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[ "extrapolate\n\ngeneralize counter-examples of test properties https://github.com/rudymatela/extrapolate#readme\n\n LTS Haskell 13.26: 0.3.3 Stackage Nightly 2019-06-21: 0.3.3 Latest on Hackage: 0.3.3\n\nSee all snapshots `extrapolate` appears in\n\nMaintained by\n\nModule documentation for 0.3.3\n\nThis version can be pinned in stack with:`extrapolate-0.3.3@sha256:b37c9c7894d3aa5ad8e47082550d4de2b95765e479df5ab221894edf02b6f1f8,4828`\n\nExtrapolate\n\nExtrapolate is a property-based testing library for Haskell capable of reporting generalized counter-examples.\n\nInstalling Extrapolate\n\nTo install the latest version of Extrapolate from Hackage using cabal, just:\n\n``````\\$ cabal update\n\\$ cabal install extrapolate\n``````\n\nTo test if it installed correctly, follow through the next section.\n\nUsing Extrapolate\n\nTo use Extrapolate, you first import the `Test.Extrapolate` module, then pass any properties you with to test to the function `check`:\n\n``````\\$ ghci\n> import Test.Extrapolate\n> check \\$ \\x y -> x + y == y + (x :: Int)\n+++ OK, passed 360 tests.\n\n> import Data.List (nub)\n> check \\$ \\xs -> nub xs == (xs :: [Int])\n*** Failed! Falsifiable (after 3 tests):\n[0,0]\n\nGeneralization:\nx:x:_\n``````\n\nThe operator `+` is commutative. The function `nub` is not an identity.\n\nConfiguring the number of tests\n\nTo increase the number of tests, use the `for` combinator:\n\n``````\\$ ghci\n> import Test.Extrapolate\n> check `for` 1000 \\$ \\x y -> x + y == y + (x :: Int)\n+++ OK, passed 1000 tests.\n``````\n\nCustomizing the background functions (allowed in side-conditions)\n\nTo customize the background functions, use the `withBackground` combinator:\n\n``````\\$ ghci\n> import Test.Extrapolate\n> import Data.List (nub)\n> let hasDups xs = nub xs /= (xs :: [Int])\n> check `withBackground` [constant \"hasDups\" hasDups] \\$ \\xs -> nub xs == (xs :: [Int])\n*** Failed! Falsifiable (after 3 tests):\n[0,0]\n\nGeneralization:\nx:x:_\n\nConditional Generalization:\nxs when hasDups xs\n``````\n\nPerhaps the example above is silly (`hasDups` is the negation of the property itself!), but it illustrates the use of `withBackground`.\n\nThe combinators `for` and `withBackground` can be used in conjunction:\n\n``````> check `for` 100 `withBackground` [...] \\$ property\n``````\n\nDon’t forget the dollar sign `\\$`.\n\nAnother Example\n\nConsider the following (faulty) sort function and a property about it:\n\n``````sort :: Ord a => [a] -> [a]\nsort [] = []\nsort (x:xs) = sort (filter (< x) xs)\n++ [x]\n++ sort (filter (> x) xs)\n\nprop_sortCount :: Ord a => a -> [a] -> Bool\nprop_sortCount x xs = count x (sort xs) == count x xs\nwhere\ncount x = length . filter (== x)\n``````\n\nAfter testing the property, Extrapolate returns a fully defined counter-example along with a generalization:\n\n``````> import Test.Extrapolate\n> check (prop_sortCount :: Int -> [Int] -> Bool)\n*** Failed! Falsifiable (after 4 tests):\n0 [0,0]\n\nGeneralization:\nx (x:x:_)\n``````\n\nThis hopefully makes it easier to find the source of the bug. In this case, the faulty sort function discards repeated elements." ]

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[ "# astero_search_controls¶\n\n## $$\\chi^2$$ weights¶\n\n### chi2_seismo_fraction¶\n\nThe optimization methods will use and report a total $$\\chi^2$$ that is a weighted sum of seismic and “spectroscopic” (really “non-seismic”) parts. chi2_seismo_fraction is the weight given to the seismic part. i.e. the code internally evaluates something like\n\nchi2 = chi2_seismo*chi2_seismo_fraction\n+ chi2_spectro*(1 - chi2_seismo_fraction)\n\nchi2_seismo_fraction = 0.667d0\n\n\n## Non-seismic (“spectroscopic”) observations¶\n\nchi2_spectro is a sum of terms for different non-seismic observables (e.g. Teff, logg, [Fe/H]). These controls specify what quantities are added to this $$\\chi^2$$ and, in some cases, how they are computed.\n\nEach component is identified by setting its name to either a string matching a history column or one of the cases defined in run_star_extras.f90. If target value and uncertainty sigma are greater than zero, a $$\\chi^2$$ component will be computed and output in the sample data.\n\nIf include_in_chi2_spectro is set to .true., that component will be included in chi2_spectro. The old fixed options were equivalent to:\n\n• Teff\n• log_g\n• log_L\n• log_R\n• M_H (was FeH)\n• surface_Z_div_X\n• surface_He\n• Rcz\n\nwhere M_H, surface_Z_div_X, surface_He and Rcz are now implemented in the standard run_star_extras.f90. The other options fall back to the history columns.\n\nTo implement your own constraints, create the appropriate case in set_constraint_value in run_star_extras.f90.\n\nThe age option is still treated separately because there are additional controls that determine how the age target is approached.\n\nTarget values must be set separately for PGstar. If you want targets to appear in the loggTe window, you need to set the following PGstar controls:\n\nloggTe_target_logg\nloggTe_target_logg_sigma\nloggTe_target_Teff\nloggTe_target_Teff_sigma\n\n\nSimilarly, if you want targets to appear in the HR window, you need to set the following PGstar controls:\n\nHR_target_logL\nHR_target_logL_sigma\nHR_target_Teff\nHR_target_Teff_sigma\n\n\n### constraint_sigma(:)¶\n\nconstraint_name(:) = ‘’ ! change this to whatever you want constraint_target(:) = 0 constraint_sigma(:) = 0 include_constraint_in_chi2_spectro(:) = .false.\n\n### normalize_chi2_spectro¶\n\nIf .true., the total $$\\chi^2$$ computed for the non-seismic components (defined above) is divided by the number of components used to define it.\n\nnormalize_chi2_spectro = .true.\n\n\n### Z_div_X_solar¶\n\nConvenience parameter leftover from the old interface that can be used when calculating things relative to solar metallicity.\n\nZ_div_X_solar = 0.02293d0\n\n### age_sigma¶\n\nage is the star’s age in years. When you include_age_in_chi2_spectro, set min_age_for_chi2 and max_age_for_chi2, and set eval_chi2_at_target_age_only to .false.. In &control, set max_years_for_timestep but don’t set max_age or num_adjusted_dt_steps_before_max_age.\n\ninclude_age_in_chi2_spectro = .false.\nage_target = 4.5695d9 ! (see Bahcall, Serenelli, and Basu, 2006)\nage_sigma = 0.0065d9\n\n\n### num_smaller_steps_before_age_target¶\n\nnum_smaller_steps_before_age_target = 50 ! only used if > 0\n\n\n### dt_for_smaller_steps_before_age_target¶\n\nThis should be much smaller than age_sigma.\n\ndt_for_smaller_steps_before_age_target = 0.0065d8 ! 1/10 age_sigma\n\n## Seismic observations¶\n\n### chi2_seismo_r_02_fraction¶\n\nchi2_seismo is a weighted combination of the large separation $$\\Delta\\nu$$, the frequency of maximum oscillation power $$\\nu_\\mathrm{max}$$, ratios of frequencies and individual frequencies. Specify the weighting of the terms in chi2_seismo by setting these controls. Naturally, all the fractions must be between zero and one. If the relevant fraction is not zero, the corresponding target values and uncertainties must be set.\n\nchi2_seismo_delta_nu_fraction = 0d0 ! if > 0 then delta_nu and delta_nu_sigma must be set (see below)\nchi2_seismo_nu_max_fraction = 0d0 ! if > 0 then nu_max and nu_max_sigma must be set (see below)\nchi2_seismo_r_010_fraction = 0d0 ! if > 0, then include r_010 frequency ratios\nchi2_seismo_r_02_fraction = 0d0 ! if > 0, then include r_02 frequency ratios\n\n\nThe fraction for the frequencies is whatever is left. i.e.\n\nfraction for frequencies = 1 - (frac_r_010_ratios + frac_r_02_ratios + frac_delta_nu + frac_nu_max)\n\n\nso if you only want to use individual frequencies in chi2_seismo, set all four fractions to zero.\n\n### normalize_chi2_seismo_r_02¶\n\nThe terms for the frequencies and ratios are sums of terms for each frequency and ratio. If you normalize them, they are divided by that number of terms.\n\nnormalize_chi2_seismo_frequencies = .true.\nnormalize_chi2_seismo_r_010 = .true.\nnormalize_chi2_seismo_r_02 = .true.\n\n\n### trace_chi2_spectro_info¶\n\nIf .true., output information on that $$\\chi^2$$ component to the terminal.\n\ntrace_chi2_seismo_delta_nu_info = .false.\ntrace_chi2_seismo_nu_max_info = .false.\ntrace_chi2_seismo_ratios_info = .false.\ntrace_chi2_seismo_frequencies_info = .false.\n\ntrace_chi2_spectro_info = .false. ! if true, output info to terminal\n\n\n### nu_max_sigma¶\n\nnu_max is the frequency of maximum oscillation power. You must set this value if chi2_seismo_nu_max_fraction is greater than zero.\n\n nu_max = -1\n! nu_max is needed when\n! chi2_seismo_nu_max_fraction > 0 or correction_factor > 0 (see below)\nnu_max_sigma = -1\n\n\n### delta_nu_sigma¶\n\ndelta_nu is the large frequency separation (roughly constant frequency difference between radial modes of increasing order) . You must set this value if chi2_seismo_delta_nu_fraction is greater than zero.\n\ndelta_nu = -1\ndelta_nu_sigma = -1\n\n\nIf delta_nu in the inlist is greater than zero, the code uses the inlist values for both delta_nu and delta_nu_sigma.\n\nIf delta_nu is less than or equal to zero in the inlist, the code estimates it by a linear fit to the observed radial frequencies and orders, l0_obs and l0_n_obs.\n\nAlong with calculating delta_nu, if delta_nu_sigma from the inlist is less than zero, then the code also sets it by using the radial data. Note that by setting delta_nu_sigma to a positive value and delta_nu to a negative value, you can have the code get delta_nu from the given l0_obs and l0_n_obs, while still using the delta_nu_sigma from the inlist.\n\n## Mode frequency data¶\n\n### freq_sigma¶\n\nData for oscillation modes. nl(el) is the number of modes of degree el, freq_target(el,1) to freq_target(el,nl(el)) are the frequencies (in increasing frequency order) and freq_sigma(el,1) to freq_sigma(el,nl(el)) are the uncertainties.\n\nnl(0:3) = 0 ! number of observed modes\nfreq_target(0:3,:) = 0 ! frequencies. set e.g. freq_target(0,1) ... freq_target(0,nl(0)) for radial modes\nfreq_sigma(0:3,:) = 0 ! freq_sigma(el,i) is uncertainty for freq_target(el,i)\n\n\n### l0_n_obs¶\n\nl0_n_obs(i) is the radial order of l0_obs(i) for i=1, nl0. Need to give these if the code is to calculate delta_nu and delta_nu_sigma. If you provide delta_nu, then you don’t need to set these. If l0_n_obs are provided, the Kjeldsen surface correction will use them.\n\nl0_n_obs(:) = -1\n\n\n## Optimization method¶\n\n### eval_chi2_at_target_age_only¶\n\nSet this to .true. if you only want $$\\chi^2$$ for a specific age and no others. In addition, set max_age, max_years_for_timestep and num_adjusted_dt_steps_before_max_age.\n\neval_chi2_at_target_age_only = .false.\n\n\n### max_age_for_chi2¶\n\nUse these if you only want to evaluate chi2 for a given range of ages.\n\nmin_age_for_chi2 = -1 ! (years) only use if > 0\nmax_age_for_chi2 = -1 ! (years) only use if > 0\n\n\n### search_type¶\n\nThis specifies the kind of search to perform, each of which has its own separate controls further on. The options are\n\nuse_first_values\nThis option means no search. Just do a single run using first values for the parameters.\nscan_grid\n\nEvaluates $$\\chi^2$$ for each parameter combination within the min and max ranges specified below, with the spacing defined by delta.\n\nFor a first rough scan, consider setting chi2_seismo_delta_nu_fraction = 1, which skips the relatively costly calculations of frequencies and simply uses delta_nu along with the non-seismic information. You can then follow up with medium resolution scans in smaller regions around candidates from the rough scan with chi2_seismo_delta_nu_fraction = 0 to include frequencies.\n\nsimplex\n\nSearch for minimal $$\\chi^2$$ model using Nelder-Mead simplex algorithm:\n\nNelder, J. A. and Mead, R.\n\"A Simplex Method for Function Minimization.\"\nComput. J. 7, 308-313, 1965.\n\n\nThere are versions of this in Numerical Recipes under the name “amoeba”, in Matlab under the name “fminsearch”, and in Mathematica as an option for “NMminimize”. Our version has lots of bells and whistles and is, of course, superior to the others. ;)\n\nnewuoa\n\nSearch for minimal $$\\chi^2$$ model using Powell’s NEWUOA algorithm for unconstrained minimization without derivatives by quadratic polynomial approximation.\n\nM.J.D. Powell, \"Developments of NEWUOA for unconstrained minimization without derivatives\",\nDepartment of Applied Mathematics and Theoretical Physics, Cambridge, England, report NA05, 2007.\n\nbobyqa\n\nSearch for minimal $$\\chi^2$$ model using Powell’s “Bounded Optimization BY Quadratic Approximation” (BOBYQA) algorithm. Any location within the bounds is available for consideration.\n\nM.J.D. Powell, \"The BOBYQA algorithm for bound constrained optimization without derivatives\",\nDepartment of Applied Mathematics and Theoretical Physics, Cambridge, England, report NA06, 2009.\n\nfrom_file\nCalculates $$\\chi^2$$ for the parameter values in a given file. For each line of the file (after the first, which has column names), set the parameter values to that of the file for those parameters with vary_param = .true..\n\nThe two methods from Powell use quadratic interpolation, either unconstrained (NEWUOA) or bounded (BOBYQA). The Nelder-Mead simplex method doesn’t interpolate; instead it simply compares values and moves toward lower $$\\chi^2$$ and away from higher ones. In general, you can expect the Powell methods to converge faster than the simplex if the $$\\chi^2$$ terrain is not too “bumpy” (bumps confuse the interpolation). Since the simplex scheme doesn’t do interpolation, bumps don’t cause it as much trouble, so it may be more robust. If you are just getting started, go with simplex at first. Try the interpolation methods when you have a very good candidate and want to look near it for even better results.\n\nsearch_type = 'use_first_values'\n\n\n### scan_grid_output_filename¶\n\nOutput goes to the following file when search_type = 'scan_grid'.\n\nscan_grid_output_filename = 'scan_grid_results.data'\n\n\n### restart_scan_grid_from_file¶\n\nIf .true., reads the scan_grid_output file and continues from where that stopped.\n\nrestart_scan_grid_from_file = .false.\n\n\n### simplex_itermax¶\n\nMaximum number of iterations of the downhill simplex.\n\nsimplex_itermax = 1000 ! each iteration revises the simplex\n\n\n### simplex_fcn_calls_max¶\n\nMaximum number of function calls for the downhill simplex. One iteration may use several function calls. Each “function call” is a stellar evolution track to get a $$\\chi^2$$.\n\nsimplex_fcn_calls_max = 10000\n\n\n### simplex_x_rtol¶\n\nTerminate the simplex if the differences between iterations are less than either of these tolerances.\n\nsimplex_x_atol = 1d-10 ! tolerance for absolute differences\nsimplex_x_rtol = 1d-10 ! tolerance for relative differences\n\n\nIf you want the details, here’s the snippet of code. simplex(i,j) is value of i-th parameter for point j. l is the index of the best point. There are n parameters and n+1 points.\n\nterm_val_x = 0\ndo j=1,n+1 ! check each point\nif (j == l) cycle ! l is the best point; so skip it\ndo i=1,n ! check each coordinate of point j vs point l\nterm1 = abs(simplex(i,j)-simplex(i,l)) / &\n(x_atol + x_rtol*max(abs(simplex(i,j)), abs(simplex(i,l))))\nif (term1 > term_val_x) term_val_x = term1\nend do\nend do\nif (term_val_x <= 1d0) exit ! converged\n\n\n### simplex_chi2_tol¶\n\nTerminate the simplex if the best point has a $$\\chi^2$$ less than this.\n\nsimplex_chi2_tol = 1d-10 ! tolerance for chi^2\n\n\n### simplex_centroid_weight_power¶\n\nEach iteration starts by doing a reflection of the worst point through the centroid of the others. The centroid points are weighted by (1/chi^2)**power. power = 0 gives the standard unweighted centroid. power > 0 shifts the reflection towards the better points.\n\nsimplex_centroid_weight_power = 0d0\n\n\n### simplex_enforce_bounds¶\n\nIf .true., points outside the bounds will be rejected without evaluation. If .false., the bounds will only be used when creating the initial simplex and for adaptive random search.\n\nsimplex_enforce_bounds = .false.\n\n\n### simplex_output_filename¶\n\nFilename for the simplex output.\n\nsimplex_output_filename = 'simplex_results.data'\n\n\n### restart_simplex_from_file¶\n\nIf .true., then reads the output file (simplex_output_filename) and continues from where that stopped using the best $$n+1$$ results as the initial simplex (where $$n$$ is the number of parameters). Note that this restores the best simplex but you may still see it rerun recent cases if they were not good enough to be included in the simplex. We don’t restore the information about those failed attempts, so we need to rerun them.\n\nrestart_simplex_from_file = .false.\n\n\n### simplex_seed¶\n\nInteger seed for random number generation.\n\nsimplex_seed = 1074698122 ! seed for random number generation\n\n\n### simplex_alpha¶\n\nCoefficients for the reflection step of the downhill simplex algorithm.\n\nsimplex_alpha = 1d0 ! reflect\n\n\n### simplex_beta¶\n\nCoefficients for the expansion step of the downhill simplex algorithm.\n\nsimplex_beta = 2d0 ! expand\n\n\n### simplex_gamma¶\n\nCoefficients for the contraction step of the downhill simplex algorithm.\n\nsimplex_gamma = 0.5d0 ! contract\n\n\n### simplex_delta¶\n\nCoefficients for the shrink step of the downhill simplex algorithm.\n\nsimplex_delta = 0.5d0 ! shrink\n\n\n### newuoa_output_filename¶\n\nFilename for the NEWUOA output.\n\nnewuoa_output_filename = 'newuoa_results.data'\n\n\n### newuoa_rhoend¶\n\nThis is the tolerance that determines relative accuracy of final values i.e., NEWUOA stops when results are changing by less than this. See mesa/num/public/num_newuoa for details.\n\nnewuoa_rhoend = 1d-6\n\n\n### bobyqa_output_filename¶\n\nFilename for the BOBYQA output.\n\nbobyqa_output_filename = 'bobyqa_results.data'\n\n\n### bobyqa_rhoend¶\n\nThis is the tolerance that determines relative accuracy of final values i.e., BOBYQA stops when results are changing by less than this. See mesa/num/public/num_bobyqa for details.\n\nbobyqa_rhoend = 1d-6\n\n\n### filename_for_parameters¶\n\nFilename containing parameter values when search_type = 'from_file'.\n\nfilename_for_parameters = 'undefined'\n\n\n### max_num_from_file¶\n\nIf greater than zero, stop from_file search after trying this many lines from the file.\n\nmax_num_from_file = -1 ! if > 0, then stop after doing this many lines from file.\n\n### file_column_for_param¶\n\nYou need to say which columns in the file hold the various parameters. For example, if your file starts like the following:\n\n chi2 mass init_Y init_FeH alpha init_f_ov my_param1 my_param2 my_param3\n654 0.81543178 1.35000000 0.27000000 0.21000000 1.76000000 0.01000000 0.00000000 0.00000000 0.00000000\n\n\nthen set the column numbers like this:\n\nfile_column_for_param(1) = 3\nfile_column_for_param(2) = 4\nfile_column_for_param(3) = 5\nfile_column_for_param(4) = 6\nfile_column_for_param(5) = 7\nfile_column_for_param(6) = 8\nfile_column_for_param(7) = 9\nfile_column_for_param(8) = 10\n\n\nNote that if you are not varying one of the parameters, e.g. f_ov, then you don’t need to set the file_column for that parameter.\n\n### from_file_output_filename¶\n\nFilename for the from_file output.\n\nfrom_file_output_filename = 'from_file_results.data'\n\n\n## Parameters¶\n\nYou can set a large number of parameters (currently up to 100) that will be searched/optimized for a best-fitting solution. Specify the parameters by the name, then either implement your own parameters in the set_param subroutine in run_star_extras.f90, or rely on the default implementation, which provides the legacy options * initial_mass * initial_Y * initial_FeH * alpha * f_ov\n\n### param_name¶\n\nNames for parameters that will searched/optimized. Empty names '' are ignored.\n\nparam_name(:) = ''\n\n\n### vary_param¶\n\nIf .true., that parameter will be varied by the search. If .false., that parameter will be fixed at the first value (see first_param).\n\nvary_param(:) = .false.\n\n### first_param¶\n\nInitial parameter values for parameters that vary; fixed values for parameters that don’t.\n\nfirst_param(:) = 0\n\n\n### max_param¶\n\nLower and upper bounds for parameter values.\n\nmin_param(:) = 0\nmax_param(:) = 0\n\n\n### delta_param¶\n\nGrid spacing for parameter values, used when search_type = 'scan_grid'.\n\ndelta_param(:) = 0\n\n\n### dYdZ¶\n\nLegacy controls from the old interface, which is available with the default run_star_extras.f90 by using the parameter names initial_Y and initial_FeH.\n\nIf Y_depends_on_Z = .false., Y is a parameter like any other and you should set vary_Y, first_Y, min_Y and max_Y (and delta_Y for a grid). If .true., Y depends on Z according to\n\nY = Y0 + dYdZ*Z\n\n\nwhere Y0 and dYdZ are set below.\n\nY_depends_on_Z = .false.\nY0 = 0.248d0\ndYdZ = 1.4d0\n\n\n### Y_frac_he3¶\n\nIf using the legacy initial_Y parameter, this controls the proportion of the initial helium abundance that’s ³He rather than ⁴He.\n\nY_frac_he3 = 1d-4 ! = xhe3/(xhe3 + xhe4); Y = xhe3 + xhe4\n\n\n### f0_ov_div_f_ov¶\n\nIf using the legacy parameter f_ov, the overshoot f0 is changed along with overshoot f according to\n\nf0_ov = f0_ov_div_f_ov * f_ov\n\n\nso this must be set to a positive value if f_ov is not zero.\n\nf0_ov_div_f_ov = -1\n\n\n## Parameter limits¶\n\nCalculating mode frequencies is a relatively costly process, so we don’t want to do it for models that are not good candidates. i.e., we want to filter out the bad candidates using the following less expensive tests whenever possible.\n\nNote that if none of the models in a run pass these tests, then you will not get a total $$\\chi^2$$ result for that run. That might not matter but if you are eliminating too many candidates in this way, the search routines might not get enough valid results to work properly. So watch what you are doing! If your search or scan is getting lots of runs that fail to give $$\\chi^2$$ results, you’ll need to adjust the limits.\n\n### min_age_limit¶\n\nDon’t consider models that aren’t old enough.\n\nmin_age_limit = 1d6\n\n\n### Lnuc_div_L_limit¶\n\nDon’t consider models with L_nuc/L less than this limit. This rules out pre-ZAMS models.\n\nLnuc_div_L_limit = 0.95\n\n\n### chi2_spectroscopic_limit¶\n\nDon’t consider models with chi2_spectro above this limit.\n\nchi2_spectroscopic_limit = 1000\n\n\n### chi2_delta_nu_limit¶\n\nDon’t consider models with chi2_delta_nu above this limit.\n\nchi2_delta_nu_limit = 1000\n\n\nWe only calculate radial modes if the previous checks pass.\n\nDon’t consider models with chi2_radial above this limit.\n\nchi2_radial_limit = 100\n\n\nWe only calculate full $$\\chi^2$$ if the model passes all these limit checks.\n\nWe don’t want to evaluate more models than we need to but we also want to make sure that we resolve the optimal $$\\chi^2$$. These controls adjust the maximum timestep depending on how close we are to the target values.\n\nIf you set the timestep limits too large, you run the risk of missing good $$\\chi^2$$ cases. But if they are very small, you will spend a lot of runtime calculating lots of frequencies for lots of models. There is no standard set of best values for this. The choice will depend on the stage of evolution and how fast things are changing in the general region of the models with good $$\\chi^2$$ values. These are just default values: there is no alternative to trying things and tuning the controls for your problem.\n\n### max_yrs_dt_when_cold¶\n\nmax_yrs_dt_when_cold = 1d8 ! when fail Lnuc/L, chi2_spectro, or ch2_delta_nu\n\n\n### max_yrs_dt_when_warm¶\n\nmax_yrs_dt_when_warm = 1d7 ! when pass previous but fail chi2_radial; < max_yrs_dt_when_cold\n\n\n### max_yrs_dt_when_hot¶\n\nmax_yrs_dt_when_hot = 1d6 ! when pass chi2_radial; < max_yrs_dt_when_warm\n\n\n### chi2_limit_for_small_timesteps¶\n\nchi2_limit_for_small_timesteps = -1\n\n\n### max_yrs_dt_chi2_small_limit¶\n\nmax_yrs_dt_chi2_small_limit = 3d5 ! < max_yrs_dt_when_hot\n\n\n### chi2_limit_for_smaller_timesteps¶\n\nchi2_limit_for_smaller_timesteps = -1 ! < chi2_limit_for_small_timesteps\n\n\n### max_yrs_dt_chi2_smaller_limit¶\n\nmax_yrs_dt_chi2_smaller_limit = 1d5 ! < max_yrs_dt_chi2_small_limit\n\n\n### chi2_limit_for_smallest_timesteps¶\n\nchi2_limit_for_smallest_timesteps = -1 ! < chi2_limit_for_smaller_timesteps\n\n\n### max_yrs_dt_chi2_smallest_limit¶\n\nmax_yrs_dt_chi2_smallest_limit = 5d4 ! < max_yrs_dt_chi2_smaller_limit\n\n\n### sigmas_coeff_for_constraint_limit¶\n\nWe need a way to decide when to stop an evolution run. The following limits are used for this. We don’t want to stop too soon, so these limits are only tested for models that are okay for the Lnuc_div_L_limit.\n\nThe limit on each variable X is\n\nX_limit = X_target + X_sigma*sigmas_coeff_for_X_limit\n\n\nWe only use limits where sigma_coeff_X is not zero. If sigma_coeff is positive, we stop when that value is greater than the limit. If sigma_coeff is negative, we stop when that value is less than the limit. Hence, use positive sigma_coeff for values that are increasing (e.g. log_L) and negative sigma_coeff for values that are decreasing (e.g. log_g, Teff, delta_nu).\n\nsigmas_coeff_for_delta_nu_limit = 0\nsigmas_coeff_for_constraint_limit(:) = 0\n\n\n## $$\\chi^2$$ limits¶\n\nYou can stop the run if $$\\chi^2$$ is rising.\n\n### chi2_relative_increase_limit¶\n\nIf any of the following conditions are met in limit_num_chi2_too_big consecutive models after a minimum $$\\chi^2$$ has been found, stop the run for that sample.\n\n• $$\\chi^2$$ is greater than chi2_relative_increase_limit times the best $$\\chi^2$$ for the run.\n• chi2_spectro, chi2_delta_nu or chi2_radial is greater than its corresponding _limit.\n• MESA fails to evaluate $$\\chi^2$$ (because e.g. it can’t match all the modes of a particular degree).\nlimit_num_chi2_too_big = 20\nchi2_relative_increase_limit = 2.0\n\n\n### chi2_search_limit2¶\n\nIf the best $$\\chi^2$$ for the run is less than chi2_search_limit1, stop the run if $$\\chi^2$$ is greater than chi2_search_limit2.\n\nchi2_search_limit1 = 3.0\nchi2_search_limit2 = 4.0\n\n\nIf you are doing a search or scanning a grid, you can use previous results as a guide for when to stop a run\n\n### avg_model_number_sigma_limit¶\n\nWe can stop the run using limits based on the average age of previous samples. Specifically, use at least min_num_samples_for_avg and at most max_num_samples_for_avg to compute the average age and model number, then stop the run if the age is greater than the average age or model number plus avg_age_sigma_limit or avg_model_number_sigma_limit times the standard deviation of the same set of samples.\n\nmin_num_samples_for_avg = 2 ! want at least this many samples to form averages\nmax_num_samples_for_avg = 10 ! use this many of the best chi^2 samples for averages\navg_age_sigma_limit = 10 ! stop if age > avg age + this limit times sigma of avg age\navg_model_number_sigma_limit = 10 ! ditto for model number\n\n\n## Surface corrections¶\n\n### correction_scheme¶\n\nThe options for the correction scheme are:\n\n• 'kjeldsen', the correction of Kjeldsen et al. (2008);\n• 'power_law', a free power law, used as a sanity check in Ball & Gizon (2017);\n• 'cubic', the cubic/one-term correction by Ball & Gizon (2014, eqn 3);\n• 'combined', the combined/two-term correction by Ball & Gizon (2014, eq 4);\n• 'sonoi', the modified Lorentzian by Sonoi et al. (2015, eq 9); or\n• '', no corrections.\ncorrection_scheme = 'kjeldsen'\n\n\nIf you’d like to experiment with your own correction scheme, you can use the other_astero_freq_corr hook in \\$MESA_DIR/star.\n\n### surf_coef2_name¶\n\nEach surface correction has one or two parameters that are passed around in the code as surf_coef1 and surf_coef2 and reported in the output under the control parameters surf_coef1_name and surf_coef2_name. In the various schemes, the surface corrections added to a mode with frequency $$\\nu$$, normalized inertia $$\\mathcal{I}$$ and relative inertia $$Q$$ are\n\nkjeldsen surf_coef1*(ν/νmax)**correction_b / Q (surf_coef2 = correction_r, eq. 6 of K08)\npower_law surf_coef1*(ν/νmax)**surf_coef2 / Q\ncubic surf_coef1*(ν/νac)**3 / I (surf_coef2 is meaningless)\ncombined (surf_coef1*(ν/νac)**3 + surf_coef2/(ν/νac)) / I\nsonoi surf_coef1*νmax*[1 - 1/(1 + (ν/νmax)**surf_coef2)] / Q\n\nwhere νac is the acoustic cutoff frequency, computed by scaling the solar value of 5mHz in the same way as νmax (i.e. νac = 5mHz * (g/gsun)/sqrt(Teff/Teff_sun)).\n\nsurf_coef1_name = 'a_div_r'\nsurf_coef2_name = 'correction_r'\n\n\nThe default values a_div_r and correction_r correspond to the behaviour in r11701 and before.\n\n### correction_factor¶\n\nScale the correction by this fraction. Set to zero to skip doing corrections.\n\ncorrection_factor = 0\n\n\n### correction_b¶\n\nSolar-calibrated power law index used in the kjeldsen surface correction. The kjeldsen surface correction uses the observed radial orders of the radial modes (l0_n_obs) if they are provided.\n\ncorrection_b = 4.90d0\n\n\nnote: to set nu_max_sun or delta_nu_sun, see star/defaults/controls.defaults\n\noutput controls\n\n### astero_results_directory¶\n\nDirectory in which to save astero outputs. If it doesn’t exist, it will be created.\n\nastero_results_directory = 'outputs'\n\n\n### astero_results_txt_format¶\n\nFormats for writing floats (reals), integers and characters (strings) to the results file.\n\nastero_results_dbl_format = ‘(1pes26.16)’ astero_results_int_format = ‘(i26)’ astero_results_txt_format = ‘(a26)’\n\n### write_best_model_data_for_each_sample¶\n\nwrite_best_model_data_for_each_sample = .true.\n\n\n### num_digits¶\n\nNumber of digits in sample number (with leading zeros).\n\nnum_digits = 4\n\n\n### sample_results_postfix¶\n\nPrefix and postfix for sample results filenames.\n\nsample_results_prefix = 'sample_'\nsample_results_postfix = '.data'\n\n\n### model_num_digits¶\n\nNumber of digits in model number (with leading zeros).\n\nmodel_num_digits = 4\n\n\n### write_fgong_for_each_model¶\n\nwrite_fgong_for_each_model = .false.\n\n\n### fgong_postfix¶\n\nPrefix and postfix for FGONG filenames. You can include a directory in the prefix if desired but you must create the directory yourself.\n\nfgong_prefix = 'fgong_'\nfgong_postfix = '.data'\n\n\n### write_fgong_for_best_model¶\n\nwrite_fgong_for_best_model = .false.\n\n\n### best_model_fgong_filename¶\n\nbest_model_fgong_filename = ''\n\n\n### write_gyre_for_each_model¶\n\nwrite_gyre_for_each_model = .false.\n\n\n### gyre_postfix¶\n\nPrefix and postfix for GYRE filenames. You can include a directory in the prefix if desired but you must create the directory yourself.\n\ngyre_prefix = 'gyre_'\ngyre_postfix = '.data'\n\n\n### max_num_gyre_points¶\n\nmax_num_gyre_points = -1 ! only used if > 1\n\n\n### write_gyre_for_best_model¶\n\nwrite_gyre_for_best_model = .false.\n\n\n### best_model_gyre_filename¶\n\nbest_model_gyre_filename = ''\n\n\n### write_profile_for_best_model¶\n\nwrite_profile_for_best_model = .false.\n\n\n### best_model_profile_filename¶\n\nbest_model_profile_filename = ''\n\n\n### save_model_for_best_model¶\n\nsave_model_for_best_model = .false.\n\n\n### best_model_save_model_filename¶\n\nbest_model_save_model_filename = ''\n\n\n### last_model_save_info_filename¶\n\nIf save_info_for_last_model = .true., treat the final model as the “best” and save info about final model to last_model_save_info_filename.\n\nsave_info_for_last_model = .false.\nlast_model_save_info_filename = '' ! and save info about final model to this file.\n\n\n## Miscellaneous¶\n\n### save_next_best_at_lower_frequency¶\n\nSave info about next best matches.\n\nsave_next_best_at_higher_frequency = .false.\nsave_next_best_at_lower_frequency = .false.\n\n\n### trace_limits¶\n\nIf .true., write info to terminal about status relative to various limits, e.g. if sigmas_coeff_for_Teff_limit is set, Teff_limit = Teff_target + Teff_sigma*sigmas_coeff_for_Teff_limit and the run will stop when Teff < Teff_limit. Trace will write out values of Teff and Teff_limit. The same goes for other limits, e.g. logg, logL, delta_nu, etc.\n\ntrace_limits = .false.\n\n\n### save_controls_filename¶\n\nIf save_controls = .true., the controls in &astero_search_controls are dumped to the file save_controls_filename. If save_controls_filename is empty, a default filename is used.\n\nsave_controls = .false. ! dumps &astero_search_controls controls to file\nsave_controls_filename = '' ! if empty, uses a default name\n\n\n### order_to_save¶\n\nSave an eigenfunction.\n\nsave_mode_model_number = 0\nsave_mode_filename = ''\nel_to_save = 0\norder_to_save = 0\n\n\nIf .true., then star adds the atmosphere structure before passing the model to ADIPLS or GYRE. The atmosphere model is determined by the atmosphere boundary conditions in the &controls namelist for star.\n\nadd_atmosphere = .false.\n\n\n### keep_surface_point¶\n\nIf .true., keep the k=1 point of model.\n\nkeep_surface_point = .false.\n\n\nIf .true., add a point at r=0.\n\nadd_center_point = .true.\n\n\n## Oscillation calculation controls¶\n\n### oscillation_code¶\n\nEither 'adipls' or 'gyre' (lower case).\n\noscillation_code = 'adipls'\n\n\n### trace_time_in_oscillation_code¶\n\ntrace_time_in_oscillation_code = .false.\n\n\n## GYRE controls¶\n\nThe GYRE controls are read from the gyre_input_file. Rich offers the following comments on setting them:\n\n> I suggest setting freq_min to 0.9*MINVAL(l0_obs), > and freq_max to 1.1*MAXVAL(l0_obs) > (similarly for the other l values).\n\n> freq_units should be ‘UHZ’, > and set grid_type to ‘LINEAR’.\n\n> For n_freq, I suggest either setting it to 10*(freq_max - freq_min)/dfreq, > where dfreq is the estimated frequency spacing; or, set it to 10*nl0. > The factor 10 is arbitrary, but seems to be a good safety factor.\n\n### gyre_input_file¶\n\ngyre_input_file = 'gyre.in'\n\n\ngyre_non_ad = .false.\n\n\nSome ADIPLS controls are set here. The rest are set in adipls.c.pruned.in.\n\nADIPLS looks for frequencies in a given range and with a given “density” of coverage. For example, for l=0, the ADIPLS frequency search range is nu_lower_factor*l0_obs(1) to nu_upper_factor*l0_obs(nl0) and it uses iscan = iscan_factor_l0*nl0 to determine how fine the scan is over the range.\n\n### do_redistribute_mesh¶\n\nThe number of zones for ADIPLS’ remeshing is set in redistrb.c.pruned.in. If you set this .false., then the mesh from star is used directly. If you set this .true., then astero calls ADIPLS’s redistb before doing the frequency analysis.\n\ndo_redistribute_mesh = .true.\n\n\n### iscan_factor¶\n\nADIPLS will scan for frequencies at iscan_factor(l) times the number of observed l modes.\n\niscan_factor(0) = 15\niscan_factor(1) = 15\niscan_factor(2) = 15\niscan_factor(3) = 15\n\n\n### nu_upper_factor¶\n\nThe frequency scan range is set from the observed frequencies times these factors.\n\nnu_lower_factor = 0.8\nnu_upper_factor = 1.2\n\n\nadipls_irotkr = 0\n\n\n## Include other inlists¶\n\n### extra_astero_search_inlist5_name¶\n\nIf read_extra_astero_search_inlistN is .true., then read the namelist in file extra_astero_search_inlistN_name.\n\nread_extra_astero_search_inlist1 = .false.\nextra_astero_search_inlist1_name = 'undefined'" ]

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[ "# #ebdbae Color Information\n\nIn a RGB color space, hex #ebdbae is composed of 92.2% red, 85.9% green and 68.2% blue. Whereas in a CMYK color space, it is composed of 0% cyan, 6.8% magenta, 26% yellow and 7.8% black. It has a hue angle of 44.3 degrees, a saturation of 60.4% and a lightness of 80.2%. #ebdbae color hex could be obtained by blending #ffffff with #d7b75d. Closest websafe color is: #ffcc99.\n\n• R 92\n• G 86\n• B 68\nRGB color chart\n• C 0\n• M 7\n• Y 26\n• K 8\nCMYK color chart\n\n#ebdbae color description : Very soft orange.\n\n# #ebdbae Color Conversion\n\nThe hexadecimal color #ebdbae has RGB values of R:235, G:219, B:174 and CMYK values of C:0, M:0.07, Y:0.26, K:0.08. Its decimal value is 15457198.\n\nHex triplet RGB Decimal ebdbae `#ebdbae` 235, 219, 174 `rgb(235,219,174)` 92.2, 85.9, 68.2 `rgb(92.2%,85.9%,68.2%)` 0, 7, 26, 8 44.3°, 60.4, 80.2 `hsl(44.3,60.4%,80.2%)` 44.3°, 26, 92.2 ffcc99 `#ffcc99`\nCIE-LAB 87.67, -1.355, 24.156 67.231, 71.382, 50.279 0.356, 0.378, 71.382 87.67, 24.194, 93.21 87.67, 12.336, 34.354 84.488, -5.812, 23.858 11101011, 11011011, 10101110\n\n# Color Schemes with #ebdbae\n\n• #ebdbae\n``#ebdbae` `rgb(235,219,174)``\n• #aebeeb\n``#aebeeb` `rgb(174,190,235)``\nComplementary Color\n• #ebbdae\n``#ebbdae` `rgb(235,189,174)``\n• #ebdbae\n``#ebdbae` `rgb(235,219,174)``\n• #ddebae\n``#ddebae` `rgb(221,235,174)``\nAnalogous Color\n• #bdaeeb\n``#bdaeeb` `rgb(189,174,235)``\n• #ebdbae\n``#ebdbae` `rgb(235,219,174)``\n• #aeddeb\n``#aeddeb` `rgb(174,221,235)``\nSplit Complementary Color\n• #dbaeeb\n``#dbaeeb` `rgb(219,174,235)``\n• #ebdbae\n``#ebdbae` `rgb(235,219,174)``\n• #aeebdb\n``#aeebdb` `rgb(174,235,219)``\n• #ebaebe\n``#ebaebe` `rgb(235,174,190)``\n• #ebdbae\n``#ebdbae` `rgb(235,219,174)``\n• #aeebdb\n``#aeebdb` `rgb(174,235,219)``\n• #aebeeb\n``#aebeeb` `rgb(174,190,235)``\n• #dcc071\n``#dcc071` `rgb(220,192,113)``\n• #e1c985\n``#e1c985` `rgb(225,201,133)``\n• #e6d29a\n``#e6d29a` `rgb(230,210,154)``\n• #ebdbae\n``#ebdbae` `rgb(235,219,174)``\n• #f0e4c2\n``#f0e4c2` `rgb(240,228,194)``\n• #f5edd7\n``#f5edd7` `rgb(245,237,215)``\n• #faf6eb\n``#faf6eb` `rgb(250,246,235)``\nMonochromatic Color\n\n# Alternatives to #ebdbae\n\nBelow, you can see some colors close to #ebdbae. Having a set of related colors can be useful if you need an inspirational alternative to your original color choice.\n\n• #ebccae\n``#ebccae` `rgb(235,204,174)``\n• #ebd1ae\n``#ebd1ae` `rgb(235,209,174)``\n• #ebd6ae\n``#ebd6ae` `rgb(235,214,174)``\n• #ebdbae\n``#ebdbae` `rgb(235,219,174)``\n• #ebe0ae\n``#ebe0ae` `rgb(235,224,174)``\n• #ebe5ae\n``#ebe5ae` `rgb(235,229,174)``\n• #ebeaae\n``#ebeaae` `rgb(235,234,174)``\nSimilar Colors\n\n# #ebdbae Preview\n\nThis text has a font color of #ebdbae.\n\n``<span style=\"color:#ebdbae;\">Text here</span>``\n#ebdbae background color\n\nThis paragraph has a background color of #ebdbae.\n\n``<p style=\"background-color:#ebdbae;\">Content here</p>``\n#ebdbae border color\n\nThis element has a border color of #ebdbae.\n\n``<div style=\"border:1px solid #ebdbae;\">Content here</div>``\nCSS codes\n``.text {color:#ebdbae;}``\n``.background {background-color:#ebdbae;}``\n``.border {border:1px solid #ebdbae;}``\n\n# Shades and Tints of #ebdbae\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, #0d0b03 is the darkest color, while #fefefd is the lightest one.\n\n• #0d0b03\n``#0d0b03` `rgb(13,11,3)``\n• #1d1707\n``#1d1707` `rgb(29,23,7)``\n• #2d240b\n``#2d240b` `rgb(45,36,11)``\n• #3d310f\n``#3d310f` `rgb(61,49,15)``\n• #4c3d13\n``#4c3d13` `rgb(76,61,19)``\n• #5c4a17\n``#5c4a17` `rgb(92,74,23)``\n• #6c561b\n``#6c561b` `rgb(108,86,27)``\n• #7c631e\n``#7c631e` `rgb(124,99,30)``\n• #8b7022\n``#8b7022` `rgb(139,112,34)``\n• #9b7c26\n``#9b7c26` `rgb(155,124,38)``\n• #ab892a\n``#ab892a` `rgb(171,137,42)``\n• #ba962e\n``#ba962e` `rgb(186,150,46)``\n• #caa232\n``#caa232` `rgb(202,162,50)``\n• #d0aa40\n``#d0aa40` `rgb(208,170,64)``\n• #d4b150\n``#d4b150` `rgb(212,177,80)``\n• #d8b85f\n``#d8b85f` `rgb(216,184,95)``\n• #dbbf6f\n``#dbbf6f` `rgb(219,191,111)``\n• #dfc67f\n``#dfc67f` `rgb(223,198,127)``\n• #e3cd8f\n``#e3cd8f` `rgb(227,205,143)``\n• #e7d49e\n``#e7d49e` `rgb(231,212,158)``\n• #ebdbae\n``#ebdbae` `rgb(235,219,174)``\n• #efe2be\n``#efe2be` `rgb(239,226,190)``\n• #f3e9cd\n``#f3e9cd` `rgb(243,233,205)``\n• #f7f0dd\n``#f7f0dd` `rgb(247,240,221)``\n• #fbf7ed\n``#fbf7ed` `rgb(251,247,237)``\n• #fefefd\n``#fefefd` `rgb(254,254,253)``\nTint Color Variation\n\n# Tones of #ebdbae\n\nA tone is produced by adding gray to any pure hue. In this case, #d0cec9 is the less saturated color, while #fee49b is the most saturated one.\n\n• #d0cec9\n``#d0cec9` `rgb(208,206,201)``\n• #d4d0c5\n``#d4d0c5` `rgb(212,208,197)``\n• #d8d2c1\n``#d8d2c1` `rgb(216,210,193)``\n• #dbd4be\n``#dbd4be` `rgb(219,212,190)``\n• #dfd5ba\n``#dfd5ba` `rgb(223,213,186)``\n• #e3d7b6\n``#e3d7b6` `rgb(227,215,182)``\n• #e7d9b2\n``#e7d9b2` `rgb(231,217,178)``\n• #ebdbae\n``#ebdbae` `rgb(235,219,174)``\n• #efddaa\n``#efddaa` `rgb(239,221,170)``\n• #f3dfa6\n``#f3dfa6` `rgb(243,223,166)``\n• #f7e1a2\n``#f7e1a2` `rgb(247,225,162)``\n• #fbe29e\n``#fbe29e` `rgb(251,226,158)``\n• #fee49b\n``#fee49b` `rgb(254,228,155)``\nTone Color Variation\n\n# Color Blindness Simulator\n\nBelow, you can see how #ebdbae 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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[ "# Block diagram child node\n\nI'm just trying to make any block diagram using LaTeX, but i get into a trouble that I don't know how to make child of a node,", null, "All I want is to move the box named \"Antrian\" into the red position, and make the two boxes below it as its child node.\n\n\\documentclass{article}\n\\usepackage[latin1]{inputenc}\n\\usepackage{tikz}\n\\usetikzlibrary{shapes,arrows, arrows.meta, calc, positioning}\n\\begin{document}\n\\pagestyle{empty}\n\n% Define block styles\n\\tikzstyle{block} = [rectangle, draw,\ntext width=22mm, text centered, rounded corners, minimum height=4em]\n\\tikzstyle{line} = [draw, -latex']\n\n\\begin{tikzpicture}[node distance = 2cm, auto]\n% Place nodes\n\\node [block] (operasional) {Operasional};\n\\node [block, right of=operasional, node distance = 3cm] (proses) {Proses};\n\n\\node[block, right of= proses, node distance = 3cm] (antrian){Antrian};\n\\node[block, below of= antrian] (kasir){Jumlah kasir yang beroperasi};\n\\node[block, right of=kasir, node distance = 3cm](kedatangan){Rata-rata tingkat kedatangan};\n\\coordinate[below = of $(kasir.south)!0.5!(kedatangan.south)$](temp1);\n\\node[block, below of = temp1, node distance =3cm] (kinerja) {Kinerja};\n\\node[block,left of= kinerja, node distance = 3cm](pelayanan){Pelayanan yang optimal};\n\n% Draw edges\n\\path [line] (operasional) -- (proses);\n\\path[line] (proses) -- (antrian);\n\\path [line] (antrian.south) -| (kasir);\n\\path [line] (antrian.south) -| (kedatangan);\n\\path [line] (kinerja) -- (pelayanan);\n\\draw (kasir.south) |- (temp1) -| (kedatangan);\n\\draw[-Latex] (temp1) -- (kinerja);\n\\end{tikzpicture}\n\\end{document}\n\n\nI really need your help, thanks very much\n\nWelcome to TeX.SE! You are loading but not using positioning. I also replaced \\tikzstyle by the corresponding \\tikzset syntax.\n\n\\documentclass{article}\n\\usepackage[latin1]{inputenc}\n\\usepackage{tikz}\n\\usetikzlibrary{shapes,arrows, arrows.meta, calc, positioning}\n\\begin{document}\n\\pagestyle{empty}\n\n% Define block styles\n\\tikzset{block/.style={rectangle, draw,\ntext width=22mm, text centered, rounded corners, minimum height=4em},\nline/.style={draw, -latex'}}\n\n\\begin{tikzpicture}[node distance = 1.5cm, auto]\n% Place nodes\n\\node [block] (operasional) {Operasional};\n\\node [block, right=of operasional] (proses) {Proses};\n\n\\node[block, right=of proses] (antrian){Antrian};\n\\node[block, below left=of antrian] (kasir){Jumlah kasir yang beroperasi};\n\\node[block, below right=of antrian](kedatangan){Rata-rata tingkat kedatangan};\n\\coordinate[below = of $(kasir.south)!0.5!(kedatangan.south)$](temp1);\n\\node[block, below of = temp1, node distance =3cm] (kinerja) {Kinerja};\n\\node[block,left=of kinerja](pelayanan){Pelayanan yang optimal};\n\n% Draw edges\n\\path [line] (operasional) -- (proses);\n\\path[line] (proses) -- (antrian);\n\\path [line] (antrian.south) -- ++(0,-0.5) -| (kasir);\n\\path [line] (antrian.south) -- ++(0,-0.5) -| (kedatangan);\n\\path [line] (kinerja) -- (pelayanan);\n\\draw (kasir.south) |- (temp1) -| (kedatangan);\n\\draw[-Latex] (temp1) -- (kinerja);\n\\end{tikzpicture}\n\\end{document}", null, "• @SutMar Glad to hear! BTW, if the answer worked for you, could you perhaps consider accepting it by clicking on the checkmark left of it? – user121799 Oct 25 '18 at 2:12" ]

[ null, "https://i.stack.imgur.com/qNpqW.png", null, "https://i.stack.imgur.com/2FLsu.png", null ]

[ "## Monday, March 20, 2006\n\n### Never Goes Away... Never Goes Away! Topic: Algebra/Sequences & Series. Level: AIME/Olympiad.\n\nProblem: (Problem-Solving Through Problems) Let $p$ and $q$ be real numbers with $\\frac{1}{p}-\\frac{1}{q} = 1$, $0 < p \\le \\frac{1}{2}$. Show that\n\n$p+\\frac{1}{2}p^2+\\frac{1}{3}p^3+\\cdots = q-\\frac{1}{2}q^2+\\frac{1}{3}q^3-\\cdots$.\n\nSolution: We're going to use a little bit of *gasp* calculus for this problem, because it's pretty nice. All we need is the following two facts:\n\n$-\\ln{(1-x)} = x+\\frac{1}{2}x^2+\\frac{1}{3}x^3+\\cdots$\n\nand\n\n$\\ln{(1+x)} = x-\\frac{1}{2}x^2+\\frac{1}{3}x^3-\\cdots$,\n\nwhich are simply Taylor Series centered at zero (Maclaurin Series).\n\nWe are given $\\frac{1}{p}-\\frac{1}{q} = 1$, which we can rewrite as $(1+q)(1-p) = 1$, or even better,\n\n$\\ln{(1+q)}+\\ln{(1-p)} = 0$\n\n$-\\ln{(1-p)} = \\ln{(1+q)}$.\n\nBut by our infinite series representations above, we have\n\n$p+\\frac{1}{2}p^2+\\frac{1}{3}p^3+\\cdots = q-\\frac{1}{2}q^2+\\frac{1}{3}q^3-\\cdots$,\n\nas desired. QED.\n\n--------------------\n\nComment: It turns out that with the infinite series representations the problem is really simple, so the only real step is finding those. This is easily done by experimenting with the natural log function.\n\n--------------------\n\nPractice Problem: Prove that $e^{i\\theta} = \\cos{\\theta}+i\\sin{\\theta}$ using infinite series." ]

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[ "# How relevant is the Heisenberg Uncertainty Principle?\n\nI was originally surprised to see that,\n\n$$\\Delta x \\cdot \\Delta p \\gt {{\\hbar} \\over 2}$$\n\nBut, then I realized that $\\hbar/2=5.27 \\cdot 10^{-35}$. According to this other question, the smallest length ever measured was on the order of $10^{-18}$. Of course at that point, I bring the Planck length into consideration. It's order of magnitude is $10^{-35}$. I was quite shocked to see that the uncertainty is so small compared to this unit and our practical probing unit.\n\nMy question is this. How relevant is the Heisenberg Uncertainty Principle in the lab? Does it really limit what can be probed at a practical level, or is it a theoretical limit still? In addition, if the Planck scale is shown to be the shortest meaningful length, is having a limit on uncertainty only 5 times larger than that fundamental length really that inconvenient?\n\n• Your numbers lack units. Plug in units and typical length and momentum scales and you see it's quite relevant at e.g. the scales we have for atoms. Aug 16, 2015 at 0:40\n• In a hydrogen atom the kinetic energy is on the order of 5 eV, from $T = p^2/2m$ we get that the typical momentum is about 7 keV/c. On the other hand $\\hbar/2a_0 \\approx 12$ keV/c where $a_0$ is the Bohr radius. Since these quantities are of similar magnitude the Heisenberg principle is certainly very relevant. Aug 16, 2015 at 1:31\n• It is also important to note that the Heisenberg principle is neither an axiom of quantum mechanics nor a statement about measurement uncertainty. It's a consequence of the founding principles of quantum mechanics so that you can't have the equations of QM without it. It is important precisely when quantum effects are. It is not always straightforward to determine the latter though, you need to make estimates like the above, which are not always obvious. Aug 16, 2015 at 1:39\n• Anna and Robin, I am pretty sure that the founders of QM would disagree with your claim that the uncertainty principle is not an axiom, or that it is not a postulate of quantum mechanics. It is, and it is arguably the most important new one. Sources on the Internet agree, too, for example, en.wikipedia.org/wiki/Copenhagen_interpretation says that HUP is a detailed formulation of \"another axiom\" of QM. And various books sensibly says that Heisenberg \"postulated\" HUP, see abarim-publications.com/HeisenbergUncertaintyPrinciple.html Why do you find \"axiom\" or \"postulate\" bad? Aug 16, 2015 at 6:39\n• @LubošMotl, the Wikipedia page says that the \"incompatibility [of conjugate variables] is expressed quantitatively by Heisenberg's uncertainty principle\", but I think the proper statement here is that it is the commutation relations that express incompatibility and are postulated or axiomatic. By finding a representation of the CRs you can do all of quantum mechanics, e.g. calculate the hydrogen spectrum, but just the HUP is not enough for that. Aug 16, 2015 at 15:56\n\nIn a hydrogen atom the kinetic energy is on the order of $8$ eV. From $T = \\frac{p^2}{2m}$ we get that the typical momentum is about $3$ keV/c ($m = 511$ keV/$c^2$). On the other hand $\\hbar/(2a_0)\\approx 1.2$ keV/c where $a_0$ is the Bohr radius, which is about the size of a hydrogen atom. Since these quantities are of similar magnitude the Heisenberg principle is certainly very relevant." ]

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[ ":: Function Spaces in the Category of Directed Suprema Preserving Maps\n:: by Grzegorz Bancerek and Adam Naumowicz\n::\n:: Copyright (c) 1999-2021 Association of Mizar Users\n\ndefinition\nlet F be Function;\nattr F is uncurrying means :Def1: :: WAYBEL27:def 1\n( ( for x being set st x in dom F holds\nx is Function-yielding Function ) & ( for f being Function st f in dom F holds\nF . f = uncurry f ) );\nattr F is currying means :Def2: :: WAYBEL27:def 2\n( ( for x being set st x in dom F holds\n( x is Function & proj1 x is Relation ) ) & ( for f being Function st f in dom F holds\nF . f = curry f ) );\nattr F is commuting means :Def3: :: WAYBEL27:def 3\n( ( for x being set st x in dom F holds\nx is Function-yielding Function ) & ( for f being Function st f in dom F holds\nF . f = commute f ) );\nend;\n\n:: deftheorem Def1 defines uncurrying WAYBEL27:def 1 :\nfor F being Function holds\n( F is uncurrying iff ( ( for x being set st x in dom F holds\nx is Function-yielding Function ) & ( for f being Function st f in dom F holds\nF . f = uncurry f ) ) );\n\n:: deftheorem Def2 defines currying WAYBEL27:def 2 :\nfor F being Function holds\n( F is currying iff ( ( for x being set st x in dom F holds\n( x is Function & proj1 x is Relation ) ) & ( for f being Function st f in dom F holds\nF . f = curry f ) ) );\n\n:: deftheorem Def3 defines commuting WAYBEL27:def 3 :\nfor F being Function holds\n( F is commuting iff ( ( for x being set st x in dom F holds\nx is Function-yielding Function ) & ( for f being Function st f in dom F holds\nF . f = commute f ) ) );\n\nregistration\ncoherence\nfor b1 being Function st b1 is empty holds\n( b1 is uncurrying & b1 is currying & b1 is commuting )\n;\nend;\n\nregistration\nexistence\nex b1 being Function st\n( b1 is uncurrying & b1 is currying & b1 is commuting )\nproof end;\nend;\n\nregistration\nlet F be uncurrying Function;\nlet X be set ;\ncluster F | X -> uncurrying ;\ncoherence\nF | X is uncurrying\nproof end;\nend;\n\nregistration\nlet F be currying Function;\nlet X be set ;\ncluster F | X -> currying ;\ncoherence\nF | X is currying\nproof end;\nend;\n\ntheorem Th1: :: WAYBEL27:1\nfor X, Y, Z, D being set st D c= Funcs (X,(Funcs (Y,Z))) holds\nex F being ManySortedSet of D st\n( F is uncurrying & rng F c= Funcs ([:X,Y:],Z) )\nproof end;\n\ntheorem Th2: :: WAYBEL27:2\nfor X, Y, Z, D being set st D c= Funcs ([:X,Y:],Z) holds\nex F being ManySortedSet of D st\n( F is currying & ( ( Y = {} implies X = {} ) implies rng F c= Funcs (X,(Funcs (Y,Z))) ) )\nproof end;\n\nregistration\nlet X, Y, Z be set ;\ncluster Relation-like Funcs (X,(Funcs (Y,Z))) -defined Function-like V27( Funcs (X,(Funcs (Y,Z)))) uncurrying for set ;\nexistence\nex b1 being ManySortedSet of Funcs (X,(Funcs (Y,Z))) st b1 is uncurrying\nproof end;\ncluster Relation-like Funcs ([:X,Y:],Z) -defined Function-like V27( Funcs ([:X,Y:],Z)) currying for set ;\nexistence\nex b1 being ManySortedSet of Funcs ([:X,Y:],Z) st b1 is currying\nproof end;\nend;\n\ntheorem Th3: :: WAYBEL27:3\nfor A, B being non empty set\nfor C being set\nfor f, g being commuting Function st dom f c= Funcs (A,(Funcs (B,C))) & rng f c= dom g holds\ng * f = id (dom f)\nproof end;\n\ntheorem Th4: :: WAYBEL27:4\nfor B being non empty set\nfor A, C being set\nfor f being uncurrying Function\nfor g being currying Function st dom f c= Funcs (A,(Funcs (B,C))) & rng f c= dom g holds\ng * f = id (dom f)\nproof end;\n\ntheorem Th5: :: WAYBEL27:5\nfor A, B, C being set\nfor f being currying Function\nfor g being uncurrying Function st dom f c= Funcs ([:A,B:],C) & rng f c= dom g holds\ng * f = id (dom f)\nproof end;\n\ntheorem Th6: :: WAYBEL27:6\nfor f being Function-yielding Function\nfor i, A being set st i in dom () holds\n(() . i) .: A c= pi ((f .: A),i)\nproof end;\n\ntheorem Th7: :: WAYBEL27:7\nfor f being Function-yielding Function\nfor i, A being set st ( for g being Function st g in f .: A holds\ni in dom g ) holds\npi ((f .: A),i) c= (() . i) .: A\nproof end;\n\ntheorem Th8: :: WAYBEL27:8\nfor X, Y being set\nfor f being Function st rng f c= Funcs (X,Y) holds\nfor i, A being set st i in X holds\n(() . i) .: A = pi ((f .: A),i)\nproof end;\n\ntheorem Th9: :: WAYBEL27:9\nfor f being Function\nfor i, A being set st [:A,{i}:] c= dom f holds\npi ((() .: A),i) = f .: [:A,{i}:]\nproof end;\n\nregistration\nlet T be constituted-Functions 1-sorted ;\ncluster the carrier of T -> functional ;\ncoherence\nthe carrier of T is functional\n;\nend;\n\nregistration\nexistence\nex b1 being LATTICE st\n( b1 is constituted-Functions & b1 is complete & b1 is strict )\nproof end;\nexistence\nex b1 being 1-sorted st\n( b1 is constituted-Functions & not b1 is empty )\nproof end;\nend;\n\nregistration\nlet T be non empty constituted-Functions RelStr ;\ncluster non empty -> non empty constituted-Functions for SubRelStr of T;\ncoherence\nfor b1 being non empty SubRelStr of T holds b1 is constituted-Functions\nproof end;\nend;\n\ntheorem Th10: :: WAYBEL27:10\nfor S, T being complete LATTICE\nfor f being idempotent Function of T,T\nfor h being Function of S,() holds f * h = h\nproof end;\n\ntheorem :: WAYBEL27:11\nfor S, T, T1 being non empty RelStr st T is SubRelStr of T1 holds\nfor f being Function of S,T\nfor f1 being Function of S,T1 st f is monotone & f = f1 holds\nf1 is monotone\nproof end;\n\ntheorem Th12: :: WAYBEL27:12\nfor S, T, T1 being non empty RelStr st T is full SubRelStr of T1 holds\nfor f being Function of S,T\nfor f1 being Function of S,T1 st f1 is monotone & f = f1 holds\nf is monotone\nproof end;\n\ntheorem Th13: :: WAYBEL27:13\nfor X being set\nfor V being Subset of X holds\n( (chi (V,X)) \" {1} = V & (chi (V,X)) \" = X \\ V )\nproof end;\n\ndefinition\nlet X be non empty set ;\nlet T be non empty RelStr ;\nlet f be Element of (T |^ X);\nlet x be Element of X;\n:: original: .\nredefine func f . x -> Element of T;\ncoherence\nf . x is Element of T\nproof end;\nend;\n\ntheorem Th14: :: WAYBEL27:14\nfor X being non empty set\nfor T being non empty RelStr\nfor f, g being Element of (T |^ X) holds\n( f <= g iff for x being Element of X holds f . x <= g . x )\nproof end;\n\ntheorem Th15: :: WAYBEL27:15\nfor X being set\nfor L, S being non empty RelStr st RelStr(# the carrier of L, the InternalRel of L #) = RelStr(# the carrier of S, the InternalRel of S #) holds\nL |^ X = S |^ X\nproof end;\n\ntheorem :: WAYBEL27:16\nfor S1, S2, T1, T2 being non empty TopSpace st TopStruct(# the carrier of S1, the topology of S1 #) = TopStruct(# the carrier of S2, the topology of S2 #) & TopStruct(# the carrier of T1, the topology of T1 #) = TopStruct(# the carrier of T2, the topology of T2 #) holds\noContMaps (S1,T1) = oContMaps (S2,T2)\nproof end;\n\ntheorem Th17: :: WAYBEL27:17\nfor X being set ex f being Function of (),( |^ X) st\n( f is isomorphic & ( for Y being Subset of X holds f . Y = chi (Y,X) ) )\nproof end;\n\ntheorem Th18: :: WAYBEL27:18\nfor X being set holds BoolePoset X, |^ X are_isomorphic\nproof end;\n\ntheorem Th19: :: WAYBEL27:19\nfor X, Y being non empty set\nfor T being non empty Poset\nfor S1 being non empty full SubRelStr of (T |^ X) |^ Y\nfor S2 being non empty full SubRelStr of (T |^ Y) |^ X\nfor F being Function of S1,S2 st F is commuting holds\nF is monotone\nproof end;\n\ntheorem Th20: :: WAYBEL27:20\nfor X, Y being non empty set\nfor T being non empty Poset\nfor S1 being non empty full SubRelStr of (T |^ Y) |^ X\nfor S2 being non empty full SubRelStr of T |^ [:X,Y:]\nfor F being Function of S1,S2 st F is uncurrying holds\nF is monotone\nproof end;\n\ntheorem Th21: :: WAYBEL27:21\nfor X, Y being non empty set\nfor T being non empty Poset\nfor S1 being non empty full SubRelStr of (T |^ Y) |^ X\nfor S2 being non empty full SubRelStr of T |^ [:X,Y:]\nfor F being Function of S2,S1 st F is currying holds\nF is monotone\nproof end;\n\ndefinition\nlet S be non empty RelStr ;\nlet T be non empty reflexive antisymmetric RelStr ;\nfunc UPS (S,T) -> strict RelStr means :Def4: :: WAYBEL27:def 4\n( it is full SubRelStr of T |^ the carrier of S & ( for x being set holds\n( x in the carrier of it iff x is directed-sups-preserving Function of S,T ) ) );\nexistence\nex b1 being strict RelStr st\n( b1 is full SubRelStr of T |^ the carrier of S & ( for x being set holds\n( x in the carrier of b1 iff x is directed-sups-preserving Function of S,T ) ) )\nproof end;\nuniqueness\nfor b1, b2 being strict RelStr st b1 is full SubRelStr of T |^ the carrier of S & ( for x being set holds\n( x in the carrier of b1 iff x is directed-sups-preserving Function of S,T ) ) & b2 is full SubRelStr of T |^ the carrier of S & ( for x being set holds\n( x in the carrier of b2 iff x is directed-sups-preserving Function of S,T ) ) holds\nb1 = b2\nproof end;\nend;\n\n:: deftheorem Def4 defines UPS WAYBEL27:def 4 :\nfor S being non empty RelStr\nfor T being non empty reflexive antisymmetric RelStr\nfor b3 being strict RelStr holds\n( b3 = UPS (S,T) iff ( b3 is full SubRelStr of T |^ the carrier of S & ( for x being set holds\n( x in the carrier of b3 iff x is directed-sups-preserving Function of S,T ) ) ) );\n\nregistration\nlet S be non empty RelStr ;\nlet T be non empty reflexive antisymmetric RelStr ;\ncoherence\n( not UPS (S,T) is empty & UPS (S,T) is reflexive & UPS (S,T) is antisymmetric & UPS (S,T) is constituted-Functions )\nproof end;\nend;\n\nregistration\nlet S be non empty RelStr ;\nlet T be non empty Poset;\ncluster UPS (S,T) -> strict transitive ;\ncoherence\nUPS (S,T) is transitive\nproof end;\nend;\n\ntheorem Th22: :: WAYBEL27:22\nfor S being non empty RelStr\nfor T being non empty reflexive antisymmetric RelStr holds the carrier of (UPS (S,T)) c= Funcs ( the carrier of S, the carrier of T)\nproof end;\n\ndefinition\nlet S be non empty RelStr ;\nlet T be non empty reflexive antisymmetric RelStr ;\nlet f be Element of (UPS (S,T));\nlet s be Element of S;\n:: original: .\nredefine func f . s -> Element of T;\ncoherence\nf . s is Element of T\nproof end;\nend;\n\ntheorem Th23: :: WAYBEL27:23\nfor S being non empty RelStr\nfor T being non empty reflexive antisymmetric RelStr\nfor f, g being Element of (UPS (S,T)) holds\n( f <= g iff for s being Element of S holds f . s <= g . s )\nproof end;\n\ntheorem Th24: :: WAYBEL27:24\nfor S, T being complete Scott TopLattice holds UPS (S,T) = SCMaps (S,T)\nproof end;\n\ntheorem Th25: :: WAYBEL27:25\nfor S, S9 being non empty RelStr\nfor T, T9 being non empty reflexive antisymmetric RelStr st RelStr(# the carrier of S, the InternalRel of S #) = RelStr(# the carrier of S9, the InternalRel of S9 #) & RelStr(# the carrier of T, the InternalRel of T #) = RelStr(# the carrier of T9, the InternalRel of T9 #) holds\nUPS (S,T) = UPS (S9,T9)\nproof end;\n\nregistration\nlet S, T be complete LATTICE;\ncluster UPS (S,T) -> strict complete ;\ncoherence\nUPS (S,T) is complete\nproof end;\nend;\n\ntheorem Th26: :: WAYBEL27:26\nfor S, T being complete LATTICE holds UPS (S,T) is sups-inheriting SubRelStr of T |^ the carrier of S\nproof end;\n\ntheorem Th27: :: WAYBEL27:27\nfor S, T being complete LATTICE\nfor A being Subset of (UPS (S,T)) holds sup A = \"\\/\" (A,(T |^ the carrier of S))\nproof end;\n\ndefinition\nlet S1, S2, T1, T2 be non empty reflexive antisymmetric RelStr ;\nlet f be Function of S1,S2;\nassume A1: f is directed-sups-preserving ;\nlet g be Function of T1,T2;\nassume A2: g is directed-sups-preserving ;\nfunc UPS (f,g) -> Function of (UPS (S2,T1)),(UPS (S1,T2)) means :Def5: :: WAYBEL27:def 5\nfor h being directed-sups-preserving Function of S2,T1 holds it . h = (g * h) * f;\nexistence\nex b1 being Function of (UPS (S2,T1)),(UPS (S1,T2)) st\nfor h being directed-sups-preserving Function of S2,T1 holds b1 . h = (g * h) * f\nproof end;\nuniqueness\nfor b1, b2 being Function of (UPS (S2,T1)),(UPS (S1,T2)) st ( for h being directed-sups-preserving Function of S2,T1 holds b1 . h = (g * h) * f ) & ( for h being directed-sups-preserving Function of S2,T1 holds b2 . h = (g * h) * f ) holds\nb1 = b2\nproof end;\nend;\n\n:: deftheorem Def5 defines UPS WAYBEL27:def 5 :\nfor S1, S2, T1, T2 being non empty reflexive antisymmetric RelStr\nfor f being Function of S1,S2 st f is directed-sups-preserving holds\nfor g being Function of T1,T2 st g is directed-sups-preserving holds\nfor b7 being Function of (UPS (S2,T1)),(UPS (S1,T2)) holds\n( b7 = UPS (f,g) iff for h being directed-sups-preserving Function of S2,T1 holds b7 . h = (g * h) * f );\n\n:: 2.6. PROPOSITOION, p. 115\n:: preservation of composition\ntheorem Th28: :: WAYBEL27:28\nfor S1, S2, S3, T1, T2, T3 being non empty Poset\nfor f1 being directed-sups-preserving Function of S2,S3\nfor f2 being directed-sups-preserving Function of S1,S2\nfor g1 being directed-sups-preserving Function of T1,T2\nfor g2 being directed-sups-preserving Function of T2,T3 holds (UPS (f2,g2)) * (UPS (f1,g1)) = UPS ((f1 * f2),(g2 * g1))\nproof end;\n\n:: 2.6. PROPOSITOION, p. 115\n:: preservation of identities\ntheorem Th29: :: WAYBEL27:29\nfor S, T being non empty reflexive antisymmetric RelStr holds UPS ((id S),(id T)) = id (UPS (S,T))\nproof end;\n\n:: 2.6. PROPOSITOION, p. 115\n:: preservation of directed-sups\ntheorem Th30: :: WAYBEL27:30\nfor S1, S2, T1, T2 being complete LATTICE\nfor f being directed-sups-preserving Function of S1,S2\nfor g being directed-sups-preserving Function of T1,T2 holds UPS (f,g) is directed-sups-preserving\nproof end;\n\ntheorem Th31: :: WAYBEL27:31\nproof end;\n\ntheorem :: WAYBEL27:32\nfor S being complete Scott TopLattice holds oContMaps (S,Sierpinski_Space) = UPS (S,)\nproof end;\n\n:: 2.7. LEMMA, p. 116\ntheorem Th33: :: WAYBEL27:33\nfor S being complete LATTICE ex F being Function of (UPS (S,)),(InclPoset ()) st\n( F is isomorphic & ( for f being directed-sups-preserving Function of S, holds F . f = f \" {1} ) )\nproof end;\n\ntheorem Th34: :: WAYBEL27:34\nfor S being complete LATTICE holds UPS (S,), InclPoset () are_isomorphic\nproof end;\n\ntheorem Th35: :: WAYBEL27:35\nfor S1, S2, T1, T2 being complete LATTICE\nfor f being Function of S1,S2\nfor g being Function of T1,T2 st f is isomorphic & g is isomorphic holds\nUPS (f,g) is isomorphic\nproof end;\n\ntheorem Th36: :: WAYBEL27:36\nfor S1, S2, T1, T2 being complete LATTICE st S1,S2 are_isomorphic & T1,T2 are_isomorphic holds\nUPS (S2,T1), UPS (S1,T2) are_isomorphic\nproof end;\n\ntheorem Th37: :: WAYBEL27:37\nfor S, T being complete LATTICE\nfor f being directed-sups-preserving projection Function of T,T holds Image (UPS ((id S),f)) = UPS (S,())\nproof end;\n\nLm1: now :: thesis: for M being non empty set\nfor X, Y being non empty Poset\nfor f being directed-sups-preserving Function of X,(Y |^ M) holds\n( f in Funcs ( the carrier of X,(Funcs (M, the carrier of Y))) & rng () c= Funcs ( the carrier of X, the carrier of Y) & dom () = M )\nlet M be non empty set ; :: thesis: for X, Y being non empty Poset\nfor f being directed-sups-preserving Function of X,(Y |^ M) holds\n( f in Funcs ( the carrier of X,(Funcs (M, the carrier of Y))) & rng () c= Funcs ( the carrier of X, the carrier of Y) & dom () = M )\n\nlet X, Y be non empty Poset; :: thesis: for f being directed-sups-preserving Function of X,(Y |^ M) holds\n( f in Funcs ( the carrier of X,(Funcs (M, the carrier of Y))) & rng () c= Funcs ( the carrier of X, the carrier of Y) & dom () = M )\n\nlet f be directed-sups-preserving Function of X,(Y |^ M); :: thesis: ( f in Funcs ( the carrier of X,(Funcs (M, the carrier of Y))) & rng () c= Funcs ( the carrier of X, the carrier of Y) & dom () = M )\nthe carrier of (Y |^ M) = Funcs (M, the carrier of Y) by YELLOW_1:28;\nthen A1: rng f c= Funcs (M, the carrier of Y) ;\ndom f = the carrier of X by FUNCT_2:def 1;\nhence f in Funcs ( the carrier of X,(Funcs (M, the carrier of Y))) by ; :: thesis: ( rng () c= Funcs ( the carrier of X, the carrier of Y) & dom () = M )\nthen commute f in Funcs (M,(Funcs ( the carrier of X, the carrier of Y))) by FUNCT_6:55;\nthen ex g being Function st\n( commute f = g & dom g = M & rng g c= Funcs ( the carrier of X, the carrier of Y) ) by FUNCT_2:def 2;\nhence ( rng () c= Funcs ( the carrier of X, the carrier of Y) & dom () = M ) ; :: thesis: verum\nend;\n\ntheorem Th38: :: WAYBEL27:38\nfor X being non empty set\nfor S, T being non empty Poset\nfor f being directed-sups-preserving Function of S,(T |^ X)\nfor i being Element of X holds () . i is directed-sups-preserving Function of S,T\nproof end;\n\ntheorem Th39: :: WAYBEL27:39\nfor X being non empty set\nfor S, T being non empty Poset\nfor f being directed-sups-preserving Function of S,(T |^ X) holds commute f is Function of X, the carrier of (UPS (S,T))\nproof end;\n\ntheorem Th40: :: WAYBEL27:40\nfor X being non empty set\nfor S, T being non empty Poset\nfor f being Function of X, the carrier of (UPS (S,T)) holds commute f is directed-sups-preserving Function of S,(T |^ X)\nproof end;\n\ntheorem Th41: :: WAYBEL27:41\nfor X being non empty set\nfor S, T being non empty Poset ex F being Function of (UPS (S,(T |^ X))),((UPS (S,T)) |^ X) st\n( F is commuting & F is isomorphic )\nproof end;\n\ntheorem Th42: :: WAYBEL27:42\nfor X being non empty set\nfor S, T being non empty Poset holds UPS (S,(T |^ X)),(UPS (S,T)) |^ X are_isomorphic\nproof end;\n\n:: 2.8. THEOREM, p. 116\n:: [CONT -> CONT] is into CONT (so [CONT -> CONT] is functor)\ntheorem :: WAYBEL27:43\nfor S, T being complete continuous LATTICE holds UPS (S,T) is continuous\nproof end;\n\n:: 2.8. THEOREM, p. 116\n:: [ALG -> ALG] is into ALG (so [ALG -> ALG] is functor)\ntheorem :: WAYBEL27:44\nfor S, T being complete algebraic LATTICE holds UPS (S,T) is algebraic\nproof end;\n\ntheorem Th45: :: WAYBEL27:45\nfor R, S, T being complete LATTICE\nfor f being directed-sups-preserving Function of R,(UPS (S,T)) holds uncurry f is directed-sups-preserving Function of [:R,S:],T\nproof end;\n\ntheorem Th46: :: WAYBEL27:46\nfor R, S, T being complete LATTICE\nfor f being directed-sups-preserving Function of [:R,S:],T holds curry f is directed-sups-preserving Function of R,(UPS (S,T))\nproof end;\n\n:: 2.10. THEOREM, p. 117\ntheorem :: WAYBEL27:47\nfor R, S, T being complete LATTICE ex f being Function of (UPS (R,(UPS (S,T)))),(UPS ([:R,S:],T)) st\n( f is uncurrying & f is isomorphic )\nproof end;" ]

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[ "This standard deviation example questions can help you to calculate mean, variance, SD easily. To find the standard deviation, first write the computational formula for the standard deviation of the sample. Solution: Question 2: A health researcher read that a 200-pound male can burn an average of 524 calories per hour playing tennis. Conclusion – Standard Deviation Examples. Example: Standard deviation in a normal distribution You administer a memory recall test to a group of students. The below are some of the solved examples with solutions to help users to know how to estimate reliable sample size by using stanadrd deviation or proportion method. Population Standard Deviation. A slightly more complex calculation is called sample standard deviation. The standard deviation for discrete series can be calculated by approaches stated below: Direct method. Example 8.14. Keep reading for standard deviation examples and the different ways it appears in daily life. Population standard deviation looks at the square root of the variance of the set of numbers. of the mean. A small standard deviation can be a goal in certain situations where the results are restricted, for example, in product manufacturing and quality control. N is the number of terms in the population. This number is the standard deviation of the sample. Evaluate the standard deviation. The data follows a normal distribution with a mean score of 50 and a standard deviation of 10. Scroll down the page for more examples and solutions on using the normal distribution formula. Sample and population standard deviation . The sample mean is 5.343 with a sample standard deviation of 0.397. Example 1: Let us consider a data sample : 10,13,7,9,6 . The normal distribution is described by two parameters: the mean, μ, and the standard deviation, σ. The probability distribution is: x-152 154 156 158 160 162 164 P (x-) 1 16 2 16 3 16 4 16 3 16 2 16 1 16. The standard deviation measures the dispersion of the dataset which is relative to its mean. The mean and standard deviation of 15 observations are found to be 10 and 5 respectively. Variance and Standard Deviation for Grouped Data Calculator. For example, the mean of the following two is the same: 15, 15, 15, 14, 16 and 2, 7, 14, 22, 30. Check that this is a valid PDF and calculate the standard deviation of X.. Figure 14. To verify that f(x) is a valid PDF, we must check that it is everywhere nonnegative and that it integrates to 1.. We see that 2(1-x) = 2 - 2x ≥ 0 precisely when x ≤ 1; thus f(x) is everywhere nonnegative. In Note 6.5 \"Example 1\" in Section 6.1 \"The Mean and Standard Deviation of the Sample Mean\" we constructed the probability distribution of the sample mean for samples of size two drawn from the population of four rowers. Population Standard Deviation Formula . If a set has a low standard deviation, the values are not spread out too much. Khan Academy is a 501(c)(3) nonprofit organization. This means that we can report what proportion the standard deviation is of the mean. However, as we are often presented with data from a sample only, we can estimate the population standard deviation from a sample standard deviation. As in discrete series another column of frequency gets added, the formula for calculation of standard deviation using direct approach is altered to incorporate frequency is stated below: Standard deviation(σ)= √(∑fD²)/N) Standard Deviation Problems with Solutions. For example, if we have a standard deviation of 1.5 and a mean of 5, the ratio of the standard deviation to the mean is 0.3. Suppose random samples of size n are drawn from a population with mean μ and standard deviation σ. Standard Deviation formula to calculate the value of standard deviation is given below: (Image to be added soon) Standard Deviation Formulas For Both Sample and Population. It is symbolized by ${s}$ . If you are studying the post metric syllabus of the stats then you are most probably going to come across this measure and it will form the significant part of your exams as well. A normal probability plot for Example 9. Sample standard deviation and bias. The coefficient of variation can be reported as a percentage. Find solutions for your homework or get textbooks Search. Standard Deviation Example. Formula : Mean : Mean = Sum of X values / N(Number of Values) Variance : … Our mission is to provide … However, the second is clearly more spread out. Let the assumed mean, A = 35, c = 10 . Around 99.7% of scores are between 20 and 80. The standard deviation measures how concentrated the data are around the mean; the more concentrated, the smaller the standard deviation. It is calculated as the square root of variance. Example: 3, 8, 14, 18, 25, 22, 15, 9, 5 The sample size is small and we don’t know anything about the distribution of the population, so we examine a normal probability plot. home / study / math / statistics and probability / statistics and probability solutions manuals / Statistics for the Behavioral Sciences / 7th edition / chapter 5 / problem 15P. A low standard deviation means that the data is very closely related to the average, thus very reliable. The mean profit earning for a sample of 41 businesses is 19, and the S.D. The distribution looks normal so we will continue with our test. Home . Sample Standard Deviation Calculator This calculator allows you to compute the sample standard deviation of a given set of numerical value and learn a step-by-step solution with a formula. This is the currently selected item. It is given by the formula: The capital Greek letter sigma is commonly used in mathematics to represent a summation of all the numbers in a grouping. σ = $\\sqrt{\\sum (X-\\mu)^{2/n}}$ Sample Standard Deviation Formula. Standard deviation (by mean method) σ = If d i = x i – are the deviations, then . Up Next. The sum of all variances gives a, which is the square of the standard deviation. 37 males were randomly selected and the mean number of calories burned per hour playing squash was 534.8 with a standard deviation of 45.9 calories. Sample Standard Deviation. We will reject… Statistics for the Behavioral Sciences (7th Edition) Edit edition. It is useful in comparing sets of data which may have the same mean but a different range. Sample standard deviation and bias. Sample and population standard deviation. Question: During a survey, 6 students were asked how many hours per day they study on an average? In other words, the standard deviation is 30% of the mean. Solution = (175+170+177+183+169)/5; Sample Mean = 174.8; Calculation of Sample Standard Deviation =SQRT(128.80) Sample Standard Deviation =5.67450438 =5.67450438/SQRT(5) = 2.538; Example #3. For example, if the mean is 40 and the standard deviation is 5, then a value x that is 1 standard deviation from the mean is in the range that you see below: 40 - 5 < x < 40 + 5 35 < x < 45 If the mean is 40 and the standard deviation is 5, then a value x that is 2 standard deviations from the mean is in the range you see below: 40 - 2 × 5 < x < 40 + 2 × 5 30 < x < 50. This is why, in many cases, the standard deviation is a preferred measure of variability. Solution (ii) Mean method. Find its standard deviation. The population deviation is denoted by . The random variable X is given by the following PDF. Usually, we are interested in the standard deviation of a population. Recall that the formula for the standard deviation is simply the square root of the variance. The standard deviation of the sample mean X-that we have just computed is the standard deviation of the population divided by the square root of the sample size: 10 = 20 / 2. The following diagram shows the formula for Normal Distribution. More on standard deviation. It's used to determine a confidence interval for drawing conclusions (such as accepting or rejecting a hypothesis). Around 95% of scores are between 30 and 70. $${s}= \\sqrt{\\frac{{\\sum}{x^2} - \\frac{({\\sum}{x})^2}{n}}{n - 1}}$$ Take the square root of the answer found in step 7 above. Solution for QUESTION 1 A random sample of size n = 64, from a population with standard deviation g=74, is used to test Ho:H=70 against H:=65.5. This Statistics video tutorial explains how to calculate the standard deviation using 2 examples problems. Population and sample standard deviation review. December 12, 2019 by self Leave a Comment. Popular Course in this category. Example 8.5 The amount of rainfall in a particular season for 6 days are given as 17.8 cm, 19.2 cm, 16.3 cm, 12.5 cm, 12.8 cm and 11.4 cm. Find the S.E. Financial Modeling Course (with … In statistical terms this means we have a sample size of 5 and in this case we use the standard deviation equation for a sample of a population: The rest of this example will be done in the case where we have a sample size of 5 pirates, therefore we will be using the standard deviation equation for a sample of a population. Standard Deviation. Practice: Sample and population standard deviation. Solution. Refer the below normal distribution examples and solutions and calculate gaussian distribution to compute the cumulative probability for any value. Example Question based on Standard Deviation Formula. A high standard deviation means that there is a large variance between the data and the statistical average, and is not as reliable. Their answers were as follows: 2, 6, 5, 3, 2, 3. We write X - N(μ, σ 2. Simple Example. Solution Population and sample standard deviation review Our mission is to provide a free, world-class education to anyone, anywhere. Solution: We can calculate the mean, variance and standard deviation of the given sample data using the given formula. Literally translating from the example above, the variance signals an approximate square difference of 118.7 per data point. These relationships are not coincidences, but are illustrations of the following formulas. Standard deviation is a very well known measure of dispersion in the fields of statistics. The Central Limit Theorem. Standard Deviation Example Problems. Another convenient way of finding standard deviation is to use the following formula. The standard deviation is a measure of the spread of scores within a set of data. Practice: Variance. Formula to calculate sample size by using standard deviation method Formula to calculate sample size by using population proportion method . Following the empirical rule: Around 68% of scores are between 40 and 60. Given a normal distribution with \\mu = 204.7 and \\sigma = 45.5, if we have a sample with sample size = 160, calculate the standard deviation of sample means. Next lesson. The equation for calculating variance is the same as the one provided above, except that we don’t take the square root. Solution Part 1. The standard deviation measures the spread of the data about the mean value. of the customers is 6.6. Population Standard Deviation. Also Check: Difference Between Variance and Standard Deviation. Sample Standard Deviation. Gaussian Distribution Examples . Solved Example Problems with Steps. e d The sample mean and standard deviation will both increase 1 Solution We from ACTL 5101 at University of New South Wales The empirical rule … Find its standard deviation. On rechecking it was found that one of the observation with value 8 was incorrect. During a survey, 6, 5, 3, 2, 3, 2, 6, 5 3... 118.7 per data point of scores are between 20 and 80 ( Edition... There is a 501 ( c ) ( 3 ) nonprofit organization and... A high standard deviation of 10 usually, we are interested in the population population deviation... ( 7th Edition ) Edit Edition get textbooks Search a memory recall test to group! The square of the observation with value 8 was incorrect a hypothesis ) variance is the standard deviation, values., 5, 3, 2, 6 students were asked how many hours per day they study an. Literally translating from the example above, the variance assumed mean, μ, 2... Be 10 and 5 respectively male can burn an average 6 students were asked many... Preferred measure of variability on using the given sample data using the normal distribution examples and solutions and gaussian! A slightly more complex calculation is called sample standard deviation ( by mean method ) σ = \\ [ {. With mean μ and standard deviation is a 501 ( c ) ( 3 nonprofit! 41 businesses is 19, and is not as reliable ’ t take the square of! Behavioral Sciences ( 7th Edition ) Edit Edition between 30 and 70 so we reject…... The data follows a normal distribution ) nonprofit organization a percentage which may have the same but. With Our test as the square root of the standard deviation is a well. The observation with value 8 was incorrect = 35, c = 10 PDF and calculate the standard deviation a! Of 0.397 example 1: let us consider a data sample: 10,13,7,9,6 is a preferred measure of variability looks... Relationships are not spread out too much for normal distribution formula, μ, σ different range solution: can. Cases, the variance of the spread of scores are between 40 and 60 that there is a well. Solution variance and standard deviation is of the variance are illustrations of mean! Data using the normal distribution with a mean score of 50 and a standard deviation is a (! Same mean but a different range a preferred measure of variability Direct method number. For standard deviation of 10 dispersion in the standard deviation formula variances gives a, which is the root. Of size n are drawn from a population more concentrated, the values are not coincidences, are. Textbooks Search computational formula for normal distribution You administer a memory recall test to group. 40 and 60 closely related to the average, thus very reliable in... The dataset which is relative to its mean and solutions and calculate the standard deviation examples solutions. A = 35, c = 10 the data about the mean variance! Deviations, then a different range around the mean ; the more concentrated, the smaller standard! And the S.D, except that we can calculate the standard deviation using 2 examples problems, world-class to! Shows the formula for the standard deviation means that there is a preferred measure of the observation with value was... Square difference of 118.7 per data point different ways it appears in daily life to its mean looks! Provide a free, world-class education to anyone, anywhere n are drawn from population... I = X i – are the deviations, then interested in the population the following.. In the fields of statistics, then variance is the same as the square root of sample... Average of 524 calories per hour playing tennis following formula to anyone,.... Is of the mean and 5 respectively examples and the statistical average, thus very reliable we! 50 and a standard deviation is of the following formula mean μ standard... Per day they study on an average of 524 calories per hour playing tennis scores are 20... Preferred measure of the sample data sample: 10,13,7,9,6 another convenient way of finding standard,... We can report what proportion the standard deviation is 30 % of scores are between 20 and 80 more! = 10 20 and 80 distribution is described by two parameters: the mean,,. ] sample standard deviation of 0.397 the average, and the statistical average, thus very reliable deviation.! Data about the mean and standard deviation using 2 examples problems given the! In daily life using 2 examples problems a hypothesis ) the sum of all variances gives a, which relative. Researcher read that a 200-pound male standard deviation example with solution burn an average of 524 calories per playing!" ]

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[ "## User Interface Design Part 4: The Model-View-Controller paradigm\n\nThus far we’ve talked about designing a user interface: the visual language that tells users what they can do, the elements of the language (nouns and verbs), the way these things contribute to consistency and discoverability and how they can be used to create a user interface that seems… inevitable. Simple. Invisible.\n\nWhere this attention to detail shines is when it comes to building our interface.\n\nBut before we can do this, let’s talk a little bit about a common way used to organize interfaces, and discuss the non-interface elements of our code.\n\nThe single most common used design pattern in application development today is the Model-View-Controller. In fact, it has been called “the seminal insight of the whole field of graphical user interfaces.” Originally invented at Xerox Parc in the 1970’s, it was first used to help build Smalltalk-80 user interface applications, and it was first described in the paper A Cookbook for Using the Model-View-Controller User Interface Paradigm in Smalltalk-80 by Glenn Krasner and Stephen Pope.\n\nThis paradigm is so important that nearly every modern application API (from iOS to Android to MacOS to Microsoft Windows) either explicitly provides interfaces that build on this model, or implicitly provides a way by which programmers can use this model.\n\nThe key idea is that we can think of our programs as being made up of three major parts:", null, "At the bottom is our “Model.” This is the “domain-specific software” which implements our application–and is the collection of code which does stuff like load a file, or provide in-memory editing of a text file, or (in our case) connects to the physical hardware that controls an HVAC system or which provides the time.\n\nOur user interface is then built on top of this model code.\n\nOur user interface roughly comprises of two parts. The first part are the “views”; these deal with everything graphical. They grab data from the model and display the contents of the screen–and they can contain “subviews” and be contained in “superviews.”\n\nThe second part is the “control” code. Controllers contain the logic which coordinates our views; they deal with user interface interactions and translate those actions into meaningful action. (So, for example, a view may represent a button, but it is the control code which determines what pressing that button “means” in terms of updating the interface and updating the model.)\n\nFor our Arduino application we don’t fully implement “views,” since some of the overhead may not quite fit in our small memory footprint. But they can help us segregate our code and help organize our thinking about the code, by thinking of the graphical parts as being separate from the control code and from our model code.\n\nAnd notice one nice property about our visual language–about our thinking of a visual language in terms of the ‘nouns’ and ‘verbs’ and the consistent use of visual designs. All of this maps very nicely on our Model-View-Controller paradigm.\n\nOur “Model” is the concrete implementation of our “nouns”: the thermostat settings. The schedule. The current time.\n\nOur “Views” represents our visual language: how we show the user our “nouns”, our “verbs”, how we represent the actions a user can take.\n\nAnd our “Control” code represents how our visual language works: how a sequence of taps on a bunch of icons translates into the user “changing the time” or “turning down the heater.”\n\nLet’s give a concrete example of one of these “nouns”; the actual thermostat control code which determines if we need to turn on or off the heater, air conditioner or fan.\n\nOur `AdaThermostat`, declared as a single `GThermostat` object, defines the actual code for setting the temperature, for getting the current temperature, and for turning on and off the three control lines that control our HVAC unit.\n\n```class AdaThermostat\n{\npublic:\n\nvoid periodicUpdate();\n\nuint8_t heatSetting; /* Target heat to temperature */\nuint8_t coolSetting; /* Target cool to temperature */\nuint8_t curTemperature; /* Current interior temperature */\nuint8_t lastSet; /* Last schedule used or 0xFF */\n\nbool heat; /* True if heater runs */\nbool cool; /* True if aircon runs */\nbool fan; /* True if fan runs */\n};```\n\nUltimately, when you change the thermostat setting–say, to heat the room to 70°–at the very core of our software, this sets `GThermostat.heatSetting` to 70.\n\nOur class has only one entry point, `periodicUpdate`, which must be called periodically in order to run our logic: to periodically read the temperature sensor and set the `curTemperature` variable, and to decide when to turn on the heater and the air conditioner.\n\nAll of our model code looks like this: a loose collection of classes which control the hardware on our thermostat.\n\nFor example, our model includes code for getting and setting the time in `AdaTime.h`:\n\n```extern void AdaTimeInitialize();\n\nextern void AdaSetTime(uint32_t t); // Time (seconds elapsed since 1/1/2017)\nextern uint32_t AdaGetTime(); // Time (seconds elapsed since 1/1/2017)\nextern uint32_t AdaGetElapsedTime(); // Time (seconds elapsed since power on)```\n\nThis uses the TIMER2 hardware module on the ATmega 328 CPU. This sets the timer to create an interrupt every 1/125th of a second, and the interrupt itself counts until 1 full second has elapsed before updating two global variables–one with the current time (measured as seconds from January 1, 2017), and the other with the number of seconds since the device was turned on.\n\nOur code also provides support routines for converting the time to a Gregorian calendar date and for quickly finding the day of the week. This allows us to determine quickly which day of the week it is when we update the thermostat according to the schedule.\n\nWe also provide a class for getting and setting the thermostat’s schedule, which is stored in EEPROM on the ATmega 328.\n\n```class AdaSchedule\n{\npublic:\n\nvoid periodicUpdate();\n\nvoid setCurSchedule(uint8_t value);\nuint8_t getCurSchedule();\n\nvoid setSchedule(uint8_t schedule, uint8_t dow, const AdaScheduleDay &item);\n};```\n\nThis allows us to get the schedule for a single day, which is an array of four temperature settings and four times:\n\n```struct AdaScheduleItem\n{\nuint8_t hour; /* 0-23 = hour of day, 0xFF = skip */\nuint8_t minute; /* 0-59 = minute */\nuint8_t heat; /* temperature to heat to */\nuint8_t cool; /* temperature to cool to */\n};\n\n{\nstruct AdaScheduleItem setting; // 4 settings per day\n};```\n\nOur user interface code for running the main display of our thermostat then makes use of these model classes to get the current state of the system.\n\nFor example, in the part of our home page which displays the temperature buttons, our code is:\n\n``` FormatNumber(buffer,GThermostat.heatSetting);\nGC.drawButton(RECT(160,200,40,37),buffer,28,KCornerUL | KCornerLL,KCenterAlign);\nFormatNumber(buffer,GThermostat.coolSetting);\nGC.drawButton(RECT(201,200,40,37),buffer,28,KCornerUR | KCornerLR,KCenterAlign);```\n\nThis will read and display the temperature on our device.\n\nBy using the Model-View-Controller paradigm and separating out the parts which control our hardware from the parts that control our display, and make creating our user interface code that much easier. It also means we can test those parts of our code that controls our hardware separately from the rest of our application.\n\nThe complete source code, which strips out most of our user interface software and illustrates the models of our thermostat software, is checked into GitHub and is available for download.\n\nNext time we’ll discuss the code we use for managing screens and the controller code which draws our home screen.\n\n## Building a 3D Graphics Pipeline on an Arduino, Part 5\n\nOne of the advantages of our design is that we can rapidly port our 3D drawing code to another platform. In fact, we could port our code off the Arduino entirely and move it to a different microprocessor or even to your desktop computer.\n\nAll we have to do is to alter how we initialize our 3D library, change our begin/end calls, and change our low-level drawing routines. While we’re at it, we can also refine our 3D library to only draw in a subset of our screen. We may wish to do this, for example, if we’re creating a game and we want a separate status area.\n\nSo let’s begin.\n\nFor this exercise we’ll be porting the code to an Arduboy, a small hand held Arduino game unit with built-in game control buttons and a small black and white screen.\n\nOur current constructor for the `G3D` class contains a reference to our library, and we assume we use the entire screen. Let’s change the code two ways. First, we want to change the drawing library we pass in. Second, we want to pass in the area of the screen we’re drawing in.\n\nSo let’s update our class. I’m going to define a preprocessor token:\n\n`#define USELIBRARY\t 2 // 1 = Adafruit, 2 = Arduboy`\n\nNow let’s update our G3D class. In our header file we’ll alter our constructor and create the appropriate definitions:\n\n```class G3D\n{\npublic:\n#if USELIBRARY == 1\nG3D(Adafruit_GFX &lib, uint16_t x, uint16_t y, uint16_t width, uint16_t height);\n#elif USELIBRARY == 2\nG3D(Arduboy &lib, uint16_t x, uint16_t y, uint16_t w, uint16_t h);\n#endif\n~G3D();\n\nvoid setColor(uint16_t c)\n{\ncolor = c;\n}\n\nvoid begin();\nvoid end();```\n\nWe also want to update the color declaration, since the Arduboy uses an 8-bit color value (which is set to either BLACK or WHITE), while the Adafruit library uses a 16-bit color value.\n\n```class G3D\n{\npublic:\n#if USELIBRARY == 1\nG3D(Adafruit_GFX &lib, uint16_t x, uint16_t y, uint16_t width, uint16_t height);\n#elif USELIBRARY == 2\nG3D(Arduboy &lib, uint16_t x, uint16_t y, uint16_t width, uint16_t height);\n#endif\n~G3D();\n\n#if USELIBRARY == 1\nvoid setColor(uint16_t c)\n{\ncolor = c;\n}\n#elif USELIBRARY == 2\nvoid setColor(uint8_t c)\n{\ncolor = c;\n}\n#endif\n\nvoid begin();\nvoid end();```\n\nThese two changes require corresponding changes to the private values in our class. We need to update the library we reference, update the screen size parameters and the color:\n\n``` private:\n/*\n* Internal state\n*/\n#if USELIBRARY == 1\n#elif USELIBRARY == 2\nArduboy &lib;\n#endif\n\nuint16_t xoffset;\nuint16_t yoffset;\nuint16_t width;\nuint16_t height;\n\n/*\n* Current drawing color\n*/\n\n#if USELIBRARY == 1\nuint16_t color;\n#elif USELIBRARY == 2\nuint8_t color;\n#endif\n\n/*\n* Stage 4 pipeline; 3D transformation\n*/```\n\nThese are all the changes we need to make to our header. From a public interface perspective, we simply update the value of USELIBRARY, and create our class linked to the correct library–and that’s it. There is no step 3.\n\n## G3D code changes.\n\nFor the body of our code, the changes are also similarly easy.\n\nFirst, of course, we need to update our constructor. The body of the constructor is the same; we simply need to change the library name of the library passed in. We also stash the screen size parameters that we’ve changed.\n\n```/* G3D::G3D\n*\n* Construct our pipeline\n*/\n\n#if USELIBRARY == 1\nG3D::G3D(Adafruit_GFX &l, uint16_t x, uint16_t y, uint16_t w, uint16_t h) : lib(l)\n#elif USELIBRARY == 2\nG3D::G3D(Arduboy &l, uint16_t x, uint16_t y, uint16_t width, uint16_t height) : lib(l)\n#endif\n{\nxoffset = x;\nyoffset = y;\nwidth = w;\nheight = h;\n\n/* Initialize components of pipeline */\np1init();\np2init();\np3init();\n}```\n\nNow our implementation on the Adafruit GFX library called the internal `startWrite` and `endWrite` calls in order to reduce the number of times we had to turn on and off the Adafrut drawing engine. But the Arduboy library doesn’t require the same thing–so these calls effectively become ‘no-ops’:\n\n```void G3D::begin()\n{\n#if USELIBRARY == 1\nlib.startWrite();\n#endif\n}\n\nvoid G3D::end()\n{\n#if USELIBRARY == 1\nlib.endWrite();\n#endif\n}```\n\nFinally we need to update our p1 move/draw and point routines. We need to do two things here: first, we need to offset our drawing to the pixel coordinates we supplied in our initializer. Second, we need to actually call the correct line drawing routines.\n\n```void G3D::p1movedraw(bool drawFlag, uint16_t x, uint16_t y)\n{\n/*\n* We use the p1drawflag and the drawFlag objects to determine the\n* way to draw our line. For us, we're always drawing single\n* segments, but we theoretically could roll up our lines into a\n* collection of line segments and send them on close. (This\n* requires hooking G3D::end().)\n*/\n\nif (drawFlag) {\n#if USELIBRARY == 1\nlib.writeLine(xoffset + p1x,xoffset + p1y,x,y,color);\n#elif USELIBRARY == 2\nlib.drawLine(xoffset + p1x,xoffset + p1y,x,y,color);\n#endif\n}\n\np1draw = drawFlag;\np1x = x;\np1y = y;\n}\n\nvoid G3D::p1point(uint16_t x, uint16_t y)\n{\n#if USELIBRARY == 1\nlib.writePixel(xoffset + x,xoffset + y,color);\n#elif USELIBRARY == 2\nlib.drawPixel(xoffset + x,xoffset + y,color);\n#endif\n}```\n\nAnd that’s it. Now our library can be used to generate 3D drawings on either an Arduboy or using the Adafruit GFX library.\n\n## Testing it out.\n\nWe can redo our cube test from the last time but on the Arduboy. Because the Arduboy has double-buffering the animation is smooth and quick–though it lacks the color of the Adafruit color display.\n\nOur example needs to be rewritten to use the double-buffer drawing support of the Arduboy. We can also make use of the viewport feature to only draw on part of the screen; we’d do this if we wanted to make use of only part of the screen.\n\nMost of the changes are relatively simple: we switch libraries and change how we start up our drawing system. The loop has the most changes, since we need to handle the double-buffering of the Arduboy API.\n\nFor the record, the complete source kit (just showing the Arduboy elements and their changes) is:\n\n```#include \"G3D.h\"\n\n/*\n* Drawing globals\n*/\n\nArduboy arduboy;\n\n// Graphics setup\nG3D draw(arduboy,0,0,100,64);\n\n// Rotation\nstatic float GXAngle;\nstatic float GYAngle;\n\nvoid setup()\n{\narduboy.beginNoLogo();\narduboy.setFrameRate(50);\narduboy.clear();\narduboy.display();\n\nGXAngle = 0;\nGYAngle = 0;\n}\n\nvoid transform()\n{\ndraw.transformation.setIdentity();\ndraw.perspective(1.0f,0.5f);\ndraw.translate(0,0,-3.5);\ndraw.rotate(AXIS_X,GXAngle);\ndraw.rotate(AXIS_Y,GYAngle);\n}\n\nvoid drawBox(int x, int y, int z)\n{\ndraw.move(x-1,y-1,z-1);\ndraw.draw(x+1,y-1,z-1);\ndraw.draw(x+1,y+1,z-1);\ndraw.draw(x-1,y+1,z-1);\ndraw.draw(x-1,y-1,z-1);\ndraw.draw(x-1,y-1,z+1);\ndraw.draw(x+1,y-1,z+1);\ndraw.draw(x+1,y+1,z+1);\ndraw.draw(x-1,y+1,z+1);\ndraw.draw(x-1,y-1,z+1);\ndraw.move(x+1,y-1,z-1);\ndraw.draw(x+1,y-1,z+1);\ndraw.move(x+1,y+1,z-1);\ndraw.draw(x+1,y+1,z+1);\ndraw.move(x-1,y+1,z-1);\ndraw.draw(x-1,y+1,z+1);\n}\n\nvoid loop()\n{\nif (!arduboy.nextFrame()) return;\n\narduboy.clear();\n\ndraw.begin();\ndraw.setColor(WHITE);\ntransform();\ndrawBox(0,0,0);\ndraw.end();\n\narduboy.display();\n\nGXAngle += 0.01;\nGYAngle += 0.02;\n}```\n\nWhen we run this we get the following:\n\nAll the code here is at GitHub, so feel free to check it out. The main branch is the latest and greatest, and can be compiled (by changing the flag in G3D.h) on either the Arduboy or on any of Adafruit’s GFX compatible displays.\n\nA quick note about the cube demo. If you watch it spin, sometimes the point of the cube nearest to you “disappears.” This is expected, and demonstrates clipping not just to the four boundaries of our screen, but to the near clipping plane. Think of it as the cube being really small and “clipping” against the camera lens of our virtual camera.\n\nThis has been a somewhat hastily written adventure into 3D, but I hope it has been an instructive one. If you found this series useful, please let me know. And if you find any problems with any of the code, or if you want to share any games you put together with this code, please let me know!\n\n## Building a 3D Graphics Pipeline on an Arduino, Part 4\n\nWe have all the pieces. Let’s put them together, shall we?\n\nWith the last article we had a clipping algorithm which could clip lines so that only the visible lines are displayed. Now let’s put the whole thing together.\n\n## Some Math in Code.\n\nOn our second article we introduced the concept of Homogeneous Coordinates. This is how we manipulate points in 3D, and you see the same matrix math used everywhere in 3D graphics, including OpenGL.\n\nFirst, let’s create the methods for manipulating our matrices and vectors.\n\nFirst, we need a 3D matrix, which we define as:\n\n```class G3DMatrix\n{\npublic:\nfloat a;\n};```\n\nRecall our 3D matrices are 4×4 matrices:", null, "Our convention is that items in our matrix across is the first index, and down is the second.\n\nSince we’re using C++, we can also pack our matrix full of all those declarations which make our life easier. Specifically we can add routines for initializing our matrix as a translation matrix, a rotation matrix, or a scale matrix, as well as our perspective matrix. We can also add our multiply method as well, just to keep things tidy.\n\n```class G3DMatrix\n{\npublic:\nG3DMatrix();\n\n// Initialize matrix\nvoid setIdentity();\nvoid setTranslate(float x, float y, float z);\nvoid setScale(float x, float y, float z);\nvoid setScale(float x);\nvoid setRotate(uint8_t axis, float angle);\nvoid setPerspective(float fov, float near);\n\n// Inline multiply transformation matrix\nvoid multiply(const G3DMatrix &m);\n\n// Raw contents of the matrix\nfloat a;\n};```\n\nWe also need our 3D vector as a homogeneous coordinate, and we include our vector/matrix multiply routines as well.\n\n```struct G3DVector {\nfloat x;\nfloat y;\nfloat z;\nfloat w;\n\n// Math support\nvoid multiply(const G3DMatrix &m, const G3DVector &v);\n};```\n\nEverything we’re writing here should seem familiar if you made it to the end of my article on homogeneous coordinates. Basically, the idea is that a 3D point (x,y,z) becomes our 3D homogeneous vector (x,y,z,1), and any 3D homogeneous coordinate (x,y,z,w) becomes (x/w, y/w, z/w).\n\nThe implementation of each of the matrix initialization methods follows pretty quickly from our homogeneous matrices in the prior article. For example, our implementation of `setTranslate` which creates a translation matrix:", null, "looks like:\n\n```void G3DMatrix::setTranslate(float x, float y, float z)\n{\nsetIdentity();\na = x;\na = y;\na = z;\n}```\n\n## Our Transformations\n\nTo our G3D class, we create methods for transformations, scaling, and the transformation matrix used to translate points in our 3D coordinate space.\n\n```class G3D\n{\npublic:\n~G3D();\n\nvoid setColor(int16_t c)\n{\ncolor = c;\n}\n\nvoid begin();\nvoid end();\nvoid move(float x, float y, float z)\n{\np4movedraw(false,x,y,z);\n}\nvoid draw(float x, float y, float z)\n{\np4movedraw(true,x,y,z);\n}\nvoid point(float x, float y, float z)\n{\np4point(x,y,z);\n}\n\nvoid translate(float x, float y, float z);\nvoid scale(float x, float y, float z);\nvoid scale(float s);\nvoid rotate(uint8_t axis, float angle);\nvoid perspective(float fov, float nclip);\nvoid orthographic(void);\n\nG3DMatrix transformation;\nprivate:```\n\nWe expose all of our transformation stuff here to simplify our work, by keeping the pieces all in one place. For example, our translate method looks like:\n\n```void G3D::translate(float x, float y, float z)\n{\nG3DMatrix m;\nm.setTranslate(x,y,z);\ntransformation.multiply(m);\n}```\n\nThis will post-multiply our matrix into our current transformation matrix. We’ll talk about the ramification of this below.\n\nFinally we need our p4movedraw and p4point methods for our class:\n\n```void G3D::p4movedraw(bool drawFlag, float x, float y, float z)\n{\nG3DVector t;\n\nt.x = transformation.a * x + transformation.a * y + transformation.a * z + transformation.a;\nt.y = transformation.a * x + transformation.a * y + transformation.a * z + transformation.a;\nt.z = transformation.a * x + transformation.a * y + transformation.a * z + transformation.a;\nt.w = transformation.a * x + transformation.a * y + transformation.a * z + transformation.a;\n\np3movedraw(drawFlag,t);\n}\n\nvoid G3D::p4point(float x, float y, float z)\n{\nG3DVector t;\n\nt.x = transformation.a * x + transformation.a * y + transformation.a * z + transformation.a;\nt.y = transformation.a * x + transformation.a * y + transformation.a * z + transformation.a;\nt.z = transformation.a * x + transformation.a * y + transformation.a * z + transformation.a;\nt.w = transformation.a * x + transformation.a * y + transformation.a * z + transformation.a;\n\np3point(t);\n}```\n\nThis simply does our matrix/vector multiplication, then passes the resulting point into our 3D clipping engine.\n\n## Testing the whole thing.\n\nOur test code is very simple.\n\nFirst, we need to set up our drawing system. We first need to set up our drawing engine, and then pass it as a construction parameter to our 3D drawing engine:\n\n```// For the Adafruit shield, these are the default.\n#define TFT_DC 9\n#define TFT_CS 10\n\n// Use hardware SPI (on Uno, #13, #12, #11) and the above for CS/DC\n\n// Graphics setup\nG3D draw(tft);```\n\nAnd finally we set up a couple of global variables for the rotation angle.\n\n```// Rotation\nstatic float GXAngle;\nstatic float GYAngle;```\n\nOur setup code simply erases the screen to black and resets our globals.\n\n```void setup()\n{\ntft.begin();\ntft.fillScreen(ILI9341_BLACK);\n\nGXAngle = 0;\nGYAngle = 0;\n}```\n\nAnd our loop draws our box at the origin, after setting up the transformation matrix. We wait a little bit, then we redraw the box in black; this is faster than erasing the screen to black. We rotate our box, and try again.\n\n```void loop()\n{\ndraw.begin();\ndraw.setColor(ILI9341_RED);\ntransform();\ndrawBox(0,0,0);\ndraw.end();\n\ndelay(100);\n\ndraw.begin();\ndraw.setColor(ILI9341_BLACK);\ndrawBox(0,0,0);\ndraw.end();\n\nGXAngle += 0.01;\nGYAngle += 0.02;\n}```\n\nOur transformation matrix.\n\nThe code we use to set up our transformation matrix is:\n\n```void transform()\n{\ndraw.transformation.setIdentity();\ndraw.perspective(1.0f,0.5f);\ndraw.translate(0,0,-6);\ndraw.rotate(AXIS_X,GXAngle);\ndraw.rotate(AXIS_Y,GYAngle);\n}```\n\nWe initialize the matrix in the opposite direction in which we use the transformations. In other words, our transformation matrix first rotates our cube around the Y axis. Then it rotates around the X axis. We then move the object 6 units away from our view, and finally set up the perspective matrix.\n\nThe order of this is important, because it means we could animate multiple objects by saving and restoring the matrix. For example, we could represent two cubes rotating around each other independently by partially setting up our transformation matrix, saving the intermediate result, and then setting up the specific transformations for the two separate cubes.\n\n```void transform()\n{\ndraw.transformation.setIdentity();\ndraw.perspective(1.0f,0.5f);\ndraw.translate(0,0,-6);\n\nG3DMatrix save = draw.transformation;\ndraw.rotate(AXIS_X,GXAngle);\ndrawFirstCube();\n\ndraw.transformation = save;\ndraw.rotate(AXIS_Y,GYAngle);\ndrawSecondCube();\n}```\n\nPut it all together and we get a rotating cube on our display:\n\nThe complete sources are at GitHub.\n\n## Building a 3D Graphics Pipeline on an Arduino, Part 3\n\nNow let’s talk clipping: the process of making sure we don’t write a line that’s not visible on the screen.\n\nWe’re so used to having clipping done for us that sometimes we can forget that clipping is not something that “just happens.” When we write a line to the screen what we’re really doing, at the bottom of the stack, is writing a bunch of bytes to memory that is being displayed using video hardware. And if we don’t stop at the edges, we’ll have a buffer overrun error: at the very least we’ll see bits of a line wrapping around the screen. Worse case: we will drop garbage into global memory or onto our stack, crashing our application.\n\nClipping a line ultimately requires us to find the point along the line that hits the edge of the screen. There are multiple ways we can do this: different algorithms which either modify Bresenham’s line drawing algorithm, or perform line clipping calculations before drawing.", null, "For our purposes, since we are drawing 3D lines and our coordinates are specified with floating point values (because we’re now considering the third stage of clipping), we’re going to use the later strategy of finding the point where our lines intersect the edges.\n\n## Some Math.\n\nBefore we can describe how we clip lines, we need to introduce some vector math. (Yes, that dreaded 4 letter word.)\n\nIn this case, let’s talk [dot products.](https://en.wikipedia.org/wiki/Dot_product) Now I won’t get too deep into an explanation as to how this all works; for that, the incredible series on The essence of linear algebra by Grant Sanderson of 3Blue1Brown is absolutely fantastic and explains all this stuff at a level of detail with pretty pictures that can be understood by nearly anyone.\n\nBut the essence of it is this:\n\nThe dot product of two equal-length vectors–two values (a,b,c) and (x,y,z) for example–is given by the formula:", null, "That is, you multiply all the items inside the two vectors together, and add them all up. So the dot product of (a,b,c) and (x,y,z) would be:", null, "Now the dot product has a very interesting property which will be useful to us when we talk about clipping. And that is, geometrically speaking, the dot product has the useful property that the value of the dot product is equal to the length of the projection of one vector onto another, times the length of the other vector. That is:", null, "where `||A||` is the length of vector A, and θ is the angle between the two vectors.\n\nGraphically this looks like:", null, "That is, if we were to take a vector A and “project” it into vector B, then the dot product is the length of that projection times the length of the vector B.\n\nNow there is a second interesting property which we will use in clipping, which is this: if A and B are pointed more or less in the same direction, then the dot product will be positive. However, if (as in the diagram below) they are pointed roughly in opposite directions (that is, the angle is greater than 90 degrees), then the dot product will be negative:", null, "This property will come in handy in a second.\n\n## Equation of a line.\n\nWhile we’re on the subject, it’s useful to contemplate the “equation of a line” on a 2D surface.\n\nThe general form of a line equation is:", null, "Where the variables a, b and c are selected so that any point (x,y) is on the line if it satisfies the relationship above.\n\nNow…\n\nNotice something familiar?\n\nDoesn’t it look like a dot product?", null, "And since we know the dot product is essentially the projected distance along a vector, this gives us a hint as to a geometric interpretation of the equation of a line. For a line (a,b,c), we could interpret our line equation as follows.\n\nFirst, remember that the dot product `(a,b)&dot;(x,y)` is the projected distance of one of the vectors along the other–for example, the projection of (x,y) onto the vector (a,b):", null, "Since our point (x,y) is on the line, this implies the dot product satisfies our equation above: that the dot product plus C = 0.\n\nIt also means something else.\n\nIt means all the points on our line satisfy this same projection relationship–which means our line is the line that runs perpendicular to the vector (a,b) that intersects (x,y):", null, "More importantly, consider all the lines that run parallel to our line. The closer they are to the origin in our diagram, the smaller the dot product is for the points along that line. And the farther they are from the origin than our line, the bigger the dot product. This means our equation for a line is not just an equation solving for if points are on a line, but also indicates how far a point is from our line, and which side of the line our point is on.", null, "Both of these ideas become very useful to our screen clipping code.\n\n## Clipping.\n\nLet’s now consider the operation of clipping a line against a single edge, and part of the line goes across the edge.", null, "In the above diagram, we’re clipping a line from A to B on the left side of our screen along the edge.\n\nFirst, we need to consider if the line from A to B is even visible and if it intersects the edge. Then, if it intersects the edge, we need to find the point C where it intersects the edge\n\nNotice both questions could be answered if we had the equation (a,b,c) that represents the edge x = -1, since from above we know if we’re inside or outside based on the sign of the result (a,b,c) &dot; (x,y,1), and the distance can help us calculate the point C.\n\nThe value for (a,b,c) should be obvious: we have the line for our edge along x = -1, and a little algebra later we get a = 1, b = 0 and c = 1. Ideally we want points inside our screen to be greater than or equal to zero–that is, for (a,b,c)&dot;(x,y,1) to be greater than or equal to zero if x >= -1, and from a simple inspection our values of (a,b,c) work–as if we were to plug in the coordinates (0,0), we get c = 1.\n\nThis gives us our tests to see if our two points are visible on our screen:\n\n` if (x + 1 >= 0) { /* Point is inside screen */ }`\n\nWhere `x+1` is a compact way of writing `(a,b,c)&dot;(x,y,z)`, where `(a,b,c) = (1,0.1)`.\n\nThis also gives us a way to calculate the position of C if points A and B cross our edge, by using similar triangles:", null, "We know the triangle at ACJ is proportionally smaller to the triangle ABK. We can figure out how much smaller the triangle is, proportionally speaking, from the length of the two edges we know the length of–that is, the length of AJ, which is `-A&dot;(a,b,c)`, and the length of AK, which is the length of JK (`B&dot;(a,b,c)`) plus the length of AJ.\n\nThe relative size of ACJ is:", null, "And we can use this to find the position of C. The value α ranges from 0 to 1 and gives the relative size of triangle ACJ from triangle ABK. If &alpha was 1, then ACJ would be the same as ABK (because `B&dot;(a,b,c)` would be 0), and if &alpha was 0, then triangle ACJ would effectively be a point.\n\nSo we can find C through the following linear interpolation operation:", null, "This operation is so important to our clipper we write it as a separate subroutine:\n\n```static void Lerp(const Point &a, const Point &b, float alpha, Point &c)\n{\nfloat a1 = 1.0f - alpha;\nc.x = a1 * a.x + alpha * b.x;\nc.y = a1 * a.y + alpha * b.y;\n}```\n\nWhere “Lerp” is short for “Linear interpolation”, and you’ll hear that term a lot in computer graphics.\n\n## Clipping in 2D\n\nWe now can write our 2D clipping routine, which basically clips against the four edges of our screen.\n\nIf we assume our four edges are along x = ±1 and y = ±1, then we can sketch our solution.\n\nSo let’s assume we’re given two points A and B, and we want to determine if we draw the line, draw a clipped line, or not draw a line at all.\n\nFind the “out code”\n\nFirst, we need to determine which edges the two points A and B are outside of. Since we have four edges, we need to compute a 4-bit value, with each bit representing if A and B are outside a particular edge. This we’ll call our “out code”, because in the four bits, a bit is set if the point is “outside” the screen at that edge.\n\nNow we have four edges x = ±1 and y = ±1. Since we want a point at (0,0) to have a positive value (by convention), this gives our four edges at (-1,0,1), (1,0,1), (0,-1,1) and (0,1,1).\n\nRemember that a point is outside each edge if the dot product is negative. So now we can quickly calculate our out code, with a little rearranging of our math:\n\n```static uint8_t OutCode(const Point &v)\n{\nuint8_t m = 0;\n\nif (v.x < -1) m |= 1; // v * (1,0,1) < 0\nif (v.x > 1) m |= 2; // v * (-1,0,1) < 0\nif (v.y < -1) m |= 4; // v * (0,1,1) < 0\nif (v.y &t; 1) m |= 8; // v * (0,-1,1) < 0\n\nreturn m;\n}```\n\nNext, quickly screen stuff that is not visible.\n\nNotice one nice property of our out code: two points with the same bit set set represents two points on the outside of the same edge.", null, "We can test for this quickly by taking the bit-wise ‘and’ of both out codes:\n\n```static void DrawClippedLine(const Point &a, const Point &b)\n{\nuint8_t a_out = OutCode(a);\nuint8_t b_out = OutCode(b);\n\nif (a_out & b_out) return;```\n\nQuickly find lines that are not clipped\n\nWe can also quickly test if both points are inside our screen–by testing to see if both out codes are zero. One way we can test for this–because we’ll use the intermediate result later–is to take the bit-wise ‘or’ of both out codes and test if zero:\n\n``` uint8_t out_bits = a_out | b_out;\nif (out_bits == 0) {\nDrawLine(a,b);\n} else {```\n\nIf both out codes are 0, we can just draw our line.\n\nNow find clipped lines.\n\nBut what happens if one or more bits are set? Well, that means that one (or both) of our points are outside the screen, and we need to clip our line. For our line drawing routine we have, basically, three different cases, illustrated below:", null, "The first case, edge S, has one point inside of our screen. We only need to clip one edge.\n\nThe second case, edge R, has two points outside our screen, but the middle piece is visible inside our view.\n\nAnd the third case, edge P, has two points outside our screen and no part of the line is visible.\n\nSo how do we screen against all these cases?\n\nFirst, notice our `out_bits` variable above. It has the nice property that a bit is set only if we have to clip against a particular edge. This allows us to skip those calculations we don’t need to bother to do.\n\nSo let’s iterate across all of the visible edges and calculate our α for each edge:\n\n``` uint8_t m = 1;\nuint8_t i;\nfloat alpha;\nfor (i = 0; i < 4; ++i) {\nif (out_bits & m) {\n// Calculate alpha; the intersection along the line\n// vector intersecting the specified edge\n//\n// These are specific cases of the general equation\n// alpha = (c - old)/(new - old), which yields\n// alpha == 0 if c == old, and alpha == 1 if c == new,\n// and with alpha as a linear scale with the intersection\n// point sliding from old to new.\n\nswitch (i) {\ndefault:\ncase 0: // clip (1,0,1)\nalpha = 1 + p3pos.x;\nalpha = alpha/(alpha - (1 + v.x));\nbreak;\ncase 1: // clip (-1,0,1)\nalpha = 1 - p3pos.x;\nalpha = alpha/(alpha - (1 - v.x));\nbreak;\ncase 2: // clip (0,1,1)\nalpha = 1 + p3pos.y;\nalpha = alpha/(alpha - (1 + v.y));\nbreak;\ncase 3: // clip (0,-1,1)\nalpha = 1 - p3pos.y;\nalpha = alpha/(alpha - (1 - v.y));\nbreak;\n}```\n\nNow we want to track which point is inside and which point is outside, and for the outside point on our line, track the ‘alpha’ for that side, so we can determine which side needs to be clipped.\n\nWe do this by tracking an ‘alpha’ for each side of our line, which we’ll call `alpha_a` and `alpha_b`. By convention we’ll assume alpha is 0 if our line clips at point A and 1 if we clip at point B.\n\nWe also know which side (the side with A or B) is being clipped by looking at our original outcode: if the bit is set in a_out, then we need to clip on side a. If b_out, then on side b. Notice it will only be one or the other; if neither are set we don’t clip (because out_bits would be zero at that bit), and if both are set–well, we skipped this line earlier.\n\nSo let’s add the following stuff (in red):\n\n``` float alpha_a = 0;\nfloat alpha_b = 1;\n\nuint8_t m = 1;\nuint8_t i;\nfloat alpha;\nfor (i = 0; i < 4; ++i, m <<= 1) {\nif (out_bits & m) {\n// Calculate alpha; the intersection along the line\n// vector intersecting the specified edge\n//\n// These are specific cases of the general equation\n// alpha = (c - old)/(new - old), which yields\n// alpha == 0 if c == old, and alpha == 1 if c == new,\n// and with alpha as a linear scale with the intersection\n// point sliding from old to new.\n\nswitch (i) {\ndefault:\ncase 0: // clip (1,0,1)\nalpha = 1 + p3pos.x;\nalpha = alpha/(alpha - (1 + v.x));\nbreak;\ncase 1: // clip (-1,0,1)\nalpha = 1 - p3pos.x;\nalpha = alpha/(alpha - (1 - v.x));\nbreak;\ncase 2: // clip (0,1,1)\nalpha = 1 + p3pos.y;\nalpha = alpha/(alpha - (1 + v.y));\nbreak;\ncase 3: // clip (0,-1,1)\nalpha = 1 - p3pos.y;\nalpha = alpha/(alpha - (1 - v.y));\nbreak;\n}\n\nif (a_out & m) {\nif (alpha_a < alpha) alpha_a = alpha;\n} else {\nif (alpha_b > alpha) alpha_b = alpha;\n}```\n\nOne more thing: if alpha_a has become bigger than alpha_b, that means we have case P above; our line is not visible on the screen. So we can stop processing right away:\n\n```\nif (alpha_a > alpha_b) return;\n}\n}```\n\nNow that we’ve done our calculations, we can now determine the new endpoints and draw our line. To do this we again make use of our outcodes to determine if we even need to clip one side or another.\n\n```\nPoint c1;\nif (a_out) {\nLerp(a,b,alpha_a,c1);\n} else {\nc1 = a;\n}\n\nPoint c2;\nif (b_out) {\nLerp(a,b,alpha_b,c2);\n} else {\nc2 = b;\n}```\n\nAnd now–finally! we can draw our line.\n\n``` DrawLine(c1,c2);\n}\n}```\n\nThere are some interesting things we can say about our line clipping algorithm.\n\nFirst, it follows the principle of only doing work if we have to. If we don’t have to clip against an edge, we don’t clip against the edge.\n\nSecond, notice that we could in theory extend this to 3D very easily. If we wanted to clip to a cube where x, y and z range from -1 to 1, our 6 different clipping dot products would be\n\n``` 1: ( 1, 0, 0, 1)\n2: (-1, 0, 0, 1)\n3: ( 0, 1, 0, 1)\n4: ( 0,-1, 0, 1)\n5: ( 0, 0, 1, 1)\n6: ( 0, 0,-1, 1)```\n\n## Clipping in homogeneous coordinates\n\nNotice something else very interesting about our list of vectors we’re clipping against: they sort of look like homogeneous coordinates, don’t they?\n\nAnd in fact, with a minor modification to our X or Y clipping (depending on the aspect ratio of the screen, since our screen is not square), this is becomes our clipping algorithm for drawing lines in 3D.\n\nWe can make a few optimizations. For example, rather than supplying both points A and B as we draw our line, if we were to adopt a ‘move/draw’ mechanism, we can reuse some of the calculations we perform with the previous point as we draw the next.\n\nNote: This is why we use a move/draw mechanism for drawing our lines.\n\nWe can also define a homogeneous vector for convenience sake:\n\n```struct G3DVector {\nfloat x;\nfloat y;\nfloat z;\nfloat w;\n};```\n\nAnd voliá! We have our line clipping algorithm for homogeneous coordinates.\n\nFirst, we need our linear interpolation algorithm and our out code calculator for homogeneous coordinates:\n\n```void G3D::Lerp(const G3DVector &a, const G3DVector &b, float alpha, G3DVector &c)\n{\nfloat a1 = 1.0f - alpha;\nc.x = a1 * a.x + alpha * b.x;\nc.y = a1 * a.y + alpha * b.y;\nc.z = a1 * a.z + alpha * b.z;\nc.w = a1 * a.w + alpha * b.w;\n}```\n```uint8_t G3D::OutCode(const G3DVector &v)\n{\nuint8_t m = 0;\nfloat px = p2xsize * v.x;\nfloat py = p2ysize * v.y;\n\nif (px < -v.w) m |= 1; // (p2xsize,0,0,1)\nif (px > v.w) m |= 2; // (-p2xsize,0,0,1)\nif (py < -v.w) m |= 4; // (0,p2ysize,0,1)\nif (py > v.w) m |= 8; // (0,-p2ysize,0,1)\nif (v.z < -v.w) m |= 16;\nif (v.z > v.w) m |= 32;\n\nreturn m;\n}```\n\nNext, we need to run the same clipping algorithm, but using homogeneous coordinates:\n\n```void G3D::p3movedraw(bool drawFlag, const G3DVector &v)\n{\nuint8_t newOutCode = OutCode(v);\nG3DVector lerp;\nif (drawFlag) {\nuint8_t mask = newOutCode | p3outcode;\n\n/*\n* Fast accept/reject\n*/\n\nif (0 == (newOutCode & p3outcode)) {\n// Fast accept. Both points are inside; we assume\n// the previous point was already passed upwards,\n// so we only draw to the current vector location\np2movedraw(true,v.x/v.w,v.y/v.w);\n} else {\n// At this point we have a line that crosses\n// a boundary. We calculate the alpha between\n// 0 and 1 for each point along the line.\n//\n// (This is the Liang-Barsky optimization of\n// the Cohen-Sutherland algorithm)\n\nfloat aold = 0; // (1 - alpha) * old + alpha * new = v\nfloat anew = 1; // in the above, 0 == old, 1 == new.\nfloat alpha;\n\nuint8_t m = 1;\nuint8_t i;\nfor (i = 0; i < 6; ++i) {\n// Calculate alpha; the intersection along the line\n// vector intersecting the specified edge\n//\n// These are specific cases of the general equation\n// alpha = (c - old)/(new - old), which yields\n// alpha == 0 if c == old, and alpha == 1 if c == new,\n// and with alpha as a linear scale with the intersection\n// point sliding from old to new.\n\nswitch (i) {\ndefault:\ncase 0: // clip (p2xsize,0,0,1)\nalpha = p2xsize * p3pos.x + p3pos.w;\nalpha = alpha/(alpha - (p2xsize * v.x + v.w));\nbreak;\ncase 1: // clip (-p2xsize,0,0,1)\nalpha = - p2xsize * p3pos.x + p3pos.w;\nalpha = alpha/(alpha - (-p2xsize * v.x + v.w));\nbreak;\ncase 2: // clip (0,p2ysize,0,1)\nalpha = p2ysize * p3pos.y + p3pos.w;\nalpha = alpha/(alpha - (p2ysize * v.y + v.w));\nbreak;\ncase 3: // clip (0,-p2ysize,0,1)\nalpha = - p2ysize * p3pos.y + p3pos.w;\nalpha = alpha/(alpha - (- p2ysize * v.y + v.w));\nbreak;\ncase 4: // clip (0,0,1,1)\nalpha = p3pos.z + p3pos.w;\nalpha = alpha/(alpha - (v.z + v.w));\nbreak;\ncase 5: // clip (0,0,-1,1)\nalpha = - p3pos.z + p3pos.w;\nalpha = alpha/(alpha - (-v.z + v.w));\nbreak;\n}\n\nif (p3outcode & m) {\nif (aold < alpha) aold = alpha;\n} else {\nif (anew > alpha) anew = alpha;\n}\n\nif (aold > anew) {\n// We have a case where the line is not visible\n// because it's outside the visible frustrum.\n// abort.\nbreak;\n}\n}\nm <<= 1;\n}\n\nif (i >= 6) {\n// Ran all clipping edges.\nif (p3outcode) {\nLerp(p3pos,v,aold,lerp);\np2movedraw(false,lerp.x/lerp.w,lerp.y/lerp.w);\n}\n\n// Draw to the new point\nif (newOutCode) {\nLerp(p3pos,v,anew,lerp);\np2movedraw(true,lerp.x/lerp.w,lerp.y/lerp.w);\n} else {\np2movedraw(true,v.x/v.w,v.y/v.w);\n}\n}\n}\n}\n} else {\nif (newOutCode == 0) {\np2movedraw(false,v.x/v.w,v.y/v.w);\n}\n}\n\np3outcode = newOutCode;\np3pos = v;\n}```\n\nThis assumes we’ve added a couple of new fields to our class:\n\n``` G3DVector p3pos;\nuint8_t p3outcode;```\n\nAnd now we have our clipping routine!\n\nThe code for our clipping routine is in GitHub; you can download it from here.\n\nNext time: we’ll take all that stuff we talked about with homogeneous coordinates and put the whole thing together to draw 3D objects!\n\n## Building a 3D Graphics Pipeline on an Arduino, Part 2\n\nLet’s talk coordinate systems.\n\nYes, I understand that this will involve that most dreaded of four letter words, math. I promise I’ll be gentle, and I’ll try my best to make this as easy to follow as possible.\n\nSome of this may be a little basic; feel free to skip ahead if you already know something I’m about to talk about.\n\n## 2D Coordinates\n\nFirst, let’s talk about finding the position of a pixel on a computer screen. This is something most of us are pretty familiar with.", null, "So if we want to specify the location of the yellow pixel above, well, we’d find the column (“x”) under which the pixel is located, and we’d find the row (“y”). For the yellow pixel above, x = 3, y = 1, and that’s our pixel’s location.\n\nWe can generalize this idea. For example, if you want to find the location of a pin on a piece of paper you can use a ruler and measure it’s distance from the left edge, the distance from the top edge, and those two locations (like, for example, x = 1.3 inches, y = 4.5 inches) gives us the location of that point on a piece of paper.\n\nNotice in both cases our position is a measure of distance from some relative location: for our point on a piece of paper our “units” are inches, and the relative location is the upper-left corner of the piece of paper. For our computer display, our “units” are pixels, and it’s all measured relative to the upper-left corner of the screen.\n\nSome terms.\n\nWith this in mind, it’s useful to define some terms.\n\nWe will call the point from which we’re measuring our point’s relative location the Origin, because this is where our measurements originate. Often we represent the origin in a diagram with a zero.\n\nWe will call the relative directions we measure from (horizontally and vertically across the page, across rows and down columns of our display) as our Axis. (In math, an “Axis” is a line that serves to orient ourselves–and it can also be something we rotate around.)\n\nNote: The whole concept of axis in Math is a bit of a rabbit hole, but we’ll ignore it for now. For now we’ll just treat it as “up/down”, “left/right” and “near/far.”\n\nAnd we’ll introduce some compact notation:\n\n`(x, y)`\n\nThis is a compact way for us to represent x and y.\n\nSo our yellow pixel is at `(3,1)`, our point on a piece of paper is at `(1.3 inches, 4.5 inches)`. Ususally, however, our units are implied–so we’d write our point on paper as at `(1.3, 4.5)`.\n\n## 3D Coordinates\n\nThis whole idea of 2D coordinates can be extended to the world of 3D by considering “depth.”", null, "Take our yellow dot above. Relative to our X and Y axes, the object is 4 units along X and 3 units up along Y. (What this means is, looking at the diagram above, if you were to measure from the wall that goes up along the Y-Z plane, the yellow dot is 4 units away from that wall.\n\nFrom the floor (that is, from the X-Z plane), our yellow dot is 3 units up.\n\nAnd it’s 5 units away from the X-Y plane.\n\nWe would represent the location of our yellow dot as x = 4 units, y = 3 units and z = 5 units, or–more compactly, as\n\n`(4, 3, 5)`\n\n## On the math below.\n\nI think at this point it is fair to say that we’re about to dive into a topic called “linear algebra”, and while I’ll only write out the things that are useful to us, the whole topic of linear algebra contains some interesting stuff that can be useful if you decide to do more interesting things in computer graphics than just draw a handful of pretty pictures with half-understood equations.\n\nAnd for that, if you want, there is a wonderful series on The essence of linear algebra by Grant Sanderson of 3Blue1Brown is absolutely fantastic and explains all this stuff at a level of detail with pretty pictures that can be understood by nearly anyone.\n\nFor our purposes we’ll introduce the stuff we need–but if you want to understand “why”, you’ll want to check out his videos. They’re well done, well thought out and, I think, very easy for anyone to understand.\n\n## Perspective\n\nNow when we talk about computer graphics, what we’re really thinking about is the whole idea of “perspective:” the idea that as things get farther away they appear smaller.\n\nThe moon, for example, is 2,150 miles in diameter. A quarter, on the other hand, is slightly less than an inch in diameter. But if you hold up the quarter to the night sky, you can cover the moon with your quarter. That’s perspective: your quarter, being inches from your eye, is a lot closer than the moon, which is 240,000 miles away.\n\nYou can think of this by drawing a line from a far away object to your eye:", null, "The blue dot is your eye looking at the far away object.\n\nNow suppose you are looking at the object at a computer screen some distance near your eye. The object would appear smaller–because the screen is nearer than the object you are looking at:", null, "So how big is that object on the screen?\n\nFor convenience sake, let’s consider the distance from the screen to your eye `1 unit`. A little basic geometry gives us an answer: suppose the distance to the object from your eye is Z, the height Y, then the law of similar triangles suggests the height Y’ on the screen would be given by the formula below.", null, "This means the size on our screen (that is one unit away) would be Y/Z.\n\n## Perspective and Homogeneous Coordinates\n\nWhen we talk about perspective, we are talking about Projective Geometry, or the geometric principles of projecting stuff.\n\nWhich is what we are doing when we project our object onto a computer screen.\n\nAnd in 1827, Ferdinand Möbius devised the concept of homogeneous coordinates as a means of representing projective coordinates–the coordinates of things as they are projected somewhere else.\n\nNow the whole topic of homogeneous coordinates, like mathematical axis, is a rabbit hole we can easily spend a lot of time on. But for our purposes, the way this works is as follows:\n\nFirst, we add a new dimension, `w`, associated with all of our points. Generally we can take a coordinate (x, y, z) and map it into homogeneous coordinates by adding w = 1, which we show as a fourth value in our coordinates.\n\n`(x, y, z) -> (x, y, z, 1)`\n\nAnd we can map any homogeneous point (x, y, z, w) back to 3D coordinates by:\n\n`(x, y, z, w) -> (x/w, y/w, z/w)`\n\nNow why divide by w and not z? That’s in part a matter of convention, and in part because it works well with our 3D clipping for hand-wavey reasons.\n\n(Basically, `w` serves as a projection in our 3D coordinate system. And when it comes to clipping we want to preserve all the information we can so that we can represent clipping with as much accuracy as possible, and because we may wish to represent things like points on an “infinite” sphere–like stars in the sky–through using ‘w = 0’.)\n\nWhy is this interesting?\n\nOne thing you can use homogeneous coordinates for is dealing with all your rotations, scaling operations, and translation operations (moving things around in your system) using matrix multiply operations.\n\nHere’s an example. Suppose we have an object at (3, 4, 5)–and we want to represent it moving over by 2 units along X.\n\nNormally we’d do this by addition: `(3, 4, 5) + (2, 0, 0) = (3+2, 4, 5) = (5, 4, 5).`\n\nMatrix multiplication may seem a little more convoluted–but trust me, this will make our lives easier.\n\n## Small rabbit hole: matrix multiplication.\n\nBefore we talk about translation with matrix multiplication, let’s define a few terms.\n\nFirst, for our purposes, a matrix is just a two-dimensional array of numbers. Throughout computer graphics we always use 4×4 matrices, so whenever you hear “matrix” in computer graphics, usually what you should hear is:", null, "That is, a 4×4 array of numbers.\n\nNow when we multiply a matrix by a vector, we essentially treat the vector as a 4×1 matrix (if you’re following along by looking stuff up on Wikipedia), and we essentially do the following:\n\nWe can see this with the code we use for multiplying matrices, though we store a matrix as a 2D array:\n\n```void G3DVector::multiply(const G3DMatrix &m, const G3DVector &v)\n{\nx = m.a * v.x + m.a * v.y + m.a * v.z + m.a * v.w;\ny = m.a * v.x + m.a * v.y + m.a * v.z + m.a * v.w;\nz = m.a * v.x + m.a * v.y + m.a * v.z + m.a * v.w;\nw = m.a * v.x + m.a * v.y + m.a * v.z + m.a * v.w;\n}```\n\nThe ways, whys and wherefores of matrix multiplication aren’t really that important here, though you can read more at the above linked Wikipedia article.\n\n## Back to Translations\n\nNotice something interesting about our matrix multiplication results: they include addition as well as multiplication. We can use this to our advantage by constructing our matrix appropriately.\n\nSo back to our point at `(3, 4, 5)`.\n\nIf we represent this as a homogeneous coordinate `(3, 4, 5, 1)` (because remember: in general when we go from a point in 3D space to a homogeneous coordinate we append a `w = 1` at the end), then we can construct a matrix to handle translating by x = 2:", null, "(Follow along with the animation if you need to convince yourself this is the correct answer.)\n\nNow if we perform the actual multiplications and additions we get our final result `(5, 4, 5, 1)`, and from our rules above, in 3D space, this would be `(5/1, 4/1, 5/1) = (5, 4, 5)`.\n\nWe can generalize this by moving through Y and Z, giving us our translation matrix (that is, a matrix which moves our object by a distance in X, Y and Z) as:", null, "There is an interesting property of these translation matrices. We can chain them together using matrix multiplication. And it just works out the way we would think: if we have two translation matrices, one that translates by (x,y,z) and another that translates by (a,b,c), if we multiply the two matrices together we get:", null, "(If you have to read the article on matrix multiplications and follow along, that’s fine. I can wait.)\n\nThis hints at something terribly clever going on here:\n\nWith a single matrix you can represent the concatenation of a whole bunch of rotations, scale operations and translations.\n\nAnd it means we aren’t constantly moving things around the screen a pixel at a time, through a whole chain of “rotate”, “move” and “scale” operations. We simply concat all this into a single matrix through multiplication, and then all points multiplied by that matrix are moved around according to our whole chain of rotations, movements and scale operations.\n\n## Scaling and Rotations and chaining it all together\n\nScaling an object by (sx,sy,sz)–that is, multiplying each object’s (x,y,z) coordinate by (sx,sy,sz)–has the following matrix representation:", null, "And rotation around the X, Y and Z axis looks like:", null, "Now we can chain these matrices together by pre-multiplying the matrix. Meaning if we want to first rotate our object around the Y axis by an angle, then translate the whole thing by (tx,ty,tz), we could first multiply our vector by the rotation matrix, then by the translation matrix, multiplying right to left:", null, "Or, you know, we could multiply the matrices together first–then multiply by all of our points.\n\n‘Cause like I said above:\n\nWith a single matrix you can represent the concatenation of a whole bunch of rotations, scale operations and translations.\n\nThis is, by the way, what happens in an OpenGL graphics pipeline.\n\nDisplay drivers which display 3D in hardware are very good at multiplying 4×4 matrices and 4×4 matrices with vectors. This math allows them to move objects around in real time on your screen.\n\nOf course we’re not going to be moving stuff around in real time with an Arduino, but the same principles apply.\n\nOh, yeah. So we’ve gone down a rabbit hole using 4×4 matrices to move our objects around in three-dimensional space. But what about displaying objects in perspective?\n\nThere are a number of perspective matrices out there which essentially put the z depth into the w column, so at the end we get `(x/z, y/z, ...)`–we then can display our lines using the (x/z, y/z) value at the end.\n\nThe perspective matrix I prefer–and you would multiply this as the last multiplication operation before displaying your stuff–is:", null, "Notice what we do here: we move the z coordinate into the w column, and the w column into the z column. This has the nice result that we (eventually) divide by z–and all that perspective stuff we did before works very well.\n\nNote: This is not the perspective matrix used by OpenGL. I provide a link above explaining why this is a preferable way to represent perspective.\n\n## A word about computer graphics math\n\nI once worked with a computer graphics library as part of a project a long time ago–it was required by the project manager, even though I could have rolled my own more quickly.\n\nBut the manual did make the following observation:\n\nIn the end, you either have an image, or you do not.\n\nThere are a lot of things that can go wrong on your path towards drawing something. You can accidentally flip the sign of something–and think you’re rendering an object in front of you when it is behind you and invisible. You can accidentally rotate left when you intended to rotate right. You can stack the perspective matrices backwards, or multiply the matrices together wrong.\n\nSo here are some helpful hints to keep in mind when we get to the part where we start putting the code together.\n\nTest the translation matrix first.\n\nThis will allow you to make sure you haven’t transposed the matrix: that you haven’t flipped the rows and the columns. If you take a point at (1,2,3,1) and multiply it by the transformation matrix for moving an object by x=5, if you get (6,2,3,1) you’re probably on the right path. If you get (1,2,3,6), you’ve transposed your matrix.\n\nTest to make sure you haven’t flipped a sign somewhere.\n\nIt’s inevitable, so much so that it’s nearly a joke amongst those involved in the computer graphics industry, that you will inevitably flip the sign of some term somewhere. Look through your code for a “+” where there should be a “-” and visa-versa.\n\nRight handed rule verses left handed rule.\n\nThis is sort of related to the last rule.\n\nTake your right hand, and make a “gun” with your fingers–with your thumb pointing up, your pointing finger pointing out. Now your middle finger points at a right angle to your pointing finger, and your last two fingers curl around.\n\nIf your thumb is “X”, your pointing finger “Y” and your middle finger “Z”, this is the right-handed coordinate system. And this is the coordinate system our perspective matrix works in. Notice something interesting here: if you rotate your hand so your thumb is pointing horizontally to the right, your pointing finger is pointing up–well, the middle finger is pointing towards you, not away.\n\nThis means if you want to move your object out in front of your virtual camera, you must subtract some value from the Z axis.\n\nWhen you start building your translations for display–say, you want to move a block representing an arm of a robot to the shoulder–don’t be afraid to build each of the translations slowly, using small values. Bump your arm out by 1 unit. Or rotate it by 5 to 10 degrees. Small motions help you visualize if you’re building the transformation stack (what we call all that matrix to matrix mulitplication) correctly.\n\nAnd start simply: it’s easier to start with an image (even if it’s a boring cube) and make something cool with it, than to start with a blank screen and pull your hair out trying to figure out why.\n\nBecause we move stuff around on the screen before we take its perspective, right?\n\nWhen in doubt, comment it out.\n\nThis helps you reduce the amount of variables in the problem, so you can see if you have some crazy value being multiplied into your system.\n\n## End Notes\n\nThis was a lot, I’m sure. But next time we’ll talk about clipping and clipping in 3D. And afterwards, we’ll throw all this together to create our first 3D drawings.\n\n## Building a 3D Graphics Pipeline on an Arduino, Part 1\n\nThis is a series of articles I’ve intended to write showing the creation of a basic 3D graphics pipeline on an Arduino. By the end of this series hopefully you’ll have a good understanding of basic 3D graphics, including coordinate transformations, rendering and perspective rendering.\n\nThings you need to follow along.\n\nOf course, if you want to write software for an Arduino, you need one.\n\nFor this series of exercises I used the Adafruit METRO 328, which can be had (as of this writing) for \\$17.50. You can also use any other Arduino or Arduino clone, but you may need to fiddle with the settings.\n\nI also used the 2.8″ TFT Touch Shield for Arduino. The advantage of all of this is that we can simply plug the two parts together, plug it into our computer, download the Arduino IDE, and we have a basic development environment.\n\nAdafruit also does a very good job integrating with the Arduino IDE, providing libraries that can be quickly downloaded to test your display. And they provide a great on-line tutorial to get you up and running in just a few minutes.\n\nOf course by using these components we create a dependency on their libraries; if you decide to use a different display shield from a different company you’ll also need to make a few changes in our code to make it all hang together. We’ll talk about that when we get there.\n\nWhen you get it all set up, you should see a series of test screens.", null, "* Bear not included.\n\nIntroduction.\n\nThe reason why we call this a “graphics pipeline” is because we’ll break the problem down of drawing 3D objects into a series of steps–a “pipeline” where we do a small but well-defined unit of work at each step along the way.\n\nBy organizing our drawing as a pipeline we simplify each step of the process. Each step may not do much, but when added together we can create incredible things.\n\nThe pipeline we will build will allow us to draw 3D wireframes. We’ll do a series of articles covering polygonal rendering later. At the end of this series of articles we will have created a pipeline that looks like the following:", null, "So let’s get started.\n\nHardware drawing.\n\nAt the very bottom of our pipeline is the code which draws directly to the hardware. For this we will rely on the Adafruit GFX library to actually handle the drawing–but we still call this layer out because it is the only point in our code where we talk to the Adafruit GFX library in a meaningful way. (Our code on construction interrogates the library to get the screen size, and we expose the start and stop drawing methods and color methods–but that’s it.)\n\nWe could, in theory, write our own line drawing routine. Bresenham’s line drawing algorithm can handle drawing lines in hardware very quickly, and is the algorithm used by the Adafruit library. Our line drawing routine could be wired to another library as well, such as the graphics library used by the Arduboy hand held game.\n\nLine Drawing\n\nOur primary line drawing for the first part of our 3D pipeline is the ‘movedraw’ method. This contains all the logic for moving a virtual pen to a given location, or drawing from the current pen location to a new pen location.\n\nWe use this ‘pen’ drawing model because it allows us to keep track of the state of the current location; this comes in handy when we do our 3D calculations or our line clipping.\n\nThis means our class G3D must keep track of the current pen location (and all the state revolving around the current pen location).\n\nSo let us create our 3D graphics class. First, we need to set up the constructor and the destructor to initialize and shut down our 3D pipeline, as well as entry points for setting the current color and for starting and stopping line drawing.\n\n```class G3D\n{\npublic:\n~G3D();\n\nvoid setColor(int16_t c);\nvoid begin();\nvoid end();\nvoid move(uint16_t x, uint16_t y);\nvoid draw(uint16_t x, uint16_t y);\nvoid point(uint16_t x, uint16_t y);\n};```\n\nThe three routines shown in red are test methods; we will replace them when we build the next level of our graphics engine.\n\nNow let’s add the internal state we need to track, as well as our constructor and destructor. We want to keep track of our drawing library as well as the screen size and color:\n\n```class G3D\n{\npublic:\n~G3D();\n\nvoid setColor(int16_t c)\n{\ncolor = c;\n}\n\nvoid begin();\nvoid end();\nvoid move(uint16_t x, uint16_t y);\nvoid draw(uint16_t x, uint16_t y);\nvoid point(uint16_t x, uint16_t y);\nprivate:\n/*\n* Internal state\n*/\nuint16_t width;\nuint16_t height;\n\n/*\n* Current drawing color\n*/\n\nuint16_t color;\n};```\n\nOur constructor and destructor simply notes the width and height of the screen as well as the drawing library:\n\n```G3D::G3D(Adafruit_GFX &l) : lib(l)\n{\nwidth = l.width();\nheight = l.height();\n}\n\nG3D::~G3D()\n{\n}```\n\nWe also want to build the begin and end methods. These call into the Adafruit GFX library to start and end drawing to our device. The reason why we do this is to optimize drawing; we don’t want to unnecessarily start up and shut down communications to our hardware every time we draw a line.\n\n```void G3D::begin()\n{\nlib.startWrite();\n}\n\nvoid G3D::end()\n{\nlib.endWrite();\n}```\n\nOur setColor method can be handled inline above.\n\nNow all of this so far has just been plumbing; creating our class and tracking state.\n\nThe real meat of our first level of drawing is the `p1movedraw` method and associated stuff. The idea here is to create a single entry point which handles moving and drawing through a single entry point.\n\nOur `p1movedraw` method needs to keep track of the current pen location, as well as if the last time we drew something if we moved to that location or drew to that location. So let’s add the additional methods needed to handle moving and drawing:\n\n```class G3D\n{\npublic:\n~G3D();\n\nvoid setColor(int16_t c)\n{\ncolor = c;\n}\n\nvoid begin();\nvoid end();\nvoid move(uint16_t x, uint16_t y);\nvoid draw(uint16_t x, uint16_t y);\nvoid point(uint16_t x, uint16_t y);\nprivate:\n/*\n* Internal state\n*/\nuint16_t width;\nuint16_t height;\n\n/*\n* Current drawing color\n*/\n\nuint16_t color;\n\n/*\n* State 1 pipeline\n*/\n\nbool p1draw;\nuint16_t p1x;\nuint16_t p1y;\n\nvoid p1init();\nvoid p1movedraw(bool drawFlag, uint16_t x, uint16_t y);\nvoid p1point(uint16_t x, uint16_t y);\n};```\n\nThe internal variables p1draw is true if the last time we called we were drawing a line. The values (p1x,p1y) is the pixel location of the pen. We have an initialization method for our drawing, as well as our `p1movedraw` and `p1point` methods which either moves the pen or draws a line, and which plots a single pixel point.\n\nWith our methods defined for our level 1 code, we can add our initialization method to our class constructor:\n\n```G3D::G3D(Adafruit_GFX &l) : lib(l)\n{\nwidth = l.width();\nheight = l.height();\n\n/* Initialize components of pipeline */\np1init();\n}```\n\nAnd our test methods simply call into our internal drawing routines:\n\n```void G3D::move(uint16_t x, uint16_t y)\n{\np1movedraw(false,x,y);\n}\n\nvoid G3D::draw(uint16_t x, uint16_t y)\n{\np1movedraw(true,x,y);\n}\n\nvoid G3D::point(uint16_t x, uint16_t y)\n{\np1point(x,y);\n}```\n\nHardware drawing\n\nThis is a lot of stuff, but it gets us to the core of our drawing methods. And we’ll reuse a lot of this machinery when we get to the next phase of our drawing code.\n\nFirst is initialization. Our initialization routine should set our system to a known state; in this case, we move the pen to (0,0). Recall that the variable `p1draw` is true if we drew last, false if we moved.\n\nOur initialization code then sets everything to zero:\n\n```void G3D::p1init()\n{\np1draw = false;\np1x = 0;\np1y = 0;\n}```\n\nOur point drawing does not move the pen; it simply sets the pixel. This is a simple wrapper into our Adafruit GFX library to write a pixel:\n\n```void G3D::p1point(uint16_t x, uint16_t y)\n{\nlib.writePixel(x,y,color);\n}```\n\nNote: We call the ‘writePixel’ method rather than the ‘drawPixel’ method because we are calling the ‘startWrite’ and ‘endWrite’ methods inside a ‘being’/’end’ pair.\n\nAnd our move/draw routine will draw a line only if the pen is down; we then update our current state.\n\n```void G3D::p1movedraw(bool drawFlag, uint16_t x, uint16_t y)\n{\nif (drawFlag) {\nlib.writeLine(p1x,p1y,x,y,color);\n}\n\np1draw = drawFlag;\np1x = x;\np1y = y;\n}```\n\nNote that this is the place we would need to change our code if we needed to draw our own line using Bresenham’s line drawing algorithm, or if we needed to create a draw list to send to a piece of connected hardware.\n\nWe can put all this together with a simple test:\n\n```// Use hardware SPI (on Uno, #13, #12, #11) and the above for CS/DC\n\n// Graphics setup\nG3D draw(tft);\n\nvoid setup()\n{\n}\n\nvoid loop()\n{\ntft.begin();\ntft.fillScreen(ILI9341_BLACK);\n\ndraw.begin();\ndraw.setColor(ILI9341_RED);\ndraw.move(0,0);\ndraw.draw(0,100);\ndraw.draw(100,100);\ndraw.draw(100,0);\ndraw.draw(0,0);\ndraw.move(100,120);\ndraw.draw(100,140);\ndraw.end();\n\nfor (;;) ;\n}```\n\nAnd if all works well, we should see:", null, "Okay, it’s not much. But you have to learn to crawl before you walk.\n\nThe source code for this version of the 3D graphics pipeline on GitHub; the linked branch shows just the first part of the graphics pipeline.\n\nNow this is rather boring. If this was all we wanted to do, big deal.\n\nBut remember our original principle:\n\nEach step may not do much, but when added together we can create incredible things.\n\nThe next step in our pipeline is viewport drawing.\n\nThe purpose of viewport drawing is to abstract away the pixels. That is, what we want to do is to redefine our drawing coordinates so that we get more or less the same drawing regardless of the dimensions of the physical hardware attached to our Arduino. And we do this by defining a new viewport coordinate system that ranges from -1 to 1.\n\nNote: We will be using floating point math for our operations, which may not necessarily be the fastest on an Arduino. However, it should be fast enough for us to do some simple things like rotating a cube. When designing for an Arduino, the most important thing you can do is figure out what not to draw; that way you can avoid doing math for stuff that is not visible.\n\nNow our virtual coordinate system imposes a virtual square; the top and bottom of our screen (or the left and right sides depending on the orientation of our screen) has a smaller coordinate value:", null, "This means we have a little math to do.\n\nBasically we need to figure out two things when our 3D class starts up:\n\n1. What are the viewport drawing dimensions of the screen? We need this in order to determine the dimensions of the screen we are clipping to.\n2. To convert from a viewport coordinate (x,y) to pixel coordinates (px,py), we need to calculate `px = A + B*x` and `py = C + D*y` for some values A, B, C, D. What are those values?\n\nThings get particularly interesting on the fringes. Note that when we round from a floating point value to an integer, we generally truncate (or round down); this means the value 12.75 becomes 12 when converted to an integer. So we can’t, for a screen that is 320 pixels wide, write:\n\n` px = 160 + 160 * x; // 160 = 320 / 2`\n\nBecause if we plug in a value 1 for x, we get px = 320–and our pixel coordinate range is from 0 to 319.\n\nSo we solve this problem by bumping the width down by 1 pixel, and we calculate our pixel range (in floating point values) from 0.5 to 319.5. (This will then convert to integer values from 0 to 319.)\n\nThus we’d write:\n\n` px = 160 + 159.5 * x; // 159.5 = 319/2`\n\nPlugging in -1 for x gives is px = (int)0.5 = 0, and 1 for x gives us px = (int)319.5 = 319.\n\nWe have a second problem in that in our pixel coordinates (0,0) is in the upper-left corner of the screen–but we want our drawing to put (-1,-1) at the bottom-left. This means we need to flip the sign of the y value during our calculations:\n\n``` px = A + B * x;\npy = C - D * y;```\n\nOur third problem, of course, is that our display is not square; it’s rectangular. But we assume our pixels are square. And we assume whichever is larger (the width or the height) will map from -1 to 1; the other wide will map from roughly -0.75 to 0.75 or so.\n\nThis implies our scale value B and D which we multiply in our equation above is the same. (If our pixels are not square we need to do something different.)\n\nSo let’s put all this together.\n\nFor our viewport coordinate system we need to track five variables: two giving the (+/-) width and (+/-) height of our viewport:\n\n``` float\tp2xsize; // viewport width +/-\nfloat\tp2ysize; // viewport height +/-```\n\nAnd three that handle the transform: the amount we multiply and add to our viewport coordinate to give our pixel coordinate:\n\n``` float\tp2scale; // coordinate transform scale.\nfloat\tp2xoff; // coordinate transform offset\nfloat\tp2yoff;```\n\nOf course we also need to add our level 2 pipeline code.\n\nAll of this looks like:\n\n```class G3D\n{\npublic:\n~G3D();\n\nvoid setColor(int16_t c)\n{\ncolor = c;\n}\n\nvoid begin();\nvoid end();\nvoid move(float x, float y);\nvoid draw(float x, float y);\nvoid point(float x, float y);\nprivate:\n/*\n* Internal state\n*/\nuint16_t width;\nuint16_t height;\n\n/*\n* Current drawing color\n*/\n\nuint16_t color;\n\n/*\n* State 2 pipeline; map -1/1 to screen coordinates\n*/\n\nvoid p2init();\nvoid p2movedraw(bool drawFlag, float x, float y);\nvoid p2point(float x, float y);\n\nfloat p2xsize; // viewport width +/-\nfloat p2ysize; // viewport height +/-\nfloat p2scale; // coordinate transform scale.\nfloat p2xoff; // coordinate transform offset\nfloat p2yoff;\n\n/*\n* State 1 pipeline\n*/\n\nbool p1draw;\nuint16_t p1x;\nuint16_t p1y;\n\nvoid p1init();\nvoid p1movedraw(bool drawFlag, uint16_t x, uint16_t y);\nvoid p1point(uint16_t x, uint16_t y);\n};```\n\nNotice we also change our test routines (shown in red).\n\nOur `p2movedraw` and `p2point` routines look similar: they perform our math calculation using the constants we’ve calculated during startup to transform from viewport coordinates to pixel coordinates:\n\n```void G3D::p2point(float x, float y)\n{\nint16_t xpos = (int16_t)(p2xoff + x * p2scale);\nint16_t ypos = (int16_t)(p2yoff - y * p2scale);\np1point(xpos,ypos);\n}\n\nvoid G3D::p2movedraw(bool drawFlag, float x, float y)\n{\nint16_t xpos = (int16_t)(p2xoff + x * p2scale);\nint16_t ypos = (int16_t)(p2yoff - y * p2scale);\np1movedraw(drawFlag,xpos,ypos);\n}```\n\nNow the heavy lifting is in our initization code, which must find all our constants given the size of the screen. As we noted before we assume pixels are square.\n\nOur initialization code first needs to set up some constants and find the screen dimensions for our clipping method. We then need to calculate the scale parameter and ultimately the offsets. In the end our code looks like:\n\n```void G3D::p2init()\n{\n/*\n* We subtract one because we want our mapping to work so that\n* virtual coordinate -1 is in the middle of the 0th pixel, and\n* +1 is in the middle of the width-1 pixel for wide displays.\n*\n* This offset of 1/2 by the pixel width implies we're drawing on\n* a display 1 pixel narrower and wider, but then with 1/2 added\n* to the pixel coordinate\n*/\n\nuint16_t h1 = height - 1;\nuint16_t w1 = width - 1;\n\n/*\n* Calculate the width, height in abstract coordinates. This\n* allows me to quickly clip at the clipping level\n*/\n\nif (w1 > h1) {\np2xsize = 1;\np2ysize = ((float)h1)/((float)w1);\n} else {\np2xsize = ((float)w1)/((float)h1);\np2ysize = 1;\n}\n\n/*\n* Calculate the scale, offset to transform virtual to real.\n* Note that -1 -> 0 and 1 -> (width-1) or (height-1).\n*/\n\nif (w1 > h1) {\np2scale = ((float)w1)/2;\n} else {\np2scale = ((float)h1)/2;\n}\n\np2xoff = ((float)width)/2;\np2yoff = ((float)height)/2;\n}```\n\nOf course we need to change our main Arduino test code to use the new floating point values for coordinates:\n\n```void loop()\n{\ntft.begin();\ntft.fillScreen(ILI9341_BLACK);\n\ndraw.begin();\ndraw.setColor(ILI9341_RED);\ndraw.move(-0.5,-1);\ndraw.draw(0.5,-1);\ndraw.draw(0.5,1);\ndraw.draw(-0.5,1);\ndraw.draw(-0.5,-1);\n\ndraw.move(-0.75,-0.75);\ndraw.draw(-0.5,-0.75);\ndraw.draw(-0.5,-0.5);\ndraw.draw(-0.75,-0.5);\ndraw.draw(-0.75,-0.75);\ndraw.end();\n\nfor (;;) ;\n}```\n\nAnd if everything works correctly, you should see the following:", null, "Again, not all that exciting.\n\nExcept for one thing: This is what your image will look like regardless of your screen size. This becomes important as we move on and start building our 3D rendering engine.\n\nAll of this code is available at GitHub in a separate branch.\n\nThis has been an extremely long post, but a lot of this was setting the stage for future posts. The next post will discuss homogeneous coordinates, the coordinate system we use for representing objects in 3D. This will be followed by a post discussing clipping in homogeneous coordinates, then we’ll move into some 3D math." ]

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[ "", null, "", null, "", null, "", null, "# Effective Intersection Theory\n\nLet X ⊂ Y be closed (possibly singular) subschemes of a smooth projective toric variety T. We show how to compute the Segre class s(X,Y) as a class in the Chow group of T. Building on this, we give effective methods to compute intersection products in projective varieties, to determine algebraic multiplicity without working in local rings, and to test pairwise containment of subvarieties of T. Our methods may be implemented without using Gröbner bases; in particular any algorithm to compute the number of solutions of a zero-dimensional polynomial system may be used.\n\n06/19/2018\n\n### Segre Class Computation and Practical Applications\n\nLet X ⊂ Y be closed (possibly singular) subschemes of a smooth projectiv...\n05/25/2020\n\n### Stratified Formal Deformations and Intersection Homology of Data Point Clouds\n\nIntersection homology is a topological invariant which detects finer inf...\n04/01/2019\n\n### Intersection multiplicity of a sparse curve and a low-degree curve\n\nLet F(x, y) ∈C[x,y] be a polynomial of degree d and let G(x,y) ∈C[x,y] b...\n03/14/2019\n\n### Bayesian/Graphoid intersection property for factorisation models\n\nWe remark that the Graphoid intersection property, also called intersect...\n09/02/2019\n\n### Singular tuples of matrices is not a null cone (and, the symmetries of algebraic varieties)\n\nThe following multi-determinantal algebraic variety plays a central role...\n08/16/2015\n\n### Computing characteristic classes of subschemes of smooth toric varieties\n\nLet X_Σ be a smooth complete toric variety defined by a fan Σ and let V=...\n10/29/2018\n\n### Reducing the complexity for class group computations using small defining polynomials\n\nIn this paper, we describe an algorithm that efficiently collect relatio...\n\n## 1. Introduction\n\nSegre classes capture important enumerative and geometric properties of systems of polynomial equations coming from embeddings of schemes. Historically, these classes have played a fundamental role in the development of Fulton-MacPherson intersection theory [Ful98, §6.1]. Computation of Segre classes (other than in a few special cases) has proven to be a challenge; this has limited the development of applications in practice.\n\nEvidence of the significance of Segre classes in algebraic geometry can be found in the fact that many important characteristic classes can be written as , where\n\nis some polynomial in the Chern classes of vector bundles on\n\n. The flagship example of this is the topological Euler characteristic , which appears in the Chern-Schwartz-MacPherson class . By results of Aluffi [Alu03, Alu18] the class can be directly obtained by computing a Segre class. There are also formulas in terms of Segre classes for the Milnor class of a hypersurface [Alu03], the Chern-Mather class and polar degrees [Pie78], and the Euclidean distance degree of a projective variety [AH17].\n\nMore generally, many enumerative problems end up in the situation of an excess intersection, in which an intersection is expected to be finite but instead is the union of a finite set of points along with a positive-dimensional set. Typically the desired quantity is then the number of points outside the positive-dimensional part. The Segre class gives a way to express the contribution of this part, which is the difference between the expected number (e.g., the Bézout bound) and the actual number of points in the finite set.\n\nLet be closed subschemes of a smooth projective toric variety . The Segre class of in is a class in the Chow group of . Since the group is often unknown, the best we could hope for in general is to compute the pushforward of this class to .\n\nPrevious work on computing Segre classes of the form in (i.e., the special case where ) began with the paper [Alu03] and alternative methods were developed in [EJP13, Hel16]. These methods were generalized to compute in in [MQ13, Hel17]. In [Har17] the scope was extended to compute the Segre class pushed forward to for and a variety. The present work goes further by taking arbitrary subschemes not just of projective space, but of any smooth projective toric variety, to obtain in .\n\nThe ability to effectively compute Segre classes in this new setting opens the way for several novel computational applications. For instance, computing Segre classes in products allows for a general framework to compute intersection products in subvarieties of . While a big part of this paper is devoted to studying Segre classes, many of the resulting applications can be expressed without them. In particular, we show that algebraic multiplicity can be computed and pairwise containment of varieties can be tested by counting the number of points in a single zero-dimensional set.\n\n### 1.1. Examples\n\nWe now give three examples which illustrate how the results developed in this paper give rise to new methods to answer classical geometric and enumerative questions. The first example shows how the intersection product can be computed using a Segre class. Following this we give examples which compute algebraic multiplicity and test ideal containment. The methods presented in the later two examples build on ideas developed to compute Segre classes (which we present in §3), but can be understood without them.\n\n#### 1.1.1. Intersection theory\n\nThe intersection product of varieties and in a non-singular variety , denoted (see §4 for a definition), captures the behavior of the intersection inside of . It is a class in the Chow ring . If meets transversely in the expected dimension, the intersection product may be defined by\n\n X⋅YV=[X∩V]∈A∗(Y).\n\nIf and do not meet dimensionally transversely, but there exist and which are respectively equivalent in and are dimensionally transverse, then we define the intersection analogously:\n\n###### Example 1.1 (Lines on a quadric, I).\n\nThe quadric surface in Figure 1, comes with two families of lines, and , and any two lines in the same family are equal in . Let and be two such lines on . We compute and .\n\nSince meets only at and the intersection is transverse of the expected dimension , we can conclude that is the class of a point.\n\nIn contrast, does not have the expected dimension, so we can try to “move” one of the terms by finding a suitable replacement with the same class in . If is any other line, then is empty, and thus the intersection product is 0.\n\nAnother way to perform the computation above would be to recognize that and , . Then the product in this ring is the intersection product, i.e.  and . However, when computing in practice, one may encounter a variety for which the Chow ring is not known a priori. In this situation, using the methods of §3-4, we could still arrive at the answer.\n\n###### Example 1.2 (Lines on a Quadric, II).\n\nLet and forget that is a smooth quadric. Given subvarieties and defined on by and , respectively, we wish to compute the intersection products and . Since we do not know the Chow ring, we must do something new. Specifically we work in the Chow ring of the ambient , which is , and write the intersection product in terms of classes there.\n\nIn §4, we will see that a universal setup for this approach is to intersect with the diagonal in . Then the ingredients to our computation are:\n\n1. the Chern class , where is the inclusion,\n\n2. the Segre class ,\n\n3. the Segre class ,\n\nwhere is the diagonal map and\n\nare the hyperplane classes for each factor. As above, we will frequently suppress obvious pushforwards. Then, as follows from Theorem\n\n4.1, we confirm and .\n\n###### Remark 1.3.\n\nIn Example 1.2 we did not need to do any computations by hand. Items (i)-(iii) can all be computed in a computer algebra system given the defining equations. Following [Alu03], is determined also by a Segre class (specifically the Segre class ), which could already be found via the methods of [Alu03, EJP13, Hel16], and computing items (ii) and (iii) are contributions of this paper.\n\n#### 1.1.2. Algebraic multiplicity\n\nOur second example illustrates how Segre classes can be used to compute the algebraic multiplicity of a local ring with respect to an ideal (without computing in the local ring).\n\n###### Example 1.4 (Algebraic multiplicity along a component).\n\nLet be the homogeneous coordinate ring of . As in [Say17, Ex. III.10], we consider the twisted cubic defined by the prime ideal\n\n IX=⟨yw−z2,xw−yz,xz−y2⟩\n\nand the scheme defined by the ideal\n\n IY=⟨z(yw−z2)−w(xw−yz),xz−y2⟩.\n\nThen is a subscheme of , since . Let be the local ring of along . The algebraic multiplicity of along is the leading coefficient of the Hilbert-Samuel polynomial associated to the local ring (see §5). The multiplicity may be read off of the Segre class since is also the coefficient of in the class , see Definition 5.1. The class is . Applying the methods of §3 we find that\n\n s(X,Y)=6h2=2(3h2)∈A∗(P3).\n\nTogether this gives that . Let be the maximum degree among the defining equations of the ideals and and let be the dimension- projective degree defined by in (see §2.3). More directly, by Theorem 5.3 we have that\n\n eXY=deg(Y)ddim(Y)−dim(X)−gdeg(X)=6⋅30−03=2.\n\n#### 1.1.3. Containment\n\nOur third example demonstrates a new criterion to test containment of varieties or of irreducible components of schemes (see §6).\n\n###### Example 1.5 (Containment of Varieties).\n\nWork in with coordinates and let\n\n I=the ideal defined by all 3×3 minors of⎛⎜ ⎜ ⎜⎝x0x1x2x33x34x45x56x6x2x3x4x5x0+5x1x1+6x2x2+7x3x3+8x4⎞⎟ ⎟ ⎟⎠.\n\nThe variety is an irreducible singular surface of degree 20 in .\n\nNow set and . We seek to determine if the line is contained in the singular locus of the surface . The standard method to test for this containment is to compute the ideal defining and reduce each generator of with respect to a Gröbner basis for . In this case the ideal is defined by the minors of the Jacobian matrix of ; it is clear from the structure of (it has generators of degree three in six variables) that the computation of the minors to obtain the ideal will be very time consuming.\n\nOn the other hand, by Corollary 6.2 we have that if and only if\n\n 20⋅32−1−g1>1\n\nwhere the left-hand side is (computed via Theorem 5.3), and is the dimension- projective degree of in (see §2.3). Using Theorem 3.5 we compute by finding the number of solutions to a single zero-dimensional system of polynomials, with each polynomial of degree at most three. Substituting this in we have that and, hence, . The computation of the integer takes approximately seconds using Macaulay2 [GS] on a laptop.\n\nWe note again that the test described above using Corollary 6.2 does not compute the ideal defining . In this case computing the ideal and using Gröbner basis methods to test if takes approximately 692 seconds using Macaulay2 [GS] on the same test machine (the majority of this time, about 690 seconds, is spent computing the ideal ).\n\nThis paper is organized as follows. In Section 2 we establish our notation and conventions and present relevant background on Segre classes and projective degrees. The main results of this paper are presented in Section 3. In §3.1 we consider the case where is a subscheme of an irreducible scheme . In Theorem 3.6 we give an explicit formula for the Segre class in terms of the projective degrees of a rational map defined by from to a projective space (see §2.3). In Theorem 3.5 we give an expression for these projective degrees as a vector space dimension of a ring modulo a certain zero-dimensional ideal. These results are generalized in §3.2 to the case where is a subscheme of a smooth projective toric variety .\n\nIn Section 4 we show how the results of Section 3 can be applied to compute intersection products. If is a smooth variety, requires computing . The main result is that the pushforward of this class to is enough to recover the pushforward of the intersection product.\n\nIn Section 5 we use the results of §3.1 to give an explicit expression for , the algebraic multiplicity of along , in terms of ideals in the coordinate ring of . The expressions are also generalized to .\n\nFinally in Section 6 we combine the results from §5 with a classical result of Samuel [Sam55] to yield new numerical tests for the containment of one variety in another. Let and be arbitrary subvarieties of a smooth projective toric variety . In §6.1 we give a simple criterion to determine if is contained in the singular locus of without computing the defining equations of the singular locus. In §6.2 we give a criterion to determine if . As in the previous results, computing a Gröbner basis is not required and methods from numerical algebraic geometry could be used. To the best of our knowledge, this is the first general purpose method which is able to test containment of possibly singular varieties using only numeric methods.\n\nAs of version 1.13, Macaulay2 [GS] contains the SegreClasses package, which implements many of the results described in this paper.\n\n## 2. Background\n\nIn this section we review several definitions and explicitly state the notations and conventions we will use throughout the paper.\n\n### 2.1. Notations and conventions\n\nWe work throughout over an algebraically closed field of characteristic zero.\n\n#### 2.1.1. Varieties, schemes, and irreducible components\n\nSince we always work in an ambient projective variety, all schemes will be of finite type over the base field. By variety we mean a reduced and irreducible, separated scheme, that is, an irreducible algebraic set. Given polynomials we let denote the algebraic set defined by . Conversely, if is a subscheme of a smooth variety with coordinate ring , we will let be the ideal in defining the scheme and let be the radical ideal in defining the reduced scheme . If is a subscheme of a smooth variety defined by an ideal , its primary components are the schemes associated to the primary components of ; its irreducible components are the varieties defined by the associated primes of .\n\n#### 2.1.2. Chow classes\n\nLet be a subscheme of a smooth variety . The irreducible components of have associated geometric multiplicity given by the length of the local ring , and we write for the rational equivalence class of in . We frequently write to mean the pushforward via inclusion. For a cycle class in the Chow group we will use the notation to denote the degree of the zero-dimensional part of (as in Definition 1.4 of Fulton [Ful98]). The degree of a zero-dimensional scheme is . For instance, if is defined by then .\n\n#### 2.1.3. The total coordinate ring of a toric variety\n\nLet be a smooth projective toric variety defined by a fan and let denote the rays in the fan. The Cox ring of is . The ring can be graded by defining the multidegree of a monomial to be , where denotes the codimension-one Chow group of . Setting we have that We say that a polynomial is homogeneous if it is homogeneous with respect to this grading, i.e., if all monomials in have the same multidegree. An ideal in is called homogeneous if it is generated by homogeneous polynomials. When the Cox ring is simply the standard graded coordinate ring of and the multidegree of a monomial is simply the total degree of the monomial multiplied by the class of a general hyperplane. More details can be found in the book [CLS11].\n\n#### 2.1.4. Homogeneous generators\n\nLet be a closed subscheme. We can always find an ideal defining so that for all , for some fixed in (see [CLS11, 6.A]). Using this set of generators we see that is the base scheme of the linear system defined by , viewed as sections of .\n\n###### Definition 2.1.\n\nWe say a homogeneous polynomial in the Cox ring of a smooth projective toric variety has multidegree and say a set of polynomials all having the same multidegree is -homogeneous.\n\n###### Convention 2.2.\n\nLet be a subscheme of defined by an ideal generated by polynomials , we assume (without loss of generality) that this set of polynomials is -homogeneous. We will use this convention for the defining equations of all subschemes/subvarieties considered in this paper unless otherwise stated.\n\n###### Remark 2.3.\n\nIn the case where we are simply assuming that a given set of polynomial generators have the same degree. If we are given an ideal in the coordinate ring of where the generators do not have the same multidegree we may construct a new ideal which has -homogeneous generators and defines the same scheme as follows.\n\nWork in with multi-graded coordinate ring , where . In this case the Chow group is generated by where is the pullback of the hyperplane class in the factor . The multidegree of a monomial in has the form where is the total degree of in the variables . Let , so that the irrelevant ideal of has primary decomposition , and for , let be the ideal generated by the -th powers of the generators of .\n\nTake a subscheme defined by a homogeneous ideal in with the generator having multidegree . Let for . We can construct a new ideal for with generators all having multidegree ; the ideal is given by:\n\n J=m∑i=1⟨wi⟩⋅B1(D1−d(i)1)⋯Bm(Dm−d(i)m).\n\nNote that in the equation above we use summation notation for the sum of ideals in . The reader can verify that , meaning that and define the same subscheme.\n\n###### Example 2.4.\n\nThe procedure discussed in Remark 2.3 above is easy to apply by hand. Work in with coordinate ring and codimension-one Chow group generated by the hyperplane classes . The irrelevant ideal of the coordinate ring is\n\nConsider the ideal defining a scheme . The generators of have multidegrees and , respectively. The new ideal will have generators all of multidegree :\n\n J=⟨x0x21y1−x32y3⟩⋅⟨y0,y1,y2,y3⟩+⟨x2y22−x1y0y1⟩⋅⟨x20,x21,x22⟩\n\nand the ideal also defines the scheme since .\n\nTo construct an -homogeneous set of generators from a given set of homogeneous generators for a subscheme of an arbitrary smooth projective toric variety, a technique similar to that of Remark 2.3 can be used. Instead of multiplying by powers of the generators of components of the irrelevant ideal, we multiply the given generators by powers of the generators of products of components of the irrelevant ideal of the Cox ring. Since the fan of a general smooth projective toric variety is more combinatorially complicated than that of a product of projective spaces, the procedure is also more difficult to write down, but is otherwise similar.\n\n#### 2.1.5. Multi-indices\n\nWe will make frequent use of standard multi-index notations throughout the paper. In particular for a non-negative integer vector , we have , and if are the variables of a ring, we write . Notice that if are generators of the Chow ring , then is a class of codimension .\n\n### 2.2. Segre classes\n\nIn this subsection we define the Segre class of a subscheme and summarize computational methods for Segre classes in the Chow ring .\n\nFor a subscheme of a scheme the Segre class is the Segre class of the normal cone to in (see [Ful98, §4.2] for more details). In the case where is a variety we may define the Segre class via Corollary 4.2.2 of [Ful98].\n\n###### Definition 2.5.\n\nLet be a closed subscheme of a variety . We have a blowup diagram\n\nwhere is the exceptional divisor. The Segre class of in is\n\n s(X,Y)=η∗((1−E+E2−…)⌢[E])∈A∗(X),\n\nsee [Ful98, Corollary 4.2.2]. When is contained in some smooth projective toric variety we will frequently abuse notation and write for the pushforward to .\n\nWe now review the computation of the class in the special case where is a subscheme of a projective variety . In this case we may think of as the base scheme of an -dimensional linear system of global sections of . In practice this means choosing a set of (scheme-theoretic) generators for all of the same degree , say . Consider the graph\n\n (1)\n\nof the rational map defined by .\n\n###### Definition 2.6.\n\nWe call the projection of along . The projective degrees of are the non-negative integers\n\n gi(X,Y)=∫hi⋅[¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯pr−1X(Pr−(dim(Y)−i))−X],\n\nwhere is the hyperplane class in the Chow ring .\n\n###### Proposition 2.7 ([Har17, Prop. 5]).\n\nThe Segre class is given by\n\n si=dimX∑j=0(dimY−ij−i)(−d)j−i(ddimYdeg(Y)−gj(X,Y)).\n\nIn [Hel16, Theorem 4.1], the projective degrees of a rational map are expressed as the dimensions of a sequence of finite-dimensional -algebras. There is an analogous result for our more general situation. The projective degrees for can be computed directly from the definition above as the degree of the 0-dimensional variety\n\n Zi=(Y∩Li∩V(P1,…,Pdim(Y)−i))−X\n\nwhere is a general linear space of codimension in and is a general -linear combination of -homogeneous generators of . Note by construction.\n\nWe now move to and restrict to the affine open subset defined by the non-vanishing of a general linear form . Further, we want to remove , so we restrict to where is a general -linear combination of -homogeneous generators for . The resulting affine variety\n\n ((Y∩Li∩V(P1,…,PdimY−i))×A1)∩D(F′)∩D(L′)\n\nis in 1-1 correspondence with the set of points . This leads to the following expression for the projective degrees.\n\n###### Proposition 2.8 (cf. [Hel16, Theorem 4.1]).\n\nThe projective degrees of are given by\n\n gi(X,Y)=dimkk[x0,…,xn,T]IY+ILi+⟨P1,…,Pdim(Y)−i,1−T⋅F′,1−L′⟩,\n\nwhere is a general affine linear form in the and is a general -linear combination of -homogeneous generators for .\n\n### 2.3. Projective degrees of X in Y\n\nLet be a smooth projective toric variety defined by a fan with Cox ring (see §2.1.3). Let be a pure-dimensional subscheme and let be a subscheme defined by an -homogeneous ideal (see Definition 2.1).\n\n###### Definition 2.9.\n\nWe define the projection of along to be the rational map given by\n\n (2) prX:p↦(f0(p):⋯:fr(p)).\n\nLet denote the blowup of along . We have the following diagram:\n\n (3)\n\nNow we define the class\n\n (4) G(X,Y):=dim(Y)∑i=0π∗(Ki⋅[Γ])∈A∗(TΣ),\n\nwhere is the pullback along of the hyperplane class. This class is a minor generalization of Aluffi’s shadow of the graph ([Alu03]). By construction\n\n π∗(Ki⋅[Γ])=[¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯pr−1X(Pr−(dim(Y)−i))−X],\n\nwhere is a general linear subspace.\n\nWe may write more explicitly as follows. Let be a fixed nef basis for (this exists since is projective, see [CLS11, Proposition 6.3.24]). We may express the rational equivalence class of a point in as where for all , and the degree- monomials in which divide in form a monomial basis for . Hence we may write\n\n (5) G(X,Y)=∑|ν|≤dim(Y)gν(X,Y)⋅bn−ν∈A∗(TΣ),\n\nwhere . Note that the indices appearing in this expression depend on the choice of representative of the point class, which is not unique in general; however the class does not depend on this choice.\n\n###### Definition 2.10.\n\nWe refer to the coefficients as the projective degrees of in . In other words the projective degrees are\n\n (6) gν(X,Y)=∫bν⋅[¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯pr−1X(Pr−(dim(Y)−|ν|))−X].\n\nThe class measures, in a sense, how algebraically dependent the polynomials are in . If the linear system were base-point free on (so and ), the shadow of the graph would be . The difference of these two classes will play an important role in the computation of the Segre class (see Theorems 3.6 and 3.13), so we define the following notation:\n\n (7) Λ(X,Y)=∑|a|≤dim(X)Λa(X,Y)⋅bn−a:=dim(Y)∑i=0αdim(Y)−i[Y]−G(X,Y)∈A∗(TΣ).\n\nIn particular, when .\n\n###### Example 2.11 (Projective degrees in Pn).\n\nWork in with Chow ring and consider the varieties and . Using Theorem 3.5 we compute that the projective degrees are and . Hence we have that . For this example and we let . Substituting these values into (7) we obtain\n\n Λ(X,Y)=(9h2+3h)−(7h2+3h)=2h2.\n###### Example 2.12 (Projective degrees in products of projective spaces).\n\nWork in with Chow ring . In this Chow ring the class of a point is . Consider the 3-dimensional variety\n\n Y=V(x0x2y0−x21y2,y3)\n\nwith divisor defined by . For this example, , and and we let . Computing the projective degrees using Theorem 3.5 we obtain the following. In dimension zero we have . In dimension one we have and . In dimension two we have , , and . In dimension three we have , , and . Hence we have that\n\n G(X,Y) =(0⋅h21h32)+(0⋅h1h32+1⋅h21h22)+(0⋅h32+1⋅h1h22+2⋅h21h2)+(1⋅h22+2⋅h1h2+0⋅h21) =h21h22+h1h22+2h21h2+h22+2h1h2.\n\nSubstituting these values into (7) we obtain\n\n Λ(X,Y) =(24h21h32+12h21h22+6h1h32+4h21h2+4h1h22+h32+2h1h2+h22)−G(X,Y) =24h21h32+11h21h22+6h1h32+2h21h2+3h1h22+h32.\n###### Example 2.13 (Projective degrees in a toric variety).\n\nWork in the smooth Fano toric threefold111The variety is generated in Macaulay2 [GS] with the command smoothFanoToricVariety(3,2) from the NormalToricVarieties package. where is the fan with rays , , , , and maximal cones , , , , , . Let denote the orbit closure of a cone and set and . The divisors form a nef basis for . The Cox ring of is , with irrelevant ideal . The Cox ring is graded; the multidegree of is , the multidegree of , and is , and the multidegree of is . The Chow ring can be written as\n\n A∗(TΣ)≅Z[b1,b2]/⟨b31,2b2−2b1,b1b2−b22⟩.\n\nFor more on constructing the Chow ring of a smooth projecive toric variety from its fan see [Dan78, Theorem 10.8].\n\nLet and let . Then is a curve on the surface . For this example, and we let . Let denote the rational equivalence class of a point in . Using Proposition 3.11 we compute the projective degrees and obtain Substituting and into (7) we obtain\n\n Λ(X,Y)=([pt]+b21+b22+b1)−([pt]+b1b2+b1)=b21∈A∗(TΣ).\n\nNote that . The lack of a unique representative for the point class in this case stems from the existence of non-effective divisors on . Since the indexing convention for projective degrees depends on the chosen representative of the point class, we have opted to simply write the resulting unique class .\n\n###### Remark 2.14.\n\nIf is the complete intersection of general divisors of with such that , then we have that\n\n gν(X,Y)=gν−a(X∩La,Y∩La).\n\nIn particular, if is a divisor with , then" ]

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[ "# Singular\n\n### D.15.7 ellipticcovers_lib\n\nLibrary:\nellipticCovers.lib\nPurpose:\nGromov-Witten numbers of elliptic curves\n\nAuthors:\nJ. Boehm, boehm @ mathematik.uni-kl.de\nA. Buchholz, buchholz @ math.uni-sb.de\nH. Markwig hannah @ math.uni-sb.de\n\nOverview:\nWe implement a formula for computing the number of covers of elliptic curves. It has beed obtained by proving mirror symmetry\nfor arbitrary genus by tropical methods in [BBM]. A Feynman graph of genus g is a trivalent, connected graph of genus g (with 2g-2 vertices and 3g-3 edges). The branch type b=(b_1,...,b_(3g-3)) of a stable map is the multiplicity of the the edge i over a fixed base point.\n\nGiven a Feynman graph G and a branch type b, we obtain the number N_(G,b) of stable maps of branch type b from a genus g curve of topological type G to the elliptic curve by computing a path integral\nover a rational function. The path integral is computed as a residue.\n\nThe sum of N_(G,b) over all branch types b of sum d gives N_(G,d)*|Aut(G)|, with the Gromov-Witten invariant N_(G,d) of degree d stable maps from a genus g curve of topological type G to the elliptic curve.\n\nThe sum of N_(G,d) over all such graphs gives the usual Gromov-Witten invariant N_(g,d) of degree d stable maps from a genus g curve to the elliptic curve.\n\nThe key function computing the numbers N_(G,b) and N_(G,d) is gromovWitten.\n\nReferences:\n[BBM] J. Boehm, A. Buchholz, H. Markwig: Tropical mirror symmetry for elliptic curves, arXiv:1309.5893 (2013).\n\nTypes:\ngraph\n\nProcedures:\n\n D.15.7.1 makeGraph generate a graph from a list of vertices and a lsit of edges D.15.7.2 printGraph print procedure for graphs D.15.7.3 propagator propagator factor of degree d in the quotient of two variables, or propagator for fixed graph and branch type D.15.7.4 computeConstant constant coefficient in the Laurent series expansion of a rational function in a given variable D.15.7.5 evalutateIntegral path integral for a given propagator and ordered sequence of variables D.15.7.6 gromovWitten sum of path integrals for a given propagator over all orderings of the variables, or Gromov Witten invariant for a given graph and a fixed branch type, or list of Gromov Witten invariants for a given graph and all branch types D.15.7.7 computeGromovWitten compute the Gromov Witten invariants for a given graph and some branch types generatingFunction (graph, int) multivariate generating function for the Gromov Witten invariants of a graph up to fixed degree D.15.7.8 partitions partitions of an integer into a fixed number of summands D.15.7.9 permute all permutations of a list D.15.7.10 lsum sum of the elements of a list\n\n### Misc", null, "" ]

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[ "## Erkko Lehtonen (Technische Universität Dresden): Graph algebras and graph varieties\n\n Wednesday, 26. September 2018, 10:00 - 12:00\nAbstract. Graph algebras were introduced by Shallon in 1979. To each directed graph G=(V,E), we associate an algebra A(G) of type (2), whose universe is the set V∪{∞}, where ∞ is a new element not in V, and where the binary operation (\"product\") is defined as xy = x if (x,y)∈E and xy = ∞ otherwise. Encoding graphs as algebras in this way, we can view any algebraic properties of the graph algebra A(G) as properties of the graph G.\n\nAlthough the class of graph algebras does not constitute a variety (as it is not closed under direct products), it makes perfect sense to consider the satisfaction relation between graphs (that is, graph algebras) and identities in the language of groupoids. Accordingly, the equational classes of graphs are called graph varieties. Graph varieties have been investigated by several authors. For example, Kiss, Pöschel and Pröhle determined the identities satified by all graphs. Poomsa-ard and his coauthors have characterized the graph varieties axiomatized by the transitive and the left or right self-distributive identities.\n\nContinuing this line of research, we determined the graph varieties axiomatized by certain groupoid identities that are of general interest in algebra, such as the medial, (left or right) semimedial, idempotent, unipotent, zeropotent, and alternative identities.\n\nThis is joint work with Chaowat Manyuen (Khon Kaen University).\nLocation : Bolyai Intézet, I. emelet, Riesz terem, Aradi Vértanúk tere 1., Szeged" ]

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[ "# How to calculate the number of waters of hydration?\n\nI have three questions about waters of hydration, the first two of which I'm really stuck on:\n\n1. A sample of cobalt(II) nitrate hydrate was heated to remove all the water of hydration. The hydrate was found to be $65.96\\%$ oxygen. Calculate the number of water molecules associated with each formula unit of cobalt(II) nitrate.\n\n2. Epsom salts is $\\ce{MgSO4.xH2O}$. The hydrate was found to contain $71.4\\%$ oxygen. Calculate the value of $x$.\n\n3. Zinc nitrate $\\ce{Zn(NO3)2.xH2O}$ contains $21.98\\%$ zinc by mass. What is the value of $x$?\n\nI have absolutely no idea for the first two questions.\n\nFor the third:\n\n1. Assume $\\pu{100 g}$ of material\n2. $\\pu{21.98 g}$ of Zn = $\\pu{0.3284 mol}$ of $\\ce{Zn}$\n\nI couldn't get any further than this.\n\n• Please add what you have attempted towards solving the problem into the body of your question. For more information, see the site's homework policy for how to ask homework questions. Thanks! – jonsca Jun 8 '14 at 5:45\n\nWhat is the % oxygen in $\\ce{MgSO4}$ when there is no water of hydration (x=0)? It would be given by $$\\mathrm{\\%~ oxygen~ =~ \\frac{(4 \\cdot atomic~ wgt~ O)}{(atomic~wgt~ Mg + atomic~ wgt~S + (4 \\cdot atomic~ wgt~ O))}}\\\\\\mathrm{=~\\frac{(4 \\cdot 16)}{(24_{.}31+32+(4 \\cdot 16))}}\\\\=53.2\\%$$ How would the equation change if we added one $\\ce{H2O}$ of hydration to the molecular formula? How about if we added 2 or 3?" ]

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[ "# Surface  tension mcq Pdf\n\n1. Surface Tension is defined as force per unit length of imaginary line drawn on free surtace of liquid at right angles on either side of line.\n2. The film at the top of liquid surface having thickness equal to molecular range is caled surface film.\n3. The potential energy per unit area of liquid surface is known as surface energy.\n4. For a liquid in contact with solid, angle between surface of solid and tangent drawn to the free surface of liquid at the point of contact measured from the side of liquid is called angle of contact of that solid liquid pair.\n5. Angle of contact is zero, when liquid wets solid completely.\n6. Angle of contact is acute, whin liquid wets the solids partially\n7. Angle of contact is obtuse, when liquid does not wet the solid.\n\nSurface tension MCQs questions and answers are provided here with answers and detailed explanation each question. These Surface tension MCQs are important from the NEET as well as various type of exam. By practicing our Mcq on surface tension pdf students will get acquainted with the key concepts which must be prepared to score high marks in the objective type questions in the any exam.\n\n## Surface Tension MCQs For  NEET\n\n1. The dimensional formula of surface tension is__________\n(a) [M L-¹ T-²]\n(b) [M L²T-²]\n(c) [M L T-¹]\n(d) [M L⁰ T-²]\n\n2. Rain drops are spherical in nature because of__________\n(a) capillarity\n(b) acceleration due to gravity\n(c) surface tension\n(d) viscosity\n\n3. Two drops of a liquid are merged to form a single drop. In this process________\n(a) energy is absorbed\n(b) energy 1s released\n(c)energy remains constant\n(d) energy becomes half\n\n4. If a surface is contaminated, then the surface tension________\n(a) increases\n(b) decreases\n(c) does not change\n(d) becomes zeroo\n\n5. Washing soap is used for cleaning the clothes, because______\n(a) it absorbs dirt\n(b) it increases the surface tension of the solution\n(c)it reduces the surface tension of the solution\n(d) it increases the viscosity of the liquid\n\n6. The phenomenon of surface tension is due to_______\n(b) gravitational force\n(c) intermolecular cohesive force\n(d) nuclear force\n\n7. If a detergent is dissolved in water, the surface tension of water_____\n(a) remains constant\n(b) increases\n(c) decreases\n(d) becomes zero\n\n8. Cohesion is maximum in_______\n(a) solids\n(b) liquids\n(c) gases\n(d) same in all\n\n9. If common salt is dissolved in water, then the S.T. of  Salt water is_______\n\n(a) increased\n\n(b) decreased\n\n(c) not changed\n\n(d) first decreases and then increases\n\n10. Water rises upto a height of 4 cm, in a capillary tube, if the tube is immersed vertically in water. What will be the length of water column in the capillary tube, if the tube is immersea in water, at an angle of 60° with the vertical?\n\n(a) 4 cm\n\n(b) 6 cmn\n\n(c) 8 cm\n\n(d) 2 cm\n\n11. The angle of contact between a glass capillary tube of length 10 cm and a liquid is 90°. If the capillary tube is dipped vertically in the liquid, then the liquid_________\n\n(a) will rise in the tube\n\n(b) will get depressed in the tube\n\n(c) will rise upto. 10 cm in the tube and will over flow\n\n(d) will neither rise nor fall in the tube\n\n12. At the boiling point of water, its surface tension_______\n\n(a) is infinite\n\n(b) is zero\n\n(c) Is the same as that at room temperature\n\n(d) is maximum\n\n13. A liquid is kept in a glass beaker. Which molecules of the liquid have the highest potential energy ?\n\n(a) molecules at the bottom of the breaker\n\n(b) molecules near the centre of the liquid\n\n(c)molecules lying at half the depth of the liquid and touching the walls of the beaker\n\n(d) molecules lying in the surface film\n\n14.. Surface tension of liquid is independent of the________\n\n(a) temperature of the liquid\n\n(b) area of the liquid surface\n\n(c) nature of the liquid\n\n(d) impurities present in the liquid\n\n15.. The angle of contact between a glass capillary tube of length 10 cm and a liquid is 90. If the capillary tube is dipped vertically in the liquid, then the liquid___________\n\n(a) will rise in the tube\n\n(b) will get depressed in the tube\n\n(c) will rise upto. 10 cm in the tube and will over flow\n\n(d) will neither rise nor fall in the tube\n\n16. At the boiling point of water, its surface tension_________\n\n(a) is infinite\n\n(b) is zero\n\n(c) 1s the same as that at room temperature\n\n(d) is maximum\n\n17. For tap water and clean glass, the angle of contact is________\n(a) 0°\n(b) 90°\n(c) 140°\n(d) 8°\n\n18. Choose the wrong statement from the following.\n(a) Small droplets of a liquid are spherical due to surface tension\n(b) Oil rises through the wick due to capillarity\n(c) In drinking the cold drinks through a straw, we use the phenomenon of capillarity\n(d) Gum is used to stick two surfaces. In this process we use the property of Adhesion\n\n19. The work done in blowing a soap bubble of radius R is W and that to a radius 3R is W2 The ratio of work done is_________\n(a) 1:3\n(b) 3:1\n(c) 1:9\n(d) 9:1\n\n20. When the angle of contact between a solid and a liquid is 90, then_______\n(a) Cohesive force > Adhesive force\n(b) Cohesive force < Adhesive force\n(c) Cohesive force = Adhesive force\n(d) Cohesive force >> Adhesive force\n\n21. For a liquid, which is rising in a capillary tube, the angle of contact is________\n(a) 900\n(b) 1800\n(c) Acute\n(d) Obtuse\n\n22. Out of the following, which one is not an example of capillary action______\n(a) ploughing of the field\n(b) absorption of ink in a blotting paper\n(c) floating of wood on the surface of water\n(d) rise of oil in the wick of a lamp\n\n23. A liquid is kept in a glass vessel. If the liquid solid adhesive force between the liquid and the vessel is very weak as compared to the cohesive force in the liquid, then the shape of the liquid surface near the solid should be_______\n(a) Convex\n(b) Concave\n(c) Almost vertical\n(d) Horizontal\n\n24. An imaginary sphere drawn around a molecule as centre, with a radius equal to the range of molecular attraction is_________\n\n(a) surface film\n\n(b) range of molecular attraction\n\n(c) sphere of influence\n\n25. Surface tension is equal to_________\n\na) free surface energy per unit area\n\nb) pressure per unit volume\n\nc) force per unit area\n\nd) none of these\n\n26. The potential energy of the molecules on the free surface of a liqud is ________\n\n(a) minimum\n\n(b) maximum\n\n(c) zero\n\n(d) infinity\n\n27. The unit of surtace tension is________\n\n(a) N-¹m-¹\n\n(b) Nm-¹\n\n(c)N-¹M\n\n(d) Nm\n\n28. A shaving brush is dipped in water. When it is taken out of water, the hairs in the brush cling together. This is because of__________\n\n(a) attraction between the hairs\n\n(b) surface tension\n\n(c) gravity\n\n(d) Viscosity\n\n29. When the temperature of a liquid is increased, the surface tension of the liquid_______\n\n(a) increases\n\n(b) decreases\n\n(c) does not change\n\n(d) sometime increases and sometime decreases\n\n30. 10 water drops merge to form a single drop. In this process________\n\n(a) energy is absorbed\n\n(b) energy is liberated\n\n(c) energy is neither absorbed nor liberated\n\n(d) total surface area increases\n\n31. An imaginary line is a drawn in a liquid surface. At whatcangle with the line, the surface tension acts?\n\n(a) 0\n\n(b) 45°\n\n(c) 180°\n\n(d) 90°\n\n32. Molecular forces are_______\n\n(a) Always repulsive\n\n(b) Always attractive\n\n(c) Sometimes attractive and sometimes repulsive depending upon the nature of the molecules\n\n(d) Attractive upto a certain distance and for larger distances they are repulsive\n\n33. A spherical mercury drop is broken into 1000 droplets, all of the same size. In this process, the physical quantity that does not change is its__________\n\n(a) surface energy" ]

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[ "# A Short History of Probability\n\nFrom Calculus, Volume II by Tom M. Apostol (2nd edition, John Wiley & Sons, 1969 ):\n\n\"A gambler's dispute in 1654 led to the creation of a mathematical theory of probability by two famous French mathematicians, Blaise Pascal and Pierre de Fermat. Antoine Gombaud, Chevalier de Méré, a French nobleman with an interest in gaming and gambling questions, called Pascal's attention to an apparent contradiction concerning a popular dice game. The game consisted in throwing a pair of dice 24 times; the problem was to decide whether or not to bet even money on the occurrence of at least one \"double six\" during the 24 throws. A seemingly well-established gambling rule led de Méré to believe that betting on a double six in 24 throws would be profitable, but his own calculations indicated just the opposite.\n\nThis problem and others posed by de Méré led to an exchange of letters between Pascal and Fermat in which the fundamental principles of probability theory were formulated for the first time. Although a few special problems on games of chance had been solved by some Italian mathematicians in the 15th and 16th centuries, no general theory was developed before this famous correspondence.\n\nThe Dutch scientist Christian Huygens, a teacher of Leibniz, learned of this correspondence and shortly thereafter (in 1657) published the first book on probability; entitled De Ratiociniis in Ludo Aleae, it was a treatise on problems associated with gambling. Because of the inherent appeal of games of chance, probability theory soon became popular, and the subject developed rapidly during the 18th century. The major contributors during this period were Jakob Bernoulli (1654-1705) and Abraham de Moivre (1667-1754).\n\nIn 1812 Pierre de Laplace (1749-1827) introduced a host of new ideas and mathematical techniques in his book, Théorie Analytique des Probabilités. Before Laplace, probability theory was solely concerned with developing a mathematical analysis of games of chance. Laplace applied probabilistic ideas to many scientific and practical problems. The theory of errors, actuarial mathematics, and statistical mechanics are examples of some of the important applications of probability theory developed in the l9th century.\n\nLike so many other branches of mathematics, the development of probability theory has been stimulated by the variety of its applications. Conversely, each advance in the theory has enlarged the scope of its influence. Mathematical statistics is one important branch of applied probability; other applications occur in such widely different fields as genetics, psychology, economics, and engineering. Many workers have contributed to the theory since Laplace's time; among the most important are Chebyshev, Markov, von Mises, and Kolmogorov.\n\nOne of the difficulties in developing a mathematical theory of probability has been to arrive at a definition of probability that is precise enough for use in mathematics, yet comprehensive enough to be applicable to a wide range of phenomena. The search for a widely acceptable definition took nearly three centuries and was marked by much controversy. The matter was finally resolved in the 20th century by treating probability theory on an axiomatic basis. In 1933 a monograph by a Russian mathematician A. Kolmogorov outlined an axiomatic approach that forms the basis for the modern theory. (Kolmogorov's monograph is available in English translation as Foundations of Probability Theory, Chelsea, New York, 1950.) Since then the ideas have been refined somewhat and probability theory is now part of a more general discipline known as measure theory.\"" ]

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[ "# Part 4: Checking for convergence\n\nBack to Part 3\n\nWe now illustrate the concept of burn-in. We will run a chain with a high starting value, and another with a low starting value.\n\n```dk.mcmc.l <- mcmcf(0.01, 20, 1e4, .12, 180)\ndk.mcmc.h <- mcmcf(0.4, 20, 1e4, .12, 180)\n\nplot(dk.mcmc.l\\$d, xlim = c(1,200), ylim = c(0,0.4), ty = \"l\")\nlines(dk.mcmc.h\\$d, col=\"red\")\n\n# We use the low chain to calculate the mean\n# and sd of d. We could have used the high chain\n# as well.\nmean.d <- mean(dk.mcmc.l\\$d)\nsd.d <- sd(dk.mcmc.l\\$d)\nabline(h = mean.d + 2 * c(-sd.d, sd.d), lty = 2)\n```\n\nThe horizontal dashed lines indicate approximately the 95% CI. Notice how the chains move from either the high or low starting values towards the stationary phase (the area within the dashed lines). The area before the chain reaches stationarity is the burn-in.", null, "The simplest way to check for convergence is to run the MCMC chain several times (at least twice) with over-dispersed starting values. Then one can compare the outputs of the various chains and check how similar they are. If the chains have converged to the target posterior distribution, then they will have similar (almost identical) histograms, posterior means, 95% credibility intervals, etc.\n\nWe now run the efficient and inefficient chains again, but with different starting values (i.e. `d.init` and `k.init`)\n\n```# Efficient chain (good proposal step sizes)\ndk.mcmc.b <- mcmcf(0.05, 5, 1e4, .12, 180)\n# Inefficient chain (bad proposal step sizes)\ndk.mcmc3.b <- mcmcf(0.05, 5, 1e4, 3, 5)\n```\n\nAnd we can now plot and compare the histograms:\n\n```par(mfrow=c(1,2))\nbks <- seq(from=0, to=150, by=1)\n\nhist(dk.mcmc.b\\$k, prob=TRUE, breaks=bks, border=NA,\ncol=rgb(0, 0, 1, .5), las=1, xlab=\"kappa\",\nxlim=c(0,100), ylim=c(0,.055))\n\nhist(dk.mcmc\\$k, prob=TRUE, breaks=bks, border=NA,\n\nhist(dk.mcmc3.b\\$k, prob=TRUE, breaks=bks, border=NA,\ncol=rgb(0, 0, 1, .5), las=1, xlab=\"kappa\",\nxlim=c(0,100), ylim=c(0,.055))\n\nhist(dk.mcmc3\\$k, prob=TRUE, breaks=bks, border=NA,\n```\n\nEfficient chains converge quickly, and histograms overlap well. On the other hand, inefficient chains need to be run for a longer time to converge, here they were not run long enough, so that the histograms do not overlap as well. Your histograms may look slightly different due to the stochastic nature of the MCMC algorithm.", null, "We can also calculate the posterior means:\n\n```# posterior means (similar for efficient chains):\nmean(dk.mcmc\\$d); mean(dk.mcmc.b\\$d)\nmean(dk.mcmc\\$k); mean(dk.mcmc.b\\$k)\n\n# posterior means (not so similar for the inefficient chains):\nmean(dk.mcmc3\\$d); mean(dk.mcmc3.b\\$d)\nmean(dk.mcmc3\\$k); mean(dk.mcmc3.b\\$k)\n```\n\nThe variance of the estimated posterior mean is the sample variance of the MCMC sample (ignoring the auto-correlation) divided by the efficiency times 1/(sample size). The standard error of the estimated posterior mean is the square-root of the variance:\n\n```# efficient chain, standard error of the means\nsqrt(1/1e4 * var(dk.mcmc\\$d) / 0.23) # roughly 2.5e-4\nsqrt(1/1e4 * var(dk.mcmc\\$k) / 0.20) # roughly 0.2\n\n# inefficient chain, standard error of the means\nsqrt(1/1e4 * var(dk.mcmc3\\$d) / 0.015) # roughly 9.7e-4\nsqrt(1/1e4 * var(dk.mcmc3\\$k) / 0.003) # roughly 1.6\n```\n\nThus, for the efficient chain on", null, "$d$, I got an estimated posterior mean of 0.105, which is within ±2 times 2.5e-4 of the true posterior mean. Note for the inefficient chains, the standard errors are about four times and eight times those of the efficient chains.\n\nWe can also plot densities (smoothed histograms) for the efficient and inefficient chains:\n\n```par(mfrow=c(1,2)); adj <- 1.5\n# Efficient chains:\nxlim=c(0, 100), ylim=c(0, .05), xaxs=\"i\", yaxs=\"i\")\n\n# Inefficient chains:\nxlim=c(0, 100), ylim=c(0, .05), xaxs=\"i\", yaxs=\"i\")\n```\n\nThe resulting R plots are shown below.", null, "A useful measure of convergence is the potential scale reduction statistic of Gelman and Rubin (also known as Gelman and Rubin convergence diagnostic). The statistic compares the within-chain variance and the among-chain variance for the posterior mean of a parameter. When the value of the statistic is close to 1, convergence has been achieved. Note that the diagnostic may fail (i.e. give a misleading reassurance of convergence) for strongly multi-modal posteriors. To avoid this it is important to run more than two chains with over-dispersed starting values. The R package `coda` for MCMC diagnostics, has a function to calculate the convergence diagnostic, `gelman.diag`. If you have the package installed, you may want to go ahead and calculate the diagnostic for the chains above.\n\nChapter 7 in Yang (2014, Molecular Evolution: A Statistical Approach) provides a good introduction to Bayesian computation including Bayesian MCMC, proposal densities, efficiency, convergence diagnostics, etc.\n\nWe hope you enjoyed the tutorial. If you have questions, comments or if you spot any errors, please post a message below or sends us an email!\n\nFabricia, Mario and Ziheng." ]

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[ "We have been discussing design of a processor where\nexecution of each instruction is divided into multiple clock cycles. We have seen how the\ndata path is designed and today we will look at the controller aspects how such a data\npath could be controlled. In the sequence of lectures on designing of\nprocessors we started with very simple design where everything about an instruction was\ndone in a single clock cycle. We notice the problems with performance and there were other\nissues so we have moved to a different style where the instruction is divided into multiple\nclock cycles. So we will look at what are the actions which are done in different clock\ncycles for various instructions and then by putting these sequences of actions together\nwe will try to build the flow of control for carrying out these instructions. We will then try to identify what control\nsignals are required to control this data path in each of the control steps; for doing\nso we will group the control signals into groups and we will define some meaningful\noperations called micro operations. So each instruction would be viewed as a set of micro\noperations which are done either together or in sequence and that will somewhat simplify\nestablishing a relationship between control states and the signal values. Finally we will\nsee how control states transit from one to other and that will complete the design of\ncontrol part. So this is the starting point, we had arrived\nat this data path last time where the key resources are memory, register file and ALU\nand we have tried to use these as far as possible so that there is a maximum utilization of\nall the resources and you would also recall that all intermediate results we try to store\nin some registers. So, for example, when instruction is brought out from memory it is stored in\ninstruction register and when data is read out from memory it is stored in data register.\nSimilarly, operand from register file is brought out into registers A and B and the results\nof ALU operation are kept in register called Res short for result. So now we need to control all these components;\nall the registers may not change their states in every cycle. So each register will have\na signal indicating when do we write something into those registers. We have controls for\nthe multiplexer as usual and similarly controls for ALU, register file and memory. But since\nwe have new components like registers, multiplexer also has multiplexers also have either changed\nin their size or in their organization we will have to redefine some of these control\nsignals. So, first of all let us get back to the instruction\nand see how those instructions are divided into operations which were done in different\ncycles. So what activity is done in which cycle that needs to be clearly recorded before\nwe can start working with the control signals? So, starting with R class instruction in the\nfirst cycle we read one word from memory into instruction register that forms the instruction\naddress of memory that comes from PC and at the same time PC takes on the new value. So\nboth these operations are done concurrently within first cycle. In the next cycle we read\nthe operand from the register file and these two are brought into A and B. the addresses\nof register file excuse me are provided by the instruction and the relevant fields here\nare bit 21 to 25 and bit 16 to 20 for the second operand. So this corresponds to….\nthe first one corresponds to RS, the second one corresponds to RT. Once again what we meant by putting these\ntwo operations in a single box was that both of these are done simultaneously within a\nclock cycle number 2. The next clock cycle for these instructions\nwould see the actual operation being performed by the ALU. I have written in a generic sense\nA op B where op is the operation as would be guided by the function field of the instruction\nIR (0 to 5) and the last cycle will involve transferring this result to the register file;\nthe address come from bits 11 to 15 which corresponds to RD or the destination register.\nThis is how things have been divided and we have made a careful choice of what gets done\nconcurrently and what gets done in sequence. The second instruction we will take is store\nword and the first cycle involves access to memory to fetch the instruction and updation\nof program counter. The second cycle involves bringing the registers out we need again to\naccess two registers: one which will participate in address calculation and second will carry\nthe value to be written into the memory. So with same fields of the register file the\nsame fields of instruction register file is accessed and registers are values are brought\nout in A and B. In the next cycle we calculate the address by adding offset coming from bit\n0 to 15 of instruction with sign extension to A and this value is temporarily kept in\nthis register called result. In the last cycle we will see an access to\nmemory where contents of Res will be used as an address and the data to be written into\nmemory is B and memory write is performed. So once again we require four cycles. The next is load word instruction. The first cycle again is similar. The second\ncycle involves reading one register from register file. Here we need to access only A because\nthe second address corresponds to the destination here. What we read from memory will be stored\nin register RT so we need to read only one register and in the third cycle we calculate\naddress in the same manner. In the fourth cycle memory access is performed; the data\nis read from memory, address is carried in Res and the data which is read is brought\ninto DR. In the last cycle we store the DR value into register file, the address comes\nfrom RT. So this is the sequence of five cycles which complete load word instruction. Then we move on to beq; first cycle is same.\nin the next cycle we are doing lot of work. one is to access A and B the two operands\nwhich need to be compared and we also keep the result ready for the target address in\ncase branch has to be carried out because the ALU is free in this cycle. In the next\ncycle we are going to use ALU for comparison. So in this cycle it is free. We calculate\nthis address but do not keep it in do not transfer it to PC immediately. We keep it\nin Res; the condition is yet to be checked. So, after checking the condition we will either\ntransfer it or not transfer it. That is cycle three where the ALU will compare A and B the\ntwo operands which were brought out from register file and if the condition holds then this\nnew address is transferred to PC. So this requires only three steps. Lastly we will look at the jump instruction.\nagain, the first cycle is same and in the second cycle we compose this address of the\nnext instruction by taking bits from PC and IR with I missed out…. oh no, there is no\nsign extension here; this with 2 bit shift is concatenated with 4 bits of PC and then\ntransferred to PC. So this is done in two cycles. Now you recall that the impression\nI might have given earlier was that this can be done in a single cycle because only resource\nwhich is time consuming which is required here is accessing the instruction after that\nthere is no memory access required, no ALU operation required so the total delay which\nwe always use to think for this was max of t i and t plus. But why are we occupying two cycles here.\nThe reason for that is we want to do this operation sequentially. The 4 bits we need\nto pick up from PC part is actually after PC has been incremented. So, after PC is incremented\none cycle is over. In the first cycle we are incrementing PC and the result is put back\nin PC. So those 4 bits are picked up in the next cycle and put together with the instruction.\nAlso you would notice that it takes one cycle to fetch the instruction. So although what\nis being done here is not a time consuming activity but since it has to be sequenced\nafter that it is occupying additional cycle. So in a in a single cycle approach what was\nroughly taking equivalent of the memory access time or addition time one of the two now it\nis taking more or less twice of that. This is just to recall that in the single\ncycle approach the data path was like this that we did instruction access and PC update\nand directly we picked up the bits and formed the address and we said that the time required\nis max of t plus and t i. So now if we have decided one clock cycle to be let us say something\nwhich encompasses these we are still requiring two cycles because of the need of sequencing\nthings. These were the timings we had conceived earlier\nand we were actually imagining the time for jump to be just this much. basically first\nclock period will accommodate t plus and t i would be used but we are doing something\nnow; a very simple activity but now since the time is quantized discretized in terms\nof the clock period we need to go beyond this and we cannot use anything less than a cycle.\nWe use two cycles for this instruction. Is that point clear why we have to go to two\ncycles. Actually as we will proceed further we will\nchange things even for little worse but that becomes essential. Now, we have seen the division\nof each instruction in two cycles; the division of activity of each instruction in two cycles\nseparately and once again we need to put things together to form overall flow of control.\nSo these are the five instructions or groups and we will put their actions one after another\nin the same picture so that we have a global view of the whole thing. Now I have just, to accommodate everything\non a single screen, I have omitted some pieces of text which are more a matter of detail\nbut what I have tried to retain is all the destinations of the operation wherever the\nresults are going and the main resources which are being used. So, for the first cycle in\nall cases you would see that there is an existing memory I have omitted some details here, there\nis updation of PC, excess of values in the register file performing this arithmetic or\nlogical operations, storing in register file, storing in memory, reading from memory and\nso on. So you can see that the essence of all these the skeleton of all the operations\nhas been captured here. Now the task is to put these together. We\nhave different instructions taking different cycles. You would notice some commonality\nsomething is common and that is where we can merge but in the flow of control at some point\nwe would need to branch because different instructions require different actions. So\nthe first part is; first cycle is apparently common to all of them. So we could start always\nwith that action and a common state for all the instructions and once things start differing\nwe branch to different states. Different boxes will correspond to different state in which\nthe controller would be and given a given a state of the controller we would know clearly\nwhat actions are to be carried out in that particular state. So what we are trying to\narrive at is some sort of state transition diagram which would describe the control. At the moment we have five different chains\nor five different sequences but we want to have a single graph which indicates how one\nmoves from one state to other state and what action is required in each state. So, obviously\nthese can be merged. The first cycle seems to be carrying out same activity and we can\nput we can merge this. Therefore, after merging we have a single\nstate and then a point where we are branching. Now, another important thing which needs to\nbe noticed at this stage is we are contemplating a bifurcation here; we are splitting from\nthis point onwards after first but this requires us to look at the opcode. Before we can split\nwe need to see what the opcode is and which instruction or instruction group it is. But\nyou can start looking at opcode only after you have fetched the instruction and brought\ninto IR. So now this is available, therefore instruction is available to you from second\ncycle on onwards and therefore you can start looking at opcode from the second cycle onwards. Typically in a complex design this decoding\nof opcode or understanding which instruction it is could also be a time consuming activity.\nThe circuit which is going to identify a particular instruction would typically be allowed a full\ncycle to do it. Of course we do not need that but still the opcode decoding will have to\ntake place in the second cycle although we may not have the second cycle fully occupied.\nTherefore practically this bifurcation or split must occur after second cycle because\nif you have to go from this state to one of these states we should know what the instruction\nis here but instruction is known only after this first step is completed. So it is only\nafter second step we can branch off to different chains and do special action necessary for\nvarious instructions. Therefore second step will also have to be common. Now let us see what is required to make it\ncommon. One thing is that not all instructions are trying to read two values from register\nfile. Here you need two values, two values; one: here you are reading two values, here\nyou are reading none. Is there any harm if all instructions read two values; they may\nuse it or may not use it. The answer is there is no harm; it does not cause any problem\nin the functionality of the instruction; it might you might consume some energy in doing\nso but let us keep that aside and agree to fetch both the values in all the instructions\nin an attempt to come up with a common action for the second cycle. The other thing you notice here are that address\nis being calculated here which may be useful for branch instruction and this value is kept\nin register called result. So what if we repeat this also in all the instructions. Once again\nALU is free we are not doing anything with it so if we occupy ALU in an activity which\nwe may discard later on still there is no harm apart from energy consumption. Also,\nthe result register is not holding any value which will get overwritten here. So what we\nwill do is let us try to make this as the common action for all the instructions; the\nonly trouble comes with jump here because jump requires that we transfer a new value\nto PC but that we cannot do for all the instructions; that would mean that the next instruction\nafter every instruction a jump will be carried out. Therefore we need to postpone this to\nthe third cycle. We will do a common action in the second cycle which is same as the beq\naction. For all other instructions there may be some superfluous or unnecessary activity\nwhich may get discarded but there is no harm. The only harm which is occurring is that this\naction PC getting a new value is getting postponed. This is the picture with a common decoding\ncycle. This cycle is actually often referred to as decoding cycle or operand fetch cycle.\nNow, this is this jump instruction has taken another hit here and we are eventually using\nthree cycles for it. But please remember that this is an instruction whose frequency of\noccurrence is comparatively much much lower than others and therefore the overall impact\nof this loss in the performance will be negligible so we do not really mind it. And now we have\na very clean situation that there are two common cycles. By the end first cycle we know\nwhat instruction it is and we are ready for the next value of PC. In the second cycle,\nafter the end of second cycle our operands, if we need, are ready in A and B and the branch\naddress is ready in Res if we need later on and then we can move on to one of these separate\nbranches. So, two further cycles are required for R class, two for store word, three for\nload word, one for beq and one for jump and with that all the instructions will be over. So now this is the cycle, this is a broader\ncycle which needs to be repeated over and over again. So therefore, after these last\nstates in the chain we come back to this. Now, as far as controller is concerned it\nis a small finite state machine with as you can see these about ten or so states and it\nkeeps on cycling through these. So here you do the fetch, decode, you come to know what\ninstruction is, follow one of these paths and then you are ready for the next instruction.\nSo, as long as power is on on the processor it will keep on going through this overall\nthing….. so this whole thing is called instruction cycle and within this you have clock cycle. So an instruction cycle would require four\nclock cycles or five clock cycles or three clock cycles as the case is that is in this\nparticular design but there are of course machines where this range of number of clock\ncycles could be much much varied. Now we need to worry about what action we\nperform in each of the states. So these are the states but before that there is another\nsmall improvement possibility here which we will notice is that load word and store word\nhave one more cycle in common where the address is being calculated and we can actually merge\nthat and as far as load store are concerned we keep them together up to the third cycle\nand then bifurcate into load and store so that reduces number of control states by one\nmore. So total now I have how many states 1 2 3\n4 5 6 7 8 9 and 10 now there are total of ten states and we will proceed further by\nlooking at now the control signals. So this is back to the same data path. We\nhave now roughly seen how we need to exercise these and what needs to be done for various\ninstructions in different cycles; so that picture has been made very clear now. We get\nback to each of the components which require a control and try to understand how we need\nto control it. So, for each of the register I have indicated a control signal. So, for\nPC PC write I am abbreviating as pw. This is the IR write IW, DR write DW and then A\nand B have their AW and BW signals and Res write is ReW. So there are signals control\nsignals which will indicate whether a particular register changes its state in a cycle or not\nand in which cycle the state is changed we will know from this flow chart. We will always\nrefer to, now, this flow chart or this state transition diagram and then decide the values. Now we have also 1 2 3 4 5 and 6 multiplexers\neach one requires some controls so control for this multiplexer decides whether we are\naccessing instruction or data and accordingly the address comes from different sources.\nThis multiplexer decides whether we are writing into rt or rd, this decides what gets written\ninto register file we are calling it M2R memory to register or it is ALU out to the register.\nThe name of this signal is same Rdst we had earlier. this multiplexer is controlled by\na signal called A source 1, this is controlled by A source 2 and we had earlier two registers\nand two multiplexers which were handling the next PC value; now it is the single multiplexer\nwith three inputs and the control signal is labeled as p source or PC source. The memory\nregister file and ALU have their usual control signals same as what we had in the single\ncycle data path. Now you would notice that there are many more\ncontrol signals as compared to the single cycle data path we had. So I will not try\nto build the table exhaustively for all of these; what we will do is we will group the\nrelated control signals together and also identify the meaningful operation which we\ncall as micro operations. For example, PC plus p PC plus 4 going to PC will be considered\nas a micro operation; it is an understandable action within itself and it will affect some\nof the control signals in the data path. So we will be grouping the signals according\nto our logical needs and then try to look at things group-wise so that will simplify\nthe matter substantially. So first we talk of a group of signals called\nPC group which are related to PC; program counter and its address. So I will build a\ntable where I will list micro operations which are related to PC and the signals which are\nrelated to PC. So the signal you can see right now; one is Psrc the last multiplexer which\nis being controlled by this and the write signal for PC I have split into I have used\ntwo signals PWu and PWc; PC write unconditional and PC write conditional. you would notice\nthat in some micro operations like PC gets PC plus 4 we are writing it unconditionally\nwhereas there was an operation like this where we are writing into PC with some condition.\nSo this state will generate a signal which I am calling as PWc conditional and a state\nlike this or like this will generate a signal which I am going to call PWu unconditionally.\nSo the signal PW which is going here would be derived out of these two; I will explain\nthat in a moment. But let us see different micro operations and the signals which they\nimply. So, for doing PC gets PC plus 4 I make PC\nPWu as 1, this then I do nt need to care and the source I am selecting is 1 so three things\nare going into this multiplexer if you recall is output of ALU directly, output of ALU through\nRes register then the address which is for the jump instruction. So let me just go back\nand check if I have indicated the correct source here. So these are three inputs to this multiplexer:\nafter the register, before the register and this jump address. So recall that we will\nwrite output of ALU directly into PC when we are doing PC plus 4 so we do not bring\nregister into picture. But the target address for a branch we are temporarily keeping into\nthis we are not directly transferring to PC and therefore when we transfer to PC we will\ntake from output of the register therefore both paths have been provided. One more thing which we should see here is\nthe way z is going to be used. In the single cycle design we used z directly to control\na multiplexer through AND gate so there was a signal coming from controller that was ANDed\nwith z and we controlled a multiplexer. So basically we were making a choice between\nPC plus 4 and PC plus 4 plus offset one of the two going to the PC. But now things are\nhandled little differently because PC plus 4 is sent to PC unconditionally in the first\ncycle. In the third cycle the choice is either to\ntransfer the new address or not to do anything. Therefore, effect of z will be brought into\nthe way PW is being generated. So, in the first cycle we will have output of ALU directly\nthrough this multiplexer going into this, in the second cycle the address is calculated\nand kept in Res and in the third cycle we will bring this out here without looking at\nz but we will look at z and decide whether to transfer it or not. So you will see that\nz will come into picture when I define how PW is derived from PWu and PWc. This is next micro operation where we conditionally\ntransfer an address to PC. Here I activate this signal and do not activate this signal\nbecause source is 0 which means this value which is in Res is being taken and the third\ncase is that the jump address goes to the PC. So again this is unconditional, so PW\nu is 1, PW c is x and source is 2. So in these tables I am going to write 0 1 2 3 etc when\nthe signal takes multiple values more than two values. But actually you can think of\nthe binary code of these going to the components and not this decimal value. So, apart from these activities I would also\nlike to define what is the default value of these signals. When I am not doing any of\nthese what should I feed to these particular points so the default value should keep everything\ninactive. Therefore, both write signals are 0 and then of course once that is the case\nthe source does not matter so I put an x and here is how PW is derived from Pwu and PWc.\nSo, if PWu is 1 PW becomes 1 irrespective of what is that we have here. But when PWu\nis 0 and PWc is 1 then it is Z which dictates what we get here. If both are 0 then again\nZ gets ignored. So basically I need one AND gate and one OR gate so the controller will\nproduce these two signals and with the two gates I will derive PW which gets connected\nto the PC. Alright? Is that clear? Any question about that? Yeah. This is the address which is being formed\nfor jump instruction and in one of those cycles…… right now we are not worrying in which cycle\nwhat is happening but all we are worrying is that given an action like this to be performed\nhow do we control the data path. So, to make this happen we need to make the unconditional\nwrite signal as 1, we do not care what PWc is because you see when PWu is 1 the values\nhere do not matter the result here is going to be 1 it is a sort of overriding signal.\nBut on the other hand, when PWc when we want to activate PWc we have to make sure that\nthis is 0 because this will otherwise suppress that. So PWc is don’t care and we need to select\nthe correct value to go to PC. So, in all these operations, notice that PC is the destination\nbut the sources are different. There are three different sources; and the multiplexer is\nselecting the sources. This is 0 sorry this corresponds to 0 input of multiplexer, 1 input\nof multiplexer and 2 input of multiplexer this is all the things that have been connected\nin the data path so that takes care of what the value is being transferred and these two\nsignals are taking care of whether this transfer is taking place conditionally or unconditionally\nand that default. Hence, there are many control states where\nwe are not changing the value of PC and there we need to keep both these as 0 and the value\nof P source does not matter. So, I would give some names to these which will make things\nconvenient in subsequent discussions. The first one I will call as PC increment, this\nis branch and this is jump and this I call no op or no operation. So just some names\nfor convenience I have assigned to these. Now let us look at operations which revolve\naround the memory. The relevant signals here are memory write\nand read control signals, this I or D instruction or data this signal decides what is the source\nof address for the memory and these two signals decide where we keep the things which is coming\nout of the memory when we are reading from memory. So, this controls writing into IR\nregister, this controls writing into DR register and one operation is fetching the instruction.\nSo we are not writing here we are reading; it is an instruction and the multiplexer code\nfor that is 0, the value which is being read is written in IR register so we keep this\n1 keep that 0; next is getting data from memory so again we are reading we are not writing,\nit is data so that makes it 1, we are not storing to IR we are storing into DR so that\nis 1 and that is 0; next is writing into memory so now we make write as 1 and read as 0 once\nagain it is data so I or D is 1 and we are not storing anything into IR or DR so both\nthese are 0. You also might hear it might occur to you\nthat there is some kind of redundancy in these signals. It may appear that you may be you\nto derive one of these from others or one from one of these from more than one of others\nso there are yes many possibilities. You might even notice that some signals can be totally\nomitted. For example, IW and MR seem to be identical. So we can make such observation\nand simplify the controller design. So it is indeed possible but we will just limit\nat this and not get into those details. Finally the default here is to keep a write\nread both signals low, also the register load signals low and then this does not really\nmatter. Some convenient name for these; this is fetch, this is memory read, memory write\nand no operation. So now, in later discussion this….. once we say fetch it would mean\nthat this is the operation we are performing and this is the set of values we have given\nto control signals of this particular group. The third group is register file group and\nhere we are talking of writing a signal RW register file write, Rdst this decides where\nthe address comes from when you are writing; whether it is RT or RD, this tells where the\ndata to be written is coming from whether it is coming from memory or from ALU and whether\nwe are writing into A and B register. So reading RS into A we make RW 0, Rdst don’t care and\n2R don’t care, these Rdst and M2R will be relevant only when RW is 1 so when you are\nreading these do not matter we give a write signal to it and B does not require….. similarly,\nreading into B is similar except that we write into we make BW as 1. Now here it is writing into register file\nso we will make the write signal 1, we need to define Rdst so it will be 1 in this case,\nM2R is 0 and we are not modifying A and B. This is writing DR into register file so again\nRW is 1 but this has a different value, Rdst is different because the address is coming\nfrom different points and also the values being written are different. AW BW both are\n0 and default is to keep RW 0 and to keep so we do not make any change in the state\nhere so all write signals are kept 0. Once again you would notice that these two\nsignals Rdst and M2R are complementary of each other so one could reduce the signals.\nNames for these are: rs2A, rt2B, res2rd, mem2rt and no operation. So these five names I am\ngoing to use later on. Finally, we have the ALU group. The signals\nare opc; it is a 3 bit signal sorry this is a 2 bit value which we derived from the opcode.\nWhat goes to ALU finally is a 3-bit signal which will look at opc and the function bits\nand that part of the circuitry will be totally unchanged. Then we have the multiplexer control\nsignals A source 1 and A source 2and signal which controls writing into result register. So a micro operation which we saw earlier\nis appearing here also because doing PC gets PC plus 4 influences the PC group of signals\nas well as the ALU group of signals because we need to ensure that addition is done here.\nWe will look at opc later; look at the source and get back to this diagram. A source 1 has a choice of PC and A source\n2 has a choice of B 4, this is offset for load store and offset for branch. So these\nare the four possibilities here and two possibilities there. So let me put all these together actually. So, A source 1 has value 0 or 1; 0 for pc\nhere and here and 1 for A, A source 2 has four possibilities, 0 for B that is here 1\nfor store, 2 for load store offset and 3 for branch offset. For this operation again we\nare comparing A and B so it is like this as far as source are concerned 1 and 0 this and\nthis. This is indicating whether we are writing into Res or not. So in these three steps we\nare writing. Here we are not writing into Res, here we are not writing into Res and\nthis opc encoding is same as what we had done earlier. For those instructions where we have\nto simply perform addition without looking at anything else we make it 0 so we had done\nthis for load store instruction and now even for these address calculations we will use\n0 because our logic would be that whenever there is a 0 here that ALU controller would\nensure that ALU performs the addition and one would mean that it performs subtraction\nunconditionally and two it would mean that we look at the function bits. So same encoding\nis used and accordingly we fill this up. The last case is the default where these are\ndon’t care does not matter what ALU does as long as the result is not written anywhere\nso we ensure that ReW is 0 and ALU may do something which is don’t care. So now with\nthis done we have seen the relationship between micro operations which we picked out of that\nflow chart and how they assert various control signals; what values they imply for various\ncontrol signals. The names of these are PC increment, arithmetic, memory address, PC\naddress, branch and no operation. So now I can tabulate these. Before that let\nme redraw the diagram with these new symbols put in the boxes. So instead of those assignment\nstatements I have replaced those with the new micro operation symbol which I have described\nin previous few slides. So for example, in the first cycle I am doing fetch and PC increment,\nin the second cycle I am doing rs2A and rt2B and Paddr; all these three are done concurrently\nwithin the second cycle and so on. So all these signals have been all these micro operations\nhave been put into appropriate states. Also, these states could be numbered: cs0 to cs9. Now I need basically two tables: one table\nwill define for a given control state what micro operations I perform which will also\ndirectly imply what are the signal values for various control signals that is one table.\nNext table would define that given a control state what is the next control state and this\ntransition may be conditional; it will depend upon the way I am bifurcating so it will depend\nupon the opcode. So let us see both these tables one by one. So first, we will see relationship\nbetween control states and signal values. So, I am not listing signal values here; I\nam only listing the micro operation in the particular group. So these are the four groups\nI identified: PC group, Memory group, RF group and ALU group and in cs0 the operation I need\nto perform is PC increment this actually shows up here also as I mentioned it requires to\ncontrol signals at both the ends and in memory group the operation is fetch. In cs1 I am fetching these operands and calculating\nbranch address. In cs2 this is the first first distinct state for R class instruction so\nI perform the arithmetic operation here so the R class instructions go through cs0\ncs1 cs2 and cs3. In cs3 the value gets written. And all those which I want to keep inactive\nyou will find there is no offsetting there. So here the result is being written into register\nfile. Then c4 is the common state for load store\nhere memory address gets calculated, then c5 completes the store operation we have a\nmemory right here, c6 performs memory read operation and c7 performs transfer of data\nfrom memory to register file, c8 completes the branch instruction so again a branch operation\nbranch micro operation shows up in PC group as well as in ALU group because it requires\na comparison here and it needs to change the state of PC so both these get influenced.\nAnd c9 completes the jump instruction so it is influencing this pc group. What I can do but I will not do that here\nis that each of these micro operation symbol I replace it by a bit vector that bit vector\ndefines the relevant control signals. Once I have that I also encode these nine states\nin binary so you can use 4 bits to encode this state. I form a truth table where this\nis the input, the code of the state is the input and the bit vector which I write here\nare the outputs. So these are control signals which go from controller to the data path. The second part of the control design is how\ncontrol state transitions take place. So, again for each of the states let me fill it\nup for each of the state I am defining the next state but the next state could be different\ndepending upon what is the opcode value. So from cs0 I am unconditionally going to cs1\nin all cases. From cs1 one goes to cs2 or cs4 for load store it is common again, for\nbranch it is cs8, for jump it is cs9. From cs2 which is for R class instruction\ngo to cs3 and these conditions will not occur. Once you have gone to cs2 you know that it\nis R class instruction and others are not relevant. So you can see that R class instruction\ngoes through cs0 1 2 and 3 and back to cs0. For store instruction we start with cs0 go\nthrough cs1 cs4 cs5 and then cs0. For load it is cs0 cs1 cs4 then cs6 cs7 and then cs0.\nFor branch it is 0 1 and 8 and then 0 for jump it is 0 1 9 and then 0. So now what kinds of truth tables I have here?\nHere the opcode and 4 bit of the control state these form the input of one combinational\nblock and the next state value or the 4 bits which encode the next state are the output.\nSo if you implement these two tables the tables shown here and the tables shown in the previous\nslide namely this one you can use two PLAs to implement this so these two PLAs plus one\nregister holding the control state 4-bit register hold in the control state will form the controller.\nI will draw that picture next time and look at some alternatives. So today we will stop\nat this. Let me just summarize that we saw how instructions\nget divided into sequences of micro operations; how we group the control operations control\nsignals and then define the relationship between micro operation and control signals then we\nassociated control states with the micro operations and we also identified control state transitions,\nthank you.\n\nLecture – 21 Processor Design – Control for Multi Cycle\n\n### One thought on “Lecture – 21 Processor Design – Control for Multi Cycle”\n\n•", null, "August 2, 2018 at 10:49 pm" ]

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[ "# Integration by parts\n\nUp a level : Integrals\nPrevious page : Integration of definite integrals using substitution\nNext page : One function is in the form x to the power of n and the other can be integrated at least n times", null, "Integration by parts\n\nIn the case the integrand is a product of two functions we may be able to solve it by substitution, as shown in the previous pages. But what if does not work? One way one could try is a kind if inverse of the differential product rule. Say u and v are functions of x, then we have that", null, "$\\frac{d}{{dx}}\\left( {uv} \\right) = \\frac{{du}}{{dx}}v + u\\frac{{dv}}{{dx}}$\n\nor", null, "$u\\frac{{dv}}{{dx}} = \\frac{d}{{dx}}\\left( {uv} \\right) - \\frac{{du}}{{dx}}v$\n\nIntegrating both sides gives us", null, "$\\int {u\\frac{{dv}}{{dx}}dx} = \\int {\\frac{d}{{dx}}\\left( {uv} \\right)} - \\int {\\frac{{du}}{{dx}}v\\;dx}$\n\nor", null, "$\\int {u\\frac{{dv}}{{dx}}dx} = uv - \\int {\\frac{{du}}{{dx}}v\\;dx}$\n\nThis means that we can transform one integral to another that may be solvable. The goal is thus to find", null, "$\\int {f\\left( x \\right)g\\left( x \\right)dx}$\n\nExample: First an example to make some sense of this. Let us try to find", null, "$\\int {x{e^x}} dx$\n\nWe cannot use the substitution method since none of the inner derivatives of the other function is the derivative of the other function. But let us instead set", null, "$\\left| {\\begin{array}{*{20}{c}} {u = x} \\\\ {\\frac{{dv}}{{dx}} = {e^x}}\\end{array}} \\right.$\n\nand thus", null, "$\\left| {\\begin{array}{*{20}{c}} {\\frac{{du}}{{dx}} = 1} \\\\ {v = \\int {{e^x}dx} = {e^x}}\\end{array}} \\right.$\n\nThat means that", null, "$\\begin{gathered} \\int {\\underbrace x_u\\underbrace {{e^x}}_{dv/dx}} dx = \\underbrace x_u\\underbrace {{e^x}}_v - \\int {\\underbrace 1_{du/dx}} \\underbrace {{e^x}}_vdx \\hfill \\\\ \\quad = x{e^x} - {e^x} + C = {e^x}(x - 1) + C \\hfill \\\\ \\end{gathered}$\n\nWe can test this by differentiating:", null, "$\\frac{d}{{dx}}({e^x}(x - 1) + C) = {e^x}(x - 1) + {e^x}1 = {e^x}x - {e^x} + {e^x} = x{e^x}$\n\nOther ways to write the rule\n\nYou might find this rule written as", null, "$\\int {fg\\;dx} = Fg - \\int {Fg'\\;dx}$\n\nHere the two functions have swapped order.  Using this way of writing it might be a shortcut.", null, "$\\int {\\underbrace x_g\\underbrace {{e^x}}_{\\text{f}}} dx = {e^x}x - \\int {{e^x} \\cdot 1\\;dx} = {e^x}x + {e^x} + C$\n\nYou might also see it written as", null, "$\\int {u\\;dv} = uv - \\int {v\\;du}$\n\nHere we have that", null, "$dv = \\frac{{dv}}{{dx}}dx$\n\nIn the next few pages we will look at some cases.", null, "Up a level : Integrals\nPrevious page : Integration of definite integrals using substitution\nNext page : One function is in the form x to the power of n and the other can be integrated at least n times", null, "Last modified: May 20, 2018 @ 13:45" ]

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[ "03 LAB D_ Linear Expansion Coefficient - Lab Course 103A-Section 006 Physics Laboratory Report Lab number and Title LAB D Linear Expansion Coefficient\n\n# 03 LAB D_ Linear Expansion Coefficient - Lab Course...\n\nThis preview shows page 1 - 2 out of 2 pages.\n\nLab Course 103A-Section 006 Physics Laboratory Report Lab number and Title: LAB D: Linear Expansion Coefficient Name: Kris Edquilag Group members: Ernessa Francois, Andrew Rivera Experiment Date : Feburary 10, 2017 Instructor’s name: Sizhan Liu 1. OBJECTIVES To measure the coefficient of linear expansion for different metal rods. 2. THEORETICAL BACKGROUND Linear dimensions (length and diameter) for most substances increase as the temperature increases in a range that does not produce a change of phase. The added heat increases the vibrational kinetic energy of the atoms in the material, which increases the distance between the atoms. To the first approximation, the increase in a linear dimension ΔL is proportional to the increase in temperature ΔT and the initial dimension L o : ΔL/L = αΔT Where the proportionality α is the coefficient of linear expansion. This coefficient depends on many factors, such as the composition and purity of the material, temperature, etc.", null, "", null, "#### You've reached the end of your free preview.\n\n•", null, "•", null, "•", null, "" ]

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[ "# How to extract the values from a matrix 282x1 if I know the rows number?\n\n2 views (last 30 days)\nFlavia Aluisi on 28 May 2017\nEdited: Guillaume on 28 May 2017\nDear all, I have the following problem: starting from a matrix A (282x1) I extract, with the command 'find', all the rows included in a given rage of values. So I got a second matrix B with all the rows numbers. Now my question is, how I can get a third matrix C with the Values of A corresponding to the rows in B?\n\nKSSV on 28 May 2017\nFind gives you indices of the range of values you sought in A. B is your result from A. To get C, use\nC=A(B);\n\nGuillaume on 28 May 2017\nEdited: Guillaume on 28 May 2017\nIf you have the rows indices, then it's just a matter of simple indexing. To extract the row numbers contained in the variable B, then for any arbitrary matrix it's simply:\nC = A(B, :);\nIf your A has only one column, as in your case, then you don't even need the colon, so:\nC = A(B);\nHowever, if your B is the result of find, then you don't even need to use find nor generate B. You can use logical indexing instead. e.g.:\n%B = find(A == 5)\n%C = A(B, :)\nC = A(A == 5, :) %no need for find\nUsing logical indexing will be (negligibly for arrays of your size) faster than using find + normal indexing." ]

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[ "# Solve this and you'll win \\$1 million...\n\nThe Millennium Maths Problems, if solved, could revolutionise our World but what will it take to crack them? And what will the benefits be?\n21 July 2015\n\n## Interview with\n\nTom Crawford, University of Cambridge\n\n## 14999534034_b31bf32808_k.jpg", null, "The Millennium Prize Problems are a set of 7 maths problems that have been deemed", null, "so important that if you can solve any of them, you'll be awarded 1 million Dollars. Graihagh Jackson spoke to Cambridge University's Tom Crawford to find out what these problems are...\n\nTom - So, yeah. These are sort of a second reincarnation of the idea of important maths problems. So, in the year 1900, there was a mathematician called Hilbert who actually came up with 23 problems that he considered to be the most important in the year 1900. In fact, one of these is actually one of the millennium problems as well. And then the year 2000 came around, it's a big event, lots of things happened, and maths thought, \"Let's get involved with this.\"\n\nGraihagh - Jump on the bandwagon.\n\nTom - Exactly. They sort of sat down and said, \"Right. These are the biggest unsolved problems in mathematics at this time.\n\nGraihagh - Did they say 7 because 7 is a prime number and an interesting maths number?\n\nTom - I'm not sure exactly why they chose 7, but that sounds like a good idea to me.\n\nGraihagh - And you say 'they', these mathematicians, who are they and I assume, they're the ones fitting the bill of a million dollars, right?\n\nTom - Yes. So, this is the Clay Institute. It's an institute based in America. They had money left to them and thought, \"Let's use this as prize money as a sort of extra motivation.\" But if you ask most mathematicians - I won't say all - if you ask most mathematicians, they wouldn't actually try and tackle these problems for the money. It's more just for the fun of doing the maths in some sense.\n\nGraihagh - So, can you give me a quick rundown of the final 7.\n\nTom - I'll start with the one I'm doing my PhD on, this is the Navier-Stokes equations. This is fluid dynamics. This is a set of equations that model the flow of every fluid - so water, air, anything. Then we've got the Mass Gap Hypothesis. You can think of this one as asking the question \"why do things have mass?\"\n\nGraihagh - Sounds interesting. So, you've mentioned 2 of them there. There's 4 more by my count, by my math.\n\nTom - 5\n\nGraihagh - 5? Oh no! Okay, what are the other 5?\n\nTom - We've got the Poincare Conjecture which is an interesting one as we'll find out later. We've got the Riemann Hypothesis which looks at prime numbers. We have P versus NP - one of the more famous ones. So, this is basically looking at how computers work. And then the last two are quite abstract. So, you have the Birch and Swinnerton-Dyer Conjecture.\n\nGraihagh - Wow! Great names.\n\nTom - And then finally, the Hodge Conjecture. So, the Hodge Conjecture is quite an interesting one because depending on which expert in the field you ask about this problem, they will give you a different definition of what this problem is.\n\nGraihagh - So, no one can agree what the actual problem is in the first place?\n\nTom - Yeah. That's how complex some of these problems are.\n\nGraihagh - Wow! So, how are we ever going to go about solving them if we can't even agree what they are in the first place?\n\nTom - Well, I mean some of them are a bit more approachable and there's a bit more general consensus about what this problem is. But I think the main idea of these problems it's not necessarily getting a solution, it's understanding more. So, by trying to conquer this giant mountain of a problem, you'll scale smaller peaks along the way and make new advances in other areas of maths.\n\nGraihagh - A million pounds sounds like an awful lot, but I suppose if you spend your entire life doing this then actually, it's not that much.\n\nTom - No. Sort of estimates of the amount of time it would take to solve one of these problems, it actually works out as being paid below minimum wage for the amount of time you would need to spend to solve one of these problems.\n\nGraihagh - Wow! We've got some very dedicated mathematicians out there.\n\nTom - Yeah, most mathematicians I think actually do it for fun, if you can believe that.\n\nGraihagh - I'm just thinking about my maths class back when I was 16 and maths certainly wasn't my favourite subject but maybe I should revisit that.\n\nTom - Yes, sounds good." ]

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[ "Want to share your content on R-bloggers? click here if you have a blog, or here if you don't.", null, "In the previous set we started with arithmetic operations on vectors. We’ll take this a step further now, by practising functions to summarize, sort and round the elements of a vector.\n\nSofar, the functions we have practised (`log`, `sqrt`, `exp`, `sin`, `cos`, and `acos`) always return a vector with the same length as the input vector. In other words, the function is applied element by element to the elements of the input vector. Not all functions behave this way though. For example, the function `min(x)` returns a single value (the minimum of all values in `x`), regardless of whether x has length 1, 100 or 100,000.\n\nBefore starting the exercises, please note this is the third set in a series of five: In the first two sets, we practised creating vectors and vector arithmetics. In the fourth set (posted next week) we will practise regular sequences and replications.\n\nYou can find all sets right now in our ebook Start Here To Learn R – vol. 1: Vectors, arithmetic, and regular sequences. The book also includes all solutions (carefully explained), and the fifth and final set of the series. This final set focuses on the application of the concepts you learned in the first four sets, to real-world data.\n\nOne more thing: I would really appreciate your feedback on these exercises: Which ones did you like? Which ones were too easy or too difficult? Please let me know what you think here!\n\n### Exercise 1\n\nDid you know R has actually lots of built-in datasets that we can use to practise? For example, the `rivers` data “gives the lengths (in miles) of 141 “major” rivers in North America, as compiled by the US Geological Survey” (you can find this description, and additonal information, if you enter `help(rivers)` in R. Also, for an overview of all built-in datasets, enter `data()`.\n\nHave a look at the `rivers` data by simply entering `rivers` at the R prompt. Create a vector `v` with 7 elements, containing the number of elements (`length`) in `rivers`, their sum (`sum`), mean (`mean`), median (`median`), variance (`var`), standard deviation (`sd`), minimum (`min`) and maximum (`max`).\n\n(Solution)\n\n### Exercise 2\n\nFor many functions, we can tweak their result through additional arguments. For example, the `mean` function accepts a `trim` argument, which trims a fraction of observations from both the low and high end of the vector the function is applied to.\n\n1. What is the result of `mean(c(-100, 0, 1, 2, 3, 6, 50, 73), trim=0.25)`? Don’t use R, but try to infer the result from the explanation of the `trim` argument I just gave. Then check your answer with R.\n2. Calculate the mean of `rivers` after trimming the 10 highest and lowest observations. Hint: first calculate the trim fraction, using the `length` function.\n\n(Solution)\n\n### Exercise 3\n\nSome functions accept multiple vectors as inputs. For example, the `cor` function accepts two vectors and returns their correlation coefficient. The `women` data “gives the average heights and weights for American women aged 30-39”. It contains two vectors `height` and `weight`, which we access after entering `attach(women)` (we’ll discuss the details of `attach` in a later chapter).\n\n1. Using the `cor` function, show that the average height and weight of these women are almost perfectly correlated.\n2. Calculate their covariance, using the `cov` function.\n3. The `cor` function accepts a third argument `method` which allows for three distinct methods (“pearson”, “kendall”, “spearman”) to calculate the correlation. Repeat part (a) of this exercise for each of these methods. Which is the method chosen by the default (i.e. without specifying the method explicitly?)\n\n(Solution)\n\n### Exercise 4\n\nIn the previous three exercises, we practised functions that accept one or more vectors of any length as input, but return a single value as output. We’re now returning to functions that return a vector of the same length as their input vector. Specifically, we’ll practise rounding functions. R has several functions for rounding. Let’s start with `floor`, `ceiling`, and `trunc`:\n\n• `floor(x)` rounds to the largest integer not greater than `x`\n• `ceiling(x)` rounds to the smallest integer not less than `x`\n• `trunc(x)` returns the integer part of `x`\n\nTo appreciate the difference between the three, I suggest you first play around a bit in R with them. Just pick any number (with or without a decimal point, positive and negative values), and see the result each of these functions gives you. Then make it somewwat closer to the next integer (either above or below), or flip the sign, and see what happens. Then continue with the following exercise:\n\nBelow you will find a series of arguments (x), and results (y), that can be obtained by choosing one or more of the 3 functions above (e.g. `y <- floor(x)`). Which of the above 3 functions could have been used in each case? First, choose your answer without using R, then check with R.\n\n1. `x <- c(300.99, 1.6, 583, 42.10)`\n`y <- c(300, 1, 583, 42)`\n2. `x <- c(152.34, 1940.63, 1.0001, -2.4, sqrt(26))`\n`y <- c(152, 1940, 1, 5, -2)`\n3. `x <- -c(3.2, 444.35, 1/9, 100)`\n`y <- c(-3, -444, 0, -100)`\n4. `x <- c(35.6, 670, -5.4, 3^3)`\n`y <- c(36, 670, -5, 27)`\n\n(Solution)\n\n### Exercise 5\n\nIn addition to `trunc`, `floor`, and `ceiling`, R also has `round` and `signif` rounding functions. The latter two accept a second argument `digits`. In case of `round`, this is the number of decimal places, and in case of `signif`, the number of significant digits. As with the previous exercise, first play around a little, and see how these functions behave. Then continue with the exercise below:\n\nBelow you will find a series of arguments (x), and results (y), that can be obtained by choosing one, or both, of the 2 functions above (e.g. `y <- round(x, digits=d)`). Which of these functions could have been used in each case, and what should the value of `d` be? First, choose your answer without using R, then check with R.\n\n1. `x <- c(35.63, 300.20, 0.39, -57.8)`\n`y <- c(36, 300, 0, -58)`\n2. `x <- c(153, 8642, 10, 39.842)`\n`y <- c(153.0, 8640.0, 10.0, 39.8)`\n3. `x <- c(3.8, 0.983, -23, 7.1)`\n`y <- c(3.80, 0.98, -23.00, 7.10)`\n\n(Solution)\n\n### Exercise 6\n\nOk, let’s continue with a really interesting function: `cumsum`. This function returns a vector of the same length as its input vector. But contrary to the previous functions, the value of an element in the output vector depends not only on its corresponding element in the input vector, but on all previous elements in the input vector. So, its results are cumulative, hence the `cum` prefix. Take for example: `cumsum(c(0, 1, 2, 3, 4, 5))`, which returns: 0, 1, 3, 6, 10, 15. Do you notice the pattern?\n\nFunctions that are similar in their behavior to `cumsum`, are: `cumprod`, `cummax` and `cummin`. From just their naming, you might already have an idea how they work, and I suggest you play around a bit with them in R before continuing with the exercise.\n\n1. The `nhtemp` data contain “the mean annual temperature in degrees Fahrenheit in New Haven, Connecticut, from 1912 to 1971”. (Although `nhtemp` is not a vector, but a timeseries object (which we’ll learn the details of later), for the purpose of this exercise this doesn’t really matter.) Use one of the four functions above to calculate the maximum mean annual temperature in New Haven observed since 1912, for each of the years 1912-1971.\n2. Suppose you put \\$1,000 in an investment fund that will exhibit the following annual returns in the next 10 years: 9% 18% 10% 7% 2% 17% -8% 5% 9% 33%. Using one of the four functions above, show how much money your investment will be worth at the end of each year for the next 10 years, assuming returns are re-invested every year. Hint: If an investment returns e.g. 4% per year, it will be worth 1.04 times as much after one year, 1.04 * 1.04 times as much after two years, 1.04 * 1.04 * 1.04 times as much after three years, etc.\n\n(Solution)\n\n### Exercise 7\n\nR has several functions for sorting data: `sort` takes a vector as input, and returns the same vector with its elements sorted in increasing order. To reverse the order, you can add a second argument: `decreasing=TRUE`.\n\n1. Use the `women` data (exercise 3) and create a vector `x` with the elements of the `height` vector sorted in decreasing order.\n2. Let’s look at the `rivers` data (exercise 1) from another perspective. Looking at the 141 data points in `rivers`, at first glance it seems quite a lot have zero as their last digit. Let’s examine this a bit closer. Using the modulo operator you practised in exercise 9 of the previous exercise set, to isolate the last digit of the `rivers` vector, sort the digits in increasing order, and look at the sorted vector on your screen. How many are zero?\n3. What is the total length of the 4 largest rivers combined? Hint: Sort the rivers vector from longest to shortest, and use one of the `cum...` functions to show their combined length. Read off the appropriate answer from your screen.\n\n(Solution)\n\n### Exercise 8\n\nAnother sorting function is `rank`, which returns the ranks of the values of a vector. Have a look at the following output:\n\n```x <- c(100465, -300, 67.1, 1, 1, 0)\nrank(x)\n## 6.0 1.0 5.0 3.5 3.5 2.0```\n1. Can you describe in your own words what `rank` does?\n2. In exercise 3(c) you estimated the correlation between `height` and `weight`, using Spearman’s rho statistic. Try to replicate this using the `cor` function, without the `method` argument (i.e., using its default Pearson method, and using `rank` to first obtain the ranks of `height` and `weight`.\n\n(Solution)\n\n### Exercise 9\n\nA third sorting function is `order`. Have a look again at the vector `x` introduced in the previous exercise, and the output of `order` applied to this vector:\n\n```x <- c(100465, -300, 67.1, 1, 1, 0)\norder(x)\n## 2 6 4 5 3 1```\n1. Can you describe in your own words what `order` does? Hint: look at the output of `sort(x)` if you run into trouble.\n2. Remember the time series of mean annual temperature in New Haven, Connecticut, in exercise 6? Have a look at the output of `order(nhtemp)`:\n```order(nhtemp)\n## 6 15 29 9 13 3 5 12 7 23 1 24 47 25 51 14 18 32 16 11 56 8 17\n## 28 45 52 31 37 4 22 36 39 54 19 34 26 49 30 33 53 55 21 27 58 10 50\n## 57 59 43 44 35 2 46 48 40 20 60 41 38 42```\n\nGiven that the starting year for this series is 1912, in which years did the lowest and highest mean annual temperature occur?\n\n1. What is the result of order(sort(x)), if x is a vector of length 100, and all of its elements are numbers? Explain your answer.\n\n(Solution)\n\n### Exercise 10\n\nIn exercise 1 of this set, we practised the `max` function, followed by the `cummax` function in exercise 6. In the final exercise of this set, we’re returning to this topic, and will practise yet another function to find a maximum. While the former two functions applied to a single vector, it’s also possible to find a maximum across multiple vectors.\n\n1. First let’s see how `max` deals with multiple vectors. Create two vectors `x` and `y`, where `x` contains the first 5 even numbers greater than zero, and `y` contains the first 5 uneven numbers greater than zero. Then see what `max` does, as in `max(x, y)`. Is there a difference with `max(y, x)`?\n2. Now, try `pmax(x, y)`, where p stands for “parallel”. Without using R, what do you think intuitively, what it will return? Then, check, and perhaps refine, your answer with R.\n3. Now try to find the parallel minimum of x and y. Again, first try to write down the output you expect. Then check with R (I assume, you can guess the appropriate name of the function).\n4. Let’s move from two to three vectors. In addition to `x` and `y`, add `-x` as a third vector. Write down the expected output for the parallel minima and maxima, then check your answer with R.\n5. Finally, let’s find out how `pmax` handles vectors of different lenghts Write down the expected output for the following statements, then check your answer with R.\n• `pmax(x, 6)`\n• `pmax(c(x, x), y)`\n• `pmin(x, c(y, y), 3)`\n\n(Solution)" ]

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[ "# Are slopes of calibration experiment different?\n\nI'm running a calibration experiment for an instrument that measures soil moisture content. I have 6 buckets filled with different soils that I weigh daily and at the same time take a measurement with the soil moisture probe.\n\nI now have 6 different XY linear models all with different slopes/intercepts/R2 etc. and want to test to see if the slopes for the different buckets are different.\n\nTo be clear, and I think the reason other answers aren't appropriate, is that n = 1, as each bucket represents a different soil type (sand, clay, silt, mixes of those for the other threee); however, the calibration curves have ~18 points each (18 different moisture contents and their corresponding instrument measurement).\n\nI've been using R, so my data are in 12 columns AX, AY, BX, BY, ... FX, FY and would appreciate the code to run the test as well. (A = sample A (sand), B = clay etc. with X and Y being moisture content and probe reading, respectively, taken at that same time (e.g., AX is moisture content for sand)). Moisture contents are different between A to F (i.e., not the same value at each weighing time).\n\nThoughts?\n\n• Could you please edit your question to clarify your objective? It seems that you might be trying to obtain a different response curve for the instrument for six different soil types, but the role of time in your data collection is not at all clear and makes one wonder what your interest in that variable might be.\n– whuber\nApr 28 '14 at 21:06\n• Sorry for the delay - the change in moisture content is, I believe, just an artifact of the time. The time is irrelevant (it's just that it takes time to dry the samples out). The plot is y vs. x with y = instrument reading of moisture content vs. x = actual moisture content determined by weighing. Is that more clear? May 4 '14 at 0:53\n\nHow about analysis of covariance, factor A runs through 6 soil types, numeric covariate is time. The interaction term will tell you if the slopes are different:\n\nnt <- 10 # length of time series\nfactorA <- c(rep('A',nt), rep('B',nt), rep('C',nt))\ntime <- rep(1:nt,3)\nn <- length(time)\nmeant <- mean(1:nt)\ntime0 <- time[1:nt] - meant # centered time, I will make the means all equal to 0\nY <- c(time0*1 + 9, time0*1 - 4, time0*1 + 3) + rnorm(n)*0.4\n# can also use different slopes\nfcA <- as.factor(factorA)\n\nlm1 <- lm(Y ~ fcA + time + fcA*time)\nanova(lm1)\n\n• In a calibration context, \"slope\" typically refers to the relationship between the instrument's response and the known (or more accurately measured) value to which it is responding. Your code seems to interpret \"slope\" differently in terms of a change over time. To avoid possible misunderstandings, your answer should explain what the code is doing and why you think it is relevant to the question.\n– whuber\nApr 28 '14 at 21:04" ]

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[ "# How to save a plot in pdf in R?\n\nR ProgrammingServer Side ProgrammingProgramming\n\nTo save a plot in pdf, we can use the pdf function in base R. For example, if we want to save a plot with the name PDF then it can be done using the below command −\n\npdf(\"PDF.pdf\")\n\nAfter this we can create the plot and use dev.off().\n\n## Example\n\npdf(\"pdfExample.pdf\")\nplot(1:10)\n\n## Output", null, "## Example\n\nLive Demo\n\ndev.off()\n\n## Output", null, "To check where it is saved, find the working directory using getwd()." ]

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[ "# تأثیر جداسازی لرزه‌یی بر شکنندگی لرزه‌یی تجهیزات صلب بلوکی\n\nنوع مقاله : پژوهشی\n\nنویسندگان\n\n1 پژوهشگاه بین‌المللی زلزله‌شناسی و مهندسی زلزله\n\n2 پژوهشگاه بین المللی زلزله شناسی و مهندسی زلزله\n\nچکیده\n\nمنحنی‌های شکنندگی لرزه‌یی برای تجهیزات غیرسازه‌یی بلوکی مهارنشده در سه حالت متفاوت: «تجهیزات روی بستر زمین و مهارنشده»، «تجهیزات روی طبقات ساختمان و بدون سیستم‌های جداساز لرزه‌یی» و «تجهیزات روی طبقات ساختمان و با جداسازی لرزه‌یی کف» به دست آمده‌اند. به‌منظور ارزیابی آسیب‌پذیری لرزه‌یی تجهیزات بلوکی‌تحت اثر حرکت گهواره‌یی آنها به هنگام زلزله، منحنی‌های شکنندگی در حالات سه‌گانه‌ی یادشده رسم شده است. تجهیزات بلوکی با نسبت ابعاد مختلف مورد مطالعه قرار گرفته‌اند. در این پژوهش، برای محاسبه‌ی منحنی‌های شکنندگی لرزه‌یی از روش شبیه‌سازی مونتـکارلو استفاده شده است. نتایج حاصل از مطالعه‌ی پاسخ دورانی تجهیزات بلوکی تحت اثر زلزله، و نیز رسم منحنی‌های شکنندگی برای آنها نشان می‌دهد که جداسازی لرزه‌یی کف طبقاتی که تجهیزات در آنها قرار دارند در کاهش میزان آسیب‌پذیری بلوک‌های صلب تأثیر به‌سزایی دارد به‌نحوی که در بلوک‌های مشابه، آنها که در طبقاتی با کف جداسازی شده قرار دارند، نسبت به بلوک‌هایی در سطح همان طبقات و بدون جداسازی لرزه‌یی، از سطح آسیب‌پذیری پایین‌تری برخوردارند. همچنین آسیب‌پذیری این تجهیزات با افزایش نسبت ابعاد بلوک‌های صلب کاهش می‌یابد؛ از دو بلوک با نسبت ابعاد یکسان، بلوکی که اندازه‌اش بزرگ‌تر است آسیب‌پذیری پایین‌تری دارد.\n\nکلیدواژه‌ها\n\nعنوان مقاله [English]\n\n### A S‌T‌U‌D‌Y O‌N T‌H‌E E‌F‌F‌E‌C‌T‌I‌V‌E‌N‌E‌S‌S O‌F S‌E‌I‌S‌M‌I‌C F‌L‌O‌O‌R I‌S‌O‌L‌A‌T‌I‌O‌N O‌F R‌I‌G‌I‌D B‌L‌O‌C‌K T‌Y‌P‌E E‌Q‌U‌I‌P‌M‌E‌N‌T U‌S‌I‌N‌G S‌E‌I‌S‌M‌I‌C F‌R‌A‌G‌I‌L‌I‌T‌Y C‌U‌R‌V‌E‌S\n\nنویسندگان [English]\n\n• A. Kalantari 1\n• V. S‌h‌a‌h‌b‌a‌z‌i 2\n1 I‌n‌t‌e‌r‌n‌a‌t‌i‌o‌n‌a‌l I‌n‌s‌t‌i‌t‌u‌t‌e o‌f E‌a‌r‌t‌h‌q‌u‌a‌k‌e E‌n‌g‌i‌n‌e‌e‌r‌i‌n‌g a‌n‌d S‌c‌i‌s‌m‌o‌l‌o‌g‌y\n2 I‌n‌t‌e‌r‌n‌a‌t‌i‌o‌n‌a‌l I‌n‌s‌t‌i‌t‌u‌t‌e o‌f E‌a‌r‌t‌h‌q‌u‌a‌k‌e E‌n‌g‌i‌n‌e‌e‌r‌i‌n‌g a‌n‌d S‌c‌i‌s‌m‌o‌l‌o‌g‌y\nچکیده [English]\n\nT‌h‌e e‌f‌f‌e‌c‌t‌i‌v‌e‌n‌e‌s‌s o‌f t‌h‌e f‌l‌o‌o‌r i‌s‌o‌l‌a‌t‌i‌o‌n o‌f b‌l‌o‌c‌k t‌y‌p‌e r‌i‌g‌i‌d e‌q‌u‌i‌p‌m‌e‌n‌t i‌s i‌n‌v‌e‌s‌t‌i‌g‌a‌t‌e‌d u‌s‌i‌n‌g s‌e‌i‌s‌m‌i‌c f‌r‌a‌g‌i‌l‌i‌t‌y c‌u‌r‌v‌e‌s. A f‌l‌o‌o‌r i‌s‌o‌l‌a‌t‌i‌o‌n s‌y‌s‌t‌e‌m i‌s p‌r‌o‌p‌o‌s‌e‌d f‌o‌r t‌h‌e e‌q‌u‌i‌p‌m‌e‌n‌t. T‌h‌e e‌q‌u‌i‌p‌m‌e‌n‌t i‌s c‌o‌n‌s‌i‌d‌e‌r‌e‌d i‌n t‌h‌r‌e‌e d‌i‌f‌f‌e‌r‌e‌n‌t c‌o‌n‌d‌i‌t‌i‌o‌n‌s r‌e‌f‌e‌r‌r‌e‌d t‌o a‌s; e‌q‌u‌i‌p‌m‌e‌n‌t o‌n t‌h‌e g‌r‌o‌u‌n‌d'', e‌q‌u‌i‌p‌m‌e‌n‌t o‌n t‌h‌e b‌u‌i‌l‌d‌i‌n‌g f‌l‌o‌o‌r a‌n‌d w‌i‌t‌h‌o‌u‌t f‌l‌o‌o‌r i‌s‌o‌l‌a‌t‌i‌o‌n'' a‌n‌d e‌q‌u‌i‌p‌m‌e‌n‌t o‌n t‌h‌e b‌u‌i‌l‌d‌i‌n‌g f‌l‌o‌o‌r w‌i‌t‌h s‌e‌i‌s‌m‌i‌c f‌l‌o‌o‌r i‌s‌o‌l‌a‌t‌i‌o‌n''. T‌h‌e s‌e‌i‌s‌m‌i‌c f‌r‌a‌g‌i‌l‌i‌t‌y c‌u‌r‌v‌e‌s a‌r‌e c‌a‌l‌c‌u‌l‌a‌t‌e‌d f‌o‌r e‌a‌c‌h c‌a‌s‌e t‌o e‌v‌a‌l‌u‌a‌t‌e t‌h‌e v‌u‌l‌n‌e‌r‌a‌b‌i‌l‌i‌t‌y o‌f t‌h‌e e‌q‌u‌i‌p‌m‌e‌n‌t t‌o r‌o‌c‌k‌i‌n‌g m‌o‌t‌i‌o‌n a‌n‌d t‌o‌p‌p‌l‌i‌n‌g. D‌i‌f‌f‌e‌r‌e‌n‌t a‌s‌p‌e‌c‌t r‌a‌t‌i‌o‌s a‌r‌e c‌o‌n‌s‌i‌d‌e‌r‌e‌d f‌o‌r t‌h‌e e‌q‌u‌i‌p‌m‌e‌n‌t. T‌h‌e r‌e‌s‌u‌l‌t‌s o‌f t‌h‌e s‌t‌u‌d‌y o‌n t‌h‌e r‌o‌c‌k‌i‌n‌g r‌e‌s‌p‌o‌n‌s‌e o‌f t‌h‌e e‌q‌u‌i‌p‌m‌e‌n‌t d‌u‌r‌i‌n‌g e‌a‌r‌t‌h‌q‌u‌a‌k‌e s‌h‌o‌w t‌h‌a‌t f‌l‌o‌o‌r i‌s‌o‌l‌a‌t‌i‌o‌n i‌s e‌f‌f‌e‌c‌t‌i‌v‌e i‌n r‌e‌d‌u‌c‌t‌i‌o‌n o‌f s‌e‌i‌s‌m‌i‌c f‌r‌a‌g‌i‌l‌i‌t‌y o‌f t‌h‌e e‌q‌u‌i‌p‌m‌e‌n‌t o‌n t‌h‌e f‌l‌o‌o‌r, c‌o‌m‌p‌a‌r‌e‌d t‌o t‌h‌e n‌o‌n-i‌s‌o‌l‌a‌t‌e‌d e‌q‌u‌i‌p‌m‌e‌n‌t. F‌u‌r‌t‌h‌e‌r‌m‌o‌r‌e, t‌h‌e v‌u‌l‌n‌e‌r‌a‌b‌i‌l‌i‌t‌y o‌f s‌i‌m‌i‌l‌a‌r e‌q‌u‌i‌p‌m‌e‌n‌t i‌n‌c‌r‌e‌a‌s‌e‌s w‌i‌t‌h a r‌e‌d‌u‌c‌t‌i‌o‌n i‌n a‌s‌p‌e‌c‌t r‌a‌t‌i‌o.\n\nکلیدواژه‌ها [English]\n\n• b‌l‌o‌c‌k t‌y‌p‌e e‌q‌u‌i‌p‌m‌e‌n‌t\n• s‌e‌i‌s‌m‌i‌c f‌r‌a‌g‌i‌l‌i‌t‌y\n• s‌e‌i‌s‌m‌i‌c f‌l‌o‌o‌r i‌s‌o‌l‌a‌t‌i‌o‌n" ]

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[ "# Squeeze-and-Excitation Networks\n\nSqueeze-and-Excitation Networks（SENet)是CVPR2018公布的一种全新的图像识别结构，它通过对特征通道间的相关性进行建模，把重要的特征进行强化来提升准确率。是目前流行的一种子结构\n\n## SQUEEZE-AND-EXCITATION BLOCKS", null, "### Squeeze: Global Information Embedding\n\n$$z_c = F_s(U_c) = \\frac{1}{H ✖️ W}\\sum^H_i\\sum^W_j u_c(i,j)$$\n\n• 能够学得各个 channel 之间的非线性相互关系\n• 必须学的是非互斥得关系，因为可以有多个 channel 被强调，而不是 one-hot 激活\n\n$$s = F_e(z, W) = \\sigma(g(z, W)) = \\sigma(W_2\\delta(W_1z))$$\n\n$W_1$, $W_2$为全连接， $\\delta$ 为 relu 激活. 对于激励模块，作者首先使用一个全连接层对 GAP 结果以 $\\frac{C}{r}$降维, 然后接 relu 激活函数学习非线性相关性，之后再跟一个全连接恢复 channel 维度.", null, "### usage\n\n• SE-block 可用于分类与检测. 对于标准网络结构如VGGNet，SE-block 可以用在每一个卷积+非线性操作之后. $Fig.2$和$Fig.3$分别是SE-Inception module和SE-Resnet module.\n• ratio. 在ImageNet上，作者得到的最佳ratio为16,可做相对参考\n\n### 简单代码(keras)\n\n\ndef squeeze_excitation_layer(x, ratio, out_dim):\n\nsqueeze = GlobalAveragePooling2D()(x)\nexcitation = Dense(units=out_dim // ratio)(squeeze)\nexcitation = Activation('relu')(excitation)\nexcitation = Dense(units=out_dim)(excitation)\nexcitation = Activation('sigmoid')(excitation)\nexcitation = Reshape((1, 1, out_dim))(excitation)\nscale = multiply([x, excitation])\n\nreturn scale" ]

[ null, "http://note4lin.top/img/SE-net/block.jpg", null, "http://note4lin.top/img/SE-net/se_module.jpg", null ]

[ "# Nonconservative forces\n\nI am really having a hard time with this there is something I am missing can someone help\na 2100 kg car starts from rest at the top of a 5.0 m long driveway that is sloped at 20 Deg with the horizontal . if an average friction force of 4000 N impedes the motion find the speed of the car at the bottom of the driveway. The answer is 3.8m/s\n\nnbo10\nUse newton's law F_total = m*a.\n\nJMD\n\nAstrophysics\nThat's not entirely correct nbo10, first u need to calculate the Gravitational force on the truck, then you calculate the Fx, which will help you further.\n\np.s. don't forget the 20 degrees\n\nLast edited:\nnbo10\nOriginally posted by Astrophysics\nThat's not entirely correct nbo10, first u need to calculate the Gravitational force on the truck, then you calculate the Fx, which will help you further.\n\np.s. don't forget the 20 degrees\n\nNo, I'm entirely correct. F_total includes gravitational foce and friction.\n\nJMD\n\nAstrophysics\nI think, I think more like a physics teacher, since I don't think F_total = m*a is very precise. I think emaqueen already knew about this equation." ]

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[ "# GFM\n\n##### Sections\nYou are here: Home Introduction to random matrix theory\n\n# Introduction to random matrix theory\n\nGFM seminar\nFCUL, C6, room 6.2.33\n2016-02-24 14:30 .. 15:30\n\nby Miguel Tierz (GFMUL)\n\nWe will give a simple introduction to random matrix theory. We will start by presenting some of the motivations for the study of random matrices, giving some historical perspective on the subject while also highlighting some of the more prevalent current applications. The definition of a random matrix ensemble will be given and its implications analyzed by showing how to work with the joint probability distribution function of the eigenvalues." ]

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[ "Cody\n\n# Problem 43084. Combine the digits to output numbers\n\nSolution 999989\n\nSubmitted on 7 Oct 2016 by Lorwin Delos Santos\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; b = 2; y_correct = 12; assert(isequal(your_fcn_name(x,b),y_correct))\n\nans = 12\n\n2   Pass\nx = 2; b = 1; c = 0; y_correct = 210; assert(isequal(your_fcn_name(x,b,c),y_correct))\n\nans = 210" ]

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[ "## Intresting Factorial!\n\nFor students of class 11-12 (age 16+)\nsamiul_samin\nPosts: 999\nJoined: Sat Dec 09, 2017 1:32 pm\n\n### Intresting Factorial!\n\nFor which lowest positive integer $n$, the last six digits of $n!$ is $0$ ?\n\nToky\nPosts: 3\nJoined: Mon Jan 14, 2019 11:36 pm\n\n### Re: Intresting Factorial!\n\nThe number is $25$. First we have to think how $0$ produces at last position. It must be done by multiplying by $10$. So we need to determine the factorial of lowest number that produces $6$, 10s...\n\n$25! = 1\\times2\\times3\\times4\\times... ...\\times24\\times25$\nHere\ni. $2\\times5 = 10$\nii. $10 = 1\\times10$\niii. $12\\times15 = 180 = 18\\times10$\niv. $20 = 2\\times10$\nv. $22\\times25 = 550 = 55\\times10$\nvi. $55 \\times 4 = 220 = 22 \\times 10$ (55 got from $v$.)\n\nThis is how we get $6$ , 10s for factorial $25$. So the answer is $25$.\n\nsamiul_samin\nPosts: 999\nJoined: Sat Dec 09, 2017 1:32 pm\n\n### Re: Intresting Factorial!\n\nI have got another answer by using floor function." ]

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[ "# Convert 190.5 CM to Inches\n\n## What is cm in inches?\n\nDo you wish to convert 190.5 centimeters into the equivalent of inches? first you should know how many inches 1 cm represents.\n\nYou may use the formula for cm to inches to calculate the conversion.\n\n## Centimeter Basis\n\nCentimeters, or Cm are the length unit used in measurement used in the metric system. Its symbol is cm. The length unit “meter” is widely used internationally to the “International System of Units”, but the unit the unit cm is not. But a cm is equal to 100 meters. It measures also 39.37 in.\n\n## Definition: Inch\n\nAn inch is an Anglo-American measurement of length measurement. The symbol is in. In bulk of European local languages, “inch” can be utilized interchangeably with “thumb” or from “thumb”. Because a man’s thumb is about an inch wide.\n\n• Electronic components such as the size of the PC screen.\n• Dimensions of tires for cars and trucks.\n\n## How Convert 190.5 cm in inches?\n\nThe cm to inches converter can help me to convert cm into inches. This simple principle could be used to convert cm into inches.\n\nBy the above, you now fully understand of cm into inches. The formula will allow you to answer the following questions:\n\n• What’s the formula to convert inches from 190.5 cm?\n• How do you convert cm to inches?\n• How do you change cm to inches?\n• How to measure cm into inches?\n• Is 190.5 cm equal to how many inches?\n\n cm inches 189.7 cm 74.68489 inches 189.8 cm 74.72426 inches 189.9 cm 74.76363 inches 190 cm 74.803 inches 190.1 cm 74.84237 inches 190.2 cm 74.88174 inches 190.3 cm 74.92111 inches 190.4 cm 74.96048 inches 190.5 cm 74.99985 inches 190.6 cm 75.03922 inches 190.7 cm 75.07859 inches 190.8 cm 75.11796 inches 190.9 cm 75.15733 inches 191 cm 75.1967 inches 191.1 cm 75.23607 inches 191.2 cm 75.27544 inches 191.3 cm 75.31481 inches" ]

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[ "# What is ggplot2 ?\n\nggplot2 is a powerful R package, implemented by Hadley Wickham, for producing nice graphs. The gg in ggplot2 means Grammar of Graphics, a graphic concept which describes plots by using a “grammar”. Plot can be divided in different fundamental parts : Plot <- data + Aesthetics + Geometry.\n\ndata are a data.frame Aesthetics is used to specify x and y variables. It can also be used to control the color, the size or the shape of a point, the height of a bar, etc….. Geometry corresponds to the type of graphs (histogram, box plot, line plot, density plot, dot plot, ….)\n\nRead more —-> ggplot2 online documentation\n\n# Introduction to easyGgplot2 package\n\n“ggplot2 is too complicated: it is hard for me to improve, and hard for new developers to understand.” — Hadley Wickham 2011 (http://vita.had.co.nz/papers/future-ig.pdf)\n\nThe idea behind ggplot2 is seductively simple but the detail is, yes, difficult. To customize a plot, the syntax is sometimes a tiny bit opaque and this raises the level of difficulty.\n\nFor these reasons, I created the easyGgplot2 package (which depends on ggplot2) to make and customize quickly plots including box plot, dot plot, strip chart, violin plot, histogram, density plot, scatter plot, bar plot, line plot, etc, ….\n\nIn this chapter, you will find many tutorials on ggplot2 plotting systems and on how to personalize quickly graphical parameters including title, axis labels, legend, background and colors.\n\nThe syntax used to call easyGgplot2 functions is simple and it’s a kind of an easy interface between users and ggplot2 package. An example to illustrate the difference between the syntax of ggplot2 and easyGgplot2 is shown hereafter.\n\n# Install and load easyGgplot2 package\n\nTo install the package, use the following R code from an R console (R >=3.1.0) :\n\n``````install.packages(\"devtools\")\nlibrary(devtools)\ninstall_github(\"easyGgplot2\", \"kassambara\")``````\n\nLoad the package using the R code:\n\n``library(easyGgplot2)``\n\n# Syntax of easyGgplot2 and ggplot2 packages\n\nIn the following R code, a box plot is drawn using either ggplot2 or easyGgplot2.\n\nggplot2 syntax :\n\n``````ggplot(data=ToothGrowth, aes(x=dose, y=len, fill=dose))+\ngeom_boxplot()+\n# Customize main title, x and y axis labels\nggtitle(\"Plot of length \\n by dose\") + xlab(\"Dose (mg)\")+\ntheme(\n# customize main title\nplot.title = element_text( size=14, face=\"bold.italic\",\ncolour=\"red\", lineheight=1.0),\n# customize x and y axis labels\naxis.title.x=element_text(size=14, face=\"bold\",\ncolour=\"blue\", hjust=0.4),\naxis.title.y=element_text(size=14, face=\"bold\",\ncolour=\"blue\", vjust=0.4, angle=90))+\n#change group colors\nscale_fill_manual(values=c('#999999','#E69F00','#56B4E9'))``````\n\neasyGgplot2 syntax :\n\n``````# Change main title and axis titles\n# Customize title styles. Possible values for the font style :\n# 'plain', 'italic', 'bold', 'bold.italic'.\nggplot2.boxplot(data=ToothGrowth, xName='dose',yName='len',\ngroupName=\"dose\",\ngroupColors=c('#999999','#E69F00','#56B4E9'),\nxtitle=\"Dose (mg)\", ytitle=\"Length\",\nmainTitle=\"Plot of length \\n by dose\",\nmainTitleFont=c(14,\"bold.italic\", \"red\"),\nxtitleFont=c(14,\"bold\", \"blue\"),\nytitleFont=c(14,\"bold\", \"blue\"))``````", null, "Note that the outputs of the two packages are the same, and easyGgplot2 is just a wrapper for ggplot2 package\n\n# Data visualizations\n\nClick on the following articles to visualize your data using easyGgplot2 R package.\n\n# ggplot2.boxplot", null, "ggplot2.boxplot is an easy to use function from easyGgplot2 package to make and customize quickly a box plot using R software and ggplot2 plotting methods.\n\n# ggplot2.dotplot", null, "ggplot2.dotplot is an easy to use function from easyGgplot2 package to make and customize quickly a dot plot using R software and ggplot2 plotting methods.\n\n# ggplot2.violinplot", null, "ggplot2.violinplot is an easy to use function from easyGgplot2 package to make and customize quickly a violin plot using R software and ggplot2 plotting methods.\n\n# ggplot2.stripchart", null, "ggplot2.stripchart is an easy to use function from easyGgplot2 package to make and customize one dimensional scatter plots (or dot plots) using R software and ggplot2 package.\n\n# ggplot2.histogram", null, "ggplot2.histogram is an easy to use function from easyGgplot2 package to make and customize quickly histogram using R software and ggplot2 plotting method.\n\n# ggplot2.density", null, "ggplot2.density is an easy to use function from easyGgplot2 package to make and customize quickly density curve using R software and ggplot2 package.\n\n# ggplot2.customize", null, "ggplot2.customize is an easy to use function from easyGgplot2 package to customize plots (e.g : box and whisker plot, histogram, density plot, etc, …) generated with R easyGgplot2 or ggplot2 package.\n\n# ggplot2.scatterplot", null, "ggplot2.scatterplot is an easy to use function from easyGgplot2 package to make and customize quickly scatter plots using R software and ggplot2 package.\n\n# ggplot2.multiplot", null, "ggplot2.multiplot is a function, from easyGgplot2 package, to put multiple graphs on the same page.\n\n# ggplot2.lineplot", null, "ggplot2.lineplot is a function, from easyGgplot2 package, to easily draw and customize line plots.\n\n# ggplot2.barplot", null, "ggplot2.barplot is a function, from easyGgplot2 package, to easily draw and customize bar graphs using R software and ggplot2 plotting methods.\n\n# Easy ggplot2 ebook\n\nNote that an eBook is available on easyGgplot2 package here.\n\nSeptember 2014 : First edition.\n\nLicence : This document is under creative commons licence (http://creativecommons.org/licenses/by-nc-sa/3.0/).\n\n# Infos\n\nThis analysis was performed using R (ver. 3.1.0), easyGgplot2 (ver 1.0.0) and ggplot2 (ver 1.0.0)." ]

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[ "# Derivative of Secx\n\nBy | July 10, 2020\n\n# Derivative of Secx\n\n### Sec(x) Derivative Rule\n\nSecant is the reciprocal of the cosine. The secant of an angle designated by a variable x is notated as sec(x).\nThe derivative rule for sec(x) is given as:\n\nFind the derivative of the following functions (it is to be understood that a, b, c, d, p, q, r and s are fixed non-zero constants sec x —1 and m and n are integers): Sec X + sec x —1 Let f (x) = sec x +1 Let u — — sec x So, f'(x) = Using quotient rule – sec x +1\n\nlet’s find the derivative…\n\nof the secant of x.\n\nHow would we go about doing this?\n\nWell sec(x) is another way of writing 1/cos(x)…\n\nand since we have a quotient here,\n\nthe quotient rule would aptly apply…\n\nso written in short hand, the quotient rule is:\n\ny-dash or y-prime… I’m going to call it y-dash…\nis equal to (vu’ – uv’) / v^2.\n\nAnd in this case, we have y = sec(x), and…\n\nI’m going to let u equal the numerator of 1\n\nand v equal the denominator: cos(x).\n\nSo now if we apply the quotient rule…\n\nfor the first part, we have v, so cos(x) remains unchanged…\n\nmultiplied by the derivative of u.\n\nSo the derivative of 1, a constant, is zero…\n\nthen minus u, which is one, times the derivative of v…\n\nwhich is the derivative of cos(x)…\n\nand that’s equal to -sin(x)…\n\nall over v^2, which is cos^2(x).\n\nSo the result is positive sin(x)…\n\ndivided by the cos^2(x).\n\nAnd I can rewrite this as…\n\n[sin(x)/cos(x)]*1/cos(x).\n\nAnd then I can realise that this part is equal to…\n\nthe tangent of x… and this part is equal to again…\n\nthe secant of x!\n\nSo the derivative of sec(x) is equal to…\n\ntan(x)*sec(x)\n\n## Derivative of sec(x)\n\n### Sec(x) Derivative Rule\n\nSecant is the reciprocal of the cosine. The secant of an angle designated by a variable x is notated as sec(x).\nThe derivative rule for sec(x) is given as:\n\nddxsec(x) = tan(x)sec(x)\n\nThis derivative rule gives us the ability to quickly and directly differentiate sec(x).\nX may be substituted for any other variable.\nFor example, the derivative ddysec(y) = tan(y)sec(y), and the derivative ddzsec(z) = tan(z)sec(z).\n\n### Proof of the Derivative Rule\n\nThe sec(x) derivative rule is originates from the relation that sec(x) = 1/cos(x). Now, the first step of finding the derivative of 1/cos(x) is using the quotient rule.\n\nUsing the quotient rule on 1/cos(x) gives us:\n[sin(x)/cos(x)][1/cos(x)]\nsin(x)/cos(x) = tan(x), and 1/cos(x) = sec(x)\nTherefore, it simplifies to tan(x)sec(x), resulting in:\nddxsec(x) = tan(x)sec(x)\n\n### Derivative Rules of the other Trigonometry Functions\n\nDerivative Rules Here’s the derivative rules for the other five major trig functions:\nddxsin(x) = cos(x)\nddxcos(x) = -sin(x)\nddxtan(x) = sec2(x)\nddxcot(x) = -csc2(x)\nddxcsc(x) = -cot(x)csc(x)\n\nAntiderivative of secxtanx\nAntiderivative of secxtanx\n\nWe have to find the antiderivative of secx tanx. To find the antiderivative of secx\ntanx we must first understand the meaning or definition of antiderivative.\nAntiderivative is an operation which is opposite of derivation operation that\nmeans antiderivative calculates the integral of a derivative.\nSuppose we have a function f(x), its derivative is g(x) means d(f(x))= g(x) than\nantiderivative of g(x) is f(x) that is J g(x) dx = f(x) + c dx. So we get f(x) as\nantiderivative of g(x) is a constant.\nWe have understood the meaning of antiderivative, so now we will calculate the\nantiderivative of secx tanx.\nKnow More About Rational Number Multiplicative Inverse Worksheets\nThe process of finding antiderivative of sec(x) tan(x) is shown in steps below-\nStep 1 : Find the derivative of sec(x), that is d(sec x) / dx = sec(x) tan(x) dx.\nTherefore g(x) sec(x) tan(x).\nStep 2 : Now to find the antiderivative of sec(x) tan(x) , we will have to find the\nintegral of sec(x) tan(x).\nThat is Jsec(x) tan(x) dx = sec(x).\nProof of above operation- => Let g(x)= sec x tan x dx => g(x) = J sin x / cos 2 x\ndx ( sec x = I / cos x , tan x = sin x / cos x) Let u = cosx\nTherefore du – sin x dx g(x) – f 1 / u 2 du g(x) – u-2 du g(x) = 1/\nu + C => g(x) = 1/ cos x + C => g(x) sec x + C Hence proved. Here we have\nused method of integration by parts to calculate the antiderivative of sec x tan x.\nHence to find the antiderivative of secxtanx we have to calculate the integral of" ]

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[ "# displs\n\nDisplay lifting scheme\n\n## Syntax\n\n``S = displs(LS,FRM)``\n\n## Description\n\nexample\n\n````S = displs(LS,FRM)` returns the character array describing the lifting scheme `LS`. The formatting operator `FRM` (see `sprintf`) builds `S`.`displs(LS)` is equivalent to `displs(LS,'%12.8f')`.```\n\n## Examples\n\ncollapse all\n\n`lshaar = liftwave('haar');`\n\nVisualized the lifting scheme.\n\n`displs(lshaar);`\n```lshaar = {... 'd' [ -1.00000000] 'p' [ 0.50000000] [ 1.41421356] [ 0.70710678] [] }; ```\n\nAdd a primal elementary lifting step to the lifting scheme. Display the resulting scheme.\n\n```els = {'p',[-0.125 0.125],0}; lsnew = addlift(lshaar,els); displs(lsnew);```\n```lsnew = {... 'd' [ -1.00000000] 'p' [ 0.50000000] 'p' [ -0.12500000 0.12500000] [ 1.41421356] [ 0.70710678] [] }; ```\n\n## Input Arguments\n\ncollapse all\n\nLifting scheme associated with a wavelet. See `liftwave`.\n\nExample: `LS = liftwave('db4')` returns the lifting scheme associated with the Daubechies wavelet `db4`.\n\nFormatting operator used to build `LS`. See `sprintf`.\n\nExample: `'%12.3f'`", null, "" ]

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[ "## H3PO4 (aq) + 3 NaHCO3 (s) → Na3PO4 (aq) + 3 H2O (ℓ) + 3 CO2 (g)\n\nBack to reactions list\n\n## Reaction Type:\n\nDouble Displacement/Decomposition of a Carbonate\n\n## Stoichiometry\n\nEnter a mass or volume in one of the boxes below. Upon hitting submit, the stoichiometric equivalents will be calculated for the remaining reactants and products. All gases are assumed to be at STP.\n\n Heat Added: kJ H3PO4          Mass: g or Solution Volume: mL of Concentration: mol/L NaHCO3         Mass: g Na3PO4         Mass: g H2O            Mass: g CO2            Mass: g or Gas Volume: L\n\n## Enthalpy of Reaction\n\n[1ΔHf(Na3PO4 (aq)) + 3ΔHf(H2O (ℓ)) + 3ΔHf(CO2 (g))] - [1ΔHf(H3PO4 (aq)) + 3ΔHf(NaHCO3 (s))]\n[1(-1997.68) + 3(-285.83) + 3(-393.51)] - [1(-1277.38) + 3(-947.68)] = 84.7200000000003 kJ\n84.72 kJ     (endothermic)\n\n## Entropy Change\n\n[1ΔSf(Na3PO4 (aq)) + 3ΔSf(H2O (ℓ)) + 3ΔSf(CO2 (g))] - [1ΔSf(H3PO4 (aq)) + 3ΔSf(NaHCO3 (s))]\n[1(-44.75) + 3(69.91) + 3(213.68)] - [1(-221.75) + 3(102.09)] = 721.5 J/K\n721.50 J/K     (increase in entropy)\n\n## Free Energy of Reaction (at 298.15 K)\n\nFrom ΔGf° values:\n[1ΔGf(Na3PO4 (aq)) + 3ΔGf(H2O (ℓ)) + 3ΔGf(CO2 (g))] - [1ΔGf(H3PO4 (aq)) + 3ΔGf(NaHCO3 (s))]\n[1(-1804.5) + 3(-237.18) + 3(-394.38)] - [1(-1018.8) + 3(-851.86)] = -124.8 kJ\n-124.80 kJ     (spontaneous)\n\nFrom ΔG = ΔH - TΔS:\n-130.40 kJ     (spontaneous)\n\n## Equilibrium Constant, K (at 298.15 K)\n\n7.3321381986e+021\nThis process is favorable at 25°C." ]

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[ "# What are holonomic and non-holonomic constraints?\n\nI was reading Herbert Goldstein's Classical Mechanics. Its first chapter explains holonomic and non-holonomic constraints, but I still don’t understand the underlying concept. Can anyone explain it to me in detail and in simple language?\n\nIf you have a mechanical system with $N$ particles, you'd technically need $n = 3N$ coordinates to describe it completely.\n\nBut often it is possible to express one coordinate in terms of others: for example of two points are connected by a rigid rod, their relative distance does not vary. Such a condition of the system can be expressed as an equation that involves only the spatial coordinates $q_i$ of the system and the time $t$, but not on momenta $p_i$ or higher derivatives wrt time. These are called holonomic constraints: $$f(q_i, t) = 0.$$ The cool thing about them is that they reduce the degrees of freedom of the system. If you have $s$ constraints, you end up with $n' = 3N-s < n$ degrees of freedom.\n\nAn example of a holonomic constraint can be seen in a mathematical pendulum. The swinging point on the pendulum has two degrees of freedom ($x$ and $y$). The length $l$ of the pendulum is constant, so that we can write the constraint as $$x^2 + y^2 - l^2 = 0.$$ This is an equation that only depends on the coordinates. Furthermore, it does not explicitly depend on time, and is therefore also a scleronomous constraint. With this constraint, the number of degrees of freedom is now 1.\n\nNon-holonomic constraints are basically just all other cases: when the constraints cannot be written as an equation between coordinates (but often as an inequality).\n\nAn example of a system with non-holonomic constraints is a particle trapped in a spherical shell. In three spatial dimensions, the particle then has 3 degrees of freedom. The constraint says that the distance of the particle from the center of the sphere is always less than $R$: $$\\sqrt{x^2 + y^2 + z^2} < R.$$ We cannot rewrite this to an equality, so this is a non-holonomic, scleronomous constraint.", null, "• \"scleronomous\" .... now there's a word! – GreenAsJade Jun 4 '18 at 11:06\n• skleronom in German, Google told me this is how you say it in English – ahemmetter Jun 4 '18 at 11:07\n• Note that there are also non-holonomic constraints that cannot be written as inequalities between coordinates. Instead, they're only expressible as equalities involving differentials of the coordinates, rather than the coordinates themselves. – Michael Seifert Jun 4 '18 at 13:39\n• @ahemmetter You have written only spatial coordinates but have included the time coordinate as well in f(qi,t) = 0. Why ? – gansub Jun 4 '18 at 13:43\n• @gansub You're right, I forgot to mention that. If constraints are holonomic or not depends on whether or not they can be expressed as the total differential of a function - $t$ is here also included. There is another distinction of constraints that deals with whether or not time is explicitly included: scleronomous (if they don't depend on time) or rheonomous (if they do). – ahemmetter Jun 4 '18 at 13:51\n\nThe question has been well-answered several times. I'll just add some geometrical context.\n\nIn geometry, the holonomy group of a connection is the set of transformations an object can experience when it is parallel transported in a loop. Many constraints can be phrased in terms of forcing something to be parallel transported. If the associated holonomy groups are not trivial, then the constraint cannot be holonomic, because the orientation of the object will depend on the loop traversed, not just the current state. So, rather confusingly, you get holonomic constraints from trivial holonomy groups.\n\nHere are some examples:\n\n• Suppose a coin is rolling without slipping in 2D. This is a holonomic constraint, because if you roll the coin forward and back to where you started, it'll end up in the same orientation. Formally this is described by parallel transport in a $$U(1)$$ bundle over $$\\mathbb{R}$$, where the $$U(1)$$ describes the orientation of the coin.\n• Suppose a ball is rolling without slipping in 3D. This is not a holonomic constraint, because if you wiggle the ball around, you can make it return to where it started, turned over. (Try it!) Formally this is described by nontrivial holonomy in an $$SO(3)$$ bundle over $$\\mathbb{R}^2$$, where the $$SO(3)$$ describes the orientation of the ball.\n• Suppose a cat is floating in space, with zero total angular momentum. This is not a holonomic constraint, because it's possible for the cat to wiggle a bit, then return to its original shape but turned around. Formally this is described by nontrivial holonomy in an $$SO(3)$$ bundle over $$S$$, where $$S$$ is the space of shapes of the cat.\n• +1 for the mention of feline configuration space. – Michael Seifert Jun 5 '18 at 12:50\n• What you say is true, but I doubt the origin of the word is connection theoretic. \"Holonomic\" is of greek origin and afaik it means something like \"entire-law\", and is usually used as a synonym of \"integrable\". I think in mechanics it is just that a holonomic constraint is \"integrable\" in the sense that one can find a definite submanifold of the config space, which satisfies the constraint, whereas a nonholonomic constraint (like a semi-holonomic one) cannot be integrated this way. I feel bringing up connections here is anachronistic to say the least, but definitely interesting. – Bence Racskó Nov 17 '18 at 10:29\n• @Uldreth Absolutely, they're both probably named after something much older, but I don't know any Greek. – knzhou Nov 17 '18 at 10:30\n\nFor completeness: There is also a notion of semi-holonomic constraints.\n\n1. Recall that a holonomic constraint$$^1$$ $$f(q,t)~=~0\\tag{H}$$ only depends on the generalized coordinates$$^2$$ $$q^j$$ and time $$t$$, but not the generalized velocities $$\\dot{q}^j$$.\n\n2. A non-holonomic constraint is unsurprisingly a constraint that is not holonomic.\n\n3. A semi-holonomic constraint $$a(q,\\dot{q},t)~\\equiv~ \\sum_{j=1}^na_j(q,t)~\\dot{q}^j+a_0(q,t)~=~0\\tag{S1}$$ is a non-holonomic constraint that depends affinely on the generalized velocities $$\\dot{q}^j$$. Equivalently, it can be written via a one-form $$\\omega~\\equiv~\\sum_{j=1}^na_j(q,t)~\\mathrm{d}q^j+a_0(q,t)\\mathrm{d}t~=~0,\\tag{S2}$$ that does not have an integrating factor $$\\lambda(q,t)\\neq 0$$ such that $$\\lambda\\omega=\\mathrm{d}f$$ becomes an exact one-form. (If $$\\lambda$$ exists, the constraint (S2) is holonomic.)\n\n--\n\n$$^1$$ There are various technical regularity conditions implicitly assumed, cf. e.g. this Phys.SE post.\n\n$$^2$$ In this answer, we also call the original point particle position variables $${\\bf r}_1, \\ldots, {\\bf r}_N$$ for generalized coordinates to be as general as possible.\n\nSuppose you've written down either a system's Lagrangian in terms of $q_i,\\,\\dot{q}_i$, or its Hamiltonian in terms of $q_i,\\,p_i$. There are some subtleties to the analysis if a function $f$ exists for which $f(q_i,\\,\\dot{q}_j,\\,t)=0$, or $f(q_i,\\,p_j,\\,t)$. Either way, you've constrained that system. We call this holonomic if $f$ is a function of the $q_i,\\,t$ alone.\n\nLet's look at some examples. A rollercoaster follows the shape of its track, so the constraint is holonomic. On the other hand, electromagnetism has $p_{A_0}=0$, which is a non-holonomic constraint. (In fact, this one doesn't even depend on the $q_i$.)\n\nA holonomic constraint is a constraint that places a definite relationship between the coordinates you're using. For example, consider a cylinder of radius $R$ rolling along a table in 1-D. The system can be described by a coordinate $x$, denoting the position of the cylinder, and a coordinate $\\theta$, describing the angle of rotation of the cylinder. If the cylinder is rolling without slipping, though, then for every infinitesimal distance $dx$ the cylinder moves, it must move a distance $d\\theta$ given by $$d\\theta = \\frac{d x}{R} \\quad \\Rightarrow \\quad dx - R d \\theta = 0.$$ But this equation can be integrated to yield $$f(x, \\theta) = x - R \\theta = 0.$$ Since the constraint can be integrated (i.e. the differential constraint in the first equation is equivalent to saying that $df = 0$ for some function $f$ of the coordinates), then this constraint is holonomic. Frequently, when writing down such a constraint, we'll just skip the step with the infinitesimal coordinate changes and simply write down a relationship $f(q_i) = C$ between the coordinates $q_i$. Note that this also implies that $x$ determines $\\theta$: if I know where the cylinder is along the table, I know what its angular orientation is, because given a value of $x$, I can solve the equation $f = 0$ for $\\theta$.\n\nA non-holonomic constraint is a system for which this integration can't be performed. The classical example of this is a sphere rolling without slipping on a table in 2D. In this case, the state of the system is described by the position of the sphere along the table (needing two coordinates, $x$ and $y$) and the angular orientation of the sphere in 3D (needing three coordinates, such as the Euler angles $\\theta$, $\\phi$, $\\psi$.)\n\nNow, suppose I displace the sphere along the table by some infinitesimal displacement $dx$ and $dy$. The values of $dx$ and $dy$, combined with the values of $\\theta$, $\\phi$, $\\psi$ before the displacement, will determine the infinitesimal changes $d\\theta$, $d\\phi$, $d\\psi$. In other words, there must be some kind of relationships of the form $$d \\theta = ( \\sim ) dx + (\\sim) dy, \\quad d \\phi = ( \\sim ) dx + (\\sim) dy, \\quad d \\psi = ( \\sim ) dx + (\\sim) dy$$ where the quantities in brackets are functions of the coordinates themselves. (Their precise form is not important for this argument.)\n\nOne might hope that we could integrate these relationships between the differentials to obtain constraints between the coordinates themselves, expressed as a set of functions $f_j(q_i) = 0$. But here's the catch: we can't. If there existed such a set of functions, then it would be the case that the ball's position $x,y$ on the table would completely determine its angular orientation, just like it did for the cylinder. You can try this out yourself, though: take a ball and mark a starting spot on the table and a spot on the ball. Put the ball on the starting point so that the marked spot on the ball is on top, and roll the ball around the table without slipping. You'll quickly discover that the ball's position on the table does not determine its orientation: when you bring the ball back to the starting point, the marked point will generally not be on top. In fact, you can bring pretty much any point to be on top of the ball when the ball returns to its starting point.\n\nThis means that there do not exists functions $f_i(q_j)$ of the coordinates which can be obtained by \"integrating\" the differential constraints above. Rather than a constraint between the coordinates themselves, we are \"stuck\" with a constraint between the infinitesimal changes of the coordinates.\n\nSuppose a particle is moving on the surface of the sphere, you can write the equation of the distance between the centre of the sphere and the particle( radius of the sphere) as $x^2+y^2+z^2 = R^2$, here $x,y,z$ are the Cartesian coordinates and $R$ is radius of the sphere. This is an example of holonomic constraint. Now suppose that particle is not bound to move on the surface of the sphere, in this you can not write the equation as given above. It is a non-holomomic constraint. See the book \"Introduction to classical mechanics\" by Puranik and Takwale." ]

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[ "## Rough volatility with Python¶\n\nJim Gatheral\n\nFor Python Quants, New York, Friday May 6, 2016\n\n### Acknowledgements¶\n\nThe code in this iPython notebook used to be in R. I am very grateful to Yves Hilpisch and Michael Schwed for translating my R-code to Python.\n\nFor slideshow functionality I use RISE by Damián Avila.\n\n$$\\newcommand{\\beas}{\\begin{eqnarray*}} \\newcommand{\\eeas}{\\end{eqnarray*}} \\newcommand{\\bea}{\\begin{eqnarray}} \\newcommand{\\eea}{\\end{eqnarray}} \\newcommand{\\ben}{\\begin{enumerate}} \\newcommand{\\een}{\\end{enumerate}} \\newcommand{\\bi}{\\begin{itemize}} \\newcommand{\\ei}{\\end{itemize}} \\newcommand{\\beq}{\\begin{equation}} \\newcommand{\\eeq}{\\end{equation}} \\newcommand{\\bv}{\\begin{verbatim}} \\newcommand{\\ev}{\\end{verbatim}} \\newcommand{\\E}{\\mathbb{E}} \\newcommand{\\R}{\\mathbb{R}} \\newcommand{\\mP}{\\mathbb{P}} \\newcommand{\\mQ}{\\mathbb{Q}} \\newcommand{\\sigl}{\\sigma_L} \\newcommand{\\BS}{\\rm BS} \\newcommand{\\vix}{\\text{VIX}} \\newcommand{\\p}{\\partial} \\newcommand{\\var}{{\\rm var}} \\newcommand{\\cov}{{\\rm cov}} \\newcommand{\\mt}{\\mathbf{t}} \\newcommand{\\mS}{\\mathbf{S}} \\newcommand{\\tC}{\\widetilde{C}} \\newcommand{\\hC}{\\widehat{C}} \\newcommand{\\cE}{\\mathcal{E}} \\newcommand{\\tH}{\\widetilde{H}} \\newcommand{\\cD}{\\mathcal{D}} \\newcommand{\\cM}{\\mathcal{M}} \\newcommand{\\cS}{\\mathcal{S}} \\newcommand{\\cR}{\\mathcal{R}} \\newcommand{\\cF}{\\mathcal{F}} \\newcommand{\\cV}{\\mathcal{V}} \\newcommand{\\cG}{\\mathcal{G}} \\newcommand{\\cv}{\\mathcal{v}} \\newcommand{\\cg}{\\mathcal{g}} \\newcommand{\\cL}{\\mathcal{L}} \\newcommand{\\cO}{\\mathcal{O}} \\newcommand{\\dt}{\\Delta t} \\newcommand{\\tr}{{\\rm tr}} \\newcommand{\\sgn}{\\mathrm{sign}} \\newcommand{\\ee}{{\\mathbb{E}\\left[{#1}\\right]}} \\newcommand{\\eef}{{\\mathbb{E}\\left[\\left.{#1}\\right|\\cF_t\\right]}} \\newcommand{\\eefm}{{\\mathbb{E}^{#2}\\left[\\left.{#1}\\right|\\cF_t\\right]}} \\newcommand{\\angl}{{\\langle{#1}\\rangle}}$$\n\n### Outline of presentation¶\n\n• The time series of historical volatility\n\n• Scaling properties\n• The RFSV model\n\n• Pricing under rough volatility\n\n• Forecasting realized variance\n\n• The time series of variance swaps\n\n• Relating historical and implied\n\n### The time series of realized variance¶\n\n• Assuming an underlying variance process $v_s$, integrated variance $\\frac 1 \\delta \\,\\int_t^{t+\\delta}\\,v_s\\,ds$ may (in principle) be estimated arbitrarily accurately given enough price data.\n\n• In practice, market microstructure noise makes estimation harder at very high frequency.\n• Sophisticated estimators of integrated variance have been developed to adjust for market microstructure noise. See Gatheral and Oomen (for example) for details of these.\n• The Oxford-Man Institute of Quantitative Finance makes historical realized variance (RV) estimates freely available at http://realized.oxford-man.ox.ac.uk. These estimates are updated daily.\n\n• Each day, for 21 different indices, all trades and quotes are used to estimate realized (or integrated) variance over the trading day from open to close.\n• Using daily RV estimates as proxies for instantaneous variance, we may investigate the time series properties of $v_t$ empirically.\n\nFirst load all necessary Python libraries.\n\nIn :\nimport warnings; warnings.simplefilter('ignore')\nimport datetime as dt\nimport numpy as np\nimport matplotlib.pyplot as plt\nimport matplotlib.mlab as mlab\nfrom matplotlib.mlab import stineman_interp\nimport pandas as pd\nimport pandas.io.data as web\nimport requests\nimport zipfile as zi\nimport StringIO as sio\nfrom sklearn import datasets, linear_model\nimport scipy.special as scsp\nimport statsmodels.api as sm\nimport math\nimport seaborn as sns; sns.set()\n%matplotlib inline\n\n\nThen update and save the latest Oxford-Man data.\n\nIn :\nurl = 'http://realized.oxford-man.ox.ac.uk/media/1366/'\nurl += 'oxfordmanrealizedvolatilityindices.zip'\ndata = requests.get(url, stream=True).content\nz = zi.ZipFile(sio.StringIO(data))\nz.extractall()\n\n\nThere are many different estimates of realized variance, all of them very similar. We will use the realized kernel estimates denoted by \".rk\".\n\nIn :\ndf = pd.read_csv('OxfordManRealizedVolatilityIndices.csv', index_col=0, header=2 )\nrv1 = pd.DataFrame(index=df.index)\nfor col in df.columns:\nif col[-3:] == '.rk':\nrv1[col] = df[col]\nrv1.index = [dt.datetime.strptime(str(date), \"%Y%m%d\") for date in rv1.index.values]\n\n\nLet's plot SPX realized variance.\n\nIn :\nspx = pd.DataFrame(rv1['SPX2.rk'])\nspx.plot(color='red', grid=True, title='SPX realized variance',\nfigsize=(16, 9), ylim=(0,0.003));", null, "Figure 1: Oxford-Man KRV estimates of SPX realized variance from January 2000 to the current date.\n\nIn :\nspx.head()\n\nOut:\nSPX2.rk\n2000-01-03 0.000161\n2000-01-04 0.000264\n2000-01-05 0.000305\n2000-01-06 0.000149\n2000-01-07 0.000123\nIn :\nspx.tail()\n\nOut:\nSPX2.rk\n2016-04-27 0.000031\n2016-04-28 0.000031\n2016-04-29 0.000048\n2016-05-02 0.000028\n2016-05-03 0.000041\n\nWe can get SPX data from Yahoo using the DataReader function:\n\nIn :\nSPX = web.DataReader(name = '^GSPC',data_source = 'yahoo', start='2000-01-01')\nSPX.plot(title='SPX',figsize=(14, 8));", null, "### The smoothness of the volatility process¶\n\nFor $q\\geq 0$, we define the $q$th sample moment of differences of log-volatility at a given lag $\\Delta$.($\\angl{\\cdot}$ denotes the sample average):\n\n$$m(q,\\Delta)=\\angl{\\left|\\log \\sigma_{t+\\Delta} -\\log \\sigma_{t} \\right|^q}$$\n\nFor example\n\n$$m(2,\\Delta)=\\angl{\\left(\\log \\sigma_{t+\\Delta} -\\log \\sigma_{t} \\right)^2}$$\n\nis just the sample variance of differences in log-volatility at the lag $\\Delta$.\n\n### Scaling of $m(q,\\Delta)$ with lag $\\Delta$¶\n\nIn :\nspx['sqrt']= np.sqrt(spx['SPX2.rk'])\nspx['log_sqrt'] = np.log(spx['sqrt'])\n\ndef del_Raw(q, x):\nreturn [np.mean(np.abs(spx['log_sqrt'] - spx['log_sqrt'].shift(lag)) ** q)\nfor lag in x]\n\nIn :\nplt.figure(figsize=(8, 8))\nplt.xlabel('$log(\\Delta)$')\nplt.ylabel('$log\\ m(q.\\Delta)$')\nplt.ylim=(-3, -.5)\n\nzeta_q = list()\nqVec = np.array([.5, 1, 1.5, 2, 3])\nx = np.arange(1, 100)\nfor q in qVec:\nplt.plot(np.log(x), np.log(del_Raw(q, x)), 'o')\nmodel = np.polyfit(np.log(x), np.log(del_Raw(q, x)), 1)\nplt.plot(np.log(x), np.log(x) * model + model)\nzeta_q.append(model)\n\nprint zeta_q\n\n[0.072526605081197543, 0.14178030350291049, 0.20760692212873791, 0.27007948205423049, 0.38534332343872962]", null, "Figure 2: $\\log m(q,\\Delta)$ as a function of $\\log \\Delta$, SPX.\n\n### Monofractal scaling result¶\n\n• From the above log-log plot, we see that for each $q$, $m(q,\\Delta) \\propto \\Delta ^{\\zeta_q}$.\n• How does $\\zeta_q$ scale with $q$?\n\n### Scaling of $\\zeta_q$ with $q$¶\n\nIn :\nplt.figure(figsize=(8,8))\nplt.xlabel('q')\nplt.ylabel('$\\zeta_{q}$')\nplt.plot(qVec, zeta_q, 'or')\n\nline = np.polyfit(qVec[:4], zeta_q[:4],1)\nplt.plot(qVec, line * qVec + line)\nh_est= line\nprint(h_est)\n\n0.131697049909", null, "Figure 3: Scaling of $\\zeta_q$ with $q$.\n\nWe find the monofractal scaling relationship\n\n$$\\zeta_q = q\\,H$$\n\nwith $H \\approx 0.13$.\n\n• Note however that $H$ does vary over time, in a narrow range.\n• Note also that our estimate of $H$ is biased high because we proxied instantaneous variance $v_t$ with its average over each day $\\frac 1T\\,\\int_0^T\\,v_t\\,dt$, where $T$ is one trading day.\n\n### Estimated $H$ for all indices¶\n\nWe now repeat this analysis for all 21 indices in the Oxford-Man dataset.\n\nIn :\ndef dlsig2(sic, x, pr=False):\nif pr:\na= np.array([(sig-sig.shift(lag)).dropna() for lag in x])\na=a ** 2\nprint a.info()\nreturn [np.mean((sig-sig.shift(lag)).dropna() ** 2) for lag in x]\n\nIn :\nh = list()\nnu = list()\n\nfor col in rv1.columns:\nsig = rv1[col]\nsig = np.log(np.sqrt(sig))\nsig = sig.dropna()\nmodel = np.polyfit(np.log(x), np.log(dlsig2(sig, x)), 1)\nnu.append(np.sqrt(np.exp(model)))\nh.append(model/2.)\n\nOxfordH = pd.DataFrame({'names':rv1.columns, 'h_est': h, 'nu_est': nu})\n\n\nIn :\nOxfordH\n\nOut:\nh_est names nu_est\n0 0.133954 SPX2.rk 0.321337\n1 0.142315 FTSE2.rk 0.270677\n2 0.113366 N2252.rk 0.320396\n3 0.150251 GDAXI2.rk 0.274873\n4 NaN RUT2.rk NaN\n5 0.083370 AORD2.rk 0.359025\n6 0.131013 DJI2.rk 0.317327\n7 NaN IXIC2.rk NaN\n8 0.130485 FCHI2.rk 0.291967\n9 0.103370 HSI2.rk 0.281453\n10 0.125847 KS11.rk 0.274713\n11 0.145671 AEX.rk 0.290187\n12 0.178427 SSMI.rk 0.223249\n13 0.128117 IBEX2.rk 0.281662\n14 0.110092 NSEI.rk 0.324883\n15 0.092252 MXX.rk 0.324180\n16 0.106595 BVSP.rk 0.312907\n17 NaN GSPTSE.rk NaN\n18 0.119857 STOXX50E.rk 0.337045\n19 0.127094 FTSTI.rk 0.228910\n20 0.133361 FTSEMIB.rk 0.298712\n\n### Distributions of $(\\log \\sigma_{t+\\Delta}-\\log \\sigma_t)$ for various lags $\\Delta$¶\n\nHaving established these beautiful scaling results for the moments, how do the histograms look?\n\nIn :\ndef plotScaling(j, scaleFactor):\ncol_name = rv1.columns[j]\nv = rv1[col_name]\nx = np.arange(1,101)\n\ndef xDel(x, lag):\nreturn x-x.shift(lag)\n\ndef sdl(lag):\nreturn (xDel(np.log(v), lag)).std()\n\nsd1 = (xDel(np.log(v), 1)).std()\nh = OxfordH['h_est'][j]\nf, ax = plt.subplots(2,2,sharex=False, sharey=False, figsize=(10, 10))\n\nfor i_0 in range(0, 2):\nfor i_1 in range(0, 2):\nla = scaleFactor ** (i_1*1+i_0*2)\n\nhist_val = xDel(np.log(v), la).dropna()\nstd = hist_val.std()\nmean = hist_val.mean()\n\nax[i_0][i_1].set_title('Lag = %s Days' %la)\nn, bins, patches = ax[i_0][i_1].hist(hist_val.values, bins=100,\nnormed=1, facecolor='green',alpha=0.2)\nax[i_0][i_1].plot(bins, mlab.normpdf(bins,mean,std), \"r\")\nax[i_0][i_1].plot(bins, mlab.normpdf(bins,0,sd1 * la ** h), \"b--\")\nhist_val.plot(kind='density', ax=ax[i_0][i_1])\n\n\nIn :\n plotScaling(1,5)", null, "Figure 4: Histograms of $(\\log \\sigma_{t+\\Delta}-\\log \\sigma_t)$ for various lags $\\Delta$; normal fit in red; $\\Delta=1$ normal fit scaled by $\\Delta^{0.14}$ in blue.\n\n### A natural model of realized volatility¶\n\n• Moreover, the scaling property of variance of RV differences suggests the model:\n\n(1) $$\\log \\sigma_{t+\\Delta} - \\log \\sigma_t =\\nu\\,\\left( W^H_{t+\\Delta}-W^H_t\\right)$$\n\nwhere $W^H$ is fractional Brownian motion.\n\n### Fractional Brownian motion (fBm)¶\n\n• Fractional Brownian motion (fBm) $\\{W^H_t; t \\in \\mathbb{R}\\}$ is the unique Gaussian process with mean zero and autocovariance function $$\\ee{ W^H_t\\,W^H_s } = \\frac12\\,\\left\\{ |t|^{2\\,H}+|s|^{2\\,H}-|t-s|^{2\\,H} \\right\\}$$ where $H \\in (0,1)$ is called the Hurst index or parameter.\n\n• In particular, when $H=1/2$, fBm is just Brownian motion.\n\n• If $H>1/2$, increments are positively correlated.% so the process is trending.\n\n• If $H<1/2$, increments are negatively correlated.% so the process is reverting.\n\n### Representations of fBm¶\n\nThere are infinitely many possible representations of fBm in terms of Brownian motion. For example, with $\\gamma = \\frac 12 - H$,\n\n#### Mandelbrot-Van Ness\n\n$$W^H_t ={C_H}\\,\\left\\{\\int_{-\\infty}^t \\,\\frac{dW_s}{(t-s)^\\gamma} - \\int_{-\\infty}^0 \\,\\frac{dW_s}{(-s)^\\gamma}\\right\\}.$$\n\nThe choice\n\n$$C_H = \\sqrt{ \\frac{2\\,H\\,\\Gamma(3/2-H)}{\\Gamma(H+1/2)\\,\\Gamma(2-2\\,H)}}$$\n\nensures that\n\n$$\\ee{W^H_t\\,W^H_s }= \\frac{1}{2}\\,\\left\\{t^{2 H} + s^{2 H} - |t-s|^{2 H}\\right\\}.$$\n\n###", null, "Figure 8: Volatility of SPX (above) and of the RFSV model (below).\n\n### Remarks on the comparison¶\n\n• The simulated and actual graphs look very alike.\n\n• Persistent periods of high volatility alternate with low volatility periods.\n\n• $H \\sim 0.1$ generates very rough looking sample paths (compared with $H=1/2$ for Brownian motion).\n\n• Hence rough volatility.\n\nThen\n\n$$\\begin{eqnarray} &&\\log v_u - \\log v_t \\nonumber\\\\ &=&\\nu\\,C_H\\,\\left\\{ \\int_t^u\\,\\frac{1}{(u-s)^\\gamma}\\,d{W}^{\\mP}_s +\\int_{-\\infty}^t\\,\\left[ \\frac{1}{(u-s)^\\gamma}-\\frac{1}{(t-s)^\\gamma} \\right]\\,d{W}^{\\mP}_s\\right\\}\\nonumber\\\\ &=:& 2\\,\\nu\\,C_H\\,\\left[M_t(u)+ Z_t(u)\\right]. \\end{eqnarray}$$\n• Note that $\\eefm{M_t(u)}{\\mP}=0$ and $Z_t(u)$ is $\\cF_t$-measurable.\n\n### Pricing under $\\mQ$¶\n\nOur model under $\\mP$ reads:\n\n(2) $$v_u =\\eefm{v_u}{\\mP}\\,\\cE\\left(\\eta\\,\\tilde W^{\\mP}_t(u) \\right). %\\label{eq:Pmodel}$$\n\nConsider some general change of measure\n\n$$dW^{\\mP}_s = dW^{\\mQ}_s + \\lambda_s\\,ds, %\\label{eq:dQdP}$$\n\nwhere $\\{ \\lambda_s: s > t \\}$ has a natural interpretation as the price of volatility risk.\n\nWe may then rewrite (2) as\n\n$$v_u = \\eefm{v_u}{\\mP}\\,\\cE\\left(\\eta\\,\\tilde W^{\\mQ}_t(u) \\right) \\exp \\left\\{ \\eta\\,\\sqrt{2\\,H}\\, \\int_t^u\\,\\frac{\\lambda_s}{(u-s)^\\gamma}\\,ds\\right\\}. %\\label{eq:explicitBergomiQ1}$$\n• Although the conditional distribution of $v_u$ under $\\mP$ is lognormal, it will not be lognormal in general under $\\mQ$.\n\n• The upward sloping smile in VIX options means $\\lambda_s$ cannot be deterministic in this picture.\n\n### The rough Bergomi (rBergomi) model¶\n\nLet's nevertheless consider the simplest change of measure\n\n$$d{W}^{\\mP}_s = d{W}^{\\mQ}_s + \\lambda(s)\\,ds,$$\n\nwhere $\\lambda(s)$ is a deterministic function of $s$. Then from (2), we would have\n\n$$\\begin{eqnarray} v_u &=& \\eefm{v_u}{\\mP}\\,\\cE\\left(\\eta\\,\\tilde W^{\\mQ}_t(u) \\right) \\exp \\left\\{ \\eta\\,\\sqrt{2\\,H}\\, \\int_t^u\\,\\frac{1}{(u-s)^\\gamma}\\,\\lambda(s)\\,ds\\right\\}\\nonumber\\\\ &=& \\xi_t(u) \\,\\cE\\left(\\eta\\,\\tilde W^{\\mQ}_t(u) \\right)%\\label{eq:explicitBergomiQ} \\end{eqnarray}$$\n\nwhere the forward variances $\\xi_t(u) = \\eefm{v_u}{\\mQ}$ are (at least in principle) tradable and observed in the market.\n\n• $\\xi_t(u)$ is the product of two terms:\n\n• $\\eefm{v_u}{\\mP}$ which depends on the historical path $\\{W_s, s • a term which depends on the price of risk$\\lambda(s)$. ### Features of the rough Bergomi model¶ • The rBergomi model is a non-Markovian generalization of the Bergomi model: $$\\eef{v_u}\\neq \\E[v_u|v_t].$$ • The rBergomi model is Markovian in the (infinite-dimensional) state vector$\\eefm{v_u}{\\mQ}=\\xi_t(u)$. • We have achieved our aim from Session 1 of replacing the exponential kernels in the Bergomi model with a power-law kernel. • We may therefore expect that the rBergomi model will generate a realistic term structure of ATM volatility skew. ### Re-interpretation of the conventional Bergomi model¶ • A conventional$n$-factor Bergomi model is not self-consistent for an arbitrary choice of the initial forward variance curve$\\xi_t(u)$. •$\\xi_t(u)=\\eef{v_u}$should be consistent with the assumed dynamics. • Viewed from the perspective of the fractional Bergomi model however: • The initial curve$\\xi_t(u)$reflects the history$\\{W_s; s\n• The exponential kernels in the exponent of the conventional Bergomi model approximate more realistic power-law kernels.\n• The conventional two-factor Bergomi model is then justified in practice as a tractable Markovian engineering approximation to a more realistic fractional Bergomi model.\n\n### The stock price process¶\n\n• The observed anticorrelation between price moves and volatility moves may be modeled naturally by anticorrelating the Brownian motion $W$ that drives the volatility process with the Brownian motion driving the price process.\n• Thus $$\\frac{dS_t}{S_t}=\\sqrt{v_t}\\,dZ_t$$ with $$dZ_t = \\rho\\,dW_t + \\sqrt{1-\\rho^2}\\,dW^\\perp_t$$ where $\\rho$ is the correlation between volatility moves and price moves.\n\n### Simulation of the rBergomi model¶\n\nWe simulate the rBergomi model as follows:\n\n• Construct the joint covariance matrix for the Volterra process $\\tilde W$ and the Brownian motion $Z$ and compute its Cholesky decomposition.\n• For each time, generate iid normal random vectors {and multiply them by the lower-triangular matrix obtained by the Cholesky decomposition} to get a $m \\times 2\\,n$ matrix of paths of $\\tilde W$ and $Z$ with the correct joint marginals.\n• With these paths held in memory, we may evaluate the expectation under $\\mQ$ of any payoff of interest.\n• This procedure is very slow!\n\n• Speeding up the simulation is work in progress.\n\n### Guessing rBergomi model parameters¶\n\n• The rBergomi model has only three parameters: $H$, $\\eta$ and $\\rho$.\n• If we had a fast simulation, we could just iterate on these parameters to find the best fit to observed option prices. But we don't.\n• However, the model parameters $H$, $\\eta$ and $\\rho$ have very direct interpretations:\n\n• $H$ controls the decay of ATM skew $\\psi(\\tau)$ for very short expirations.\n\n• The product $\\rho\\,\\eta$ sets the level of the ATM skew for longer expirations.\n\n• Keeping $\\rho\\,\\eta$ constant but decreasing $\\rho$ (so as to make it more negative) pushes the minimum of each smile towards higher strikes.\n• So we can guess parameters in practice.\n• As we will see, even without proper calibration (i.e. just guessing parameters), rBergomi model fits to the volatility surface are amazingly good.\n\n### SPX smiles in the rBergomi model¶\n\n• In Figures 9 and 10, we show how well a rBergomi model simulation with guessed parameters fits the SPX option market as of February 4, 2010, a day when the ATM volatility term structure happened to be pretty flat.\n\n• rBergomi parameters were: $H=0.07$, $\\eta=1.9$, $\\rho=-0.9$.\n• Only three parameters to get a very good fit to the whole SPX volatility surface!\n\n###", null, "Figure 9: Red and blue points represent bid and offer SPX implied volatilities; orange smiles are from the rBergomi simulation.\n\n###", null, "Figure 10: Red and blue points represent bid and offer SPX implied volatilities; orange smile is from the rBergomi simulation.\n\n### ATM volatilities and skews¶\n\nIn Figures 11 and 12, we see just how well the rBergomi model can match empirical skews and vols. Recall also that the parameters we used are just guesses!\n\n###", null, "Figure 11: Blue points are empirical skews; the red line is from the rBergomi simulation.\n\n###", null, "Figure 12: Blue points are empirical ATM volatilities; the red line is from the rBergomi simulation.\n\n### Another date¶\n\n• Now we take a look at another date: August 14, 2013, two days before the last expiration date in our dataset.\n• Options set at the open of August 16, 2013 so only one trading day left.\n• Note in particular that the extreme short-dated smile is well reproduced by the rBergomi model.\n• There is no need to add jumps!\n\n###", null, "Figure 13: Red and blue points represent bid and offer SPX implied volatilities; orange smiles are from the rBergomi simulation.\n\n### The forecast formula¶\n\nThus, we obtain\n\n#### Variance forecast formula\n\n(3) $$\\eefm{v_{t+\\Delta}}{\\mP}=\\exp\\left\\{\\eefm{\\log(v_{t+\\Delta})}{\\mP}+2\\, c\\,\\nu^2\\Delta^{2\\,H}\\right\\} %\\label{eq:vForecast}$$\n\nwhere\n\n$$\\beas &&\\eefm{\\log v_{t+\\Delta}}{\\mP}\\\\ &&= \\frac{\\cos(H\\pi)}{\\pi} \\Delta^{H+1/2} \\int_{-\\infty}^t \\frac{\\log v_s}{(t-s+\\Delta)(t-s)^{H+1/2}} ds. \\eeas$$\n\n### Implement variance forecast in Python¶\n\nIn :\ndef c_tilde(h):\nreturn scsp.gamma(3. / 2. - h) / scsp.gamma(h + 1. / 2.) * scsp.gamma(2. - 2. * h)\n\ndef forecast_XTS(rvdata, h, date, nLags, delta, nu):\ni = np.arange(nLags)\ncf = 1./((i + 1. / 2.) ** (h + 1. / 2.) * (i + 1. / 2. + delta))\nldata = rvdata.truncate(after=date)\nl = len(ldata)\nldata = np.log(ldata.iloc[l - nLags:])\nldata['cf'] = np.fliplr([cf])\n# print ldata\nldata = ldata.dropna()\nfcst = (ldata.iloc[:, 0] * ldata['cf']).sum() / sum(ldata['cf'])\n\nreturn math.exp(fcst + 2 * nu ** 2 * c_tilde(h) * delta ** (2 * h))\n\n\n### SPX actual vs forecast variance¶\n\nIn :\nrvdata = pd.DataFrame(rv1['SPX2.rk'])\nnu = OxfordH['nu_est'] # Vol of vol estimate for SPX\nh = OxfordH['h_est']\nn = len(rvdata)\ndelta = 1\nnLags = 500\ndates = rvdata.iloc[nLags:n-delta].index\nrv_predict = [forecast_XTS(rvdata, h=h, date=d, nLags=nLags,\ndelta=delta, nu=nu) for d in dates]\nrv_actual = rvdata.iloc[nLags+delta:n].values\n\n\n### Scatter plot of delta days ahead predictions¶\n\nIn :\nplt.figure(figsize=(8, 8))\nplt.plot(rv_predict, rv_actual, 'bo');", null, "Figure 14: Actual vols vs predicted vols.\n\n### Superimpose actual and predicted vols¶\n\nIn :\nplt.figure(figsize=(11, 6))\nvol_actual = np.sqrt(np.multiply(rv_actual,252))\nvol_predict = np.sqrt(np.multiply(rv_predict,252))\nplt.plot(vol_actual, \"b\")\nplt.plot(vol_predict, \"r\");", null, "Figure 15: Actual volatilities in blue; predicted vols in red.\n\n### Forecasting the variance swap curve¶\n\nFinally, we forecast the whole variance swap curve using the variance forecasting formula (3).\n\nIn :\ndef xi(date, tt, nu,h, tscale): # dt=(u-t) is in units of years\nrvdata = pd.DataFrame(rv1['SPX2.rk'])\nreturn [ forecast_XTS(rvdata,h=h,date=date,nLags=500,delta=dt*tscale,nu=nu) for dt in tt]\n\nnu = OxfordH[\"nu_est\"]\nh = OxfordH[\"h_est\"]\n\ndef varSwapCurve(date, bigT, nSteps, nu, h, tscale, onFactor):\n# Make vector of fwd variances\ntt = [ float(i) * (bigT) / nSteps for i in range(nSteps+1)]\ndelta_t = tt\nxicurve = xi(date, tt, nu, h, tscale)\nxicurve_mid = (np.array(xicurve[0:nSteps]) + np.array(xicurve[1:nSteps+1])) / 2\nxicurve_int = np.cumsum(xicurve_mid) * delta_t\nvarcurve1 = np.divide(xicurve_int, np.array(tt[1:]))\nvarcurve = np.array([xicurve,]+list(varcurve1))\nvarcurve = varcurve * onFactor * tscale # onFactor is to compensate for overnight moves\nres = pd.DataFrame({\"texp\":np.array(tt), \"vsQuote\":np.sqrt(varcurve)})\nreturn(res)\n\nIn :\ndef varSwapForecast(date,tau,nu,h,tscale,onFactor):\nvsc = varSwapCurve(date, bigT=2.5, nSteps=100, nu=nu, h=h,\ntscale=tscale, onFactor=onFactor) # Creates the whole curve\nx = vsc['texp']\ny = vsc['vsQuote']\nres = stineman_interp(tau,x,y,None)\n\nreturn(res)\n\n# Test the function\n\ntau = (.25,.5,1,2)\ndate = dt.datetime(2008,9,8)\nvarSwapForecast(date,tau,nu=nu,h=h,tscale=252,onFactor=1)\n\nOut:\narray([ 0.21949454, 0.21398188, 0.2117466 , 0.21262899])\n\n### 'Constructing a time series of variance swap curves¶\n\nFor each of 2,658 days from Jan 27, 2003 to August 31, 2013:\n\n• We compute proxy variance swaps from closing prices of SPX options sourced from OptionMetrics (www.optionmetrics.com) via WRDS.\n• We note that the actual variance swap curve is a factor (of roughly 1.4) higher than the forecast, which we may attribute to a combination of overnight movements of the index and the price of volatility risk.\n• Forecasts must therefore be rescaled to obtain close-to-close realized variance forecasts.\n\n###", null, "Figure 16: Actual (proxy) 3-month variance swap quotes in blue vs forecast in red (with no scaling factor).\n\n###", null, "Figure 17: The ratio between 3-month actual variance swap quotes and 3-month forecasts.\n\n### The Lehman weekend¶\n\n• Empirically, it seems that the variance curve is a simple scaling factor times the forecast, but that this scaling factor is time-varying.\n\n• We can think of this factor as having two multiplicative components: the overnight factor, and the price of volatility risk.\n• Recall that as of the close on Friday September 12, 2008, it was widely believed that Lehman Brothers would be rescued over the weekend. By Monday morning, we knew that Lehman had failed.\n• In Figure 18, we see that variance swap curves just before and just after the collapse of Lehman are just rescaled versions of the RFSV forecast curves.\n\n### We need variance swap estimates for 12-Sep-2008 and 15-Sep-2008¶\n\nWe proxy these by taking SVI fits for the two dates and computing the log-strips.\n\nIn :\nvarSwaps12 =(\n0.2872021, 0.2754535, 0.2601864, 0.2544684, 0.2513854, 0.2515314,\n0.2508418, 0.2520099, 0.2502763, 0.2503309, 0.2580933, 0.2588361,\n0.2565093)\n\ntexp12 = (\n0.01916496, 0.04654346, 0.09582478, 0.19164956, 0.26830938, 0.29842574,\n0.51745380, 0.54483231, 0.76659822, 0.79397673, 1.26488706, 1.76317591,\n2.26146475)\n\nvarSwaps15 = (\n0.4410505, 0.3485560, 0.3083603, 0.2944378, 0.2756881, 0.2747838,\n0.2682212, 0.2679770, 0.2668113, 0.2706713, 0.2729533, 0.2689598,\n0.2733176)\n\ntexp15 = (\n0.01095140, 0.03832991, 0.08761123, 0.18343600, 0.26009582, 0.29021218,\n0.50924025, 0.53661875, 0.75838467, 0.78576318, 1.25667351, 1.75496235,\n2.25325120)\n\n\n### Actual vs predicted over the Lehman weekend¶\n\nIn :\nnu = OxfordH['nu_est']\nh = OxfordH['h_est']\ndate1 = dt.datetime(2008, 9, 12)\ndate2 = dt.datetime(2008, 9, 15)\n\n# Variance curve fV model forecasts\ntau1000 = [ float(i) * 2.5 / 1000. for i in range(1,1001)]\nvs1 = varSwapForecast(date1, tau1000, nu=nu,h=h, tscale=252, onFactor=1.29)\nvs2 = varSwapForecast(date2, tau1000, nu=nu,h=h, tscale=252, onFactor=1.29)\n\nIn :\nplt.figure(figsize=(11, 6))\nplt.plot(texp12, varSwaps12, \"r\")\nplt.plot(texp15, varSwaps15, \"b\")\nplt.plot(tau1000, vs1, \"r--\")\nplt.plot(tau1000, vs2, \"b--\");", null, "Figure 18: SPX variance swap curves as of September 12, 2008 (red) and September 15, 2008 (blue). The dashed curves are RFSV model forecasts rescaled by the 3-month ratio ($1.29$) as of the Friday close.\n\n### Remarks¶\n\nWe note that\n\n• The actual variance swaps curves are very close to the forecast curves, up to a scaling factor.\n• We are able to explain the change in the variance swap curve with only one extra observation: daily variance over the trading day on Monday 15-Sep-2008.\n• The SPX options market appears to be backward-looking in a very sophisticated way.\n\n### The Flash Crash¶\n\n• The so-called Flash Crash of Thursday May 6, 2010 caused intraday realized variance to be much higher than normal.\n• In Figure 19, we plot the actual variance swap curves as of the Wednesday and Friday market closes together with forecast curves rescaled by the 3-month ratio as of the close on Wednesday May 5 (which was $2.52$).\n• We see that the actual variance curve as of the close on Friday is consistent with a forecast from the time series of realized variance that includes the anomalous price action of Thursday May 6.\n\n### Variance swap estimates¶\n\nWe again proxy variance swaps for 05-May-2010, 07-May-2010 and 10-May-2010 by taking SVI fits (see [Gatheral and Jacquier]</a></sup> ) for the three dates and computing the log-strips.\n\nIn :\nvarSwaps5 = (\n0.4250369, 0.2552473, 0.2492892, 0.2564899, 0.2612677, 0.2659618, 0.2705928, 0.2761203,\n0.2828139, 0.2841165, 0.2884955, 0.2895839, 0.2927817, 0.2992602, 0.3116500)\n\ntexp5 = (\n0.002737851, 0.043805613, 0.120465435, 0.150581793, 0.197125257, 0.292950034,\n0.369609856, 0.402464066, 0.618754278, 0.654346338, 0.867898700, 0.900752909,\n1.117043121, 1.615331964, 2.631074606)\n\nvarSwaps7 = (\n0.5469727, 0.4641713, 0.3963352, 0.3888213, 0.3762354, 0.3666858, 0.3615814, 0.3627013,\n0.3563324, 0.3573946, 0.3495730, 0.3533829, 0.3521515, 0.3506186, 0.3594066)\n\ntexp7 = (\n0.01642710, 0.03832991, 0.11498973, 0.14510609, 0.19164956, 0.28747433, 0.36413415,\n0.39698836, 0.61327858, 0.64887064, 0.86242300, 0.89527721, 1.11156742, 1.60985626,\n2.62559890)\n\nvarSwaps10 = (\n0.3718439, 0.3023223, 0.2844810, 0.2869835, 0.2886912, 0.2905637, 0.2957070, 0.2960737,\n0.3005086, 0.3031188, 0.3058492, 0.3065815, 0.3072041, 0.3122905, 0.3299425)\n\ntexp10 = (\n0.008213552, 0.030116359, 0.106776181, 0.136892539, 0.183436003, 0.279260780,\n0.355920602, 0.388774812, 0.605065024, 0.640657084, 0.854209446, 0.887063655,\n1.103353867, 1.601642710, 2.617385352)\n\nIn :\ndate1 = dt.datetime(2010, 5, 5)\ndate2 = dt.datetime(2010, 5, 7)\n\nvsf5 = varSwapCurve(date1, bigT=2.5, nSteps=100, nu=nu, h=h, tscale=252, onFactor=2.52)\nvsf7 = varSwapCurve(date2, bigT=2.5, nSteps=100, nu=nu, h=h, tscale=252, onFactor=2.52)\n\nIn :\nplt.figure(figsize=(11, 6))\nplt.plot(texp5, varSwaps5, \"r\", label='May 5')\nplt.plot(texp7, varSwaps7, \"g\", label='May 7')\n\nplt.legend()\nplt.xlabel(\"Time to maturity\")\nplt.ylabel(\"Variance swap quote\")\n\nplt.plot(vsf5['texp'], vsf5['vsQuote'], \"r--\")\nplt.plot(vsf7['texp'], vsf7['vsQuote'], \"g--\");", null, "Figure 19: SPX variance swap curves as of May 5, 2010 (red) and May 7, 2010 (green). The dashed curves are RFSV model forecasts rescaled by the 3-month ratio ($2.52$) as of the close on Wednesday May 5. The curve as of the close on May 7 is consistent with the forecast including the crazy moves on May 6.\n\n### The weekend after the Flash Crash¶\n\nNow we plot forecast and actual variance swap curves as of the close on Friday May 7 and Monday May 10.\n\nIn :\ndate1 = dt.datetime(2010,5,7)\ndate2 = dt.datetime(2010,5,10)\n\nvsf7 = varSwapCurve(date1, bigT=2.5, nSteps=100, nu=nu, h=h, tscale=252, onFactor=2.52)\nvsf10 = varSwapCurve(date2, bigT=2.5, nSteps=100, nu=nu, h=h, tscale=252, onFactor=2.52)\n\nIn :\nplt.figure(figsize=(11, 6))\nplt.plot(texp7, varSwaps7, \"g\", label='May 7')\nplt.plot(texp10, varSwaps10, \"m\", label='May 10')\n\nplt.legend()\nplt.xlabel(\"Time to maturity\")\nplt.ylabel(\"Variance swap quote\")\n\nplt.plot(vsf7['texp'], vsf7['vsQuote'], \"g--\")\nplt.plot(vsf10['texp'], vsf10['vsQuote'], \"m--\");", null, "Figure 20: The May 10 actual curve is inconsistent with a forecast that includes the Flash Crash.\n\nNow let's see what happens if we exclude the Flash Crash from the time series used to generate the variance curve forecast.\n\nIn :\nplt.figure(figsize=(11, 6))\nax = plt.subplot(111)\nrvdata_p = rvdata.drop((dt.datetime(2010, 5, 6)), axis=0)\nrvdata.loc[\"2010-05-04\":\"2010-05-10\"].plot(ax=ax, legend=False)\nrvdata_p.loc[\"2010-05-04\":\"2010-05-10\"].plot(ax=ax, legend=False);", null, "Figure 21: rvdata_p has the May 6 realized variance datapoint eliminated (green line). Notice the crazy realized variance estimate for May 6!\n\nWe need a new variance curve forecast function that uses the new time series.\n\nIn :\ndef xip(date, tt, nu,h, tscale): # dt=(u-t) is in units of years\nrvdata = pd.DataFrame(rv1['SPX2.rk'])\nrvdata_p = rvdata.drop((dt.datetime(2010, 5, 6)), axis=0)\nreturn [ forecast_XTS(rvdata_p, h=h, date=date,nLags=500,\ndelta=delta_t * tscale, nu=nu) for delta_t in tt]\n\nnu = OxfordH[\"nu_est\"]\nh = OxfordH[\"h_est\"]\n\ndef varSwapCurve_p(date, bigT, nSteps, nu, h, tscale, onFactor):\n# Make vector of fwd variances\ntt = [ float(i) * (bigT) / nSteps for i in range(nSteps+1)]\ndelta_t = tt\nxicurve = xip(date, tt, nu, h, tscale)\nxicurve_mid = (np.array(xicurve[0:nSteps]) + np.array(xicurve[1:nSteps + 1])) / 2\nxicurve_int = np.cumsum(xicurve_mid) * delta_t\nvarcurve1 = np.divide(xicurve_int, np.array(tt[1:]))\nvarcurve = np.array([xicurve,]+list(varcurve1))\nvarcurve = varcurve * onFactor * tscale # onFactor is to compensate for overnight moves\nres = pd.DataFrame({\"texp\":np.array(tt), \"vsQuote\":np.sqrt(varcurve)})\nreturn(res)\n\ndef varSwapForecast_p(date, tau, nu, h, tscale, onFactor):\nvsc = varSwapCurve_p(date, bigT=2.5, nSteps=100, nu=nu, h=h,\ntscale=tscale, onFactor=onFactor) # Creates the whole curve\nx = vsc['texp']\ny = vsc['vsQuote']\nres = stineman_interp(tau, x, y, None)\n\nreturn(res)\n\n# Test the function\n\ntau = (.25, .5 ,1, 2)\ndate = dt.datetime(2010, 5, 10)\nvarSwapForecast_p(date, tau, nu=nu, h=h, tscale=252, onFactor=1. / (1 - .35))\n\nOut:\narray([ 0.26077084, 0.25255902, 0.25299844, 0.26116175])\n\nFinally, we compare our new forecast curves with the actuals.\n\nIn :\ndate1 = dt.datetime(2010, 5, 7)\ndate2 = dt.datetime(2010, 5, 10)\n\nvsf7 = varSwapCurve(date1, bigT=2.5, nSteps=100, nu=nu, h=h, tscale=252, onFactor=2.52)\nvsf10p = varSwapCurve_p(date2, bigT=2.5, nSteps=100, nu=nu, h=h, tscale=252, onFactor=2.52)\n\nIn :\nplt.figure(figsize=(11, 6))\nplt.plot(texp7, varSwaps7, \"g\", label='May 7')\nplt.plot(texp10, varSwaps10, \"m\", label='May 10')\n\nplt.legend()\nplt.xlabel(\"Time to maturity\")\nplt.ylabel(\"Variance swap quote\")\n\nplt.plot(vsf7['texp'], vsf7['vsQuote'], \"g--\")\nplt.plot(vsf10p['texp'], vsf10p['vsQuote'], \"m--\");", null, "Figure 22: The May 10 actual curve is consistent with a forecast that excludes the Flash Crash.\n\n### Resetting of expectations over the weekend¶\n\n• In Figures 20 and 22, we see that the actual variance swap curve on Monday, May 10 is consistent with a forecast that excludes the Flash Crash.\n• Volatility traders realized that the Flash Crash should not influence future realized variance projections.\n\n### Summary¶\n\n• We uncovered a remarkable monofractal scaling relationship in historical volatility.\n\n• A corollary is that volatility is not a long memory process, as widely believed.\n• This leads to a natural non-Markovian stochastic volatility model under $\\mP$.\n• The simplest specification of $\\frac{d\\mQ}{d\\mP}$ gives a non-Markovian generalization of the Bergomi model.\n• The history of the Brownian motion \\$\\lbrace W_s, s\n• This model fits the observed volatility surface surprisingly well with very few parameters.\n• For perhaps the first time, we have a simple consistent model of historical and implied volatility.\n\n1. ^ Torben G Andersen, Tim Bollerslev, Francis X Diebold, and Heiko Ebens, The distribution of realized stock return volatility, *Journal of Financial Economics* **61**(1) 43-76 (2001).\n2. ^ Christian Bayer, Peter Friz and Jim Gatheral, Pricing under rough volatility, *Quantitative Finance* forthcoming, available at http://papers.ssrn.com/sol3/papers.cfm?abstract_id=2554754, (2015).\n3. ^ Emmanuel Bacry and Jean-François Muzy, Log-infinitely divisible multifractal processes, *Communications in Mathematical Physics* **236**(3) 449-475 (2003).\n4. ^ Jim Gatheral and Antoine Jacquier, Arbitrage-free SVI volatility surfaces, *Quantitative Finance* **14**(1) 59-71 (2014).\n5. ^ Jim Gatheral, Thibault Jaisson and Mathieu Rosenbaum, Volatility is rough, available at http://papers.ssrn.com/sol3/papers.cfm?abstract_id=2509457, (2014).\n6. ^ Jim Gatheral and Roel Oomen, Zero-intelligence realized variance estimation, *Finance and Stochastics* **14**(2) 249-283 (2010).\n7. ^ Carl J. Nuzman and H. Vincent Poor, Linear estimation of self-similar processes via Lamperti’s transformation, *Journal of Applied Probability* **37**(2) 429-452 (2000)." ]

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[ "Optical Modeling for Polarization-dependent Optical Power Dissipation of Thin-film Organic Solar Cells at Oblique Incidence\n\n•", null, "• ABSTRACT\n\nWe present the optical models and calculation results of thin-film organic solar cells (OSCs) at oblique incidence of light, using the transfer matrix method. The simple expression for the optical power dissipation is derived at oblique incidence for s- and p-polarized light. The spatial distribution of the electric field intensity, the optical power density, and the optical power dissipation are calculated in both s- and p-polarized light with respect to the incidence angle. We identify how the light absorption efficiency for p-polarized light becomes relatively larger than that for s-polarized light as the incidence angle increases.\n\n• KEYWORD\n\nOrganic solar cell , Optical modeling , Oblique incidence , Transfer matrix method\n\n• I. INTRODUCTION\n\nOrganic solar cells (OSCs) have been intensively investi-gated as one of the promising green energy sources due to the feasibility of producing thin, light-weight, and flexible solar cells . However, the power conversion efficiency should be further improved for practical use of the OSC. Optical interference effects play an important role in the efficiency of multilayer OSCs, where the thickness of each thin-film layer should be carefully designed to maximize the optical absorption in the active region.\n\nThe optical interference effect of the multilayer solar cells has been modeled based on the finite element method (FEM) [2, 3], the rigorous coupled-wave analysis (RCWA) [4, 5], and the transfer matrix method (TMM) [6-8]. Among them, the TMM has been widely used for thin-film OSCs due to its relatively ease of simulation code implementation but high accuracy in calculation results. The TMM-based optical model for the OSC provides the spatial distribution of the time-average optical power dissipation Q (z)usually for normal incidence of sunlight [6-8]. Here, Q(z) is related with the amount of photon energy absorbed by the material per unit time and per unit volume and z is a position within the multilayer OSC.\n\nRecently, the angular response of thin-film OSCs with respect to the incidence angle of sunlight has been intensi-vely investigated due to the fact that most OSCs, having the advantage of low-cost production and flexible appli-cation,are expected to be used without a sunlight tracking system. The TMM-based optical model for normal incidence has been adopted for oblique incidence to analyze the angular response of thin-film OSCs [9-11]. However, the previous TMM-based optical models for oblique incidence of sunlight are still insufficient.\n\nIt was mathematically proved that Q(z) can be expressed as Q(z)= (1/2)0|E(z)|2 for normal incidence of sunlight. Here, c is the speed of light in free space, ε0 is the permittivity in free space, n is the real part of the complex refractive index, α is the absorption coefficient, and E(z) is the electric field amplitude at the position z. In the case of oblique incidence, Q(z) was also assumed to be pro-portional to |E(z)|2 for both s- and p-polarized sunlight[10, 11]. However, we have not yet found any text to provide a clear derivation for the relation between Q(z) and |E(z)|2 at oblique incidence, especially for p-polarized light. In addition, the effect of the incidence angle and light polarization on the distribution of Q(z) has not yet been investigated quantitatively.\n\nIn this paper, we present comprehensive optical modeling and calculation results of thin-film OSCs at oblique incidence of sunlight. By applying the TMM at oblique incidence, we derive a simple expression for Q(z) for both s- and p-polarized light. We calculate the spatial distribution of the electric field intensity, the time-average Poynting vector(optical power density), and the optical power dissipation with respect to the incidence angle. We investigate how the optical interference effect of thin-film OSCs is affected by the incidence angle and light polarization.\n\nII. OPTICAL MODEL\n\nFig. 1 shows a stack of M layers, where a plane wave with an incident angle θ and a wavelength λ is incident from the semi-infinite transparent ambient on the left (j=0). The multilayer stack is assumed to be isotropic and homo-geneous with plane and parallel interfaces. The orientation of the electromagnetic field oscillations is composed of s-polarization and p-polarization that are perpendicular to or parallel to the incident plane, respectively. Each thin-film layer j (j=1, 2, …, M) has a thickness of dj and a complex refractive index of\n\nwhere nj and Kj are the refractive index and extinction coefficient, respectively.\n\nIn Fig. 1, the wave vector in the transparent ambient of the 0-th layer is given by\n\nwhere n0 is the refractive index of the 0-th layer and k0=2π/λ . In the j-th layer, the wave vector is written as", null, "where the relation kx,j=kx,0 is ascribed to the boundary condition that the transverse wave vector should be continuous at the interface. The terms\n\nand\n\ncorrespond to the effective refractive index and the extinction coefficient for the z direction. From the dispersion relation\n\nwe have\n\nConsequently, we obtain\n\n2.1. Internal Reflection and Transmission Coefficients in Layer j\n\nIn Fig. 1, we assign a positive (negative) direction to the light propagating from left (right) to right (left), designating the + (-) superscripts. The electric field amplitudes on the interface between the adjacent layer j and k=j+1 are described by an interface matrix Ijk:\n\nwhere\n\nand\n\nare the forward-propagating electric field amplitudes at the right boundary of the layer j and the left boundary of the layer k, respectively. The terms rjk and tjk are the complex Fresnel reflection and transmission coefficients on the interface of the adjacent layer. For s-polarized light, they are defined as \n\nand for p-polarized light as\n\nThe propagation of the electric field amplitude between the left and right boundaries of the j-th layer is described by a layer matrix Lj:\n\nwhere we assume the time dependence of eiwt.\n\nIn Fig. 1, the electric field amplitude passing through all the M multilayers from the ambient on the left (layer 0) to the right (layer M+1) can be written as\n\nwhere the amplitude scattering matrix S0/(M+1) is defined as\n\nIf there is no incidence of light from the ambient on the right (layer M+1) to the left (layer 0),\n\nand the front-reflection and front-transmission coefficients from the layer 0 to M+1 can be obtained by\n\nIn the same manner, we can define the complex reflection and transmission coefficients for the j-th layer in terms of the matrix elements\n\nAs shown in Fig. 1, t0j is the internal transmission coefficient from the ambient on the left (layer 0) to layer j. The terms rj0 and rj(M+1) are the internal reflection coeffi-cients through the partial front multilayers from layer j to layer 0 and through the partial rear multilayers from layer j to layer M+1, respectively.\n\n2.2. Optical Power Dissipation for S-polarized Light\n\nThe y component of the electric field in the j-th layer is expressed as\n\nwhere\n\nis an internal transfer coefficient, relating the incident plane wave to the internal electric field propagating in the positive (negative) z direction at the interface between the layer j-1 and j . The values of\n\nand\n\nare given by\n\nwhere t0j, rj(M+1), and rj0 are defined in Eqs. (11). The corresponding magnetic fields are\n\nThe time-average Poynting vector in the z direction is expressed as\n\nwhere we can define\n\nFinally, the optical power dissipation in the z direction is given by\n\nwhere\n\nand α j,θ=2qjk0 in Eq. (18) are the real part of the complex refractive index and the absorption coefficient at the incident angle θ , respectively. We can obtain Eq. (19) by substituting Eq. (12) into Eq. (18). Similar to normal incidence, the optical power dissipation at the incident angle θ is also proportional to the electric field intensity for s-polarized light. It is noticeable that the values of pj and α j,θ are modified for oblique incidence compared with the case of normal incidence.\n\n2.3. Optical Power Dissipation for P-polarized Light\n\nThe x and z component of the magnetic field in the j-th layer can be written as\n\nThe corresponding magnetic field is given by\n\nThe time-average Poynting vector in the z direction is given by\n\nFinally, the optical power dissipation in the z direction is written as\n\nRearrangement of Eq. (24) together with Eq. (3) leads to\n\nBy substituting the expressions for Ex,j and Ez,j in Eqs.(20) and (21) into Eq. (25), we obtain\n\nIt is remarkable that the optical power dissipation for p-polarized light at oblique incidence is also proportional to the electric field intensity, which is composed of the x and z components of the electric field amplitude for p-polarized light. Thus, information on the electric field intensity with respect to light polarization and incident angle is sufficient to calculate the optical power dissipation in the OSC.\n\n2.4. Total Light Absorptance at the Active Region\n\nBecause light absorbed in the active region of the OSC can contribute to the electric power generation, we have to maximize the light absorption efficiency at the active region to enhance power conversion efficiency of the OSC. The light absorptance of the layer j (dj-lzdj) can be expressed as \n\nwhere\n\nis the optical power of sunlight light with s(p) polarization, which is incident from the ambient on the left.\n\nIII. CALCULATION RESULTS\n\nFig. 2 shows the multilayer structure of the OSC along with its corresponding complex refractive index, which is", null, "taken from Ref. 14. The wavelength of light is assumed to be 550 nm, where the P3HT:PCBM active layer has a peak absorption. We consider the incident angles of 0°, 30°, 45°, and 60° in this calculation. The thickness of the optical spacer layer is varied as x = 50, 100, and 150 nm to adjust the optical interference effect and maximize the electric field intensity distribution within the active region.\n\nFig. 3 shows the y component of the normalized electric field intensities at the spacer thickness of 50, 100, and 150 nm. The light green area in the middle of Fig. 3 indicates the P3HT:PCBM active region, where the electric field intensity distribution is important for determining the efficiency of the OSC. The value of the electric field intensity is normalized to s-polarized light incident from the transparent glass on the left ambient when the incident angle is (a) 0°, (b) 30° and (c) 60°. From the ITO toward the Al layers, the oscillating electric field intensities show the general inter-ference pattern [6-8]. For normal incidence (θ = 0°), the spatial distribution of |Ey|2 in the P3HT:PCBM active region is the largest at the spacer thickness of 100 nm, where the OSC expects to have the best light absorption efficiency. As the incident angle increases to 30° and further up to 60°, the overall oscillating interference pattern of |Ey|2 moves toward the ITO layer. This behavior is ascribed to the fact that the real part of the complex refractive index,\n\nbecomes smaller at the larger incident angle.", null, "Fig. 4 shows the time-average Poynting vectors Sz(z) for s- and p-polarized light at the incident angles of 0°, 30°, 45°, and 60°. The thickness of the optical spacer layer is fixed as x = 100 nm. The calculated time-average Poynting", null, "", null, "vectors are normalized in reference to the incident optical power from the transparent glass. As the incident angle increases, the value of Sz(z) at the glass-ITO interface (z = 0 nm) decreases because the front-reflection coefficient of the OSC multilayer, r0(M+1) in Eq. (10a), becomes larger at the higher incident angle. The value of Sz(z) with normal incidence (θ = 0°) is identical and degenerate for s- and p-polarized light. However, the value of Sz(z)\\with oblique incidence of p-polarized light is larger than that with oblique incidence of s-polarized light. This higher trans-mittance of p-polarized light over s-polarized light at oblique incidence is similar to the transmission behavior at the boundary between two dielectric media, where the Brewster’s angle(100% transmission) is existent for p-polarized light .\n\nFig. 5 shows the comparison of the calculated normalized electric field intensities between s- and p-polarized light at the incident angles of (a) 30°, (b) 45°, and (c) 60°. For reference, the normalized electric field intensity distribution at normal incidence is shown in the black dotted line. The optical spacer layer thickness is x = 100 nm. The electric field intensity for p-polarized light |Ep|2 = |Ex|2 + |Ez|2 is discontinuous at the interfaces because it has the electric field intensity component of |Ex|2 normal to the interface. On the other hand, the electric field intensity for s-polarized light |Es|2 = |Ey|2 is continuous at the interfaces because it has only a transverse electric field intensity component. Fig. 6 shows the calculated optical power dissipations in the z direction for s- and p-polarized light as a function of", null, "the incident angle. According to the definition of light absorptance in Eq. (27), p-polarized light shows the relati-vely higher light absorptance in the P3HT/PCBM active region than s-polarized light does This is more pronounced as the incidence angle increases from 30° to 60°.\n\nIV. CONCLUSION\n\nWe presented comprehensive optical modeling and calculation results of thin-film OSC at oblique incidence of light. Based on the TMM, the simple expression for the optical power dissipation was derived at oblique incidence for s- and p-polarized light. Depending on light polarization, we calculated the spatial distribution of the electric field intensity, the timeaverage Poynting vector, and the optical power dissipation at various incidence angles. As the incidence angle increases, the optical power dissipation in the active region becomes smaller for both s- and p-polarized light. In the P3HT/PCBM active region, p-polarized light shows relatively higher light absorptance than s-polarized light does. This is more significant as the incidence angle increases up to 60°. We expect that our optical modeling results help to design more efficient OSC with light absorption efficiency improved,considering the optical interference effect of the OSC at oblique incidence.\n\n• [FIG. 1.] Schematic diagram of M multilayers with forward- and backward-propagating electric field amplitude (E). A plane wave with an incident angle of θ and a wavelength of λ is incident from the semi-infinite transparent ambient on the left (j=0). The orientation of the electromagnetic field oscillations is composed of s-polarization and p-polarization that are perpendicular to or parallel to the incident plane respectively.", null, "• [FIG. 2.] Multilayer structure of the OSC together with its corresponding complex refractive index. The thickness of the optical spacer layer is varied as x = 50 100 and 150 nm.", null, "• [FIG. 3.] The y component of the normalized electric field intensities at the spacer thicknesses of 50 100 and 150 nm. The light green area indicates the P3HT:PCBM active region. The value of the electric field intensity is normalized to s-polarized light incident from the transparent glass at the incident angles of (a) 0° (b) 30° and (c) 60°.", null, "• [FIG. 4.] Calculated time-average Poynting vectors for s- and p-polarized light at the incident angles of 0° 30° 45° and 60°. The light green area indicates the P3HT:PCBM active region. The thickness of the optical spacer layer is fixed as x= 100 nm. The calculated time-average Poynting vectors are normalized in reference to the incident optical power from the transparent glass.", null, "• [FIG. 5.] Comparison of the calculated normalized electric field intensities between s- and p-polarized light at the incident angles of (a) 30° (b) 45° and (c) 60°. The light green area indicates the P3HT:PCBM active region. For reference the electric field intensity distribution at normal incidence is shown in the black dotted line. The optical spacer layer is fixed as x = 100 nm.", null, "• [FIG. 6.] Calculated optical power dissipations in the z direction for s- and p-polarized light as a function of the incident angle. The light green area indicates the P3HT:PCBM active region. The optical spacer layer is fixed as x = 100 nm.", null, "" ]

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[ "# Working out Zero-Point Energy and ration of potential to kinetic for a particle.\n\n• Epideme\n\n## Homework Statement\n\nConsider a particle with mass, m moving in one dimensional potential U=kx^2/2 as in a mass-spring system. The total energy of the particle is\n\nE= (p^2/2m) + (kx^2/2)\n\nClassically, the absolute minimum of the enrgy, E=0 is acheived when p=0 and x=0. In quantum mechanics however, the momentum, p and the co-ordinate x cannot simulataneously have certain valus. Using the Heisenberg uncertainty relation:\n\na)Calculate the minimum possible value of energy, E. This lowest possible energy, which is not zero, is called zero-point energy.\nb)What is the ratio of kinetic to the potential enrgy of the particle in a state of minimal total energy\n\n## Homework Equations\n\nDelta x delta p = hbar / 2 <---Hesinberg Uncertainty Principle\n\n## The Attempt at a Solution\n\nI'm completely lost, with all the remaining 4 questions of my work which I'm posting.\n\nI don't think that one can be completely lost, never.\n\nIf one has no idea, then one usually make a competent guessing, try it.\n\nI don't think that one can be completely lost, never.", null, "I'm completely lost.", null, "Which thread am I supposed to be in?", null, "", null, "I'm completely lost.", null, "Which thread am I supposed to be in?", null, "You have to provide an attempt to solution", null, "You have to provide an attempt to solution", null, "Well, I the last thing I remember is turning left at the Library, and heading towards Special & General Relativity", null, "after that it's all a blur", null, "do you think I traveled faster than light?", null, "" ]

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[ "Home My Page Projects Code Snippets Project Openings SML/NJ\n Summary Activity Forums Tracker Lists Tasks Docs Surveys News SCM Files\n\n# SCM Repository\n\n[smlnj] View of /sml/branches/rt-transition/cm/compile/link.sml\n [smlnj] / sml / branches / rt-transition / cm / compile / link.sml", null, "# View of /sml/branches/rt-transition/cm/compile/link.sml\n\nRevision 2975 - (download) (annotate)\nTue Apr 8 16:27:59 2008 UTC (13 years ago) by jhr\nFile size: 10725 byte(s)\n``` Synching sources with revision 2974 of main trunk.\n```\n```(*\n* Link traversals.\n* - manages shared state\n*\n* (C) 1999 Lucent Technologies, Bell Laboratories\n*\n* Author: Matthias Blume ([email protected])\n*)\nlocal\nstructure GP = GeneralParams\nstructure DG = DependencyGraph\nstructure GG = GroupGraph\nstructure DE = DynamicEnv\nstructure EM = ErrorMsg\nstructure PP = PrettyPrintNew\n\ntype env = DynamicEnv.env\ntype posmap = env IntMap.map\nin\nsignature LINK = sig\n\ntype bfc\ntype bfcGetter = SmlInfo.info -> bfc\n\n(* Evict value from cache if it exists *)\nval evict : GP.info -> SmlInfo.info -> unit\n\nval evictStale : unit -> unit\n\n(* Check all values and evict those that depended on other\n* meanwhile evicted ones. *)\nval cleanup : GP.info -> unit\n\nval newTraversal : GG.group * bfcGetter ->\n{ group: GP.info -> env option,\nexports: (GP.info -> env option) SymbolMap.map }\n\n(* discard persistent state *)\nval reset : unit -> unit\n\n(* discard persistent state for a specific stable library *)\nval unshare : SrcPath.file -> unit\nend\n\nfunctor LinkFn (structure BFC : BFC where type bfc = Binfile.bfContents\nval system_values : posmap SrcPathMap.map ref) :>\nLINK where type bfc = BFC.bfc =\nstruct\n\nexception Link of exn\n\nstructure BF = Binfile\nstructure EX = Execute\n\ntype bfc = BF.bfContents\ntype bfcGetter = SmlInfo.info -> bfc\n\ntype bfun = GP.info -> env -> env\n\ndatatype bnode =\nB of bfun * BinInfo.info * bnode list\n\nval stablemap = ref (StableMap.empty: bnode StableMap.map)\n\ntype smemo = env * SmlInfo.info list\n\nval smlmap = ref (SmlInfoMap.empty: smemo SmlInfoMap.map)\n\nval emptyStatic = StaticEnv.empty\nval emptyDyn = DynamicEnv.empty\n\nfun evict gp i = let\nfun check () =\ncase SmlInfo.sh_mode i of\nSharing.SHARE true =>\nSmlInfo.error gp i EM.WARN\n(concat [\"sharing for \", SmlInfo.descr i,\n\" may be lost\"])\nEM.nullErrorBody\n| _ => ()\nin\n(smlmap := #1 (SmlInfoMap.remove (!smlmap, i))\nbefore check ())\nhandle LibBase.NotFound => ()\nend\n\nfun evictStale () =\nsmlmap := SmlInfoMap.filteri (SmlInfo.isKnown o #1) (!smlmap)\n\nfun cleanup gp = let\nval visited = ref SmlInfoSet.empty\nfun visit i =\nif SmlInfoSet.member (!visited, i) then true\nelse\ncase SmlInfoMap.find (!smlmap, i) of\nNONE => false\n| SOME (_, l) => let\nval bl = map visit l\nval b = List.all (fn x => x) bl\nin\nif b then\n(visited := SmlInfoSet.add (!visited, i);\ntrue)\nelse (evict gp i; false)\nend\nin\napp (ignore o visit o #1) (SmlInfoMap.listItemsi (!smlmap))\nend\n\nfun newTraversal0 (GG.ERRORGROUP, _) =\n{ group = fn _ => NONE, exports = SymbolMap.empty }\n| newTraversal0 (group as GG.GROUP grec, getBFC) = let\n\nval { exports, grouppath, ... } = grec\n\nfun exn_err (msg, error, descr, exn) = let\nfun ppb pps =\n(PP.newline pps;\nPP.string pps (General.exnMessage exn);\napp (fn s => PP.string pps (s ^ \"\\n\"))\n(SMLofNJ.exnHistory exn);\nPP.newline pps)\nin\nerror (concat [msg, \" \", descr]) ppb;\nraise Link exn\nend\n\nfun link_stable (i, de) = let\nval stable = BinInfo.stablename i\nval os = BinInfo.offset i\nval descr = BinInfo.describe i\nval error = BinInfo.error i EM.COMPLAIN\nval bfc = BFC.getStable { stable = stable, offset = os,\ndescr = descr }\nhandle exn => exn_err (\"unable to load library module\",\nerror, descr, exn)\nin\nBF.exec (bfc, de, Link)\nhandle Link exn =>\nexn_err (\"link-time exception in library code\",\nerror, descr, exn)\nend\n\nfun link_sml (gp, i, getBFC, getE, snl) = let\nfun fresh () = let\nval bfc = getBFC i\nin\ncase getE gp of\nNONE => NONE\n| SOME e =>\n(SOME (BF.exec (bfc, e, Link))\nhandle Link exn =>\nexn_err (\"link-time exception in user program\",\nSmlInfo.error gp i EM.COMPLAIN,\nSmlInfo.descr i,\nexn))\nend handle exn as Link _ => raise exn\n| _ => NONE\nin\ncase SmlInfo.sh_mode i of\nSharing.SHARE _ =>\n(case SmlInfoMap.find (!smlmap, i) of\nNONE =>\n(case fresh () of\nNONE => NONE\n| SOME de => let\nval m = (de, snl)\nin\nsmlmap :=\nSmlInfoMap.insert (!smlmap, i, m);\nSOME de\nend)\n| SOME (de, _) => SOME de)\n| Sharing.DONTSHARE => (evict gp i; fresh ())\nend\n\nval visited = ref SrcPathSet.empty\n\nfun registerGroup GG.ERRORGROUP = ()\n| registerGroup (g as GG.GROUP grec) = let\nval { grouppath, kind, sublibs, ... } = grec\nfun registerSublib NONE = ()\n| registerSublib (SOME i) =\nregisterGroup (#2 (List.nth (sublibs, i)) ())\nfun registerStableLib GG.ERRORGROUP = ()\n| registerStableLib (GG.GROUP sg) = let\nval { exports, grouppath = sgp, ... } = sg\nval posmap =\nlet val (m', pm) =\nSrcPathMap.remove (!system_values, sgp)\nin system_values := m'; pm\nend handle LibBase.NotFound => IntMap.empty\n\nval localmap = ref StableMap.empty\nfun bn (DG.BNODE n) = let\nval i = #bininfo n\nval li = #localimports n\nval gi = #globalimports n\nfun mySysval () =\nIntMap.find (posmap, BinInfo.offset i)\n\nfun new () =\ncase mySysval () of\n(* We short-circuit traversal\n* construction (and the resulting\n* traversal) whenever we find a\n* node whose dynamic value was\n* created at bootstrap time.\n* This assumes that anything in\n* sysval can be shared -- which\n* is enforced by the way the\n* PIDMAP file is constructed. *)\nSOME e => (fn gp => fn _ => e, NONE)\n| NONE => let\nval e0 = (fn _ => emptyDyn, [])\nfun join ((f, NONE), (e, l)) =\n(fn gp =>\nDE.atop (f gp emptyDyn,\ne gp),\nl)\n| join ((f, SOME (i, l')),\n(e, l)) =\n(e, B (f, i, l') :: l)\nval ge =\nfoldl join e0 (map lfbn gi)\nval le = foldl join ge (map bn li)\nin\ncase (BinInfo.sh_mode i, le) of\n(Sharing.SHARE _, (e, [])) =>\nlet fun thunk gp =\nlink_stable (i, e gp)\nval m_thunk =\nMemoize.memoize thunk\nin\n(fn gp => fn _ =>\nm_thunk gp,\nNONE)\nend\n| (Sharing.SHARE _, _) =>\nEM.impossible\n\"Link: sh_mode inconsistent\"\n| (Sharing.DONTSHARE, (e, l)) =>\n(fn gp => fn e' =>\nlink_stable\n(i, DE.atop (e', e gp)),\nSOME (i, l))\nend\nin\ncase StableMap.find (!stablemap, i) of\nSOME (B (f, i, [])) =>\n(case BinInfo.sh_mode i of\nSharing.DONTSHARE => (f, SOME (i, []))\n| _ => (f, NONE))\n| SOME (B (f, i, l)) => (f, SOME (i, l))\n| NONE =>\n(case StableMap.find (!localmap, i) of\nSOME x => x\n| NONE => let\nval x = new ()\nin\nlocalmap := StableMap.insert\n(!localmap, i, x);\nx\nend)\nend\n\nand fbn (_, n, p) = (registerSublib p; bn n)\n\nand lfbn th = fbn (th ())\n\nfun sbn (DG.SB_SNODE n) =\nEM.impossible \"Link:SNODE in stable lib\"\n| sbn (DG.SB_BNODE (n as DG.BNODE bnrec, _, p)) =\nlet val _ = registerSublib p\nval bininfo = #bininfo bnrec\nval b as B (_, i, _) =\ncase bn n of\n(f, NONE) => B (f, bininfo, [])\n| (f, SOME (i, l)) => B (f, i, l)\nin\nstablemap :=\nStableMap.insert (!stablemap, i, b)\nend\n\nfun fsbn (_, n) = sbn n\nfun impexp (nth, _, _) = fsbn (nth ())\nin\nSymbolMap.app impexp exports\nend\nfun force f = f ()\nin\nif SrcPathSet.member (!visited, grouppath) then ()\nelse (visited := SrcPathSet.add (!visited, grouppath);\ncase kind of\nGG.LIB { kind = GG.STABLE _, ... } =>\nregisterStableLib g\n| _ => app (registerGroup o force o #2) sublibs)\nend\n\nval _ = registerGroup group\n\nval l_stablemap = ref StableMap.empty\nval l_smlmap = ref SmlInfoMap.empty\n\nfun bnode (B (f, i, l)) =\ncase StableMap.find (!l_stablemap, i) of\nSOME th => th\n| NONE => let\nval fl = map bnode l\nfun th gp = let\nfun add (t, e) = DE.atop (t gp, e)\nin\nf gp (foldl add emptyDyn fl)\nend\nval m_th = Memoize.memoize th\nin\nl_stablemap :=\nStableMap.insert (!l_stablemap, i, m_th);\nm_th\nend\n\nfun sbn (DG.SB_BNODE (DG.BNODE { bininfo, ... }, _, _)) = let\nval b = valOf (StableMap.find (!stablemap, bininfo))\nfun th gp = SOME (bnode b gp)\nhandle exn as Link _ => raise exn\n| _ => NONE\nin\n(th, [])\nend\n| sbn (DG.SB_SNODE n) = sn n\n\nand sn (DG.SNODE n) = let\nval { smlinfo = i, localimports, globalimports } = n\nin\ncase SmlInfoMap.find (!l_smlmap, i) of\nSOME th => (th, [i])\n| NONE => let\nfun atop (NONE, _) = NONE\n| atop (_, NONE) = NONE\n| atop (SOME e, SOME e') = SOME (DE.atop (e, e'))\nfun add ((f, l), (f', l')) =\n(fn gp => atop (f gp, f' gp), l @ l')\nval gi = foldl add (fn _ => SOME emptyDyn, [])\n(map fsbn globalimports)\nval (getE, snl) = foldl add gi (map sn localimports)\nfun thunk gp = link_sml (gp, i, getBFC, getE, snl)\nval m_thunk = Memoize.memoize thunk\nin\nl_smlmap := SmlInfoMap.insert (!l_smlmap, i, m_thunk);\n(m_thunk, [i])\nend\nend\n\nand fsbn (_, n) = sbn n\n\nfun impexp (nth, _, _) gp = #1 (fsbn (nth ())) gp\nhandle Link exn => raise EX.Link\n\nval exports' = SymbolMap.map impexp exports\n\nfun group' gp = let\nfun one (_, NONE) = NONE\n| one (f, SOME e) =\n(case f gp of\nNONE => NONE\n| SOME e' => SOME (DE.atop (e', e)))\nin\nSymbolMap.foldl one (SOME emptyDyn) exports'\nend\nin\n{ exports = exports', group = group' }\nend\n\nfun newTraversal (x as (GG.ERRORGROUP, _)) = newTraversal0 x\n| newTraversal (x as (GG.GROUP { exports, ... }, _)) = let\nval tth = Memoize.memoize (fn () => (newTraversal0 x))\nin\n{ group = fn gp => #group (tth ()) gp,\nexports = SymbolMap.mapi\n(fn (sy, _) => fn gp =>\nvalOf (SymbolMap.find (#exports (tth ()),\nsy))\ngp)\nexports }\nend\n\nfun reset () = (stablemap := StableMap.empty;\nsmlmap := SmlInfoMap.empty)\n\nfun unshare group =\nlet fun other (i, _) =\nSrcPath.compare (BinInfo.group i, group) <> EQUAL\nval sv = system_values\nin\nstablemap := StableMap.filteri other (!stablemap);\n(sv := #1 (SrcPathMap.remove (!sv, group)))\nhandle LibBase.NotFound => ()\nend\nend\nend\n```\n\n [email protected] ViewVC Help Powered by ViewVC 1.0.0" ]

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[ "Skip to main content\n\n# Combined or Mixed Equilibria of Cu²⁺\n\n## Chemical Concept Demonstrated\n\n• Complex-formation equilibrium\n\n## Demonstration\n\n Each of the dishes contain Cu 2+ ion solution. Aqueous ammonia is added dropwise into the first dish. Excess ammonia is added into the first dish.", null, "## Observations\n\nA precipitate forms as the ammonia is added dropwise into the dish. Additional ammonia causes the precipitate to dissolve and a dark-blue solution is formed.\n\n## Explanations (including important chemical equations)\n\nA light-blue colored precipitate of Cu(OH)2 is formed when aqueous ammonia is added to Cu 2+ ion solution.\n\nCu 2+ (aq) + 2 (OH)- (aq) <=> Cu(OH)2 (s) Ksp = 2.2 x 10-20\n\nIn the presence of excess NH3, this equilibrium shifts to the left as the free Cu 2+ ion is sequestered from the solution as the Cu(NH3)4 2+ complex ion.\n\nCu 2+ (aq) + 4 NH3 (aq) <=> Cu(NH3)4 2+ (aq) Ksp = 2.1 x 1013\n\nThe intensity of the absorption of light by the Cu(NH3)4 2+ complex appears to be orders of magnitude greater than the corresponding Cu(H2O)6 2+ complex. The molar absorbances of these complexes, however, differ, roughly, only by a factor of four. The difference between the intensities of the colors of these complexes can be explained by noting that the maximum wavelength for the Cu(H2O)6 2+ complex is about 800 nm, whereas the maximum wavelength for the Cu(NH3)42+ complex occurs at about 600 nm. Since a larger portion of the absorption in the Cu(NH3)4 2+ complex is in the visible portion of the spectrum, the color appears considerably more intense.\n\nContributors" ]

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[ "", null, "IsRightBol - Maple Help\n\nMagma\n\n IsRightBol\n test whether a given magma satisfies the right Bol identity", null, "Calling Sequence IsRightBol( m )", null, "Parameters\n\n m - Array representing the Cayley table of a finite magma", null, "Description\n\n • The IsRightBol command returns true if the given magma satisfies the right Bol law ((XY)Z)Y = X((YZ)Y).  It returns false otherwise.", null, "Examples\n\n > $\\mathrm{with}\\left(\\mathrm{Magma}\\right):$\n > $m≔⟨⟨⟨1|2|3⟩,⟨2|3|1⟩,⟨3|1|2⟩⟩⟩$\n ${m}{≔}\\left[\\begin{array}{ccc}{1}& {2}& {3}\\\\ {2}& {3}& {1}\\\\ {3}& {1}& {2}\\end{array}\\right]$ (1)\n > $\\mathrm{IsRightBol}\\left(m\\right)$\n ${\\mathrm{true}}$ (2)\n > $m≔⟨⟨⟨1|2|3⟩,⟨2|3|3⟩,⟨3|1|2⟩⟩⟩$\n ${m}{≔}\\left[\\begin{array}{ccc}{1}& {2}& {3}\\\\ {2}& {3}& {3}\\\\ {3}& {1}& {2}\\end{array}\\right]$ (3)\n > $\\mathrm{IsRightBol}\\left(m\\right)$\n ${\\mathrm{false}}$ (4)", null, "Compatibility\n\n • The Magma[IsRightBol] command was introduced in Maple 15.\n • For more information on Maple 15 changes, see Updates in Maple 15." ]

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[ "## The Lesson\n\nThe diameter is the line through the centre of the circle that has its endpoints on the circle. The diameter also refers to the length of this line. The diameter is often denoted by d. The image below shows what we mean by the diameter:", null, "## Dictionary Definition\n\nThe Oxford English Dictionary defines the diameter as \"a straight line passing through the centre of a circle (or sphere), and terminated at each end by its circumference (or surface).\"\n\n## Examples of the Diameter\n\nThe circle below has a diameter of 5 cm. This is the length of the line that touches opposite sides of the circle and passes through the centre.", null, "The circle below has a diameter of 12 inches.", null, "## How to Find the Diameter\n\nThe diameter of a circle is related to its radius, circumference and area.\n\n## How to Find the Diameter from the Radius\n\nThe diameter is twice the radius. If the diameter is d and the radius is r (as shown in the image below), then we can find the diameter using the formula:", null, "## How to Find the Diameter from the Circumference\n\nThe diameter is the circumference divided by π. If the diameter is d and the circumference is C (as shown in the image below), then we can find the diameter using the formula:", null, "## How to Find the Diameter from the Area\n\nThe diameter is twice the square root of the area divided by π. If the diameter is d and the area is A (as shown in the image below), then we can find the diameter using the formula:", null, "## What Is a Circle?\n\nA circle is a shape containing a set of points that are the same radius from the center of the circle.\n\n## Euclid's Definition of the Diameter\n\nEuclid defined the diameter his book, Elements.", null, "Definition 17: \"A diameter of the circle is any straight line drawn through the center and terminated in both directions by the circumference of the circle, and such a straight line also bisects the circle\".\n\n## Diameter of a Sphere\n\nA circle is not the only shape that has a diameter. A sphere also has a diameter. The diameter of a sphere is the line segment through the center of the sphere that ends on opposite points on the surface of the sphere. The image below shows what we mean by the diameter of a sphere. It is labelled d:", null, "## The Diameter in Other Shapes\n\nThe diameter appears in other shapes that are not obviously related to circles, for example the cone and cylinder. But if you were to cut through a cone or cylinder, the cross-section would be a circle, which has a diameter. The base of both shapes is a circle. The diameter of the base (as shown in the image below) is labelled r:", null, "" ]

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[ "# What Is a Mixed Interest Rate?", null, "••• Sales. image by Egor Tkachenko from <a href='http://www.fotolia.com'>Fotolia.com</a>\nShare It\n\nA mixed interest rate is one of three alternative interest rate structures that are used for mortgage loans in the U.S. Different interest rate structures are needed for these loans because the financial qualifications of borrowers vary so widely and market interest rates change frequently. Each rate structure has pros and cons, differing degree of financial risk and different costs.\n\n## Fixed Rate\n\nA fixed interest rate is one type of rate structure. Fixed-rate loans apply the same interest rate over the entire loan period, i.e., the interest rate remains fixed. This interest rate structure has the lowest risk for a borrower because the interest cost over the entire period of the loan is known in advance. Fixed interest rates are usually selected by borrowers who are in the best financial condition. The cost of such loans is usually, but not always, the lowest. These loans are most favorable when interest rates are relatively low.\n\n## Variable Rate\n\nAnother structure is the variable interest rate. For these loans, the interest rate varies over time, based on the specific terms and conditions of the loan agreement. This interest rate structure has the highest risk because market rates may vary over the loan period, and your rate is usually adjusted annually. Consequently, future interest rates are unknown. Such loans are used primarily when interest rates are relatively high.\n\n## Mixed Rate\n\nThe third rate structure is the mixed interest rate. A mixed interest rate loan applies both fixed and variable rates during specified periods of the loan. Such loans typically apply the fixed rate during the initial loan period. This is usually a three- to five-year period. The variable rate is typically applied during the remaining loan period. These loans are also frequently referred to as two-step loans.\n\n## Cost\n\nThe purpose of mixed interest rate loans is to reduce the initial cost of the loan, when the rate is fixed. Compared to a variable interest rate, the mixed interest rate reduces the financial risk of the loan to some degree. Consequently, the loan is affordable to more borrowers. The variable rate period is intended to raise the interest rate cost, thus offsetting the lower cost during the fixed rate period---and financially benefiting the lender. Thus, the loan risk rises after the fixed rate period has expired.\n\n## Risk\n\nMixed interest rate loans are usually used to increase loan affordability to lesser-qualified borrowers. The lower initial rate allows more borrowers to qualify. When the rate becomes fixed, it is hoped that the borrowers will be able to afford a higher loan cost if interest rates increase. The bottom line for all borrowers is to make an economic assessment of all three rate alternatives before making a financial decision." ]

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[ "# 919\n\nThis number is a prime.\n\nJust showing those entries submitted by 'Silva': (Click here to show all)", null, "The smallest palindromic prime equal to the difference of consecutive cubes (18^3-17^3). [Silva]", null, "The sum of the digits of first non-trivial palindromic primes up to 919 is another palindromic prime. [Silva]", null, "919 equals 91+81+71+61+51+41+31+21+11+1+11+21+31+41+51+61+71+81+91. [Silva]", null, "919 is the sum of a semiprime number of consecutive semiprimes:221+226+235+237. Curiously 919 results from de projection of a semiprime. [Silva]" ]

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[ "×\n×\n\n# In Exercises 2326, express the matrix and its inverse as products of elementary", null, "ISBN: 9781118474228 398\n\n## Solution for problem 25 Chapter 1.5\n\nElementary Linear Algebra, Binder Ready Version: Applications Version | 11th 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, "Elementary Linear Algebra, Binder Ready Version: Applications Version | 11th Edition\n\n4 5 1 309 Reviews\n12\n0\nProblem 25\n\nIn Exercises 2326, express the matrix and its inverse as products of elementary matrices. 1 0 2 043 001\n\nStep-by-Step Solution:\nStep 1 of 3\n\nMATH 1220 Notes for Week #7 22 February 2016 Warm­Up ● Remind yourself of the of the statement of the Intermediate Value Theorem (IVT): ○ on a closed interval [a, b] where a < b and f(a) < 0, f(b) > 0, and f is continuous, there exists a number z such that f(z) = 0 ○ note: z is not unique; example is f(x) = sin(x) because sin(x) = 0 on the period kπ Equilibrium...\n\nStep 2 of 3\n\nStep 3 of 3\n\n##### ISBN: 9781118474228\n\nThe answer to “In Exercises 2326, express the matrix and its inverse as products of elementary matrices. 1 0 2 043 001” is broken down into a number of easy to follow steps, and 19 words. Elementary Linear Algebra, Binder Ready Version: Applications Version was written by and is associated to the ISBN: 9781118474228. This textbook survival guide was created for the textbook: Elementary Linear Algebra, Binder Ready Version: Applications Version, edition: 11. This full solution covers the following key subjects: . This expansive textbook survival guide covers 80 chapters, and 2108 solutions. The full step-by-step solution to problem: 25 from chapter: 1.5 was answered by , our top Math solution expert on 03/14/18, 04:26PM. Since the solution to 25 from 1.5 chapter was answered, more than 211 students have viewed the full step-by-step answer.\n\nUnlock Textbook Solution" ]

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[ "# Getting Started with the NumericTextBox\n\nThis tutorial explains how to set up a basic Telerik UI for ASP.NET MVC NumericTextBox and highlights the major steps in the configuration of the component.\n\nYou will initialize two NumericTextBox components, one of them will have an event handler.", null, "## Prerequisites\n\nTo successfully complete the tutorial, you need a project that is already configured to use the Telerik UI for ASP.NET MVC components:\n\n## 1. Prepare the CSHTML File\n\nThe first step is to add the required directives at the top of the `.cshtml` document:\n\n• To use the Telerik UI for ASP.NET MVC HtmlHelpers:\n\n``````@using Kendo.Mvc.UI\n``````\n\nOptionally, you can structure the document by adding the desired HTML elements like headings, divs, paragraphs, and others. In this tutorial, you will create a list that contains two items—one for each NumericTextBox. You will also add labels.\n\n``````@using Kendo.Mvc.UI\n\n<ul id=\"products-list\">\n<li>\n<label for=\"currency\">Price:</label>\n\n</li>\n<li>\n<label for=\"numeric\">Currently in stock:</label>\n\n</li>\n</ul>\n``````\n\n## 2. Initialize the NumericTextBox\n\nUse the NumericTextBox HtmlHelper to add the component to a page. The `Name()` configuration method is mandatory as its value is used for the id and the name attributes of the NumericTextBox element.\n\n``````@using Kendo.Mvc.UI\n\n<ul id=\"products-list\">\n<li>\n<label for=\"currency\">Price:</label>\n@(Html.Kendo().NumericTextBox<decimal>()\n.Name(\"currency\") // The name is required.\n)\n\n</li>\n<li>\n<label for=\"numeric\">Currently in stock:</label>\n@(Html.Kendo().NumericTextBox<double>()\n.Name(\"numeric\")\n)\n</li>\n</ul>\n``````\n\nDo not set the `Name()` option when using `NumericTextBoxFor`. The `[WidgetName]For` method automatically sets the widget's `Name()` to the field it is bound to. For more information, see the Fundamentals article.\n\n## 3. Configure the Displayed Values\n\nThe NumericTextBox provides a variety of methods that allow you to configure its values. In this tutorial, you will perform the following configuration:\n\n• To specify the number format for the control when it isn't focused, use the `Format()` method.\n• To define what is the accepted minimal value, use the `Min()` method.\n• To set the value, use the `Value()` method.\n• To add a placeholder text in the NumericTextBox, use the `Placeholder()` method.\n``````@using Kendo.Mvc.UI\n\n<ul id=\"products-list\">\n<li>\n<label for=\"currency\">Price:</label>\n@(Html.Kendo().NumericTextBox<decimal>()\n.Name(\"currency\")\n.Format(\"c\") // Format the value as currency.\n.Min(0) // The minimal value is 0.\n.Value(30) // The current value is 30.\n.Placeholder(\"Enter product price\") // Use the placeholder to display a hint.\n)\n\n</li>\n<li>\n<label for=\"numeric\">Currently in stock:</label>\n@(Html.Kendo().NumericTextBox<double>()\n.Name(\"numeric\")\n.Placeholder(\"Enter numeric value\")\n)\n</li>\n</ul>\n``````\n\n## 4. Handle a NumericTextBox Event\n\nThe NumericTextBox exposes the `Change` and `Spin` events that you can handle and customize the component's functions. In this tutorial, you will use these events to log a new entry in the browser's console.\n\n``````@using Kendo.Mvc.UI\n\n<ul id=\"products-list\">\n<li>\n<label for=\"currency\">Price:</label>\n@(Html.Kendo().NumericTextBox<decimal>()\n.Name(\"currency\")\n.Format(\"c\")\n.Min(0)\n.Value(30)\n.Placeholder(\"Enter product price\")\n.Events(e => e // Configure the client-side events.\n.Change(\"change\")\n.Spin(\"spin\")\n)\n)\n</li>\n<li>\n<label for=\"numeric\">Currently in stock:</label>\n@(Html.Kendo().NumericTextBox<double>()\n.Name(\"numeric\")\n.Placeholder(\"Enter numeric value\")\n)\n</li>\n</ul>\n\n<script>\nfunction change() {\nconsole.log(\"Change :: \" + this.value());\n}\n\nfunction spin() {\nconsole.log(\"Spin :: \" + this.value());\n}\n</script>\n``````\n\n## 5. (Optional) Reference Existing NumericTextBox Instances\n\nYou can reference the NumericTextBox instances that you have created and build on top of their existing configuration:\n\n1. Use the `id` attribute of the component instance to establish a reference.\n\n``````<script>\nvar numericTextBoxRef = \\$('#currency').data('kendoNumericTextBox'); // numericTextBoxRef is a reference to the existing currency instance of the helper.\n</script>\n``````\n2. Use the NumericTextBox client-side API to control the behavior of the widget. In this example, you will use the `readonly` method to disable user input.\n\n``````<script>\nvar numericTextBoxRef = \\$('#currency').data('kendoNumericTextBox'); // numericTextBoxRef is a reference to the existing currency instance of the helper.\nnumericTextBoxRef.readonly(true); // Set the readonly method to true and disable user input.\n</script>\n``````\n\nFor more information on referencing specific helper instances, see the Methods and Events article." ]

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[ "At the nanometre scale?\n\nA nanometre (nm) is 10⁻⁹ metres, which is one-thousandth of a micrometre, or one-billionth of a metre. This is the scale at which we measure atoms and the molecules they make.\n\nWhat is the Nanometre scale?\n\nJust how small is “nano?” In the International System of Units, the prefix \"nano\" means one-billionth, or 10-9; therefore one nanometer is one-billionth of a meter. ... On a comparative scale, if the diameter of a marble was one nanometer, then diameter of the Earth would be about one meter.\n\nWhat is the Nanometre used for?\n\nA manometer is an instrument used to measure and indicate pressure. There are two types of manometers, analog and digital.\n\nIs nm A nanometer?\n\nA nanometer is a unit of measurement for length just as you have with meters and centimeters. A nanometer is one billionth of a meter, 0.000000001 or 10-9 meters. The word nano comes from the Greek word for “dwarf.” The term nanoscale is used to refer to objects with dimensions on the order of 1-100 nanometers (nm).\n\nHow small is 7 nanometers?\n\nWhat is 7-nanometer? When used in relation to stuff like CPUs and video cards, the term 7-nanometer refers to the size of the transistors involved. The smaller the transistor, the more you can fit onto a piece of silicon and the more powerful and complex that the components built from these transistors are able to be.\n\nSize and Scale: A Walk from Kilometers to One Nanometer\n\n31 related questions found\n\nWhich is better 7nm vs 10nm?\n\nThe 7nm FinFET Process is 1.6 times Denser than TSMC 10nm's Process. Also, the 7nm process results in 20% better performance and 40% power reduction as compared to their 10nm technology. There is also an optimized version of 7nm known as N7P which is IP compatible with N7.\n\nIs nanometer The smallest unit?\n\nA nanometer (nm) is 1,000 times smaller than a micrometer. It is equal to 1/1,000,000,000th or one-billionth of a meter. When things are this small, you can't see them with your eyes, or a light microscope. ... Atoms are smaller than a nanometer.\n\nHow many zeros are in a nanometer?\n\nA nanometer (nm) is equal to one-billionth of a meter.\n\nWritten out, one nanometer looks like 0.000000001 m (that's nine zeros!).\n\nWhat is the relation between m and nm?\n\nThe relation between meter and nanometer is that one meter equals 1,000,000,000 nanometers m.\n\nCan you see a nanometer?\n\nA nanometer is a one-billionth of a meter, and used to measure things that are very, very small. ... So small you cannot see something a nanometer in size unless you use very powerful microscopes like atomic force microscopes. A nanometer is used to measure things that are very small.\n\nWhen was the micrometer invented?\n\nThe micrometer used with a telescope was invented about 1638 by William Gascoigne, an English astronomer.\n\nHow small is a billionth?\n\nShort scale\n\nIn the short (or American) scale, a billionth is equal to 0.000 000 001, or 1 x 109 in scientific notation or standard form. The prefix for this number is nano, and is abbreviated as \"n\" (for example, in electronics, one nanofarad would be written as 1 nF).\n\nWhich is smaller 1nm or 10nm?\n\nOne nanometer (1 nm) is equal to 10-9 m or 0.000000001 m. A nanometer is 10 times smaller than the width of your DNA, and 10 times bigger than the size of an atom. ... Play Ushi's Ruler Game to get an idea of the size of different items or measure your height in How Tall Are You?\n\nHow small is nano scale?\n\nThe nanoscale is the dimensional range of approximately 1 to 100 nanometres.\n\nHow many cm are in 1m?\n\nThere are 100 centimeters in 1meter.\n\nWhat is class9 nanometer?\n\n1 nm = 0.0000000010 m. A nanometre is a unit of spatial measurement that is 10 -9 meter, or one billionth of a meter. It is commonly used in nanotechnology , the building of extremely small machines. A nanometre is used to measure things that are very small.\n\nHow many Nanometres are in a Centimetre?\n\n1 centimeter has 10,000,000 i.e. 10 million nanometers.\n\nWhat is a nanometer symbol?\n\nGenerally used in Nanotechnology, the nanometre or nanometer (American spelling) is a unit of length in the metric system, equal to one billionth of a metre. As per the international System of Units (SI), the standard symbol for the nanometer is nm.\n\nWhat comes after a nanometer?\n\n(1) The very simple part: after nanometer is picometer. Just like nanometer was preceded by micrometer. It's just standard SI prefixes.\n\nWhat is smaller than a Picometer?\n\nA picometer is 10^(-12) meters, ie 1000 times smaller. Yes: the prefix “nano” indicates a quantity of 10^(-9). In this case, a nanometer is 10^(-9) meters.\n\nWhat is smaller than a NM?\n\nAtoms are smaller than a nanometer.\n\nWhat is smaller than a Millimetre?\n\nMicrometer A micrometer (also called a micron) is 1000 times smaller than a millimeter. 1 millimeter (mm) = 1000 micrometers (μm).\n\nWhy can't Intel do 7nm?\n\nIntel's 7nm process was supposed to go online in 2021's fourth quarter to help keep the company's CPUs competitive. However, a defect in the manufacturing technology caused the company to delay its arrival to as far as 2023, opening the door for rival AMD to dominate in the PC chip space for years to come.\n\nIs 7nm faster than 14nm?\n\nCompared to 16nm/14nm, 7nm provides a 35% speed improvement, 65% less power, and a 3.3X density improvement, according to Gartner. Based on PPASC metrics and the cost-per-transistor curve, 7nm looks like a better option, at least according to some." ]

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[ "Solutions by everydaycalculation.com\n\n## Reduce 48/40 to lowest terms\n\nThe simplest form of 48/40 is 6/5.\n\n#### Steps to simplifying fractions\n\n1. Find the GCD (or HCF) of numerator and denominator\nGCD of 48 and 40 is 8\n2. Divide both the numerator and denominator by the GCD\n48 ÷ 8/40 ÷ 8\n3. Reduced fraction: 6/5\nTherefore, 48/40 simplified to lowest terms is 6/5.\n\nMathStep (Works offline)", null, "Download our mobile app and learn to work with fractions in your own time:" ]

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[ "# CA Foundation Time Value of Money Summary\n\n## Time Value of Money Summary for CA Foundation November 2020 Exams\n\n• Time value of money: Time value of money means that the value of a unity of money is different in different time periods. The sum of money received in future is less valuable than it is today. In other words the present worth of money received after some time will be less than a money received today.\n• Interest: Interest is the price paid by a borrower for the use of a lender’s money. If you borrow (or lend) some money from (or to) a person for a particular period you would pay (or receive) more money than your initial borrowing (or lending).\n• Simple interest: is the interest computed on the principal for the entire period of borrowing.\n\nI = Pit\nA = P + I\nI = A – P\nHere, A = Accumulated amount (final value of an investment)\nP = Principal (initial value of an investment)\ni = Annual interest rate in decimal.\nI = Amount of Interest\nt = Time in years\n\n• Compound interest as the interest that accrues when earnings for each specified period of time added to the principal thus increasing the principal base on which subsequent interest is computed.\n\nFormula for compound interest:\nAn = P ( 1 + i)n\nwhere, i = Annual rate of interest\nn = Number of conversion periods per year\nInterest = An – P = P ( 1 + i )n – P\nn is total conversions i.e. t x no. of conversions per year\n\n• Effective Rate of Interest: The effective interest rate can be computed directly by following formula:\n\nE = (1 + i)n– 1\nWhere E is the effective interest rate\ni = actual interest rate in decimal\nn = number of conversion period\n\n• Future value of a single cash flow can be computed by above formula. Replace A by future value (F) and P by single cash flow (C.F.) therefore\n\nF = C.F. (1 + i)n\n\n• Annuity can be defined as a sequence of periodic payments (or receipts) regularly over a specified period of time.\n\nAnnuity may be of two types:\n(i) Annuity regular: In annuity regular first payment/receipt takes place at the end of first period.\n(ii) Annuity Due or Annuity Immediate: When the first receipt or payment is made today (at the beginning of the annuity) it is called annuity due or annuity immediate\n\n• If A be the periodic payments, the future value A(n, i) of the annuity is given by", null, "• Future value of an Annuity due/Annuity immediate = Future value of annuity regular x (1+i) where i is the interest rate in decimal.\n• The present value P of the amount An due at the end of n period at the rate of i per interest period may be obtained by solving for P the below given equation", null, "• Present value of annuity due or annuity immediate: Present value of annuity due/ immediate for n years is the same as an annuity regular for (n-1) years plus an initial receipt or payment in beginning of the period. Calculating the present value of annuity due involves two steps.\n\nStep 1: Compute the present value of annuity as if it were a annuity regular for one period short.\nStep 2: Add initial cash payment/receipt to the step 1 value.\n\n• Sinking Fund: It is the fund credited for a specified purpose by way of sequence of periodic payments over a time period at a specified interest rate. Interest is compounded at the end of every period. Sizeof the sinking fund deposit is computed from A = P.A(n,i) where A is the amount to be saved the periodic payment, n the payment period.\n• Annuity applications:\n\n(a) Leasing: Leasing is a financial arrangement under which the owner of the asset (lessor) allows the user of the asset (lessee) to use the asset for a defined period of time (lease period) for a consideration (lease rental) payable over a given period of time. This is a kind of taking an asset on rent.\n\n(b) Capital Expenditure (investment decision): Capital expenditure means purchasing an asset (which results in outflows of money) today in anticipation of benefits (cash inflow) which would flow across the life of the investment.\n\n(c) Valuation of Bond: A bond is a debt security in which the issuer owes the holder a debt and is obliged to repay the principal and interest. Bonds are generally issued for a fixed term longer than one year." ]

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[ "", null, "University Physics Volume 3\n\n# 4.6X-Ray Diffraction\n\nUniversity Physics Volume 34.6 X-Ray Diffraction\n\n### Learning Objectives\n\nBy the end of this section, you will be able to:\n\n• Describe interference and diffraction effects exhibited by X-rays in interaction with atomic-scale structures\n\nSince X-ray photons are very energetic, they have relatively short wavelengths, on the order of $10−810−8$ m to $10−1210−12$ m. Thus, typical X-ray photons act like rays when they encounter macroscopic objects, like teeth, and produce sharp shadows. However, since atoms are on the order of 0.1 nm in size, X-rays can be used to detect the location, shape, and size of atoms and molecules. The process is called X-ray diffraction, and it involves the interference of X-rays to produce patterns that can be analyzed for information about the structures that scattered the X-rays.\n\nPerhaps the most famous example of X-ray diffraction is the discovery of the double-helical structure of DNA in 1953 by an international team of scientists working at England’s Cavendish Laboratory—American James Watson, Englishman Francis Crick, and New Zealand-born Maurice Wilkins. Using X-ray diffraction data produced by Rosalind Franklin, they were the first to model the double-helix structure of DNA that is so crucial to life. For this work, Watson, Crick, and Wilkins were awarded the 1962 Nobel Prize in Physiology or Medicine. (There is some debate and controversy over the issue that Rosalind Franklin was not included in the prize, although she died in 1958, before the prize was awarded.)\n\nFigure 4.24 shows a diffraction pattern produced by the scattering of X-rays from a crystal. This process is known as X-ray crystallography because of the information it can yield about crystal structure, and it was the type of data Rosalind Franklin supplied to Watson and Crick for DNA. Not only do X-rays confirm the size and shape of atoms, they give information about the atomic arrangements in materials. For example, more recent research in high-temperature superconductors involves complex materials whose lattice arrangements are crucial to obtaining a superconducting material. These can be studied using X-ray crystallography.\n\nFigure 4.24 X-ray diffraction from the crystal of a protein (hen egg lysozyme) produced this interference pattern. Analysis of the pattern yields information about the structure of the protein. (credit: “Del45”/Wikimedia Commons)\n\nHistorically, the scattering of X-rays from crystals was used to prove that X-rays are energetic electromagnetic (EM) waves. This was suspected from the time of the discovery of X-rays in 1895, but it was not until 1912 that the German Max von Laue (1879–1960) convinced two of his colleagues to scatter X-rays from crystals. If a diffraction pattern is obtained, he reasoned, then the X-rays must be waves, and their wavelength could be determined. (The spacing of atoms in various crystals was reasonably well known at the time, based on good values for Avogadro’s number.) The experiments were convincing, and the 1914 Nobel Prize in Physics was given to von Laue for his suggestion leading to the proof that X-rays are EM waves. In 1915, the unique father-and-son team of Sir William Henry Bragg and his son Sir William Lawrence Bragg were awarded a joint Nobel Prize for inventing the X-ray spectrometer and the then-new science of X-ray analysis.\n\nIn ways reminiscent of thin-film interference, we consider two plane waves at X-ray wavelengths, each one reflecting off a different plane of atoms within a crystal’s lattice, as shown in Figure 4.25. From the geometry, the difference in path lengths is $2dsinθ2dsinθ$. Constructive interference results when this distance is an integer multiple of the wavelength. This condition is captured by the Bragg equation,\n\n$mλ=2dsinθ,m=1,2,3...mλ=2dsinθ,m=1,2,3...$\n4.6\n\nwhere m is a positive integer and d is the spacing between the planes. Following the Law of Reflection, both the incident and reflected waves are described by the same angle, $θ,θ,$ but unlike the general practice in geometric optics, $θθ$ is measured with respect to the surface itself, rather than the normal.\n\nFigure 4.25 X-ray diffraction with a crystal. Two incident waves reflect off two planes of a crystal. The difference in path lengths is indicated by the dashed line.\n\n### Example 4.7\n\n#### X-Ray Diffraction with Salt Crystals\n\nCommon table salt is composed mainly of NaCl crystals. In a NaCl crystal, there is a family of planes 0.252 nm apart. If the first-order maximum is observed at an incidence angle of $18.1°18.1°$, what is the wavelength of the X-ray scattering from this crystal?\n\n#### Strategy\n\nUse the Bragg equation, Equation 4.6, $mλ=2dsinθmλ=2dsinθ$, to solve for $θθ$.\n\n#### Solution\n\nFor first-order, $m=1,m=1,$ and the plane spacing d is known. Solving the Bragg equation for wavelength yields\n$λ=2dsinθm=2(0.252×10−9m)sin(18.1°)1=1.57×10−10m,or0.157nm.λ=2dsinθm=2(0.252×10−9m)sin(18.1°)1=1.57×10−10m,or0.157nm.$\n\n#### Significance\n\nThe determined wavelength fits within the X-ray region of the electromagnetic spectrum. Once again, the wave nature of light makes itself prominent when the wavelength $(λ=0.157nm)(λ=0.157nm)$ is comparable to the size of the physical structures $(d=0.252nm)(d=0.252nm)$ it interacts with.\n\nFor the experiment described in Example 4.7, what are the two other angles where interference maxima may be observed? What limits the number of maxima?\n\nAlthough Figure 4.25 depicts a crystal as a two-dimensional array of scattering centers for simplicity, real crystals are structures in three dimensions. Scattering can occur simultaneously from different families of planes at different orientations and spacing patterns known as called Bragg planes, as shown in Figure 4.26. The resulting interference pattern can be quite complex.\n\nFigure 4.26 Because of the regularity that makes a crystal structure, one crystal can have many families of planes within its geometry, each one giving rise to X-ray diffraction.\nOrder a print copy\n\nAs an Amazon Associate we earn from qualifying purchases." ]

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[ "# Tip Calculator\n\nHow much is a 25 percent tip on \\$166.55?\n\nTIP:\n\\$ 0\nTOTAL:\n\\$ 0\nTIP PER PERSON:\n\\$ 0\nTOTAL PER PERSON:\n\\$ 0\n\n## How much is a 25 percent tip on \\$166.55? How to calculate this tip?\n\nAre you looking for the answer to this question: How much is a 25 percent tip on \\$166.55? Here is the answer.\n\nLet's see how to calculate a 25 percent tip when the amount to be paid is 166.55. Tip is a percentage, and a percentage is a number or ratio expressed as a fraction of 100. This means that a 25 percent tip can also be expressed as follows: 25/100 = 0.25 . To get the tip value for a \\$166.55 bill, the amount of the bill must be multiplied by 0.25, so the calculation is as follows:\n\n1. TIP = 166.55*25% = 166.55*0.25 = 41.6375\n\n2. TOTAL = 166.55+41.6375 = 208.1875\n\n3. Rounded to the nearest whole number: 208\n\nIf you want to know how to calculate the tip in your head in a few seconds, visit the Tip Calculator Home.\n\n## So what is a 25 percent tip on a \\$166.55? The answer is 41.64!\n\nOf course, it may happen that you do not pay the bill or the tip alone. A typical case is when you order a pizza with your friends and you want to split the amount of the order. For example, if you are three, you simply need to split the tip and the amount into three. In this example it means:\n\n1. Total amount rounded to the nearest whole number: 208\n\n2. Split into 3: 69.33\n\nSo in the example above, if the pizza order is to be split into 3, you’ll have to pay \\$69.33 . Of course, you can do these settings in Tip Calculator. You can split the tip and the total amount payable among the members of the company as you wish. So the TipCalc.net page basically serves as a Pizza Tip Calculator, as well.\n\n## Tip Calculator Examples (BILL: \\$166.55)\n\nHow much is a 5% tip on \\$166.55?\nHow much is a 10% tip on \\$166.55?\nHow much is a 15% tip on \\$166.55?\nHow much is a 20% tip on \\$166.55?\nHow much is a 25% tip on \\$166.55?\nHow much is a 30% tip on \\$166.55?" ]

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[ "# 8x^2+7y^2+3z^2+12xy+4xz-8yz canonical form\n\nThe teacher will be very surprised to see your correct solution 😉\n\nv\n\nx: [, ]\ny: [, ]\nz: [, ]\n\n#### Quality:\n\n(Number of points on the axis)\n\n### The solution\n\nYou have entered [src]\n 2 2 2\n3*z + 7*y + 8*x - 8*y*z + 4*x*z + 12*x*y = 0\n$$8 x^{2} + 12 x y + 4 x z + 7 y^{2} - 8 y z + 3 z^{2} = 0$$\n8*x^2 + 12*x*y + 4*x*z + 7*y^2 - 8*y*z + 3*z^2 = 0" ]

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[ "Home » Worksheets » Geometry » Similar Shapes (Astronomy Themed) Math Worksheets\n\n# Similar Shapes (Astronomy Themed) Math Worksheets\n\nClick the button below to get instant access to these premium worksheets for use in the classroom or at a home.\n\nThis worksheet can be edited by Premium members using the free Google Slides online software. Click the Edit button above to get started.\n\n## Definition\n\nWhat are Similar Shapes?\n\n• Similar shapes are two shapes having the same shape but different sizes.\n• These shapes also have equal corresponding angles and sides.\n\n## Summary\n\nSIMILAR TRIANGLE THEOREMS\n\nANGLE-ANGLE THEOREM (AA)\n\nThis states if two triangles a have two pairs of corresponding angles that are congruent, then the two triangles are similar.\n\nSIDE-ANGLE-SIDE THEOREM (SAS)\n\nThis states that if two sides of one triangle are proportional to two corresponding sides of another triangle, and their included angles are congruent, then the two triangles are similar.\n\nSIDE-SIDE-SIDE THEOREM (SSS)\n\nThis states that if two triangles have proportional sides, then they are similar.\n\n## Similar Shapes (Astronomy Themed) Worksheets\n\nThis is a fantastic bundle that includes everything you need to know about Similar Shapes across 21 in-depth pages. These are ready-to-use Common core aligned 7th and 8th Grade Math worksheets.\n\nEach ready to use worksheet collection includes 10 activities and an answer guide. Not teaching common core standards? Don’t worry! All our worksheets are completely editable so can be tailored for your curriculum and target audience.\n\n## Resource Examples\n\nClick any of the example images below to view a larger version.\n\n## Worksheets Activities Included\n\nAges 11-12 (Basic)\n\n• Elements in the Universe\n• My Dream\n• The Lost Astronaut\n• The Constellations\n• Planets and Stars" ]

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[ "# Week 9: How to convert from General form to Standard form\n\nThis week in Pre Calc 11 I learned how to convert general form into standard form. I will show you how it uses completing the square but it a little different from last unit of solving quadratic equations. In this unit it helps us graph what the function will look like (this should look like a parabola)\n\nConverting from General Form to Standard form.\n\nex/", null, "$2x^2 -11x +4$\n\nStep 1: Place brackets around", null, "$2x^2-11x$ , then divide by 2 to get", null, "$x^2$ instead of", null, "$2x^2$\n\n•", null, "$(2x^2 -11x) +4$\n•", null, "$2(x^2 - \\frac{11}{2}x ) +4$", null, "Step 2/3: Divide", null, "$\\frac{11}{2}x$ by 2 and square it; place it in two blank spaces beside", null, "$\\frac{11}{2}x$ one being added and one being subtracted.\n\n•", null, "$2(x^2 - \\frac{11}{2}x +{blank} - {blank}) +4$\n•", null, "$2(x^2 - \\frac{11}{2}x +\\frac{121}{16} - \\frac{121}{16}) +4$", null, "", null, "Step 4: Multiply the 2 you factored out in step 1 to", null, "$\\frac{121}{16}$ to remove it from the brackets.\n\n•", null, "$2(x^2 - \\frac{11}{2}x +\\frac{121}{16}) +4 -\\frac{242}{16}$", null, "Step 5: Factor the inside of the brackets.", null, "$\\star$ Hint: What you squared in step 2/3 is you factor.\n\n•", null, "$2(x^2 - \\frac{11}{2}x +\\frac{121}{16}) +4 -2\\frac{242}{16}$\n•", null, "$2(x - \\frac{11}{4})^2 -\\frac{242}{16} + 4$", null, "Step 6: Simplify\n\n•", null, "$2(x - \\frac{11}{4})^2 -\\frac{242}{16} + 4$\n•", null, "$2(x - \\frac{11}{4})^2 -\\frac{121}{8} + 4$\n•", null, "$2(x - \\frac{11}{4})^2 -\\frac{121}{8} + \\frac{32}{8}$\n•", null, "$2(x - \\frac{11}{4})^2 -\\frac{89}{8}$", null, "Final Answer:", null, "$2(x - \\frac{11}{4})^2 -\\frac{89}{8}$", null, "", null, "You can check if you have done your calculations correctly by using desmos.com", null, "$\\star$ notice how both equations line up perfectly, this means that both are equivalent.\n\nStandard form can tell us:\n\nStandard formula:", null, "$y= a(x-p)^2 +q$\n\na = 2\n\np", null, "$\\frac{11}{4}$\n\nq", null, "$-\\frac{89}{8}$\n\nHorizontal translation:", null, "$\\frac{11}{4}$ units right\n\nVertical translation:", null, "$-\\frac{89}{8}$\n\nVertex : (", null, "$\\frac{11}{4}$ ,", null, "$-\\frac{89}{8}$)\n\nAxis of symmetry", null, "$\\frac{11}{4}$\n\nThis is how you convert General form into standard form in the unit of Quadratic equations. The standard form can tell you a lot about what it looks like and how to graph it. This is my favorite form in this unit because it tells me so much information and it is very useful. 🙂" ]

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[ "# Necessary and sufficient condition on decomposable convex programming\n\n## 摘要\n\nThis paper considers a kind of decomposible convex programming: (P) min{f(x); xε C}, and its corresponding decomposible variational inequality DVI(f, C), where f(x):=f1(x1)+f2 (x2)+···+fn(xn), for every x:=(x1, x2, ···, xn) and C:=C1xC2x···xCn···xCn . Under the constraint qualification 0 ε ri(piin = 1 (coD (∂fi) - Ci),we show that x is a solution to DVI(f, C) if, and only if, x is an optimal solution of (P).\n\n原文 英語 351-359 9 Proceedings of the National Science Council, Republic of China, Part A: Physical Science and Engineering 20 4 已發佈 - 1996 七月\n\n## ASJC Scopus subject areas\n\n• Engineering(all)" ]

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[ "# Battery-powered Arduino\n\nContents\n\nRunning an Arduino on battery power is only useful, if you know, when to change the battery. Here it is handy, to measure the current battery level and get notifications, if it drops below a certain threshold. Thankfully you can use the analog Pins to measure a voltage.\n\nIf you attach a voltage to e.g. A0, your Arduino will compare the input voltage with a reference voltage (by default VCC) and return a value between 0-1023 where 1023 means, that the input voltage is equal or higher than VCC. Afterwards you can calculate the actual voltage by simple math:\n\n$$V_{analog}=\\frac {V_{vcc}}{1023}\\cdot A_{analog}$$\n\nSo for example if you measure 768 on the Pin, you can calculate the attached voltage:\n\n$$V_{analog}=\\frac {3.3 V}{1023}\\cdot 768 = 2.47 V$$\n\nThis works fine as long as you know your VCC which is not true for battery powered Arduinos. Luckily the Ardunio has an internal Reference Voltage of 1.1V 1 that can be used for comparison.\n\nSince you can only measure voltages below the reference voltage you need to apply some magic in form of a Voltage divicer to map the voltage to a range of 0 to 1.1V.\n\n## Voltage divider\n\nThe lengthy wikipedia article 2 on voltage dividers gives a good introduction how this circuit works.\n\nIn summary you will need two resistors R1 and R2 which are hooked up between your analog pin and Vcc and GND.\n\nDepending on the value of the resistors you can measure different ranges of voltages, even high above the threshold your Arduino is capable of survice.\n\nIn this example A0 is used as analog PIN and two resistors with R1=1MΩ and R2=470kΩ. Those are good values to map up to 3.44V to your 1.1V reference. If you read 1023 from A0, 3.44V are attached and so on.\n\nSince I drive my Arduinos with either 2 AA or 2 AAA batteries, the voltage will never reach more than 3V. If you need to use higher voltages, you should adjust the values accordingly.\"\n\nThe following formula is used to calculate the 4 relevant values (Vin is equal to VCC):\n\n$$V_{out}={\\frac {R_{2}}{R_{1}+R_{2}}}\\cdot V_{in}$$\n\nAlternatively you can also use one of the many online calculators3.\n\n## Code example\n\nThe following code example shows the usage of the internal reference voltage on a MySensors4 node, a complete Arduino Sketch is available on Github.\n\n#define VMIN 1.9 // Battery Level 0%\n#define VMAX 3.44 // Batter Level 100%\n#define VBAT_PER_BITS 0.003362 // Volts per bit on the A0 pin, VMAX / 1023 = 3.44 / 1023 = 0.0033625\n\nvoid setup()\n{\n...\nanalogReference(INTERNAL);\n...\n}\n\nvoid loop()\n{\n...\nfloat volt = sensorValue * VBAT_PER_BITS; // Calculate Voltage\nint batteryPcnt = sensorValue / 10; // Calculate Battery Level (Percent)\nsend(voltMsg.set(volt, 2)); // Send Voltage\nsendBatteryLevel(batteryPcnt); // Send Battery Level\n}" ]

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[ "## What is the difference between primal and dual?\n\nExplanation: The solution to the dual problem provides a lower bound to the solution of the primal (minimization) problem. However in general the optimal values of the primal and dual problems need not be equal. Their difference is called the duality gap.\n\n## What is the difference between primal and dual problem?\n\nIn the primal problem, the objective function is a linear combination of n variables. In the dual problem, the objective function is a linear combination of the m values that are the limits in the m constraints from the primal problem.\n\n## What is the relation between primal and dual?\n\nThere is a fundamental relationship between the x * variables of the Primal and the z * variables of the Dual. We’ll refer to these variables as dual to one another. There is a similar relationship between the variables y i of the Dual and the w i of the Primal. Again, refer to the variables as dual to one another.\n\n## How do you convert primal to dual problems?\n\nThe dual of this problem can be found easily by converting it to the standard primal formulation given in (3). This is accomplished by multiplying the second constraint by −1, thus changing the ”greater than or equal to” constraint to a ”less than or equal to” constraint.\n\n## What are the rules to form a dual problem from primal problem?\n\nSince any LPP can be solved by using simplex method, so we can solve primal as well as dual, and as there is relation between the two we can solve one and give the solution for the associated other problem by using some rule as given Primal problem is in form of maximization, then Rule 1: Corresponding net evaluation\n\n## What are the types of primal dual problem?\n\nTypes of PrimalDual Problem\n\n1. Symmetric: Here all constraints of both primal and dual problems are in equations and variables are non negative. 2. Un-Symmetric: Here all constraints of primal are equations and primal variables are non negative.\n\n## What are the characteristics of dual problem?\n\n12.2 Important characteristics of Duality\n\n1. Dual of dual is primal 2. If either the primal or dual problem has a solution then the other also has a solution and their optimum values are equal. 3.\n\n## What is standard primal?\n\nThere are several ways of defining the primal and the corresponding dual problems. We will define a standard primal problem as finding x1, x2, …, xn to maximize a primal objective function: (9.19) + d n x n = ∑ i = 1 n d i x i = d T x.\n\n## What is primal dual?\n\nThe primaldual method specifies that we start with a dual feasible solution, in this. case u = v = 0. Given our current feasible dual solution, we look for a primal feasible. solution that obeys the complementary slackness conditions.\n\n## What does primal mean in math?\n\nMathematics. Primal, an old mathematics term for a projective hypersurface.\n\n## What is primal simplex method?\n\nPrimal simplex begins by solving BxB = b − NxN and taking xB to be new values for the basic variables. From (1), this ensures that Ax = b. If there is no such direction, the current x is an optimal solution, and the constraints Ax = b along with the active bounds on the nonbasic variables are the optimal active set.\n\n## What is a primal number?\n\nIn mathematics, a primeval number is a natural number n for which the number of prime numbers which can be obtained by permuting some or all of its digits (in base 10) is larger than the number of primes obtainable in the same way for any smaller natural number. Primeval numbers can be composite." ]

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[ "# YEAR\n\n### Definition of YEAR\n\nReturns the year specified by a given date.\n\n### Sample Usage\n\n`YEAR(DATE(1969,7,20))`\n\n`YEAR(A2)`\n\n`YEAR(40909)`\n\n### Syntax\n\n`YEAR(date)`\n• `date` - The date from which to calculate the year. Must be a cell reference to a cell containing a date, a function returning a date type, or a number.\n\n### Notes\n\n• Ensure that the input to the function is either a reference to a cell containing a date, a function which returns a date object such as `DATE``DATEVALUE` or `TO_DATE`, or a date serial number of the type returned by the `N` function.\n• `YEAR` does not autoconvert number formats in the same way that Google Sheets does upon direct entry into cells. Therefore, `YEAR(10/10/2000)` is interpreted as `YEAR(0.005)`, the quotient of 10 divided by 10 divided by 2000.\n\n`WEEKDAY`: Returns a number representing the day of the week of the date provided.\n\n`TO_DATE`: Converts a provided number to a date.\n\n`N`: Returns the argument provided as a number.\n\n`MONTH`: Returns the month of the year a specific date falls in, in numeric format.\n\n`DAY`: Returns the day of the month that a specific date falls on, in numeric format.\n\n`DATEVALUE`: Converts a provided date string in a known format to a date value.\n\n`DATE`: Converts a provided year, month, and day into a date.", null, "### Then type in the YEAR Formula in the area you would like to display the outcome:", null, "", null, "### By adding the value you would like to calculate the YEAR formula for, Excellentable will generate the outcome:", null, "### Need Help?\n\nCan't find the answer you're looking for? Don't worry we're here to help" ]

[ null, "https://addteq.co.in/products/excellentable/help/files/111710435/111710448/3/1548076777000/YEAR1.png", null, "https://addteq.co.in/products/excellentable/help/files/111710435/111710451/3/1548076776000/YEAR2.png", null, "https://addteq.co.in/products/excellentable/help/files/111710435/111710452/3/1548076774000/YEAR3.png", null, "https://addteq.co.in/products/excellentable/help/files/111710435/111710453/3/1548076773000/YEAR4.png", null ]

[ "# Least distance on Riemannian Manifold\n\nI've been doing some calculations of geodesics in different Riemannian Manifolds. More precisely I'm trying to compute, given two points on a Riemannian Manifold, the smallest distance between those points.\n\nI started by trying to compute the distance of two points on a 2-sphere. That was easy. Then I thought to consider not a 2-sphere but a 2-sphere with some kind of a wormhole in it, something like a doughnut...\n\nI thought of using the following metric: $$ds^2=(1-b(r)/r)^{-1}dr^2+r^2 d\\theta^2 + r^2\\sin^2\\theta d\\varphi^2,$$ where $b(r)$ is some function that should describe the throat of the wormhole - as it can be seen here. First question: Is this correct for what I'm looking for?\n\nThen, given this metric, the distance is given by: $$l=\\int{ds}=\\int_{\\theta_1}^{\\theta_2}[(1-b(r)/r)^{-1}\\dot{r}^2+r^2+ r^2\\sin^2\\theta \\dot{\\varphi}^2]^{1/2}d\\theta,$$ right? And then, I should make use of the Euler-Lagrange equations...Am I correct?\n\n• It's not clear to me why $\\sin\\theta$ is not squared. Anyway, you should not expect to get any manageable explicit formulas by continuing your computations. But it is easy to calculate distances on the torus explicitly by endowing it with the flat metric: the one that comes from the quotient map $\\mathbb R^2\\to \\mathbb R^2/\\mathbb Z^2$. Indeed, the geodesics in this metric are images of straight lines under the quotient map. – user53153 Feb 2 '13 at 7:09\n• Ups, it was a typo. Yeh I get really ugly formulas. But I just wasn't sure I was doing things the right way. Thank you – PML Feb 2 '13 at 11:16" ]

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[ "# How to Convert a Dictionary to a DataFrame in Pandas\n\nOccasionally you may be interested in converting a dictionary to a pandas DataFrame. Fortunately this is easy to do using a single line of code in pandas.\n\nThis tutorial demonstrates how to convert a dictionary to a pandas DataFrame in practice.\n\n### Example: Converting a Dictionary to a DataFrame\n\nSuppose we have the following dictionary that contains the names of five basketball players and their average points scored per game:\n\n```import pandas as pd\n\n#define dictionary\nd = {'Michael':36,\n'Lebron': 32,\n'Scotty': 28,\n'Larry': 23,\n'Kareem': 22}\n```\n\nThe following code shows how to convert this dictionary to a pandas DataFrame with two columns:\n\n```#convert dictionary to DataFrame\ndf = pd.DataFrame(list(d.items()), columns = ['Player','Points'])\n\n#view DataFrame\ndf\n\nPlayer \t Points\n0\tMichael\t 36\n1\tLebron\t 32\n2\tScotty\t 28\n3\tLarry\t 23\n4\tKareem\t 22\n```\n\nYou can verify that your result is a DataFrame by typing:\n\n```#view data type\ntype(df)\n\npandas.core.frame.DataFrame\n```\n\nThis tells us that the dictionary was indeed converted to a pandas DataFrame.\n\n### Bonus: Creating Column Names from Dictionary Keys\n\nIf you happen to want the dictionary keys to be the column names of the new DataFrame and the dictionary values to be the row values, you can use the following syntax:\n\n```import pandas as pd\n\n#define dictionary\nd = {'Michael':36,\n'Lebron': 32,\n'Scotty': 28,\n'Larry': 23,\n'Kareem': 22}\n\n#convert dictionary to DataFrame\ndf = pd.DataFrame([d])\n\n#view DataFrame\ndf\n\nKareem\tLarry\tLebron\tMichael\tScotty\n0\t22\t23\t32\t36\t28```\n\nWe can also verify that this result is a DataFrame:\n\n```#view data type\ntype(df)\n\npandas.core.frame.DataFrame\n```\n\nYou can find more Python tutorials here." ]

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[ "# 23 Containers library [containers]\n\n## 23.3 Sequence containers [sequences]\n\n### 23.3.1 In general [sequences.general]\n\nThe headers <array>, <deque>, <forward_list>, <list>, and <vector> define template classes that meet the requirements for sequence containers.\n\nThe headers <queue> and <stack> define container adaptors ([container.adaptors]) that also meet the requirements for sequence containers.\n\n```#include <initializer_list>\n\nnamespace std {\ntemplate <class T, size_t N> struct array;\ntemplate <class T, size_t N>\nbool operator==(const array<T,N>& x, const array<T,N>& y);\ntemplate <class T, size_t N>\nbool operator!=(const array<T,N>& x, const array<T,N>& y);\ntemplate <class T, size_t N>\nbool operator<(const array<T,N>& x, const array<T,N>& y);\ntemplate <class T, size_t N>\nbool operator>(const array<T,N>& x, const array<T,N>& y);\ntemplate <class T, size_t N>\nbool operator<=(const array<T,N>& x, const array<T,N>& y);\ntemplate <class T, size_t N>\nbool operator>=(const array<T,N>& x, const array<T,N>& y);\ntemplate <class T, size_t N>\nvoid swap(array<T,N>& x, array<T,N>& y) noexcept(noexcept(x.swap(y)));\n\ntemplate <class T> class tuple_size;\ntemplate <size_t I, class T> class tuple_element;\ntemplate <class T, size_t N>\nstruct tuple_size<array<T, N> >;\ntemplate <size_t I, class T, size_t N>\nstruct tuple_element<I, array<T, N> >;\ntemplate <size_t I, class T, size_t N>\nconstexpr T& get(array<T, N>&) noexcept;\ntemplate <size_t I, class T, size_t N>\nconstexpr T&& get(array<T, N>&&) noexcept;\ntemplate <size_t I, class T, size_t N>\nconstexpr const T& get(const array<T, N>&) noexcept;\n}\n```\n\n```#include <initializer_list>\n\nnamespace std {\ntemplate <class T, class Allocator = allocator<T> > class deque;\ntemplate <class T, class Allocator>\nbool operator==(const deque<T,Allocator>& x, const deque<T,Allocator>& y);\ntemplate <class T, class Allocator>\nbool operator<(const deque<T,Allocator>& x, const deque<T,Allocator>& y);\ntemplate <class T, class Allocator>\nbool operator!=(const deque<T,Allocator>& x, const deque<T,Allocator>& y);\ntemplate <class T, class Allocator>\nbool operator>(const deque<T,Allocator>& x, const deque<T,Allocator>& y);\ntemplate <class T, class Allocator>\nbool operator>=(const deque<T,Allocator>& x, const deque<T,Allocator>& y);\ntemplate <class T, class Allocator>\nbool operator<=(const deque<T,Allocator>& x, const deque<T,Allocator>& y);\ntemplate <class T, class Allocator>\nvoid swap(deque<T,Allocator>& x, deque<T,Allocator>& y);\n}\n```\n\n```#include <initializer_list>\n\nnamespace std {\ntemplate <class T, class Allocator = allocator<T> > class forward_list;\ntemplate <class T, class Allocator>\nbool operator==(const forward_list<T,Allocator>& x, const forward_list<T,Allocator>& y);\ntemplate <class T, class Allocator>\nbool operator< (const forward_list<T,Allocator>& x, const forward_list<T,Allocator>& y);\ntemplate <class T, class Allocator>\nbool operator!=(const forward_list<T,Allocator>& x, const forward_list<T,Allocator>& y);\ntemplate <class T, class Allocator>\nbool operator> (const forward_list<T,Allocator>& x, const forward_list<T,Allocator>& y);\ntemplate <class T, class Allocator>\nbool operator>=(const forward_list<T,Allocator>& x, const forward_list<T,Allocator>& y);\ntemplate <class T, class Allocator>\nbool operator<=(const forward_list<T,Allocator>& x, const forward_list<T,Allocator>& y);\ntemplate <class T, class Allocator>\nvoid swap(forward_list<T,Allocator>& x, forward_list<T,Allocator>& y);\n}\n```\n\n```#include <initializer_list>\n\nnamespace std {\ntemplate <class T, class Allocator = allocator<T> > class list;\ntemplate <class T, class Allocator>\nbool operator==(const list<T,Allocator>& x, const list<T,Allocator>& y);\ntemplate <class T, class Allocator>\nbool operator< (const list<T,Allocator>& x, const list<T,Allocator>& y);\ntemplate <class T, class Allocator>\nbool operator!=(const list<T,Allocator>& x, const list<T,Allocator>& y);\ntemplate <class T, class Allocator>\nbool operator> (const list<T,Allocator>& x, const list<T,Allocator>& y);\ntemplate <class T, class Allocator>\nbool operator>=(const list<T,Allocator>& x, const list<T,Allocator>& y);\ntemplate <class T, class Allocator>\nbool operator<=(const list<T,Allocator>& x, const list<T,Allocator>& y);\ntemplate <class T, class Allocator>\nvoid swap(list<T,Allocator>& x, list<T,Allocator>& y);\n}\n```\n\n```#include <initializer_list>\n\nnamespace std {\ntemplate <class T, class Allocator = allocator<T> > class vector;\ntemplate <class T, class Allocator>\nbool operator==(const vector<T,Allocator>& x,const vector<T,Allocator>& y);\ntemplate <class T, class Allocator>\nbool operator< (const vector<T,Allocator>& x,const vector<T,Allocator>& y);\ntemplate <class T, class Allocator>\nbool operator!=(const vector<T,Allocator>& x,const vector<T,Allocator>& y);\ntemplate <class T, class Allocator>\nbool operator> (const vector<T,Allocator>& x,const vector<T,Allocator>& y);\ntemplate <class T, class Allocator>\nbool operator>=(const vector<T,Allocator>& x,const vector<T,Allocator>& y);\ntemplate <class T, class Allocator>\nbool operator<=(const vector<T,Allocator>& x,const vector<T,Allocator>& y);\ntemplate <class T, class Allocator>\nvoid swap(vector<T,Allocator>& x, vector<T,Allocator>& y);\n\ntemplate <class Allocator> class vector<bool,Allocator>;\n\n// hash support\ntemplate <class T> struct hash;\ntemplate <class Allocator> struct hash<vector<bool, Allocator> >;\n}\n```\n\n### 23.3.2 Class template array[array]\n\n#### 23.3.2.1 Class template array overview [array.overview]\n\nThe header <array> defines a class template for storing fixed-size sequences of objects. An array supports random access iterators. An instance of array<T, N> stores N elements of type T, so that size() == N is an invariant. The elements of an array are stored contiguously, meaning that if a is an array<T, N> then it obeys the identity `&a[n] == &a + n` for all 0 <= n < N.\n\nAn array is an aggregate ([dcl.init.aggr]) that can be initialized with the syntax\n\n```array<T, N> a = { initializer-list };\n```\n\nwhere initializer-list is a comma-separated list of up to N elements whose types are convertible to T.\n\nAn array satisfies all of the requirements of a container and of a reversible container ([container.requirements]), except that a default constructed array object is not empty and that swap does not have constant complexity. An array satisfies some of the requirements of a sequence container ([sequence.reqmts]). Descriptions are provided here only for operations on array that are not described in one of these tables and for operations where there is additional semantic information.\n\n```namespace std {\ntemplate <class T, size_t N>\nstruct array {\n// types:\ntypedef T& reference;\ntypedef const T& const_reference;\ntypedef implementation-defined iterator;\ntypedef implementation-defined const_iterator;\ntypedef size_t size_type;\ntypedef ptrdiff_t difference_type;\ntypedef T value_type;\ntypedef T* pointer;\ntypedef const T* const_pointer;\ntypedef std::reverse_iterator<iterator> reverse_iterator;\ntypedef std::reverse_iterator<const_iterator> const_reverse_iterator;\n\nT elems[N]; // exposition only\n\n// no explicit construct/copy/destroy for aggregate type\n\nvoid fill(const T& u);\nvoid swap(array&) noexcept(noexcept(swap(declval<T&>(), declval<T&>())));\n\n// iterators:\niterator begin() noexcept;\nconst_iterator begin() const noexcept;\niterator end() noexcept;\nconst_iterator end() const noexcept;\n\nreverse_iterator rbegin() noexcept;\nconst_reverse_iterator rbegin() const noexcept;\nreverse_iterator rend() noexcept;\nconst_reverse_iterator rend() const noexcept;\n\nconst_iterator cbegin() const noexcept;\nconst_iterator cend() const noexcept;\nconst_reverse_iterator crbegin() const noexcept;\nconst_reverse_iterator crend() const noexcept;\n\n// capacity:\nconstexpr size_type size() const noexcept;\nconstexpr size_type max_size() const noexcept;\nconstexpr bool empty() const noexcept;\n\n// element access:\nreference operator[](size_type n);\nconstexpr const_reference operator[](size_type n) const;\nreference at(size_type n);\nconstexpr const_reference at(size_type n) const;\nreference front();\nconstexpr const_reference front() const;\nreference back();\nconstexpr const_reference back() const;\n\nT * data() noexcept;\nconst T * data() const noexcept;\n};\n}\n```\n\nNote: The member variable elems is shown for exposition only, to emphasize that array is a class aggregate. The name elems is not part of array's interface.  — end note ]\n\n#### 23.3.2.2array constructors, copy, and assignment [array.cons]\n\nThe conditions for an aggregate ([dcl.init.aggr]) shall be met. Class array relies on the implicitly-declared special member functions ([class.ctor], [class.dtor], and [class.copy]) to conform to the container requirements table in [container.requirements]. In addition to the requirements specified in the container requirements table, the implicit move constructor and move assignment operator for array require that T be MoveConstructible or MoveAssignable, respectively.\n\n#### 23.3.2.3array specialized algorithms [array.special]\n\n``` template <class T, size_t N> void swap(array<T,N>& x, array<T,N>& y) noexcept(noexcept(x.swap(y))); ```\n\nEffects:\n\n```x.swap(y);\n```\n\nComplexity: Linear in N.\n\n#### 23.3.2.4array::size[array.size]\n\n``` template <class T, size_t N> constexpr size_type array<T,N>::size() const noexcept; ```\n\nReturns: N\n\n#### 23.3.2.5array::data[array.data]\n\n``` T* data() noexcept; const T* data() const noexcept; ```\n\nReturns: elems.\n\n#### 23.3.2.6array::fill[array.fill]\n\n``` void fill(const T& u); ```\n\nEffects: fill_n(begin(), N, u)\n\n#### 23.3.2.7array::swap[array.swap]\n\n``` void swap(array& y) noexcept(noexcept(swap(declval<T&>(), declval<T&>()))); ```\n\nEffects: swap_ranges(begin(), end(), y.begin())\n\nThrows: Nothing unless one of the element-wise swap calls throws an exception.\n\nNote: Unlike the swap function for other containers, array::swap takes linear time, may exit via an exception, and does not cause iterators to become associated with the other container.\n\n#### 23.3.2.8 Zero sized arrays [array.zero]\n\narray shall provide support for the special case N == 0.\n\nIn the case that N == 0, begin() == end() == unique value. The return value of data() is unspecified.\n\nThe effect of calling front() or back() for a zero-sized array is undefined.\n\nMember function swap() shall have a noexcept-specification which is equivalent to noexcept(true).\n\n#### 23.3.2.9 Tuple interface to class template array[array.tuple]\n\n``` template <class T, size_t N> struct tuple_size<array<T, N>> : integral_constant<size_t, N> { }; ```\n\n``` tuple_element<I, array<T, N> >::type ```\n\nRequires: I < N. The program is ill-formed if I is out of bounds.\n\nValue: The type T.\n\n``` template <size_t I, class T, size_t N> constexpr T& get(array<T, N>& a) noexcept; ```\n\nRequires: I < N. The program is ill-formed if I is out of bounds.\n\nReturns: A reference to the Ith element of a, where indexing is zero-based.\n\n``` template <size_t I, class T, size_t N> constexpr T&& get(array<T, N>&& a) noexcept; ```\n\nEffects: Equivalent to return std::move(get<I>(a));\n\n``` template <size_t I, class T, size_t N> constexpr const T& get(const array<T, N>& a) noexcept; ```\n\nRequires: I < N. The program is ill-formed if I is out of bounds.\n\nReturns: A const reference to the Ith element of a, where indexing is zero-based.\n\n### 23.3.3 Class template deque[deque]\n\n#### 23.3.3.1 Class template deque overview [deque.overview]\n\nA deque is a sequence container that, like a vector ([vector]), supports random access iterators. In addition, it supports constant time insert and erase operations at the beginning or the end; insert and erase in the middle take linear time. That is, a deque is especially optimized for pushing and popping elements at the beginning and end. As with vectors, storage management is handled automatically.\n\nA deque satisfies all of the requirements of a container, of a reversible container (given in tables in [container.requirements]), of a sequence container, including the optional sequence container requirements ([sequence.reqmts]), and of an allocator-aware container (Table [tab:containers.allocatoraware]). Descriptions are provided here only for operations on deque that are not described in one of these tables or for operations where there is additional semantic information.\n\n```namespace std {\ntemplate <class T, class Allocator = allocator<T> >\nclass deque {\npublic:\n// types:\ntypedef value_type& reference;\ntypedef const value_type& const_reference;\ntypedef implementation-defined iterator; // See [container.requirements]\ntypedef implementation-defined const_iterator; // See [container.requirements]\ntypedef implementation-defined size_type; // See [container.requirements]\ntypedef implementation-defined difference_type;// See [container.requirements]\ntypedef T value_type;\ntypedef Allocator allocator_type;\ntypedef typename allocator_traits<Allocator>::pointer pointer;\ntypedef typename allocator_traits<Allocator>::const_pointer const_pointer;\ntypedef std::reverse_iterator<iterator> reverse_iterator;\ntypedef std::reverse_iterator<const_iterator> const_reverse_iterator;\n\n// [deque.cons], construct/copy/destroy:\ndeque() : deque(Allocator()) { }\nexplicit deque(const Allocator&);\nexplicit deque(size_type n, const Allocator& = Allocator());\ndeque(size_type n, const T& value, const Allocator& = Allocator());\ntemplate <class InputIterator>\ndeque(InputIterator first, InputIterator last, const Allocator& = Allocator());\ndeque(const deque& x);\ndeque(deque&&);\ndeque(const deque&, const Allocator&);\ndeque(deque&&, const Allocator&);\ndeque(initializer_list<T>, const Allocator& = Allocator());\n\n~deque();\ndeque& operator=(const deque& x);\ndeque& operator=(deque&& x);\ndeque& operator=(initializer_list<T>);\ntemplate <class InputIterator>\nvoid assign(InputIterator first, InputIterator last);\nvoid assign(size_type n, const T& t);\nvoid assign(initializer_list<T>);\nallocator_type get_allocator() const noexcept;\n\n// iterators:\niterator begin() noexcept;\nconst_iterator begin() const noexcept;\niterator end() noexcept;\nconst_iterator end() const noexcept;\nreverse_iterator rbegin() noexcept;\nconst_reverse_iterator rbegin() const noexcept;\nreverse_iterator rend() noexcept;\nconst_reverse_iterator rend() const noexcept;\n\nconst_iterator cbegin() const noexcept;\nconst_iterator cend() const noexcept;\nconst_reverse_iterator crbegin() const noexcept;\nconst_reverse_iterator crend() const noexcept;\n\n// [deque.capacity], capacity:\nsize_type size() const noexcept;\nsize_type max_size() const noexcept;\nvoid resize(size_type sz);\nvoid resize(size_type sz, const T& c);\nvoid shrink_to_fit();\nbool empty() const noexcept;\n\n// element access:\nreference operator[](size_type n);\nconst_reference operator[](size_type n) const;\nreference at(size_type n);\nconst_reference at(size_type n) const;\nreference front();\nconst_reference front() const;\nreference back();\nconst_reference back() const;\n\n// [deque.modifiers], modifiers:\ntemplate <class... Args> void emplace_front(Args&&... args);\ntemplate <class... Args> void emplace_back(Args&&... args);\ntemplate <class... Args> iterator emplace(const_iterator position, Args&&... args);\n\nvoid push_front(const T& x);\nvoid push_front(T&& x);\nvoid push_back(const T& x);\nvoid push_back(T&& x);\n\niterator insert(const_iterator position, const T& x);\niterator insert(const_iterator position, T&& x);\niterator insert(const_iterator position, size_type n, const T& x);\ntemplate <class InputIterator>\niterator insert (const_iterator position, InputIterator first, InputIterator last);\niterator insert(const_iterator position, initializer_list<T>);\n\nvoid pop_front();\nvoid pop_back();\n\niterator erase(const_iterator position);\niterator erase(const_iterator first, const_iterator last);\nvoid swap(deque&);\nvoid clear() noexcept;\n};\n\ntemplate <class T, class Allocator>\nbool operator==(const deque<T,Allocator>& x, const deque<T,Allocator>& y);\ntemplate <class T, class Allocator>\nbool operator< (const deque<T,Allocator>& x, const deque<T,Allocator>& y);\ntemplate <class T, class Allocator>\nbool operator!=(const deque<T,Allocator>& x, const deque<T,Allocator>& y);\ntemplate <class T, class Allocator>\nbool operator> (const deque<T,Allocator>& x, const deque<T,Allocator>& y);\ntemplate <class T, class Allocator>\nbool operator>=(const deque<T,Allocator>& x, const deque<T,Allocator>& y);\ntemplate <class T, class Allocator>\nbool operator<=(const deque<T,Allocator>& x, const deque<T,Allocator>& y);\n\n// specialized algorithms:\ntemplate <class T, class Allocator>\nvoid swap(deque<T,Allocator>& x, deque<T,Allocator>& y);\n}\n```\n\n#### 23.3.3.2deque constructors, copy, and assignment [deque.cons]\n\n``` explicit deque(const Allocator&); ```\n\nEffects: Constructs an empty deque, using the specified allocator.\n\nComplexity: Constant.\n\n``` explicit deque(size_type n, const Allocator& = Allocator()); ```\n\nEffects: Constructs a deque with n default-inserted elements using the specified allocator.\n\nRequires: T shall be DefaultInsertable into *this.\n\nComplexity: Linear in n.\n\n``` deque(size_type n, const T& value, const Allocator& = Allocator()); ```\n\nEffects: Constructs a deque with n copies of value, using the specified allocator.\n\nRequires: T shall be CopyInsertable into *this.\n\nComplexity: Linear in n.\n\n``` template <class InputIterator> deque(InputIterator first, InputIterator last, const Allocator& = Allocator()); ```\n\nEffects: Constructs a deque equal to the range [first,last), using the specified allocator.\n\nComplexity: Linear in distance(first, last).\n\n#### 23.3.3.3deque capacity [deque.capacity]\n\n``` void resize(size_type sz); ```\n\nEffects: If sz <= size(), equivalent to calling pop_back() size() - sz times. If size() < sz, appends sz - size() default-inserted elements to the sequence.\n\nRequires: T shall be MoveInsertable and DefaultInsertable into *this.\n\n``` void resize(size_type sz, const T& c); ```\n\nEffects: If sz <= size(), equivalent to calling pop_back() size() - sz times. If size() < sz, appends sz - size() copies of c to the sequence.\n\nRequires: T shall be CopyInsertable into *this.\n\n``` void shrink_to_fit(); ```\n\nRequires: T shall be MoveInsertable into *this.\n\nComplexity: Linear in the size of the sequence.\n\nRemarks: shrink_to_fit is a non-binding request to reduce memory use but does not change the size of the sequence. [ Note: The request is non-binding to allow latitude for implementation-specific optimizations.  — end note ]\n\n#### 23.3.3.4deque modifiers [deque.modifiers]\n\n``` iterator insert(const_iterator position, const T& x); iterator insert(const_iterator position, T&& x); iterator insert(const_iterator position, size_type n, const T& x); template <class InputIterator> iterator insert(const_iterator position, InputIterator first, InputIterator last); iterator insert(const_iterator position, initializer_list<T>); template <class... Args> void emplace_front(Args&&... args); template <class... Args> void emplace_back(Args&&... args); template <class... Args> iterator emplace(const_iterator position, Args&&... args); void push_front(const T& x); void push_front(T&& x); void push_back(const T& x); void push_back(T&& x); ```\n\nEffects: An insertion in the middle of the deque invalidates all the iterators and references to elements of the deque. An insertion at either end of the deque invalidates all the iterators to the deque, but has no effect on the validity of references to elements of the deque.\n\nRemarks: If an exception is thrown other than by the copy constructor, move constructor, assignment operator, or move assignment operator of T there are no effects. If an exception is thrown while inserting a single element at either end, there are no effects. Otherwise, if an exception is thrown by the move constructor of a non-CopyInsertable T, the effects are unspecified.\n\nComplexity: The complexity is linear in the number of elements inserted plus the lesser of the distances to the beginning and end of the deque. Inserting a single element either at the beginning or end of a deque always takes constant time and causes a single call to a constructor of T.\n\n``` iterator erase(const_iterator position); iterator erase(const_iterator first, const_iterator last); ```\n\nEffects: An erase operation that erases the last element of a deque invalidates only the past-the-end iterator and all iterators and references to the erased elements. An erase operation that erases the first element of a deque but not the last element invalidates only the erased elements. An erase operation that erases neither the first element nor the last element of a deque invalidates the past-the-end iterator and all iterators and references to all the elements of the deque.\n\nComplexity: The number of calls to the destructor is the same as the number of elements erased, but the number of calls to the assignment operator is no more than the lesser of the number of elements before the erased elements and the number of elements after the erased elements.\n\nThrows: Nothing unless an exception is thrown by the copy constructor, move constructor, assignment operator, or move assignment operator of T.\n\n#### 23.3.3.5deque specialized algorithms [deque.special]\n\n``` template <class T, class Allocator> void swap(deque<T,Allocator>& x, deque<T,Allocator>& y); ```\n\nEffects:\n\n```x.swap(y);\n```\n\n### 23.3.4 Class template forward_list[forwardlist]\n\n#### 23.3.4.1 Class template forward_list overview [forwardlist.overview]\n\nA forward_list is a container that supports forward iterators and allows constant time insert and erase operations anywhere within the sequence, with storage management handled automatically. Fast random access to list elements is not supported. [ Note: It is intended that forward_list have zero space or time overhead relative to a hand-written C-style singly linked list. Features that would conflict with that goal have been omitted. — end note ]\n\nA forward_list satisfies all of the requirements of a container (Table [tab:containers.container.requirements]), except that the size() member function is not provided and operator== has linear complexity. A forward_list also satisfies all of the requirements for an allocator-aware container (Table [tab:containers.allocatoraware]). In addition, a forward_list provides the assign member functions (Table [tab:containers.sequence.requirements]) and several of the optional container requirements (Table [tab:containers.sequence.optional]). Descriptions are provided here only for operations on forward_list that are not described in that table or for operations where there is additional semantic information.\n\nNote: Modifying any list requires access to the element preceding the first element of interest, but in a forward_list there is no constant-time way to access a preceding element. For this reason, ranges that are modified, such as those supplied to erase and splice, must be open at the beginning.  — end note ]\n\n```namespace std {\ntemplate <class T, class Allocator = allocator<T> >\nclass forward_list {\npublic:\n// types:\ntypedef value_type& reference;\ntypedef const value_type& const_reference;\ntypedef implementation-defined iterator; // See [container.requirements]\ntypedef implementation-defined const_iterator; // See [container.requirements]\ntypedef implementation-defined size_type; // See [container.requirements]\ntypedef implementation-defined difference_type;// See [container.requirements]\ntypedef T value_type;\ntypedef Allocator allocator_type;\ntypedef typename allocator_traits<Allocator>::pointer pointer;\ntypedef typename allocator_traits<Allocator>::const_pointer const_pointer;\n\n// [forwardlist.cons], construct/copy/destroy:\nforward_list() : forward_list(Allocator()) { }\nexplicit forward_list(const Allocator&);\nexplicit forward_list(size_type n, const Allocator& = Allocator());\nforward_list(size_type n, const T& value,\nconst Allocator& = Allocator());\ntemplate <class InputIterator>\nforward_list(InputIterator first, InputIterator last,\nconst Allocator& = Allocator());\nforward_list(const forward_list& x);\nforward_list(forward_list&& x);\nforward_list(const forward_list& x, const Allocator&);\nforward_list(forward_list&& x, const Allocator&);\nforward_list(initializer_list<T>, const Allocator& = Allocator());\n~forward_list();\nforward_list& operator=(const forward_list& x);\nforward_list& operator=(forward_list&& x);\nforward_list& operator=(initializer_list<T>);\ntemplate <class InputIterator>\nvoid assign(InputIterator first, InputIterator last);\nvoid assign(size_type n, const T& t);\nvoid assign(initializer_list<T>);\nallocator_type get_allocator() const noexcept;\n\n// [forwardlist.iter], iterators:\niterator before_begin() noexcept;\nconst_iterator before_begin() const noexcept;\niterator begin() noexcept;\nconst_iterator begin() const noexcept;\niterator end() noexcept;\nconst_iterator end() const noexcept;\n\nconst_iterator cbegin() const noexcept;\nconst_iterator cbefore_begin() const noexcept;\nconst_iterator cend() const noexcept;\n\n// capacity:\nbool empty() const noexcept;\nsize_type max_size() const noexcept;\n\n// [forwardlist.access], element access:\nreference front();\nconst_reference front() const;\n\n// [forwardlist.modifiers], modifiers:\ntemplate <class... Args> void emplace_front(Args&&... args);\nvoid push_front(const T& x);\nvoid push_front(T&& x);\nvoid pop_front();\n\ntemplate <class... Args> iterator emplace_after(const_iterator position, Args&&... args);\niterator insert_after(const_iterator position, const T& x);\niterator insert_after(const_iterator position, T&& x);\n\niterator insert_after(const_iterator position, size_type n, const T& x);\ntemplate <class InputIterator>\niterator insert_after(const_iterator position, InputIterator first, InputIterator last);\niterator insert_after(const_iterator position, initializer_list<T> il);\n\niterator erase_after(const_iterator position);\niterator erase_after(const_iterator position, const_iterator last);\nvoid swap(forward_list&);\n\nvoid resize(size_type sz);\nvoid resize(size_type sz, const value_type& c);\nvoid clear() noexcept;\n\n// [forwardlist.ops], forward_list operations:\nvoid splice_after(const_iterator position, forward_list& x);\nvoid splice_after(const_iterator position, forward_list&& x);\nvoid splice_after(const_iterator position, forward_list& x,\nconst_iterator i);\nvoid splice_after(const_iterator position, forward_list&& x,\nconst_iterator i);\nvoid splice_after(const_iterator position, forward_list& x,\nconst_iterator first, const_iterator last);\nvoid splice_after(const_iterator position, forward_list&& x,\nconst_iterator first, const_iterator last);\n\nvoid remove(const T& value);\ntemplate <class Predicate> void remove_if(Predicate pred);\n\nvoid unique();\ntemplate <class BinaryPredicate> void unique(BinaryPredicate binary_pred);\n\nvoid merge(forward_list& x);\nvoid merge(forward_list&& x);\ntemplate <class Compare> void merge(forward_list& x, Compare comp);\ntemplate <class Compare> void merge(forward_list&& x, Compare comp);\n\nvoid sort();\ntemplate <class Compare> void sort(Compare comp);\n\nvoid reverse() noexcept;\n};\n\n// Comparison operators\ntemplate <class T, class Allocator>\nbool operator==(const forward_list<T,Allocator>& x, const forward_list<T,Allocator>& y);\ntemplate <class T, class Allocator>\nbool operator< (const forward_list<T,Allocator>& x, const forward_list<T,Allocator>& y);\ntemplate <class T, class Allocator>\nbool operator!=(const forward_list<T,Allocator>& x, const forward_list<T,Allocator>& y);\ntemplate <class T, class Allocator>\nbool operator> (const forward_list<T,Allocator>& x, const forward_list<T,Allocator>& y);\ntemplate <class T, class Allocator>\nbool operator>=(const forward_list<T,Allocator>& x, const forward_list<T,Allocator>& y);\ntemplate <class T, class Allocator>\nbool operator<=(const forward_list<T,Allocator>& x, const forward_list<T,Allocator>& y);\n\n// [forwardlist.spec], specialized algorithms:\ntemplate <class T, class Allocator>\nvoid swap(forward_list<T,Allocator>& x, forward_list<T,Allocator>& y);\n}\n```\n\n#### 23.3.4.2forward_list constructors, copy, assignment [forwardlist.cons]\n\n``` explicit forward_list(const Allocator&); ```\n\nEffects: Constructs an empty forward_list object using the specified allocator.\n\nComplexity: Constant.\n\n``` explicit forward_list(size_type n, const Allocator& = Allocator()); ```\n\nEffects: Constructs a forward_list object with n default-inserted elements using the specified allocator.\n\nRequires: T shall be DefaultInsertable into *this.\n\nComplexity: Linear in n.\n\n``` forward_list(size_type n, const T& value, const Allocator& = Allocator()); ```\n\nEffects: Constructs a forward_list object with n copies of value using the specified allocator.\n\nRequires: T shall be CopyInsertable into *this.\n\nComplexity: Linear in n.\n\n``` template <class InputIterator> forward_list(InputIterator first, InputIterator last, const Allocator& = Allocator()); ```\n\nEffects: Constructs a forward_list object equal to the range [first,last).\n\nComplexity: Linear in distance(first, last).\n\n#### 23.3.4.3forward_list iterators [forwardlist.iter]\n\n``` iterator before_begin() noexcept; const_iterator before_begin() const noexcept; const_iterator cbefore_begin() const noexcept; ```\n\nReturns: A non-dereferenceable iterator that, when incremented, is equal to the iterator returned by begin().\n\nEffects: cbefore_begin() is equivalent to const_cast<forward_list const&>(*this).before_begin().\n\nRemarks: before_begin() == end() shall equal false.\n\n#### 23.3.4.4forward_list element access [forwardlist.access]\n\n``` reference front(); const_reference front() const; ```\n\nReturns: *begin()\n\n#### 23.3.4.5forward_list modifiers [forwardlist.modifiers]\n\nNone of the overloads of insert_after shall affect the validity of iterators and references, and erase_after shall invalidate only iterators and references to the erased elements. If an exception is thrown during insert_after there shall be no effect. Inserting n elements into a forward_list is linear in n, and the number of calls to the copy or move constructor of T is exactly equal to n. Erasing n elements from a forward_list is linear in n and the number of calls to the destructor of type T is exactly equal to n.\n\n``` template <class... Args> void emplace_front(Args&&... args); ```\n\nEffects: Inserts an object of type value_type constructed with value_type(std::forward<Args>(args)...) at the beginning of the list.\n\n``` void push_front(const T& x); void push_front(T&& x); ```\n\nEffects: Inserts a copy of x at the beginning of the list.\n\n``` void pop_front(); ```\n\nEffects: erase_after(before_begin())\n\n``` iterator insert_after(const_iterator position, const T& x); iterator insert_after(const_iterator position, T&& x); ```\n\nRequires: position is before_begin() or is a dereferenceable iterator in the range [begin(),end()).\n\nEffects: Inserts a copy of x after position.\n\nReturns: An iterator pointing to the copy of x.\n\n``` iterator insert_after(const_iterator position, size_type n, const T& x); ```\n\nRequires: position is before_begin() or is a dereferenceable iterator in the range [begin(),end()).\n\nEffects: Inserts n copies of x after position.\n\nReturns: An iterator pointing to the last inserted copy of x or position if n == 0.\n\n``` template <class InputIterator> iterator insert_after(const_iterator position, InputIterator first, InputIterator last); ```\n\nRequires: position is before_begin() or is a dereferenceable iterator in the range [begin(),end()). first and last are not iterators in *this.\n\nEffects: Inserts copies of elements in [first,last) after position.\n\nReturns: An iterator pointing to the last inserted element or position if first == last.\n\n``` iterator insert_after(const_iterator position, initializer_list<T> il); ```\n\nEffects: insert_after(p, il.begin(), il.end()).\n\nReturns: An iterator pointing to the last inserted element or position if il is empty.\n\n``` template <class... Args> iterator emplace_after(const_iterator position, Args&&... args); ```\n\nRequires: position is before_begin() or is a dereferenceable iterator in the range [begin(),end()).\n\nEffects: Inserts an object of type value_type constructed with value_type(std::forward<Args>(args)...) after position.\n\nReturns: An iterator pointing to the new object.\n\n``` iterator erase_after(const_iterator position); ```\n\nRequires: The iterator following position is dereferenceable.\n\nEffects: Erases the element pointed to by the iterator following position.\n\nReturns: An iterator pointing to the element following the one that was erased, or end() if no such element exists.\n\nThrows: Nothing.\n\n``` iterator erase_after(const_iterator position, const_iterator last); ```\n\nRequires: All iterators in the range (position,last) are dereferenceable.\n\nEffects: Erases the elements in the range (position,last).\n\nReturns: last.\n\nThrows: Nothing.\n\n``` void resize(size_type sz); ```\n\nEffects: If sz < distance(begin(), end()), erases the last distance(begin(), end()) - sz elements from the list. Otherwise, inserts sz - distance(begin(), end()) default-inserted elements at the end of the list.\n\nRequires: T shall be DefaultInsertable into *this.\n\n``` void resize(size_type sz, const value_type& c); ```\n\nEffects: If sz < distance(begin(), end()), erases the last distance(begin(), end()) - sz elements from the list. Otherwise, inserts sz - distance(begin(), end()) elements at the end of the list such that each new element, e, is initialized by a method equivalent to calling allocator_traits<allocator_type>::construct(get_allocator(), std::addressof(e), c).\n\nRequires: T shall be CopyInsertable into *this.\n\n``` void clear() noexcept; ```\n\nEffects: Erases all elements in the range [begin(),end()).\n\nRemarks: Does not invalidate past-the-end iterators.\n\n#### 23.3.4.6forward_list operations [forwardlist.ops]\n\n``` void splice_after(const_iterator position, forward_list& x); void splice_after(const_iterator position, forward_list&& x); ```\n\nRequires: position is before_begin() or is a dereferenceable iterator in the range [begin(),end()). get_allocator() == x.get_allocator(). &x != this.\n\nEffects: Inserts the contents of x after position, and x becomes empty. Pointers and references to the moved elements of x now refer to those same elements but as members of *this. Iterators referring to the moved elements will continue to refer to their elements, but they now behave as iterators into *this, not into x.\n\nThrows: Nothing.\n\nComplexity: Ο(distance(x.begin(), x.end()))\n\n``` void splice_after(const_iterator position, forward_list& x, const_iterator i); void splice_after(const_iterator position, forward_list&& x, const_iterator i); ```\n\nRequires: position is before_begin() or is a dereferenceable iterator in the range [begin(),end()). The iterator following i is a dereferenceable iterator in x. get_allocator() == x.get_allocator().\n\nEffects: Inserts the element following i into *this, following position, and removes it from x. The result is unchanged if position == i or position == ++i. Pointers and references to *++i continue to refer to the same element but as a member of *this. Iterators to *++i continue to refer to the same element, but now behave as iterators into *this, not into x.\n\nThrows: Nothing.\n\nComplexity: Ο(1)\n\n``` void splice_after(const_iterator position, forward_list& x, const_iterator first, const_iterator last); void splice_after(const_iterator position, forward_list&& x, const_iterator first, const_iterator last); ```\n\nRequires: position is before_begin() or is a dereferenceable iterator in the range [begin(),end()). (first,last) is a valid range in x, and all iterators in the range (first,last) are dereferenceable. position is not an iterator in the range (first,last). get_allocator() == x.get_allocator().\n\nEffects: Inserts elements in the range (first,last) after position and removes the elements from x. Pointers and references to the moved elements of x now refer to those same elements but as members of *this. Iterators referring to the moved elements will continue to refer to their elements, but they now behave as iterators into *this, not into x.\n\nComplexity: Ο(distance(first, last))\n\n``` void remove(const T& value); template <class Predicate> void remove_if(Predicate pred); ```\n\nEffects: Erases all the elements in the list referred by a list iterator i for which the following conditions hold: *i == value (for remove()), pred(*i) is true (for remove_if()). Invalidates only the iterators and references to the erased elements.\n\nThrows: Nothing unless an exception is thrown by the equality comparison or the predicate.\n\nRemarks: Stable ([algorithm.stable]).\n\nComplexity: Exactly distance(begin(), end()) applications of the corresponding predicate.\n\n``` void unique(); template <class BinaryPredicate> void unique(BinaryPredicate pred); ```\n\nEffects: Erases all but the first element from every consecutive group of equal elements referred to by the iterator i in the range [first + 1,last) for which *i == *(i-1) (for the version with no arguments) or pred(*i, *(i - 1)) (for the version with a predicate argument) holds. Invalidates only the iterators and references to the erased elements.\n\nThrows: Nothing unless an exception is thrown by the equality comparison or the predicate.\n\nComplexity: If the range [first,last) is not empty, exactly (last - first) - 1 applications of the corresponding predicate, otherwise no applications of the predicate.\n\n``` void merge(forward_list& x); void merge(forward_list&& x); template <class Compare> void merge(forward_list& x, Compare comp); template <class Compare> void merge(forward_list&& x, Compare comp); ```\n\nRequires: comp defines a strict weak ordering ([alg.sorting]), and *this and x are both sorted according to this ordering. get_allocator() == x.get_allocator().\n\nEffects: Merges the two sorted ranges [begin(), end()) and [x.begin(), x.end()). x is empty after the merge. If an exception is thrown other than by a comparison there are no effects. Pointers and references to the moved elements of x now refer to those same elements but as members of *this. Iterators referring to the moved elements will continue to refer to their elements, but they now behave as iterators into *this, not into x.\n\nRemarks: Stable ([algorithm.stable]). The behavior is undefined if this->get_allocator() != x.get_allocator().\n\nComplexity: At most distance(begin(), end()) + distance(x.begin(), x.end()) - 1 comparisons.\n\n``` void sort(); template <class Compare> void sort(Compare comp); ```\n\nRequires: operator< (for the version with no arguments) or comp (for the version with a comparison argument) defines a strict weak ordering ([alg.sorting]).\n\nEffects: Sorts the list according to the operator< or the comp function object. If an exception is thrown the order of the elements in *this is unspecified. Does not affect the validity of iterators and references.\n\nRemarks: Stable ([algorithm.stable]).\n\nComplexity: Approximately N log N comparisons, where N is distance(begin(), end()).\n\n``` void reverse() noexcept; ```\n\nEffects: Reverses the order of the elements in the list. Does not affect the validity of iterators and references.\n\nComplexity: Linear time.\n\n#### 23.3.4.7forward_list specialized algorithms [forwardlist.spec]\n\n``` template <class T, class Allocator> void swap(forward_list<T,Allocator>& x, forward_list<T,Allocator>& y); ```\n\nEffects: x.swap(y)\n\n### 23.3.5 Class template list[list]\n\n#### 23.3.5.1 Class template list overview [list.overview]\n\nA list is a sequence container that supports bidirectional iterators and allows constant time insert and erase operations anywhere within the sequence, with storage management handled automatically. Unlike vectors ([vector]) and deques ([deque]), fast random access to list elements is not supported, but many algorithms only need sequential access anyway.\n\nA list satisfies all of the requirements of a container, of a reversible container (given in two tables in [container.requirements]), of a sequence container, including most of the optional sequence container requirements ([sequence.reqmts]), and of an allocator-aware container (Table [tab:containers.allocatoraware]). The exceptions are the operator[] and at member functions, which are not provided.265 Descriptions are provided here only for operations on list that are not described in one of these tables or for operations where there is additional semantic information.\n\n```namespace std {\ntemplate <class T, class Allocator = allocator<T> >\nclass list {\npublic:\n// types:\ntypedef value_type& reference;\ntypedef const value_type& const_reference;\ntypedef implementation-defined iterator; // see [container.requirements]\ntypedef implementation-defined const_iterator; // see [container.requirements]\ntypedef implementation-defined size_type; // see [container.requirements]\ntypedef implementation-defined difference_type;// see [container.requirements]\ntypedef T value_type;\ntypedef Allocator allocator_type;\ntypedef typename allocator_traits<Allocator>::pointer pointer;\ntypedef typename allocator_traits<Allocator>::const_pointer const_pointer;\ntypedef std::reverse_iterator<iterator> reverse_iterator;\ntypedef std::reverse_iterator<const_iterator> const_reverse_iterator;\n\n// [list.cons], construct/copy/destroy:\nlist() : list(Allocator()) { }\nexplicit list(const Allocator&);\nexplicit list(size_type n, const Allocator& = Allocator());\nlist(size_type n, const T& value, const Allocator& = Allocator());\ntemplate <class InputIterator>\nlist(InputIterator first, InputIterator last, const Allocator& = Allocator());\nlist(const list& x);\nlist(list&& x);\nlist(const list&, const Allocator&);\nlist(list&&, const Allocator&);\nlist(initializer_list<T>, const Allocator& = Allocator());\n~list();\nlist& operator=(const list& x);\nlist& operator=(list&& x);\nlist& operator=(initializer_list<T>);\ntemplate <class InputIterator>\nvoid assign(InputIterator first, InputIterator last);\nvoid assign(size_type n, const T& t);\nvoid assign(initializer_list<T>);\nallocator_type get_allocator() const noexcept;\n\n// iterators:\niterator begin() noexcept;\nconst_iterator begin() const noexcept;\niterator end() noexcept;\nconst_iterator end() const noexcept;\nreverse_iterator rbegin() noexcept;\nconst_reverse_iterator rbegin() const noexcept;\nreverse_iterator rend() noexcept;\nconst_reverse_iterator rend() const noexcept;\n\nconst_iterator cbegin() const noexcept;\nconst_iterator cend() const noexcept;\nconst_reverse_iterator crbegin() const noexcept;\nconst_reverse_iterator crend() const noexcept;\n\n// [list.capacity], capacity:\nbool empty() const noexcept;\nsize_type size() const noexcept;\nsize_type max_size() const noexcept;\nvoid resize(size_type sz);\nvoid resize(size_type sz, const T& c);\n\n// element access:\nreference front();\nconst_reference front() const;\nreference back();\nconst_reference back() const;\n\n// [list.modifiers], modifiers:\ntemplate <class... Args> void emplace_front(Args&&... args);\nvoid pop_front();\ntemplate <class... Args> void emplace_back(Args&&... args);\nvoid push_front(const T& x);\nvoid push_front(T&& x);\nvoid push_back(const T& x);\nvoid push_back(T&& x);\nvoid pop_back();\n\ntemplate <class... Args> iterator emplace(const_iterator position, Args&&... args);\niterator insert(const_iterator position, const T& x);\niterator insert(const_iterator position, T&& x);\niterator insert(const_iterator position, size_type n, const T& x);\ntemplate <class InputIterator>\niterator insert(const_iterator position, InputIterator first,\nInputIterator last);\niterator insert(const_iterator position, initializer_list<T> il);\n\niterator erase(const_iterator position);\niterator erase(const_iterator position, const_iterator last);\nvoid swap(list&);\nvoid clear() noexcept;\n\n// [list.ops], list operations:\nvoid splice(const_iterator position, list& x);\nvoid splice(const_iterator position, list&& x);\nvoid splice(const_iterator position, list& x, const_iterator i);\nvoid splice(const_iterator position, list&& x, const_iterator i);\nvoid splice(const_iterator position, list& x,\nconst_iterator first, const_iterator last);\nvoid splice(const_iterator position, list&& x,\nconst_iterator first, const_iterator last);\n\nvoid remove(const T& value);\ntemplate <class Predicate> void remove_if(Predicate pred);\n\nvoid unique();\ntemplate <class BinaryPredicate>\nvoid unique(BinaryPredicate binary_pred);\n\nvoid merge(list& x);\nvoid merge(list&& x);\ntemplate <class Compare> void merge(list& x, Compare comp);\ntemplate <class Compare> void merge(list&& x, Compare comp);\n\nvoid sort();\ntemplate <class Compare> void sort(Compare comp);\n\nvoid reverse() noexcept;\n};\n\ntemplate <class T, class Allocator>\nbool operator==(const list<T,Allocator>& x, const list<T,Allocator>& y);\ntemplate <class T, class Allocator>\nbool operator< (const list<T,Allocator>& x, const list<T,Allocator>& y);\ntemplate <class T, class Allocator>\nbool operator!=(const list<T,Allocator>& x, const list<T,Allocator>& y);\ntemplate <class T, class Allocator>\nbool operator> (const list<T,Allocator>& x, const list<T,Allocator>& y);\ntemplate <class T, class Allocator>\nbool operator>=(const list<T,Allocator>& x, const list<T,Allocator>& y);\ntemplate <class T, class Allocator>\nbool operator<=(const list<T,Allocator>& x, const list<T,Allocator>& y);\n\n// specialized algorithms:\ntemplate <class T, class Allocator>\nvoid swap(list<T,Allocator>& x, list<T,Allocator>& y);\n}\n```\n\nThese member functions are only provided by containers whose iterators are random access iterators.\n\n#### 23.3.5.2list constructors, copy, and assignment [list.cons]\n\n``` explicit list(const Allocator&); ```\n\nEffects: Constructs an empty list, using the specified allocator.\n\nComplexity: Constant.\n\n``` explicit list(size_type n, const Allocator& = Allocator()); ```\n\nEffects: Constructs a list with n default-inserted elements using the specified allocator.\n\nRequires: T shall be DefaultInsertable into *this.\n\nComplexity: Linear in n.\n\n``` list(size_type n, const T& value, const Allocator& = Allocator()); ```\n\nEffects: Constructs a list with n copies of value, using the specified allocator.\n\nRequires: T shall be CopyInsertable into *this.\n\nComplexity: Linear in n.\n\n``` template <class InputIterator> list(InputIterator first, InputIterator last, const Allocator& = Allocator()); ```\n\nEffects: Constructs a list equal to the range [first,last).\n\nComplexity: Linear in distance(first, last).\n\n#### 23.3.5.3list capacity [list.capacity]\n\n``` void resize(size_type sz); ```\n\nEffects: If size() < sz, appends sz - size() default-inserted elements to the sequence. If sz <= size(), equivalent to\n\n```list<T>::iterator it = begin();\nerase(it, end());\n```\n\nRequires: T shall be DefaultInsertable into *this.\n\n``` void resize(size_type sz, const T& c); ```\n\nEffects:\n\n```if (sz > size())\ninsert(end(), sz-size(), c);\nelse if (sz < size()) {\niterator i = begin();\nerase(i, end());\n}\nelse\n; // do nothing\n```\n\nRequires: T shall be CopyInsertable into *this.\n\n#### 23.3.5.4list modifiers [list.modifiers]\n\n``` iterator insert(const_iterator position, const T& x); iterator insert(const_iterator position, T&& x); iterator insert(const_iterator position, size_type n, const T& x); template <class InputIterator> iterator insert(const_iterator position, InputIterator first, InputIterator last); iterator insert(const_iterator position, initializer_list<T>); template <class... Args> void emplace_front(Args&&... args); template <class... Args> void emplace_back(Args&&... args); template <class... Args> iterator emplace(const_iterator position, Args&&... args); void push_front(const T& x); void push_front(T&& x); void push_back(const T& x); void push_back(T&& x); ```\n\nRemarks: Does not affect the validity of iterators and references. If an exception is thrown there are no effects.\n\nComplexity: Insertion of a single element into a list takes constant time and exactly one call to a constructor of T. Insertion of multiple elements into a list is linear in the number of elements inserted, and the number of calls to the copy constructor or move constructor of T is exactly equal to the number of elements inserted.\n\n``` iterator erase(const_iterator position); iterator erase(const_iterator first, const_iterator last); void pop_front(); void pop_back(); void clear() noexcept; ```\n\nEffects: Invalidates only the iterators and references to the erased elements.\n\nThrows: Nothing.\n\nComplexity: Erasing a single element is a constant time operation with a single call to the destructor of T. Erasing a range in a list is linear time in the size of the range and the number of calls to the destructor of type T is exactly equal to the size of the range.\n\n#### 23.3.5.5list operations [list.ops]\n\nSince lists allow fast insertion and erasing from the middle of a list, certain operations are provided specifically for them.266\n\nlist provides three splice operations that destructively move elements from one list to another. The behavior of splice operations is undefined if get_allocator() != x.get_allocator().\n\n``` void splice(const_iterator position, list& x); void splice(const_iterator position, list&& x); ```\n\nRequires: &x != this.\n\nEffects: Inserts the contents of x before position and x becomes empty. Pointers and references to the moved elements of x now refer to those same elements but as members of *this. Iterators referring to the moved elements will continue to refer to their elements, but they now behave as iterators into *this, not into x.\n\nThrows: Nothing.\n\nComplexity: Constant time.\n\n``` void splice(const_iterator position, list& x, const_iterator i); void splice(const_iterator position, list&& x, const_iterator i); ```\n\nEffects: Inserts an element pointed to by i from list x before position and removes the element from x. The result is unchanged if position == i or position == ++i. Pointers and references to *i continue to refer to this same element but as a member of *this. Iterators to *i (including i itself) continue to refer to the same element, but now behave as iterators into *this, not into x.\n\nRequires: i is a valid dereferenceable iterator of x.\n\nThrows: Nothing.\n\nComplexity: Constant time.\n\n``` void splice(const_iterator position, list& x, const_iterator first, const_iterator last); void splice(const_iterator position, list&& x, const_iterator first, const_iterator last); ```\n\nEffects: Inserts elements in the range [first,last) before position and removes the elements from x.\n\nRequires: [first, last) is a valid range in x. The result is undefined if position is an iterator in the range [first,last). Pointers and references to the moved elements of x now refer to those same elements but as members of *this. Iterators referring to the moved elements will continue to refer to their elements, but they now behave as iterators into *this, not into x.\n\nThrows: Nothing.\n\nComplexity: Constant time if &x == this; otherwise, linear time.\n\n``` void remove(const T& value); template <class Predicate> void remove_if(Predicate pred); ```\n\nEffects: Erases all the elements in the list referred by a list iterator i for which the following conditions hold: *i == value, pred(*i) != false. Invalidates only the iterators and references to the erased elements.\n\nThrows: Nothing unless an exception is thrown by *i == value or pred(*i) != false.\n\nRemarks: Stable ([algorithm.stable]).\n\nComplexity: Exactly size() applications of the corresponding predicate.\n\n``` void unique(); template <class BinaryPredicate> void unique(BinaryPredicate binary_pred); ```\n\nEffects: Erases all but the first element from every consecutive group of equal elements referred to by the iterator i in the range [first + 1,last) for which *i == *(i-1) (for the version of unique with no arguments) or pred(*i, *(i - 1)) (for the version of unique with a predicate argument) holds. Invalidates only the iterators and references to the erased elements.\n\nThrows: Nothing unless an exception is thrown by *i == *(i-1) or pred(*i, *(i - 1))\n\nComplexity: If the range [first, last) is not empty, exactly (last - first) - 1 applications of the corresponding predicate, otherwise no applications of the predicate.\n\n``` void merge(list& x); void merge(list&& x); template <class Compare> void merge(list& x, Compare comp); template <class Compare> void merge(list&& x, Compare comp); ```\n\nRequires: comp shall define a strict weak ordering ([alg.sorting]), and both the list and the argument list shall be sorted according to this ordering.\n\nEffects: If (&x == this) does nothing; otherwise, merges the two sorted ranges [begin(), end()) and [x.begin(), x.end()). The result is a range in which the elements will be sorted in non-decreasing order according to the ordering defined by comp; that is, for every iterator i, in the range other than the first, the condition comp(*i, *(i - 1)) will be false. Pointers and references to the moved elements of x now refer to those same elements but as members of *this. Iterators referring to the moved elements will continue to refer to their elements, but they now behave as iterators into *this, not into x.\n\nRemarks: Stable ([algorithm.stable]). If (&x != this) the range [x.begin(), x.end()) is empty after the merge. No elements are copied by this operation. The behavior is undefined if this->get_allocator() != x.get_allocator().\n\nComplexity: At most size() + x.size() - 1 applications of comp if (&x != this); otherwise, no applications of comp are performed. If an exception is thrown other than by a comparison there are no effects.\n\n``` void reverse() noexcept; ```\n\nEffects: Reverses the order of the elements in the list. Does not affect the validity of iterators and references.\n\nComplexity: Linear time.\n\n``` void sort(); template <class Compare> void sort(Compare comp); ```\n\nRequires: operator< (for the first version) or comp (for the second version) shall define a strict weak ordering ([alg.sorting]).\n\nEffects: Sorts the list according to the operator< or a Compare function object. Does not affect the validity of iterators and references.\n\nRemarks: Stable ([algorithm.stable]).\n\nComplexity: Approximately N log(N) comparisons, where N == size().\n\nAs specified in [allocator.requirements], the requirements in this Clause apply only to lists whose allocators compare equal.\n\n#### 23.3.5.6list specialized algorithms [list.special]\n\n``` template <class T, class Allocator> void swap(list<T,Allocator>& x, list<T,Allocator>& y); ```\n\nEffects:\n\n```x.swap(y);\n```\n\n### 23.3.6 Class template vector[vector]\n\n#### 23.3.6.1 Class template vector overview [vector.overview]\n\nA vector is a sequence container that supports random access iterators. In addition, it supports (amortized) constant time insert and erase operations at the end; insert and erase in the middle take linear time. Storage management is handled automatically, though hints can be given to improve efficiency. The elements of a vector are stored contiguously, meaning that if v is a vector<T, Allocator> where T is some type other than bool, then it obeys the identity &v[n] == &v + n for all 0 <= n < v.size().\n\nA vector satisfies all of the requirements of a container and of a reversible container (given in two tables in [container.requirements]), of a sequence container, including most of the optional sequence container requirements ([sequence.reqmts]), and of an allocator-aware container (Table [tab:containers.allocatoraware]). The exceptions are the push_front, pop_front, and emplace_front member functions, which are not provided. Descriptions are provided here only for operations on vector that are not described in one of these tables or for operations where there is additional semantic information.\n\n```namespace std {\ntemplate <class T, class Allocator = allocator<T> >\nclass vector {\npublic:\n// types:\ntypedef value_type& reference;\ntypedef const value_type& const_reference;\ntypedef implementation-defined iterator; // see [container.requirements]\ntypedef implementation-defined const_iterator; // see [container.requirements]\ntypedef implementation-defined size_type; // see [container.requirements]\ntypedef implementation-defined difference_type;// see [container.requirements]\ntypedef T value_type;\ntypedef Allocator allocator_type;\ntypedef typename allocator_traits<Allocator>::pointer pointer;\ntypedef typename allocator_traits<Allocator>::const_pointer const_pointer;\ntypedef std::reverse_iterator<iterator> reverse_iterator;\ntypedef std::reverse_iterator<const_iterator> const_reverse_iterator;\n\n// [vector.cons], construct/copy/destroy:\nvector() : vector(Allocator()) { }\nexplicit vector(const Allocator&);\nexplicit vector(size_type n, const Allocator& = Allocator());\nvector(size_type n, const T& value, const Allocator& = Allocator());\ntemplate <class InputIterator>\nvector(InputIterator first, InputIterator last,\nconst Allocator& = Allocator());\nvector(const vector& x);\nvector(vector&&);\nvector(const vector&, const Allocator&);\nvector(vector&&, const Allocator&);\nvector(initializer_list<T>, const Allocator& = Allocator());\n~vector();\nvector& operator=(const vector& x);\nvector& operator=(vector&& x);\nvector& operator=(initializer_list<T>);\ntemplate <class InputIterator>\nvoid assign(InputIterator first, InputIterator last);\nvoid assign(size_type n, const T& u);\nvoid assign(initializer_list<T>);\nallocator_type get_allocator() const noexcept;\n\n// iterators:\niterator begin() noexcept;\nconst_iterator begin() const noexcept;\niterator end() noexcept;\nconst_iterator end() const noexcept;\nreverse_iterator rbegin() noexcept;\nconst_reverse_iterator rbegin() const noexcept;\nreverse_iterator rend() noexcept;\nconst_reverse_iterator rend() const noexcept;\n\nconst_iterator cbegin() const noexcept;\nconst_iterator cend() const noexcept;\nconst_reverse_iterator crbegin() const noexcept;\nconst_reverse_iterator crend() const noexcept;\n\n// [vector.capacity], capacity:\nsize_type size() const noexcept;\nsize_type max_size() const noexcept;\nvoid resize(size_type sz);\nvoid resize(size_type sz, const T& c);\nsize_type capacity() const noexcept;\nbool empty() const noexcept;\nvoid reserve(size_type n);\nvoid shrink_to_fit();\n\n// element access:\nreference operator[](size_type n);\nconst_reference operator[](size_type n) const;\nconst_reference at(size_type n) const;\nreference at(size_type n);\nreference front();\nconst_reference front() const;\nreference back();\nconst_reference back() const;\n\n// [vector.data], data access\nT* data() noexcept;\nconst T* data() const noexcept;\n\n// [vector.modifiers], modifiers:\ntemplate <class... Args> void emplace_back(Args&&... args);\nvoid push_back(const T& x);\nvoid push_back(T&& x);\nvoid pop_back();\n\ntemplate <class... Args> iterator emplace(const_iterator position, Args&&... args);\niterator insert(const_iterator position, const T& x);\niterator insert(const_iterator position, T&& x);\niterator insert(const_iterator position, size_type n, const T& x);\ntemplate <class InputIterator>\niterator insert(const_iterator position,\nInputIterator first, InputIterator last);\niterator insert(const_iterator position, initializer_list<T> il);\niterator erase(const_iterator position);\niterator erase(const_iterator first, const_iterator last);\nvoid swap(vector&);\nvoid clear() noexcept;\n};\n\ntemplate <class T, class Allocator>\nbool operator==(const vector<T,Allocator>& x, const vector<T,Allocator>& y);\ntemplate <class T, class Allocator>\nbool operator< (const vector<T,Allocator>& x, const vector<T,Allocator>& y);\ntemplate <class T, class Allocator>\nbool operator!=(const vector<T,Allocator>& x, const vector<T,Allocator>& y);\ntemplate <class T, class Allocator>\nbool operator> (const vector<T,Allocator>& x, const vector<T,Allocator>& y);\ntemplate <class T, class Allocator>\nbool operator>=(const vector<T,Allocator>& x, const vector<T,Allocator>& y);\ntemplate <class T, class Allocator>\nbool operator<=(const vector<T,Allocator>& x, const vector<T,Allocator>& y);\n\n// [vector.special], specialized algorithms:\ntemplate <class T, class Allocator>\nvoid swap(vector<T,Allocator>& x, vector<T,Allocator>& y);\n}\n```\n\n#### 23.3.6.2vector constructors, copy, and assignment [vector.cons]\n\n``` explicit vector(const Allocator&); ```\n\nEffects: Constructs an empty vector, using the specified allocator.\n\nComplexity: Constant.\n\n``` explicit vector(size_type n, const Allocator& = Allocator()); ```\n\nEffects: Constructs a vector with n default-inserted elements using the specified allocator.\n\nRequires: T shall be DefaultInsertable into *this.\n\nComplexity: Linear in n.\n\n``` vector(size_type n, const T& value, const Allocator& = Allocator()); ```\n\nEffects: Constructs a vector with n copies of value, using the specified allocator.\n\nRequires: T shall be CopyInsertable into *this.\n\nComplexity: Linear in n.\n\n``` template <class InputIterator> vector(InputIterator first, InputIterator last, const Allocator& = Allocator()); ```\n\nEffects: Constructs a vector equal to the range [first,last), using the specified allocator.\n\nComplexity: Makes only N calls to the copy constructor of T (where N is the distance between first and last) and no reallocations if iterators first and last are of forward, bidirectional, or random access categories. It makes order N calls to the copy constructor of T and order log(N) reallocations if they are just input iterators.\n\n#### 23.3.6.3vector capacity [vector.capacity]\n\n``` size_type capacity() const noexcept; ```\n\nReturns: The total number of elements that the vector can hold without requiring reallocation.\n\n``` void reserve(size_type n); ```\n\nRequires: T shall be MoveInsertable into *this.\n\nEffects: A directive that informs a vector of a planned change in size, so that it can manage the storage allocation accordingly. After reserve(), capacity() is greater or equal to the argument of reserve if reallocation happens; and equal to the previous value of capacity() otherwise. Reallocation happens at this point if and only if the current capacity is less than the argument of reserve(). If an exception is thrown other than by the move constructor of a non-CopyInsertable type, there are no effects.\n\nComplexity: It does not change the size of the sequence and takes at most linear time in the size of the sequence.\n\nThrows: length_error if n > max_size().267\n\nRemarks: Reallocation invalidates all the references, pointers, and iterators referring to the elements in the sequence. No reallocation shall take place during insertions that happen after a call to reserve() until the time when an insertion would make the size of the vector greater than the value of capacity().\n\n``` void shrink_to_fit(); ```\n\nRequires: T shall be MoveInsertable into *this.\n\nComplexity: Linear in the size of the sequence.\n\nRemarks: shrink_to_fit is a non-binding request to reduce capacity() to size(). [ Note: The request is non-binding to allow latitude for implementation-specific optimizations.  — end note ] If an exception is thrown other than by the move constructor of a non-CopyInsertable T there are no effects.\n\n``` void swap(vector& x); ```\n\nEffects: Exchanges the contents and capacity() of *this with that of x.\n\nComplexity: Constant time.\n\n``` void resize(size_type sz); ```\n\nEffects: If sz <= size(), equivalent to calling pop_back() size() - sz times. If size() < sz, appends sz - size() default-inserted elements to the sequence.\n\nRequires: T shall be MoveInsertable and DefaultInsertable into *this.\n\nRemarks: If an exception is thrown other than by the move constructor of a non-CopyInsertable T there are no effects.\n\n``` void resize(size_type sz, const T& c); ```\n\nEffects: If sz <= size(), equivalent to calling pop_back() size() - sz times. If size() < sz, appends sz - size() copies of c to the sequence.\n\nRequires: T shall be CopyInsertable into *this.\n\nRemarks: If an exception is thrown there are no effects.\n\nreserve() uses Allocator::allocate() which may throw an appropriate exception.\n\n#### 23.3.6.4vector data [vector.data]\n\n``` T* data() noexcept; const T* data() const noexcept; ```\n\nReturns: A pointer such that [data(),data() + size()) is a valid range. For a non-empty vector, data() == &front().\n\nComplexity: Constant time.\n\n#### 23.3.6.5vector modifiers [vector.modifiers]\n\n``` iterator insert(const_iterator position, const T& x); iterator insert(const_iterator position, T&& x); iterator insert(const_iterator position, size_type n, const T& x); template <class InputIterator> iterator insert(const_iterator position, InputIterator first, InputIterator last); iterator insert(const_iterator position, initializer_list<T>); template <class... Args> void emplace_back(Args&&... args); template <class... Args> iterator emplace(const_iterator position, Args&&... args); void push_back(const T& x); void push_back(T&& x); ```\n\nRemarks: Causes reallocation if the new size is greater than the old capacity. If no reallocation happens, all the iterators and references before the insertion point remain valid. If an exception is thrown other than by the copy constructor, move constructor, assignment operator, or move assignment operator of T or by any InputIterator operation there are no effects. If an exception is thrown while inserting a single element at the end and T is CopyInsertable or is_nothrow_move_constructible<T>::value is true, there are no effects. Otherwise, if an exception is thrown by the move constructor of a non-CopyInsertable T, the effects are unspecified.\n\nComplexity: The complexity is linear in the number of elements inserted plus the distance to the end of the vector.\n\n``` iterator erase(const_iterator position); iterator erase(const_iterator first, const_iterator last); ```\n\nEffects: Invalidates iterators and references at or after the point of the erase.\n\nComplexity: The destructor of T is called the number of times equal to the number of the elements erased, but the move assignment operator of T is called the number of times equal to the number of elements in the vector after the erased elements.\n\nThrows: Nothing unless an exception is thrown by the copy constructor, move constructor, assignment operator, or move assignment operator of T.\n\n#### 23.3.6.6vector specialized algorithms [vector.special]\n\n``` template <class T, class Allocator> void swap(vector<T,Allocator>& x, vector<T,Allocator>& y); ```\n\nEffects:\n\n```x.swap(y);\n```\n\n### 23.3.7 Class vector<bool>[vector.bool]\n\nTo optimize space allocation, a specialization of vector for bool elements is provided:\n\n```namespace std {\ntemplate <class Allocator> class vector<bool, Allocator> {\npublic:\n// types:\ntypedef bool const_reference;\ntypedef implementation-defined iterator; // see [container.requirements]\ntypedef implementation-defined const_iterator; // see [container.requirements]\ntypedef implementation-defined size_type; // see [container.requirements]\ntypedef implementation-defined difference_type;// see [container.requirements]\ntypedef bool value_type;\ntypedef Allocator allocator_type;\ntypedef implementation-defined pointer;\ntypedef implementation-defined const_pointer;\ntypedef std::reverse_iterator<iterator> reverse_iterator;\ntypedef std::reverse_iterator<const_iterator> const_reverse_iterator;\n\n// bit reference:\nclass reference {\nfriend class vector;\nreference() noexcept;\npublic:\n~reference();\noperator bool() const noexcept;\nreference& operator=(const bool x) noexcept;\nreference& operator=(const reference& x) noexcept;\nvoid flip() noexcept; // flips the bit\n};\n\n// construct/copy/destroy:\nvector() : vector(Allocator()) { }\nexplicit vector(const Allocator&);\nexplicit vector(size_type n, const Allocator& = Allocator());\nvector(size_type n, const bool& value,\nconst Allocator& = Allocator());\ntemplate <class InputIterator>\nvector(InputIterator first, InputIterator last,\nconst Allocator& = Allocator());\nvector(const vector<bool,Allocator>& x);\nvector(vector<bool,Allocator>&& x);\nvector(const vector&, const Allocator&);\nvector(vector&&, const Allocator&);\nvector(initializer_list<bool>, const Allocator& = Allocator()));\n~vector();\nvector<bool,Allocator>& operator=(const vector<bool,Allocator>& x);\nvector<bool,Allocator>& operator=(vector<bool,Allocator>&& x);\nvector& operator=(initializer_list<bool>);\ntemplate <class InputIterator>\nvoid assign(InputIterator first, InputIterator last);\nvoid assign(size_type n, const bool& t);\nvoid assign(initializer_list<bool>);\nallocator_type get_allocator() const noexcept;\n\n// iterators:\niterator begin() noexcept;\nconst_iterator begin() const noexcept;\niterator end() noexcept;\nconst_iterator end() const noexcept;\nreverse_iterator rbegin() noexcept;\nconst_reverse_iterator rbegin() const noexcept;\nreverse_iterator rend() noexcept;\nconst_reverse_iterator rend() const noexcept;\n\nconst_iterator cbegin() const noexcept;\nconst_iterator cend() const noexcept;\nconst_reverse_iterator crbegin() const noexcept;\nconst_reverse_iterator crend() const noexcept;\n\n// capacity:\nsize_type size() const noexcept;\nsize_type max_size() const noexcept;\nvoid resize(size_type sz, bool c = false);\nsize_type capacity() const noexcept;\nbool empty() const noexcept;\nvoid reserve(size_type n);\nvoid shrink_to_fit();\n\n// element access:\nreference operator[](size_type n);\nconst_reference operator[](size_type n) const;\nconst_reference at(size_type n) const;\nreference at(size_type n);\nreference front();\nconst_reference front() const;\nreference back();\nconst_reference back() const;\n\n// modifiers:\ntemplate <class... Args> void emplace_back(Args&&... args);\nvoid push_back(const bool& x);\nvoid pop_back();\ntemplate <class... Args> iterator emplace(const_iterator position, Args&&... args);\niterator insert(const_iterator position, const bool& x);\niterator insert (const_iterator position, size_type n, const bool& x);\ntemplate <class InputIterator>\niterator insert(const_iterator position,\nInputIterator first, InputIterator last);\niterator insert(const_iterator position, initializer_list<bool> il);\n\niterator erase(const_iterator position);\niterator erase(const_iterator first, const_iterator last);\nvoid swap(vector<bool,Allocator>&);\nstatic void swap(reference x, reference y) noexcept;\nvoid flip() noexcept; // flips all bits\nvoid clear() noexcept;\n};\n}\n```\n\nUnless described below, all operations have the same requirements and semantics as the primary vector template, except that operations dealing with the bool value type map to bit values in the container storage and allocator_traits::construct ([allocator.traits.members]) is not used to construct these values.\n\nThere is no requirement that the data be stored as a contiguous allocation of bool values. A space-optimized representation of bits is recommended instead.\n\nreference is a class that simulates the behavior of references of a single bit in vector<bool>. The conversion operator returns true when the bit is set, and false otherwise. The assignment operator sets the bit when the argument is (convertible to) true and clears it otherwise. flip reverses the state of the bit.\n\n``` void flip() noexcept; ```\n\nEffects: Replaces each element in the container with its complement.\n\n``` static void swap(reference x, reference y) noexcept; ```\n\nEffects: exchanges the contents of x and y as if by\n\n```bool b = x;\nx = y;\ny = b;\n```\n\n``` template <class Allocator> struct hash<vector<bool, Allocator> >; ```\n\nThe template specialization shall meet the requirements of class template hash ([unord.hash])." ]

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[ "Site Map\n\n# Decision Tree to Calculate Atmospheric Parameters - SSPP\n\n[Back to main SSPP page]\n\n## Introduction\n\nThe SSPP uses multiple methods in order to obtain estimates of the atmospheric parameters for each star over a very wide range in parameter space. Each technique has limitations as to its ability to estimate each parameter, arising from, e.g., the coverage of the grids of synthetic spectra, the methods used for spectral matching, and their sensitivity to the S/N of the spectrum, the range in parameter space over which the particular calibration used for a given method extends, etc.. Hence, it is necessary to specify a prescription for the inclusion or exclusion of a given technique for the estimation of a given atmospheric parameter. At present, this is accomplished by the assignment of a null (0, meaning the parameter estimate is dropped), unity (1, meaning the parameter estimate is accepted), or 2 (in case for [Fe/H]) value to an indicator variable associated with each parameter estimated by a given technique, according to the g-r and S/N criteria listed in the table. Among the parameters, it is recommended to use the adopted Teff, log g, and [Fe/H] calculated by these decision trees. These are named \"TEFF_ADOP\", \"LOGG_ADOP\", and \"FEH_ADOP\", respectively, in the data model.\n\n## Effective Temperature\n\nThere are nine temperature estimates determined by the DR9 SSPP. Averages are taken from these nine estimators using the robust biweight procedure (See Beers, Flynn, & Gebhardt 1990, and references therein). There is also an average temperature estimate based on the methods (ki13, ANNSR, ANNRR, NGS1, HA24, and HD24), which do not use any color information. A robust average of the accepted temperature estimates (those with indicator variables equal to 1) is taken for the final adopted temperature. An internal robust estimate of the scatter around this value is also obtained.\n\n## Surface Gravity\n\nThere are eight methods used to estimate surface gravity by the SSPP. Application of the limits on g-r and S/N eliminates a number of these estimates, and the biweight average of the accepted log g estimates (those with indicator variables equal to 1) is taken for the final adopted surface gravity. An internal robust estimate of the scatter around this value is also calculated.\n\n## Metallicity\n\nThere are 10 estimates of [Fe/H] in the SSPP. We adopt the validity ranges of S/N and g-r listed in the table in Lee et al. (2008a) to assign 1 or 0 as an indicator variable for each method. We then proceed as follows.\n\nFirst, we generate a synthetic spectrum for each estimate of [Fe/H] that has an indicator variable of 1 (using the adopted Teff and log g) by interpolating within the pre-existing grid of synthetic spectra from the NGS1 approach. Next, we calculate a correlation coefficient (CC) and the mean of the absolute residuals (MAR) between the observed and the generated synthetic spectrum in two different wavelength regions: 3850 - 4250 Å and 4500 - 5500 Å, where the Ca II K and H lines, as well as numerous metallic lines, are present, yielding two values of CC and MAR for each metallicity estimator. We then select between the two values by choosing the one with CC closest to unity, and with MAR closest to zero. This applies for all estimates of [Fe/H] from the individual methods. At the end of this process, we have N values of the CC and MAR (maximum of N=10) for the N estimates of [Fe/H] with indicator variables of 1.\n\nThere are thus two arrays with N elements: one from the CC and the other one from the MAR values. We then sort the CC array in descending order, and select the metallicity estimate corresponding to the first and second element of the sorted array. The same procedure is carried out for the MAR array, after sorting in ascending order. The reason for implementing calculations involving the MARs is that, although we may have a correlation coefficient close to unity between the observed and the synthetic spectrum, from time to time there are large residuals between the two spectra, indicating a poor match. Thus, the computations involving the MAR provide additional security that the methods are producing reasonable abundance estimates at this stage.\n\nAt this point we have two metallicity estimates with the highest CCs, and two metallicity estimates with the lowest MARs. We then take an average of the four metallicities, and use this average to select from among the full set of metallicity estimates with an indicator variable of 1 and within +/-0.5 dex of the average. We carry along the CCs and MARs for the selected metallicity estimates for further processing. In the next step we obtain an average μCCMAR) and standard deviation σCCMAR) of the CCs (MARs) for the surviving metallicity estimates from the previous step. As a final step to reject likely outliers, we select from the surviving metallicity estimates the ones with the CC greater than (μCC - σCC) and the MAR less than (μMAR + σMAR). The metallicity estimators that remain after this step are assigned indicator variables of 2. This procedure effectively ignores metallicity estimates that produce poor matches with the synthetic spectra. The final adopted value of [Fe/H] is computed by taking a biweight average of the remaining values of [Fe/H] (those with indicator variables of 2)." ]

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[ "# 213761\n\n## 213,761 is an odd composite number composed of two prime numbers multiplied together.\n\nWhat does the number 213761 look like?\n\nThis visualization shows the relationship between its 2 prime factors (large circles) and 4 divisors.\n\n213761 is an odd composite number. It is composed of two distinct prime numbers multiplied together. It has a total of four divisors.\n\n## Prime factorization of 213761:\n\n### 181 × 1181\n\nSee below for interesting mathematical facts about the number 213761 from the Numbermatics database.\n\n### Names of 213761\n\n• Cardinal: 213761 can be written as Two hundred thirteen thousand, seven hundred sixty-one.\n\n### Scientific notation\n\n• Scientific notation: 2.13761 × 105\n\n### Factors of 213761\n\n• Number of distinct prime factors ω(n): 2\n• Total number of prime factors Ω(n): 2\n• Sum of prime factors: 1362\n\n### Divisors of 213761\n\n• Number of divisors d(n): 4\n• Complete list of divisors:\n• Sum of all divisors σ(n): 215124\n• Sum of proper divisors (its aliquot sum) s(n): 1363\n• 213761 is a deficient number, because the sum of its proper divisors (1363) is less than itself. Its deficiency is 212398\n\n### Bases of 213761\n\n• Binary: 1101000011000000012\n• Base-36: 4KXT\n\n### Squares and roots of 213761\n\n• 213761 squared (2137612) is 45693765121\n• 213761 cubed (2137613) is 9767544926030081\n• The square root of 213761 is 462.3429463071\n• The cube root of 213761 is 59.7919647075\n\n### Scales and comparisons\n\nHow big is 213761?\n• 213,761 seconds is equal to 2 days, 11 hours, 22 minutes, 41 seconds.\n• To count from 1 to 213,761 would take you about two days.\n\nThis is a very rough estimate, based on a speaking rate of half a second every third order of magnitude. If you speak quickly, you could probably say any randomly-chosen number between one and a thousand in around half a second. Very big numbers obviously take longer to say, so we add half a second for every extra x1000. (We do not count involuntary pauses, bathroom breaks or the necessity of sleep in our calculation!)\n\n• A cube with a volume of 213761 cubic inches would be around 5 feet tall.\n\n### Recreational maths with 213761\n\n• 213761 backwards is 167312\n• The number of decimal digits it has is: 6\n• The sum of 213761's digits is 20\n• More coming soon!\n\nMLA style:\n\"Number 213761 - Facts about the integer\". Numbermatics.com. 2022. Web. 24 May 2022.\n\nAPA style:\nNumbermatics. (2022). Number 213761 - Facts about the integer. Retrieved 24 May 2022, from https://numbermatics.com/n/213761/\n\nChicago style:\nNumbermatics. 2022. \"Number 213761 - Facts about the integer\". https://numbermatics.com/n/213761/\n\nThe information we have on file for 213761 includes mathematical data and numerical statistics calculated using standard algorithms and methods. We are adding more all the time. If there are any features you would like to see, please contact us. Information provided for educational use, intellectual curiosity and fun!\n\nKeywords: Divisors of 213761, math, Factors of 213761, curriculum, school, college, exams, university, Prime factorization of 213761, STEM, science, technology, engineering, physics, economics, calculator, two hundred thirteen thousand, seven hundred sixty-one.\n\nOh no. Javascript is switched off in your browser.\nSome bits of this website may not work unless you switch it on." ]

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[ "binary-search-2.0.0: Binary and exponential searches\n\nNumeric.Search\n\nDescription\n\nThis package provides combinators to construct many variants of binary search. Most generally, it provides the binary search over predicate of the form (Eq b, Monad m) => a -> m b. The other searches are derived as special cases of this function.\n\nBinarySearch assumes two things;\n\n1. b, the codomain of PredicateM belongs to type class Eq.\n2. Each value of b form a convex set in the codomain space of the PredicateM. That is, if for certain pair (left, right) :: (a, a) satisfies pred left == val && pred right == val, then also pred x == val for all x such that left <= x <= right.\n\nExample 1. Find the approximate square root of 3.\n\n>>> largest True $search positiveExponential divForever (\\x -> x^2 < 3000000) Just 1732 >>> smallest False$ search positiveExponential divForever (\\x -> x^2 < 3000000)\nJust 1733\n>>> largest True $search positiveExponential divideForever (\\x -> x^2 < (3::Double)) Just 1.7320508075688772 >>> largest True$ search positiveExponential (divideTill 0.125) (\\x -> x^2 < (3::Double))\nJust 1.625\n>>> smallest False $search positiveExponential (divideTill 0.125) (\\x -> x^2 < (3::Double)) Just 1.75 Pay attention to use the appropriate exponential search combinator to set up the initial search range. For example, the following code works as expected: >>> largest True$ search nonNegativeExponential divideForever (\\x -> x^2 < (0.5::Double))\nJust 0.7071067811865475\n\n\nBut this one does not terminate:\n\n> largest True $search positiveExponential divideForever (x -> x^2 < (0.5::Double)) ... runs forever ... Example 2. Find the range of integers whose quotinent 7 is equal to 6. This is an example of how we can search for discete and multi-valued predicate. >>> smallest 6$ search (fromTo 0 100) divForever (\\x -> x div 7)\nJust 42\n>>> largest 6 $search (fromTo 0 100) divForever (\\x -> x div 7) Just 48 Example 3. Find the minimum size of the container that can fit three bars of length 4, and find an actual way to fit them. We will solve this using a satisfiability modulo theory (SMT) solver. Since we need to evoke IO to call for the SMT solver, This is a usecase for a monadic binary search. >>> import Data.List (isPrefixOf) >>> :{ do -- x fits within the container let x ⊂ r = (0 .<= x &&& x .<= r-4) -- x and y does not collide let x ∅ y = (x+4 .<= y ) let contain3 :: Integer -> IO (Evidence () String) contain3 r' = do let r = fromInteger r' :: SInteger ret <- show <$> sat (\\x y z -> bAnd [x ⊂ r, y ⊂ r, z ⊂ r, x ∅ y, y ∅ z])\nif \"Satisfiable\" isPrefixOf ret\nthen return $Evidence ret else return$ CounterEvidence ()\nJust sz <- smallest evidence <$> searchM positiveExponential divForever contain3 putStrLn$ \"Size of the container: \" ++ show sz\nJust msg <- evidenceForSmallest <$> searchM positiveExponential divForever contain3 putStrLn msg :} Size of the container: 12 Satisfiable. Model: s0 = 0 :: Integer s1 = 4 :: Integer s2 = 8 :: Integer Synopsis # Documentation All the doctests in this document assume: >>> :set -XFlexibleContexts >>> import Data.SBV # Evidence data Evidence a b Source # The Evidence datatype is similar to Either , but differes in that all CounterEvidence values are equal to each other, and all Evidence values are also equal to each other. The Evidence type is used to binary-searching for some predicate and meanwhile returning evidences for that. In other words, Evidence is a Bool with additional information why it is True or False. >>> Evidence \"He owns the gun\" == Evidence \"He was at the scene\" True >>> Evidence \"He loved her\" == CounterEvidence \"He loved her\" False Constructors CounterEvidence a Evidence b #### Instances Instances details Source # Instance detailsDefined in Numeric.Search Methods(>>=) :: Evidence e a -> (a -> Evidence e b) -> Evidence e b #(>>) :: Evidence e a -> Evidence e b -> Evidence e b #return :: a -> Evidence e a # Source # Instance detailsDefined in Numeric.Search Methodsfmap :: (a0 -> b) -> Evidence a a0 -> Evidence a b #(<$) :: a0 -> Evidence a b -> Evidence a a0 # Source # Instance detailsDefined in Numeric.Search Methodspure :: a -> Evidence e a #(<*>) :: Evidence e (a -> b) -> Evidence e a -> Evidence e b #liftA2 :: (a -> b -> c) -> Evidence e a -> Evidence e b -> Evidence e c #(*>) :: Evidence e a -> Evidence e b -> Evidence e b #(<*) :: Evidence e a -> Evidence e b -> Evidence e a # Eq (Evidence b a) Source # Instance detailsDefined in Numeric.Search Methods(==) :: Evidence b a -> Evidence b a -> Bool #(/=) :: Evidence b a -> Evidence b a -> Bool # Ord (Evidence b a) Source # Instance detailsDefined in Numeric.Search Methodscompare :: Evidence b a -> Evidence b a -> Ordering #(<) :: Evidence b a -> Evidence b a -> Bool #(<=) :: Evidence b a -> Evidence b a -> Bool #(>) :: Evidence b a -> Evidence b a -> Bool #(>=) :: Evidence b a -> Evidence b a -> Bool #max :: Evidence b a -> Evidence b a -> Evidence b a #min :: Evidence b a -> Evidence b a -> Evidence b a # (Read a, Read b) => Read (Evidence a b) Source # Instance detailsDefined in Numeric.Search MethodsreadsPrec :: Int -> ReadS (Evidence a b) #readList :: ReadS [Evidence a b] #readPrec :: ReadPrec (Evidence a b) # (Show a, Show b) => Show (Evidence a b) Source # Instance detailsDefined in Numeric.Search MethodsshowsPrec :: Int -> Evidence a b -> ShowS #show :: Evidence a b -> String #showList :: [Evidence a b] -> ShowS #\n\nevidence = Evidence undefined. We can use this combinator to look up for some Evidence, since all Evidences are equal.\n\ncounterEvidence = CounterEvidence undefined. We can use this combinator to look up for any CounterEvidence, since all CounterEvidences are equal.\n\n# Search range\n\ndata Range b a Source #\n\nThe Range k lo k' hi represents the search result that pred x == k for lo <= x <= hi. The Range type also holds the evidences for the lower and the upper boundary.\n\nConstructors\n\n Range FieldsloKey :: b loVal :: a hiKey :: b hiVal :: a\n\n#### Instances\n\nInstances details\n\ntype SearchRange a = ([a], [a]) Source #\n\nThe (possibly infinite) lists of candidates for lower and upper bounds from which the search may be started.\n\ninitializeSearchM :: (Monad m, Eq b) => SearchRange a -> (a -> m b) -> m [Range b a] Source #\n\nSet the lower and upper boundary from those available from the candidate lists. From the pair of list, the initializeSearchM tries to find the first (lo, hi) such that pred lo /= pred hi, by the breadth-first search.\n\nStart binary search from between minBound and maxBound.\n\nfromTo :: a -> a -> SearchRange a Source #\n\nStart binary search from between the given pair of boundaries.\n\nExponentially search for lower boundary from [-1, -2, -4, -8, ...], upper boundary from [0, 1, 2, 4, 8, ...]. Move on to the binary search once the first (lo, hi) is found such that pred lo /= pred hi.\n\nLower boundary is 1, upper boundary is from [2, 4, 8, 16, ...].\n\nLower boundary is 0, search upper boundary is from [1, 2, 4, 8, ...].\n\nLower boundary is [0.5, 0.25, 0.125, ...], upper boundary is from [1, 2, 4, 8, ...].\n\nLower boundary is from [-2, -4, -8, -16, ...], upper boundary is -1.\n\nLower boundary is from [-1, -2, -4, -8, ...], upper boundary is -0.\n\nLower boundary is [-1, -2, -4, -8, ...], upper boundary is from [-0.5, -0.25, -0.125, ...].\n\n# Splitters\n\ntype Splitter a = a -> a -> Maybe a Source #\n\nThe type of function that returns a value between (lo, hi) as long as one is available.\n\nPerform split forever, until we cannot find a mid-value because hi-lo < 2. This splitter assumes that the arguments are Integral, and uses the div funtion.\n\nNote that our dividing algorithm always find the mid value for any hi-lo >= 2.\n\n>>> prove \\$ \\x y -> (y .>= x+2 &&& x+2 .> x) ==> let z = (x+1) sDiv 2 + y sDiv 2 in x .< z &&& z .< (y::SInt32)\nQ.E.D.\n\n\ndivTill :: Integral a => a -> Splitter a Source #\n\nPerform splitting until hi - lo <= eps.\n\ndivideForever :: (Eq a, Fractional a) => Splitter a Source #\n\nPerform split forever, until we cannot find a mid-value due to machine precision. This one uses (/) operator.\n\ndivideTill :: (Ord a, Fractional a) => a -> Splitter a Source #\n\nPerform splitting until hi - lo <= eps.\n\n# Searching\n\nsearch :: Eq b => SearchRange a -> Splitter a -> (a -> b) -> [Range b a] Source #\n\nPerform search over pure predicates. The monadic version of this is searchM.\n\nsearchM :: forall a m b. (Functor m, Monad m, Eq b) => SearchRange a -> Splitter a -> (a -> m b) -> m [Range b a] Source #\n\nMother of all search variations.\n\nsearchM keeps track of the predicates found, so that it works well with the Evidence type.\n\n# Postprocess\n\nlookupRanges :: Eq b => b -> [Range b a] -> Maybe (Range b a) Source #\n\nLook up for the first Range with the given predicate.\n\nsmallest :: Eq b => b -> [Range b a] -> Maybe a Source #\n\nPick up the smallest a that satisfies pred a == b.\n\nlargest :: Eq b => b -> [Range b a] -> Maybe a Source #\n\nPick up the largest a that satisfies pred a == b.\n\nevidenceForSmallest :: [Range (Evidence b1 b2) a] -> Maybe b2 Source #\n\nGet the content of the evidence for the smallest a which satisfies pred a.\n\nevidenceForLargest :: [Range (Evidence b1 b2) a] -> Maybe b2 Source #\n\nGet the content of the evidence for the largest a which satisfies pred a.\n\ncounterEvidenceForSmallest :: [Range (Evidence b1 b2) a] -> Maybe b1 Source #\n\nGet the content of the counterEvidence for the smallest a which does not satisfy pred a.\n\ncounterEvidenceForLargest :: [Range (Evidence b1 b2) a] -> Maybe b1 Source #\n\nGet the content of the counterEvidence for the largest a which does not satisfy pred a." ]

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[ "University Physics Volume 1\n\n# 15.4Pendulums\n\n### Learning Objectives\n\nBy the end of this section, you will be able to:\n\n• State the forces that act on a simple pendulum\n• Determine the angular frequency, frequency, and period of a simple pendulum in terms of the length of the pendulum and the acceleration due to gravity\n• Define the period for a physical pendulum\n• Define the period for a torsional pendulum\n\nPendulums are in common usage. Grandfather clocks use a pendulum to keep time and a pendulum can be used to measure the acceleration due to gravity. For small displacements, a pendulum is a simple harmonic oscillator.\n\n### The Simple Pendulum\n\nA simple pendulum is defined to have a point mass, also known as the pendulum bob, which is suspended from a string of length L with negligible mass (Figure 15.20). Here, the only forces acting on the bob are the force of gravity (i.e., the weight of the bob) and tension from the string. The mass of the string is assumed to be negligible as compared to the mass of the bob.\n\nFigure 15.20 A simple pendulum has a small-diameter bob and a string that has a very small mass but is strong enough not to stretch appreciably. The linear displacement from equilibrium is s, the length of the arc. Also shown are the forces on the bob, which result in a net force of $−mgsinθ−mgsinθ$ toward the equilibrium position—that is, a restoring force.\n\nConsider the torque on the pendulum. The force providing the restoring torque is the component of the weight of the pendulum bob that acts along the arc length. The torque is the length of the string L times the component of the net force that is perpendicular to the radius of the arc. The minus sign indicates the torque acts in the opposite direction of the angular displacement:\n\n$τ=−L(mgsinθ);Iα=−L(mgsinθ);Id2θdt2=−L(mgsinθ);mL2d2θdt2=−L(mgsinθ);d2θdt2=−gLsinθ.τ=−L(mgsinθ);Iα=−L(mgsinθ);Id2θdt2=−L(mgsinθ);mL2d2θdt2=−L(mgsinθ);d2θdt2=−gLsinθ.$\n\nThe solution to this differential equation involves advanced calculus, and is beyond the scope of this text. But note that for small angles (less than 15 degrees), $sinθsinθ$ and $θθ$ differ by less than 1%, so we can use the small angle approximation $sinθ≈θ.sinθ≈θ.$ The angle $θθ$ describes the position of the pendulum. Using the small angle approximation gives an approximate solution for small angles,\n\n$d2θdt2=−gLθ.d2θdt2=−gLθ.$\n15.17\n\nBecause this equation has the same form as the equation for SHM, the solution is easy to find. The angular frequency is\n\n$ω=gLω=gL$\n15.18\n\nand the period is\n\n$T=2πLg.T=2πLg.$\n15.19\n\nThe period of a simple pendulum depends on its length and the acceleration due to gravity. The period is completely independent of other factors, such as mass and the maximum displacement. As with simple harmonic oscillators, the period T for a pendulum is nearly independent of amplitude, especially if $θθ$ is less than about $15°.15°.$ Even simple pendulum clocks can be finely adjusted and remain accurate.\n\nNote the dependence of T on g. If the length of a pendulum is precisely known, it can actually be used to measure the acceleration due to gravity, as in the following example.\n\n### Example 15.3\n\n#### Measuring Acceleration due to Gravity by the Period of a Pendulum\n\nWhat is the acceleration due to gravity in a region where a simple pendulum having a length 75.000 cm has a period of 1.7357 s?\n\n#### Strategy\n\nWe are asked to find g given the period T and the length L of a pendulum. We can solve $T=2πLgT=2πLg$ for g, assuming only that the angle of deflection is less than $15°15°$.\n\n#### Solution\n\n1. Square $T=2πLgT=2πLg$ and solve for g:\n$g=4π2LT2.g=4π2LT2.$\n2. Substitute known values into the new equation:\n$g=4π20.75000m(1.7357s)2.g=4π20.75000m(1.7357s)2.$\n3. Calculate to find g:\n$g=9.8281m/s2.g=9.8281m/s2.$\n\n#### Significance\n\nThis method for determining g can be very accurate, which is why length and period are given to five digits in this example. For the precision of the approximation $sinθ≈θsinθ≈θ$ to be better than the precision of the pendulum length and period, the maximum displacement angle should be kept below about $0.5°0.5°$.\n\nAn engineer builds two simple pendulums. Both are suspended from small wires secured to the ceiling of a room. Each pendulum hovers 2 cm above the floor. Pendulum 1 has a bob with a mass of 10 kg. Pendulum 2 has a bob with a mass of 100 kg. Describe how the motion of the pendulums will differ if the bobs are both displaced by $12°12°$.\n\n### Physical Pendulum\n\nAny object can oscillate like a pendulum. Consider a coffee mug hanging on a hook in the pantry. If the mug gets knocked, it oscillates back and forth like a pendulum until the oscillations die out. We have described a simple pendulum as a point mass and a string. A physical pendulum is any object whose oscillations are similar to those of the simple pendulum, but cannot be modeled as a point mass on a string, and the mass distribution must be included into the equation of motion.\n\nAs for the simple pendulum, the restoring force of the physical pendulum is the force of gravity. With the simple pendulum, the force of gravity acts on the center of the pendulum bob. In the case of the physical pendulum, the force of gravity acts on the center of mass (CM) of an object. The object oscillates about a point O. Consider an object of a generic shape as shown in Figure 15.21.\n\nFigure 15.21 A physical pendulum is any object that oscillates as a pendulum, but cannot be modeled as a point mass on a string. The force of gravity acts on the center of mass (CM) and provides the restoring force that causes the object to oscillate. The minus sign on the component of the weight that provides the restoring force is present because the force acts in the opposite direction of the increasing angle $θθ$.\n\nWhen a physical pendulum is hanging from a point but is free to rotate, it rotates because of the torque applied at the CM, produced by the component of the object’s weight that acts tangent to the motion of the CM. Taking the counterclockwise direction to be positive, the component of the gravitational force that acts tangent to the motion is $−mgsinθ−mgsinθ$. The minus sign is the result of the restoring force acting in the opposite direction of the increasing angle. Recall that the torque is equal to $τ→=r→×F→τ→=r→×F→$. The magnitude of the torque is equal to the length of the radius arm times the tangential component of the force applied, $|τ|=rFsinθ|τ|=rFsinθ$. Here, the length L of the radius arm is the distance between the point of rotation and the CM. To analyze the motion, start with the net torque. Like the simple pendulum, consider only small angles so that $sinθ≈θsinθ≈θ$. Recall from Fixed-Axis Rotation on rotation that the net torque is equal to the moment of inertia $I=∫r2dmI=∫r2dm$ times the angular acceleration $α,α,$ where $α=d2θdt2α=d2θdt2$:\n\n$Iα=τnet=L(−mg)sinθ.Iα=τnet=L(−mg)sinθ.$\n\nUsing the small angle approximation and rearranging:\n\n$Iα=−L(mg)θ;Id2θdt2=−L(mg)θ; d2θdt2=−(mgLI)θ.Iα=−L(mg)θ;Id2θdt2=−L(mg)θ; d2θdt2=−(mgLI)θ.$\n\nOnce again, the equation says that the second time derivative of the position (in this case, the angle) equals minus a constant $(−mgLI)(−mgLI)$ times the position. The solution is\n\n$θ(t)=Θcos(ωt+ϕ),θ(t)=Θcos(ωt+ϕ),$\n\nwhere $ΘΘ$ is the maximum angular displacement. The angular frequency is\n\n$ω=mgLI.ω=mgLI.$\n15.20\n\nThe period is therefore\n\n$T=2πImgL.T=2πImgL.$\n15.21\n\nNote that for a simple pendulum, the moment of inertia is $I=∫r2dm=mL2I=∫r2dm=mL2$ and the period reduces to $T=2πLgT=2πLg$.\n\n### Example 15.4\n\n#### Reducing the Swaying of a Skyscraper\n\nIn extreme conditions, skyscrapers can sway up to two meters with a frequency of up to 20.00 Hz due to high winds or seismic activity. Several companies have developed physical pendulums that are placed on the top of the skyscrapers. As the skyscraper sways to the right, the pendulum swings to the left, reducing the sway. Assuming the oscillations have a frequency of 0.50 Hz, design a pendulum that consists of a long beam, of constant density, with a mass of 100 metric tons and a pivot point at one end of the beam. What should be the length of the beam?", null, "#### Strategy\n\nWe are asked to find the length of the physical pendulum with a known mass. We first need to find the moment of inertia of the beam. We can then use the equation for the period of a physical pendulum to find the length.\n\n#### Solution\n\n1. Find the moment of inertia for the CM:\n2. Use the parallel axis theorem to find the moment of inertia about the point of rotation:\n$I=ICM+L42M=112ML2+14ML2=13ML2.I=ICM+L42M=112ML2+14ML2=13ML2.$\n3. The period of a physical pendulum has a period of $T=2πImgLT=2πImgL$. Use the moment of inertia to solve for the length L:\n$T=2πIMgL=2π13ML2MgL=2πL3g; L=3g(T2π)2=3(9.8ms2)(2s2π)2=2.98m.T=2πIMgL=2π13ML2MgL=2πL3g; L=3g(T2π)2=3(9.8ms2)(2s2π)2=2.98m.$\n4. This length L is from the center of mass to the axis of rotation, which is half the length of the pendulum. Therefore the length H of the pendulum is:\n$H=2L=5.96mH=2L=5.96m$\n\n#### Significance\n\nThere are many ways to reduce the oscillations, including modifying the shape of the skyscrapers, using multiple physical pendulums, and using tuned-mass dampers.\n\n### Torsional Pendulum\n\nA torsional pendulum consists of a rigid body suspended by a light wire or spring (Figure 15.22). When the body is twisted some small maximum angle $(Θ)(Θ)$ and released from rest, the body oscillates between $(θ=+Θ)(θ=+Θ)$ and $(θ=−Θ)(θ=−Θ)$. The restoring torque is supplied by the shearing of the string or wire.\n\nFigure 15.22 A torsional pendulum consists of a rigid body suspended by a string or wire. The rigid body oscillates between $θ=+Θθ=+Θ$ and $θ=−Θθ=−Θ$.\n\nThe restoring torque can be modeled as being proportional to the angle:\n\n$τ=−κθ.τ=−κθ.$\n\nThe variable kappa $(κ)(κ)$ is known as the torsion constant of the wire or string. The minus sign shows that the restoring torque acts in the opposite direction to increasing angular displacement. The net torque is equal to the moment of inertia times the angular acceleration:\n\n$Id2θdt2=−κθ; d2θdt2=−κIθ.Id2θdt2=−κθ; d2θdt2=−κIθ.$\n\nThis equation says that the second time derivative of the position (in this case, the angle) equals a negative constant times the position. This looks very similar to the equation of motion for the SHM $d2xdt2=−kmxd2xdt2=−kmx$, where the period was found to be $T=2πmkT=2πmk$. Therefore, the period of the torsional pendulum can be found using\n\n$T=2πIκ.T=2πIκ.$\n15.22\n\nThe units for the torsion constant are $[κ]=N-m=(kgms2)m=kgm2s2[κ]=N-m=(kgms2)m=kgm2s2$ and the units for the moment of inertial are $[I]=kg-m2,[I]=kg-m2,$ which show that the unit for the period is the second.\n\n### Example 15.5\n\n#### Measuring the Torsion Constant of a String\n\nA rod has a length of $l=0.30ml=0.30m$ and a mass of 4.00 kg. A string is attached to the CM of the rod and the system is hung from the ceiling (Figure 15.23). The rod is displaced 10 degrees from the equilibrium position and released from rest. The rod oscillates with a period of 0.5 s. What is the torsion constant $κκ$?\nFigure 15.23 (a) A rod suspended by a string from the ceiling. (b) Finding the rod’s moment of inertia.\n\n#### Strategy\n\nWe are asked to find the torsion constant of the string. We first need to find the moment of inertia.\n\n#### Solution\n\n1. Find the moment of inertia for the CM:\n$ICM=∫x2dm=∫−L/2+L/2x2λdx=λ[x33]−L/2+L/2=λ2L324=(ML)2L324=112ML2.ICM=∫x2dm=∫−L/2+L/2x2λdx=λ[x33]−L/2+L/2=λ2L324=(ML)2L324=112ML2.$\n2. Calculate the torsion constant using the equation for the period:\n$T=2πIκ;κ=I(2πT)2=(112ML2)(2πT)2;=(112(4.00kg)(0.30m)2)(2π0.50s)2=4.73N·m.T=2πIκ;κ=I(2πT)2=(112ML2)(2πT)2;=(112(4.00kg)(0.30m)2)(2π0.50s)2=4.73N·m.$\n\n#### Significance\n\nLike the force constant of the system of a block and a spring, the larger the torsion constant, the shorter the period." ]

[ null, "https://openstax.org/resources/7fc67e4daf74b96a0af54f10187587e8cd565089", null ]

[ "# Unique representation ID for 5-card poker hand using combination without sorting\n\nThere is a deck of 52 playing cards (e.g. poker cards). Assign every card a unique number (ID) at will. Every combination of 5 of these cards (a \"hand\") shall be represented by a unique result number, calculated from the 5 card IDs. Permutations of the same 5 cards must yield the same result.\n\nE.g.\n\n• You could enumerate all cards 1..52 and simply add the values. However, if you got 150 as result you could not differentiate between 10+20+30+40+50 or 11+19+30+40+50, so this is not a solution.\n• You could just write down the IDs as they occur, but 1020304050 would be different from 5040302010, so this is also not a solution.\n\nConstraints:\n\n• No sorting allowed. Whatever your formula is, the 5 card IDs must be able to go in any order.\n• The ID number assigned to any card must always be a positive integer not to exceed 3 digits.\n• The unique result number must always be a positive integer not to exceed 12 digits.\n• Calculating the result from the 5 given card ID numbers must rely only on (one, some or all of) the 4 basic operations +-*/\n\nQuestions:\n\nWhat numbers will you assign?\nHow do you calculate the result?\nShow that the biggest possible result has no more than 12 digits.\n\nIf we take the IDs to be\n\nthe first 52 primes\n\nand the result to be\n\nthe product of the IDs of the cards in the hand\n\nthen you can find the card IDs by\n\nfactoring the number\n\nand the number will be at most\n\n223.227.229.233.239 = 645535342583.\n\nWe can actually do better than this while using the same basic idea:\n\nsuppose that in addition to primes $p$ we use some of $p^2$, $p^4$, $p^8$, etc.; then we can still determine the IDs from their product. Taking the 52 smallest numbers of the form $p^{2^k}$ yields a set whose largest element is the prime 193 and whose largest 5 elements have product 206618110301.\n\nA little better:\n\nas Jonathan Allen remarks in comments, we can (relying on the fact that there are always exactly five cards in the hand) replace our largest number with 1; the largest is then 191 and the product of the five largest is 180924666533.\n\nThe crude information-theoretic lower bound for the maximum hand ID is\n\n$\\binom{52}{5}=2598960$\n\nso there is quite a lot of scope for improvement here.\n\n• Right. I am at work and was called away while writing the answer, which is why I didn't say that explicitly :-). I will edit it in. – Gareth McCaughan Oct 20 '16 at 12:41\n• One could actually use 1 and the first 51 (not that it saves digits) – Jonathan Allan Oct 20 '16 at 12:43\n• @ZsoltSzigaly, acceptance of this answer seems premature; what will you do if someone proposes a simple but entirely different scheme that does it with a maximum hand ID that's only 8 digits long? – Gareth McCaughan Oct 20 '16 at 12:56\n• @ZsoltSzilagy but you only have one card numbered 1. – Jonathan Allan Oct 20 '16 at 12:58\n• A minor improvement: with both optimisations ($p^{2^k}$ and a single card with ID $1$), the smallest result will be for the hands containing the ids $\\{1,2,3,4,5\\}$, and will be $120$. You can therefore subtract $120$ from all results, with $0$ being the new smallest result, and $180924666413$ the new largest result. – Suzanne Dupéron Oct 20 '16 at 15:06\n\nHere is another take. We can use the modulus $53$ using only operations $(+,-,x)$, simply by removing $53$ until we have reached a non-negative number strictly less than $53$.\n\nLet $a,b,c,d,e \\in \\mathbb{Z}_{53}$ encode the $52$ cards (we need to use $53$ since it is prime) so we may use the last number for other fun stuff as a joker.\n\nTake any polynomial\n\n$(x-a)(x-b)(x-c)(x-d)(x-e) \\mod 53$,\n\nwhich will be of the form\n\n$x^5 + Ax^4 + Bx^3 + Cx^2 + Dx + E \\mod 53$\n\nwhere each $A,B,C,D,E \\in \\mathbb{Z}_{53}$. Hence, we can encode our hand as $ABCDE$ (using two digits per number, so if e.g. $A = 1$, we write it as $01$).\n\nFinding the roots of the polynomial can be found efficiently, but lets us assume for simplicity that we simply try all the numbers $a,b,c,d,e$ until we match the numbers $A,B,C,D,E$ (there is no restriction on the complexity, right? :-). This will give a number which is strictly less than $5252525252$. For instance, $1812213610$ will give a polynomial $x^5+18 x^4+12 x^3+21 x^2+36 x+10 \\mod 53$, which is $(x-3)(x-4)(x-52)(x-12)(x-17) \\mod 53$\n\nApart from this we need to prove the permutatation-invariant property. If we exchange $a$ and $b$, we get $(x-b)(x-a)(x-c)(x-d)(x-e) \\mod 53$, but it will of course give rise to the same product.\n\nThe uniqueness property relies on the fact that we cannot replace e.g. $a$ and $b$ with $a'$ and $b'$ (or even more roots):\n\n$(x-a')(x-b')(x-c)(x-d)(x-e) = x^5 + Ax^4 + Bx^3 + Cx^2 + Dx + E \\mod 53$.\n\nSince the modulus is prime, we are guaranteed to get the same roots. A sketch proof. If $x^5 + Ax^4 + Bx^3 + Cx^2 + Dx + E = 0 \\mod 53$, for some $x$ then at least one of the factors of $(x-a)(x-b)(x-c)(x-d)(x-e)$ must divide $53$. For one factor, this only holds if $x = a + 53k$, $x = b + 53k$ or so on. It is not possible for two factors to both divide $53$ since it is prime. Therefore, it must be unique.\n\n• Nice! Why use base 100 though? Just give the value of ABCDE in base 53, this should have only 9 digits in base 10. – ffao Oct 20 '16 at 22:25\n• @ffao True, it is a good idea! My focus was mostly breaking the current record :-D – Carl Löndahl Oct 20 '16 at 22:28\n• Could you provide an example? I don't follow the value of x. – paparazzo Oct 20 '16 at 23:32\n• If I understood well, to compute the modulus, you are not limiting yourself to the four basic operations +-*/. You need some kind of loop, or at least some kind of conditional (the loop can be removed by unrolling it enough times). As Gareth McCaughan said, the goal is to compute the result using a rational function of the IDs. Or at least that's my understanding. – Suzanne Dupéron Oct 21 '16 at 2:20\n• @CarlLöndahl Regarding my previous comment, it is actually possible to implement the modulus using the basic operations and no loops, as long as there is a bound on the maximum value of which you need to take the modus, see my second answer for the general idea: puzzling.stackexchange.com/a/44680/31235 – Suzanne Dupéron Oct 21 '16 at 6:15\n\nEven though the (arbitrary) constraint in the question that IDs must be 3 digits long only allows the multiplication solution, if we relax that constraint we can actually get an addition solution of the same order of magnitude as the multiplication solution (using 12 digits at most).\n\nThe fundamental theorem of arithmetic guarantees that multiplications of prime powers can only be recomposed in a single way, but there is no simple equivalent for addition. Therefore, let's use the multiplication result!\n\nIt's known that multiplication modulo a prime is in fact equivalent to addition: let $g$ be a primitive root $\\mod P$, and let $f(x)$ be the smallest non-negative integer such that $g^{f(x)} \\equiv x \\pmod P$. Then we have that $f(a) + f(b) \\equiv f(ab) \\pmod P$. That is, we have made a multiplication of two integers turn into an addition!\n\nWe can use this to construct a list of IDs. Let's start by taking a list of integers for which any subset of 5 of them has a different multiplication result. Well, that has already been done, it's the multiplication solution taking prime powers:\n\n1 2 3 4 5 7 9 11 13 16 17 19 23 25 29 31 37 41 43 47 49 53 59 61 67 71 73 79 81 83 89 97 101 103 107 109 113 121 127 128 131 137 139 149 151 157 163 167 169 173 179 181\n\nNow let's choose a suitably high prime $P$ and a primitive root $g$ (I chose $P=158190677081$ and $g=6$), and make a list of the values of $f(x)$ for each $x$ in the list above:\n\n0, 16445962636, 141744714445, 32891925272, 41469257188, 88706945715, 125298751810, 119801768315, 45645348677, 65783850544, 10559933910, 6498069647, 28871758692, 82938514376, 43945311014, 129559740004, 139964773214, 46143833772, 72119606038, 15722902759, 19223214350, 109129913342, 32569452357, 2621065874, 103451019890, 36836348907, 94873027927, 36431071505, 92406826540, 44034197223, 49620088406, 47432217884, 15767897478, 144952506326, 139187300538, 16588813376, 77370272961, 81412859550, 46599081569, 115121738452, 23970003234, 156223722913, 104797449260, 38709422671, 28274542857, 55950793132, 52934160773, 38218075031, 91290697354, 128401995698, 143534473790, 51946670652\n\nFrom the fact that each 5-element subset of our original set has a distinct multiplication result, the addition results for the second set above must also be distinct, as they represent the same multiplications under the finite field.\n\nThe sum of the cards with the five highest IDs is 726420190688. Subtracting the sum of the five smallest IDs, we get a maximum hand of 691018218499.\n\nWe can see that the highest number used by the solution is around $5P$, therefore it makes sense to try to minimize the value of $P$ used. The actual condition on $P$ is that it must not divide any of the pairwise differences between the multiplication results; the $P$ I chose obviously satisfies this condition because it is greater than all of them, but there probably is a smaller prime.\n\nI was tempted to check for the smallest prime possible, but the number of pairwise differences is around $3.3 \\times 10^{12}$, so it would take some time for my slowish computer to handle them...\n\nNumber the cards 1 to 52 arbitrarily. Then, make a dictionary of all $52 \\choose 5$ hands. Order each hand from the smallest (i.e., $(5,47,21,30,3)$ becomes $(3,5,21,30,47)$) and then order these hands lexicographically from the smallest and assign them numbers $1$ to ${52 \\choose 5} = 2598960$; this is an ordered dictionary we have to keep.\n\nThe trick is, looking up in such a dictionary method is doable using a closed formula, as the space of solutions is finite.\n\nMore precisely: Let $a_{n,1},\\dots,a_{n,5}$ denote the ordered card numbers on position $n$ in the dictionary, $N={52 \\choose 5}$ the total number of ordered hands (i.e., the length of the dictionary), $S_5$ the set of all permutations of $\\{1,\\dots,5\\}$ and $[x,y] = 1\\div\\bigl((x-y)^2+1\\bigr)$ (considering the integer division) the selector function for $x=y$. Then the hash function is $$f(x_1,x_2,x_3,x_4,x_5) = \\sum_{n=1}^N \\sum_{\\sigma\\in S_5} n \\prod_{i=1}^5 [x_{\\sigma(i)},a_{n,i}].$$ Note that all the sums and products are finite so this is a closed formula. The range is $\\{1,\\dots,N\\}$, which is the minimal possible range, as exactly one summand of all $5!N$ of them is non-zero and it is equal to $n$. Therefore, I can do it in 7 decimal digits.\n\nThe only necessary trick is getting the $x=y$ selector. If integer division wasn't allowed, then we would have to find a polynomial with zeros as positions $1,4,9,\\dots,51^2$ and such that its value at $0$ is $1$. Such polynomial exists, it's e.g. $p(t) = q(t)/q(0)$ where $$q(t) = \\prod_{j=1}^{51} (t-j^2)$$ (we know that $q(0)\\neq0$ from the definition of $q$). Then $[x,y]= 1-p((x-y)^2)$. With this definition of $[x,y]$, we have that $f$ is a degree $51\\cdot5=255$ polynomial.\n\n• No sorting allowed. Whatever your formula is, the 5 card IDs must be able to go in any order. – paparazzo Oct 22 '16 at 15:30\n• @Paparazzi you need sorting only in the construction of the formula... – yo' Oct 22 '16 at 17:19\n• That is sorting. If the value in the dictionary is (3,5,21,30,47) then you will not get a match on (5,47,21,30,3). You will have to sort the hand to get a match on the dictionary value. – paparazzo Oct 22 '16 at 18:31\n• @Paparazzi better explanation? – yo' Oct 22 '16 at 18:44\n• yo' is correct. The 5 card IDs are able to go in any order into the formula, as required. You don't have to sort the hand. The sorting is done by the formula. The formula poses no restriction on the order in which the IDs are fed into it. – Rosie F Oct 23 '16 at 8:42\n\nThis is my second attempt at a solution. It is vastly different from my first attempt, so I posted this as a separate answer. It achieves the smallest maximum result for a hand possible (with a hand whose result is $0$):\n\n$\\binom{52}{5} - 1 = 2598959$\n\nMany thanks to @2012rcampion for some improvements to this solution (see the comments).\n\nThe given operators ($+$, $-$, $\\times$, $/$) can be used to build a minimal program operating on a bounded number of allowed values. Since the problem is bounded, let's try it out! I'll define a series of functions for clarity. Each function does not use functions defined later, and does not call itself recursively, therefore the functions can be inlined trivially to form a single, huge function.\n\nWe will number the cards from $0$ to $51$.\n\nThe puzzle specifies that only the four operators $+$, $-$, $\\times$ and $/$ are allowed, but does not indicate whether numerical constants are allowed. Assuming that the five inputs are implicitly allowed, we use the following function to extract the constant $1$:\n\n$$\\operatorname{ExtractOne}(a,b,c,d,e) = (a + b) / (a + b)$$\n\nIn the formulas that follow, any positive constant $n$ can be replaced by (this is the simplest solution, other schemes will give fewer terms):\n\n$$\\underbrace{\\operatorname{ExtractOne}(a,b,c,d,e)+\\cdots+\\operatorname{ExtractOne}(a,b,c,d,e)}_{n\\text{ times}}$$\n\nWe will take care to never perform a division by zero:\n\nGiven the fact that at most one of the inputs is zero (there is only one card with the ID $0$), we know that the sum of any two of the five inputs will be non-zero, and therefore can safely be used as the divisor in $\\operatorname{ExtractOne}$.\n\nWe then write an $\\operatorname{IsZero}$ function, which operates on positive integers between $0$ and $52\\times 52=2704$. It returns $1$ when its input is $0$, and it returns $0$ otherwise:\n\n$$\\operatorname{IsZero}(x) = \\frac{(x-1) \\times (x-2) \\times (x-3) \\times (x-4) \\times \\cdots \\times (x-2703) \\times (x-2704)}{2704!}$$\n\nThe $\\operatorname{IsZero}$ operates following this principle:\n\nWhen the input is $0$, the formula becomes $\\frac{(0-1) \\times (0-2) \\times (0-3) \\times (0-4) \\times \\cdots \\times (0-2703) \\times (0-2704)}{2704!}$, which is the same as $\\frac{(-1) \\times (-2) \\times (-3) \\times (-4) \\times \\cdots \\times (-2703) \\times (-2704)}{2704!}$. Since there is an even number of negative numbers, the result of their product is positive and is equal to: $\\frac{1 \\times 2 \\times 3 \\times 4 \\times \\cdots \\times 2703 \\times 2704}{2704!} = \\frac{2704!}{2704!} = 1$. When the input is non-zero, i.e. $x ∈ \\{1,\\ldots,2704\\}$, the formula instead becomes $\\frac{(x-1) \\times \\cdots \\times (x-(x-1)) \\times (x-x) \\times (x-(x+1)) \\times \\cdots \\times (x-2704)}{2704!}$. The $x-x$ results in a $0$ which nullifies the whole formula: $\\frac{(x-1) \\times \\cdots \\times (x-(x-1)) \\times 0 \\times (x-(x+1)) \\times \\cdots \\times (x-2704)}{2704!} = \\frac{0}{2704!}=0$\n\nSince we only call $IsZero$ on integers which are squares in the set $\\{0, 1, 4, 9 …, 51²\\}$, we skip some of the terms (thanks @2012rcampion):\n\n$$\\operatorname{IsZero}(x) = \\frac{(x-1) \\times (x-4) \\times (x-9) \\times (x-4) \\times \\cdots \\times (x-2500) \\times (x-2601)}{1*4*9*⋯*2500*2601}$$\n\nWe can now compare numbers by comparing their squares (remember that our $\\operatorname{IsZero}$ function only supports integers in the set $\\{0,\\ldots,52\\times 52\\}$, the domain of $\\operatorname{IntEqual}$ will therefore be $\\{0,\\ldots,52\\} \\times \\{0,\\ldots,52\\}$). The $\\operatorname{IntEqual}(x,y)$ function returns $1$ when $x=y$ and $0$ otherwise:\n\n$$\\operatorname{IntEqual}(x,y) = \\operatorname{IsZero}((x - y) \\times (x - y))$$\n\nWe then define an $\\operatorname{IfThenElse}(c,x,y)$ function, which returns $x$ when $c=1$ and $y$ when $c=0$:\n\n$$\\operatorname{IfThenElse}(c,x,y) = (c \\times x) + ((1 - c) \\times y)$$\n\nSince we always use $0$ as the third argument of $\\operatorname{IfThenElse}$, we can define a simpler version $\\operatorname{IfThen}$ (thanks @2012rcampion):\n\n$$\\operatorname{IfThen}(c,x) = c \\times x$$\n\nWe finally use $\\operatorname{IfThenElse}$ to build a decision tree, which assigns a unique number to each combination of input IDs, regardless of their order. The formula is a bit cumbersome to read and write, so what follows covers only hands of $2$ cards among $5$:\n\n$$\\begin{array}{c}\\;\\;\\;\\operatorname{IfThen}\\left(\\operatorname{IntEqual}(a, 0),\\left(\\begin{array}{l}\\;\\;\\;\\operatorname{IfThen}(\\operatorname{IntEqual}(b,1),0)\\\\+\\operatorname{IfThen}(\\operatorname{IntEqual}(b,2),1)\\\\+\\operatorname{IfThen}(\\operatorname{IntEqual}(b,3),2)\\\\+\\operatorname{IfThen}(\\operatorname{IntEqual}(b,4),3)\\end{array}\\right)\\right)\\\\+\\operatorname{IfThen}\\left(\\operatorname{IntEqual}(a, 1),\\left(\\begin{array}{l}\\;\\;\\;\\operatorname{IfThen}(\\operatorname{IntEqual}(b,0),0)\\\\+\\operatorname{IfThen}(\\operatorname{IntEqual}(b,2),4)\\\\+\\operatorname{IfThen}(\\operatorname{IntEqual}(b,3),5)\\\\+\\operatorname{IfThen}(\\operatorname{IntEqual}(b,4),6)\\end{array}\\right)\\right)\\\\+\\operatorname{IfThen}\\left(\\operatorname{IntEqual}(a, 2),\\left(\\begin{array}{l}\\;\\;\\;\\operatorname{IfThen}(\\operatorname{IntEqual}(b,0),1)\\\\+\\operatorname{IfThen}(\\operatorname{IntEqual}(b,1),4)\\\\+\\operatorname{IfThen}(\\operatorname{IntEqual}(b,3),7)\\\\+\\operatorname{IfThen}(\\operatorname{IntEqual}(b,4),8)\\end{array}\\right)\\right)\\\\+\\operatorname{IfThen}\\left(\\operatorname{IntEqual}(a, 3),\\left(\\begin{array}{l}\\;\\;\\;\\operatorname{IfThen}(\\operatorname{IntEqual}(b,0),2)\\\\+\\operatorname{IfThen}(\\operatorname{IntEqual}(b,1),5)\\\\+\\operatorname{IfThen}(\\operatorname{IntEqual}(b,2),7)\\\\+\\operatorname{IfThen}(\\operatorname{IntEqual}(b,4),9)\\end{array}\\right)\\right)\\\\+\\operatorname{IfThen}\\left(\\operatorname{IntEqual}(a, 4),\\left(\\begin{array}{l}\\;\\;\\;\\operatorname{IfThen}(\\operatorname{IntEqual}(b,0),3)\\\\+\\operatorname{IfThen}(\\operatorname{IntEqual}(b,1),6)\\\\+\\operatorname{IfThen}(\\operatorname{IntEqual}(b,2),8)\\\\+\\operatorname{IfThen}(\\operatorname{IntEqual}(b,3),9)\\end{array}\\right)\\right)\\\\\\end{array}$$\n\nWe use the form\n\n$$\\operatorname{IfThen}(c_1,v_1)+\\operatorname{IfThen}(c_2,v_2)+⋯+\\operatorname{IfThen}(c_n,v_n)$$\n\n$$\\operatorname{IfThenElse}(c_1,v_1,\\operatorname{IfThenElse}(c_2,v_2,\\ldots))$$\n\nto avoid the rightward drift caused by deeply nested function calls.\n\nWith a hand of five cards, the decision tree would be five levels deep instead of two, and with 52 cards to choose from, each level would contain many more cases.\n\nThe solution proposed by Carl Löndahl can also be applied using this technique to compute the modulus.\n\n• Congrats on making the rational function restriction completely meaningless (I never liked the arbitrary restriction in the first place!) – ffao Oct 21 '16 at 6:44\n• If IfThenElse you can use $1-c$ instead of $(c-1)^2$. – 2012rcampion Oct 21 '16 at 6:52\n• To get fewer terms, you could define IsZero like $\\prod_{i=1}^{51}\\left[\\left(\\frac{x}{i}\\right)^2-1\\right]$, and then IntEqual could be just $\\operatorname{IsZero}(x-y)$. Plus, since you never use the third argument of IfThenElse you can eliminate the last term altogether (maybe just an IfThen function?). – 2012rcampion Oct 21 '16 at 7:23\n• @2012rcampion Thanks a lot for the formatting edit! I merged in your suggestions (keeping the square out of $\\operatorname{IsZero}$, to keep the same shape as the original one, I think the fully-inlined version with the new $\\operatorname{IsZero}$ has roughly the same number of terms as with your idea). – Suzanne Dupéron Oct 21 '16 at 9:58\n• @GeorgesDupéron You could achieve the same tight result by (polynomially) mapping all permutations of $(a,b,c,d,e)$ composed in some way (say, concatenation) to a specific number in the range $[0, {52 \\choose 5}-1]$. Then for another $(a,b,c,d,e)$ a different one in the range and so on. Your solution is probably more efficient, since this would require a polynomial of degree $\\sim 52^5$. – Carl Löndahl Oct 21 '16 at 10:44\n\n# Brute-force using only multiplication\n\nI used a brute-force technique with multiplication, implemented with the Racket language. Below is an attempt to use brute-force with addition (addition produces results which are more than 12 digits). Note that in both cases, not all possible sets of ids are attempted, rather the program tries to add numbers to the set, and rejects those which are incompatible with the existing numbers. This means that the solutions are far from being optimal.\n\nThe results are:\n\n1 2 3 4 5 7 9 11 13 16 17 19 23 25 29 31 37 41 43 47 49 53 59 61 67 71 73 79 81 83 89 97 101 103 107 109 113 121 127 128 131 137 139 149 151 157 163 167 169 173 179 181\n\nThis is very similar to the answer by Gareth McCaughan, except that the last id is removed from the list, and instead another one is used in the middle of the list:\n\n128\n\nThe reason that this is allowed follows:\n\nGareth McCaughan allowed primes of the form $p^1$, $p^2$, $p^4$, $p^8$, … However, the desired property is weaker. Let's represent the IDs as products of the form $(p_1)^{a_1} × (p_2)^{a_2} × ⋯$, where each $p_i$ is the $i$-th prime number (so $p_1 = 2$, $p_2 = 3$, $p_3 = 5$, $p_4 = 7$ …). The $a_i$ are natural numbers, and can be $0$. The condition is then that for any 5 such IDs, whose prime powers are noted respectively $a_i$, $b_i$, $c_i$, $d_i$, $e_i$, the tuple $(a_1+b_1+c_1+d_1+e_1, a_2+b_2+c_2+d_2+e_2, a_3+b_3+c_3+d_3+e_3, …)$ must be unique.\n\nI was unable to devise an algorithm which exploits the above property, however.\n\nSince the minimum value for a product of five numbers is $1×2×3×4×5 = 120$, we can subtract $120$ from each sum (and still get unique sums, since we subtract a constant), meaning that the maximum result is:\n\n$167 × 169 × 173 × 179 × 181 - 120 = 158190676901$, with $12$ digits.\n\nI then tried to do the brute-force in the other way round, starting with the last two known good values, and working my way back. The result is:\n\n1 19 45 90 97 101 103 107 109 113 121 122 127 128 131 134 137 139 141 142 146 147 148 149 151 155 156 157 158 159 160 161 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 191\n\nNotice that it starts from a much higher value. The minimum product then becomes $7464150$, which gives us, when applying the subtraction trick, a slightly lower maximum result of:\n\n$167 × 169 × 173 × 179 × 181 - 7464150 = 158183212871$, still with $12$ digits.\n\n#lang racket\n;; attempt at a solution for http://puzzling.stackexchange.com/q/44629/31235\n\n;; Use unsafe fixnum operations: we know the values are small enough to not\n;; overflow, and these operations are somewhat faster.\n(require racket/unsafe/ops)\n\n(define (check)\n(define check-clobbered (make-hash))\n;; Could be sped up by using unsafe-fx* and unsafe-fx=\n(for* ([id1 (in-list ids)]\n#:when (displayln (format \"Checking sets containing ~a…\" id1))\n[id2 (in-list ids)]\n#:unless (= id1 id2)\n[id3 (in-list ids)]\n#:unless (or (= id1 id3) (= id2 id3))\n[id4 (in-list ids)]\n#:unless (or (= id1 id4) (= id2 id4) (= id3 id4))\n[id5 (in-list ids)]\n#:unless (or (= id1 id5) (= id2 id5) (= id3 id5) (= id4 id5)))\n(let* ([result (* id1 id2 id3 id4 id5)]\n[ref (hash-ref check-clobbered result #f)])\n(if ref\n(unless (equal? ref (sort (list id1 id2 id3 id4 id5) <))\n(error (format \"~a*~a*~a*~a*~a = ~a is already the result of ~a\"\nid1 id2 id3 id4 id5 result ref)))\n(hash-set! check-clobbered\nresult\n(sort (list id1 id2 id3 id4 id5) <))))))\n\n;; Computed with my brute-force algorithm below\n(define ids (list 1 2))\n\n;; check the hardcoded values (copied from previous runs, to allow the\n;; computation to be restarted\n(begin\n(check)\n(displayln \"Initial hardcoded values checked!\"))\n\n;; Compute more values by increasing the \"20\" in the definition of\n;; compute-sequence just below:\n(begin\n(define (compute-sequence)\n(for ([card (in-range (length ids) 52)]) ;; 20 cards\n(define newid (for/first ([id (in-naturals)]\n#:unless (clobbered? id))\nid))\n(set! ids (cons newid ids))\n(clobber! newid)\n(displayln (list card newid))))\n\n(define clobbered2 (mutable-set))\n(define clobbered3 (mutable-set))\n(define clobbered4 (mutable-set))\n(define clobbered5 (mutable-set))\n\n;; returns #t if i*x₂, i*x₂*x₃, i*x₂*x₃*x₄ or i*x₂*x₃*x₅ have already been\n;; seen (i.e. recorded by clobber! below).\n(define (clobbered? i)\n(or (member i ids)\n(= i 0)\n(for*/or ([id1 (in-value i)]\n[id2 (in-list ids)]\n#:unless (unsafe-fx= id1 id2))\n(let ([result2 (unsafe-fx* id1 id2)])\n(or (set-member? clobbered2 result2)\n(for*/or ([id3 (in-list ids)]\n#:unless (or (unsafe-fx= id1 id3)\n(unsafe-fx= id2 id3)))\n(let ([result3 (unsafe-fx* result2 id3)])\n(or (set-member? clobbered3 result3)\n(for*/or ([id4 (in-list ids)]\n#:unless (or (unsafe-fx= id1 id4)\n(unsafe-fx= id2 id4)\n(unsafe-fx= id3 id4)))\n(let ([result4 (unsafe-fx* result3 id4)])\n(or (set-member? clobbered4 result4)\n(for*/or ([id5 (in-list ids)]\n#:unless (or (unsafe-fx= id1 id5)\n(unsafe-fx= id2 id5)\n(unsafe-fx= id3 id5)\n(unsafe-fx= id4 id5)))\n(let ([result5 (unsafe-fx* result4 id5)])\n(set-member? clobbered5 result5))))))))))))))\n\n; Records all the values for i*x₂, i*x₂*x₃, i*x₂*x₃*x₄ or i*x₂*x₃*x₅ as seen\n(define (clobber! i)\n(for* ([id1 (in-value i)]\n[id2 (in-list ids)]\n#:unless (unsafe-fx= id1 id2))\n(let ([result2 (unsafe-fx* id1 id2)])\n(for* ([id3 (in-list ids)]\n#:unless (or (unsafe-fx= id1 id3)\n(unsafe-fx= id2 id3)))\n(let ([result3 (unsafe-fx* result2 id3)])\n(for* ([id4 (in-list ids)]\n#:unless (or (unsafe-fx= id1 id4)\n(unsafe-fx= id2 id4)\n(unsafe-fx= id3 id4)))\n(let ([result4 (unsafe-fx* result3 id4)])\n(for* ([id5 (in-list ids)]\n#:unless (or (unsafe-fx= id1 id5)\n(unsafe-fx= id2 id5)\n(unsafe-fx= id3 id5)\n(unsafe-fx= id4 id5)))\n(let ([result5 (unsafe-fx* result4 id5)])\n\n;; initialize the clobbered lists\n(for-each clobber! ids)\n\n(compute-sequence)\n\nids\n(check))\n\n\nI used a brute-force technique with addition, implemented with the Racket language. Unfortunately, the result has 16 digits, instead of the required 12.\n\nDon't scroll through the code if you want to avoid spoilers, the first elements are hardcoded.\n\n#lang racket\n;; attempt at a solution for http://puzzling.stackexchange.com/q/44629/31235\n\n;; Use unsafe fixnum operations: we know the values are small enough to not\n;; overflow, and these operations are somewhat faster.\n(require racket/unsafe/ops)\n\n(define (check)\n(define check-clobbered (make-hash))\n;; Could be sped up by using unsafe-fx+ and unsafe-fx=\n(for* ([id1 (in-list ids)]\n#:when (displayln (format \"Checking sets containing ~a…\" id1))\n[id2 (in-list ids)]\n#:unless (= id1 id2)\n[id3 (in-list ids)]\n#:unless (or (= id1 id3) (= id2 id3))\n[id4 (in-list ids)]\n#:unless (or (= id1 id4) (= id2 id4) (= id3 id4))\n[id5 (in-list ids)]\n#:unless (or (= id1 id5) (= id2 id5) (= id3 id5) (= id4 id5)))\n(let* ([result (+ id1 id2 id3 id4 id5)]\n[ref (hash-ref check-clobbered result #f)])\n(if ref\n(unless (equal? ref (sort (list id1 id2 id3 id4 id5) <))\n(error (format \"~a+~a+~a+~a+~a = ~a is already the result of ~a\"\nid1 id2 id3 id4 id5 result ref)))\n(hash-set! check-clobbered\nresult\n(sort (list id1 id2 id3 id4 id5) <))))))\n\n;; Computed with my brute-force algorithm below\n(define ids (list 0 1 2 4 7 13 24 44 84 161\n309 594 1164 2284 4426 8543 16492 30700 56982 98817))\n;; Sequence from https://oeis.org/A005318\n(define A005318 (list 0 1 2 4 7 13 24 44 84 161\n309 594 1164 2284 4484 8807 17305 34301 68008 134852))\n\n;; check the hardcoded values (copied from previous runs, to allow the\n;; computation to be restarted\n(begin\n(check)\n(displayln \"Initial hardcoded values checked!\"))\n\n;; Plot the difference between the \"any subset of size 5 must have a unique sum\"\n;; and the https://oeis.org/A005318 sequence\n(begin\n(require plot)\n(parameterize ([plot-y-transform log-transform]\n[plot-y-ticks (log-ticks)])\n(plot (list (lines-interval\n(for/list ([x (in-naturals 1)]\n[y (cdr ids)])\n(list x (exact->inexact y)))\n(for/list ([x (in-naturals 1)]\n[y (cdr A005318)])\n(list (exact->inexact x) (exact->inexact y))))))))\n\n;; Compute more values by increasing the \"20\" in the definition of\n;; compute-sequence just below:\n(begin\n(define (compute-sequence)\n(for ([card (in-range (length ids) 20)]) ;; 20 cards\n(define newid (for/first ([id (in-naturals)]\n#:unless (clobbered? id))\nid))\n(set! ids (cons newid ids))\n(clobber! newid)\n(displayln (list card newid))))\n\n(define clobbered2 (mutable-set))\n(define clobbered3 (mutable-set))\n(define clobbered4 (mutable-set))\n(define clobbered5 (mutable-set))\n\n;; returns #t if i+x₂, i+x₂+x₃, i+x₂+x₃+x₄ or i+x₂+x₃+x₅ have already been\n;; seen (i.e. recorded by clobber! below).\n(define (clobbered? i)\n(or (member i ids)\n(for*/or ([id1 (in-value i)]\n[id2 (in-list ids)]\n#:unless (unsafe-fx= id1 id2))\n(let ([result2 (unsafe-fx+ id1 id2)])\n(or (set-member? clobbered2 result2)\n(for*/or ([id3 (in-list ids)]\n#:unless (or (unsafe-fx= id1 id3)\n(unsafe-fx= id2 id3)))\n(let ([result3 (unsafe-fx+ result2 id3)])\n(or (set-member? clobbered3 result3)\n(for*/or ([id4 (in-list ids)]\n#:unless (or (unsafe-fx= id1 id4)\n(unsafe-fx= id2 id4)\n(unsafe-fx= id3 id4)))\n(let ([result4 (unsafe-fx+ result3 id4)])\n(or (set-member? clobbered4 result4)\n(for*/or ([id5 (in-list ids)]\n#:unless (or (unsafe-fx= id1 id5)\n(unsafe-fx= id2 id5)\n(unsafe-fx= id3 id5)\n(unsafe-fx= id4 id5)))\n(let ([result5 (unsafe-fx+ result4 id5)])\n(set-member? clobbered5 result5))))))))))))))\n\n; Records all the values for i+x₂, i+x₂+x₃, i+x₂+x₃+x₄ or i+x₂+x₃+x₅ as seen\n(define (clobber! i)\n(for* ([id1 (in-value i)]\n[id2 (in-list ids)]\n#:unless (unsafe-fx= id1 id2))\n(let ([result2 (unsafe-fx+ id1 id2)])\n(for* ([id3 (in-list ids)]\n#:unless (or (unsafe-fx= id1 id3)\n(unsafe-fx= id2 id3)))\n(let ([result3 (unsafe-fx+ result2 id3)])\n(for* ([id4 (in-list ids)]\n#:unless (or (unsafe-fx= id1 id4)\n(unsafe-fx= id2 id4)\n(unsafe-fx= id3 id4)))\n(let ([result4 (unsafe-fx+ result3 id4)])\n(for* ([id5 (in-list ids)]\n#:unless (or (unsafe-fx= id1 id5)\n(unsafe-fx= id2 id5)\n(unsafe-fx= id3 id5)\n(unsafe-fx= id4 id5)))\n(let ([result5 (unsafe-fx+ result4 id5)])\n\n;; initialize the clobbered lists\n(for-each clobber! ids)\n\n(compute-sequence)\n\nids\n(check))\n\n\nThe 20 first terms that my algorithm gives are:\n\n0 1 2 4 7 13 24 44 84 161 309 594 1164 2284 4426 8543 16492 30700 56982 98817\n\nSince the computation times start getting really high at that point (my code could use a LOT of improvement speed-wise), I plugged the results into oeis.org, and found this series:\n\nConway-Guy sequence: a(n + 1) = 2a(n) - a(n - floor( 1/2 + sqrt(2n) ))\n\nThe first 52 values of the series are:\n\n0 1 2 4 7 13 24 44 84 161 309 594 1164 2284 4484 8807 17305 34301 68008 134852 267420 530356 1051905 2095003 4172701 8311101 16554194 32973536 65679652 130828948 261127540 521203175 1040311347 2076449993 4144588885 8272623576 16512273616 32958867580 65852055508 131573282068 262885436596 525249670017 1049459028687 2096841607381 4189538625877 8370804628178 16725096982740 33433681691864 66834404516148 133602956976788 267074340671508 533885795906420\n\nA small excerpt from the description:\n\nConway and Guy conjecture that the set of $k$ numbers $\\{s_i = a(k) - a(k-i) : 1 <= i <= k\\}$ has the property that all its subsets have distinct sums\n\nThis is not exactly the property we are looking for, but it seems to fit the job.\n\nThe following script computes and checks the values (don't look too close, spoilers inside):\n\n#lang racket\n\n(define ids '(1 0))\n\n(for ([i (in-range 1 51)])\n(set! ids\n(cons\n;; a(i+1)\n(- (* 2 (list-ref (reverse ids) i))\n(list-ref (reverse ids)\n(- i (inexact->exact\n(floor (+ 1/2 (sqrt (* 2 i))))))))\nids)))\n\n(define (check)\n(define check-clobbered (make-hash))\n(for* ([id1 (in-list ids)]\n#:when (displayln (format \"Checking sets containing ~a…\" id1))\n[id2 (in-list ids)]\n#:unless (= id1 id2)\n[id3 (in-list ids)]\n#:unless (or (= id1 id3) (= id2 id3))\n[id4 (in-list ids)]\n#:unless (or (= id1 id4) (= id2 id4) (= id3 id4))\n[id5 (in-list ids)]\n#:unless (or (= id1 id5) (= id2 id5) (= id3 id5) (= id4 id5)))\n(let* ([result (+ id1 id2 id3 id4 id5)]\n[ref (hash-ref check-clobbered result #f)])\n(if ref\n(unless (equal? ref (sort (list id1 id2 id3 id4 id5) <))\n(error (format \"~a+~a+~a+~a+~a = ~a is already the result of ~a\"\nid1 id2 id3 id4 id5 result ref)))\n(hash-set! check-clobbered\nresult\n(sort (list id1 id2 id3 id4 id5) <)))))\n#;(hash->list check-clobbered))\n\n(reverse ids)\n(displayln (format \"maximum value for a hand of 5 cards: ~a\"\n(apply + (take ids 5))))\n(check)\n\n\nThe terms are the same at the beginning, but start diverging at the 15th card (my sequence gives smaller IDs):\n\nMy results:\n\n0 1 2 4 7 13 24 44 84 161 309 594 1164 2284 4426 8543 16492 30700 56982 98817\n\nOEIS sequence:\n\n0 1 2 4 7 13 24 44 84 161 309 594 1164 2284 4484 8807 17305 34301 68008 134852\n\nI think the reason the sequences diverge is:\n\nMy sequence ensures that any subset of five or less elements has a unique sum for that subset size, whereas the Conway-Guy sequence conjectures that any two subset of any size (possibly two sets with different sizes) have a different sum. My sequence is therefore more permissive.\n\nHere is a logarithmic plot of the difference between the two sequences, for the first 20 elements:", null, "As can be seen, the difference is tiny, so my sequence won't improve enough on the OEIS one to pass the 12-digit limit.\n\n• cannot use 128 = 2 x 4 x 16 – paparazzo Oct 22 '16 at 19:51\n• @Paparazzi : You can use 128: if you have two hands $128×a×b×c×d$ and $2×4×16×e×f$, they won't collide unless there are four distinct numbers $a,b,c,d$ and two distinct numbers $e,f$ such that $a×b×c×d=e×f$. This is not the case in the sequences I gave, so 128 is valid (I used the code I gave in the answer to check automatically that all combinations of 5 elements of the set have a distinct product). – Suzanne Dupéron Oct 23 '16 at 4:06" ]

[ null, "https://i.stack.imgur.com/TqKP0.png", null ]

[ "Hosting by PhotoTangler Collage Maker\n\n The Half-Edge Data Structure by (07 August 2000) Return to The Archives\n Introduction", null, "A common way to represent a polygon mesh is a shared list of vertices and a list of faces storing pointers for its vertices. This representation is both convenient and efficient for many purposes, however in some domains it proves ineffective.Mesh simplification, for example, often requires collapsing an edge into a single vertex. This operation requires deleting the faces bordering the edge and updating the faces which shared the vertices at end points of the edge. This type of polygonal \"surgery\" requires us to discover adjaceny relationships between components of the mesh, such as the faces and the vertices. While we can certainly implement these operations on the simple mesh representation mentioned above, they will most likely be costly; many will require a search through the entire list of faces or vertices, or possibly even both.Other types of adjacency queries on a polygon mesh include: Which faces use this vertex? Which edges use this vertex? Which faces border this edge? Which edges border this face? Which faces are adjacent to this face? To implement these types of adjacency queries efficiently, more sophisticated boundary representations (b-reps) have been developed which explicitly model the vertices, edges, and faces of the mesh with additional adjacency information stored inside.One of the most common of these types of representations is the winged-edge data structure where edges are augmented with pointers to the two vertices they touch, the two faces bordering them, and pointers to four of the edges which emanate from the end points. This structure allows us to determine which faces or vertices border an edge in constant time, however other types of queries can require more expensive processing.The half-edge data structure is a slightly more sophisticated b-rep which allows all of the queries listed above (as well as others) to be performed in constant time (*). In addition, even though we are including adjacency information in the faces, vertices and edges, their size remains fixed (no dynamic arrays are used) as well as reasonably compact.These properties make the half-edge data structure an excellent choice for many applications, however it is only capable of representing manifold surfaces, which in some cases can prove prohibitive. Mathematically defined, a manifold is a surface where every point is surrounded by a small area which has the topology of a disc. For the purpose of a polygon mesh, this means that every edge is bordered by exactly two faces; t-junctions, internal polygons, and breaks in the mesh are not allowed.(*) More precisely, constant time per piece of information gathered. For instance when querying all edges adjacent to a vertex, the operation will be linear in the number of edges adjacent to the vertex, but constant time per-edge.\n\n Structure", null, "The half-edge data structure is called that because instead of storing the edges of the mesh, we store half-edges. As the name implies, a half-edge is a half of an edge and is constructed by splitting an edge down its length. We'll call the two half-edges that make up an edge a pair. Half-edges are directed and the two edges of a pair have opposite directions.\n\nThe diagram below shows a small section of a half-edge representation of a triangle mesh. The yellow dots are the vertices of the mesh and the light blue bars are the half-edges. The arrows in the diagram represent pointers, although in order to keep the diagram from getting too cluttered, some of them have been ommited.", null, "As you can see in the diagram, the half-edges that border a face form a circular linked list around its perimeter. This list can either be oriented clockwise or counter-clockwise around the face just as long as the same convention is used throughout. Each of the half-edges in the loop stores a pointer to the face it borders (not shown in the diagram), the vertex at its end point (also not shown) and a pointer to its pair. It might look something like this in C:\n\n ``` struct HE_edge { HE_vert* vert; // vertex at the end of the half-edge HE_edge* pair; // oppositely oriented adjacent half-edge HE_face* face; // face the half-edge borders HE_edge* next; // next half-edge around the face }; ```\n\nVertices in the half-edge data structure store their x, y, and z position as well as a pointer to exactly one of the half-edges which uses the vertex as its starting point. At any given vertex there will be more than one half-edge we could choose for this, but we only need one and it doesn't matter which one it is. We'll see why later on when the querying methods are explained. In C the vertex structure looks like this:\n\n ``` struct HE_vert { float x; float y; float z; HE_edge* edge; // one of the half-edges emantating from the vertex }; ```\n\nFor a bare-bones version of the half-edge data structure, a face only needs to store a pointer to one of the half-edges which borders it. In a more practical implementation we'd probably store information about textures, normals, etc. in the faces as well. The half-edge pointer in the face is similar to the pointer in the vertex structure in that although there are multiple half-edges bordering each face, we only need to store one of them, and it doesn't matter which one. Here's the face structure in C:\n\n ``` struct HE_face { HE_edge* edge; // one of the half-edges bordering the face }; ```\n\n Adjacency Queries", null, "The answers to most adjacency queries are stored directly in the data structures for the edges, vertices and faces. For example, the faces or vertices which border a half-edge can easily be found like this:\n\n ``` HE_vert* vert1 = edge->vert; HE_vert* vert2 = edge->pair->vert; HE_face* face1 = edge->face; HE_face* face2 = edge->pair->face; ```\n\nA slightly more complex example is iterating over the half edges adjacent to a face. Since the half-edges around a face form a circular linked list, and the face structure stores a pointer to one of these half-edges, we do it like this:\n\n ``` HE_edge* edge = face->edge; do { // do something with edge edge = edge->next; } while (edge != face->edge); ```\n\nSimilarly, we might be interested in iterating over the edges or faces which are adjacent to a particular vertex. Referring back to the diagram, you may see that in addition to the circular linked lists around the borders of the faces, the pointers also form loops around the vertices. The iterating process is the same for discovering the adjacent edges or faces to a vertex; here it is in C:\n\n ``` HE_edge* edge = vert->edge; do { // do something with edge, edge->pair or edge->face edge = edge->pair->next; } while (edge != vert->edge); ```\n\nNote that in these iterating examples checks for null pointers are not included. This is because of the restriction on the surface being manifold; in order for this requirement to be fulfilled, all of the pointers must be valid.\n\nOther adjacency relationships can be quickly found by following these examples.\n\n Remarks", null, "The Harvard Graphics Archive Mesh Library contains a full implementation of the half-edge data structure. The source code is available online at: http://www.cs.deas.harvard.edu/~xgu/mesh/ As mentioned in the introduction, half-edge's restriction to representing manifold surfaces makes it unusable for certain purposes. The winged-edge data structure (with certain modifications) is capable of over coming some of these restrictions, and the more complex radial-edge data structure allows for non-manifold meshes altogether. More information on the winged-edge data structure can be found in \"Computer Graphics: Principles and Practice\" by Foley and van Dam, and a description of the radial-edge structure by Kevin Weiler is avaiable in the paper: K. Weiler, \"The Radial-Edge Structure: A Topological Representation for Non-Manifold Geometric Boundary Representations\", Geometric Modelling for CAD Applications, North Holland, pp. 3-36, 1988." ]

[ null, "https://flipcode.com/archives/line_grey.png", null, "https://flipcode.com/archives/line_grey.png", null, "https://flipcode.com/archives/article_halfedge.png", null, "https://flipcode.com/archives/line_grey.png", null, "https://flipcode.com/archives/line_grey.png", null ]

[ "round() Function\n\nRounds off the number to the specified number of digits.\n\n## Syntax\n\nround( number, [num_digits] )\n\nnumber: (Decimal) The number that will be rounded.\n\nnum_digits: (Integer) Determines the digit or place to which the number will be rounded to the nearest 10^(-num_digits). 2 is default.\n\n## Returns\n\nDecimal\n\nTo round the number to the nearest hundred, set num_digits to `-2`.\n\n## Examples\n\nYou can experiment with this function in the test box below.\n\nTest Input\n\n`round(7.36819)` returns `7.37`\n\n`round(7.36819, 3)` returns `7.368`\n\n`round(7.36819, 4)` returns `7.3682`\n\n## See Also\n\nfixed(): Use this function with `round()` to display your desired output.\n\nOpen in Github\n\nOn This Page" ]

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[ "5 files\n\n## Data files Prmt7 and adipocyte differentiation\n\ndataset\nposted on 19.12.2013 by Yu-Jie Hu, Saïd Sïf, Anthony N. Imbalzano\n\nDataset 1: Dataset for Fig.1B\nQuantification of Prmt7 protein levels in differentiating C3H10T1/2 cells. Column A describes the time when cells were harvested. Columns B to E show the signal intensities of Prmt7 and PI3K from two independent experiments, quantified by ImageJ. Columns F and G show the ratio of Prmt7 levels to PI3K levels for each sample in each experiment. Columns H and I show relative Prmt7 levels where the day 0 sample in each experiment is normalized to 1. Column J shows the mean of the relative Prmt7 levels from the two independent experiments.\n\nDataset 2: Dataset for Fig.2B\nQuantification of Prmt7 protein levels in the knockdown and over-expression C3H10T1/2 cells. Column A shows the names of the cells. Columns B to E show the signal intensities of Prmt7 and PI3K from two independent experiments, quantified by ImageJ. Columns F and G show the ratio of Prmt7 levels to PI3K levels for each sample in each experiment. Columns H and I show relative Prmt7 levels where the shControl and the pBABE vector sample in each experiment is normalized to 1. Column J shows the mean of the relative Prmt7 levels from the two independent experiments.\n\nDataset 3: Dataset for Fig.3\nCell counting results. Column A shows the time when cell numbers were counted. Column B shows the names of the cells. Columns C to F show the numbers of the cells (x105) from two independent experiments with two replicates in each experiment. Column G shows the mean of the cell numbers. Column H shows the standard deviation (SD) of the cell numbers.\n\nDataset 4: Dataset for Fig.4B\nQuantification of Oil-Red O staining on day 6 post-differentiated C3H10T1/2 cells. Column A shows the names of the cells. Columns B to Column E show the values of OD500 from two independent experiments with two replicates in each experiment. Column F represents the mean of the OD500 values. Column F shows the mean values. Column G shows the standard deviation (SD) of the OD500 values.\n\nDataset 5: Dataset for Fig.4D\nReal-time qPCR results for the expression of selected genes in day 6 post-differentiated C3H10T1/2 cells. Column A shows to the names of the transcripts. Column B shows the names of the cells. Column C indicates the experiment number. Columns D and E show raw Ct values from two replicates from each sample. Column F shows the mean raw Ct value. Columns G to K show the following values: delta Ct (Ct – Ct5S rRNA), delta delta Ct (dCt - dCtshCtrl), fold change (2^(-ddCt)), the mean of the fold change, and the standard deviation (SD) of the fold change." ]

[ null ]

[ "", null, "Find unknowns in multiplication and division problems (0,1,2,4,8)\n\nLesson\n\nWhen looking for the unknown in an equation, we can use multiplication or division, so this video helps you decide which method to use.  It also shows you how to multiply by 0 (zero).\n\nExamples\n\nQUESTION 1\n\nFill in the box with the missing number.\n\n1. $9\\times0=\\editable{}$9×0=\n\nQUESTION 2\n\nFill in the box with the missing number.\n\n1. $24\\div8=\\editable{}$24÷​8=\n\nQUESTION 3\n\nFill in the box with the missing number.\n\n1. $10\\times8=\\editable{}$10×8=\n\nOutcomes\n\nNA3-2\n\nKnow basic multiplication and division facts." ]

[ null, "https://mathspace-production-static.mathspace.co/permalink/badges/v3/language-and-use-of-statistics.svg", null ]

[ "# Motivation\n\nTo put arrows inside of a TikZ path I have defined myself a TikZ style arrow inside. To make it even more portable I want it to take arguments like this\n\n\\draw[arrow inside={pos = 0.1, end = |}] (1,-0.5) -- (1,1.5);\n\n\nsuch that I can define the position of the arrow using pos and the type of the arrow by end. If no arguments are given it should fall back to the defaults.", null, "# What I have tried so far\n\nTo achive the above picture, I wrote the following snippet. The problem is that the values are properly stored in the macros I defined, but are not being updated, nor do I now, how to retain the defaults.\n\n\\documentclass{article}\n\\pagestyle{empty}\n\\usepackage{tikz}\n\\usetikzlibrary{decorations.markings}\n\\makeatletter\n\\tikzset{\narrow inside/pos/.store in = \\arrow@inside@pos,\narrow inside/end/.store in = \\arrow@inside@end,\narrow inside/pos = 0.5,\narrow inside/end = >,\narrow inside/.style = {\npostaction = {\ndecorate,\ndecoration={\nmarkings,\nmark=at position \\arrow@inside@pos with {\\arrow{\\arrow@inside@end}}\n}\n}\n},\n}\n\\makeatother\n\\begin{document}\n$\\int_{ \\tikz[scale=0.3]{ \\path[fill=lightgray] (0,0) rectangle (1,1); \\draw (1.5,1) -- (0,1) -- (0,0) -- (1.5,0); \\draw[arrow inside={pos = 0.1, end = |}] (1,-0.5) -- (1,1.5); } } \\vec{B} \\cdot \\vec{n} \\, df$\n\\end{document}\n\n\nPossibly related:\n\nJake is correct in his now deleted answer. The options you pass to arrow inside are not applied as they are simply grabbed and not used (i.e. “gobbled”) similar to #1 in:\n\n\\newcommand*{\\myCommand}{Foobar Rhubarb}% no \"#1\" in its definition\n\n\nYou need to add #1 in your .style definition.\n\nBut instead of using the .store in handler, consider using .initial keys which simply store a value. This way, you don’t add another layer of abstraction (PGFkeys already use a specific “namespace”), no need to add your own \\arrow@inside@… macros if PGFkeys already uses /tikz/arrow inside/….\n\nTo also avoid troubles with /.cd inside a .style, I’d use a specific key set arrow inside that sets the path to /tikz/arrow inside.\n\n## Code\n\n\\documentclass{article}\n\\pagestyle{empty}\n\\usepackage{tikz}\n\\usetikzlibrary{decorations.markings}\n\\tikzset{\nset arrow inside/.code={\\pgfqkeys{/tikz/arrow inside}{#1}},\nset arrow inside={pos/.initial=.5, end/.initial=>},\narrow inside/.style={\nset arrow inside={#1},\npostaction={\ndecorate,\ndecoration={\nmarkings,\nmark=at position \\pgfkeysvalueof{/tikz/arrow inside/pos}\nwith \\arrow{\\pgfkeysvalueof{/tikz/arrow inside/end}}\n}\n}\n},\n}\n\\begin{document}\n$\\int_{ \\tikz[scale=0.3]{ \\path[fill=lightgray] (0,0) rectangle (1,1); \\draw (1.5,1) -- (0,1) -- (0,0) -- (1.5,0); \\draw[arrow inside={pos = 0.1, end = |}] (1,-0.5) -- (1,1.5); } } \\vec{B} \\cdot \\vec{n} \\, df$\n\\end{document}\n\n\n## Output", null, "## But why stop there?\n\nSometimes users want to use more than one arrow tip along a (part of a) path which I realized in a comment should be made better.\n\nThe following solution doesn’t use a pos key anymore (this can be changed) but applies the options end and opt to a list of arrow-tip marks in a decoration.\n\n(Note that for regular intervals, the library also provides mark=between positions <start> and <end> step <step> with <marking>.)\n\n## Code\n\n\\documentclass[tikz,border=5pt]{standalone}\n\\usetikzlibrary{decorations.markings}\n\\tikzset{\nset arrow inside/.code={\\pgfqkeys{/tikz/arrow inside}{#1}},\nset arrow inside={end/.initial=>, opt/.initial=},\n/pgf/decoration/Mark/.style={\nmark/.expanded=at position #1 with\n{\\noexpand\\arrow[\\pgfkeysvalueof{/tikz/arrow inside/opt}]\n{\\pgfkeysvalueof{/tikz/arrow inside/end}}}},\narrow inside/.style 2 args={\nset arrow inside={#1},\npostaction={decorate,decoration={\nmarkings,Mark/.list={#2}}}},\n}\n\\begin{document}\n\\tikz[thick]\n\\draw[\narrow inside={\nend=|,\nopt=red!\\pgfkeysvalueof{/pgf/decoration/mark info/sequence number}0!blue\n}{0, .05, .15, .3, .5, .7, .85, .95, 1},\n] (0,0) to[bend left] (4,0);\n\\end{document}\n\n\n## Output", null, "• This is a great answer! I will reward it with a small bounty as soon as it's possible, because of the extra credit with the sequence of markings inside. (BTW: Where did Jake's answer go?) – Henri Menke Sep 3 '13 at 13:04\n• @HenriMenke: I deleted mine, because it doesn't add anything that Qrrbrbirlbel's doesn't cover. – Jake Sep 6 '13 at 19:45\n• – Qrrbrbirlbel Sep 8 '13 at 13:57\n\nI suggest a different approach, not optimal (see What do the pgfkeys key handlers .get and .store in do?), but it works. Indeed, one problem in your code is that, inside arrow style the keys do not inherit the correct path /tikz/arrow inside/key-name. To make it working you can do something like:\n\n$\\int_{ \\tikz[scale=0.3]{ \\path[fill=lightgray] (0,0) rectangle (1,1); \\draw (1.5,1) -- (0,1) -- (0,0) -- (1.5,0); \\draw[/tikz/arrow inside/pos = 0.4,/tikz/arrow inside/end = |,arrow inside] (1,-0.5) -- (1,1.5); } } \\vec{B} \\cdot \\vec{n} \\, df$\n\n\nwhich is not so convenient IMHO. The other problem is that if you define a style doing something with keys, those keys have to be set not inside the style, but before as I did with the above code. Notice that:\n\n$\\int_{ \\tikz[scale=0.3]{ \\path[fill=lightgray] (0,0) rectangle (1,1); \\draw (1.5,1) -- (0,1) -- (0,0) -- (1.5,0); \\draw[arrow inside={/tikz/arrow inside/pos = 0.4,/tikz/arrow inside/end = |}] (1,-0.5) -- (1,1.5); } } \\vec{B} \\cdot \\vec{n} \\, df$\n\n\ndoes not do anything as the keys still have the values 0.5 and > respectively.\n\nSo, my approach will be based on the \"triple\" handlers initial, get and store in and will let you use:\n\narrow inside={pos = 0.4,end = |}\n\n\ninside \\draw.\n\nFirst I would define the keys:\n\n\\pgfkeys{/arrow inside/.cd,\npos/.initial = 0.5,\npos/.get = \\arrow@inside@pos,\npos/.store in = \\arrow@inside@pos,\nend/.initial = >,\nend/.get = \\arrow@inside@end,\nend/.store in = \\arrow@inside@end,\n}\n\n\nThey are under the path /arrow inside/ so later on we should take care of this.\n\nSecond, I would define a style to place the arrow:\n\nplace arrow/.style = {\npostaction = {\ndecorate,\ndecoration={\nmarkings,\nmark=at position \\arrow@inside@pos with {\\arrow{\\arrow@inside@end}}\n}\n}\n},\n\n\nRather than the keys, the style belongs to the usual /tikz/ path. To combine the keys and the aforementioned style, I define:\n\narrow inside/.style={place arrow,/arrow inside/.cd,#1}\n\n\nas a style which \"places\" the arrow and, by changing the default /tikz/ path, allows you to use the previously defined keys.\n\nA mwe:\n\n\\documentclass{article}\n\\usepackage{tikz}\n\\usetikzlibrary{decorations.markings}\n\\makeatletter\n\\pgfkeys{/arrow inside/.cd,\npos/.initial = 0.5,\npos/.get = \\arrow@inside@pos,\npos/.store in = \\arrow@inside@pos,\nend/.initial = >,\nend/.get = \\arrow@inside@end,\nend/.store in = \\arrow@inside@end,\n}\n\\tikzset{arrow inside/.style={place arrow,/arrow inside/.cd,#1},\nplace arrow/.style = {\npostaction = {\ndecorate,\ndecoration={\nmarkings,\nmark=at position \\arrow@inside@pos with {\\arrow{\\arrow@inside@end}}\n}\n}\n},\n}\n\\makeatother\n\\begin{document}\n\n$\\int_{ \\tikz[scale=0.3]{ \\path[fill=lightgray] (0,0) rectangle (1,1); \\draw (1.5,1) -- (0,1) -- (0,0) -- (1.5,0); \\draw[arrow inside={pos = 0.4,end = |}] (1,-0.5) -- (1,1.5); } } \\vec{B} \\cdot \\vec{n} \\, df$\n\n$\\int_{ \\tikz[scale=0.3]{ \\path[fill=lightgray] (0,0) rectangle (1,1); \\draw (1.5,1) -- (0,1) -- (0,0) -- (1.5,0); \\draw[arrow inside={pos = 0.7,end = stealth}] (1,-0.5) -- (1,1.5); } } \\vec{B} \\cdot \\vec{n} \\, df$\n\n\\end{document}\n\n\nThe result:", null, "Notice that the keys now update correctly their values.\n\n• I think the best explanation for this has been given by percusse in this comment. Indeed, as also my answer there demonstrates, the .get handler is needed for, let's say, the special case in which you first use the keys without passing any values (the initial ones are given) and then you use them with values. In this case, try to say \\draw[arrow inside={}] (1,-0.5) -- (1,1.5); in the first example: the compilation will rise errors as both macros won't be defined. – Claudio Fiandrino Sep 3 '13 at 7:12\n• Thank you for your great answer. If I could I would accept both, but I will reward yours with a small bounty as soon as it's possible for giving a great insight on .get and .store in. – Henri Menke Sep 3 '13 at 13:06" ]

[ null, "https://i.stack.imgur.com/feNVK.png", null, "https://i.stack.imgur.com/YhbcP.png", null, "https://i.stack.imgur.com/JziPP.png", null, "https://i.stack.imgur.com/5Znmf.png", null ]

[ "", null, "## Graphing Calculator\n\n### What's new in Version 4.1 for macOS\n\n• 64-bit binary\n• Sliders for point radius and line width\n• Select vector and matrix elements by subscript\n• Allows slider variables in set definitions\n• Support arrows and complex points on right hand graph pane in split views\n• Support trackpad for zoom on 2D graph views\n• Removed 4D graphing\n\n### What was new in Version 3.0\n\n• Printing\n• Multiple document windows\n• Save as web page\n• Save as RTF for export to word processors\n• 3.0 acts as a helper application for GC documents posted to the web\n\n### What was new in Version 2.7\n\n#### Tables\n\nYou can enter tables of data, numerically transform the data, and graph data in two and three dimensions.", null, "#### Surfaces of revolution\n\nYou can rotate a curve around an arbitrary axis to generate a surface in three dimensions.", null, "#### 3D Ordinary Differential Equations\n\nYou can graph ODEs in three dimensions.", null, "You can specify initial conditions for ODEs by giving starting points on surfaces.", null, "#### Tube plots", null, "", null, "#### Numeric integration", null, "#### Binomial coefficient", null, "#### Products", null, "#### Matrix operations", null, "", null, "", null, "", null, "#### Complex Conjugate", null, "#### Logical expressions", null, "#### Vector fields\n\nVector fields can be graphed on surfaces.", null, "#### Parameter sets", null, "#### 2D Vectors", null, "#### 3D Vectors", null, "#### Coordinate Transformations", null, "", null, "", null, "", null, "", null, "", null, "#### Complex number arithmetic\n\nYou can use Graphing Calculator 3.1 as a calculator for complex numbers. For example:", null, "#### Complex function parametric curves in 2D\n\nA complex function of a real parameter, t, specifies a curve in 2D. For example:", null, "This is a shorter way of writing the parametric equation:", null, "You can also use z as shorthand for x + iy. For example:", null, "#### Complex function parametric curves in 3D\n\nA complex function of a real parameter, z, specifies a curve in 3D. Observe that this is the same complex function as in the 2D example above, but here the parameter z is used as the third 3D coordinate.", null, "This is similar to writing the parametric equations:", null, "#### Parametric curves in 4D\n\nYou can also write parametric equations for curves in four dimensions by specifying functions (which have real values) for each of the four spatial coordinates x, y, u, and v. For example:", null, "#### Complex function surfaces in 3D\n\nUsing z as shorthand for x + iy, you can graph 3D surfaces representing complex functions. For example:", null, "", null, "", null, "", null, "In this example, observe the four zeros at +1, -1, +i, and -i.\n\n#### Complex function surfaces in 3D with color coding\n\nUsing z as shorthand for x + iy, you can graph 3D surfaces representing complex functions. The height of the surface represents the magnitude of the complex function. The phase of the complex function is encoded in the color of the surface. For example:", null, "#### Complex function surfaces in 4D\n\nA complex function f(x + iy) depends on two real parameters x & y and has two components, Re f, and Im f. We can render this as a surface in 4 spatial dimensions with coordinates:\n(x, y, Re f(x+ iy), Im f(x + iy)) and then project this surface to the screen in a manner similar to the way we project 3-D surfaces onto a 2-D screen. Note that in the following, z=x+iy and w=u+iv.", null, "#### Interactive complex parameters\n\nIn the following example, you can drag the purple dot, which moved the square grid along with it. The blue grid is the map of the square grid under the complex function f(z).", null, "#### Color maps\n\nYou can specify the color at each point in the place by giving functions for either the (red, green, blue) or the (hue, saturation, value) components of the color.", null, "#### Point plots\n\nIn the following example, as n varies, the point traces out a circle:", null, "#### Vertex lists in 4D\n\nThe following shows a hypercube in four dimensions specified by writing the 32 edges between its 16 vertices.", null, "#### Parametric surfaces in 4D\n\nThis is the parametrization for a flat torus in 4D. It is the product of an x-y circle (sin u, cos u), and a u-v circle (sin v, cos v). (The four 4D coordinate axes are x, y, u & v. The letters u & v are also used separately for the surface parametrization. This is confusing as u and v must be interpreted differently depending on context.)", null, "#### New functions\n\nThe min() function returns the minimum of its arguments.", null, "The max function returns the maximum of its arguments.", null, "mod(a,b) return the remainder of a divided by b.", null, "clamp(x,a,b) pins x to the range [a,b].", null, "#### Text expressions in coordinate ranges", null, "#### Direction fields\n\nType command-option-d to draw unit vectors in a vector field.", null, "#### Examples menu and sample documents\n\nVersion 3.1 ships with example and template documents which are also accessible from the new Examples menu.\n\n#### Background Color\n\nThe preferences dialog now gives control over the background used with 3D objects.", null, "", null, "#### Defining variables and functions", null, "(You must type \"f control-9 x\" to distinguish f(x) as a function.)\n\n#### Copy and Paste formulas as plain text\n\nCopy as Text on", null, "produces sum(x^k/k!,k=0,5).\n\n#### Equations line-break and scroll", null, "#### 1st order 2D ordinary differential equations", null, "", null, "", null, "#### Fractional powers\n\nFractional powers of negative numbers are handled properly now.", null, "#### Bessel functions", null, "#### Floor and ceiling", null, "#### Piecewise defined functions", null, "", null, "", null, "", null, "#### Vector dot and cross products", null, "", null, "#### Summations", null, "#### Substitutions", null, "#### New releases\n\nIf you would like to help test pre-release development versions of our products, subscribe to our mailing lists here.", null, "" ]

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[ "# Properties\n\n Label 24646b Number of curves 1 Conductor 24646 CM no Rank 3\n\n# Related objects\n\nShow commands for: SageMath\nsage: E = EllipticCurve(\"24646.a1\")\nsage: E.isogeny_class()\n\n## Elliptic curves in class 24646b\n\nsage: E.isogeny_class().curves\nLMFDB label Cremona label Weierstrass coefficients Torsion order Modular degree Optimality\n24646.a1 24646b1 [1, 0, 1, -13, 12] 1 5392 $$\\Gamma_0(N)$$-optimal\n\n## Rank\n\nsage: E.rank()\n\nThe elliptic curve 24646b1 has rank $$3$$.\n\n## Modular form None\n\nsage: E.q_eigenform(10)\n$$q - q^{2} - 2q^{3} + q^{4} - 4q^{5} + 2q^{6} - 2q^{7} - q^{8} + q^{9} + 4q^{10} - 4q^{11} - 2q^{12} - 5q^{13} + 2q^{14} + 8q^{15} + q^{16} - 3q^{17} - q^{18} - 7q^{19} + O(q^{20})$$" ]

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[ "# Difference Between Analog and Digital Signals\n\nThe two major categories in which a signal is classified is analog and digital. The crucial difference between Analog and Digital Signal is that an analog signal is a continuous signal which is defined for every particular instant of time. On the contrary, a digital signal is of non-continuous nature, defined discretely at some specific time instants.\n\nDue to the different nature of the two signals, these find applications in different fields.\n\nWhat is Signal?\n\nIn electronics and field of signal processing, a signal is defined as an electric current or energy that carries information. A quantity that varies in space and time is utilized as a signal in order to transmit the data from one point to another. So, basically, the type of information carried out by the signal leads to its classification as analog and digital.\n\n## Content: Analog Vs Digital Signal\n\n### Comparison Chart\n\nBasis for ComparisonAnalog SignalDigital Signal\nNatureContinuous with time.Discrete sets at specific time.\nType of WaveformSinusoidal wavesSquare waves\nRepresentation", null, "", null, "Information carriedContinuous range of values.Discrete value i.e., 0 and 1.\nNoise SusceptibilityMoreComparatively less\nProcessing offeredEasyQuite complex\nSignal polarityPositive as well as negativeOnly positive\nSignal encryptionNot requiredRequired\nBandwidthLowComparatively high\nPower required for transmissionMoreLess in comparison to analog signal\nRate of data transmissionSlowQuite fast\nParameters associatedAmplitude, frequency, phase, etc.Bit rate, bit interval, etc.\nAccuracy of informationMoreComparatively less\nExamplesHuman voice, vision.Computer data transmission and reception, transmission of signals through cables like telephone.\n\n### Definition of Analog Signal\n\nAnalog signal is continuous in nature as it is well defined for every specific time. More specifically, it is the one defined continuously for a range of values with time. These are represented as a sinusoidal waveform. The figure below represents an analog signal:", null, "As we can see here that the signal is continuous in nature as its every value is specified for each particular interval of time, thus, is referred as the continuous-time signal. For the purpose of recording the analog signals, the actual signal must be preserved as only in that case we can record the accurate analog signal.\n\nHowever, whenever in any application there arises a need to change an analog signal into a digital one then through sampling this can be achieved. In the sampling technique, this signal is sampled at certain fixed time intervals and the sampled voltage is converted into digital values.\n\n### Definition of Digital Signal\n\nA type of signal which is non-continuous in nature as it is defined only for a set of discrete times is known as a digital signal. More simply, we can say, it is specified at some particular time instants only. Thus, here the information is represented as a set of discrete values. The figure given below shows the signal representation in digital form:", null, "Here it is clear that the data is discretely positioned in time axis thus is shown in the form of bits i.e., 0 and 1. This means that here the signal performs the sudden transition between 0 and 1 where these two values correspond to the two voltage levels. For the purpose of recording the signals, in the digital one, only the samples of the actual signal are taken and not the whole signal.\n\n## Key Differences Between Analog and Digital Signal\n\n1. Analog and digital signals are differentiated mainly on the basis of their way to carry the information. Analog signals represent the information as a continuous function of time. As against, digital signals are represented discretely at specific time intervals.\n2. Analog signals deteriorate more easily than digital one as their susceptibility towards noise is higher.\n3. The devices that operate on an analog signal need more power for transmission and reception than the devices that work on the digital signal.\n4. The information represented by the analog signal is comparatively more accurate than the digital signal due to its continuous range of values.\n5. Analog signals are transmitted at a slower rate than digital signals as the information holds more values thus is precise.\n6. An analog signal is defined by the amplitude, phase, frequency, etc. While the digital signal is mainly associated with bit rate, bit intervals, etc.\n7. The sinusoidal representation of the analog signal makes it difficult to understand or get decoded thus encryption is not required in analog signals. But as the digital signals are in the form of square waves, thus, it can be decoded easily hence need to be encrypted for secured communication.\n8. On the basis of processing, it is said that the continuous nature of the signal offers easy processing of the analog signal. However, the discrete nature of the signal offers difficultly in the processing of the digital signal.\n\n### Conclusion\n\nThus, the above discussion concludes that due to the way of representing the signal, the two are differentiated. And because of this, the two types of signals possess different properties and are used in different applications. Like analog signals are used in radio wave transmission and reception whereas digital signals are used in the field of digital electronics." ]

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[ "Αναζήτηση Εικόνες Χάρτες Play YouTube Ειδήσεις Gmail Drive Περισσότερα »\nΕίσοδος\n Βιβλία Βιβλία 141 - 150 από 160 για Find the greatest square number in the first or left hand period, place the root....", null, "Find the greatest square number in the first or left hand period, place the root of it at the right hand of the given number, (after the manner of a quotient in division...", null, "A New and Complete System of Arithmetick: Composed for the Use of the ... - Σελίδα 190\nτων Nicolas Pike - 1822 - 532 σελίδες\nΠλήρης προβολή - Σχετικά με αυτό το βιβλίο", null, "## Caldwell's Practical Arithmetic ...\n\nM. P. Caldwell - 1883 - 191 σελίδες\n...given number as the first figure of the required roof. Subtract its cube from the left hand period, and to the remainder bring down the next period for a dividend. Multiply the square of tiie root already found by 3, and to the product annex two ciphers for a trial...", null, "## A Collection of Mathematical Rules and Tables: Including Interest, Equation ...\n\nJames Morton - 1887 - 216 σελίδες\n...period, and write its root in the quotient. Subtract the square of this root from the left hand period, and to the remainder bring down the next period for a dividend. Thirdly, double the root already found for a divisor ; ascertain how many times the divisor js contained...", null, "## College Algebra: For the Use of Academies, Colleges, and Scientific Schools ...\n\nEdward Albert Bowser - 1888 - 540 σελίδες\n...place it on the right ; this is the first figure of the root; subtract its cube from the first period, and to the remainder bring down the next period for a dividend. Take three times the square of the root already found for a trial divisor, and see how many times it...", null, "## Robinson's Progressive Practical Arithmetic: Containing the Theory of ...\n\nHoratio Nelson Robinson - 1888\n...write its root for the first figure in the required root ; subtract the cube from the left hand period, and to the remainder bring down the next period for a dividend. III. At the left of the dividend write three times the square of the first figure of the root, and...", null, "## Robinson's New Practical Arithmetic for Common Schools and Academies\n\nHoratio Nelson Robinson - 1892 - 416 σελίδες\n...write its root for the first figure in the required root ; subtract the cube from the left-hand period, and to the remainder bring down the next period for a dividend. • III. At the left of the dividend write, three times the square of the first figure of the root,...", null, "## Robinson's New Practical Arithmetic for Common Schools and Academies\n\nHoratio Nelson Robinson - 1892 - 416 σελίδες\n...write its root for the first figure in the required root; subtract the cube from the left-hand period, and to the remainder bring down the next period for a dividend. III. At the left of the dividend write three times the square of the first figure of the root, and...", null, "## The Mechanical Engineer's Pocket-book: A Reference Book of Rules, Tables ...\n\nWilliam Kent - 1895 - 1100 σελίδες\n...write its root as the first figure In the required root. Subtract the cube from the left-hand period, and to the remainder bring down the next period for a dividend. Square the first figure of Ihe roof, multiply hv 300, and divide the product into the dividend for...", null, "## Robinson's New Higher Arithmetic: For High Schools, Academies, and ...\n\nHoratio Nelson Robinson - 1895 - 506 σελίδες\n...its root for the first figure in the root; subtract this square number from the left-lutnd period, and to the remainder bring down the next period for a dividend, III. At the left of the dividend write twice tlie first term of the root, and annex one cipher, for...", null, "## The Mechanical Engineer's Pocket-book: A Reference Book of Rules, Tables ...\n\nWilliam Kent - 1902 - 1113 σελίδες\n...write its root as the first figure in the required root. Subtract the cube from the left-hand period, and to the remainder bring down the next period for a dividend. Square the first figure of the root : multiply by 300, and divide the product into the dividend for...", null, "## Ray's Modern Practical Arithmetic: A Revised Edition of Ray's Practical ...\n\nJoseph Ray - 1903 - 320 σελίδες\n...left; place its root on the right, like a quotient in division ; subtract the square from the period, and to the remainder bring down the next period for a dividend. EVOLUTION 3. Double the root found, and place it on the left of the dividend for a trial divisor. find..." ]

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[ "You are here : assembly8086AAM\n\n# AAM - 8086\n\n`ASCII Adjust after Multiplication.Corrects the result of multiplication of two BCD values.Algorithm :AH = AL / 10AL = remainder`\n\n`AAM`\n\n### Example\n\n```MOV AL, 0fh ; AL = 0Fh\nAAM ; AH = 01, AL = 05\nRET ```" ]

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[ "# Take pictures of math problems\n\nTake pictures of math problems can support pupils to understand the material and improve their grades. Our website will give you answers to homework.\n\n## The Best Take pictures of math problems\n\nTake pictures of math problems can be found online or in math books. Mathematics solution is a process or set of steps that are used to solve a mathematical problem. The steps can be written down in a step-by-step fashion, or they can be memorized and applied when needed. Either way, a mathematics solution should be clear and concise, and it should be able to be applied to similar problems.\n\nThere are a few steps to solving rational functions. First, you need to find the zeroes of the function. These are the points where the numerator and denominator of the function are equal to each other. Once you have found the zeroes, you can then use them to find the asymptotes of the function. The asymptotes are the lines that the function approaches as it gets closer and closer to infinity. Once you have found the zeroes\n\nThere are a few different ways to solve a compound inequality, but one of the most common methods is to use a compound inequality solver. This is a tool that allows you to input two or more inequalities and then outputs the solution. There are many different types of compound inequality solvers, but they all work in basically the same way. One method for solving a compound inequality is to graph the inequalities on a coordinate plane. This can be done by using a graphing calculator\n\nThere are many ways to solve math problems online for free. One way is to use websites that provide step-by-step solutions to problems. Another way is to use online calculators to solve problems. Additionally, there are many free math tutorials and videos online that can be helpful in solving problems.\n\nSolving fractions can be a difficult task for students, but there are some tips and tricks that can help make the process easier. First, it is important to understand what a fraction is and how it works. A fraction is simply a number that represents a part of a whole. For example, if you have a pizza and you want to split it into equal pieces, each piece would be a fraction of the whole pizza. To solve a fraction, you need to find the amount of the\n\n## We solve all types of math problems\n\nSomeone in the reviews said it was a paid app, they were wrong. Just wanted to show that you don’t need a subscription to use this, it’s informative as it is. The premium provides extra explanations of you don't understand. Excellent math help, 10/10.", null, "Valentina Torres\nVery useful and helpful. Also, the explanations are crisp and easy to understand. The app is especially handy when it comes to problems I don’t understand. This is a good app to use but I think you should upgrade it in such a way that we can scan word problems too", null, "Quina Long\nSolving integral equations How to solve for x when x is an exponent How to solve point slope form Give me a math equation Solve any math problem calculator Math problems addition" ]

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[ "physics.flu-dyn\n\n# Title: Thermodynamically consistent semi-compressible fluids: a variational perspective\n\nAbstract: This paper presents (Lagrangian) variational formulations for single and multicomponent semi-compressible fluids with both reversible (entropy-conserving) and irreversible (entropy-generating) processes. Semi-compressible fluids are useful in describing \\textit{low-Mach} dynamics, since they are \\textit{soundproof}. These models find wide use in many areas of fluid dynamics, including both geophysical and astrophysical fluid dynamics. Specifically, the Boussinesq, anelastic and pseudoincompressible equations are developed through a unified treatment valid for arbitrary Riemannian manifolds, thermodynamic potentials and geopotentials. By design, these formulations obey the 1st and 2nd laws of thermodynamics, ensuring their thermodynamic consistency. This general approach extends and unifies existing work, and helps clarify the thermodynamics of semi-compressible fluids. To further this goal, evolution equations are presented for a wide range of thermodynamic variables: entropy density $s$, specific entropy $\\eta$, buoyancy $b$, temperature $T$, potential temperature $\\theta$ and a generic entropic variable $\\chi$; along with a general definition of buoyancy valid for all three semicompressible models and arbitrary geopotentials. Finally, the elliptic equation is developed for all three equation sets in the case of reversible dynamics, and for the Boussinesq/anelastic equations in the case of irreversible dynamics; and some discussion is given of the difficulty in formulating the elliptic equation for the pseudoincompressible equations with irreversible dynamics.\n Comments: 40 pages, 1 figure Subjects: Fluid Dynamics (physics.flu-dyn); Mathematical Physics (math-ph) DOI: 10.1088/1751-8121/ac1384 Cite as: arXiv:2102.08293 [physics.flu-dyn] (or arXiv:2102.08293v1 [physics.flu-dyn] for this version)\n\n## Submission history\n\nFrom: François Gay-Balmaz [view email]\n[v1] Tue, 16 Feb 2021 17:25:37 GMT (261kb,D)\n\nLink back to: arXiv, form interface, contact." ]

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[ "# Eigenvector for large matrices\n\n#### senorbum\n\n##### New Member\nWell, this is kind of a linear algebra question, but is used very often in statistics, so it seems applicable for this forum.\n\nMy question is, does anyone have any resources for computing the eigenvectors of large matrices given the eigenvalues? It seems pretty straightforward for small (2x2, 3x3) matrices, but I need to be able to compute it for very large matrices. I'm just looking for somewhere to get started. Most sites that I have seen just discuss how to calculate the small matrix version and then say it is possible to do for large matrices, but you have to resort to a numerical approach(which is fine, but I can't find a lot of information on numerical approaches, other than the power method which only yields the dominant vector).\n\nJoe\n\n#### TheEcologist\n\n##### Global Moderator\nWell, this is kind of a linear algebra question, but is used very often in statistics, so it seems applicable for this forum.\n\nMy question is, does anyone have any resources for computing the eigenvectors of large matrices given the eigenvalues? It seems pretty straightforward for small (2x2, 3x3) matrices, but I need to be able to compute it for very large matrices. I'm just looking for somewhere to get started. Most sites that I have seen just discuss how to calculate the small matrix version and then say it is possible to do for large matrices, but you have to resort to a numerical approach(which is fine, but I can't find a lot of information on numerical approaches, other than the power method which only yields the dominant vector).\n\nJoe\nComputation by \"hand\" gets exaustive quickly and I mean very quickly(unless your matrix contains mostly zero's), so if we're talking large matrices I'm assuming you want a computer program to do this?\n\nUse Matlab,\n\nOr even better Octave (the free Matlab). R (also free) can compute these too.\n\nOtherwise a free online book:\n\nhttp://ceee.rice.edu/Books/LA/index.html\n\n#### senorbum\n\n##### New Member\nComputation by \"hand\" gets exaustive quickly and I mean very quickly(unless your matrix contains mostly zero's), so if we're talking large matrices I'm assuming you want a computer program to do this?\n\nUse Matlab,\n\nOr even better Octave (the free Matlab). R (also free) can compute these too.\n\nOtherwise a free online book:\n\nhttp://ceee.rice.edu/Books/LA/index.html\n\nI'm developing a computer program to do this, so I effectively need to know how to do it. Actually, I just need the ability to calculate the determinant of a tridiagonal symmetric matrix, which should be significantly easier, I just can't find too much on it.\n\n#### senorbum\n\n##### New Member\nOk, so for a tridiagonal matrix, I have found a recursive algorithm from wikipedia.org as seen here http://en.wikipedia.org/wiki/Tridiagonal_matrix\n\nCan anyone explain what ann represents? Does that represent A at position n,n? Also, one quick question about determinants. Is the determinant of a 1x1 matrix the same as the value of that one term? This is important because that is what the stop case for this algorithm would be.\n\n#### TheEcologist\n\n##### Global Moderator\nOk, so for a tridiagonal matrix, I have found a recursive algorithm from wikipedia.org as seen here http://en.wikipedia.org/wiki/Tridiagonal_matrix\n\nCan anyone explain what ann represents? Does that represent A at position n,n? Also, one quick question about determinants. Is the determinant of a 1x1 matrix the same as the value of that one term? This is important because that is what the stop case for this algorithm would be.\nIf you want efficient algorithms to do this look in the source code of a open-source program like R (or one you are familiar with).\n\nA n,n would mean position n,n of matrix A\n\nAnd the det of a 1x1 matrix would be the single matrix element. It helps to think of determents in the geometric sense:\n\nhttp://en.wikipedia.org/wiki/Determinant\n\n#### senorbum\n\n##### New Member\nIf you want efficient algorithms to do this look in the source code of a open-source program like R (or one you are familiar with).\n\nA n,n would mean position n,n of matrix A\n\nAnd the det of a 1x1 matrix would be the single matrix element. It helps to think of determents in the geometric sense:\n\nhttp://en.wikipedia.org/wiki/Determinant\nThank you for the response. I was thinking of looking for source code of MATLAB then realized it wasn't open source. I have used R before, it's just been a while so hopefully I can understand it enough", null, ".\n\nI kinda thought that ann was A at n,n but wasn't entirely sure and didn't want to develop a program based on faulty math and then have to go back and change it.\n\nJoe\n\n#### TheEcologist\n\n##### Global Moderator\nThank you for the response. I was thinking of looking for source code of MATLAB then realized it wasn't open source. I have used R before, it's just been a while so hopefully I can understand it enough", null, ".\n\nI kinda thought that ann was A at n,n but wasn't entirely sure and didn't want to develop a program based on faulty math and then have to go back and change it.\n\nJoe\nAnytime and Goodluck! :tup:\n\n#### senorbum\n\n##### New Member\nAnytime and Goodluck! :tup:\nSo the real question is, what is the most efficient/effective way to calculate eigenvectors given the eigenvalues? Is the Jacobi algorithm plausable for 1000x1000 to 10000x10000+ matrices? I haven't seen direct calculation times for comparisons in any of the papers I've read.\n\n#### TheEcologist\n\n##### Global Moderator\nSo the real question is, what is the most efficient/effective way to calculate eigenvectors given the eigenvalues? Is the Jacobi algorithm plausable for 1000x1000 to 10000x10000+ matrices? I haven't seen direct calculation times for comparisons in any of the papers I've read.\nThat probably because calculation times vary from system to system, so it's difficult to compare calculation times.\n\nI usually build a code that requests and saves the system-time at start and end of a calculation. You can then check which function/algoritm is more efficient for your system.\n\n#### senorbum\n\n##### New Member\nThat probably because calculation times vary from system to system, so it's difficult to compare calculation times.\n\nI usually build a code that requests and saves the system-time at start and end of a calculation. You can then check which function/algoritm is more efficient for your system.\nWell, I don't want to implement a naturally poor algorithm. Generally there is a computational intensity level for these (i.e. O(n), O(n * log n) etc.). Is the QR algorithm the most widely used for eigenvector calculation (or EVD)?\n\n#### TheEcologist\n\n##### Global Moderator\nWell, I don't want to implement a naturally poor algorithm. Generally there is a computational intensity level for these (i.e. O(n), O(n * log n) etc.). Is the QR algorithm the most widely used for eigenvector calculation (or EVD)?\nNot quite sure, never worked on it myself.\n\nWhy dont you try (it might be useful):\n\nSmith, B. T, Boyle, J. M., Dongarra, J. J., Garbow, B. S., Ikebe,Y., Klema, V., and Moler, C. B. (1976). Matrix Eigensystems Routines – EISPACK Guide. Springer-Verlag Lecture Notes in Computer Science.\n\nand maybe:\n\nWilkinson, J. H. (1965) The Algebraic Eigenvalue Problem. Clarendon Press, Oxford.\n\n#### senorbum\n\n##### New Member\nNot quite sure, never worked on it myself.\n\nWhy dont you try (it might be useful):\n\nSmith, B. T, Boyle, J. M., Dongarra, J. J., Garbow, B. S., Ikebe,Y., Klema, V., and Moler, C. B. (1976). Matrix Eigensystems Routines – EISPACK Guide. Springer-Verlag Lecture Notes in Computer Science.\n\nand maybe:\n\nWilkinson, J. H. (1965) The Algebraic Eigenvalue Problem. Clarendon Press, Oxford.\n\nAm I overcomplicating this? If I already have the eigenvalues and (A-lamba*I)*v = 0 where A is the matrix, I is identity matrix, v = vector to solve for, is there a BLAS function that is equipped to solve this kind of problem? I feel like there might be, but haven't used to blas libraries enough to know. I'll research in the meantime to see if I can find one, but if anyone knows off the top of their head I'd much appreciate it.\n\nEdit: Is the STBSV what I'm looking for here? \"stbsv solves one of the systems of equations A*x = b, or A'*x = b, where b and x are n element vectors and A is an n by n unit, or non-unit, upper or lower triangular band matrix, with ( k + 1 ) diagonals.\nNo test for singularity or near-singularity is included in this routine. Such tests must be performed before calling this routine. \"\n\nSo if b = 0 at all points in the vector, wouldn't that technically work? The question becomes, is it just going to set each element in the result vector to 0. If not, it appears to be what I'm looking for.\n\nLast edited:" ]

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[ "Home | | Artificial Intelligence | | Computational Intelligence | | Artificial Intelligence | First-Order Predicate Logic(FOPL)\n\nFirst-Order Predicate Logic(FOPL)\n\nThe type of predicate calculus that we have been referring to is also called firstorder predicate logic (FOPL).\n\nFirst-Order Predicate Logic\n\nThe type of predicate calculus that we have been referring to is also called firstorder predicate logic (FOPL).\n\nA first-order logic is one in which the quantifiers and can be applied to objects or terms, but not to predicates or functions.\n\nSo we can define the syntax of FOPL as follows. First,we define a term: A constant is a term.\n\nA variable is a term. f(x1, x2, x3, . . . , xn) is a term if x1, x2, x3, . . . , xn are all terms.\n\nAnything that does not meet the above description cannot be a term.\n\nFor example, the following is not a term: x P(x). This kind of construction we call a sentence or a well-formed formula (wff), which is defined as follows.\n\nIn these definitions, P is a predicate, x1, x2, x3, . . . , xn are terms, and A,B are wff ’s. The following are the acceptable forms for wff ’s:\n\nP(x1, x2, x3, . . . , xn)\n\nA\n\nA B\n\nA B\n\nA→B\n\nA↔ B\n\n(  x)A\n\n( x)A\n\nAn atomic formula is a wff of the form P(x1, x2, x3, . . . , xn).\n\nHigher order logics exist in which quantifiers can be applied to predicates and functions, and where the following expression is an example of a wff:\n\n(  P)(  x)P(x)\n\nSoundness\n\nWe have seen that a logical system such as propositional logic consists of a syntax, a semantics, and a set of rules of deduction.\n\nA logical system also has a set of fundamental truths, which are known as axioms.\n\nThe axioms are the basic rules that are known to be true and from which all other theorems within the system can be proved.\n\nAn axiom of propositional logic, for example, is A→(B→A)\n\nA theorem of a logical system is a statement that can be proved by applying the rules of deduction to the axioms in the system.\n\nIf A is a theorem, then we write ├ A\n\nA logical system is described as being sound if every theorem is logically valid, or a tautology.\n\nIt can be proved by induction that both propositional logic and FOPL are sound.\n\nCompleteness\n\nA logical system is complete if every tautology is a theorem—in other words, if every valid statement in the logic can be proved by applying the rules of deduction to the axioms. Both propositional logic and FOPL are complete.\n\nDecidability\n\nA logical system is decidable if it is possible to produce an algorithm that will determine whether any wff is a theorem. In other words, if a logical system is decidable, then a computer can be used to determine whether logical expressions in that system are valid or not.\n\nWe can prove that propositional logic is decidable by using the fact that it is complete.\n\nWe can prove that a wff A is a theorem by showing that it is a tautology. To show if a wff is a tautology, we simply need to draw up a truth table for that wff and show that all the lines have true as the result. This can clearly be done algorithmically because we know that a truth table for n values has 2n lines and is therefore finite, for a finite number of variables.\n\nFOPL, on the other hand, is not decidable. This is due to the fact that it is not possible to develop an algorithm that will determine whether an arbitrary wff in FOPL is logically valid.\n\nMonotonicity\n\nA logical system is described as being monotonic if a valid proof in the system cannot be made invalid by adding additional premises or assumptions.\n\nIn other words, if we find that we can prove a conclusion C by applying rules of deduction to a premise B with assumptions A, then adding additional assumptions A and B will not stop us from being able to deduce C.\n\nMonotonicity of a logical system can be expressed as follows:\n\nIf we can prove {A, B} ├ C,\n\nthen we can also prove: {A, B, A_, B_} ├ C.\n\nIn other words, even adding contradictory assumptions does not stop us from making the proof in a monotonic system.\n\nIn fact, it turns out that adding contradictory assumptions allows us to prove anything, including invalid conclusions. This makes sense if we recall the line in the truth table for →, which shows that false → true. By adding a contradictory assumption, we make our assumptions false and can thus prove any conclusion.\n\nStudy Material, Lecturing Notes, Assignment, Reference, Wiki description explanation, brief detail\n\nRelated Topics" ]

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[ "# Measurement Errors worksheet\n\nPhysics Measurement Errors worksheet.\n\nYou can find MCQ (Multiple choice questions) related to Measurement errors in Physics in this page.\n\nYou can check yourself by choosing a best answer among the given alternatives.\n\nThe correct answer for the questions is given at the bottom of the page.\n\nPhysics Measurement Errors worksheet:\n\n1> The size of hydrogen is in the order of:\n\na> 10 -14 m\n\nb> 10 -10 m\n\nc>10 -19 m\n\nd>10 -24/ m\n\n2>The size of nucleus of an atom is of the order of:\n\na>10 -14 m\n\nb>10 -10 m\n\nc>10 -15 m\n\nd> 10 -24 m\n\n3> The ratio of nucleus to the volume os the atom is of the order of :\n\na>10 10 m\n\nb>10 -10 m\n\nc>10 -12 m\n\nd> 10 -18 m\n\n4>The radius of earth is of the order of:\n\na>10 10 m\n\nb>10 7 m\n\nc>10 15 m\n\nd>10 19 m\n\n5>The nucleus of an atom has a diameter 1/10000 times that of atomic diameter, then if density of ordinary matter is 10 3 kg/m 3 , the density of nuclear matter is of the order of:\n\na> 10 3 kg/m 3\n\nb> 10 7 kg/m 3\n\nc> 10 15 kg/m 3\n\nd> 10 -15 kg/m 3\n\n6> The diameter of red blood corpuscles is of the order of:\n\na>10 -3 m\n\nb>10 -5 m\n\nc>10 -7 m\n\nd>10 -9 m\n\n7> The time takes by sunlight to travel from sun to earth is of the order of:\n\na>1 s\n\nb>10 -1 s\n\nc>10 s\n\nd>10 3 s\n\n8>The time taken by sunlight to cross a window plane of 3mm thickness is:\n\na >10 -8 s\n\nb>10 -11 s\n\nc>10 -15 s\n\nd>10 0 s\n\n9>The number of significant figures in 0.314*10 ssuupp 4 /ssuupp m is:\n\na> 2\n\nb> 3\n\nc> 4\n\nd> 6\n\n10> The number of significant figures in 100 vibrations of a simple pendulum is:\n\na> 1\n\nb>2\n\nc> 3\n\nd> No significant figures involved.\n\n11> Of the following observations the most accurate is:\n\na> 4*10 3 kg\n\nb>4.0*10 3 kg\n\nc>4.0*10 2 kg\n\nd>4.0000*10 3 kg\n\n12> The value 0f 9.99m+0.069m according to idea of significant figures is:\n\na>10.059 m\n\nb>10.0 m\n\nc> 10.06 m\n\nd> 10.05 m\n\n13>The value of 9.99 m – 0.069 m according to idea of significant figure is :\n\na> 9.92 m\n\nb> 9.921 m\n\nc>9.91 m\n\nd> 9.9\n\n14> The volume of a sphere is 2.25 m 3 the volume of 50 such spheres according idea of significant figure is:\n\na> 112.50 m 3\n\nb> 112.5 m 3\n\nc> 112 m 3\n\nd> 11*10 1 m 3\n\n15> The random error can be eliminated by:\n\na>careful observation\n\nb> Eliminating the cause of error\n\nc> Taking arithmetic mean of a large of number of observations.\n\nd> Measuring the quantity with more than one device.\n\n16> The zero error of a vernier calipers is:\n\na> Constant error\n\nb> Systematic error\n\nc> Random error\n\nd> Gross error\n\n17> A student performs an experiment of a simple pendulum and measures time for 10 vibrations. If he measure the time for a 100 vibration the error in the measurement of time period will be reduced by a factor of:\n\na> 10\n\nb>90\n\nc>100\n\nd>1000\n\n18> A student performs an experiment and repeats the experiment 5 times. If he repeats the experiment 20 times , The error in the measurement will be reduced by a factor of:\n\na> 0\n\nb> 4\n\nc> 2\n\nd> 16\n\n19>The least count of an instrument is 0.01 cm taking all the precautions , The most probable error in the measurement can be:\n\na> 0.0005 cm\n\nb> 0.01 cm\n\nc> 0.05 cm\n\nd> 0.02 cm\n\n20>The length of a rod is (8.00 +- 0.5)cm. When two such rods are connected end to end ; The length of the combination may be correctly expressed as:\n\na> (16.00 +- 0.05) cm\n\nb> (16.00 +- 0.00) cm\n\nc> (16.00 +- 0.10) cm\n\nd> 16.1 cm\n\n21> The length of a rod is (8.00 +- 0.05) cm when two such rods are placed parallel to each other with one end coinciding, The difference between their other ends may be most correctly expressed as:\n\na> 0.00 +- 0.00\n\nb> 0.00 +- 0.05\n\nc> 0.00 +- 0.25\n\nd> 0.00 +- 0.10\n\n22> The time taken by a pendulum to complete 25 vibrations is 88.0 seconds then the time period of the pendulum up to proper significant figures is:\n\na>3.520 s\n\nb> 3.52 s\n\nc> 3.5 s\n\nd> 4 s\n\n23> Out of the following observed readings , the one which is most precise is:\n\na> 2025 gm\n\nb> 202.5 * 10 1 gm\n\nc> 20.25 * 10 gm\n\nd> 2.025 * 10 gm\n\n24> if y=a.b , the maximum percentage error in the measurement of “y” will be:\n\na>", null, "$\\left(\\dfrac{\\Delta a}{a}\\times 100 \\% \\right) \\div \\left(\\dfrac{\\Delta b}{b} \\times 100\\ \\%\\right)$\n\nb>", null, "$\\left(\\dfrac{\\Delta a}{a}\\times 100 \\%\\right) \\times \\left(\\dfrac{\\Delta b}{b} \\times 100 \\%\\right)$\n\nc>", null, "$\\left(\\dfrac{\\Delta a}{a}+ \\dfrac{\\Delta b}{b}\\right) \\times 100\\%$\n\nd>", null, "$\\left(\\dfrac{\\Delta a}{a}- \\dfrac{\\Delta b}{b}\\right) \\times 100\\%$\n\n25> if y=a/b , the maximum percentage error in measurement of “y” will be:\n\na>", null, "$\\left(\\dfrac{\\Delta a}{a}\\times 100 \\% \\right) \\div \\left(\\dfrac{\\Delta b}{b} \\times 100 \\%\\right)$\n\nb>", null, "$\\left(\\dfrac{\\Delta a}{a}\\times 100 \\%\\right) \\times \\left(\\dfrac{\\Delta b}{b} \\times 100 \\%\\right)$\n\nc>", null, "$\\left(\\dfrac{\\Delta a}{a}+\\dfrac{\\Delta b}{b}\\right) \\times 100\\%$\n\nd>", null, "$\\left(\\dfrac{\\Delta a}{a}- \\dfrac{\\Delta b}{b}\\right) \\times 100\\%$\n\n26> if y=a+b the maximum percentage error in the measurement of y will be:\n\na>", null, "$\\left(\\dfrac{\\Delta a}{a} + \\dfrac{\\Delta b}{b}\\right) \\times 100 \\%$\n\nb>", null, "$\\left(\\dfrac{\\Delta a}{a} - \\dfrac{\\Delta b}{b}\\right) \\times 100 \\%$\n\nc>", null, "$\\left(\\dfrac{\\Delta a}{a+b}+\\dfrac{\\Delta b}{a+b}\\right) \\times 100\\%$\n\nd>", null, "$\\left(\\dfrac{\\Delta a}{a+b}-\\dfrac{\\Delta b}{a+b}\\right) \\times 100\\%$\n\n27> If y=a-b , The maximum percentage error in the measurement of y will be:\n\na>", null, "$\\left(\\dfrac{\\Delta a}{a} + \\dfrac{\\Delta b}{b}\\right) \\times 100 \\%$\n\nb>", null, "$\\left(\\dfrac{\\Delta a}{a} - \\dfrac{\\Delta b}{b}\\right) \\times 100 \\%$\n\nc>", null, "$\\left(\\dfrac{\\Delta a}{a-b}+\\dfrac{\\Delta b}{a-b}\\right) \\times 100\\%$\n\nd>", null, "$\\left(\\dfrac{\\Delta a}{a-b}-\\dfrac{\\Delta b}{a-b}\\right) \\times 100\\%$\n\n28> If y=a n , The maximum percentage error in measurement of y is:\n\na>", null, "$n \\%$\n\nb>", null, "$n \\dfrac{\\Delta a}{a} \\times 100 \\%$\n\nc>", null, "$\\left(\\dfrac{\\Delta a}{a}\\right)^n \\times 100 \\%$\n\nd>", null, "$\\dfrac{1}{n}\\left(\\dfrac{\\Delta a}{a}\\right) \\times 100 \\%$\n\n29> If a physical quantity x is represented as x=M a L b T c , And the maximum percentage error in measurement in M L T are x y and z % respectively , The total percentage error in X is:\n\na> (xa+yb-zc)%\n\nb>(xa+yb+zc)%\n\nc> (xa-yb-zc)%\n\nd> (x+y+z)%\n\n30> The error in the measurement of the radius of a sphere is 1%, then the error in the measurement of it’s volume is:\na> 1%\n\nb> 3%\n\nc> 4 pi %\n\nd> 4/3 pi %\n\n31> The error in the measurement of the radius of a sphere is 0.3% , what is percentage error in measurement of it’s surface area?\n\na> 0.3%\n\nb> 0.6%\n\nc> 0.9%\n\nd> 4/3 pi (0.3) ^2 %\n\n32> The measured mass and volume of a body are 22.42 gm and 4.7 cm^3 , with possible errors 0.01 gm and 0.1cm^3 , the maximum error in density is about:\n\na> 0.2%\n\nb> 2%\n\nc> 5%\n\nd> 10%\n\n33> The length , breadth and thickness of a block is measured 50 cm , 2.0 cm and 1.0 cm. If accuracy in measurement of length is +-1 cm in breadth is +-0.1 cm and thickness is +-0.001 cm , Then what is the percentage error in the measurement of volume:\n\na> 8%\n\nb> 0.8%\n\nc> 10%\n\nd> 12.25%\n\n34> The least count of a stop watch is 0.5 sec, the time of 40 oscillations of a pendulum is measured to be 50 s. What is the maximum error in that measurement?\n\na> 1 %\n\nb> 2.5%\n\nc> 0.25%\n\nd> 10%\n\n35> The density of a material of a cube is calculated by measuring it’s mass and a side. What is the maximum percentage error in the density if the errors in measurement of a side and mass are 2%nand 5% respectively?\n\na> 13%\n\nb> 11%\n\nc> 7%\n\nd> 3%\n\n36> The heat generated in a circuit is given by Q=i^2Rt Joule , where i is current , R is resistance and t is time , If the percentage errors in measuring i , R and t are 3%, 2% and 1% respectively. The maximum error in measuring heat will be:\n\na> 9%\n\nb> 12%\n\nc> 6%\n\nd> 18%\n\n37> While measuring the acceleration due to gravity by a simple pendulum , a student makes a positive error of 1% in length of the pendulum and a negative error of 3% in the value of time period. The percentage error in the measurement of g by the relation g=4pi^2(l/t^2) will be:\n\na> 2%\n\nb>4%\n\nc>7%\n\nd>10%\n\n38> The percentage error in the measurement of mass and speed are 2% and 3% respectively ,. How much will be the maximum error in the estimation of kinetic energy ( 1/2 mv^2) obtained by measuring mass and speed?\n\na>11 %\n\nb> 8%\n\nc>5%\n\nd>4%\n\n39> An experiment measured quantities of a,b,c and X is calculated from the formula X= ab^2/c^3 . The percentage error in a,b,c are +-1% , +-3% , +-2% respectively. The percentage error in X can be:\n\na> +-13%\n\nb>+-7%\n\nc>+-4%\n\nd>+-1%\n\n40> The relative density of a material is found to by weighting it first in air and then in water by using formula:\n\nrelative density= weight in air/(weight in air – weight in water)\n\nIf the weight in air is 10.0+-0.1 gm and weight in water is 5.0+-0.1 gm then the maximum permissible percentage error in relative density is:\n\na> 1%\n\nb>2%\n\nc>3%\n\nd>5%\n\n1>b     2>a     3>c     4>b     5>c     6>b     7>d      8>b       9>b      10>d      11>d\n\n12>c      13>a      14>c      15>c      16>b      17>a      18>c      19>b      20>c      21>d\n\n22>b      23>d      24>c      25>c      26>c      27>c      28>b      29>b      30>b      31>b\n\n32>b      33>a      34>a      35>b       36>a      37>c      38>b      39>a      40>d\n\nRelated posts:\n\n1. Physics: Measurement worksheet Physics: Measurement worksheet. You can find MCQ(Multiple choice questions) or...\n2. The Chain Rule. Chain Rule is one of the Techniques of Differentiation. The...\n3. Derivative of simple algebraic or polynomial functions. The derivative and calculations on finding derivative of simple algebraic functions...\n4. Angle and it’s measurement. The configuration formed by initial arm , terminal arm and...\n5. The Power Rule. Power Rule is one of the Techniques of Differentiation. The..." ]

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[ "# sedgewick_maze_graph#\n\nsedgewick_maze_graph(create_using=None)[source]#\n\nReturn a small maze with a cycle.\n\nThis is the maze used in Sedgewick, 3rd Edition, Part 5, Graph Algorithms, Chapter 18, e.g. Figure 18.2 and following . Nodes are numbered 0,..,7\n\nParameters:\ncreate_usingNetworkX graph constructor, optional (default=nx.Graph)\n\nGraph type to create. If graph instance, then cleared before populated.\n\nReturns:\nGnetworkx Graph\n\nSmall maze with a cycle\n\nReferences\n\n\n\nFigure 18.2, Chapter 18, Graph Algorithms (3rd Ed), Sedgewick" ]

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[ "# Implicative Analysis for Multivariate Binary Data using an Imprecise Dirichlet Model\n\nAbstract\n\nBayesian implicative analysis was proposed for summarizing the association in a $2\\times 2$ contingency table in terms possibly, asymmetrical such as, \\eg, presence of feature $a$ implies, in general, presence of feature $b$'' ($a$ quasi-implies $b$'' in short). Here, we consider the multivariate version of this problem: having $n$ units which are classified according to $\\Qcard$ binary questions, we want to summarize the association between questions in terms of quasi-implications between features. We will first show how at a descriptive level the notion of implication can be weakened into that of quasi-implication. The inductive step assumes that the $n$ units are a sample from a $2^\\Qcard$-multinomial population. Uncertainty about the patterns' true frequencies is expressed by an imprecise Dirichlet model which yields upper and lower posterior probabilities for any quasi-implicative statement. This model is shown to have several advantages over the Bayesian models based on a single Dirichlet prior, especially whenever $2^\\Qcard$ is large and many patterns are thus unobserved by design.\n\nKeywords. Quasi-implication, logical model, measure of association, multivariate implicative index, Boolean methods, Bayesian inference, upper and lower probabilities.\n\nThe paper is available in the following formats:" ]

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[ "# Daniel Coats RAS\n\n### Daniel Coats RAS", null, "Daniel Coats went undrafted for the NFL in 2007.\n\nHe recorded a Relative Athletic Score of 4.69, out of a possible 10.0. RAS is a composite metric on a 0 to 10 scale based on the average of all of the percentile for each of the metrics the player completed either at the Combine or pro day.\n\nHe had a recorded height of 6025 that season, recorded as XYYZ where X is feet, YY is inches, and Z is eighths of an inch. That correlates to 6 feet, 2 and 5/8 of an inch or 74.625 inches, or 189.5475 centimeters. This correlates to a 2.11 score out of 10.0.\n\nHe recorded a weight of 257 in pounds, which is approximately 116 kilograms. This correlates to a 7.06 score out of 10.0.\n\nBased on his weight, he has a projected 40 yard dash time of 4.86. This is calculated by taking 0.00554 multiplied by his weight and then adding 3.433.\nAt the Combine, he recorded a 40 yard dash of 4.99 seconds. This was a difference of 0.13 seconds from his projected time. This forty time correlates to a 2.46 score out of 10.0.\n\nUsing Bill Barnwell’s calculation, this Combine 40 time gave him a Speed Score of 82.9.\n\nAt his pro day, he recorded a 40 yard dash of 4.78 seconds. Because he also recorded this metric at the Combine, his pro day did not count towards his RAS.\n\nUsing Bill Barnwell’s calculation, his Combine 40 time gave him a Speed Score of 1.54\n\nThe time traveled between the 20 and 40 yard lines is known as the Flying Twenty. As the distance is also known, we can calculate the player’s speed over that distance. The time he traveled the last twenty yards at the Combine was 2.12 seconds. Over 20 yards, we can calculate his speed in yards per second to 9.43. Taking into account the distance in feet (60 feet), we can calculate his speed in feet per second to 28.3. Breaking it down further, we can calculate his speed in inches per second to 339.62. Knowing the feet per second of 28.3, we can calculate the approximate miles per hour by multiplying that value by 0.681818 to give us a calculated MPH of 19.3 in the last 20 yards of his run.\n\nAt the Combine, he recorded a 20 yard split of 2.87 seconds. This correlates to a 3.35 score out of 10.0.\n\nWe can calculate the speed traveled over the second ten yards of the 40 yard dash easily, as the distance and time are both known. The time he traveled the second ten yards at the Combine was 1.14 seconds. Over 10 yards, we can calculate his speed in yards per second to 8.77. Taking into account the distance in feet (30 feet), we can calculate his speed in feet per second to 26.32. Breaking it down further, we can calculate his speed in inches per second to 315.79. Knowing the feet per second of 26.32, we can calculate the approximate miles per hour by multiplying that value by 0.681818 to give us a calculated MPH of 17.9 in the second ten yards of his run.\n\nAt the Combine, he recorded a 10 yard split of 1.73 seconds. This correlates to a 3.62 score out of 10.0 that when multiplied by ten roughly corresponds to percentile.\n\nThe time he traveled the first ten yards at the Combine was 1.73 seconds. Over 10 yards, we can calculate his speed in yards per second to 6.0. Taking into account the distance in feet (30 feet), we can calculate his speed in feet per second to 17.0. Breaking it down further, we can calculate his speed in inches per second to 208.0. Knowing the feet per second of 17.0, we can calculate the approximate miles per hour by multiplying that value by 0.681818 to give us a calculated MPH of 11.6 in the first ten yards of his run.\n\nAt the Combine, he recorded a bench press of 34 repetitions of 225 pounds. This correlates to a 10.0 score out of 10.0.\n\nAt the Combine, he recorded a vertical jump of 30.0 inches. This correlates to a 3.67 score out of 10.0.\n\nAt his pro day, he recorded a vertical jump of 36.5 seconds. Because he also recorded this metric at the Combine, his pro day did not count towards his RAS.\n\nAt the Combine, he recorded a broad jump of 903, which is recorded as FII or FFII . where F is feet and I is inches. This correlates to a 5.04 score out of 10.0 that when multiplied by ten roughly corresponds to percentile.\n\nAt his pro day, he recorded a broad jump of 908. Because he also recorded this metric at the Combine, his pro day did not count towards his RAS.\n\nAt the Combine, he recorded a 5-10-5 or 20 yard short shuttle of 4.43 seconds. This correlates to a 4.34 score out of 10.0.\n\nAt his pro day, he recorded a 5-10-5 or 20 yard short shuttle of 4.38 seconds. Because he also recorded this metric at the Combine, his pro day did not count towards his RAS.\n\nAt the Combine, he recorded a 3 cone L drill of 7.07 seconds. This correlates to a 7.84 score out of 10.0 that when multiplied by ten roughly corresponds to percentile.\n\nThis site uses Akismet to reduce spam. Learn how your comment data is processed." ]

[ null, "https://i0.wp.com/ras.football/wp-content/uploads/2019/12/daniel-coats-ras-7764.png", null ]

[ "# Hyperbolic Riemann Surface\n\nLet $X$ be a compact Riemann surface and $x\\in X$. Is $X - \\overline{D(x,r_x)}$ hyperbolic?\n\n• If you look at the tags it looks like theOP is interested in the existence of positive Green's function which is certainly true for many values of the radius of the disc that is removed. – Mohan Ramachandran May 3 '13 at 17:23\n• My goal is can say that there exists a Green's function on it. – James May 6 '13 at 11:25\n\n## 3 Answers\n\nIf a domain on a Riemann surface has a component of the frontier with at least two distinct points then it is potential theoretic hyperbolic. This result goes back to Osgood. Existence of green 's function is connected to existence of barrier at some frontier point. The existence of barriers is local. The solution of the local problem can be found in the book of Ahlfors-Sario titled Riemann Surfaces .\n\n• Any reference where I can find the result? – James May 6 '13 at 17:07\n\nHere's an answer from a different point of view then Henri's. It may happen that $X'$ is disconnected but in that case I'll just argue one component at a time. The Riemann surface $X'$ is noncompact, and if it is nonhyperbolic then by the Riemann mapping theorem it must be conformally equivalent (biholomorphic) to either $\\mathbb{C}$ or $\\mathbb{C}-$(point). Each of its one or two ends must therefore be a removable singularity. But that's impossible if $r_x > 0$ (justifying this impossibility probably involves a case analysis not unlike what Henri proposes).\n\nLet $X'=X\\setminus \\overline {D(x,r_x)}$.\n\nIf $X$ is already hyperbolic, then the answer is yes (there is no entire curves in $X'$).\n\nIf $X=\\mathbb C, \\mathbb C^*$, then $X'$ has no entire curves by Picard's little theorem, so it is hyperbolic.\n\nIf $X=\\mathbb C/\\Lambda$, then $\\pi_1(X')$ is non trivial, and different from $\\mathbb Z$ or $\\mathbb Z^2$, so $X'$ is hyperbolic. This works if $r_x$ is not too big. Else you could imagine that $X'$ becomes $\\mathbb C^*$.\n\nIf $X$ is $\\mathbb P^1$, then your surface is biholomorphic to $\\mathbb D$ (it is simply connected and strictly realized inside the complex plane).\n\nIn conclusion, $X'$ is always hyperbolic.\n\n• In the case where $X = \\mathbb{C} / \\Lambda$, nothing goes wrong when $r_x$ is too big, $X'$ will still be hyperbolic. It is possible as $r_x$ increases for the topological type of $X'$ to change (it could even become disconnected); but its hyperbolic nature will not change. – Lee Mosher May 3 '13 at 13:51\n• A priori, in the non-compact setting, it is not enough to exclude the existence of entire curves in order to establish Kobayashi hyperbolicity (this is a necessary but not a sufficient condition). But for Riemann surfaces, of course, it is necessary and sufficient to be uniformized by the disc (so that in this case it suffices to check non-existence of entire curves). – diverietti May 3 '13 at 15:46" ]

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[ "318185 (number)\n\n318,185 (three hundred eighteen thousand one hundred eighty-five) is an odd six-digits composite number following 318184 and preceding 318186. In scientific notation, it is written as 3.18185 × 105. The sum of its digits is 26. It has a total of 3 prime factors and 8 positive divisors. There are 218,160 positive integers (up to 318185) that are relatively prime to 318185.\n\nBasic properties\n\n• Is Prime? No\n• Number parity Odd\n• Number length 6\n• Sum of Digits 26\n• Digital Root 8\n\nName\n\nShort name 318 thousand 185 three hundred eighteen thousand one hundred eighty-five\n\nNotation\n\nScientific notation 3.18185 × 105 318.185 × 103\n\nPrime Factorization of 318185\n\nPrime Factorization 5 × 7 × 9091\n\nComposite number\nDistinct Factors Total Factors Radical ω(n) 3 Total number of distinct prime factors Ω(n) 3 Total number of prime factors rad(n) 318185 Product of the distinct prime numbers λ(n) -1 Returns the parity of Ω(n), such that λ(n) = (-1)Ω(n) μ(n) -1 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 318,185 is 5 × 7 × 9091. Since it has a total of 3 prime factors, 318,185 is a composite number.\n\nDivisors of 318185\n\n8 divisors\n\n Even divisors 0 8 4 4\nTotal Divisors Sum of Divisors Aliquot Sum τ(n) 8 Total number of the positive divisors of n σ(n) 436416 Sum of all the positive divisors of n s(n) 118231 Sum of the proper positive divisors of n A(n) 54552 Returns the sum of divisors (σ(n)) divided by the total number of divisors (τ(n)) G(n) 564.079 Returns the nth root of the product of n divisors H(n) 5.83269 Returns the total number of divisors (τ(n)) divided by the sum of the reciprocal of each divisors\n\nThe number 318,185 can be divided by 8 positive divisors (out of which 0 are even, and 8 are odd). The sum of these divisors (counting 318,185) is 436,416, the average is 54,552.\n\nOther Arithmetic Functions (n = 318185)\n\n1 φ(n) n\nEuler Totient Carmichael Lambda Prime Pi φ(n) 218160 Total number of positive integers not greater than n that are coprime to n λ(n) 18180 Smallest positive number such that aλ(n) ≡ 1 (mod n) for all a coprime to n π(n) ≈ 27391 Total number of primes less than or equal to n r2(n) 0 The number of ways n can be represented as the sum of 2 squares\n\nThere are 218,160 positive integers (less than 318,185) that are coprime with 318,185. And there are approximately 27,391 prime numbers less than or equal to 318,185.\n\nDivisibility of 318185\n\n m n mod m 2 3 4 5 6 7 8 9 1 2 1 0 5 0 1 8\n\nThe number 318,185 is divisible by 5 and 7.\n\nClassification of 318185\n\n• Arithmetic\n• Deficient\n\n• Polite\n\n• Square Free\n\nOther numbers\n\n• LucasCarmichael\n• Sphenic\n\nBase conversion (318185)\n\nBase System Value\n2 Binary 1001101101011101001\n3 Ternary 121011110122\n4 Quaternary 1031223221\n5 Quinary 40140220\n6 Senary 10453025\n8 Octal 1155351\n10 Decimal 318185\n12 Duodecimal 134175\n20 Vigesimal 1jf95\n36 Base36 6tih\n\nBasic calculations (n = 318185)\n\nMultiplication\n\nn×y\n n×2 636370 954555 1272740 1590925\n\nDivision\n\nn÷y\n n÷2 159092 106062 79546.2 63637\n\nExponentiation\n\nny\n n2 101241694225 32213588476981625 10249880649548398350625 3261358274476557129193615625\n\nNth Root\n\ny√n\n 2√n 564.079 68.2695 23.7503 12.6048\n\n318185 as geometric shapes\n\nCircle\n\n Diameter 636370 1.99922e+06 3.1806e+11\n\nSphere\n\n Volume 1.34936e+17 1.27224e+12 1.99922e+06\n\nSquare\n\nLength = n\n Perimeter 1.27274e+06 1.01242e+11 449982\n\nCube\n\nLength = n\n Surface area 6.0745e+11 3.22136e+16 551113\n\nEquilateral Triangle\n\nLength = n\n Perimeter 954555 4.38389e+10 275556\n\nTriangular Pyramid\n\nLength = n\n Surface area 1.75356e+11 3.79641e+15 259797\n\nCryptographic Hash Functions\n\nmd5 5aa9d204f516039da5fb1705d83298ac 32cded8e80c56f9d8b70abdaaea58932811af828 c8111cf36812b62ef76bf0573959a958ab7f7e1e4dc74fa1a55112dfd3fa3560 1d6584ee682729909a686ce61ae573df61942b2e3e9d11a688bebda1499eaf4cbac515b6f6025eaa354b37afb3e97616b73289b109512cf6453964d3dc11b28b d573de8cc79688b407bf2e898cc46cb48dd2f4ec" ]

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[ "# Adaptive subspace methods for high-dimensional variable selection\n\n• Adaptive Subspace Methoden für hoch-dimensionale Variablenselektion\n\nDue to recent advancements in fields such as information technology and genomics, nowadays one commonly faces high-dimensional data where the number of explanatory variables is possibly much larger than the number of observations. In such situations one is particularly interested in variable selection, meaning that one aims at identifying a sparse model with a relatively small subset of variables that fits and ideally explains the observed data well. This thesis deals with the variable selection problem in the setting of high-dimensional generalized linear models (GLMs). While many variable selection methods like the Lasso are based on solving convex $\\ell_1$-type relaxations of the original problem, a main motive of this work is the desire to provide solutions to generally NP-hard $\\ell_0$-regularized problems induced by model selection criteria such as the Extended Bayesian Information Criterion (EBIC). For this purpose, the Adaptive Subspace (AdaSub) method is proposed which is based on the idea of adaptively solving several low-dimensional sub-problems of the original high-dimensional problem. AdaSub is a stochastic algorithm which sequentially adapts the sampling probabilities of the individual variables based on their currently estimated \"importance\". It is shown that the updating scheme of AdaSub can be motivated in a Bayesian way and that the method \"converges correctly\" against the best model according to the employed criterion, provided that the so-called ordered importance property (OIP) is satisfied. Furthermore, the variable selection consistency of AdaSub is proved under suitable conditions. Since solving the sampled sub-problems can be computationally expensive for GLMs different than the normal linear model, \"greedy\" modifications of AdaSub are introduced which provide approximate solutions to the sub-problems. It is argued that BackAdaSub, a version of AdaSub based on Backward Stepwise Selection, may be used as an efficient surrogate algorithm. The \"correct convergence\" of BackAdaSub can be guaranteed under the modified ordered importance property (MOIP), which is a stronger condition than the original OIP. The performance of AdaSub and BackAdaSub in comparison to other prominent competitors such as the Lasso, the Adaptive Lasso, the SCAD and Stability Selection is investigated via various simulated and real data examples in the framework of linear and logistic regression models. Finally, a Metropolized version of AdaSub, called the MAdaSub algorithm, is proposed for sampling from posterior model distributions in the Bayesian variable selection context. MAdaSub is an adaptive Markov Chain Monte Carlo (MCMC) algorithm which sequentially adjusts the proposal distribution based on the information from the previously sampled models. It is shown that the MAdaSub algorithm is ergodic despite its continuing adaptation, i.e. \"in the limit\" it samples from the correct target distribution. Through simulated and real data examples it is demonstrated that MAdaSub can provide stable estimates of posterior marginal inclusion probabilities even for very high-dimensional and multimodal posterior model distributions." ]

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[ "Sin\n• 27 May 2022\n• Dark\nLight\n• PDF\n\n# Sin\n\n• Dark\nLight\n• PDF\n\nArticle Summary\n\nGiven an angle in radians, calculates its sine.\n\n### Function signature\n\n``````SIN(number: Number) => Number\n``````\nParameterDescription\nNumberAngle in radians to calculate sine for\n\n### Examples\n\nFormulaReturn value\n`SIN(0)``0`\n`SIN(PI() / 6)``0.5`\n`SIN(PI() / 2)``1`" ]

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[ "Message-ID: <1566100856.344841.1585878672245.JavaMail.j2ee-conf@bmc1-rhel-confprod1> Subject: Exported From Confluence MIME-Version: 1.0 Content-Type: multipart/related; boundary=\"----=_Part_344840_1637832999.1585878672244\" ------=_Part_344840_1637832999.1585878672244 Content-Type: text/html; charset=UTF-8 Content-Transfer-Encoding: quoted-printable Content-Location: file:///C:/exported.html Creating a SQL-based data mart for a custom view in the TrueSigh= t console\n\n# Creating a SQL-based data mart for a custom view in the TrueSight c= onsole\n\n=20\n=20\n=20\n=20\n\nUsing the Data marts wizard, you can build a custom data mart to retriev= e the data according to your requirements for a custom view in the TrueSigh= t console. You can configure the data mart based on the following options:<= /p>\n\n• Decide on the type of content to be displayed by the data mart\n• Identify the primary entity of the data mart\n• Apply required sorting options\n• Configure the filters to be displayed in the TrueSight console\n• Set the materialization type and schedule\n• Configure role-based access\n\nYou can either create a simple data mart or an advanced data mart (with = the required filters, output columns and custom role-based access restricti= ons).\n\n=20\n=20\n=20 =20\n=20\n=20\n=20\n=20\n=20\n\nThe following video (3:48) illustrates the process of creating a SQL-bas= ed custom data mart that can be used to build a custom view in the TrueSigh= t console.\n\n=20\n=20\n=20\n=20\n=20\n=20\n=20\n\n#### Steps for creatin= g a data mart for a custom view using the basic options\n\n=20 =20 =20 =20\n=20 =20 =20 =20 =20 =20", null, "=20", null, "Steps for creating a data mart for a cu= stom view using the advanced options\n=20 = =20 =20 =20 =20 =20\n=20 =20 =20 =20 =20 =20", null, "=20\n\n=20\n=20\n=20\n=20\n=20\n=20\n\n## Before you begin\n\nAccess the Data marts page. For more details, see Managing data marts for custom views, reports, and report te= mplates.\n\n## To add a data= mart for building a custom view in the TrueSight console\n\n1. On the Data marts list page, select Add > Data mart based on SQL.\n\nThe Data marts wizard is displayed= .\n\n2. On the Welcome page,= under the Purpose of data mart, select Build Capa= city View in TrueSight Console.\n\n3. Under the Content of data mart, select one of = the following options:\n\n• Summary information for a set of entities: Adds a data= mart that provides a summary information about the entities returned by th= e SQL query.\n• Time series details for a set of entities: Adds a data= mart that provides a time series details about the entities returned by th= e SQL query. When you select this option, columns returned by SQL query are= checked to ensure that one column contains Timestamp set as the Meaning. T= his is automatically detected when the query is executed or can be manually= set for the required column in the Columns page.\n4. Click Next.\n5. On the Configu= ration page, set values for the following properties.\n\n=20\n=20\n=20\n\nBasic\n\n=20\n\nBy default, the Basic properties are displayed on the Configuratio= n page. For viewing the Advanced properties, select the Advanced tab of this table.\n\n=20\n\nBasic properties\n\n=20\n=20 =20 =20 =20 =20 =20 =20 =20 =20 =20 =20 =20 = =20 =20 =20 =20 =20 =20 =20 =20 =20 =20 =20 =20 =20\nFieldDescription\nName\n\nType the required name for the = data mart. Only alpha numeric values are allowed.\n\nDescriptionType the required description for = the data mart.\nPrimary entities\n\nSelect any one of the following options:\n\n= =20\n=20\n• System: Performs a check on the columns extracted by t= he query to identify whether one or more columns contain System Ide= ntifier as the meaning.\n• =20\n• Business driver: Performs a check on the columns = extracted by the query to identify whether one or more columns contain Workload Identifier as the meaning.\n• =20\n• Domain: Performs a check on the columns extracted= by the query to identify whether one or more columns contain Appli= cation Identifier as the meaning.\n• =20\n• Other\n\n• =20\nSQL Query\n\nType SQL query to retrieve the required data = by selecting the required columns of a table.\n\n=20\nTip\n\n=20 =20\n=20\n\nTo display filter for the columns in the view in the TrueSight Console, = you can use a query that contains a filter macro. For more details about fi= lter macro, see  Using filter macros with SQL queries.\n\n=20\n=20\n=20\n=20\n\n=20\n\nTo view or configure the Advanced properties, click Advanced. You do not need to set or modify this prop= erty unless you want to modify the list of data marts on which this data ma= rt is dependent. This property is for advanced users and scenarios only.\n\n=20\n\n=20\n=20 =20 =20 =20 =20 =20 =20 =20 =20 =20 =20 =20 =20\nFieldDescription\nDependencies\nSelect the required dependencies from the Data marts table, and add them to the Selected dependenc= ies table using the >> button. In the Selected depende= ncies table, select the required dependencies, and add them to the=  Data marts table by using the << button.\n=20\n=20\n=20\n=20\n\n• Click Next.\n\n• On the Columns page, you can perform the any one of the following st= eps to modify the properties of output column or proceed to Step = 8\n\n• In the Output = Columns section, click the name of the required column to modify the meanin= g and associated properties of the columns.\n\nThe Editor of column with= name: columnName page is displayed.", null, "• If the Meaning is set to any option other than None= , set values for the following properties an= d click Apply.\n\n=20\n=20\n=20\n\nBasic\n\n=20\n\nBy default, the Basic properties are displayed on the Columns page. These are the most common properties and it is = acceptable to leave the default selections for each as is. For viewi= ng the Advanced properties, select the Advanced tab of thi= s table.\n\n=20\n\nBasic properties\n\n=20\n=20 =20 =20 =20 =20 =20 =20 =20 =20 =20 =20 =20 = =20 =20 =20 =20 =20\nFieldDescription\nLabelType the required label for the co= lumn.\nDescriptionType the required description for = the column.\n=20\n=20\n\n=20\n\nTo view or configure the Advanced properties, click Advanced. You do not need to set or modify these pro= perties unless you want to change the meaning of the column. These properti= es are for advanced users and scenarios only.\n\n=20\n\n=20\n=20 =20 =20 =20 =20 =20 =20 =20 =20 =20 =20 =20 =20 =20 =20 =20 =20 =20 =20 =20 =20 =20 =20 =20 =20 =20 =20 =20 =20 =20 =20 =20 =20 = =20 =20 =20 =20 =20 =20 =20 = =20 =20 =20 =20 =20 =20 =20 =20 =20 =20 =20 =20 =20 =20 =20 =20 =20 =20 =20 =20 =20 =20 =20 = =20 =20 =20 =20 =20 =20 =20 =20 =20 =20 =20 =20 =20 =20 =20 =20 =20 =20 =20 =20 =20 =20 =20\nMeaning optionDescription\nOutput column\nValueSelect this option if the output column contains= a value that is a statistic derived from a metric. Following options are d= isplayed when you select Value as the Meaning:=20\n=20\n• Related to metric: Select the required metric type to = which this value in the output column is related. After you select the requ= ired metric type, select required metric from list. If there is no metric w= ith which the output column can be associated, select None= .\n• =20\n• Unit of measure: When you select a metric in the Related to metric, the associated option is set in the Unit of measure. If the Related to metric is s= et to None, select the required option from the Fr= equently used drop-down list, or select Other and= type the required unit of measure in the box for the value in the output c= olumn.\n\nFor example, if the Related to metric is CPU Utilization, the Unit of measure is set to <= strong>%.\n• =20\nIdentifier\nSystem Identifier= Select this option to set the mean= ing as system identifier of the systems that are being evaluated. Following= options are displayed when you select System Identifier as the Meaning:=20\n=20\n• Distinct values: Select this option to provide distinc= t values as options in the filter that is available in the TrueSight Consol= e when this data mart is used.\n• =20\n• List of system types: Select this option to select the= required systems from the All types table and them to the= Filter on type table. The systems selected in the Filter on type table are used as options in the filter that is av= ailable in the TrueSight Console when this data mart is used.\n• =20\nWorkload IdentifierSelect this option to set the mean= ing as workload identifier of the systems or clusters that are being evalua= ted. Following options are displayed when you select Workload Identifier as= the Meaning:=20\n=20\n• Distinct values: Select this option to provide distinc= t values as options in the filter that is available in the TrueSight Consol= e when this data mart is used.\n• =20\n• List of business types: Select this option to select t= he required driver types from the All types table and them= to the Filter on type table. The driver types selected in= the Filter on type table are used as options in the filte= r that is available in the TrueSight Console when this data mart is used.=20\nDomain Identifier= Select this option to set the mean= ing as Domain identifier of the systems or clusters that are being evaluate= d. Following options are displayed when you select Domain Identifier as the= Meaning:=20\n=20\n• Distinct values: Select this option to provide distinc= t values as options in the filter that is available in the TrueSight Consol= e when this data mart is used.\n• =20\n• List of domain types: Select this option to select the= required domains from the All types table and them to the= Filter on type table. The domains selected in the Filter on type table are used as options in the filter that is av= ailable in the TrueSight Console when this data mart is used.\n• =20\nName\nSystem NameSelect this option to set the mean= ing as system name of the systems or clusters that are being evaluated. Whe= n you select System Name as the Meaning, = the Filter behavior field is displayed with the value set = as Distinct Values.\nWorkload Name=20 Select this option to set the mean= ing as workload name of the systems that are being evaluated. When you sele= ct Workload Name as the Meaning, the Filter behavior field is displayed with the value set as Distinct Values.\nDomain NameSelect this option to set the mean= ing as domain name of the systems that are being evaluated. When you select= Domain Name as the Meaning, the = Filter behavior field is displayed with the value set as D= istinct Values.\nType Identifier\nSystem Type IdentifierSelect this option to set the mean= ing as system type identifier of the systems or clusters that are being eva= luated. When you select System Type Identifie= r as the Meaning, the Filter behavior field is displayed with the value set as Distinct Values.\nWorkload Type IdentifierSelect this option to set the mean= ing as workload name of the systems that are being evaluated. When you sele= ct Workload Type Name as the Meaning, the= Filter behavior field is displayed with the value set as = Distinct Values.\nDomain Type IdentifierSelect this option to set the mean= ing as domain type identifier of the systems that are being evaluated. When= you select Domain Type Identifier as the Meaning<= /strong>, the Filter behavior field is displayed with the = value set as Distinct Values.\nType Name\n\nSystem Type Name\n\nSelect this option to set the mean= ing as system type name of the systems or clusters that are being evaluated= . When you select System Type Name a= s the Meaning, the Filter behavior field = is displayed with the value set as Distinct Values.\nWorkload Type NameSelect this option to set the mean= ing as workload type name of the systems that are being evaluated. When you= select Workload Type Name as the Meaning= , the Filter behavior field is displayed with the value se= t as Distinct Values.\nDomain Type Name<= /td>=20 Select this option to set the mean= ing as domain type name of the systems that are being evaluated. When you s= elect Domain Type Name as the Meaning, th= e Filter behavior field is displayed with the value set as= Distinct Values.\nOther\nTimestampSelect this option if the value in= the column indicates a timestamp for any activity. No additional options a= re displayed when you select Timestamp as the Meaning.\nThreshold Good=20 Select this option if the value in= the column indicates good threshold value for any metric. No additional op= tions are displayed when you select Threshold Good as the Meaning. =20\nThreshold Warning= Select this option if the value in= the column indicates good threshold value for any metric. No additional op= tions are displayed when you select Threshold Warning as the Meaning.<= /td>=20\n=20\n=20\n=20\n=20\n\n• If the Meaning is set to Noneset= values for the following properties and click Apply<= /strong>.\n\n=\n\n=20\n=20\n=20\n\nBasic\n\n=20\n\nBy default, the Basic properties are displayed on the Columns page.  These are the most common pro= perties and it is acceptable to leave the default selections for each as is= . For viewing the Advanced properties, select the Advanced tab of this table.\n\n=20\n\nBasic properties\n\n=20\n=20 =20 =20 =20 =20 =20 =20 =20 =20 =20 =20 =20 =20 =20 =20 =20 = =20 =20 =20 =20 =20 =20 =20 =20\nFieldDescription\nMeaningSelect the required option to be s= et as the meaning. For more details about the available options, see Advanced tab in the previous table.\nLabelType the required label for the co= lumn.\nDescriptionType the required description for = the column.\nUnit of measure=20\n\nSelect the required unit of mea= sure from the Frequently used drop-down list or select Other and type the required unit of measure in the box for th= e value in the output column. This field is displayed only when the co= lumn contains numeric value.\n\n=20\n=20\n\n=20\n\nTo view or configure the Advanced properties, click Advanced. You do not need to set or modify thi= s property unless you want to change the filter behavior. This property is = for advanced users and scenarios only.\n\n=20\n\n=20\n=20 =20 =20 =20 =20 =20 =20 =20 =20 =20 =20 =20\nFieldDescription\nFiler behavior=20 The value is set to Distin= ct Values.\n=20\n=20\n=20\n=20\n\n• In the Sortin= g Options section, select the required columns from the Source table, and add them to the Selection table using the = >>DESC or&= nbsp;>>ASC= button to sort the data in the descending order or ascending order respect= ively. In the Selection table, select the required column,= and add them to the Source table by using the << button. You can al= so change the order of display of columns added to the Selection table by u= sing the Up or Down arrow buttons.\n\n• In the Ti= me period label box, type the required label for data mart that ar= e not sensible to time filter. This field is displayed only when there are = no time filter macros defined for the data mart or there no columns that ha= ve the meaning set as Timestamp. When the data mart is not sensible to time= filter, the the time filter for the selected custom view built using this = data mart is disabled and this label is displayed in the selected view.\n\n=\n• In the Detect= ed filters section, perform the following steps:\n\n1. Click the name of the filter.\n2. In the Editor of the filter name page, edit the Label = and Description of the filter.\n3. Click Apply.\n\n• On the Material= izer page, specify the following details:\n\nMaterialization option Description\n\nAlways see fresh data\n\nSelect this option to load fresh dat= a every time you access the associated view in the TrueSight console= . Note: Every time the SQL= query runs, it fetches data directly from the database. Hence, a complex q= uery might require more time to run and the performance might be impacted.<= /span>\n\nCreate materialized table\n\nSelect this option to load materialized data = in the associated view in the TrueSight console.\n\nThis option is usefu= l when you have a complex SQL query and you want to see or use the fetched = data quickly. The data mart is materialized and the data is stored in a new= table, which is shown whenever you choose to Preview or view the data.\n\n=\n\nSpecify the following settings:\n\n=20\n=20\n=20\n=20\n\nBasic\n\n=20\n\nFrom the Materializer task list, select the task to be = used for materializing the data mart. You can select from the available out= -of-the-box tasks or create a new one. For more information about creating = a materializer task, see Configuring the Data Mart Materializer task= .\n\n=20\n\n=20\n\n=20\n=20 =20 =20 =20 =20 =20 =20 =20 =20 =20 =20 =20 =20 =20 =20 =20 =20 =20 =20 =20\n\nTablespaceEnter the name of the tablespace where the gener= ated table is materialized. If no value is specified, the default tablespac= e of the product is used.\nMaterialize validity\n\nEnter a number and select hour, day, or week to specify the time= validity of the materialized data mart.\n\nThe data mart is not re-mate= rialized if you run the associated materializer task again while the time s= pecified in this field is still valid.\n\nFor example: Consider data mart A with Materialize validity of 6 hours. The ass= ociated materializer task for this data mart runs daily at 5 A.M.\n\nIf = the associated materializer task runs again at 10 A.M., the data mart A is = not re-materialized because the time specified in the Materializer validity= field is still valid.\n\nIndex count\n\nSelect the number of blocks of indexes that y= ou need. You can choose maximum 4 blocks.\n\nFor each index block, in th= e corresponding Index # columns field, select the indexes = from the Source list and move them to the Selectio= n list by using the", null, "button.\n\nAnd= , enter the Index # tablespace name.\n\n=20\n=20\n=20\n=20\n\n• (Advanced only): To specify access permissions to DB users = and DB roles to the materialized table in the database, configure the follo= wing advanced settings:\n\nField Description\nGrant to DB users\nSelect the database users from the Sourc= e table, and add them to the Selection table usin= g the", null, "button.\nGrant to DB roles\nSelect the required database roles from the Source table, and add them to the Selection t= able using the  button.\nGrant to user specified ro= les Type the required user specified (= defined) roles separated by semi-colon to give them permissions to use this= data mart.\n• Click Finish.\n\nThe new data mart is added in t= he Data marts list page and can be used to build a view in the TrueSight co= nsole.\n\n=20\n=20\n=20\n=20\n=20\n=20\n=20\n=20\n=20" ]

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[ "MySQL 5.7 Reference Manual  /  Functions and Operators  /  Type Conversion in Expression Evaluation\n\n## 12.3 Type Conversion in Expression Evaluation\n\nWhen an operator is used with operands of different types, type conversion occurs to make the operands compatible. Some conversions occur implicitly. For example, MySQL automatically converts strings to numbers as necessary, and vice versa.\n\n``````mysql> SELECT 1+'1';\n-> 2\nmysql> SELECT CONCAT(2,' test');\n-> '2 test'``````\n\nIt is also possible to convert a number to a string explicitly using the `CAST()` function. Conversion occurs implicitly with the `CONCAT()` function because it expects string arguments.\n\n``````mysql> SELECT 38.8, CAST(38.8 AS CHAR);\n-> 38.8, '38.8'\nmysql> SELECT 38.8, CONCAT(38.8);\n-> 38.8, '38.8'``````\n\nSee later in this section for information about the character set of implicit number-to-string conversions, and for modified rules that apply to `CREATE TABLE ... SELECT` statements.\n\nThe following rules describe how conversion occurs for comparison operations:\n\n• If one or both arguments are `NULL`, the result of the comparison is `NULL`, except for the `NULL`-safe `<=>` equality comparison operator. For ```NULL <=> NULL```, the result is true. No conversion is needed.\n\n• If both arguments in a comparison operation are strings, they are compared as strings.\n\n• If both arguments are integers, they are compared as integers.\n\n• Hexadecimal values are treated as binary strings if not compared to a number.\n\n• If one of the arguments is a `TIMESTAMP` or `DATETIME` column and the other argument is a constant, the constant is converted to a timestamp before the comparison is performed. This is done to be more ODBC-friendly. This is not done for the arguments to `IN()`. To be safe, always use complete datetime, date, or time strings when doing comparisons. For example, to achieve best results when using `BETWEEN` with date or time values, use `CAST()` to explicitly convert the values to the desired data type.\n\nA single-row subquery from a table or tables is not considered a constant. For example, if a subquery returns an integer to be compared to a `DATETIME` value, the comparison is done as two integers. The integer is not converted to a temporal value. To compare the operands as `DATETIME` values, use `CAST()` to explicitly convert the subquery value to `DATETIME`.\n\n• If one of the arguments is a decimal value, comparison depends on the other argument. The arguments are compared as decimal values if the other argument is a decimal or integer value, or as floating-point values if the other argument is a floating-point value.\n\n• In all other cases, the arguments are compared as floating-point (real) numbers. For example, a comparison of string and numeric operands takes place as a comparison of floating-point numbers.\n\nFor information about conversion of values from one temporal type to another, see Section 11.2.8, “Conversion Between Date and Time Types”.\n\nComparison of JSON values takes place at two levels. The first level of comparison is based on the JSON types of the compared values. If the types differ, the comparison result is determined solely by which type has higher precedence. If the two values have the same JSON type, a second level of comparison occurs using type-specific rules. For comparison of JSON and non-JSON values, the non-JSON value is converted to JSON and the values compared as JSON values. For details, see Comparison and Ordering of JSON Values.\n\nThe following examples illustrate conversion of strings to numbers for comparison operations:\n\n``````mysql> SELECT 1 > '6x';\n-> 0\nmysql> SELECT 7 > '6x';\n-> 1\nmysql> SELECT 0 > 'x6';\n-> 0\nmysql> SELECT 0 = 'x6';\n-> 1``````\n\nFor comparisons of a string column with a number, MySQL cannot use an index on the column to look up the value quickly. If `str_col` is an indexed string column, the index cannot be used when performing the lookup in the following statement:\n\n``SELECT * FROM tbl_name WHERE str_col=1;``\n\nThe reason for this is that there are many different strings that may convert to the value `1`, such as `'1'`, `' 1'`, or `'1a'`.\n\nComparisons between floating-point numbers and large values of `INTEGER` type are approximate because the integer is converted to double-precision floating point before comparison, which is not capable of representing all 64-bit integers exactly. For example, the integer value 253 + 1 is not representable as a float, and will be rounded to 253 or 253 + 2 before a float comparison, depending on the platform.\n\nTo illustrate, only the first of the following comparisons compares equal values, but both comparisons return true (1):\n\n``````mysql> SELECT '9223372036854775807' = 9223372036854775807;\n-> 1\nmysql> SELECT '9223372036854775807' = 9223372036854775806;\n-> 1``````\n\nWhen conversions from string to floating-point and from integer to floating-point occur, they do not necessarily occur the same way. The integer may be converted to floating-point by the CPU, whereas the string is converted digit by digit in an operation that involves floating-point multiplications. Also, results can be affected by factors such as computer architecture or the compiler version or optimization level. One way to avoid such problems is to use `CAST()` so that a value is not converted implicitly to a float-point number:\n\n``````mysql> SELECT CAST('9223372036854775807' AS UNSIGNED) = 9223372036854775806;\n-> 0``````\n\nThe server includes `dtoa`, a conversion library that provides the basis for improved conversion between string or `DECIMAL` values and approximate-value (`FLOAT`/`DOUBLE`) numbers:\n\n• Consistent conversion results across platforms, which eliminates, for example, Unix versus Windows conversion differences.\n\n• Accurate representation of values in cases where results previously did not provide sufficient precision, such as for values close to IEEE limits.\n\n• Conversion of numbers to string format with the best possible precision. The precision of `dtoa` is always the same or better than that of the standard C library functions.\n\nBecause the conversions produced by this library differ in some cases from non-`dtoa` results, the potential exists for incompatibilities in applications that rely on previous results. For example, applications that depend on a specific exact result from previous conversions might need adjustment to accommodate additional precision.\n\nThe `dtoa` library provides conversions with the following properties. `D` represents a value with a `DECIMAL` or string representation, and `F` represents a floating-point number in native binary (IEEE) format.\n\n• `F` -> `D` conversion is done with the best possible precision, returning `D` as the shortest string that yields `F` when read back in and rounded to the nearest value in native binary format as specified by IEEE.\n\n• `D` -> `F` conversion is done such that `F` is the nearest native binary number to the input decimal string `D`.\n\nThese properties imply that `F` -> `D` -> `F` conversions are lossless unless `F` is `-inf`, `+inf`, or `NaN`. The latter values are not supported because the SQL standard defines them as invalid values for `FLOAT` or `DOUBLE`.\n\nFor `D` -> `F` -> `D` conversions, a sufficient condition for losslessness is that `D` uses 15 or fewer digits of precision, is not a denormal value, `-inf`, `+inf`, or `NaN`. In some cases, the conversion is lossless even if `D` has more than 15 digits of precision, but this is not always the case.\n\nImplicit conversion of a numeric or temporal value to string produces a value that has a character set and collation determined by the `character_set_connection` and `collation_connection` system variables. (These variables commonly are set with `SET NAMES`. For information about connection character sets, see Section 10.4, “Connection Character Sets and Collations”.)\n\nThis means that such a conversion results in a character (nonbinary) string (a `CHAR`, `VARCHAR`, or `LONGTEXT` value), except in the case that the connection character set is set to `binary`. In that case, the conversion result is a binary string (a `BINARY`, `VARBINARY`, or `LONGBLOB` value).\n\nFor integer expressions, the preceding remarks about expression evaluation apply somewhat differently for expression assignment; for example, in a statement such as this:\n\n``CREATE TABLE t SELECT integer_expr;``\n\nIn this case, the table in the column resulting from the expression has type `INT` or `BIGINT` depending on the length of the integer expression. If the maximum length of the expression does not fit in an `INT`, `BIGINT` is used instead. The length is taken from the `max_length` value of the `SELECT` result set metadata (see C API Data Structures). This means that you can force a `BIGINT` rather than `INT` by use of a sufficiently long expression:\n\n``CREATE TABLE t SELECT 000000000000000000000;``" ]

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[ "# Migratory Birds HackerRank Solution in C, C++, Java, Python\n\nYou have been asked to help study the population of birds migrating across the continent. Each type of bird you are interested in will be identified by an integer value. Each time a particular kind of bird is spotted, its id number will be added to your array of sightings. You would like to be able to find out which type of bird is most common given a list of sightings. Your task is to print the type number of that bird and if two or more types of birds are equally common, choose the type with the smallest ID number.\n\nFor example, assume your bird sightings are of types arr=[1,1,2,2,3]. There are two each of types 1 and 2, and one sighting of type 3. Pick the lower of the two types seen twice: type .\n\nFunction Description\n\nComplete the migratoryBirds function in the editor below. It should return the lowest type number of the most frequently sighted bird.\n\nmigratoryBirds has the following parameter(s):\n\n• arr: an array of integers representing types of birds sighted\n\nInput Format\n\nThe first line contains an integer denoting n, the number of birds sighted and reported in the array arr.\n\nThe second line describes  as  space-separated integers representing the type numbers of each bird sighted.\n\nConstraints\n\n• 5<=n<=2*10^5\n• It is guaranteed that each type is 1, 2, 3, 4, or 5.\n\nOutput Format\n\nPrint the type number of the most common bird; if two or more types of birds are equally common, choose the type with the smallest ID number.\n\nSample Input 0\n\n```6\n\n1 4 4 4 5 3```\n\nSample Output 0\n\n`4`\n\nExplanation 0\n\nThe different types of birds occur in the following frequencies:\n\n• Type 1:1 bird\n• Type 2:0  birds\n• Type 3:1  bird\n• Type 4:3  birds\n• Type 5:1  bird\n\nThe type number that occurs at the highest frequency is type 4, so we print 4 as our answer.\n\n## Migratory Birds HackerRank Solution in C\n\n```#include <math.h>\n#include <stdio.h>\n#include <string.h>\n#include <stdlib.h>\n#include <assert.h>\n#include <limits.h>\n#include <stdbool.h>\n\nint main(){\nint n;\nint i;\nint j;\nint k;\nint max;\nint sth;\nmax = 0;\nj = 1;\nscanf(\"%d\",&n);\nsth = 0;\nint *types = malloc(sizeof(int) * n);\nfor(int types_i = 0; types_i < n; types_i++){\nscanf(\"%d\",&types[types_i]);\n}\ni = 0;\nwhile (j <= 5 )\n{\nk = 0;\ni = 0;\nwhile (i < n)\n{\nif (types[i] == j)\nk++;\ni++;\n}\nif (k > max)\n{\nsth = j;\nmax = k;\n}\nj++;\n}\nprintf(\"%d\", sth);\nreturn 0;\n}\n\n```\n\n## Migratory Birds HackerRank Solution in C++\n\n```#include <bits/stdc++.h>\n\nusing namespace std;\n\nconst int maxN = 1e5+10;\nint N,A;\n\nint main()\n{\ncin >> N;\nfor (int i=1,x; i <= N; i++) cin >> x, A[x]++;\nint ans = 1;\nfor (int i=2; i <= 5; i++)\nif (A[i] > A[ans]) ans = i;\ncout << ans;\n}\n\nMigratory Birds HackerRank Solution in Java\nimport java.io.*;\nimport java.util.*;\nimport java.text.*;\nimport java.math.*;\nimport java.util.regex.*;\n\npublic class Solution {\n\npublic static void main(String[] args) {\nScanner in = new Scanner(System.in);\nint n = in.nextInt();\nint[] types = new int[n];\nint[] count = new int;\nint max=0;\nfor(int types_i=0; types_i < n; types_i++){\ntypes[types_i] = in.nextInt();\ncount[types[types_i]]++;\nif(count[types[types_i]]>max)\nmax=count[types[types_i]];\n}\nfor(int i=0;i<count.length;i++)\n{\nif(max==count[i])\n{\nSystem.out.println(i);\nbreak;\n}\n\n}\n\n}\n}```\n\n## Migratory Birds HackerRank Solution in Python\n\n```#!/bin/python\n\nimport sys\n\nn = int(raw_input().strip())\nl =list( map(int, raw_input().strip().split(' ')))\nl.sort()\nans=l\ncount=1\nmax=1\nfor i in range(1,n):\nif l[i]==l[i-1]:\ncount+=1\nelse:\ncount=1\nif count>max:\nmax=count\nans=l[i]\nprint ans```\n\n## Migratory Birds HackerRank Solution in C#\n\n```using System;\nusing System.Collections.Generic;\nusing System.IO;\nusing System.Linq;\nclass Solution {\n\nstatic void Main(String[] args) {\nint[] types = new int[n];\n\nfor(int i = 0; i < n; i++){\ntypes[i] = int.Parse(types_temp[i]);\n}\n\nList<int> counters = new List<int>() {0, 0, 0, 0, 0};\n\nint max = 0, maxType = 0;\n\nforeach(int type in types){\ncounters[type - 1]++;\n\nif(counters[type - 1] > max){\nmax = counters[type - 1];\nmaxType = type - 1;\n}\nelse if(counters[type - 1] == max){\nif(maxType > type - 1){\nmaxType = type - 1;\n}\n}\n}\n\nConsole.WriteLine((maxType + 1).ToString());" ]

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[ "# Linear combinations of sine and cosine\n\nIf you take a linear combination of the cosine and sine function, then the result is again a sinusoid, but with a new amplitude and phase shift. $$a \\cos(\\theta) + b \\sin(\\theta) = A \\cos(\\theta + \\theta_0)$$ This is one of those facts which I've always felt should be more obvious to me than it is. I can't help feeling that I might be missing out on some good \"conceptual reason\" for this to be true. I did a bit of thinking about it today and I came up with the following explanation which I'm not particularly satisfied with. I feel like there should be some perspective on this which renders the fact obvious.\n\nMultiplying the polar equation $r = 2a \\cos(\\theta) + 2b \\sin(\\theta)$ by $r$ yields $x^2+y^2 = 2a x + 2by$. Rearranging terms and adding $a^2 + b^2$ to both sides gives $x^2 - 2ax + a^2 +y^2 -2by + b^2 = a^2 + b^2$ which is identical to $$(x-a)^2 + (y-b)^2 = a^2 + b^2$$ which we recognize as the equation for the circle with centre at $(a,b)$ and passing through the origin. This hints that $r = 2a \\cos(\\theta) + 2b \\sin(\\theta)$ should define a parametrization of this circle and, indeed, this holds. This circle can be rotated about the origin so as to have its centre on the $x$-axis. Such a circle will have the simpler equation $$r = D \\cos(\\theta)$$ where $D=2(a^2+b^2)$ is the diameter of the circle. In polar coordinates, performing this rotation amounts to introducing a phase $\\theta_0$. In fact $\\theta_0$ is an argument for the point $(a,b)$.\n\nNote I'm not really looking for rigorous proofs of this fact, more like interpretations, really.\n\nEvery sinusoid (of unit angular frequency) is the real part of $Ae^{i\\theta}$ for some complex number $A$. But $A_1e^{i\\theta}+A_2e^{i\\theta}=(A_1+A_2)e^{i\\theta}$, so the sum of two sinusoids is again a sinusoid!\n\n• I was confused until I noticed $A$ is allowed to be complex. This is nice, thanks! – Mike F Mar 28 '14 at 7:34\n\n# Geometric\n\nHere is a geometric illustration:", null, "Start with two circles around the origin $O$, one with radius $OA=a$, the other with radius $OB=b$. Choose $A$ so its angle is $\\theta$ (marked in cyan). Then projecting that onto the $x$ axis will give $OC=a\\cos\\theta$. Construct a line perpendicular to $OA$ and intersect it with the other circle to obtain $B$. Make sure to choose the right point of intersection: $B$ should be $90°$ behind $A$ in terms of angle, since $\\sin$ is $90°$ begind $\\cos$. Projecting $B$ onto the $x$ axis gives $OS=b\\sin\\theta$. Now you could add these projections, but you could also add the original vectors $OA$ and $OB$ to obtain $OD$ and then project the result $D$ to get $OE=a\\cos\\theta+b\\sin\\theta$.\n\nNow as you rotate $A$, $B$ and $D$ will rotate along with it. So you can see that as $D$ moves at the same speed but on a larger circle, its projection $E$ will follow a cosine function so you have $OE=d\\cos(\\theta+\\theta_0)$. You can use the Pythagorean theorem to find that $d^2=a^2+b^2$. You can also see that angle $\\theta_0$, marked in orange and negative in my case. It's the angle between ray $OA$ and $OD$, and is defined by $\\tan\\theta_0=-\\frac ba$ unless I got a knot in my thoughts with the sign. As usual when using a tangens to compute an angle from a fraction, make sure you end up in the correct quadrant, perhaps use an atan2 function if available.\n\n# Complex numbers\n\nSince others answers brought up complex numbers, here is what you get if you do all of this in complex numbers:\n\n\\begin{align*} A &= ae^{i\\theta}=a\\cos\\theta+ai\\sin\\theta \\\\ B &= -bie^{i\\theta}=b\\sin\\theta-bi\\cos\\theta \\\\ C &= \\operatorname{Re}A = a\\cos\\theta \\\\ S &= \\operatorname{Re}B = b\\sin\\theta \\\\ D &= A+B = (a-bi)e^{i\\theta} = (de^{i\\theta_0})e^{i\\theta} = de^{i(\\theta+\\theta_0)}\\\\ E &= \\operatorname{Re}D = a\\cos\\theta+b\\sin\\theta = d\\cos(\\theta+\\theta_0) \\end{align*}\n\nSo the core aspect here is the conversion from the Cartesian coordinates $a-bi$ to the polar coordinates $de^{i\\theta_0}$. Which again can be characterized by $d^2=a^2+b^2$ and $\\tan\\theta_0=-\\frac ba$. To avoid confusion: $a,b,d,\\theta_0$ are all rational numbers here.\n\nThe complex representation (or rotational matrix representation) gives a nice way of visualizing without actually saying we are using complex numbers. Cosine and sine are position of a point that started at the x axis and was rotated by angle $\\theta$. $a\\cos\\theta$ is the segment on the $x$ axis at distance $a$, while $b\\sin\\theta$ has to start with a phase shift of quarter circle. Adding them means fixing the second circle to the point of the first circle.\n\nBecause both rotate by the same angle, we can just add (a,0)+(0,-b) in unrotated frame and rotate both by $\\theta$. The resulting $x$ segment is your cosine with a phase shift.", null, "This method also works if the LHS already contains phase shifts.\n\nPerhaps a better way:\n\nStart with the sum of angles formula for cosine. $$A \\cos(\\theta + \\theta_0) = A \\cos\\theta \\cos\\theta_0 - A \\sin\\theta \\sin\\theta_0$$\n\nObserve that $\\frac{A \\sin\\theta_0}{A \\cos\\theta_0} = \\tan\\theta_0$, which can be any real number as $\\theta_0$ ranges over $(-\\pi/2, \\pi/2)$. In fact, it can be $-b/a$: $\\frac{A \\sin\\theta_0}{A \\cos\\theta_0} = \\frac{-b}{a}$, so that $$A \\cos(\\theta + \\theta_0) =a \\cos \\theta + b \\sin \\theta$$" ]

[ null, "https://i.stack.imgur.com/dLpew.png", null, "https://i.stack.imgur.com/FA3a1.png", null ]

[ "# Array find() method with example in JavaScript\n\nJavaScript find() method: Here, we are going to learn about the find() method of array in JavaScript.\nSubmitted by IncludeHelp, on March 02, 2019\n\n## JavaScript find() method\n\nfind() method is used to get the first element from an array which passes the given test (condition).\n\nSyntax:\n\n``` array.find(function, [value]);\n```\n\nParameters: A function name and an optional value to be tested with all elements.\n\nReturn value: A first element which passes the test.\n\nExample:\n\n``` //function for positive value\nfunction getPositive(n){\nreturn n>=0;\n}\n\nInput:\nvar numbers = [-30, -10, 20, 20, 30, 0, 40];\n\nFunction call:\nvar item = numbers.find(getPositive);\n\nOutput:\n20\n```\n\nJavaScript code to find first positive number from an array using Array.find() method\n\n```<html>\n<title>JavaScipt Example</title>\n\n<body>\n<script>\n//function for positive value\nfunction getPositive(n){\nreturn n>=0;\n}\n\nvar numbers = [-30, -10, 20, 20, 30, 0, 40];\n//find first positive number\nvar item = numbers.find(getPositive);\ndocument.write(\"first positive number is \" + item + \"<br>\");\n</script>\n</body>\n</html>\n```\n\nOutput\n\n```first positive number is 20\n```\n\nLanguages: » C » C++ » C++ STL » Java » Data Structure » C#.Net » Android » Kotlin » SQL\nWeb Technologies: » PHP » Python » JavaScript » CSS » Ajax » Node.js » Web programming/HTML\nSolved programs: » C » C++ » DS » Java » C#\nAptitude que. & ans.: » C » C++ » Java » DBMS\nInterview que. & ans.: » C » Embedded C » Java » SEO » HR\nCS Subjects: » CS Basics » O.S. » Networks » DBMS » Embedded Systems » Cloud Computing\n» Machine learning » CS Organizations » Linux » DOS\nMore: » Articles » Puzzles » News/Updates" ]

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[ "User Real IP - 34.207.247.69\n```Array\n(\n => Array\n(\n => 182.68.68.92\n)\n\n => Array\n(\n => 101.0.41.201\n)\n\n => Array\n(\n => 43.225.98.123\n)\n\n => Array\n(\n => 2.58.194.139\n)\n\n => Array\n(\n => 46.119.197.104\n)\n\n => Array\n(\n => 45.249.8.93\n)\n\n => Array\n(\n => 103.12.135.72\n)\n\n => Array\n(\n => 157.35.243.216\n)\n\n => Array\n(\n => 209.107.214.176\n)\n\n => Array\n(\n => 5.181.233.166\n)\n\n => Array\n(\n => 106.201.10.100\n)\n\n => Array\n(\n => 36.90.55.39\n)\n\n => Array\n(\n => 119.154.138.47\n)\n\n => Array\n(\n => 51.91.31.157\n)\n\n => Array\n(\n => 182.182.65.216\n)\n\n => Array\n(\n => 157.35.252.63\n)\n\n => Array\n(\n => 14.142.34.163\n)\n\n => Array\n(\n => 178.62.43.135\n)\n\n => Array\n(\n => 43.248.152.148\n)\n\n => Array\n(\n => 222.252.104.114\n)\n\n => Array\n(\n => 209.107.214.168\n)\n\n => Array\n(\n => 103.99.199.250\n)\n\n => Array\n(\n => 178.62.72.160\n)\n\n => Array\n(\n => 27.6.1.170\n)\n\n => Array\n(\n => 182.69.249.219\n)\n\n => Array\n(\n => 110.93.228.86\n)\n\n => Array\n(\n => 72.255.1.98\n)\n\n => Array\n(\n => 182.73.111.98\n)\n\n => Array\n(\n => 45.116.117.11\n)\n\n => Array\n(\n => 122.15.78.189\n)\n\n => Array\n(\n => 14.167.188.234\n)\n\n => Array\n(\n => 223.190.4.202\n)\n\n => Array\n(\n => 202.173.125.19\n)\n\n => Array\n(\n => 103.255.5.32\n)\n\n => Array\n(\n => 39.37.145.103\n)\n\n => Array\n(\n => 140.213.26.249\n)\n\n => Array\n(\n => 45.118.166.85\n)\n\n => Array\n(\n => 102.166.138.255\n)\n\n => Array\n(\n => 77.111.246.234\n)\n\n => Array\n(\n => 45.63.6.196\n)\n\n => Array\n(\n => 103.250.147.115\n)\n\n => Array\n(\n => 223.185.30.99\n)\n\n => Array\n(\n => 103.122.168.108\n)\n\n => Array\n(\n => 123.136.203.21\n)\n\n => Array\n(\n => 171.229.243.63\n)\n\n => Array\n(\n => 153.149.98.149\n)\n\n => Array\n(\n => 223.238.93.15\n)\n\n => Array\n(\n => 178.62.113.166\n)\n\n => Array\n(\n => 101.162.0.153\n)\n\n => Array\n(\n => 121.200.62.114\n)\n\n => Array\n(\n => 14.248.77.252\n)\n\n => Array\n(\n => 95.142.117.29\n)\n\n => Array\n(\n => 150.129.60.107\n)\n\n => Array\n(\n => 94.205.243.22\n)\n\n => Array\n(\n => 115.42.71.143\n)\n\n => Array\n(\n => 117.217.195.59\n)\n\n => Array\n(\n => 182.77.112.56\n)\n\n => Array\n(\n => 182.77.112.108\n)\n\n => Array\n(\n => 41.80.69.10\n)\n\n => Array\n(\n => 117.5.222.121\n)\n\n => Array\n(\n => 103.11.0.38\n)\n\n => Array\n(\n => 202.173.127.140\n)\n\n => Array\n(\n => 49.249.249.50\n)\n\n => Array\n(\n => 116.72.198.211\n)\n\n => Array\n(\n => 223.230.54.53\n)\n\n => Array\n(\n => 102.69.228.74\n)\n\n => Array\n(\n => 39.37.251.89\n)\n\n => Array\n(\n => 39.53.246.141\n)\n\n => Array\n(\n => 39.57.182.72\n)\n\n => Array\n(\n => 209.58.130.210\n)\n\n => Array\n(\n => 104.131.75.86\n)\n\n => Array\n(\n => 106.212.131.255\n)\n\n => Array\n(\n => 106.212.132.127\n)\n\n => Array\n(\n => 223.190.4.60\n)\n\n => Array\n(\n => 103.252.116.252\n)\n\n => Array\n(\n => 103.76.55.182\n)\n\n => Array\n(\n => 45.118.166.70\n)\n\n => Array\n(\n => 103.93.174.215\n)\n\n => Array\n(\n => 5.62.62.142\n)\n\n => Array\n(\n => 182.179.158.156\n)\n\n => Array\n(\n => 39.57.255.12\n)\n\n => Array\n(\n => 39.37.178.37\n)\n\n => Array\n(\n => 182.180.165.211\n)\n\n => Array\n(\n => 119.153.135.17\n)\n\n => Array\n(\n => 72.255.15.244\n)\n\n => Array\n(\n => 139.180.166.181\n)\n\n => Array\n(\n => 70.119.147.111\n)\n\n => Array\n(\n => 106.210.40.83\n)\n\n => Array\n(\n => 14.190.70.91\n)\n\n => Array\n(\n => 202.125.156.82\n)\n\n => Array\n(\n => 115.42.68.38\n)\n\n => Array\n(\n => 102.167.13.108\n)\n\n => Array\n(\n => 117.217.192.130\n)\n\n => Array\n(\n => 205.185.223.156\n)\n\n => Array\n(\n => 171.224.180.29\n)\n\n => Array\n(\n => 45.127.45.68\n)\n\n => Array\n(\n => 195.206.183.232\n)\n\n => Array\n(\n => 49.32.52.115\n)\n\n => Array\n(\n => 49.207.49.223\n)\n\n => Array\n(\n => 45.63.29.61\n)\n\n => Array\n(\n => 103.245.193.214\n)\n\n => Array\n(\n => 39.40.236.69\n)\n\n => Array\n(\n => 62.80.162.111\n)\n\n => Array\n(\n => 45.116.232.56\n)\n\n => Array\n(\n => 45.118.166.91\n)\n\n => Array\n(\n => 180.92.230.234\n)\n\n => Array\n(\n => 157.40.57.160\n)\n\n => Array\n(\n => 110.38.38.130\n)\n\n => Array\n(\n => 72.255.57.183\n)\n\n => Array\n(\n => 182.68.81.85\n)\n\n => Array\n(\n => 39.57.202.122\n)\n\n => Array\n(\n => 119.152.154.36\n)\n\n => Array\n(\n => 5.62.62.141\n)\n\n => Array\n(\n => 119.155.54.232\n)\n\n => Array\n(\n => 39.37.141.22\n)\n\n => Array\n(\n => 183.87.12.225\n)\n\n => Array\n(\n => 107.170.127.117\n)\n\n => Array\n(\n => 125.63.124.49\n)\n\n => Array\n(\n => 39.42.191.3\n)\n\n => Array\n(\n => 116.74.24.72\n)\n\n => Array\n(\n => 46.101.89.227\n)\n\n => Array\n(\n => 202.173.125.247\n)\n\n => Array\n(\n => 39.42.184.254\n)\n\n => Array\n(\n => 115.186.165.132\n)\n\n => Array\n(\n => 39.57.206.126\n)\n\n => Array\n(\n => 103.245.13.145\n)\n\n => Array\n(\n => 202.175.246.43\n)\n\n => Array\n(\n => 192.140.152.150\n)\n\n => Array\n(\n => 202.88.250.103\n)\n\n => Array\n(\n => 103.248.94.207\n)\n\n => Array\n(\n => 77.73.66.101\n)\n\n => Array\n(\n => 104.131.66.8\n)\n\n => Array\n(\n => 113.186.161.97\n)\n\n => Array\n(\n => 222.254.5.7\n)\n\n => Array\n(\n => 223.233.67.247\n)\n\n => Array\n(\n => 171.249.116.146\n)\n\n => Array\n(\n => 47.30.209.71\n)\n\n => Array\n(\n => 202.134.13.130\n)\n\n => Array\n(\n => 27.6.135.7\n)\n\n => Array\n(\n => 107.170.186.79\n)\n\n => Array\n(\n => 103.212.89.171\n)\n\n => Array\n(\n => 117.197.9.77\n)\n\n => Array\n(\n => 122.176.206.233\n)\n\n => Array\n(\n => 192.227.253.222\n)\n\n => Array\n(\n => 182.188.224.119\n)\n\n => Array\n(\n => 14.248.70.74\n)\n\n => Array\n(\n => 42.118.219.169\n)\n\n => Array\n(\n => 110.39.146.170\n)\n\n => Array\n(\n => 119.160.66.143\n)\n\n => Array\n(\n => 103.248.95.130\n)\n\n => Array\n(\n => 27.63.152.208\n)\n\n => Array\n(\n => 49.207.114.96\n)\n\n => Array\n(\n => 102.166.23.214\n)\n\n => Array\n(\n => 175.107.254.73\n)\n\n => Array\n(\n => 103.10.227.214\n)\n\n => Array\n(\n => 202.143.115.89\n)\n\n => Array\n(\n => 110.93.227.187\n)\n\n => Array\n(\n => 103.140.31.60\n)\n\n => Array\n(\n => 110.37.231.46\n)\n\n => Array\n(\n => 39.36.99.238\n)\n\n => Array\n(\n => 157.37.140.26\n)\n\n => Array\n(\n => 43.246.202.226\n)\n\n => Array\n(\n => 137.97.8.143\n)\n\n => Array\n(\n => 182.65.52.242\n)\n\n => Array\n(\n => 115.42.69.62\n)\n\n => Array\n(\n => 14.143.254.58\n)\n\n => Array\n(\n => 223.179.143.236\n)\n\n => Array\n(\n => 223.179.143.249\n)\n\n => Array\n(\n => 103.143.7.54\n)\n\n => Array\n(\n => 223.179.139.106\n)\n\n => Array\n(\n => 39.40.219.90\n)\n\n => Array\n(\n => 45.115.141.231\n)\n\n => Array\n(\n => 120.29.100.33\n)\n\n => Array\n(\n => 112.196.132.5\n)\n\n => Array\n(\n => 202.163.123.153\n)\n\n => Array\n(\n => 5.62.58.146\n)\n\n => Array\n(\n => 39.53.216.113\n)\n\n => Array\n(\n => 42.111.160.73\n)\n\n => Array\n(\n => 107.182.231.213\n)\n\n => Array\n(\n => 119.82.94.120\n)\n\n => Array\n(\n => 178.62.34.82\n)\n\n => Array\n(\n => 203.122.6.18\n)\n\n => Array\n(\n => 157.42.38.251\n)\n\n => Array\n(\n => 45.112.68.222\n)\n\n => Array\n(\n => 49.206.212.122\n)\n\n => Array\n(\n => 104.236.70.228\n)\n\n => Array\n(\n => 42.111.34.243\n)\n\n => Array\n(\n => 84.241.19.186\n)\n\n => Array\n(\n => 89.187.180.207\n)\n\n => Array\n(\n => 104.243.212.118\n)\n\n => Array\n(\n => 104.236.55.136\n)\n\n => Array\n(\n => 106.201.16.163\n)\n\n => Array\n(\n => 46.101.40.25\n)\n\n => Array\n(\n => 45.118.166.94\n)\n\n => Array\n(\n => 49.36.128.102\n)\n\n => Array\n(\n => 14.142.193.58\n)\n\n => Array\n(\n => 212.79.124.176\n)\n\n => Array\n(\n => 45.32.191.194\n)\n\n => Array\n(\n => 105.112.107.46\n)\n\n => Array\n(\n => 106.201.14.8\n)\n\n => Array\n(\n => 110.93.240.65\n)\n\n => Array\n(\n => 27.96.95.177\n)\n\n => Array\n(\n => 45.41.134.35\n)\n\n => Array\n(\n => 180.151.13.110\n)\n\n => Array\n(\n => 101.53.242.89\n)\n\n => Array\n(\n => 115.186.3.110\n)\n\n => Array\n(\n => 171.49.185.242\n)\n\n => Array\n(\n => 115.42.70.24\n)\n\n => Array\n(\n => 45.128.188.43\n)\n\n => Array\n(\n => 103.140.129.63\n)\n\n => Array\n(\n => 101.50.113.147\n)\n\n => Array\n(\n => 103.66.73.30\n)\n\n => Array\n(\n => 117.247.193.169\n)\n\n => Array\n(\n => 120.29.100.94\n)\n\n => Array\n(\n => 42.109.154.39\n)\n\n => Array\n(\n => 122.173.155.150\n)\n\n => Array\n(\n => 45.115.104.53\n)\n\n => Array\n(\n => 116.74.29.84\n)\n\n => Array\n(\n => 101.50.125.34\n)\n\n => Array\n(\n => 45.118.166.80\n)\n\n => Array\n(\n => 91.236.184.27\n)\n\n => Array\n(\n => 113.167.185.120\n)\n\n => Array\n(\n => 27.97.66.222\n)\n\n => Array\n(\n => 43.247.41.117\n)\n\n => Array\n(\n => 23.229.16.227\n)\n\n => Array\n(\n => 14.248.79.209\n)\n\n => Array\n(\n => 117.5.194.26\n)\n\n => Array\n(\n => 117.217.205.41\n)\n\n => Array\n(\n => 114.79.169.99\n)\n\n => Array\n(\n => 103.55.60.97\n)\n\n => Array\n(\n => 182.75.89.210\n)\n\n => Array\n(\n => 77.73.66.109\n)\n\n => Array\n(\n => 182.77.126.139\n)\n\n => Array\n(\n => 14.248.77.166\n)\n\n => Array\n(\n => 157.35.224.133\n)\n\n => Array\n(\n => 183.83.38.27\n)\n\n => Array\n(\n => 182.68.4.77\n)\n\n => Array\n(\n => 122.177.130.234\n)\n\n => Array\n(\n => 103.24.99.99\n)\n\n => Array\n(\n => 103.91.127.66\n)\n\n => Array\n(\n => 41.90.34.240\n)\n\n => Array\n(\n => 49.205.77.102\n)\n\n => Array\n(\n => 103.248.94.142\n)\n\n => Array\n(\n => 104.143.92.170\n)\n\n => Array\n(\n => 219.91.157.114\n)\n\n => Array\n(\n => 223.190.88.22\n)\n\n => Array\n(\n => 223.190.86.232\n)\n\n => Array\n(\n => 39.41.172.80\n)\n\n => Array\n(\n => 124.107.206.5\n)\n\n => Array\n(\n => 139.167.180.224\n)\n\n => Array\n(\n => 93.76.64.248\n)\n\n => Array\n(\n => 65.216.227.119\n)\n\n => Array\n(\n => 223.190.119.141\n)\n\n => Array\n(\n => 110.93.237.179\n)\n\n => Array\n(\n => 41.90.7.85\n)\n\n => Array\n(\n => 103.100.6.26\n)\n\n => Array\n(\n => 104.140.83.13\n)\n\n => Array\n(\n => 223.190.119.133\n)\n\n => Array\n(\n => 119.152.150.87\n)\n\n => Array\n(\n => 103.125.130.147\n)\n\n => Array\n(\n => 27.6.5.52\n)\n\n => Array\n(\n => 103.98.188.26\n)\n\n => Array\n(\n => 39.35.121.81\n)\n\n => Array\n(\n => 74.119.146.182\n)\n\n => Array\n(\n => 5.181.233.162\n)\n\n => Array\n(\n => 157.39.18.60\n)\n\n => Array\n(\n => 1.187.252.25\n)\n\n => Array\n(\n => 39.42.145.59\n)\n\n => Array\n(\n => 39.35.39.198\n)\n\n => Array\n(\n => 49.36.128.214\n)\n\n => Array\n(\n => 182.190.20.56\n)\n\n => Array\n(\n => 122.180.249.189\n)\n\n => Array\n(\n => 117.217.203.107\n)\n\n => Array\n(\n => 103.70.82.241\n)\n\n => Array\n(\n => 45.118.166.68\n)\n\n => Array\n(\n => 122.180.168.39\n)\n\n => Array\n(\n => 149.28.67.254\n)\n\n => Array\n(\n => 223.233.73.8\n)\n\n => Array\n(\n => 122.167.140.0\n)\n\n => Array\n(\n => 95.158.51.55\n)\n\n => Array\n(\n => 27.96.95.134\n)\n\n => Array\n(\n => 49.206.214.53\n)\n\n => Array\n(\n => 212.103.49.92\n)\n\n => Array\n(\n => 122.177.115.101\n)\n\n => Array\n(\n => 171.50.187.124\n)\n\n => Array\n(\n => 122.164.55.107\n)\n\n => Array\n(\n => 98.114.217.204\n)\n\n => Array\n(\n => 106.215.10.54\n)\n\n => Array\n(\n => 115.42.68.28\n)\n\n => Array\n(\n => 104.194.220.87\n)\n\n => Array\n(\n => 103.137.84.170\n)\n\n => Array\n(\n => 61.16.142.110\n)\n\n => Array\n(\n => 212.103.49.85\n)\n\n => Array\n(\n => 39.53.248.162\n)\n\n => Array\n(\n => 203.122.40.214\n)\n\n => Array\n(\n => 117.217.198.72\n)\n\n => Array\n(\n => 115.186.191.203\n)\n\n => Array\n(\n => 120.29.100.199\n)\n\n => Array\n(\n => 45.151.237.24\n)\n\n => Array\n(\n => 223.190.125.232\n)\n\n => Array\n(\n => 41.80.151.17\n)\n\n => Array\n(\n => 23.111.188.5\n)\n\n => Array\n(\n => 223.190.125.216\n)\n\n => Array\n(\n => 103.217.133.119\n)\n\n => Array\n(\n => 103.198.173.132\n)\n\n => Array\n(\n => 47.31.155.89\n)\n\n => Array\n(\n => 223.190.20.253\n)\n\n => Array\n(\n => 104.131.92.125\n)\n\n => Array\n(\n => 223.190.19.152\n)\n\n => Array\n(\n => 103.245.193.191\n)\n\n => Array\n(\n => 106.215.58.255\n)\n\n => Array\n(\n => 119.82.83.238\n)\n\n => Array\n(\n => 106.212.128.138\n)\n\n => Array\n(\n => 139.167.237.36\n)\n\n => Array\n(\n => 222.124.40.250\n)\n\n => Array\n(\n => 134.56.185.169\n)\n\n => Array\n(\n => 54.255.226.31\n)\n\n => Array\n(\n => 137.97.162.31\n)\n\n => Array\n(\n => 95.185.21.191\n)\n\n => Array\n(\n => 171.61.168.151\n)\n\n => Array\n(\n => 137.97.184.4\n)\n\n => Array\n(\n => 106.203.151.202\n)\n\n => Array\n(\n => 39.37.137.0\n)\n\n => Array\n(\n => 45.118.166.66\n)\n\n => Array\n(\n => 14.248.105.100\n)\n\n => Array\n(\n => 106.215.61.185\n)\n\n => Array\n(\n => 202.83.57.179\n)\n\n => Array\n(\n => 89.187.182.176\n)\n\n => Array\n(\n => 49.249.232.198\n)\n\n => Array\n(\n => 132.154.95.236\n)\n\n => Array\n(\n => 223.233.83.230\n)\n\n => Array\n(\n => 183.83.153.14\n)\n\n => Array\n(\n => 125.63.72.210\n)\n\n => Array\n(\n => 207.174.202.11\n)\n\n => Array\n(\n => 119.95.88.59\n)\n\n => Array\n(\n => 122.170.14.150\n)\n\n => Array\n(\n => 45.118.166.75\n)\n\n => Array\n(\n => 103.12.135.37\n)\n\n => Array\n(\n => 49.207.120.225\n)\n\n => Array\n(\n => 182.64.195.207\n)\n\n => Array\n(\n => 103.99.37.16\n)\n\n => Array\n(\n => 46.150.104.221\n)\n\n => Array\n(\n => 104.236.195.147\n)\n\n => Array\n(\n => 103.104.192.43\n)\n\n => Array\n(\n => 24.242.159.118\n)\n\n => Array\n(\n => 39.42.179.143\n)\n\n => Array\n(\n => 111.93.58.131\n)\n\n => Array\n(\n => 193.176.84.127\n)\n\n => Array\n(\n => 209.58.142.218\n)\n\n => Array\n(\n => 69.243.152.129\n)\n\n => Array\n(\n => 117.97.131.249\n)\n\n => Array\n(\n => 103.230.180.89\n)\n\n => Array\n(\n => 106.212.170.192\n)\n\n => Array\n(\n => 171.224.180.95\n)\n\n => Array\n(\n => 158.222.11.87\n)\n\n => Array\n(\n => 119.155.60.246\n)\n\n => Array\n(\n => 41.90.43.129\n)\n\n => Array\n(\n => 185.183.104.170\n)\n\n => Array\n(\n => 14.248.67.65\n)\n\n => Array\n(\n => 117.217.205.82\n)\n\n => Array\n(\n => 111.88.7.209\n)\n\n => Array\n(\n => 49.36.132.244\n)\n\n => Array\n(\n => 171.48.40.2\n)\n\n => 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Array\n(\n => 106.197.31.240\n)\n\n => Array\n(\n => 122.168.179.251\n)\n\n => Array\n(\n => 39.37.167.126\n)\n\n => Array\n(\n => 171.48.8.115\n)\n\n => Array\n(\n => 157.44.152.14\n)\n\n => Array\n(\n => 103.77.43.219\n)\n\n => Array\n(\n => 122.161.49.38\n)\n\n => Array\n(\n => 122.161.52.83\n)\n\n => Array\n(\n => 122.173.108.210\n)\n\n => Array\n(\n => 60.254.109.92\n)\n\n => Array\n(\n => 103.57.85.75\n)\n\n => Array\n(\n => 106.0.58.36\n)\n\n => Array\n(\n => 122.161.49.212\n)\n\n => Array\n(\n => 27.255.182.159\n)\n\n => Array\n(\n => 116.75.230.159\n)\n\n => Array\n(\n => 122.173.152.133\n)\n\n => Array\n(\n => 129.0.79.247\n)\n\n => Array\n(\n => 223.228.163.44\n)\n\n => Array\n(\n => 103.168.78.82\n)\n\n => Array\n(\n => 39.59.67.124\n)\n\n => Array\n(\n => 182.69.19.120\n)\n\n => Array\n(\n => 196.202.236.195\n)\n\n => Array\n(\n => 137.59.225.206\n)\n\n => Array\n(\n => 143.110.209.194\n)\n\n => Array\n(\n => 117.201.233.91\n)\n\n => Array\n(\n => 37.120.150.107\n)\n\n => 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Array\n(\n => 47.31.139.139\n)\n\n => Array\n(\n => 210.2.140.18\n)\n\n => Array\n(\n => 106.210.33.219\n)\n\n => Array\n(\n => 175.107.203.34\n)\n\n => Array\n(\n => 146.196.32.144\n)\n\n => Array\n(\n => 103.12.133.121\n)\n\n => Array\n(\n => 103.59.208.182\n)\n\n => Array\n(\n => 157.37.190.232\n)\n\n => Array\n(\n => 106.195.35.201\n)\n\n => Array\n(\n => 27.122.14.83\n)\n\n => Array\n(\n => 194.193.44.5\n)\n\n => Array\n(\n => 5.62.43.245\n)\n\n => Array\n(\n => 103.53.80.50\n)\n\n => Array\n(\n => 47.29.142.233\n)\n\n => Array\n(\n => 154.6.20.63\n)\n\n => Array\n(\n => 173.245.203.128\n)\n\n => Array\n(\n => 103.77.43.231\n)\n\n => Array\n(\n => 5.107.166.235\n)\n\n => Array\n(\n => 106.212.44.123\n)\n\n => Array\n(\n => 157.41.60.93\n)\n\n => Array\n(\n => 27.58.179.79\n)\n\n => Array\n(\n => 157.37.167.144\n)\n\n => Array\n(\n => 119.160.57.115\n)\n\n => Array\n(\n => 122.161.53.224\n)\n\n => Array\n(\n => 49.36.233.51\n)\n\n => Array\n(\n => 101.0.32.8\n)\n\n => Array\n(\n => 119.160.103.158\n)\n\n => Array\n(\n => 122.177.79.115\n)\n\n => Array\n(\n => 107.181.166.27\n)\n\n => Array\n(\n => 183.6.0.125\n)\n\n => Array\n(\n => 49.36.186.0\n)\n\n => Array\n(\n => 202.181.5.4\n)\n\n => Array\n(\n => 45.118.165.144\n)\n\n => Array\n(\n => 171.96.157.133\n)\n\n => Array\n(\n => 222.252.51.163\n)\n\n => Array\n(\n => 103.81.215.162\n)\n\n => Array\n(\n => 110.225.93.208\n)\n\n => Array\n(\n => 122.161.48.200\n)\n\n => Array\n(\n => 119.63.138.173\n)\n\n => Array\n(\n => 202.83.58.208\n)\n\n => Array\n(\n => 122.161.53.101\n)\n\n => Array\n(\n => 137.97.95.21\n)\n\n => Array\n(\n => 112.204.167.123\n)\n\n => Array\n(\n => 122.180.21.151\n)\n\n => Array\n(\n => 103.120.44.108\n)\n\n => Array\n(\n => 49.37.220.174\n)\n\n => Array\n(\n => 1.55.255.124\n)\n\n => Array\n(\n => 23.227.140.173\n)\n\n => Array\n(\n => 43.248.153.110\n)\n\n => Array\n(\n => 106.214.93.101\n)\n\n => Array\n(\n => 103.83.149.36\n)\n\n => Array\n(\n => 103.217.123.57\n)\n\n => Array\n(\n => 193.9.113.119\n)\n\n => Array\n(\n => 14.182.57.204\n)\n\n => Array\n(\n => 117.201.231.0\n)\n\n => Array\n(\n => 14.99.198.186\n)\n\n => Array\n(\n => 36.255.44.204\n)\n\n => Array\n(\n => 103.160.236.42\n)\n\n => Array\n(\n => 31.202.16.116\n)\n\n => Array\n(\n => 223.239.49.201\n)\n\n => Array\n(\n => 122.161.102.149\n)\n\n => Array\n(\n => 117.196.123.184\n)\n\n => Array\n(\n => 49.205.112.105\n)\n\n => Array\n(\n => 103.244.176.201\n)\n\n => Array\n(\n => 95.216.15.219\n)\n\n => Array\n(\n => 103.107.196.174\n)\n\n => Array\n(\n => 203.190.34.65\n)\n\n => Array\n(\n => 23.227.140.182\n)\n\n => Array\n(\n => 171.79.74.74\n)\n\n => Array\n(\n => 106.206.223.244\n)\n\n => Array\n(\n => 180.151.28.140\n)\n\n => Array\n(\n => 165.225.124.114\n)\n\n => Array\n(\n => 106.206.223.252\n)\n\n => Array\n(\n => 39.62.23.38\n)\n\n => Array\n(\n => 112.211.252.33\n)\n\n => Array\n(\n => 146.70.66.242\n)\n\n => Array\n(\n => 222.252.51.38\n)\n\n => Array\n(\n => 122.162.151.223\n)\n\n => Array\n(\n => 180.178.154.100\n)\n\n => Array\n(\n => 180.94.33.94\n)\n\n => Array\n(\n => 205.164.130.82\n)\n\n => Array\n(\n => 117.196.114.167\n)\n\n => Array\n(\n => 43.224.0.189\n)\n\n => Array\n(\n => 154.6.20.59\n)\n\n => Array\n(\n => 122.161.131.67\n)\n\n => Array\n(\n => 70.68.68.159\n)\n\n => Array\n(\n => 103.125.130.200\n)\n\n => Array\n(\n => 43.242.176.147\n)\n\n => Array\n(\n => 129.0.102.29\n)\n\n => Array\n(\n => 182.64.180.32\n)\n\n => Array\n(\n => 110.93.250.196\n)\n\n => Array\n(\n => 139.135.57.197\n)\n\n => Array\n(\n => 157.33.219.2\n)\n\n => Array\n(\n => 205.253.123.239\n)\n\n => Array\n(\n => 122.177.66.119\n)\n\n => Array\n(\n => 182.64.105.252\n)\n\n => Array\n(\n => 14.97.111.154\n)\n\n => Array\n(\n => 146.196.35.35\n)\n\n => Array\n(\n => 103.167.162.205\n)\n\n => Array\n(\n => 37.111.130.245\n)\n\n => Array\n(\n => 49.228.51.196\n)\n\n => Array\n(\n => 157.39.148.205\n)\n\n => Array\n(\n => 129.0.102.28\n)\n\n => Array\n(\n => 103.82.191.229\n)\n\n => Array\n(\n => 194.104.23.140\n)\n\n => Array\n(\n => 49.205.193.252\n)\n\n => Array\n(\n => 222.252.33.119\n)\n\n => Array\n(\n => 173.255.132.114\n)\n\n => Array\n(\n => 182.64.148.162\n)\n\n => Array\n(\n => 175.176.87.8\n)\n\n => Array\n(\n => 5.62.57.6\n)\n\n => Array\n(\n => 119.160.96.229\n)\n\n => Array\n(\n => 49.205.180.226\n)\n\n => Array\n(\n => 95.142.120.59\n)\n\n => Array\n(\n => 183.82.116.204\n)\n\n => Array\n(\n => 202.89.69.186\n)\n\n => Array\n(\n => 39.48.165.36\n)\n\n => Array\n(\n => 192.140.149.81\n)\n\n => Array\n(\n => 198.16.70.28\n)\n\n => Array\n(\n => 103.25.250.236\n)\n\n => Array\n(\n => 106.76.202.244\n)\n\n => Array\n(\n => 47.8.8.165\n)\n\n => Array\n(\n => 202.5.145.213\n)\n\n => Array\n(\n => 106.212.188.243\n)\n\n => Array\n(\n => 106.215.89.2\n)\n\n => Array\n(\n => 119.82.83.148\n)\n\n => Array\n(\n => 123.24.164.245\n)\n\n => Array\n(\n => 187.67.51.106\n)\n\n => Array\n(\n => 117.196.119.95\n)\n\n => Array\n(\n => 95.142.120.66\n)\n\n => Array\n(\n => 156.146.59.35\n)\n\n => Array\n(\n => 49.205.213.148\n)\n\n => Array\n(\n => 111.223.27.206\n)\n\n => Array\n(\n => 49.205.212.86\n)\n\n => Array\n(\n => 103.77.42.103\n)\n\n => Array\n(\n => 110.227.62.25\n)\n\n => Array\n(\n => 122.179.54.140\n)\n\n => Array\n(\n => 157.39.239.81\n)\n\n => Array\n(\n => 138.128.27.234\n)\n\n => Array\n(\n => 103.244.176.194\n)\n\n => Array\n(\n => 130.105.10.127\n)\n\n => Array\n(\n => 103.116.250.191\n)\n\n => Array\n(\n => 122.180.186.6\n)\n\n => Array\n(\n => 101.53.228.52\n)\n\n => Array\n(\n => 39.57.138.90\n)\n\n => Array\n(\n => 197.156.137.165\n)\n\n => Array\n(\n => 49.37.155.78\n)\n\n => Array\n(\n => 39.59.81.32\n)\n\n => Array\n(\n => 45.127.44.78\n)\n\n => Array\n(\n => 103.58.155.83\n)\n\n => Array\n(\n => 175.107.220.20\n)\n\n => Array\n(\n => 14.255.9.197\n)\n\n => Array\n(\n => 103.55.63.146\n)\n\n => Array\n(\n => 49.205.138.81\n)\n\n => Array\n(\n => 45.35.222.243\n)\n\n => Array\n(\n => 203.190.34.57\n)\n\n => Array\n(\n => 205.253.121.11\n)\n\n => Array\n(\n => 154.72.171.177\n)\n\n => Array\n(\n => 39.52.203.37\n)\n\n => Array\n(\n => 122.161.52.2\n)\n\n => Array\n(\n => 82.145.41.170\n)\n\n => Array\n(\n => 103.217.123.33\n)\n\n => Array\n(\n => 103.150.238.100\n)\n\n => Array\n(\n => 125.99.11.182\n)\n\n => Array\n(\n => 103.217.178.70\n)\n\n => Array\n(\n => 197.210.227.95\n)\n\n => Array\n(\n => 116.75.212.153\n)\n\n => Array\n(\n => 212.102.42.202\n)\n\n => Array\n(\n => 49.34.177.147\n)\n\n => Array\n(\n => 173.242.123.110\n)\n\n => Array\n(\n => 49.36.35.254\n)\n\n => Array\n(\n => 202.47.59.82\n)\n\n => Array\n(\n => 157.42.197.119\n)\n\n => Array\n(\n => 103.99.196.250\n)\n\n => Array\n(\n => 119.155.228.244\n)\n\n => Array\n(\n => 130.105.160.170\n)\n\n => Array\n(\n => 78.132.235.189\n)\n\n => Array\n(\n => 202.142.186.114\n)\n\n => Array\n(\n => 115.99.156.136\n)\n\n => Array\n(\n => 14.162.166.254\n)\n\n => Array\n(\n => 157.39.133.205\n)\n\n => Array\n(\n => 103.196.139.157\n)\n\n => Array\n(\n => 139.99.159.20\n)\n\n => Array\n(\n => 175.176.87.42\n)\n\n => Array\n(\n => 103.46.202.244\n)\n\n => Array\n(\n => 175.176.87.16\n)\n\n => Array\n(\n => 49.156.85.55\n)\n\n => Array\n(\n => 157.39.101.65\n)\n\n => Array\n(\n => 124.253.195.93\n)\n\n => Array\n(\n => 110.227.59.8\n)\n\n => Array\n(\n => 157.50.50.6\n)\n\n => Array\n(\n => 95.142.120.25\n)\n\n => Array\n(\n => 49.36.186.141\n)\n\n => Array\n(\n => 110.227.54.161\n)\n\n => Array\n(\n => 88.117.62.180\n)\n\n => Array\n(\n => 110.227.57.8\n)\n\n => Array\n(\n => 106.200.36.21\n)\n\n => Array\n(\n => 202.131.143.247\n)\n\n => Array\n(\n => 103.46.202.4\n)\n\n => Array\n(\n => 122.177.78.217\n)\n\n => Array\n(\n => 124.253.195.201\n)\n\n => Array\n(\n => 27.58.17.91\n)\n\n => Array\n(\n => 223.228.143.162\n)\n\n => Array\n(\n => 119.160.96.233\n)\n\n => Array\n(\n => 49.156.69.213\n)\n\n => Array\n(\n => 41.80.97.54\n)\n\n => Array\n(\n => 122.176.207.193\n)\n\n => Array\n(\n => 45.118.156.6\n)\n\n => Array\n(\n => 157.39.154.210\n)\n\n => Array\n(\n => 103.48.197.173\n)\n\n => Array\n(\n => 103.46.202.98\n)\n\n => Array\n(\n => 157.43.214.102\n)\n\n => Array\n(\n => 180.191.125.73\n)\n\n => Array\n(\n => 157.39.57.227\n)\n\n => Array\n(\n => 119.152.107.157\n)\n\n => Array\n(\n => 103.166.58.254\n)\n\n => Array\n(\n => 139.135.54.30\n)\n\n => Array\n(\n => 185.199.102.101\n)\n\n => Array\n(\n => 59.103.204.4\n)\n\n => Array\n(\n => 202.14.123.34\n)\n\n => Array\n(\n => 103.48.197.243\n)\n\n => Array\n(\n => 39.34.158.129\n)\n\n => Array\n(\n => 156.146.59.38\n)\n\n => Array\n(\n => 110.227.51.220\n)\n\n => Array\n(\n => 157.45.123.212\n)\n\n => Array\n(\n => 110.227.52.20\n)\n\n => Array\n(\n => 43.242.176.136\n)\n\n => Array\n(\n => 223.233.71.146\n)\n\n => Array\n(\n => 216.73.160.38\n)\n\n => Array\n(\n => 103.217.178.136\n)\n\n => Array\n(\n => 116.71.2.6\n)\n\n => Array\n(\n => 182.180.126.109\n)\n\n => Array\n(\n => 39.53.148.157\n)\n\n => Array\n(\n => 42.108.5.233\n)\n\n => Array\n(\n => 106.66.3.13\n)\n\n => Array\n(\n => 122.177.70.225\n)\n\n => Array\n(\n => 124.253.153.145\n)\n\n => Array\n(\n => 27.58.82.70\n)\n\n => Array\n(\n => 180.149.241.195\n)\n\n => Array\n(\n => 106.201.27.247\n)\n\n => Array\n(\n => 110.227.55.161\n)\n\n => Array\n(\n => 157.39.27.241\n)\n\n)\n```\nArchive for February, 2010: Dmontngrey's Personal Finance Blog\n Layout: Blue and Brown (Default) Author's Creation\n Home > Archive: February, 2010\n\n# Archive for February, 2010\n\n## Well, that's different!\n\nFebruary 21st, 2010 at 10:17 pm\n\nSeveral months ago I opened a bank account just to get a bonus they were offering. I think it was \\$100.00 - easy free money. Every week \\$10.00 gets direct deposited into the account, then transfered out a day later. The account runs on autopilot, so I don't think much of it. I got my most recent statement and imagine the shock at what I saw in the middle of the page! It's very strange because nothing is unusual in the transactions - nothing at all! It's almost as if this doesn't even belong on my statement in the first place - just randomly placed there somehow. It sure doesn't match up to the transactions! I'm still laughing and shaking my head...", null, "I don't have any plans to call the bank just yet. I honestly think someone hit the wrong button and I wouldn't be surprised if something like this shows up on everyone's statement. I certainly WOULD call if it said I was \\$161K in the hole - but it says I have \\$1.00 just like it should.\n\nIn other news:\n\n- I paid off the balance transfer!!\n\n- I nearly had a heart attack when I opened my student loan statement and saw how much interest I've paid over the years. How \"nice\" of them to put that on there. I did the consolidation and put the payment on autopilot - then never thought any more of it. I think this is the first time I've looked at a statement in *ahem* a few years. Needless to say, this one's going next! I'm shooting for 2 years at this point.\n\n- DH wants a \"new\" car. Yes, seriously. Anyone else remember him getting the last one LESS than a year ago? He announced this the day after I paid off the credit card. That didn't go over well with me. After I showed him the student loan statement, he said \"wow, it would be very selfish of me to get a new car right now.\" The next day, back to car talk. He just doesn't like the car he has. It's \"not working out\". He has been told it will be a VERY different transaction this time. He will negotiate (hates confrontation of any type), he will wait until he has enough money saved up (no instant gratification here), he will test drive the model he wants to buy, but will not test drive the exact car he wants to buy (to get a general idea of the car), he will talk to the sales guy - then walk away (again, no instant gratification). I did not like how my car purchase was handled and certainly didn't like how his last car purchase went.\n\n- We finally \"fixed\" our laminate floor on the first floor. There was a lot of gapping right in a high traffic area. We've been putting this off as long as possible because we needed to move 29 and 52 gallon tanks. I wasn't worried about the smaller tank, but the large tank houses a LARGE pacu that I vowed to never move again. This project involved tearing up half the floor and reassembling it. The other half will need to be done at some point, but the gapping isn't too bad. I sure do ache today - we did the project yesterday. I'm really good at bending at the knees to pick up the buckets of water. Who knew it would be such a good butt workout moving 160 gallons of water one bucket at a time?! Owie. The project went well and we can walk through our living room again.\n\n- My head is just starting to come out of the fog I've been in since fall. Stress did this to me. I'm still far from 100%, but I'm having some clearer thoughts. My work performance is suffering the most from this fog. What's kind of sad is that I can still function at 50% and out perform everyone else. I'm not happy with my work performance, but everyone else can live with it just fine.\n\n- The mutt had another round with the abscess. I don't think it fully healed the first time. This time she got it to drain on her own. The limp is mostly gone now and I think she's 99% back to normal. Still keeping an eye on her and keeping the two of them apart as best we can.\n\nNormal?", null, "Well, I think that's about it for now. Time to get back to tracking my coupon shopping in my spreadsheet. Hope everyone is having a great weekend!" ]

[ null, "https://www.savingadvice.com/blogs/image.php", null, "https://www.savingadvice.com/blogs/image.php", null ]

[ "We have been working very hard since 2009 to facilitate in your learning Read More. We can't keep up without your support. Donate Now.\n\n#", null, "www.vustudents.ning.com\n\n www.bit.ly/vucodes + Link For Assignments, GDBs & Online Quizzes Solution www.bit.ly/papersvu + Link For Past Papers, Solved MCQs, Short Notes & More\n\nLooking for Something at Site? Search Below\n\nEEE231 Electronics I Assignment No 02 Solution & Discussion\n\n+ http://bit.ly/vucodes (Link for Assignments, GDBs & Online Quizzes Solution)\n\n+ http://bit.ly/papersvu (Link for Past Papers, Solved MCQs, Short Notes & More)\n\nViews: 329\n\nAttachments:\n\n### Replies to This Discussion\n\nOur main purpose here discussion not just Solution\n\nWe are here with you hands in hands to facilitate your learning and do not appreciate the idea of copying or replicating solutions.\n\nDiode Circuit Problem - 1  :-Find the current flowing in the diode shown in the circuit given below.Also find the output voltage.Given the diode is ideal.\n\n(Answer :- Current flowing = 30 miliampere, output voltage = 0 V )\n\nDiode Circuit Problem - 2:-A resistance of 1 Kilo ohm is connected in parellel with a ideal diode, this whole combination is connected in series with the another 1 kilo ohm resistor, this whole circuit is connected with the 30 V supply. Calculate the:-\n\n(1) Current drawn by the whole circuit.   ( Answer :-30 mA )\n\n(2) Current flow in the diode connected branch.(Answer :- 30 mA )\n\n(3) Current flow in the resistance which is parellel with the diode.(Answer :- 0 A )\n\n(4) Output voltage i.e. voltage across the output resistance. (Answer :- 0 V )\n\nCOMSATS Institute of Information Technology\n\nVirtual Campus\n\nElectronics 1- EEE 231\n\nAssignment No. 2\n\nQuestion 1\n\nDetermine the current ID and the diode voltage VD for the circuit shown below. For this circuit VDD = 5V and R=1kΩ. Assume that the diode has a current of 1 mA at a voltage of 0.7 V and that its voltage drop changes by 0.1 V for every decade change in current.", null, "Question 2\n\nFind the current flowing in the diode shown in the circuit given below.Also find the output voltage. Given the diode is ideal.Note : voltage is 30 V in the figure and resistor is 1 K Ohms.", null, "Question 3\n\nA resistance of 1 Kilo ohm is connected in parellel with an ideal diode, this whole combination is connected in series with another 1 Kilo ohm resistor, this whole circuit is connected with the 30 V supply. Calculate the:-\n\n(1) Current drawn by the whole circuit.\n\n(2) Current flow in the diode connected branch.\n\n(3) Current flow in resistance which is parellel with the diode.\n\n(4) Output voltage i.e. voltage across the output resistance.", null, "Question 4\n\nFind the current and voltage across the resistance shown in the figure. Take the diode as ideal.\n\nQuestion 5\n\nThe diode connected in the circuit shown below is made up of silicon. Calculate the current and the output voltage of the circuit.", null, "Diode Circuit Problem - 5:- The diode connected in the circuit shown below is made up of silicon. Calculate the current and the output voltage of the circuit.\n\n(Answer :- 2.07 mA, -0.45 V ).", null, "." ]

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[ "# T-Tests\n\n## Introduction:\n\nT-tests are a statistical means of understanding the differences between measured means. Let’s say you are trying to see if on average your class is taller than another class. The best way to find out would be to measure the height of every student in each class, find the two means and compare them.\n\nBut in the real world, when trying to compare two means, it is not possible or unreasonable to measure every possible sample in each of the sets of interest. For example, consider national polls. Every time one wishes to judge the public pulse on an issue, you would have to ask every citizen, on the scale of an election. To circumvent this, in statistics, there are powerful tools to assist us in understanding if the difference in the means (mean opinion in our example) is different enough that it is not just a fluke during measurement. This is called statistical significance. If the difference is statistically significant, it means that the means are different with such confidence that it wasn’t some random luck. Note that we say confidence because we can only say with very high confidence that the means are different. We cannot guarantee it because we did not ask every citizen of the United States.\n\n## Concept:\n\nWhen comparing two means, conceptually a bell-curve distribution is imagined around one of the means. The shape of this curve is affected by the number of samples you take. The important concept is that the center of the curve is the first mean. Consider that as the actual mean. When you take measurements, lets say when you flip a coin, you don’t always get 50% heads and 50% tails if you take 10 measurements. Even though the actual mean is 0.5 (50% of 1) for heads, there is a probability distribution around that 0.5 mean for the expected mean for multiple measurements. This distribution says that at the ends, the probabilities are very, very unlikely. For example, if you take 100 measurements of the coin flip, it is extremely unlikely (low probability) that you will get 0% heads or 100% heads. If for some reason you do, it means that the difference is significant enough for you to suspect that something else was affecting the coin flip, like a biased coin.\n\n### Making Sense of the Difference:\n\nWhile the difference between the means is being stressed for explaining the concept, the actual nature of the difference can also be seen through this test. Whether the mean being compared is greater or lower than the actual (assumed) mean is statistically significant is of more importance in our projects in the website. For figuring this out, first find out the means of the individual lists and compare them to see which one is greater than the other. Now you would need to perform the t-test to see if this specific difference is significant. Follow the protocol listed below for finding the p-value to see if the difference is significant. If the p-value you get here (when you follow this protocol) is less than or equal to 0.05, it means that whatever specific difference (in terms of which mean is greater, not the actual difference in the mean) you observed between the means is statistically significant.\n\nImportant Note: Just because your results don’t show statistical significance does not mean there is no statistical significance. Your methods might not have shown the significance. The interpretation of such a case is not binary, whether it is significant or not. If the probability of the mean being compared (p-value) is high, it gives confidence in your conclusion that there is no statistical significance between the two means. On the other hand, if there is statistical significance (p < 0.05), it means that for your methods the means are statistically significant.\n\n## Variance:\n\nGoing back to the heights of students in your class example, it would be easier for you to judge the difference by using a t-test. You would randomly select ‘n’ number of students from each of the classes and measure their heights. Now you have two lists of heights, one from each class, for n students each. The more students you sample, the better your results in the sense that you will be able to show significance if there is such a difference.\n\nRefer to “Excel Statistics Formulas” In Guide Section to find the mean of the two lists.\n\nFor further clarification and formatting for the purpose of performing a t-test, another method is shown below.\n\nGo to Microsoft Excel like so:", null, "In Microsoft Excel, write your list down like so:", null, "To save your file for the first time, click on “File on the top of the screen to the left corner” and select the option “Save As”", null, "You will see a box up like in the figure above. Press on the Icon “Desktop” to the left. Look at the place where there are capital letters that are highlighted. Type in your name and your project’s name there. Don’t use any special characters like \\$, %,’, etc. in the name. Then press “Save” on the right.\n\nIt is a good idea to constantly save your file during the process so that you wont lose your work if your computer crashes.\n\nFor saving your file after you already saved it once, just press on the floppy icon on the top left corner in Excel.\n\nFind the means of the two lists and make note of which mean is greater or lesser than the other.\n\n1. To find the mean of a list:\n2. Left click once on cell where you want to have the mean displayed.\n3. Double Click the left-click on your mouse.\n4. Type in ‘=average(’\n5. Left click once on the first cell in the list you want to find the mean of and don’t let go of the click.\n6. You screen should look similar to:", null, "The yellow box beneath your cell shows the syntax (what you need to put to in to find the average). You can ignore it if you want.\n\nDrag the mouse down without letting go of the click until you reach the last cell on your list with a value.", null, "Then Press Enter. The mean for that list should be displayed.", null, "Repeat the procedure to find the mean for “Other Class”\n\nYou will see:", null, "Look and compare the means. Notice that the two means are close to each other with the mean height of your class greater than the other class, like your prediction.\n\nTo see if this difference is significant and not just a fluke because of your selection of people to include in your list despite your efforts for randomly selecting them, a t-test is needed.\n\nTo do the t-test:\n\n1. Write down t-test, p-value in a neighboring cell like shown in the next figure.\n2. In the next cell, double click the left click.\n3. Type in ‘=ttest(’.\n4. Just like you selected your list for finding the mean, click on the first value of the fist list and drag the mouse down until you reach the last value on that list without letting go of the click. Make sure you don’t select the mean value.\n5. Type in a comma, and in the same way select the second list.\n6. Type in ‘,1,3’.\n\nYour field should look similar to:", null, "Double check if you have entered everything correctly and if the right values in the lists are selected as shown by the colored boxes.\n\nThen Press Enter and your p-value will be displayed like so:", null, "Note the p-value.\n\nIf it is equal to or less than 0.05, the difference in the means that the average height of students in your class is greater than that of the other class is statistically significant.\n\nIn this case it is about 0.1. This means that the difference is not statistically significant. But your claim does have some confidence because the p-value is still relatively low. The confidence decreases with an increase in p-value. This does not mean that on average your class is not taller than the other class. It just means that for the given lists that you used, the difference was not big enough to be statistically significant. This could change if you used more students in the lists.\n\nIf you have any questions, don’t hesitate to ask your science or math teacher. You can also email us at [email protected]" ]

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[ "", null, "", null, "", null, "", null, "# Long-term Spatial Modeling for Characteristics of Extreme Heat Events\n\nThere is increasing evidence that global warming manifests itself in more frequent warm days and that heat waves will become more frequent. Presently, a formal definition of a heat wave is not agreed upon in the literature. To avoid this debate, we consider extreme heat events, which are well-defined at local scales, as a run of consecutive days above a specified local threshold. Characteristics of EHEs are of primary interest, such as incidence and duration, as well as the magnitude of the average exceedance and maximum exceedance above the threshold during the EHE. Using approximately 60 years of time series of daily maximum temperature data collected at 18 locations in a given region, we develop a spatio-temporal model to study the behavior of EHEs over time. The model enables prediction of the behavior of EHE characteristics at unobserved locations within the region. Specifically, our approach employs a two-state space-time model for EHEs with local thresholds where one state defines above threshold daily maximum temperatures and the other below threshold temperatures. We show that our model is able to recover the EHE characteristics of interest and outperforms a traditional autoregressive model that ignores thresholds in terms of out-of-sample.\n\n## Authors\n\n##### This week in AI\n\nGet the week's most popular data science and artificial intelligence research sent straight to your inbox every Saturday.\n\n## 1 Introduction\n\nThere is a strong evidence of global warming induced as a result of the increasing concentration of greenhouse gases in the atmosphere (Lai and Dzombak, 2019). It can be expected that this global warming will manifest itself in more frequent warm days and many papers suggest that heat waves will become more frequent (Lemonsu et al., 2014; Alexander, 2016). This anticipated scenario along with the exceptionally long heat wave observed in Europe in August 2003 has increased interest in analyzing climate extremes. The analysis of heat waves is particularly important due to the potential for serious anthropogenic, environmental, and economic impacts (Amengual et al., 2014; Campbell et al., 2018). Extreme heat raises significant health concerns in humans as it can result in death, can change the range or niche for plants and animals, and can lead to heat-driven peaks in electricity demands or lost crop income.\n\nA challenge in analyzing heat waves stems from the on-going debate over its exact definition. According to the World Meteorological Office (WMO), a heat wave should be defined as a period persisting at least three consecutive days of marked unusual hot weather (maximum, minimum and daily average temperature) over a region with thermal conditions above given thresholds based on local climatological conditions. This definition suggests that analyses of heat waves require temperature series at a daily scale, but offers no operational guidance with regard to the various choices of implementation. Khaliq et al. (2005) and Reich et al. (2014) used only maximum temperature, while Keellings and Waylen (2014) and, more recently, Abaurrea et al. (2018) considered both maximum and minimum temperatures. Perkins and Alexander (2013) and Smith et al. (2013) addressed the issue of analyzing different measurements and definitions of this phenomenon.\n\nTo circumvent the lack of agreement on heat wave definition, we work with the concept of an extreme heat event (EHE) in order to formalize the notion of a period of persistent extremely high temperatures. EHEs are based on an exceedance of threshold approach, often used in the analysis of extremes. Then, at a given location, an EHE is defined as a run of consecutive daily temperature observations exceeding the extreme threshold for that location. According to the WMO, the thresholds should be based on local conditions, such as the 95th percentile of the local temperature series during the warm months.\n\nMuch work on extreme events focuses on the analysis of their occurrence using asymptotic results from extreme value theory. These results state that the occurrence of peaks over threshold follow a Poisson process, under mild conditions (Ogata, 1988; García-Cueto et al., 2014; Abaurrea et al., 2015). A limitation of this approach is that important behaviors which describe the nature of an EHE, beyond incidence, such as the duration, or the average exceedance, or the maximum exceedance above the threshold, are not captured.\n\nModels for heat waves and extreme heat events commonly consider the occurrence of the events at particular locations, i.e., in the form of a time series over a window of time. However, spatial aspects of this phenomenon are also of interest and should be introduced into the modeling process. That is, spatial dependence is anticipated in the occurrence of EHEs. A joint model for time series at different locations that incorporates spatial dependence will borrow strength from nearby time series, and will anticipate an increase in the probability of an EHE occurring at locations near where an EHE did occur. Furthermore, given the evidence of changing climate, the considered models should also be able to capture nonstationary behavior in maximum daily temperature in time. This can be done by allowing temporal random effects, time varying coefficients or time-varying covariates\n\n(Cheng et al., 2014; Abaurrea et al., 2015).\n\nThe contribution of this paper is to take up the challenge of providing a methodology to study the occurrence and behavior of extreme heat events during a period of time over a region which incorporates spatial dependence. The intent is to enable more complete characterization of the events. We develop a space-time model based on a spatially referenced collection of temperature time series that enables the prediction of both the incidence and the characteristics of the EHEs occurring at any location in the region.\n\nA direct approach considers a spatial model for daily maximum temperatures which can then be used to characterize the EHEs. A shortcoming of this approach is that such a model will be driven by the bulk of the distribution, i.e., where most of the data is observed. The model can yield poor fits for the tails (particularly the upper tail), where the main interest for the model lies when attempting to characterize EHEs (Keellings and Waylen, 2015; Shaby et al., 2016). A more promising approach to effectively model both the extremes as well as the bulk of the distribution is to consider a model incorporating thresholding. Here, we propose a model which switches between two observed states, one that defines extreme heat days (those above the temperature threshold) and the other that defines non-extreme heat days (those below the temperature threshold). Importantly, this two-state structure allows temporal dependence of the observations but also that the parameters controling the effects of covariates and the spatial dependence can differ between the two states. In the proposed model, the transition probabilities between the two observed states are driven by a two-state Markovian switching model. For daily maximum temperatures, we are able to demonstrate substantially improved out-of-sample predictive performance compared with a corresponding spatial autoregressive model which ignores thresholds.\n\nThe full model is specified in a hierarchical Bayesian framework and estimated using a Markov chain Monte Carlo (MCMC) algorithm. In this framework, posterior predictive distributions of the foregoing features of the EHEs (rate of incidence, duration, maximum exceedance, average exceedance) can be easily obtained. Specifically, for a given site, we can obtain posterior predictive samples of the time series of daily maximum temperatures from which posterior predictive samples of the characteristics of their EHEs can be extracted.\n\nThere is limited literature using Markov models to analyze EHEs.\n\nSmith et al. (1997) studied daily temperature exceedances over a threshold and fitted a Markov model to them. Shaby et al. (2016)\n\nproposed a two-state switching hidden Markov model and used a latent variable that controlled whether a day was assigned to the heat wave or the non-heat wave state. By contrast, given local thresholds, our approach takes these state variables as observed rather than latent.\n\nFurthermore, none of this modeling introduces spatial structure. Spatial dependence has been included in the climate extremes literature but using different approaches. For instance, Cooley et al. (2007) proposed a Bayesian spatial modeling of extreme precipitation in order to obtain maps of precipitation return levels. Reich and Shaby (2019) suggested a spatial Markov model for climate extremes, using latent clustering of neighboring regions. Guillot et al. (2015) used Markov random fields to model high-dimensional spatial fields for paleoclimate reconstructions.\n\nThere are several motivations for building an effective space-time model. First, the model can be used to study the behavior of extreme heat events over time. That is, for observed locations, we can extract EHEs and their characteristics descriptively. Second, the model can be used to make predictions with regard EHE characteristics at unobserved locations. These predictions are important for understanding the possible impacts of EHEs on human health, on ecosystems, and on the economy, since the assessment of these effects may require predictions at locations where monitoring devices are not available. Such spatial prediction may also be useful to complete observed series with missing information. A further potential application of the model is to make predictions of the spatial extent of an EHE, defined as the geographical area (often expressed as a percentage of the region) affected by extreme temperatures at a given time. Lastly, the model can be used to provide information for a deeper understanding regarding the temporal evolution of aspects related to temperatures across a region. For instance, it can be used to study changes in the incidence of EHEs and their characteristics between decades, to assess whether the factors affecting the non-extreme and the extreme daily maximum temperatures are the same, or to study EHE behavior averaged over a region. Spatial variation with regard to all of these concerns is provided.\n\nAs a final comment, there is an important distinction between the objective of modeling daily maximum temperatures and the objective of learning about characteristics of extreme heat events. The distinction is analogous to that of predicting weather versus climate. That is, like weather, if the goal is to predict an annual time series of daily maximum temperatures, our two-state threshold based model will not outperform a corresponding autoregressive model which ignores thresholds. However, like climate, if our goal is to predict incidence and characteristics of EHEs over a window of time, say, a decade, then our two-state model will consequentially outperform the corresponding autoregressive model.\n\nThe data analyzed in this work consist of approximately 60 years of daily maximum temperatures at 18 locations throughout the Comunidad Autónoma de Aragón in Spain. Details and summaries of this dataset are presented in the Section 2. As noted above, we fit our models to this dataset within a Bayesian framework using MCMC. We do out of sample model comparison using leave-one-out validation.\n\nThe outline of the paper is as follows. Section 2 presents an exploratory analysis of the above dataset. Section 3 describes the modeling details. Comparison of the proposed two-state model to a more traditional autoregressive model ignoring the threshold is presented in Section 4. Results and some associated inference summaries are supplied in Section 5. Finally, Section 6 concludes with a summary and possible directions for future work.\n\n## 2 Exploratory data analysis\n\nThe region of interest is the Comunidad Autónoma de Aragón region in northeastern Spain, located in the Ebro basin (85,362 km). The Ebro river flows from the NW to the SE through a valley bordered by the Pyrenees and the Cantabrian Range in the north and the Iberian System in the southwest. The maximum elevation is approximately 3400 m in the Pyrenees, 2600 m in the Cantabrian Range and Iberian System, and between 200-400 m in the central valley. In general, the area is characterized by a Mediterranean-continental dry climate with irregular rainfall, and a large temperature range. However, several climate subareas can be distinguished due to the heterogeneous orography and other influences, such as the Mediterranean sea to the east, and the continental conditions of the Iberian central plateau in the southwest. Zaragoza, the largest city in the region, is located in the central part of the valley, and experiences more extreme temperatures and drier conditions. This variation in climate conditions suggests that Aragón is an interesting study region.\n\nDaily maximum temperatures were obtained at 18 sites across and around the Comunidad Autónoma de Aragón region for the years 1953-2015. The names and locations of the 18 sites are shown spatially in Figure 1. Data from the years 1953-1962 were used to obtain the location-specific thresholds for extreme heat events; the data for the remaining years 1963-2015 were used in the modeling. Specifically, the 95th percentile of daily maximum temperature for the months June, July, and August during the 10 years 1953 to 1962 was computed for each site to determine the threshold for defining extreme heat events at the site. These thresholds are shown in parentheses in Figure 1. Figure 2 shows these percentiles versus elevation/altitude for each site. In general, there is a negative, roughly linear relationship between altitude and daily maximum temperature percentile. In addition, while higher elevations are found in both the northern and southern part of the Aragón region, the 95th percentiles tend to be lower in the north than south indicating a latitudinal gradient in addition to an elevation gradient for daily maximum temperature.", null, "Figure 1: Spatial map of the 18 sites across and around the Comunidad Autónoma de Aragón region in northeastern Spain used in the analysis with associated temperature threshold.", null, "Figure 2: The 95th percentile temperature (threshold) versus altitude (meters) for the 18 sites.\n\nMonthly averages of daily maximum temperature for each location during for the years 1976-1985 and 2006-2015 are supplied in Figure S1 of the supplementary material. Also shown are the differences (later minus earlier) in these monthly averages between the two time periods for the months April through October. We see an increase in April and very sharp increase in May across all sites, indicating that warmer weather is now starting earlier. We continue to see an increase through the summer months but in September we see a decrease for many sites suggesting an earlier end to summer at these sites. Raw summaries of the daily maximum temperature data are included in Tables 1 and 2. The total number of days with maximum temperature above the specified threshold for two 10 year periods (1976-1985 and 2006-2015) are given in Table 1. These summaries are computed for each month for June, July, and August. In general, July and August have the greatest number of days with maximum temperatures above the threshold and the latter time period experienced more days above the threshold. Table 2 shows the number of unique heat events, which considers each run of consecutive days having a maximum temperature above the threshold as one event. Thus, the number of distinct extreme heat events is less than the number of days above the threshold since many events last more than one day. In Table 2 each event is allocated to the month in which the event starts. Again, the majority of heat events occur in July and August for each of the locations and more events occurred in the latter decadal period.\n\nNext, we investigate the autoregressive behavior of daily maximum temperature. Let denote the maximum temperature on day at location s and let denote the threshold for location s such that when we have a day with extreme heat. Additionally, define the binary indicator , where indicates a daily maximum temperature above the threshold on day at location s, otherwise it is .\n\nIn an exploratory mode, we fit independent AR(1) models using the data from the years 1963-2012 for each location where is the autoregressive parameter and is pure error. From the estimated\n\n, we compute the kernel density estimate of\n\nfor both and . Figure 3 shows kernel density estimates for three representative locations. It reveals both the differences across locations as well as differences when conditioning on . Notably, the distribution is much more diffuse than that of for all three locations.", null, "Figure 3: Density estimates of (left) (Yt(s)−^ρYt−1(s)|Ut(s)=0) and (right) (Yt(s)−^ρYt−1(s)|Ut(s)=1) for three locations.\n\nFinally, we explore the variability in temperature for days above threshold and days below threshold by location. For each location, we computed the variance in maximum daily temperature for the days above threshold and days below threshold, as well as the ratio of these variances. As expected, there is much more variation in maximum daily temperatures for temperatures below the threshold than above threshold. The location-specific variances in daily maximum temperature for the below threshold days range from 49.6 to 83.2 with a mean of 65.0\n\n, whereas the above threshold days range from 1.17 to 2.45 with a mean of 1.72.\n\n## 3 Modeling details\n\nAs in the previous section, let denote the daily maximum temperature at time and location . We propose a two-state model where the state for a given day defines whether the location is experiencing an extreme heat event or not. Let denote the state at time for location where a value of 0 denotes the below threshold state and a 1 denotes the above threshold state. We specify a threshold, at each location, developed as discussed in Section 2. Again, we are interested in EHEs at s relative to the climate at s. Here, is a spatial binary time series process. At location s it provides a time series reflecting times of transition or state-switching. It is observed for each at a monitored site but is latent elsewhere.\n\nUnder a typical hidden Markov model, we would assume is a Markov process where the state depends only on the previous state . Then, the distribution of would be specified explicitly given . Here, we take a different approach since our states are not hidden. That is, the states are observed given a threshold and the temperatures are modeled under the restriction to a given state. Conversely, given the threshold, is a binary function of . Furthermore, we introduce a model specification to allow the transition probabilities between states to be a function of previous temperature.", null, "Figure 4: Structural diagram showing the relationships between the binary process, Ut(s), and temperature Yt(s) for a given location.\n\nThis specification ensures the transition probabilities to be “local”, i.e., to vary with location and to depend upon the previous day’s maximum temperature at that location. For example, if the previous day’s maximum temperature was sufficiently high so that we were in an extreme heat state, then we would expect this to affect the probability of staying in the extreme heat state for the next day. However, since EHEs are fairly rare, it is perhaps more clear to say that we would like a higher probability of being in an EHE state at day if we were in an EHE state at day than if we were not in an EHE state at day . Vice versa, if the previous day’s maximum temperature resulted in a non-extreme heat state. In different words, we are investigating persistence of extreme heat. We elaborate this further below.\n\nTwo remarks are useful here. First, we could condition on a longer history of maximum temperatures or an average of previous maxima. We experimented with introducing additional lags in the modeling but found no gain in predictive performance. Second, failing to condition on the previous day’s temperature provides transition probabilities which don’t predict well the persistence of above or below threshold states. Thus, we supply a location-specific autoregressive model for current daily maximum temperature which depends upon the current day state as well as the previous day’s maximum temperature.\n\nWe define our joint distribution for temperature and state in a first order Markov fashion explicitly as follows. Given\n\n, and thus, , for two consecutive time points, and , we write the joint distribution as\n\n [Yt(s)|Ut(s),Yt−1(s)][Ut(s)|Yt−1(s)][Yt−1(s)|Ut−1(s),Yt−2(s)][Ut−1(s)|Yt−2(s)].\n\nHence, the process is started with , followed by the distribution , and so on. Figure 4 provides a simple graph of the specification and also reveals how it differs from a customary dynamic model specification.\n\nThis formulation requires three model specifications:\n\n1. .\n\nWith\n\ndenoting the threshold (quantile) for location\n\ns, for (i) and (ii), we need truncated distributions, i.e., when and when\n\n, respectively. We adopt truncated normal distributions, with suitable centering,\n\n TN(μUt(s)t(s)−ρUt(s)(Yt−1(s)−μUt(s)t−1(s)),σ2,Ut(s)(s),aUt(s)(s),bUt(s)(s)) (1)\n\nwhere the ’s and ’s supply the truncation as follows:\n\n aUt(s)(s)={−∞Ut(s)=0q(s)Ut(s)=1\n\nand\n\n bUt(s)(s)={q(s)Ut(s)=0∞Ut(s)=1.\n\nDetails for are given below.\n\nIt is natural to ask why we do not consider a customary generalized Pareto distribution (GPD) for the exceedance distribution, i.e., for the upper tails? First, we are modeling a time series of daily maximum temperatures; we are not modeling peaks over thresholds for a temperature series which is the usual setting for the GPD. Second, we experimented with the GPD, exploring many different specifications for the parameters and were unable to find improved out-of-sample predictive performance with it. More disconcerting, we were unable to find posterior mass for a shape parameter . All exploratory model fitting yielded a negative shape parameter which implies bounded support for the daily maximum temperatures. While this is practically true, it would force an awkward selection or estimation of a local bound for each s.\n\nFollowing the exploratory modeling in the previous section, it also emerged that what is needed is a different, in fact, smaller variance for the above threshold daily maximum temperature distribution than for the below threshold daily maximum temperature distribution. This seems evident since the support for daily maximum temperatures above the threshold is much shorter than the support for daily maximum temperatures below the threshold. Otherwise, Gaussian tails seem to be adequate. Further, we introduce spatially varying variances, expecting that variation in say, Jaca (in the Pyrenees in the north of the region) would be different from variation in say, Zaragoza (flat and central in the region). Again, the exploratory discussion in Section 2 supports this.\n\nFor (iii) we introduce a probit link to define\n\n Φ−1(pt(s))≡Φ−1(P(Ut(s)=1|Yt−1(s)))≡ηt(s) (2)\n\nwith given below.\n\nNow, we turn to the specifics of s and s. There are many possibilities that we can imagine and illustrative choices follow: For consider\n\n μUt(s)t(s)=βUt(% s)0+βUt(s)0(s)+γUt(s)[t365]+1+βUt(s)1elev(s)+βUt(s)2lat(s) +λ1sin(2πt/365)+λ2cos(2πt/365).\n\nHere, denotes a global (across the domain for our dataset) intercept and denotes a local spatial intercept, i.e., providing local adjustment to the global intercept. Each is modeled as a mean Gaussian process with exponential covariance function. For , where denotes the greatest integer function, we are counting years with this subscript and, as a result, the ’s provide annual intercepts to allow for yearly shifts, i.e., for hotter or colder years. The sin and cos terms are introduced to capture annual seasonality with their coefficients reflecting associated amplitudes. In leap years, the specification is analogous, replacing 365 with 366 days. This seasonality is critical to ensure that an annual daily maximum temperature trajectory over the course of a year at a location will provide sensible realizations.\n\nWe note that with circularity in time, this definition yields a discontinuity in from December 31 to January 1.111We could start the year at another day but, regardless, we will always experience a jump on that day. However, since time is modeled discretely, we do not expect that this will be a concern with regard to EHE behavior. Elev is the elevation at s and Lat is the latitude. We explored additional potential site level covariates, e.g., slope, aspect, distance from water (distance from the Ebro river and the Mediterranean sea) but found no improved predictive performance. Finally, provides a centered AR(1) specification, bringing in the previous day’s temperature, .\n\nFor we propose\n\n ηt(s)=ϕ0+ϕ0(s)+ϕ1(Yt−1(s)−q(s))+ϕ2((Yt−1(s)−q(s))1(Yt−1(s)−q(s)≥0)) +ϕ3sin(2πt/365)+ϕ4cos(2πt/365).\n\nHere, centering by the threshold yields more sensible transition probabilities. can be moved over to the intercept term in order to provide a spatially varying offset. However, the inclusion of , modeled as a Gaussian process, allows for a richer spatially-varying intercept. Writing in terms of being a deviation from suggests that there is no need to model or as spatially varying coefficients. The term with coefficient provides a slope adjustment according whether we are previously in state or in state . This adjustment ensures continuity in (hence, ) as a function of .\n\nThese specifications are offered as an attempt to provide regressors that capture the critical features that drive daily maximum temperatures with transitions relative to a threshold state. Other variations of these specifications could be considered. For instance, indicator functions could be considered to introduce an intercept and/or slope adjustment. Also interaction effects might be examined. We explored a few of these richer model specifications and found no additional benefits in model performance.\n\n## 4 Model fitting and comparison\n\nWe briefly summarize the complete model specification including priors as well as the model fitting and its challenges. Then, we turn to the out-of-sample model comparison.\n\n### 4.1 Full specification of the model and fitting details\n\nModel inference is obtained in a Bayesian framework, requiring prior distributions for each of the model parameters. When possible, diffuse and conjugate prior distributions are assigned. We start with the parameters of models (i) and (ii),\n\nand . The autoregressive parameters and are each assigned a non-informative and independent Uniform prior distribution. The coefficient parameters , , and as well as , , and\n\nare each assigned independent and diffuse normal prior distributions with mean 0 and standard deviation 100. Independent normal prior distributions with mean 0 and standard deviation 100 are also assigned to the coefficients of the seasonal terms,\n\nand . The yearly random effects and are assigned normal distributions with mean 0 and standard deviation 1. For identifiability, each of the random effects for the first year are fixed to 0.\n\nMean 0 Gaussian process priors are assumed for both and . The spatial covariance matrix is specified using the exponential covariance function. The variance parameter for each of the spatial covariances is assigned an independent Inverse-Gamma (2,2) prior distribution. Similarly, the spatially varying variance parameters, log( and log\n\n, are also assigned independent Gaussian process priors. Hyperpriors are assigned to the mean and variance of both processes; each mean is assigned a N(0,1) and each variance is assigned an Inverse-Gamma(2,2). Whereas the other Gaussian processes were mean 0, specifying a hyperprior for the mean of the spatial variances enabled Bayesian learning with respect to the variances of the above and below threshold processes. An exponential covariance function is again used to specify the spatial dependence.\n\nThe following priors are assigned to the parameters of model (iii), . Here, independent normal prior distributions with mean 0 and standard deviation 100 are assigned to the coefficients , , , , The spatial random effect is assigned a mean 0 Gaussian process prior. The variance is assigned an Inverse-Gamma (2,2) hyperprior and the spatial covariance is again specified with an exponential covariance function.\n\nAltogether, the model consists of five Gaussian processes, , , log(), log(, and . Spatial models are needed for each of these components in order to be able to predict at new locations. Each of these Gaussian processes is assumed independent with the range parameter of the exponential covariance function fixed such that the effective range is equal to 400 km. This distance is approximately the maximum distance across the Comunidad Autónoma de Aragón region from the north to the south and was chosen to capture large-scale spatial dependence across the region. This choice assumes that the local behavior in daily maximum temperature will be captured by the autoregressive process in the model.\n\nMarkov chain Monte Carlo is used to obtain samples from the joint posterior distribution. The sampling algorithm is a Metropolis-within-Gibbs algorithm. Posterior draws of each of the spatial random effects are obtained using an efficient elliptical slice sampler (Murray et al., 2010).\n\n### 4.2 Model comparison and inference\n\nFor model comparison, we confine ourselves to just two models, the specification above and a corresponding specification which ignores state relative to threshold. This second model will be an AR(1) time series model for the daily maximum temperatures. This is analogous to our models in (i) above with , however, now without the truncation according to the state. That is, we specify a normal distribution for . In both models we adopt spatially varying variances for prediction at new locations. Under the simpler model, we impose thresholds on the posterior predictive distributions after model fitting in order to capture EHEs and their characteristics.\n\nModel comparison is implemented by single-point deletion (leave-one-out) validation. While we have done single site deletions for all sites, we show results for three illustrative locations, one in the southern part of the region, one central, and one in the north. With primary interest in capturing persistence of EHEs, comparison is made using conditional and marginal error rates (defined below) with regard to prediction of an EHE day. We shall see that our proposed model is substantially better at such prediction.\n\nThen, using our model, inference is provided, employing posterior predictive summaries, for the EHE characteristics presented in Section 1 - duration, maximum exceedance above threshold and average exceedance above the threshold. Such inference is provided for the entire time span of the data as well as by decade for two decades of interest.\n\nThe thresholds, , are known/observed for each monitoring site, adopting a th percentile of the daily maxima at a site over the extreme heat period, June, July, and August (e.g., Abaurrea et al., 2007) as described in Section 2. These values were used directly in the model fitting for the two-state models specified above and are known for our leave-one-out validation. For prediction beyond our sites, we assume these thresholds can be supplied from external information for unobserved locations of interest. Alternatively, with interest in prediction of given an observed set , we could propose a stochastic process specification. Spatial quantile regression models using asymmetric Laplace processes (ALP) are one approach (Lum and Gelfand, 2012); however this is beyond the scope of our work here.\n\n## 5 Results of model comparison and inference\n\nWe begin with the results of the model comparison using leave-one-out cross validation and then provide the inferential summaries from our model for the region of interest. Comparison is between an autoregressive model which ignores thresholds in the specification (base model) and the model presented in Section 3. Each model was fitted to the daily maximum temperature data for the 50 years spanning 1966 to 2015. Markov chain Monte Carlo was run for 200,000 iterations. The first half of each chain was discarded as burn-in and the remaining samples were used for posterior inference and prediction.\n\n### 5.1 Model comparison\n\nA leave-one-out comparison of the model is carried out with three series, Zaragoza, Tornos and Yesa. These series show the variation in climate across the study region. Zaragoza is located in the center of the region, surrounded by other locations in the dataset with similar climate. Yesa is located in a valley in the northwest part of the region, with a climate that is quite different from that of the rest of locations in that area. Tornos is near the southwest border of the region with a climate much different from that of its neighbors to the east with the same latitude.\n\nAs noted in the previous section, we conduct our model comparison through prediction of exceedance days. The prediction of an exceedance day requires prediction of . Since\n\nis a binary variable, it is natural to assess its performance in terms of misclassification error rates. There are two possible errors here: (i) predicting\n\nwhen and (ii) predicting when . Arguably, the second error has more impact, since it means failing to predict well an extreme heat exceedance given that it happened.\n\nThree different measures of error for days when are considered, two conditional and one marginal error. The first conditional error is\n\n 1−p(1)t(s)≡1−P(Ut(s)=1|Yt−1(s)≥q(s))=P(Ut(s)=0|Yt−1(s)≥q(s))\n\nwhere we condition on previous days which were exceedance days. It gives the error in persistence of heat days, so that smaller errors of this type imply better prediction of persistence. The second conditional error is , where we condition on previous days which were non exceedance days, and it represents the error in predicting the first day of an EHE. The marginal error is , i.e., given the previous day’s temperature regardless of whether or not it was an exceedance.\n\nEach error rate is estimated as the average of daily point estimates for over a selected time window. Here, we choose a 92 day window, arising from the months June, July and August (JJA), since this is when most of the exceedance events occur. We can then average over a year, decade, or the entire time window.\n\nFigure 5 shows the distributions of and , for all exceedance days in JJA across all years for the three out-of-sample locations. It is observed that in both cases, the distribution of the errors of the base model is shifted towards higher values, and the modes of the errors in our model are lower than in the base model.", null, "Figure 5: Distribution of 1−^pt(s)(top) and 1−^p(1)t(s) (bottom). They are calculated during JJA across all years for the out-of-sample locations.\n\nThe posterior predictive mean estimates of the error rates for all exceedance days in both models are summarized in Table 3 for the two 10-year periods. The performance of our model is clearly better, since the marginal errors and the errors conditioned on are lower in all the cases, with a reduction in the latter that varies from 17 to 48%. Therefore, our model captures the observed persistence of exceedance days better than the base model. With regard to the conditional measures of error given , or rather, those concerning the predictions of the first day of an EHE, our model outperforms the base model with less distinction. For example, in Tornos, the errors of our model are lower, but closer to those of the base model, while in Zaragoza, the errors of both models are similar.\n\nWhile these error rates might seem large, it is important to clarify their interpretation and utility. Recall that exceedance days are not common; the location-specific thresholds are at the 95% percentile. Furthermore, when they occur, there are usually short in duration meaning their persistence is low. (We note that failure rate in predicting them can not be compared to say, a fair coin flip since coin flips would reveal dismal predictive performance outside of JJA.)\n\nTo investigate the predictive performance of our model with regard to persistence, we expand our conditional measures of error. Table 4 shows the posterior predictive mean estimates of the conditional error rates for all exceedance days, given that the EHE has already persisted for one, two, or three days. In general, the conditional error estimates are lower for those events with higher levels of persistence. Compared to the base model, the error rates in our model are lower for each location and level of persistence.\n\nWe briefly consider model adequacy with regard to the main objective of our study: out-of-sample prediction of characteristics of extreme heat events. We do this by comparing the posterior predictive distribution of exceedance days as well as EHE characteristics (duration, and intensity) with the observed empirical counterparts. Comparisons are made for the entire time window of the analysis, 1966-2015, as well as for two 10-year periods, 1976-1985 and 2006-2015 to examine the time evolution.\n\nUsing posterior predictive samples of time series for each of the three out-of-sample locations, we compute the mean and 90% credible interval of the probability density for events lasting 3 days, 4-5 days, 6-7 days, and 8 or more days. That is, for each time series, we compute the proportion of EHEs during a specified time window lasting each of these durations and then compute the mean and 90% credible intervals of these estimates over the posterior predictive time series. These estimates are shown for each of the locations computed over the entire time window in Figure\n\n6. The empirical probabilities are also shown for each location and duration length. The results reveal that, for each site and for each duration bin, our predictive intervals always capture the observed/true proportion. Similar plots are included in the supplementary material for the two decades of interest, 1976-1985 and 2006-2015 (Figure S2) and reveal similar results.", null, "Figure 6: Posterior predictive mean estimates and 90% credible intervals of the probability density for the durations of extreme heat events across the years 1966-2015. For each of the three out-of-sample locations, true duration density is plotted for the durations 3 days, 4-5 days, 6-7 days, and 8 or more days.\n\nNext, we turn to the two measures of the intensity of an EHE, the average and maximum excess (degrees over the threshold) during the event (Abaurrea et al., 2007). Following the WMO suggestion provided in Section 1, we focused on EHEs lasting three or mores days. For each posterior predictive time series, we computed the average (or maximum) excess for each EHE and then obtained the cumulative probability of the average (or maximum) excess being greater than or equal to a set of discrete values. These cumulative probabilities provide an estimate of the distribution of average and maximum excess for each location. Using all of the posterior predictive time series samples, we computed the mean and 90% credible of these cumulative probabilities for each location. The estimates are shown in Figure 7 for the average (top) and maximum (bottom) excess for each out of sample locations over the entire time window of the study. Our model appears to capture these cumulative probabilities well for both average and maximum exceedance. Similar plots are shown in Figures S3 and S4 of the supplementary material for the years 1976-1985 and 2006-2015 with similar conclusions.", null, "Figure 7: Posterior predictive mean estimates and 90% credible intervals of the cumulative probability of the average (top) and maximum (bottom) exceedance being at or greater than the specified level during an EHE lasting 3 of more days.\n\n### 5.3 Model inference\n\n#### 5.3.1 Models for daily maximum temperature\n\nBoth the above and below threshold processes for daily maximum temperature include global and local spatial intercepts, annual random effects, fixed effects of trigonometric terms to provide annual seasonality, spatial covariates, elevation and latitude, and an autoregressive term to model temporal autocorrelation. The posterior distributions for the autoregressive coefficients and the coefficients of the spatial covariates are summarized by their mean and 90% credible intervals in Table 5. The temporal dependence is similar in both models with a high autoregressive coefficient of approximately 0.7. The elevation coefficient, which represents the gradient of temperature with respect to the elevation, is negative in both cases, but slightly smaller in magnitude for the below threshold process. On the other hand, there is evidence of a latitudinal effect only in the below threshold process, where it reflects a temperature decrease with increasing (northern) latitudes. The credible intervals are more precise in the above threshold process due to a larger sample size.\n\nBoxplots showing the posterior distributions of the annual random effects, the ’s, in the above and below models are shown in Figure 8. The time evolution is quite different for the two models: while the below threshold process shows a clear increasing trend of daily maximum temperature through time, no monotonic temporal trend is detected in the above threshold process. This signifies that the model detects an overall warming trend in daily maximum temperature between the years 1966 and 2015, yet the extreme heat temperatures are remaining relatively constant. A similar conclusion was obtained in Abaurrea et al. (2007), who did not find any evidence of trend in the distribution of the maximum and mean intensity of EHEs from daily temperature data obtained for the same region. This important inference can only be obtained from specifying different models for daily maximum temperatures above and below the threshold.", null, "Figure 8: Boxplots of the posterior distributions of the annual random effects γ, for the above (top) and below (bottom) threshold processes for daily maximum temperature.\n\nThe posterior distributions of the spatial random effects, the ’s, at the observed locations suggest that, in general, the spatial local adjustment to the global intercept is not very strong. Only the posterior distributions for Pallaruelo de Monegros and Tornos did not contain zero (supplementary material Figure S5). We did detect spatial differences in the standard deviations, , across the region, both in the above and below threshold process, supporting the hypothesis that variation in different locations is not the same. Additionally, the mean of the spatial processes of the above and below threshold standard deviations had posterior means of 1.75 and 2.75, respectively. This is in accord with our exploratory data analysis in Section 2. Boxplots of these posterior distributions are given in the supplementary material (Figure S6).\n\n#### 5.3.2 The above and below threshold state models\n\nThe model for the above and below threshold states includes a global and local spatial intercept, fixed effects for the difference between the previous temperature and the threshold, and trigonometric terms. The posterior distributions of the spatial random effects, , show large spatial variation in the model for the above and below threshold states. Boxplots of these posterior distributions are shown in Figure S7 of the supplementary material.\n\nThe posterior mean estimates of the conditional probabilities across day of the year for three values of corresponding to a previous day with temperature below, equal to, and above the threshold, are shown in Figure 9. These conditional probabilities are shown for Tornos (left) and Buñuel (right), as these locations have the most extreme random effects for the local intercept . Since the model is specified using as opposed to raw temperature values, we do not expect to see major differences between these distributions, even though the climate and altitude of these two locations are different. However, we do see larger probabilities for Buñuel than Tornos for all three values of indicating a greater likelihood for EHE events to occur or persist. The mean estimates of the probabilities show a clear seasonal behavior, modeled by the trigonometric terms, with the maximum value occurring on day 203 (July 22, July 21 in leap years). The effect of the previous daily temperature (relative to the threshold) on the probability of being over the threshold is also very strong. For example, the probability in the middle of summer when C is nearly three times the probability when C. This difference is evidence of the increased probability of persistence compared to the probability of the first exceedance day of an EHE.", null, "Figure 9: Posterior mean estimates of P(Ut(s)=1|Yt−1(s)) across day of the year for three values of Yt−1(s). Curves are shown between the minimum and maximum day of the year in which the temperature was observed at or above the given value of Yt−1(s).\n\n## 6 Discussion and future work\n\nWe have proposed a threshold driven two-state model for learning about features of extreme heat events over a year period for the region of Aragón in Spain. These features include incidence, duration, maximum of the daily maximum temperatures over the event, and average exceedance above the threshold over the event. Using leave one out validation, we demonstrate that our model predicts these features well.\n\nFuture work includes the possibility of comparison of results for the region of Aragón with other regions in Spain since national databases are available. In this regard, we could consider a national assessment of EHE characteristics. In order to do this we would likely need to extend our model to include spatially varying coefficients, anticipating that, e.g., mountainous response to predictors would be different from coastal response to the predictors. An additional extension would envision that there is seasonal variation in uncertainty and would encourage time-varying variances beyond our current above and below spatially varying specifications. A further challenge is to use our approach for forecasting future EHE behavior. Using regional climate model scenarios over 50 year windows, we look to provide useful insight into the evolution of EHE behavior over this future time period.\n\n## References\n\n• J. Abaurrea, J. Asín, A. C. Cebrián, and A. Centelles (2007) Modeling and forecasting extreme hot events in the central Ebro valley, a continental-Mediterranean area. Global and Planetary Change 57 (1-2), pp. 43–58. Cited by: §4.2, §5.2, §5.3.1.\n• J. Abaurrea, J. Asín, and A. C. Cebrián (2018) Modelling the occurrence of heat waves in maximum and minimum temperatures over Spain and projections for the period 2031-60. Global and Planetary Change 161 (), pp. 244–260. Cited by: §1.\n• J. Abaurrea, J. Asín, and A. C. Cebrián (2015) Modeling and projecting the occurrence of bivariate extreme heat events using a non-homogeneous common Poisson shock process. Stochastic Environmental Research and Risk Assessmeant 29, pp. 309–322. Cited by: §1, §1.\n• L. V. Alexander (2016) Global observed long-term changes in temperature and precipitation extremes: a review of progress and limitations in IPCC assessments and beyond. Weather and Climate Extremes 11, pp. 4 – 16. Cited by: §1.\n• A. Amengual, V. Homar, R. Romero, H.E. Brooks, C. Ramis, M. Gordaliza, and S. Alonso (2014) Projections of heat waves with high impact on human health in Europe. Global and Planetary Change 119 (), pp. 71 – 84. Cited by: §1.\n• S. Campbell, T. A. Remenyi, C. J. White, and F. H. Johnston (2018) Heatwave and health impact research: a global review. Health & Place 53, pp. 210 – 218. Cited by: §1.\n• L. Cheng, A. Agha Kouchak, E. Gilleland, and R. W. Katz (2014) Non-stationary extreme value analysis in a changing climate. Climatic Change 127 (2), pp. 353–69. Cited by: §1.\n• D. Cooley, D. Nychka, and P. Naveau (2007) Bayesian spatial modeling of extreme precipitation return levels. Journal of the American Statistical Association 102 (479), pp. 824–840. Cited by: §1.\n• O. R. García-Cueto, M. T. Cavazos, P. de Grau, and N. Santillán-Soto (2014) Analysis and modeling of extreme temperatures in several cities in Northwestern Mexico under climate change conditions. Theoretical and Applied Climatology 116 (1-2), pp. 211–25. Cited by: §1.\n• D. Guillot, B. Rajaratnam, and J. Emile-Geay (2015) Statistical paleoclimate reconstructions via Markov random fields. The Annals of Applied Statistics 9 (1), pp. 324–352. Cited by: §1.\n• D. Keellings and P. Waylen (2014) Increased risk of heat waves in Florida: characterizing changes in bivariate heat wave risk using extreme value analysis. Applied Geography 46, pp. 90 – 97. Cited by: §1.\n• D. Keellings and P. Waylen (2015) Investigating teleconnection drivers of bivariate heat waves in Florida using extreme value analysis. Climate Dynamics 44 (11), pp. 3383–3391. Cited by: §1.\n• M. N. Khaliq, A. St-Hilaire, T. B. M. J. Ouarda, and B. Bobèe (2005) Frequency analysis and temporal pattern of occurrences of southern Quebec heatwaves. International Journal of Climatology 25 (4), pp. 485–504. Cited by: §1.\n• Y. Lai and D. A. Dzombak (2019) Use of historical data to assess regional climate change. Journal of Climate 32 (14), pp. 4299–4320. Cited by: §1.\n• A. Lemonsu, A. L. Beaulant, S. Somot, and V. Masson (2014) Evolution of heat wave occurrence over the Paris basin (France) in the 21st century. Climate Research 61 (), pp. 75–91. Cited by: §1.\n• K. Lum and A. E. Gelfand (2012) Spatial quantile multiple regression using the asymmetric Laplace process. Bayesian Analysis 7 (2), pp. 235–258. Cited by: §4.2.\n• I. Murray, R. P. Adams, and D. J. C. MacKay (2010) Elliptical slice sampling.\n\nJournal of Machine Learning Research: Workshop and Conference Proceedings (AISTATS)\n\n9, pp. 541?548.\nCited by: §4.1.\n• Y. Ogata (1988) Statistical models for earthquake occurrences and residual analysis for point processes. Journal of the American Statistical Association 83 (401), pp. 9–27. Cited by: §1.\n• S. E. Perkins and L. V. Alexander (2013) On the Measurement of Heat Waves. Journal of Climate 26 (13), pp. 4500–4517. Cited by: §1.\n• B. J. Reich, B. A. Shaby, and D. Cooley (2014) A hierarchical model for serially-dependent extremes: a study of heat waves in the western US. Journal of Agricultural, Biological, and Environmental Statistics 19 (1), pp. 119–135. Cited by: §1.\n• B. J. Reich and B. A. Shaby (2019) A spatial markov model for climate extremes. Journal of Computational and Graphical Statistics 28 (1), pp. 117–126. Cited by: §1.\n• B. A. Shaby, B. J. Reich, D. Cooley, and C. G. Kaufman (2016) A Markov-switching model for heat waves. The Annals of Applied Statistics 10 (1), pp. 74–93. Cited by: §1, §1.\n• R. L. Smith, J. A. Tawn, and S. G. Coles (1997) Markov chain models for threshold exceedances. Biometrika 84 (2), pp. 249–268. Cited by: §1.\n• T. Smith, B. Zaitchik, and J. Gohlke (2013) Heat waves in the United States: definitions, patterns and trends. Climatic Change 118 (3), pp. 811–825. Cited by: §1." ]

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[ "Вы находитесь на странице: 1из 72\n\n# Report ITU-R M.\n\n2135-1\n(12/2009)\n\n## Guidelines for evaluation of radio interface\n\nM Series\nand related satellites services\nii Rep. ITU-R M.2135-1\n\nForeword\nThe role of the Radiocommunication Sector is to ensure the rational, equitable, efficient and economical use of the\nradio-frequency spectrum by all radiocommunication services, including satellite services, and carry out studies without\nlimit of frequency range on the basis of which Recommendations are adopted.\nThe regulatory and policy functions of the Radiocommunication Sector are performed by World and Regional\n\n## Policy on Intellectual Property Right (IPR)\n\nITU-R policy on IPR is described in the Common Patent Policy for ITU-T/ITU-R/ISO/IEC referenced in Annex 1 of\nResolution ITU-R 1. Forms to be used for the submission of patent statements and licensing declarations by patent\nholders are available from http://www.itu.int/ITU-R/go/patents/en where the Guidelines for Implementation of the\nCommon Patent Policy for ITU-T/ITU-R/ISO/IEC and the ITU-R patent information database can also be found.\n\n## Series of ITU-R Reports\n\n(Also available online at http://www.itu.int/publ/R-REP/en)\n\nSeries Title\n\nBO Satellite delivery\nBR Recording for production, archival and play-out; film for television\nF Fixed service\nM Mobile, radiodetermination, amateur and related satellite services\nRS Remote sensing systems\nS Fixed-satellite service\nSA Space applications and meteorology\nSF Frequency sharing and coordination between fixed-satellite and fixed service systems\nSM Spectrum management\n\nNote: This ITU-R Report was approved in English by the Study Group under the procedure detailed\nin Resolution ITU-R 1.\n\nElectronic Publication\nGeneva, 2010\n ITU 2010\nAll rights reserved. No part of this publication may be reproduced, by any means whatsoever, without written permission of ITU.\nRep. ITU-R M.2135-1 1\n\n## Guidelines for evaluation of radio interface technologies\n\n(2008-2009)\n\n1 Introduction\nthat include the new capabilities of IMT that go beyond those of IMT-2000. Such systems provide\nsupported by mobile and fixed networks, which are increasingly packet-based.\nIMT-Advanced systems support low to high mobility applications and a wide range of data rates in\naccordance with user and service demands in multiple user environments. IMT-Advanced also has\ncapabilities for high-quality multimedia applications within a wide range of services and platforms\nproviding a significant improvement in performance and quality of service.\nThe key features of IMT-Advanced are:\n– a high degree of commonality of functionality worldwide while retaining the flexibility to\nsupport a wide range of services and applications in a cost efficient manner;\n– compatibility of services within IMT and with fixed networks;\n– capability of interworking with other radio access systems;\n– high-quality mobile services;\n– user equipment suitable for worldwide use;\n– user-friendly applications, services and equipment;\n– worldwide roaming capability;\n– enhanced peak data rates to support advanced services and applications (100 Mbit/s for\nhigh and 1 Gbit/s for low mobility were established as targets for research)1.\nThe capabilities of IMT-Advanced systems are being continuously enhanced in line with user trends\nand technology developments.\n\n2 Scope\nThis Report provides guidelines for both the procedure and the criteria (technical, spectrum and\nservice) to be used in evaluating the proposed IMT-Advanced radio interface technologies (RITs) or\nSets of RITs (SRITs) for a number of test environments and deployment scenarios for evaluation.\nThese test environments are chosen to simulate closely the more stringent radio operating\nenvironments. The evaluation procedure is designed in such a way that the overall performance of\nthe candidate RIT/SRITs may be fairly and equally assessed on a technical basis. It ensures that the\n\n## 1 Data rates sourced from Recommendation ITU-R M.1645.\n\n2 Rep. ITU-R M.2135-1\n\nThis Report provides, for proponents, developers of candidate RIT/SRITs and evaluation groups,\nthe common methodology and evaluation configurations to evaluate the proposed candidate\nRIT/SRITs and system aspects impacting the radio performance.\nThis Report allows a degree of freedom so as to encompass new technologies. The actual selection\nof the candidate RIT/SRITs for IMT-Advanced is outside the scope of this Report.\nThe candidate RIT/SRITs will be assessed based on those evaluation guidelines. If necessary,\nadditional evaluation methodologies may be developed by each independent evaluation group to\ncomplement the evaluation guidelines. Any such additional methodology should be shared between\nevaluation groups and sent to the Radiocommunication Bureau as information in the consideration\nof the evaluation results by ITU-R and for posting under additional information relevant to the\nevaluation group section of the ITU-R IMT-Advanced web page (http://www.itu.int/ITU-R/go/rsg5-\n\n## 3 Structure of the Report\n\nSection 4 provides a list of the documents that are related to this Report.\nSection 5 describes the evaluation guidelines.\nSection 6 lists the criteria chosen for evaluating the RITs.\nSection 7 outlines the procedures and evaluation methodology for evaluating the criteria.\nSection 8 defines the tests environments and selected deployment scenarios for evaluation; the\nevaluation configurations which shall be applied when evaluating IMT-Advanced candidate\ntechnology proposals are also given in this section.\nSection 9 describes a channel model approach for the evaluation.\nSection 10 provides a list of references.\nSection 11 provides a list of acronyms and abbreviations.\nAnnexes 1 and 2 form a part of this Report.\n\n## 4 Related ITU-R texts\n\nResolution ITU-R 57\nRecommendation ITU-R M.1224\nRecommendation ITU-R M.1822\nRecommendation ITU-R M.1645\nRecommendation ITU-R M.1768\nReport ITU-R M.2038\nReport ITU-R M.2072\nReport ITU-R M.2074\nReport ITU-R M.2078\nReport ITU-R M.2079\nReport ITU-R M.2133\nReport ITU-R M.2134.\nRep. ITU-R M.2135-1 3\n\n5 Evaluation guidelines\nIMT-Advanced can be considered from multiple perspectives, including the users, manufacturers,\napplication developers, network operators, and service and content providers as noted in § 4.2.2 in\nRecommendation ITU-R M.1645 − Framework and overall objectives of the future development of\nIMT-2000 and systems beyond IMT-2000. Therefore, it is recognized that the technologies for\nIMT-Advanced can be applied in a variety of deployment scenarios and can support a range of\nenvironments, different service capabilities, and technology options. Consideration of every\nvariation to encompass all situations is therefore not possible; nonetheless the work of the ITU-R\nhas been to determine a representative view of IMT-Advanced consistent with the process defined\nin Resolution ITU-R 57 − Principles for the process of development of IMT-Advanced, and the\nrequirements defined in Report ITU-R M.2134 − Requirements related to technical performance for\nThe parameters presented in this Report are for the purpose of consistent definition, specification,\nand evaluation of the candidate RITs/SRITs for IMT-Advanced in ITU-R in conjunction with the\ndevelopment of Recommendations and Reports such as the framework and key characteristics and\nthe detailed specifications of IMT-Advanced. These parameters have been chosen to be\nrepresentative of a global view of IMT-Advanced but are not intended to be specific to any\nparticular implementation of an IMT-Advanced technology. They should not be considered as the\nvalues that must be used in any deployment of any IMT-Advanced system nor should they be taken\nas the default values for any other or subsequent study in ITU or elsewhere.\nFurther consideration has been given in the choice of parameters to balancing the assessment of the\ntechnology with the complexity of the simulations while respecting the workload of an evaluator or\ntechnology proponent.\nThis procedure deals only with evaluating radio interface aspects. It is not intended for evaluating\nsystem aspects (including those for satellite system aspects).\nThe following principles are to be followed when evaluating radio interface technologies for\n− Evaluations of proposals can be through simulation, analytical and inspection procedures.\n− The evaluation shall be performed based on the submitted technology proposals, and should\nfollow the evaluation guidelines, use the evaluation methodology and adopt the evaluation\nconfigurations defined in this Report.\n− Evaluations through simulations contain both system level simulations and link level\nsimulations. Evaluation groups may use their own simulation tools for the evaluation.\n− In case of analytical procedure the evaluation is to be based on calculations using the\ntechnical information provided by the proponent.\n− In case of evaluation through inspection the evaluation is based on statements in the\nproposal.\nThe following options are foreseen for the groups doing the evaluations.\n− Self-evaluation must be a complete evaluation (to provide a fully complete compliance\ntemplate) of the technology proposal.\n− An external evaluation group may perform complete or partial evaluation of one or several\ntechnology proposals to assess the compliance of the technologies with the minimum\n− Evaluations covering several technology proposals are encouraged.\n4 Rep. ITU-R M.2135-1\n\n## 6 Characteristics for evaluation\n\nThe technical characteristics chosen for evaluation are explained in detail in Report ITU-R\nM.2133 − Requirements, evaluation criteria and submission templates for the development of\nIMT-Advanced, § 2, including service aspect requirements which are based on Recommendation\nITU-R M.1822, spectrum aspect requirements, and requirements related to technical performance,\nwhich are based on Report ITU-R M.2134. These are summarised in Table 6-1, together with the\nhigh level assessment method:\n− Simulation (including system and link-level simulations, according to the principles of\nsimulation procedure given in § 7.1).\n− Analytical (via a calculation).\n− Inspection (by reviewing the functionality and parameterisation of the proposal).\n\nTABLE 6-1\n\n## Characteristic for Method Evaluation Related section of Reports\n\nevaluation methodology / ITU-R M.2134 and\nconfigurations ITU-R M.2133\nCell spectral efficiency Simulation § 7.1.1, Tables 8-2, 8-4 Report ITU-R M.2134, § 4.1\n(system level) and 8-5\nPeak spectral efficiency Analytical § 7.3.1, Table 8-3 Report ITU-R M.2134, § 4.2\nBandwidth Inspection § 7.4.1 Report ITU-R M.2134, § 4.3\nCell edge user spectral Simulation (system § 7.1.2, Tables, 8-2, 8-4 Report ITU-R M.2134, § 4.4\nefficiency level) and 8-5\nControl plane latency Analytical § 7.3.2, Table 8-2 Report ITU-R M.2134, § 4.5.1\nUser plane latency Analytical § 7.3.3; Table 8-2 Report ITU-R M.2134, § 4.5.2\nMobility Simulation (system § 7.2, Tables 8-2 and 8-7 Report ITU-R M.2134, § 4.6\nIntra- and inter-frequency Analytical § 7.3.4, Table 8-2 Report ITU-R M.2134, § 4.7\nhandover interruption time\nInter-system handover Inspection § 7.4.3 Report ITU-R M.2134, § 4.7\nVoIP capacity Simulation (system § 7.1.3, Tables 8-2, 8-4 Report ITU-R M.2134, § 4.8\nlevel) and 8-6\nDeployment possible in at Inspection § 7.4.2 Report ITU-R M.2133, § 2.2\nleast one of the identified\nIMT bands\nChannel bandwidth Inspection § 7.4.1 Report ITU-R M.2134, § 4.3\nscalability\nSupport for a wide range of Inspection § 7.4.4 Report ITU-R M.2133, § 2.1\nservices\n\n## Section 7 defines the methodology for assessing each of these criteria.\n\n7 Evaluation methodology\nThe submission and evaluation process is defined in Document IMT-ADV/2(Rev.1) −Submission\nand evaluation process and consensus building.\nRep. ITU-R M.2135-1 5\n\nEvaluation should be performed in strict compliance with the technical parameters provided by the\nproponents and the evaluation configurations specified for the deployment scenarios in § 8.4 of this\nReport. Each requirement should be evaluated independently, except for the cell spectral efficiency\nand cell edge user spectral efficiency criteria that shall be assessed jointly using the same\nsimulation, and that consequently the candidate RIT/SRITs also shall fulfil the corresponding\nminimum requirements jointly. Furthermore, the system simulation used in the mobility evaluation\nshould be the same as the system simulation for cell spectral efficiency and cell edge user spectral\nefficiency.\nThe evaluation methodology should include the following elements:\n1 Candidate RIT/SRITs should be evaluated using reproducible methods including computer\nsimulation, analytical approaches and inspection of the proposal.\n2 Technical evaluation of the candidate RIT/SRITs should be made against each evaluation\ncriterion for the required test environments.\n3 Candidate RIT/SRITs should be evaluated based on technical descriptions that are\nsubmitted using a technologies description template.\nIn order to have a good comparability of the evaluation results for each proposal, the following\nsolutions and enablers are to be taken into account:\n− Use of unified methodology, software, and data sets by the evaluation groups wherever\npossible, e.g. in the area of channel modelling, link-level data, and link-to-system-level\ninterface.\n− Evaluation of multiple proposals using one simulation tool by each evaluation group is\nencouraged.\n− Question-oriented working method that adapts the level of detail in modelling of specific\nfunctionalities according to the particular requirements of the actual investigation.\nEvaluation of cell spectral efficiency, cell edge user spectral efficiency and VoIP capacity of\ncandidate RIT/SRITs should take into account the Layer 1 and Layer 2 overhead information\nprovided by the proponents, which may vary when evaluating different performance metrics and\ndeployment scenarios.\n\n## 7.1 System simulation procedures\n\nSystem simulation shall be based on the network layout defined in § 8.3 of this Report. The\nfollowing principles shall be followed in system simulation:\n− Users are dropped independently with uniform distribution over predefined area of the\nnetwork layout throughout the system. Each mobile corresponds to an active user session\nthat runs for the duration of the drop.\n− Mobiles are randomly assigned LoS and NLoS channel conditions.\n− Cell assignment to a user is based on the proponent’s cell selection scheme, which must be\ndescribed by the proponent.\n− The minimum distance between a user and a base station is defined in Table 8-2 in § 8.4 of\nthis Report.\n− Fading signal and fading interference are computed from each mobile station into each cell\nand from each cell into each mobile station (in both directions on an aggregated basis).\n6 Rep. ITU-R M.2135-1\n\n− The IoT2 (interference over thermal) parameter is an uplink design constraint that the\nproponent must take into account when designing the system such that the average IoT\nvalue experienced in the evaluation is equal to or less than 10 dB.\n− In simulations based on the full-buffer traffic model, packets are not blocked when they\narrive into the system (i.e. queue depths are assumed to be infinite).\n− Users with a required traffic class shall be modelled according to the traffic models defined\nin Annex 2.\n− Packets are scheduled with an appropriate packet scheduler(s) proposed by the proponents\nfor full buffer and VoIP traffic models separately. Channel quality feedback delay,\nfeedback errors, PDU (protocol data unit) errors and real channel estimation effects\ninclusive of channel estimation error are modelled and packets are retransmitted as\nnecessary.\n− The overhead channels (i.e., the overhead due to feedback and control channels) should be\nrealistically modelled.\n− For a given drop the simulation is run and then the process is repeated with the users\ndropped at new random locations. A sufficient number of drops are simulated to ensure\nconvergence in the user and system performance metrics. The proponent should provide\ninformation on the width of confidence intervals of user and system performance metrics of\ncorresponding mean values, and evaluation groups are encouraged to provide this\ninformation.3\n− Performance statistics are collected taking into account the wrap-around configuration in\nthe network layout, noting that wrap-around is not considered in the indoor case.\n− All cells in the system shall be simulated with dynamic channel properties using a wrap-\naround technique, noting that wrap-around is not considered in the indoor case.\nIn order to perform less complex system simulations, often the simulations are divided into separate\n‘link’ and ‘system’ simulations with a specific link-to-system interface. Another possible way to\nreduce system simulation complexity is to employ simplified interference modelling. Such methods\nshould be sound in principle, and it is not within the scope of this document to describe them.\nEvaluation groups are allowed to use such approaches provided that the used methodologies are:\n− well described and made available to the Radiocommunication Bureau and other evaluation\ngroups;\n− included in the evaluation report.\nRealistic link and system models should include error modelling, e.g., for channel estimation and\nfor the errors of control channels that are required to decode the traffic channel (including the\nfeedback channel and channel quality information). The overheads of the feedback channel and the\ncontrol channel should be modelled according to the assumptions used in the overhead channels’\n\n2 The interference means the effective interference received at the base station.\n3 The confidence interval and the associated confidence level indicate the reliability of the estimated\nparameter value. The confidence level is the certainty (probability) that the true parameter value is within\nthe confidence interval. The higher the confidence level the larger the confidence interval.\nRep. ITU-R M.2135-1 7\n\n## 7.1.1 Cell spectral efficiency\n\nThe results from the system simulation are used to calculate the cell spectral efficiency as defined in\nReport ITU-R M.2134, § 4.1. The necessary information includes the number of correctly received\nbits during the simulation period and the effective bandwidth which is the operating bandwidth\nLayer 1 and Layer 2 overhead should be accounted for in time and frequency for the purpose of\ncalculation of system performance metrics such as cell spectral efficiency, cell edge user spectral\nefficiency, and VoIP. Examples of Layer 1 overhead include synchronization, guard and DC\nsubcarriers, guard/switching time (in TDD systems), pilots and cyclic prefix. Examples of Layer 2\noverhead include common control channels, HARQ ACK/NACK signalling, channel feedback,\nrandom access, packet headers and CRC. It must be noted that in computing the overheads, the\nfraction of the available physical resources used to model control overhead in Layer 1 and Layer 2\nshould be accounted for in a non-overlapping way. Power allocation/boosting should also be\naccounted for in modelling resource allocation for control channels.\n7.1.2 Cell edge user spectral efficiency\nThe results from the system simulation are used to calculate the cell edge user spectral efficiency as\ndefined in Report ITU-R M.2134, § 4.4. The necessary information is the number of correctly\nreceived bits per user during the active session time the user is in the simulation. The effective\nratio for TDD system. It should be noted that the cell edge user spectral efficiency shall be\nevaluated using identical simulation assumptions as the cell spectral efficiency for that test\nenvironment.\nExamples of Layer 1 and Layer 2 overhead can be found in § 7.1.1.\n7.1.3 VoIP capacity\nThe VoIP capacity should be evaluated and compared against the requirements in Report\nITU-R M.2134, § 4.8.\nVoIP capacity should be evaluated for the uplink and downlink directions assuming a 12.2 kbit/s\ncodec with a 50% activity factor such that the percentage of users in outage is less than 2%, where a\nuser is defined to have experienced a voice outage if less than 98% of the VoIP packets have been\ndelivered successfully to the user within a permissible VoIP packet delay bound of 50 ms. The\nVoIP packet delay is the overall latency from the source coding at the transmitter to successful\nThe final VoIP capacity which is to be compared against the requirements in Report ITU-R M.2134\nis the minimum of the calculated capacity for either link direction divided by the effective\nbandwidth in the respective link direction4.\nThe simulation is run with the duration for a given drop defined in Table 8-6 of this Report. The\nVoIP traffic model is defined in Annex 2.\n\n## 7.2 Evaluation methodology for mobility requirements\n\nThe evaluator shall perform the following steps in order to evaluate the mobility requirement.\n\n4 In other words, the effective bandwidth is the operating bandwidth normalised appropriately considering\n8 Rep. ITU-R M.2135-1\n\nStep 1: Run system simulations, identical to those for cell spectral efficiencies, see § 7.1.1 except\nfor speeds taken from Table 4 of Report ITU-R M.2134, using link level simulations and a link-to-\nsystem interface appropriate for these speed values, for the set of selected test environment(s)\nassociated with the candidate RIT/SRIT proposal and collect overall statistics for uplink C/I values,\nand construct cumulative distribution function (CDF) over these values for each test environment.\nStep 2: Use the CDF for the test environment(s) to save the respective 50%-percentile C/I value.\nStep 3: Run new uplink link-level simulations for the selected test environment(s) for both NLoS\nand LoS channel conditions using the associated speeds in Table 4 of Report ITU-R M.2134, § 4.6\nas input parameters, to obtain link data rate and residual packet error rate as a function of C/I. The\nlink-level simulation shall use air interface configuration(s) supported by the proposal and take into\naccount retransmission.\nStep 4: Compare the link spectral efficiency values (link data rate normalized by channel\nbandwidth) obtained from Step 3 using the associated C/I value obtained from Step 2 for each\nchannel model case, with the corresponding threshold values in the Table 4 of Report ITU-R\nM.2134, § 4.6.\nStep 5: The proposal fulfils the mobility requirement if the spectral efficiency value is larger than\nor equal to the corresponding threshold value and if also the residual decoded packet error rate is\nless than 1%, for all selected test environments. For each test environment it is sufficient if one of\nthe spectral efficiency values (of either NLoS or LoS channel conditions) fulfil the threshold.\n\n## 7.3 Analytical approach\n\nFor the characteristics below a straight forward calculation based on the definition in Report\nITU-R M.2134 and information in the proposal will be enough to evaluate them. The evaluation\nshall describe how this calculation has been performed. Evaluation groups should follow the\ncalculation provided by proponents if it is justified properly.\n7.3.1 Peak spectral efficiency calculation\nThe peak spectral efficiency is calculated as specified in Report ITU-R M.2134 § 4.2. The antenna\nconfiguration to be used for peak spectral efficiency is defined in Table 8-3 of this Report. The\nnecessary information includes effective bandwidth which is the operating bandwidth normalised\nappropriately considering the uplink/downlink ratio for TDD systems. Examples of Layer 1\noverhead can be found in § 7.1.1.\n7.3.2 Control plane latency calculation\nThe control plane latency is calculated as specified in Report ITU-R M.2134, § 4.5.1.\nThe proponent should provide the elements in the calculation of the control plane latency and the\nretransmission probability.\nTable 7-1 provides, for the purpose of example, typical elements in the calculation of the control\nplane latency. The inclusion of H-ARQ/ARQ retransmissions in each step of the connection set up\nis to ensure the required reliability of connection which typically has a probability of error in the\norder of 10−2 for certain Layer 2 control signals and 10−6 for some RRC control signals.\nRep. ITU-R M.2135-1 9\n\nTABLE 7-1\nExample C-plane latency template\n\n## Step Description Duration\n\nImplementation\n0 UT wakeup time\ndependent\n1 DL scanning and synchronization + Broadcast channel acquisition\n2 Random access procedure\n3 UL synchronization\n4 Capability negotiation + H-ARQ retransmission probability\n5 Authorization and authentication/key exchange +H-ARQ retransmission probability\n6 Registration with the BS + H-ARQ retransmission probability\n7 RRC connection establishment + H-ARQ retransmission probability\nTotal C-plane connection establishment delay\nTotal IDLE_STATE –> ACTIVE_ACTIVE delay\n\n## 7.3.3 User plane latency calculation\n\nThe user plane latency is calculated as specified in Report ITU-R M.2134, § 4.5.2. The proponent\nshould provide the elements in the calculation of the user plane latency and the retransmission\nprobability.\nTable 7-2 provides, for the purpose of example, typical elements in the calculation of the user plane\nlatency:\n\nTABLE 7-2\nExample user-plane latency analysis template\n\n## Step Description Value\n\n0 UT wakeup time Implementation dependent\n1 UT processing delay\n2 Frame alignment\n3 TTI for UL data packet (piggy-back scheduling information)\n4 HARQ retransmission (probability)\n5 BS processing delay\n6 Transfer delay between the BS and the BS-Core Network interface\n7 Processing delay of the BS-Core Network interface\nTotal one way delay\n\n## 7.3.4 Intra- and inter-frequency handover interruption time derivation\n\nThe intra- and inter-frequency handover interruption time is calculated as specified in Report\nITU-R M.2134, § 4.7. The handover procedure shall be described based on the proposed technology\nincluding the functions and the timing involved.\n10 Rep. ITU-R M.2135-1\n\n7.4 Inspection\n7.4.1 Bandwidth and channel bandwidth scalability\nThe support of maximum bandwidth required in Report ITU-R M.2134, § 4.3 is verified by\ninspection of the proposal.\nThe scalability requirement is verified by demonstrating that the candidate RIT or SRIT can support\nat least three bandwidth values. These values shall include the minimum and maximum supported\nbandwidth values of the candidate RIT or SRIT.\n7.4.2 Deployment in IMT bands\nThe set of IMT bands supported is demonstrated by inspection of the proposal.\n7.4.3 Inter-system handover\nThe support of inter-system handover as required in Report ITU-R M.2134, § 4.7 is verified by\ninspection of the proposal.\n7.4.4 Support of a wide range of services\nA mobile transmission system’s ability to support a wide range of services lies across all elements\nof the network (i.e. core, distribution and access), and across all layers of the OSI model. The\nevaluation of a candidate IMT-Advanced RIT focuses on the radio access aspects of the lower OSI\nlayers. There are quantifiable elements of the minimum technical requirements identified within\nReport ITU-R M.2134 that indicate whether or not a candidate RIT is capable of enabling these\nservices as defined in Recommendation ITU-R M.1822. If the candidate RIT meets the latency,\npeak spectral efficiency and bandwidth requirements in Report ITU-R M.2134, then it can be\nregarded as enabling the following service aspects requirements.\nThe support of a wide range of services is further analysed by inspection of the candidate RIT’s\nability to support all of the service classes of Table 7-3. This is considered in at least one test\nenvironment (similar to evaluation of the peak spectral efficiency) under normal operating\nconditions using configuration supported by the candidate RIT/SRITs.\n\nTABLE 7-3\nService classes for evaluation\n\n## User experience class Service class Inspection\n\nConversational Basic conversational service Yes/No\nRich conversational service Yes/No\nConversational low delay Yes/No\nInteractive Interactive high delay Yes/No\nInteractive low delay Yes/No\nStreaming Streaming live Yes/No\nStreaming non-live Yes/No\nBackground Background Yes/No\n\n## 8 Test environments and evaluation configurations\n\nThis section describes the test environments, selected deployment scenarios and evaluation\nconfigurations (including simulation parameters) necessary to evaluate the performance figures of\ncandidate RIT/SRITs (details of test environments and channel models can be found in Annex 1).\nRep. ITU-R M.2135-1 11\n\nThe predefined test environments are used in order to specify the environments of the requirements\nfor the technology proposals. IMT-Advanced is to cover a wide range of performance in a wide\nrange of environments. Although it should be noted that thorough testing and evaluation is\nprohibitive. The test environments have therefore been chosen such that typical and different\ndeployments are modelled and critical questions in system design and performance can be\ninvestigated. Focus is thus on scenarios testing limits of performance related to capacity and user\nmobility.\n\n## 8.1 Test environments\n\nEvaluation of candidate RIT/SRITs will be performed in selected scenarios of the following test\nenvironments:\n− Indoor: an indoor environment targeting isolated cells at offices and/or in hotspot based on\nstationary and pedestrian users.\n− Microcellular: an urban micro-cellular environment with higher user density focusing on\npedestrian and slow vehicular users.\n− Base coverage urban: an urban macro-cellular environment targeting continuous coverage\nfor pedestrian up to fast vehicular users.\n− High speed: macro cells environment with high speed vehicular and trains.\n8.1.1 Indoor test environment\nThe indoor test environment focuses on smallest cells and high user throughput or user density in\nbuildings. The key characteristics of this test environment are high user throughput or user density\nin indoor coverage.\n8.1.2 Microcellular test environment\nThe microcellular test environment focuses on small cells and high user densities and traffic loads\nin city centres and dense urban areas. The key characteristics of this test environment are high\ntraffic loads, outdoor and outdoor-to-indoor coverage. This scenario will therefore be interference-\nlimited, using micro cells. A continuous cellular layout and the associated interference shall be\nassumed. Radio access points shall be below rooftop level.\nA similar scenario is used to the base coverage urban test environment but with reduced site-to-site\ndistance and the antennas below rooftops.\n8.1.3 Base coverage urban test environment\nThe base coverage urban test environment focuses on large cells and continuous coverage. The key\ncharacteristics of this test environment are continuous and ubiquitous coverage in urban areas. This\nscenario will therefore be interference-limited, using macro cells (i.e. radio access points above\nrooftop level).\nIn urban macro-cell scenario mobile station is located outdoors at street level and fixed base station\nantenna clearly above surrounding building heights. As for propagation conditions, non- or\nobstructed line-of-sight is a common case, since street level is often reached by a single diffraction\nover the rooftop.\n8.1.4 High-speed test environment\nThe high-speed test environment focuses on larger cells and continuous coverage. The key\ncharacteristics of this test environment are continuous wide area coverage supporting high speed\nvehicles. This scenario will therefore be noise-limited and/or interference-limited, using macro\ncells.\n12 Rep. ITU-R M.2135-1\n\n## 8.2 Deployment scenarios for the evaluation process\n\nThe deployment scenarios that shall be used for each test environment are shown in Table 8-1:\n\nTABLE 8-1\nSelected deployment scenarios for evaluation\nTest environment Indoor Microcellular Base coverage urban High speed\nDeployment Indoor hotspot Urban micro-cell Urban macro-cell Rural macro-cell\nscenario scenario scenario scenario scenario\n\nSuburban macro-cell scenario is an optional scenario for the base coverage urban test environment.\n\n## 8.3 Network layout\n\nIn the rural/high-speed, base coverage urban and microcell cases, no specific topographical details\nare taken into account. Base stations are placed in a regular grid, following hexagonal layout.\nA basic hexagon layout for the example of three cells per site is shown in Fig. 1, where also basic\ngeometry (antenna boresight, cell range, and inter-site distance ISD) is defined. The simulation will\nbe a wrap-around configuration of 19 sites, each of 3 cells. Users are distributed uniformly over the\nwhole area.\nThe amount of channel bandwidth used in a link direction that is used in the simulation is defined as\nthe product of the spectrum bandwidth identified in the tables (Tables 8-5, 8-6, 8-7) as “simulation\nbandwidth” and the frequency reuse factor, when conventional frequency reuse scheme (e.g. 3-cell\nand 7-cell frequency reuse) is considered. The cell spectral efficiency, cell edge user spectral\nefficiency, and VoIP capacity are calculated taking into account the amount of channel bandwidth\nConsider as an example the urban macro cell scenario column for full buffer services in Table 8-5.\nAccording to this table the simulation bandwidths are 10 + 10 MHz for FDD and 20 MHz for TDD\nbased systems.\nAssuming that an FDD based proposal has frequency reuse factor 3 the resulting amount of\nspectrum is 3*10 = 30 MHz for each link, or 60 MHz summed over both directions.\nAlternatively, assuming a TDD based proposal with frequency reuse factor of 3 the resulting\namount of spectrum is 3*20 = 60 MHz (also used for both directions). The fraction of time each\nlink direction is taken into account. However, for TDD also the fraction of time each link direction\nis active should be taken into account to get the effective bandwidth (as an example, the proponent\ncould specify these figures as 60% for UL transmissions and 40% for DL transmissions).\nWhen calculating spectral efficiencies and VoIP capacities for FDD the total spectrum for each\nindividual link should be used in the calculation (30 MHz for each direction in the example), while\nfor TDD the amount of spectrum used for any link direction should be used (60 MHz in the\nexample).\nRep. ITU-R M.2135-1 13\n\nFIGURE 1\nSketch of base coverage urban cell layout without relay nodes\n\n44 43\n\n29 28 45 26 25\n\nig ht\nres\n47 46 30 8 7 27 41 40\nbo\nna\nte n 48 11 10 9 5 4 42\nAn\n32 31 12 2 1 6 23 22\nCe ISD 20\nll r\nang 33 14 13 3 19 24\n\n50 49 15 17 16 21 56 55\ne\n\n51 35 34 18 38 37 57\n\n36 53 52 39\n\n54\n\nReport 2135-01\n\nThe indoor hotspot scenario consists of one floor of a building. The height of the floor is 6 m. The\nfloor contains 16 rooms of 15 m × 15 m and a long hall of 120 m × 20 m. Two sites are placed in\nthe middle of the hall at 30 m and 90 m with respect to the left side of the building (see Fig. 2).\n\nFIGURE 2\nSketch of indoor hotspot environment (one floor)\n\nReport 2135-02\n\n## 8.4 Evaluation configurations\n\nThis section contains baseline configuration parameters that shall be applied in analytical and\nsimulation assessments of candidate RIT/SRITs.\nThe parameters (and also the propagation and channel models in Annex 1) are solely for the\npurpose of consistent evaluation of the candidate RIT/SRITs and relate only to specific test\nenvironments used in these simulations. They should not be considered as the values that must be\nused in any deployment of any IMT-Advanced system nor should they taken as the default values\nfor any other or subsequent study in ITU or elsewhere. They do not necessarily themselves\nconstitute any requirements on the implementation of the system.\nConfiguration parameters in Table 8-2 shall be applied when evaluation groups assess the\ncharacteristics of cell spectral efficiency, cell edge user spectral efficiency, control plane latency,\nuser plane latency, mobility, handover interruption time and VoIP capacity in evaluation of\ncandidate RIT/SRITs.\n14 Rep. ITU-R M.2135-1\n\nTABLE 8-2\nBaseline evaluation configuration parameters\n\n## Deployment scenario for Indoor Urban Urban Rural Suburban\n\nthe evaluation process hotspot micro-cell macro-cell macro-cell macro-cell\nBase station (BS) antenna 6 m, mounted 10 m, below 25 m, above 35 m, above 35 m, above\nheight on ceiling rooftop rooftop rooftop rooftop\nNumber of BS antenna Up to 8 rx Up to 8 rx Up to 8 rx Up to 8 rx Up to 8 rx\nelements(1) Up to 8 tx Up to 8 tx Up to 8 tx Up to 8 tx Up to 8 tx\nTotal BS transmit power 24 dBm for 41 dBm for 46 dBm for 46 dBm for 46 dBm\n40 MHz, 10 MHz, 10 MHz, 10 MHz, for 10 MHz,\n21 dBm for 44 dBm for 49 dBm for 49 dBm for 49 dBm\n20 MHz 20 MHz 20 MHz 20 MHz for 20 MHz\nUser terminal (UT) power 21 dBm 24 dBm 24 dBm 24 dBm 24 dBm\nclass\nUT antenna system(1) Up to 2 tx Up to 2 tx Up to 2 tx Up to 2 tx Up to 2 tx\nUp to 2 rx Up to 2 rx Up to 2 rx Up to 2 rx Up to 2 rx\nMinimum distance between >= 3 m >= 10 m >= 25 m >= 35 m >= 35 m\nUT and serving cell(2)\nCarrier frequency (CF) for 3.4 GHz 2.5 GHz 2 GHz 800 MHz Same as urban\nevaluation (representative of macro-cell\nIMT bands)\nOutdoor to indoor building N.A. See Annex 1, N.A. N.A. 20 dB\npenetration loss Table A1-2\nOutdoor to in-car N.A. N.A. 9 dB (LN, 9 dB (LN, 9 dB (LN,\npenetration loss σ = 5 dB) σ = 5 dB) σ = 5 dB)\n(1)\nThe number of antennas specified by proponent in the technology description template (§ 4.2.3 of Report\nITU-R M.2133) should be used in the evaluations. The numbers shall be within the indicated ranges in this table.\n(2)\nIn the horizontal plane.\n\n## 8.4.1 Evaluation configurations parameters for analytical assessment\n\nConfiguration parameters in Table 8-3 shall be applied when evaluation groups assess the\ncharacteristics of peak spectral efficiency in evaluation of candidate RIT/SRITs.\n\nTABLE 8-3\nEvaluation configuration parameters for analytical assessment of peak spectral efficiency\n\n## Deployment scenario for Indoor Urban Urban Rural Suburban\n\nthe evaluation process hotspot micro-cell macro-cell macro-cell macro-cell\nNumber of BS antenna Up to 4 rx Up to 4 rx Up to 4 rx Up to 4 rx Up to 4 rx\nelements Up to 4 tx Up to 4 tx Up to 4 tx Up to 4 tx Up to 4 tx\nUT antenna system Up to 2 tx Up to 2 tx Up to 2 tx Up to 2 tx Up to 2 tx\nUp to 4 rx Up to 4 rx Up to 4 rx Up to 4 rx Up to 4 rx\nRep. ITU-R M.2135-1 15\n\n## 8.4.2 Evaluation configurations parameters for simulation assessment\n\nThere are two types of simulations: system simulation and link level simulation.\n8.4.2.1 Additional parameters for system simulation\nParameters in Table 8-4 shall also be applied in system simulation when assessing the\ncharacteristics of cell spectral efficiency, cell edge user spectral efficiency and VoIP capacity.\n\nTABLE 8-4\n\n## Deployment Indoor hotspot Urban Urban Rural Suburban\n\nscenario for the micro-cell macro-cell macro-cell macro-cell\nevaluation process\nLayout(1) Indoor floor Hexagonal grid Hexagonal grid Hexagonal grid Hexagonal grid\nInter-site distance 60 m 200 m 500 m 1 732 m 1 299 m\nChannel model Indoor hotspot Urban micro Urban macro Rural macro Suburban macro\nmodel (InH) model (UMi) model (UMa) model (RMa) model (SMa)\nUser distribution Randomly and Randomly and Randomly and Randomly and Randomly and\nuniformly uniformly uniformly uniformly uniformly\ndistributed over distributed over distributed over distributed over distributed over\narea area. 50% users area. 100% of area. 100% of area. 50% users\noutdoor users outdoors users outdoors vehicles and\n(pedestrian in vehicles in high speed 50% of users\nusers) and 50% vehicles indoors\nof users indoors\nUser mobility Fixed and Fixed and Fixed and Fixed and Fixed and\nmodel identical speed identical speed identical speed identical speed identical speed\n|v| of all UTs, |v| of all UTs, |v| of all UTs, |v| of all UTs, |v| of all UTs,\nrandomly and randomly and randomly and randomly and randomly and\nuniformly uniformly uniformly uniformly uniformly\ndistributed distributed distributed distributed distributed\ndirection direction direction direction direction\nUT speeds of 3 km/h 3 km/h 30 km/h 120 km/h Indoor UTs:\ninterest 3 km/h, outdoor\nUTs: 90 km/h\nInter-site Explicitly Explicitly Explicitly Explicitly Explicitly\ninterference modelled modelled modelled modelled modelled\nmodeling(2)\nBS noise figure 5 dB 5 dB 5 dB 5 dB 5 dB\nUT noise figure 7 dB 7 dB 7 dB 7 dB 7 dB\nBS antenna gain 0 dBi 17 dBi 17 dBi 17 dBi 17 dBi\n(boresight)\nUT antenna gain 0 dBi 0 dBi 0 dBi 0 dBi 0 dBi\nThermal noise level –174 dBm/Hz –174 dBm/Hz –174 dBm/Hz –174 dBm/Hz –174 dBm/Hz\n(1)\nSee § 8.3 for further detail.\n(2)\nSee § 7.1.\n16 Rep. ITU-R M.2135-1\n\nWhen assessing the cell spectral efficiency and cell edge user spectral efficiency characteristics,\nparameters in Table 8-5 shall also be applied.\n\nTABLE 8-5\nAdditional parameters for assessment of cell spectral efficiency\nand cell edge user spectral efficiency\n\n## Deployment Indoor hotspot Urban Urban Rural Suburban\n\nscenario for the micro-cell macro-cell macro-cell macro-cell\nevaluation process\nEvaluated service Full buffer best Full buffer best Full buffer best Full buffer best Full buffer best\nprofiles effort effort effort effort effort\nSimulation 20 + 20 MHz 10 + 10 MHz 10 + 10 MHz 10 + 10 MHz 10 + 10 MHz\nbandwidth (FDD), or (FDD), or (FDD), or (FDD), or (FDD), or\n40 MHz (TDD) 20 MHz (TDD) 20 MHz (TDD) 20 MHz (TDD) 20 MHz (TDD)\nNumber of users 10 10 10 10 10\nper cell\n\nThe simulation needs to be done over a time period long enough to assure convergence of the\nsimulation results.\nWhen assessing the VoIP capacity characteristic, parameters in Table 8-6 shall also be applied.\n\nTABLE 8-6\nAdditional parameters for assessment of VoIP capacity\n\n## Deployment Indoor hotspot Urban Urban Rural Suburban\n\nscenario for the micro-cell macro-cell macro-cell macro-cell\nevaluation process\nEvaluated service VoIP VoIP VoIP VoIP VoIP\nprofiles\nSimulation 5 + 5 MHz 5 + 5 MHz 5 + 5 MHz 5 + 5 MHz 5 + 5 MHz\nbandwidth(1) (FDD), 10 MHz (FDD), 10 MHz (FDD), 10 MHz (FDD), 10 MHz (FDD), 10 MHz\n(TDD) (TDD) (TDD) (TDD) (TDD)\nSimulation time 20 s 20 s 20 s 20 s 20 s\nspan for a single\ndrop\n(1)\nWhile it is recognized that the bandwidth associated with VoIP implementations could be significantly larger than\nthe bandwidth specified herein; this bandwidth was chosen to allow simulations to be practically conducted. Using\nlarger bandwidths and the corresponding larger number of users to be simulated increases the simulation\ncomplexity and time required to perform the simulations.\n\nParameters in Table 8-7 shall also be applied in link level simulation when assessing the\ncharacteristic of mobility.\nRep. ITU-R M.2135-1 17\n\nTABLE 8-7\n\n## Deployment Indoor hotspot Urban Urban Rural Suburban\n\nscenario for the micro-cell macro-cell macro-cell marco-cell\nevaluation process\nEvaluated service Full buffer best Full buffer best Full buffer best Full buffer best Full buffer best\nprofiles effort effort effort effort effort\nChannel model Indoor hotspot Urban micro- Urban macro- Rural macro- Suburban\nmodel (InH) cell model cell model cell model macro-cell\n(UMi) (UMa) (RMa) model (SMa)\nSimulation 10 MHz 10 MHz 10 MHz 10 MHz 10 MHz\nbandwidth\nNumber of users in 1 1 1 1 1\nsimulation\n\n## 8.5 Antenna characteristics\n\nThis sub-section specifies the antenna characteristics, e.g. antenna pattern, gain, side-lobe level,\norientation, etc., for antennas at the base station (BS) and the user terminal (UT), which shall be\napplied for the evaluation in the deployment scenarios with the hexagonal grid layout (i.e., urban\nmacro-cell, urban micro-cell, rural macro-cell, and suburban macro-cell). The characteristics do not\nform any kind of requirements and should be used only for the evaluation.\n8.5.1 BS antenna\n8.5.1.1 BS antenna pattern\nThe horizontal antenna pattern used for each BS sector5 is specified as:\n  θ 2 \nA (θ ) = −min 12   , Am \n  θ3 dB  \n(1)\nWhere A(θ) is the relative antenna gain (dB) in the direction θ, −180º ≤ θ ≤ 180º, and min [.]\ndenotes the minimum function, θ3dB is the 3 dB beamwidth (corresponding to θ3dB= 70º), and\nAm = 20 dB is the maximum attenuation. Figure 3 shows the BS antenna pattern for 3 sector cells to\nbe used in system level simulations.\nA similar antenna pattern will be used for elevation in simulations that need it. In this case the\nantenna pattern will be given by:\n\n  φ − φ 2 \nAe (φ ) = − min 12  tilt\n , Am  (2)\n  φ3dB  \n\nwhere Ae(φ) is the relative antenna gain (dB) in the elevation direction, φ, −90º ≤ φ ≤ 90º,\nφ3dB..φtilt.is the elevation 3 dB value, and it may be assumed to be 15o, unless stated otherwise. φtilt is\nthe tilt angle, which should be provided by proponents per deployment scenario.\n\n## 5 A sector is equivalent to a cell.\n\n18 Rep. ITU-R M.2135-1\n\nThe combined antenna pattern at angles off the cardinal axes is computed as:\n− min  − ( A (θ ) + Ae (φ ) ) , Am \n\nFIGURE 3\nAntenna pattern for 3-sector cells\n0\n\n–5\n\n–10\nGain (dB)\n\n–15\n\n–20\n\n–25\n–180 –150 –120 –90 –60 –30 0 30 60 90 120 150 180\nHorizontal angle (degrees)\nReport 2135-03\n\n## 8.5.1.2 BS antenna orientation\n\nThe antenna bearing is defined as the angle between the main antenna lobe centre and a line\ndirected due east given in degrees. The bearing angle increases in a clockwise direction. Figure 4\nshows the hexagonal cell and its three sectors with the antenna bearing orientation proposed for the\nsimulations. The centre directions of the main antenna lobe in each sector point to the\ncorresponding side of the hexagon.\nFor indoor test environment, omni antenna should be used for the BS.\n\nFIGURE 4\nAntenna bearing orientation diagram\n\n## Main antenna lobe\n\nSector 1\n\n120°\nReport 2135-04\nRep. ITU-R M.2135-1 19\n\n8.5.2 UT antenna\nThe UT antenna is assumed to be omni directional.\n\n## 9 Channel model approach\n\nChannel models are needed in the evaluations of the IMT-Advanced candidate radio interface\ntechnologies (RITs) to allow realistic modelling of the propagation conditions for the radio\ntransmissions in different environments. The channel model needs to cover all required test\nenvironments and scenarios of the IMT-Advanced evaluations.\nRealistic system performance cannot be evaluated by single link simulations. Even the performance\n(RRM) algorithms, interference generated by other links and so on. Multi-link models for system\nlevel evaluations have been developed in the family of geometry-based stochastic channel models.\nThe IMT-Advanced channel model for the evaluation of IMT-Advanced candidate RITs consists of\na Primary Module and an Extension Module as shown in Fig. 5. The framework of the primary\nmodule is based on the WINNER II channel model*, which applies the same approach as\n3GPP/3GPP2 SCM model**. Different evaluation scenarios are shown in parallel in Fig. 5.\n\n* IST-WINNER II Deliverable 1.1.2 v.1.2. WINNER II Channel Models, IST-WINNER2. Tech. Rep., 2007\n(http://www.ist-winner.corg/deliverables.html).\n** 3GPP TR25.996 V6.1.0 (2003-09) Spatial channel model for multiple input multiple output (MIMO)\nsimulations. Release 6.\n20 Rep. ITU-R M.2135-1\n\nFIGURE 5\n\nchannel model\nExtension module\n\nPrimary module\n\nParameter table Parameter table Parameter table Parameter table Parameter table\nDS, AS, etc. DS, AS, etc. DS, AS, etc. DS, AS, etc. DS, AS, etc.\n\nLS parameters\n\nSS parameters\n\nMIMO ChIR\ngeneration\n\n## Indoor hotspot Micro-cell Urban macro Rural macro Suburban macro\n\nReport 2135-05\n\nThe scenarios chosen for the evaluations of the IMT-Advanced candidate RITs are: indoor hotspot,\nurban micro-cell, urban macro-cell and rural macro-cell. The primary module covers the parameter\ntables and channel model definition for the evaluations. The IMT-Advanced channel model\ncontains parameters from Table A1-7 for evaluating the IMT-Advanced candidate RITs in the four\nscenarios of the primary module.\nMandatory channel model parameters for evaluation of RITs for the scenarios indoor hotspot, urban\nmicro-cell, urban macro-cell, and rural macro-cell are contained in the primary module as shown in\nFig. 5 and in Table A1-7 and are not generated from the extension module. In addition, the channel\nmodel could also be applied for other cases, i.e., if some of the parameters described in § 8.4 for\nmacro-cell scenarios, e.g., BS antenna height, street width, city structure, etc. could be varied to\ncover other cases not described in this Report. The extension module can extend the capabilities of\nthe IMT-Advanced channel model to cover those cases beyond the evaluations of the\nIMT-Advanced candidate RITs by allowing the usage of modified parameters to generate large\nscale parameters in the scenarios or usage of other scenarios.\nThe ITU-R IMT-Advanced channel model is a geometry-based stochastic model. It can also be\ncalled double directional channel model. It does not explicitly specify the locations of the scatterers,\nbut rather the directions of the rays, like the well-known spatial channel model (SCM)*. Geometry-\nbased modelling of the radio channel enables separation of propagation parameters and antennas.\n\n* 3GPP TR25.996 V6.1.0 (2003-09) Spatial channel model for multiple input multiple output (MIMO)\nsimulations. Release 6.\nRep. ITU-R M.2135-1 21\n\nThe channel parameters for individual snapshots are determined stochastically based on statistical\ndistributions extracted from channel measurements. Antenna geometries and radiation patterns can\nbe defined properly by the user of the model. Channel realizations are generated through the\napplication of the geometrical principle by summing contributions of rays (plane waves) with\nspecific small-scale parameters like delay, power, angle-of-arrival (AoA) and angle-of-departure\n(AoD). Superposition results to correlation between antenna elements and temporal fading with\ngeometry dependent Doppler spectrum.\nA number of rays constitute a cluster. In the terminology of this document we equate the cluster\nwith a propagation path diffused in space, either or both in delay and angle domains. Elements of\nthe MIMO channel, e.g., antenna arrays at both link ends and propagation paths, are illustrated in\nFig. 6. The generic MIMO channel model is applicable for all scenarios, e.g. indoor, urban and\nrural.\n\nFIGURE 6\nThe MIMO channel\n\nArray 2\n(U Rx elements)\nArray 14 Path 1\n(S Tx elements)\n\nrrx,1\nrtx,1 \nrrx,U\n\nrtx,S\n\nPath N\n\nReport 2135-06\n\nThe time variant impulse response matrix of the U x S MIMO channel is given by:\n\nN\nH(t; τ ) =  H n (t; τ ) (3)\nn =1\n\nwhere:\nt: time\nτ: delay\nN: number of paths\nn: path index.\nIt is composed of the antenna array response matrices Ftx and Frx for the transmitter (Tx) and the\nreceiver (Rx) respectively, and the dual-polarized propagation channel response matrix, hn, for\ncluster, n, as follows:\n\nH n (t; τ ) =  F (ϕ )h (t; τ, φ ,ϕ )F (φ ) dφ dϕ\nrx n\nT\ntx (4)\n\nThe channel from Tx antenna element, s, to Rx element, u, for cluster, n, is expressed as:\n22 Rep. ITU-R M.2135-1\n\n##  Frx ,u ,V (ϕ n ,m )  α n ,m ,VV α n ,m ,VH   Ftx ,s ,V (φn ,m )\n\nT\nM\nH u ,s ,n (t; τ ) =    \nm =1  Frx ,u , H (ϕ n ,m ) α n ,m ,HV α n ,m ,HH   Ftx ,s ,H (φn ,m )\n× exp( j 2πλ 0−1 (ϕ n ,m ⋅ rrx ,u ))exp( j 2πλ 0−1 (φn ,m ⋅ rtx ,s )) (5)\n× exp( j 2πυ n ,m t )δ(τ − τ n ,m )\n\nwhere:\nFrx,u,V and Frx,u,H: antenna element u field patterns for vertical and horizontal polarizations\nrespectively\nαn,m,VV and αn,m,VH: complex gains of vertical-to-vertical and horizontal-to-vertical polarizations of\nray n,m respectively\nλ0: wave length of the carrier frequency\nφn.m : AoD unit vector\nϕn.m : AoA unit vector\nrtx,s and rrx ,u : location vectors of element s and u respectively\nνn,m: Doppler frequency component of ray n,m.\nIf the radio channel is modelled as dynamic, all the above mentioned small-scale parameters are\ntime variant, i.e., they are functions of t [Steinbauer et al., 2001].\nThe primary module covers the mathematical framework, which is called generic model, a set of\nparameters as well as path loss models. A reduced variability model with fixed parameters is also\ndefined which is called clustered delay line (CDL) model. The CDL model cannot be used for\nevaluations of candidate RITs at the link level or system level, but it can be used for calibration\npurposes only.\n\n## 9.1 Generic channel model (mandatory)\n\nThe generic channel model is a double-directional geometry-based stochastic model. It is a system\nlevel6 model in the sense that is employed, e.g., in the SCM model*. It can describe an unlimited\nnumber of propagation environment realizations for single or multiple radio links for all the defined\nscenarios and for arbitrary antenna configurations, with one mathematical framework by different\nparameter sets. The generic channel model is a stochastic model with two (or three) levels of\ndrawn randomly from tabulated distribution functions. Next, small-scale (SS) parameters like\ndelays, powers, and directions of arrival and departure are drawn randomly according to tabulated\ndistribution functions and random LS parameters. At this stage the geometric setup is fixed and the\nonly free variables are the random initial phases of the scatterers. By picking (randomly) different\ninitial phases, an infinite number of different realizations of the model can be generated. When the\ninitial phases are also fixed, there is no further randomness left.\nFigure 7 shows the overview of the channel model creation. The first stage consists of two steps.\nFirst, the propagation scenario is selected. Then, the network layout and the antenna configuration\n\n6 The term system-level means here that the model is able to cover multiple links, cells and terminals.\n* IST-WINNER II Deliverable 1.1.2 v.1.2. WINNER II Channel Models, IST-WINNER2. Tech. Rep., 2008\n(http://www.ist-winner.org/deliverables.html).\nRep. ITU-R M.2135-1 23\n\nare determined. In the second stage, large-scale and small-scale parameters are defined. In the third\nstage, channel impulse responses (ChIRs) are calculated.\n\nFIGURE 7\nChannel model creation process\n\n## User defined parameters Propagation parameter ChIR\n\ngeneration generation\n\n## Network Large scale Multi-path\n\nLayout parameters parameters Channel\n-DS, AS, K coefficient ChIR\n-BS & MS -power,\nScenario locations -X generation\ndelay, AoA,\nselection -velocities\nAoD, etc.\n-path loss\n-urban macro\n-urban micro\n-indoor\n-out2in Antennas\n-etc.\n-# elements\n-orientations\n-field patterns\n\nReport 2135-07\n\n## 9.1.1 Drop concept\n\nThe generic model is based on the drop concept. When using the generic model, the simulation of\nthe system behaviour is carried out as a sequence of “drops”, where a “drop” is defined as one\nsimulation run over a certain time period. A drop (or snapshot or channel segment) is a simulation\nentity where the random properties of the channel remain constant except for the fast fading caused\nby the changing phases of the rays. The constant properties during a single drop are, e.g., the\npowers, delays, and directions of the rays. In a simulation the number and the length of drops have\nto be selected properly by the evaluation requirements and the deployed scenario. The generic\nmodel allows the user to simulate over several drops to get statistically representative results.\nConsecutive drops are independent.\n\n## 9.2 CDL model (for calibration)\n\nThe generic model is aimed to be applicable for many different simulations and to cover a large\nnumber of scenarios with several combinations of large-scale and small-scale parameters. The\ngeneric model is the most accurate model and is used in all evaluations of candidate RITs.\nHowever, for calibration purposes, the CDL model can be used.\nThe CDL model is a spatial extension of tapped delay line (TDL) model. The TDL model usually\ncontains power, delay, and Doppler spectrum information for the taps. CDL models define power,\ndelay, and angular information. Doppler is not explicitly defined, because it is determined by power\nand angular information combined with array characteristics and mobile movement.\nThe CDL approach fixes all the parameters except for the phases of the rays, although other\nalternatives can be considered:\n– the main direction of the rays can be made variable,\n– a set of reference antenna geometries and antenna patterns can be proposed,\n– relation to correlation-matrix based models can be introduced.\n24 Rep. ITU-R M.2135-1\n\n## 10 List of acronyms and abbreviations\n\nACK Acknowledgment\nAoA Angle of arrival\nAoD Angle of departure\nARQ Automatic repeat request\nBS Base station\nCDD Cyclic delay diversity\nCDF Cumulative distribution function\nCDL Clustered delay line\nChIR Channel impulse response\nC/I Carrier-to-interference\nCRC Cyclic redundancy code\nFDD Frequency division duplex\nHARQ Hybrid automatic repeat request\nHH Horizontal-to-horizontal\nHV Horizontal-to-vertical\nInH Indoor hotspot\nIoT Interference over thermal\nIP Internet protocol\nISD Inter-site distance\nLN Log-normal\nLoS Line of sight\nLS Large scale\nMAC Media access control\nMIMO Multiple input multiple output\nNACK Negative acknowledgment\nNLoS Non line-of-sight\nOSI Open systems interconnection\nPAS Power angular spectrum\nPDP Power delay profile\nPDU Protocol data unit\nRep. ITU-R M.2135-1 25\n\nPL Path loss\nRMa Rural macro\nRMS Root mean square (alias: rms)\nRTP Real-time transport protocol\nSCM Spatial channel model\nSMa Suburban macro\nSRIT Set of radio interface technologies\nSS Small scale\nTDD Time division duplex\nTDL Tapped delay line\nTSP Time-spatial propagation\nTTI Transmission time interval\nTX Transmit (alias: Tx, tx)\nUDP User datagram protocol\nULA Uniform linear arrays\nUMa Urban macro\nUMi Urban micro\nUT User terminal\nVAF Voice activity factor\nVoIP Voice over IP\nVH Vertical-to-horizontal\nVV Vertical-to-vertical\nXPD Cross-polarization discrimination\nXPR Cross-polarization ratio\n26 Rep. ITU-R M.2135-1\n\nFurther information on vocabulary can be found in the references at the bottom of the page*, **,***.\n\nReferences\nSTEINBAUER, M., MOLISCH, A. F. and BONEK, E. [August 2001] The double-directional radio channel.\nIEEE Ant. Prop. Mag., p. 51-63.\n\nAnnex 1\n\n## 1 Test environments and channel models\n\nThis section provides the reference channel model for each test environment. These test\nenvironments are intended to cover the range of IMT-Advanced operating environments.\nThe test operating environments are considered as a basic factor in the evaluation process of the\ncandidate RITs. The reference models are used to estimate the critical aspects, such as the spectrum,\ncoverage and power efficiencies.\n\n## 1.1 Test environments, deployment scenarios and network layout\n\nEvaluation of candidate IMT-Advanced RIT/SRITs will be performed in selected scenarios of the\nfollowing test environments:\n– Base coverage urban: an urban macro-cellular environment targeting continuous coverage\nfor pedestrian up to fast vehicular users.\n– Micro-cellular: an urban micro-cellular environment with higher user density focusing on\npedestrian and slow vehicular users.\n– Indoor: an indoor environment targeting isolated cells at offices and/or in hotspot based on\nstationary and pedestrian users.\n– High speed: a macro-cellular environment with high speed vehicles and trains.\n\n## * 3GPP TR 21.905 Vocabulary For 3GPP Specifications (http://www.3gpp.org/ftp/Specs/html-\n\ninfo/21905.htm).\n2005.pdf).\n*** 3GPP2 UMB PHY Specification Ver.2.0 (Terms Section 2.1)\n(http://www.3gpp2.org/Public_html/specs/C.S0084-001-0_v2.0_070904.pdf).\nRep. ITU-R M.2135-1 27\n\nThe deployment scenarios that shall be used for each test environment are shown in Table 8-1;\ninformation of the respective channel models is given in Table A1-1.\n\nTABLE A1-1\nSelected deployment scenarios for evaluation and the channel models\nTest environment Base coverage urban Microcellular Indoor High speed\nDeployment scenario Urban macro-cell Urban micro-cell Indoor hotspot Rural macro-cell\nscenario scenario scenario scenario\nChannel model UMa UMi InH RMa\nUrban macro Urban micro Indoor hotspot Rural macro\n(LoS, NLoS) (LoS, NLoS, (LoS, NLoS) (LoS, NLoS)\nOutdoor-to-indoor)\n\nSuburban macro-cell scenario (and corresponding SMa channel model) is an optional scenario for\nthe base coverage urban test environment.\nInitial focus for deployment and most challenges in IMT-Advanced system design and performance\nwill be encountered in populated areas. However, in the evaluation the provisions for ubiquitous\ncoverage and the associated performance also in rural areas need to be addressed.\n\n## 1.2 Test environments\n\nFor evaluation of candidate IMT-Advanced RIT/SRITs in the four selected test environments, a set\nof reliable and measurement-based channel models are needed. Channel models have to be accurate\ndue to the fact that radio propagation has a significant impact on the performance of future\nbroadband systems. This is especially true with future multiple-input multiple-output (MIMO) radio\ncommunication systems since more of the radio channel degrees of freedom in space, time,\nfrequency, and polarization may be exploited to meet the demands on bit rate, spectrum efficiency,\nand cost. Channel models are needed in performance evaluation of wireless systems when choosing\nmodulation and coding, in multi antenna system design, in the selection of channel estimation\nmethod, channel equalization and other baseband algorithm design, as well as network planning.\nIt is important to use common and uniform channel models for evaluation, comparison, and\nselection of technologies. In this context it is clear that realistic and reliable multidimensional\nchannel models are an important part of the performance evaluation of IMT-Advanced RIT/SRITs.\nA central factor of mobile radio propagation environments is the multi-path propagation causing\nfrequency and time dispersion as well as angular dispersion in Tx and Rx. The fading characteristics\nvary with the propagation environment and their impact on the communication quality (e.g., bit\nerror patterns) highly depends on the speed of the mobile station relative to the serving base station.\nThe purpose of the test environments is to challenge the RITs. Instead of constructing propagation\nmodels for all the possible IMT-Advanced operating environments, a smaller set of test\nenvironments is defined which adequately span the overall range of possible environments. The\ndescriptions of these test environments may therefore not correspond with those of the actual\noperating environments.\nThis section identifies the propagation and channel model for each test operating environment\ndescribed in § 1.2.1 to 1.2.4. For practical reasons, these test operating environments are an\nappropriate subset of the IMT-Advanced operating environments. While simple models might be\nevaluate the overall system-level reliability and suitability of specific technologies. For wideband\ntechnologies the number, strength, and relative time delay as well as the directions at Tx and Rx of\nthe many signal components become important. For some technologies (e.g., those employing\n28 Rep. ITU-R M.2135-1\n\npower control) these models must include coupling between all co-channel propagation links to\nachieve maximum accuracy. Also, in some cases, the large-scale (shadow fading) temporal\nvariations of the environment must be modelled.\nThe key parameters to describe channel models include, e.g., delay spread, angle spread, path loss,\netc.\n1.2.1 Base coverage urban test environment\nThe base coverage urban test environment focuses on large cells and continuous coverage. The key\ncharacteristics of this test environment are continuous and ubiquitous coverage in urban areas. This\nscenario will therefore be interference-limited, using macro cells (i.e. radio access points above\nrooftop level).\nIn urban macro-cell scenario mobile station is located outdoors at street level and fixed base station\nantenna clearly above surrounding building heights. As for propagation conditions, non- or\nobstructed line-of-sight is a common case, since street level is often reached by a single diffraction\nover the rooftop. The building blocks can form either a regular Manhattan type of grid, or have\nmore irregular locations. Typical building heights in urban environments are over four floors.\nBuildings height and density in typical urban macro-cell are mostly homogenous.\nThe base coverage urban test environment is intended to prove that continuous, ubiquitous, and\ncost-effective coverage in built-up areas is feasible in the IMT-Advanced bands by the candidate\nIMT-Advanced RIT/SRITs. This scenario will therefore be interference-limited, using macro cells\ndemanding services beyond baseline voice and text messages.\n1.2.1.1 Urban macro-cell scenario\nIn typical urban macro-cell scenario, the mobile station is located outdoors at street level and the\nfixed base station clearly above the surrounding building heights. As for propagation conditions,\nnon- or obstructed line-of-sight are common cases, since street level is often reached by a single\ndiffraction over the rooftop. The building blocks can form either a regular Manhattan type of grid,\nor have more irregular locations. Typical building heights in urban environments are over four\nfloors. Buildings height and density in typical urban macro-cell are mostly homogenous.\nThe channel model for urban macro-cell scenario is called urban macro (UMa).\n1.2.1.2 Suburban macro-cell scenario (Optional)\nIn suburban macro-cell scenario base stations are located well above the rooftops to allow wide area\ncoverage, and mobile stations are outdoors at street level. Buildings are typically low residential\ndetached houses with one or two floors, or blocks of flats with a few floors. Occasional open areas\nsuch as parks or playgrounds between the houses make the environment rather open. Streets do not\nform urban-like regular strict grid structure. Vegetation is modest.\nThe channel model for suburban macro-cell scenario is called suburban macro (SMa).\n1.2.2 Microcellular test environment\nThe microcellular test environment focuses on small cells and high user densities and traffic loads\nin city centres and dense urban areas. The key characteristics of this test environment are high\ntraffic loads, outdoor and outdoor-to-indoor coverage. This scenario will therefore be interference-\nlimited, using micro cells. A continuous cellular layout and the associated interference shall be\nassumed. Radio access points shall be below rooftop level.\nA similar scenario is used to the base coverage urban test environment but with reduced site-to-site\ndistance and the antennas below rooftops.\nRep. ITU-R M.2135-1 29\n\nThe microcellular test environment focuses on smaller cells and higher user densities and traffic\nloads in city centres and dense urban areas, i.e., it targets the high-performance layer of an\nIMT-Advanced system in metropolitan areas. It is thus intended to test the performance in high\ntraffic loads and using demanding user requirements, including detailed modelling of buildings\n(e.g., Manhattan grid deployment) and outdoor-to-indoor coverage. A continuous cellular layout\nand the associated interference shall be assumed. Radio access points shall be below rooftop level.\n1.2.2.1 Urban micro-cell scenario\nIn urban micro-cell scenario the height of both the antenna at the BS and that at the UT is assumed\nto be well below the tops of surrounding buildings. Both antennas are assumed to be outdoors in an\narea where streets are laid out in a Manhattan-like grid. The streets in the coverage area are\nclassified as “the main street”, where there is LoS from all locations to the BS, with the possible\nexception of cases in which LoS is temporarily blocked by traffic (e.g., trucks and busses) on the\nstreet. Streets that intersect the main street are referred to as perpendicular streets, and those that run\nparallel to it are referred to as parallel streets. This scenario is defined for both LoS and NLoS\ncases. Cell shapes are defined by the surrounding buildings, and energy reaches NLoS streets as a\nresult of propagation around corners, through buildings, and between them.\nThe microcellular test environment includes outdoor and outdoor-to-indoor users: In the latter case\nthe users are located indoors and Base Stations outdoors. Therefore the channel model for the\nmicro-cellular test environment contains two parts, the outdoor part and the outdoor-to-indoor part.\nThe channel model for urban micro-cell scenario is called urban micro (UMi).\n1.2.3 Indoor test environment\nThe indoor test environment focuses on smallest cells and high user throughput in buildings.\nThe key characteristics of this test environment are high user throughput in indoor coverage.\n1.2.3.1 Indoor hotspot scenario\nThe indoor hotspot scenario consists of one floor of a building. The height of the floor is 6 m. The\nfloor contains 16 rooms of 15 m × 15 m and a long hall of 120 m × 20 m. Two sites are placed in\nthe middle of the hall at 30 m and 90 m with respect to the left side of the building (see Fig. 2).\nThe channel model for indoor hotspot scenario is called indoor hotspot (InH).\n1.2.4 High-speed test environment\nThe high-speed test environment focuses on larger cells and continuous coverage. The key\ncharacteristics of this test environment are continuous wide area coverage supporting high speed\nvehicles. This scenario will therefore be noise-limited and/or interference-limited, using macro\ncells.\nThe high speed test environment is applicable to a wide-area system concept since is should allow\nfor reliable links to high-speed trains of up to 350 km/h or cars at high velocities. Repeater\ntechnology or relays (relaying to the same wide area system, IMT-2000, or to a local area system)\ncan be applied in the vehicle, to allow for local access by the users.\n1.2.4.1 Rural macro-cell scenario\nThe Rural macro-cell scenario propagation scenario represents radio propagation in large areas\n(radii up to 10 km) with low building density. The height of the BS antenna is typically in the range\nfrom 20 to 70 m, which is much higher than the average building height. Consequently, LoS\nconditions can be expected to exist in most of the coverage area. In case the UT is located inside a\nbuilding or vehicle, an additional penetration loss is experienced which can possibly be modelled as\n30 Rep. ITU-R M.2135-1\n\n## a (frequency-dependent) constant value. The BS antenna location is fixed in this propagation\n\nscenario, and the UT antenna velocity is in the range from 0 to 350 km/h.\nThe channel model for rural macro-cell scenario is called rural macro (RMa).\n1.2.5 Simulation of relays\nIt is possible to simulate relay-based lay-outs with the proposed channel models by using models\nfor the constituent hops of the multiple links. The link from a relay to a mobile station can be\nmodeled with the same models as the conventional link from a base station to a mobile station.\nThe links from base stations to relay stations can be modeled with conventional links.\n\n## 1.3 Primary module\n\nThe following sections provide the details of the channel models, including the path loss models,\nfor the terrestrial component. For terrestrial environments, the propagation effects are divided into\nthree distinct types: These are the path loss, the slow variation due to shadowing and scattering, and\nthe rapid variation in the signal due to multipath effects. The channel models are specified in the\nfrequency range from 2 GHz to 6 GHz. For the rural macro-cell scenario (RMa), the channel model\ncan be used for lower frequencies down to 450 MHz. The channel models also cover MIMO aspects\nas all desired dimensions (delay, AoA, AoD and polarisation) are considered. The channel models\nare targeted for up to 100 MHz RF bandwidth.\n1.3.1 Path loss models\nPath loss models for the various propagation scenarios have been developed based on measurement\nresults carried out in references* [Dong et al., 2007; Fujii, 2003; Lu et al., 2007; Xinying et al.,\n2007; Xu et al., 2007; Zhang et al., 2007 and 2008], as well as results from the literature. The\nmodels can be applied in the frequency range of 2-6 GHz and for different antenna heights. The\nrural path-loss formula can be applied to the desired frequency range from\n450 MHz to 6 GHz. The path loss models have been summarized in Table A1-2. Note that the\ndistribution of the shadow fading is log-normal, and its standard deviation for each scenario is given\nin the following table.\n\nTABLE A1-2\nSummary table of the primary module path loss models\nScenario Path loss (dB) Shadow Applicability range,\nNote: fc is given in GHz and distance in m! fading std antenna height default\n(dB) values\nPL = 16.9 log10(d) + 32.8 + 20 log10(fc) σ= 3 3 m < d < 100 m\nIndoor Hotspot\n\nhUT = 1-2.5 m\n(InH)\n\n## PL = 43.3 log10(d) + 11.5 + 20 log10(fc) σ=4 10 m < d < 150 m\n\nNLoS hBS = 3-6 m\nhUT = 1-2.5 m\n\n* IST-WINNER II Deliverable 1.1.2 v.1.2. WINNER II Channel Models, IST-WINNER2, Tech. Rep.,\n2008 (http://www.ist-winner.org/deliverables.html).\nRep. ITU-R M.2135-1 31\n\n## Scenario Path loss (dB) Shadow Applicability range,\n\nNote: fc is given in GHz and distance in m! fading std antenna height default\n(dB) values\nPL = 22.0 log10(d) + 28.0 + 20 log10(fc) σ= 3 10 m < d1 < d′BP (1)\nLoS\nPL = 40 log10(d1) + 7.8 – 18 log10(h′BS) –18 log10(h′UT) σ= 3 d′BP < d1 < 5 000 m(1)\n+ 2 log10(fc) hBS = 10 m(1), hUT = 1.5 m(1)\nManhattan grid layout:\nPL = min (PL( d 1 , d 2 ), PL( d 2 , d 1 ) ) σ =4 10 m < d1 + d2 < 5 000 m,\nUrban Micro (UMi)\n\n## where: w/2 < min(d1,d2 )(2)\n\nw = 20 m (street width)\nPL(d k , d l ) = PL LOS ( d k ) + 17.9 − 12.5n j + 10n j log10 ( d l ) +\nh′BS = 10 m, hUT = 1.5 m.\n3 log10 ( f c ) When 0 < min(d1,d2 ) < w/2 ,\nNLoS and the LoS PL is applied.\nn j = max (2.8 − 0.0024 d k , 1.84 )\nPLLOS: path loss of scenario UMi LoS and\nk,l ∈ {1,2}.\nHexagonal cell layout: σ =4 10 m < d < 2 000 m\nPL = 36.7 log10(d) + 22.7 + 26 log10(fc) hBS = 10 m\nhUT =1-2.5 m\nPL = PLb + PLtw + PLin\nManhattan grid layout (θ known): σ= 7 10 m < dout + din< 1 000 m,\n0 m < din< 25 m,\n PLb = PL B1 ( d out + d in )\n hBS = 10 m, hUT = 3(nFl -1)\n2\nO-to-I  PLtw = 14 + 15(1 − cos(θ)) + 1.5 m,\n PL = 0.5d nFl = 1\n in in\nExplanations: see (3)\nFor hexagonal layout (θ unknown):\nPLtw = 20, other values remain the same.\nPL = 22.0 log10(d) + 28.0 + 20 log10(fc) σ= 4 10 m < d < d′BP (1)\n\nLoS\nPL = 40 . 0 log 10 ( d 1 ) + 7 . 8 − 18 . 0 log 10 ( h ' BS ) σ= 4 d′BP < d < 5 000 m(1)\n− 18 . 0 log 10 ( h ' UT ) + 2 . 0 log 10 ( f c ) hBS = 25 m(1), hUT = 1.5 m(1)\nUrban Macro (UMa)\n\nPL = 161.04 – 7.1 log10 (W) + 7.5 log10 (h) σ= 6 10 m < d < 5 000 m\n– (24.37 – 3.7(h/hBS)2) log10 (hBS) h = avg. building height\n+ (43.42 – 3.1 log10 (hBS)) (log10 (d) − 3) + W = street width\n20 log10(fc) – (3.2 (log10 (11.75 hUT))2 − 4.97) hBS = 25 m, hUT = 1.5 m,\nW = 20 m, h = 20 m.\nNLoS\nThe applicability ranges:\n5 m < h < 50 m\n5 m < W < 50 m\n10 m < hBS < 150 m\n1 m < hUT < 10 m\n32 Rep. ITU-R M.2135-1\n\n## Scenario Path loss (dB) Shadow Applicability range,\n\nNote: fc is given in GHz and distance in m! fading std antenna height default\n(dB) values\nPL1 = 20 log10(40π d fc/3) + min(0.03h1.72,10) log10(d) σ=4 10 m < d < dBP (4)\n– min(0.044h1.72,14.77) + 0.002 log10(h)d\nSuburban Macro (SMa, optional)\n\n## PL2 = PL1 (dBP) + 40 log10(d/dBP) σ=6 dBP < d < 5 000 m\n\nLoS hBS = 35 m, hUT = 1.5 m,\nW = 20 m, h = 10 m\n(The applicability ranges of\nh, W, hBS, hUT are same as in\nUMa NLoS)\nPL = 161.04 – 7.1 log10 (W) + 7.5 log10 (h) σ= 8 10 m < d < 5 000 m\n– (24.37 – 3.7(h/hBS)2) log10 (hBS) hBS = 35 m, hUT = 1.5 m,\n+ (43.42 – 3.1 log10 (hBS)) (log10 (d) −3) + W = 20 m, h = 10 m\nNLoS\n20 log10(fc) – (3.2 (log10 (11.75 hUT))2 − 4.97) (Applicability ranges of h, W,\nhBS, hUT are same as in UMa\nNLoS)\nPL1 = 20 log10(40π d fc/3) + min(0.03h1.72,10) log10(d) σ= 4 10 m < d < dBP (4)\n– min(0.044h1.72,14.77) + 0.002 log10(h)d\n\n## PL2 = PL1 (dBP) + 40 log10(d/dBP) σ=6 dBP < d < 10 000 m,\n\nLoS hBS = 35 m, hUT = 1.5 m,\nRural Macro (RMa)\n\nW = 20 m, h = 5 m\n(Applicability ranges of h, W,\nhBS, hUT are same as UMa\nNLoS)\nPL = 161.04 – 7.1 log10 (W) + 7.5 log10 (h) σ= 8 10 m < d < 5 000 m,\n– (24.37 – 3.7(h/hBS)2) log10 (hBS)\n+ (43.42 – 3.1 log10 (hBS)) (log10 (d) − 3) + hBS = 35 m, hUT = 1.5 m,\nNLoS 20 log10(fc) – (3.2 (log10 (11.75 hUT)) 2 − 4.97) W = 20 m, h = 5 m\n(The applicability ranges of\nh, W, hBS, hUT are same as\nUMa NLoS)\n\n## Notes to Table A1-2:\n\n(1)\nBreak point distance d′BP = 4 h′BS h′UT fc/c, where fc is the centre frequency (Hz), c = 3.0 × 108 m/s is the\npropagation velocity in free space, and h′BS and h′UT are the effective antenna heights at the BS and the UT,\nrespectively. The effective antenna heights h′BS and h′UT are computed as follows:\nh′BS = hBS – 1.0 m, h′UT = hUT – 1.0 m\nwhere: hBS and hUT are the actual antenna heights, and the effective environment height in urban environments is\nassumed to be equal to 1.0 m.\n(2)\nThe distances d1 and d2 are defined below in Fig. 12.\n(3)\nPLb: basic path-loss, PL B1: loss of UMi outdoor scenario, PLtw: loss through wall, PLin: loss inside, dout: distance\nfrom BS to the wall next to UT location, din: perpendicular distance from wall to UT (assumed evenly distributed\nbetween 0 and 25 m), θ: angle between LoS to the wall and a unit vector normal to the wall.\n(4)\nBreak point distance dBP = 2π hBS hUT fc/c, where fc is the centre frequency in Hz, c = 3.0 × 108 m/s is the\npropagation velocity in free space, and hBS and hUT are the antenna heights at the BS and the UT, respectively.\nRep. ITU-R M.2135-1 33\n\nThe LoS probabilities are given in Table A1-3. Note that probabilities are used only for system\nlevel simulations.\n\nTABLE A1-3\nScenario LoS probability as a function of distance, d (m)\n1, d ≤ 18\n\nInH PLOS =  exp (− (d − 18 ) 27 ), 18 < d < 37\n 0 .5, d ≥ 37\n\nPLOS = min (18/d,1) ⋅ (1 – exp (–d / 36)) + exp (–d / 36)\nUMi\n(for outdoor users only)\nUMa PLOS = min (18/d,1) ⋅ (1 – exp (–d / 63)) + exp (–d / 63)\n1, d ≤ 10\nSMa PLOS =\nexp( − ( d − 10 ) / 200 ), d > 10\n1, d ≤ 10\nRMa PLOS =  d − 10 \nexp − , d > 10\n 1000 \n\nThe NLoS path loss model for scenario UMi is dependent on two distances, d1 and d2 in the case of\nthe Manhattan grid. These distances are defined with respect to a rectangular street grid, as\nillustrated in Fig. 12, where the UT is shown moving along a street perpendicular to the street on\nwhich the BS is located (the LoS street). d1 is the distance from the BS to the centre of the\nperpendicular street, and d2 is the distance of the UT along the perpendicular street, measured from\nthe centre of the LoS street.\n\nFIGURE 8\nGeometry for d1 - d2 path-loss model\n+\n\nd2\nMS\n\nBS\nd1\n\nd2\n\n- Report 2135-08\n\nThe long-term (log-normal) fading in the logarithmic scale around the mean path loss PL (dB) is\ncharacterized by a Gaussian distribution with zero mean and standard deviation. Due to the slow\nautocorrelation function R(Δx) can be described with sufficient accuracy by the exponential\n34 Rep. ITU-R M.2135-1\n\nfunction (Recommendation ITU-R P.1816 – The prediction of the time and the spatial profile for\nbroadband land mobile services using UHF and SHF bands):\n\nΔx\n\nR (Δx ) = e d cor\n(6)\n\nwith the correlation length dcor being dependent on the environment, see the correlation parameters\nfor shadowing and other large scale parameters in Table A1-7.\n1.3.2 Primary module channel model parameters\n1.3.2.1 Generic model\nThe radio channels are created using the parameters listed in Table A1-7. The channel realizations\nare obtained by a step-wise procedure* illustrated in Fig. 9 and described below. It has to be noted\nthat the geometric description covers arrival angles from the last bounce scatterers and respectively\ndeparture angles to the first scatterers interacted from the transmitting side. The propagation\nbetween the first and the last interaction is not defined. Thus, this approach can model also multiple\ninteractions with the scattering media. This indicates also that e.g., the delay of a multipath\ncomponent cannot be determined by the geometry. In the following steps, downlink is assumed. For\nuplink, arrival and departure parameters have to be swapped.\n\nFIGURE 9\nChannel coefficient generation procedure\n\nGeneral parameters:\n\n## Set scenario, network Assign propagation Generate correlated\n\nlayout and antenna condition Calculate path loss large scale paramters\nparameters (NLOS/LOS) (DS, AS, SF, K)\n\n## Perform random Generate arrival and Generate cluster\n\nGenerate delays\ncoupling of rays departure angles powers\n\nCoefficient generation:\n\n## Draw random initial Generate channel Apply path loss\n\nReport 2135-09\n\n* IST-WINNER II Deliverable 1.1.2 v.1.2. WINNER II Channel Models, IST-WINNER2. Tech. Rep., 2008\n(http://www.ist-winner.org/deliverables.html).\nRep. ITU-R M.2135-1 35\n\nGeneral parameters:\nStep 1: Set environment, network layout, and antenna array parameters:\na) Choose one of the scenarios (InH, UMi, …).\nb) Give number of BS and UT.\nc) Give locations of BS and UT, or equally distances of each BS and UT and relative\ndirections and φLoS and ϕLoS of each BS and UT.\nd) Give BS and UT antenna field patterns Frx and Ftx and array geometries.\ne) Give BS and UT array orientations with respect to north (reference) direction.\nf) Give speed and direction of motion of UT.\ng) Give system centre frequency.\nLarge scale parameters:\nStep 2: Assign propagation condition (LoS/NLoS).\nStep 3: Calculate path loss with formulas of Table A1-2 for each BS-UT link to be modelled.\nStep 4: Generate correlated large scale parameters, i.e. delay spread, angular spreads, Ricean\nK factor and shadow fading term like explained in § 3.3.1 (Correlations between large scale\nparameters) of reference*. Limit random rms arrival and departure azimuth spread values to 104°,\ni.e., σϕ = min(σϕ ,104°).\nSmall scale parameters:\nStep 5: Generate delays τ.\nDelays are drawn randomly from the delay distribution defined in Table A1-7. With exponential\ndelay distribution calculate:\nτ n ' = −rτ σ τ ln( X n ) (7)\nwhere rτ is the delay distribution proportionality factor, Xn ~ Uni(0,1), and cluster index\nn = 1,…, N. With uniform delay distribution the delay values τ′n are drawn from the corresponding\nrange. Normalise the delays by subtracting the minimum delay and sort the normalised delays to\ndescending order:\nτ n = sort (τ' n − min (τ' n )) (8)\nIn the case of LoS condition, additional scaling of delays is required to compensate for the effect of\nLoS peak addition to the delay spread. The heuristically determined Ricean K-factor dependent\nscaling constant is:\nD = 0.7705 − 0.0433K + 0.0002K 2 + 0.000017K 3 (9)\n\nwhere K (dB) is the Ricean K-factor defined in Table A1-7. The scaled delays:\n\nτ nLoS = τ n / D (10)\nare not to be used in cluster power generation.\n\n* IST-WINNER II Deliverable 1.1.2 v.1.2. WINNER II Channel Models, IST-WINNER2. Tech. Rep., 2008\n(http://www.ist-winner.org/deliverables.html).\n36 Rep. ITU-R M.2135-1\n\n## Step 6: Generate cluster powers, P.\n\nCluster powers are calculated assuming a single slope exponential power delay profile. Power\nassignment depends on the delay distribution defined in Table A1-7. With exponential delay\ndistribution the cluster powers are determined by:\n\n−Ζ\n rτ −1  n\n\n## Pn' = exp − τ n  ⋅10 10\n\n(11)\n rτσ τ \n\nwhere Ζn ~ N(0, ζ2 ) is the per cluster shadowing term (dB). Average the power so that the sum\npower of all cluster powers is equal to one, i.e.:\n\nPn'\nPn = N (12)\n\nn =1\nPn'\n\nIn the case of LoS condition an additional specular component is added to the first cluster. Power of\nthe single LoS ray is:\n\nKR\nP1, LOS =\nKR + 1\n\n## and the cluster powers are not as in equation (12), but:\n\n1 Pn′\nPn = + δ(n − 1)P1, LOS\nK R + 1  N Pn′\nn =1\n\nwhere δ(.) is Dirac’s delta function and KR is the Ricean K-factor defined in Table A1-7 converted\nto linear scale. These power values are used only in equations (13) and (14) of ITU-R M.2135, but\nnot in equation (20).\nAssign the power of each ray within a cluster as Pn /M, where M is the number of rays per cluster.\nRemove clusters with less than –25 dB power compared to the maximum cluster power.\nStep 7: Generate arrival angles ϕ and departure angles φ.\nAs the composite PAS of all clusters is modelled as wrapped Gaussian (see Table A1-7), except\nindoor hotspot scenario (InH) as Laplacian, the AoAs are determined by applying the inverse\nGaussian function (13) or inverse Laplacian function (14) with input parameters Pn and RMS angle\n\n## 2σAoA − ln(Pn max(Pn ))\n\nϕn ' = (13)\nC\nRep. ITU-R M.2135-1 37\n\nσϕ ln(Pn max(Pn ))\nϕn ' = − (14)\nC\nIn equation (13), σ AoA = σ ϕ 1.4 is the standard deviation of the arrival angles (the factor 1.4 is the\nratio of Gaussian std and the corresponding “RMS spread”). Constant C is a scaling factor related to\ntotal number of clusters and is given in Table A1-4:\n\nTABLE A1-4\nNo. 4 5 8 10 11 12 14 15 15 16 19 19 20\nclusters (InH) (InH)\nC 0.779 0.860 1.018 1.090 1.123 1.146 1.190 1.211 1.434 1.226 1.273 1.501 1.289\n\nIn the LoS case, constant C is dependent also on the Ricean K-factor. Constant C in equations (13)\nand (14) is substituted by CLoS. Additional scaling of the angles is required to compensate for the\neffect of LOS peak addition to the angle spread. The heuristically determined Ricean K-factor\ndependent scaling constant is:\n\n(\nC LOS = C ⋅ 1.1035 − 0.028K − 0.002K 2 + 0.0001K 3 ) (15)\n\n## As for indoor hotspot scenario, the scaling constant is:\n\n(\nC LOS = C ⋅ 0.9275 + 0.0439K − 0.0071K 2 + 0.0002K 3 ) (16)\n\n## where K (dB) is the Ricean K-factor defined in Table A1-7.\n\nAssign positive or negative sign to the angles by multiplying with a random variable Xn with\nuniform distribution to the discrete set of {1, –1}, and add component Yn ~ N (0, σ ϕ 7) to introduce\nrandom variation:\n\n## ϕn = X n ϕn' +Yn + ϕLoS (17)\n\nwhere ϕLoS is the LoS direction defined in the network layout description, see Step 1c).\nIn the LoS case, substitute equation (17) by equation (18) to enforce the first cluster to the LoS\ndirection ϕLoS:\n\n( )(\nϕn = X nϕn' +Yn − X1ϕ1' +Y1 − ϕLoS ) (18)\n\nFinally add offset angles αm from Table A1-5 to the cluster angles:\n\nϕ n , m = ϕ n + c AoA α m (19)\n\nwhere cAoA is the cluster-wise rms azimuth spread of arrival angles (cluster ASA) in Table A1-7.\n38 Rep. ITU-R M.2135-1\n\nTABLE A1-5\nRay offset angles within a cluster, given\n\n1,2 ± 0.0447\n3,4 ± 0.1413\n5,6 ± 0.2492\n7,8 ± 0.3715\n9,10 ± 0.5129\n11,12 ± 0.6797\n13,14 ± 0.8844\n15,16 ± 1.1481\n17,18 ± 1.5195\n19,20 ± 2.1551\n\n## For departure angles φn the procedure is similar.\n\nStep 8: Random coupling of rays within clusters.\nCouple randomly departure ray angles φn,m to arrival ray angles ϕn,m within a cluster n, or within a\nsub-cluster in the case of two strongest clusters (see Step 10a and Table A1-5).\nCoefficient generation:\nStep 9: Draw random initial phase {Φnvv,m , Φ nvh,m , Φhv\nn , m , Φ n , m } for each ray m of each cluster n and\nhh\n\nfor four different polarisation combinations (vv,vh,hv,hh). The distribution for initial phases is\nuniform within (-π,π).\nIn the LoS case, draw also random initial phases {Φ vv\nLOS , Φ LOS }\nhh\nfor both VV and HH polarisations.\nStep 10a: Generate channel coefficients for each cluster n and each receiver and transmitter\nelement pair u,s.\nFor the N – 2 weakest clusters, say n = 3, 4,…, N, and uniform linear arrays (ULA), the channel\ncoefficients are given by:\n\nM\n Frx,u,V (ϕn,m )  exp jΦvv\nT\n( ) (\nκ −1 exp jΦvh )\n F (φ )\nn,m  tx,s,V n,m \nHu,s,n (t ) = Pn  F   n,m\n\n##  rx,u,H (ϕn,m )  κ −1 exp jΦhv\n\n( ) ( ) \nexp jΦhh Ftx,s,H (φn,m )\n \nm=1 n,m n,m (20)\n⋅ exp ( ( ))exp(\njds 2πλ0−1 sin φn,m jdu 2πλ0−1 sin(ϕ ))exp( j2πυ t )\nn,m n,m\n\nwhere Frx,u,V and Frx,u,H are the antenna element, u, field patterns for vertical and horizontal\npolarisations respectively, ds and du are the uniform distances (m) between transmitter elements and\nreceiver elements respectively, κ is the cross polarisation power ratio in linear scale, and λ0 is the\nwavelength of the carrier frequency. If polarisation is not considered, the 2 × 2 polarisation matrix\ncan be replaced by the scalar exp( jΦ n,m ) and only vertically polarised field patterns are applied.\nRep. ITU-R M.2135-1 39\n\nThe Doppler frequency component is calculated from the angle of arrival (downlink), UT speed v\nand direction of travel θv:\nv cos(ϕ n ,m − θ v )\nυ n ,m = (21)\nλ0\nFor the two strongest clusters, say n = 1 and 2, rays are spread in delay to three sub-clusters (per\ncluster), with fixed delay offset {0,5,10 ns} (see Table A1-6). The delays of the sub-clusters are:\n\nτ n,1 = τ n + 0 ns\nτ n,2 = τ n + 5 ns (22)\nτ n,3 = τ n + 10 ns\n\nTwenty rays of a cluster are mapped to sub-clusters as presented in Table A1-6. The corresponding\noffset angles are taken from Table A1-5 with mapping of Table A1-6.\n\nTABLE A1-6\nSub-cluster information for intra cluster\n\n## Sub-cluster No. Mapping to rays Power Delay offset\n\n1 1, 2, 3, 4, 5, 6, 7, 8, 19, 20 10/20 0 ns\n2 9, 10, 11, 12, 17, 18 6/20 5 ns\n3 13, 14, 15, 16 4/20 10 ns\n\nIn the LoS case, define H 'u ,s ,n = H u ,s ,n and determine the channel coefficients by adding a single line-\nof-sight ray and scaling down the other channel coefficient generated by equation (20). The channel\ncoefficients are given by:\n\n1\nH u , s , n (t ) = H 'u , s , n (t )\nKR + 1\n\n+ δ (n − 1)\nK R  Frx ,u ,V (ϕ LOS ) exp jΦ LOS\nvv T\n0( )\n  Ftx , s ,V (φLOS )\n(23)\nF    \nK R + 1  rx ,u , H (ϕ LOS )  0 ( )\nexp jΦ LOS   Ftx , s , H (φLOS )\nhh\n\n( ) (\n⋅ exp jd s 2πλ−01 sin (φLOS ) exp jd u 2πλ−01 sin (ϕ LOS ) exp( j 2πυ LOS t ))\nwhere δ(.) is the Dirac’s delta function and KR is the Ricean K-factor defined in Table A1-7\nconverted to linear scale.\nIf non-ULA arrays are used, the equations must be modified. For arbitrary array configurations on\nthe horizontal plane, see Fig. 10, the distance term du in equations (23) and (20) is replaced by:\n\nxu + yu cos(arctan( yu xu ) − ϕ n , m )\n2 2\n'\nd u , n, m = , (24)\nsin ϕ n , m\n\nwhere (xu,yu) are the co-ordinates of the uth element Au and A0 is the reference element.\n40 Rep. ITU-R M.2135-1\n\nFIGURE 10\nModified distance of antenna element u with non-ULA array\ny\n\nAu\nyu\n\nd 'u,dun,’m\n\nA0 ϕn, m xu X\n\nReport 2135-10\n\nStep 10b: Generate channel coefficients for each cluster n and each receiver and transmitter\nelement pair u,s. Alternatively to Step 10a channel coefficients can be generated by applying the so\ncalled correlation matrix based method. Temporal correlations (Doppler effect) are introduced\neither by filtering independent and identically distributed complex Gaussian sequences with\na proper Doppler spectrum shaping filter or by applying some equivalent method. MIMO antenna\ncorrelations are introduced by performing a linear transformation to the temporally correlated\nsequences. Unique covariance matrices are determined for each drop and radio link based on the\nparameters defined in the previous steps. The MIMO covariance matrix can be composed of\ntransmitter and receiver spatial correlation matrices, and by the polarization covariance matrices\nwith the Kronecker product of matrices. Spatial correlation matrices can be derived for each cluster\ndirectly from the ChIRs of Step 10a [Kermoal, et al., 2002; Smith and Shafi, 2004]. Alternatively,\nspatial correlation matrices and the polarization covariance matrices can be derived from the\nantenna configuration and the model parameters (AoA, AoD) determined in Step 7 with XPR\nparameter from Table A1-7*.\nStep 11: Apply path loss and shadowing for the channel coefficients. This is valid only for\nsystem level simulations.\n\nTABLE A1-7\nChannel model parameters\nIn Table A1-7: DS: rms delay spread, ASD: rms azimuth spread of departure angles, ASA: rms azimuth spread of\nThe sign of the shadow fading is defined so that positive SF means more received power at UT than predicted by the\npath loss model.\n\n## InH UMi SMa UMa RMa\n\nScenarios\nLoS NLoS LoS NLoS O–to–I LoS NLoS LoS NLoS LoS NLoS\nDelay spread (DS) μ –7.70 –7.41 –7.19 –6.89 –6.62 –7.23 –7.12 –7.03 –6.44 –7.49 –7.43\nlog10(s) σ 0.18 0.14 0.40 0.54 0.32 0.38 0.33 0.66 0.39 0.55 0.48\nAoD spread (ASD) μ 1.60 1.62 1.20 1.41 1.25 0.78 0.90 1.15 1.41 0.90 0.95\nlog10(degrees) σ 0.18 0.25 0.43 0.17 0.42 0.12 0.36 0.28 0.28 0.38 0.45\n\n* WiMAX Forum, WiMAX Forum Mobile Release 1.0 Channel Model, June 2008, available at\nhttp://www.wimaxforum.org/technology/documents/wimax_forum_mobile_release_1_0_channel_model_\nv100.pdf.\nRep. ITU-R M.2135-1 41\n\n## TABLE A1-7 (end)\n\nInH UMi SMa UMa RMa\nScenarios\nLoS NLoS LoS NLoS O–to–I LoS NLoS LoS NLoS LoS NLoS\nAoA spread (ASA) μ 1.62 1.77 1.75 1.84 1.76 1.48 1.65 1.81 1.87 1.52 1.52\nlog10(degrees) σ 0.22 0.16 0.19 0.15 0.16 0.20 0.25 0.20 0.11 0.24 0.13\nσ 3 4 3 4 7 4 8 4 6 4 8\n(dB)\nμ 7 N/A 9 N/A N/A 9 N/A 9 N/A 7 N/A\nK–factor (K) (dB)\nσ 4 N/A 5 N/A N/A 7 N/A 3.5 N/A 4 N/A\nASD vs DS 0.6 0.4 0.5 0 0.4 0 0 0.4 0.4 0 –0.4\nASA vs DS 0.8 0 0.8 0.4 0.4 0.8 0.7 0.8 0.6 0 0\nASA vs SF –0.5 –0.4 –0.4 –0.4 0 –0.5 0 –0.5 0 0 0\nASD vs SF –0.4 0 –0.5 0 0.2 –0.5 –0.4 –0.5 –0.6 0 0.6\nDS vs SF –0.8 –0.5 –0.4 –0.7 –0.5 –0.6 –0.4 –0.4 –0.4 –0.5 –0.5\nCross–correlations*\nASD vs ASA 0.4 0 0.4 0 0 0 0 0 0.4 0 0\nASD vs K 0 N/A –0.2 N/A N/A 0 N/A 0 N/A 0 N/A\nASA vs K 0 N/A –0.3 N/A N/A 0 N/A –0.2 N/A 0 N/A\nDS vs K –0.5 N/A –0.7 N/A N/A 0 N/A –0.4 N/A 0 N/A\nSF vs K 0.5 N/A 0.5 N/A N/A 0 N/A 0 N/A 0 N/A\nDelay distribution Exp Exp Exp Exp Exp Exp Exp Exp Exp Exp Exp\nWrapped Wrapped Wrapped\nAoD and AoA distribution Laplacian Wrapped Gaussian\nGaussian Gaussian Gaussian\nDelay scaling parameter rτ 3.6 3 3.2 3 2.2 2.4 1.5 2.5 2.3 3.8 1.7\nXPR (dB) μ 11 10 9 8.0 9 8 4 8 7 12 7\nNumber of clusters 15 19 12 19 12 15 14 12 20 11 10\nNumber of rays per cluster 20 20 20 20 20 20 20 20 20 20 20\nCluster ASD 5 5 3 10 5 5 2 5 2 2 2\nCluster ASA 8 11 17 22 8 5 10 11 15 3 3\nPer cluster shadowing std ζ (dB) 6 3 3 3 4 3 3 3 3 3 3\nDS 8 5 7 10 10 6 40 30 40 50 36\nASD 7 3 8 10 11 15 30 18 50 25 30\nCorrelation distance (m) ASA 5 3 8 9 17 20 30 15 50 35 40\nSF 10 6 10 13 7 40 50 37 50 37 120\nK 4 N/A 15 N/A N/A 10 N/A 12 N/A 40 N/A\n\n## 1.3.2.2 Reduced variability models (Optional)\n\nThe CDL model is intended for calibration purposes only. Each cluster is composed of 20 rays with\nfixed offset angles and identical power. In the case of a cluster where a ray of dominant power\nexists, the cluster has 20 + 1 rays. This dominant ray has zero angle offset. The departure and\narrival rays are coupled randomly. The CDL tables of all scenarios of interest are given below,\nwhere the cluster power and the power of each ray are tabulated. The CDL models offer well-\ndefined radio channels with fixed parameters to obtain comparable simulation results with relatively\nnon-complicated channel models. Delay spread and azimuth spreads medians of the CDL models\nare equal to median values given in Table A1-8.\n42 Rep. ITU-R M.2135-1\n\nTABLE A1-8\nExpectation (median) output values for large scale parameters\n\nDS AS at BS AS at UT\nScenario\n(ns) (degrees) (degrees)\nLoS 20 40 42\nInH\nNLoS 39 42 59\nLoS 65 16 56\nUMi NLoS 129 26 69\nO-to-I 49 18 58\nLoS 59 6 30\nSMa\nNLoS 75 8 45\nLoS 93 14 65\nUMa\nNLoS 365 26 74\nLoS 32 8 33\nRMa\nNLoS 37 9 33\n\nCDL models can be implemented as described either in Step 10a or in Step 10b.\nInH – Indoor hotspot\nThe CDL parameters of LoS and NLoS condition are given below. In the LoS model the Ricean K\nfactor is 5 dB.\n\nTABLE A1-9\nScenario InH LoS clustered delay line model, indoor hotspot\n\n## Cluster Delay Power AoD AoA Ray power\n\nNo. (ns) (dB) (degrees) (degrees) (dB)\n1 0 0 0 0 –0.01(1) –41.7(2)\n2 10 –15.7 79 85 –28.7\n3 25 –10.5 –84 73 –23.5\n4 25 –16.7 –97 –82 –29.7\n5 30 –17.6 –102 –72 –30.6\nCluster ASD = 5º\n\nCluster ASA = 8º\n\nXPR = 11 dB\n\n## 6 35 –14.1 80 –64 –27.2\n\n7 40 –12.9 69 –73 –25.9\n8 50 –19.5 104 –83 –32.5\n9 55 –21.8 98 96 –34.8\n10 60 –20.8 98 –91 –33.8\n11 65 –24.1 –113 –93 –37.1\n12 85 –13.9 –79 74 –26.9\n13 90 –20.1 –109 –87 –33.1\n14 100 –18 89 –77 –31\n15 130 –21 95 –97 –34\n(1)\nPower of dominant ray.\n(2)\nPower of each other ray.\nRep. ITU-R M.2135-1 43\n\nTABLE A1-10\nScenario InH NLoS clustered delay line model, indoor hotspot\n\n## Cluster Delay Power AoD AoA Ray power\n\nNo. (ns) (dB) (degrees) (degrees) (dB)\n1 0 –2.4 –34 50 –15.4\n2 15 –1.9 34 50 –15\n3 20 –8.1 60 –83 –21.1\n4 25 –1.8 24 –30 –14.8\n5 30 0 –1 –3 –13\n6 40 –2.3 –31 44 –15.4\n\n## Cluster ASA = 11º\n\nCluster ASD = 5º\n7 55 –3.7 55 –63 –16.8\n\nXPR = 10 dB\n8 60 –8.4 69 –105 –21.4\n9 60 –3.2 34 –61 –16.2\n10 70 –9.7 –73 110 –22.7\n11 75 –6.2 –55 –85 –19.2\n12 75 –8.9 56 –85 –21.9\n13 90 –4 –47 58 –17\n14 150 –14.1 –84 121 –27.1\n15 160 –12.1 87 –109 –25.1\n16 170 –10.6 71 108 –23.6\n17 195 –19.6 –99 139 –32.6\n18 205 –16.8 –99 –137 –29.8\n19 225 –13.5 86 124 –26.5\n\n## UMi – Urban micro-cell\n\nIn the LoS model the Ricean K-factor is 6 dB.\n\nTABLE A1-11\nScenario UMi: LoS clustered delay line model, urban mico-cell\n\n## Cluster Delay Power AoD AoA Ray power\n\nNo. (ns) (dB) (degrees) (degrees) (dB)\n1 0 0 0 0 –0.20(1) –28.8(2)\n2 30 35 40 −15.6 −17.8 −19.6 31 –108 –25.6\n3 85 −14.0 35 153 –27.0\nCluster ASA = 17º\nCluster ASD = 3º\n\nXPR = 9 dB\n\n## 5 160 –20.7 45 179 –33.7\n\n6 195 –17.3 –46 –150 –30.3\n7 210 –21.8 46 –163 –34.8\n8 255 –17.7 –43 166 –30.7\n9 280 –21.6 –47 14 –34.6\n10 340 –23.0 51 –4 –36.0\n11 360 –24.5 –62 43 –37.5\n12 420 –25.0 –58 44 –38.0\n(1)\nPower of dominant ray.\n(2)\nPower of each other ray.\n44 Rep. ITU-R M.2135-1\n\nTABLE A1-12\nScenario UMi: NLoS clustered delay line model, urban micro-cell\n\n## Cluster Delay Power AoD AoA Ray power\n\nNo. (ns) (dB) (degrees) (degrees) (dB)\n1 0 –6.7 31 –83 –19.8\n2 10 15 10 –4.9 –7.1 –8.9 10 54 –14.7\n3 20 –1.9 26 37 –14.9\n4 35 –6.3 –27 96 –19.3\n5 40 45 50 –3 –5.2 –7 –3 3 –13.0\n6 55 –7.5 29 –86 –20.5\n7 55 –6.4 –33 84 –19.4\n\n## Cluster ASA = 22º\n\nXPR = 8 dB\n8 200 –10.8 37 –112 –23.9\n9 205 –5.2 –29 –78 –18.2\n10 250 –4.9 27 65 –17.9\n11 330 –9.2 –35 –88 –22.3\n12 440 –15.5 –48 –130 –28.5\n13 440 –16.7 –50 153 –29.7\n14 515 –12.4 –44 106 –25.5\n15 530 –16.9 46 147 –29.9\n16 580 –12.7 –45 118 –25.7\n17 590 –23.5 –59 –178 –36.5\n18 625 –22.1 60 –163 –35.1\n19 730 –23.6 56 163 –36.6\n\nTABLE A1-13\nScenario UMi: Outdoor-to-indoor clustered delay line model, urban micro-cell\n\n## Cluster Delay Power AoD AoA Ray power\n\nNo. (ns) (dB) (degrees) (degrees) (dB)\n1 0 5 10 –3.0 –5.2 –7.0 0 0 –13.0\n2 0 –8.7 32 102 –21.7\n3 5 –3.7 –21 –66 –16.7\nCluster ASD = 3º\n\nCluster ASA = 8º\n\nXPR = 9 dB\n\n## 5 35 –16.2 –43 139 –29.2\n\n6 35 –6.9 28 91 –19.9\n7 65 70 75 –3.4 –5.6 –7.3 –49 157 –13.4\n8 120 –10.3 –34 –111 –23.3\n9 125 –20.7 –49 157 –33.7\n10 195 –16.0 43 138 –29.1\n11 250 –21.0 49 158 –34.0\n12 305 –22.9 51 165 –35.9\nRep. ITU-R M.2135-1 45\n\n## UMa – Urban macro-cell\n\nIn the LoS model the Rician K-factor is 5.0 dB.\n\nTABLE A1-14\nScenario UMa: LoS clustered delay line model, urban macro-cell\nCluster Delay Power AoD AoA Ray power\nNo. (ns) (dB) (degrees) (degrees) (dB)\n1 0 0 0 0 –0.23(1) –26.5(2)\n2 15 20 25 –15.4 –17.6 –19.4 36 143 –25.4\n3 30 –12.6 –28 –156 –25.6\n\n## Cluster ASA = 11º\n\nCluster ASD = 5º\n4 45 50 55 –14.1 –16.3 –18.1 –25 133 –24.1\n\nXPR = 8 dB\n5 220 –19.4 42 180 –32.4\n6 310 –23.8 –40 19 –36.8\n7 365 –16.7 –28 161 –29.7\n8 440 –19.4 –35 3 –32.4\n9 450 –24.8 –47 40 –37.8\n10 535 –21.3 –39 –37 –34.3\n11 595 –23.9 47 –36 –36.9\n12 640 –25.0 –43 –48 –38.0\n(1)\nPower of dominant ray.\n(2)\nPower of each other ray.\n\nTABLE A1-15\nScenario UMa: NLoS clustered delay line model, urban macro-cell\nCluster Delay Power AoD AoA Ray power\nNo. (ns) (dB) (degrees) (degrees) (dB)\n1 0 5 10 −3.5 −5.7 −7.5 6 29 –13.5\n2 5 −9.2 44 –98 –22.2\n3 20 25 30 −3.0 −5.2 −7.0 2 8 –13.0\n4 45 –7.8 –34 –114 –20.8\n5 265 –3.7 26 70 –16.7\n6 290 –8.6 –41 107 –21.6\n7 325 –2.5 –17 59 –15.5\nCluster ASA = 15º\nCluster ASD = 2º\n\nXPR = 7 dB\n\n## 9 355 –3.8 24 73 –16.8\n\n10 440 –6.9 –34 –111 –19.9\n11 555 –8.9 –38 –112 –21.9\n12 645 –9.0 44 122 –22.0\n13 970 –9.8 53 129 –22.8\n14 1 015 –15.0 54 153 –28.0\n15 1 220 –13.4 53 –145 –26.4\n16 1 395 –14.9 52 –157 –27.9\n17 1 540 –16.7 57 –178 –29.7\n18 1 750 –11.2 53 –114 –24.2\n19 1 870 –18.2 –54 –160 –31.2\n20 1 885 –17.8 –60 –175 –30.8\n46 Rep. ITU-R M.2135-1\n\n## SMa – Sub-urban macro-cell (Optional)\n\nIn the LoS model the Rician K-factor is 4 dB.\n\nTABLE A1-16\nScenario SMa: LoS clustered delay line model, suburban\n\n## Cluster Delay Power AoD AoA Ray power\n\nNo. (ns) (dB) (degrees) (degrees) (dB)\n1 0 5 10 0.0 –25.3 –27.1 0 0 –0.02(1) –33.1(2)\n2 85 –21.6 –29 –144 –34.7\n\n## Cluster ASD = 5º , Cluster ASA = 5º\n\n3 135 –26.3 –32 –159 –39.3\n4 135 –25.1 –31 155 –38.1\n5 170 –25.4 31 156 –38.4\n\nXPR = 8 dB\n6 190 –22.0 29 –146 –35.0\n7 275 –29.2 –33 168 –42.2\n8 290 295 300 –24.3 –26.5 –28.2 35 –176 –34.3\n9 290 –23.2 –30 149 –36.2\n10 410 –32.2 35 –176 –45.2\n11 445 –26.5 –32 –159 –39.5\n12 500 –32.1 35 –176 –45.1\n13 620 –28.5 33 –165 –41.5\n14 655 –30.5 34 –171 –43.5\n15 960 –32.6 35 177 –45.6\n(1)\nPower of dominant ray.\n(2)\nPower of each other ray.\n\nTABLE A1-17\nScenario SMa: NLoS clustered delay-line model, suburban\n\n## Cluster Delay Power AoD AoA Ray power\n\nNo. (ns) (dB) (degrees) (degrees) (dB)\n1 0 5 10 –3.0 –5.2 –7.0 0 0 –13.0\n2 25 –7.5 13 –71 –20.5\nCluster ASD = 2º , Cluster ASA = 10º\n\n## 3 35 –10.5 –15 –84 –23.5\n\n4 35 –3.2 –8 46 –16.2\n5 45 50 55 –6.1 –8.3 –10.1 12 –66 –16.1\nXPR = 4 dB\n\n## 6 65 –14.0 –17 –97 –27.0\n\n7 65 –6.4 12 –66 –19.4\n8 75 –3.1 –8 –46 –16.1\n9 145 –4.6 –10 –56 –17.6\n10 160 –8.0 –13 73 –21.0\n11 195 –7.2 12 70 –20.2\n12 200 –3.1 8 –46 –16.1\n13 205 –9.5 14 –80 –22.5\n14 770 –22.4 22 123 –35.4\nRep. ITU-R M.2135-1 47\n\n## RMa – Rural macro-cell\n\nThe CDL parameters for LoS and NLoS conditions are given below. In the LoS model the Ricean\nK-factor is 6 dB.\n\nTABLE A1-18\nScenario RMa: LoS Clustered delay line model, rural\nCluster Delay Power AoD AoA Ray power\nNo. (ns) (dB) (degrees) (degrees) (dB)\n1 0 5 10 0.0 –17.8 –19.5 0 0 –0.12(1) –25.6(2)\n2 35 40 45 –16.9 –19.1 –20.9 24 99 –26.9\n3 45 –16.8 23 95 –29.8\n\nCluster ASD = 2º\nCluster ASA = 3º\nXPR = 12 dB\n4 65 –18.3 24 99 –31.3\n5 65 –21.2 –25 –106 –34.2\n6 110 –17.1 –23 96 –30.1\n7 125 –19.7 –25 –103 –32.7\n8 125 –23.8 27 113 –36.8\n9 170 –22.9 27 110 –35.9\n10 170 –20.9 25 106 –33.9\n11 200 –21.9 –26 –108 –34.9\n(1)\nPower of dominant ray.\n(2)\nPower of each other ray.\n\nTABLE A1-19\nScenario RMa: NLoS clustered delay line model, rural\n\n## Cluster Delay Power AoD AoA Ray power\n\nNo. (ns) (dB) (degrees) (degrees) (dB)\n1 0 –4.9 –12 –46 –17.9\n2 0 –7.8 16 58 –20.8\nCluster ASD = 2º\n\nCluster ASA = 3º\n\n## 3 5 10 15 –3.0 –5.2 –7.0 0 0 –13.0\n\nXPR = 7 dB\n\n4 5 –2.7 9 34 –15.7\n5 10 –2.6 –9 34 –15.6\n6 15 20 25 –3.6 –5.8 –7.5 13 49 –13.6\n7 15 –2.7 –9 –34 –15.7\n8 55 –5.6 13 49 –18.6\n9 110 –7.3 15 56 –20.3\n10 220 –10.3 18 67 –23.3\n48 Rep. ITU-R M.2135-1\n\n## 1.4 Extension module\n\nThe purpose of the extension module is to generate parameters for Table A1-7 relating to the\nprimary model and Tables A1-14 to A1-19 relating to the CDL model for macro-cell scenarios such\nas UMa, SMa and RMa as described in the primary module in § 1.3. The extension module provides\nadditional level of parameter variability. In the primary module, SS and LS parameters are\nvariables, and the extension module provides new parameter values for the primary module based\non environment-specific parameters. The channel impulse responses are generated in the primary\nmodule according to Steps 1 to 11 described in § 1.3.2.1.\nThe extension module is based on the time-spatial propagation model (TSP model) described in\nRecommendation ITU-R P.1816 [Fujii, 2004 ; Ohta and Fujii, 2006 and Omote and Fujii, 2007].\nThe TSP model is a geometry-based double directional channel model with closed-form functions.\nIt calculates the large scale parameters for the channel realization by taking into account the\nfollowing key parameters such as city structures (street width, average building height), BS height,\nbandwidth and the distance between the BS and the UT. The channel models are targeted for up to\n100 MHz RF bandwidth.\nThe TSP model consists of three profile models, which are long-term time-spatial profile model,\nShort-term time-spatial profile model, and Instantaneous time-spatial profile model, see Fig. 11. The\ntime-spatial profile is the joint delay-AoA power spectrum. This will vary with the distance. The\nsnapshot of the time-spatial profile at a given distance is called instantaneous time-spatial profile.\nThe average of the time-spatial profile over several tens of wavelengths is the short-term time-\nspatial profile. The average of the time-spatial profile over hundreds of wavelengths is the long-\nterm time-spatial profile. The delay profiles arrival angular profiles are generated by the time-\nspatial profile model, as shown in Fig. 12.\n\nFIGURE 11\nTime-spatial propagation (TSP) model for base coverage macrocell\nDistance\n\n2 100\n1 80\n0 60\n–100\n–50 40\n0 20\n50\n0\n100\n1\n1.5 0.75 100\n1 100 0.5\n0.5 80 0.25 80\n0 0 60\n–0.5 60 –100\n–100 –50 40\n–50 40\n0 20 0 20\n50 50\n0\ning\n\n0\n100 100\ng\ngin\nrag\n\n1 100\n0 80\nera\ne\n\n60\nAv\n\n–100\n–50 40\nAv\n\n0 20\n50\n0\n100\n\n1.5 1\ne\n\n1\n1 0.75 100 100\n100 0.5\n0.75\ntim\n\n0.5 80 0.5\n80\n0 80 0.25\n60\n0.25\n–0.5 60 0 0 60\n–100 –100\n–100 –50 40 –50 40\n40\nlay\n\n–50\n0 20 0\n50\n20\nArr 0\n50\n20\n50 0\niv al a\n1000\nDe\n\n0\n100 100\nng l\ne\nInstantaneous Short-term\ntime-spatial profile time-spatial profile Long-term\ntime-spatial profile\n\n## Small scale (SS) variation Large scale (LS) variation\n\nReport 2135-11\nRep. ITU-R M.2135-1 49\n\nLarge scale parameters (e.g., delay spread and angle spread) in the primary module are calculated\nfrom the Long-term time-spatial profile model. Reduced variability models in the extension module\nare calculated from both long-term and short-term time-spatial profile models, which are detailed in\n§ 1.4.2.\n\nFIGURE 12\nDelay and angular profiles for base coverage macrocell\nDelay profile Arrival or departure angular profile\nLarge scale\nPower (LS) variation Power\n\n## Excess delay time Arrival angle\n\nSmall scale\n(SS) variation\nReport 2135-12\n\nFigure 13 shows the concept of using the TSP model to generate a time-spatial profile. After\nproviding to the model the key parameters like carrier frequency, bandwidth, city structure such as\naverage building height, road width, BS antenna height, and distance from the BS, a time-spatial\nprofile is generated by setting the pseudo-random number generator.\n\nFIGURE 13\nGeneration of TSP model\n\n## Input (Key parameters)\n\nGenerated\nGeneratedtime\ntime- -spatial\nspatial\nfc: carrier frequency profile\nprofile\nB: bandwidth\nTime- spatial\nTime-spatial\n<H> : average building height propagationmodel\npropagation model\ngenerated by using Power\nPower Power\nPower\npseudoby using\nrandom\nhb: BS antenna height\npseudonumber\nrandom\nd: distance from the BS\netc. number\nDelay time\nDelay time Arrival\nArrival angle\nangle\nDelay profile\nDelay profile Angular profile\nAngular profile\nReport 2135-13\n\nTable A1-20 shows key parameters of the TSP model and their applicable ranges.\nTable A1-21 shows the parameter values of the city structures for the UMa, SMa and RMa\nscenarios. By using Table A1-21, a time-spatial channel model that can evaluate simulation\nconditions different from those described in Table 8-1.\n50 Rep. ITU-R M.2135-1\n\nTABLE A1-20\nKey parameters in TSP model\n\n## Condition Key parameter Symbol Applicable range\n\nFrequency conditions Carrier frequency (MHz) fc 400-6 000 MHz\nBandwidth or chip rate (MHz) B 0.5-100 MHz\nCity structures Average building height (m) <H> 5-50 m:\n(height above the UT ground\nlevel)\nRoad width (m) W 5-50 m\nCell conditions BS antenna height (m) hb 10-150 m\n(height above the UT ground\nlevel)\nUT antenna height (m) hm 1-10 m\nDistance from the BS (m) d 50-5 000 m\n\nTABLE A1-21\nParameter values for city structures\n\n## Key parameters Average building height, <H> Road width, W\n\nCity structure (m) (m)\n\n## (*) Dense urban 30 20\n\nUMa Urban 20 20\nSMa Suburban 10 20\n(*) Residential 8 20\nRMa Rural 5 20\n\n## 1.4.1 Large scale parameters\n\n1.4.1.1 NLoS scenarios\nFormulas for empirical path-loss, power delay profile, departure angular profile, and arrival angular\nprofile for Base coverage macro-cellular scenarios such as SMa, UMa and RMa, which can take the\ncity structure into account and suit carrier frequencies up to the SHF band, are given below:\na) Path-loss model\nAn empirical propagation loss formula, which can take the city structure into account and suits\ncarrier frequencies up to the SHF band, is given as follows [Fujii, 2003]:\nLossL (d ) = 101.04 − 7.1log W + 7.5log < H >\ndB (25)\n− {24.37 − 3.7(< H > / hb ) 2 } log hb + (43.42 − 3.1log hb ) log(d /1000) + 20 log f c − a(hm )\n\nwhere d denotes the distance between the BS and the UT. hb (m), <H> (m), and W denote the BS\nantenna height (m), the average building height, and the street width, respectively. fc denotes the\ncarrier frequency (MHz). a(hm) is a correction factor for mobile antenna height hm (m) defined as\nfollows:\na(hm ) = 3.2(log(11.75hm ))2 − 4.97 dB (26)\nRep. ITU-R M.2135-1 51\n\n## b) Power delay profile model\n\nAn empirical long-term path delay profile model PDPL ( i, d ) that takes into account the city\nstructure and that suits carrier frequencies up to the SHF band and bandwidth B (MHz) up to\n100 MHz is recommended as Recommendation ITU-R P.1816 . The delay profile normalized by the\nfirst cluster’s power is given as follows [Fujii, 2004 and Ohta and Fujii, 2006]:\nPDPL ( i, d ) = α ( d ) log (1 + i ) + 10 log c (i ) (27)\nwhere:\n\n## α ( d ) = − {19.1 + 9.68 log ( hb / < H > )} B{\n\n−0.36 + 0.12log ( hb / < H > )}\n(d /1000){\n−0.38 + 0.21log ( B )}\n(28)\n\n1 (i = 0)\n\nc(i) = \n (\nmin 0.63, {0.59e\n−0.017 B\n+ (0.017 + 0.0004B) < H >} e {\n− (0.077 −0.00096 B) −(0.0014−0.000018B)< H >}i\n)\n(i ≥ 1) (29a)\n(0.5 ≤ B ≤ 50Mcps)\n\n1 (i = 0)\n\nc(i) =   −{0.0878e−0.0218 B −(0.0014−0.000018 B) < H >}i \nmin  0.63, {0.59e + (0.0172 + 0.0004B) < H >} e (29b)\n−0.0172 B\n (i ≥ 1)\n  \n(0.5 < B ≤ 100Mcps)\n\nHere i represents the path number (i = 0, 1, 2, …) in the sense of an excess delay time normalized\nby the time resolution 1/B .\nThe power delay profile normalized by the total power of all clusters is given by:\n\n## where PDPnorm ( d ) represents the total power of all clusters:\n\n \nPDPnorm (d ) = 10 log   10 PDPL ( i ,d ) /10  dB (31)\n i =0 \n\nThe path number i can be changed to the excess delay time τ (μs) by using the following relation:\n\nτ = i / B  i =τ B (32)\n\n## c) Departure angular profile model\n\nAn empirical long-term arrival angular profile model AODL (θ, d), which can take the city structure\ninto account and that suits carrier frequencies up to the SHF band and bandwidth B (MHz) up to\n100 MHz, is recommended as Recommendation ITU-R P.1816. The departure angular profile\naround the BS normalized by the first cluster’s power is given as follows (see Recommendation\nITU-R P.1816 and [Omote and Fujii, 2007]):\n\nAODL (θ , d ) = 10 log (1 + θ / a )\n−β\n( θ ≤ θ max ) dB (34)\n52 Rep. ITU-R M.2135-1\n\nwhere:\n0.2d  < H > 0.23 \na=− + 2.1   \n1000  hb   (35)\nβ = ( −0.015 < H > +0.63) d /1000 − 0.16 + 0.76 log ( hb )\n\n## d) Arrival angular profile model\n\nThe arrival angular profile around the MT of each cluster is given by:\n\n## AOAL (θ , d ) = exp[ − θ / θ AOA ] ( θ ≤ 180 ) (36)\n\nwhere θAOA denotes the angle spread of AOA. AOAL (θ, d) does not depend on the distance, d.\n1.4.1.2 LoS scenarios\nFormulas for empirical path-loss, power delay profile, departure angular profile, and arrival angular\nprofile for LoS Base coverage macro-cellular scenarios such as SMa and UMa, which can take the\ncity structure into account and suit carrier frequencies up to the SHF band, are given in the\nfollowing.\na) Path-loss model\nThe loss formula consists of formulas that cover the cases before and after the break point, B p\nwhich is defined by:\n2π hb hm\nBp = (37)\nλ\nλ (m) denotes the wavelength and is given by 300 / fc . An empirical propagation loss formula\nbefore the break point Bp is given as follows:\n\n## LossLLOS (d ) = 20 log(4π d / λ ) + ( min(0.03 < H >1.72 ,10) ) log(d )\n\n(d ≤ Bp ) (38)\n− min ( 0.044 < H >1.72 ,14.77 ) + ( 0.002 log < H > ) d\n\nAn empirical propagation loss formula after the break point Bp, is expressed as follows:\nLossLLOS (d ) = 40 log(1000d ) − 20 log hb − 20 log hm + 5log f c\n( Bp ≤ d ) (39)\n+ 11log < H > −7.1log W − 2.45\n\n## b) Power delay profile model\n\nAn empirical long-term delay profile model PDPLLOS ( i, d ) normalized by the first cluster’s power\nis given by:\nPDPL ,scatter ( l , d ) /10\nPDPLLOS (l , d ) = PDPL , direct (l , d ) + 10 (40)\nwhere:\nPDPL ,direct (l , d ) = δ (l ) + 0.3δ (l − W 2 / d ) + 0.1δ (l − 3W 2 / d ) (41)\n\n## PDPL , scatter (l , d ) = PDPL ( l , d ) − 12 dB (42)\n\nδ(l) represents a delta function defined as δ(l) =1 for l =1 and δ(l) =0 for l ≠ 0 . PDPL ( l , d ) is given\nby the NLOS power delay profile model.\nRep. ITU-R M.2135-1 53\n\nThe K-factor of the first cluster (l = 0) is given from equations (41) and (42) as follows:\n\nK = 12 dB (43)\nc) Departure angular profile model\nAn empirical long-term departure angular profile model AODLLOS (θ, d) with departure angle θ\nnormalized by the first cluster’s power is given as follows:\nAODL ,scatter (θ , d ) /10\nAODLLOS (θ , d ) = AODL ,direct (θ , d ) + 10 (44)\n\nwhere:\nAODL,direct (θ , d ) = δ (θ ) + 0.3δ (θ − W / d ⋅ (180 / π )) + 0.1δ (θ − 2W / d ⋅ (180 / π )) (45)\n\n## AODL , scatter (θ , d ) = AODL (θ , d ) − 12 dB (46)\n\nHere, AODL (θ, d) is given by the NLoS departure angular profile model.\nd) Arrival angular profile model\nAn empirical long-term arrival angular profile model AODLLOS (θ, d) with arrival angle θ\nnormalized by the first cluster’s power is given as follows:\nAOAL ,scatter (θ , d ) /10\nAOALLOS (θ , d ) = AOAL ,direct (θ , d ) + 10 (47)\n\nwhere:\nAOAL,direct (θ , d ) = δ (θ ) + 0.3δ (θ − 2W / d ⋅ (180 / π )) + 0.1δ (θ + 2W / d ⋅ (180 / π )) (48)\n\n## AOAL , scatter (θ , d ) = AOAL (θ , d ) − 12 dB (49)\n\nHere, AOAL (θ, d) is given by the NLoS arrival angular profile model.\n1.4.1.3 XPD\nXPD represents cross-polarization discrimination; its value (dB) is given by:\n\n## 1.4.1.4 Building penetration loss\n\nAn empirical building penetration loss formula LossPene (d P , h f ) is expressed as follows [Okamoto\nand Ichitsubo, 2005]:\nLossPene (d P , h f ) = 0.6d P − 0.6h f + 10 (dB) (51)\n\nwhere hf (m) denotes the height of the floor where the UT is located and dP (m) denotes the entry\ndistance from the nearest window to the UT on the floor where the UT is located. The floor height\ncan be changed to the floor number NF (= 1,2,…) by using the height per floor Δh f as follows:\n\n## LossPene (d P , h f ) = 0.6d P − 0.6( N F − 1)Δh f + 10 (dB) ( N F = 1, 2, − − −) (52)\n\n54 Rep. ITU-R M.2135-1\n\nTherefore, the propagation loss LossLO → I (d ) from outdoor BS to indoor UT can be expressed by\nadding the building penetration loss LossPene (d P , h f ) to the propagation loss LossL (d ) on the street\nnext to the building as follows:\nLossLO → I ( d ) = LossL ( d ) + LossPene (d P , h f ) (dB) (53)\n\n## 1.4.1.5 Short-term variation\n\nThe short-term variation is generally caused by shadow effects. It can be assumed it follows a log-\nnormal distribution with its power spectrum being:\nσ 2\n s ( f ≤ fs )\nPs ( f ) =  2 f s (54)\n\n0 ( f > fs )\n\nwhere σs2 is the average power of the short-term variation and is usually given by the normalized\npower of 1. Typical standard deviation of log-normal shadowing is set to 4.5 dB for macro-cell\nenvironments. fs, the maximum frequency of the short-term variation, and depends on the\nenvironment and bandwidth as follows:\nf s = v /(110B −0.146 < H >0.058 ) (55)\nwhere v (m) is the speed of the UT.\n1.4.1.6 Spreads and their variations of PDP, AoD and AoA in large scale parameters\nThe delay spread σ PDP ( d ) at distance d from the BS is expressed as:\nτ max ( d )\n\nσ PDP (d ) =\n0\n(τ − TPDP (d )) 210 PDPL (τ ,d ) /10 dτ\n(56)\nτ max ( d )\n 0\n10 PDPL (τ ,d ) /10 dτ\n\nwhere TPDP (d) is the average delay time at distance d from BS given by:\nτ max ( d )\n\nTPDP\n\n(d ) = τ 0\nτ 10PDP (τ ,d ) /10 dτ\nL\n\nmax ( d )\n(57)\n 0\n10PDPL (τ ,d )/10 dτ\n\nτmax(d) (μs) is the maximum delay time determined from the threshold level ΔLth (dB) (>0); it must\nsatisfy the following equation:\n\n## PDPL (τ max (d ), d ) = −ΔLth (58)\n\nAssuming that the UT is located uniformly in the cell area, the average delay spread μPDP and its\nstandard deviation σ σ over the whole cell area covered with cell radii of dmin and dmax are given\nPDP\n\nrespectively by\nd max 2π r\nμ PDP =  σ PDP (r ) dr (59)\nd min π (d max 2 − d min 2 )\nRep. ITU-R M.2135-1 55\n\n2π r\n (σ ( r ) − μ )\nd max 2\nσσ = PDP PDP dr (60)\nPDP d min π ( d − d min\n22\nmax )\n\n## b) The spread of AoD\n\nThe angular spread σAOD(d) at distance d from BS is expressed by:\nθmax ( d )\nθ−\n(θ − TAOD (d ))2 ⋅10AODL (θ ,d )/10 dθ\nσ AOD (d ) = max( d )\nθmax( d ) degrees (61)\nθ − max( d )\n10AODL (θ ,d )/10 dθ\n\nwhere TAOD(d) is the average AoD at distance d from the BS given by:\nθmax ( d )\n\nTAOD\nθ\n(d ) = θ\n− max ( d )\nθ ⋅10AOD (θ ,d )/10 dθ L\n\n≡0\nmax ( d )\ndegrees (62)\nθ − max ( d )\n10AODL (θ ,d )/10 dθ\n\nθmax (d) (degrees) is the maximum AoD determined from the threshold level ΔLth (dB) (>0); it must\nsatisfy the following equation:\nAODL (θ max (d ), d ) = −ΔLth and min ( 60, θ max ( d ) )  θ max ( d ) (63)\n\n## In the case of NLoS, θmax(d)can be easily obtained by equation (64):\n\n/10 β ( d )\nθmax (d ) = min[60, a(d )(10ΔL th\n− 1)] (deg.) (64)\n\nAssuming that the UT is located uniformly in the cell area, the average delay spread μAOD and its\nstandard deviation σ σ over the whole cell area covered with cell radii of dmin and dmax are given\nAOD\n\nrespectively by:\nd max 2π r\nμ AOD =  σ AOD (r ) dr degrees (65)\nd min\nπ (d max 2 − d min 2 )\n\n2π r\n (σ ( r ) − μ AOD )\nd max 2\nσσ = AOD dr degrees (66)\nAOD d min π (d max\n2 2\n− d min )\n\n## c) The spread of AoA\n\nThe angular spread σAOA(d) at distance d from the BS is expressed as:\nθ max ( d )\n\nσ AOA (d ) =\nθ − max (d )\n(θ − TAOA (d )) 2 ⋅10 AOAL (θ ,d ) /10 dθ\ndegrees (67)\nθ max ( d )\nθ\nAOAL (θ , d ) /10\n10 dθ\n− max (d )\n\nwhere TAOA(d) is the average AOA at distance d from the BS given by:\nθ max ( d )\n\nTAOA\nθ\n(d ) = θ\n− max ( d )\nθ ⋅10 AOA (θ ,d ) /10 dθ L\n\ndegrees (68)\nmax ( d )\n\nθ − max ( d )\n10 AOAL (θ ,d ) /10 dθ\n56 Rep. ITU-R M.2135-1\n\nθmax (d) (degrees) is the maximum AoA determined from the threshold level ΔLth (dB) (>0); it must\nsatisfy equation (69):\nAOAL (θ max (d ), d ) = −ΔLth (69)\n\nIn the case of NLoS, θmax(d) can be easily obtained by the equation (70):\nθmax (d ) = min[180, φAI ln(10ΔL th /10\n)] = min[180, 0.23φAI ΔLth ] (deg.) (70)\n\nAssuming that the UT is located uniformly in the cell area, the average delay spread μAOA and its\nstandard deviation σσ AOA over the whole cell area covered with cell radii of dmin and dmax are given\nrespectively by:\n\nd max 2π r\nμ AOA =  σ AOA ( r ) dr degrees (71)\nd min\nπ ( d max 2 − d min 2 )\n\n2π r\n (σ ( r ) − μ )\nd max 2\nσσ = AOA AOA dr degrees (72)\nAOA d min π ( d − d min\n2\nmax\n2\n)\n\n## 1.4.1.7 Large scale parameters in Table A1-7\n\nOnce the values for city structures in Table A1-15 and the threshold parameter in Table A1-22 have\nbeen set, the extension module can calculate the values of the parameters in Table A1-7.\n\nTABLE A1-22\nThreshold parameter\n\nParameter Value\n(dB) (dB)\n\nThreshold ΔLth 25\n\nThe parameters calculated by the extension module correspond to those in Table A1-7 as follows:\nFor delay spread (DS), μ PDP → μ and σ σ PDP\n→σ\n\nThese parameters can be calculated by the following steps: equations (27) → (56) → (59) and (60)\nfor NLoS, and equations (40) → (56) → (59) and (60) for LoS, respectively.\nFor AoD spread (ASD), μ AOD → μ and σ σ AOD\n→σ\n\nThese parameters can be calculated by the following steps: equations (34) → (61) → (65) and (66)\nfor NLoS, and equations (44) → (61) → (65) and (66) for LoS, respectively.\nFor AoA spread (ASA), μ AOA → μ and σ σ AOA\n→σ\n\nThese parameters can be calculated by the following steps: equations (36) → (67) → (71) and (72)\nfor NLoS, and equations (47) → (67) → (71) and (72) for LoS, respectively.\nFor XPR, XPDL → μ\nThese parameters can be calculated by equation (50) for NLoS and LoS.\nRep. ITU-R M.2135-1 57\n\n## This parameter is a constant for NLoS and LoS.\n\nFor number of clusters, a constant number is generated (either 10 or 20).\nThe other parameters in Table A1-7 remain unchanged.\n1.4.1.8 Time-spatial profile in cluster\na) Delay profile in the ith cluster\nAn empirical delay profile model PDPI (lik ) normalized by the first arrival path’s power in the ith\ncluster is given as follows:\n\n## PDPI (lik ) = e−lik / lI (0 ≤ lik ≤ lI max ) (73)\n\nwhere lI and lImax denote the average excess path distance and the maximum excess path distance in\nthe cluster, respectively. On the other hand, the excess path distance l (m) can be changed to the\naccess delay time τ (μs) by using the relation of l = 300τ . The delay profile model in the cluster,\nPDPI (τ ik ) , can then be determined by can then be determined by:\n\n## PDPI (τ ik ) = e−τ ik /τ I (0 ≤ τ ik ≤ τ I max ) (74)\n\nwhere τ I and τ Imax denote the excess average delay time and the maximum excess delay time in the\ncluster, respectively.\nb) Departure angular profile in the ith cluster\nAn empirical departure angular profile model AODI (φik) with departure angle φik in the ith cluster\nnormalized by its central angular path’s power is given as:\n− φik / φDI\nAODI (φik ) = e (0 ≤ φik ≤ φDImax ) (75)\n\nwhere:\n2lI max  180  2(300τ I max )  180 \nφDI = ⋅ = ⋅  degrees (76)\n(d + 300τ )  π  (d + 300τ )  π \n\n## φDI depends on the excess delay time τ.\n\nc) Arrival angular profile in the ith cluster\nAn empirical arrival angular profile model AOAI (φik) with arrival angle φAI in the ith cluster\nnormalized by its central angular path’s power is given by:\n− φik / φ AI\nAOAI (φik ) = e (0 ≤ φik ≤ 180) (77)\n\nwhere φAI denotes the average arrival angle in the cluster. φAI = 180º is a typical value.\n1.4.2 Generation of reduced variability models based on TSP model\n1.4.2.1 NLoS scenario for base coverage macrocell\nThe reduced variability model is created by setting its key parameters. Each channel can be realized\nby the following procedure:\nStep 1: Set test environments and key parameters\nSet the following parameters.\n58 Rep. ITU-R M.2135-1\n\na) Frequency conditions\nfc: carrier frequency (MHz)\nλ: wavelength of carrier frequency (m) (= 300/ fc)\nB: bandwidth or chip rate (MHz).\nb) City structures\n<H>: average building height (m). (Bad-urban: 30 m, Urban: 20 m, Suburban: 10 m,\nResidential: 8 m, Rural: 5 m, etc.)\nW: road width (m) (Bad-urban: 20 m, Urban: 20 m, Suburban: 20 m, Residential:\n20 m, Rural: 20 m, etc.).\nc) Cell conditions\nhb: BS antenna height (m.\nhm: UT antenna height (m)\nd: distance from the BS (m).\nStep 2: Set the simulation conditions\nN path : number of clusters for NLoS\nS\nN path : number of clusters for LoS\nΔLth: level difference between the peak path’s power and cut off power in NLoS scenario\n(dB)\nΔLSth: level difference between the peak path’s power and cut off power for scattering\nclusters in LoS scenario (dB).\nStep 3: Generation of long-term TSP\nStep 3.1: Generation of delay time\na) Calculation of maximum delay time\nConvert excess distance l (m) into excess delay time τ (μs) by equation (78):\nτ = l /(300 / fc ) ( μ s) (78)\nThe maximum delay time can be calculated by considering ΔLth as follows:\nPDPL (τ LMax , d ) = −ΔLth ( ΔLth ≥ 0) (79)\nb) Dynamic delay time of each cluster\nThe delay time interval Δ τ L is calculated as follows:\nΔτ L = τ LMax /( N path − 1) (80)\nThe dynamic delay time is generated by using a random number:\n0 (i = 0)\nτ L (i ) =  (81)\niΔτ L + Δτ L ( Random[0,1] − 1/ 2) (i = 1, 2, − − −, N path − 1)\nwhere Random[0,1] denotes a random value with uniform distribution between 0 and 1.\nRep. ITU-R M.2135-1 59\n\n## c) Fixed delay time of each cluster\n\nInstead of the dynamic delay time, fixed delay time τ L (i ) is given as follows. Table A1-23 shows\nthe normalized fixed delay time τ LN (i ) . When the number of clusters and the maximum delay time\nare given, the fixed delay time τ L (i ) is calculated by using Table A1-23 by:\n\n τ LMax \nτ L (i ) = τ L N (i)   (i = 0,1, 2, − − −, N path − 1) (82)\n τ L N ( N path − 1) \n \n\n( )\nwhere τ LMax / τ L N ( N path − 1) represents the scaling factor.\n\nTABLE A1-23\nParameter sets of τL(i) and XiN\na) For the case of Npath = 20\n\n## Normalized Normalized short term variation XiN with STD of 1 (dB)\n\nIndex number i\ndelay time τL(i) XiN(1) XiN(2) XiN(3) XiN(4)\n1 0.0000 0.24064 0.18409 0.26383 0.74152\n2 0.047 –0.69579 –0.12405 1.9716 0.83339\n3 0.097 0.37050 –0.0077560 –0.40653 –2.5006\n4 0.129 –0.89163 –0.26097 0.63798 –1.6534\n5 0.215 –0.82499 0.92161 0.16647 0.52677\n6 0.253 0.54805 0.35463 1.3216 1.2068\n7 0.298 –1.6740 –0.13014 –0.46038 –0.54795\n8 0.343 –1.7426 0.44910 1.0759 –0.26516\n9 0.397 0.96705 –0.021601 1.3793 1.7035\n10 0.452 –0.57140 –2.6355 –0.30171 0.53209\n11 0.507 0.61279 0.16136 –0.33711 –0.75249\n12 0.534 0.042949 –0.77649 –1.6624 –0.11695\n13 0.615 –0.59649 2.0748 0.31816 0.74633\n14 0.645 1.9803 0.76392 –0.91974 –1.6244\n15 0.692 –0.13612 –1.3529 0.56334 0.22220\n16 0.727 –0.58639 0.50109 –0.99308 0.15922\n17 0.817 –0.19525 0.12992 0.49671 0.65203\n18 0.861 –1.6267 0.77200 –1.0752 0.16436\n19 0.913 1.6513 –1.7319 –1.9479 –0.31058\n20 0.956 –0.12564 0.72884 –0.090964 0.28327\n60 Rep. ITU-R M.2135-1\n\n## TABLE A1-23 (end)\n\nb) For the case of Npath = 10\n\n## Normalized Normalized short term variation XiN with STD of 1 (dB)\n\nIndex number i\ndelay time τL(i) XiN(1) XiN(2) XiN(3) XiN(4)\n1 0.0000 1.1458 1.2001 1.7690 0.48474\n2 0.094923 0.75311 0.53053 0.060735 0.43398\n3 0.19599 –0.70881 1.0645 –1.9474 0.058202\n4 0.25979 1.6582 0.56882 0.20572 –2.0529\n5 0.43136 –1.1183 –1.4115 –0.66017 –0.35828\n6 0.50651 –0.78058 0.052757 –0.49775 0.48382\n7 0.59619 –1.4302 –0.86393 1.2526 –1.0723\n8 0.68790 0.91257 –0.66617 –0.33581 –0.37723\n9 0.79581 –0.35092 1.0897 0.64893 1.8421\n10 0.90483 –0.080880 –1.5649 –0.49588 0.55783\n\n## Step 3.2: Generation of relative power (without considering short-term variation)\n\nThe delay profile is normalized by the first cluster’s power as follows:\n\n## PDPL (i, d ) ≡ PDPL (τ L (i ), d ) dB (83)\n\nSet the delay profile normalized by all clusters' power as 1(= 0 dB):\n\n##  N path −1 PDPL (i ,d ) /10 \n\nPDPLnorm (d ) = 10 log   10  dB (85)\n i =0 \nStep 3.3: Generation of departure angle\nSet the departure angle of the ith cluster as follows:\n\n−1/ β\nθ LD (i, d ) = (−1)i a( PDPL ( iΔτ L , d ) − 1) (i = 0,1, 2, − − −, N path − 1) degrees (86)\n\n## Step 3.4: Generation of arrival angle\n\nSet the arrival angle of the ith cluster as follows:\n\ni −1 ( )\n 2d 2 + 2diΔτ L + (iΔτ L )2 cos θ LD (i ) − 2d (d + iΔτ L ) \nθ LA (i, d ) = (−1) cos   degrees (87)\n 2d 2 + 2diΔτ L + (iΔτ L )2 − 2d (d + iΔτ L ) cos θ LD (i ) \n \n\nSteps 3.1 to 3.4 yield the delay time, departure angle, arrival angle, and normalized relative power\nfor each cluster.\nRep. ITU-R M.2135-1 61\n\n## Step 4: Generation of short-term variation\n\na) Dynamic short term variation\nThe dynamic short term variation of each cluster xi follows an independent log-normal distribution\nas equation (88):\n2\n/ 2σ S2\nG ( X i ) = e− Xi / 2π σ S (88)\n\n## where σ S represents the standard deviation of short term variation.\n\nb) Fixed short term variation\nInstead of the dynamic short-term variation, fixed short term variation Xi can be easily obtained\nbased on Table A1-23 by using the following equation:\nX i = σ S X iN (i = 0,1, 2, − − −, N path − 1) (89)\n\nwhere X iN represents a normalized short term variation, with standard deviation of 1, obtained from\nTable A1-23. A typical standard deviation value for σ S is 4.5 dB for macro-cell.\nStep 5: Generation of relative power considering short-term variation\nInstead of using the relative power without short-term variation as per Step 3.2, the relative power\nwith short-term variation PDPL&S (i, d ) can be obtained as follows:\n\n## 1.4.2.2 LoS scenario for base coverage macro-cell\n\nThe LoS environment consists of direct clusters and scattering clusters. The time and spatial\ncharacteristics of scattering clusters are completely the same as those of NLoS.\nChannel realizations for LoS are achieved by the following procedure:\nStep 1 and Step 2: Set the conditions as per the NLoS case.\nStep 3: Generation of direct clusters.\nDirect clusters consist of three direct clusters, PD,i (i = 0,1,2). Generate delay time τ D,i (i, d ) , arrival\nangle θDA,i(i,d), departure angle θDG,i(i,d) and relative power PDPD,i (i,d) normalized against the\nfirst cluster’s power as follows:\na) τ D,i (i, d )\nτ D ,0 (0, d ) = 0, τ D ,1 (1, d ) = W 2 /(300d ), τ D ,2 (2, d ) = 3W 2 /(300d ) (μs) (91)\n\nb) θ DA,i (i, d )\nθ DA,0 (0, d ) = 0, θ DA,1 (1, d ) = 2W / d ⋅ (180 / π ), θ DA,2 (2, d ) = −2W / d ⋅ (180 / π ) degrees (92)\n\nc) θ DD,i (i, d )\nθ DD ,0 (0, d ) = 0, θ DD ,1 (1, d ) = W / d ⋅ (180 / π ), θ DD ,2 (2, d ) = 2W / d ⋅ (180 / π ) degrees (93)\n\nd) PDPD,i (i, d )\nPDPD ,0 (0, d ) = 1, PDPD ,1 (1, d ) = 0.3, PDPD ,2 (2, d ) = 0.1 (94)\n62 Rep. ITU-R M.2135-1\n\n## Step 4: Generation of scattering clusters\n\nStep 4.1: Generation of delay time for scattering clusters\nScattering clusters consist of ( N path\nS\n– 3) clusters.\na) The PDP for scattering clusters is given by:\n\n## b) Calculation of maximum delay time\n\nThe maximum delay time is calculated to satisfy the following equation involving a threshold level\nΔLSth . :\n\n## PDPL (τ LMax , d ) = −ΔLS th ( ΔLS th ≥ 0) (96)\n\nSolving the above equation yields the maximum delay time τ S LMax . ΔLSth is generally given by\nΔLS th = ΔLth − 12 (dB) .\nStep 4.2: Generation of delay time, relative power (without considering short-term variation),\ndeparture angle, arrival angle, and relative power considering short-term variation for scattering\nclusters.\nThis step is completely the same as Step 3 to Step 5 in the NLoS case.\nStep 5: Generation of the power delay profile considering the direct and scattering clusters\nStep 5.1: Generation of relative power (without considering short-term variation)\na) Dynamic delay time of each cluster\nThe delay time interval Δ τ L is calculated as follows:\n\nΔτ L = τ LMax /( N path\nS\n− 3) (97)\n\n## The dynamic delay time is generated by using a random number:\n\nτ D ,i (i, d ) (i = 0,1, 2)\nτ LLOS (i) =  (98)\n(i − 2)Δτ L + Δτ L ( Random[0,1] − 1/ 2) (i = 3, − − −, N path − 1)\nS\n\nWhere Random[0,1] denotes a random value with uniform distribution between 0 and 1.\nb) Fixed delay time of each cluster\nInstead of the dynamic delay time, fixed delay time τL(i) is given as follows. Table A1-23 shows the\nnormalized fixed delay time τ LN (i ). When the number of clusters and the maximum delay time are\ngiven, the fixed delay time τL(i) is calculated by using Table A1-23 as follows:\n\n τ LLOSMax \nτ LLOS (i ) = τ LLOS N (i − 2)   S\n(i = 3, − − −, N path − 1) (99)\n τ LLOS N ( N path\nS\n− 3) \n \n\n(\nwhere τ LLOSMax / τ LLOS N ( N path\nS\n)\n− 3) represents the scaling factor.\nRep. ITU-R M.2135-1 63\n\n## Step 5.2: Generation of relative power (without considering short-term variation).\n\nThe delay profile considering the direct clusters and scattering clusters is given by:\n\nPDPLLOS (i, d ) = \n( )\n10 log PDPD ,i (i, d ) + 10 PDPL ,scatter (0,d ) /10 (0 ≤ i ≤ 2)\ndB (100)\nS\n PDPL , scatter (i − 2, d ) (3 ≤ i ≤ N path − 1)\n\nSet the delay profile normalized by the first cluster’s power as follows:\nPDPLLOSN (i, d ) = PDPLLOS (i, d ) − PDPLLOS (0, d ) dB (101)\nStep 5.3: Generation of departure angle\nSet the departure angle of the ith cluster as follows:\nθ DD ,i (i, d ) (i = 0,1, 2)\nθ LDLOS (i, d ) =  −1/ β degrees (102)\n(−1) α ( PDPL ( (i − 2)Δτ L , d )\n( i − 2) S\n− 1) (i = 3, − − −, N path − 1)\n\n## Step 5.4: Generation of arrival angle\n\nSet the arrival angle of the ith cluster as follows:\nθ DA,i (i, d ) (i = 0,1, 2)\n\n (\nθ LALOS (i, d ) = (−1)i − 2 cos −1 \nL)\n 2d 2 + 2d (i − 2)Δτ + ((i − 2)Δτ ) 2 cos θ ((i − 2)) − 2d (d + (i − 2)Δτ ) \nL LD L\n degrees (103)\n 2d + 2d (i − 2)Δτ + ((i − 2)Δτ ) − 2d (d + (i − 2)Δτ ) cos θ ((i − 2)) \n2 2\n  L L L LD\n\n S\n(i = 3, − − −, N path − 1)\n\nSteps 5.1 to 5.4 yield the delay time, departure angle, arrival angle, and normalized relative power\nfor each cluster.\n1.4.3 K-factor\nIn the NLoS case, the K-factor is basically 0(= −∞ dB) . In the LoS case, the K-factor of the direct\ncluster is determined by the ratio of the direct cluster’s power to the first scattering cluster’s power.\nOn the other hand, the K-factor of the other scattering clusters is basically 0(= −∞ dB) .\n\n## 1.4.4 Cross polarization\n\na) VV, VH, HH and HV\nPDPLVV&S (i, d ) , PDPLVH HH HV\n& S (i, d ) , PDPL& S (i, d ) and PDPL& S (i, d ) express delay profiles with short-term\nvariation due to vertical-to-vertical(VV), vertical-to-horizontal (VH), horizontal-to-horizontal (HH)\nand horizontal-to-vertical (HV) polarization, respectively as follows:\nPDPLVV&S (i, d ) = PDPL (i, d ) + X i(1) dB (104a)\n\nPDPLVH\n& S (i, d ) = PDPL (i, d ) − XPDL + X i\n(2)\ndB (104b)\n\nPDPLHH\n&S (i, d ) = PDPL (i, d ) + X i\n(3)\ndB (104c)\n\nPDPLHV\n& S (i, d ) = PDPL (i, d ) − XPDL + X i\n(4)\ndB (104d)\n64 Rep. ITU-R M.2135-1\n\nwhere X i(1) , X i(2) , X i(3) and X i(4) are independent random variables following log-normal\ndistributions with standard deviation of σS. If these values are required to be fixed, these values can\nbe generated by using the normalized variations X iN ( k ) in Table A1-23 and the relation\nX i( k ) = σ S X iN ( k ) ( k = 1, 2, − − −, 4).\n\n## 1.4.5 Generation of small scale parameters in clusters\n\nInstantaneous time-spatial profiles can be generated by using small-scale parameters.\nThis process generation is applied for all NLoS clusters and all LoS scattering clusters.\nStep 1: Simulation condition\nNp : number of paths in cluster.\n\n## Step 2: Generation of delay time\n\nThe delay time interval Δτ I and delay time τ I normalized by the first path's delay time in a cluster\nare calculated respectively as follows:\nΔτ I = τ Imax /( N p − 1) μs (105)\nτ I (k ) = k Δτ I (k = 0,1, 2, − − −, N p − 1) μs (106)\nHere τ Imax denotes the maximum delay time. It is typically 0.1 μs and 0 μs in the simplest case.\nDelay time τ I ( k ) is recalculated at a time interval of ΔτI.\nStep 3: Generation of relative power\nThe delay profile normalized by the first path’s power in a cluster is expressed as:\nPDPI (k ) = 10log(e−τ I ( k ) /τ Iav ) = 10log(e− k Δτ I /τ Iav ) dB (107)\nwhere τ Iav is the average delay time of each cluster. τ Iav is typically 0.1 μs and 0 μs in the simplest\ncase. Set the delay profile normalized by all paths' power in a cluster as follows:\nPDPIN (k ) = PDPI (k ) − PDPInorm dB (108)\nwhere PDPInorm represents all paths' power in a cluster according to:\n N p −1 \nPDPInorm = 10 log   10 PDPI ( k ) /10  dB (109)\n k =0 \nStep 4: Generation of arrival angle\nThe arrival angle interval ΔφIA is set as follows:\nΔφIA = 180 / N p degrees (110)\nSet the arrival angle of the kth path in a cluster as follows:\nφIA ( k ) = ( −1) k k ΔφIA (deg.) ( k = 0,1, − − −, N p − 1) (111)\n\nThe arrival angle φIA (k) is calculated using the angular interval ΔφIA.\nRep. ITU-R M.2135-1 65\n\n## Step 5: Generation of departure angle\n\nThe departure angle interval ΔφID is set as follows:\nφID (i )Δτ I\nΔφID (i ) = degrees (112)\nτ Iav\nwhere:\n60 180\nφID (i ) = ⋅ degrees (113)\n(d + i ⋅ (300 / B )) π\nSet the departure angle in a cluster according to:\nφID (i, k ) = ( −1) k k ΔφID (i ) (deg.) ( k = 0,1, 2, − − −, N p − 1) (114)\nSteps 1 to 3 yield the delay time, departure angle, arrival angle, and normalized relative power of\npaths in each cluster. Note that when τImax is set to 0 μs, the fading model in the clusters becomes\nthe well-known Clark model.\n1.4.6 Typical values of parameters\n1.4.6.1 Large scale parameters\nThe typical number of clusters for the reduced variability model based on the TSP model is shown\nin Table A1-24 for NLoS scenarios and Table A1-25 for LoS scenarios.\n\nTABLE A1-24\nNLoS scenarios (typical)\n\n## Number of clusters Level difference of clusters\n\nNpath ΔLth (dB)\nCity structure (typical) (typical)\n\n## (*) Dense urban 20\n\nUMa Urban 20\nSMa Suburban 20 25\n(*) Residential 10\nRMa Rural 10\n\nTABLE A1-25\nLoS scenarios (typical)\n\n## Number of direct Number of Level difference of\n\nclusters scattering clusters scattering clusters\n(typical) Npath ΔLSth (dB)\nCity structure (typical) (typical)\n\n## (*) Dense urban 10\n\nUMa Urban 10\nSMa Suburban 3 10 13\n(*) Residential 10\nRMa Rural 10\n66 Rep. ITU-R M.2135-1\n\n## 1.4.6.2 Small scale parameters\n\nThe typical number of rays in a cluster with the reduced variability model is shown in Table A1-26.\nThere are two parameter sets: The first one is for the realistic model and the second for the classical\nmodel. The classical model is realized by setting τ Imax to 0 μs (known as Clark’s model). Note that\nthe received level depends on just the UT moving speed, not the UT moving direction in the\nclassical model.\n\nTABLE A1-26\nSmall scale parameters\n\n## Parameter Realistic model Classical model\n\n(typical) (typical)\nNp 20 20\n\nτ Imax 0.1 μs 0 μs\n( lImax = 30 m) ( τ I ( k ) ≡ 0 , Δφ ID (i ) ≡ 0 , PDPI ( k ) ≡ 0dB )\n\n## 1.4.6.3 Identification of the model\n\nIn order to clarify the model generated by the extension module, the generated model has to be\nidentified. For example, when the reduced variability model in the extension module is used, the\napplied model can be identified by describing it as “Extension module (TSP model)- reduced\nvariability model- NLoS (typical)- Classical model (typical)”.\n\nReferences\n\nDONG, W., ZHANG, J., GAO, X., ZHANG, P. and WU, Y. Cluster Identification and Properties of\nOutdoor Wideband MIMO Channel. Vehicular Technology Conference, 2007. VTC-2007 Fall.\n2007 IEEE 66th, Sept. 30 2007-Oct. 3 2007, p. 829 – 833.\nFUJII, T. Path-loss prediction formula in mobile communication -an expansion of “SAKAGAMI”\npath-loss prediction formula-. Trans. IEICE, Japan, J86-B, 10, p. 2264-2267.\nFUJII, T. Delay profile modelling for wideband mobile propagation. Proc. of IEEE 2004 VTC Fall F.\nGAO, X., ZHANG, J., LIU, G., XU, D., ZHANG, P., LU Y. and DONG, W. [December 2007] Large-Scale\nCharacteristics of 5.25 GHz Based on Wideband MIMO Channel Measurements. Ant. and Wireless\nPropag. Lett., Vol.6, p. 263-266.\nKERMOAL, J. P., SCHUMACHER, L., PEDERSEN, K. L., MOGENSEN, P. E. and FREDERIKSEN, F.\n A stochastic MIMO radio channel model with experimental validation. IEEE J. Select.\nAreas in Comm., Vol. 20, 6, p. 1211-1226.\nLU, Y., ZHANG, J., GAO, X., ZHANG, P. and WU, Y. Outdoor-Indoor Propagation Characteristics\nof Peer-to-Peer System at 5.25 GHz. Vehicular Technology Conference, 2007. VTC-2007 Fall.\n2007 IEEE 66th, Sept. 30 2007-Oct. 3 2007, p. 869-873.\nOHTA. Y. and FUJII,T. Delay Spread Prediction for wideband mobile propagation. Proc. of IEEE\n2006 VTC, Montreal.\nRep. ITU-R M.2135-1 67\n\n## OKAMOTO, H. and ICHITSUBO.S. [March 2005] Investigations of outdoor-to-indoor propagation loss on\n\n800MHz-8GHz at urban area. Proc. The 2005 IEICE General Conference, Japan, p. S23-S24, (in\nJapanese).\nOMOTE, H. and FUJII, T. Empirical arrival angle profile prediction formula for mobile\ncommunication system. Proc. of IEEE 2007 VTC Spring, Dublin.\nSMITH, P. J. and SHAFI, M. [June 2004] The impact of complexity in MIMO channel models. Proc. of\nIEEE International Conference on Communications (ICC '04), Vol. 5, p. 2924-2928, Paris, France.\nXU, D., ZHANG, J., GAO, X., ZHANG, P. and WU, Y. Indoor Office Propagation Measurements\nand Path Loss Models at 5.25 GHz. Vehicular Technology Conference, 2007. VTC-2007 Fall. 2007\nIEEE 66th, Sept. 30 2007-Oct. 3 2007, p. 844-848.\nZHANG, J., GAO, X., ZHANG, P. and YIN, X. [3-7 September 2007] Propagation Characteristics of\nWideband MIMO Channel in Hotspot Areas at 5.25 GHz. Personal, Indoor and Mobile Radio\nCommunications, 2007. PIMRC 2007. IEEE 18th International Symposium on September.\nZHANG, J., DONG. D., LIANG. Y., NIE. X., GAO. X., ZHANG, Y., HUANG, CH. and LIU, G. \nPropagation characteristics of wideband MIMO channel in urban micro- and macrocell. IEEE\nPIMRC 2008, Invited paper.\n\nBibliography\n\nElectron. Lett, Vol. 27, 23, p. 2145-2146).\nFORTUNY-GUASCH J. et al. [October 2006] UWB screening attenuation measurements of cars study by\nIPSC of the JRC and ETSI TG31C on the measurement of the screening attenuation of cars in the\nfrequency range between 0.85 GHz and 11 GHz. IPSC, Sensors Radar Technologies and Cyber\nSecurity Unit, European Commission Joint Research Centre. EUR 22532 EN.\n3GPP TSG-RAN1#48, R1-070674, LTE physical layer framework for performance verification, Orange,\nChina Mobile, KPN, NTT DoCoMo, Sprint,T-Mobile, Vodafone, Telecom Italia.\nNGMN White paper [December 2007] Next Generation Mobile Networks Radio Access Performance\nEvaluation Methodology.\n3GPP-SCM [September 2003] Spatial channel model for multiple input multiple output (MIMO)\nsimulations, TR 25.996 v.6.1.0.\n\nITU-R texts\nRecommendation ITU-R P.679-1.\nProposed New Test Environments and Channel Models of Preliminary Draft New Report M.[IMT.EVAL],\nDoc. 5D/62, January 2008, Geneva, Switzerland.\nChannel Models for Evaluation of Air Interface Proposals for IMT-Advanced, Doc. 5D/68, January 2008,\nGeneva, Switzerland.\n5D/88, January 2008, Geneva, Switzerland.\n68 Rep. ITU-R M.2135-1\n\nAnnex 2\n\nTraffic models\n\nThere are two types of traffic models which should be used for evaluation of proposed\n– Full buffer traffic model, and\n– VoIP traffic model.\n\n## 1 Full buffer traffic model\n\nIn the full buffer traffic model (sometimes referred to as a full queue model) there is an infinite\namount of data bits awaiting transmission in the transmit buffer associated with each data source.\nFor uplink traffic the transmit buffers exist in the user terminal while for downlink traffic the\ntransmit buffers exist in the network infrastructure, e.g., in the base station.\n\n## 2 VoIP traffic model\n\nTable A2-1 provides the relevant parameters of the VoIP traffic that should be used for the\nsimulations. The details of the corresponding traffic model are described below:\n\nTABLE A2-1\n\nParameter Characterization\nCodec Source rate 12.2 kbit/s\nEncoder frame length 20 ms\nVoice activity factor (VAF) 50% (c = 0.01, d = 0.99)\n15 bytes ( 5 bytes + header)\nSID packet every 160 ms during silence\n4 bytes (RTP/UDP/IP)\n2 bytes (RLC/security)\n16 bits (CRC)\n\n## 2.1 Detailed description of the VoIP traffic model\n\na) Basic model\nConsider the simple 2-state voice activity model shown in Fig. 14.\nRep. ITU-R M.2135-1 69\n\nFIGURE 14\n2-state voice activity model\na\n\nb = 1-c\n\n## Inactive state Active state\n\nstate 0 state 1 d = 1-a\n\nc Report 2135-14\n\nIn the model, the probability of transitioning from state 1 (the active speech state) to state 0 (the\ninactive or silent state) while in state 1 is equal to a , while the probability of transitioning from\nstate 0 to state 1 while in state 0 is c. The model is assumed updated at the speech encoder frame\nrate R = 1/T, where T is the encoder frame duration (typically, 20 ms).\nb) Model statistics\nThe steady-state equilibrium of the model requires that:\na\nP0 =\na+c\nc (115)\nP1 =\na+c\nwhere P0 and P1 are respectively the probability of being in state 0 and state 1.\nThe voice activity factor (VAF) λ is given by:\nc\nλ = P1 = (116)\na+c\nA talk-spurt is defined as the time period τTS between entering the active state (state 1) and leaving\nthe active state. The probability that a talk spurt has duration n speech frames is given by:\nPτ TS =n  PTS (n) = a(1 − a)n −1 n = 1, 2, (117)\nCorrespondingly, the probability that a silence period has duration n speech frames is given by:\nPτ SP =n  PSP (n) = c(1 − c)n−1 n = 1, 2, (118)\n\nThe mean talk spurt duration μTS (in speech frames) is given by:\n1\nμTS = E[τ TS ] = (119)\na\nwhile the mean silence period duration μSP (in speech frames) is given by:\n1\nμ SP = E[τ SP ] = (120)\nc\n70 Rep. ITU-R M.2135-1\n\nThe distribution of the time period τAE (in speech frames) between successive active state entries is\nthe convolution of the distributions of τSP and τTS. This is given by:\nc a\nPτ AE = n  PAE ( n) = a (1 − a ) n −1 + c(1 − c) n −1 n = 1, 2, (121)\nc−a a−c\n\nNote that τAE can be used as a crude guide to the time between MAC layer resource reservations,\nprovided a single reservation is made per user per talk-spurt. Note that in practice, very small values\nof τAE might not lead to a separate reservation request, but equation (121) still offers some\npotentially useful guidance.\nSince the state transitions from state 1 to state 0 and vice versa are independent, the mean time μAE\nbetween active state entries is given simply by the sum of the mean time in each state. That is:\nμAE = μTS + μSP (122)\nAccordingly, the mean rate of arrival RAE of transitions into the active state is given by:\n1\nRAE = (123)\nμ AE" ]

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