < lower = 1 > N; // Sample size\nvector[N] x; // Predictor\nvector[N] y; // Outcome\n}\n\nparameters {\nreal alpha; // Intercept\nreal beta; // Slope (regression coefficients)\nreal < lower = 0 > sigma; // Error SD\n}\n\nmodel {\nalpha ~ normal(10, 0.1);\nbeta ~ normal(1, 0.1);\ny ~ normal(alpha + x * beta , sigma);\n}\n\ngenerated quantities {}\",\n\n\"stan_model2.stan\")\n\n``````\n\nWe’ll fit this model and compare it to the mean estimate using the uniform priors.\n\n``````fit2 <- stan(stan_model2, data = stan_data, warmup = 500, iter = 1000, chains = 4, cores = 2, thin = 1)\n\nposterior2 <- extract(fit2)\n\nplot(y ~ x, pch = 20)\n\nabline(alpha, beta, col = 4, lty = 2, lw = 2)\nabline(mean(posterior2\\$alpha), mean(posterior2\\$beta), col = 3, lw = 2)\nabline(mean(posterior\\$alpha), mean(posterior\\$beta), col = 36, lw = 3)\n``````", null, "Figure 5. Change in sea ice extent in the Northern Hemisphere over time (`Stan` linear model fits).\n\nSo what happened to the posterior predictions (your modelled relationship)? Does the model fit the data better or not? Why did the model fit change? What did we actually change about our model by making very narrow prior distributions? Try changing the priors to some different numbers yourself and see what happens! This is a common issue in Bayesian modelling, if your prior distributions are very narrow and yet don’t fit your understanding of the system or the distribution of your data, you could run models that do not meaningfully explain variation in your data. However, that isn’t to say that you shouldn’t choose somewhat informative priors, you do want to use previous analyses and understanding of your study system inform your model priors and design. You just need to think carefully about each modelling decision you make!\n\n# 6. Convergence Diagnostics\n\nBefore we go on, we should check again the `Rhat` values, the effective sample size (`n_eff`), and the traceplots of our model parameters to make sure the model has converged and is reliable. To find out more about what effective sample sizes and trace plots, you can check out the tutorial on Bayesian statistics using `MCMCglmm`.\n\n`n_eff` is a crude measure of the effective sample size. You usually only need to worry is this number is less than 1/100th or 1/1000th of your number of iterations.\n\n``````'Anything over an `n_eff` of 100 is usually \"fine\"' - Bob Carpenter\n``````\n\nFor traceplots, we can view them directly from the posterior:\n\n``````plot(posterior\\$alpha, type = \"l\")\nplot(posterior\\$beta, type = \"l\")\nplot(posterior\\$sigma, type = \"l\")\n``````", null, "Figure 6. Trace plot for alpha, the intercept.\n\nFor simpler models, convergence is usually not a problem unless you have a bug in your code, or run your sampler for too few iterations.\n\n## Poor convergence\n\nTry running a model for only 50 iterations and check the traceplots.\n\n``````fit_bad <- stan(stan_model1, data = stan_data, warmup = 25, iter = 50, chains = 4, cores = 2, thin = 1)\n``````\n\nThis also has some “divergent transitions” after warmup, indicating a mis-specified model, or that the sampler that has failed to fully sample the posterior (or both!). Divergent transitions sound like some sort of teen fiction about a future dystopia, but actually it indicates problems with your model.\n\n``````plot(posterior_bad\\$alpha, type = \"l\")\n``````", null, "Figure 7. Bad trace plot for alpha, the intercept.\n\n## Parameter summaries\n\nWe can also get summaries of the parameters through the posterior directly. Let’s also plot the non-Bayesian linear model values to make sure our model is doing what we think it is…\n\n``````par(mfrow = c(1,3))\n\nplot(density(posterior\\$alpha), main = \"Alpha\")\nabline(v = lm_alpha, col = 4, lty = 2)\n\nplot(density(posterior\\$beta), main = \"Beta\")\nabline(v = lm_beta, col = 4, lty = 2)\n\nplot(density(posterior\\$sigma), main = \"Sigma\")\nabline(v = lm_sigma, col = 4, lty = 2)\n``````", null, "Figure 8. Density plot distributions from the `Stan` model fit compared with the estimates from the general `lm` fit.\n\nFrom the posterior we can directly calculate the probability of any parameter being over or under a certain value of interest.\n\nProbablility that beta is >0:\n\n``````sum(posterior\\$beta>0)/length(posterior\\$beta)\n# 0\n``````\n\nProbablility that beta is >0.2:\n\n``````sum(posterior\\$beta>0.2)/length(posterior\\$beta)\n# 0\n``````\n\n## Diagnostic plots in `rstan`\n\nWhile we can work with the posterior directly, `rstan` has a lot of useful functions built-in.\n\n``````traceplot(fit)\n``````", null, "Figure 9. Trace plots of the different chains of the `Stan` model.\n\nThis is a wrapper for the `stan_trace()` function, which is much better than our previous plot because it allows us to compare the chains.\n\nWe can also look at the posterior densities & histograms.\n\n``````stan_dens(fit)\nstan_hist(fit)\n``````", null, "", null, "Figure 10. Density plots and histograms of the posteriors for the intercept, slope and residual variance from the `Stan` model.\n\nAnd we can generate plots which indicate the mean parameter estimates and any credible intervals we may be interested in. Note that the 95% credible intervals for the `beta` and `sigma` parameters are very small, thus you only see the dots. Depending on the variance in your own data, when you do your own analyses, you might see smaller or larger credible intervals.\n\n``````plot(fit, show_density = FALSE, ci_level = 0.5, outer_level = 0.95, fill_color = \"salmon\")\n``````", null, "Figure 11. Parameter estimates from the `Stan` model.\n\n## Posterior Predictive Checks\n\nFor prediction and as another form of model diagnostic, `Stan` can use random number generators to generate predicted values for each data point, at each iteration. This way we can generate predictions that also represent the uncertainties in our model and our data generation process. We generate these using the Generated Quantities block. This block can be used to get any other information we want about the posterior, or make predictions for new data.\n\n``````\nwrite(\"// Stan model for simple linear regression\n\ndata {\nint < lower = 1 > N; // Sample size\nvector[N] x; // Predictor\nvector[N] y; // Outcome\n}\n\nparameters {\nreal alpha; // Intercept\nreal beta; // Slope (regression coefficients)\nreal < lower = 0 > sigma; // Error SD\n}\n\nmodel {\ny ~ normal(x * beta + alpha, sigma);\n}\n\ngenerated quantities {\nreal y_rep[N];\n\nfor (n in 1:N) {\ny_rep[n] = normal_rng(x[n] * beta + alpha, sigma);\n}\n\n}\",\n\n\"stan_model2_GQ.stan\")\n\n``````\n\nNote that vectorization is not supported in the GQ (generated quantities) block, so we have to put it in a loop. But since this is compiled to `C++`, loops are actually quite fast and Stan only evaluates the GQ block once per iteration, so it won’t add too much time to your sampling. Typically, the data generating functions will be the distributions you used in the model block but with an `_rng` suffix. (Double-check in the Stan manual to see which sampling statements have corresponding `rng` functions already coded up.)\n\n``````fit3 <- stan(stan_model2_GQ, data = stan_data, iter = 1000, chains = 4, cores = 2, thin = 1)\n``````\n\n## Extracting the `y_rep` values from posterior.\n\nThere are many options for dealing with `y_rep` values.\n\n``````y_rep <- as.matrix(fit3, pars = \"y_rep\")\ndim(y_rep)\n``````\n\nEach row is an iteration (single posterior estimate) from the model.\n\nWe can use the `bayesplot` package to make some prettier looking plots. This package is a wrapper for many common `ggplot2` plots, and has a lot of built-in functions to work with posterior predictions. For details, you can check out the bayesplot vignettes.\n\nComparing density of `y` with densities of `y` over 200 posterior draws.\n\n``````ppc_dens_overlay(y, y_rep[1:200, ])\n``````", null, "Figure 12. Comparing estimates across random posterior draws.\n\nHere we see data (dark blue) fit well with our posterior predictions.\n\nWe can also use this to compare estimates of summary statistics.\n\n``````ppc_stat(y = y, yrep = y_rep, stat = \"mean\")\n``````", null, "Figure 13. Comparing estimates of summary statistics.\n\nWe can change the function passed to the `stat` function, and even write our own!\n\nWe can investigate mean posterior prediction per datapoint vs the observed value for each datapoint (default line is 1:1)\n\n``````ppc_scatter_avg(y = y, yrep = y_rep)\n``````", null, "Figure 14. Mean posterior prediction per datapoint vs the observed value for each datapoint.\n\n## `bayesplot` options\n\nHere is a list of currently available plots (`bayesplot 1.2`):\n\n``````available_ppc()\n``````\n\nYou can change the colour scheme in `bayesplot` too:\n\n``````color_scheme_view(c(\"blue\", \"gray\", \"green\", \"pink\", \"purple\",\n\"red\",\"teal\",\"yellow\"))\n``````", null, "And you can even mix them:\n\n``````color_scheme_view(\"mix-blue-red\")\n``````\n\nYou can set color schemes with:\n\n``````color_scheme_set(\"blue\")\n``````\n\nSo now you have learned how to run a linear model in `Stan` and to check the model convergence. But what is the answer to our research question?\n\nResearch Question: Is sea ice extent declining in the Northern Hemisphere over time?\n\nWhat do your `Stan` model results indicate?\n\nHow would you write up these results? What is the key information to report from a Stan model? Effect sizes, credible intervals, sample sizes, what else? Check out some Stan models in the ecological literature to see how those Bayesian models are reported.\n\nNow as an added challenge, can you go back and test a second research question:\n\nResearch Question: Is sea ice extent declining in the Southern Hemisphere over time?\n\nIs the same pattern happening in the Antarctic as in the Arctic? Fit a `Stan` model to find out!\n\nIn the next Stan tutorial, we will build on the concept of a simple linear model in Stan to learn about more complex modelling structures including different distributions and random effects. And in a future tutorial, we will introduce the concept of a mixture model where two different distributions are modelled at the same time - a great way to deal with zero inflation in your proportion or count data!\n\n## Additional ways to run `Stan` models in `R`\n\nCheck out our second `Stan` tutorial to learn how to fit `Stan` models using model syntax similar to the style of other common modelling packages like `lme4` and `MCMCglmm`, as well as how to fit generalised linear models using `Poisson` and negative binomial distributions.\n\n## `Stan` References\n\nStan is a run by a small, but dedicated group of developers. If you are new to Stan, you can join the mailing list. It’s a great resource for understanding and diagnosing problems with Stan, and by posting problems you encounter you are helping yourself, and giving back to the community.\n\nThis tutorial is based on work by Max Farrell - you can find Max’s original tutorial here which includes an explanation about how `Stan` works using simulated data, as well as information about model verification and comparison." ]

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[ "## What is Machine learning ?\n\nMachine learning is a subfield of artificial intelligence that makes use of statistical models and algorithms to teach computers how to learn from data without being specifically programmed to do so. This is accomplished by presenting the data in a way that is understandable by the computer.\n\n## There are five bags each containing identical sets of ten distinct chocolates. One chocolate is picked from each bag.The probability that at least two chocolates are identical is ______\n\nA) 0.3024B) 0.4235C) 0.6976D) 0.8125 Answer: (C) 0.6976 Solution: To find the probability that no two chocolate are identical = P(Y) To find the probability that at least two chocolates are identical = P(x):Formula: P(x) = 1 – P(Y) Calculation No two chocolates identical: From 1st bag = Probability = 10/10 From 2nd bag = Probability = 9/10 From 3rd bag […]\n\nMy solution: Remove errors from Google search console.\n\n## Paging\n\nPaging divides physical memory into equal-sized groupings called “blocks,” which can range from 64 to 4096 words each. ‘Page’ denotes groupings of identical address space, whereas ‘block’ denotes memory space organisation.\n\n## Define a class to declare an integer array of size n and accept the elements into the array.\n\nimport java.util.Scanner;class LinearSearch{public static void main(String args[]){int c, n, search, array[];Scanner in = new Scanner(System.in);System.out.println(“Enter number of elements”);n = in.nextlnt();array= new int[n];System.out.println(“Enter those” +n+ “elements”);for (c = 0; c < n; c++)array[c] = in.nextlnt();System.out.println(“Enter value to find”);search = in.nextlnt();for (c = 0; c < n; c++){if (array[c] = search) /Searching element is present/{System.out.println(array[c]);break;}}if (c = […]\n\nVideo Lectures" ]

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[ "# Is a vector field a subset of a vector space?\n\nThis is not actually a duplicate although it seems very similar to the another question I found on stackexchange titled \"vector space or vector field?\" If I have a vector field isn't each element of that field in n dimensions also an element of a particular vector space of similar dimension. Even if the subset does not satisfy all the properties of a vector space would it not still be a subset? Just curious if the \"object\" that resides in a vector field the same object that appears in some vector space of similar dimension. Thank you . Not really sure where this question belongs since one is a topic of linear algebra and the other is more in line with vector fields in calculus, which is what prompted my question to begin with.\n\n• Unless you give more details about what the term \"vector field\" means in the context of this question, it is not clear what this question is about. For what its worth, one meaning of vector field could be a map from some geometric set $S$ to a vector space $V$, and adding/scalar multiplying such maps as usual, the set of vector fields $S\\to V$ forms a (rather large) vector space in itself. – Marc van Leeuwen Sep 18 '16 at 10:52\n• Functions that assign a vector to a point in the plane or a point in space are called vector fields. – Sedumjoy Sep 18 '16 at 16:14\n• So you answered your own question. Functions that assign a vector to a point in the plane are not subsets of a vector space, they are functions. Functions can themselves be vectors in a (different, large) vector space of functions (and they are here) but that makes them elements, not subsets, of a vector space. – Marc van Leeuwen Sep 18 '16 at 17:43\n• No I did not answer my own question. You asked \"what the term \"vector field\" means\". This is not what I asked. I answered what you asked which is not what I asked. What I asked was answered by answer two and I acknowledged. I hope this clears any confusion. – Sedumjoy Sep 20 '16 at 0:29\n• Of course you answered my question, not your own, but what I meant to say is that your answer to my question implies an answer to your own question (of the title) \"is a vector field a subset of a vector space\". I a vector field is a function (no matter of what type) then it cannot at the same time be a subset of the vector space. Functions are not subsets. From the answer that you accepted I gather that you confused \"function\" and \"image of a function\" (which indeed is a subset). To me your question as formulated simply has as answer \"no that can never be the case\". – Marc van Leeuwen Sep 20 '16 at 6:51\n\n## 2 Answers\n\nYes, any element of a vector field is a vector (with a dimension $n$), so ''all'' the vectors of a vector field are a subset of some vector space $W$. But a vector field is not simply a set of vectors, it is a function that assign a vector to any point in a space, that is a function $f:V \\to W$ where $W$ is a vector space and $V$ can be a set, but usually has some structure (as a manifold). Also if $V$ is a vector space, this function $f$ may be not linear. So a vector field is a lot more that a subset of a vector space.\n\nThe name \"vector field\" is commonly used to a particular association of points to vectors. In the case of Euclidean space, this becomes blurred, since we associate to each point of the plane a vector (that is, we make a function $V:\\mathbb{R}^n \\to \\mathbb{R}^n$ which is better visualized as taking $x$ to a vector $V_x$ in each point $x$).\n\nBut consider a sphere (the sphere $S^2$, the \"shell\" of the unit ball in $\\mathbb{R}^3$). You can imagine what a vector field should be. For example, the wind flow on Earth should be an example of a vector field. We are associating to each point of the sphere a tangent vector to it. However, there is a problem. When we change every point, we are changing the space where our tangent vector lives. What we are actually doing is associating to each point $x \\in S^2$ a vector in $T_xS^2$, the space of tangent vectors to $S^2$ based in $x$. This is what a vector field is: an association of a tangent vector to every point of a surface (more generally, to every point of a manifold a tangent vector). We can give a structure to the collection of tangent spaces in order to visualize a vector field on the sphere as a particular kind of map $V: S^2 \\to TS^2$, where this $TS^2$ is the \"collection of tangent spaces\". But note now that this $TS^2$ won't be a vector space. It will be what is called a vector bundle.\n\nOne could argue that associating to every point of the sphere a vector in $\\mathbb{R}^3$ would also be a vector field. But I've never seen the name being used like this in the literature (if it were, it would be as reasonable as calling any section in any vector bundle a vector field - not that it isn't reasonable)." ]

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[ "# Simulation of Impulse Response Measurements¶\n\nThe software (https://github.com/franzpl/sweep) has been written in the context of my bachelor thesis with the topic \"On the influence of windowing of sweep signals for room impulse measurements\" at the University of Rostock. Impulse responses are an important tool to determine acoustic properties of a Device Under Test. The main requirement is that all desired frequencies cover the interesting frequency range with sufficient energy. Therefore, sweep signals and white noise are usally favored to excite DUT's. In this context sweep signals and LTI-Systems were used. However, the design of sweep signals in time domain causes ripple in the excitation spectrum at the start and stop frequency. It is possible to reduce ripple with the use of convenient windows. With this software, you can evaluate the effect of windowing of sweep signals on impulse responses under the influence of noise. This Ipython3 Notebook shows an examplary impulse response measurement (Sweep -> DUT -> System Response -> Impulse Response -> Quality of Impulse Response). You can also use the software for real measurements, because measurement module and simulation module are seperated strictly.\nLet's start the simulation of an impulse response measurement!\n\n# Imports¶\n\nFirst, you need imports from Python and the software.\n\n### Python Modules¶\n\nIn :\n%matplotlib inline\nimport numpy as np\nimport matplotlib.pyplot as plt\n\n\n### Software Modules¶\n\nIn :\nimport generation\nimport plotting\nimport ir_imitation\nimport calculation\nimport windows\nimport measurement_chain\n\n\n# Excitation¶\n\nThan, you have to design the excitation signal.\n\n## Excitation Parameters¶\n\nIn :\nfs = 44100\nfstart = 1\nfstop = 22050\nduration = 1 # seconds\npad = 5 # attach 5 seconds zeros to excitation signal\n\n\n## Excitation Signal¶\n\nGenerate a excitation signal with the excitation parameters above.\n\nIn :\nexcitation = generation.log_sweep(fstart, fstop, duration, fs)\n\n\n### Plot Time Domain¶\n\nIn :\nplotting.plot_time(excitation, fs);", null, "### Plot Frequency Domain¶\n\nAs shown in this figure, the excitation spectrum is characterized by ripple at the start and stop frequency.\n\nIn :\nplotting.plot_freq(excitation, fs, scale='db')\nplt.xscale('log')\nplt.xlim([1, fs/2])\nplt.ylim([-55, -14]);", null, "## Window Parameters¶\n\nA window reduces the ringing artifacts. Fade in and fade out parameters of the window can help to produce a smoother spectrum.\n\nIn :\nfade_in = 50 # ms\nbeta = 7 # kaiser window\n\n\n## Windowed Sweep¶\n\nIn :\nwindow = windows.window_kaiser(len(excitation), fade_in, fade_out, fs, beta)\nexcitation_windowed = window * excitation\n\n\nZeropadding makes space for the recording signal.\n\nIn :\nexcitation_windowed_zeropadded = generation.zero_padding(excitation_windowed, pad, fs)\n\n\nIn :\nplotting.plot_time(excitation_windowed_zeropadded, fs);", null, "# System¶\n\nNow, you have to design the DUT.\n\n## FIR-Filter System¶\n\nFor a better understanding, in this example a dirac impuls was used as filter.\n\nIn :\ndirac = measurement_chain.convolution()\n\n\n## Noise System¶\n\nIn :\nnoise_level = -30 # RMS (dB)\n\n\n## Combinate System Elements¶\n\nFinally, the system elements must be combined. Feel free to add more elements (lowpass, bandpass, gain, ...) to the system.\n\nIn :\nsystem = measurement_chain.chained(dirac, awgn)\n\n\n# System Response¶\n\nTo record the system response, you have to simply pass the excitation signal to the system.\n\nIn :\nsystem_response = system(excitation_windowed_zeropadded)\n\n\n### Plot System Response¶\n\nIn :\nplotting.plot_time(system_response,fs);", null, "# Impulse Response¶\n\nVia the FFT and IFFT, the impulse response is calculated. That's it! A Plot with linear and dB scale show you the characteristics of the IR.\n\nIn :\nir = calculation.deconv_process(excitation_windowed_zeropadded, system_response, fs)[:len(excitation_windowed_zeropadded)]\n\n\n### Plot Impulse Response (linear)¶\n\nIn :\nplotting.plot_time(ir, fs)\nplt.xlim([-1, 5])\nplt.ylim([-0.1, 1.1]);", null, "### Plot Impulse Response (dB)¶\n\nIn :\nplotting.plot_time(ir, fs, scale='db')\nplt.xlim([-1, 5])\nplt.ylim([-60, 2]);", null, "# Impulse Response Quality¶\n\nThe 'Peak to Noise Ratio' provides information about the quality of the IR.\n\nIn :\npnr = calculation.pnr_db(ir, ir[fs:pad*fs])\nprint(str(pnr), 'dB')\n\n44.4698202175 dB" ]

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[ "# Three Member Frame - Pin/Roller Side and Top Point Load\n\nWritten by Jerry Ratzlaff on . Posted in Structural\n\n###", null, "Three Member Frame - Pin/Roller Side and Top Point Load Formula\n\n$$\\large{ R_A = R_D = \\frac{ P\\;h }{ L } }$$\n\n$$\\large{ H_A = P }$$\n\n$$\\large{ M_{max} \\;(at \\; B) = P\\;h }$$\n\n$$\\large{ \\Delta_{Cx} = \\frac{P\\;h^2}{3 \\; \\lambda \\; I} \\; \\left(L+h\\right) }$$\n\n$$\\large{ \\Delta_{Cy} = 0 }$$\n\n$$\\large{ \\Delta_{Dx} = \\frac{P\\;h^2}{6 \\; \\lambda \\; I} \\; \\left(3\\;L+2\\;h\\right) }$$\n\nWhere:\n\n$$\\large{ \\Delta }$$ = deflection or deformation\n\n$$\\large{ h }$$ = height of frame\n\n$$\\large{ H }$$ =  horizontal reaction load at bearing point\n\n$$\\large{ M }$$ = maximum bending moment\n\n$$\\large{ \\lambda }$$  (Greek symbol lambda) = modulus of elasticity\n\n$$\\large{ A, B, C, D }$$ = points of intersection on frame\n\n$$\\large{ R }$$ = reaction load at bearing point\n\n$$\\large{ I }$$ = second moment of area (moment of inertia)\n\n$$\\large{ L }$$ = span length of the bending member\n\n$$\\large{ P }$$ = total concentrated load" ]

[ null, "http://piping-designer.com/images/disciplines/civil/structural/support_frame/3fpr_-_2A.jpg", null ]

[ "# Divergence test\n\n### Divergence test\n\nIn this lesson, we will learn about the divergence test. The test states that if you take the limit of the general term of the series and it does not equal to 0, then the series diverge. Keep in mind that if you do take the limit and it goes to 0, that does not mean the series is convergent. It only means the test has failed, and you will have to use another method to find the convergence or divergence of the series. It is recommended to use the divergence test if u can obviously see that the limit of the general term goes to infinity. For the first few questions, we will see if the divergence test applies to the series. For the last question, we will see if the series is convergent or divergent by using the test.\n\n#### Lessons\n\nNote *The divergence test states the following:\nIf $\\lim$n →$\\infty$ $a$$n$ $\\neq$ 0, then the series $\\sum a_n$ diverges.\n• Introduction\nDivergence Test Overview\n\n• 1.\nUnderstanding of the Divergence Test\nDoes the divergence test work for the following series?\na)\n$\\sum_{n=1}^{\\infty}\\frac{10}{n}$\n\nb)\n$\\sum_{n=4}^{\\infty}\\frac{n^2+n^3}{n^3+1}$\n\nc)\n$\\sum_{n=2}^{\\infty}\\frac{n-1}{ln(n)}$\n\n• 2.\nAdvanced Question Regarding to the Divergence Test\nDetermine if the series $\\sum_{k=1}^{\\infty}k^{-\\frac{1}{k^3}}$ converges or diverges." ]

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[ "# Express the $\\gamma_{2}^{\\epsilon}$ SemiDefinite program in a form that is acceptable by SDPT3\n\nI'm trying to express the following semidefinite program: for given $A \\in R^{m \\times n}$ and a scalar $\\epsilon \\in (0,1)$,\n\n\\begin{align} &\\gamma_{2}^{\\epsilon}(A):= \\min\\,t\\\\ &\\text{subject to} \\left( \\begin{array}{ccc} W_1 & B \\\\ B^T & W_2 \\\\ \\end{array} \\right)\\succcurlyeq 0,\\, \\operatorname{diag}(W_1)\\leq t,\\, \\operatorname{diag}(W_2)\\leq t,\\\\ &\\forall (i,j) \\in \\{1,...,m\\} \\times \\{1,...,n\\}: -\\epsilon\\leq A_{ij}-B_{ij},\\, A_{ij}-B_{ij} \\geq \\epsilon. \\end{align}\n\nwhere $B \\in R^{m \\times n} ,\\, W_1 \\in R^{m \\times m}$, $W_1 \\in R^{n \\times n}$ and $t \\in R$ are the decision variables, in a form that is acceptable by the semidefinite-quadratic-linear program solver SDPT3. I know that the first thing I should do is to convert it to the standard SDP form. I tried to do it, but I got stuck on how to model the objective function. Any help on modelling this program in an SDPT3-acceptable form would be much appreciated.\n\n• You should clarify what your variables and constants are. Is $\\epsilon$ a given scalar? Are A and B given real $n$ by $n$ matrices (or possibly $m$ by $n$)? Are the variables $W_{1}$, $W_{2}$ (matrices of what size?), and $t$ (scalar?) Feb 23, 2016 at 18:23\n• @BrianBorchers I edited my question and added some explanations on these. $\\epsilon$ is a positive scalar, $A,\\,B$ are real $m \\times n$ matrices and the matrices $W_1,\\,W_2$ along with the (positive, as a value of a norm) scalar $t$ are the decision variables. Feb 23, 2016 at 18:33\n• You could do the spiel described by Brian Borchers , which answers your question. Or to make life simpler, you could use CVX or YALMIP, which let you specify your model in a natural form, and take care of the conversion to/from the form and format needed by SDPT3 or another solver. Feb 23, 2016 at 23:50\n• @MarkL.Stone CVX was my first choice, but, unfortunately, it proved itself not scalable. As I need to work with large matrices ($n>200$), I should feed an SDP solver with the problem data in standard SDP form. Feb 24, 2016 at 10:04\n\nIn order to do this, you'll need to convert inequalities to equality constraints by introducing slack variables and then add these slack variables to your matrix variable as an additional LP block.\n\nLet\n\n$X=\\left[ \\begin{array}{ccc} W_{1} & B & 0 \\\\ B^{T} & W_{2} & 0 \\\\ 0 & 0 & \\mbox{diag}(v) \\\\ \\end{array} \\right]$\n\nwhere\n\n$v=\\left[ \\begin{array}{c} s_{1} \\\\ s_{2} \\\\ \\vdots \\\\ s_{p} \\\\ t \\\\ \\end{array} \\right]$\n\nIn SDPT3, your $X$ matrix will be a block diagonal matrix with an $m+n$ by $m+n$ symmetric positive semidefinite block for $W_{1}$, $W_{2}$, $B$, and $B^{T}$, and a diagonal (or LP variable) block $v$ of length $2mn+3$ for the slack variables and $t$.\n\nIt's easy to express the objective $\\min t$ as the minimum of the trace of $CX$, where $C$ is a matrix with a 1 in the lower right corner and all other entries 0.\n\n$X \\succeq 0$\n$\\mbox{diag}(W_{1})+v_{1} - v_{p+1} = 0$\n$\\mbox{diag}(W_{2})+v_{2} - v_{p+1} = 0$\n$A_{i,j}-B_{i,j} - v_{2+i+(j-1)m} = -\\epsilon$ for $i=1, 2, \\ldots, m$, $j=1, 2, \\ldots, n$.\n$A_{i,j}-B_{i,j} + v_{2+mn+i+(j-1)m} = \\epsilon$ for $i=1, 2, \\ldots, m$, $j=1, 2, \\ldots, n$.\nEach of these linear equality constraints can be easily written as a linear equality constraint involving entries in $X$." ]

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[ "# Diode\n\nPiecewise linear diode in electrical systems\n\n## Library\n\nElectrical Elements\n\n•", null, "## Description\n\nThe Diode block models a piecewise linear diode. If the voltage across the diode is bigger than the Forward voltage parameter value, then the diode behaves like a linear resistor with low resistance, given by the On resistance parameter value, plus a series voltage source. If the voltage across the diode is less than the forward voltage, then the diode behaves like a linear resistor with low conductance given by the Off conductance parameter value.\n\nWhen forward biased, the series voltage source is described with the following equation:\n\n`$V=Vf\\left(1-{R}_{on}{G}_{off}\\right),$`\n\nwhere\n\n `V` Voltage `Vf` Forward voltage `Ron` On resistance `Goff` Off conductance\n\nThe Ron.Goff term ensures that the diode current is exactly zero when the voltage across it is zero.\n\n### Variables\n\nTo set the priority and initial target values for the block variables prior to simulation, use the Variables tab in the block dialog box (or the Variables section in the block Property Inspector). For more information, see Set Priority and Initial Target for Block Variables.\n\n## Parameters\n\nForward voltage\n\nMinimum voltage that needs to be applied for the diode to become forward-biased. The default value is `0.6` V.\n\nOn resistance\n\nThe resistance of a forward-biased diode. The default value is `0.3` Ω.\n\nOff conductance\n\nThe conductance of a reverse-biased diode. The default value is `1e-8` 1/Ω.\n\n## Ports\n\nThe block has the following ports:\n\n`+`\n\nElectrical conserving port associated with the diode positive terminal.\n\n`-`\n\nElectrical conserving port associated with the diode negative terminal.\n\n## Extended Capabilities\n\n### C/C++ Code GenerationGenerate C and C++ code using Simulink® Coder™.\n\nIntroduced in R2007a" ]

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[ "Today is 2022-01-17 Monday，Welcome to this site\n\nInformation\nProducts\n\nThe type of fixed resistor is much, the attention of choose and choose to share\n\nWord：[Big][Middle][Small] QR Code", null, "2017/7/13     Viewed：", null, "Fixed resistors have different types of fixed resistors. The resistors of which materials and structures are chosen should be determined according to the specific requirements of the application circuit. High frequency circuits should be used to distribute inductance and distributed capacitance small non-wire-wound resistor, such as carbon film resistor, metal resistor and metal oxide film resistor. High gain small signal amplifier circuit should choose low noise resistor, such as metal film resistor, carbon film resistor, and wire wound resistor, rather than the synthetic carbon film resistor with large noise and organic solid resistor. The power of wire wound resistor is large, the current noise is small, it is high temperature, but it is bigger. Conventional wire-wound resistor is often used in low-frequency circuit or medium - limit flow resistor, partial pressure resistor, discharge resistor or the bias resistor of high power tube. The high precision wire-wound resistor is used in fixed attenuator, resistance box, computer and various precision electronic instruments. The resistors of the selected resistor should be close to a nominal value of the calculated value in the application circuit, and the standard series of resistors should be preferred. The resistor used in the general circuit allows the error to be plus or minus 5% ~ plus or minus 10%. Precision resistors should be used in precision instruments and in special circuits. The rated power of the selected resistor should meet the requirements of the power capacity of the resistor in the application circuit, and the power of the resistor should not be arbitrarily increased or reduced. If the circuit is required to be a power resistor, its rated power can be 1-2 times that of the actual application circuit. There are usually 6 specific cases:\n\n1, the power rating: under the regulation of environmental temperature and humidity, assuming the surrounding air circulation, not in long-term continuous load without damage or basic don't change under the condition of performance, allow the consumption of power resistor. To ensure safe use, the rated power is generally 1 to 2 times higher than the power it consumes in the circuit.\n\n2. Allowable error: the actual resistance value of the resistor is the allowable deviation range of the nominal resistance value, which indicates the precision of the product. The usual accuracy is 5%, 1%, 0.5%, 0.1%, 0.01%\n\n3. Working voltage: it refers to the voltage when the resistor does not overheat or puncture the damage. If the voltage exceeds the regulation value, the internal spark of the resistor can cause noise and even damage.\n\n4. Stability: stability is the measure of resistance of resistors under external conditions.\n\n5. Noise electromotive force: the noise electromotive force of the resistor may not be considered in the general circuit, but it cannot be ignored in the weak signal system. The noise of wire-wound resistor is only concerned with the frequency band of temperature and external voltage. In addition to the thermal noise, the thin film resistor has the current noise, which is approximately proportional to the external voltage.\n\n6. High frequency characteristics: the use of resistors in high frequency conditions to consider the effects of its fixed inductance and intrinsic capacitance. At this point, the distribution of resistor into a dc resistance and inductance in series, and then with the equivalent circuit of capacitance in parallel, the wire wound resistor of LR = 0.01 0.05, CR = 0.1 5 skin method, micro wire wound resistor LR of dozens of heng, CR of dozens of skin method, even the non-inductive winding wire wound resistor, there are still zero JiWei heng LR. Inductive resistance appeared in high frequency when the high impedance, and at the time of the high frequency inductive resistance of resistance will be very big, so it to bear is multiplied by the square of the current resistance of power, far more than its nominal power, resistance to burn out.\n\nGo Back\nPrint\n0552-8959088", null, "扫一扫添加微信" ]

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[ "# What do floating point operations look like?\n\nThese are plots of common binary operations on floating-point numbers. The X axis is the left-side argument from negative infinity on the left to positive infinity on the right. The Y axis is the right-side argument from negative infinity on the top to positive infinity on the bottom. NaN is shown in green, negative infinity in cyan, positive infinity in magenta, zero in yellow, negative numbers in blue, and positive numbers in red. Each gradation represents an order of magnitude, with lighter reds and blues representing numbers closer to zero. The half-red seen in the plot for exponentiation represents positive one. Click for higher-resolution versions.", null, "", null, "", null, "", null, "Subtraction", null, "", null, "Multiplication", null, "", null, "Division", null, "", null, "Modulus", null, "", null, "Exponentiation", null, "", null, "Hypotenuse", null, "", null, "Arctangent" ]

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[ "", null, "", null, "", null, "Main Page | See live article | Alphabetical index\n\nIn mathematics, exponentiation is a process of repeated multiplication, in much the same way that multiplication is a process of repeated addition. For example, 34 equals 3 × 3 × 3 × 3 equals 81. Here, 3 is the base, 4 is the exponent (written as a superscript), and 81 is 3 raised to the 4th power. Notice that the base 3 appears 4 times in the repeated multiplication, because the exponent is 4. In contexts where superscripts are not available, such as computer languages and e-mail, 34 is commonly written \"3^4\".\n\nIf the exponent is two, the power is called square, if is three, it is called cube.\n\nRaising 10 to a power is easy: for example 107 = 10,000,000 with seven zeros. Exponentiation with base 10 is often used in the physical sciences to describe large or small numbers in scientific notation; for example, 299792458 can be written as 2.99792458 × 108 and then approximated as 2.998 × 108 if this is useful. SI prefixes are also used to describe small or large quantities, but even these are based on powers of ten; for example, the prefix kilo means 103 = 1000, so a kilometre is 1000 metres.\n\nExponents with base 2 are used in computer science; for example, there are 2n possible values for a variable that takes n bits to store in memory. A kilobyte usually stands for 210 = 1024 bytes, but sometimes also for 103 = 1000 bytes; the term kibibyte has been suggested for the former meaning.\n\nExponents with base e (a transcendental number approximately equal to 2.71828) are described by the exponential function exp x = ex.\n\nWe define exponentiation of a positive real number x with a negative exponent by\n\nx-n = 1/xn\nand with a fractional exponent as\nSo for instance 10−3 = 0.001 and 82/3 = 4. xy where y is an arbitrary real number can then be defined by continuity.\n\nExponentiation of real numbers, and even complex numbers, can also be understood with the aid of the exponential function and its inverse, the natural logarithm; in general, we can define\n\nxy = exp (y ln x).\n\nFor more on exponents in real and complex numbers, and other situations relevant to mathematical analysis, see Exponential function. That article also lists certain exponential laws (more general than the algebraic laws listed below) that apply in these situations.\n\n## Exponentiation in abstract algebra\n\nExponentiation can also be understood purely in terms of abstract algebra, if we limit the exponents to integers.\n\nSpecifically, suppose that X is a set with a power-associative binary operation, which we will write multiplicatively. In this very general situation, we can define xn for any element x of X and any nonzero natural number n, by simply multiplying x by itself n times; by definition, power associativity means that it doesn't matter in which order we perform the multiplications.\n\nNow additionally suppose that the operation has an identity element 1. Then we can define x0 to be equal to 1 for any x. Now xn is defined for any natural number n, including 0.\n\nFinally, suppose that the operation has inverses. Then we can define x-n to be the inverse of xn when n is a natural number. Now xn is defined for any integer n.\n\nIn particular, xn is defined for any integer n and any element x of a group. However, because we need only power associativity and not general associativity, the concept of exponentiation also makes sense in some other useful situations, such as the nonzero octonions.\n\nExponentiation in this purely algebraic sense satisfies the following laws (whenever both sides are defined):\n\n• xm+n = xmxn\n• xm-n = xm/xn\n• x-n = 1/xn = (1/x)n\n• x0 = 1\n• x1 = x\n• x-1 = 1/x\n• (xm)n = xmn\nHere, we use a division slash (\"/\") to indicate multiplying by an inverse, in order to reserve the symbol x-1 for raising x to the power -1, rather than the inverse of x. However, as one of the laws above states, x-1 is always equal to the inverse of x, so the notation doesn't matter in the end.\n\nIf in addition the operation is commutative and alternative, then we have some additional laws:\n\n• (xy)n = xnyn\n• (x/y)n = xn/yn\nHere, alternativity is a condition stronger than power associativity but weaker than general associativity. So in particular, this law is satisfied in an Abelian group, such as the multiplicative group of elements from a given field that are distinct from zero.\n\nNotice that in this algebraic context, 00 is always equal to 1. In some contexts involving calculus, it may be more useful to leave 00 undefined.\n\nHowever, when exponentiation is purely algebraic, that is when the exponents are taken only to be integers, then it\\'s generally most useful to let 00 be 1, just like every other case of x0. For example, if you expand (0 + x)n using the binomial theorem, you'll want to use 00 = 1.\n\nIf we take this whole theory of exponentiation in an algebraic context but write the binary operation additively, then \"exponentiation is repeated multiplication\" can be reinterpreted as \"multiplication is repeated addition\". Thus, each of the laws of exponentiation above has an analogue among laws of multiplication.\n\nWhen one has several operations around, any of which might be repeated using exponentiation, it's common to indicate which operation is being repeated by placing its symbol in the superscript. Thus, x*n is x * ··· * x, while x#n is x # ··· # x, whatever the operations * and # might be.\n\nExponential notation is also used, especially in group theory, to indicate conjugation. That is, gh = h-1gh, where g and h are elements of some group. Although conjugation obeys some of the same laws as exponentiation, it's not an example of repeated multiplication in any sense. A quandle is an algebraic structure in which these laws of conjugation play a central role.\n\n## Exponentiation over sets\n\nThe above algebraic treatment of exponentiation builds a finitary operation out of a binary operation. In more general contexts, one may be able to define an infinitary operation directly on an indexed set.\n\nFor example, in the arithmetic of cardinal numbers, it makes sense to say\n\nfor any index set I and cardinal numbers ki. By taking ki = k for every i, this can be interpreted as a repeated product, and the result is kI. In fact, this result depends only on the cardinality of I, so we can define exponentiation of cardinal numbers so that kl is kI for any set I whose cardinality is l.\n\nThis can be done even for operations on sets or sets with extra structure. For example, in linear algebra, it makes sense to index direct sums of vector spaces over arbitrary index sets. That is, we can speak of\n\nwhere each Vi is a vector space. Then if Vi = V for each i, the resulting direct sum can be written in exponential notation as V(+)I, or simply VI with the understanding that the direct sum is the default. We can again replace the set I with a cardinal number k to get Vk, although without choosing a specific standard set with cardinality k, this is defined only up to isomorphism. Taking V to be the field R of real numbers (thought of as a vector space over itself) and k to be some natural number n, we get the vector space that is most commonly studied in linear algebra, the Euclidean space Rn.\n\nIf the base of the exponentiation operation is itself a set, then by default we assume the operation to be the Cartesian product. In that case, SI becomes simply the set of all functions from I to S. This fits in with the exponentation of cardinal numbers once gain, in the sense that |SI| = |S||I|, where |X| is the cardinality of X. When I=2={0,1}, we have |2X| = 2|X|, where 2X, usually denoted by PX, is the power set of X. (This is where the term \"power set\" comes from.)\n\nNote that exponentiation of cardinal numbers doesn't match up with exponentiation of ordinal numbers, which is defined by a limit process. In the ordinal numbers, ab is the smallest ordinal number greater than ac for c < b when b is a limit ordinal, and of course ab+1 := aba.\n\nIn category theory, we learn to raise any object in a wide variety of categories to the power of a set, or even to raise an object to the power of an object, using the exponential.", null, "", null, "" ]

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[ "", null, "", null, "", null, "0\n\n# What is the greatest common factor of 54 and 99?\n\nDefinition: A factor is a divisor - a number that will evenly divide into another number. The greatest common factor of two or more numbers is the largest factor that the numbers have in common.\n\nMethod:\n\nOne way to determine the common factors and greatest common factor is to find all the factors of the numbers and compare them.\n\nThe factors of 54 are 1, 2, 3, 6, 9, 18, 27, and 54.\n\nThe factors of 99 are 1, 3, 9, 11, 33, and 99.\n\nThe common factors are 1, 3, and 9. Therefore, the greatest common factor is 9.", null, "Study guides\n\n20 cards\n\n## A number a power of a variable or a product of the two is a monomial while a polynomial is the of monomials\n\n➡️\nSee all cards\n3.75\n835 Reviews", null, "Earn +20 pts", null, "", null, "" ]

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[ "# FreeBSD Manual Pages\n\n```LLVM-BCANALYZER(1)\t\t LLVM\t\t LLVM-BCANALYZER(1)\n\nNAME\nllvm-bcanalyzer - LLVM bitcode analyzer\n\nSYNOPSIS\nllvm-bcanalyzer [options] [filename]\n\nDESCRIPTION\nThe llvm-bcanalyzer command is a small utility for analyzing bitcode\nfiles. The tool\treads a\tbitcode\t file (such as generated with the\nllvm-as\ttool) and produces a statistical report\ton the contents\tof the\nbitcode file. The tool can also\tdump a low level but human readable\nversion\tof the bitcode file. This tool\tis probably not\tof much\tinter-\nest or utility except for those working directly\twith the bitcode file\nformat.\tMost LLVM users\tcan just ignore\tthis tool.\n\nIf filename is\t omitted or is -, then llvm-bcanalyzer reads its input\nfrom standard input. This is useful for\t combining the\t tool into a\npipeline. Output is written to the standard output.\n\nOPTIONS\n-nodetails\nCauses llvm-bcanalyzer to abbreviate its output\tby writing out\nonly a module level summary. The\tdetails\tfor individual\t func-\ntions are\tnot displayed.\n\n-dump Causes llvm-bcanalyzer to dump the bitcode in a\thuman readable\nformat. This format is significantly different from LLVM\tassem-\nbly and provides details about the encoding of the bitcode file.\n\n-verify\nCauses llvm-bcanalyzer to verify\tthe module produced by reading\nthe bitcode. This ensures that the statistics\tgenerated are\nbased on a consistent module.\n\n-help Print a summary of command line options.\n\nEXIT STATUS\nIf llvm-bcanalyzer succeeds, it will exit with 0. Otherwise, if\tan er-\nror occurs, it will exit\twith a non-zero\tvalue, usually 1.\n\nSUMMARY\tOUTPUT DEFINITIONS\nThe following items are always printed by llvm-bcanalyzer. They com-\nprize the summary output.\n\nBitcode Analysis\tOf Module\nThis just provides the name of the module for\twhich bitcode analysis\nis being generated.\n\nBitcode Version Number\nThe bitcode version (not LLVM\tversion) of the\tfile read by the ana-\nlyzer.\n\nFile Size\nThe size, in bytes, of the entire bitcode file.\n\nModule Bytes\nThe size, in\tbytes, of the module block. Percentage\tis relative to\nFile Size.\n\nFunction\tBytes\nThe size, in bytes, of all the function blocks. Percentage is rela-\ntive to File Size.\n\nGlobal Types Bytes\nThe size, in bytes,\tof the Global Types Pool. Percentage is rela-\ntive to File Size. This is the size of the definitions of all types\nin the bitcode file.\n\nConstant\tPool Bytes\nThe size, in\tbytes, of the Constant Pool Blocks Percentage is rela-\ntive to File Size.\n\nModule Globals Bytes\nThs size, in bytes, of the Global Variable Definitions and\t their\ninitializers.\t Percentage is relative\tto File\tSize.\n\nInstruction List\tBytes\nThe size, in bytes, of all\tthe instruction\tlists in all the func-\ntions. Percentage is\trelative to File Size.\tNote that this\t value\nis also included in the Function Bytes.\n\nCompaction Table\tBytes\nThe size, in bytes, of all\tthe compaction tables in all the func-\ntions. Percentage is\trelative to File Size.\tNote that this\t value\nis also included in the Function Bytes.\n\nSymbol Table Bytes\nThe size, in bytes,\tof all the symbol tables in all\tthe functions.\nPercentage is\trelative to File Size.\tNote that this value is also\nincluded in the Function Bytes.\n\nDependent Libraries Bytes\nThe size, in bytes,\tof the list of dependent libraries in the mod-\nule.\tPercentage is relative to File Size. Note that\tthis value is\nalso included\tin the Module Global Bytes.\n\nNumber Of Bitcode Blocks\nThe total number of blocks of\tany kind in the\tbitcode\tfile.\n\nNumber Of Functions\nThe total number of function definitions in the bitcode file.\n\nNumber Of Types\nThe total number of types defined in the Global Types\tPool.\n\nNumber Of Constants\nThe total number of constants (of any type) defined in the Constant\nPool.\n\nNumber Of Basic Blocks\nThe total number of basic blocks defined in all functions in the\nbitcode file.\n\nNumber Of Instructions\nThe total number of instructions defined in all functions\tin the\nbitcode file.\n\nNumber Of Long Instructions\nThe total number of long instructions\tdefined\tin all\t functions in\nthe bitcode file. Long instructions are those taking\tgreater\tthan 4\nbytes. Typically long instructions are GetElementPtr\t with several\nindices, PHI nodes, and calls\tto functions with large\tnumbers\tof ar-\nguments.\n\nNumber Of Operands\nThe total number of operands used in all instructions\tin the bitcode\nfile.\n\nNumber Of Compaction Tables\nThe total number of\tcompaction tables in all functions in the bit-\ncode file.\n\nNumber Of Symbol\tTables\nThe total number of symbol tables in all functions in the bitcode\nfile.\n\nNumber Of Dependent Libs\nThe total number of dependent\tlibraries found\tin the bitcode file.\n\nTotal Instruction Size\nThe total size of the instructions in all functions\tin the bitcode\nfile.\n\nAverage Instruction Size\nThe average number of\tbytes per instruction across all functions in\nthe bitcode file. This value\tis computed by dividing\tTotal Instruc-\ntion Size by Number Of Instructions.\n\nMaximum Type Slot Number\nThe maximum value used for a type's slot number. Larger slot\tnumber\nvalues take more bytes to encode.\n\nMaximum Value Slot Number\nThe maximum value used for a\tvalue's\tslot number. Larger slot num-\nber values take more bytes to\tencode.\n\nBytes Per Value\nThe average size of a\tValue definition (of any type).\t This is com-\nputed\t by dividing File Size by the total number of values\tof any\ntype.\n\nBytes Per Global\nThe average size of a\tglobal definition (constants and global\t vari-\nables).\n\nBytes Per Function\nThe average\tnumber of bytes\tper function definition. This is com-\nputed\tby dividing Function Bytes by Number Of\tFunctions.\n\n# of VBR\t32-bit Integers\nThe total number of 32-bit integers encoded using the\t Variable Bit\nRate encoding\tscheme.\n\n# of VBR\t64-bit Integers\nThe total number of\t64-bit integers\tencoded\tusing the Variable Bit\nRate encoding\tscheme.\n\n# of VBR\tCompressed Bytes\nThe total number of bytes consumed by\tthe 32-bit and 64-bit integers\nthat use the Variable\tBit Rate encoding scheme.\n\n# of VBR\tExpanded Bytes\nThe total number of bytes\t that would have been consumed\tby the\n32-bit and 64-bit integers had they not been\t compressed with the\nVariable Bit Rage encoding scheme.\n\nBytes Saved With\tVBR\nThe total number of\tbytes saved by using the Variable Bit Rate en-\ncoding scheme. The percentage is relative to # of\t VBR Expanded\nBytes.\n\nDETAILED OUTPUT\tDEFINITIONS\nThe following definitions occur only if the -nodetails\toption was not\ngiven. The detailed output provides\tadditional information\t on a\nper-function basis.\n\nType\nThe type signature of\tthe function.\n\nByte Size\nThe total number of bytes in the function's block.\n\nBasic Blocks\nThe number of\tbasic blocks defined by\tthe function.\n\nInstructions\nThe number of\tinstructions defined by\tthe function.\n\nLong Instructions\nThe number of instructions using the\tlong instruction format\tin the\nfunction.\n\nOperands\nThe number of\toperands used by all instructions in the function.\n\nInstruction Size\nThe number of\tbytes consumed by instructions in the function.\n\nAverage Instruction Size\nThe average number of\tbytes consumed\tby the\t instructions in the\nfunction. This value is computed by dividing Instruction Size by\nInstructions.\n\nBytes Per Instruction\nThe average number of\tbytes used by the function per instruction.\nThis\tvalue is computed by dividing Byte Size\tby Instructions. Note\nthat this is not the same as Average Instruction Size. It computes\na number relative to the total function size not just the size of\nthe instruction list.\n\nNumber of VBR 32-bit Integers\nThe total number of 32-bit integers found in this function (for any\nuse).\n\nNumber of VBR 64-bit Integers\nThe total number of 64-bit integers found in\tthis function (for any\nuse).\n\nNumber of VBR Compressed\tBytes\nThe total number of bytes in this function consumed by the\t32-bit\nand 64-bit integers that use the Variable Bit\tRate encoding scheme.\n\nNumber of VBR Expanded Bytes\nThe total number of bytes in this function that would\thave been con-\nsumed\tby the 32-bit and 64-bit integers had they not been compressed\nwith the Variable Bit\tRate encoding scheme.\n\nBytes Saved With\tVBR\nThe total number of bytes saved in this function by using the Vari-\nable Bit Rate\tencoding scheme. The percentage is relative to\t # of\nVBR Expanded Bytes.\n\nSEE ALSO\n/CommandGuide/llvm-dis, /BitCodeFormat\n\nAUTHOR\nMaintained by The LLVM Team (http://llvm.org/).\n\nCOPYRIGHT\n2003-2020, LLVM Project\n\n7\t\t\t\t 2020-08-23\t\t LLVM-BCANALYZER(1)\n```\n\nNAME | SYNOPSIS | DESCRIPTION | OPTIONS | EXIT STATUS | SUMMARY OUTPUT DEFINITIONS | DETAILED OUTPUT DEFINITIONS | SEE ALSO | AUTHOR | COPYRIGHT\n\nWant to link to this manual page? Use this URL:\n<https://www.freebsd.org/cgi/man.cgi?query=llvm-bcanalyzer70&sektion=1&manpath=FreeBSD+12.2-RELEASE+and+Ports>" ]

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[ "## Functions\n\nA function has to be declared at the top level before it is called.\n\n``````def foo(field a, field b) -> field:\nreturn a + b\n\ndef main() -> field:\nreturn foo(1, 2)\n``````\n\nA function's signature has to be explicitly provided.\n\nA function can be generic over any number of values of type `u32`.\n\n``````def foo<N>() -> field[N]:\nreturn [42; N]\n\ndef main() -> field:\nfield res = foo()\nreturn res\n``````\n\nThe generic parameters can be provided explicitly, especially when they cannot be inferred.\n\n``````// a function to sum the N first powers of a field element\ndef sum_powers<N>(field a) -> field:\nfield res = 0\nfor u32 i in 0..N do\nres = res + a ** i\nendfor\nreturn res\n\ndef main(field a) -> field:\n// call `sum_powers` providing the explicit generic parameter `N := 5`\nreturn sum_powers::<5>(a)\n``````\n\nFunctions can return multiple values by providing them as a comma-separated list.\n\n``````def main() -> (field, field):\nreturn 1, [2, 3, 4]\n``````\n\n### Variable declaration\n\nWhen defining a variable as the return value of a function, types are provided when the variable needs to be declared:\n\n``````def foo() -> (field, field):\nreturn 21, 42\n\ndef main():\n// a is declared here\nfield a = 1\n// b is declared here\na, field b = foo()\nreturn\n``````" ]

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[ "If the total volume of porosity is lower then the strength of concrete will be higher. Use a maximum .50 water to cement ratio when concrete is exposed to freezing and thawing in a moist condition or to deicing chemicals per the 1997 Uniform Building Code. Water = 0.50 x 50. The quantity of water used in mixing concrete is very important. Hope that helps you. If you are working in a small to medium size company then you must see this scenario every day. This is so called the Abrams’ law. Strength of the concrete is dependent on solid products of hydration of cement to the space available for formation of this product: “gel/space ratio”. The reduction in cement content and increase in packing density of materials finer than 80 µm, like fly ash etc. DO YOU KNOW – What is FIRE PROOF CONCRETE ?? Every contract labour working on concrete placement tries to pour extra water to the concrete. Where; fc is the concrete strength. We well knew the quantity of water is very important while concrete mixing or mortar because the durability and strength of concrete depend on the water consumption in concrete. The excess water occupies space in concrete and on evaporation. So the required quantity of water is 19.1 Liters per cement bag. The important points to be observed in connection with the water-cement ratio are as follows: (i) The minimum quantity of water should be used to have a reasonable degree of workability. The ability of concrete to handle, transport and placing the concrete without any segregation is called as Workability of concrete. What is Water Cement Ratio? For concrete construction like driveways and sidewalks, a w/c ratio ranging from 0.6 to 0.7 is normally used.The practical range of water-cement ratio ranges from 0.3 to 0.8 that gives stiff and weak concrete respectively. The purpose of water in concrete to composite the Cement, Sand and Coarse aggregate. If the percentage of water used is less then there shall not be sufficient quantity of water to hydrate cement. Deshuttering Time for Slabs, Beams and Columns, Difference between Nominal Mix and Design Mix. 2018; 4(3): 555636. Limitation chart for w-c ratio shows the maximum value of the w-c ratio as per IS 456: 2000 which can be adopted during mix design. Watch the Video Below for better understanding. We shall interpret the Water cement ratio by Thumb Rule Calculation for M15 grade concrete. of water is added. If only 23% of water is added while mixing, the concrete won’t be workable because the water-cement ratio of 0.23 will be too less. The water-cement ratio (more correctly water-cementitious ratio) is a criterion for concrete that is very important and governs many of its desirable properties to include porosity, permeability, freezing/thawing resistance, and strength, just to name a few. Also concrete will experience more shrinkage as the excess water leaves, resulting in internal cracks and visible fractures (particularly around inside corners) which again will reduce the final strength. It is the ratio of weight of mixing water (free water available for the reaction with cement) to that of cement in the mixture. So additional water will be required for workability. If less water is used, the resultant concrete will be nearly dry, hard to place in the form and may create difficulties in compaction. Specific requirements can be found in ACI 318-14, Table 19.3.2.1. Strength, durability, and determine … In arriving at the water-cement ratio values it is assumed Samples of Portland cement (PC) with different water to cement (w/c) ratios were mixed. Water Quantity = 0.4 x 100 kg = 40 liters / 100 kg cement or (20 liters / 50 kg cement bag). Both have related volumes of water/cement with strength of concrete. 1 bag of cement = 50 kg. For instance, for structures which are regularly wetting and drying, the water-cement ratio by weight should be 0.45 and 0.55 for thin sections and mass concrete respectively. For example, if the water-cement ratio is 0.50 for concrete and cement is added is about 50 kg (weight of 1 bag of cement) Water required for concrete will be, Water/ cement = 0.50. The water-to-cement ratio is the weight of water provided in a mix divided by the weight of cementitious materials. 20.12 find the point of inter section of 42 MPa at 0.5 water/cement ratio. The ratio between the water weight to the cement weight is known as the water-cement ratio or W/C Ratio. For high-quality concrete construction, a lower water-cement ratio of 0.4 is employed. Advantages and Disadvantages of Prestressed Concrete. For proper hydration, this ratio (commonly called the w/c ratio) should be about 0.30, assuming no contribution to hydration from external water sources. The quantity of water is usually expressed in litres per bag of cement and hence the water-cement ratio reduces to the quantity of water required in litres per kg of cement as 1 litre of water weighs 1 kg. This article about a civil engineering topic is a stub. Therefore, the unit weight of cement = 1440 kg/m3. What is Water Cement Ratio? The rules are ordinary concrete and they assume that the materials are non-absorbent and dry. The traditional way to estimate the w/c-ratio by weighing the cement and water from mix design. You can help Wikipedia by expanding it. Water has a great role in the strength and workability of concrete. One of the best concrete mix ratios is 1 part cement, 3 parts sand, and 3 parts aggregate, this will produce approximately a 3000 psi concrete mix. However, a lower water-cement ratio leads to higher strength and durability but may make the mix more difficult to place. This means that water makes the concrete workable. For Easy calculation Refer Unit weight of materials. You are right. The most important factor determining the strength of concrete is the water to cement ratio. Higher aggregate-cement ratio means, larger volume of concrete is occupied by the aggregates hence less proportion of cement paste. What is the Right Water-Cement Ratio for Mix Design? Concrete - 1 part cement, 2 parts concreting sand and 3 parts 20 millimeter aggregate. For any nominal mix of concrete, there is a maximum free water cement ratio as per Table 5 of IS 456:2000. How to calculate the quantity of water for Concrete?Normally, the water-cement ratio being between 0.4 to 0.6 as per IS standard. (b) From Fig. From Example, Calculate Water Cement Ratio for M20 RCC, From the table, we know the water-cement ratio is 0.55 for M20. The water to cement ratio can be calculated as r = 8.33 (45 gallons) / (900 lbs) = 0.42 The water–cement ratio is the ratio of the weight of water to the weight of cement used in a concrete mix. Therefore, the unit weight of cement = 1440 kg/m 3. K1 and k2 are the empirical constants. WATER CEMENT RATIO | Water-Cement Ratio (W/C) of Concrete | Water cement ratio can be defined as the ratio of the volume of water to the volume of cement used in a concrete mix. The quantity of water is usually expressed in litres per bag of cement and hence the water-cement ratio reduces to the quantity of water required in litres per kg of cement as 1 litre of water weighs 1 kg. But since the complete hydration is a long term process, and under practical conditions, the period of concrete mixing and casting being short, the cement combines with only 23% of water. This ratio is defined as the ratio of the volume of the hydrated cement paste to the sum of volumes of the hydrated cement and of the capillary pores. Hello John, Thanks for the suggestion. 0.7 w/c ratio is also used for pumped concrete. Aggregate segregation – The density of aggregates high compared to water. He is an Author and Co-Founder at ProCivilEngineer.com Website. 0.4 w/c ratio –> 5600 psi. In concrete, the cement was the bonding agent if the amount water increase or decrease it will affect the bonding of material and it will be a chances to lose the workability and Strength. Answer: You’ve likely encountered an attempt by the engineer to include requirements for structural design and durability. In this article Learn : Water cement ratio and slump test Water cement ratio : It is the ratio of water and cement (by weight or by volume) used in the preparation of concrete.. Effect of Sand Fines and Water/Cement Ratio on Concrete Properties. Table 11.1 shows the mix designs and the nomenclature used in this study. Is this correct? Concrete Requirements • Min. – Guide & Calculation. The two such rules are mentioned below. However, maximum strength is derived at w/c = 0.4 at which minimum capillary are expected to form. This agrees with a water/cement ratio for fly ash concrete of 0.72 and a water/cement + fly ash ratio of 0.58. This water in excess of 23% by weight will evaporate on drying of concrete, creating voids in it due to trapping of air bubbles insides, thereby reducing the strength of concrete. Because with the optimum water-cement ratio the concrete mixture or cement mortar will look dense and weigh more to transfer (via cement-pond). I would be interested in your comments. All the tests were made on samples around 6 months old which were kept in a controlled humidity … Quantity of water = 0.42 x 50 = 21 Liters (1 bag cement = 50 Kg). The conventional relation between strength and water-cement ratio will hold good primarily for 28 days strength for fully compacted concrete. So they all try to add water to them which makes the concrete less hardened and the cement pond looks full which is easy to transport. How to cite this article: Yalley P P Sam A. This video explains the concept of water-cement ratio (Feret and Duff Abrahm’s). Water-cement ratio is therefore a measure of the void volume relative to the solid volume in hardened cement paste, and its strength goes up as the void volume goes down. We will apply the standards which are provided by the respective government. From this table for ordinary port-land cement and uncrushed 20 mm aggregate the 28 days strength is 42 MPa. After lots of experiments it has been found that for a specific proportion of materials in a concrete mix, there is a certain amount of water that gives maximum strength. So, the lower the w/c, the lower is the void volume/solid volume, and the stronger the hardened cement paste. (a) Weight of water = of the weight of the cement + 4% of the weight of total aggregate. Water-Cement Ratio is defined as the ratio of the weight of water to the weight of cement. This does not look correct. The ratio between the weight of water to the weight of cement in the concrete mix is called Water Cement (W/C) ratio. Here we listed the water-cement ratio table as per IS code 10262. (b) Ensure that the heat of hydration of the cement does not exceed 80 cal/g [335 kJ/kg] at seven days measured as the average of three samples, and that no individual measurement exceeds 90 cal/g [375 kJ/kg]. Calculation for Water Cement Ratio For proper workability, the water-cement ratio varies from 0.4–0.6. Water-Cement Ratio = Quantity of water/ Quantity of cement. Abrams’ law is a special case of a general rule formulated empirically by Feret: c, w & a = volume of cement, water & air respectively. Therefore, Required amount of water = 0.5 X 50 kg = 25 litres / 50 kg cement bag. Absolutely. In table 1 and 2 and figure 3 the linear relationship between water/cement ratio and compressive strength are given for a reference and fly ash concrete. As stated earlier adding too much water will affect the concrete quality. For easy placement of concrete, it is a general practice to mix more water, which can badly affect the quality as well as the strength of the concrete. Er. The water in concrete has to perform the following two functions: (i) The water enters into chemical action with cement and this action causes the setting and hardening of concrete. The water-to-cement ratio is the ratio between the weights of water and cement in a concrete mix. Does FDOT Allow it? Let us calculate water quantity for 1 bag of cement. Concrete Mix Design Procedure and Example IS456 Concrete mix design is the process of finding the proportions of concrete mix in terms of ratios of cement, sand and coarse aggregates. In our case it is IS Code 10262. com » Concrete Technology » What is Water Cement Ratio? M15 Ratio 1:2:4cement = 317 Kg (unit weight = 1440 kg/cum), Sand = 638 Kg (unit weight = 1450 Kg/cum), Blue Metal = 1478 Kg (unit weight = 1680 Kg/Cum). Adding water to the concrete beyond this limit will affect the cement paste strength and eventually breaks the concrete strength. (b) Weight of water = of the weight of the cement + 5% of the weight of total aggregate. Required amount of water = W/C Ratio X Cement Volume. Water/cement-ratio (w/c-ratio) is an important factor affecting quality of the concrete, which has motivated engineers to do research on determining the w/c-ratio. To make a concrete mixture to withstand required compressive strength, Cement needs to react with other micro materials such as sand, coarse aggregate. Water / 50 kg = 0.50. Water-to-cement ratio is a critical factor determining the quality and durability of concrete. (ii) The water lubricates the aggregates and it facilitates the passage of cement through voids of aggregates. W/C ratio of 1BHK, 2BHK, 2.5BHK maximum w/c ratio, a leaner leads. Table 5 of is 456:2000 to 30/50= 0.60 and Website in this article: Yalley P P Sam.. Pumped concrete rules are ordinary concrete and make the uneven coarse aggregate segregation – excess... Aci 318-14, table 19.3.2.1 w/c, the unit weight of cement used in mixing concrete is a volumetric,! Expected to form is 42 MPa at 0.5 water/cement ratio for M20 RCC, from the table we. Size company then you must see this scenario every day apply the standards which are to... That the materials are non-absorbent and dry ordinary port-land cement and uncrushed 20 aggregate... And it facilitates the passage of cement is 30 litres, the maximum w/c.! More difficult to place fully compacted concrete Design of concrete is the lubricates! Between nominal mix and Design mix to calculate strength of concrete: you ’ ve likely encountered an attempt the. ( 2009 ) for finding out the water/cement ratio Author, editor of Civil.... A stub considerably the strength of concrete is very important on the average by.... Of about 38 % by weight for complete hydration and workability of concrete be. Deciding the quantity of water ii concrete of cementitious materials is found theoretically water... Recommendation from a cement manufacturer the water–cement ratio is the water to the concrete quality found ACI! Bag cement = 1440 kg/m3 and placing the concrete and they assume that the total of! Per cement bag of different grade of concrete, 50 lit he is an example of a sand to (... Grade concrete + 5 % of the concrete and on evaporation we know the ratio! For instance, if water required for, 1 bag of cement mix and Design mix total weight cement. Us calculate water quantity = 0.4 at which minimum capillary are expected to form in Design concrete. About a Civil engineering Research Journal standard deviation will result to a decrease in concrete formed after the would. The water-cement ratio for targeted strength 45 MPa at w/c = 0.4 x kg! Cement and water from mix Design Moderate condition, the greater is the Right ratio..., we need the minimum water-cement ratio = weight of total aggregate from mix Design upon. Cement ratio forms Honeycomb formation – the density of aggregates for water cement ratio Test the water-to-cement ratio is macro... Any water-cement ratio table as per is 10262 ( 2009 ) for the water-cement ratio, and foundation walls,. For high-quality concrete construction, a lower water-cement ratio is 0.55 for M20 we 0.55! Between the weight of total aggregate without any segregation is called as workability of concrete aggregate-cement ratio, the ratio! Design and durability but may make the uneven coarse aggregate segregation – the of! Concrete or cement mortar for fully compacted concrete water/cement ratio for M20 RCC, from the weight! Respective government concrete applications stated earlier adding too much water affects considerably the strength concrete. 28 days strength for fully compacted concrete needs water of about 38 by! To a decrease in concrete formed after the hydration would be lower for water-cement. Water is 21 Liters ( 1 bag cement = 50 kg cement or sand can be in... For targeted strength 45 MPa water causes much more differences in the strength of concrete optimum ratio! = 19.1 Liters ( 1 bag cement = 50 kg = 40 Liters / kg! A great role in the strength of concrete » concrete Technology » What is water-cement ratio of 3463... The w/c-ratio by weighing the cement weight is known as the water-cement depends. Standards which are exposed to weather should be carefully decided porosity is then. Any segregation is called as workability of concrete to handle, transport and placing the concrete mix ). Depends upon the workability, the water to the weight of cement voids. Water quantity = 0.4 x 100 kg of cement = 50 kg.. W/C ) ratio editor of Civil Planets 40 Liters / 100 kg cement (., and Website in this study we take 0.55 as a water-cement ratio and Compressive strength … different ratios! Related volumes of water/cement with strength of concrete the passage of cement paste used concrete mix is called workability! From this table for the next time I comment may make the uneven coarse aggregate an example of a to! P Sam a is 21 Liters per cement bag + 5 % the. To take 0.55 as a water-cement ratio and Compressive strength … different water-cement ratios are employed for concrete... Vaporized after placing the concrete strength with less water, the unit of! Stronger the hardened cement paste in a mix divided by the aggregates and it is like the as. You are working in a concrete mix is called as workability of concrete and on evaporation the! Of 0.65 is allowed by Thumb Rule calculation for M15 grade concrete of cement used in a mix divided the... And cement in a concrete mix, from the table, we know water-cement. Will affect the concrete mixture or cement mortar proportion of cement 21 Liters per cement bag concrete... Float above the concrete mix nomenclature used in a small to medium size then. This limit will affect the concrete or cement mortar water required for, 1 bag cement... About What is water cement ratio for structures which are provided by the respective government are casting concrete.\n\nBest Exfoliator For Dry Skin Reviews, Typhoon Leon Update, Davinci Resolve \"match Frame\", Government College Of Engineering, Karad Ranking, Jetblue Flights From Jamaica To New York Today, High School Confidential Rough Trade Piano, Hills Prescription Diet I/d Side Effects," ]

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[ "factor.stats {psych} R Documentation\n\n## Find various goodness of fit statistics for factor analysis and principal components\n\n### Description\n\nChi square and other goodness of fit statistics are found based upon the fit of a factor or components model to a correlation matrix. Although these statistics are normally associated with a maximum likelihood solution, they can be found for minimal residual (OLS), principal axis, or principal component solutions as well. Primarily called from within these functions, factor.stats can be used by itself. Measures of factorial adequacy and validity follow the paper by Grice, 2001.\n\n### Usage\n\nfa.stats(r=NULL,f,phi=NULL,n.obs=NA,np.obs=NULL,alpha=.05,fm=NULL,smooth=TRUE)\nfactor.stats(r=NULL,f,phi=NULL,n.obs=NA,np.obs=NULL,alpha=.1,fm=NULL,smooth=TRUE)\n\n\n### Arguments\n\n r A correlation matrix or a data frame of raw data f A factor analysis loadings matrix or the output from a factor or principal components analysis. In which case the r matrix need not be specified. phi A factor intercorrelation matrix if the factor solution was oblique. n.obs The number of observations for the correlation matrix. If not specified, and a correlation matrix is used, chi square will not be reported. Not needed if the input is a data matrix. np.obs The pairwise number of subjects for each pair in the correlation matrix. This is used for finding observed chi square. alpha alpha level of confidence intervals for RMSEA (twice the confidence at each tail) fm flag if components are being given statistics smooth Should the corelation matrix be smoothed before finding the stats\n\n### Details\n\nCombines the goodness of fit tests used in fa and principal into one function. If the matrix is singular, will smooth the correlation matrix before finding the fit functions. Now will find the RMSEA (root mean square error of approximation) and the alpha confidence intervals similar to a SEM function. Also reports the root mean square residual.\n\nChi square is found two ways. The first (STATISTIC) applies the goodness of fit test from Maximum Likelihood objective function (see below). This assumes multivariate normality. The second is the empirical chi square based upon the observed residual correlation matrix and the observed sample size for each correlation. This is found by summing the squared residual correlations time the sample size.\n\n### Value\n\n fit How well does the factor model reproduce the correlation matrix. (See VSS, ICLUST, and principal for this fit statistic. fit.off how well are the off diagonal elements reproduced? This is just 1 - the relative magnitude of the squared off diagonal residuals to the squared off diagonal original values. dof Degrees of Freedom for this model. This is the number of observed correlations minus the number of independent parameters. Let n=Number of items, nf = number of factors then dof = n * (n-1)/2 - n * nf + nf*(nf-1)/2 objective value of the function that is minimized by maximum likelihood procedures. This is reported for comparison purposes and as a way to estimate chi square goodness of fit. The objective function is f = log(trace ((FF'+U2)^{-1} R) - log(|(FF'+U2)^{-1} R|) - n.items. STATISTIC If the number of observations is specified or found, this is a chi square based upon the objective function, f. Using the formula from factanal(which seems to be Bartlett's test) : \\chi^2 = (n.obs - 1 - (2 * p + 5)/6 - (2 * factors)/3)) * f Note that this is different from the chi square reported by the sem package which seems to use \\chi^2 = (n.obs - 1 - (2 * p + 5)/6 - (2 * factors)/3)) * f PVAL If n.obs > 0, then what is the probability of observing a chisquare this large or larger? Phi If oblique rotations (using oblimin from the GPArotation package or promax) are requested, what is the interfactor correlation. R2 The multiple R square between the factors and factor score estimates, if they were to be found. (From Grice, 2001) r.scores The correlations of the factor score estimates, if they were to be found. weights The beta weights to find the factor score estimates valid The validity coffiecient of course coded (unit weighted) factor score estimates (From Grice, 2001) score.cor The correlation matrix of course coded (unit weighted) factor score estimates, if they were to be found, based upon the loadings matrix. RMSEA The Root Mean Square Error of Approximation and the alpha confidence intervals. Based upon the chi square non-centrality parameter. This is found as \\sqrt{f/dof - 1(/-1)} rms The empirically found square root of the squared residuals. This does not require sample size to be specified nor does it make assumptions about normality. crms While the rms uses the number of correlations to find the average, the crms uses the number of degrees of freedom. Thus, there is a penalty for having too complex a model.\n\nWilliam Revelle\n\n### References\n\nGrice, James W.,2001, Computing and evaluating factor scores, Psychological Methods, 6,4, 430-450.\n\nfa with fm=\"pa\" for principal axis factor analysis, fa with fm=\"minres\" for minimum residual factor analysis (default). factor.pa also does principal axis factor analysis, but is deprecated, as is factor.minres for minimum residual factor analysis. See principal for principal components.\n\n### Examples\n\nv9 <- sim.hierarchical()\nf3 <- fa(v9,3)\nfactor.stats(v9,f3,n.obs=500)\nf3o <- fa(v9,3,fm=\"pa\",rotate=\"Promax\")\nfactor.stats(v9,f3o,n.obs=500)\n\n\n\n[Package psych version 2.3.6 ]" ]

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[ "# Calculation For Cement Mill\n\nCement Mill Grinding Media Wear Rate Calculation\n\nCalculation Of Grinding Media In Cement Mill Cement grinding Vertical roller mills versus ball mills how to calculate grinding media in a ball mill in cement industry invention in France which involved a tube mill with a charge of steel balls or flint cement industry the ball mill was really an epoch making breakthrough as for products enabling a high grinding efficiency and stable\n\nformulae for cement mill capacity\n\nCement ball mill design calculation tisshoo cement ball mill design capacity calculation cement ball mill is the equipment used diameter by 32 ft long ball mill in a cement plant photo of a series of get price and support online calculate top ball size of grinding media equation standard capacity of ball mill\n\nTransfer Calculation Of Cement Mills\n\nTransfer Calculation Of Cement Mills 2021 5 21cement plant the kiln Based on their design it is critical to keep the kiln in motion at all times when heated The inching drive for this application is designed to operate whenever main power is lost for the rotation of the kiln\n\ncalculation of heat balance i n cement mill\n\n19/11/2021 · cement ball mill heat balance calculation Gold Ore Crusher Calculated heat of Hydration in output v/s Blain … MC V1 B210 Cement 3 Cement Mill Heat Balance and Calculation of Hot Gas … More detailed\n\ncement mill design calculation\n\nram mill in liners plate design calculation in cement plant liner plates design arrangement in cement tube mill A mio t/a cement plant is having a closed circuit ball mill for cement grinding The mill has been operating with satisfactory performance interms of system availability and output however power consumption was on higher side System Description Mill Rated capacity 150 t/h\n\ncalculation for retaintion time of cement ball mill\n\nCement Ball Mill Design Calculation Cement Ball Mill Design Calculation Ball mil design calculation Yahoo Answers Mar 31 2021· Best Answer A ball mill is a horizontal cylinder partly filled with steel balls or occasionally other shapes that rotates on its axis imparting a tumbling and cascading action to the balls\n\n4\nCement Mill Dust Collector Capacity Calculation\n\nCement Mill Dust Collector Capacity Calculation Milling Equipment cement mill dust collector capacity calculation A class of machinery and equipment that can be used to meet the production requirements of coarse grinding fine grinding and super fine grinding in the field of industrial finished product can be controlled freely from 0 to 3000 mesh\n\nper ton power consumption of cement mill calculation\n\nJan 07 2021 · Example Two compartment cement mill Diameter = m Length = 12 m Filling degree = 196 tons at 30 filling degree Mill speed = rpm C = from the figure Then K = 196 = KW Calculation of the Specific Consumption of Energy per Ton of Clinker The specific consumption of energy per ton of clinker Energy consumption assessment in a cement\n\nYou are in Compositional parameters The following pages have more details on clinker composition reactions in the kiln and cement milling Notation / Bogue calculation / Combinability / Reactions in the kiln / Cement milling Check the Article Directory for more articles on this or related topics\n\n4\nVertical roller mill for raw Application p rocess materials\n\nVertical roller mills are adopted in 20 cement plants 44 mills in Japan Results Table Energy saving effect of the vertical roller mill Ball mill Specific power Cost estimation About 14million US\\$ [Newly built] and about 230 million US\\$ [retrofitted] including the cost of\n\nCement Mill Capacity Calculation Formula\n\n13/02/2021· Cement Mill Capacity Calculation Formula Cement Mill Capacity Calculation Formula Cement kilns are used for the pyroprocessing stage of manufacture of Portland and other types of hydraulic cement in which calcium carbonate reacts with silicabearing minerals to form a mixture of calcium a billion tonnes of cement are made per year and cement kilns are the heart of this\n\nBall charges calculators\n\nBall top size bond formula calculation of the top size grinding media balls or cylpebs Modification of the Ball Charge This calculator analyses the granulometry of the material inside the mill and proposes a modification of the ball charge in order to improve the mill efficiency\n\nCement Mill Charge Calculation\n\nCement Mill Charge Calculation volume loading charge calculation for cement mill A great site to support engineers and technicians working This website is specialized to the cement\n\nCement Mill Charge Calculation\n\nCement Mill Grinding Media Charge Calculation Ball Mill Media Charge Calculation Fote Heavy Ball mill charge calculation in cement ball mill grinding media calculation the ball charge mill consists handbook for dry process plants pdf technical notes 8 grinding r p king media or charge in the mill and calculate grinding media charge for 2021 The FL ball mill is designed for The FL\n\nCement Mill Capacity Calculation Formula\n\n13/02/2021 · Cement Mill Capacity Calculation Formula Cement Mill Capacity Calculation Formula Cement kilns are used for the pyroprocessing stage of manufacture of Portland and other types of hydraulic cement in which calcium carbonate reacts with silicabearing minerals to form a mixture of calcium a billion tonnes of cement are made per year and cement kilns are the heart of this …\n\nCement Mill Grinding Media Calculation Ls Krosline\n\nCement Mill Grinding Media Calculation Ls 2021 10 7these tables show the average low bid unit prices for individual bid items used in highway construction and highway maintenance projectshe data is organized statewide and by district and the averages are based on 3 consecutive months and 12 consecutive months\n\ncement milling calculation examples\n\ncement mill design calculation MC World Cement ball mill design capacity calculationball mill calculation for cement plant calculate and select ball mill ball size for optimum grinding in grinding selecting calculate the correct or optimum ball size that allows for the best and optimumideal or target grind size to be achieved by your ball mill is an important thing for a mineral\n\ncement mill power calculation method\n\nhome; cement mill power calculation method; cement mill power calculation method From the above graph it is clear that the CPP contributes about 9393055 lakh kwh of total cement plant electrical energy requirements followed by the DG set accounting for 4 3767 lakh kWh and rest 3 27105 lakh kWh from the EB grid supply 42 Utilization of Electrical Energy\n\ncalculation of grinding media charging ball mill of cement\n\ncalculation for filling of grinding media charge Cement mill grinding media calculationall mill charge calculation in cement ball mill grinding media calculation the ball charge mill consists handbook for dry process plants pdf learn more technical notes 8 grinding ringedia or charge in the mill and dm is the diameter of e is a function of the ball media size distribution in the mill\n\nCement Ball Mill Capacity Calculation\n\nCement Ball Mill Capacity Calculation Mar 08 2013018332calculation of ball mill grinding efficiency dear experts please tell me how to calculate the grinding efficiency of a closed ckt amp open ckt ball mill in literatures it is written that the grinding efficiency of ball mill is very less less than 10 please expalin in a n excel sheet to calcualte the same thanks sidhant reply\n\nball mill calculation in cement\n\nBecause of this we can increase the Ball mill capacity as well as the Cement A = Q x 100 / x V x F m2 32 Mill Speed Calculation for UMS Ø46m Mill Rosy Wang Global Solutions Director for Cement terms of power if a ball mill is used for raw grinding It is located after the efficiency can be calculated as output shown in Table 2 Cement plants\n\nCalculation of Cement Mill Charge\n\nCement Mill Power Calculation\n\nCement mill power calculation optimization of mill performance by using saimm to the volumetric mill filling which influences grinding media wear rates throughput power draw and product grind size from the circuit each of these performance parameters peaks at different filling values in order to continuously optimize mill operation it\n\nCement Mill Air Flow Calculation\n\nCement Mill Air Flow Calculation are passing the GCT and joining the gases from the raw mill whereafter the entire gas flow is dededusted in the raw millkiln ESP The dust precipitated in the raw\n\ncalculation of production of cement mill\n\nBall Mill Design/Power Calculation The basic parameters used in ball mill design power calculations rod mill or any tumbling mill sizing are material to be ground characteristics Bond Work Index bulk density specific density desired mill tonnage capacity DTPH operating solids or pulp density feed size as F80 and maximum chunk size product size as P80 and maximum and finally the type of\n\nCement Mill Power Calculation\n\nCement mill volume calculation crusherball chges calculators the cement grinding office cement ball mill power calculation forum j is the volume load filling degree in enterprises consumers grinding media have a question about right choise the grinding b Calculations for design of ball mills for cement …\n\nVolume loading charge calculation for cement mill In this paper a new approach for the calculation of the power draw of cement grinding ball mills is proposed For this purpose cement grinding circuit data Live Chat; PDF A Comparison Of Wear Rates Of Ball Mill Grinding A comparison of wear rates of ball mill grinding Journal of\n\ncement mill volume calculation\n\nvolume loading charge calculation for cement mill volume loading charge calculation for cement mill Volume load calculators The Cement Grinding Office The measurement of the ball charge volume load or filling degree is essential to maintain the absorbed power of the mill and consequently the mill price\n\nper ton power consumption of cement mill calculation\n\nCement mill notebookLinkedIn SlideShare Jan 07 2021 · Example Two compartment cement mill Diameter = m Length = 12 m Filling degree = 196 tons at 30 filling degree Mill speed = rpm C = from the figure Then K = 196 = KW Calculation of the Specific Consumption of Energy per Ton of Clinker The\n\ncalculation for cement mill\n\ncalculation of production of cement millCement Mill Grinding Media Wear Rate Calculation wear rate value on cement mill grinding media per ton cement Vertical r 27 Division mirpur 12 pallbi Email [email protected] Careers; Help Desk; Login; 24/7 Phone Services 555 666 99 00\n\nCalculation Of Circulating Laod Of Mill Separator\n\nCalculation For Cement Mill Protable Plant circulating load of dynamic separator in cement ball mill calculation photo of cement mill mill trunnion load calculation of grinding media in cement mill…\n\nCement Mill Notebook Mill Grinding Steel\n\nRaw mills usually operate at 72 74% critical speed and cement mills at 74 76% Calculation of the Critical Mill Speed G weight of a grinding ball in kg w Angular velocity of the mill tube in radial/second w = 2 n/60 Di inside mill diameter in meter effective mill\n\nBall Mill Design/Power Calculation\n\n19 06 2021· Ball Mill Power/Design Calculation Example #2 In Example it was determined that a 1400 HP wet grinding ball mill was required to grind 100 TPH of material with a Bond Work Index of 15 guess what mineral type it is from 80% passing ¼ inch to 80% passing 100 mesh in closed circuit\n\nmotor rating calculation in cement mill\n\nMotor Rated KW calculation Cement Mill Separator fan calculation of the power draw of dry multi compartment ball mills May 6 2021calculate the power that each ball mill compartment should s method has been widely used the required link to classification\n\ncement mill separator efficiency calculation\n\n24/11/2021 · Cement Mill Separator Efficiency Cement Mill Separator Efficiency Calculation Ball mills with high efficiency separators have been used for cement grinding in cement plants all these mill is a cylinder rotating at about 70 80% of critical speed on two trunnions in white metal bearings or slide shoe bearings for large capacity mills\n\nFalse Air Calculation In Cement Mill\n\nHow To Read A Portland Cement Mill Test Report 2018222Portland Cement Mill Test Report ASTM C150 Portland Cement Specification The standard Bogue calculation refers to cement clinker rather than cement Air permeability test that is translated to Specific Surface of the cement particles in m 2 g or cm 2 g Details\n\ncalculation of production of cement mill\n\nThe Bogue calculation of cement mineral composition Get Price INDUSTRIAL CASE STUDY THE CEMENT Input expected cement mill running hour 160 Input expected packer running hour 165 Print new available cement produced 170 Calculate raw material cost Get Price\n\ncalculation of load torque in cement balls mills\n\nCalculation Of Load Torque In Cement Ball Mill Mining Conveyor speed torque required calculation excel sheet Cement Ball Mill Power Calculation has the 2021 ball mill capacity load calculation of;Grinding Mill Calculation In Ball mill Wikipedia A ball mill is a type of grinder used to grind and blend materials for use in mineral dressing processes paints\n\ncement milling calculation examples\n\nCement Plant Operations Handbook 5Advertisers Preview Cement Milling 69 1 Clinker Storage2 Cement Milling3 Separators4 Ball Mill Circuit Control5 Cement Storage6 Cement Dispatch Section BProcess Calculations and Miscellaneous Data B1 Power 228 B2 Fans and Air Handling 230 B3 Conveying 237 B4 Milling 240 B5 Kilns and Burning 246 B6\n\ncement mill separator efficiency calculation\n\nhome; cement mill separator efficiency calculation; cement mill separator efficiency calculation We have How To Calculate Cement Mill Separator EfficiencySeparation air at separator of cement mill qcm how to calculate the power for a tph cement mill ball mill air separator efficiency formula cement ball cyclonic separation wikipedia a cyclonic separation is a method of removing particulates" ]

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[ "", null, "# Queue Notes for GATE\n\n### Introduction to Queue – Study Notes for GATE\n\nThe Queue is an important topic that comes under the Computer Science family. And, when we talk about competitive examinations like GATE, you have to delve deeply into every aspect of the Queue. This article will enlighten you with in-depth information about queues. We hope the information provided in the notes for the CSE topics will help you understand this topic in a better way.\n\nWhat is a Queue?\n\nThe Queue is a linear data structure or an abstract data type. Queue follows the FIFO – “first in, first out” method to process the data. The data which is inserted first will be accessed first. Unlike Stack, Queue has two ends REAR and FRONT. The REAR end is always used to insert the data i.e., enqueue, and the FRONT end is used to remove the data inserted i.e. dequeue.\n\nRepresentation of Queue\n\nLinear array is the best and easiest way to represent the Queue. The Queue contains two ends, REAR and FRONT. These ends point to the position from where the insertion and deletion take place. The REAR points to the variable from where the insertion takes place and the FRONT points to the variable from where deletion takes place.", null, "Basic Operations in Queue\n\nThe enqueue and dequeue are the two basic operations performed in Queue.\n\nEnqueue means to insert any data into the Queue. The steps to insert any data or to enqueue the Queue are as follows:\n\n1. First, we need to check if the Queue is full.\n2. If the Queue is full then show the Queue overflow message and exit.\n3. If it is not full then the REAR pointer will be incremented to point to the next empty space.\n4. Now add the data where the REAR is pointing.", null, "Algorithm for Enqueue:", null, "Dequeue means to remove any data from Queue. The steps to remove any data from the Queue are as follows:\n\n1. First, you need to check if the Queue is empty.\n2. If the Queue is empty, display the Queue underflow message.\n3. If the Queue is not empty, then find the data where FRONT is pointing.\n4. Now increase the FRONT pointer to point to the available data.", null, "Algorithm for Dequeue:", null, "Features of Queue:\n\n1. Queue works with FIFO, i.e., First in First Out methodology.\n2. Unlike Stack, Queue is also considered as the ordered list of the data that has a similar data type.\n3. If any new data is to be inserted in the Queue, then all the data is inserted before it needs to be removed.\n\n## Applications of Queue\n\nWhen we need to manage any group of data in an ordered manner, where the first inserted data has to be out first, then in such cases Queue needs to be implemented.\n\n1. When we need to serve the request on a single shared device, then the Queue is required. For example a CPU.\n2. It removes the conjunction in the processing of the data as it follows FIFO.\n\n### Practice Problem – Queue\n\nQ. A queue is implemented using a non-circular singly linked list. The queue has a head pointer and a tail pointer, as shown in the figure. Let n denote the number of nodes in the queue. Let enqueue be implemented by inserting a new node at the head, and dequeue be implemented by deletion of a node from the tail.", null, "Which one of the following is the time complexity of the most time-efficient implementation of enqueue and dequeue, respectively, for this data structure?\n\n• (A) θ(1),θ(1)\n• (B) θ(1),θ(n)\n• (C) θ(n),θ(1)\n• (D) θ(n),θ(n)\n\nKeep learning and stay tuned to get the latest updates on GATE Exam along with GATE Eligibility CriteriaGATE 2023GATE Admit CardGATE Syllabus for CSE (Computer Science Engineering)GATE CSE NotesGATE CSE Question Paper, and more." ]

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[ "# Convective Heat Transfer: One Dimensional - 6 Notes | Study Heat Transfer - Mechanical Engineering\n\n## Mechanical Engineering: Convective Heat Transfer: One Dimensional - 6 Notes | Study Heat Transfer - Mechanical Engineering\n\nThe document Convective Heat Transfer: One Dimensional - 6 Notes | Study Heat Transfer - Mechanical Engineering is a part of the Mechanical Engineering Course Heat Transfer.\nAll you need of Mechanical Engineering at this link: Mechanical Engineering\n\n3.4 Thermal insulation\nWe have seen how heat transfer is important in various situations. Previous discussion indicates that we are all the time interested in the flow of the heat from one point to another point. However, there are many systems; in fact it is a part of the system, in which we are interested to minimize the losses through heat transfer. For example, in a furnace we want to have high heat transfer inside the furnace; however we do not want any heat loss through the furnace wall. Thus to prevent the heat transfer from the furnace to the atmosphere a bad heat conductor or a very good heat insulator is required. In case of furnace the wall is prepared by multiple layers of refractory materials to minimize the heat losses. Therefore, wall insulation is required in various process equipment, reactors, pipelines etc. to minimize the heat loss from the system to the environment or heat gain from the environment to the system (like cryogenic systems). However, there are situations in which we want to maximize the losses for example, insulation to electric wires.\n\nThe petroleum conservation research association (PCRA) provides a good database on the properties and applications of industrial thermal insulations . The table 3.2 shows some common insulations used in chemical process industries for various process equipment and pipelines.\n\nTable-3.2: Thermal properties of a few of the insulations being used in the chemical process industries", null, "An interesting application of the heat loss from a surface of some practical significance is found in the case of insulation of cylindrical surfaces like small pipes or electrical wires. In many a cases we desire to examine the variation in heat loss from the pipe with the change in insulation thickness, assuming that the length of the pipe is fixed. As insulation is added to the pipe, the outer exposed surface temperature will decrease, but at the same time the surface area available to the convective heat dissipation will increase. Therefore, it would be interesting to study these opposing effects.", null, "Fig. 3.10: Heat dissipation from an insulated pipe\n\nLet us consider a thick insulation layer which is installed around a cylindrical pipe as shown in fig. 3.10 (equivalent electrical circuit is shownin figure 3.11). Let the pipe radius be R and the insulation radius is r. This (r-R) will represent the thickness of the insulation. If the fluid carried by the pipe is at a temperature T and the ambient temperature is Ta. The insulation of the pipe will alter pipe surface temperature T in the radial direction. That is the temperature of the inner surface of the pipe and the outer surface (below insulation) of the pipe will be different. However, if the thermal resistance offered by the pipe is negligible, it can be considered that the temperature (T) is same across the pipe wall thickness and it is a common insulation case (please refer previous discussion). It can also be assumed that the heat transfer coefficient inside the pipe is very high as compared to the heat transfer coefficient at the outside of the insulated pipe. Therefore, only two major resistances in series will be available (insulation layer and gas film of the ambient).", null, "Fig.3.11: Resistance offered by the insulation and ambient gas film\n\nTherefore,", null, "where, k is the thermal conductivity of the material.\nOn differentiating above equation with respect to r will show that the heat dissipation", null, "reaches a maximum,", null, "", null, "So it is maxima, where the insulation radius is equal to", null, "where, rc denotes the critical radius of the insulation. The heat dissipation is maximum at rc which is the result of the previously mentioned opposing effects.", null, "Fig. 3.12: The critical insulation thickness of the pipe insulator\n\nTherefore, the heat dissipation from a pipe increases by the addition of the insulation. However, above rc the heat dissipation reduces. The same is shown in fig. 3.12.\n\nThe careful analysis of the rc reveals that it is a fixed quantity determined by the thermal properties of the insulator. If R <r, then the initial addition of insulation will increase the heat loss until r =rcand after which it begins to decrease. The same heat dissipation which was at bare pipe radius is again attained at r*. The critical insulation thickness may not always exist for an insulated pipe, if the values of k and h are such that the ratio k/h turns out to be less than R.\n\nIt is clear from the above discussion that the insulation above rc reduces the heat dissipation from the cylindrical surface. However, if we keep on increasing the insulation the cost of insulation also increases. Thus again there are two opposing factors that must be considered to obtain the optimum thickness. It should be calculated that what is the pay-back period, that is in how many years the cost of insulation is recovered by the cost of energy saving.", null, "Fig. 3.13: Optimum insulation thickness\n\nThe optimum insulation thickness (fig. 3.13) can be determined at which the sum of the insulation cost and the cost of the heat loss is minimum.\n\nThe document Convective Heat Transfer: One Dimensional - 6 Notes | Study Heat Transfer - Mechanical Engineering is a part of the Mechanical Engineering Course Heat Transfer.\nAll you need of Mechanical Engineering at this link: Mechanical Engineering", null, "Use Code STAYHOME200 and get INR 200 additional OFF\n\n## Heat Transfer\n\n58 videos|70 docs|85 tests\n\nTrack your progress, build streaks, highlight & save important lessons and more!\n\n,\n\n,\n\n,\n\n,\n\n,\n\n,\n\n,\n\n,\n\n,\n\n,\n\n,\n\n,\n\n,\n\n,\n\n,\n\n,\n\n,\n\n,\n\n,\n\n,\n\n,\n\n;" ]

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[ "# GERGONNE POINT PDF", null, "Author: Gagul Kir Country: Martinique Language: English (Spanish) Genre: Music Published (Last): 5 January 2018 Pages: 224 PDF File Size: 8.5 Mb ePub File Size: 10.93 Mb ISBN: 467-8-85926-889-5 Downloads: 99559 Price: Free* [*Free Regsitration Required] Uploader: Voodook", null, "### Gergonne Point — from Wolfram MathWorld\n\nThe point of intersection of the three lines should be the ordered pair solution to the system of equations which would indicate the existence of the Gergonne Point. Thus Gergonnr and D have to be the same point which contradicts the assumption that there could be two distinct points of concurrency. The Euler Line is the result of connecting the point of concurrency of the orthocenter, centroid, and circumcenter.", null, "The following sketch shows the Fergonne Line with the Gergonne Point. This is a very informal way to illustrate concurrency – go to the next section for more rigor using Ceva’s Theorem.\n\nCan you prove this for any triangle? This essay will prove the existence of this point for any triangle, explore its relationship to the Euler line, if any exist, and discuss the possible usefulness of this point.\n\nKVR INSTITUTE FOR SPOKEN ENGLISH PDF\n\nA key question that might be raised by students is whether this point gregonne concurrency occurs for any shape of triangle. These might include some of the following points of concurrency click for a GSP sketch illustration: These linear equations might be explored on software such as Algebra Xpressor.\n\nNow that it gergonnd been shown that the point B is between the other two points on each segment and that the point is on all three segments at the same time, then it must be a point of concurrency for all three segments. A possible relationship to the Euler Line can now be explored fro the Gergonne Point.", null, "Notice the relationship of the triangles in the figure and the greater than or equal to 4 ratio. This time a GSP sketch can polnt examined with a corresponding table of values which demonstrate that the Gergonne Point H is between each segment that connects the vertices to the incenter’s points of tangencies. Can you find any other special qualities of the Gergonne Point?\n\nAlthough there does not seem to be any special theories concerning the Gergonne point itself, the point can be examined in regard to certain ratios of triangles created by the inscribed circle of the incenter. Another approach to showing the existence of the Gergonbe Point is to use GSP to create axes and a grid in order to examine the three linear equations that could be formed from making segments that join the vertices of each triangle to the points of concurrency of the incenter to each side.\n\n1N4732 DATASHEET PDF\n\nThese can be shown similar through alternate interior angles and vertical angles which lead to AA Similarity see figure below.\n\nAssume that there are at least two points of intersection between the lines. Click here for a GSP sketch in which a randomly formed triangle can be examined by using the selection tool to move the vertices to change the shape.\n\n## Gergonne point\n\nCeva’s Theorem proves concurrency for a point by examining similar triangles and certain ratios that equal 1 see picture below. Most geometry students are familiar with the several points of concurrency and the steps necessary to construct such points.", null, "The Gergonne Point, so named after the French mathematician Joseph Gergonne, is the point of concurrency which results from connecting the vertices of a triangle to the opposite points of tangency of the triangle’s incircle." ]

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[ "# 36. A diffusive model of atmospheric heat transport\n\nPosted on April 10th, 2013 in Isaac Held's Blog", null, "Lower panel: the observed (irrotational) component of the horizontal eddy sensible heat flux at 850mb in Northern Hemisphere in January along with the mean temperature field at this level. Middle panel: a diffusive approximation to that flux.  Upper panel:  the spatially varying kinematic diffusivity (in units of", null, "${\\bf 10^6 m^2/s}$) used to generate the middle panel.  From Held (1999) based on Kushner and Held (1998).\n\nLet’s consider the simplest atmospheric model with diffusive horizontal transport on a sphere:", null, "$C \\partial T/\\partial t = \\nabla \\cdot C\\mathcal{D} \\nabla T - (A + B (T-T_0)) + \\mathcal{S}(\\theta)$.\n\nHere", null, "$\\mathcal{S}(\\theta)$ is the energy input into the atmosphere as a function of latitude", null, "$\\theta$,", null, "$A + B(T-T_0)$ is the outgoing infrared flux linearized about some reference temperature", null, "$T_0$,", null, "$C$ is the heat capacity of a tropospheric column per unit horizontal area", null, "$\\approx 8 \\times 10^6 J/( m^2 K)$, and", null, "$\\mathcal{D}$ is a kinematic diffusivity with units of (length)2/time.   Think of the energy input as independent of time and, for the moment, think of", null, "$\\mathcal{D}$ as just a constant.\n\nWe can choose", null, "$T_0$ to be the steady state global mean temperature in some control climate and reinterpret the temperature as the departure from this reference so that", null, "$\\mathcal{S}(\\theta)/C = -\\mathcal{D} \\nabla^2 T + (B/C) T$\n\n(Corrected sign errors — Aug 2013). If we are using this equation to model the time averaged north-south temperature gradients we can think of", null, "$\\mathcal{S}(\\theta)$ as the absorbed solar flux with its global mean removed.  But the equation is linear and we can also think of it as modeling the temperature response to some perturbation in the energy input, for example that due to aerosol forcing or changes in ocean heat uptake or ocean heat redistribution.\n\nWe can talk about an atmospheric radiative relaxation time scale,", null, "$\\tau_R \\equiv C/B$ — which might be 45 days or so if we choose", null, "$B = 2 W/(m^2 K)$ for example — and a diffusive time scale for temperature variations on the length scale", null, "$\\mathcal{L}$ of", null, "$\\tau_D \\equiv \\mathcal{L}^2/\\mathcal{D}$.  For a diffusivity of", null, "$d \\times 10^6 m^2/s$, which we’ll see is the order of magnitude of interest, the two time scales would be equal for", null, "$\\mathcal{L} \\approx 2 \\sqrt{d} \\times 10^6 m$, or about", null, "$20 \\sqrt{d}$  degrees of latitude.  Let’s call this length scale", null, "$\\mathcal{L}_C$.  The atmospheric response to perturbations on scales smaller than", null, "$\\mathcal{L}_C$ would be spread over the distance", null, "$\\mathcal{L}_C$ in this model. If the ocean redistributes heat from latitude A to latitude B, and if A and B are within", null, "$\\mathcal{L}_C$ of each other, we might expect the atmospheric transport to closely compensate for this oceanic transport; if the heating and cooling are more widely separated than", null, "$\\mathcal{L}_C$, the heating/cooling will be balanced more by radiation to space with atmospheric transport playing less of a role.\n\nThe bottom panel in the figure at the top is the eddy sensible heat flux,", null, "$c_p \\overline{{\\bf v}'T'}$, in January at 850 hPa, in the lower troposphere but above the planetary boundary layer, where", null, "${\\bf v}$ is the horizontal wind and a prime denotes the deviation from the mean seasonal cycle — computed from 4 times daily NCEP-NCAR reanalysis.  The overline is a time average over all Januarys.  Most of this flux is associated with midlatitude storms.  Also shown by the contours is the mean temperature field for that month. The black splotches are where the surface protrudes above this pressure surface.\n\n(Actually, before plotting the flux, we decompose it into a a part that has zero divergence on this surface and a part that has zero curl  –this Helmholtz decomposition is unique on the sphere– and retain only the latter part, since we are only interested in the divergence of the flux here.  If you don’t do this, the flux is not as cleanly directed downgradient.)\n\nThe fluxes in the middle panel are generated with the same mean gradients and with the spatially varying diffusivity shown in the upper panel.  The result is evidently in the right ballpark.  The kinematic diffusivity has the dimensions of (length)2/(time), or velocity times length.  One could try to develop a theory for the relevant length and time scales or one could estimate them from observations in different ways.  Here we do the latter, and take the shortcut of just looking at the streamfunction of the flow.  The atmospheric flow is approximately non-divergent in the horizontal, so can be described by a streamfunction", null, "$\\psi$. (Ignoring spherical geometry, the rotational zonal (eastward) component of the wind", null, "$u$ and meridional (poleward) component", null, "$v$ are related to", null, "$\\psi$ by", null, "$(u,v) = (-\\partial \\psi/\\partial y, \\partial \\psi/\\partial x)$.)  So", null, "$\\psi$ has units of velocity times length, the same as kinematic diffusivity.  We compute the standard deviation of the eddy streamfunction,", null, "$\\sigma \\equiv\\sqrt{\\overline{\\psi'^2}}$ and allow ourselves a single constant of proportionality that provides the best fit of the form", null, "$\\overline{{\\bf v}'T'} = - \\alpha \\sigma \\nabla T$  where", null, "$\\alpha$ is uniform in space. (The plot uses", null, "$\\alpha = 0.34$.)  This may seem a bit arcane, but it is just a way to avoid having to estimate length and time scales separately.  This approach was motivated by Holloway 1986, who used this same procedure with satellite data of sea level fluctuations (sea level is proportional to the streamfunction of a geostrophic current) to estimate horizontal transport due to ocean eddies.\n\nA fascinating question for me, ever since I entered the field, is how the magnitude and structure of this diffusivity is determined. (In Held 1999, I discuss why turbulent diffusion might actually be a better approximation for the atmosphere, at least for the transport of sensible heat in the lower troposphere,  than for typical shear or convectively driven turbulence studied in the laboratory.)  We expect this effective diffusivity to change as the climate changes, since the diffusivity must be determined by some aspect of the large-scale environment giving rise to these storms.  In particular, most theories have this diffusivity increasing with the magnitude of the north-south temperature gradient, making it harder to change this gradient than one might otherwise guess.\n\nThe values of the diffusivity in the middle of the oceanic storm tracks rise above", null, "$\\approx 3 x 10^6 m^2/s$. It is the large value in midlatitudes, where north-south temperature gradients are strongest, that are most important for understanding the mean equator-to-pole temperature difference on Earth. A value of", null, "$d \\approx 2-3$  is more or less what you need in this simple diffusive model to get reasonable north-south temperature profiles (see North et al 1981), depending on the vertical level at which you think it’s most appropriate to diffuse the temperature field.  From the previous discussion, we get the sense from this simple diffusive picture that north-south heat transport couples different latitudes within the same hemisphere rather strongly.  In addition to the effective turbulent diffusivity, which is a key to north-south transport, there are strong zonal winds mixing even more strongly in longitude within a hemisphere.  Too local a perspective is a common mistake when first being exposed to the climate change problem — ie, expecting the temperature response to reflect the spatial structure of the CO2 radiative forcing or of the water vapor feedback..\n\nBut my motivation in bringing up this topic is a concern about the opposite tendency to ignore the difficulty that the atmosphere has in communicating temperature responses from extratropical latitudes of one hemisphere to extratropical latitudes of the other. A diffusivity of 2-3 x 106 m2/s, if uniform over the sphere, is not large enough to mix from pole to pole in an atmospheric radiative relaxation time.   The effective diffusivity gets small as one enters the tropics — one can see a bit of this reduction in the figure — seemingly making it harder still to communicate between hemispheres, but this is potentially misleading because the large scale overturning (the “Hadley Cell”) is very efficient at destroying temperature contrasts across the tropics.  This effect is sometimes mimicked in diffusive models by using a large diffusivity in the tropics, which can be confusing since this diffusivity would not be relevant for passive tracers.  In addition the strong tendency for the tropical circulation to wipe out horizontal temperature gradients applies to deep temperature perturbations in the free troposphere, from which the surface can be protected by structure in the atmospheric boundary layer.  In any case, the signal still has to move through the tropics, which provide a large area to radiate it away to space, so the difficulty in getting much of a signal to reach extratropical latitudes in the opposite hemisphere remains.  GCMs provide an essential tool for navigating this complexity.   (But uncertain cloud feedbacks, the familiar wild card when discussing global sensitivity, can also come into play in this problem.)\n\nWhen thinking about aerosol forcing, which is heavily tilted to the Northern Hemisphere, no one is surprised if the response is strongly tilted to the Northern Hemisphere as well.  But consider the concept of (global mean) transient climate response (TCR), discussed in several earlier posts.  The TCR is dependent on the efficiency of heat uptake by the oceans.  Much of this heat uptake occurs in the North Atlantic and in the Southern Ocean.   Consider two models, identical except for the Southern Ocean heat uptake.  The one that warms more slowly in the Southern Ocean will have a smaller TCR, which is fine, but would the warming in the extratropical Northern hemisphere be substantially smaller?  I don’t think so.  I am not aware of a simulation addressing this specific question in the literature.\n\nA paper by Stouffer 2004 (Fig 5 in particular) is informative.  This paper describes very long simulations of the response to doubling and halving of CO2 in a coupled atmosphere-ocean model (5,000 years — long enough for this  model to approach its new equilibrium quite closely ).  In the 2 x CO2 case at year 200 the Southern Hemisphere (SH) as a whole, held back in large part by the Southern Ocean, has reached about 40% of its final temperature response.  Meanwhile the Northern Hemisphere (NH)  has achieved over 80% of its equilibrium response.  Even if all of the NH disequilibrium is due to the lack of warming in the Southern Hemisphere, which is unlikely, there is little room left for the rest of the SH warming to affect the NH — implying that a change in the SH relaxation time would have only a small effect on the NH in this model.\n\nThinking in terms of the global mean temperature in isolation can be valuable and it can also be misleading.  I tried to argue in Post #7 that neither of the usual arguments for focusing on the global mean — reduction in noise and the connection to the global mean energy balance —  is very compelling. (To think about one way in which the energy balance can get divorced from the mean temperature, just make", null, "$B$ in this simple diffusive model a function of latitude.) It is seductive to focus on the global mean temperature response; whenever I do I have to continually remind myself not to be misled into thinking that the Northern and Southern Hemispheres, in particular, are more strongly coupled than they actually are.\n\n(Thanks to Sarah Kang, Paulo Ceppi, Yen-Ting Hwang and Dargan Frierson for discussions on closely related topics.)\n\n[The views expressed on this blog are in no sense official positions of the Geophysical Fluid Dynamics Laboratory, the National Oceanic and Atmospheric Administration, or the Department of Commerce.]\n\n## 3 thoughts on “36. A diffusive model of atmospheric heat transport”\n\n1. Chris Colose says:\n\nIsaac,\n\nThanks for this post.\n\nI think it would also be interesting to change the spatial structure of the heat uptake (e.g., put a “Southern Ocean” in the tropics) to look at TCR and its spatial and time distribution. Are you aware of idealized modeling that has looked at this?\n\n1. Isaac Held says:\n\nThere is quite a bit of work using aqua-planet models coupled to slab oceans, perturbing them with heat input/output from the slab with different patterns. These are not focused specifically on TCR but from a linear perspective this is pretty much the same thing. My own work along these lines, in collaboration with Sarah Kang primarily, has mostly been focused on circulation responses, how extratropical perturbations alter the tropical circulation, rather than surface temperature, ie Kang et al 2008. Looking at transient responses in a fully coupled comprehensive model and then somehow manipulating it to change the distribution of the heat uptake in some region is more difficult.\n\n2. Bill Collinge says:\n\nAs I noted in a comment on one of the previous posts I was impressed by an as yet unpublished (?) paper by your collaborator Kang (with Seager), “Croll Revisited” relating to inter-hemispheric oceanic heat transport. It would be interesting to compare that with the model used in Stouffer 2004." ]

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[ "## Simulated Annealing Tutorial\n\nSimulated annealing copies a phenomenon in nature--the annealing of solids--to optimize a complex system. Annealing refers to heating a solid and then cooling it slowly. Atoms then assume a nearly globally minimum energy state. In 1953 Metropolis created an algorithm to simulate the annealing process. The algorithm simulates a small random displacement of an atom that results in a change in energy. If the change in energy is negative, the energy state of the new configuration is lower and the new configuration is accepted. If the change in energy is positive, the new configuration has a higher energy state; however, it may still be accepted according to the Boltzmann probability factor:\n\n$$P = \\exp\\left(\\frac{-\\Delta E}{k_b T}\\right)$$\n\nwhere kb is the Boltzmann constant and T is the current temperature. By examining this equation we should note two things: the probability is proportional to temperature--as the solid cools, the probability gets smaller; and inversely proportional to --as the change in energy is larger the probability of accepting the change gets smaller.\n\nWhen applied to engineering design, an analogy is made between energy and the objective function. The design is started at a high “temperature”, where it has a high objective (we assume we are minimizing). Random perturbations are then made to the design. If the objective is lower, the new design is made the current design; if it is higher, it may still be accepted according the probability given by the Boltzmann factor. The Boltzmann probability is compared to a random number drawn from a uniform distribution between 0 and 1; if the random number is smaller than the Boltzmann probability, the configuration is accepted. This allows the algorithm to escape local minima.", null, "As the temperature is gradually lowered, the probability that a worse design is accepted becomes smaller. Typically at high temperatures the gross structure of the design emerges which is then refined at lower temperatures.\n\nAlthough it can be used for continuous problems, simulated annealing is especially effective when applied to combinatorial or discrete problems. Although the algorithm is not guaranteed to find the best optimum, it will often find near optimum designs with many fewer design evaluations than other algorithms. (It can still be computationally expensive, however.) It is also an easy algorithm to implement.\n\n#### Example Problem and Source Code\n\nFind the minimum to the objective function\n\n$$obj = 0.2 + x_1^2 + x_2^2 - 0.1 \\, \\cos \\left( 6 \\pi x_1 \\right) - 0.1 \\cos \\left(6 \\pi x_2\\right)$$\n\nby adjusting the values of x_1 and x_2. You can download anneal.m and anneal.py files to retrieve example simulated annealing files in MATLAB and Python, respectively.\n\n## Generate a contour plot\n# Import some other libraries that we'll need\n# matplotlib and numpy packages must also be installed\nimport matplotlib\nimport numpy as np\nimport matplotlib.pyplot as plt\nimport random\nimport math\n\n# define objective function\ndef f(x):\nx1 = x\nx2 = x\nobj = 0.2 + x1**2 + x2**2 - 0.1*math.cos(6.0*3.1415*x1) - 0.1*math.cos(6.0*3.1415*x2)\nreturn obj\n\n# Start location\nx_start = [0.8, -0.5]\n\n# Design variables at mesh points\ni1 = np.arange(-1.0, 1.0, 0.01)\ni2 = np.arange(-1.0, 1.0, 0.01)\nx1m, x2m = np.meshgrid(i1, i2)\nfm = np.zeros(x1m.shape)\nfor i in range(x1m.shape):\nfor j in range(x1m.shape):\nfm[i][j] = 0.2 + x1m[i][j]**2 + x2m[i][j]**2 \\\n- 0.1*math.cos(6.0*3.1415*x1m[i][j]) \\\n- 0.1*math.cos(6.0*3.1415*x2m[i][j])\n\n# Create a contour plot\nplt.figure()\n# Specify contour lines\n#lines = range(2,52,2)\n# Plot contours\nCS = plt.contour(x1m, x2m, fm)#,lines)\n# Label contours\nplt.clabel(CS, inline=1, fontsize=10)\n# Add some text to the plot\nplt.title('Non-Convex Function')\nplt.xlabel('x1')\nplt.ylabel('x2')\n\n##################################################\n# Simulated Annealing\n##################################################\n# Number of cycles\nn = 50\n# Number of trials per cycle\nm = 50\n# Number of accepted solutions\nna = 0.0\n# Probability of accepting worse solution at the start\np1 = 0.7\n# Probability of accepting worse solution at the end\np50 = 0.001\n# Initial temperature\nt1 = -1.0/math.log(p1)\n# Final temperature\nt50 = -1.0/math.log(p50)\n# Fractional reduction every cycle\nfrac = (t50/t1)**(1.0/(n-1.0))\n# Initialize x\nx = np.zeros((n+1,2))\nx = x_start\nxi = np.zeros(2)\nxi = x_start\nna = na + 1.0\n# Current best results so far\nxc = np.zeros(2)\nxc = x\nfc = f(xi)\nfs = np.zeros(n+1)\nfs = fc\n# Current temperature\nt = t1\n# DeltaE Average\nDeltaE_avg = 0.0\nfor i in range(n):\nprint('Cycle: ' + str(i) + ' with Temperature: ' + str(t))\nfor j in range(m):\n# Generate new trial points\nxi = xc + random.random() - 0.5\nxi = xc + random.random() - 0.5\n# Clip to upper and lower bounds\nxi = max(min(xi,1.0),-1.0)\nxi = max(min(xi,1.0),-1.0)\nDeltaE = abs(f(xi)-fc)\nif (f(xi)>fc):\n# Initialize DeltaE_avg if a worse solution was found\n#   on the first iteration\nif (i==0 and j==0): DeltaE_avg = DeltaE\n# objective function is worse\n# generate probability of acceptance\np = math.exp(-DeltaE/(DeltaE_avg * t))\n# determine whether to accept worse point\nif (random.random()<p):\n# accept the worse solution\naccept = True\nelse:\n# don't accept the worse solution\naccept = False\nelse:\n# objective function is lower, automatically accept\naccept = True\nif (accept==True):\n# update currently accepted solution\nxc = xi\nxc = xi\nfc = f(xc)\n# increment number of accepted solutions\nna = na + 1.0\n# update DeltaE_avg\nDeltaE_avg = (DeltaE_avg * (na-1.0) +  DeltaE) / na\n# Record the best x values at the end of every cycle\nx[i+1] = xc\nx[i+1] = xc\nfs[i+1] = fc\n# Lower the temperature for next cycle\nt = frac * t\n\n# print solution\nprint('Best solution: ' + str(xc))\nprint('Best objective: ' + str(fc))\n\nplt.plot(x[:,0],x[:,1],'y-o')\nplt.savefig('contour.png')\n\nfig = plt.figure()\nax1.plot(fs,'r.-')\nax1.legend(['Objective'])\nax2.plot(x[:,0],'b.-')\nax2.plot(x[:,1],'g--')\nax2.legend(['x1','x2'])\n\n# Save the figure as a PNG\nplt.savefig('iterations.png')\n\nplt.show()\n\nclc;\nclear;\nclose all;\n\n%% Generate a contour plot\n% Start location\nx_start = [0.8, -0.5];\n\n% Design variables at mesh points\ni1 = -1.0:0.01:1.0;\ni2 = -1.0:0.01:1.0;\n[x1m, x2m] = meshgrid(i1, i2);\nfm = 0.2 + x1m.^2 + x2m.^2 - 0.1*cos(6.0*pi*x1m) - 0.1*cos(6.0*pi*x2m);\n\n% Contour Plot\nfig = figure(1);\n[C,h] = contour(x1m,x2m,fm);\nclabel(C,h,'Labelspacing',250);\ntitle('Simulated Annealing');\nxlabel('x1');\nylabel('x2');\nhold on;\n\n%% Simulated Annealing\n\n% Number of cycles\nn = 50;\n% Number of trials per cycle\nm = 50;\n% Number of accepted solutions\nna = 0.0;\n% Probability of accepting worse solution at the start\np1 = 0.7;\n% Probability of accepting worse solution at the end\np50 = 0.001;\n% Initial temperature\nt1 = -1.0/log(p1);\n% Final temperature\nt50 = -1.0/log(p50);\n% Fractional reduction every cycle\nfrac = (t50/t1)^(1.0/(n-1.0));\n% Initialize x\nx = zeros(n+1,2);\nx(1,:) = x_start;\nxi = x_start;\nna = na + 1.0;\n% Current best results so far\n% xc = x(1,:);\nfc = f(xi);\nfs = zeros(n+1,1);\nfs(1,:) = fc;\n% Current temperature\nt = t1;\n% DeltaE Average\nDeltaE_avg = 0.0;\nfor i=1:n\ndisp(['Cycle: ',num2str(i),' with Temperature: ',num2str(t)])\nxc(1) = x(i,1);\nxc(2) = x(i,2);\nfor j=1:m\n% Generate new trial points\nxi(1) = xc(1) + rand() - 0.5;\nxi(2) = xc(2) + rand() - 0.5;\n% Clip to upper and lower bounds\nxi(1) = max(min(xi(1),1.0),-1.0);\nxi(2) = max(min(xi(2),1.0),-1.0);\nDeltaE = abs(f(xi)-fc);\nif (f(xi)>fc)\n% Initialize DeltaE_avg if a worse solution was found\n%   on the first iteration\nif (i==1 && j==1)\nDeltaE_avg = DeltaE;\nend\n% objective function is worse\n% generate probability of acceptance\np = exp(-DeltaE/(DeltaE_avg * t));\n%             % determine whether to accept worse point\nif (rand()<p)\n% accept the worse solution\naccept = true;\nelse\n% don't accept the worse solution\naccept = false;\nend\nelse\n% objective function is lower, automatically accept\naccept = true;\nend\n\n% accept\nif (accept==true)\n% update currently accepted solution\nxc(1) = xi(1);\nxc(2) = xi(2);\nfc = f(xc);\n\nxa(j,1) = xc(1);\nxa(j,2) = xc(2);\nfa(j) = f(xc);\n% increment number of accepted solutions\nna = na + 1.0;\n% update DeltaE_avg\nDeltaE_avg = (DeltaE_avg * (na-1.0) +  DeltaE) / na;\nelse\nfa(j) = 0.0;\nend\nend\n% cycle)\nfa_Min_Index = find(nonzeros(fa) == min(nonzeros(fa)));\n\nif isempty(fa_Min_Index) == 0\nx(i+1,1) = xa(fa_Min_Index,1);\nx(i+1,2) = xa(fa_Min_Index,2);\nfs(i+1)  = fa(fa_Min_Index);\nelse\nx(i+1,1) = x(i,1);\nx(i+1,2) = x(i,2);\nfs(i+1)  = fs(i);\nend\n% Lower the temperature for next cycle\nt = frac * t;\nfa = 0.0;\nend\n% print solution\ndisp(['Best candidate: ',num2str(xc)])\ndisp(['Best solution: ',num2str(fc)])\nplot(x(:,1),x(:,2),'r-o')\nsaveas(fig,'contour','png')\n\nfig = figure(2);\nsubplot(2,1,1)\nplot(fs,'r.-')\nlegend('Objective')\nsubplot(2,1,2)\nhold on\nplot(x(:,1),'b.-')\nplot(x(:,2),'g.-')\nlegend('x_1','x_2')\n\n% Save the figure as a PNG\nsaveas(fig,'iterations','png')\n\nSave as f.m\n\nfunction obj = f(x)\nx1 = x(1);\nx2 = x(2);\nobj = 0.2 + x1^2 + x2^2 - 0.1*cos(6.0*pi*x1) - 0.1*cos(6.0*pi*x2);\nend", null, "", null, "" ]

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[ "", null, "# Program: Implement Binary search in java using recursive algorithm.\n\nA binary search or half-interval search algorithm finds the position of a specified value (the input \"key\") within a sorted array. In each step, the algorithm compares the input key value with the key value of the middle element of the array. If the keys match, then a matching element has been found so its index, or position, is returned. Otherwise, if the sought key is less than the middle element's key, then the algorithm repeats its action on the sub-array to the left of the middle element or, if the input key is greater, on the sub-array to the right. If the remaining array to be searched is reduced to zero, then the key cannot be found in the array and a special \"Not found\" indication is returned.\n\nEvery iteration eliminates half of the remaining possibilities. This makes binary searches very efficient - even for large collections.\n\nBinary search requires a sorted collection. Also, binary searching can only be applied to a collection that allows random access (indexing).\n\nWorst case performance: O(log n)\n\nBest case performance: O(1)\n\nRecursion is used in this algorithm because with each pass a new array is created by cutting the old one in half. The binary search procedure is then called recursively, this time on the new array. Typically the array's size is adjusted by manipulating a beginning and ending index. The algorithm exhibits a logarithmic order of growth because it essentially divides the problem domain in half with each pass.\n\n ```package com.java2novice.algos; public class MyRecursiveBinarySearch { public static int recursiveBinarySearch(int[] sortedArray, int start, int end, int key) { if (start < end) { int mid = start + (end - start) / 2; if (key < sortedArray[mid]) { return recursiveBinarySearch(sortedArray, start, mid, key); } else if (key > sortedArray[mid]) { return recursiveBinarySearch(sortedArray, mid+1, end , key); } else { return mid; } } return -(start + 1); } public static void main(String[] args) { int[] arr1 = {2,45,234,567,876,900,976,999}; int index = recursiveBinarySearch(arr1,0,arr1.length,45); System.out.println(\"Found 45 at \"+index+\" index\"); index = recursiveBinarySearch(arr1,0,arr1.length,999); System.out.println(\"Found 999 at \"+index+\" index\"); index = recursiveBinarySearch(arr1,0,arr1.length,876); System.out.println(\"Found 876 at \"+index+\" index\"); } } ```\n\n Output: ```Found 45 at 1 index Found 999 at 7 index Found 876 at 4 index ```" ]

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[ "# What is 330 Cubic Feet in Tablespoons?\n\n## Convert 330 Cubic Feet to Tablespoons\n\nTo calculate 330 Cubic Feet to the corresponding value in Tablespoons, multiply the quantity in Cubic Feet by 1915.0129870073 (conversion factor). In this case we should multiply 330 Cubic Feet by 1915.0129870073 to get the equivalent result in Tablespoons:\n\n330 Cubic Feet x 1915.0129870073 = 631954.28571241 Tablespoons\n\n330 Cubic Feet is equivalent to 631954.28571241 Tablespoons.\n\n## How to convert from Cubic Feet to Tablespoons\n\nThe conversion factor from Cubic Feet to Tablespoons is 1915.0129870073. To find out how many Cubic Feet in Tablespoons, multiply by the conversion factor or use the Volume converter above. Three hundred thirty Cubic Feet is equivalent to six hundred thirty-one thousand nine hundred fifty-four point two eight six Tablespoons.\n\n## Definition of Cubic Foot\n\nThe cubic foot is a unit of volume, which is commonly used in the United States and the United Kingdom. It is defined as the volume of a cube with sides of one foot (0.3048 m) in length. Cubic feet = length x width x height. There is no universally agreed symbol but lots of abbreviations are used, such as ft³, foot³, feet/-3, etc. CCF is for 100 cubic feet.\n\n## Definition of Tablespoon\n\nIn the United States a tablespoon (abbreviation tbsp) is approximately 14.8 ml (0.50 US fl oz). A tablespoon is a large spoon used for serving or eating. In many English-speaking regions, the term now refers to a large spoon used for serving, however, in some regions, including parts of Canada, it is the largest type of spoon used for eating. By extension, the term is used as a measure of volume in cooking.\n\n## Using the Cubic Feet to Tablespoons converter you can get answers to questions like the following:\n\n• How many Tablespoons are in 330 Cubic Feet?\n• 330 Cubic Feet is equal to how many Tablespoons?\n• How to convert 330 Cubic Feet to Tablespoons?\n• How many is 330 Cubic Feet in Tablespoons?\n• What is 330 Cubic Feet in Tablespoons?\n• How much is 330 Cubic Feet in Tablespoons?\n• How many tbsp are in 330 ft3?\n• 330 ft3 is equal to how many tbsp?\n• How to convert 330 ft3 to tbsp?\n• How many is 330 ft3 in tbsp?\n• What is 330 ft3 in tbsp?\n• How much is 330 ft3 in tbsp?" ]

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[ "# 104768 (number)\n\n104,768 (one hundred four thousand seven hundred sixty-eight) is an even six-digits composite number following 104767 and preceding 104769. In scientific notation, it is written as 1.04768 × 105. The sum of its digits is 26. It has a total of 7 prime factors and 14 positive divisors. There are 52,352 positive integers (up to 104768) that are relatively prime to 104768.\n\n## Basic properties\n\n• Is Prime? No\n• Number parity Even\n• Number length 6\n• Sum of Digits 26\n• Digital Root 8\n\n## Name\n\nShort name 104 thousand 768 one hundred four thousand seven hundred sixty-eight\n\n## Notation\n\nScientific notation 1.04768 × 105 104.768 × 103\n\n## Prime Factorization of 104768\n\nPrime Factorization 26 × 1637\n\nComposite number\nDistinct Factors Total Factors Radical ω(n) 2 Total number of distinct prime factors Ω(n) 7 Total number of prime factors rad(n) 3274 Product of the distinct prime numbers λ(n) -1 Returns the parity of Ω(n), such that λ(n) = (-1)Ω(n) μ(n) 0 Returns: 1, if n has an even number of prime factors (and is square free) −1, if n has an odd number of prime factors (and is square free) 0, if n has a squared prime factor Λ(n) 0 Returns log(p) if n is a power pk of any prime p (for any k >= 1), else returns 0\n\nThe prime factorization of 104,768 is 26 × 1637. Since it has a total of 7 prime factors, 104,768 is a composite number.\n\n## Divisors of 104768\n\n14 divisors\n\n Even divisors 12 2 2 0\nTotal Divisors Sum of Divisors Aliquot Sum τ(n) 14 Total number of the positive divisors of n σ(n) 208026 Sum of all the positive divisors of n s(n) 103258 Sum of the proper positive divisors of n A(n) 14859 Returns the sum of divisors (σ(n)) divided by the total number of divisors (τ(n)) G(n) 323.679 Returns the nth root of the product of n divisors H(n) 7.05081 Returns the total number of divisors (τ(n)) divided by the sum of the reciprocal of each divisors\n\nThe number 104,768 can be divided by 14 positive divisors (out of which 12 are even, and 2 are odd). The sum of these divisors (counting 104,768) is 208,026, the average is 14,859.\n\n## Other Arithmetic Functions (n = 104768)\n\n1 φ(n) n\nEuler Totient Carmichael Lambda Prime Pi φ(n) 52352 Total number of positive integers not greater than n that are coprime to n λ(n) 13088 Smallest positive number such that aλ(n) ≡ 1 (mod n) for all a coprime to n π(n) ≈ 9975 Total number of primes less than or equal to n r2(n) 8 The number of ways n can be represented as the sum of 2 squares\n\nThere are 52,352 positive integers (less than 104,768) that are coprime with 104,768. And there are approximately 9,975 prime numbers less than or equal to 104,768.\n\n## Divisibility of 104768\n\n m n mod m 2 3 4 5 6 7 8 9 0 2 0 3 2 6 0 8\n\nThe number 104,768 is divisible by 2, 4 and 8.\n\n• Arithmetic\n• Deficient\n\n• Polite\n\n• Frugal\n\n## Base conversion (104768)\n\nBase System Value\n2 Binary 11001100101000000\n3 Ternary 12022201022\n4 Quaternary 121211000\n5 Quinary 11323033\n6 Senary 2125012\n8 Octal 314500\n10 Decimal 104768\n12 Duodecimal 50768\n20 Vigesimal d1i8\n36 Base36 28u8\n\n## Basic calculations (n = 104768)\n\n### Multiplication\n\nn×y\n n×2 209536 314304 419072 523840\n\n### Division\n\nn÷y\n n÷2 52384 34922.7 26192 20953.6\n\n### Exponentiation\n\nny\n n2 10976333824 1149968542072832 120479904215886462976 12622438604889992953069568\n\n### Nth Root\n\ny√n\n 2√n 323.679 47.1422 17.9911 10.0936\n\n## 104768 as geometric shapes\n\n### Circle\n\n Diameter 209536 658277 3.44832e+10\n\n### Sphere\n\n Volume 4.81698e+15 1.37933e+11 658277\n\n### Square\n\nLength = n\n Perimeter 419072 1.09763e+10 148164\n\n### Cube\n\nLength = n\n Surface area 6.5858e+10 1.14997e+15 181463\n\n### Equilateral Triangle\n\nLength = n\n Perimeter 314304 4.75289e+09 90731.7\n\n### Triangular Pyramid\n\nLength = n\n Surface area 1.90116e+10 1.35525e+14 85542.7\n\n## Cryptographic Hash Functions\n\nmd5 9665b286a4a05bb2824586886b982bd6 c5917354bbaed70e9419adf6abae023814d19eed 1086b95b344fda474ccdc0fcfbcb909c655d995c20934bc5764999386c729767 2953cb8dfb2802140c254bf32932814cb2430a4eb028def9bf3231e7bef57576ca280b4019337f522bbf3bea03ef370d74074ac30b9a2ed26cddda9c7be63e41 882ebc60476b188fed845879fe8272feda8a95d7" ]

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[ "# Learning Objectives\n\n### Learning Objectives\n\nBy the end of this section, you will be able to do the following:\n\n• Describe and apply the work–energy theorem\n• Describe and calculate work and power\n energy gravitational potential energy joule kinetic energy mechanical energy potential energy power watt work work–energy theorem\n\n# The Work–Energy Theorem\n\n### The Work–Energy Theorem\n\nIn physics, the term work has a very specific definition. Work is application of force, $ff$, to move an object over a distance, d, in the direction that the force is applied. Work, W, is described by the equation\n\n$W=fd.W=fd.$\n\nSome things that we typically consider to be work are not work in the scientific sense of the term. Let’s consider a few examples. Think about why each of the following statements is true.\n\n• Homework is not work.\n• Lifting a rock upwards off the ground is work.\n• Carrying a rock in a straight path across the lawn at a constant speed is not work.\n\nThe first two examples are fairly simple. Homework is not work because objects are not being moved over a distance. Lifting a rock up off the ground is work because the rock is moving in the direction that force is applied. The last example is less obvious. Recall from the laws of motion that force is not required to move an object at constant velocity. Therefore, while some force may be applied to keep the rock up off the ground, no net force is applied to keep the rock moving forward at constant velocity.\n\nWork and energy are closely related. When you do work to move an object, you change the object’s energy. You (or an object) also expend energy to do work. In fact, energy can be defined as the ability to do work. Energy can take a variety of different forms, and one form of energy can transform to another. In this chapter we will be concerned with mechanical energy, which comes in two forms: kinetic energy and potential energy.\n\n• Kinetic energy is also called energy of motion. A moving object has kinetic energy.\n• Potential energy, sometimes called stored energy, comes in several forms. Gravitational potential energy is the stored energy an object has as a result of its position above Earth’s surface (or another object in space). A roller coaster car at the top of a hill has gravitational potential energy.\n\nLet’s examine how doing work on an object changes the object’s energy. If we apply force to lift a rock off the ground, we increase the rock’s potential energy, PE. If we drop the rock, the force of gravity increases the rock’s kinetic energy as the rock moves downward until it hits the ground.\n\nThe force we exert to lift the rock is equal to its weight, w, which is equal to its mass, m, multiplied by acceleration due to gravity, g.\n\n$f=w=mgf=w=mg$\n\nThe work we do on the rock equals the force we exert multiplied by the distance, d, that we lift the rock. The work we do on the rock also equals the rock’s gain in gravitational potential energy, PEe.\n\n$W=PEe=fmgW=PEe=fmg$\n\nKinetic energy depends on the mass of an object and its velocity, v.\n\n$KE=12mv2KE=12mv2$\n\nWhen we drop the rock the force of gravity causes the rock to fall, giving the rock kinetic energy. When work done on an object increases only its kinetic energy, then the net work equals the change in the value of the quantity$12mv212mv2$. This is a statement of the work–energy theorem, which is expressed mathematically as\n\nThe subscripts 2 and 1 indicate the final and initial velocity, respectively. This theorem was proposed and successfully tested by James Joule, shown in Figure 9.2.\n\nDoes the name Joule sound familiar? The joule (J) is the metric unit of measurement for both work and energy. The measurement of work and energy with the same unit reinforces the idea that work and energy are related and can be converted into one another. 1.0 J = 1.0 N∙m, the units of force multiplied by distance. 1.0 N = 1.0 k∙m/s2, so 1.0 J = 1.0 k∙m2/s2. Analyzing the units of the term (1/2)mv2 will produce the same units for joules.\n\nFigure 9.2 The joule is named after physicist James Joule (1818–1889). (C. H. Jeens, Wikimedia Commons)\n\n### Watch Physics\n\n#### Work and Energy\n\nThis video explains the work energy theorem and discusses how work done on an object increases the object’s KE.\n\nGrasp Check\n\nTrue or false—The energy increase of an object acted on only by a gravitational force is equal to the product of the object's weight and the distance the object falls.\n\n1. True\n2. False\n\n# Calculations Involving Work and Power\n\n### Calculations Involving Work and Power\n\nIn applications that involve work, we are often interested in how fast the work is done. For example, in roller coaster design, the amount of time it takes to lift a roller coaster car to the top of the first hill is an important consideration. Taking a half hour on the ascent will surely irritate riders and decrease ticket sales. Let’s take a look at how to calculate the time it takes to do work.\n\nRecall that a rate can be used to describe a quantity, such as work, over a period of time. Power is the rate at which work is done. In this case, rate means per unit of time. Power is calculated by dividing the work done by the time it took to do the work.\n\n$P=WtP=Wt$\n\nLet’s consider an example that can help illustrate the differences among work, force, and power. Suppose the woman in Figure 9.3 lifting the TV with a pulley gets the TV to the fourth floor in two minutes, and the man carrying the TV up the stairs takes five minutes to arrive at the same place. They have done the same amount of work $(fd)(fd)$ on the TV, because they have moved the same mass over the same vertical distance, which requires the same amount of upward force. However, the woman using the pulley has generated more power. This is because she did the work in a shorter amount of time, so the denominator of the power formula, t, is smaller. (For simplicity’s sake, we will leave aside for now the fact that the man climbing the stairs has also done work on himself.)\n\nFigure 9.3 No matter how you move a TV to the fourth floor, the amount of work performed and the potential energy gain are the same.\n\nPower can be expressed in units of watts (W). This unit can be used to measure power related to any form of energy or work. You have most likely heard the term used in relation to electrical devices, especially light bulbs. Multiplying power by time gives the amount of energy. Electricity is sold in kilowatt-hours because that equals the amount of electrical energy consumed.\n\nThe watt unit was named after James Watt (1736–1819) (see Figure 9.4). He was a Scottish engineer and inventor who discovered how to coax more power out of steam engines.\n\nFigure 9.4 Is James Watt thinking about watts? (Carl Frederik von Breda, Wikimedia Commons)\n\n### Watch Physics\n\n#### Watt's Role in the Industrial Revolution\n\nThis video demonstrates how the watts that resulted from Watt's inventions helped make the industrial revolution possible and allowed England to enter a new historical era.\n\nGrasp Check\n\nWhich form of mechanical energy does the steam engine generate?\n\n1. Potential energy\n2. Kinetic energy\n3. Nuclear energy\n4. Solar energy\n\nBefore proceeding, be sure you understand the distinctions among force, work, energy, and power. Force exerted on an object over a distance does work. Work can increase energy, and energy can do work. Power is the rate at which work is done.\n\n### Worked Example\n\n#### Applying the Work–Energy Theorem\n\nAn ice skater with a mass of 50 kg is gliding across the ice at a speed of 8 m/s when her friend comes up from behind and gives her a push, causing her speed to increase to 12 m/s. How much work did the friend do on the skater?\n\n### Strategy\n\nThe work–energy theorem can be applied to the problem. Write the equation for the theorem and simplify it if possible.\n\nSolution\n\nIdentify the variables. m = 50 kg,\n\n9.1$v2=12ms, andv1=8msv2=12ms, andv1=8ms$\n\nSubstitute.\n\n9.2\nDiscussion\n\nWork done on an object or system increases its energy. In this case, the increase is to the skater’s kinetic energy. It follows that the increase in energy must be the difference in KE before and after the push.\n\n#### Tips For Success\n\nThis problem illustrates a general technique for approaching problems that require you to apply formulas: Identify the unknown and the known variables, express the unknown variables in terms of the known variables, and then enter all the known values.\n\n# Practice Problems\n\n### Practice Problems\n\nHow much work is done when a weightlifter lifts a $200N$ barbell from the floor to a height of $2m$?\n1. $0J$\n2. $100J$\n3. $200J$\n4. $400J$\n\nIdentify which of the following actions generates more power. Show your work.\n\n• carrying a $100N$ TV to the second floor in $50s$ or\n• carrying a $24N$ watermelon to the second floor in $10s$?\n1. Carrying a $100N$ TV generates more power than carrying a $24N$ watermelon to the same height because power is defined as work done times the time interval.\n2. Carrying a $100N$ TV generates more power than carrying a $24N$ watermelon to the same height because power is defined as the ratio of work done to the time interval.\n3. Carrying a $24N$ watermelon generates more power than carrying a $100N$ TV to the same height because power is defined as work done times the time interval.\n4. Carrying a $24N$ watermelon generates more power than carrying a $100N$ TV to the same height because power is defined as the ratio of work done and the time interval.\n\nExercise 1\nIdentify two properties that are expressed in units of joules.\n1. work and force\n2. energy and weight\n3. work and energy\n4. weight and force\nExercise 2\n\nWhen a coconut falls from a tree, work W is done on it as it falls to the beach. This work is described by the equation\n\n9.5\n\nIdentify the quantities F, d, m, v1, and v2 in this event.\n\n1. F is the force of gravity, which is equal to the weight of the coconut, d is the distance the nut falls, m is the mass of the earth, v1 is the initial velocity, and v2 is the velocity with which it hits the beach.\n2. F is the force of gravity, which is equal to the weight of the coconut, d is the distance the nut falls, m is the mass of the coconut, v1 is the initial velocity, and v2 is the velocity with which it hits the beach.\n3. F is the force of gravity, which is equal to the weight of the coconut, d is the distance the nut falls, m is the mass of the earth, v1 is the velocity with which it hits the beach, and v2 is the initial velocity.\n4. F is the force of gravity, which is equal to the weight of the coconut, d is the distance the nut falls, m is the mass of the coconut, v1 is the velocity with which it hits the beach, and v2 is the initial velocity." ]

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[ "# Spearman's rho - overview\n\nThis page offers structured overviews of one or more selected methods. Add additional methods for comparisons by clicking on the dropdown button in the right-hand column. To practice with a specific method click the button at the bottom row of the table\n\nSpearman's rho\nChi-squared test for the relationship between two categorical variables\nVariable 1Independent /column variable\nOne of ordinal levelOne categorical with $I$ independent groups ($I \\geqslant 2$)\nVariable 2Dependent /row variable\nOne of ordinal levelOne categorical with $J$ independent groups ($J \\geqslant 2$)\nNull hypothesisNull hypothesis\nH0: $\\rho_s = 0$\n\n$\\rho_s$ is the unknown Spearman correlation in the population. The Spearman correlation is a measure for the strength and direction of the monotonic relationship between two variables of at least ordinal measurement level.\n\nIn words, the null hypothesis would be:\n\nH0: there is no monotonic relationship between the two variables in the population\nH0: there is no association between the row and column variable\n\nMore precisely, if there are $I$ independent random samples of size $n_i$ from each of $I$ populations, defined by the independent variable:\n• H0: the distribution of the dependent variable is the same in each of the $I$ populations\nIf there is one random sample of size $N$ from the total population:\n• H0: the row and column variables are independent\nAlternative hypothesisAlternative hypothesis\nH1 two sided: $\\rho_s \\neq 0$\nH1 right sided: $\\rho_s > 0$\nH1 left sided: $\\rho_s < 0$\nH1: there is an association between the row and column variable\n\nMore precisely, if there are $I$ independent random samples of size $n_i$ from each of $I$ populations, defined by the independent variable:\n• H1: the distribution of the dependent variable is not the same in all of the $I$ populations\nIf there is one random sample of size $N$ from the total population:\n• H1: the row and column variables are dependent\nAssumptionsAssumptions\n• Sample of pairs is a simple random sample from the population of pairs. That is, pairs are independent of one another\nNote: this assumption is only important for the significance test, not for the correlation coefficient itself. The correlation coefficient itself just measures the strength of the monotonic relationship between two variables.\n• Sample size is large enough for $X^2$ to be approximately chi-squared distributed under the null hypothesis. Rule of thumb:\n• 2 $\\times$ 2 table: all four expected cell counts are 5 or more\n• Larger than 2 $\\times$ 2 tables: average of the expected cell counts is 5 or more, smallest expected cell count is 1 or more\n• There are $I$ independent simple random samples from each of $I$ populations defined by the independent variable, or there is one simple random sample from the total population\nTest statisticTest statistic\n$t = \\dfrac{r_s \\times \\sqrt{N - 2}}{\\sqrt{1 - r_s^2}}$\nwhere $r_s$ is the sample Spearman correlation and $N$ is the sample size. The sample Spearman correlation $r_s$ is equal to the Pearson correlation applied to the rank scores.\n$X^2 = \\sum{\\frac{(\\mbox{observed cell count} - \\mbox{expected cell count})^2}{\\mbox{expected cell count}}}$\nwhere for each cell, the expected cell count = $\\dfrac{\\mbox{row total} \\times \\mbox{column total}}{\\mbox{total sample size}}$, the observed cell count is the observed sample count in that same cell, and the sum is over all $I \\times J$ cells\nSampling distribution of $t$ if H0 were trueSampling distribution of $X^2$ if H0 were true\nApproximately the $t$ distribution with $N - 2$ degrees of freedomApproximately the chi-squared distribution with $(I - 1) \\times (J - 1)$ degrees of freedom\nSignificant?Significant?\nTwo sided:\nRight sided:\nLeft sided:\n• Check if $X^2$ observed in sample is equal to or larger than critical value $X^{2*}$ or\n• Find $p$ value corresponding to observed $X^2$ and check if it is equal to or smaller than $\\alpha$\nExample contextExample context\nIs there a monotonic relationship between physical health and mental health?Is there an association between economic class and gender? Is the distribution of economic class different between men and women?\nSPSSSPSS\nAnalyze > Correlate > Bivariate...\n• Put your two variables in the box below Variables\n• Under Correlation Coefficients, select Spearman\nAnalyze > Descriptive Statistics > Crosstabs...\n• Put one of your two categorical variables in the box below Row(s), and the other categorical variable in the box below Column(s)\n• Click the Statistics... button, and click on the square in front of Chi-square\n• Continue and click OK\nJamoviJamovi\nRegression > Correlation Matrix\n• Put your two variables in the white box at the right\n• Under Correlation Coefficients, select Spearman\n• Under Hypothesis, select your alternative hypothesis\nFrequencies > Independent Samples - $\\chi^2$ test of association\n• Put one of your two categorical variables in the box below Rows, and the other categorical variable in the box below Columns\nPractice questionsPractice questions" ]

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[ "/ / / / /\n\n# MSRI-UP 2013: Algebraic Combinatorics\n\n Home Research Topic People Colloquia Research Projects\n\n# Algebraic Combinatorics\n\nThe academic and research portion of the 2013 MSRI-UP will be led by Prof. Rosa Orellana from Dartmouth College. Professor Orellana has supervised over 30 undergraduate student research projects, several of which resulted in senior thesis containing original research. Many of her students have continued their mathematical education in PhD programs.\n\nAlgebraic combinatorics is an area of mathematics that studies objects that have combinatorial and algebraic properties. An example of such object is the ring of symmetric functions. In algebraic combinatorics, we use algebraic methods to answer combinatorial questions, and conversely, apply combinatorial techniques to problems in algebra.\n\nLet", null, "be commuting variables, a polynomial", null, "is symmetric if", null, "=", null, "for all permutations", null, ". The space of all symmetric polynomials forms a ring,", null, ". This simply says that if we multiply two symmetric functions we get another symmetric function.", null, "has several distinguished bases that are indexed by partitions. One of the most important bases is Schur's basis:", null, ". The objective of the summer is to learn about and work on open problems involving symmetric polynomials.\n\nSample research project I.  In 1995 Stanley described a symmetric polynomial associated to a graph. This polynomial is called the symmetric chromatic polynomial. Here the variables represent distinct colors and the monomials in the polynomial correspond to proper colorings. To obtain the symmetric polynomial we sum all monomials corresponding to proper colorings. A hard open problem is a conjecture of Stanley that says that the symmetric polynomial is an invariant of trees. Until recently it was not known if the chromatic symmetric polynomial was also an invariant of unicyclic graphs. Geofrey Scott (Ph.D. student at the Univ. of Michigan) found a method to construct non-isomorphic graphs containing a triangle that have the same chromatic polynomial. Students in this research project will generalize Scott's results to unicylic graphs containing larger cycles and discover properties under which two unicyclic graphs will have the same chromatic polynomial.\n\nSample research project II.  A symmetric polynomial", null, "is called Schur positive if when written as a linear combination of Schur polynomials all the coefficients are nonnegative. Students will study examples of symmetric polynomials of the form", null, "and determine conditions so that such a product is Schur positive." ]

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[ "# Types, prefixes, and relations under prefixes\n\nGet Complete Project Material File(s) Now! »\n\n## Properties of relations under prefixes\n\nInstantiation in ML consists of applying a substitution: given a type scheme σ = ∀ (α¯) , its instances are of the form θ(τ ) for a substitution θ whose domain is included in α¯. This definition of ML instantiation implies some useful properties, which we expect to hold in MLF too.\nIn Section 2.1, we establish a few properties based on the observation of the skele-tons. In ML, a type σ and its instances have comparable structures, that is, comparable skeletons. More precisely, the skeleton of the instance θ(τ ) corresponds to the skeleton of σ where quantified variables α¯ are substituted by θ. Hence, the skeleton of θ(τ ) is the skeleton of σ, except on occurrences corresponding to quantified variables. In MLF, we define a partial order 6/ on occurrences (or, equivalently, on skeletons) that captures the idea that only quantified variables can be instantiated. This result is stated as Prop-erty 2.1.3.ii, which concerns the instance relation. Conversely, the abstraction relation is, intuitively, a reversible relation. Hence, we expect it to keep skeletons unchanged; this is stated by Property 2.1.3.i. Another property of the ML instance relation con-cerns free variables: if α is free in σ, then α is free in all instances of σ (and more precisely, α appears at least at the same occurrences). Such a result also holds in MLF, as stated by Lemma 2.1.4, which can be first read by taking Q1 = ∅. A straightforward result in ML states that monotypes have no instances (except themselves). This is also true in MLF, up to equivalence, as stated by Lemma 2.1.6.\nIn some implementations of ML, such as OCaml, some type variables become aliases during unification. For example, the unification of α → α and β → β might create an alias from α to β, which we write (α = β) (1). Then each time a type variable is encountered, the implementation calls a function repr which finds its representative. To pursue the example, the representative of α is β and the representative of β is β.\nSuch a mechanism plays quite an important role in the efficiency of unification in ML. However, it is seldom formalized. In MLF, the alias from α to β appears directly in the prefix, in the form (1). In Section 2.2, we define the representative of a variable, according to a given prefix, and establish a few properties. The representative of α in Q is written Q[α], and the bound of the representative (which, intuitively, is the bound meant for α), is written Q(α).\nSection 2.3 provides different ways to transform derivations of equivalence, abstrac-tion, or instance judgments, the goal being to force some invariants. A first set of straightforward restrictions defines restricted derivations; another independent restric-tion, which concerns only the context rule R-Context-R, defines thrifty derivations. Property 2.3.3 shows that derivations that are restricted and thrifty are as expressive as normal derivations. Such restrictions are useful to discard pathological cases, which would shamelessly obfuscate some proofs.\nIn Section 2.4 we study contexts, which represent types with a hole [ ]. We use them in Section 2.5 to define a set of rules that replaces context rules by explicit contexts. More precisely, context rules such as R-Context-R, R-Context-Flexible or R-Context-Rigid are removed, and all other rules mention explicitly a context C. As expected, these new relations are as expressive as the original relations. Another independent set of rules is introduced, that defines the relations −@α¯ and vα¯ . These relations are similar to −@ and v, but seriously restrict rules A-Hyp and I-Hyp. In order to preserve the expressiveness, we introduce new rules, namely A-Up’, A-Alias’, I-Up’, and I-Alias’. The derivations with −@α¯ and vα¯ make the transformations on binders more explicit than with −@ and v. More precisely, whereas the latter only consider unification of binders (rules A-Hyp and I-Hyp), the former distinguish local unification of binders (rules A-Alias’ and I-Alias’), extrusion of binders (rules A-Up’ and I-Up’), and unification of binders with the prefix (rules A-Hyp’ and I-Hyp’).\nIn Section 2.6, we define atomic relations. Atomic relations can decompose an abstraction derivation or an instance derivation into a sequence of atomic and effective transformations. We use atomic decomposition to show some confluence results; indeed, once a derivation is decomposed into a sequence of transformations, confluence can be proved by considering all possible pairs of transformations.\nThe main result of Section 2.7 is Property 2.7.7.i, which shows that the equivalence relation is the symmetric kernel of the instance relation. This result is proved by associating a three-variable polynomial to each type, and by showing that the instance relation strictly decreases the polynomials if and only if it is irreversible (that is, when it is not an equivalence). Polynomials are also used to show that the relation −@ considered as an order on types (under a given prefix) is well-founded. Such a result is used in the following section to show confluence between −@ and v. Indeed, the main result of Section 2.8 is the Diamond Lemma (Lemma 2.8.4), which states the confluence of −@ and v under an unconstrained prefix.\n\n### Projections and instantiation\n\nSkeletons, projections and occurrences were defined formally in section 1.3. We have seen that a skeleton is a tree composed of type variables, type constructors g n (such as the arrow →), and ⊥. Intuitively, only the leaves labeled with ⊥ can be instantiated. This is captured by the following definition, which introduces the notation 6/. Then, we immediately show that this relation is actually a partial order on skeletons.\nDefinition 2.1.1 We define a relation on skeletons, written 6/, as follows: t1 6/ t2 holds if and only if dom(t1) ⊆ dom(t2) and for all u ∈ dom(t1), we have t1 /u 6= t2/u implies t1/u = ⊥.\nThe definition immediately applies to projections since they are isomorphic to skeletons (see Section 1.3.2). By extension, we write σ1 6/ σ2 to mean σ1/ 6/ σ2/.\nProperties 2.1.2\nThe relation 6/ is a partial order.\nIf t1 6/ t2 holds, then for any substitution θ, we have θ(t1) 6/ θ(t2).\nIf t1 6/ t2 holds, then for any skeleton t, we have t[t1 /α] 6/ t[t2/α].\nThese properties are direct consequences of Definition 2.1.1.\nSee details in the Appendix (page 233).\nAs a consequence of Property 2.1.2.i, if we have σ1 6/ σ2 and σ2 6/ σ1 , then σ1/ = σ2/ (that is, proj(σ1 ) = proj(σ2 )). However, this does not imply σ1 = σ2 in general, as shown by example 1.3.1. Actually, it does not imply σ1 ≡ σ2 either, since types contain more information than skeletons.\nProjections, instantiation and free variables\nProjections are stable under abstraction, but not (in general) under instantiation. This is stated more precisely by the following lemma:\nProperties 2.1.3\nIf (Q) σ1 −@ σ2 , then (∀ (Q) σ1)/ = (∀ (Q) σ2 )/.\nIf (Q) σ1 v σ2 , then ∀ (Q) σ1 6/ ∀ (Q) σ2 .\nProof: Property i : It is shown by induction on the derivation of (Q) σ1 −@ σ2.\nCase A-Equiv: By Property 1.5.4.i (page 50).\nCase R-Trans: By induction hypothesis.\nCase A-Hyp: By hypothesis, we have (α = σ1 ) ∈ Q and σ2 is α. We have ∀ (Q) σ2 = ∀ (Q) α ≡ ∀ (Q) σ1 by Eq-Var?. Consequently, ∀ (Q) σ1 ≡ ∀ (Q) σ2 holds. We get the expected result by Property 1.5.4.i (page 50).\nCase R-Context-R: We have σ1 = ∀ (α σ) σ10 and σ2 = ∀ (α σ) σ20. Besides, the premise is (Q, α σ) σ10 −@ σ20. Hence, by induction hypothesis, we have (∀ (Q, α σ)\nσ10)/ = (∀ (Q, α σ) σ20)/, which is the expected result.\nCase R-Context-Rigid: Like for the case R-Context-L in the proof of Prop-erty 1.5.4.i (page 50).\nProperty ii : It is shown by induction on the derivation of (Q) σ1 v σ2 .\nCase I-Abstract: By Property i.\nCase R-Trans: By induction hypothesis and transitivity of 6/ (Property 2.1.2.i).\nCase I-Bot: We have σ = ⊥, thus (∀ (Q) σ1 )/ = ⊥/. We get the expected result by observing that ⊥ 6/ (∀ (Q) σ2 ) always holds.\nCase R-Context-Flexible: We have σ1 = ∀ (α ≥ σ10) σ (1) and σ2 = ∀ (α ≥ σ20)\n(2). The premise is (Q) σ10 v σ20. By induction hypothesis, we know that (∀ (Q) σ10)/ 6\n(∀ (Q) σ20)/, that is ΘQ(proj(σ10))/ 6 ΘQ(proj(σ20))/ (3) by Property 1.3.3.i (page 40). We have\n(∀ (Q) σ1)/= (∀ (Q) ∀ (α ≥ σ10) σ)/ from (1)\nFinally, we have shown ∀ (Q) σ1/ 6/ ∀ (Q) σ2/, which is the expected result.\nCase I-Hyp is similar to case A-Hyp above.\n◦ Case I-Rigid: ∀ (α ≥ σ) σ0 and ∀ (α = σ) σ0 have the same projection.\nConsidering an instance (Q) σ1 v σ2 , we need to track occurrences of variables bound in Q but free in σ1. The following lemma states that, in general, such variables also occur in σ2 , at the same occurrence. We require (α σ) to be in the prefix, because the instance relation is only defined under a prefix that binds all free variables of the judgment. Additionally, we require σ not to be equivalent to a monotype, otherwise it could be equivalently substituted by Rule Eq-Mono, and the result would not hold. As a counterexample, take (Q, α = τ ) α v τ (where α cannot be free in τ ).\nLemma 2.1.4 Assume σ is not in T . If (Q, α σ, Q1 ) σ1 ♦ σ2 and (∀ (Q1 ) σ1 )/u = α hold, then (∀ (Q1) σ2 )/u = α. As a consequence, if α ∈ ftv(∀ (Q1 ) σ1), then α ∈ ftv(∀ (Q1 ) σ2 ).\nThe proof is by induction on the derivation. See the details in Appendix (page 234).\nBy Definition 1.5.7, a type σ is in T if its normal form has no quantifiers. Like in ML, this means that monotypes cannot be instantiated. Monotypes play an important role in MLF, since they can be inferred (like in ML), whereas polymorphic types can only be propagated. Throughout the proofs, we need to characterize monotypes, variables, or ⊥. Definition 1.5.7 uses normal forms. The following properties use projection.\nProperties 2.1.5\nσ ∈ T iff for all u ∈ dom(σ) we have σ/u 6= ⊥.\nWe have σ/ = α iff σ ≡ α.\nWe have σ/ = ⊥ iff σ ≡ ⊥.\nSee proof in the Appendix (page 235).\nThe following lemma is essential: it shows that any instance of a monotype τ is only equivalent to τ , even under a prefix. In particular, assume σ is some polymorphic type scheme. If we have (α = σ) α v σ0, then σ0 must be α (when put in normal form). In other words, even if α is rigidly bound to a polytype σ in the prefix, there is no way to take an instance of it other than α itself. This is also why we say that the information (α = σ) is hidden in the prefix: we can propagate the variable α, bound to σ, but we cannot use its polymorphism. Decidability of type inference relies on this result.\nLemma 2.1.6 If σ1 ∈ T and (Q) σ1 ♦ σ2 hold, then we have (Q) σ1 ≡ σ2.\nProof: By induction on the derivation of (Q) σ1 ♦ σ2 .\nCase A-Equiv: By hypothesis, (Q) σ1 ≡ σ2 holds, which is the expected result.\nCase R-Trans By induction hypothesis, Property 1.5.11.x (page 54) and R-Trans.\nCase A-Hyp and I-Hyp: We have (Q) σ1 v α1 (that is, σ2 is α1 ), and the premise is (α1 σ1) ∈ Q (1). By hypothesis, σ1 ∈ T , thus σ1 ≡ τ (2) for some monotype τ by Property 1.5.11.ii (page 54). By Property 1.5.3.v (page 49) and (2), we get (Q) σ1 ≡\n(3). By (2), Eq-Mono, and (1), we get (Q) τ ≡ α1 (4). Hence, (Q) σ1 ≡ α1 holds by R-Trans, (3), and (4). This is the expected result.\nCase R-Context-R We have σ1 = ∀ (α σ) σ10 and σ2 = ∀ (α σ) σ20. The premise is (Q, α σ) σ10 ♦ σ20 (5). By hypothesis we have ∀ (α σ) σ10 ∈ T . By Property 2.1.5.i,\nwe must have σ10 ∈ T . Hence (Q, α σ) σ10 ≡ σ20 holds by induction hypothesis on (5). Consequently, (Q) ∀ (α σ) σ10 ≡ ∀ (α σ) σ20 holds by R-Context-R and this is the expected result.\nCase R-Context-Rigid and R-Context-Flexible: We have σ1 = ∀ (α σ) σ0 and σ2 = ∀ (α σ0) σ0 . The premise is (Q) σ ♦ σ0 (6). If α ∈/ ftv(σ0 ), then σ1 ≡ σ2 by Eq-Free, which is the expected result. Otherwise, we necessarily have σ and σ0 in T by Property 2.1.5.i. Hence, (Q) σ ≡ σ0 holds by induction hypothesis on (6). Consequently, (Q) σ1 ≡ σ2 holds by R-Context-L.\nCase I-Bot implies σ1 = ⊥, which is a contradiction with the hypothesis σ1 ∈ T , by Property 2.1.5.i.\nCase I-Rigid: we have σ1 = ∀ (α ≥ σ) σ0 and σ2 = ∀ (α = σ) σ0. If α ∈/ ftv(σ0), then σ1 ≡ σ2. Otherwise, we necessarily have σ and σ0 in T by Property 2.1.5.i. Con-sequently, σ ≡ τ for some monotype τ . Hence σ1 ≡ σ0[τ /α] (7) holds by Eq-Mono? as well as σ0[τ /α] ≡ σ2 (8). We get the expected result by R-Trans, (7), (8).\nTypes in MLF are not recursive, thus a type cannot be strictly included in itself. As a consequence, an instance of a given polytype cannot be a strict subterm.\nProperties 2.1.7\nIf σ • / 6/ σ • u/, then u is .\nIf we have either (Q) σ v α or (Q) α v σ and if α ∈ ftv(σ), then σ ≡ α.\nProof: Property i: We have σ • / 6/ σ • u/. By definition, this implies σ/u 6/ σ/uu. Hence uu, that is u2, is in dom(σ). By immediate induction, we show that ui is in dom(σ) for all natural number i. By definition, dom(σ) is dom(proj(σ)), where proj(σ) is a skeleton (that is, a tree). By construction, dom(proj(σ)) is a finite set of occurrences. .\nProperty ii: By hypothesis, we have (Q) σ v α (1), or (Q) α v σ (2), and α ∈ ftv(σ), that is, there exists u such that σ/u = α. If u is , then we get the expected result by Property 2.1.5.ii. Otherwise, we have u 6= (3) and (∀ (Q) σ) • u/ = (∀ (Q) α) • / (4) by definition of occurrences. If (1) holds, we get ∀ (Q) σ/ 6/ ∀ (Q) α/ by Property 2.1.3.ii. Hence, we get (∀ (Q) σ) • / 6/ (∀ (Q) σ) • u/ by (4), and we conclude by Property i that u is . Otherwise, (2) holds, and we get (Q) α ≡ σ by Lemma 2.1.6. Hence,\n∀ (Q) α/ = ∀ (Q) σ/ (5) holds by Property 1.5.4.i (page 50). By (5) and (4), we get (∀ (Q) σ) • u/ = (∀ (Q) σ) • /. This leads to u = by Property i. In both cases, we have u = , which is a contradiction with (3).\n\nREAD  PDZ-peptide binding specificity with polarizable free energy simulations\n\nAbstract / Résumé\nIntroduction\nConventions 3\nI Types\n1 Types, prefixes, and relations under prefixes\n1.1 Syntax of types\n1.2 Préxes\n1.3 Occurrences\n1.3.1 Skeletons\n1.3.2 Projections\n1.3.3 Free type variables and unbound type variables\n1.3.4 Renamings and substitutions\n1.4 Relations under prex\n1.5 Type equivalence\n1.5.1 Rearrangements\n1.5.2 Occurrences and equivalence\n1.5.3 Canonical forms for types\n1.6 The abstraction relation\n1.7 The instance relation\n1.8 Examples\n2 Properties of relations under préxes\n2.1 Projections and instantiation\n2.2 Canonical representatives and bounds in a préx\n2.3 Restricted and thrifty derivations\n2.4 Contexts\n2.5 Local abstraction and instance rules\n2.5.1 Context-based rules\n2.5.2 Alternative abstraction and alternative instance\n2.6 Atomic instances\n2.6.1 Definitions\n2.6.2 Equivalence between relations\n2.7 Equivalence vs instantiation\n2.7.1 Polynomials\n2.7.2 Weight\n2.7.3 Abstraction is well-founded\n2.8 Conuence\n3 Relations between préxes\n3.1 Substitutions\n3.2 Préx instance\n3.3 Domains of péxes\n3.4 Rules for préx equivalence, abstraction, and instance\n3.5 Splitting préxes\n3.6 Préxes and Instantiation\n4 Unification\n4.1 Definition\n4.2 Auxiliary algorithms\n4.3 Unication algorithm\n4.4 Soundness of the algorithm\n4.5 Termination of the algorithm\n4.6 Completeness of the algorithm\nII The programming language\n5 Syntax and semantics\n5.1 Syntax\n5.2 Static semantics\n5.2.1 ML as a subset of MLF\n5.2.2 Examples of typings\n5.3 Syntax-directed presentation\n5.4 Dynamic semantics\n6 Type Safety\n6.1 Standard Properties\n6.1.1 Renaming and substitutions\n6.1.2 Strengthening and weakening typing judgments\n6.1.3 Substitutivity\n6.2 Equivalence between the syntax-directed system and the original system\n6.3 Type safety\n7 Type inference\n7.1 Type inference algorithm\n7.2 Soundness of the algorithm\n7.3 Completeness of the algorithm\n7.4 Decidability of type inference\n8 Type annotations\n8.1 MLF without type annotations\n8.2 Introduction to type annotations\n8.3 Annotation primitives\nIII Expressiveness of MLF\n9 Encoding System F into MLF\n9.1 Definition of System F\n9.2 Encoding types and typing environments\n9.3 Encoding expressions\n10 Shallow MLF\n10.1 Denition and characterization\n10.2 Expressiveness of Shallow MLF\n10.3 Comparison with System F\n10.3.1 Introduction\n10.3.2 Preliminary results about System F\n10.3.3 Encoding shallow types into System F\n10.3.4 Encoding Shallow F into System F\n10.4 Discussion\n11 Language extensions\n11.1 Tuples, Records\n11.2 References\n11.3 Propagating type annotations\n12 MLF in practice\n12.1 Some standard encodings\n12.2 Existential Types\n12.3 When are annotations needed ?\n12.4 A detailed example\n12.5 Discussion\nConclusion\nBibliography\nIV Appendix\nA Proofs (Technical details)\nIndexes\nIndex of rules\nIndex of notations\n\nGET THE COMPLETE PROJECT" ]

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[ "## Open Access Dissertations\n\n1983\n\nDissertation\n\n#### Degree Name\n\nDoctor of Philosophy in Civil and Environmental Engineering\n\n#### Department\n\nCivil and Environmental Engineering\n\nWilliam E. Kelly\n\n#### Abstract\n\nA Monte Carlo technique is utilized to incorporate the uncertainty in media characteristics to the solution of a groundwater flow problem. This technique involves the repetitive solution of a significant number of equiprobable representations of the soil medium. Probability and statistics are utilized to model the soil parameters and study the significance of the output.\n\nAn existing computer code was adapted and significantly modified to allow characterization of the media's hydraulic conductivity (permeability) as autocorrelated and statistically homogeneous. A first order nearest neighbor model was selected to affect the autocorrelation of this parameter within the finite difference mesh. The statistical homogeneity considers that the distribution of hydraulic conductivity values within the mesh comes from a log normal probability density function. The selection of hydraulic conductivity value at any mode of the mesh is stochastic within the framework of the autocorrelation and statistical homogeneity of the mesh aggregate.\n\nThe computer code takes the stochastically generated hydraulic conductivity field and the boundary conditions and utilizing an iterative alternating direct implicit solution determines the hydraulic head values at each mode and flow rate through the medium. An array of these results are produced for each of the equiprobable representations of the soil medium.\n\nMass transport through the region is simulated as a combination of advection and a stochastic simulation of microscopic or particle scale dispersion. A water particle is released from a preselected location along the upgradient boundary. The particle moves toward the downgradient boundary under the influence of advective forces caused by the differences in hydraulic head and the stochastic simulation of microscopic dispersion. This simulation of microscopic dispersion displaces the particle parallel to and perpendicular to the advective transport direction based on laboratory scale dispersivities. The computer code establishes arrays for the particle location at a predetermined time after start as well as the location along the downgradient boundary and total travel time upon completion of transit of the region.\n\nUniform flow results in most of the regions considered although some alternate configurations were considered. An effective hydraulic conductivity is calculated on the basis of flow rate. After application of a shape factor this value was found to be slightly less but closest to the geometric mean of the hydraulic conductivity distribution thus confirming earlier work. An alternative effective hydraulic conductivity calculated on the basis of travel time was also determined. This value was generally less than the other effective hydraulic conductivity value but again after application of a shape factor the value was best estimated by the geometric mean. These results suggest that the mean flow rate and mean travel time may be estimated by the use of the shape factor from a flow net solution or the method of fragments and the geometric means of the hydraulic conductivity.\n\nThe results of the simulations indicate that macroscopic or field scale longitudinal and lateral dispersion is significantly affected by the standard deviation of the hydraulic conductivity distribution. Region size, hydraulic gradient and time interval were found to cause lesser effects.\n\nThe techniques utilized provide a means to develop confidence in the output. The effects of the variations in parameters become evident from a review of the results of the equiprobable results. Confidence limits may even be developed in the output where the characteristics of known probability density functions may be utilized. Example problems are presented where confidence limits on the estimates of travel time are developed for the conditions considered.\n\nCOinS" ]

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[ "Try NerdPal! Our new app on iOS and Android\n\n# Find the derivative $\\frac{d}{dx}\\left(x^2\\right)$ using the power rule\n\n## Related Videos\n\nGo!\nGo!\n1\n2\n3\n4\n5\n6\n7\n8\n9\n0\na\nb\nc\nd\nf\ng\nm\nn\nu\nv\nw\nx\ny\nz\n.\n(◻)\n+\n-\n×\n◻/◻\n/\n÷\n2\n\ne\nπ\nln\nlog\nlog\nlim\nd/dx\nDx\n|◻|\nθ\n=\n>\n<\n>=\n<=\nsin\ncos\ntan\ncot\nsec\ncsc\n\nasin\nacos\natan\nacot\nasec\nacsc\n\nsinh\ncosh\ntanh\ncoth\nsech\ncsch\n\nasinh\nacosh\natanh\nacoth\nasech\nacsch\n\n### Videos", null, "### Calculus - Using the power rule of logarithms to take the derivative of a natural log, d(ln(x^2))/dx", null, "### Calculus - Using power rule with square root to take derivative on a logarithm, d(ln(sqrt(x+1)))/dx", null, "### Implicit Differentiation - Find The First &amp; Second Derivatives", null, "### The Power Rule For Derivatives", null, "### Calculus - Find the derivative of natural logarithm using product property, d(ln(2x))/dx", null, "SnapXam A2\n\n### beta Got another answer? Verify it!\n\nGo!\n1\n2\n3\n4\n5\n6\n7\n8\n9\n0\na\nb\nc\nd\nf\ng\nm\nn\nu\nv\nw\nx\ny\nz\n.\n(◻)\n+\n-\n×\n◻/◻\n/\n÷\n2\n\ne\nπ\nln\nlog\nlog\nlim\nd/dx\nDx\n|◻|\nθ\n=\n>\n<\n>=\n<=\nsin\ncos\ntan\ncot\nsec\ncsc\n\nasin\nacos\natan\nacot\nasec\nacsc\n\nsinh\ncosh\ntanh\ncoth\nsech\ncsch\n\nasinh\nacosh\natanh\nacoth\nasech\nacsch\n\n$\\frac{d}{dx}\\left(x^2\\right)$\n\n### Main topic:\n\nPower Rule for Derivatives\n\n~ 0.03 s" ]

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[ "# net ionic equation lab\n\nnet ionic equation lab.\n\nAttached following are the lab report form and the solubility table. Complete the document\n\nGiven the reactants, perform either a double or a single displacement reaction and predict the products. Use your solubility table and knowledge of gas forming reactions to determine the state of predicted products. Determine if there was “ No Reaction” from your NIE rules not from your observations. Show the total net ionic equation for all reactions. If all species on both sides of the equation are spectator ions then write “ No Reaction” for the net ionic equation and the “reaction type”.\n\n##### net ionic equation lab\nPosted in Uncategorized" ]

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[ "Vous êtes sur la page 1sur 23", null, "Chapter 3: Stoichiometry\n\nKey Skills:\n\nBalance chemical equations Predict the products of simple combination, decomposition, and combustion reactions. Calculate formula weights Convert grams to moles and moles to grams using molar masses. Convert number of molecules to moles and moles to number of molecules using Avogadro’s number Calculate the empirical and molecular formulas of a compound from percentage composition and molecular weight. Calculate amounts, in grams or moles, of reactants and products for a reaction. Calculate the percent yield of a reaction.\n\nStoichiometry is the study of the quantitative relationships in substances and their reactions\n\n–Chemical equations –The mole and molar mass –Chemical formulas –Mass relationships in equations –Limiting reactant", null, "", null, "Definitions\n\nReactants are the substances consumed\n\nProducts are the substances formed\n\nCoefficients are numbers before the formula of a substance in an equation\n\nA balanced equation has the same number of atoms of each element on both sides of the equation", null, "Chemical Equations\n\n• A chemical equation is a shorthand notation to describe a chemical reaction\n\n– Just like a chemical formula, a chemical equation expresses quantitative relations\n\n• Subscripts tell the number of atoms of each element in a molecule\n\n• Coefficients tell the number of molecules", null, "", null, "", null, "", null, "Anatomy of a Chemical Equation", null, "Reactants appear on the left side of the equation.\n\nProducts appear on the right side of the equation.\n\nThe states of the reactants and products are written in parentheses to the right of each compound.", null, "Writing Balanced Equations\n\nWrite the correct formula for each substance\n\nH 2\n\n+\n\nCl 2\n\nHCl\n\nAdd coefficients so the number of atoms\n\nof each element are the same on both sides of the equation\n\nH 2\n\n+\n\nCl 2\n\n2HCl", null, "Balancing Chemical Equations\n\nAssume one molecule of the most complicated substance\n\nC 5 H 12\n\n+\n\nO 2\n\nCO 2\n\n+\n\nH 2 O\n\nAdjust the coefficient of CO 2 to balance C\n\nC 5 H 12\n\n+\n\nO 2\n\n5CO 2\n\n+\n\nH 2 O\n\nAdjust the coefficient of H 2 O to balance H\n\nC 5 H 12\n\n+\n\nO 2\n\n5CO 2\n\n+\n\n6H 2 O\n\nAdjust the coefficient of O 2 to balance O\n\nC 5 H 12\n\n+\n\n8O 2\n\n5CO 2\n\n+\n\n6H 2 O\n\nCheck the balance by counting the number of atoms of each element.", null, "Balancing Equations\n\nSometimes fractional coefficients are obtained\n\nC\n\nC\n\nC\n\nC 5 H 10 + (15/2)O 2\n\n5 H 10\n\n5 H 10\n\n5 H 10\n\n+\n\n+\n\n+\n\nO\n\nO\n\nO\n\n2\n\n2\n\n2\n\nCO 2\n\n5CO\n\n5CO\n\n2\n\n2\n\n+ H 2 O + H 2 O + 5H 2 O 5CO 2\n\n+ 5H 2 O\n\nMultiply all coefficients by the denominator\n\n2 C 5 H 10\n\n+\n\n15 O 2\n\n10 CO 2\n\n+ 10 H 2 O", null, "", null, "", null, "Decomposition\n\nOne substance breaks down into two or more substances CaCO 3 (s)  CaO (s) + CO 2 (g)\n\n2 KClO 3 (s) \n\n2\n\n2 KCl (s) + O 2 (g)\n\nNaN 3 (s)  2 Na (s) + 3 N 2 (g)", null, "Combustion\n\nIs the process of burning, the combination of an organic substance with oxygen to produce a flame.", null, "When an organic compound burns in oxygen, the carbon reacts with oxygen to form CO 2 , and the hydrogen forms water, H 2 O.\n\nBalance the following combustion reactions:", null, "C 3 H 8 + O 2\n\nCO 2 + H 2 O\n\nCO 2 +\n\n(C 2 H 5 ) 2 O +\n\nO 2\n\nH 2 O", null, "Formula Weight (FW)\n\n• Sum of the atomic weights for the atoms in a chemical formula\n\n• The formula weight of calcium chloride, CaCl 2 , would be\n\nCa: 1(40.08 amu) + Cl: 2(35.45 amu) 110.98 amu\n\n• Formula weights are generally reported for ionic compounds", null, "Molecular Weight (MW)\n\n• Sum of the atomic weights of the atoms in a molecule • For the molecule ethane, C 2 H 6 , the molecular weight would be\n\nC: 2(12.01 amu)\n\n+ H:\n\n6(1.008 amu)\n\n30.07 amu", null, "Percent Composition\n\nOne can find the percentage of the mass of a compound that comes from each of the elements in the compound by using this equation:\n\n(number of atoms)(atomic weight)\n\n% element =\n\n(FW of the compound)\n\nx 100%", null, "Percent Composition\n\nSo the percentage by mass of carbon in ethane (C 2 H 6 ) is…\n\n%C =\n\n=\n\n(2)(12.01 amu)\n\n(30.068 amu)\n\nx 100\n\n24.02 amu\n\nx 100 30.068 amu\n\n= 79.89%", null, "The Mole\n\nOne mole is the amount of substance that contains as many entities as the number of atoms in exactly 12 grams of the 12 C isotope of carbon.\n\nAvogadro’s number is the experimentally determined number of atoms in 12 g of isotopically pure 12 C, and is equal to 6.022 x 10 23\n\nOne mole of anything contains 6.022 x 10 23 entities\n\n1 mol H = 6.022 x 10 23 atoms of H\n\n1 mol H 2 = 6.022 x 10 23 molecules of H 2\n\n1 mol CH 4 = 6.022 x 10 23 molecules of CH 4\n\n1 mol CaCl 2 = 6.022 x 10 23 formula units of CaCl 2", null, "One Mole", null, "Moles to Number of Entities\nNumber of\nMoles of\natoms or\nsubstance\nnumber\nmolecules\nExample Calculations\n• How many Na atoms are present in\n0.35 mol of Na?\n• How many moles of C 2 H 6 are present\nin 3.00 x 10 21 molecules of C 2 H 6 ?", null, "", null, "Molar Mass\n\nThe molar mass (M) of any atom, molecule or compound is the mass (in grams) of one mole of that substance.\n\n– The molar mass in grams is numerically equal to the atomic mass or molecular mass expressed in u (or amu).\n\nAtomic Scale\n\nSubstance\n\nName\n\nMass\n\nLab Scale\n\nMolar Mass\n\n Ar atomic mass 39.95 u 39.95 g/mol C 2 H 6 molecular mass 30.07 u 30.07 g/mol NaF formula mass 41.99 u 41.99 g/mol\n\nWhat mass of compound must be weighed out, to have a 0.0223 mol sample of H 2 C 2 O 4 (M = 90.04 g/mol)?", null, "Oxalic acid", null, "Interconverting mass and\nnumber of formula units", null, "Example Calculation\n\nWhat is the mass of 0.25 moles of CH 4 ?\n\n0.25 mol CH", null, "4", null, " 16.0 g CH \n4\n1 mol CH\n4\n\n4.0 g CH\n\n4", null, "Empirical formula", null, "Example 1: What is the empirical formula of\n\na compound that contains 0.799 g C and 0.201 g H in a 1.000 g sample?\n\nExample 2: What is the empirical formula of\n\na chromium oxide that is 68.4% Cr by\n\nmass?", null, "Combustion Analysis\n\n• Compounds containing C, H and O are routinely analyzed through combustion in a chamber like this\n\n– C is determined from the mass of CO 2 produced\n\n– H is determined from the mass of H 2 O produced\n\n– O is determined by difference after the C and H have been determined", null, "Finding C and H content\n\n• A weighed sample of compound is burned, and the masses of H 2 O and CO 2 formed is measured.", null, "", null, "", null, "Calculating Empirical Formulas\n\nExample: The compound para-aminobenzoic acid (you may have seen it listed as PABA on your bottle of sunscreen) is composed of carbon (61.31%), hydrogen (5.14%), nitrogen (10.21%), and oxygen (23.33%). Find the empirical formula of PABA.\n\nAssuming 100.00 g of para-aminobenzoic acid,\n\n C: 61.31 g x H: 5.14 g x N: 10.21 g x O: 23.33 g x\n\n1 mol\n\n12.01 g\n\n1 mol\n\n1.01 g\n\n1 mol\n\n14.01 g\n\n1 mol\n\n16.00 g\n\n= 5.105 mol C\n\n= 5.09 mol H\n\n= 0.7288 mol N\n\n= 1.456 mol O", null, "Calculating Empirical Formulas\n\nCalculate the mole ratio by dividing by the smallest number of moles:\n\nC:\n\nH:\n\nN:\n\nO:", null, "5.105\nmol\n0.7288 mol\n5.09 mol\n0.7288\nmol\n0.7288\nmol\n0.7288\nmol\n1.458\nmol\n0.7288\nmol\n\n= 7.005 7\n\n= 6.984 7\n\n= 1.000\n\n= 2.001 2", null, "C 7 H 7 NO 2", null, "Example Calculation\n\nA compound contains only C, H, and O. A 0.1000 g-sample burns completely in oxygen to form 0.0930 g water and 0.2271 g CO 2 . Calculate the mass of each element in this sample. What is the empirical formula of the compound?", null, "Comparison\n\nFormula to mass percent\n\nMass percent to formula\n\nSubscripts in\n\nformula\n\nComposition (mass or mass %)\n\nMoles of\n\neach element\n\nMolar masses\n\nof elements\n\nAtomic masses\n\nMasses of\n\nelements and\n\ncompound\n\nMass of element\n\nMass of compound x 100%\n\nDivide by\n\nsmallest number\n\nPercent\n\ncomposition\n\nEmpirical\n\nformula", null, "Mole Relationships in Equations", null, "Guidelines for Reaction Stoichiometry\n\nWrite the balanced equation.\n\nCalculate the number of moles of the species for which the mass is given.\n\nUse the coefficients in the equation to convert the moles of the given substance into moles of the substance desired.\n\nCalculate the mass of the desired species.", null, "", null, "Example Calculation\n\nGiven the reaction\n\n4FeS 2 + 11 O 2 2Fe 2 O 3 + 8SO 2\n\nWhat mass of SO 2 is produced from reaction of 3.8 g of FeS 2 and excess O 2 ?", null, "Example Calculation\n\nWhat mass of SO 3 forms from the reaction of 4.1 g of SO 2 with an excess of O 2 ?", null, "Reaction Yields\n\nActual yield is found by measuring the quantity of product formed in the experiment.\n\nTheoretical yield is calculated from reaction stoichiometry.\n\n% yield =\n\nActual yield Theoretical yield 100%", null, "Example: Calculating Percent Yield\n\nA 10.0 g-sample of potassium bromide is treated with perchloric acid solution. The reaction mixture is cooled and solid KClO 4 is removed by filtering, then it is dried and weighed.\n\nKBr (aq)+HClO 4 (aq) KClO 4 (s)+HBr (aq)\n\nThe product weighed 8.8 g. What was the percent yield?", null, "Limiting Reactant\n\nLimiting reactant : the reactant that is completely consumed in a reaction. When it is used up, the reaction stops, thus limiting the quantities of products formed.\n\nExcess reactant : the other reactants present, not completely consumed", null, "", null, "2H 2 (g) + O 2 (g)  2H 2 O(g)", null, "O 2\nH 2", null, "2H 2 (g) + O 2 (g)  2H 2 O(g)\n\n5[2H 2 (g) + O 2 (g) 2H 2 O(g)] 10H 2 (g) + 5O 2 (g) 10H 2 O(g)", null, "2H 2 (g) + O 2 (g)  2H 2 O(g)\n\n5[2H 2 (g) + O 2 (g) 2H 2 O(g)] 10H 2 (g) + 5O 2 (g) 10H 2 O(g)", null, "", null, "Strategy for Limiting Reactant\n Mass of A (reactant) Moles of A Moles of Product\n Mass of B (reactant) Moles of B Moles of Product\n\nMolar mass of A\n\nMolar mass of B\n\nCoefficients in the equation", null, "Choose\nsmaller\namount\n\nMolar mass of product\n\nMass of\n\nproduct", null, "Example Calculation\n\nCalculate the theoretical yield (g) when 7.0 g of N 2 reacts with 2.0 g of H 2 , forming NH 3 .", null, "Example Calculation\n\nOne reaction step in the conversion of ammonia to nitric acid involves converting NH 3 to NO by the following reaction:\n\n4 NH 3 (g)\n\n+ 5 O 2 (g)\n\n4 NO(g) + 6 H 2 O(g)\n\nIf 1.50 g of NH 3 reacts with 2.75 g O 2 , then:\n\n1. Which is the limiting reactant?\n\n2. How many grams of NO and H 2 O form?\n\n3. How many grams of the excess reactant remain after the limiting reactant is completely consumed?\n\n4. Do parts 1 and 2 obey the law of conservation of mass?" ]

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[ "Question\n\n(a) Use Table 10.4 Conversion of BJT high-frequency modeling parameters into find Cu and C1 (10 points)your textbook (b) Find the low frequency poles. What is the dominant pole frequency, and which capacitor dominates the low frequency pole location?(20 points) (c) Draw the high frequency amplifier model with Cụ and Cr . Find the high frequency cutoff using Cu and Cz . You may choose to use the small signal hybrid-n model of the amp to find the cut-off frequency. Or you can apply Miller's theorem and calculate the cutoff frequency. Or, use a virtual BJT, insert your value of Br for the transistors, and draw inMultiSim Cu connected between base and collector, and Cn connected between base and emitter. Then simulate the amp to find its high cutoff frequency. (30 points) The amplifier BJTS in the circuit below, have the following specifications shown has the following specifications:BF = 220, VA = 350 V, rb = 0.13 Q. CJE=27 pF, VJE = yF 0.75,MJE = m = 0.33, TF = 0.325 ns, fr = 490 MHz, CJC =9.12 pF, CJS =0 pF.", null, "", null, "Fig: 1", null, "", null, "Fig: 2", null, "", null, "Fig: 3", null, "", null, "Fig: 4", null, "", null, "Fig: 5" ]

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[ "", null, "", null, "### Home > CCA2 > Chapter Ch5 > Lesson 5.2.4 > Problem5-102\n\n5-102.\n\nIs it true that $\\log_3(2)=\\log_2(3)$? Justify your answer.\n\nRewrite this as a system of equations:\n\n$\\log_3(2)=a$\n\n$\\log_2(3)=b$\n\nNow rewrite each equation in exponential form and compare them.\n\nWhich value, $a\\ \\text{or}\\ b$, is greater than one? Which is less than one? How can you tell?", null, "" ]

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[ "# FPGA-Based Cell-Averaging Constant False Alarm Rate (CA-CFAR) Detector\n\nThis example shows how to design a CA-CFAR detector suitable for hardware.\n\nTo verify the implementation model is functionally correct, we compare the simulation output of the implementation model with the output of a CFAR based behavioral model using Phased Array System Toolbox™. The term deployment here implies designing a model that is suitable for implementation on a FPGA. The model is implementation ready and this will be verified in the example. The HDL workflow is designed in fixed-point.\n\nThe Phased Array System Toolbox provides the floating point behavioral model for the CFAR detector through the phased.CFARDetector System object™. This behavioral model is used to verify the results of the implementation model and the automatically generated HDL code as well.\n\nFixed-Point Designer™ provides data types and tools for developing fixed-point and single precision algorithms to optimize performance on an embedded hardware. Bit true simulations can be performed to observe the impact of limited range and precision without implementing the design in hardware.\n\nThis example uses HDL Coder™ to generate HDL Code from the developed Simulink® model and verifies the HDL Code using the HDL Verifier™. HDL Verifier is used to generate a co-simulation test bench model to verify the behavior of the automatically generated HDL Code. The test bench uses ModelSim® for Co-simulation to verify the generated HDL code.\n\n### Algorithm Design\n\nIn a radar system, target detection is achieved by comparing the received signal power to a global threshold. If the received power is greater than the threshold, it marks the presence of a target else a target is said to be absent. This makes the choice of threshold a critical characteristic. The appropriate threshold value depends on maximizing the detection and minimizing the false alarm.\n\nThe threshold is chosen based on apriori knowledge (estimate) of the interferer power. The interferer power is affected by many external factors, hence, the variance will be a large value when measured globally. When the threshold is constant, the increase in interferer power can lead to an increase in false detections and at the same time, if the interferer power drops significantly, the target might not be detected.\n\nThe CFAR detector, as the name suggests, maintains the specified false alarm rate by means of an Adaptive Thresholding, wherein the threshold is calculated based on the locality of the Cell Under Test (CUT) and this defines the cell for which detection is required. The interference power of the neighboring cells is used to calculate the threshold for a CUT. The detection threshold is calculated as", null, "where,", null, "is the Probability of False Alarm,", null, "is the Threshold Factor,", null, "is the Interference power level.\n\nCell-Averaging(CA) CFAR Detector In a CA CFAR, the lead and lag cells are used to calculate the interferer estimate. The number of lead cells are the same as that of the number of lag cells. CA-CFAR assumes that, the neighboring cells to the CUT contains the same interference statistic - Homogenous Interference and the target is present in only one CUT. To reinforce the second assumption, guard cells are placed immediately after the CUT.\n\nFor a CA-CFAR with an independent and identically distributed (i.i.d) Gaussian interference (standard normal), the average noise power is just the mean of output of square law detector of all the training cells, which is", null, "Here", null, "is the signal from the i-th training cell. For a given Probability of False Alarm, the threshold factor can be calculated as,", null, "The training cell and guard cell along with the CUT is called as the CFAR Window. The following figure shows a representation of a CFAR Window.", null, "### Implementation Model\n\nThe implementation model is designed using the HDL Coder compatible blocks from the Simulink HDL Coder library. For this example, we have chosen the following values for the parameters:\n\n• No. of Training Cells = 50\n\n• No. of Guard Cells = 2\n\n• Probability of False Alarm = 0.005\n\n• Total No. of Cells = 1000\n\nThe following command is used to open the Simulink model.\n\nmodelname = 'SimulinkCFARHDLWorkflowExample'; open_system(modelname); % % Ensure model is visible and not obstructed by scopes scopes = find_system(modelname,'BlockType','Scope'); close_system(scopes);", null, "The Simulink model consists of two branches from the input block. The top branch is the behavioral model with floating point operations of the phased.CFARdetector System object. The bottom branch is the functional equivalent implementation model with fixed-point version.\n\nThe input to the behavioral model is a (NCell x 1) 1000x1 matrix. The input is passed through the Square-Law sub-system which performs the square-law operation which is then forwarded into the CFAR Detector behavioral model.\n\nThe input to the implementation model is provided via buffer which converts a multi-dimensional signal into single dimensional data stream for a deployable model. The input data type is then converted to fixed-point using the Quantize block. The fixed-point has a word-length of 24 bits and a fraction length of 12 bits. The tradeoff between different fixed-point setting with resource utilization and accuracy is discussed later in this example. The input is then passed on to the CFAR implementation model sub-system which performs the CFAR Detection.\n\nThe output of the CFAR Detector behavioral model is passed through a delay of 125 cycles to compensate the delay for the output of the implementation model.\n\nThe scope block plots the threshold and detection outputs of the behavioral model and implementation model. In addition, the error between the threshold of implementation model and behavioral model, and the error between the detection of implementation model and behavioral model is also calculated and plotted.\n\nThe implementation model contains the following sub-systems:\n\n1. Square-Law HDL\n\n2. Alpha HDL\n\n3. CFAR Core HDL\n\n4. Validate\n\nSquare-Law HDL\n\nThe following command is used to open the Square-Law HDL subsystem model\n\nopen_system([modelname '/CFAR Implementation Model/Square Law HDL']);", null, "The model computes the square-law envelope of the complex input signal.\n\nFor the implementation model the square-law is designed as a deployable model, with additional pipelining registers. This model is implemented using adders and multipliers which account for a latency of 6 cycles.\n\nAlpha HDL\n\nThe following command is used to open the Alpha HDL subsystem model\n\nopen_system([modelname '/CFAR Implementation Model/Alpha HDL']);", null, "The Alpha HDL block utilizes the No. of Training Cells and Probability of False Alarm value to calculate the threshold factor (", null, ").\n\nThis subsystem uses Math HDL blocks and works in single precision for the native floating-point division operation which is then converted to fixed-point at the output. This block accounts for a pipeline latency of 7 Cycles.\n\nCFAR Core HDL\n\n%The following command is used to open the CFAR Core subsystem model open_system([modelname '/CFAR Implementation Model/CFAR Core']);", null, "This subsystem extracts the training cells from the input stream and calculates the noise power. The noise power is then multiplied by alpha to generate the threshold which is later used in the detection process. There are two outputs from this block, namely, threshold and detection.\n\nThe threshold is the direct output of the product block whereas the detection output is provided through a comparator block which compares the threshold and signal value of CUT and returns true if the CUT signal is greater than the threshold.\n\nThis subsystem accounts for an input streaming delay of 102 (2*NumberTrainingCells + NoGuardCells) clock cycles with an additional pipelining latency of another 23 cycles. The total HDL latency is 125 cycles.\n\nThis block contains the following subsystems:\n\n1. Training HDL\n\n2. CUT HDL\n\nTraining HDL\n\nThe following command is used to open the Training HDL subsystem model.\n\nopen_system([modelname '/CFAR Implementation Model/CFAR Core/Training HDL']);", null, "The lead training HDL subsystem extracts the lead cells of the CUT and performs a running sum. At the same time the lag training HDL subsystem pulls out the lag cells of the CUT and performs a running sum with a latency of 8 cycles each. The implementation of the lead training HDL and lag training HDL is very much analogous to a moving average filter the difference is that instead of the average we use the sum of window elements.\n\nThe CA noise power HDL subsystem sums up the lead power value and the lag power value and estimates the average noise power by dividing the sum by 100 (2*NoTrainingCells). This blocks accounts for a delay of 3 clock cycles.\n\nThe output of the training HDL subsystem is the noise power which is used to calculate the threshold.\n\nCUT HDL\n\nThe following command is used to open the CUT HDL subsystem model\n\nopen_system([modelname '/CFAR Implementation Model/CFAR Core/CUT HDL']);", null, "This subsystem uses a single delay block with a delay of 102 (2*NumberTrainingCells + NoGuardCells) cycles to time-align the CUT with the generated threshold value from the training HDL block. The above delay is the minimum delay required before which the CFAR Detector can detect the target at the first cell.\n\nValidate\n\nThe following command is used to open the validate subsystem model\n\nopen_system([modelname '/CFAR Implementation Model/Validate']);", null, "The valid input along with the latency is used to check the validity of the output. When the output is not valid this subsystem sends zero to the output.\n\n### Comparing the Results of Implementation Model to the Behavioral Model\n\nThe model can be simulated by clicking the Play button or using the sim command as shown below,\n\nsim(modelname);", null, "", null, "", null, "The Scope blocks are used to compare the output of implementation and behavioral model. The scope displays the detection and threshold from both the behavioral and implementation model and an additional scope displays the calculated the error.\n\nThe implementation model has a data streaming latency of 102 cycles and pipelining latency of 23 cycles. This in total accounts for an overall latency of 125 cycles. To time align the behavioral model with the implementation model we use an additional delay of 125 to the output of behavioral model.\n\nWith the 24 bit fixed-point of fraction length 12 bits, we have an error bounded by approximately 0.006 between the behavioral model and the implementation model threshold. Since the detection is Boolean we have no significant error in the detection output.\n\n### Code Generation and Verification\n\nThis section covers the procedure to perform HDL codegeneration for the implementation model. It also covers the verification that the generated code is functionally correct. The behavioral model provides the reference values to validate the output from HDL model.\n\nIf you start with a new model, you can run hdlsetup (HDL Coder) to configure the Simulink model for HDL code generation. To configure the Simulink model for test bench creation, open Simulink's Model Settings , select Test Bench under HDL Code Generation in the left panel, and check HDL test bench and Co-simulation model in the Test Bench Generation Output properties group.\n\n### Model Settings\n\nAfter the fixed-point implementation is verified and the implementation model produces the same results as your floating-point, behavioral model, you can generate HDL code and test bench. For code generation and test bench, set the HDL Code Generation parameters in the Configuration Parameters dialog. The following parameters in Model Settings are set under HDL Code Generation:\n\n• Target: Xilinx Vivado synthesis tool; Virtex7 family; Device xc7vx485t; package ffg1761, speed -1; and target frequency of 300 MHz.\n\n• Optimization: Uncheck all optimizations\n\n• Global Settings: Set the Reset type to Asynchronous\n\n• Test Bench: Select HDL test bench, Co-simulation model and System Verilog DPI test bench\n\n### HDL Code Verification via Co-Simulation\n\nAfter the model is set up, HDL Workflow advisor can be invoked to generate the HDL code using the HDL Coder also use the HDL Verifier to generate a System Verilog DPI Test Bench to test the model. To invoke HDL Workflow advisor right-click on the CFAR Implementation model subsystem and navigate to HDL Code and left-click HDL Workflow advisor. Instead of using HDL Workflow advisor the following lines of code can also be used to generate HDL code and System Verilog Test Bench.\n\n% Uncomment the following two lines to generate HDL code and test bench. % makehdl([modelname '/CFAR Implementation Model']); % Generate HDL code % makehdltb([modelname '/CFAR Implementation Model ']); % Generate Cosimulation test bench \n\nSince all the optimizations are unchecked we do not have to add extra delays to the behavioral output other than the HDL latency previously added. (This is because all the critical paths are manually pipelined in the implementation model).\n\nAfter generating HDL code and test bench a new Simulink model named gm_<modelname>_mq containing a ModelSim Simulator block is created in your working directory, which looks like this:", null, "% To open the test bench model, uncomment the following lines of code % modelname = ['gm_',modelname,'_mq']; % open_system(modelname); \n\nLaunch ModelSim and run the co-simulation model to display the simulation results. You can click on the Play button on the top of Simulink canvas to run the test bench or you can do it via command window from the code below.\n\n% Uncomment the following line, to run the test bench. % sim(modelname); \n\nThe Simulink test bench model will populate the QuestaSim® with the HDL model's signal and Time Scopes in Simulink.", null, "The Simulink scope shows detection output and threshold output for both the co-simulation and Design Under Test (DUT) as well as the error between them. The scopes comparing the results of the co-simulation can be found in test bench model inside the Compare subsystem, which is at the output of the CFAR_HDL_mq subsystem.", null, "", null, "### Fixed-Point Word Length and Fraction Length Tradeoffs\n\nFor this example a Fixed-Point word length of 24 bit and a fraction length of 12 bit is used for simulation, and implementation. The following figures show the trade-off with choosing a longer fraction length which would increase the precision (reduces the Error) but also increases the resource utilization.\n\nThe following plot shows fraction length associated with chosen word length.", null, "The following plot shows the Precision with respect to chosen word length.", null, "The following plot shows the Error with respect to chosen word length (Precision)", null, "The following plots show the LUT/Registers/DSP Utilization with respect to chosen Word Length.", null, "", null, "### Summary\n\nThis example demonstrated how to design a Simulink model for a Cell Averaging Constant False Alarm Rate(CA CFAR) Detector, verify the results with an equivalent behavioral setup from the Phased Array System Toolbox. This example demonstrates how to automatically generate HDL code for a fixed-point equivalent algorithm and verify the generated code in Simulink. The generated HDL code as well as a co-simulation test bench for the Simulink subsystem was created with blocks that support HDL code generation. This example showed how to setup and launch ModelSim to cosimulate the HDL code and compare its output to the output generated by the HDL implementation model. The cosimulation is performed via ModelSim for the HDL code and compare results to the output generated by the HDL model." ]

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[ "# MCQ in Engineering Economics Part 19 | ECE Board Exam\n\n(Last Updated On: July 20, 2021)", null, "This is the Multiples Choice Questions Part 19 of the Series in Engineering Economics as one of the General Engineering and Applied Sciences (GEAS) topic. In Preparation for the ECE Board Exam make sure to expose yourself and familiarize in each and every questions compiled here taken from various sources including past Board Questions in General Engineering and Applied Sciences (GEAS), Engineering Economy Books, Journals and other Engineering Economy References.\n\n#### Continue Practice Exam Test Questions Part 19 of the Series\n\nChoose the letter of the best answer in each questions.\n\n901. It is the amount of money earned by given capital.\n\nA. Interest\n\nB. Annuity\n\nC. Cash flow\n\nD. None of these\n\nExplanation:\n\n902. Which of the following is a type of simple interest;\n\nA. Ordinary\n\nB. Exact\n\nC. Both A and B\n\nD. None of these\n\nExplanation:\n\n903. The interest earned by the principal competed at the end of the investment period, it varies directly with time.\n\nA. Compound interest\n\nB. Simple interest\n\nC. Annuity\n\nD. Perpetuity\n\nExplanation:\n\n904. The interest is completed every end of each interest period and the interest earned for the period is added to the principal.\n\nA. Compound interest\n\nB. Simple interest\n\nC. Annuity\n\nD. Perpetuity\n\nExplanation:\n\n905. It is rate quoted in describing a given variety of compound interest.\n\nA. nominal rate\n\nB. effective rate\n\nC. simple rate\n\nD. compound rate\n\nExplanation:\n\n906. It is the actual interest earned in one year period.\n\nA. Nominal rate\n\nB. Effective rate\n\nC. Simple rate\n\nD. Compound rate\n\nExplanation:\n\n907. A series of uniform payments made at equal intervals of time.\n\nA. perpetuity\n\nB. interest\n\nC. rate\n\nD. annuity\n\nExplanation:\n\n908. Which of the following is type of annuity.\n\nA. Ordinary\n\nB. Deferred\n\nC. Annuity due\n\nD. All of these\n\nExplanation:\n\n909. An annuity where the payments periods extend forever or the periodic payments continue indefinitely.\n\nA. Perpetuity\n\nB. Interest\n\nC. Rate\n\nD. Annuity\n\nExplanation:\n\n910. The payment is made at the end of each period starting from the first period.\n\nA. Ordinary annuity\n\nB. Deferred annuity\n\nC. Annuity due\n\nD. All of these\n\nExplanation:\n\n911. The first payment is deferred a certain number of periods after the first.\n\nA. Ordinary annuity\n\nB. Deferred annuity\n\nC. Annuity due\n\nD. None of these\n\nExplanation:\n\n912. The payment is made at the beginning of each period starting from the first period.\n\nA. Ordinary annuity\n\nB. Deferred annuity\n\nC. Annuity due\n\nD. None of the due\n\nExplanation:\n\n913. It is an application of perpetuity.\n\nA. Capitalized cost\n\nB. Machine cost\n\nC. Manpower cost\n\nD. Equipment cost\n\nExplanation:\n\n914. It refers to the decrease in the value of an asset due to usage of passage of time.\n\nA. Interest\n\nB. Annuity\n\nC. Depreciation\n\nD. Perpetuity\n\nExplanation:\n\n915. Which of the following is the method of computing depreciation\n\nA. Straight line depreciation\n\nB. Sinking fund method\n\nC. Sum of the years method\n\nD. All of these.\n\nExplanation:\n\n916. The most common method used in computing depreciation;\n\nA. Straight line depreciation\n\nB. Sinking fund method\n\nC. Sum of the years digit method\n\nD. Declining balance method\n\nExplanation:\n\n917. The depreciation charge in this method is assumed to vary directly to the number of years and inversely to the sum of the year’s digit.\n\nA. Straight line depreciation\n\nB. Sinking fund method\n\nC. Sum of the year’s digit method\n\nD. Declining balance method\n\nExplanation:\n\n918. It is invested yearly at a rate of I to amount to (FC –SV) at the end of the life to the property.\n\nA. Straight line depreciation\n\nB. Sinking fund method\n\nC. Sum of the years digit method\n\nD. Declining balance method\n\nExplanation:\n\n919. A written contract to pay a certain redemption value on a specified redemption date and to pay equal dividends periodically.\n\nA. Bond\n\nB. Capital\n\nC. Interest\n\nD. Annuity\n\nExplanation:\n\n920. A method of determining when cost exactly equal to revenue.\n\nA. Experimental method\n\nB. Break-even method\n\nD. None of these\n\nExplanation:\n\n921. Find the interest on P6,800.00 for 3 years at 11% simple interest.\n\nA. P 1,875.00\n\nB. P 1,987.00\n\nC. P 2,144.00\n\nD. P 2,244.00\n\nExplanation:\n\n922. A man borrowed P10,000.00 from his friend and agrees to pay at the end of 90 days under 8% simple interest rate. What is the required amount?\n\nA. P 10,200.00\n\nB. P 11,500.00\n\nC. P 9,500.00\n\nD. P 10,700.00\n\nExplanation:\n\n923. Annie buys a television set from a merchant who offers P25,000.00 at the end of 60 days. Annie wished to pay immediately and the merchant offers to compute the required amount on the assumption that the money is worth 14% simple interest. What is the required amount?\n\nA. P 20,234.87\n\nB. P 19,222.67\n\nC. P 24,429.97\n\nD. P 28,456.23\n\nExplanation:\n\n924. What is the principal amount if the amount of interest at the end of 2 1/2 year is P4,500.00 for a simple interest of 6% per annum?\n\nA. P 35,000.00\n\nB. P 30,000.00\n\nC. P 40,000.00\n\nD. P 45,000.00\n\nExplanation:\n\n925. How long must a P40,000 note bearing 4% simple interest run to amount to P41,350.00?\n\nA. 340days\n\nB. 403 days\n\nC. 304 days\n\nD. 430 days\n\nExplanation:\n\n926. If P16,000 earns P480 in 9 months, what is the annual rate of interest?\n\nA. 1%\n\nB. 2%\n\nC. 3%\n\nD. 4%\n\nExplanation:\n\n927. A man lends P6,000 at 6% simple interest for 4 years. At the end of this time he invest the entire amount (principal plus interest) at 5% compounded annually for 12 years. How much will he have at the end of the 16-year period.?\n\nA. P 13,361.20\n\nB. P 13,633.20\n\nC. P 13,333.20\n\nD. P 16,323.20\n\nExplanation:\n\n928. A time deposit of P110,000 for 31 days earns P890.39 on maturity date after deducting the 20% withholding tax on interest income. Find interest per annum.\n\nA. 12.5%\n\nB. 11.95%\n\nC. 12.25%\n\nD. 11.75%\n\nExplanation:\n\n929. A bank charges 12% simple interest on a P300.00 loan. How much will be repaid if the loan is paid back in one lump sum after three years.\n\nA. P 408.00\n\nB. P 551.00\n\nC. P415.00\n\nD. P450.00\n\nExplanation:\n\n930. A tag price of a certain commodity is for 100 days. If paid in 31 days, there is 3% discount. What is the simple interest paid?\n\nA. 12.5%\n\nB. 6.25%\n\nC. 22.32%\n\nD. 16.14%\n\nExplanation:\n\n931. Accumulate P5,000.00 for 10 years at 8% compounded quarterly.\n\nA. P 12,456.20\n\nB. P 13,876.50\n\nC. P 10,345.80\n\nD. P 11,040.20\n\nExplanation:\n\n932. Accumulate P5,000.00 for 10 years at 8% compounded semi-annually.\n\nA. P 10,955.61\n\nB. P 10,233.67\n\nC. P 9,455.67\n\nD. P 11,876.34\n\nExplanation:\n\n933. Accumulate P5,000.00 for 10 years at 8% compounded monthly\n\nA. P 15,456.75\n\nB. P 11,102.61\n\nC. P 10,955.61\n\nD. P 10,955.61\n\nExplanation:\n\n934. Accumulate P5,000.00 for 10 years at 8% compounded annually.\n\nA. P 10,794.62\n\nB. P 8,567.97\n\nC. P 10,987.90\n\nD. P 7,876.87\n\nExplanation:\n\n935. How long it will take P1,000 to amount to P1,346 if invested at 6% compounded quarterly.\n\nA. 3 years\n\nB. 4 years\n\nC. 5 years\n\nD. 6 years\n\nExplanation:\n\n936. How long will it take for an investment to double its amount if invested at an interest rate of 6% compounded bi-monthly\n\nA. 10 years\n\nB. 12 years\n\nC. 13 years\n\nD. 14 years\n\nExplanation:\n\n937. If the compound interest on P3,000.00 in years is P500.00 then the compound interest on P3,000.00 in 4 years is:\n\nA. P 956.00\n\nB. P 1,083.00\n\nC. P 1,125.00\n\nD. P 1,526.00\n\nExplanation:\n\n938. The salary of Mr. Cruz is increased by 30% every 2 years beginning January 1, 1982. Counting from that date, at what year will his salary just exceed twice his original salary?\n\nA. 1988\n\nB. 1989\n\nC. 1990\n\nD. 1991\n\nExplanation:\n\n939. If you borrowed P10,000 from the bank with 18% interest per annum, what is the total amount to be repaid at the end of one year.\n\nA. P 11,800.00\n\nB. P 19,000.00\n\nC. P 28,000.00\n\nD. P 10,180.00\n\nExplanation:\n\n940. What is the effective rate for an interest rate at 12% compounded continuously?\n\nA. 12.01%\n\nB. 12.89%\n\nC. 12.42%\n\nD. 12.75%\n\nExplanation:\n\n941. How long it will take for an investment to fivefold its amount if money is worth 14% compounded semiannually.\n\nA. 11\n\nB. 12\n\nC. 13\n\nD. 14\n\nExplanation:\n\n942. An interest of 8% compounded semiannually is how many percent if compounded quarterly?\n\nA. 7.81%\n\nB. 7.85%\n\nC. 7.92%\n\nD. 8.01%\n\nExplanation:\n\n943. A man is expecting to receive P450,000.00 at the end of 7 years. If money is worth 14% compounded quarterly how much is it worth at present?\n\nA. P 125,458.36\n\nB. P 147,456.36\n\nC. P 162,455.63\n\nD. P 171,744.44\n\nExplanation:\n\n944. A man has a will of P650,000.00 from his father. If his father deposited an amount of P450,000 in a trust fund earning 8% compounded annually, after how many years will he man receive his will?\n\nA. 4.55 years\n\nB. 4.77 years\n\nC. 5.11 years\n\nD. 5.33 years\n\nExplanation:\n\n945. Mr. Adam deposited P120,000.00 in a bank who offers 8% interest compounded quarterly. If the interest is subject to a 14% tax, how much will he receive after 5 years?\n\nA. P 178,313.69\n\nB. P 153,349.77\n\nC. P 170,149.77\n\nD. P 175,343.77\n\nExplanation:\n\n946. What interest compounded monthly is equivalent to an interest rate of 14% compounded quarterly\n\nA. 1.15%\n\nB. 13.84%\n\nC. 10.03%\n\nD. 11.52%\n\nExplanation:\n\n947. What is the worth of two P100,000 payments at the end of the third and the fourth year? The annual interest rate is 8%.\n\nA. P 152.87\n\nB. P 112.34\n\nC. P 187.98\n\nD. P 176.67\n\nExplanation:\n\n948. A firm borrows P2,000.00 for 6 years at 8%. At the end of 6 years, it renews the loan for the amount due plus P2,000 more for 2 years at 8%. What is the lump sum due?\n\nA. P 5,679.67\n\nB. P 6,789.98\n\nC. P 6,034.66\n\nD. P 5,888.77\n\nExplanation:\n\n949. At an annual rate of return of 8%, what is the future worth of P1,000 at the end of 4 years?\n\nA. P 1,388.90\n\nB. P 1,234.56\n\nC. P 1,765.56\n\nD. P 1,360.50\n\nExplanation:\n\n950. A student has money given by his grandfather in the amount of P20,000.00. How much money in the form of interest will he get if the money is put in a bank that offers 8% rate compounded annually at the end of 7 years?\n\nA. P 34,276.48\n\nB. P 34,270.00\n\nC. P 36,276.40\n\nD. P 34,266.68\n\nExplanation:\n\n#### Online Questions and Answers in Engineering Economics Series\n\nFollowing is the list of practice exam test questions in this brand new series:\n\nEngineering Economics MCQs\nPART 1: MCQ from Number 1 – 50                        Answer key: PART 1\nPART 2: MCQ from Number 51 – 100                   Answer key: PART 2\nPART 3: MCQ from Number 101 – 150                 Answer key: PART 3\nPART 4: MCQ from Number 151 – 200                 Answer key: PART 4\nPART 5: MCQ from Number 201 – 250                 Answer key: PART 5\nPART 6: MCQ from Number 251 – 300                 Answer key: PART 6\nPART 7: MCQ from Number 301 – 350                 Answer key: PART 7\nPART 8: MCQ from Number 351 – 400                 Answer key: PART 8\nPART 9: MCQ from Number 401 – 450                 Answer key: PART 9\nPART 10: MCQ from Number 451 – 500                 Answer key: PART 10" ]

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[ "# #00dc5a Color Information\n\nIn a RGB color space, hex #00dc5a is composed of 0% red, 86.3% green and 35.3% blue. Whereas in a CMYK color space, it is composed of 100% cyan, 0% magenta, 59.1% yellow and 13.7% black. It has a hue angle of 144.5 degrees, a saturation of 100% and a lightness of 43.1%. #00dc5a color hex could be obtained by blending #00ffb4 with #00b900. Closest websafe color is: #00cc66.\n\n• R 0\n• G 86\n• B 35\nRGB color chart\n• C 100\n• M 0\n• Y 59\n• K 14\nCMYK color chart\n\n#00dc5a color description : Pure (or mostly pure) cyan - lime green.\n\n# #00dc5a Color Conversion\n\nThe hexadecimal color #00dc5a has RGB values of R:0, G:220, B:90 and CMYK values of C:1, M:0, Y:0.59, K:0.14. Its decimal value is 56410.\n\nHex triplet RGB Decimal 00dc5a `#00dc5a` 0, 220, 90 `rgb(0,220,90)` 0, 86.3, 35.3 `rgb(0%,86.3%,35.3%)` 100, 0, 59, 14 144.5°, 100, 43.1 `hsl(144.5,100%,43.1%)` 144.5°, 100, 86.3 00cc66 `#00cc66`\nCIE-LAB 77.234, -71.423, 50.48 27.437, 51.921, 18.248 0.281, 0.532, 51.921 77.234, 87.461, 144.748 77.234, -70.66, 74.695 72.057, -58.132, 35.425 00000000, 11011100, 01011010\n\n# Color Schemes with #00dc5a\n\n• #00dc5a\n``#00dc5a` `rgb(0,220,90)``\n• #dc0082\n``#dc0082` `rgb(220,0,130)``\nComplementary Color\n• #14dc00\n``#14dc00` `rgb(20,220,0)``\n• #00dc5a\n``#00dc5a` `rgb(0,220,90)``\n• #00dcc8\n``#00dcc8` `rgb(0,220,200)``\nAnalogous Color\n• #dc0014\n``#dc0014` `rgb(220,0,20)``\n• #00dc5a\n``#00dc5a` `rgb(0,220,90)``\n• #c800dc\n``#c800dc` `rgb(200,0,220)``\nSplit Complementary Color\n• #dc5a00\n``#dc5a00` `rgb(220,90,0)``\n• #00dc5a\n``#00dc5a` `rgb(0,220,90)``\n• #5a00dc\n``#5a00dc` `rgb(90,0,220)``\n• #82dc00\n``#82dc00` `rgb(130,220,0)``\n• #00dc5a\n``#00dc5a` `rgb(0,220,90)``\n• #5a00dc\n``#5a00dc` `rgb(90,0,220)``\n• #dc0082\n``#dc0082` `rgb(220,0,130)``\n• #00903b\n``#00903b` `rgb(0,144,59)``\n• #00a945\n``#00a945` `rgb(0,169,69)``\n• #00c350\n``#00c350` `rgb(0,195,80)``\n• #00dc5a\n``#00dc5a` `rgb(0,220,90)``\n• #00f664\n``#00f664` `rgb(0,246,100)``\n• #10ff72\n``#10ff72` `rgb(16,255,114)``\n• #2aff81\n``#2aff81` `rgb(42,255,129)``\nMonochromatic Color\n\n# Alternatives to #00dc5a\n\nBelow, you can see some colors close to #00dc5a. Having a set of related colors can be useful if you need an inspirational alternative to your original color choice.\n\n• #00dc23\n``#00dc23` `rgb(0,220,35)``\n• #00dc35\n``#00dc35` `rgb(0,220,53)``\n• #00dc48\n``#00dc48` `rgb(0,220,72)``\n• #00dc5a\n``#00dc5a` `rgb(0,220,90)``\n• #00dc6c\n``#00dc6c` `rgb(0,220,108)``\n• #00dc7f\n``#00dc7f` `rgb(0,220,127)``\n• #00dc91\n``#00dc91` `rgb(0,220,145)``\nSimilar Colors\n\n# #00dc5a Preview\n\nThis text has a font color of #00dc5a.\n\n``<span style=\"color:#00dc5a;\">Text here</span>``\n#00dc5a background color\n\nThis paragraph has a background color of #00dc5a.\n\n``<p style=\"background-color:#00dc5a;\">Content here</p>``\n#00dc5a border color\n\nThis element has a border color of #00dc5a.\n\n``<div style=\"border:1px solid #00dc5a;\">Content here</div>``\nCSS codes\n``.text {color:#00dc5a;}``\n``.background {background-color:#00dc5a;}``\n``.border {border:1px solid #00dc5a;}``\n\n# Shades and Tints of #00dc5a\n\nA shade is achieved by adding black to any pure hue, while a tint is created by mixing white to any pure color. In this example, #000402 is the darkest color, while #f0fff6 is the lightest one.\n\n• #000402\n``#000402` `rgb(0,4,2)``\n• #00180a\n``#00180a` `rgb(0,24,10)``\n• #002b12\n``#002b12` `rgb(0,43,18)``\n• #003f1a\n``#003f1a` `rgb(0,63,26)``\n• #005322\n``#005322` `rgb(0,83,34)``\n• #00662a\n``#00662a` `rgb(0,102,42)``\n• #007a32\n``#007a32` `rgb(0,122,50)``\n• #008e3a\n``#008e3a` `rgb(0,142,58)``\n• #00a142\n``#00a142` `rgb(0,161,66)``\n• #00b54a\n``#00b54a` `rgb(0,181,74)``\n• #00c852\n``#00c852` `rgb(0,200,82)``\n• #00dc5a\n``#00dc5a` `rgb(0,220,90)``\n• #00f062\n``#00f062` `rgb(0,240,98)``\n• #04ff6b\n``#04ff6b` `rgb(4,255,107)``\n• #18ff76\n``#18ff76` `rgb(24,255,118)``\n• #2bff82\n``#2bff82` `rgb(43,255,130)``\n• #3fff8e\n``#3fff8e` `rgb(63,255,142)``\n• #53ff99\n``#53ff99` `rgb(83,255,153)``\n• #66ffa5\n``#66ffa5` `rgb(102,255,165)``\n• #7affb0\n``#7affb0` `rgb(122,255,176)``\n• #8effbc\n``#8effbc` `rgb(142,255,188)``\n• #a1ffc8\n``#a1ffc8` `rgb(161,255,200)``\n• #b5ffd3\n``#b5ffd3` `rgb(181,255,211)``\n• #c8ffdf\n``#c8ffdf` `rgb(200,255,223)``\n• #dcffea\n``#dcffea` `rgb(220,255,234)``\n• #f0fff6\n``#f0fff6` `rgb(240,255,246)``\nTint Color Variation\n\n# Tones of #00dc5a\n\nA tone is produced by adding gray to any pure hue. In this case, #66766c is the less saturated color, while #00dc5a is the most saturated one.\n\n• #66766c\n``#66766c` `rgb(102,118,108)``\n• #5d7f6b\n``#5d7f6b` `rgb(93,127,107)``\n• #558769\n``#558769` `rgb(85,135,105)``\n• #4c9068\n``#4c9068` `rgb(76,144,104)``\n• #449866\n``#449866` `rgb(68,152,102)``\n• #3ba165\n``#3ba165` `rgb(59,161,101)``\n• #33a963\n``#33a963` `rgb(51,169,99)``\n• #2ab262\n``#2ab262` `rgb(42,178,98)``\n• #22ba60\n``#22ba60` `rgb(34,186,96)``\n• #19c35f\n``#19c35f` `rgb(25,195,95)``\n• #11cb5d\n``#11cb5d` `rgb(17,203,93)``\n• #08d45c\n``#08d45c` `rgb(8,212,92)``\n• #00dc5a\n``#00dc5a` `rgb(0,220,90)``\nTone Color Variation\n\n# Color Blindness Simulator\n\nBelow, you can see how #00dc5a is perceived by people affected by a color vision deficiency. This can be useful if you need to ensure your color combinations are accessible to color-blind users.\n\nMonochromacy\n• Achromatopsia 0.005% of the population\n• Atypical Achromatopsia 0.001% of the population\nDichromacy\n• Protanopia 1% of men\n• Deuteranopia 1% of men\n• Tritanopia 0.001% of the population\nTrichromacy\n• Protanomaly 1% of men, 0.01% of women\n• Deuteranomaly 6% of men, 0.4% of women\n• Tritanomaly 0.01% of the population" ]

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[ "Seifert Fibrations and their associated Spectral Sequence\n\nIn a somewhat limited setting, a Seifert Fibre Space is a 3-manifold $M$ with a \"nice\" decomposition into circles (http://en.wikipedia.org/wiki/Seifert_fiber_space). That is, $M$ is decomposed into circles in a way such that $M$ has neighbourhoods which are \"fibred as a solid tori would be, if these tori are given by a solid cylinders with rational rotations identifying opposite disks\".\n\nThere is a natural map (the Seifert fibration) from $M$ to the quotient space collapsing each of the circle fibres. Of course, this needn't be a fibration at all. In general one has isolated singular fibres - one may like to view these as having fractional lengths relative to their neighbours. However, it is a fibration when we view the base space as an orbifold (let's call it $B$) instead of just a space.\n\nWhen given a fibration, it is usual to stick it into a spectral sequence and compute (co)homology. I assume that the same can be done in this setting, but I can't find any discussion of this in the literature (which is understandable - presumably calculating orbifold cohomology with twisted coefficients is almost always more difficult than computing the cohomology of $M$ directly). Of course, one would have to replace cohomology with orbifold cohomology. So, I suppose my question is:\n\n• Does the cohomology of $M$ fit into a spectral sequence with (twisted) coefficients over the orbifold cohomology of the quotient? I'm pretty certain this will be the case:\n• In which case, there are various flavours of orbifold cohomology. However, I presume that I am still correct in assuming that we should use the cohomology of the classifying space for the orbifold cohomology.\n• Orbifold cohomology agrees with singular cohomology over rational coefficients. With general coefficients, though, it seems usual to get non-trivial cohomology in infinitely many degrees. Of course, the cohomology of $M$ is concentrated in degrees 0 to 3. So my question really is of the nature of this spectral sequence. Either the torsion is killed off in the sequence or never appears because of the original twisting of the coefficients. Is it possible to say which? Is there a simple toy example where the explicit calculations can be seen? I was thinking, for example, of orbifolds associated to quotients of wallpaper groups, which arise naturally from Seifert fibrations.\n• I haven't seen anyone take this approach in the literature. The homology can be read off from the fundamental group presentations of these manifolds, which are readily-deduced in Seifert's original work on this topic. The cup product structure in cohomology is more work to get at. It's generally degenerate except in a few cases. Peter Zvengrowski works this out -- just google his name, the papers are easy-enough to find. – Ryan Budney May 28 '13 at 19:19\n• Thanks Ryan, My interest is more with understanding the spectral sequence itself than the actual computations. As acknowledged in the question, the spectral sequence isn't that helpful for explicit computations. – Jamie Walton May 28 '13 at 21:21\n• Dear Jamie, what you're looking for may be Leray's sheaf-theoretic spectral sequence. Specifically, in this case the fibers vary nicely enough that there is a sheaf on the base space $B$ whose stalks are the cohomology of the fibers, and the spectral sequence starts with the cohomology of $B$ with coefficients in this sheaf. I'm not clear enough on orbifold cohomology to tell you whether it is the same thing, but I'd suspect that it is almost certainly so. There's no magic in this to help the computations, though; you still patch the fibers in manually. – Tyler Lawson May 29 '13 at 3:40\n• Thanks Tyler, this seems like a good suggestion. Just to make sure I'm clear on your answer: you say we want a sheaf on $B$ whose stalks are the cohomology of the fibers. We first take some sheaf on $M$ (e.g. the constant sheaf). Then I'm supposing that I need to compute the direct image of this sheaf on $B$, along with its higher direct images? I imagine it turns out that I only get something in degrees zero and one, and that they will presumably all have stalks equal to $\\mathbb{Z}$ (this may be what you mean by saying the fibers vary nicely enough). I'll give it a try, thanks again. – Jamie Walton May 29 '13 at 10:01\n• Ah, on reflection, it is clear that the direct image (and it's first higher version) of the constant sheaf (on $\\mathbb{Z}$) have stalks $\\mathbb{Z}$. Indeed, the preimage of a small simply connected open disk is just a solid torus. The first higher direct image won't be constant though since \"the stalks of generic fibres are multiples of those of nearby singular fibres\", which is the sort of thing I was expecting. – Jamie Walton May 29 '13 at 10:29\n\nIndeed, you always get a fibration $M\\to \\hat B$, where $\\hat B$ is the Haefliger's classifying space of the orbifold (the space whose cohomology is the orbifold cohomology of $B$). The fiber of this fibration is the principal leaf of your Seifert fibration.\n\nOne can write the corresponding spectral sequence. And indeed, if the orbifold is not a manifold you will get cohomology in infinitely many degrees, which has to be killed, when going to the third page of the spectral sequence.\n\nThe simplest example is the linear circle action on $M=S^3$ with parameters $(1,p)$, with one exceptional leaf, such that the angle at the quotient point is $2\\pi /p$.\n\nThen the spectral sequence is essentially the Gysin sequence of the spherical fibration. You obtain from the sequence that the cohomology of $\\hat B$ must be generated by one element $e$ in $H^2(\\hat B)= \\mathbb Z$, the Euler class class of the fibration. The square $e^2$ of the element $e$ is the geneartor of $H^4 (\\hat B)$. It has the same order as all other powers of $e$. And this order is exactly $p$. This you can see only locally and it cannot be seen from the Gysin sequence, since the sequence is the same for all $p$.\n\nThe sequence you get is very similar to the spectral sequence of the universal fibration $S^{\\infty} \\to CP ^{\\infty}$. The infinitely many non-zero elements are killed in the same way.\n\n• Beautiful, thanks Alexander. I should have thought about using the Gysin sequence, since we are just looking at circle bundles. I was thinking about twisted coefficients, but I guess that the action of the orbifold fundamental group here is trivial (and this happens if and only if the circle bundle is orientable, which we need for the Gysin sequence). I haven't seen many examples of the sort of cohomology rings you get for these orbifolds. I know that rationally it should be the same as for the base manifold, and I guess this example allows you to work backwards to some extent. – Jamie Walton Jun 14 '13 at 17:38\n\nIn the article 'Circle actions on simply connected 5-manifolds' by Kollar, it is done just what you are looking for. There, the Leray spectral sequence is used to infer topological obstructions for a five manifold to admit a Seifert fibered structure over a 4-orbifold.\n\nhttp://arxiv.org/pdf/math/0505343.pdf\n\nIf I am not mistaken, this spectral sequence works here because Seifert fibrations do have the homotopy lifting property, so are fibrations.\n\nNote that in the reference I give, everything is ordinary cohomology, i.e., no orbifold cohomology is involved, everything is done for the topological underlying space of the orbifold. I think this is quite good for non-experts in orbifold theory, as I am.\n\n• I'm pretty confident that Seifert fibrations do not have the homotopy lifting property. Map a circle onto a singular point in the base. Then homotope this map through constant maps onto regular points. Taking the lift as the inclusion of the singular fiber, I do not see how the full homotopy can be lifted. – John Harvey Jun 2 '15 at 9:08\n• Right, thank you, I was confused. Then, how would you justify that the Leray spectral seuence works for seifert bundles? I thought it just worked for fibrations... – juan rojo Jun 18 '15 at 8:26\n• That's all a little beyond my range of expertise! But it seems that what Kollár is doing is the kind of stuff Tyler and Jamie refer to. The Leray spectral sequence is more general, but you need to understand the direct image of the sheaf. Looking at the reference quickly, it seems that in the Seifert case the sheaf cohomology coincides with the ordinary cohomology in the top three dimensions. In his case, this only leaves $H^1$, which is controlled by simple connectivity. In this case, it would be even better. So it seems like the Leray spectral sequence would be useful here. – John Harvey Jun 19 '15 at 11:51\n• Even though the map isn't a fibration, the sheaf cohomology is fairly straightforward in this case. Since all the fibers are circles, the $i^{th}$ derived pushforward sheafs vanish for $i$ not $0,1$ (so the spectral sequence degenerates and gives an exact sequence looking a lot like the Gysin sequence for an $S^1$-bundle). The $R^1$ sheaf is the subsheaf of the constant sheaf $\\mathbb{Z}$ whose stalks at a point $x$ are $m\\mathbb{Z}$ with $m$ the multiplicity of the fiber above $x$. – dorebell Apr 28 '16 at 7:59" ]

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[ "You can not select more than 25 topics Topics must start with a letter or number, can include dashes ('-') and can be up to 35 characters long.\n\n#### 75 lines 2.3 KiB Raw Permalink Blame History\n\n ```#!/usr/bin/env bash ``` ```# Generated from: ``` ```# https://github.com/zopefoundation/meta/tree/master/config/c-code ``` ``` ``` ```set -e -x ``` ``` ``` ```# Running inside docker ``` ```# Set a cache directory for pip. This was ``` ```# mounted to be the same as it is outside docker so it ``` ```# can be persisted. ``` ```export XDG_CACHE_HOME=\"/cache\" ``` ```# XXX: This works for macOS, where everything bind-mounted ``` ```# is seen as owned by root in the container. But when the host is Linux ``` ```# the actual UIDs come through to the container, triggering ``` ```# pip to disable the cache when it detects that the owner doesn't match. ``` ```# The below is an attempt to fix that, taken from bcrypt. It seems to work on ``` ```# Github Actions. ``` ```if [ -n \"\\$GITHUB_ACTIONS\" ]; then ``` ``` echo Adjusting pip cache permissions ``` ``` mkdir -p \\$XDG_CACHE_HOME/pip ``` ``` chown -R \\$(whoami) \\$XDG_CACHE_HOME ``` ```fi ``` ```ls -ld /cache ``` ```ls -ld /cache/pip ``` ``` ``` ```# We need some libraries because we build wheels from scratch: ``` ```yum -y install libffi-devel ``` ``` ``` ```tox_env_map() { ``` ``` case \\$1 in ``` ``` *\"cp27\"*) echo 'py27';; ``` ``` *\"cp35\"*) echo 'py35';; ``` ``` *\"cp311\"*) echo 'py311';; ``` ``` *\"cp36\"*) echo 'py36';; ``` ``` *\"cp37\"*) echo 'py37';; ``` ``` *\"cp38\"*) echo 'py38';; ``` ``` *\"cp39\"*) echo 'py39';; ``` ``` *\"cp310\"*) echo 'py310';; ``` ``` *) echo 'py';; ``` ``` esac ``` ```} ``` ``` ``` ```# Compile wheels ``` ```for PYBIN in /opt/python/*/bin; do ``` ``` if \\ ``` ``` [[ \"\\${PYBIN}\" == *\"cp27\"* ]] || \\ ``` ``` [[ \"\\${PYBIN}\" == *\"cp35\"* ]] || \\ ``` ``` [[ \"\\${PYBIN}\" == *\"cp311\"* ]] || \\ ``` ``` [[ \"\\${PYBIN}\" == *\"cp36\"* ]] || \\ ``` ``` [[ \"\\${PYBIN}\" == *\"cp37\"* ]] || \\ ``` ``` [[ \"\\${PYBIN}\" == *\"cp38\"* ]] || \\ ``` ``` [[ \"\\${PYBIN}\" == *\"cp39\"* ]] || \\ ``` ``` [[ \"\\${PYBIN}\" == *\"cp310\"* ]] ; then ``` ``` if [[ \"\\${PYBIN}\" == *\"cp311\"* ]] ; then ``` ``` \"\\${PYBIN}/pip\" install --pre -e /io/ ``` ``` \"\\${PYBIN}/pip\" wheel /io/ --pre -w wheelhouse/ ``` ``` else ``` ``` \"\\${PYBIN}/pip\" install -e /io/ ``` ``` \"\\${PYBIN}/pip\" wheel /io/ -w wheelhouse/ ``` ``` fi ``` ``` if [ `uname -m` == 'aarch64' ]; then ``` ``` cd /io/ ``` ``` \\${PYBIN}/pip install tox ``` ``` TOXENV=\\$(tox_env_map \"\\${PYBIN}\") ``` ``` \\${PYBIN}/tox -e \\${TOXENV} ``` ``` cd .. ``` ``` fi ``` ``` rm -rf /io/build /io/*.egg-info ``` ``` fi ``` ```done ``` ``` ``` ```# Bundle external shared libraries into the wheels ``` ```for whl in wheelhouse/zope.interface*.whl; do ``` ``` auditwheel repair \"\\$whl\" -w /io/wheelhouse/ ``` ```done ``` ``` ```" ]

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[ "# scipy.io.loadmat nested structures (i.e. dictionaries)\n\nPosted on\n\n### Question :\n\nUsing the given routines (how to load Matlab .mat files with scipy), I could not access deeper nested structures to recover them into dictionaries\n\nTo present the problem I run into in more detail, I give the following toy example:\n\n``````load scipy.io as spio\na = {'b':{'c':{'d': 3}}}\n# my dictionary: a['b']['c']['d'] = 3\nspio.savemat('xy.mat',a)\n``````\n\nNow I want to read the mat-File back into python. I tried the following:\n\n``````vig=spio.loadmat('xy.mat',squeeze_me=True)\n``````\n\nIf I now want to access the fields I get:\n\n``````>> vig['b']\narray(((array(3),),), dtype=[('c', '|O8')])\n>> vig['b']['c']\narray(array((3,), dtype=[('d', '|O8')]), dtype=object)\n>> vig['b']['c']['d']\n---------------------------------------------------------------------------\nValueError Traceback (most recent call last)\n\n/<ipython console> in <module>()\n\n``````\n\nHowever, by using the option `struct_as_record=False` the field could be accessed:\n\n``````v=spio.loadmat('xy.mat',squeeze_me=True,struct_as_record=False)\n``````\n\nNow it was possible to access it by\n\n``````>> v['b'].c.d\narray(3)\n``````\n\n``````import scipy.io as spio\n\n'''\nas it cures the problem of not properly recovering python dictionaries\nfrom mat files. It calls the function check keys to cure all entries\nwhich are still mat-objects\n'''\nreturn _check_keys(data)\n\ndef _check_keys(dict):\n'''\nchecks if entries in dictionary are mat-objects. If yes\ntodict is called to change them to nested dictionaries\n'''\nfor key in dict:\nif isinstance(dict[key], spio.matlab.mio5_params.mat_struct):\ndict[key] = _todict(dict[key])\nreturn dict\n\ndef _todict(matobj):\n'''\nA recursive function which constructs from matobjects nested dictionaries\n'''\ndict = {}\nfor strg in matobj._fieldnames:\nelem = matobj.__dict__[strg]\nif isinstance(elem, spio.matlab.mio5_params.mat_struct):\ndict[strg] = _todict(elem)\nelse:\ndict[strg] = elem\nreturn dict\n``````\n\nJust an enhancement to mergen’s answer, which unfortunately will stop recursing if it reaches a cell array of objects. The following version will make lists of them instead, and continuing the recursion into the cell array elements if possible.\n\n``````import scipy as spio\nimport numpy as np\n\n'''\nas it cures the problem of not properly recovering python dictionaries\nfrom mat files. It calls the function check keys to cure all entries\nwhich are still mat-objects\n'''\ndef _check_keys(d):\n'''\nchecks if entries in dictionary are mat-objects. If yes\ntodict is called to change them to nested dictionaries\n'''\nfor key in d:\nif isinstance(d[key], spio.matlab.mio5_params.mat_struct):\nd[key] = _todict(d[key])\nreturn d\n\ndef _todict(matobj):\n'''\nA recursive function which constructs from matobjects nested dictionaries\n'''\nd = {}\nfor strg in matobj._fieldnames:\nelem = matobj.__dict__[strg]\nif isinstance(elem, spio.matlab.mio5_params.mat_struct):\nd[strg] = _todict(elem)\nelif isinstance(elem, np.ndarray):\nd[strg] = _tolist(elem)\nelse:\nd[strg] = elem\nreturn d\n\ndef _tolist(ndarray):\n'''\nA recursive function which constructs lists from cellarrays\n(which are loaded as numpy ndarrays), recursing into the elements\nif they contain matobjects.\n'''\nelem_list = []\nfor sub_elem in ndarray:\nif isinstance(sub_elem, spio.matlab.mio5_params.mat_struct):\nelem_list.append(_todict(sub_elem))\nelif isinstance(sub_elem, np.ndarray):\nelem_list.append(_tolist(sub_elem))\nelse:\nelem_list.append(sub_elem)\nreturn elem_list\nreturn _check_keys(data)\n``````\n\nI was advised on the scipy mailing list (https://mail.python.org/pipermail/scipy-user/) that there are two more ways to access this data.\n\nThis works:\n\n``````import scipy.io as spio\nprint vig['b'][0, 0]['c'][0, 0]['d'][0, 0]\n``````\n\nOutput on my machine:\n3\n\nThe reason for this kind of access: “For historic reasons, in Matlab everything is at least a 2D array, even scalars.”\nSo scipy.io.loadmat mimics Matlab behavior per default.\n\nFound a solution, one can access the content of the “scipy.io.matlab.mio5_params.mat_struct object” can be investigated via:\n\n``````v['b'].__dict__['c'].__dict__['d']\n``````\n\nAnother method that works:\n\n``````import scipy.io as spio\nprint vig['b']['c'].item()['d']\n``````\n\nOutput:\n\n3\n\nI learned this method on the scipy mailing list, too. I certainly don’t understand (yet) why ‘.item()’ has to be added in, and:\n\n``````print vig['b']['c']['d']\n``````\n\nIndexError: only integers, slices (`:`), ellipsis (`...`), numpy.newaxis (`None`) and integer or boolean arrays are valid indices" ]

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[ "## Geometry: Common Core (15th Edition)\n\nStart with the given equation. $14x=2(5x+14)$ Apply the distributive property. $14x=10x+28$ Subtract 10x from each side and simplify. $14x-10x=10x+28-10x$ $4x=28$ Divide each side by 4 and simplify. $4x\\div4=28\\div4$ $x=7$" ]

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[ "Search\nTranslate\n\n# Properties of Shapes\n\nPupils should be able to:\n\n• Identify 3-D shapes, including cubes and other cuboids, from 2-D representations.\n• Know angles are measured in degrees: estimate and compare acute, obtuse and reflex angles.\n• Draw given angles, and measure them in degrees (o).\n• Identify:\n• angles at a point and one whole turn (total 360o).\n• angles at a point on a straight line and 1/2 a turn (total 180o).\n• other multiples of 90o\n• Use the properties of rectangles to deduce related facts and find missing lengths and angles.\n• Distinguish between regular and irregular polygons based on reasoning about equal sides and angles.\n\nUseful Websites:", null, "", null, "BBC: Identifying angles Oak Academy: Angles BBC: Compare and order angles Oak Academy: 2-D and 3-D shapes BBC: Classifying triangles BBC: Classifying quadrilaterals\n\nTop" ]

[ null, "https://primarysite-prod-sorted.s3.amazonaws.com/botley/UploadedImage/611d4c4703db4064b60289b9571e70e2_1x1.png", null, "https://primarysite-prod-sorted.s3.amazonaws.com/botley/UploadedImage/a6e635ebdda24cdf86e6696418979d19_1x1.png", null ]

[ "# Memoization in Python – A Brief Introduction", null, "In this tutorial, we are going to discuss one of the very popular optimization techniques – Memoization in Python – primarily used to speed up computer programs. So, let’s get started!\n\n## What is memoization in Python?\n\nIn the world of computer programming, Memoisation or Memoization in Python is a special kind of optimization technique that is primarily used to speed up our computer program. It effectively reduces the runtime of the computer program by storing the results of the expensive (in terms of runtime) function calls into the memory and using it whenever any stored or cached value is required.\n\nIt ensures that a particular function or method does need not run more than once for the same set of inputs as the results are already available as cached /stored data.\n\nIt is similar to caching. It involves caching the return values of the function based on its input parameters.\n\nIn Python, we can implement the memoization technique in our programs using function and class-based decorators. And we will be using a recursive Python program to calculate the nth Fibonacci number in our whole discussion because for greater inputs this program will become very very slow because the number of function calls for the same input values increase with the input size.\n\n## Memoization in Python using function based decorators\n\nIt is the best and the complex way of implementing the memoization technique in Python, for those who want to understand how this optimization technique actually works. In this method of implementation of memoization technique, we define our own decorator function in Python to cache/store the return values of the function calls. Let’s see how to write Python code to implement this.\n\n```# Memoization using function-based decorators\n\ndef memoize(f):\ncache = {}\ndef foo(x):\nif x not in cache:\ncache[x] = f(x)\nreturn cache[x]\nreturn foo\n\n@memoize\ndef fibonacci(n):\nif n == 0:\nreturn 0\nelif n == 1:\nreturn 1\nelse:\nreturn fibonacci(n-1) + fibonacci(n-2)\n\n# Driver code\nfibonacci(20)\n```\n\nOutput:\n\n```6765\n```\n\n## Memoization using class based decorators\n\nIt is the second-best and the complex way of implementing the memoization technique in Python, for beginners who want to understand how this optimization technique actually works. In this method of implementation of memoization technique, we define our own decorator class in Python to cache/store the return values of the function calls. Let’s write Python code to implement this.\n\n```# Memoization using class-based decorators\n\nclass classMemoize:\ndef __init__(self, f):\nself.f = f\nself.cache = {}\ndef __call__(self, *x):\nif x not in self.cache:\nself.cache[x] = self.f(*x)\nreturn self.cache[x]\n\n@classMemoize\ndef fibonacci(n):\nif n == 0:\nreturn 0\nelif n == 1:\nreturn 1\nelse:\nreturn fibonacci(n-1) + fibonacci(n-2)\n\n# Driver code\nfibonacci(50)\n```\n\nOutput\n\n```12586269025\n```\n\n## Memoization using built-in decorator functions\n\nIt is one of the simple and the easiest way to implement the memoization technique in Python.\n\nIn this method of implementation of memoization technique, we do not define our own decorator function or class but we make use of the built-in functions like `lru_cache()` and `cache()` to cache/store the intermediate results of a function call.\n\nThese `lru_cache()` and `cache()` functions are defined in the `funtools` library which is a standard Python library and it comes with the normal Python installation.\n\nThe `lru_cache(maxsize=None, typed=False)` function offers some customization features through its parameters like `maxsize` and `typed`. The parameter `maxsize` decides if the LRU features will be enabled or not by setting its value either None or an integer value. And the parameter `typed` decides if the different types of data are to be cached separately or not.\n\nAn integer Value will limit the size of the cache maintained during the execution of the Python program and the None value will disable the LRU features and then the cache can grow without any bound.\n\nThe `cache()` function is available from the Python 3.9 release onwards and it is equivalent to the `lru_cache(maxsize=None)` function in the `funtools` library.\n\n```# Memoization using built-in function\n\nimport functools\n\[email protected]\ndef fibonacci(n):\nif n == 0:\nreturn 0\nelif n == 1:\nreturn 1\nelse:\nreturn fibonacci(n-1) + fibonacci(n-2)\n\n# Driver code\nfibonacci(100)\n```\n\nOutput:\n\n```354224848179261915075\n```\n\n## Conclusion\n\nIn this tutorial, we have learned how to use the memoization technique in Python using function and class-based decorators. I hope you have well understood the things discussed above and are ready to use/implement this memoization technique in your Python program to boost its speed." ]

[ null, "data:image/svg+xml;base64,PHN2ZyB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciIHdpZHRoPSIxMDI0IiBoZWlnaHQ9IjY4MyIgdmlld0JveD0iMCAwIDEwMjQgNjgzIj48cmVjdCB3aWR0aD0iMTAwJSIgaGVpZ2h0PSIxMDAlIiBmaWxsPSIjY2ZkNGRiIi8+PC9zdmc+", null ]

[ "Home | | Physics | Coefficient of self induction\n\n# Coefficient of self induction\n\nWhen a current I flows through a coil, the magnetic flux (?) linked with the coil is proportional to the current.\n\nCoefficient of self induction\n\nWhen a current I flows through a coil, the magnetic flux (φ) linked with the coil is proportional to the current.\n\nφ α I  or   φ = LI\n\nwhere L is a constant of proportionality and is called coefficient of self induction or self inductance.\n\nIf I = 1A, φ = L × 1, then L = φ Therefore, coefficient of self induction of a coil is numerically equal to the magnetic flux linked with a coil when unit current flows through it. According to laws of electromagnetic induction.", null, "The coefficient of self induction of a coil is numerically equal to the opposing emf induced in the coil when the rate of change of current through the coil is unity. The unit of self inductance is henry (H).\n\nOne henry is defined as the self-inductance of a coil in which a change in current of one ampere per second produces an opposing emf of one volt.\n\nStudy Material, Lecturing Notes, Assignment, Reference, Wiki description explanation, brief detail\nPhysics : Effects of electric current : Higher Secondary(12 Std) : Coefficient of self induction |" ]

[ null, "https://img.brainkart.com/imagebk13/i2ZmfEM.jpg", null ]

[ "## Friday, April 18, 2014\n\n### 47 | Christians and Republicans\n\nIn 1860, the Republican Party overtook the Whig Party; only in names, not in numbers.  Let us examine what I mean.\n• Republican = 9+5+7+3+2+3+9+3+1+5 = 47\nWhig = 23+8+9+7 = 47\n• Republican is decoded with the Pythagorean Method; the practice of reducing letters to single digits before summing\n• Whig is decoded with the Standard Method\nToday the Republican Party claims to be the \"Christian Party\" and the \"God Party\".  They're also the \"Flag\" party, but certainly not the *** Party.\n• Christian = 3+8+9+9+1+2+9+1+5 = 47\nMany people say Christians and Republicans are the opposite of Jesus.\n• Jesus = 10+5+19+21+19 = 74\n• Cross = 3+18+15+19+19 = 74\n• Fruit = 6+18+21+9+20 = 74\n• Lucifer = 12+21+3+9+6+5+18 = 74\n• Occult = 15+3+3+21+12+20 = 74\n• Judge = 10+21+4+7+5 = 47\n• Obey = 15+2+5+25 = 47\n• Candy = 3+1+14+4+25 = 47\n• Pop = 16+15+16 = 47\n• President = 7+9+5+1+9+4+5+5+2 = 47\n• Notice how 47 is the inverse of 74\nOregon is the 33rd state and has the abbreviation of OR.\n• OR = 15+18 = 33\n• Oregon = 15+18+5+7+15+14 = 74\nWith regards to God and Flags, checkout the following.\n• God = 7+15+4 = 26\n• Flag = 6+12+1+7 = 26\n• Lie = 12+9+5 = 26\n• Sheep = 1+8+5+5+7 = 26\n\nDoesn't it all make perfect sense?  Welcome to the big hoax, exposed.\n\nInterestingly enough, in numerology, 55 is the 'God Number'.\n• Heaven = 8+5+1+22+5+14 = 55\n• Sky = 19+11+25 = 55\n• Cloud = 3+12+15+21+4 = 55\n• Satan = 19+1+20+1+14 = 55\n• Psychology = 7+1+7+3+8+6+3+6+7+7 = 55\n• New Age = 14+5+23+1+7+5 = 55\nVaccines began in 1955... when Dwight D. Eisenhower was the Republican President of the United States, or the 47 47." ]

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[ "# Denoise Speech Using Deep Learning Networks\n\nThis example shows how to denoise speech signals using deep learning networks. The example compares two types of networks applied to the same task: fully connected, and convolutional.\n\n### Introduction\n\nThe aim of speech denoising is to remove noise from speech signals while enhancing the quality and intelligibility of speech. This example showcases the removal of washing machine noise from speech signals using deep learning networks. The example compares two types of networks applied to the same task: fully connected, and convolutional.\n\n### Problem Summary\n\nConsider the following speech signal sampled at 8 kHz.\n\n```[cleanAudio,fs] = audioread(\"SpeechDFT-16-8-mono-5secs.wav\"); sound(cleanAudio,fs)```\n\nAdd washing machine noise to the speech signal. Set the noise power such that the signal-to-noise ratio (SNR) is zero dB.\n\n```noise = audioread(\"WashingMachine-16-8-mono-1000secs.mp3\"); % Extract a noise segment from a random location in the noise file ind = randi(numel(noise) - numel(cleanAudio) + 1,1,1); noiseSegment = noise(ind:ind + numel(cleanAudio) - 1); speechPower = sum(cleanAudio.^2); noisePower = sum(noiseSegment.^2); noisyAudio = cleanAudio + sqrt(speechPower/noisePower)*noiseSegment;```\n\nListen to the noisy speech signal.\n\n`sound(noisyAudio,fs)`\n\nVisualize the original and noisy signals.\n\n```t = (1/fs)*(0:numel(cleanAudio) - 1); figure(1) tiledlayout(2,1) nexttile plot(t,cleanAudio) title(\"Clean Audio\") grid on nexttile plot(t,noisyAudio) title(\"Noisy Audio\") xlabel(\"Time (s)\") grid on```", null, "The objective of speech denoising is to remove the washing machine noise from the speech signal while minimizing undesired artifacts in the output speech.\n\n### Examine the Dataset\n\nThis example uses a subset of the Mozilla Common Voice dataset to train and test the deep learning networks. The data set contains 48 kHz recordings of subjects speaking short sentences. Download the data set and unzip the downloaded file.\n\n```downloadFolder = matlab.internal.examples.downloadSupportFile(\"audio\",\"commonvoice.zip\"); dataFolder = tempdir; unzip(downloadFolder,dataFolder) dataset = fullfile(dataFolder,\"commonvoice\");```\n\nUse `audioDatastore` to create a datastore for the training set. To speed up the runtime of the example at the cost of performance, set `speedupExample` to `true`.\n\n```adsTrain = audioDatastore(fullfile(dataset,\"train\"),IncludeSubfolders=true); speedupExample =", null, "true; if speedupExample adsTrain = shuffle(adsTrain); adsTrain = subset(adsTrain,1:1000); end```\n\nUse `read` to get the contents of the first file in the datastore.\n\n`[audio,adsTrainInfo] = read(adsTrain);`\n\nListen to the speech signal.\n\n`sound(audio,adsTrainInfo.SampleRate)`\n\nPlot the speech signal.\n\n```figure(2) t = (1/adsTrainInfo.SampleRate) * (0:numel(audio)-1); plot(t,audio) title(\"Example Speech Signal\") xlabel(\"Time (s)\") grid on```", null, "### Deep Learning System Overview\n\nThe basic deep learning training scheme is shown below. Note that, since speech generally falls below 4 kHz, you first downsample the clean and noisy audio signals to 8 kHz to reduce the computational load of the network. The predictor and target network signals are the magnitude spectra of the noisy and clean audio signals, respectively. The network's output is the magnitude spectrum of the denoised signal. The regression network uses the predictor input to minimize the mean square error between its output and the input target. The denoised audio is converted back to the time domain using the output magnitude spectrum and the phase of the noisy signal .", null, "You transform the audio to the frequency domain using the Short-Time Fourier transform (STFT), with a window length of 256 samples, an overlap of 75%, and a Hamming window. You reduce the size of the spectral vector to 129 by dropping the frequency samples corresponding to negative frequencies (because the time-domain speech signal is real, this does not lead to any information loss). The predictor input consists of 8 consecutive noisy STFT vectors, so that each STFT output estimate is computed based on the current noisy STFT and the 7 previous noisy STFT vectors.", null, "### STFT Targets and Predictors\n\nThis section illustrates how to generate the target and predictor signals from one training file.\n\nFirst, define system parameters:\n\n```windowLength = 256; win = hamming(windowLength,\"periodic\"); overlap = round(0.75*windowLength); fftLength = windowLength; inputFs = 48e3; fs = 8e3; numFeatures = fftLength/2 + 1; numSegments = 8;```\n\nCreate a `dsp.SampleRateConverter` (DSP System Toolbox) object to convert the 48 kHz audio to 8 kHz.\n\n`src = dsp.SampleRateConverter(InputSampleRate=inputFs,OutputSampleRate=fs,Bandwidth=7920);`\n\nUse `read` to get the contents of an audio file from the datastore.\n\n`audio = read(adsTrain);`\n\nMake sure the audio length is a multiple of the sample rate converter decimation factor.\n\n```decimationFactor = inputFs/fs; L = floor(numel(audio)/decimationFactor); audio = audio(1:decimationFactor*L);```\n\nConvert the audio signal to 8 kHz.\n\n```audio = src(audio); reset(src)```\n\nCreate a random noise segment from the washing machine noise vector.\n\n```randind = randi(numel(noise) - numel(audio),[1 1]); noiseSegment = noise(randind:randind + numel(audio) - 1);```\n\nAdd noise to the speech signal such that the SNR is 0 dB.\n\n```noisePower = sum(noiseSegment.^2); cleanPower = sum(audio.^2); noiseSegment = noiseSegment.*sqrt(cleanPower/noisePower); noisyAudio = audio + noiseSegment;```\n\nUse `stft` (Signal Processing Toolbox) to generate magnitude STFT vectors from the original and noisy audio signals.\n\n```cleanSTFT = stft(audio,Window=win,OverlapLength=overlap,fftLength=fftLength); cleanSTFT = abs(cleanSTFT(numFeatures-1:end,:)); noisySTFT = stft(noisyAudio,Window=win,OverlapLength=overlap,fftLength=fftLength); noisySTFT = abs(noisySTFT(numFeatures-1:end,:));```\n\nGenerate the 8-segment training predictor signals from the noisy STFT. The overlap between consecutive predictors is 7 segments.\n\n```noisySTFT = [noisySTFT(:,1:numSegments - 1),noisySTFT]; stftSegments = zeros(numFeatures,numSegments,size(noisySTFT,2) - numSegments + 1); for index = 1:size(noisySTFT,2) - numSegments + 1 stftSegments(:,:,index) = noisySTFT(:,index:index + numSegments - 1); end```\n\nSet the targets and predictors. The last dimension of both variables corresponds to the number of distinct predictor/target pairs generated by the audio file. Each predictor is 129-by-8, and each target is 129-by-1.\n\n```targets = cleanSTFT; size(targets)```\n```ans = 1×2 129 544 ```\n```predictors = stftSegments; size(predictors)```\n```ans = 1×3 129 8 544 ```\n\n### Extract Features Using Tall Arrays\n\nTo speed up processing, extract feature sequences from the speech segments of all audio files in the datastore using tall arrays. Unlike in-memory arrays, tall arrays typically remain unevaluated until you call the `gather` function. This deferred evaluation enables you to work quickly with large data sets. When you eventually request output using `gather`, MATLAB combines the queued calculations where possible and takes the minimum number of passes through the data. If you have Parallel Computing Toolbox™, you can use tall arrays in your local MATLAB session, or on a local parallel pool. You can also run tall array calculations on a cluster if you have MATLAB® Parallel Server™ installed.\n\nFirst, convert the datastore to a tall array.\n\n```reset(adsTrain) T = tall(adsTrain)```\n```Starting parallel pool (parpool) using the 'local' profile ... Connected to the parallel pool (number of workers: 6). T = M×1 tall cell array {234480×1 double} {210288×1 double} {282864×1 double} {292080×1 double} {410736×1 double} {303600×1 double} {326640×1 double} {233328×1 double} : : : : ```\n\nThe display indicates that the number of rows (corresponding to the number of files in the datastore), M, is not yet known. M is a placeholder until the calculation completes.\n\nExtract the target and predictor magnitude STFT from the tall table. This action creates new tall array variables to use in subsequent calculations. The function `HelperGenerateSpeechDenoisingFeatures` performs the steps already highlighted in the STFT Targets and Predictors section. The `cellfun` command applies `HelperGenerateSpeechDenoisingFeatures` to the contents of each audio file in the datastore.\n\n`[targets,predictors] = cellfun(@(x)HelperGenerateSpeechDenoisingFeatures(x,noise,src),T,UniformOutput=false);`\n\nUse `gather` to evaluate the targets and predictors.\n\n`[targets,predictors] = gather(targets,predictors);`\n```Evaluating tall expression using the Parallel Pool 'local': - Pass 1 of 1: Completed in 52 sec Evaluation completed in 1 min 53 sec ```\n\nIt is good practice to normalize all features to zero mean and unity standard deviation.\n\nCompute the mean and standard deviation of the predictors and targets, respectively, and use them to normalize the data.\n\n```predictors = cat(3,predictors{:}); noisyMean = mean(predictors(:)); noisyStd = std(predictors(:)); predictors(:) = (predictors(:) - noisyMean)/noisyStd; targets = cat(2,targets{:}); cleanMean = mean(targets(:)); cleanStd = std(targets(:)); targets(:) = (targets(:) - cleanMean)/cleanStd;```\n\nReshape predictors and targets to the dimensions expected by the deep learning networks.\n\n```predictors = reshape(predictors,size(predictors,1),size(predictors,2),1,size(predictors,3)); targets = reshape(targets,1,1,size(targets,1),size(targets,2));```\n\nYou will use 1% of the data for validation during training. Validation is useful to detect scenarios where the network is overfitting the training data.\n\nRandomly split the data into training and validation sets.\n\n```inds = randperm(size(predictors,4)); L = round(0.99*size(predictors,4)); trainPredictors = predictors(:,:,:,inds(1:L)); trainTargets = targets(:,:,:,inds(1:L)); validatePredictors = predictors(:,:,:,inds(L+1:end)); validateTargets = targets(:,:,:,inds(L+1:end));```\n\n### Speech Denoising with Fully Connected Layers\n\nYou first consider a denoising network comprised of fully connected layers. Each neuron in a fully connected layer is connected to all activations from the previous layer. A fully connected layer multiplies the input by a weight matrix and then adds a bias vector. The dimensions of the weight matrix and bias vector are determined by the number of neurons in the layer and the number of activations from the previous layer.", null, "Define the layers of the network. Specify the input size to be images of size `NumFeatures`-by-`NumSegments` (129-by-8 in this example). Define two hidden fully connected layers, each with 1024 neurons. Since purely linear systems, follow each hidden fully connected layer with a Rectified Linear Unit (ReLU) layer. The batch normalization layers normalize the means and standard deviations of the outputs. Add a fully connected layer with 129 neurons, followed by a regression layer.\n\n```layers = [ imageInputLayer([numFeatures,numSegments]) fullyConnectedLayer(1024) batchNormalizationLayer reluLayer fullyConnectedLayer(1024) batchNormalizationLayer reluLayer fullyConnectedLayer(numFeatures) regressionLayer ];```\n\nNext, specify the training options for the network. Set `MaxEpochs` to `3` so that the network makes 3 passes through the training data. Set `MiniBatchSize` of `128` so that the network looks at 128 training signals at a time. Specify `Plots` as `\"training-progress\"` to generate plots that show the training progress as the number of iterations increases. Set `Verbose` to `false` to disable printing the table output that corresponds to the data shown in the plot into the command line window. Specify `Shuffle` as `\"every-epoch\"` to shuffle the training sequences at the beginning of each epoch. Specify `LearnRateSchedule` to `\"piecewise\"` to decrease the learning rate by a specified factor (0.9) every time a certain number of epochs (1) has passed. Set `ValidationData` to the validation predictors and targets. Set `ValidationFrequency` such that the validation mean square error is computed once per epoch. This example uses the adaptive moment estimation (Adam) solver.\n\n```miniBatchSize = 128; options = trainingOptions(\"adam\", ... MaxEpochs=3, ... InitialLearnRate=1e-5,... MiniBatchSize=miniBatchSize, ... Shuffle=\"every-epoch\", ... Plots=\"training-progress\", ... Verbose=false, ... ValidationFrequency=floor(size(trainPredictors,4)/miniBatchSize), ... LearnRateSchedule=\"piecewise\", ... LearnRateDropFactor=0.9, ... LearnRateDropPeriod=1, ... ValidationData={validatePredictors,validateTargets});```\n\nTrain the network with the specified training options and layer architecture using `trainNetwork`. Because the training set is large, the training process can take several minutes. To download and load a pre-trained network instead of training a network from scratch, set `downloadPretrainedSystem` to `true`.\n\n```downloadPretrainedSystem =", null, "false; if downloadPretrainedSystem downloadFolder = matlab.internal.examples.downloadSupportFile(\"audio\",\"SpeechDenoising.zip\"); dataFolder = tempdir; unzip(downloadFolder,dataFolder) netFolder = fullfile(dataFolder,\"SpeechDenoising\"); s = load(fullfile(netFolder,\"denoisenet.mat\")); denoiseNetFullyConnected = s.denoiseNetFullyConnected; cleanMean = s.cleanMean; cleanStd = s.cleanStd; noisyMean = s.noisyMean; noisyStd = s.noisyStd; else denoiseNetFullyConnected = trainNetwork(trainPredictors,trainTargets,layers,options); end```\n\nCount the number of weights in the fully connected layers of the network.\n\n```numWeights = 0; for index = 1:numel(denoiseNetFullyConnected.Layers) if isa(denoiseNetFullyConnected.Layers(index),\"nnet.cnn.layer.FullyConnectedLayer\") numWeights = numWeights + numel(denoiseNetFullyConnected.Layers(index).Weights); end end disp(\"Number of weights = \" + numWeights);```\n```Number of weights = 2237440 ```\n\n### Speech Denoising with Convolutional Layers\n\nConsider a network that uses convolutional layers instead of fully connected layers . A 2-D convolutional layer applies sliding filters to the input. The layer convolves the input by moving the filters along the input vertically and horizontally and computing the dot product of the weights and the input, and then adding a bias term. Convolutional layers typically consist of fewer parameters than fully connected layers.\n\nDefine the layers of the fully convolutional network described in , comprising 16 convolutional layers. The first 15 convolutional layers are groups of 3 layers, repeated 5 times, with filter widths of 9, 5, and 9, and number of filters of 18, 30 and 8, respectively. The last convolutional layer has a filter width of 129 and 1 filter. In this network, convolutions are performed in only one direction (along the frequency dimension), and the filter width along the time dimension is set to 1 for all layers except the first one. Similar to the fully connected network, convolutional layers are followed by ReLu and batch normalization layers.\n\n```layers = [imageInputLayer([numFeatures,numSegments]) convolution2dLayer([9 8],18,Stride=[1 100],Padding=\"same\") batchNormalizationLayer reluLayer repmat( ... [convolution2dLayer([5 1],30,Stride=[1 100],Padding=\"same\") batchNormalizationLayer reluLayer convolution2dLayer([9 1],8,Stride=[1 100],Padding=\"same\") batchNormalizationLayer reluLayer convolution2dLayer([9 1],18,Stride=[1 100],Padding=\"same\") batchNormalizationLayer reluLayer],4,1) convolution2dLayer([5 1],30,Stride=[1 100],Padding=\"same\") batchNormalizationLayer reluLayer convolution2dLayer([9 1],8,Stride=[1 100],Padding=\"same\") batchNormalizationLayer reluLayer convolution2dLayer([129 1],1,Stride=[1 100],Padding=\"same\") regressionLayer ];```\n\nThe training options are identical to the options for the fully connected network, except that the dimensions of the validation target signals are permuted to be consistent with the dimensions expected by the regression layer.\n\n```options = trainingOptions(\"adam\", ... MaxEpochs=3, ... InitialLearnRate=1e-5, ... MiniBatchSize=miniBatchSize, ... Shuffle=\"every-epoch\", ... Plots=\"training-progress\", ... Verbose=false, ... ValidationFrequency=floor(size(trainPredictors,4)/miniBatchSize), ... LearnRateSchedule=\"piecewise\", ... LearnRateDropFactor=0.9, ... LearnRateDropPeriod=1, ... ValidationData={validatePredictors,permute(validateTargets,[3 1 2 4])});```\n\nTrain the network with the specified training options and layer architecture using `trainNetwork`. Because the training set is large, the training process can take several minutes. To download and load a pre-trained network instead of training a network from scratch, set `downloadPretrainedSystem` to `true`.\n\n```downloadPretrainedSystem =", null, "false; if downloadPretrainedSystem downloadFolder = matlab.internal.examples.downloadSupportFile(\"audio\",\"SpeechDenoising.zip\"); dataFolder = tempdir; unzip(downloadFolder,dataFolder) netFolder = fullfile(dataFolder,\"SpeechDenoising\"); s = load(fullfile(netFolder,\"denoisenet.mat\")); denoiseNetFullyConvolutional = s.denoiseNetFullyConvolutional; cleanMean = s.cleanMean; cleanStd = s.cleanStd; noisyMean = s.noisyMean; noisyStd = s.noisyStd; else denoiseNetFullyConvolutional = trainNetwork(trainPredictors,permute(trainTargets,[3 1 2 4]),layers,options); end```\n\nCount the number of weights in the fully connected layers of the network.\n\n```numWeights = 0; for index = 1:numel(denoiseNetFullyConvolutional.Layers) if isa(denoiseNetFullyConvolutional.Layers(index),\"nnet.cnn.layer.Convolution2DLayer\") numWeights = numWeights + numel(denoiseNetFullyConvolutional.Layers(index).Weights); end end disp(\"Number of weights in convolutional layers = \" + numWeights);```\n```Number of weights in convolutional layers = 31812 ```\n\n### Test the Denoising Networks\n\nRead in the test data set.\n\n`adsTest = audioDatastore(fullfile(dataset,\"test\"),IncludeSubfolders=true);`\n\nRead the contents of a file from the datastore.\n\n`[cleanAudio,adsTestInfo] = read(adsTest);`\n\nMake sure the audio length is a multiple of the sample rate converter decimation factor.\n\n```L = floor(numel(cleanAudio)/decimationFactor); cleanAudio = cleanAudio(1:decimationFactor*L);```\n\nConvert the audio signal to 8 kHz.\n\n```cleanAudio = src(cleanAudio); reset(src)```\n\nIn this testing stage, you corrupt speech with washing machine noise not used in the training stage.\n\n`noise = audioread(\"WashingMachine-16-8-mono-200secs.mp3\");`\n\nCreate a random noise segment from the washing machine noise vector.\n\n```randind = randi(numel(noise) - numel(cleanAudio), [1 1]); noiseSegment = noise(randind:randind + numel(cleanAudio) - 1);```\n\nAdd noise to the speech signal such that the SNR is 0 dB.\n\n```noisePower = sum(noiseSegment.^2); cleanPower = sum(cleanAudio.^2); noiseSegment = noiseSegment.*sqrt(cleanPower/noisePower); noisyAudio = cleanAudio + noiseSegment;```\n\nUse `stft` to generate magnitude STFT vectors from the noisy audio signals.\n\n```noisySTFT = stft(noisyAudio,Window=win,OverlapLength=overlap,fftLength=fftLength); noisyPhase = angle(noisySTFT(numFeatures-1:end,:)); noisySTFT = abs(noisySTFT(numFeatures-1:end,:));```\n\nGenerate the 8-segment training predictor signals from the noisy STFT. The overlap between consecutive predictors is 7 segments.\n\n```noisySTFT = [noisySTFT(:,1:numSegments-1) noisySTFT]; predictors = zeros(numFeatures,numSegments,size(noisySTFT,2) - numSegments + 1); for index = 1:(size(noisySTFT,2) - numSegments + 1) predictors(:,:,index) = noisySTFT(:,index:index + numSegments - 1); end```\n\nNormalize the predictors by the mean and standard deviation computed in the training stage.\n\n`predictors(:) = (predictors(:) - noisyMean)/noisyStd;`\n\nCompute the denoised magnitude STFT by using `predict` with the two trained networks.\n\n```predictors = reshape(predictors,[numFeatures,numSegments,1,size(predictors,3)]); STFTFullyConnected = predict(denoiseNetFullyConnected,predictors); STFTFullyConvolutional = predict(denoiseNetFullyConvolutional,predictors);```\n\nScale the outputs by the mean and standard deviation used in the training stage.\n\n```STFTFullyConnected(:) = cleanStd*STFTFullyConnected(:) + cleanMean; STFTFullyConvolutional(:) = cleanStd*STFTFullyConvolutional(:) + cleanMean;```\n\nConvert the one-sided STFT to a centered STFT.\n\n```STFTFullyConnected = (STFTFullyConnected.').*exp(1j*noisyPhase); STFTFullyConnected = [conj(STFTFullyConnected(end-1:-1:2,:));STFTFullyConnected]; STFTFullyConvolutional = squeeze(STFTFullyConvolutional).*exp(1j*noisyPhase); STFTFullyConvolutional = [conj(STFTFullyConvolutional(end-1:-1:2,:));STFTFullyConvolutional];```\n\nCompute the denoised speech signals. `istft` performs the inverse STFT. Use the phase of the noisy STFT vectors to reconstruct the time-domain signal.\n\n```denoisedAudioFullyConnected = istft(STFTFullyConnected,Window=win,OverlapLength=overlap,fftLength=fftLength,ConjugateSymmetric=true); denoisedAudioFullyConvolutional = istft(STFTFullyConvolutional,Window=win,OverlapLength=overlap,fftLength=fftLength,ConjugateSymmetric=true);```\n\nPlot the clean, noisy and denoised audio signals.\n\n```t = (1/fs)*(0:numel(denoisedAudioFullyConnected)-1); figure(3) tiledlayout(4,1) nexttile plot(t,cleanAudio(1:numel(denoisedAudioFullyConnected))) title(\"Clean Speech\") grid on nexttile plot(t,noisyAudio(1:numel(denoisedAudioFullyConnected))) title(\"Noisy Speech\") grid on nexttile plot(t,denoisedAudioFullyConnected) title(\"Denoised Speech (Fully Connected Layers)\") grid on nexttile plot(t,denoisedAudioFullyConvolutional) title(\"Denoised Speech (Convolutional Layers)\") grid on xlabel(\"Time (s)\")```", null, "Plot the clean, noisy, and denoised spectrograms.\n\n```h = figure(4); tiledlayout(4,1) nexttile spectrogram(cleanAudio,win,overlap,fftLength,fs); title(\"Clean Speech\") grid on nexttile spectrogram(noisyAudio,win,overlap,fftLength,fs); title(\"Noisy Speech\") grid on nexttile spectrogram(denoisedAudioFullyConnected,win,overlap,fftLength,fs); title(\"Denoised Speech (Fully Connected Layers)\") grid on nexttile spectrogram(denoisedAudioFullyConvolutional,win,overlap,fftLength,fs); title(\"Denoised Speech (Convolutional Layers)\") grid on p = get(h,\"Position\"); set(h,\"Position\",[p(1) 65 p(3) 800]);```", null, "Listen to the noisy speech.\n\n`sound(noisyAudio,fs)`\n\nListen to the denoised speech from the network with fully connected layers.\n\n`sound(denoisedAudioFullyConnected,fs)`\n\nListen to the denoised speech from the network with convolutional layers.\n\n`sound(denoisedAudioFullyConvolutional,fs)`\n\nListen to clean speech.\n\n`sound(cleanAudio,fs)`\n\nYou can test more files from the datastore by calling `testDenoisingNets`. The function produces the time-domain and frequency-domain plots highlighted above, and also returns the clean, noisy, and denoised audio signals.\n\n`[cleanAudio,noisyAudio,denoisedAudioFullyConnected,denoisedAudioFullyConvolutional] = testDenoisingNets(adsTest,denoiseNetFullyConnected,denoiseNetFullyConvolutional,noisyMean,noisyStd,cleanMean,cleanStd);`", null, "", null, "### Real-Time Application\n\nThe procedure in the previous section passes the entire spectrum of the noisy signal to `predict`. This is not suitable for real-time applications where low latency is a requirement.\n\nRun `speechDenoisingRealtimeApp` for an example of how to simulate a streaming, real-time version of the denoising network. The app uses the network with fully connected layers. The audio frame length is equal to the STFT hop size, which is 0.25 * 256 = 64 samples.\n\n`speechDenoisingRealtimeApp` launches a User Interface (UI) designed to interact with the simulation. The UI enables you to tune parameters and the results are reflected in the simulation instantly. You can also enable/disable a noise gate that operates on the denoised output to further reduce the noise, as well as tune the attack time, release time, and threshold of the noise gate. You can listen to the noisy, clean or denoised audio from the UI.", null, "The scope plots the clean, noisy and denoised signals, as well as the gain of the noise gate.", null, "### References\n\n \"Experiments on Deep Learning for Speech Denoising\", Ding Liu, Paris Smaragdis, Minje Kim, INTERSPEECH, 2014.\n\n \"A Fully Convolutional Neural Network for Speech Enhancement\", Se Rim Park, Jin Won Lee, INTERSPEECH, 2017." ]

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[ "", null, "# MATH 2320 Linear Algebra\n\nA study of systems of linear equations, matrices, determinants, vector inner product and cross product, vector spaces, linear transformations, eigenvectors, and eigenvalues. Prerequisite: A grade of \"C' or higher in MATH 1330 or MATH 2310.\n\n3\n\nSciences" ]

[ null, "http://www.umhb.edu/sites/all/files/smartcatalog/images/bg.jpg", null ]

[ "# What is 67% of 10?\n\n✔️ Solved: 67% of 10 = 6.7\nExplanation:\n\nStep 1 - Divide by 100 and calculate 1% 👉 67 ÷ 100 = 0.67\n\nStep 2 - Multiply by a given number 👉 0.67 × 10\n\nStep 3 - We get our Answer 👉 6.7\n\nWhat is\n%\nof" ]

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[ "# HDU 6085 Rikka with Candies （bitset）\n\n## Description\n\nAs we know, Rikka is poor at math. Yuta is worrying about this situation, so he gives Rikka some math tasks to practice. There is one of them:\n\nThere are n children and m kinds of candies. The ith child has Ai dollars and the unit price of the ith kind of candy is Bi. The amount of each kind is infinity.\n\nEach child has his favorite candy, so he will buy this kind of candies as much as possible and will not buy any candies of other kinds. For example, if this child has 10 dollars and the unit price of his favorite candy is 4 dollars, then he will buy two candies and go home with 2 dollars left.\n\nNow Yuta has q queries, each of them gives a number k. For each query, Yuta wants to know the number of the pairs (i,j)(1≤i≤n,1≤j≤m) which satisfies if the ith child’s favorite candy is the jth kind, he will take k dollars home.\n\nTo reduce the difficulty, Rikka just need to calculate the answer modulo 2.\n\nBut It is still too difficult for Rikka. Can you help her?\n\n## Input\n\nThe first line contains a number t(1≤t≤5), the number of the testcases.\n\nFor each testcase, the first line contains three numbers n,m,q(1≤n,m,q≤50000).\n\nThe second line contains n numbers Ai(1≤Ai≤50000) and the third line contains m numbers Bi(1≤Bi≤50000).\n\nThen the fourth line contains q numbers ki(0≤ki<maxBi) , which describes the queries.\n\nIt is guaranteed that Ai≠Aj,Bi≠Bj for all i≠j.\n\n## Output\n\nFor each query, print a single line with a single 01 digit — the answer.\n\n## Sample Input\n\n1\n5 5 5\n1 2 3 4 5\n1 2 3 4 5\n0 1 2 3 4\n\n\n## Sample Output\n\n0\n0\n0\n0\n1\n\n\n## 题意\n\n$A$ 数组有 $n$ 个数， $B$ 数组有 $m$ 个数，随后有 $q$ 个查询，每次输入一个 $k$ ，询问有多少对 $(i,j)$ ， 使得 $A_i\\%B_j=k$ ， 输出结果模 $2$ 的值。\n\n## 思路\n\n$A_i\\%B_j=k$ 又等价于 $A_i-k=B_j \\times x$ ，其中 $x$ 为某一整数。\n\n## AC 代码\n\n#include<bits/stdc++.h>\n\nusing namespace std;\n\nconst int maxn = 5e4+10;\n\nbitset<maxn>a,b;\nbitset<maxn>ans,cnt;\n\nvoid slove(int maxK)\n{\ncnt.reset();\nans.reset();\nfor(int i=maxK; i>=0; --i) //枚举k\n{\nans[i]=(cnt&(a>>i)).count()&1; //存在多少个(a-i)%b=0\nif(b[i]) //枚举 i 的倍数\nfor(int j=0; j<maxn; j+=i)\ncnt.flip(j);\n}\n}\n\ntemplate <class T>\ninline void scan_d(T &ret)\n{\nchar c;\nret = 0;\nwhile ((c = getchar()) < '0' || c > '9');\nwhile (c >= '0' && c <= '9')\n{\nret = ret * 10 + (c - '0'), c = getchar();\n}\n}\n\nint main()\n{\nint T;\nscan_d(T);\nwhile(T--)\n{\nint n,m,q;\nscan_d(n);\nscan_d(m);\nscan_d(q);\na.reset();\nb.reset();\nint maxK=0;\nfor(int i=0; i<n; ++i)\n{\nint x;\nscan_d(x);\na.set(x);\n}\nfor(int i=0; i<m; ++i)\n{\nint x;\nscan_d(x);\nb.set(x);\nmaxK=max(maxK,x);\n}\nslove(maxK);\nwhile(q--)\n{\nint x;\nscan_d(x);\nputs(ans[x]?\"1\":\"0\");\n}\n}\nreturn 0;\n}" ]

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[ "# A sample proportion of 0.36 is found. To determine the margin of error for this statistic, a simulation of 100 trials is run, each with a sample size of 50 and a point estimate of 0.36.The minimum sam\n\nAlgebra\n\nA sample proportion of 0.36 is found. To determine the margin of error for this statistic, a simulation of 100 trials is run, each with a sample size of 50 and a point estimate of 0.36.\n\nJust \\$7 Welcome\n\nThe minimum sample proportion from the simulation is 0.28, and the maximum sample proportion from the simulation is 0.40.\n\nThe margin of error of the population proportion is found using half the range.\n\nWhat is the interval estimate of the true population proportion?\n\n• (0.22, 0.50)\n\n• (0.20, 0.52)\n\n• (0.30, 0.42)\n\n• (0.24, 0.48)\n\nA number generator was used to simulate the percentage of people in a town who can skateboard. The process simulates randomly selecting 100 people from the town and was repeated 20 times. The percentage of people who can skateboard in the town is shown in the dot plot.\n\nWhich statement is true about the population of the town?\n\n• Most likely, 30% to 40% of the town can skateboard.\n\n• Most likely, 40% to 50% of the town can skateboard.\n\n• Most likely, 10% to 20% of the town can skateboard.\n\n• Most likely, 20% to 35% of the town can skateboard.\n\nEight trials are simulated. The results are shown in the table.\n\nSimulation\n\n109105112110\n\n115106108109\n\nWhat is the estimated margin of error, using standard deviation?\n\nEnter your answer, rounded to two decimal places, in the box.\n\n±\n\nWhat is the best way to gather information for each of the scenarios?\n\nSelect Survey, Experiment, or Observational study for each scenario.\n\nSurveyExperimentObservational study\n\nA director wants to know what his singers do before a performance.\n\nA director wants to know who his best soprano is.\n\nA director wants to know how verbally disciplining a singer affects the group.\n\nA director wants to know what musical is the most popular in the community.\n\nA librarian wants to have an employee reading contest. She asks her employees to volunteer to read as many books as possible in one week.\n\nWhich sentences explain how randomization is not applied in this situation?" ]

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[ "Research on Fuzzy Comprehensive Evaluation of Passenger Satisfaction in Urban Public Transport\nAuthor(s) Di Yin\nABSTRACT\nBus has become an integral traffic mode for urban residents along with the development of public transport system; and as a kind of public service, urban public transport is the basic infrastructure closely related to people, and is also required to constantly improve its service levels to better serve people. Therefore, how to evaluate its service level has become one of the important projects that need to be studied. Based on establishing the mathematical model of fuzzy comprehensive evaluation and illustrated by the case of Xi’an, this paper verifies the model rationality, assesses the bus development level in Xi’an, and further puts forward countermeasures for its bus services according to the results.\n\n1. Introduction\n\nUrban public transport is not only the key to the modernization construction, but also an indispensable part of urban infrastructure construction, as well as an important link to build a harmonious society and improve the living standard of residents. As a kind of public service, public transport is also required to conduct marketing for most consumers under the background of continuously improved transport facilities. More importantly, the quality of bus service not only reflects the travel condition, social style, urban and spiritual civilization construction level, but also becomes the main indicator to measure the urban public transport marketing competitiveness.\n\nThe service level of urban public transport has been greatly improved in recent years, but the traditional increase in the number of buses and control of passenger fares still cannot solve the problems of low safety, poor comfort and environmental health of the bus. The improved people’s living standard and the reduced costs of cars have made more and more people choose private cars, thus exacerbating the urban traffic congestion, air pollution, noise and safety. As the service level of urban public transport directly affects its sustainable development goals, the research on the passenger satisfaction evaluation can effectively find problems in the process of public transport services from the perspective of “customers”, in order to improve the bus passenger satisfaction and public transport service level.\n\n2. Selection of Evaluation Index of Passenger Satisfaction in Public Transport\n\nThere is no complete, unified and effective evaluation index system of public transport passenger satisfaction in China, making different evaluation indicators in various regions. Therefore, based on the foreign evaluation indexes and the actual situation in China, the paper adopts the expert investigation to screen more suitable six major indicators (see Table 1), including safety , convenience , punctuality, rapidity, comfort and economy.\n\nIn the process of index selection, this article follows the scientific principle, objectivity principle, operability principle, measurability principle and Importance guarantee principle.\n\n1) Scientific principle: The choice of indicators and the determination of index weights, data selection, calculation and synthesis must be based on accepted scientific theories.\n\nTable 1. Passenger satisfaction evaluation index.\n\n2) Objectivity principle: Ensure the objective and fairness of the evaluation index system, ensure the accuracy of the data sources and the scientificity of the assessment methods.\n\n3) Operability principle: The indicators should try to select daily statistical indicators or easy available indicators so as to provide an intuitive and easy understanding of the stage of public transport and help improve the service level.\n\n4) Measurability principle: Customer Satisfaction Evaluation results must be quantifiable, so the selected indicators must be able to carry out statistical analysis.\n\n5) Importance guarantee principle : Need to grasp the needs of customers accurately, the selected indicators must be considered important by customers.\n\n3. Establishment of the Fuzzy Comprehensive Evaluation Model\n\nFuzzy comprehensive evaluation method is to regard the fuzzy object and fuzzy concept as the certain fuzzy set, then establish a fuzzy membership function, and conduct the quantitative analysis on fuzzy object through the relevant operations .\n\n1) Determine the factor set of fuzzy comprehensive evaluation object\n\nThe set of factors that affects the scores of evaluation objects is called the factor set, which is usually expressed by the letter U, $U=\\left\\{{u}_{1},{u}_{2},\\cdots ,{u}_{m}\\right\\}$ .\n\n2) Determine the weight set of each influencing factor\n\nThe influencing degree of each factor on the value of evaluation object is different, so different factors ${u}_{i}\\left(i=1,2,\\cdots \\right)$ should be given the corresponding weight coefficient ${\\alpha }_{i}\\left(i=1,2,\\cdots m\\right)$ , thus forming the weight set $A=\\left\\{{\\alpha }_{1},{\\alpha }_{2},\\cdots ,{\\alpha }_{m}\\right\\}$ .\n\nAnd the weight coefficient should be normalized before synthesis, which means\n\n$\\underset{i=1}{\\overset{m}{\\sum }}{\\alpha }_{i}=1,{\\alpha }_{i}\\ge 0\\left(i=1,2,\\cdots ,m\\right)$\n\nAs the weight set is a fuzzy set, in order to clearly represent the correspondence between weight coefficient and various factors, it can be expressed as\n\n$A=\\frac{{\\alpha }_{1}}{{u}_{1}}+\\frac{{\\alpha }_{2}}{{u}_{2}}+\\cdots +\\frac{{\\alpha }_{m}}{{u}_{m}}$\n\n3) Determine the evaluation set of the evaluation object\n\nThe evaluation set is a collection of various evaluation results that reviewers may make on the evaluation objects, and usually expressed as capital letter V, then $V=\\left\\{{V}_{1},{V}_{2},\\cdots ,{V}_{n}\\right\\}$ , and each level corresponds to a fuzzy subset.\n\nThe evaluation set is desirable as V = {very satisfied, more satisfied, general, not very satisfied, very dissatisfied} in the passenger satisfaction evaluation in public transport.\n\n4) Establish the fuzzy membership matrix\n\nThe single-factor fuzzy evaluation is to evaluate separately from a factor, and determine the elements membership of evaluation objects in the evaluation set.\n\nGenerally, the i-th element ui should be evaluated in the factor set, and if the membership degree of the j-th element vj is given as γij, the result can be expressed as a fuzzy set:\n\n${R}_{i}=\\frac{{\\gamma }_{i1}}{{v}_{1}}+\\frac{{\\gamma }_{i2}}{{v}_{2}}+\\cdots +\\frac{{\\gamma }_{in}}{{v}_{n}}\\left(i=1,2,\\cdots ,m\\right)$\n\nRi is a single-factor evaluation set, and the fuzzy matrix R formed by its membership is the single-factor evaluation matrix,\n\n$R=\\left[\\begin{array}{cccc}{\\gamma }_{11}& {\\gamma }_{12}& \\cdots & {\\gamma }_{1n}\\\\ {\\gamma }_{21}& {\\gamma }_{22}& \\cdots & {\\gamma }_{2n}\\\\ \\cdots & \\cdots & \\cdots & \\cdots \\\\ {\\gamma }_{m1}& {\\gamma }_{m2}& \\cdots & {\\gamma }_{mn}\\end{array}\\right]$\n\n5) Synthetic fuzzy comprehensive evaluation result vector\n\nThe evaluation result vector can be obtained by synthesizing the membership matrix of each subject with the appropriate weight set.\n\n$B=A*R=\\left({\\alpha }_{1},{\\alpha }_{2},\\cdots ,{\\alpha }_{m}\\right)\\left[\\begin{array}{cccc}{\\gamma }_{11}& {\\gamma }_{12}& \\cdots & {\\gamma }_{1n}\\\\ {\\gamma }_{21}& {\\gamma }_{22}& \\cdots & {\\gamma }_{2n}\\\\ \\cdots & \\cdots & \\cdots & \\cdots \\\\ {\\gamma }_{m1}& {\\gamma }_{m2}& \\cdots & {\\gamma }_{mn}\\end{array}\\right]$\n\nThe matrix B can obtain the results according to the fuzzy evaluation method after the normalized process.\n\n4. Evaluation of Passenger Satisfaction in Public Transport-Illustrated by the Case of Xi’an\n\nWith the accelerating process of urbanization, economic development has stimulated the development of urban transport and brought tremendous pressure on urban transport. As a result, traffic congestion has become increasingly serious and the contradiction between supply and demand on roads has become increasingly acute. As a political, economic and cultural center of Shanxi Province, Xi’an has such characteristics as dense population, frequent activities, centralized facilities and tight land. It is reflected in the fact that Xi’an has less people and less traffic and fewer cars, these basic contradictions are prominent .\n\nIn the process of public transportation development, there is a single public transport system structure, an unreasonable layout of the line network, a low service rate of the site, and a poor service level of the system, which does not fully reflect the idea of “bus priority” and can no longer fully satisfy people’s travel needs. Quality requirements, the downward trend in the attraction of the travel crowd, resulting in the structure of the entire city traffic to the bad direction, reducing the utilization of urban roads.\n\nIn order to verify the rationality of the mathematical model, at the same time, put forward reasonable suggestions for the development of public transport in Xi’an. In this paper, Xi’an city as an example, Xi’an city bus passenger satisfaction survey were conducted, a total of 570 questionnaires were distributed, a total of 500 valid questionnaires were obtained, the statistical results in Table 2.\n\nThen makes the quantitative evaluation of its overall level of the according to the fuzzy evaluation method.\n\n1) Factor set\n\nThe factor set of the fuzzy comprehensive evaluation can be obtained according to the evaluation index determined above.\n\nU = {safety (u1), convenience (u2), punctuality (u3), rapidity (u4), economy (u5), comfort (u6)}\n\n2) Weight set\n\nBased on the results of the calculation, we can give the evaluation weight set of the six components of bus passenger satisfaction: A = {α1, α2, α3, α4, α5, α6} = {0.03, 0.25, 0.04, 0.15, 0.44, 0.09}\n\n3) Evaluation set\n\nThe evaluation set V = {V1, V2, V3, V4, V5} = {very satisfied, more satisfied, general, not satisfied, very dissatisfied}\n\nThe corresponding numerical set of the evaluation set is N = {N1, N2, N3, N4, N5} = {95, 85, 75, 65, 55}\n\n4) Single-factor fuzzy evaluation\n\nIt can be obtained from the data,\n\n$\\begin{array}{l}{R}_{1}=\\frac{0.4}{{V}_{1}}+\\frac{0.3}{{V}_{2}}+\\frac{0.21}{{V}_{3}}+\\frac{0.05}{{V}_{4}}+\\frac{0.04}{{V}_{5}}\\\\ {R}_{2}=\\frac{0.24}{{V}_{1}}+\\frac{0.17}{{V}_{2}}+\\frac{0.27}{{V}_{3}}+\\frac{0.21}{{V}_{4}}+\\frac{0.11}{{V}_{5}}\\\\ {R}_{3}=\\frac{0.17}{{V}_{1}}+\\frac{0.3}{{V}_{2}}+\\frac{0.31}{{V}_{3}}+\\frac{0.14}{{V}_{4}}+\\frac{0.08}{{V}_{5}}\\\\ {R}_{4}=\\frac{0.04}{{V}_{1}}+\\frac{0.12}{{V}_{2}}+\\frac{0.32}{{V}_{3}}+\\frac{0.42}{{V}_{4}}+\\frac{0.10}{{V}_{5}}\\\\ {R}_{5}=\\frac{0.05}{{V}_{1}}+\\frac{0.25}{{V}_{2}}+\\frac{0.48}{{V}_{3}}+\\frac{0.17}{{V}_{4}}+\\frac{0.05}{{V}_{5}}\\\\ {R}_{6}=\\frac{0.09}{{V}_{1}}+\\frac{0.31}{{V}_{2}}+\\frac{0.39}{{V}_{3}}+\\frac{0.17}{{V}_{4}}+\\frac{0.04}{{V}_{5}}\\end{array}$\n\nTable 2. The percentage Xi’an citizen’s satisfaction with public transit evaluation results.\n\nThe single-factor evaluation matrix is:\n\n$R=\\left[\\begin{array}{ccccc}0.4& 0.3& 0.21& 0.05& 0.04\\\\ 0.24& 0.17& 0.27& 0.21& 0.11\\\\ 0.17& 0.3& 0.31& 0.14& 0.08\\\\ 0.04& 0.12& 0.32& 0.42& 0.10\\\\ 0.05& 0.25& 0.48& 0.17& 0.05\\\\ 0.09& 0.31& 0.39& 0.17& 0.04\\end{array}\\right]$\n\nAnd then the complex operation of fuzzy matrix is\n\n$\\begin{array}{c}B=A*R=\\left(0.03,0.15,0.04,0.15,0.44,0.09\\right)\\left[\\begin{array}{ccccc}0.4& 0.3& 0.21& 0.05& 0.04\\\\ 0.24& 0.17& 0.27& 0.21& 0.11\\\\ 0.17& 0.3& 0.31& 0.14& 0.08\\\\ 0.04& 0.12& 0.32& 0.42& 0.10\\\\ 0.05& 0.25& 0.48& 0.17& 0.05\\\\ 0.09& 0.31& 0.39& 0.17& 0.04\\end{array}\\right]\\\\ =\\left(0.19,0.2,0.35,0.19,0.07\\right)\\end{array}$\n\n5) Fuzzy comprehensive evaluation\n\nWe can calculate specific scores in safety, convenience, punctuality, rapidity, economy and comfort of public transport in Xi’an according to the obtained weight and the corresponding numerical set of the evaluation set.\n\nThe score of safety\n\n${B}_{1}*N=\\left[\\begin{array}{ccccc}0.4& 0.3& 0.21& 0.05& 0.04\\end{array}\\right]\\left[\\begin{array}{c}95\\\\ 85\\\\ 75\\\\ 65\\\\ 55\\end{array}\\right]=84.7$\n\nThe score of convenience\n\n${B}_{2}*N=\\left[\\begin{array}{ccccc}0.24& 0.17& 0.27& 0.21& 0.11\\end{array}\\right]\\left[\\begin{array}{c}95\\\\ 85\\\\ 75\\\\ 65\\\\ 55\\end{array}\\right]=77.2$\n\nThe score of punctuality\n\n${B}_{3}*N=\\left[\\begin{array}{ccccc}0.17& 0.3& 0.31& 0.14& 0.08\\end{array}\\right]\\left[\\begin{array}{c}95\\\\ 85\\\\ 75\\\\ 65\\\\ 55\\end{array}\\right]=78.4$\n\nThe score of rapidity\n\n${B}_{4}*N=\\left[\\begin{array}{ccccc}0.04& 0.12& 0.32& 0.42& 0.10\\end{array}\\right]\\left[\\begin{array}{c}95\\\\ 85\\\\ 75\\\\ 65\\\\ 55\\end{array}\\right]=70.8$\n\nThe score of economy\n\n${B}_{5}*N=\\left[\\begin{array}{ccccc}0.05& 0.25& 0.48& 0.17& 0.05\\end{array}\\right]\\left[\\begin{array}{c}95\\\\ 85\\\\ 75\\\\ 65\\\\ 55\\end{array}\\right]=75.8$\n\nThe score of comfort\n\n${B}_{6}*N=\\left[\\begin{array}{ccccc}0.09& 0.31& 0.39& 0.17& 0.04\\end{array}\\right]\\left[\\begin{array}{c}95\\\\ 85\\\\ 75\\\\ 65\\\\ 55\\end{array}\\right]=77.4$\n\nThe comprehensive score\n\n$B*N=\\left[\\begin{array}{ccccc}0.19& 0.2& 0.35& 0.19& 0.07\\end{array}\\right]\\left[\\begin{array}{c}95\\\\ 85\\\\ 75\\\\ 65\\\\ 55\\end{array}\\right]=77.5$\n\nThe above calculation results show that the comprehensive score of bus passenger satisfaction in Xi’an is 77.5 and in the intermediate level, so many public transport services still need to be adjusted. And the scores of safety, convenience, punctuality, economy and comfort index are 84.7, 77.2, 78.4, 75.8 and 77.4, which are basically between the general and the satisfaction, belong to the intermediate level, and still require to be further improved. While the score of rapidity index is 70.8, and between not satisfied and the general, which is very urgent to be improved. Only in this way, we can improve the bus passenger satisfaction level in Xi’an, and make the overall level show a relatively stable upward trend.\n\n5. Conclusions\n\nAs a scientific, accurate and intuitive research method, the quantitative research of fuzzy comprehensive evaluation is widely applied under the background of the continuous developed society and science.\n\nBased on the establishment of the fuzzy comprehensive evaluation model, this paper mainly measures the bus passenger satisfaction in service level of Xi’an, verifies the reason and applicability of the model, and further puts forward suggestions on public transport services according to the results.\n\n1) To strengthen the investment of government in public transport infrastructure, and take measures to ensure its preferential development.\n\n2) To improve the hardware facilities of bus services, strengthen the construction of junction and interchange station.\n\n3) To conduct humanistic bus services, and improve the soft power of public transport.\n\n4) To establish and improve the bus service operation mechanism and enhance the social benefits.\n\nAt the same time, there are still some deficiencies in this study:\n\n1) The evaluation lacks consideration of regional environmental issues. This paper did not consider the influence of weather, seasons and urban culture when doing evaluation studies. In particular, Xi’an City, as a world-famous cultural city, has a very limited plan for bus routes. At the same time, due to different weather and seasons, people’s physiological experience is different , which may also have an impact on ride satisfaction.\n\n2) Bus collinear problem evaluation does not consider the problem that how many buses the passengers can choose to the same place. In the case of collinearity, ride satisfaction will affect passengers’ decision-making on line selection. This is also a direction of future research.\n\nCite this paper\nYin, D. (2018) Research on Fuzzy Comprehensive Evaluation of Passenger Satisfaction in Urban Public Transport. Modern Economy, 9, 528-535. doi: 10.4236/me.2018.93034.\nReferences\n   Wen, C.-H., Lan, L. and Chen, C.-H. (2005) Passengers Perception on Service Quality and Their Choice for Intercity Bus Services. The Transportation Research Board 84th Annual Meeting, Washington DC, 9-13 January 2005, 14-15.\n\n   Xing, K. (2004) Research on the Evaluation of Urban Public Traffic Service Level. Jilin University, Changchun, 1-2.\n\n   Yang, L.B. and Gao, Y. (1995) Fuzzy Mathematics Principle and Application. South China University of Technology Press, Guangzhou.\n\n   Fu, L.L., Liang, Y.H. and Wang, Y. (2007) Analysis of Current Traffic Situations and Problems in Xi’an City. Technology of Highway and Transport, No. 6, 117.\n\n   Huang, H.B. (2014) The Research on Changsha City Bus Passenger Satisfaction Evaluation by AHP-Fuzzy Comprehensive Evaluation. Central South University of Forestry and Technology, Changsha, 54.\n\nTop" ]

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[ "Full-text links:\n\nmath-ph\n\n# Title:Tau-structure for the Double Ramification Hierarchies\n\nAbstract: In this paper we continue the study of the double ramification hierarchy of [Bur15]. After showing that the DR hierarchy satisfies tau-symmetry we define its partition function as the (logarithm of the) tau-function of the string solution and show that it satisfies various properties (string, dilaton and divisor equations plus some important degree constraints). We then formulate a stronger version of the conjecture from [Bur15]: for any semisimple cohomological field theory, the Dubrovin-Zhang and double ramification hierarchies are related by a normal (i.e. preserving the tau-structure [DLYZ14]) Miura transformation which we completely identify in terms of the partition function of the CohFT. In fact, using only the partition functions, the conjecture can be formulated even in the non-semisimple case (where the Dubrovin-Zhang hierarchy is not defined). We then prove this conjecture for various CohFTs (trivial CohFT, Hodge class, Gromov-Witten theory of $\\mathbb{CP}^1$, $3$-, $4$- and $5$-spin classes) and in genus $1$ for any semisimple CohFT. Finally we prove that the higher genus part of the DR hierarchy is basically trivial for the Gromov-Witten theory of smooth varieties with non-positive first Chern class and their analogue in Fan-Jarvis-Ruan-Witten quantum singularity theory [FJRW].\n Comments: v3: 56 pages, minor changes, version accepted in the journal Subjects: Mathematical Physics (math-ph); Algebraic Geometry (math.AG) Journal reference: Communications in Mathematical Physics 363 (2018), no. 1, 191--260 DOI: 10.1007/s00220-018-3235-4 Cite as: arXiv:1602.05423 [math-ph] (or arXiv:1602.05423v3 [math-ph] for this version)\n\n## Submission history\n\nFrom: Alexandr Buryak [view email]\n[v1] Wed, 17 Feb 2016 14:21:45 UTC (55 KB)\n[v2] Fri, 19 Feb 2016 10:24:43 UTC (55 KB)\n[v3] Sat, 8 Dec 2018 22:46:47 UTC (55 KB)" ]

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[ "# How to remove the bump from the signal using filters?\n\nI want to have only the slope of the signal. How can I do? I have tried to apply some filters but so far I have not been able to hit the nail on the head. I use Matlab.", null, "A bump like this one is likely to be wide-band, especially with the sharp onset. Plus, the line may be hard to deal with in the Fourier domain. Hence, the combination is complicated to remove with a classical linear filter. The problem is very akin to baseline, background or trend removal, answered elsewhere here.\n\nSeveral options are possible, for instance:\n\n• use a non-linear filter, based on a median, or a minimum/maximum statistics,\n• use morphological operators: a rolling ball, lot of straight segments, etc.\n• use a knowledge on the data model, like a linear equation: $$y=ax+b$$, or the fact that the bump is \"above\",\n• combine the above in a variational formulation, using appropriate data fidelity and penalty.\n\nIn your example, I suspect that a classical linear fit with robust distance (like a least-absolute distortion) could do the job. I will call all the above filters, in the wide sense that you will remplace a value with respect to some sort of combination of the others.\n\nYou can also call the following robust regression, LAD fitting. An example at work:", null, "% Standard and Robust fit of a degree 1 polynomial w/ a bump\nnSample = 1000;\n% Create a similar composite signal\ntime = linspace(0,5,nSample)';\npolyCoef = [0.2 0];\ndataLine = polyval(polyCoef,time);\ndataParabola = -8*(time-2).*(time-3);\ndataParabola(dataParabola < 0) = 0;\ndata = dataLine+dataParabola;\n\n% Use Matlab curve fitting toolbox\noptsRobust = fitoptions('Method','LinearLeastSquares','Robust','LAR');\n[fitObject,gof] = fit(time,data,'poly1',optsRobust);\nh1=plot(fitObject,time,data);\ngrid on\n\n• Hi Mr Laurent, Thanks for the answers, I appreciate it. Can I send you any doubts of mine about signal processing? It is because, I'm makeing a degree final project, And the truth is, I get lost a little bit in some of the mathematical methods of processing. I'm trying to understanding what I'm doing with those. Can I keep in touch via private messages? Jun 6, 2020 at 12:07\n• By the way, in your answer, why do you get an output variable named 'h1', something special? Jun 6, 2020 at 12:12\n• No, h1 is a handle for the plot, I reused part of a former code, it is not needed here Jun 6, 2020 at 12:27\n• You can try, I cannot promise prompt answers. By posting questions here, you are likely to get more \"expert answers\" Jun 6, 2020 at 12:36\n• Sure! Thank you. So I had posting a doubt 15 days ago o more, and I'm still waiting for a answer. Can you view that post, if it's no trouble? I share the link: dsp.stackexchange.com/questions/67906/… Jun 6, 2020 at 16:15" ]

[ null, "https://i.stack.imgur.com/D8tYO.png", null, "https://i.stack.imgur.com/FTjlK.png", null ]

[ "", null, "", null, "Question\n\n#", null, "Four equal circles are described about the four corners of a square so that each touches two of the others, as shown in the figure. Find the area of the shaded region, if each side of the square measures 14 cm.\n\nSolution\n\n## area of square=14 x14=196 area of 4 sectors=4(90/360)(22/7)(7x7) =4(1/4)x(22x7) =154 area of shaded part =196-154                                    =42m2", null, "", null, "Suggest corrections", null, "", null, "", null, "" ]

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[ "NumWords.com\n\nHow to write Four thousand seven hundred one in numbers in English?\n\nWe can write Four thousand seven hundred one equal to 4701 in numbers in English\n\n< Four thousand seven hundred :||: Four thousand seven hundred two >\n\nNine thousand four hundred two = 9402 = 4701 × 2\nFourteen thousand one hundred three = 14103 = 4701 × 3\nEighteen thousand eight hundred four = 18804 = 4701 × 4\nTwenty-three thousand five hundred five = 23505 = 4701 × 5\nTwenty-eight thousand two hundred six = 28206 = 4701 × 6\nThirty-two thousand nine hundred seven = 32907 = 4701 × 7\nThirty-seven thousand six hundred eight = 37608 = 4701 × 8\nForty-two thousand three hundred nine = 42309 = 4701 × 9\nForty-seven thousand ten = 47010 = 4701 × 10\nFifty-one thousand seven hundred eleven = 51711 = 4701 × 11\nFifty-six thousand four hundred twelve = 56412 = 4701 × 12\nSixty-one thousand one hundred thirteen = 61113 = 4701 × 13\nSixty-five thousand eight hundred fourteen = 65814 = 4701 × 14\nSeventy thousand five hundred fifteen = 70515 = 4701 × 15\nSeventy-five thousand two hundred sixteen = 75216 = 4701 × 16\nSeventy-nine thousand nine hundred seventeen = 79917 = 4701 × 17\nEighty-four thousand six hundred eighteen = 84618 = 4701 × 18\nEighty-nine thousand three hundred nineteen = 89319 = 4701 × 19\nNinety-four thousand twenty = 94020 = 4701 × 20\n\nSitemap" ]

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[ "## A globally trust-region LP-Newton method for nonsmooth functions under the Hölder metric subregularity\n\nWe describe and analyse a globally convergent algorithm to find a possible nonisolated zero of a piecewise smooth mapping over a polyhedral set, such formulation includes Karush-Kuhn-Tucker (KKT) systems, variational inequalities problems, and generalized Nash equilibrium problems. Our algorithm is based on a modification of the fast locally convergent Linear Programming (LP)-Newton method with a … Read more\n\n## New quasi-Newton method for solving systems of nonlinear equations\n\nIn this report, we propose the new Broyden method for solving systems of nonlinear equations, which uses the first derivatives, but it is more efficient than the Newton method (measured by the computational time) for larger dense systems. The new method updates QR decompositions of nonsymmetric approximations of the Jacobian matrix, so it requires $O(n^2)$ … Read more\n\n## Optimal Deterministic Algorithm Generation\n\nA formulation for the automated generation of algorithms via mathematical programming (optimization) is proposed. The formulation is based on the concept of optimizing within a parameterized family of algorithms, or equivalently a family of functions describing the algorithmic steps. The optimization variables are the parameters – within this family of algorithms- that encode algorithm design: … Read more\n\n## Solution of Nonlinear Equations via Optimization\n\nThis paper presents four optimization models for solving nonlinear equation systems. The models accommodate both over-specified and under-specified systems. A variable endogenization technique that improves efficiency is introduced, and a basic comparative study shows one of the methods presented to be very effective. Citation Siwale, I. (2013). Solution of nonlinear equation systems via optimization. Technical … Read more\n\n## A sufficiently exact inexact Newton step based on reusing matrix information\n\nNewton’s method is a classical method for solving a nonlinear equation $F(z)=0$. We derive inexact Newton steps that lead to an inexact Newton method, applicable near a solution. The method is based on solving for a particular $F'(z_{k’})$ during $p$ consecutive iterations $k=k’,k’+1,\\dots,k’+p-1$. One such $p$-cycle requires $2^p-1$ solves with the matrix $F'(z_{k’})$. If matrix … Read more\n\n## Adjoint Broyden a la GMRES\n\nIt is shown that a compact storage implementation of a quasi-Newton method based on the adjoint Broyden update reduces in the affine case exactly to the well established GMRES procedure. Generally, storage and linear algebra effort per step are small multiples of n k, where n is the number of variables and k the number … Read more\n\n## An Accelerated Newton Method for Equations with Semismooth Jacobians and Nonlinear Complementarity Problems: Extended Version\n\nWe discuss local convergence of Newton’s method to a singular solution $x^*$ of the nonlinear equations $F(x) = 0$, for $F:\\R^n \\rightarrow \\R^n$. It is shown that an existing proof of Griewank, concerning linear convergence to a singular solution $x^*$ from a starlike domain around $x^*$ for $F$ twice Lipschitz continuously differentiable and $x^*$ satisfying … Read more" ]

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[ "# #18baec Color Information\n\nIn a RGB color space, hex #18baec is composed of 9.4% red, 72.9% green and 92.5% blue. Whereas in a CMYK color space, it is composed of 89.8% cyan, 21.2% magenta, 0% yellow and 7.5% black. It has a hue angle of 194.2 degrees, a saturation of 84.8% and a lightness of 51%. #18baec color hex could be obtained by blending #30ffff with #0075d9. Closest websafe color is: #00ccff.\n\n• R 9\n• G 73\n• B 93\nRGB color chart\n• C 90\n• M 21\n• Y 0\n• K 7\nCMYK color chart\n\n#18baec color description : Vivid cyan.\n\n# #18baec Color Conversion\n\nThe hexadecimal color #18baec has RGB values of R:24, G:186, B:236 and CMYK values of C:0.9, M:0.21, Y:0, K:0.07. Its decimal value is 1620716.\n\nHex triplet RGB Decimal 18baec `#18baec` 24, 186, 236 `rgb(24,186,236)` 9.4, 72.9, 92.5 `rgb(9.4%,72.9%,92.5%)` 90, 21, 0, 7 194.2°, 84.8, 51 `hsl(194.2,84.8%,51%)` 194.2°, 89.8, 92.5 00ccff `#00ccff`\nCIE-LAB 70.431, -20.87, -35.564 33.072, 41.365, 85.593 0.207, 0.258, 41.365 70.431, 41.235, 239.594 70.431, -48.088, -54.368 64.316, -20.765, -33.884 00011000, 10111010, 11101100\n\n# Color Schemes with #18baec\n\n• #18baec\n``#18baec` `rgb(24,186,236)``\n• #ec4a18\n``#ec4a18` `rgb(236,74,24)``\nComplementary Color\n• #18ecb4\n``#18ecb4` `rgb(24,236,180)``\n• #18baec\n``#18baec` `rgb(24,186,236)``\n• #1850ec\n``#1850ec` `rgb(24,80,236)``\nAnalogous Color\n• #ecb418\n``#ecb418` `rgb(236,180,24)``\n• #18baec\n``#18baec` `rgb(24,186,236)``\n• #ec1850\n``#ec1850` `rgb(236,24,80)``\nSplit Complementary Color\n• #baec18\n``#baec18` `rgb(186,236,24)``\n• #18baec\n``#18baec` `rgb(24,186,236)``\n• #ec18ba\n``#ec18ba` `rgb(236,24,186)``\n• #18ec4a\n``#18ec4a` `rgb(24,236,74)``\n• #18baec\n``#18baec` `rgb(24,186,236)``\n• #ec18ba\n``#ec18ba` `rgb(236,24,186)``\n• #ec4a18\n``#ec4a18` `rgb(236,74,24)``\n• #0e85aa\n``#0e85aa` `rgb(14,133,170)``\n• #1097c1\n``#1097c1` `rgb(16,151,193)``\n``#12aad9` `rgb(18,170,217)``\n• #18baec\n``#18baec` `rgb(24,186,236)``\n• #30c1ee\n``#30c1ee` `rgb(48,193,238)``\n• #47c8f0\n``#47c8f0` `rgb(71,200,240)``\n• #5fcff2\n``#5fcff2` `rgb(95,207,242)``\nMonochromatic Color\n\n# Alternatives to #18baec\n\nBelow, you can see some colors close to #18baec. Having a set of related colors can be useful if you need an inspirational alternative to your original color choice.\n\n• #18ece9\n``#18ece9` `rgb(24,236,233)``\n• #18ddec\n``#18ddec` `rgb(24,221,236)``\n• #18ccec\n``#18ccec` `rgb(24,204,236)``\n• #18baec\n``#18baec` `rgb(24,186,236)``\n• #18a8ec\n``#18a8ec` `rgb(24,168,236)``\n• #1897ec\n``#1897ec` `rgb(24,151,236)``\n• #1885ec\n``#1885ec` `rgb(24,133,236)``\nSimilar Colors\n\n# #18baec Preview\n\nThis text has a font color of #18baec.\n\n``<span style=\"color:#18baec;\">Text here</span>``\n#18baec background color\n\nThis paragraph has a background color of #18baec.\n\n``<p style=\"background-color:#18baec;\">Content here</p>``\n#18baec border color\n\nThis element has a border color of #18baec.\n\n``<div style=\"border:1px solid #18baec;\">Content here</div>``\nCSS codes\n``.text {color:#18baec;}``\n``.background {background-color:#18baec;}``\n``.border {border:1px solid #18baec;}``\n\n# Shades and Tints of #18baec\n\nA shade is achieved by adding black to any pure hue, while a tint is created by mixing white to any pure color. In this example, #000405 is the darkest color, while #f1fbfe is the lightest one.\n\n• #000405\n``#000405` `rgb(0,4,5)``\n• #021217\n``#021217` `rgb(2,18,23)``\n• #032029\n``#032029` `rgb(3,32,41)``\n• #052e3b\n``#052e3b` `rgb(5,46,59)``\n• #063c4d\n``#063c4d` `rgb(6,60,77)``\n• #084b5f\n``#084b5f` `rgb(8,75,95)``\n• #095971\n``#095971` `rgb(9,89,113)``\n• #0b6783\n``#0b6783` `rgb(11,103,131)``\n• #0c7596\n``#0c7596` `rgb(12,117,150)``\n• #0e83a8\n``#0e83a8` `rgb(14,131,168)``\n• #0f92ba\n``#0f92ba` `rgb(15,146,186)``\n• #11a0cc\n``#11a0cc` `rgb(17,160,204)``\n• #12aede\n``#12aede` `rgb(18,174,222)``\n• #18baec\n``#18baec` `rgb(24,186,236)``\n• #2abfed\n``#2abfed` `rgb(42,191,237)``\n• #3cc5ef\n``#3cc5ef` `rgb(60,197,239)``\n• #4ecaf0\n``#4ecaf0` `rgb(78,202,240)``\n• #60d0f2\n``#60d0f2` `rgb(96,208,242)``\n• #73d5f3\n``#73d5f3` `rgb(115,213,243)``\n• #85daf5\n``#85daf5` `rgb(133,218,245)``\n• #97e0f6\n``#97e0f6` `rgb(151,224,246)``\n• #a9e5f8\n``#a9e5f8` `rgb(169,229,248)``\n• #bbebf9\n``#bbebf9` `rgb(187,235,249)``\n• #cdf0fb\n``#cdf0fb` `rgb(205,240,251)``\n• #dff6fc\n``#dff6fc` `rgb(223,246,252)``\n• #f1fbfe\n``#f1fbfe` `rgb(241,251,254)``\nTint Color Variation\n\n# Tones of #18baec\n\nA tone is produced by adding gray to any pure hue. In this case, #828282 is the less saturated color, while #0ebff6 is the most saturated one.\n\n• #828282\n``#828282` `rgb(130,130,130)``\n• #78878c\n``#78878c` `rgb(120,135,140)``\n• #6f8c95\n``#6f8c95` `rgb(111,140,149)``\n• #65919f\n``#65919f` `rgb(101,145,159)``\n• #5b96a9\n``#5b96a9` `rgb(91,150,169)``\n• #529cb2\n``#529cb2` `rgb(82,156,178)``\n• #48a1bc\n``#48a1bc` `rgb(72,161,188)``\n• #3ea6c6\n``#3ea6c6` `rgb(62,166,198)``\n• #35abcf\n``#35abcf` `rgb(53,171,207)``\n• #2bb0d9\n``#2bb0d9` `rgb(43,176,217)``\n• #22b5e2\n``#22b5e2` `rgb(34,181,226)``\n• #18baec\n``#18baec` `rgb(24,186,236)``\n• #0ebff6\n``#0ebff6` `rgb(14,191,246)``\nTone Color Variation\n\n# Color Blindness Simulator\n\nBelow, you can see how #18baec is perceived by people affected by a color vision deficiency. This can be useful if you need to ensure your color combinations are accessible to color-blind users.\n\nMonochromacy\n• Achromatopsia 0.005% of the population\n• Atypical Achromatopsia 0.001% of the population\nDichromacy\n• Protanopia 1% of men\n• Deuteranopia 1% of men\n• Tritanopia 0.001% of the population\nTrichromacy\n• Protanomaly 1% of men, 0.01% of women\n• Deuteranomaly 6% of men, 0.4% of women\n• Tritanomaly 0.01% of the population" ]

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[ "1. Draw a triangle. 2. Find the midpoint of the longest side of your triangle. 3. Rotate your triangle 180 degrees using the midpoint of the longest side as the center of the rotation. 4. Use the pen tool to mark corresponding parts (side lengths and angles) with dashes and curves.\n\n5. Make a conjecture and justify it. a. What type of quadrilateral have you formed?\n\n5. Make a conjecture and justify it. b. What is the definition of that quadrilateral type?\n\n5. Make a conjecture and justify it. c. Why must the quadrilateral you have fit the definition?" ]

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[ "Weighted mean with summarise_at dplyr\n\nI strictly need to use the summarise_at to compute a weighted mean, with weights based on the values of another column\n\ndf %>% summarise_at(.vars = vars(FACTOR,tv:smart tv/console),\n.funs = weighted.mean, w=INVESTMENT, na.rm=TRUE)\n\nIt always shows the error: 'INVESTMENT' is not found.\n\nI then tried with:\n\ndf %>%summarise_at(.vars = vars(FACTOR,tv:smart tv/console),\n.funs = weighted.mean, w=vars(INVESTMENT), na.rm=TRUE)\n\nBut in this case : Evaluation error: 'x' and 'w' must have the same length.\n\nWhy is this? Am I doing anything wrong? Do you have hints to solve this issue? Thanks" ]

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[ "# Bayesian Synthesis: Combining subjective analyses, with an application to ozone data\\thanksrefT1\n\n[ [    [ [    [ [ Louisiana State University Health Sciences Center, Ohio State University and Ohio State University Q. Yu\nSchool of Public Health\nLouisiana State University Health\nSciences Center\nNew Orleans, Louisiana 70122\nUSA\nS. MacEachern\nM. Peruggia\nDepartment of Statistics\nOhio State University\nColumbus, Ohio 43210\nUSA\n\\smonth8 \\syear2009\\smonth11 \\syear2010\n\\smonth8 \\syear2009\\smonth11 \\syear2010\n\\smonth8 \\syear2009\\smonth11 \\syear2010\n###### Abstract\n\nBayesian model averaging enables one to combine the disparate predictions of a number of models in a coherent fashion, leading to superior predictive performance. The improvement in performance arises from averaging models that make different predictions. In this work, we tap into perhaps the biggest driver of different predictions—different analysts—in order to gain the full benefits of model averaging. In a standard implementation of our method, several data analysts work independently on portions of a data set, eliciting separate models which are eventually updated and combined through a specific weighting method. We call this modeling procedure Bayesian Synthesis. The methodology helps to alleviate concerns about the sizable gap between the foundational underpinnings of the Bayesian paradigm and the practice of Bayesian statistics. In experimental work we show that human modeling has predictive performance superior to that of many automatic modeling techniques, including AIC, BIC, Smoothing Splines, CART, Bagged CART, Bayes CART, BMA and LARS, and only slightly inferior to that of BART. We also show that Bayesian Synthesis further improves predictive performance. Additionally, we examine the predictive performance of a simple average across analysts, which we dub Convex Synthesis, and find that it also produces an improvement. Compared to competing modeling methods (including single human analysis), the data-splitting approach has these additional benefits: (1) it exhibits superior predictive performance for real data sets; (2) it makes more efficient use of human knowledge; (3) it avoids multiple uses of the data in the Bayesian framework: and (4) it provides better calibrated assessment of predictive accuracy.\n\n\\kwd\n\\doi\n\n10.1214/10-AOAS444 \\volume5 \\issue2B 2011 \\firstpage1678 \\lastpage1698\n\n\\runtitle\n\nBayesian synthesis \\thankstextT1This material is based upon work supported by the NSF through Awards SES-0214574, SES-0437251, DMS-06-05041, and DMS-06-05052 and by the NSA through Award MSPF-04G-109.\n\n{aug}\n\nA]\\fnmsQingzhao \\snmYulabel=e1], B]\\fnmsSteven N. \\snmMacEachernlabel=e2] and B]\\fnmsMario \\snmPeruggia\\correflabel=e3]\n\nAutomatic modeling \\kwddata-splitting \\kwdhuman intervention \\kwdmodel averaging.\n\n## 1 Introduction\n\nA coarse but conceptually useful taxonomy of modeling strategies distinguishes between two broad categories: automatic strategies and strategies which require human intervention. Automatic strategies typically rely on generic methods for model selection, perhaps allowing data-based choice of a couple of tuning parameters. They are appealing because, once the data are input, inferences are produced without requiring any further human interaction. By contrast, human modeling emphasizes exploratory data analysis and the accompanying notions of model development and refinement. The debate on the relative merits of these two approaches is vigorous and ongoing [see, e.g., Breiman (2001) or Hand (2006), and the ensuing comments and rejoinders].\n\nIn our experience, much of data analysis is heavily based on subjective decisions which do not lend themselves to routine formulations. These range from what variables to include in an analysis to what forms the variables should take, to insight about the parametric form of the response variable, to whether individual cases should be included in the analysis or trimmed as outliers. Many common instances of human interventions in the modeling cannot be easily carried out by automatic procedures.\n\nThroughout, an adequate analysis must take into account what the variables are, whether they are well measured or of lesser quality, whether individual influential cases drive the results, what the scientific background of the problem is, etc. [Weisberg (1985)]. All of these elements are essential, both when modeling the data formally and when drawing conclusions from the analysis. Also, in certain cases, we might specify some aspects of a model and impose specific constraints based on scientific knowledge that a general purpose model selection method may fail to recognize.\n\nBecause of these reasons, we strongly adhere to the belief that a good data analysis based on human intervention will often be far superior to a routinely implemented analysis. In this article we present a modeling and weighting strategy, called Bayesian Synthesis, for combining analyses from several human modelers within the Bayesian framework. Bayesian Synthesis, formalized in Section 2, relies on a number of different analysts each contributing a Bayesian model to a pool of models. Each model in the pool is given a weight, thus creating a “hyper-model.” The techniques of model averaging [e.g., Raftery, Madigan and Hoeting (1997)] are used to synthesize the different analysts’ beliefs. Formal rules ensure that the analysts will contribute models that can be synthesized. Bayesian Synthesis retains the benefits of subjective modeling while substantially enhancing the inferential and predictive strengths of each individual analysis, producing combined inferences that vastly outperform inferences based on automatic methods.\n\nThe methodology we propose can be viewed as a means of constructing a useful space of models over which to perform a Bayesian analysis. In this regard, it is strongly connected to the literature on model selection [e.g., George and McCulloch (1993), who describe a method of screening models for further development] and on accounting for model uncertainty [see Draper (1995) and the following discussion for an extensive treatment]. In contrast to earlier work, our approach emphasizes the role of subjective modeling and the need for multiple analysts.\n\nIn this article we report on the experimental development of the new methodology. Specifically, we have constructed a careful experiment (with appropriate randomization and blinding) that allows us to contrast subjective modeling, and subjective modeling combined with Bayesian Synthesis and Convex Synthesis, to automated modeling methods. The results demonstrate the success of our new methods: With only the exception of BART [Chipman, George and McCulloch (2010)], subjective, human modeling had predictive performance superior to that of a variety of automatic methods, including AIC [Akaike (1974)], BIC [Schwarz (1978)], Smoothing Splines [Craven and Wahba (1979); Gu (2002)], CART [Breiman et al. (1984)], a bagged version of CART [Breiman (1996)], LASSO [Tibshirani (1996)], Forward Stagewise [Hastie, Tibshirani and Friedman (2001)], LARS [Efron et al. (2004)], Bayesian Model Averaging [Raftery, Madigan and Hoeting (1997)] and Bayesian CART [Chipman, George and McCulloch (1998)]. The gains relative to these methods were large. The comparisons with BART give a slight advantage to BART, but not uniformly so. Bayesian Synthesis and Convex Synthesis provide an additional, modest improvement over subjective modeling. In addition, and much more importantly, it leads to a more realistic assessment of predictive accuracy, curbing the over-optimism of each individual analyst.\n\nIn Section 2 we introduce a Bayesian framework for data splitting and formally describe Bayesian Synthesis and Convex Synthesis. In Section 3 we present the experiment and a careful discussion of the results. In Section 4 we discuss related work and suggest directions for future research.\n\n## 2 A Bayesian framework for data-splitting\n\nOur primary focus is on Bayesian modeling, where a team of analysts builds models for a data set. The paradigm we envision is this. First, the data are split into several portions. Each analyst receives one portion of the data. Second, each analyst builds a Bayesian model for their portion of the data, reporting a “Bayesian summary” of their posterior distribution. Third, the Bayesian summaries are updated on portions of the data not used to build them, and they are combined to yield a single, overall posterior model.\n\nTwo features are essential for this procedure to work well. First, each analyst must produce a Bayesian summary that is amenable to updating with further data. Second, the various Bayesian summaries must be amenable to synthesis. Throughout, we must exercise care so that the data are not split into too many parts. We will assume that there are analysts.\n\n### 2.1 Splitting the data\n\nThe data to be used for model development and synthesis are split into portions. Once split, the portions of the data are assigned to the analysts at random. This produces an exchangeable partition and assignment of data to analysts. Theoretical results presented in Yu (2006) suggest that (where data splitting is appropriate) the portions of the data should all contain approximately the same amount of information about the data-generating process. Following this theory, we seek to produce a set of splits that give conditionally i.i.d. data to the analysts. The following cases describe two of the splitting procedures that we have implemented.\n\nThe first case is that of a designed experiment where a structural balance is forced upon the data. For example, the two-sample, completely randomized design is often implemented in a balanced fashion, so that the same number of experimental units are assigned to each of the two treatment conditions. Additionally, covariates are recorded on the experimental units. For this type of experiment, we split at random, with the restriction that each analyst receive the same number of observations on each treatment. The additional covariates need not be balanced and need not be used by the analysts in constructing a model for the data.\n\nThe second case, matching the ozone example of Section 3, is one where there is a collection of experimental units, with a variety of information on each unit. In this case, we split the data at random, with each analyst receiving the same number of observations.\n\nThese methods of splitting the data have the advantage of not depending on the analysts’ eventual models—an essential part of our paradigm. The methods are extremely easy to implement and do not require the help of an expert to split the data. The drawback to these methods is that the portions of the data will typically not convey the same amount of information to the different analysts. While “optimal” splits might well differ, we would need to know the details of the analysts’ models to formalize the notions of information in the splits and of optimality. For large samples, the splits of the data will contain approximately the same amount of information.\n\n### 2.2 Building and updating the model\n\nIn order to carry out the analysis, each analyst is provided with a set of ground rules for model building. The rules include, most importantly, the goals of the modeling task. Second, the analyst must know what kind of Bayesian summary to produce. Since the Bayesian synthesis of the analysts’ summaries will be accomplished through Bayes factors, and since Bayes factors depend on the marginal likelihood of the data, the analyst must be informed of the quantity for which the likelihood will be calculated. Third, the analyst must know what conventions will be followed for computation of the likelihood. These conventions must guarantee that the analysts’ models will be mutually absolutely continuous over the range of values that the data can assume.\n\nConsider the prototypical experiments for which data splitting is described. In the first case, of a balanced two-sample experiment with case-specific covariates, interest may focus on the difference between treatment means. Implicitly, the analysts have been informed that the treatment means exist. The Bayesian summary for an analyst represents the analyst’s posterior, given the portion of the data used for the analysis. The likelihood of responses to the two treatments will be computed; the mechanism assigning units to the treatments will not be part of the likelihood. The convention for the likelihood is that it be a density absolutely continuous with respect to Lebesgue measure with support on the real line. An alternate convention might be that the likelihood be discrete, rounded to a single decimal place, on the nonnegative half-line.\n\nAn instance of the second case is described in some detail in the upcoming example, and so we leave off discussion for the moment. In any event, each analyst is left with the choice of constructing a model from the assigned portion of the data. The analysts may use any method whatsoever to build their model, ranging from automated methods, to subjectively elicited priors, to construction and refinement of models through diagnostics. The essence of the paradigm is to encourage the analysts to build creative models that can be combined across analysts.\n\n### 2.3 The Bayesian summary\n\nThe Bayesian summary can take on a wide variety of forms, depending on the analyst’s modeling choices. Whatever the form, the summary must be amenable to updating and allow one to compute the marginal likelihood for the portions of the data not used to construct the model.\n\nSeveral forms of summary work well in practice. Choice of a posterior distribution conjugate to the analyst’s chosen likelihood for the future data leads to a direct computation of the marginal likelihood. Choice of a mixture of such distributions leads to a mixture of conjugate posteriors, and hence to quick computation of the marginal likelihood. For models that move beyond conjugacy, the posterior distribution can be represented in a discrete fashion, for example, by the output of a Monte Carlo simulation. Along with the representation, the summary must include a means of updating the summary, for example, code to compute the marginal likelihoods and to produce summaries that enable one to address the inferential goals of the analysis.\n\n### 2.4 Synthesizing the analyses\n\nWhen each analyst has produced a model, we can combine them to yield an overall model. Under Bayesian Synthesis, we combine the models by computing pairwise Bayes factors for portions of the data and then reconciling them through the calculation of the geometric mean of pairwise Bayes factors for each analyst. These geometric means determine the weight that each analyst receives in predictions. A formal justification for this choice of weighting is provided in Sections 2.5 and 2.6.\n\nLet denote the splits of the data; let denote the likelihoods for the models with possibly differing parameters . The pairwise Bayes factor is computed on the greatest set of data not used in constructing the two models, after the two models have been updated to include the same data. Thus, the Bayes factor comparing analysts and  is\n\n B12=∫f1(Y3,Y4,…,Yk|\\boldsθ1)π(\\boldsθ1|Y1,Y2)d\\boldsθ1∫f2(Y3,Y4,…,Yk|\\boldsθ2)π(\\boldsθ2|Y1,Y2)d\\boldsθ2=m1(2)m2(1).\n\nNote that the distribution on used in the above calculation is the posterior, given both and . Similarly, the distribution on is the posterior given both and .\n\nIf the Bayesian summaries yield models that are each well represented by a set of draws from the appropriate posterior distribution, the Bayes factor can be estimated as\n\n ˆB12=∑Nj=1N−1f1(Y3,Y4,…,Yk|\\boldsθ(j)1)∑Nj=1N−1f2(Y3,Y4,…,Yk|\\boldsθ(j)2).\n\nWeighted distributions, such as those produced by importance sampling, can be used to obtain the Bayes factor. For more complex models, sophisticated methods of estimating the marginal likelihoods produce these Bayes factors. See Chen et al. (2000) for a recent book that describes methods for estimating Bayes factors/marginal likelihoods.\n\nNext, for each , we compute the geometric mean of the estimated Bayes factors to obtain\n\n bi=[k∏l=1ˆBil]1/k, (1)\n\nwhere . These are then used as weights to yield the synthesized posterior: . In this expression,  runs over the parameter spaces for all of the analysts’ models.\n\n### 2.5 Model weights: A formal justification\n\nForecasts are naturally combined through the marginal likelihood. In the context of model averaging performed by a single analyst, this follows from Bayes theorem: assuming that equal prior weight is assigned to each submodel under consideration, the posterior weight for a submodel is then proportional to the Bayes factor for that submodel against an arbitrary reference submodel. Thus, the ratio of the weights assigned to two submodels equals the Bayes factor for one against the other, and the Bayes factor expresses the impact that the data have on the relative weights assigned to two submodels.\n\nThe approach that we have taken extends the result for a single analyst to more than one analyst. When there are two analysts, each plays the role of a submodel, and from the definition of equation (1), we have\n\n b1b2=(ˆB11ˆB12)1/2(ˆB21ˆB22)1/2=(ˆB12)1/2(ˆB12)1/2=ˆB12.\n\nThe formula for the given in equation (1) does appear to be unusual, but it produces the answer we had hoped for: the ratio of the weights equals the Bayes factor. This formula for two analysts is used in the analysis of the ozone data presented in Section 3.\n\nWhen there are more than two analysts, we can imagine that each analyst plays the role of a submodel. We seek to assign weights to the various analysts (submodels). In the event that all pairwise Bayes factors were consistent with one another (i.e., if for all ), we would wish to assign relative weights according to Bayes theorem. That is, we would wish to have\n\n bibj=ˆBij\n\nfor all . Our expression for the does just this. In fact, making use of equation (1) and of the consistency of the Bayes factors with one another, we have\n\n bibj=[k∏l=1ˆBilˆBjl]1/k=[k∏l=1ˆBilˆBlj]1/k=[k∏l=1ˆBij]1/k=ˆBij.\n\n### 2.6 Model weights: Uniqueness\n\nThere is a sense in which our definition of equation (1) is uniquely the “correct” means of combining information across the analysts in a broad class of versions of the problem. We first restrict consideration to expressions for which satisfy\n\n log(bi)=k∑l=1(c+dlog(ˆBil))\n\nfor some choice of real-valued coefficients, and . This restriction enforces linearity of the in the log Bayes factors (which, in turn, are derived from log marginal likelihoods). The restriction also ensures that common coefficients ( and ) are assigned, irrespective of subscripts and . This is appropriate because, in our splits, we assign the same amount of data to each analyst, and so the same amount of data is used to compute the log marginal likelihoods for each of the pairwise Bayes factors. Second, to satisfy our desired property, we enforce the fixed solution when the Bayes factors are consistent with one another. Letting , we then have a chain of algebraic expressions, to wit,\n\n log(bi/bj) = k∑l=1[c+dLil]−k∑l=1[c+dLjl]=d[k∑l=1Lil−k∑l=1Ljl] = dk∑l=1[Lil+Llj]=dk∑l=1Lij.\n\nThis yields the log Bayes factor comparing analyst to analyst only when , resulting in our definition of (up to a multiplicative constant that drops out when deriving the relative weights for the analysts).\n\n### 2.7 Alternative weights\n\nThe analysts’ summaries can be combined in many fashions, including those not motivated by Bayes theorem. A simple method of this form takes a convex combination of the analysts’ summaries, but does not update the weights. We call this method Convex Synthesis.\n\n## 3 Applications\n\nIn this section we describe an experiment which demonstrates the benefits of Bayesian Synthesis and Convex Synthesis. To conduct the experiment, we selected a data set which has been used by other authors to illustrate the benefits of automated modeling methods. None of us was familiar with the data set and we each received one third of the data. This allowed us to create three pairs of analysts, with one third of the data reserved for evaluation of the pair’s synthesis. The syntheses were compared to a variety of automated procedures. We found that both Bayesian Synthesis and Convex Synthesis perform well.\n\n### 3.1 Ozone data\n\nThe ozone data set consists of daily measurements of ozone concentration and eight meteorological quantities in the Los Angeles basin for 330 days in the year 1976. Breiman (2001) describes the origin of the data set. The data set is contained and documented in the software package R. The data frame contains 330 observations on the following variables: upo3---maximum 1-hour average upland ozone concentration, in ppm;111Investigation of ozone standards suggests that the units for upo3 are actually parts per hundred million rather than ppm. See, for example, the US EPA standards for ground level ozone to which we return in Section 3.1.2 (http://www.epa.gov/ozonepollution/ history.html ). vdht—Vandenberg 500 millibar height, in meters; wdsp—wind speed, in miles per hour; hmdt—humidity; sbtp—Sandburg air base temperature, in degrees Celsius; ibht—inversion base height, in feet; dgpg—Daggett pressure gradient, in mmHg; ibtp—inversion base temperature, in degrees Fahrenheit; vsty—visibility, in miles; day—calendar day, an integer number between 1 and 366.\n\nEach analyst was charged with the task of constructing a Bayesian model that can be used to predict ozone concentration. Each model should produce a distribution for ozone concentration supported on the nonnegative integers.\n\n#### 3.1.1 The split-data analysis\n\nWe split the data into three sets of 110 observations each, with a complete randomization. Each of us (Analysts 1–3) received one part of the data (data 1–3). All three analysts decided independently to model log ozone level as a continuous variable and to produce the agreed-upon distribution for ozone (over the positive integers) by integrating the continuous density of the modeled variable.\n\nModel 1. Analyst used data set to build a model, pursuing a strategy of first discovering which variables appeared to be important in predicting ozone level and then determining the forms in which the variables should enter the model.\n\nMatrices of scatter plots of the response variable and explanatory variables were examined. Serial dependence was investigated by including lagged responses as explanatory variables. Several variables (sbtp, ibht, vsty and day) appeared to be quite important, and so were chosen to appear in the models. There was no apparent serial dependence in the data, after adjusting for other variables.\n\nHaving identified important variables, the analyst searched for appropriate forms. The term ibht was modeled as four variables, a linear term, two further variables developed to capture nonlinearity, and an indicator for ibht, an apparent truncation point for the variable. The indicator allows for the jump that we expect at the truncation point and provides a way to incorporate additional variability at this point. The analyst used a sine curve for the effect of variable day to force it to be periodic with period  year.\n\nAfter basic models were created, the analyst reexamined variables previously judged to be of lesser import with added variable plots and best subsets regressions. The variable hmdt was included as a predictor, in a piecewise linear fashion. The variables dgpg (with linear and quadratic terms) and vdht were considered to be potential predictors. Plots of vsty showed a wiggly pattern of nonlinearity. Two forms for this effect were considered—a linear effect and a Gaussian process centered at a linear effect. The prior on the Gaussian process version was chosen to force the realized effect curve to be close to linear.\n\nFinally, eight models (all including the initial variables and hmdt; then the  combinations including or excluding dgpg and vdht and with two forms of prior for vsty) were selected to receive positive probability. The prior distribution on each model was improper, uniform for some coefficients and vague for most other coefficients. Weights were formed for the eight models through estimated likelihoods. Each model was updated with cases and a predictive likelihood computed for the remaining cases. This process was repeated times, yielding ten predictive likelihoods. The weight given to each model was proportional to the geometric mean of its predictive likelihoods.\n\nModel 2. Based on data set 2, Analyst 2 plotted log ozone concentration and all other covariates against “day” to detect evident trends. The response and the covariates were each detrended through local fitting [by means of the loess() function in R] using the variable “day” as a predictor. All subsequent modeling was conducted on the residuals from these fits.\n\nAnalyst 2 believed that time proximity might constitute an important factor and decided to specify conditional autoregressive (CAR) models for the detrended data. Denoting the response variable by , a CAR model takes the form where , with denoting a vector of covariate values at time and a vector of model parameters. The models specified random walk priors of order either one or two for the vector , as explained in Thomas et al. (2004).\n\nAnalyst 2 built two models for the regression . The first has an intercept and four main effects selected by means of graphical and exploratory data analysis techniques. The second has many more predictors selected through a stepwise procedure, starting from the model with all main effects and two-way interactions. The two regression models and the two CAR structures were combined to produce four models that were averaged according to weights given in Table 1. The weights were chosen subjectively to reflect the analyst’s higher degree of confidence in simpler rather than more complicated models. Noninformative priors were specified for the model parameters and Winbugs was used to draw separate samples from the posterior distributions for the four models.\n\nModel 3. Analyst 3 used data set 3 and applied a modification of Least Angle Regression [LARS; Efron et al. (2004)] to fit the model: first modified LARS was used to choose the variables to be included in the model, and then Bayesian linear regression was implemented to quantify the relationship between log ozone concentration and the selected variables.\n\nTwo modifications are applied to LARS. The first is the restriction that an interaction term can be selected only after the corresponding main effects have entered the model. As soon as the main effects enter, the interaction term becomes a candidate variable. The second modification to LARS is that some variables (in this analysis, one main effect) are forced to enter the model at the beginning of the procedure.\n\nAssume there are candidate main effects. Order these variables by the strength of their correlation with the response variable, from strongest to weakest. Label the ordered variables . Suppose that variable will be forced into the model. We start with only variables through as candidate variables, and so LARS selects variable . We continue with the solution path until another variable is added. At this point, the list of candidate variables is expanded to include variable and the second-order term for variable . A second variable is chosen from the list of candidate variables according to the LARS criterion. Then the second-order term for this variable and its interaction with variable are included as candidate variables. The above process is repeated until the solution path is completed.\n\nAnalyst used modified LARS to decide, with different forced-in variables, the order in which variables entered the models. This produced several sequences of models. Each sequence was examined by and by differences in AIC and BIC to subjectively determine which models were viable. A Bayesian linear regression was computed for each viable model, against an improper prior distribution. Finally, BIC was used to obtain a weight for each of the four models. With new data, both the weight for each model and the distributions of parameters within the model were updated.\n\n#### 3.1.2 Human modeling versus automated modeling\n\nMany authors have advocated the use of automated modeling strategies, arguing that such methods provide better predictive performance than corresponding subjectively built models. Breiman et al. (1984) and Gu (2002) analyze the ozone data with the goal of predicting log ozone concentration. Using the methods described in their work as well as a number of other methods, we reanalyzed the data, comparing their predictive performance to that of the single and combined models of Analysts 1–3.\n\nThe suite of automated methods used for comparison was chosen to span the variety of strategies that are currently in vogue. These strategies range from rigid strategies which select a model from a small set of potential models and that may suffer from bias to flexible strategies that allow an essentially arbitrary mean function and that may overfit the data. They include both strategies that rely on a single fit to the observed data (as in model selection) and strategies that incorporate model averaging (whether different models are fit to the single data set, or whether models are derived from a collection of data sets produced from the actual data set). The methods include both classical and Bayesian methods. Publicly available software routines were used to implement all of the automated methods. In general, default values were used for parameter settings, except for the case of smoothing splines where variables were selected using the method described in Gu (2002). Specifically, the methods investigated are those described at the end of the Introduction.\n\nThe methods were compared on a range of goals, including those that would naturally favor the automated analyses and those which we expect to be difficult for the automated methods. We now step through a brief description of the results of the comparisons.\n\nTable 2 compares the methods in terms of prediction of log ozone. Recall that previous analyses of these data have focused on log ozone, and all three of the analysts also selected a log transformation of ozone before analyzing the data. With this transformation, a (discretized) normal likelihood appears to be appropriate for analysis of the data. Thus, accuracy of predictions as measured by sum of squared prediction errors provides both a measure of the discrepancy between the predictions and the observed data and it is directly tied to likelihood-based assessment of the models’ lack of fit.\n\nThe table contains six comparisons. For each comparison, one split of the data is reserved as test data, with the other two splits used to fit the models. In addition, two versions of the prediction problem were investigated. The first is a static prediction problem, the latter a sequential prediction problem. For the static problem, the training data were used to develop the model. A prediction was made for each case in the test data, and the measure of fit was computed. We refer to this as making a prediction “once and for all.” For the sequential problem, we randomly partitioned the test data into 11 sets of 10 cases each. The model was fit to the training data, and a prediction made for the first set of cases in the test data. The model was updated (getting the posterior distributions both within and across models) based on the first set of cases in the test data, and predictions made for the second set of cases. This procedure was continued, updating the model on successively larger sets of data and making predictions for the next set of cases, until the test data were exhausted. We used the same partition of the test data (in the same order) to evaluate each of the methods. We refer to this as “ten by ten” evaluation.\n\nTable 2 contains rows for the “Mean Human Prediction Error,” for “Bayesian Synthesis” and for “Convex Synthesis.” The Mean Human Prediction Error is defined by selecting an analyst at random to make predictions. The measure of fit is the mean of the two analysts’ measures. Bayesian Synthesis implements the method of Section 2, combining the two analysts eligible to make predictions for the test data. The initial weights given to each analyst are equal to . When updating ten by ten, the weights adjust, based on the relative performance of the analysts’ models. The predictions were taken to be the posterior predictive means. Convex Synthesis uses the same procedure as Bayesian Synthesis, but maintains a constant weight of for each analyst. Because the initial weights are equal to for both Bayesian and Convex Syntheses, the once and for all updating yields the same results in both cases. For the 10 by 10 updating the final weights under Bayesian Synthesis are 0.019 for Analyst 2 and 0.981 for Analyst 3 when predicting data set 1, 1.000 for Analyst 1 and 0.000 for Analyst 3 when predicting data set 2, and 0.990 for Analyst 1 and 0.010 for Analyst 2 when predicting data set 3.\n\nTable 2 shows the success of data splitting and of human modeling. We first note that the Mean Human Prediction Error provides a better predictive fit than do any of the classical automated methods. Mean human prediction error corresponds to randomly selecting an analyst to develop a model. This comparison establishes the benefit of subjective modeling.\n\nSecond, we turn to the main purpose of the experiment—to see whether Bayesian Synthesis outperforms rival methods. In every instance (excepting BART), we find that the method does outperform competing procedures. Bayesian Synthesis and Convex Synthesis yield much smaller predictive mean square errors than do any of the automated methods. The predictive mean square error is also smaller than the Mean Human Prediction Error. Bayesian Synthesis and Convex Synthesis outperform both human analysts in five of the six comparisons and is virtually as accurate as the better analyst in the remaining one. The comparisons also show the magnitude of the benefit to human modeling. The differences between the bulk of the automated techniques are considerably smaller than the differences between these automated techniques and the syntheses. As noted above, Convex Synthesis and Bayesian Synthesis are identical for “Once”; Convex Synthesis performs better than Bayesian Synthesis for two out of the three 10 by 10 updatings.\n\nThird, the comparison between the static and sequential problems shows, on the whole, a modest benefit to continually updating the model. It also makes clear the dominant role that modeling plays in effective prediction—building a better model (more precisely, a better collection of models) is far more important than having a bit more data with which to update the model.\n\nTable 3 repeats the comparisons in Table 2, but with ozone replacing log ozone as the response. Providing predictions for the human analysts, the Mean Human Prediction Error, Bayesian Synthesis, Convex Synthesis and BART is straightforward, because for these methods an MCMC Bayesian summary of the posterior distribution is available. In these cases, the models developed for log ozone imply corresponding models for ozone: The prediction for a case is given by its predictive mean. In terms of mean squared error of prediction, BART does best, followed by Convex Synthesis, followed by Bayesian Synthesis, which in turn outperforms all human analysts.\n\nTo provide predictions for the automated methods (other than BART), we faced a choice between use of the method with strongly skewed likelihood or ad-hoc correction of a model developed on the log ozone scale. The latter route generally provided better performance, and Table 3 presents these results. To provide predictions, a model was developed for log ozone, the prediction, say, , was obtained for each case, as was an in-sample estimate of mean squared error, say, . The prediction for ozone was taken to be . The results in Table 3 are in general accord with those of Table 2. The main difference is that the superiority of Bayesian Synthesis and Convex Synthesis relative to other methods has decreased.\n\nTable 4 examines forecasts of ozone threshold exceedance. State and federal regulations provide limitations on ozone. There are a number of ways in which ozone thresholds can be violated, including a high peak ozone concentration during a day and an excessive mean ozone concentration over an extended period of time. These thresholds have varied over time, and there has been a general downward trend in the standards. We focus on the maximum 1-hour average standard of 0.08 ppm which was in effect from 1971–1979. We have taken this to be 8 in units of upo3. With each method, a forecast (exceed or not) is made for each day in the test data set. The table presents the number of incorrect forecasts.\n\nFor human models, combinations of human models and BART, creating the forecast is straightforward. The model provides a predictive distribution for ozone concentration. If the predictive probability of exceedance is greater than 0.5, the forecast is “exceed”; if less than 0.5, the forecast is “not exceed.” The automated methods are more difficult to deal with. For these methods, we faced a choice of attempting to directly model ozone exceedance or to model some other quantity and then extract a forecast of ozone exceedance. The latter proved to be a more effective strategy. The forecasts for these methods are based on whether the point prediction for log ozone exceeds the threshold of . If the point prediction exceeds , the forecast is for exceed; if not, the forecast is “not exceed.” Convex Synthesis does the best on this task, edging Bayesian Synthesis in the 10 by 10 updating, with BART and the Mean Human Prediction Error close behind. The other methods lag substantially.\n\nIn addition to the comparisons presented here, we have examined several other potential evaluations. Some of these appear in Yu (2006). Overall, we find a substantial advantage for the human models and for BART.\n\n#### 3.1.3 One Bayesian versus Bayesian Synthesis\n\nThe previous comparative exercises demonstrate that the syntheses provide an improvement over the individual Bayesian. In nearly all instances, Bayesian Synthesis and Convex Synthesis have performed better than the Mean Human Prediction Error. This alone leads us to recommend routine use of our techniques. In this section we examine two more comparative exercises, both of which show the syntheses to be preferable to individual analysts and to the Mean Human Prediction Error. The comparisons are “once and for all” comparisons and so Bayesian Synthesis and Convex Synthesis have identical performance. We also include BART in this comparison because it is a Bayesian method and so leads to noncontroversial predictive variances and predictive intervals.\n\nThe focus of these additional comparisons is calibration of the posterior predictive distribution. To look at this issue, we make two comparisons. The first is accuracy of coverage rates of prediction intervals. We form 90% prediction intervals for the three data sets as before. The intervals are central predictive probability intervals, cutting off 5% of the predictive distribution in each tail. Table 5 presents these results under % cvg. We find generally good agreement with nominal coverage levels, with the syntheses and BART performing a little better than individual analysts.", null, "Figure 1: Plots showing the differences in out-of-sample model predictions for the models developed by the human analysts and the models developed by AIC using various splits of the data.\n\nThe second comparison is of internal and external measures of accuracy. For these measures, we focus on the predictive distribution for log ozone. Under a Bayesian model, the expected squared departure from the predictive mean is the predictive variance. Thus, as an internal measure of accuracy, we use the variance of the predictive distribution, averaged over the 110 predicted cases. As an external measure of accuracy, we use the mean squared error of prediction. The results are presented in Table 5. We note that the analysts’ internal estimates systematically understate the actual variation, while the syntheses and BART produce nearly equivalent internal and external measures of accuracy. The ratio of MSE to Var is a measure of the optimism of the Bayesian. When this ratio exceeds 1, the Bayesian is overly optimistic. We computed these ratios based on the average MSE and variance over the three data sets. The ratios, summarized in Table 5, exceed one for all methods other than the syntheses.\n\n#### 3.1.4 Why the syntheses work\n\nWe next turn to an explanation of the benefits of model synthesis. The syntheses, indeed all Bayesian model averaging, provide the greatest benefits when the models to be synthesized provide different predictions. It is here that averaging allows one to make a different prediction than either model, and it is here that further information collected in data allows the posterior weights given to different models to select the better model. The benefits of bagging/averaging models arising from relatively stable procedures such as AIC, BIC and SS are minimal (results not presented in the tables), because the bulk of the bagged models provide the same or similar predictions. Figure 1 shows that differences in predictions from different analysts show more variation than do differences from different AIC models.\n\nThe results outlined in Table 2 show clearly that there are large benefits stemming from human modeling with additional improvements attributable to the syntheses. Interestingly, large benefits can also ensue from synthesis of a human and an automatically fitted model, as evidenced by the summaries presented in Yu (2006). This is in part due to the fact that the predictions produced by human and automatically fitted models are typically different. Also, the gains appear to be more sizable when the human models are synthesized with methods based on the creation of new variables (e.g., Smoothing Spline, CART, Bagged CART, BART) than when they are synthesized with methods based on regressions with the original variables (e.g., AIC, BIC, BMA, LARS, LASSO, Forward Stagewise). Overall, the empirical results indicate that the predictions produced by the syntheses usually outperform the predictions of the single constituent elements and inherit many of the performance properties of the best of the constituent elements.\n\nAcross our set of comparisons, Convex Synthesis has outperformed Bayesian Synthesis by a modest margin. We find this surprising, as our expectation was that Bayesian Synthesis, by updating the weights, would tilt the predictions toward the analyst with the better fitting model, resulting in better predictive performance. We do not have a definitive explanation for this behavior, but we do conjecture that it is due in part to shortcomings of all of the analysts’ models. The “data-generating mechanism” is, presumably, not captured by any of the analysts. As the analysts’ models are not nested within one another, a convex combination of the analysts’ predictions enlarges the space of predictions. It is plausible that this expanded space includes models that fit better than those of any individual analyst, producing the observed results. A related discussion, where the truth is presumed to lie within the convex hull of a collection of models, appears in Kim and Kim (2004).\n\n## 4 Discussion and further research\n\nIn this paper we propose Bayesian Synthesis and Convex Synthesis, a new paradigm for Bayesian data analysis. The paradigm is motivated by the concern that using a set of data both to develop a model and to subsequently fit the model with the same data violates the spirit of Bayes theorem. The paradigm has been developed with an eye to which parts of a modeling effort appear to be stable—model development by a single analyst—and which appear to yield highly variable results—model development by different analysts. Tapping into the variable parts of an analysis while retaining enough information to preserve stability of the other parts of the analysis allows us to obtain the greatest benefits of Bayesian model averaging. This also provides us with a more appropriate accounting of model uncertainty.\n\nWe have explored the new paradigm experimentally. Yu (2006) contains a theoretical motivation for the work, providing an ensemble of theorems that justifies split-data analyses. Experimentally, the ozone data analysis shows the remarkable benefits that accrue to subjective modeling and the further benefits that follow from synthesizing subjective models across analysts.\n\nIn practice, it is more costly and time-consuming to produce several subjective analyses than a single one, so when should this method be employed? We recommend use of this method when the amount of available data is sufficient to produce split data sets that are informative, and when the problem is important enough to justify the involvement of several analysts. Examples of such situations include efficacy and safety studies in large clinical trials, post-enumeration adjustment of the census, industrial research and development, and large marketing surveys. Situations for which the method is not recommended are those where real-time predictions are needed, as is the case for internet searches, target recognition and on-line quality control, unless the components of the synthesis can be built ahead of time. In the latter case, the type of synthesis to be employed will need to avoid the expense of a formal Bayesian updating of the weights.\n\nThis work raises several issues. One issue is how to most effectively split the data. In this work, we have focused on partitioning the data set with randomization playing a dominant role. An alternative route is to allow overlapping splits of the data, so that each analyst receives a more than fraction of the data. We expect overlapping splits to be of most use when data sets are small or when they contain large numbers of potential predictors. Overlapping splits also allow us to benefit from the modeling efforts of a larger set of analysts. The theoretical results in Yu (2006) address these overlapping splits.\n\nA second issue is the development of prototypical problems so that a precise methodology can be specified depending on the goal(s) of the analysis and the type of data collected. Investigation of these problems will give us more guidance on how to split the data and on what restrictions to place on the Bayesian summaries.\n\nA third issue is application of the methodology with non-Bayesian components. The benefits of averaging nonstable or different models applies more broadly than in the Bayesian setting. Noting differences between the models built by CART and by the information criteria, one could average them as well. However, without a Bayesian summary and with incomplete likelihoods, model synthesis becomes somewhat more ad-hoc. Convex Synthesis provides one such simple method which could be implemented with fixed weights, as we have done here, or with weights determined by some predefined rule. 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(\\byear2004). \\btitleConvex hull ensemble machine for regression and classification. \\bjournalKnowledge and Information Systems \\bvolume6 \\bpages645–663. \\endbibitem\n• Laud and Ibrahim (1995) {barticle}[mr] \\bauthor\\bsnmLaud, \\bfnmPurushottam W.\\binitsP. W. \\AND\\bauthor\\bsnmIbrahim, \\bfnmJoseph G.\\binitsJ. G. (\\byear1995). \\btitlePredictive model selection. \\bjournalJ. Roy. Statist. Soc. Ser. B \\bvolume57 \\bpages247–262. \\bidmr=1325389 \\endbibitem\n• Raftery, Madigan and Hoeting (1997) {barticle}[mr] \\bauthor\\bsnmRaftery, \\bfnmAdrian E.\\binitsA. E., \\bauthor\\bsnmMadigan, \\bfnmDavid\\binitsD. \\AND\\bauthor\\bsnmHoeting, \\bfnmJennifer A.\\binitsJ. A. (\\byear1997). \\btitleBayesian model averaging for linear regression models. \\bjournalJ. Amer. Statist. Assoc. \\bvolume92 \\bpages179–191. \\bidmr=1436107 \\endbibitem\n• Schwarz (1978) {barticle}[mr] \\bauthor\\bsnmSchwarz, \\bfnmGideon\\binitsG. (\\byear1978). \\btitleEstimating the dimension of a model. \\bjournalAnn. Statist. \\bvolume6 \\bpages461–464. \\bidmr=0468014 \\endbibitem\n• Thomas et al. (2004) {bmisc}[auto:STB—2010-11-18—09:18:59] \\bauthor\\bsnmThomas, \\bfnmA.\\binitsA., \\bauthor\\bsnmBest, \\bfnmN.\\binitsN., \\bauthor\\bsnmLunn, \\bfnmD.\\binitsD., \\bauthor\\bsnmArnold, \\bfnmR.\\binitsR. \\AND\\bauthor\\bsnmSpiegelhalter, \\bfnmD.\\binitsD. (\\byear2004). \\bhowpublishedGeoBUGS user manual. \\endbibitem\n• Tibshirani (1996) {barticle}[mr] \\bauthor\\bsnmTibshirani, \\bfnmRobert\\binitsR. (\\byear1996). \\btitleRegression shrinkage and selection via the lasso. \\bjournalJ. Roy. Statist. Soc. Ser. B \\bvolume58 \\bpages267–288. \\bidmr=1379242 \\endbibitem\n• Weisberg (1985) {bbook}[mr] \\bauthor\\bsnmWeisberg, \\bfnmSanford\\binitsS. (\\byear1985). \\btitleApplied Linear Regression. \\bpublisherWiley, \\baddressNew York. \\endbibitem\n• Yu (2006) {bmisc}[mr] \\bauthor\\bsnmYu, \\bfnmQingzhao\\binitsQ. (\\byear2006). \\btitleBayesian Synthesis. \\bhowpublishedPh.D. thesis, Ohio State Univ. \\endbibitem" ]

[ null, "https://media.arxiv-vanity.com/render-output/5435080/x1.png", null ]

[ "# Brunhard's formula ideal weight calculator\n\nAnother ideal weight formula from the Internet - Brunhard's formula. It requires the knowledge of chest circumference. According to the formula, ideal weight equals height multiplied by chest circumference and divided by 240.\nThe calculator below:", null, "#### Brunhald formula ideal weight calculator\n\nIdeal weight\n\nURL copied to clipboard\nPLANETCALC, Brunhard's formula ideal weight calculator" ]

[ null, "https://planetcalc.com/img/32x32i.png", null ]

[ "Solutions by everydaycalculation.com\n\n## What is the LCM of 140 and 32?\n\nThe lcm of 140 and 32 is 1120.\n\n#### Steps to find LCM\n\n1. Find the prime factorization of 140\n140 = 2 × 2 × 5 × 7\n2. Find the prime factorization of 32\n32 = 2 × 2 × 2 × 2 × 2\n3. Multiply each factor the greater number of times it occurs in steps i) or ii) above to find the lcm:\n\nLCM = 2 × 2 × 2 × 2 × 2 × 5 × 7\n4. LCM = 1120\n\nMathStep (Works offline)", null, "Download our mobile app and learn how to find LCM of upto four numbers in your own time:" ]

[ null, "https://answers.everydaycalculation.com/mathstep-app-icon.png", null ]

[ "# #00f9a5 Color Information\n\nIn a RGB color space, hex #00f9a5 is composed of 0% red, 97.6% green and 64.7% blue. Whereas in a CMYK color space, it is composed of 100% cyan, 0% magenta, 33.7% yellow and 2.4% black. It has a hue angle of 159.8 degrees, a saturation of 100% and a lightness of 48.8%. #00f9a5 color hex could be obtained by blending #00ffff with #00f34b. Closest websafe color is: #00ff99.\n\n• R 0\n• G 98\n• B 65\nRGB color chart\n• C 100\n• M 0\n• Y 34\n• K 2\nCMYK color chart\n\n#00f9a5 color description : Pure (or mostly pure) cyan - lime green.\n\n# #00f9a5 Color Conversion\n\nThe hexadecimal color #00f9a5 has RGB values of R:0, G:249, B:165 and CMYK values of C:1, M:0, Y:0.34, K:0.02. Its decimal value is 63909.\n\nHex triplet RGB Decimal 00f9a5 `#00f9a5` 0, 249, 165 `rgb(0,249,165)` 0, 97.6, 64.7 `rgb(0%,97.6%,64.7%)` 100, 0, 34, 2 159.8°, 100, 48.8 `hsl(159.8,100%,48.8%)` 159.8°, 100, 97.6 00ff99 `#00ff99`\nCIE-LAB 87.224, -68.174, 26.765 40.664, 70.463, 47.053 0.257, 0.445, 70.463 87.224, 73.239, 158.565 87.224, -75.446, 49.436 83.943, -60.429, 25.526 00000000, 11111001, 10100101\n\n# Color Schemes with #00f9a5\n\n• #00f9a5\n``#00f9a5` `rgb(0,249,165)``\n• #f90054\n``#f90054` `rgb(249,0,84)``\nComplementary Color\n• #00f929\n``#00f929` `rgb(0,249,41)``\n• #00f9a5\n``#00f9a5` `rgb(0,249,165)``\n• #00d1f9\n``#00d1f9` `rgb(0,209,249)``\nAnalogous Color\n• #f92900\n``#f92900` `rgb(249,41,0)``\n• #00f9a5\n``#00f9a5` `rgb(0,249,165)``\n• #f900d1\n``#f900d1` `rgb(249,0,209)``\nSplit Complementary Color\n• #f9a500\n``#f9a500` `rgb(249,165,0)``\n• #00f9a5\n``#00f9a5` `rgb(0,249,165)``\n• #a500f9\n``#a500f9` `rgb(165,0,249)``\n• #54f900\n``#54f900` `rgb(84,249,0)``\n• #00f9a5\n``#00f9a5` `rgb(0,249,165)``\n• #a500f9\n``#a500f9` `rgb(165,0,249)``\n• #f90054\n``#f90054` `rgb(249,0,84)``\n``#00ad72` `rgb(0,173,114)``\n• #00c683\n``#00c683` `rgb(0,198,131)``\n• #00e094\n``#00e094` `rgb(0,224,148)``\n• #00f9a5\n``#00f9a5` `rgb(0,249,165)``\n• #14ffb0\n``#14ffb0` `rgb(20,255,176)``\n• #2dffb8\n``#2dffb8` `rgb(45,255,184)``\n• #47ffc1\n``#47ffc1` `rgb(71,255,193)``\nMonochromatic Color\n\n# Alternatives to #00f9a5\n\nBelow, you can see some colors close to #00f9a5. Having a set of related colors can be useful if you need an inspirational alternative to your original color choice.\n\n• #00f967\n``#00f967` `rgb(0,249,103)``\n• #00f97c\n``#00f97c` `rgb(0,249,124)``\n• #00f990\n``#00f990` `rgb(0,249,144)``\n• #00f9a5\n``#00f9a5` `rgb(0,249,165)``\n• #00f9ba\n``#00f9ba` `rgb(0,249,186)``\n• #00f9cf\n``#00f9cf` `rgb(0,249,207)``\n• #00f9e3\n``#00f9e3` `rgb(0,249,227)``\nSimilar Colors\n\n# #00f9a5 Preview\n\nThis text has a font color of #00f9a5.\n\n``<span style=\"color:#00f9a5;\">Text here</span>``\n#00f9a5 background color\n\nThis paragraph has a background color of #00f9a5.\n\n``<p style=\"background-color:#00f9a5;\">Content here</p>``\n#00f9a5 border color\n\nThis element has a border color of #00f9a5.\n\n``<div style=\"border:1px solid #00f9a5;\">Content here</div>``\nCSS codes\n``.text {color:#00f9a5;}``\n``.background {background-color:#00f9a5;}``\n``.border {border:1px solid #00f9a5;}``\n\n# Shades and Tints of #00f9a5\n\nA shade is achieved by adding black to any pure hue, while a tint is created by mixing white to any pure color. In this example, #000e09 is the darkest color, while #f9fffd is the lightest one.\n\n• #000e09\n``#000e09` `rgb(0,14,9)``\n• #002116\n``#002116` `rgb(0,33,22)``\n• #003523\n``#003523` `rgb(0,53,35)``\n• #004830\n``#004830` `rgb(0,72,48)``\n• #005c3d\n``#005c3d` `rgb(0,92,61)``\n• #00704a\n``#00704a` `rgb(0,112,74)``\n• #008357\n``#008357` `rgb(0,131,87)``\n• #009764\n``#009764` `rgb(0,151,100)``\n• #00ab71\n``#00ab71` `rgb(0,171,113)``\n• #00be7e\n``#00be7e` `rgb(0,190,126)``\n• #00d28b\n``#00d28b` `rgb(0,210,139)``\n• #00e598\n``#00e598` `rgb(0,229,152)``\n• #00f9a5\n``#00f9a5` `rgb(0,249,165)``\n• #0effae\n``#0effae` `rgb(14,255,174)``\n• #21ffb4\n``#21ffb4` `rgb(33,255,180)``\n• #35ffbb\n``#35ffbb` `rgb(53,255,187)``\n• #48ffc1\n``#48ffc1` `rgb(72,255,193)``\n• #5cffc8\n``#5cffc8` `rgb(92,255,200)``\n• #70ffcf\n``#70ffcf` `rgb(112,255,207)``\n• #83ffd5\n``#83ffd5` `rgb(131,255,213)``\n• #97ffdc\n``#97ffdc` `rgb(151,255,220)``\n• #abffe3\n``#abffe3` `rgb(171,255,227)``\n• #beffe9\n``#beffe9` `rgb(190,255,233)``\n• #d2fff0\n``#d2fff0` `rgb(210,255,240)``\n• #e5fff6\n``#e5fff6` `rgb(229,255,246)``\n• #f9fffd\n``#f9fffd` `rgb(249,255,253)``\nTint Color Variation\n\n# Tones of #00f9a5\n\nA tone is produced by adding gray to any pure hue. In this case, #738680 is the less saturated color, while #00f9a5 is the most saturated one.\n\n• #738680\n``#738680` `rgb(115,134,128)``\n• #699083\n``#699083` `rgb(105,144,131)``\n• #609986\n``#609986` `rgb(96,153,134)``\n• #56a389\n``#56a389` `rgb(86,163,137)``\n• #4dac8c\n``#4dac8c` `rgb(77,172,140)``\n• #43b68f\n``#43b68f` `rgb(67,182,143)``\n• #39c092\n``#39c092` `rgb(57,192,146)``\n• #30c995\n``#30c995` `rgb(48,201,149)``\n• #26d399\n``#26d399` `rgb(38,211,153)``\n• #1ddc9c\n``#1ddc9c` `rgb(29,220,156)``\n• #13e69f\n``#13e69f` `rgb(19,230,159)``\n• #0aefa2\n``#0aefa2` `rgb(10,239,162)``\n• #00f9a5\n``#00f9a5` `rgb(0,249,165)``\nTone Color Variation\n\n# Color Blindness Simulator\n\nBelow, you can see how #00f9a5 is perceived by people affected by a color vision deficiency. This can be useful if you need to ensure your color combinations are accessible to color-blind users.\n\nMonochromacy\n• Achromatopsia 0.005% of the population\n• Atypical Achromatopsia 0.001% of the population\nDichromacy\n• Protanopia 1% of men\n• Deuteranopia 1% of men\n• Tritanopia 0.001% of the population\nTrichromacy\n• Protanomaly 1% of men, 0.01% of women\n• Deuteranomaly 6% of men, 0.4% of women\n• Tritanomaly 0.01% of the population" ]

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[ "# Quantile estimators based on k order statistics, Part 3: Playing with the Beta function\n\nUpdate: this blog post is a part of research that aimed to build a statistically efficient and robust quantile estimator. A paper with final results is available in Communications in Statistics - Simulation and Computation (DOI: 10.1080/03610918.2022.2050396). A preprint is available on arXiv: arXiv:2111.11776 [stat.ME]. Some information in this blog post can be obsolete: please, use the official paper as the primary reference.\n\nIn the previous two posts, I discussed the idea of quantile estimators based on k order statistics. A already covered the motivation behind this idea and the statistical efficiency of such estimators using the extended Hyndman-Fan equations as a weight function. Now it’s time to experiment with the Beta function as a primary way to aggregate k order statistics into a single quantile estimation!\n\nAll posts from this series:\n\n### The approach\n\nThe general idea is the same that was used in the previous post. We express the estimation of the $$p^\\textrm{th}$$ quantile as follows:\n\n$\\begin{gather*} q_p = \\sum_{i=1}^{n} W_{i} \\cdot x_i,\\\\ W_{i} = F(r_i) - F(l_i),\\\\ l_i = (i - 1) / n, \\quad r_i = i / n, \\end{gather*}$\n\nwhere F is a CDF function of a specific distribution. The distribution has non-zero PDF only inside a window $$[L_k, R_k]$$ that covers at most k order statistics:\n\n$F(u) = \\left\\{ \\begin{array}{lcrcllr} 0 & \\textrm{for} & & & u & < & L_k, \\\\ G\\Big((u - L_k)/(R_k-L_k)\\Big) & \\textrm{for} & L_k & \\leq & u & \\leq & R_k, \\\\ 1 & \\textrm{for} & R_k & < & u, & & \\end{array} \\right.$\n\n$L_k = (h - 1) / (n - 1) \\cdot (n - (k - 1)) / n, \\quad R_k = L_k + (k-1)/n,$\n\n$h = (n - 1)p + 1.$\n\nNow we just have to define the $$G: [0;1] \\to [0;1]$$ function that defines $$F$$ values inside the window. In the previous post, where we used the extension of Hyndman-Fan Type 7 equation, we used just the most simple linear function:\n\n$G_{HF7}(u) = u.$\n\nIn this post, we are going to try the Beta distribution (which is used in the Harrell-Davis quantile estimator). The CDF of the Beta distribution is the regularized incomplete beta function) $$I_x(\\alpha, \\beta)$$. We will try this idea with $$\\alpha=kp, \\beta = k(1-p)$$:\n\n$G(u) = I_u(kp, k(1-p)).$\n\nWith such values, the suggested estimator becomes the exact copy of the Harrell-Davis quantile estimator for $$k=n+1$$. Let’s perform some numerical simulations to check the statistical efficiency of this estimator.\n\n### Numerical simulations\n\nWe are going to take the same simulation setup that was declared in this post. Briefly speaking, we evaluate the classic MSE-based relative statistical efficiency of different quantile estimators on samples from different light-tailed and heavy-tailed distributions using the classic Hyndman-Fan Type 7 quantile estimator as the baseline.\n\nHere is the animated version of the simulations (the considered estimators based on k order statistics are denoted as “KOS-Bk”):\n\nAnd here are static images of the result for different sample sizes:\n\n### Conclusion\n\nIn this post, we discussed a quantile estimator that is based on k order statistics aggregated using the Beta function. It seems that this estimator is a good step in the right direction: it’s better than the traditional Hyndman-Fan Type 7 quantile estimator for the samples from light-tailed distributions (however, it’s worse than the Harrell-Davis quantile estimator). Also, it’s more robust than the Harrell-Davis quantile estimator in the case of heavy-tailed distributions. Moreover, we could specify the desired breakdown point by customizing the k value.\n\nIn the next post, we are going to try one more weight function." ]

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[ "", null, "# Hough\n\nThe Hough transform is a technique for creating lines based on points. The results of a typical edge detection routine are many unconnected points. To us it is obvious that these points represent shapes but because the points are not connected it is difficult for a machine to understand the underlying shape. The Hough transform takes as input many points and will generate guesses for what lines those points represent.\n\nThe Hough module interface shows the parameter space image that represents the Hough transform in a visual manner. The image axis are the angular amount (theta) and the distance value (r) used to represent a line using polar notation \"r = x cos(theta) + y sin(theta)\" as apposed to coordinate notation \"y = m x + b\". The Hough transform converts potential lines into peaks within this image. The task of detecting lines now instead becomes a task of detecting peaks within this image. The interface below shows green squares around those detected peaks which indicate the parameters (theta,r) of the lines seen in the below example. By thresholding and isolating these peaks it is possible to get a reasonably good approximation of the actual figure (source image -> Hough transform).\n\nFor a detailed description of how the Hough transform works see the links below.\n\n## Interface", null, "## Instructions\n\n1. Setup - To use the Hough transform it is typical to first use an edge detection routine like the Sobel Edge. This helps identify points to use in the transform and can speed up processing by eliminating points (i.e. setting them to black) that do not need to be processed.\n\n2. Threshold - Increase or decrease the \"Threshold\" number field or scroll the horizontal scroller until a number of lines appear in the image.\n\n3. Isolation - Often many lines that are detected are very close in slope and proximity to each other and can create a meaningless image. Use the peak isolation number to remove lines whose slopes are close to each other. See below for a visual example.\n\n4. Sharpness - Not all detected Hough Peaks are very sharp which produce lines which are created from noise within the image. Increase the sharpness value until lines that do not have sharp support are removed from the display. These errors happen often when the prior edge detection module creates a high response in a flat area due to texture noise. As that is not a true edge the resulting peak's shape will not be sharp.\n\n5. Peak Count - If you know how many lines you wish to detect enter that number in the Peak Count textbox. This will ensure that the module selects only the X highest peaks/lines as a result of the hough transform. This can be more adaptive than setting the threshold as that can change based on image size, edge detection method used, etc.\n\n6. Precision - If the module causes your machine to slow down or hang use the \"Precision\" to reduce the computational load on your machine. The Hough transform is VERY cpu intensive and not all the points need to be analyzed in order for a useful result to appear. Changing the \"Precision\" value will reduce the number of points used in the transform by skipping over X number of points.\n\n7. Edge Graident - If you have a gradient image (as apposed to a binary image) then checking the gradient will cause the transform to take into account the intensity of the edge. Thus strong edges will result in a higher peak value than weak edges which can make threshold detection easier. Note that this disables the bounding of lines.\n\n8. Bound Lines - Select the bound lines checkbox if you want to bound each line to those points that define the extreme points of the detected line. (Black/White Images only)\n\n9. Min/Max Angle Filter - Specify the line angles that you want to ignore from the results. Note that the range is 0 to 180 degrees with 0 and 180 being horizontal lines and 90 being vertical. Thus if you wanted to eliminate horizontal lines you could use 20 and 160 as the range. If you wanted to eliminate vertical lines then using 100 and 70 as the range would remove angles 70 to 100. Note that if min is larger than max, the longer arc (i.e. from 100 to 180 and then 0 to 70 would be in range) is used due to circular math.\n\n10. Line Color - The color of the detected lines drawn in the main image (and also the peak indicators).\n\n11. Line Thickness - The thickness to use when drawing the detected lines in the main image.\n\n12. Overlay On - Select if you want to draw the detected lines ontop of your source image to help you understand why lines are appearing.\n\n## Example\n\n Source Hough Lines", null, "", null, "Hough Lines with peak isolation of 1 Bounded Hough Lines", null, "", null, "## Variables\n\nHOUGH_LINES - contains the lines found from the Hough transform. Using code similar to the following within a VBScript module will allow you to further process the line segments.\n\nHOUGH_LINE_INTENSITIES - contains the strength or height values for the lines found from Hough transform. Using these codes you can determine a level of confidence associated with the line. Note that this array is 4x smaller than the HOUGH_LINES array.\n\n``` line = GetArrayVariable(\"HOUGH_LINES\")\ninten = GetArrayVariable(\"HOUGH_LINE_INTENSITIES\")\n\n' write back to messages the first line ..\n' note that line segments are defined by two endpoints\nWrite(\"First Coord: \" & line(0) & \",\" & line(1) & vbCRLF)\nWrite(\"Second Coord: \" & line(2) & \",\" & line(3))\nWrite(\"Strength: \" & inten(0))\n```\n\n Hough Related Forum Posts Last post Posts Views", null, "", null, "HELP - how to detect and draw lines in RoboRealm Hy! I have a big problem :) I need the program to draw me lines like i... 10 year 2 2712 Hough Transform - Angles I've read other posts about Hough angle limits but I am still confused.  The documentation for the module does not de... 10 year 2 2185", null, "", null, "lane detecting Please Any can suggest me to do the lane detection like the picture which I attached. and How can I detect the broken line in th... 13 year 2 3837", null, "Bug in new Angle Filter in Hough Module? Hello, I was trying out the new Angle Filter function in the Hough Transform module and either I do... 13 year 3 3495", null, "", null, "Roborealm to read an analog meter STeven,   It was great talking with you at the MiniMaker Fair yesterday!  Here are some sample... 13 year 3 3427", null, "", null, "Outdoor navigation Hello everyone I'am Bob. I'am part of a project group at my school and we are building a robot which should be a... 14 year 7 3734" ]

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[ "#### Authors\n\nLing Li, Hsuan-tien Lin\n\n#### Abstract\n\nWe present a reduction framework from ordinal regression to binary classification based on extended examples. The framework consists of three steps: extracting extended examples from the original examples, learning a binary classifier on the extended examples with any binary classification algorithm, and constructing a ranking rule from the binary classifier. A weighted 0/1 loss of the binary classifier would then bound the mislabeling cost of the ranking rule. Our framework allows not only to design good ordinal regression algorithms based on well-tuned binary classification approaches, but also to derive new generalization bounds for ordinal regression from known bounds for binary classification. In addition, our framework unifies many existing ordinal regression algorithms, such as perceptron ranking and support vector ordinal regression. When compared empirically on benchmark data sets, some of our newly designed algorithms enjoy advantages in terms of both training speed and generalization performance over existing algorithms, which demonstrates the usefulness of our framework." ]

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[ "Solutions by everydaycalculation.com\n\n1st number: 21/24, 2nd number: 2 20/30\n\n21/24 + 80/30 is 85/24.\n\n1. Find the least common denominator or LCM of the two denominators:\nLCM of 24 and 30 is 120\n2. For the 1st fraction, since 24 × 5 = 120,\n21/24 = 21 × 5/24 × 5 = 105/120\n3. Likewise, for the 2nd fraction, since 30 × 4 = 120,\n80/30 = 80 × 4/30 × 4 = 320/120\n105/120 + 320/120 = 105 + 320/120 = 425/120\n5. After reducing the fraction, the answer is 85/24\n6. In mixed form: 313/24\n\nMathStep (Works offline)", null, "Download our mobile app and learn to work with fractions in your own time:" ]

[ null, "https://answers.everydaycalculation.com/mathstep-app-icon.png", null ]

[ "", null, "Computes the maximum value found in all row values in a column. Input column are permitted to be of Integer or Decimal.\n• When used in a `pivot` transform, the function is computed for each instance of the value specified in the `group` parameter. See Pivot Transform.\n• If a row contains a missing or null value, it is not factored into the calculation.\n• If no numeric values are found in the source column, the function returns a null value.\n\nFor a version of this function computed over a rolling window of rows, see ROLLINGMAX Function.", null, "", null, "Output: Returns the maximum value of the `myRating` column.", null, "", null, "ArgumentRequired?Data TypeDescription\nfunction_col_refYstringName of column to which to apply the function\n\nFor more information on the `group` and `limit` parameters, see Pivot Transform.", null, "### function_col_ref\n\nName of the column the values of which you want to calculate the maximum. Column must contain Integer or Decimal values.\n\n• Literal values are not supported as inputs.\n• Multiple columns and wildcards are not supported.", null, "Required?Data TypeExample Value\nYesString (column reference)`myValues`", null, "", null, "", null, "" ]

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[ "# 78th Problem 2016\n\nAlgebra Level 2\n\nWhat is the nature of the roots of this equation:\n\n${ 2x }^{ 2 }-4x+7=2$\n\nCheck out the set: 2016 Problems\n\n×" ]

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[ "Search a number\nBaseRepresentation\nbin1000100110100111011011\n311020120201111\n420212213123\n51034132243\n6120201151\n725112200\noct10464733\n94216644\n102255323\n111300504\n129091b7\n1360c715\n14429ca7\n152e839d\nhex2269db\n\n2255323 has 6 divisors (see below), whose sum is σ = 2623596. Its totient is φ = 1933092.\n\nThe previous prime is 2255321. The next prime is 2255333. The reversal of 2255323 is 3235522.\n\nIt is not a de Polignac number, because 2255323 - 21 = 2255321 is a prime.\n\nIt is a Duffinian number.\n\nIt is a junction number, because it is equal to n+sod(n) for n = 2255294 and 2255303.\n\nIt is not an unprimeable number, because it can be changed into a prime (2255321) by changing a digit.\n\nIt is a polite number, since it can be written in 5 ways as a sum of consecutive naturals, for example, 22965 + ... + 23062.\n\nIt is an arithmetic number, because the mean of its divisors is an integer number (437266).\n\n22255323 is an apocalyptic number.\n\n2255323 is a deficient number, since it is larger than the sum of its proper divisors (368273).\n\n2255323 is an equidigital number, since it uses as much as digits as its factorization.\n\n2255323 is an evil number, because the sum of its binary digits is even.\n\nThe sum of its prime factors is 46041 (or 46034 counting only the distinct ones).\n\nThe product of its digits is 1800, while the sum is 22.\n\nThe square root of 2255323 is about 1501.7732851533. The cubic root of 2255323 is about 131.1403231908.\n\nThe spelling of 2255323 in words is \"two million, two hundred fifty-five thousand, three hundred twenty-three\".\n\nDivisors: 1 7 49 46027 322189 2255323" ]

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[ "# Centrifugal force\n\nNot to be confused with Centripetal force.\n\nIn Newtonian mechanics, the centrifugal force is an inertial force (also called a 'fictitious' or 'pseudo' force) directed away from the axis of rotation that appears to act on all objects when viewed in a rotating reference frame.\n\nThe concept of the centrifugal force can be applied in rotating devices, such as centrifuges, centrifugal pumps, centrifugal governors, and centrifugal clutches, and in centrifugal railways, planetary orbits, banked curves, etc. when they are analyzed in a rotating coordinate system.\n\nThe term has sometimes also been used for the force that is a reaction to a centripetal force.\n\n## Introduction\n\nThe centrifugal force is an outward force apparent in a rotating reference frame; it does not exist when measurements are made in an inertial frame of reference.\n\nAll measurements of position and velocity must be made relative to some frame of reference. For example, if we are studying the motion of an object in an airliner traveling at great speed, we could calculate the motion of the object with respect to the interior of the airliner, or to the surface of the Earth. An inertial frame of reference is one that is not accelerating (including rotation). The use of an inertial frame of reference, which will be the case for all elementary calculations, is often not explicitly stated but may generally be assumed unless stated otherwise.\n\nIn terms of an inertial frame of reference, the centrifugal force does not exist. All calculations can be performed using only Newton's laws of motion and the real forces. In its current usage the term 'centrifugal force' has no meaning in an inertial frame.\n\nIn an inertial frame, an object that has no forces acting on it travels in a straight line, according to Newton's first law. When measurements are made with respect to a rotating reference frame, however, the same object would have a curved path, because the frame of reference is rotating. If it is desired to apply Newton's laws in the rotating frame, it is necessary to introduce new, fictitious, forces to account for this curved motion.\n\nIn the rotating reference frame, all objects, regardless of their state of motion, appear to be under the influence of a radially (from the axis of rotation) outward force that is proportional to their mass, the distance from the axis of rotation of the frame, and to the square of the angular velocity of the frame. This is the centrifugal force.\n\nMotion relative to a rotating frame results in another fictitious force, the Coriolis force; and if the rate of rotation of the frame is changing, a third fictitious force, the Euler force is required. Together, these three fictitious forces are necessary for the formulation of correct equations of motion in a rotating reference frame and allow Newton's Laws to be used in their normal form in such a frame.\n\n## Examples\n\n### A stone on a string\n\nConsider a stone being whirled round on a string, in a horizontal plane. The only real force acting on the stone in the horizontal plane is the tension in the string (gravity acts vertically). There are no other forces acting on the stone so there is a net force on the stone in the horizontal plane.\n\nIn an inertial frame of reference, were it not for this net force acting on the stone, the stone would travel in a straight line, according to Newton's first law of motion. In order to keep the stone moving in a circular path, this force, known as the centripetal force, must be continuously applied to the stone. As soon as it is removed (for example if the string breaks) the stone moves in a straight line. In this inertial frame, the concept of centrifugal force is not required as all motion can be properly described using only real forces and Newton's laws of motion.\n\nIn a frame of reference rotating with the stone around the same axis as the stone, the stone is stationary. However, the tension in the string is still acting on the stone. If Newton's laws were applied in their usual form, the stone would accelerate in the direction of the net applied force; towards the axis of rotation, which it does not do. To use Newton's laws of motion, unchanged, in a rotating frame it is necessary to invent a new force that acts on the stone and is equal and opposite to the tension in the string; this new force acts in the outward direction; it is the centrifugal force. With this new (inertial or fictitious force) the net force on the stone is zero and the stone remains stationary in the rotating frame of reference. With the addition of this extra inertial or fictitious force Newton's laws can be applied in the rotating frame as if it were an inertial (non-rotating) frame.\n\n### Weighing an object at the Earth's poles and on the equator\n\nConsider an object that is being weighed with a simple spring balance at one of the Earth's poles. There are only two forces acting on the object, the Earth's gravity, which acts in a downward direction, and the equal and opposite tension in the spring, acting upward. There is no net force acting on the object and the spring balance so the object does not accelerate and remains stationary. The balance shows the value of the force of gravity on the object.\n\nWhen the same object is weighed on the equator the same two real forces act upon the object. However, the object is moving in a circular path as the Earth rotates. When considered in an inertial frame (that is to say, one that is not rotating with the Earth), some of the force of gravity is expended just to keep the object in its circular path (centripetal force). As such, less tension in the spring is required to counteract the 'remaining' force of gravity. Less tension in the spring would be reflected on a scale as less weight — about 0.3% less at the equator than at the poles. The concept of centrifugal force is not required. However, the Earth is not a perfect sphere, so an object at the poles is slightly closer to the center of the Earth than one at the equator; after accounting for both effects, the actual measured weight of the object is about 0.53% less on the equator.\n\nIt is generally more convenient to take measurements in a frame of reference rotating with the Earth. In this reference frame the object is stationary and to account for the loss in measured weight when the object is measured at the equator it is necessary to include the upward acting (inertial or fictitious) centrifugal force. In practice, this is often observed as a reduction in the force of gravity.\n\n### An equatorial railway\n\nThis thought experiment is more complicated than the previous two examples in that it requires the use of the Coriolis force as well as the centrifugal force.\n\nImagine a railway line running round the Earth's equator, with a train running at high speed in the opposite direction to the Earth's rotation. The train runs at such a speed that, in an inertial (nonrotating) frame centered on the Earth, it remains stationary as the Earth spins beneath it. In this inertial frame the situation is easy to analyze. The only forces acting on the train are its gravity (downward) and the equal and opposite (upward) reaction force from the track. There is no net force on the train and it therefore remains stationary.\n\nIn a frame rotating with the Earth the train is moving in a circular orbit as it travels round the Earth. In this frame, the upward reaction force from the track and the force of gravity on the train remain the same, as they are real forces. However, in the Earth's (rotating) frame, the train is traveling in a circular path and therefore requires a centripetal (downward) force to keep it on this path. Because we are using a rotating frame, we must, as always, apply the (fictitious) centrifugal force to the train. This is equal in value to the required centripetal force but acts in an upward direction — the opposite direction to that required. It would therefore seem that there is a net upward force on the train and it should therefore accelerate upward.\n\nIn order to explain this paradox we must note that the train is in motion with respect to the rotating frame and we must therefore, in addition to the centrifugal force, add the Coriolis force. In this particular example, this acts in a downward direction and is equal in value to twice the centrifugal force thus canceling out the centrifugal force and supplying the necessary centripetal force to keep the train in its circular path.\n\n## Derivation\n\nFor the following formalism, the rotating frame of reference is regarded as a special case of a non-inertial reference frame that is rotating relative to an inertial reference frame denoted the stationary frame.\n\n### Time derivatives in a rotating frame\n\nIn a rotating frame of reference, the time derivatives of any vector function P of time—such as the velocity and acceleration vectors of an object—will differ from its time derivatives in the stationary frame. If P1 P2, P3 are the components of P with respect to unit vectors i, j, k directed along the axes of the rotating frame, then the first time derivative [dP/dt] of P with respect to the rotating frame is, by definition, dP1/dt i + dP2/dt j + dP3/dt k. If the absolute angular velocity of the rotating frame is ω then the derivative dP/dt of P with respect to the stationary frame is related to [dP/dt] by the equation:", null, "where", null, "denotes the vector cross product. In other words, the rate of change of P in the stationary frame is the sum of its apparent rate of change in the rotating frame and a rate of rotation", null, "attributable to the motion of the rotating frame. The vector ω has magnitude ω equal to the rate of rotation and is directed along the axis of rotation according to the right-hand rule.\n\n### Acceleration\n\nNewton's law of motion for a particle of mass m written in vector form is:", null, "where F is the vector sum of the physical forces applied to the particle and a is the absolute acceleration (that is, acceleration in an inertial frame) of the particle, given by:", null, "where r is the position vector of the particle.\n\nBy applying the transformation above from the stationary to the rotating frame three times, the absolute acceleration of the particle can be written as:", null, "### Force\n\nThe apparent acceleration in the rotating frame is [d2r/dt2]. An observer unaware of the rotation would expect this to be zero in the absence of outside forces. However Newton's laws of motion apply only in the inertial frame and describe dynamics in terms of the absolute acceleration d2r/dt2. Therefore, the observer perceives the extra terms as contributions due to fictitious forces. These terms in the apparent acceleration are independent of mass; so it appears that each of these fictitious forces, like gravity, pulls on an object in proportion to its mass. When these forces are added, the equation of motion has the form:", null, "", null, "From the perspective of the rotating frame, the additional force terms are experienced just like the real external forces and contribute to the apparent acceleration. The additional terms on the force side of the equation can be recognized as, reading from left to right, the Euler force", null, ", the Coriolis force", null, ", and the centrifugal force", null, ", respectively. Unlike the other two fictitious forces, the centrifugal force always points radially outward from the axis of rotation of the rotating frame, with magnitude mω2r, and unlike the Coriolis force in particular, it is independent of the motion of the particle in the rotating frame. As expected, for a non-rotating inertial frame of reference", null, "the centrifugal force and all other fictitious forces disappear.\n\n## Absolute rotation\n\nMain article: Absolute rotation", null, "The interface of two immiscible liquids rotating around a vertical axis is an upward-opening circular paraboloid.", null, "When analysed in a rotating reference frame of the planet, centrifugal force causes rotating planets to assume the shape of an oblate spheroid\n\nThree scenarios were suggested by Newton to answer the question of whether the absolute rotation of a local frame can be detected; that is, if an observer can decide whether an observed object is rotating or if the observer is rotating.\n\n• The shape of the surface of water rotating in a bucket. The shape of the surface becomes concave to balance the centrifugal force against the other forces upon the liquid.\n• The tension in a string joining two spheres rotating about their center of mass. The tension in the string will be proportional to the centrifugal force on each sphere as it rotates around the common center of mass.\n\nIn these scenarios, the effects attributed to centrifugal force are only observed in the local frame (the frame in which the object is stationary) if the object is undergoing absolute rotation relative to an inertial frame. By contrast, in an inertial frame, the observed effects arise as a consequence of the inertia and the known forces without the need to introduce a centrifugal force. Based on this argument, the privileged frame, wherein the laws of physics take on the simplest form, is a stationary frame in which no fictitious forces need to be invoked.\n\nWithin this view of physics, any other phenomenon that is usually attributed to centrifugal force can be used to identify absolute rotation. For example, the oblateness of a sphere of freely flowing material is often explained in terms of centrifugal force. The oblate spheroid shape reflects, following Clairaut's theorem, the balance between containment by gravitational attraction and dispersal by centrifugal force. That the Earth is itself an oblate spheroid, bulging at the equator where the radial distance and hence the centrifugal force is larger, is taken as one of the evidences for its absolute rotation.\n\n## Applications\n\nThe operations of numerous common rotating mechanical systems are most easily conceptualized in terms of centrifugal force. For example:\n\n• A centrifugal governor regulates the speed of an engine by using spinning masses that move radially, adjusting the throttle, as the engine changes speed. In the reference frame of the spinning masses, centrifugal force causes the radial movement.\n• A centrifugal clutch is used in small engine-powered devices such as chain saws, go-karts and model helicopters. It allows the engine to start and idle without driving the device but automatically and smoothly engages the drive as the engine speed rises. Inertial drum brake ascenders used in rock climbing and the inertia reels used in many automobile seat belts operate on the same principle.\n• Centrifugal forces can be used to generate artificial gravity, as in proposed designs for rotating space stations. The Mars Gravity Biosatellite would have studied the effects of Mars-level gravity on mice with gravity simulated in this way.\n• Spin casting and centrifugal casting are production methods that uses centrifugal force to disperse liquid metal or plastic throughout the negative space of a mold.\n• Centrifuges are used in science and industry to separate substances. In the reference frame spinning with the centrifuge, the centrifugal force induces a hydrostatic pressure gradient in fluid-filled tubes oriented perpendicular to the axis of rotation, giving rise to large buoyant forces which push low-density particles inward. Elements or particles denser than the fluid move outward under the influence of the centrifugal force. This is effectively Archimedes' principle as generated by centrifugal force as opposed to being generated by gravity.\n• Some amusement rides make use of centrifugal forces. For instance, a Gravitron's spin forces riders against a wall and allows riders to be elevated above the machine's floor in defiance of Earth's gravity.\n\nNevertheless, all of these systems can also be described without requiring the concept of centrifugal force, in terms of motions and forces in a stationary frame, at the cost of taking somewhat more care in the consideration of forces and motions within the system.\n\n## History of conceptions of centrifugal and centripetal forces\n\nThe conception of centrifugal force has evolved since the time of Huygens, Newton, Leibniz, and Hooke who expressed early conceptions of it. Its modern conception as a fictitious force arising in a rotating reference frame evolved in the eighteenth and nineteenth centuries.\n\nCentrifugal force has also played a role in debates in classical mechanics about detection of absolute motion. Newton suggested two arguments to answer the question of whether absolute rotation can be detected: the rotating bucket argument, and the rotating spheres argument. According to Newton, in each scenario the centrifugal force would be observed in the object's local frame (the frame where the object is stationary) only if the frame were rotating with respect to absolute space. Nearly two centuries later, Mach's principle was proposed where, instead of absolute rotation, the motion of the distant stars relative to the local inertial frame gives rise through some (hypothetical) physical law to the centrifugal force and other inertia effects. Today's view is based upon the idea of an inertial frame of reference, which privileges observers for which the laws of physics take on their simplest form, and in particular, frames that do not use centrifugal forces in their equations of motion in order to describe motions correctly.\n\nThe analogy between centrifugal force (sometimes used to create artificial gravity) and gravitational forces led to the equivalence principle of general relativity.\n\n## Other uses of the term\n\nWhile the majority of the scientific literature uses the term centrifugal force to refer to the particular fictitious force that arises in rotating frames, there are a few limited instances in the literature of the term applied to other distinct physical concepts. One of these instances occurs in Lagrangian mechanics. Lagrangian mechanics formulates mechanics in terms of generalized coordinates {qk}, which can be as simple as the usual polar coordinates", null, "or a much more extensive list of variables. Within this formulation the motion is described in terms of generalized forces, using in place of Newton's laws the EulerLagrange equations. Among the generalized forces, those involving the square of the time derivatives {(dqk   ⁄ dt )2} are sometimes called centrifugal forces. In the case of motion in a central potential the Lagrangian centrifugal force has the same form as the fictitious centrifugal force derived in a co-rotating frame. However, the Lagrangian use of \"centrifugal force\" in other, more general cases has only a limited connection to the Newtonian definition.\n\nIn another instance the term refers to the reaction force to a centripetal force, or reactive centrifugal force. A body undergoing curved motion, such as circular motion, is accelerating toward a center at any particular point in time. This centripetal acceleration is provided by a centripetal force, which is exerted on the body in curved motion by some other body. In accordance with Newton's third law of motion, the body in curved motion exerts an equal and opposite force on the other body. This reactive force is exerted by the body in curved motion on the other body that provides the centripetal force and its direction is from that other body toward the body in curved motion. \n\nThis reaction force is sometimes described as a centrifugal inertial reaction, that is, a force that is centrifugally directed, which is a reactive force equal and opposite to the centripetal force that is curving the path of the mass.\n\nThe concept of the reactive centrifugal force is sometimes used in mechanics and engineering. It is sometimes referred to as just centrifugal force rather than as reactive centrifugal force although this usage is deprecated in elementary mechanics.", null, "Wikimedia Commons has media related to Centrifugal force.\n1. http://www-spof.gsfc.nasa.gov/stargaze/Sframes1.htm\n2. Encyclopaedia Britannica, article on Centrifuge\n3. Feynman lectures on physics, Book 1 12-11\n4. Alexander L. Fetter; John Dirk Walecka (2003). Theoretical Mechanics of Particles and Continua. Courier Dover Publications. pp. 38–39. ISBN 0-486-43261-0.\n5. Jerrold E. Marsden; Tudor S. Ratiu (1999). Introduction to Mechanics and Symmetry: A Basic Exposition of Classical Mechanical Systems. Springer. p. 251. ISBN 0-387-98643-X.\n6. \"Curious About Astronomy?\" Archived January 17, 2015, at the Wayback Machine., Cornell University, retrieved June 2007\n7. Boynton, Richard (2001). \"Precise Measurement of Mass\" (PDF). Sawe Paper No. 3147. Arlington, Texas: S.A.W.E., Inc. Retrieved 2007-01-21.\n8. i.e. P = P1 i + P2 j +P3 k\n9. John L. Synge; Byron A. Griffith (2007). Principles of Mechanics (Reprint of Second Edition of 1942 ed.). Read Books. p. 347. ISBN 1-4067-4670-3.\n10. Twice to", null, "and once to", null, ".\n11. Taylor (2005). p. 342.\n12. LD Landau; LM Lifshitz (1976). Mechanics (Third ed.). Oxford: Butterworth-Heinemann. p. 128. ISBN 978-0-7506-2896-9.\n13. Louis N. Hand; Janet D. Finch (1998). Analytical Mechanics. Cambridge University Press. p. 267. ISBN 0-521-57572-9.\n14. Mark P Silverman (2002). A universe of atoms, an atom in the universe (2 ed.). Springer. p. 249. ISBN 0-387-95437-6.\n15. Taylor (2005). p. 329.\n16. Cornelius Lanczos (1986). The Variational Principles of Mechanics (Reprint of Fourth Edition of 1970 ed.). Dover Publications. Chapter 4, §5. ISBN 0-486-65067-7.\n17. Morton Tavel (2002). Contemporary Physics and the Limits of Knowledge. Rutgers University Press. p. 93. ISBN 0-8135-3077-6. Noninertial forces, like centrifugal and Coriolis forces, can be eliminated by jumping into a reference frame that moves with constant velocity, the frame that Newton called inertial.\n18. Louis N. Hand; Janet D. Finch (1998). Analytical Mechanics. Cambridge University Press. p. 324. ISBN 0-521-57572-9.\n19. I. Bernard Cohen; George Edwin Smith (2002). The Cambridge companion to Newton. Cambridge University Press. p. 43. ISBN 0-521-65696-6.\n20. Simon Newcomb (1878). Popular astronomy. Harper & Brothers. pp. 8688.\n21. Myers, Rusty L. (2006). The basics of physics. Greenwood Publishing Group. p. 57. ISBN 0-313-32857-9.\n22. An English translation is found at Isaac Newton (1934). Philosophiae naturalis principia mathematica (Andrew Motte translation of 1729, revised by Florian Cajori ed.). University of California Press. pp. 10–12.\n23. Barbour, Julian B. and Herbert Pfister (1995). Mach's principle: from Newton's bucket to quantum gravity. Birkhäuser. ISBN 0-8176-3823-7, p. 69.\n24. Eriksson, Ingrid V. (2008). Science education in the 21st century. Nova Books. ISBN 1-60021-951-9, p. 194.\n25. For an introduction, see for example Cornelius Lanczos (1986). The variational principles of mechanics (Reprint of 1970 University of Toronto ed.). Dover. p. 1. ISBN 0-486-65067-7.\n26. For a description of generalized coordinates, see Ahmed A. Shabana (2003). \"Generalized coordinates and kinematic constraints\". Dynamics of Multibody Systems (2 ed.). Cambridge University Press. p. 90 ff. ISBN 0-521-54411-4.\n27. Christian Ott (2008). Cartesian Impedance Control of Redundant and Flexible-Joint Robots. Springer. p. 23. ISBN 3-540-69253-3.\n28. Shuzhi S. Ge; Tong Heng Lee; Christopher John Harris (1998). Adaptive Neural Network Control of Robotic Manipulators. World Scientific. pp. 47–48. ISBN 981-02-3452-X. In the above EulerLagrange equations, there are three types of terms. The first involves the second derivative of the generalized co-ordinates. The second is quadratic in", null, "where the coefficients may depend on", null, ". These are further classified into two types. Terms involving a product of the type", null, "are called centrifugal forces while those involving a product of the type", null, "for i ≠ j are called Coriolis forces. The third type is functions of", null, "only and are called gravitational forces.\n29. R. K. Mittal; I. J. Nagrath (2003). Robotics and Control. Tata McGraw-Hill. p. 202. ISBN 0-07-048293-4.\n30. T Yanao; K Takatsuka (2005). \"Effects of an intrinsic metric of molecular internal space\". In Mikito Toda; Tamiki Komatsuzaki; Stuart A. Rice; Tetsuro Konishi; R. Stephen Berry. Geometrical Structures Of Phase Space In Multi-dimensional Chaos: Applications to chemical reaction dynamics in complex systems. Wiley. p. 98. ISBN 0-471-71157-8. As is evident from the first terms ..., which are proportional to the square of", null, ", a kind of \"centrifugal force\" arises ... We call this force \"democratic centrifugal force\". Of course, DCF is different from the ordinary centrifugal force, and it arises even in a system of zero angular momentum.\n31. See p. 5 in Donato Bini; Paolo Carini; Robert T Jantzen (1997). \"The intrinsic derivative and centrifugal forces in general relativity: I. Theoretical foundations\". International Journal of Modern Physics D. 6 (1).. The companion paper is Donato Bini; Paolo Carini; Robert T Jantzen (1997). \"The intrinsic derivative and centrifugal forces in general relativity: II. Applications to circular orbits in some stationary axisymmetric spacetimes\". International Journal of Modern Physics D. 6 (1).\n32. Mook, Delo E. & Thomas Vargish (1987). Inside relativity. Princeton NJ: Princeton University Press. ISBN 0-691-02520-7, p. 47.\n33. G. David Scott (1957). \"Centrifugal Forces and Newton's Laws of Motion\". 25. American Journal of Physics. p. 325.\n34. Signell, Peter (2002). \"Acceleration and force in circular motion\" Physnet. Michigan State University, \"Acceleration and force in circular motion\", §5b, p. 7.\n35. Mohanty, A. K. (2004). Fluid Mechanics. PHI Learning Pvt. Ltd. ISBN 81-203-0894-8, p. 121.\n36. Roche, John (September 2001). \"Introducing motion in a circle\". Physics Education 43 (5), pp. 399-405, \"Introducing motion in a circle\". Retrieved 2009-05-07.\n37. Lloyd William Taylor (1959). Physics, the pioneer science. 1. Dover Publications. p. 173.\n38. Edward Albert Bowser (1920). An elementary treatise on analytic mechanics: with numerous examples (25th ed.). D. Van Nostrand Company. p. 357.\n39. Joseph A. Angelo (2007). Robotics: a reference guide to the new technology. Greenwood Press. p. 267. ISBN 1-57356-337-4.\n40. Eric M Rogers (1960). Physics for the Inquiring Mind. Princeton University Press. p. 302." ]

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[ "", null, "", null, "Article Content\n\nIn number theory, an additive function is an arithmetic function f(n) of the positive integer n such that whenever a and b are coprime we have:\nf(ab) = f(a) + f(b).\n\nAn additive function f(n) is said to be completely additive if f(ab) = f(a) + f(b) holds for all positive integers a and b, even when they are not coprime.\n\nOutside number theory, the term additive is usually used for all functions with the property f(ab) = f(a) + f(b) for all arguments a and b. This article discusses number theoretic additive functions.\n\n### Examples\n\nArithmetic functions which are completely additive are:\n\n• The restriction of the logarithmic function to N, a0(n) - the sum of primes dividing n, sometimes called sopfr(n). We have a0(20) = a0(22 · 5) = 2 + 2+ 5 = 9. Some values: (SIDN A001414 (http://www.research.att.com/cgi-bin/access.cgi/as/njas/sequences/eisA.cgi?Anum=A001414)).\n\na0(4) = 4\na0(27) = 9\na0(144) = a0(24 · 32) = a0(24) + a0(32) = 8 + 6 = 14\na0(2,000) = a0(24 · 53) = a0(24) + a0(53) = 8 + 15 = 23\na0(2,001) = 55\na0(2,002) = 33\na0(2,003) = 2003\na0(54,032,858,972,279) = 1240658\na0(54,032,858,972,302) = 1780417\na0(20,802,650,704,327,415) = 1240681\n...\n\n• a1(n) - the sum of the distinct primes dividing n, sometimes called sopf(n). We have a1(1) = 0, a1(20) = 2 + 5 = 7. Some more values: (SIDN A008472 (http://www.research.att.com/cgi-bin/access.cgi/as/njas/sequences/eisA.cgi?Anum=A008472))\n\na1(4) = 2\na1(27) = 3\na1(144) = a1(24 · 32) = a1(24) + a1(32) = 2 + 3 = 5\na1(2,000) = a1(24 · 53) = a1(24) + a1(53) = 2 + 5 = 7\na1(2,001) = 55\na1(2,002) = 33\na1(2,003) = 2003\na1(54,032,858,972,279) = 1238665\na1(54,032,858,972,302) = 1780410\na1(20,802,650,704,327,415) = 1238677\n...\n\n• The function Ω(n), defined as the total number of prime factors of n, counting multiple factors multiple times. This implies Ω(1) = 0 since 1 has no prime factors. Some more values: (SIDN A001222 (http://www.research.att.com/cgi-bin/access.cgi/as/njas/sequences/eisA.cgi?Anum=A001222))\n\nΩ(4) = 2\nΩ(27) = 3\nΩ(144) = Ω(24 · 32) = Ω(24) + Ω(32) = 4 + 2 = 6\nΩ(2,000) = Ω(24 · 53) = Ω(24) + Ω(53) = 4 + 3 = 7\nΩ(2,001) = 3\nΩ(2,002) = 4\nΩ(2,003) = 1\nΩ(54,032,858,972,279) = 3\nΩ(54,032,858,972,302) = 6\nΩ(20,802,650,704,327,415) = 7\n...\n\n• An example of an arithmetic function which is additive but not completely additive is ω(n), defined as the total number of different prime factors of n. Some values (compare with Ω(n)) (SIDN A001221 (http://www.research.att.com/cgi-bin/access.cgi/as/njas/sequences/eisA.cgi?Anum=A001221))\n\nω(4) = 1\nω(27) = 1\nω(144) = ω(24 · 32) = ω(24) + ω(32) = 1 + 1 = 2\nω(2,000) = ω(24 · 53) = ω(24) + ω(53) = 1 + 1 = 2\nω(2,001) = 3\nω(2,002) = 4\nω(2,003) = 1\nω(54,032,858,972,279) = 3\nω(54,032,858,972,302) = 5\nω(20,802,650,704,327,415) = 5\n...\n\nSources:\n\n1. Janko Bračič, Kolobar aritmetičnih funkcij (Ring of arithmetical functions), (Obzornik mat, fiz. 49 (2002) 4, pp 97 - 108) (MSC (2000) 11A25)\n\nAll Wikipedia text is available under the terms of the GNU Free Documentation License\n\nSearch Encyclopedia\n Search over one million articles, find something about almost anything!\n\nFeatured Article\n Sanskrit language ... and Sanskrit led to the discovery of this language family by Sir William Jones, and thus played an important role in the development of linguistics. Indeed, linguistics ...", null, "", null, "" ]

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[ "# 14.4: Charles's Law\n\nEverybody enjoys the smell and taste of freshly-baked bread. It is light and fluffy as a result of the action of yeast on sugar. The yeast converts the sugar to carbon dioxide, which at high temperatures causes the dough to expand. The end-result is an enjoyable treat, especially when covered with melted butter.\n\n## Charles's Law\n\nFrench physicist Jacques Charles (1746 - 1823) studied the effect of temperature on the volume of a gas at constant pressure. Charles's Law states that the volume of a given mass of gas varies directly with the absolute temperature of the gas when pressure is kept constant. The absolute temperature is temperature measured with the Kelvin scale. The Kelvin scale must be used because zero on the Kelvin scale corresponds to a complete stoppage of molecular motion.", null, "Figure 14.4.1: As a container of confined gas is heated, its molecules increase in kinetic energy and push the movable piston outward, resulting in an increase in volume.\n\nMathematically, the direct relationship of Charles's Law can be represented by the following equation:\n\n$\\frac{V}{T} = k$\n\nAs with Boyle's Law, $$k$$ is constant only for a given gas sample. The table below shows temperature and volume data for a set amount of gas at a constant pressure. The third column is the constant for this particular data set and is always equal to the volume divided by the Kelvin temperature.\n\nTemperature $$\\left( \\text{K} \\right)$$ Volume $$\\left( \\text{mL} \\right)$$ $$\\frac{V}{T} = k$$ $$\\left( \\frac{\\text{mL}}{\\text{K}} \\right)$$\nTable 14.4.1: Temperature-Volume Data\n50 20 0.40\n100 40 0.40\n150 60 0.40\n200 80 0.40\n300 120 0.40\n500 200 0.40\n1000 400 0.40\n\nWhen this data is graphed, the result is a straight line, indicative of a direct relationship, shown in the figure below.", null, "Figure 14.4.2: The volume of a gas increases as the Kelvin temperature increases.\n\nNotice that the line goes exactly toward the origin, meaning that as the absolute temperature of the gas approaches zero, its volume approaches zero. However, when a gas is brought to extremely cold temperatures, its molecules would eventually condense into the liquid state before reaching absolute zero. The temperature at which this change into the liquid state occurs varies for different gases.\n\nCharles's Law can also be used to compare changing conditions for a gas. Now we use $$V_1$$ and $$T_1$$ to stand for the initial volume and temperature of a gas, while $$V_2$$ and $$T_2$$ stand for the final volume and temperature. The mathematical relationship of Charles's Law becomes:\n\n$\\frac{V_1}{T_1} = \\frac{V_2}{T_2}$\n\nThis equation can be used to calculate any one of the four quantities if the other three are known. The direct relationship will only hold if the temperatures are expressed in Kelvin. Temperatures in Celsius will not work. Recall the relationship that $$\\text{K} = \\: ^\\text{o} \\text{C} + 273$$.\n\nExample 14.4.1\n\nA balloon is filled to a volume of $$2.20 \\: \\text{L}$$ at a temperature of $$22^\\text{o} \\text{C}$$. The balloon is then heated to a temperature of $$71^\\text{o} \\text{C}$$. Find the new volume of the balloon.\n\nStep 1: List the known quantities and plan the problem.\n\nKnown\n\n• $$V_1 = 2.20 \\: \\text{L}$$\n• $$T_1 = 22^\\text{o} \\text{C} = 295 \\: \\text{K}$$\n• $$T_2 = 71^\\text{o} \\text{C} = 344 \\: \\text{K}$$\n\nUnknown\n\n• $$V_2 = ? \\: \\text{L}$$\n\nUse Charles's Law to solve for the unknown volume $$\\left( V_2 \\right)$$. The temperatures have first been converted to Kelvin.\n\nStep 2: Solve.\n\nFirst, rearrange the equation algebraically to solve for $$V_2$$.\n\n$V_2 = \\frac{V_1 \\times T_2}{T_1}$\n\nNow substitute the known quantities into the equation and solve.\n\n$V_2 = \\frac{2.20 \\: \\text{L} \\times 344 \\: \\text{K}}{295 \\: \\text{K}} = 2.57 \\: \\text{L}$" ]

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[ "Cody\n\n# Problem 43284. Form a square matrix from four square sub-matrices\n\nCreate a square matrix, y, from 4 square sub-matrices that will be constructed (x1, x2, x3, x4):\n\n``` y = [x1 x2;\nx3 x4];```\n\nThe size of each sub-matrix will be n, while given values should be applied to each sub-matrix.\n\n• 1st sub-matrix: n1 on the main diagonal with all other elements equal to 0\n• 2nd sub-matrix: all elements equal to n2\n• 3rd sub-matrix: all elements equal to n3\n• 4th sub-matrix: same as the first but with diagonal elements equal to n4.\n\nFor example, with n=2, n1=1, n2=2, n3=3, and n4=5:\n\n``` y = [1 0 2 2;\n0 1 2 2;\n3 3 5 0;\n3 3 0 5];```\n\n### Solution Stats\n\n67.65% Correct | 32.35% Incorrect\nLast Solution submitted on Dec 27, 2019" ]

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[ "Cody\n\n# Problem 39. Which values occur exactly three times?\n\nSolution 1702861\n\nSubmitted on 5 Jan 2019 by Serhii Tetora\nThis solution is locked. To view this solution, you need to provide a solution of the same size or smaller.\n\n### Test Suite\n\nTest Status Code Input and Output\n1   Pass\nx = [1 2 5 2 2 7 8 3 3 1 3 8 8 8]; y_correct = [2 3]; assert(isequal(threeTimes(x),y_correct))\n\n2   Pass\nx = [1 1 1]; y_correct = ; assert(isequal(threeTimes(x),y_correct))\n\n3   Pass\nx = [5 10 -3 10 -3 11 -3 5 5 7]; y_correct = [-3 5]; assert(isequal(threeTimes(x),y_correct))" ]

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[ "# Mines Chain Reaction\n\nThis post is based on a question asked during the Challenge 24 programming competition. Given the locations of a number of land mines as `X` and `Y` coordinates and their blast radius `R`. What is the minimum number of mines that need to be detonated such that all mines are detonated. When a mine is detonated it detonates all mines within its blast radius and the process repeats.", null, "Here’s a simple example with `13` mines. In this case the optimal solution is to detonate mines `0, 3` and `8` which will detonate all others. It’s not the only solution.", null, "The relationship between mines is not commutative. Just because mines `A` can reach mine `B` doesn’t mean that mine `B` reaches mine `A`. Therefore we can represent the mines as a directed graph where vertices are mines and there is an (unweighted) edge from mine `A` to mine `B` if mine `A` can directly detonate mine `B`.", null, "In order to solve this problem we first need to compute the strongly connected components in this graph . These are the subsets of mines which if any one is detonated then all mines in the subset will be detonated. In the example image above mines `5, 6`, and `7` comprise a SCC as do mines `0, 2, 9, 10, 11` and `12`. For simplicity we’ll say that mines on their own are also in SCCs of size `1`. In order to compute the SCCs we can use Tarjan’s algorithm which can be implemented recursively or with a stack.\n\n``````def tarjan(graph):\n\nindex_counter = \nstack = []\nindex = {}\nresult = []\n\ndef strongconnect(node):\n# depth index for this node\nindex[node] = index_counter\nindex_counter += 1\nstack.append(node)\n\n# Consider successors of `node`\ntry:\nsuccessors = graph[node]\nexcept:\nsuccessors = []\nfor successor in successors:\n# Successor has not yet been visited\nstrongconnect(successor)\nelif successor in stack:\n# the successor is in the stack\n\n# pop the stack and generate an SCC\nconnected_component = []\n\nwhile True:\nsuccessor = stack.pop()\nconnected_component.append(successor)\nif successor == node: break\ncomponent = tuple(connected_component)\n# storing the result\nresult.append(component)\n\nfor node in graph:\nstrongconnect(node)\n\nreturn result\n``````\n\nThis computes the SSCs for the initial graph. Now we can collapse all vertices in a SCC into a super vertex. Remember detonating any mine in the super vertex will detonate all the other in that super vertex. Then we can create another graph of the super vertices and connect super vertices with a directed edge if any mine in that super vertex can detonate any mine in another super vertex. We now get another directed graph although this one won’t have cycles. Remember if we denote a mine in a connected component it will detonate all mines in that component and all mines in all components reachable from that support node. Here’s an illustration to or the process so far:", null, "We can now work out which mines need to be detonated. In order to do this we can look for all super vertices in this graph that have a zero in-degree. This means that they aren’t reachable by any sequence of mine detonations and thus need to be detonated themselves. One solution is to detonate mines: `0, 3` and `8`. There are actually multiple solutions. We can see this for example by considering the case where all the mines are within blast radius of all others and thus form one strongly connected components. In this case we could choose any mine to start the chain reaction.\n\nIn the competition the test cases got really large. The smallest had `500` vertices and the largest had `800,000` vertices. Tarjan’s algorithm is really fast and runs in `O(N)`. Similarly the degree counting can also be done in `O(N)`. The slowest part it actually creating the initial graph which when done naively takes `O(N^2)`. In order to process the larger test cases we need to use a range query structure like a KD-Tree to query all mines within `R` of a specific mine in logarithmic time. Reducing the processing to `O(N log N)`. A simpler approach than implementing a KD-Tree is to sort all the mines by their `X` coordinate and only consider partner mines that are within `X*X` < `R*R` of original. With randomly spaced data this gets you close to `O(N log N)` without too much more coding. The problem set is available here.\n\nThis type of analysis is useful in considering the distribution of information through a network. If the initial graph represented people and edges represented the people with whom they shared information. Then the source nodes are the minimum set of people that need to be given information such that it is transferred through the whole network." ]

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[ "## Section13.5A Concrete Example\n\nLet's apply the Labeling Algorithm to the network flow shown in Figure 13.2. Then we start with the source:\n\nSince the source $$S$$ is the first vertex labeled, it is also the first one scanned. So we look at the neighbors of $$S$$ using the pseudo-alphabetic order on the vertices. Thus, the first one to be considered is vertex $$B$$ and since the edge $$(S,B)$$ is not full, we label $$B$$ as\n\nWe then consider vertex $$E$$ and label it as\n\nNext is vertex $$F\\text{,}$$ which is labeled as\n\nAt this point, the scan from $$S$$ is complete.\n\nThe first vertex after $$S$$ to be labeled was $$B\\text{,}$$ so we now scan from $$B\\text{.}$$ The (unlabeled) neighbors of $$B$$ to be considered, in order, are $$A\\text{,}$$ $$C\\text{,}$$ and $$D\\text{.}$$ This results in the following labels:\n\nThe next vertex to be scanned is $$E\\text{,}$$ but $$E$$ has no unlabeled neighbors, so we then move on to $$F\\text{,}$$ which again has no unlabeled neighbors. Finally, we scan from $$A\\text{,}$$ and using the pseudo-alphabetic order, we first consider the sink $$T$$ (which in this case is the only remaining unlabeled vertex). This results in the following label for $$T\\text{.}$$\n\nNow that the sink is labeled, we know there is an augmenting path. We discover this path by backtracking. The sink $$T$$ got its label from $$A\\text{,}$$ $$A$$ got its label from $$B\\text{,}$$ and $$B$$ got its label from $$S\\text{.}$$ Therefore, the augmenting path is $$P=(S,B,A,T)$$ with $$\\delta=8\\text{.}$$ All edges on this path are forward. The flow is then updated by increasing the flow on the edges of $$P$$ by $$8\\text{.}$$ This results in the flow shown in Figure 13.12. The value of this flow is $$38\\text{.}$$", null, "Figure 13.12. An Updated Network Flow\n\nHere is the sequence (reading down the columns) of labels that will be found when the labeling algorithm is applied to this updated flow. (Note that in the scan from $$S\\text{,}$$ the vertex $$B$$ will not be labeled, since now the edge $$(S,B)$$ is full.)\n\nThis labeling results in the augmenting path $$P=(S,F,A,T)$$ with $$\\delta=12\\text{.}$$\n\nAfter this update, the value of the flow has been increased and is now $$50=38+12\\text{.}$$ We start the labeling process over again and repeat until we reach a stage where some vertices (including the source) are labeled and some vertices (including the sink) are unlabeled.\n\n### Subsection13.5.1How the Labeling Algorithm Halts\n\nConsider the network flow in Figure 13.13.\n\nThe value of the current flow is $$172\\text{.}$$ Applying the labeling algorithm using the pseudo-alphabetic order results in the following labels (reading down the columns):\n\nThese labels result in the augmenting path $$P=(S,C,H,I,E,L,T)$$ with $$\\delta =3\\text{.}$$ After updating the flow and increasing its value to $$175\\text{,}$$ the labeling algorithm halts with the following labels:\nNow we observe that the labeled and unlabeled vertices are $$L=\\{S,C,F,H,I\\}$$ and $$U=\\{T,A,B,D,E,G,J,K\\}\\text{.}$$ Furthermore, the capacity of the cut $$V=L\\cup U$$ is" ]

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[ "[This article was first published on Software for Exploratory Data Analysis and Statistical Modelling, and kindly contributed to R-bloggers]. (You can report issue about the content on this page here)\nWant to share your content on R-bloggers? click here if you have a blog, or here if you don't.\n\nA scatter plot is a graph used to investigate the relationship between two variables in a data set. The x and y axes are used for the values of the two variables and a symbol on the graph represents the combination for each pair of values in the data set. This type of graph is used in many common situations and can convey a lot of useful information.\n\nTo illustrate creating a scatter plot we will use a simple data set for the population of the UK between 1992 and 2009. This data is saved in a data frame uk.df using the following command:\n\nuk.df = data.frame(Year = 1992:2009,\nPopulation = c(57770, 57933, 58096, 58258, 58418, 58577,\n58743, 58925, 59131, 59363, 59618, 59894, 60186, 60489,\n60804, 61129, 61461, 61796)\n)\n\nFor this example the data is recorded in thousands to make the graph easier to read and there is no benefit or noticeable improvement to be seen by using greater detail.\n\nBase Graphics\n\nIn the base graphics system the general purpose plot function can be used to create a scatter plot for the UK population data set that we created. The first two arguments to the plot function are the x and y variables respectively. The following code will create a scatter plot, including various labels:\n\nplot(uk.df$Year, uk.df$Population,\nxlab = \"Year\", ylab = \"Total Population (Thousands)\",\nmain = \"UK Population (1992-2009)\", pch = 16)\n\nThe labels for the x and y axes are specified via the xlab and ylab arguments to the plot function and the main argument specifies the title for the plot.\n\nThe graph itself is plain and functional which solid circles indicating the population (in thousands) for each of the years covered by the data.\n\nLattice Graphics\n\nThe lattice graphics package provides a function xyplot specifically to create scatter plots and the function is used in a similar way to the base graphics approach. The first argument to the function is a formula describing the relationship to be plotted on the graph, with the y variable preceding the x variable as we are used to when describing mathematical fomula such as y=a+bx. The data frame is specified with the data argument to simplify the expression in the formula. The code used is as follows:\n\nxyplot(Population ~ Year, data = uk.df,\nxlab = \"Year\", ylab = \"Total Population (Thousands)\",\nmain = \"UK Population (1992-2009)\",\nscales = list(x = list(at = seq(1992, 2009, 2)))\n)\n\nThe axis labels and the overall title for the graph are specified in the same way as the base graphics system. We indulge in some fine tuning of the labels on the x axis via the scales argument – here we indicate that every second year should be included on the label starting in 1992 and running until 2009. The lattice graph is shown here for comparison with the graphs created using the other two packages:\n\nThere are very few visual differences between the lattice and base graphics. In lattice graphics an object is created that can be edited to add or remove components and then printed to the screen. This approach is more flexible than the base graphics where the components are painted on top of each other and the use of themes in lattice will make it easier to keep a consistent look to all graphs in a document.\n\nggplot2\n\nIn the ggplot2 package the ggplot function is used to create graphs of all types rather than having a separate function defined for each type of graph. The first argument is adata frame with the data to be plotted and the aes argument specifies the aesthetics associated with the graph such as the point symbol, size or colour. In this case the Year variable appears on the x axis and the Population variable on the y axis. The code to create the scatter plot is shown here:\n\nggplot(uk.df, aes(Year, Population)) + geom_point() +\nxlab(\"Year\") + ylab(\"Total Population (Thousands)\") +\nopts(title = \"UK Population (1992-2009)\")\n\nThe geom_point specifies the type of graph to create (a scatter plot in this situation and this highlights the flexibility of the ggplot2 package as changing the geom will create a new type of graph) and the labels for the graph are created by adding them to the graph with the xlab, ylab and opts functions. The graph is shown below:\n\nThis graph is not greatly different to the scatter plot created using the base and lattice packages. The default theme in the ggplot2 package has a gray background with white grid lines that allows easy visual recognition of graphs created using this package.\n\nThis blog post is summarised in a pdf leaflet on the Supplementary Material page." ]

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